{"id":"21720","title":"Repulsive gravity","text":"IANAP, so feel free to berate me for thinking apocryphal thoughts! Just as magnetism has two charges, in which particles of like-charge repulse and particles of dissimilar charge attract, might gravity have two charges in which particles of like-charge attract and particles of dissimilar charge repulse? In practice, the state of magnetism means that there is no system composed of many particles in which all particles attract. Rather, there is a net 0 charge if there are equal numbers of each particle type. My silly theory regarding gravity would mean in practice that there would be two (or more) \"clumps\" (or universes) in existence, which are racing away from each other. So in our clump (universe) we see only attracting particles, because all the opposing particles have long since separated out and are racing away beyond the boundary of the observable universe. Just like the alien who lands in China and assumes that all humans have slanted eyes, we only observe the attracting particles (or \"charge\") and disregard the other, unobservable, \"charge\". Is there any way to disprove this idea, or like string theory can I go one believing it as it can never be disproved? Thanks."}
{"id":"46155","title":"Attractiveness of spin 2 gauge theories","text":"> **Possible Duplicate:**   >  Why is gravitation force always attractive? I have heard that the attractiveness of gravitation is due to the fact that it is a spin 2 gauge theory. Why is this so? I find this very interesting and would love to understand this fact."}
{"id":"122190","title":"Gravitational Force (conceptual)","text":"1. Why is gravitational force always an attractive force?    2. And is the Newtonian formula of gravitational force applicable for very small particles like electrons and protons etc.? From **Formula of Gravitational Force** , I'm referring to: $$F_G = \\cfrac{GMm}{R^2}, $$ where $M$ and $m$ are the masses of objects. A logical explanation will be much appreciated."}
{"id":"78995","title":"Why gravity is an attractive force?","text":"Why gravity is an attractive force? * * * One may say that it is because of space time curvature but General Relativity is built on this law: $\\displaystyle G \\frac{m_1 \\times m_2}{r^2}$ (To be more precise, it is derived from it's potentiel form known as Gauss's law for gravity that can be written like this: $\\nabla \\cdot g = - 4\\pi Gp $). So GR can't explain why gravity is attractive. General Relativity only explains how gravity occurs in terms of space time. * * * Another may stick with the Quantum theory stating that gravitons are responsible for the force of gravity, but why a massless particle with spin 2 will exchange positive momentum?"}
{"id":"11543","title":"What happens when two smooth surfaces touch?","text":"I am wondering what will be the physics to explain how two neutral, chemically nonreactive objects stick. I know that using van der Waals formalism, we can treat neutral body electrodynamic forces and arrive with attractive forces that pull the objects together. Now, once the objects touch (say a mechanical cantiliver in a MEMS sensor like the one used in an iPhone), what happens to the forces? A quantitative answer or some estimate on how strong the attractive force is for simple cases will be very appreciated. in response to anna's comment : Let us consider what happens in vacuum for ultra smooth surfaces, with no residual electrical charge and fully chemically stablized surfaces (example, silicon crystals with stabilised surface bonds)."}
{"id":"11544","title":"vander waals and casimir forces","text":"Does one need to invoke quantum mechanics to explain casimir force or vander waals force. I see that textbooks show derivation of vander waal force with no QM but casimir force is typically described with QM. Is there a difference between vanderwaal and casimir forces ? Are there distinct examples of these two forces in real life. Is there a way to prove a given force is vanderwaal and not casimir or vice versa."}
{"id":"34214","title":"$\\lambda=\\frac{2h}{p}$?","text":"I am studying quantum physics and there is something I don't understand: I know that for any particle $E=hf$ (Einstein relation) and $v=\\lambda f$ ($v$ is the speed of the particle). I also know that the kinetic energy is $E_k=\\frac{mv^2}{2}$. Solving those 3 equations for $\\lambda$: $$h\\frac{v}{\\lambda}=\\frac{mv^2}{2}$$ I finally find $\\lambda=\\frac{2h}{mv}=\\frac{2h}{p}$ which is not consistent with the De Broglie relation $\\lambda=\\frac{h}{p}$. Where am I wrong in my development ?"}
{"id":"134357","title":"Wave Packets, Group velocity, and Phase velocity","text":"Mathematically, you find that the wave function of a particle $\\Psi (x,t)$ moves with the same velocity as the velocity of the particle ($v_{particle} = v_{group}$). Is there a reason why the particle velocity should not equal the phase velocity?"}
{"id":"36242","title":"Group Velocity and Phase Velocity of Matter Wave?","text":"In quantum mechanics, what is the difference between group velocity and phase velocity of matter wave? How can it also be that phase velocity of matter wave always exceeds the speed of light?"}
{"id":"16063","title":"Speed of a particle in quantum mechanics: phase velocity vs. group velocity","text":"Given that one usually defines two different velocities for a wave, these being the phase velocity and the group velocity, I was asking their meaning for the associated particle in quantum mechanics. And is one of them more representative for a particle?"}
{"id":"108496","title":"group and phase velocity of free particle","text":"If Schrödinger wave equation is for matter waves then for a free particle Group velocity $V_g =2$ Phase velocity $V_p$ But matter waves satisfy the relation $V_g V_p = C^2$ where $V_p>C$ Does this contradiction tell free particle should not be dealt quantum mechanically?"}
{"id":"82442","title":"Efficiency of the insulation of a house","text":"I had an argument about the most cost-effective way to keep the energy bill low in the winter (here, temperature usually have an average of -20°C (-4°F)). He thinks that it's more effective to keep a temperature at a constant temperature at say 24°C (75.2°F), because It would cost as much or more to re- heat the house if you let the temperature drop. I simply think that the lower the temperature the better, because heat from the house will dissipate less and you need less energy to maintant the temperature. Am-I right? if so, which scientific theory could I look at to my make point?"}
{"id":"134680","title":"Perfectly vertical spinning top","text":"Consider a non-spinning top. If a top is perfectly vertical, and the interface between its base and the ground is perfectly flat, it should stay in (unstable) equilibrium. I.e. it does fall. What about if the top, in addition to being perfectly vertical, is also spinning? Does this situation differ from the previous one?"}
{"id":"30978","title":"Loschmidt's paradox - really a paradox?","text":"Is Loschmidt's paradox a paradox even today? In other words, is the paradox resolved or not?"}
{"id":"20761","title":"Repulsion of the pieces of a  broken magnet","text":"> **Possible Duplicate:**   >  Why does it seem like a broken magnet's poles flip? I have experienced that if we break a bar magnet into two pieces and try to bring those broken faces together it gets repelled each other. Why is it so? consider a bar magnet, $ NN--------SS $ $NN$ for north pole and $SS$ for south pole. I broke that into two pieces, So one piece would be $ NN--SS'$ and another piece would be $'NN--SS$ .( $'$ indicates the broken part). Now if I bring those two together, $SS'$ of first should attract the $'NN$ of the other. But in the reality it is getting repelled. or is there any pole exchange occuring? $$NN--------SS$$ gives either $$NN---SS'$$ and $$'SS---NN$$ **OR** $$SS---NN'$$ and $$NN'\\---SS$$ so it will repel each other. Please explain why is this happening? Edit: The cut has been made perpendicular to the polar axis."}
{"id":"105603","title":"What happens to the bar magnet field when it breaks?","text":"References says alignment of a magnetic field of a bar magnet remains same even if we cut it into two or more pieces. According to this cut edges should attract with each other. When I tried it was working in the opposite way. why is it so ?"}
{"id":"131102","title":"Broken bar magnet repels?","text":"**![Why a bar magnet broken from between repel on joining again even if the poles are opposite, i.e. North and South.](http:\/\/i.stack.imgur.com\/bM4tl.png)** **Why a bar magnet broken from between repel on joining again even if the poles are opposite, i.e. North and South.**"}
{"id":"80590","title":"In truth, only atoms and the void","text":"I have a question about this motto used by Sean Carroll in his blog: > _In truth, only atoms and the void._ Can you explain what this sentence means? My interpretation is that the sentence does not makes sense because in physics, \"atom\" has multiple meanings, the same goes for \"void\". These words have so many meanings that, used without qualification, they mean nothing. The word truth is also a notoriously loaded and undefined word. Does he mean the absolute truth (which has no place in physics) or is he using it colloquially, we don't know. Let's assume that he is using \"atom\" to mean \"absolutely indivisible unit\" and \"void\" to mean \"absolutely empty space that contains nothing but atoms\". This view is absurd because defines \"void\" to be \"empty\" and \"not-empty\" at the same time. But besides that, if in truth, there is nothing but atoms and the void, does Sean Carroll deny the existence of fields? Here are a few meanings that physicists give to \"void\" (a search of titles in arxiv containing \"void\"): rigid void, relativistic void, magnetic fields in voids, empty voids, nano void, dynamics of void and so on. So, in physics void can mean anything but \"void\". Then, what does \"In truth, only atoms and the void\" mean? Does it really have such a deep meaning to included as a motto of a blog?"}
{"id":"91073","title":"Diffraction\/Bragg's law : how does $2\\theta$ come about?","text":"I'm trying to get my head around a problem (I should have checked whether I had the answer in class, the exams are coming up now and I don't know if I'll get a lecturer response over the holidays) I can't get figure out the relationship between $\\theta$ and $2\\theta$ in the diagrams supposedly making clear how a crystal lattice diffracts. From what I've read just now my understanding is that Bragg diffraction is actually transmission part-way into the crystal, then reflection off of an atom inside, hence the angle is twice the incident angle. The textbook we've been referred to for further reading makes no note of a $2\\theta$ (it can be read here). The example used in the lecture is below ![enter image description here](http:\/\/i.stack.imgur.com\/VmKym.png) I obtained an answer by using a right-angled triangle made by the X-ray detector at the middle arrow, as this is the only perpendicular angle I can see. I really don't get how this would give the angle at the crystal as $2\\theta$ (and I'm not at all confident that I should halve this angle to use in calculation of $\\theta$ Can someone explain why I need to do so (as despite my misgivings, this is clearly indicated as the correct procedure). I've obtained an answer and am confident with the theory, maths etc., I just don't see the derivation of the $2\\theta$ as opposed to $\\theta$."}
{"id":"93009","title":"Dynamics of circular motion","text":"If there is a disc rotating about its centre, let the surface be frictionless and if a coin is placed anywhere on the disc (not at the center) why doesn't it fly off even though there is a centrifugal force acting on it with respect to the disc's frame?"}
{"id":"63001","title":"Do we expect that the universe is simply-connected?","text":"I heard recently that the universe is expected to be essentially flat. If this is true, I believe this means (by the 3d Poincare conjecture) that the universe _cannot_ be simply-connected, since the 3-sphere isn't flat (i.e. doesn't admit a flat metric). Is any\/all of this true? If so (or even if not), what sorts of unexpected things might follow from the non-simply-connectedness of our universe?"}
{"id":"361","title":"Symmetrical twin paradox","text":"Take the following gedankenexperiment in which two astronauts meet each other again and again in a perfectly symmetrical setting - a hyperspherical (3-manifold) universe in which the 3 dimensions are curved into the 4. dimension so that they can travel without acceleration in straight opposite directions and yet meet each other time after time. On the one hand this situation is perfectly symmetrical - even in terms of homotopy and winding number. On the other hand the Lorentz invariance should break down according to GRT, so that one frame is preferred - but which one? **So the question is: Who will be older? And why?** And even if there is one prefered inertial frame - the frame of the other astronaut should be identical with respect to all relevant parameters so that both get older at the same rate. Which again seems to be a violation of SRT in which the other twin seems to be getting older faster\/slower... How should one find out what the preferred frame is when everything is symmetrical - even in terms of GRT? And if we are back to a situation with a preferred frame: what is the difference to the classical Galilean transform? Don't we get all the problems back that seemed to be solved by RT - e.g. the speed limit of light, because if there was a preferred frame you should be allowed to classically add velocities and therefore also get speeds bigger than c ?!? (I know SRT is only a local theory but I don't understand why the global preferred frame should not 'override' the local one). Could anyone please enlighten me (please in a not too technical way because otherwise I wouldn't understand!) **EDIT**   Because there are still things that are unclear to me I posted a follow-up question: Here"}
{"id":"91405","title":"General relativity and global aspects","text":"The theory of general relativity tells me something about the global structure of space-time, eg simply connected ?"}
{"id":"64415","title":"Our Universe Can't be Looped?","text":"With reference to the Twin-Paradox (I am new with this), now information of who has actually aged comes from the fact that one of the twins felt some acceleration. So if universe was like a loop, and the actually travelling twin again reached earth after completing the loop, then no such information would have biased the actually travelling twin, and the paradox will still remain (?). And universe thus can't be a loop, it must have an end point?"}
{"id":"1787","title":"What is known about the topological structure of spacetime?","text":"General relativity says that spacetime is a Lorentzian 4-manifold $M$ whose metric satisfies Einstein's field equations. I have two questions:   1. What topological restrictions do Einstein's equations put on the manifold. For instance, the existence of a Lorentz metric implies some topological things, like the euler characteristic vanishing.   2. Are there any experiments being done or even any hypothetical experiments that can give information on the topology? E.g. is there a group of graduate students out there trying to contract loops to discover the fundamental group of the universe?"}
{"id":"2175","title":"Is it possible for information to be transmitted faster than light by using a rigid pole? ","text":"Is it possible for information (like 1 and 0s) to be transmitted faster than light? For instance, take a rigid pole of several AU in length. Now say you have a person on each end, and one of them starts pulling and pushing on his\/her end. The person on the opposite end should receive the pushes and pulls instantaneously as no particle is making the full journey. Would this actually work?"}
{"id":"19873","title":"light travels a maximum speed... \/?","text":"> **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light? we know that speed of light is an unconquerable term in physics..light takes about 1 year to travel 9500000000000000 metres that is light year....suppose i have a needle of length 95000000000000000 metres long....suppose that one end of needle is at my hand and other end is at distance 95000000000000000 metres apart. . now if i push gently the end of needle at my hand then at the same time the other end at a distance of 95000000000000000 metres will move forward.....now light has taken 1 year to travel a distance of 95000000000000000 metres..and i have made a mechanical disturbance to travel the same distance with in no time from one end of the needle to other end placed at a distance of 9500000000000000 metres apart. . in other words i have made a mechanical disturbance to travel faster than light.. . ."}
{"id":"108836","title":"Send a signal from Earth to a planet billion light years away instantly","text":"If light was infinitely fast, we could just send a light signal from Earth to the planet. But I was wondering, if we made a perfectly non-elastic rope as long as the distance between earth and the far-away planet. Could somebody pulling the rope from the earth send an instant signal to a person holding the end of the rope on the far-away planet? Will the rope side on the other planet move at the exact same moment as the rope side on the Earth? Of course, we should ignore the obvious factors making this means of communication impossible -_-"}
{"id":"87835","title":"Gravitational Effect Versus the Speed of Light","text":"If, for some reason, the sun were to suddenly disappear altogether, I would like to know the following:   * would we \"feel\" it first (i.e. being thrown into outer space due to no longer having anything to orbit);   * would we see the sun disappear at precisely the same moment that we \"feel\" it?; or   * would we see it before we feel it (shortly before inevitably being subsequently thrown into outer space)? For reference: light from the sun takes ~8.5 minutes to reach us here on Earth. I have read that gravitational waves travel at the speed of light, however I personally would imagine that we feel it first for this reason: how could we possibly \"feel\" something **8.5 minutes after it has happened** if it is something as major as planet Earth being thrown into outer space? Surely we wouldn't be hurtling through space for 8.5 minutes before noticing, would we? But then that would lead us into the realm of the Universal Speed Limit and how, in theory, nothing should be able to exceed it (except, perhaps, subatomic particles being able to communicate instantly (MUCH faster than the speed of light), but that is a whole other question and not for discussion here). Any help would be great. Mods: if you feel that this question belongs in a theoretical Stack site, then please kindly move it to the appropriate one. Many thanks."}
{"id":"60790","title":"Solar Catastrophe","text":"Consider all of sudden the sun vanishes. What would happen to planetary motion. Will it continue to move in elliptical path or move in a tangential to the orbit immediately after sun vanishes or move in elliptical orbit for some time after the vanishing of sun or any other cases? If so, please explain..."}
{"id":"74029","title":"'Push' in a rigid rod travel at speed of sound or speed of light","text":"Two person, $A$ and $B$, each holding one end of a long solid rod. Now person $A$ pushes the rod on one end. **Question** : Is it correct that the information that the rod has been pushed will travel to the other end at the speed of light whereas the actual 'push' will travel at the speed of sound in the rod? i.e. If the rod has length $ ct $ , then will person $B$ feel the push in time $t$ or $ct\/v$? ($c$ is speed at which EM waves propagates in this experiment, and $v$ is the speed of sound in the rod.)"}
{"id":"13952","title":"How fast gravity force propagates?","text":"> **Possible Duplicate:**   >  The speed of gravity Lets say for example that a pertubance occured on the Sun that changes its distance to the Earth. How fast we are going to notice the gravity changes? Is it immediately? Light speed?"}
{"id":"40968","title":"Do the changes of gravity travel across distances instantly or limited to the speed of light?","text":"> **Possible Duplicate:**   >  The speed of gravity It takes a long time for a radio signal to travel to the planet Mars. What if we made a special type of radio that could detect small changes in gravity from a fall away object. For example, a satellite in orbit around Earth. That satellite had a device that allowed it to make small changes in it's gravity field. The detector orbiting Mars could detect these changes, and convert it into a communications signal. Would these two devices be able to communicate with each other instantly using gravity? There by providing a real-time communications link between Earth and Mars?"}
{"id":"13297","title":"If you make a steel rod from here to Alpha Centauri and move it, will the movement appear there instantly?","text":"> **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light? imagine this theoretical situation:      You have make an extremely long piece of steel (or anything really) one end is placed on earth the other is all the way at Alpha Centauri, 4ly away.   Here on earth the end is hanging loosely, on Alpha Centauri there is a bell at the end of the rod. Now if you move our end straight in the direction of AC will the bell there ring instantly ? Or will it be slower\/same as speed of light.      If I take make it small scale, for example moving a steel rod with a bell on the end, I can't imagine that there would be a delay, but of course the speed of light is the limit, so what could cause the delay ? Thanks for explanation.   My very sketch: http:\/\/imageshack.us\/photo\/my-images\/221\/steelrod.png\/"}
{"id":"72598","title":"Steel rod to Mars vs the speed of sound, how is supersonic travel possible?","text":"I remember reading this passage in the \"Feynman Lectures\", where Dr. Feynman describes an experiment in which a theoretical metal rod of length equal to the distance between Mars and Earth is arranged between mars and earth. Then the \"rod handler\" on earth gives the rod a push upwards. So the question posed is, will the rod instantaneously move backwards on Mars? He then proceeds to say that actually the movement will be felt roughly after the time it would take for sound to propagate from Earth to Mars i.e. at the speed of sound. This is because the atoms in the rod actually propagate their position at the speed of sound. See also this Phys.SE post and links therein. **My question is:** If this were true, how is supersonic travel possible? Won't there be a scenario where the atoms in the components of the aircraft won't \"catch up\" because the propagation of the new positions is slower than the actual speed? Won't the aircraft disintegrate?"}
{"id":"12736","title":"Does gravity spread instantly?","text":"> **Possible Duplicate:**   >  The speed of gravity I am real noob in physics, so sorry if this question is really stupid. Today in a casual conversation I claimed that if the sun were to instantly disappear, we would have felt it in any way nos ooner than after about 8 minutes(which is the distance in light minutes from Earth to Sun). My friend said that light-wise I am right, but we would instantly have felt the gravitation change. I argued that gravitation doesn't spread faster than light. Am I right or is he right? Thanks in advance and sorry again for the noob Q :)"}
{"id":"103729","title":"Does gravity act instantaneously?","text":"some one states that the earth would travel in it path for 8 min if the sun vanished. The equations of gravity show the Moon would crash into the Earth and the Earth would head for Mars the instant the Sun is gone. Can someone show how there is a delta Time and the relaxing of the bend in space-time fields is finite?"}
{"id":"30497","title":"Here's a way to transmit data faster than the speed of light","text":"> **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light by using > a rigid pole? Assume there is a long rod or a string connecting two points separated by a distance of several light years. It will take light several years to travel, but by a simple pull or push of the string or rod we can transmit date in binary format, what do you think?"}
{"id":"69839","title":"what is the speed of gravitational interactions?","text":"Example. Two bodies with masses m1 and m2 distance between them one year of light. What happens to the force of attraction between the two bodies when one body suddenly disappears? Is the force disappears at the same time or the force will act on second body for one year?"}
{"id":"55731","title":"How much time does it take to affect?","text":"We know gravitational is continuously acting on us. But let us assume that we are hanging in the space alone away from anything to affect our position. And suddenly a giant planet appear's a few million km from us then we will feel it's gravitational force on our body. But I wanted to know will it work instantly or it will take some time to take effect just like light take some time to reach us ?"}
{"id":"102106","title":"Can Information Travel Faster Than The Speed Of Light?","text":"Many believe that nothing can travel faster than speed of light, not even information. Personally, i think theoretically information can. Consider this following imaginary experiment: Imagine we are living on a planet that is big enough for a, let's say, 10-light-seconds-tall tower to erect. We hang a pendulum near the planet's surface using a long thin wire at the top of the tower. If someone at top of the tower cut the wire then the pendulum will instantly falling to the ground. In this case we can say that the information \"someone cuts the wire\" travels 10-light-seconds distance in no time. Since someone on the surface can only see the act of cutting the wire 10 seconds later, can we infer that the information travels faster than light?"}
{"id":"112666","title":"Speed\/direction of gravity for a moving source","text":"Consider the Earth, and a bowling ball held 186,000 miles (1 light second) above it. When the ball is released, it will start to fall vertically downwards towards the Earth. Now consider the case if the Earth is moving sideways at 1000 miles\/second. The bowling ball is released just as the Earth passes directly underneath. Does the ball fall a) vertically again, or b) does it fall towards where the Earth was 1 second before? Gravity propagating at the speed of light, would suggest answer b) but as most of the matter in the universe is travelling at very high speed, and planetary orbits are circularish, I think the answer is a)"}
{"id":"21122","title":"If I move a long solid stick can I send message fastest than light?","text":"> **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light? I mean by using a perfect solid stick long enough and moving it forward and backward can I send information fastest than light ? Can you imagine a solid stick long enough to reach the moon and using it to comunicate with the lunar base. Will be information faster than light? What are the theoretical reason other than technical reason for rules this as impossible? We should rule that a perfect unelastic solid exist ?"}
{"id":"26742","title":"Does gravity travel at the speed of light?","text":"Whenever I did calculations in high school physics involving gravity, it was either \"a ball falling to the earth\" type scenario, or a basic measurement of the gravitational attraction between two planetoids. I think I read somewhere recently that gravitational changes \"aren't reflected instantly across the universe but instead propagate at the speed of light\" somewhat like the ripples in a pond expanding. Is this true? Is there experimental evidence to confirm or deny this? What is the theoretical basis for it?"}
{"id":"134446","title":"Gravity Concept Question","text":"Thought experiment: Imagine the sun suddenly disappeared; lets say it some how transported to the edge of the observable universe.What will be the effects on the space-time? (1) What does General Relativity (specifically) have to say about the disappearance of mass from the portion of space-time being analyzed? I predict that the space will oscillate ,like a spring; interacting with it self (assuming the mass was spherical, and did not cause a force to act upon space in any direction), until it dampens out due to the interference, and initial \"springiness\" of space.But I do not know, I am no GR specialist. (2) Can someone also tell about what will happen if the opposite happened, (a mass suddenly appeared)?"}
{"id":"80442","title":"How faster is the effect of force exerted by gravity?","text":"We all know that Earth revolves around the Sun due to Gravitational force of the Sun. Also we know that it takes just over 8 minutes for the Sun's light to reach Earth. Now let us say, hypothetically if the sun were to suddenly extinguish (or disappear), would the Earth still not experience any effect of it until just over 8 minutes later (equivalent to time taken by light to reach from Sun to Earth) ? That is, Would the earth still continue to revolve (due to the centripetal force) for next 8 minutes after Sun disappearance and then thrown out of its revolving orbit (due to centrifugal) OR will it be immediately thrown out on the moment the Sun disappeared (i.e., not after 8 minutes) ?"}
{"id":"111067","title":"Transfer of energy faster than the speed of light","text":"In a vacuum: On my left I have a rod that is 558000 miles long (3 light seconds). A button is secured to its end which activates a laser beam sent back to me once I've pushed the rod in the direction of the button. On my right I have a laser aimed at a reflector that is 558000 miles away pointing straight back toward me. I push the rod on my left and activate the laser on my right at exactly the same time. Which laser beam will reach me first, the one on the right, or the one on the left?"}
{"id":"23466","title":"sending information over a wire--mechanically","text":"> **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light? I've thought about this since I was a little kid. I know it isn't exactly feasible, but it still bothers me. I hand you a really long wire, and we agree that \"a long tug means 1, two short tugs mean 0\", then you move off into the galaxy, a few light-years away from me. I proceed to give you information by tugging on the wire. With a really tight wire, couldn't I talk to you faster than the speed of light?"}
{"id":"68746","title":"can we break the speed of light","text":"I know this is impossible but I just want to know where I went wrong. here's my proposal: Let us imagine that we have a source of light and a switch that are far one light year from us. so it would take light one year to get to us. If we had a very long rod that connects my hand to the switch. When I move the rod a little distance I will reach the swich. So the time needed for me to close the switch is less than that made by light.right? Would I break the speed of light this way?"}
{"id":"106535","title":"Spinning theoretical object moving faster than the speed of light","text":"Let's say you have the Earth, or any celestial body, spinning as it is. What if you build a tower from the surface, and extended it out into space. If it was built far enough, could the furthest end exceed the speed of light?"}
{"id":"51646","title":"Time taken for gravity of a distant object to interact with a newly created particle?","text":"> **Possible Duplicate:**   >  The speed of gravity   >  Does gravity travel at the speed of light? Imagine there is a large mass $m_1$ (e.g. a star) 1 light-year away from us. It is stable, stationary relative to us and has been in place for a long time, much more than a year. A small mass $m_2$ (e.g. a proton) has just been created locally, 1 light-year away from $m_1$. How much time does it take for $m_2$ to feel the gravitational pull of $m_1$, and how can this be explained with the virtual-graviton theory of gravity? Some possible answers I can imagine: a) Immediately $m_2$ interacts with virtual-gravitons sent by $m_1$, a year ago. b) 1 year. It takes this long for freshly launched virtual-gravitons from $m_1$ to reach $m_2$ and vice-versa before any effect is felt on either mass c) 2 years. There needs to be an exchange of information \/ virtual-gravitons between $m_1$ and $m_2$ and this is the minimum time it could take. d) None of the above"}
{"id":"118472","title":"Will we feel the gravity of a star 10 light years away for the next 10 years if, somehow, it vanishes today from its position?","text":"I was watching a relativity video, and although I am not sure, I felt that it was trying to tell that the effect of gravitation of a body is instantaneous, in the sense that a sudden change in the mass of a body will instantaneously be felt by any object at a distance, even before light can be exchanged between them. So I want to make sure if its true. You don’t have to explain me why this is so if you don’t wish to. Also, please tell me if this has been experimentally established, or it is just a theory."}
{"id":"46553","title":"Why doesn't pushing balls in a tube propagate the movement faster than the SoL?","text":"> **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light by using > a rigid pole? On one episode of QI they asked the question, \"How fast do electrons move travelling around an electric current.\" The answer is (more or less) \"very slowly\", with the explanation that it's not the electrons that move fast, it's the force (I think). They likened it to pushing on one end of a long tube of touching marbles and observing how quickly one fell out the other end. This made me think about the tube. I realise that this tube cannot allow a force to propagate through the tube instantaneously because that would be faster than the speed of light, so my question is, why not? * * * **TL;DR;** Why doesn't a marble exit a long tube filled with touching marbles immediately when a force is applied to the marble at the other end?"}
{"id":"56319","title":"How the effect travel's?","text":"Let us assume that we placed lot's of ball touching each other in a hollow cylindrical tube, now if we push one ball at the end the ball at the other end move's instantly. So how do the information from one side of the tube travel's to the other side of the tube instantly. and at what speed at it travel's. If it travels at the speed of light so if make a long tube of the length of about $3*10^8$m it would take 1 second to see the effect at the other end assuming that nothing will bend or compress all thing's are ideal? I have read other related question but it didn't contained what I actually wanted to know. So it is not a duplicate."}
{"id":"66632","title":"Can a pushed plank beat light and break the laws of physics?","text":"Imagine you are one lightyear away from a photon sensitive (light sensitive) switch. So it is obvious that light would take one year to reach to the switch. Now I have a one lightyear long plank. I simply point the plank towards the switch and press it. Now I just did work which light would take 1 year to do in a matter of seconds. Now the question is, did I break the laws of physics? Please do not give an answer like \" it is impossible to make a plank this long \". Just think about it. Even I am thinking about this question right now."}
{"id":"2272","title":"What happens when I move a very long bar?","text":"> **Possible Duplicate:**   >  Faster than light information Hey there, I just want to make clear that I'm new here and I don't know whether this is the right place to ask such a hypothetical question. So if it's not, I'd be glad if someone could tell me where to ask. :) Imagine you and a friend of yours were very far away from each other, let's say the distance is 1 light-year. Now you're holding a bar which measures exactly 1 light-year and currently touches your friend's shoulder. What happens when you move that bar (in order to poke your friend)? Will there be a delay? And what magnitude will that delay have? A year? Several years? Looking forward to your answers. :)"}
{"id":"107798","title":"speed of gravitational waves","text":"Do gravitational waves have a certain speed? Is it the speed of light or infinite, or am I misunderstanding what a gravitational wave is? I think it is a ripple in spacetime caused by interactions between objects, but i could be wrong."}
{"id":"14390","title":"Faster than light communication","text":"> **Possible Duplicate:**   >  Is it possible for information to be transmitted faster than light? Suppose you have a very long rod, both ends are in space and the rod is 100 light years long. If you were to push the rod forward at one end, how long would it take for this be registered at the other end?. Is there theoretically a material the rod could be made of that would allow this?. Is this a possibility for faster than light communication across astronomical distances?. Also is \"intermolecular\" actually a word?."}
{"id":"116333","title":"Information faster than light?","text":"Imagine that you have two typewriters on the other side of the galaxy. Their typebars are connected by ropes (stretched to the maximum), so anything you type will instantly appear at the second typewriter. Are the sentences information? If so, why not send information faster than light? If not, I'd at least ask if such a transfer is actually immediate. ![enter image description here](http:\/\/i.stack.imgur.com\/8ft0S.png) Thank you very much."}
{"id":"106718","title":"The scissor paradox: can we pass the information faster than light?","text":"click to view the image   Before I start, I want to say that this is not a duplicate of \"Is it possible for information to be transmitted faster than light by using a rigid pole?\", Since point A is not a real object, it is possible for A to exceed the speed of light. **_Notice that I have already taken the bending effect into account and decided that if the widths of the bars are small enough, then it's safe to treat the scissor as if it weren't bending_**. In other words, when they are thin, then the motion will obey the equation in a small region around point O. Please read to the end and have a look on my analysis in the last part before you explain.    Suppose we have a \"scissor\", which is composed by two bars with a same width of length \"l\". Bring them closer to each other, during this process, we have the equation(#):$$v=-\\frac l4\\frac{cos\\frac{\\theta}2}{sin^2\\frac{\\theta}2}\\omega$$, where v is the velocity of A, \\theta is the angle between BC and BA as shown in the picture.It's also an arbitrary function of time t, or $\\theta=\\theta(t)$. $\\omega$ is the angular velocity of $\\theta$, thus $\\omega=\\frac d{dt}\\theta(t)$.Here is the derivation:   1\\. Draw a line vertical to L from A, intersecting with L at C.     1. Since $sin\\theta=\\frac{AC}{AB}$, $$AB=\\frac{AC}{sin{\\theta}}$$.   2. Since $cos\\frac{\\theta}2=\\frac{AO}{AB}$, $$AO=AB*cos\\frac{{{\\theta}}}2$$   3. from 2 and 3 we have: $$AO=\\frac{cos\\frac{\\theta}2}{sin{\\theta}}*AC$$   4. Since $AC=l$, the equation in 4 becomes: $$AO=\\frac{cos\\frac{\\theta}2}{sin\\theta}*l$$or$$AO=\\frac l{2sin\\frac {\\theta}2}$$   5. Differentiate each side with respect to t, and denote $\\frac d{dt}AO$ as \"v\", getting the equation(#): $$v=-\\frac l4\\frac{cos\\frac{\\theta}2}{sin^2\\frac{\\theta}2}\\omega$$    Again, we just look at a small area around point O where our equation is obeyed. Now, at any time $t_0$, $\\theta$ and $\\omega$ are independent of each other, thus $\\omega$ is a free variable. At t= 0, let's set the value of $\\omega$ really large and constant, so that the instantaneous speed of A is larger than even the speed of light. Later, since the factor $\\frac{cos \\frac{\\theta}2}{sin^2 \\frac {\\theta}2}$ in equation(#) is quite large when $\\theta$ is close to 0, and since large $\\omega$ leads to small $\\theta$ during a short period of time, the velocity of A will be increased by this factor. Thus overall, $\\omega$ is set to be a large constant, l is a constant, the only changing factor is increasing. Therefore, the velocity of A will increase. Meanwhile, we emit a beam of light. A is in a superluminal motion. And A will arrive at a detector first, causing the bars to touch the detector, and sending the command to the machine. Then the light beam comes later, thus being detected later. Notice that here again, we place the detector close enough to point O, so that our equation is valid. Remark: Someone may argue that we are not sure if A could attain a speed higher to that of light. So suppose at t=0, the force has already affected the area around point O, and is now traveling further.Meanwhile, ${\\theta}_0=\\frac{\\pi}3$ A is speeded up to 0.999c, then since our formula is valid at t=0 and later period of time, let's apply it and set $l=0.001m, c=3*10^8m\/s,{\\theta}_0=\\frac{\\pi}3$. Then plug them in, we get $$\\omega=-3.461*10^{11}rad\/s $$. Now we want to know its velocity at $t=1*10^{-12}s$, and we get $v=6.89062*10^8m\/s$, faster than light.    According to relativity, this shall violate casualty, and therefore could never happen. With which kind of mechanism is this paradox solved? I have taken the bending effect into account, and decided that as long as the widths of the bars are small enough, then it's safe to treat the scissor as if it weren't bending."}
{"id":"5456","title":"The speed of gravity?","text":"Sorry for the layman question, but it's not my field. Suppose this thought experiment is performed. Light takes 8 minutes to go from the surface of the Sun to Earth. Imagine the Sun is suddenly removed. Clearly, for the remaining 8 minutes, we won't see any difference. However, I am wondering about the gravitational effect of the Sun. If the propagation of the gravitational force travels with the speed of light, for 8 minutes the Earth will continue to follow an orbit around nothing. If however, gravity is due to a distortion of spacetime, this distortion will cease to exist as soon as the mass is removed, thus the Earth will leave through the orbit tangent. What is the state of the art of research for this thought experiment? I am pretty sure this is knowledge that can be inferred from observation."}
{"id":"114798","title":"Consequences of inverse square law with vast distances (Gravity); (in addition, is light speed broken)?","text":"As is well known, the gravitational force between two masses is dependent on the spatial distance between them. Therefore, even at vast distances, the masses exert equal and opposite forces on one another. I know that our theories need for this to be the case (to derive things such as escape velocity, which calls the force at r = $\\infty$ to be zero (by utilizing a limit where the distance approaches infinity)), yet the universe is so vast and complex (with an unimaginable number of masses). With so many masses in space, it seems as though this could make a huge difference (over a large distance, such as special relativity's involvement in magnetism) . Also, say there is a supernova 400 light years away from earth. The time independent theory states that we should immediately notice a difference in the net forces acting on us, yet that would mean information travelled faster than light, correct? My main point is this: Is it absolutely imperative that there is still a force infinitely far away? Would the universe still function if we were to cut it off at something like a light year? ***** (This is my main point) Does the infinite reach of the gravitational force have anything to do with the scaffolding of the universe as a whole ?"}
{"id":"64676","title":"Relativistic canonical transformation","text":"What is relativistic canonical transformation? I need every piece of information about it. Does anyone know a reference or an article about relativistic canonical transformation? For example, in classical mechanics, under one and only one condition, you can say that a transformation is canonical and that is: $J\\cdot M\\cdot J^T= M$   where $J$ and $M$ are two matrices which are represented in _Goldstein (3rd Edition) - Page 342_"}
{"id":"45448","title":"is there a way to split a black hole?","text":"Classically, black holes can merge, becoming a single black hole with an horizon area greater than the sum of both merged components. Is it thermodynamically \/ statistically possible to split a black hole in multiple black holes? If the sum of the areas of the product black holes would exceed the area of the original black hole, it seems to be a statistically favorable transition by the fact alone that would be a state with larger entropy than the initial state"}
{"id":"53738","title":"Ashcroft Mermin Solid State Physics Eq. 2.60ff","text":"I'm trying to follow the steps in Eq. 2.60 of said book. What I cant seem to figure out is how to change the integration variables from 'k' to 'E', as they state. The equation is $$\\int \\frac{d\\textbf{k}}{4\\pi^3} F(\\epsilon(\\textbf{k})) = \\int_0^\\infty \\frac{k^2 dk}{\\pi^2} F(\\epsilon(k)) = \\int_{-\\infty}^\\infty d\\epsilon \\, g(\\epsilon) F(\\epsilon)$$ I can follow the first transformation (why is $\\textbf{k}$ suddendly $k$?), $$\\int\\frac{1}{4\\pi^3} k^2 F(\\epsilon(k)) \\, dk \\int_0^\\pi \\sin \\theta \\, d\\theta \\int_0^{2\\pi} d\\phi = \\int_0^\\infty \\frac{k^2 dk}{\\pi^2} F(\\epsilon(k))$$ But what's happening in the second step is unclear to me. In the book it says, \"one often exploits the fact that the integrand depends on $\\textbf{k}$ only through the electronic energy $\\epsilon = \\hbar^2k^2\/2m$,...\", but I'm unsure how this is used. Could anybody point this out to me?"}
{"id":"63021","title":"Gas Laws And Adiabatic Process","text":"Air at 20 degrees Celsius is compressed adiabatically from 1 bar to 10 bar, what will its temperature be? With $$P_1 = 1,$$ $$P_2 = 10$$ $$T_1 = 293K,$$ $$T_2 = unknown$$ using $$\\dfrac{P_1}{P_2}=\\dfrac{T_1}{T_2}$$ my solution was $$\\dfrac{1}{10}=\\dfrac{293}{T_2}$$ giving $$T_2 = 2930$$ My physics tutor said this is wrong and I should use $P_1V_1^\\lambda=P_2V_2^\\lambda$ and then use $\\dfrac{V_1}{V_2}=\\dfrac{T_1}{T_2}$ to find the temperature. The only problem I now have is that $V_1 = V_2 = unknown$ which is making me think that he may have forgot to add that information to the question. I would like to know if there is anyway to solve this (I do not want the answer just a point in the right direction). thanks in advance!"}
{"id":"9258","title":"Why are the third generation superpartners lighter than the other sfermions in MSUGRA","text":"In the MSUGRA breaking scenario, the stop particle typically appears at energies reachable at the LHC. Other sfermions, notably the partners of up, down, strange and charm are assumed to be degenerate in mass, and also heavier than the stop. Something similar holds for the stau. Why is the third generation different in MSUGRA (not degenerate as the first two), and why is the mass hierarchy inverted wrt. the Standard Model sector (3rd generation sparticles lighter)? (I guess these features are not neccessarily specific to MSUGRA, but might apply to more general models as well.)"}
{"id":"9259","title":"Time dilation when falling into black hole","text":"I know that if one astronaut falls into a black hole, then a distant observer will see him take an infinite amount of time to reach the event horizon (provided the observer can see light of arbitrarily large wavelengths). But the falling astronaut will only take a finite amount of time to reach the horizon. My question is: What will the falling astronaut see if he \"looks backwards\" while falling. Will he see the distant observer growing old and all the stars dying by the time he reaches the horizon ?"}
{"id":"95796","title":"EQUAL TIME commutation relations","text":"Why is equal time commutation relation used in canonical quantization of free fields?"}
{"id":"10701","title":"How much force is in a keystroke? (estimated, of course)?","text":"I'm a software developer, and I need to calculate the estimated amount of force expended typing stored text. Preferrably in some interesting way. (i.e. the force exerted on keys thus far is enough to push a car 5 miles) (or: equivalent to 100 kg of TNT) Assumptions:   * We're not going to worry about deletes or moving the cursor or anything, I'm just counting characters of stored text.   * I don't really care if the space requires more force or not, this is more of a \"fun fact\" than anything.   * From what I've found online, **the mean force required for a keystroke is about 12.9N** (source).   * Hundreds of millions of characters have been typed. Questions   * What is a good way to make this something people can relate to?   * How can I calculate it? Thank you all in advance for your valuable time and input. EDIT: I thought my original post would make this pretty clear: I realize I only have the _force_ required to push each keystroke. I'm looking for a way to demonstrate that force _applied_ to something to help people quantify it, hypothetical energy in terms people can understand."}
{"id":"9252","title":"Has quantum entanglement been demonstrated to be able to take place over infinite distances?","text":"In my poor understanding of quantum physics, quantum entanglement means that certain properties of one of two 'entangled' quantum particles can lead to change over infinitely large distances when the other particles' properties are changed. Disregarding this already mind-boggling event taking place over say 10 meters distance; how have physicists been able to demonstrate, beyond reasonable doubt, that this can take place over infinitely large distances? For instance: have they done some of these tests between ISS and Earth perhaps? How can they be so sure?"}
{"id":"123156","title":"What is imaginary time?","text":"I am not professional physicist; but I am curious about Stephen Hawking's \"imaginary time\". It would be better to elaborate exactly what it is. I am not confused because of the word \"imaginary\" but I find it confusing to imagine a two dimensional \"plane time\". If we express time in a plane instead of a one dimensional axis, then what does the movement of an observer along the imaginary axis signify physically?"}
{"id":"17524","title":"Path to obtain the shortest traveling time","text":"Asume we have a particle sitting at the point A(0,0) in a gravitational field. (g=9.81) It is going to move along some path to the point B(a,b) Where a>0 and b<0\\. What is the curve the particle needs to be moving at to arrive at B in the least time possible? (Fastest curve between two points. ) ![Example image,, various possible paths.](http:\/\/i.stack.imgur.com\/D9fRA.png)"}
{"id":"90314","title":"Specify the Stress Energy Tensor and Calculate the Curvature","text":"I have a simple question about general relativity and the Einstein field equations, I wonder if you can specify the stress energy tensor, i.e. specify some mass distribution in space and then calculate the curvature to later find equations of motions etc, instead of starting out with how the geomerty would look. I am quite new to general relativity and so I am bound to have misconceptions. **Edit: 2013 December 19th** I have found this article which at page 10, Chapter 5, section 5.2 does something simillar to what I meant, apperently there is a general from for the Stress energy tensor (for what is known as a perfect fluid(?)), and from it they derive something simillar to the second component in the normal schwarzschild metric i.e $$A(r)=(1-\\frac{2U}{r})^{-1}$$ where $U$ is the energy. I do have one remaining question, the name of the general form of the stress energy tensor confuses me somewhat, \"perfect fluid\" is it just its name, and is it still fully capabable of describing the stress energy tensor in general relativity?"}
{"id":"90317","title":"Spring problem?","text":"I came across this problem in physics \"Physics for Scientists and Engineers with Modern Physics by Serway\" > _A block on the end of a spring is pulled to position $x = A$ and released > from rest. In one full cycle of its motion, through what total distance does > it travel?_ Why the answer is $4A$ instead of $2A$?"}
{"id":"90319","title":"What is the reasoning behind the Hill Sphere?","text":"According to Wikipedia, Hill Sphere is : the volume of space around an object where the gravity of that object dominates over the gravity of a more massive but distant object around which the first object orbits. True as this may be, it just mathematically supports a phenomenon that has been observed but it does not give reason or logic as to why does this happen in the first place. I mean why should the gravity of a less massive object dominate the gravity of a more massive one? I wasn't aware of the Hill Sphere until recently when I was trying to visualize the orbits of different celestial bodies. The Hill Sphere comes closest to explaining why the moon orbits the Earth, more than it orbits the Sun and why the Earth orbits the sun, more than it orbits the center of our galaxy. By this logic all celestial bodies within the Gravitational pull of the center of our galaxy should directly be orbiting the center. My argument is that if the Hill sphere of the Sun is as large as the solar system itself, any object within this sphere should be orbiting the sun. Why was the moon caught into the earth's gravitational pull in the first place when it had a much stronger pull from the sun? The answer to this would also eventually clarify why the earth orbits around the sun and not the center of the milky way."}
{"id":"134147","title":"Add air resistance to projectile motion","text":"I am given an initial x and y position and initial velocity and I was asked to graph the trajectory in 1 second intervals. This is what I have so far: If $x_0 = 1, v_{0x} = 70, y_0 = 0, v_{0y} = 80, a_x = 0, a_y = -9.8$, and time will be 0,1,2,3... and so on. Using these equations on every second you can find the plot will be a bell shaped with the highest point being ~ 325 m at about 600 seconds: $$ x = x0 + (v_{0x})t + 1\/2((a_x)t^2) $$ $$ y = y0 + (v_{0y})t + 1\/2((a_y)t^2) $$ Usually in physics, we are taught in perfect condition with no air resistance. But what if there was air resistance? How would it affect this problem? How can i add it to my calculations and see what the difference is?"}
{"id":"112950","title":"Can an object be deconstructed on an atomic Level?","text":"Can a machine deconstruct objects on an atomic level. But is that possible? Not the machine per say but the simple (not that simple) act of what it does. Ex.Taking a broken computer and separating the different atoms so it would be just a group of atoms instead of a computer. If you have any suggested readings on the topic that would be greatly appreciated. Q: Can an object be broken up to be just separate atoms?"}
{"id":"112953","title":"If I charge a battery using a much higher amperage, can it explode?","text":"If I have a 12V 4Ah lead acid battery and use a battery charger that, let's say for example, can charge 10A, 50A, or 100A. If I theoretically turned it to 100A will the battery explode? I understand that when you use a higher amperage the battery will charge quickly but due to resistance and flow of ions a lot more heat will be generated, so will this heat cause an explosion..or perhaps just a bursting of that battery spewing boiling acid? And no I am not trying this in real life..I just recall seeing the scene in the Amazing Spider-Man 2 when Parker is trying to build his web shooters to be able to resist large amounts of electricity yet they keep exploding."}
{"id":"39442","title":"Intuitive explanation of why momentum is the Fourier transform variable of position?","text":"Does anyone have a (semi-)intuitive explanation of why momentum is the Fourier transform variable of position? (By semi-intuitive I mean, I already have intuition on Fourier transform between time\/frequency domains in general, but I don't see why momentum would be the Fourier transform variable of position. E.g. I'd expect it to be a derivative instead.)"}
{"id":"118800","title":"If you were able to get rid of Hydrogen from a weather balloon as it were rising and expanding would the weather balloon rise further than usual?","text":"I know that the weather balloon will eventually be stopped because of the atmosphere no longer being buoyant but would the balloon be able to go farther than it regularly would?"}
{"id":"13052","title":"Is the earth expanding?","text":"I recently saw this video on youtube: http:\/\/www.youtube.com\/watch?v=oJfBSc6e7QQ and I don't know what to make of it. It seems as if the theory has enough evidence to be correct but where would all the water have appeared from? Would that much water have appeared over 60 million years? Also what would cause it to expand. The video suggests that since the time of dinosaurs the earths size has doubled in volume, how much of this is and can be true? [could someone please tag this, I don't know what category this should come under]"}
{"id":"118808","title":"Solving electromagnetic vector field using the Lagrangian","text":"Given an action of the form \\begin{equation}S=-\\frac{1}{4}\\int d^4x\\eta^{\\mu\\nu}\\eta^{\\lambda\\rho}F_{\\mu\\lambda}F_{\\nu\\rho}\\end{equation} where $F_{\\mu\\nu}=\\partial_{\\mu}A_{\\nu}-\\partial_{\\nu}A_{\\mu}$, $\\eta_{\\mu\\nu}=g_{\\mu\\nu}\/a^2(\\eta)$, where $g_{\\mu\\nu}$ is given by the line element: \\begin{equation}ds^2=a^2(\\eta)[d\\eta^2-(dx^i)^2]\\end{equation} I would like to solve for $A_{\\mu}$, and standard solution is \\begin{equation}A_{\\mu}^{(\\alpha)}=e_{\\mu}^{(\\alpha)}e^{ik_\\nu x^\\nu}.\\end{equation} I am interested in knowing how to derive this result. My approach is first write the Lagrangian from action and use EL eq \\begin{equation}\\frac{\\partial \\mathcal{L}}{\\partial A_{\\mu}}-\\frac{d}{d x^{\\nu}}\\frac{\\partial \\mathcal{L}}{\\partial(\\partial_{\\nu}A_{\\mu})}=0\\end{equation} My main problem is mathematical difficulty in evaluating the EL eq. Can anyone please help me on this?"}
{"id":"62088","title":"Scalar top quark (stop) pair production","text":"A rather simple question: Starting from an electrically neutral state, pairs of top quarks are produced as top and anti-top, and denoted as $t\\bar t$. Now the production of pairs of scalar top quarks, the supersymmetric partners of the top quarks, seems to be commonly denoted as $\\tilde t \\tilde t^*$ (e.g. 1, 2), rather than $\\tilde t \\bar{\\tilde t}$. Why this notational difference? What is $\\tilde t^*$?"}
{"id":"20251","title":"Electron shell bombardment","text":"If you bombard an electron shell with a photon below the critical level to promote the electron to a higher state, will the shell absorb nothing and the photon get deflected with the same amount of frequency\/energy that it came with initially?"}
{"id":"129185","title":"Normalising a wave function in parts?","text":"If we have the wave function $\\psi_{100}(r,\\theta,\\phi)=R_{10}(r)Y_{00}(\\theta,\\phi)$ when we are normalising it we do the following: $$1=\\int| \\psi_{100}(r,\\theta,\\phi)|^2sin(\\theta) r^2drd\\theta d\\phi$$ but can we also normalise the individual parts separately i.e. $$1=\\int r^2|R_{10}(r)|^2 dr $$ and $$1=\\int |Y_{00}(\\theta,\\phi)|^2 sin(\\theta)d\\theta d\\phi $$ I ask this as this is what my textbook is doing and knowing integration it seems wrong! Please if this is right can you explain it, thanks."}
{"id":"1171","title":"Resonance in a gravitational field?","text":"Assume that there are only well behaved functions as mass distributions, and there are no other forces except gravitation. Is it than possible to create an arrangement where a variation of a certain quantity (could be mass density or gravitational field or momentum) has a resonance?"}
{"id":"23508","title":"Computer game with quantum optics\/ information","text":"Is there a computer game using principles of quantum optics or quantum information? By game I don't mean just a simulation or an interactive course, but something that can be played in an enjoyable way. By 'principles' I mean actual rules, not just things bearing the same name or vaguely related. As a reference, let me use Chromatron \\- it is a puzzle game using principles of geometrical optics and color mixing. (There are some non-physical or unrealistic elements, but most most of them - like mirrors, prisms, beamsplitters and filters - are based on real physics)."}
{"id":"129232","title":"How can I describe an equation for multiple objects over time?","text":"Lets say i have $n$ different objects that effects each other only by the classic gravity force. I have their initial locations, masses and velocity's: $$ x_1(0),\\cdots,x_n(0) $$ $$ m_1,\\cdots,m_n $$ $$ v_1(0),\\cdots,v_n(0) $$ Is it possible to describe each object location with an equation? $$ x_i(t)= ? $$ I tried doing this : $$ x_i(t) = x_i(0)+\\int_0^t v_i(t') dt' $$ $$ v_i(t')=v_i(0)+\\int_0^{t'} \\frac{1}{m_i} \\sum\\limits_{j=1}^n \\frac{Gm_im_j}{r_{i,j}(t'')^2} dt'' $$ Now I'm stuck because the distance function $ r_{i,j} $ is depends on $ x_i $ and $ x_j $ which makes the whole thing circular..."}
{"id":"129231","title":"Thought experiment regarding perpetual motion machines and paradoxes","text":"A paradox (Ex.: if someone goes back in time to kill their own grandfather, then they won't exist, but then they wouldn't be able to go in time to kill their grandfather, but then they WOULD exist, but then they would go back in time...) is, really, the only \"perpetual motion machine\" (and I know I use that term loosely here) that works, since the conclusion of one argument causes the beginning of the argument to be false, which changes the conclusion, which changes the premises of the argument, ad infinitum. Could this mean that if a fully verifiable paradox were ever found in action within the physical world, the properties of said paradox might be able to power an actual, physical perpetual motion machine? Perhaps the wave\/particle duality of light might be considered such a paradox, or possibly the properties of quantum particles. If we could somehow put their inherent paradoxical qualities to practical use, we might have a solution to problems that, up until that point, have seemed to us to be, well, paradoxes. DISCLAIMER: I am not an upper-level scientist; this was merely a thought experiment. In all honesty, I probably have a number of concepts flubbered around in my head, which why I'm asking here - the home of many more- capable\/knowledgeable-than-me actual scientists."}
{"id":"129235","title":"How to numerically solve a complex equation?","text":"I want to know that if you are given a very complex equation g(x)=A(T). How could you solve for x, which is a function of variable T. To be more specific, I encounter a polylogarithmic function I need to solve numerically.."}
{"id":"12834","title":"Microsecond trading with neutrinos","text":"The Spread Networks corporation recently laid down 825 miles of fiberoptic cable between New York and Chicago, stretching across Pennsylvania, for the sole purpose of reducing the latency of microsecond trades to less than 13.33 milliseconds (http:\/\/www.spreadnetworks.com\/spread-networks\/spread- solutions\/dark-fiber-networks\/overview). The lesson I would draw from this is that, in the near future, oil and natural gas extraction won't be the only lucrative use of ocean platforms. So here's my question - since trades are occurring on the scale of tens to hundreds of microseconds, and considering the amount of money involved, can one use neutrino beams to beat the limitation due to having to travel the great-circle\/orthodromic distance between two trading hubs? I'm imagining something similar to the MINOS detector (http:\/\/en.wikipedia.org\/wiki\/MINOS), where a neutron beam was generated at Fermilab in Batavia, Illinois, and detected ~735 km away, ~700 meters under the ground in a Northern Minnesota mine. Is it possible to beat a signal traveling at the speed of light across the great-circle distance from, say, New York to Tokyo, using a neutron beam traveling the earth? Is it realistic to talk about generating these beams on a microsecond time-scale? Addendum - Over what distances can you reasonably detect a neutrino beam?"}
{"id":"12839","title":"Did classical applications of density functional theory precede its use as an electronic structure method?","text":"Density Functional Theory (DFT) is usually considered an electronic structure method, however a paper by Argaman and Makov highlights the applicability of the DFT formalism to classical systems, such as classical fluid density. This presentation by Roundy et al. calculates water properties using a classical approach, with an eye to combining it with KS DFT for calculating solvent effects. Not being too clear on the history of DFT, _was DFT invented for electronic structure problems first_ , and its applicability to classical problems incidental, or was it formulated for classical systems first and then adapted to the quantum many body problem?"}
{"id":"65883","title":"Huge buildings affect Earth's rotation?","text":"Does constructing huge buildings affect the rotation of the Earth, similar to skater whose angular rotation increases when her arms are closed comparatively than open?"}
{"id":"8540","title":"Some questions on Conformal Field Theory, Current algebras and the Sugawara construction","text":"Since I don't know how to add another question to an already existing topic, I'm opening a new thread. However I'm referring to: Beginners questions concerning Conformal Field Theory As noted, a few weeks ago I started reading about Conformal Field Theory. I'm actually from a more mathematical background, however I'm not very familiar with Quantum Field Theory. Though I'm quite familiar with Quantum Mechanics\/Classical Mechanics. Now again some questions again turned up:   1. Think of a theory with an energy-momentum-tensor that is given on the plane. Let's assume the most general form $T(z)=\\sum z^{-n-2} L_{n}$ and $L_{n} = \\frac{1}{2 \\pi i} \\oint dz z^{n+1} T(z)$. Now some of my reference (such as David Tong in the reference question above) point out that $L_{0}$ generates scalings\/rotations and $L_{1},L_{-1}$ generate translations. So let's consider the example of a rotation. The generator of a rotation is $z \\frac{\\partial}{\\partial z}$. Now in order to show that $L_{0}$ actualyy generates this rotation one needs to show that $[L_{0},\\phi]$ = $z \\frac{\\partial}{\\partial z}$ \\phi. I've shown this for the example of the free boson, however I'm not 100% sure how to prove it in the general case. Can someone help me? (Maybe it's related to Operator Product Expansions...)    2. The second question goes a bit deeper into the theory. It concern Current Algebras. I've read some articles on the Sugawara construction and there Mr Sugawara proposes and Energy-Momentum-Tensor of the form $T(z) = \\gamma \\sum_{a=1}^{dim g} : j^{a}(z) j^{a}(z):$ . However I don't really see how he comes up with it or why this seems to be a \"natural choice\" of an Energy-Momentum tensor. I've heard that it includes the Energy Momentum Tensor of the free boson (given by $T(z)=\\partial_{z} \\phi \\partial_{z} \\phi$) as a special case. For me this is not so obvious. Can someone please explain to me how he comes up that in an easy way. I don't think it's necessary to show me all the calculations. Just the basic idea would be useful to get some intuition.    3. I'm having some troubles on understanding the intuition behind current algebras. (I haven't read about WZW Models yet). The Virasoro algebra appeared to me in a kind of natural way in the example of the free boson. The generalization is then pretty much straight forward. However I don't have that kind of intuition for current algebras. I've read that they provide some \"additional symmetry structure\" which reduces the number of possible correlation functions. But I don't know any details. I'd be more than happy if someone could comment on that."}
{"id":"119041","title":"A question on lowering the total spin","text":"Is there a way to lower the total spin of the state and fixing the $S_z$ rather than lowering the $S_z$ by spin ladder operator? Or in other words, how to connect the $S=1$ state with $S=2$ or $S=0$ state? Is there such an operation though it might be unphysical?"}
{"id":"55624","title":"Do electrons have definite and single value of momentum and position?","text":"Do electrons (individually) have definite and single value of momentum and position or do they simultaneously have multiple position (a spread) at a time? In other words, according to the uncertainty principle, is it just impossible to _measure_ the exact position and momentum or is it actually impossible for an electron to _have_ a single position and momentum? Consequently, assuming it's just impossible to measure the exact position and momentum, does this mean that in atomic orbitals the electrons actually do occupy single position at a certain time (instead of somehow having 'smeared out' multiple positions at a time). How does velocity apply to atomic orbitals? Do electrons just randomly teleport according to the probability distribution or do they actually travel in 'normal' trajectories (going step by step instead of teleporting, for example footballs travel in step by step and don't randomly teleport AFAIK) Edit: Also, does $\\hbar$ in the uncertainty principle equation, $\\Delta x \\Delta p = \\frac{\\hbar}{2} $ arise from Planck length ( **From my understanding, the impossibility to measure the exact position and momentum arises from the quantization of length and time**. Is my understanding correct?)"}
{"id":"88396","title":"Are the polarization field and electric field in LIH dielectric in dynamic equilibrium?","text":"If we know that the Polarization P in LIH dielectrics is proportional to the net field inside the dielectric according to: P = ε0χeE.....(1) And we know that D = εE........(2) Does it not follow that we can ascertain the polarization directly from the applied (free charge) field, since we can relate D to E, and then E to P? The author of my electrodynamics text (Griffiths), says that we cannot. His explanation being that once we place dielectric in an external field E0, the material will polarize and create an opposing field to the applied field, which in turn modifies the polarization again, and this process repeats over and over. In actuality, are these two quantities (E and P) in some sort of dynamic equilibrium within the material? If so, how come (1) and (2) are valid?"}
{"id":"52817","title":"What is the Riemann curvature tensor contracted with the metric tensor?","text":"Can the Ricci curvature tensor be obtained by a 'double contraction' of the Riemann curvature tensor? For example $R_{\\mu\\nu}=g^{\\sigma\\rho}R_{\\sigma\\mu\\rho\\nu}$."}
{"id":"22674","title":"What's the validity of the thermodynamic definition of entropy?","text":"Thermodynamic Entropy Variation is defined as $$\\Delta S = \\int_i^f \\frac{dQ}{T},$$ where $i$ and $f$ are the initial and final states of the process. My question is: **does this equation apply to quasi-static irreversible processes, or only to reversible processes?** Obviously, it does not apply to processes that go through non-equilibrium states, since Temperature (or any state variable) is not even well defined in these states. But I'm unsure of whether it applies to irreversible processes that are quasi-static (and therefore don't go through non-equilibrium states)."}
{"id":"109776","title":"How long would it take me to travel to a distant star?","text":"Suppose I wanted to travel to one of the recently discovered potentially Earth-like planets such as Kepler 186f that is 490 light years away. Assuming I had a powerful rocket and enough fuel, how long would it take me?"}
{"id":"127402","title":"Special relativity allows arbitrarily low travel times between two locations","text":"I wish I had a good way of illustrating this, but anyway, doesn't the following travel strategy allow you to get anywhere in arbitrarily little time? You're at rest at the origin of space-time, and you'd like to meet up with an object some distance away in the positive x-axis - in other words, you'd like to reach the vertical world line of that object. You begin traveling towards the object at t = 0 in the following 3 phase trip: **I**. You accelerate nearly to the speed of light relative to the stationary frame of the origin and the object you're trying to reach. **II**. You level out your speed and coast. **III**. You decelerate right before you reach the object. As long as the observer travels on a space-time path that's always nearly lightlike, the proper time of his journey will be almost zero. Not that he sees a finite amount of distance go by him in zero time - the proper distance between the endpoints of his nearly lightlike journey is also nearly zero, so the speed of light is never exceeded by objects passed by the traveller. I realize that the travel time in the stationary reference frame is finite - it's the same as that of a light ray, and that during the acceleration phase ( **I**.), the traveller will see all of this finite time go by at his destination, but _he'll_ never experience that amount of time going by, right? If the above reasoning is correct, then there isn't really a limit on how far you can travel in a human lifetime, right? People seem to think that the finite maximum speed c is a death knell for, say, trips to Andromeda (and beyond). But if the above is correct, then the only limit on short travel times is the energy required to accelerate (a limit which exists in a Newtonian world as well). In short, isn't it possible (in principle) to go arbitrarily far in a human lifetime?"}
{"id":"122275","title":"How much time passed for the passenger traveling with at speed-of-light spaceship?","text":"Let's suppose we have a spaceship with the exact speed of light. If a traveller takes this spaceship to go to proxima centauri (approximately 4 years light away from Earth) and come back, we (as observers on Earth) will see the ship coming back after approximately 8 years. But how much time would have passed for the traveller on the ship? How can be this calculated with a formula?"}
{"id":"22670","title":"Double slit experiment and indirect measurements","text":"In the classic Young double slit experiment, with slits labeled as \"A\" and \"B\" and the detector screen \"C\", we put a detector with 100% accuracy (no particle can pass through the slit without the detector noticing) on slit B, leaving slit A unchecked. What kind of pattern should we expect on the detector C? Probably the right question is: knowing that a particle hasn't been through one of the slits makes the wavefunction collapse, precipitating in a state in which the particle passed through the other slit?"}
{"id":"25591","title":"Does the universe have a center?","text":"If the big bang was the birth of everything, and the big bang was an event in the sense that it had a location and a time (time 0), wouldn't that mean that our universe has a center? _Where_ was the big bang? _What_ is there now? Are we moving away from the center still? Are these even valid questions?"}
{"id":"57402","title":"Size of the Observable Universe","text":"I wanted to know what the observable universe is so I was thinking and I thought, it must be age of the universe times 2. Well I was wrong. I found on one website that it is 46B LY across in each direction. How does this make sense? I get how the universe has expanded since then, but we should only be able to see light that is 13.7 billion LY old. Does this mean that the Universe is expanding faster than the speed of light? Or light from other objects is travelling to us faster than speed of light?"}
{"id":"12049","title":"How can a quasar be 29 billion light-years away from Earth if Big Bang happened only 13.8 billion years ago?","text":"I was reading through the Wikipedia article on Quasars and came across the fact that the most distant Quasar is 29 Billion Light years. This is what the article exactly says > _The highest redshift quasar known (as of June 2011[update]) is > ULAS-J1120+0641, with a redshift of 7.085, which corresponds to a **proper > distance** of approximately 29 billion light-years from Earth._ Now I come to understand that the Big Bang singularity is believed to be around 13.8 Billion years ago. So how is this possible? Does the presence of such a quasar negate the Big Bang Theory? I'm not a student of Physics and was reading this out of (whimsical) curiosity. Is there something I'm missing here or the \"proper distance\" mentioned in the fact is a concept that explains this? Edit: My Bad! Here's how.. A simple google search led me to this article (sciencedaily.com\/releases\/2011\/06\/110629132527.htm) which says the farthest quasar found is 12.9 billion LYs and not 29 billion. So in the end we have just proven that wikipedia needs more moderation. Lol, what a waste of time! Although I did find out a lot of new stuff from the answers, so thanks for all your responses!"}
{"id":"65547","title":"Diameter of the universe","text":"**Should the diameter of our universe always be more than its age in light years?** As if the distance between any two points in the universe is equal to 13.5 billion light years then the light from the big bang could also reach there and certainly more length than its age in light yrs would violate the very basic laws of our universe."}
{"id":"112494","title":"Can the coordinate of the big bang point be calculated via observed universe or it is impossible?","text":"We know all galaxies spread out after Big Bang theory.The key idea is that the universe is expanding after that theory. Can we play back the scenes via observable universe (galaxies) and can we calculate the coordinate of big bang point as a fixed reference point in the universe? Is there a such calculation in literature or claim? EDIT: After answers I have some questions: If the distribution of matter in the universe is homogeneous,how should we think the geometry of universe. Is it a sphere ? Does not a line go to endless as we imagine? Is there any idea of the border of observed universe? If the universe is homogeneous and like sphere geometry, can we see the beginning of our baby galaxy while observing in the sky? I really will appreciate if someone explains what the universe geometry structure is if we have a homogeneous universe after big bang. Thanks"}
{"id":"104964","title":"If nothing can travel faster than the speed of light, how can there be parts of the universe we can't see?","text":"Assuming we originated from a single infinitely dense point in space time in the big bang, how can there be parts of the universe that we can't see as the light has not reached us yet, if nothing can travel faster than the speed of light relative to our frame of reference? Is that because the universe can expand faster than the speed of light? Or am I missing something?"}
{"id":"110491","title":"Does Fermi-Dirac Statistics explain anti-particles?","text":"I wondered whether the Fermi-Dirac Statistics describes the anti-fermion particles. Does it include the anti-particles?"}
{"id":"26562","title":"If we were to travel through space (sci-fi style), how close to the false-color images would the galaxies we see be?","text":"I understand that the black-and-white images you see looking through a household telescope are only like that due to the intensity of the light that reaches us, and that most of the astronomy images we find online have some color modification. But if we were to eventually travel between galaxies, how would those galaxies appear to the naked eye? How close to the false color images would they be, and how accurately do we know this?"}
{"id":"126461","title":"I am learning Quantum Mechanics and I have some questions about some basic concept","text":"1. What does a \"STATE\" exactly mean in quantum mechanics?    2. What is the equivalence of \"STATE\" in classical mechanics?   3. If we have a wave function $\\Psi$ , its absolute square $|\\Psi|^2$ is the probability density of finding the particle somewhere in the space, and I know it can be written as $\\langle \\Psi |\\Psi \\rangle$. But what is the physical meaning of $\\langle \\Phi | \\Psi \\rangle$, where $\\Phi$ and $\\Psi$ are two different wave functions. Why we need to take inner product of two different wave functions."}
{"id":"65268","title":"Why is this not a realisable operation on a quantum system?","text":"Let $\\rho = \\begin{bmatrix}\\ 1&0 \\\\\\ 0&0 \\end{bmatrix}$, $\\rho' = \\begin{bmatrix}\\ 0&0 \\\\\\ 0&1 \\end{bmatrix}$, $\\rho'' = \\dfrac{1}{2}\\begin{bmatrix}\\ 1&1 \\\\\\ 1&1 \\end{bmatrix}$ (all density operators). Consider a physical operation $\\phi$ such that $\\phi(\\rho) = \\rho$, $\\phi(\\rho') = \\rho'$, $\\phi(\\rho'') = \\dfrac{1}{5}\\begin{bmatrix}\\ 4&2 \\\\\\ 2&1 \\end{bmatrix}$. Why is $\\phi$ not a realisable physical operation? It certainly preserves trace and positivity..."}
{"id":"65264","title":"Leap from photon gas energy distribution to black body radiation?","text":"I remember considering in class in college, the case of a photon gas trapped in a d-dimensional box as a subject of interest, whose energy distribution, heat capacity, etc. should be calculated. This momentum\/energy distribution was then related to that emitted by a black body - that up to a constant factor, they are one in the same. I could understand the argument that making a small hole in such a box would constitute a \"black body\", as the hole can't reflect back light impinging on it, but I can't see how to make the conclusion that this is the exact same distribution. Shouldn't the atomic structure of matter be brought into account as well, or is it considered \"hollow\" on sufficiently large scales?"}
{"id":"65261","title":"Kinetic energy of two charged balls at infinite distance between them","text":"If I have two balls with masses and charges $m_1, q_1^{+}$, $m_2, q_2^{+}$, initially held at distance $d$, and then released, how can I know the kinetic energies of each of the balls at infinite distance between them? I'm quite stuck on that, because they both have the same potential energy at the beginning, and it decreases not in the same pattern, as if one of the balls was stationary. So it not only falls like $1\/R$, because at the same time, the other ball that is causing this potential energy, is also being repelled. So how can I really find out the energies? I tried to apply the conservation of energy law, because I know that at infinite distance from each other they'll have zero potential energy, thus all the initial was transformed into kinetic form, however I'm stuck with the initial potential energy (they both have it, so should I put $2U_p$?), and even so, I can't find their kinetic energies separately, without having another equation."}
{"id":"103421","title":"antimatter moving back in time","text":"This just blew my mind away! I was watching a video about imagining the fourth dimension and the narrator said that little line. Can some people elaborate on this. Also please keep it simple not too complicated. Thank you!"}
{"id":"58101","title":"Do particles travel backward and forward in time?","text":"> All these classical ideas are pointless and obsolete today, because in > quantum mechanics, the particles are completely different objects, defined > by quantum motion of fields, not by the location of classical points (at > least not in a causal field picture). The notion of a point particle was > replaced by the more subtle notion of a quantum point particle, which has a > probability amplitude to be at various places. This quantum point particle > can reproduce the quantum field if it is allowed to go backward and forward > in time. _Ron Maimon_ That was quoted by Ron Maimon. From this - Can I gather that in reality, particles travel backward and forward in time or, is this just a mathematical expression or model?"}
{"id":"92889","title":"Partition function for a microcanonical ensemble","text":"Is it possible to write down a partition function for a microcanonical ensemble?"}
{"id":"114445","title":"Easy formula for ultrarelativistic bremsstrahlung?","text":"I am very curious if an easy calculable formula for the bremstrahlung radiation of deeply relativistic, charged particles exists, if they are moving on circular orbit: $P(E,m_0,Q,r)=?$ ...where   * $P$ is the power of the Bremstrahlung radiation;   * $E$ is the total kinetic energy of the particles (we are in deeply relativistic case, thus $E\\gg{m_0}c^2$);   * $m_0$ is the total rest mass of the particles;   * $Q$ is the total charge of the particles;   * and $r$ is the radius of the orbit. If a such clean, trivial formula doesn't exist, a link were also okay, where it can be found."}
{"id":"32229","title":"squeezed radiation astronomy","text":"Squeezed electromagnetic vacuum does have a renormalized energy density smaller than the vacuum. So it makes it in my opinion a inconspicuous candidate for a dark energy carrier. Are there observatories at the moment attempting to detect squeezed radiation from astrophysical and cosmic background sources? If not, What sort of equipment do you need to measure squeezed vacuum radiation?"}
{"id":"75288","title":"electromagnetic interference","text":"If the atmosphere is filled with electromagnetic waves all oscillating at different wavelengths and speeds how is it that they don’t all interfere with each other? For example turning on your light seems to have no effect on the sound coming from your radio."}
{"id":"89993","title":"Space as a function of time?","text":"So - a little bit of background. Obviously from Einstein's equations it's shown that energy can be converted to matter and vice versa; in essence, energy and matter are different manifestations of the same \"stuff\". Now, I've done a bit of reading about the interplay between space and time. Some of what I've read would seem to indicate that just like energy and matter, space and time are different manifestations of the same \"stuff\", though the majority of what I've read seems to indicate that this isn't the case. So I got to thinking...what if space and time aren't the same thing, but there's interplay between them? What if one is a function of the other? And so I come to my question: Could space be a function of time? Or vice versa? From what I've gathered, our universe is expanding at a faster and faster rate. If space is a (seemingly exponential) function of time, then it makes complete sense that as more and more time passes, the amount of space in existence would increase. In addition (also from what I gather), nothing can go faster than the speed of light. We know that the closer to the speed of light that something is going, the more time slows down for that something, until the speed of light is reached and time stops for whatever it is. Since we're outside observers, a light year is the distance _we perceive_ light to travel in one year. But since (for the light) time stops completely, \"how far it travels in a year\" becomes meaningless to the light itself. Because no time is passing for it, wherever light goes, it effectively goes that distance in a relative time frame of zero seconds. This phenomenon would also make sense if space is a function of time; if there is no time affecting that light, there would be no space affecting it either. If from the light's perspective time has stopped completely, then from its perspective there would be no distance between it and any destination in the universe; any journey the light made would be instantaneous. Again, that's from the perspective of the light itself, not the perspective of someone who's observing the light. So does that make any sense? Could space be a function of time? Note: I am not a physicist. I am a lay-scientist, in the very loosest definition of that term. I ask this question assuming there are glaring errors and inaccuracies, which is why I'm asking it here - so you can all point me in the right direction."}
{"id":"57662","title":"Do any good theories exist on why the weak interaction is so profoundly chiral?","text":"I find the profound asymmetry in the sensitivity of left and right chiral particles to be one of the most remarkable analytical observations captured in the Standard Model. Yet for some, I've not found much in the way of discussions that worry about _why_ of such as truly remarkable fact is true. I can't help but be reminded a wee bit of views on the motions of planets before Newton... you know, \"it be Angels that do push them around, ask ye not why!\" Seriously, I know the Standard Model has a lot of givens in it... but surely _someone_ has mulled over why the universe might exhibit such a non-intuitive and thus _interesting_ asymmetry? And perhaps even developed some solid speculations or full theories on why such in-your-face chiral asymmetries exist in nature? Do such theories exist, or is this asymmetry truly just a \"given\" and nothing more?"}
{"id":"99852","title":"where does the photon go after scattering?","text":"My question is about `photo electric` but it could be applied to other daily routine phenomenon. As we know rest mass of photon is zero. When a photon strikes the metal surface it transfers its energy to the electrons. Whether electron will be emitted or not, it depends upon work function. But my question is about that photon which was hit on metal surface. Does that photon vanished? Does that photon turned into nothing? where does it go after scattering? When I study this I only find the story about the emitted electrons but not about photons after collision. Am i missing some basic concept?"}
{"id":"105631","title":"Electron degeneracy pressure","text":"Why is it that in stars undergoing gravitational collapse electron degeneracy kicks in? Why couldn't the electrons form energy bands like in semiconductors?"}
{"id":"128115","title":"Yang Mills theory and SU(N) groups","text":"Trying to get a better understanding of the relation between a SU(N) Yang Mill theory and its number of \"color\" space. Most of the description I've found so far are either way to complex\/specific. Yiannis answer on this post is almost what I'm looking for, except I was hoping someone could provide addition sources and readings corresponding precisely to what he is describing."}
{"id":"10325","title":"Graduate Physics Problems Books","text":"Need to brush up on my late-undergrad and early-grad physics and was wondering if anyone can recommend books or lecture notes (hard copy, or on-line) that also have solutions. Two that I have come across are: **Princeton Problems in Physics with Solutions** \\- Nathan Newbury **University of Chicago Graduate Problems in Physics with Solutions** \\- Jeremiah A. Cronin **Spacetime Physics** \\- Taylor & Wheeler (favorite book on special relativity; has a lot of problems with solutions at the back; a lot of the problems really enforce the material and discuss paradoxes) If possible, please also provide a reason why you like the books as opposed to just listing them."}
{"id":"117034","title":"Guides for solving E&M problems","text":"I am an undergraduate physics student in a university and I am looking for a good guide or a handbook with lots of questions and full answers in subjects such as Gauss' law and Maxwell equations to study for my test."}
{"id":"73663","title":"A set of problems for \"The Theoretical Minimum\" Stanford course","text":"I'm really excited by an opportunity to watch Leonard Susskind's course \"The Theoretical Minimum\". However, I couldn't find any physics problems on the website, and I'm a little bit wary of just watching videos. I want to think on some problems during that. Maybe there's a good collection of Physics problems on related themes, so I could try them between the videos?"}
{"id":"20282","title":"Please recommend a physics problems book similar to Demidovich","text":"> **Possible Duplicate:**   >  Graduate Physics Problems Books Does anyone have recommendation for a physics problems collection book (series) that is similar to Demidovich's _A Collection of Problems and Exercises in Mathematical Analysis_? I am looking for the encyclopediac coverage and Russian editorial style, if you like. # Update: I'd like problems for 1) senior undergraduates major in physics and\/or 2) Ph.D. candidates in physics"}
{"id":"126643","title":"Where can some worked problems in classical mechanics (and more specifically the Lagrangian and Hamiltonian formalisms) be found?","text":"I've been looking for a textbook in classical mechanics that's readily available (like can be found in the library of James Cook University of Townsville, Australia) and full of fully-answered questions in the Lagrangian\/Hamiltonian formalisms yet I can't find any. My first port of call was the Schaum's Outlines and Demystified series but the only member of these series I could find that was relevant was Lagrangian Dynamics which is difficult for me to track-down in real-life. It would be particularly helpful if one could point me to a free eBook with this material."}
{"id":"36019","title":"Problem book in Quantum mechanics with emphasis on physical(ly relevant) problems","text":"I am a second year undergraduate and studying quantum mechanics from sakurai's 'Modern Quantum Mechanics'. Is it a good idea to solve problems from sakurai, which are mostly mathematical in nature? I need a textbook that has physically relevant problems, maybe going even into condensed matter, or field theory in its exercises. This would probably help me to appreciate and understand qm better. Sorry if this question is too localised but I just had to post it."}
{"id":"133861","title":"Problem books like I.E. Irodov for advanced physics","text":"I really enjoyed doing problems from Irodov while learning introductory physics. But I am not able to find a book like that for Graduate level physics. Can you suggest me a book which has good (and hard) problems from advanced topics like Quantum Theory, QFT, Relativity, etc?"}
{"id":"130203","title":"Requesting some research study problems on Classical Mechanics","text":"Someone please tell (advice) me some research study problems on Classical Mechanics that can be tackled with Undergraduate level knowledge."}
{"id":"61212","title":"Goldstein's Classical Mechanics exercises solutions","text":"Does anyone know where I can find some (good) solution of Goldstein's book _Classical Mechanics_?"}
{"id":"113141","title":"Translations and Noether's Theorem","text":"I'm fine with $U(1)$ symmetry and Noether's Theorem, but struggling with the translations of the field; namely $$\\phi'(x^{\\mu})=\\phi(x^{\\mu}-a^{\\mu}),$$ where $a^{\\mu}$ constant four-vector $$x^{\\mu}=x^{\\mu}+a^{\\mu},$$ and the Lagrangian density $${\\cal L}=\\frac{1}{2}\\partial_{\\mu}\\phi^*\\partial^{\\mu}\\phi-V(\\phi^*\\phi).$$ So a few questions:   1. I can't show the Lagrangian is invariant under this transformation. Is it just a case that as $a^{\\mu}$ is constant then the first term in the Lagrangian will obviously stay the same? But what about $V$? How I can show that's invariant?   2. Infinitesimally, is the transformation $\\phi'(x^{\\mu})=\\phi(x^{\\mu})-a^{\\mu}\\partial_{\\mu}\\phi(x^{\\mu})?$    3. If I'm right in point 2., how can I apply Noether's Theorem to this?"}
{"id":"99994","title":"How to simulate a full-suspension bike?","text":"As a fun project, I would like to roughly simulate the suspension operation of a full-suspension mountain bike. This is _not_ another one of those \"How does a bicycle stay upright?\" questions. Follow some metrics of interest.   * Pedal bob - the amount of movement in the rear triangle upon pedaling up an incline. An estimate of the lost energy due to this.   * Sharp obstacle transient response - for example how does the bike move, when hitting a curb at high speed.   * Large obstacle transient response - for example, how does the bike move when it falls (along with the rider) 1 meter. I am somewhat familiar with python, Mathematica, ScyLab, Matlab, Ansys. I guess an analytical solution would be too difficult. _Which approach will give quickly a parameterizable model and analyze it for the abovementioned information?_ Here several common frame variations are mentioned. EDIT: More concretely:   * What model to use? Is there a widespread \"bicycle\" model for dynamic analysis?   * What software to utilize, in order to help me solve the problem?"}
{"id":"57082","title":"Gravitational distortion of an object's diameter, at a distance,","text":"Does the curvature of space-time cause objects to look smaller than they really are? What is the relationship between the optical distortion and the mass of the objects?"}
{"id":"19973","title":"what causes virtual particle pair production to not occur in the space occupied by matter?","text":"Are virtual particles only popping in and out of existence where the local energy density is below a certain point? What I wonder is, does any kind of matter prevent the pairs from appearing? Is there a shell surrounding an atom or maybe I should call it a boundary beyond which particle pair production occurs, and within the boundary it does not, I have wondered if the different orbitals around an atom are affected (set)by the influence of the virtual particles."}
{"id":"7781","title":"Does the curvature of spacetime theory assume gravity?","text":"Whenever I read about the curvature of spacetime as an explanation for gravity, I see pictures of a sheet (spacetime) with various masses indenting the sheet to form \"gravity wells.\" Objects which are gravitationally attracted are said to roll down the curved sheet of spacetime into the gravity well. This is troubling to me, because, in-order for objects on the locally slanted spacetime sheet to accelerate, gravity must be assumed. Therefore I ask; does the explanation of gravity as the curvature of spacetime assume gravity? If yes, what is the point of the theory? If No, what am I missing?"}
{"id":"75698","title":"Is the space-time deformation only a way to say how gravity works?","text":"Here we have the classical picture of the deformation of the space-time: https:\/\/blogs.stsci.edu\/livio\/files\/2012\/06\/spacetime.jpg And I would to know if this representation is only a way to say how the gravity field of the earth is shaped, and how the gravity works in the area near it, or not."}
{"id":"16532","title":"Does Spacetime have a \"This Side Up\" arrow?","text":"> **Possible Duplicate:**   >  Does the curvature of spacetime theory assume gravity? Forgive my naivete as I am not schooled in Physics or Mathematics. I was watching NOVA's \"The Fabric of the Cosmos\" last night. The subject was basically spacetime and how our conception of it moved from the Newtonian description to the Einsteinian. i.e. That spacetime is like a fabric stretched across the universe and that large objects like planets and stars create a depression in the spacetime fabric and the orbiting of satellites is really just the satellite getting caught in the \"swirl\" of the depression. The narrator used a pool table surface as an example of the fabric of spacetime and then with some CGI he dropped a bowling ball on it which caused a depression and then shot a pool ball near it which caused the pool ball to get caught in the swirl. I found myself pondering the following though:   1. I find it really hard to believe that there is a \"This Side Up\" arrow for spacetime. But if there is, what causes it? You can see a diagram of what I'm trying to describe in this Gravity Probe B article. I understand that spacetime can be twisted so I'm guessing reality is more like the depressions are at all orientations through the universe (sorry, I lack the language to describe this). But I still wonder: Why is the depression one orientation and not the other?   2. What is the effect of the other side of the depression? i.e. If the swirl on the other side pulls objects in, does the other side of the depression repulse objects? Warning: any answers that include mathematics will be lost on me. Any suggestions for further layperson reading on this fascinating subject welcome. Thanks!"}
{"id":"3324","title":"Misused physics analogies","text":"Have you noticed that many questions and misconceptions in physics arise due to misuse of analogies, which were invented to \"explain in simple words\" some physical phenomena? For example there is a \"stretching rubber band\" analogy for Hubble expansion of the Universe, which make people ask about the \"tension\" of this rubber band or speculating about \"cutting a hole\" in it. I'm pretty sure that there are a lot of such misuses. Do you have any remarkable examples of these (together with a proper treatment n of course)?"}
{"id":"65363","title":"How does relativity explain gravity, without assuming gravity","text":"I have seen the \"objects pull down on space-time\" explanations, but they assume a \"pull down\" force themselves. Could anyone explain the space-time explanation without assuming gravity in the first place? ![](http:\/\/i.stack.imgur.com\/RVFrW.png)"}
{"id":"116644","title":"Is the concept of space-time curvature a recursive one?","text":"A way some people explain (or try to explain) how gravity works is using space-time curvature: an object with high mass distorts the surrounding space- time plane like a bowling ball distorts a sheet of plastic and the surrounding objects are always trying to roll down to that other object. However, this explanation to me seems flawed, because in order to work, it needs to be influenced by the very thing it tries to explain. The reason a bowling ball distorts a sheet is because Earth's gravity is affecting it. However, the space-time curvature which this distortion is an analogy to is trying to explain gravity. The implication of this space-time curvature is that something is pulling down on the space-time plane, but this pulling itself requires gravity from a perpendicular source, it seems to me. I'm having difficulty wrapping my head around this. The method we use to explain gravity itself uses gravity to work, sort of like some kind of recursive loop. it's comparable to \"to get fuel, we need to drive, and to drive, we need fuel\", and I don't know enough about gravity to figure this out myself. Am I understanding this analogy correctly, or are there deeper things at play?"}
{"id":"51198","title":"Bowling ball on a rubber sheet analogy - what pulls the ball down","text":"> **Possible Duplicate:**   >  Does the curvature of spacetime theory assume gravity? Since I read Cosmos long ago, I see the same analogy about the balls rolling on a rubber sheet used to explain how gravity works. But a ball rolls on a surface because gravity is pulling it down. In space it will follow a straight line and go over any hole on the surface. So, in the analogy we all know, where the curvature of the rubber sheet is gravity, what is pulling the ball down? If space is curved, what keeps an object attached to that \"surface\" of space?"}
{"id":"59958","title":"If the universe is 3D, how is space-time like a \"fabric\"?","text":"I have been taught that space-time should be viewed as a fabric and that objects with a large gravitational influence indent that fabric. My question is, if the singularity of a black-hole punctures space-time, how is this accomplished if the universe is 3D? Can an object move completely around the black-hole in all directions? Would you be able to travel \"below\" a black- hole?"}
{"id":"72456","title":"Why is getting a tan in the evening less likely than getting a tan in the morning?","text":"When I was young I read one book in which is written that you get more tan on the morning than on evening even light angle from Sun to Earth is the same. Don't remember exact reason, I think because ultraviolet is more absorbed on the evening because air is more humid or something like this. Is it true?"}
{"id":"6811","title":"Can PEPS explain the holographic principle in quantum gravity?","text":"Condensed matter physicists have shown using quantum information that in many condensed matter systems, entanglement entropy only scales as the area of the boundary, and not the volume. This is the basis for the density matrix renormalization group and Projected Entangled Pair States (PEPS). Does this also explain the holographic principle in quantum gravity?"}
{"id":"6813","title":"is the sun itself moving","text":"I just thought of an interesting question.Is the sun actually moving? I have learnt that the way the lunar landers and the space ships of the Mercury and Apallo missions moved and controlled their positions through the use of thrusters.NoW apply that same concept to the sun are the constant explosions on the surface of the sun enough to move it?"}
{"id":"6817","title":"What is the formula for the relativistic ellipse?","text":"If an astronomer moves at relativistic speed, the stars and constellations are distorted. He sees the stars towards which he is moving blue shifted, while the ones he's moving away from are red shifted. In addition, the apparent direction of distant stars is modified. I think that the coolest way of representing this is the \"relativistic ellipse\". Think of a spherical shell with the inner surface reflective. Imagine a flashbulb going off at the center. The light is reflected and returns to the source. Now think of the same situation observed from a moving frame of reference. The light is still emitted and returns to the center at the same time, but that point has now moved. Since the speed of light is the same in all frames of reference, the distance traveled by each light ray has to be the same. So the paths of the light rays (and the boundary of the spherical shell) describes an ellipse, the \"relativistic ellipse\": ![RelativisticEllipse](http:\/\/i.stack.imgur.com\/9PoOJ.png) The above drawing is from the book \"Relativity in Curved Spacetime\" which I highly recommend and can be purchased here or here. Perhaps when my copy arrives it will have the equation. Should be possible to work it out using Lorentz invariance or contraction or something similar. * * * Per the answer by Helder Velez, Chapter 4 of Hans de Vries's book has the following useful diagram:   ![HansEllipse](http:\/\/i.stack.imgur.com\/gC4Gu.jpg)"}
{"id":"9756","title":"What is the extent of Earth's gravitational pull?","text":"1. Where the gravitational pull of Earth exist up to?    2. What distance from Earth it will be zero?   3. How do the skydivers fly at a same altitude?    4. Won't they feel gravitational pull?    5. What is the Earth's gravitational pull?"}
{"id":"9754","title":"Dimensonal analysis of damping constant?","text":"What are the units of the damping constant from the following equation by dimensional analysis? $$\\zeta = \\frac{c}{2\\sqrt{mk}}$$ I'm assuming the units have to be s^-1, as the damping constant is present in the exponential equation which plots damping of y=Ae^kt (which plots amplitude vs time). Is that a correct assumption? If somebody could do a quick dimensional analysis to confirm it would be great."}
{"id":"92143","title":"Net acceleration felt by a body on Earth","text":"For any body on earth will the only acceleration it has be gravity or should we take into consideration even centrifugal force due to rotation? Is there any precise formula to determine the net acceleration?"}
{"id":"107709","title":"Kepler's third law doesn't give earth's orbital period! Why?","text":"I tried to calculate earth's orbital period using Kepler's third law, but I found 365.2075 days for the orbital period instead of 365.256363004 which is the correct value. I checked everything, and I couldn't find what's the problem. I used these values for my calculation:   * Semi-major axis, a: 149,598,261 km   * Gravitational constant, G: 6.67*10-11 N·(m\/kg)2   * Solar mass, M: 1.9891*1030 kg"}
{"id":"112239","title":"How to calculate the energy freed in the reaction: $^{10}_5Be +\\space ^2_1H \\rightarrow \\space^{11}_5B + \\space ^1_1H$?","text":"I have the following reaction: $^{10}_5\\mathrm{Be} +\\space ^2_1\\mathrm{H} \\rightarrow \\space^{11}_5\\mathrm{B} + \\space ^1_1\\mathrm{H}$ And I know that I have to use the formula: $E = \\Delta m\\cdot c^2 = \\Delta m \\cdot \\frac{931,5MeV}{u}$. So I just need $\\Delta m$ which is equal to: $\\Delta m = m_b - m_a$ where $m_b$ represents the mass \"before the reaction\" and $m_a$ the mass \"after the reaction\" so we have:   $m_b = m(^{10}_5\\mathrm{Be}) + m(^2_1\\mathrm{H})$   $m_a = m(^{11}_5\\mathrm{B}) + m(^1_1\\mathrm{H})$ The book which contains this problem contains the following table: http:\/\/i.imgur.com\/esoGDVf.png but from this table, I only know $ m(^1_1\\mathrm{H})$ and $m(^2_1\\mathrm{H})$ i.e.   $m_b = m(^{10}_5\\mathrm{Be}) + 2.01410u$   $m_a = m(^{11}_5\\mathrm{B}) + 1.00783u$ How do I calculate $m(^{10}_5\\mathrm{Be})$ and $m(^{11}_5\\mathrm{B})$ ? P.S. I don't know if the tag is correct. The chapter in the book where I found this exercise is called \"Basics of nuclear physics\"."}
{"id":"11841","title":"About unitarity and R-charge in 2+1 superconformal field theory","text":"* How does unitarity require that every scalar operator in a $2+1$ SCFT will have to have a scaling dimension $\\geq \\frac{1}{2}$ ?    * Why is an operator with scaling dimension exactly equal to $\\frac{1}{2}$ said to be \"free (i.e decoupled from the rest of the theory)\" ?    * Let us say that we know in some such theory the R-charge is monotonically decreasing with increasing coupling constant (say $\\lambda$). Then define $\\lambda _n ^f$ to be that value of the coupling constant at which the R-charge of the operator $Tr[\\phi ^n]$ becomes $= \\frac{1}{2n}$. Let $\\lambda _n ^m$ be that that value of the coupling at which the R-charge of the same operator becomes $= \\frac{2}{n}$ (..marginal ?..) Then clearly $\\lambda _n ^m < \\lambda_n ^f$.  How does this imply that there has to exist a $\\lambda _c \\leq \\lambda_2^f$ where the theory might undergo a phase transition?"}
{"id":"88028","title":"Time dilation in a gravitational field and the equivalence principle","text":"A clock near the surface of the earth will run slower than one on the top of the mountain. If the equivalence principal tells us that being at rest in a gravitational field is equivalent to being in an accelerated frame of reference in free space, shouldn't the clock near the earth run _increasingly_ slower than the other clock over time? If those two clocks are considered to be in two different frames of acceleration, the one near the earth will have a greater acceleration than the one on the mountain, and thus over time, their relative velocity will increase over time which will increase the time dilation. Or is this not a proper use of the equivalence principle?"}
{"id":"26435","title":"Given that matter cannot escape a black hole, how did the big bang produce the universe we see today?","text":"Extrapolation of the expansion of the Universe backwards in time using general relativity yields an infinite density and temperature at a finite time in the past. If the matter contained within our galaxy were concentrated within a small radius wouldn't that lead to the entire universe being a black hole? If so how is it that all the matter of our observable universe could have originated from a region of infinite density?"}
{"id":"30332","title":"Heat of vaporization of water - dependence on relative humidity?","text":"Does the heat of vaporization of water depend strongly on the relative humidity of the gas into which it evaporates? Some context: If we want to calculate the dew point of water, we find the temperature at which the _partial pressure_ of the water lies on the liquid\/vapor boundary of the water phase diagram. This is why water can evaporate from our bodies even though we do not heat it to anywhere near its boiling point. The heat of vaporization should be pressure dependent (in addition to temperature dependent). Yet, when specifying the heat of vaporization, most references only specify the _total_ ambient temperature, usually 1 atmosphere. Why is the total pressure used in this case instead of the partial pressure? And if the partial pressure is what matters after all, then wouldn't relative humidity be important when calculating heats of vaporization? Of course, relative humidity governs the rate and the total amount of evaporation, which is why we can't cool ourselves by sweating in humid weather, but that's not what my question is about."}
{"id":"76166","title":"Hydrogen ground state energy calculation?","text":"We want to find the energy of a hydrogen atom ($Z=1$) in the ground state $$ \\psi_{100} = \\frac{1}{\\sqrt{\\pi}}e^{-r}\\ \\ \\ \\ \\ \\ (\\mbox{atomic units}) $$ with Hamiltonian $$ H = -\\frac{1}{2}\\nabla^2-\\frac{1}{r} $$ Then $$ \\begin{align*} \\langle \\psi_{100}|H|\\psi_{100}\\rangle &=\\int_0^\\infty \\int_0^{2\\pi} \\int_0^\\pi r^2\\sin\\theta\\frac{1}{\\sqrt{\\pi}}e^{-r}\\left(-\\frac{1}{2}\\frac{d^2}{dr^2}-\\frac{1}{r}\\right)\\frac{1}{\\sqrt{\\pi}}e^{-r} d\\theta d\\phi dr \\\\\\ &=\\int_0^\\pi \\sin\\theta d\\theta \\int_0^{2\\pi}d\\phi \\int_0^\\infty \\frac{r^2e^{-r}}{\\sqrt{\\pi}}\\left(-\\frac{1}{2\\sqrt{\\pi}}e^{-r}-\\frac{1}{r\\sqrt{\\pi}}e^{-r}\\right)dr \\\\\\ &= 4\\pi \\cdot \\frac{1}{\\pi}\\int_0^\\infty \\left(-\\frac{r^2}{2}e^{-2r}-re^{-2r}\\right)dr \\\\\\ &= 4\\left(-\\frac{3}{8}\\right) \\\\\\ &= -\\frac{3}{2} \\end{align*} $$ However, I've read everywhere that $E = -\\frac{Z^2}{2n^2}$, and so for a hydrogen atom in the ground state we should have $E=-\\frac{1}{2}$. So why am I getting $-\\frac{3}{2}$? I've double-checked with Mathematica."}
{"id":"76161","title":"Lever Counterweight of uneven weight distances","text":"So, I have a camera crane that has a pivot leaving 2 distances, 0.5mts and 1.5mts respectively. Plus a small L-shaped piece of metal and the end of d2 (1.5) which holds a camera, which I suppose I have to add it to the force applied on that side. One additional variable is that there's a piece of metal attached to the fulcrum and then connected to the second distance (d2, 1.5) at the end, which leaves me the doubt of where to add that weight (which is 580grs, let's say d2*) to the force, or add it as another variable in the equation. The current equation that i'm using (which was without the second piece of weight on distance 2) is: $$ w = \\dfrac{f * d2}{d1} $$ ![Lever example](http:\/\/i.stack.imgur.com\/1P35v.png) So i'm dubious if this is the right equation taking in count the variables I have explained above. Thanks!"}
{"id":"100133","title":"Noether's Theorem in Field Theory","text":"This question is regarding Noether's Theorem in general, but also in the application to an example. The example is: > _Find the conserved current for the Lagrangian_ > $$L=\\bar{\\psi}(\\frac{i}{2}\\gamma^{\\mu}\\partial_{\\mu}-m)\\psi.$$ Is my first step to find a transformation that leaves the action invariant? But isn't there more than one symmetry meaning this question has several answers? From Peskin and Schroeder, it says we need to find $$\\partial_{\\mu}j^{\\mu}(x)=0 \\; \\text{ for } \\; j^{\\mu}(x)=\\frac{\\partial L}{\\partial (\\partial_{\\mu}\\psi)}\\Delta\\psi-J^{\\mu}.$$ So I think I'm right in saying this $J^{\\mu}$ is dependent on the transformation we're making? e.g. if it's just $\\psi \\rightarrow \\psi +a$ then $J^{\\mu}=0$. The Lagrangian changes under the transformation of the field as $$L \\rightarrow L+\\alpha \\partial_{\\mu}J^{\\mu}.$$ But this doesn't help me because plugging $\\psi \\rightarrow \\psi +\\alpha \\Delta \\psi$ (an arbitrary transformation of the field $\\psi$) into the Lagrangian at the top gives me (after no more than one line) $$L \\rightarrow L+\\alpha \\bar{\\psi}(\\frac{i}{2}\\gamma^{\\mu}\\partial_{\\mu}-m)\\Delta \\psi$$ and how do we use this to find $J^{\\mu}$? (Something I've assumed throughout all of this is that $\\psi$ and $\\bar{\\psi}$ are treated completely separate and here we're only considering $\\psi$. I hope I am right in doing this.) I haven't really pin pointed a question here as my understanding breaks down at many points and when I think I finally understand it, I'm given a new Lagrangian and become stuck again. There must be some general procedure?"}
{"id":"119864","title":"Simple units question (School)","text":"A tank has 2000 litres of capacity (Ct). It fills at 5 l\/s. It is filled with 800 l at time 0. a) How many seconds will it need to fill half the tank. So I did this: X = time.               5 l   *  X     2000L      ----       = -------   -  800L       s           2               X =   200L        ---------          5  L            ---             s               X =   200          -------             5         ---------             s      Eventually, I get to `X = 40 \/ s`, but I guess it should be `X = 40s`. However, I fail to see what I should do to get the seconds multiplying the 40. PS: Shouldnt I give the X a unit at the beggining, or is it ok to be just X?"}
{"id":"102109","title":"Numerical Ising Model - Wolff algorithm and correlations","text":"I'm doing some numerical Monte Carlo analysis on the 2 dimensional Ising model at the critical point. I was using the Metropolis 'single flip' evolution at first with success, though it suffers from critical slow down and makes studying large lattices unlikely possible. I'm now looking at cluster flip algorithms, specifically the Wolff algorithm. I managed to implement it, and it looks to be working as it should (flipping a unique spin at $T = +\\infty$, the whole lattice at $T=0$, matches the right energy density in the thermodynamic limit...) but I don't get the right behaviour for the two point $<\\sigma_i\\sigma_j>$ correlation function. According to CFT it should behave like: $$<\\sigma_i\\sigma_j> \\;\\propto \\;\\frac{1}{|i-j|^{\\frac{1}{4}}}$$ I'm more and more convinced that it has to do with boundary conditions, I use non periodic free boundaries. The literature on the subject doesn't say much on this point. Am I missing a subtlety (or an evidence) in this procedure, or in the use of this algorithm?"}
{"id":"111954","title":"What does it mean to have infinite negative conformal time?","text":"In the context of Inflationary Cosmology, it is postulated that there was a period of shrinking Hubble Sphere radius $(aH)^{-1}$. $$ \\frac{d}{dt} (aH)^{-1} < 0 $$ Then the regions of the universe which appear to us to not be in causal contact can indeed have been in causal contact at some (conformal? physical? I'm not sure. My intuition would be physical) time in the past. This solves the Horizon Problem, namely that we observe a homogenous isotropic universe to one part in 10,000, yet it doesn't seem that all regions of the universe were in casual contact to achieve such a uniformity. $\\\\\\$ Anyway, when this shrinking Hubble sphere radius $(aH)^{-1}$ is invoked, it has the result of pushing the Hot Big Bang singularity back to negative infinity conformal time. It is this statement that I am having trouble grasping. $\\\\\\$ Explicitly $$ \\chi_{PH}(\\tau) = \\tau - \\tau_i = \\int_{t_i}^{t} \\frac{1}{a(t)} dt = \\int_{a_i}^a (a H)^{-1} d\\log(a) $$ where by Friedmann $$ (aH)^{-1} \\propto a^{\\frac{1+3w}{2}} $$ giving $$ \\chi_{PH}(a(\\tau)) = \\tau - \\tau_i \\propto \\frac{2}{1+3w} \\left[ a(\\tau)^{\\frac{2}{1+3w}} - a_i^{\\frac{2}{1+3w} } \\right] $$ So for a strong energy violating fluid such that $$ 1 + 3w < 0 $$ this becomes $$ \\tau_i \\propto \\frac{2}{1+3w} a_i^{\\frac{2}{1+3w}} \\rightarrow -\\infty $$ $\\\\\\$ I understand the maths behind it, and can't argue against that the initial conformal time goes to minus infitity, but I don't know how to think about that. In a standard Hot Big Bang the singularity occurs at $\\tau_i = 0$ conformal time. This seems fine to me. I am aware that conformal time is unphysical, but unless the scale factor goes negative, then negative conformal time suggests to me negative physical time. And a negative scale factor would bring a whole other conceptual problem! Namely, they are directly related by $$ d\\tau = \\frac{1}{a(t)} dt $$ $\\\\\\$ So my questions are; What does it mean to have negative conformal time? And what does it mean to have negative infinity conformal time? $\\\\\\$ Thanks"}
{"id":"111951","title":"Can low end hepa vaccums still clean near all of the air?","text":"In the US HEPA filters will remove 99.97% of particles sized 0.3 microns. In Europe HEPA filters do not have to meet this exact figure and there are different classes of HEPA filters e.g. h10(removes 85% at .3 microns, h11 removes 95% at .3 microns etc, h14 removes 99.97% etc.). Suppose you have a vacuum cleaner that is h10(removes 85% of particles at .3 microns), when vacuuming 15% of .3 microns will pass through the filter. If we continue to vacuum the area these will go into the vacuum again and be filtered again. Therefore even though the filter performs less, is it correct to say eventually(with continued vacuuming) the filter will remove the remaining 15% so its not a big deal if you get one with 85% or 99.9% filtration so long as you are bothered to continue vacuuming? Thanks"}
{"id":"111950","title":"The \"replica trick\" initial formula?","text":"In Spin-glass theory for pedestrians by Castellani and Cavagna, the initial formula used to introduce the replica trick is written as: $$\\overline{\\log Z}=\\lim_{n\\rightarrow0}\\frac{1}{n}\\log\\overline{Z^{n}}\\qquad(1)$$ where the overbar denotes average over quenched disorder. I don't know how to prove this formula. In other treatments I have seen of the replica method (wiki, for example), one starts from: $$\\log Z = \\lim_{n\\rightarrow 0} \\frac{Z^n-1}{n}\\qquad(2)$$ which I understand. How are (2) and (1) connected? What's the proof of (1)?"}
{"id":"91210","title":"Why is the orbital resonance of the Galilean moons stable?","text":"It is well known that the orbits of Ganymede, Europa and Io are in a 4:2:1 resonance. Most online sources (including but not limited to Wikipedia) say that such an orbital resonance, along with the 3:2 resonance, is \"stable and self-correcting\", but fail to explain why this is so. The textbook _Fundamental Astronomy_ says that this phenomenon is due to \"tidal forces\" but does not elaborate further. I presume it refers to the tidal deceleration which causes the orbits of the moons to evolve outwards, which, other sources say, caused the moons to eventually enter into resonance, but this also does not explain why the resonance is stable. I am aware of a similar question here but I'm more interested in stability rather than instability in this case. In short, why is the orbital resonance of the Galilean moons stable, and how is it different from other cases of orbital resonance that are unstable? I don't mind (and would prefer) if the answer is mathematical in nature."}
{"id":"91214","title":"Is it possible to use the parabolic shape of a rotating fluid to measure the angular frequency of the rotation of the Earth?","text":"A fluid in a rotating bucket will take on a parabolic shape (for example of some simple derivations of this result see http:\/\/en.wikipedia.org\/wiki\/Bucket_argument). The assumptions that play into the derivations that I've seen do not take into consideration a Coriolis effect. Suppose one did not know that the earth was spinning. Is it possible to measure the angular frequency of the earth using a bucket of fluid by considering how the Coriolis effect modified the curvature of the paraboloid in question? Perhaps it's possible to optically measure the curvature of the paraboloid setup by the spinning process and from that determine the angular frequency of the planets rotation. I am concerned though that the effect of the rotation of the earth is so minuscule on the parabolic shape that it would be impossible to physically measure it."}
{"id":"23267","title":"What is lambda R in Richardson's Law?","text":"I've got to calculate the thermionic emission through a diode, so I need to use Richardson's Law. However, one thing's got me confused - according to the Wikipedia page: $$J = A_GT^2e^\\frac{-W}{kt}$$ I could live with that, but for $A_G$. Apparently, I'm not the only one; Wikipedia's a bit cryptic about what $A_G$ is, mentioning that physicists have struggled with it for decades, \"but there is agreement that $A_G$ must be written in the form:\" $$A_G = \\lambda_RA_0$$ \"Where $\\lambda_R$ is a material-specific correction factor that is typically of order 0.5, and $A_0$ is a universal constant.\" That's the last real mention of $\\lambda_R$, and the only reference for it is in French. So, what is $\\lambda_R$, really? How can I figure out its numerical value so that I can actually use Richardson's Law?"}
{"id":"101703","title":"Coupling constant is turned off adiabatically?","text":"To me, adiabatic processes are idealisation. What do people mean with statements such as: \"turning off the coupling constant (in QED say) adiabatically\"?"}
{"id":"132663","title":"Conservation of energy in a different frame of reference","text":"Consider a rollercoaster that goes down a slope: ![rollercoaster](http:\/\/i.stack.imgur.com\/94VBI.png) At the higher level it has speed $v_0$, then it goes down a slope and at the end it has speed $v_0 + \\Delta v$. The carriage is not powered and has negligible friction. So by conservation of energy we should, after eliminating mass $m$ that appears in all terms, have $$ \\frac{1}{2}v_0^2 + gh = \\frac{1}{2}(v_0 + \\Delta v)^2 \\\\\\ \\frac{1}{2}v_0^2 + gh = \\frac{1}{2}v_0^2 + v_0\\Delta v + \\frac{1}{2}\\Delta v^2 \\\\\\ gh = \\frac{1}{2}\\Delta v^2 + v_0\\Delta v $$ (left side is energy at the beginning, right side energy at the end). But what if we consider the same situation in a reference frame moving left with speed $v_0$. Then we should get just $$ gh = \\frac{1}{2}\\Delta v^2 $$ These differ by a $v_0\\Delta v$ term, so solving for $\\Delta v$ gives different result. But obviously the speeds should be the same. So where is the error?"}
{"id":"87901","title":"Is uncertainty principle a technical difficulty in measurement?","text":"I have searched for an answer to this question on physics SE but I have not seen a question in which it is addressed properly. Please let me know if there is an answer already. My question briefly is, is the uncertainty principle a technical difficulty in measurement OR is it an intrinsic concept in Quantum Mechanics irrelevant of any measurement?? Everyone knows the thought experiment of measuring the position of an electron. One can detect electron's position by hitting it with a photon, due to Compton scattering the collision of the photon with electron will change electron's momentum. This experiment is used to explain uncertainty principle to layman, but it is over simplified, isn't it? It also gives an impression that if there was a better suited experimental method the uncertainty principle becomes irrelevant. I personally think it is intrinsic as it arises from the non-zero commutator of position and momentum operators irrespective of the measurement process. Am I right? **EDIT:** My question is similar to certain extent to this question and this question. The answers there are nice but they focus on explaining basics of quantum mechanics more than they comment on the technical difficulty part. In the answers of question 2 there are statements like \"So, it's not a knowledge limit\" and \"you're sort of correct when you say it's an observational limit\" without further comments To summarize, assume hypothetically we managed to find a way in the future where we can have a look at an electron without disturbing it by measurement or causing its wave function to collapse, would the uncertainty principle still hold in such a case?? Why\/Why not?"}
{"id":"63022","title":"Probability of position in linear shm?","text":"The problem that got me thinking goes like this:- > _Find $dp\/dx$ where $p$ is the probability of finding a body at a random > instant of time undergoing linear shm according to $x=a\\sin(\\omega t)$. Plot > the probability versus displacement graph. $x$=Displacement from mean._ My work: $$v=dx\/dt=\\omega \\sqrt{a^2-x^2}$$ Probability of finding within $x$ and $x+dx$ is $dt\/T$ where dt is the time it spends there and T$$ is the total period.   Therefore $$dp=dt\/T=\\frac{dx}{\\pi \\sqrt{a^2-x^2}}$$ because $t=2\\pi \/\\omega$ and the factor 2 is to account for the fact that it spends time twice in one oscillation. The answer matches the answer and also the condition that integration $-a$ to $a$ of $dp =1$. But when i try to find p as a function of x to plot the graph I get $$p=\\frac{1}{\\pi}\\arcsin(x\/a)+C.$$ But then I get stuck as there is no way to find $C$ (except the fact that for $C=0$ the probability at the mean position is $0$ and hence $C$ cannot equal 0) which I know of. So how can I get a restraint on $C$ to find its value and hence to properly graph it with the condition that the probability from $-a$ to $a$ be 1?"}
{"id":"55194","title":"Why can't we accelerate to the speed of light?","text":"Why can't we accelerate to the speed of light? It's just a speed and nothing else. Universe also crossed the speed of light at the time of big bang. Is this is just a interpretation or there is any fact behind it? I am asking it as I don't know much about this theory."}
{"id":"290","title":"What really allows airplanes to fly?","text":"What aerodynamic effects actually contribute to producing the lift on an airplane? I know there's a common belief that lift comes from the Bernoulli effect, where air moving over the wings is at reduced pressure because it's forced to travel further than air flowing under the wings. But I also know that this is wrong, or at best a minor contribution to the actual lift. The thing is, none of the many sources I've seen that discredit the Bernoulli effect explain what's actually going on, so I'm left wondering. Why do airplanes actually fly? Is this something that can be explained or summarized at a level appropriate for someone who isn't trained in fluid dynamics? (Links to further reading for more detail would also be much appreciated)"}
{"id":"112478","title":"Can the lift generated by a helicopter be justified using Bernoulli theorem?","text":"When the shaft of the helicopter rotates, it creates a low pressure. Because of the low pressure, the helicopter lifts. Is my understanding that this is just an application of Bernoulli's theorem?"}
{"id":"64114","title":"Equal Transit Time Fallacy","text":"I learned, in grade school, that lift was generated via the particles on either side of the wing having to reach the other end at the same time. Looking back, that indeed has no physicality to it. So what is the real explanation of how lift is generated? I've found references to Navier Stokes, but I don't know what that is. What is the best way to learn, from the ground up, why some curved object would generate a definite change in pressure? I can see the results, and I know that there is a pressure difference, but **why** is that pressure difference, and the two different velocities generated?"}
{"id":"59977","title":"how to explain the upright force for the plane?","text":"I remember in the high school physics, my teacher told us that the design of the plane wing is because we want the air above the wing flowing faster than the air flowing below so the pressure above and below will be different so the net force is pointing upright. I am thinking if it is possible to explain this by bernoulli equation? But I soon stuck there because I don't know if bernoulli equation applicable to the case of the air flow in the open region? If so, what cross section we should choose?"}
{"id":"118850","title":"Why air above airplane's wing moves faster?","text":"One explanation I read: Because of the wing's geometry, the \"upper\" side of the wing is longer, so the air has to travel faster: ![enter image description here](http:\/\/i.stack.imgur.com\/HDgVA.jpg) **My wondering** : Who said (and what was his\/her explanation) that air must travel the distance above and under the wing in the _same_ time? I'll try to be clearer. Considering the below image, does **m1** and **m2** meets at the same point exactly at the same time on point **T**? ![enter image description here](http:\/\/i.stack.imgur.com\/j7Icc.png)"}
{"id":"95030","title":"How does this help an aeroplane to fly?","text":"I read it somewhere on the internet that wings of an aeroplane are designed in such a way, that they increase the velocity of air above the wings and so pressure above the plane becomes less than the pressure below it and therefore the aeroplane flies well in air, but how is velocity increased above the wings of an aeroplane?"}
{"id":"128854","title":"Moment of inertia of a cylinder about its base","text":"I've tried to find the moment of inertia of a cylinder rotating about an axis parallel to its base (i.e about the 'End diameter') as one can see here . But when I checked my results with different references ,I've found that it's incorrect!.I need a help to figure out where I did it wrong. since $$I=\\int\\limits x^2.dm$$ $$dm = \\rho.dv$$ where $$dv= r.d\\theta . dr .dh$$ & $$\\rho=\\frac{M}{\\pi R^2 H}$$ $M$:cylinder total mass. $H$:total height of cylinder. $R$:the radius of the cylinder. The distance of each infinitesimal element from the axis of rotation would be : $$\\sqrt{r^2+h^2}$$ Therfore, $$I= \\frac{2M}{ R^2 H}\\int\\limits_{0}^{R} \\int\\limits_{0}^{H}\\, r(r^2+h^2)dh\\,dr$$ and this gives a result of : $$\\frac{M~H^2}{3}+\\frac{M~R^2}{2}$$ Which is obviously wrong since the second term should be multiplied by a factor of $1\/2$."}
{"id":"6771","title":"Star Surface Temperature Vs. Mass","text":"I am hoping that someone can clarify this for me. With these equations. (Boltzmann Law) and radius of Radius of star how does surface temp scale with mass > $R _x \\approx R_\\bigodot (\\frac{M_x}{M_\\bigodot})^.5$ > > $L _x \\approx L_\\bigodot (\\frac{M_x}{M_\\bigodot})^3.5$ > > $\\frac{L}{\\pi R^2} = \\sigma T^4$ > > $T_\\bigodot = 5800 K$ = Surface Temp of Sun What I have done is that I have substituted $R_x$ and $L_x$ in the 3rd equation and simplified it in terms of $T$ where my final expression is > $T = (\\frac{L_\\bigodot M_x^2.5}{\\pi\\sigma M_\\bigodot^2.5 R_\\bigodot})^0.25$ Am i done? What am I suppose to do next? Thanks"}
{"id":"26728","title":"Why don't we see solar and lunar eclipses often?","text":"Since we see the new moon at least once in a month when the Moon gets in between of the Sun and the Moon at the night and as far as I know if this happens during the day, you'll get to see a solar eclipse. Why don't we get to see this often or in the day? Does it mean that in some part of world there's a solar eclipse when we are seeing a new moon? I'm looking for a diagram or interactive way to understand this if possible as I'm not a native English speaker, but I'll try my best to do so."}
{"id":"62361","title":"Strange things about new moon","text":"I have some strange and infantile questions about new moon. I want to know how is it possible that the Moon is not visible at night and also at day it is not Sun eclipse? I will explain the problem in a few images. ![enter image description here](http:\/\/i.stack.imgur.com\/PURQc.png) ![enter image description here](http:\/\/i.stack.imgur.com\/g0pBC.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/ZijhQ.png) With these images, how it is possible that there is not solar eclipse at every new moon? Or is there somewhere on Earth?"}
{"id":"73448","title":"Diagonalization of a hamiltonian for a quantum wire with proximity-induced superconductivity","text":"I'm trying to diagonalize the Hamiltonian for a 1D wire with proximity-induced superconductivity. In the case without a superconductor it's all fine. However, with a superconductor I don't get the correct result for the energy spectrum of the Hamiltonian in > Tudor D. Stanescu and Sumanta Tewari. “Majorana fermions in semiconductor > nanowires: fundamentals, modeling, and experiment.” Journal of Physics: > Condensed Matter **25** , no. 23 (2013): 233201. arXiv:1302.5433 [cond- > mat.supr-con]. given by $$ H = \\eta_{k}\\tau_{z} + B\\sigma_{x} + \\alpha k\\sigma_{y}\\tau_{z} + \\Delta\\tau_{x} $$ Here $\\sigma$ and $\\tau$ are the Pauli matrices for the spin and particle-hole space. Now the correct result is: $E^{2}_{k} = \\Delta^{2} + \\eta_{k}^{2} + B^{2} + \\left(\\alpha k\\right)^{2} \\pm 2\\sqrt{B^{2}\\Delta^{2} + \\eta^{2}_{k}B^{2} + \\eta^{2}_{k}\\left(\\alpha k\\right)^{2}}$ Now, my problem is that I don't know how I can bring the Hamiltonian in the correct matrix form for the calculation of the eigenvalues. If I try it with the upper Hamiltonian I have completely wrong results for the energy spectrum. I believe my mistake is the interpretation of the Pauli matrices $\\tau$ but I don't know how I can write the Hamiltonian in the form to get the correct eigenvalues."}
{"id":"117190","title":"Why is gas mileage typically better when traveling on the highway than on country roads or in the city?","text":"The gas mileage of my vehicle tends to improve the more I have been driving on the interstate on that tank of gas - if I go through a tank of gas without at any point driving on the interstate, I will typically get 24-26 MPG, but when I have driven almost exclusively on the interstate, I will typically get 26-29 MPG. What confuses me is that, when traveling on the interstate, I am driving great distances at higher RPMs, which I would expect to correlate to more gas usage and thus worse gas mileage. Why does the inverse seem to be true?"}
{"id":"134363","title":"Riemann curvature tensor notation in Wald","text":"This question is entirely on tensorial notation in Wald's General Relativity. When specifying the properties of the Riemann tensor on pg39, he states: $R_{[abc]}^{\\quad \\ \\ \\ d} = 0$ and > For the derivative operator $\\nabla_a$ naturally associated with the metric, > $\\nabla_a g_{bc}=0$, we have $R_{abcd} = -R_{abdc}$. and > The Bianchi identity holds: $\\nabla_{[a}R_{bc]d}^{\\quad \\ \\ e} = 0$ Questions:   1. What do the square brackets around \"abc\" mean?   2. Why does $R_{abc}^{\\quad d}$ become $R_{abcd}$? What is the relation between the two?   3. What does having $R$ in the square brackets mean? Thank you."}
{"id":"81990","title":"How much the probability wave acts like a wave?","text":"A superposition of two probability wave creates standing wave. Well, that is convincing. Dose waves described by schrodinger's equation have other properties of wave? like reflection, refraction, polarization ect."}
{"id":"81992","title":"Intuitively, Why is Power Proportional to $I^2R$","text":"As the resistance of a circuit goes down, the power increases because the current increases, assuming constant voltage. Why is this? I feel like resistance and current are inversely proportional, so lowering one should just raise the other, and power should stay relatively constant. What is it about power that makes it work like this? Edit: Again, namely I find it confusing that decreasing resistance actually has a net increase in power used. I feel like while the current will increase, it should be countered by the lowered resistance..."}
{"id":"128144","title":"How far can one hear sound?","text":"I was thinking how far can I hear sound coming from a concert. Today I was walking at night and I could hear sound from somewhere very far. I started following the sound but sound used to disappear momentarily and then reappear. I went atleast 2 miles but could not find where it was coming from. Now I am really confused because sound is energy and because of energy losses, as I understand sound cannot go as far as 20 miles. The nearest concert was atleast 20 miles away. I came back and read articles(I cannot proof the reliability) where sound had traveled 200 miles but it was mostly sounds from explosion or volcanoes. Maybe this makes sense because explosions or volcanoes have alot of energy but how can one explain sound being heard from a concert very far away(approx 20 miles)."}
{"id":"116453","title":"Candidates for holographic QFT of 4D Einstein gravity","text":"If we are to believe that holographic principle holds over a wide number of dimensions, and gravitational theories, but specially, those that are relevant to our universe, then there must be some 3D QFT that is the dual description of our current 4D General relativity, over what it seems to be asymptotically flat de-Sitter spacetime. > What QFT theories are candidates for being the holographic dual of > gravitational theory in _our_ universe?"}
{"id":"30267","title":"What differs string theory from philosophy or religion?","text":"> **Possible Duplicate:**   >  What experiment would disprove string theory? A hypothesis without hard evidence sounds very much like philosophy or religion to me. All of them tries to establish a functional model explaining how the world works. In my understanding, string theory is an unprovable theory. What differs string theory from philosophy or religion?"}
{"id":"134097","title":"Why does time slow down the closer you are to a mass?","text":"When ever i look this up all I get is sites saying how its because general relativity says \"-\" why does it do it though? it is because there is more motion near gravity than further away? Or is it something completely different?"}
{"id":"134096","title":"Why is metallic hydrogen degenerate matter?","text":"Why is metallic hydrogen considered a form of degenerate matter, akin to neutronium and electron-degenerate matter? I can understand that for the other two, degeneracy pressure is the only force countering inward gravity for very massive stars, but how does this concept also apply to super pressurised hydrogen? Furthermore, why would being supported by degeneracy pressure make the hydrogen metallic in nature?"}
{"id":"75166","title":"What are magnetic field lines?","text":"Does a magnetic field have concentrations of magnetic force lines as seen when putting iron filings over top a bar magnet or are these imaginary? I.e. are they just an artifact of the iron being a 'conductor' of the magnetic field lines making them look like they are concentrated along these path lines but are really continuum of strength around the bar magnet when there are no filings are present? Also I learned that field lines do not cross, yet there are magnet configurations who's forces are explained as the magnetic force vectors are indeed crossing and are additive such as a Halbach Magnet Array. So what is actually happening here?"}
{"id":"36179","title":"Would a rocket burn more fuel to get from Earth's surface to LEO, or to get from LEO to GEO?","text":"Would a rocket burn more fuel to get from Earth's surface to Low Earth Orbit, or to get from LEO to Geosynchronous Earth Orbit?"}
{"id":"18768","title":"Elasticity of Space; How does the expansion of Space affect gravity?","text":"Does space have an elastic quality? What I was thinking about was if space is expanding, is it being 'stretched', like a balloon being blown up, and if so, is this causing gravity to weaken? Imagine space as a 2 dimensional sheet (got this from one of Brian Greene's books) with planetary bodies resting on it and causing a depression in it, if you were pulling this sheet from all sides over a period of time, you would cause the depression of the planetary body to decrease and eventually become flat, which if we go back to reality, would mean that the gravitational 'constant' had changed to the point where the planetary body had no influence on those objects which were previously orbiting around it (or even residing on it's surface). Is this the case in reality? Or does space not have an elastic quality? If not, can you explain to me what exactly it means for space to be expanding? In case you didn't notice, I'm a layman (hence the Brian Greene books :p), so try to keep your answers\/explanations conceptual if possible."}
{"id":"89941","title":"Do bullet\/gun physics change if the gun is traveling very fast?","text":"If I'm running at say $400\\: \\mathrm{m\/s}$ and a bullet travels at $400\\: \\mathrm{m\/s}$ and I fire the gun, will I see the bullet leave the barrel? I either see it stay in the barrel floating because we are moving at the same speed. \\-- Or -- The bullet would move and have a speed of $800\\: \\mathrm{m\/s}$. What is the correct answer?"}
{"id":"18762","title":"Locality in Quantum Mechanics","text":"We speak of locality or non-locality of an equation in QM, depending on whether it has no differential operators of order higher than two. My question is, how could one tell from looking at the concrete solutions of the equation whether the equ. was local or not...or, to put it another way, what would it mean to say that a concrete solution was non-local? edit: let me emphasise this grew from a problem in one-particle quantum mechanics. (Klein-Gordon eq.) Let me clarify that I am asking what is the _physical meaning_ of saying a solution, or space of solutions, is non-local. Answers that hinge on the _form_ of the equation are...better than nothing, but I am hoping for a more physical answer, since it is the solutions which are physical, not the way they are written down .... This question, which I had already read, is related but the relation is unclear. Why are higher order Lagrangians called 'non-local'?"}
{"id":"18764","title":"What happened to the work done by friction here?","text":"**Problem\/Solution** ![](http:\/\/img832.imageshack.us\/img832\/1313\/88791065.jpg)! **Question** What happened to the work done by friction BEFORE it touched the spring? Why was that neglected? Also they say there is no physical meaning behind the negative root, so what is the \"unphysical\" meaning behind the negative root? How are we supposed to know that the speed is constant before and just as it makes contact with the spring? **@atomSmasher** ![](http:\/\/img59.imageshack.us\/img59\/917\/friclk.jpg)! I am referring to the green region. Shouldn't it be $E_i = \\frac{1}{2}mv_A ^2 - f_k(x_b + x_{green distance})$ $E_f = \\frac{1}{2}kx_B ^2$ I realize we would have two unknowns then."}
{"id":"33523","title":"Cosmological constant of standard model of cosmology and observational data","text":"I am curious whether the current Lambda-CDM model of cosmology matches well with observational data, especially expansion of the universe. How well does Lambda-CDM defend its established status from other models, such as quintessence (quintessence can be said to extend Lambda-CDM, but there are some models against the standard model, I guess.)?"}
{"id":"71776","title":"Is there a connection between the core of the earth and heaviest element produced by stars?","text":"I was unable to answer this question from my daughter. Is it just a coincidence or is there a connection between the following two observations: (1) the core of the earth is made of an iron (-nickel alloy) as well as the existence of iron meteorites with the fact that (2) the heaviest element that can be produced by a star (prior to supernova) is iron? Related: Elements of a Planet reveals nearby supernova remnant?"}
{"id":"71775","title":"Calculating angular momentum","text":"A particle of mass $m$ is at a very large distance $p$ from origin $O$ and is moving with velocity $\\vec{V}$ which is perpendicular to $\\vec{OP}$. I have to calculate angular momentum $L$ of the particle. I know that $\\vec{L}=\\vec{r}\\times m\\vec{\\dot r}$.   Since $|\\vec{r}|=p$ and $|\\dot r|=V$ and $\\alpha=90^{\\circ}$ is angle between $\\vec{r}$ and $\\vec{\\dot r}$, I got that $L=|\\vec{r}||m\\vec{\\dot r}|sin\\alpha=pmV\\cdot 1=pmV$. In the book, it's written that $L=pV$. What happened with mass?"}
{"id":"34008","title":"Does the (relativistic) mass change? Why?","text":"I learned recently that when an object moves with a velocity comparable to the velocity of light the (relativistic) mass changes. How does this alteration take place?"}
{"id":"45320","title":"Why does an object with higher speed gain more (relativistic) mass?","text":"Today, in my high school physics class, we had an introductory class on electromagnetism. My teacher explained at some point that an object with a very high speed (he said it started to get somewhat clearly noticable when travelling at 10% of the speed of light) will gain mass, and that that's the reason why you can't go faster than light. One of my classmates then asked, why is this so? Why does an object with higher speed gain more mass? This of course is a logical question, since it is not very intuitive that a higher speed leads to a higher mass. My teacher (to my surprise (responded saying that it is a meaningless question, we don't know why, in the same way we don't know why the universe was created and those kind of philosophical questions. I, being interested in physics, couldn't believe this, I was sure that what he said wasn't true. So after a while of thinking I responded saying: _Can't we describe it with Einstein's $E=mc^2?$ If an object gains speed, he gains more (kinetic) energy. With this equality we see that the more energy an object gets, to more massive it becomes._ He then replied saying that this formula is used for different cases, whereupon he gave a vague explanation as to when it is used. He gave me an example to show what I said was incorrect; when a car goes from $10 m\/s$ to $40m\/s$, according to what I said we would see a big increase in mass, and we don't (this sounded logical to me). So here I am, with the following questions:   * Why does an object with a higher speed have more mass (than the same object with a smaller speed)?   * When is $E=mc^2$ used and why is my argument incorrect in explaining this phenomenon?"}
{"id":"76203","title":"Reduced graphs and pinch-singular surfaces","text":"I am reading a book on perturbative QCD by John Collins. In Chapter 5, the terms **_reduced graph_** and **_pinch-singular surface_** are used for the analysis of mass singularities. However, their meanings are not explainded very clearly. It says (Sect 5.1.4, page 91) > The PSSs (for the physical region, which is all that concerns us) are where > the on-shell propagators and momenta correspond to classically allowed > scattering processes treated in coordinate space. I cannot get the key points from this _definition_. For reduced graphs, it provides some examples. But I still cannot understand why they are important and how they are related to the Feynman diagrams. Could someone help me to clarify these terminologies? Many thanks!"}
{"id":"25896","title":"Are there formulae for calculating stellar luminosity and effective temperature as a function of age?","text":"> **Possible Duplicate:**   >  Are there formulae for calculating stellar luminosity and effective > temperature as a function of age? Is there a manageable formula or set of formulas or simple algorithms that approximate stellar luminosity and effective temperature (or radius) as a function of stellar age? I'm aware that accurate modeling of these attributes is complex and is determined by many factors; what I'm looking for is something that serves as a decent approximation of the sort used in numerous illustrations or applets that show example \"paths\" taken by representative stars as they trace their evolution across the H-R diagram. (From this original question posted on Physics.)"}
{"id":"44912","title":"Ball rolling down an inclined plane going in to a loop","text":"I apologize if this question is not up to par. When I was doing exercises in basic mechanics I checked the answers and I can't seem to find what I'm doing wrong. Suppose we have a ball with mass $m$ and radius $r$ on an inclined plane with height $h$. At the end of the inclined plane is a loop with a radius of $R$ and we can assume that $r<<R$. We are asked what the minimum height is the inclined plane should have so as to let the ball complete the loop. Here is my reasoning: We have $U_1\\geq K_{rot,2}+K_{trans,2}+U_2$. For the ball rotating we have $I=\\frac{1}{2}mr^2$ and $\\omega=\\frac{v}{r}$. So $mgh\\geq \\frac{1}{2}I\\omega^2+\\frac{1}{2}mv^2+mg\\cdot2R=\\frac{3}{4} mv^2+mg\\cdot 2R$. The minimal speed to complete the loop implies $F_{centripetal}=\\frac{mv^2}{R}= mg$ So $v^2= Rg$ and we have $gh\\geq \\frac{3}{4}Rg+g\\cdot 2R$ which means $h\\geq 2\\frac{3}{4}R$, while the book says that the answer should be $h\\geq 2.70R$. Can you explain what I am doing wrong? Thank you **EDIT:** Moment of inertia corrected. For the ball rotating we have $I=\\frac{2}{5}mr^2$ and $\\omega=\\frac{v}{r}$. So $mgh\\geq \\frac{1}{2}I\\omega^2+\\frac{1}{2}mv^2+mg\\cdot2R=\\frac{7}{10} mv^2+mg\\cdot 2R$. After the edit with correction of the moment of inertia I am getting the right answer. We get $gh\\geq\\frac{7}{10}Rg+g\\cdot 2R$ so $h\\geq2\\frac{7}{10}R$ in accordance with the book."}
{"id":"95963","title":"Information about fields and superfields","text":"I want some explanation of fields and superfields (types and components), and what the relationship between them and representation of a group."}
{"id":"3580","title":"Virtual photon description of B and E fields","text":"I continue to find it amazing that something as “bulky” and macroscopic as a static magnetic or electric field is actually a manifestation of virtual photons. So putting on your QFT spectacles, look closely at the space near the pole of a powerful magnet – virtual photons! Now look between the plates of a charged capacitor – virtual photons again! But if it’s all virtual photons, how do we get the difference between a magnetic and electric field? [This question needs the tag \"magnetostatics\" as well]."}
{"id":"86658","title":"How long does it take for a metal to reach equilibrium?","text":"I wonder if there is a measure of how long a piece of metal takes to reach electrostatic equilibrium. Does it depend on piece's size? Does it depend on the amount of imbalance? Lots of websites and textbooks report \"after a very short time\". But how much short? Thanks a lot"}
{"id":"89496","title":"Centripetal force in frame of reference of body moving In a circle","text":"Suppose a body is moving in a circle about a fixed point. In the frame of reference of the body, is the centripetal force felt or is only the centrifugal force felt? More generally, does a body only feel the effect of pseudo forces in an accelerated reference frame?"}
{"id":"88721","title":"Is the sum of the amplitudes of an electromagnetic wave always 1?","text":"It's been a while, and I'm trying to verify my understanding. I remember reasoning (but never being taught) that the sum of the (normalized) electric and magnetic waves in a single electromagnetic wave at any single point in time is always one. Like so:               Given A of em-wave is normalized (max = 1, min = -1)              abs(sum(A of m-wave, A of e-wave)) = 1      I haven't been able to find anywhere that says this specifically (I probably suck at searching, or am wrong). Am I correct in my understanding?"}
{"id":"88724","title":"Is the total energy of earth changing with time?","text":"Many years ego, Earth was hot. Over time, it has lost energy and has become colder. Is it now in equilibrium or is its total energy changing?"}
{"id":"88727","title":"Quantizing the Dirac Field: which commutation relations are more fundamental?","text":"When quantizing a system, what is the more (physically) fundamental commutation relation, $[q,p]$ or $[a,a^\\dagger]$? (or are they completely equivalent?) For instance, in Peskin & Schroeder's QFT, section 3.5, when trying to quantize the Dirac field, they first say what commutation relation they _expect_ to get for $[\\Psi(\\vec{x}),\\Psi^\\dagger(\\vec{y})]$ (where $i\\Psi^\\dagger$ is the conjugate momentum to $\\Psi$), in analogy to the Klein- Gordon field, then they _postulate_ a commutation relation between $[a^r_{\\vec{p}},a^{s\\dagger}_{\\vec{q}}]$ etc., and then verify that they indeed get what they expected for $[\\Psi(\\vec{x}),\\Psi^\\dagger(\\vec{y})]$. Why did we need to postulate the value of $[a^r_{\\vec{p}},a^{s\\dagger}_{\\vec{q}}]$? Couldn't we have just computed it off of $[\\Psi(\\vec{x}),\\Psi^\\dagger(\\vec{y})]$? (which, by expecting to get it, we could have just as well already _postulated_ it). I suppose that would entail explicitly writing something like: $$ a_{\\vec{p}}^r = \\frac{1}{\\sqrt{2E_{\\vec{p}}}} u^{r\\dagger}(\\vec{p})\\int_{\\mathbb{R}^3}d^3\\vec{x}\\,e^{-i\\vec{p}\\cdot\\vec{x}}\\Psi(\\vec{x})$$ and a similar expression for $b$."}
{"id":"65063","title":"When (Exactly) Does Bremsstrahlung Radiation Occur?","text":"Regarding the Bremsstrahlung Radiation emitted when a charged particle curves inside a uniform magnetic field; Is there a way of calculating the angle through which the charged particle will precess before emitting the first quantum of radiation? Also, is there a way of determining the angle through which the charged particle will precess between the first and second emissions of radiation? My investigations revealed only what I think are the angles of emission relative to the tangent of the curve here."}
{"id":"13985","title":"calculating of g force in fun ride","text":"we vistited yesterday a fun park and saw a ride which was a big centrifuge only in vertical. this one had a speed meter but it was broken and i also wanted to know how much g-force is on there... so my first thought was to calculate this with the centrifugal-accerleration. I have $\\omega = \\frac {0.5*2\\pi} 1 [\\frac {rad} s]$ and $r= 25[m]$, so the accerleration is $a_z = \\omega^2r = \\pi^2*25 = ~246$ then i can calculate the g force with $g_{force} = \\frac {a_z} g$ which gives me about $24g$ that seems fairly too high! what am i missing?"}
{"id":"22749","title":"Angle of deflection of an atomic nucleus","text":"when firing a proton (for example) to an atomic nucleus, from a distance $D$, the deflection angle of the proton $\\alpha $ to the type of changes atomic nuclei? or always constant?"}
{"id":"77662","title":"Waves produced by a mass of water on the calm sea surface","text":"Suppose we have a parallelepiped shaped box full of water on the surface of the sea. Suddenly the box disappears. What is the shape of the waves vs. time caused by the fall of the water contained in the box?"}
{"id":"128088","title":"Mathematical derivation of interference pattern for electrons?","text":"One of the most famous experiments in quantum mechanics in the context of wave-particle duality is certainly passing a beam of electrons through two slits, which results in an interference pattern being formed on a screen positioned after them. Now, **starting from the Schrödinger equation** (which we know is obeyed by electrons), how can we derive the formula describing the interference pattern? I am pretty sure we already know the answer, i.e. it is the same pattern generated by light going through two slits (so a sine squared modulated by a sinc envelope), but how do we get it **_from quantum mechanics_** ?"}
{"id":"77660","title":"What's the relationship between the energy density of a black-body and its radiant exitance?","text":"Through a bit calculation we can derive that in a cavity, the energy density $$u(f,T)=\\overline{E(f)}\\times G(f)=\\frac{8\\pi h}{c^3}\\frac{f^3}{e^{h\\nu \/kT}-1}$$ If we take the integral over all frequency, we get $$U(T)=\\frac{8πh}{c^3}\\frac{(kT)^4}{h^3}{\\frac{π^4}{15}}=C_{onst}T^4$$ And Stefan-Boltzmann Law claims that for a perfect black-body $$j^*=\\sigma T^4$$ where $j^*$ is the radiant exitance, which is defined as the total energy radiated per unit surface area of a black body across all wavelengths per unit time. And it just so happens that $\\frac{\\sigma}{C_{onst}}=\\frac{c}{4}$, why is that? P.S. The professor told me to refer to some thermodynamics book, where a more general case is discussed. But we don't have that book in our library and the professor's now out of town xD."}
{"id":"1720","title":"Ice skating, how does it really work?","text":"Okay, some textbooks I came across, and a homework assignment I had to do several years ago, suggested that the reason we can skate on ice is the peculiar $p(T)$-curve of the ice-water boundary. The reasoning is that due to the high pressure the skates put on the ice, it will melt at temperaturs below $273 K$ and thus provide a thin film of liquid on which we can skate. It was then mentioned as fun fact that you could ice-skate on a planet with lakes of frozen dioxide because that gas has the $p(T)$-curve the other way round. My calculations at that time told me that this was, pardon my french, bollocks. The pressure wasn't nearly high enough to lower the melting point to even something like $-0.5$ degrees Celsius. I suppose it is some other mechanism, probably related to the crystal structure of ice, but I'd really appreciate if someone more knowledgeable could tell something about it."}
{"id":"98835","title":"Being in a solid state, why is ice slippery?","text":"Saying that ice is slippery is like saying that water is wet -- it's something we've known for as long as we can be said to have known anything. Presumably, humans as a species knew ice was slippery before we knew fire was hot, or that it existed. But ask anyone why, and they won't be able to give you any better explanation than one of those cave people would have."}
{"id":"69693","title":"What exactly is the composition of radiation in interplanetary space?","text":"The articles I found on radiation in the solar system mostly dealt with solar wind, I wonder about other types. Is there a breakdown that tells me, withhin an order of magnitude, at least what intensity I can expect for a the different spectra - hard gamma through to radio frequencies, and chargede particles - and how this varies depending on my position?"}
{"id":"43338","title":"How to explain (pedagogically) why there is 4 spacetime dimensions while we see only the 3 spatial dimesions?","text":"I have been asked this question by a student, but I was able and in the same time incapable to give a good answer for this without equations, so do you have ideas how one can explain this in a simple way? (Answers like we can take time as an imaginary, or our space is actually pseudo-Euclidean will be hard to grasp for new students.) Note that the problem is _not_ in visualizing the 4th dimension, that an easy thing to explain. The problem is more related to why we are in 3D that moving along 1D time dimension? In Differential geometry this interpreted by fiber bundles, but how to explain it to usual student."}
{"id":"43332","title":"Why is a hole in a ball?","text":"I wanted to know why there is a hole in the ball (basketball, volleyball, handball) to fill the ball with air. Why can't the ball come with filled in air and fully sealed so that there is no loss of air?"}
{"id":"116352","title":"Doppler effect and light","text":"Approaching the speed of sound in an aircraft is relatively difficult, because the closer you get to Mach 1, the denser the pressure is around you (sound accumulates causing vibrations). Is there a similar effect as you approach the speed of light? Does the Doppler Effect apply for light as it does with sound? Will light accumulate around your aircraft causing heat? Speaking strictly hypothetical."}
{"id":"56841","title":"mixture of maximally mixed and maximally entangled state","text":"Consider the quantum system $\\mathcal{B}(\\mathbb{C}^d\\otimes\\mathbb{C}^d)$ and $|\\psi\\rangle=\\frac{1}{\\sqrt{d}}\\sum_{i=0}^{d-1}|i,i\\rangle$ be the (standard) maximally entangled state. Consider the state $\\rho_\\lambda=\\lambda \\frac{\\mathbb{I}_{d^2}}{d^2}+(1-\\lambda)|\\psi\\rangle\\langle\\psi|.$ Now for some values of $\\lambda$ this state is entangled (example $\\lambda=0$ it is $|\\psi\\rangle\\langle\\psi|$) and hence its entanglement can be detected by partial transpose operation. Can $\\rho(\\lambda)$ be an entangled state which is positive under partial transpose (known in literature as PPT entangled state) for some values $\\lambda$? My intuition tells me that this is the case. However, I was told (without reference) that this is not the case and for the values of $\\lambda$ we get only separable states or not PPT entangled states. I could not find the corresponding paper. May be I am not giving the proper string for searching. Advanced thanks for any suggestion, reference or comment."}
{"id":"116356","title":"Quick question on degeneracy of harmonic oscillator states","text":"I'm currently learning about symmetry between particles. For a simple case of two non-interacting particles at $x_1$ and $x_2$, we know that the wavefunction can be written as $\\psi_{n_1, n_2} = \\phi_{n_1}(x_1) \\phi_{n_2}(x_2)$ Let's consider two cases in ground state and first excited state. (a) Two particles are not identical, and have spin zero (b) Two identical particles, each with spin zero We know that the energy is given by $E_n = (n_1 + n_2 + 1)\\hbar \\omega$. **For case (a)** Since they are not identical, the wavefunction at ground state would be $\\phi_1(x_1) \\phi_1 (x_2)$ right? Then the degeneracy would be 0 right? Since $n_1 = n_2 = 1$. For the first excited state, the wavefunction would be $\\phi_2(x_1)\\phi_1(x_2)$, assuming $E_2(x_1) < E_2(x_2)$, meaning we just take whatever is lower. The degeneracy would be 0, since they are distinguishable? **For case (b)** Now they are identical, ground state wavefunction would still be $\\phi_1(x_1)\\phi_1(x_2)$ right? Degeneracy of ground state would be 0, since $n_1=n_2=1$? But for the first excited state, the degeneracy would be 2, since either $n_1=2$ and $n_2=1$ or $n_1=1$ and $n_2=2$? Spin 0 particles are bosons, so the wavefunction must be symmetric right? So I'm thinking $\\psi = \\frac{1}{2} \\left[ \\phi_2(x_1)\\phi_1(x_2) + \\phi_1(x_1)\\phi_2(x_2)\\right]$ The concept of symmetry and exchange symmetry is still new to me, I'd appreciate any response!"}
{"id":"94364","title":"Torque for rotation","text":"The force for revolution of earth is provided by the gravitational force of attraction between earth and the sun. What provides the force(torque) for the rotation of earth?"}
{"id":"56849","title":"Why is electron presented in books, pictures as a sphere?","text":"Why is electron presented in books, pictures as a sphere, when in fact it's not?"}
{"id":"52343","title":"What does it mean to find acceleration in terms of g?","text":"I'm having trouble understanding what a problem I have is seeking. To simplify the problem: > A particle reaches a speed of 1.6 m\/s in a 5.0 micrometer launch. The speed > is reduced to zero in 1.0 mm by the air. Assume constant acceleration and > find the acceleration in terms of g during a) the launch and b) the speed > reduction. The basic strategy to find acceleration I am using is to calculate two velocity equations: one between (0 m\/s, 0 m) and (1.6 m\/s, 5.0 micrometers); the second between (1.6 m\/s, 5.0 micrometers) and (0 m\/s, 1.0 mm). Then I will derive the acceleration value for each. Because acceleration is constant I can expect a linear velocity equation. What is confusing me is that we are to assume constant acceleration. Thus the acceleration equation will merely be some real number. So, what exactly is expected if it is to be in terms of g? Is my strategy to find acceleration incorrect?"}
{"id":"17352","title":"Nonlinear absorption coefficient and the band gap","text":"How does the nonlinear absorption coefficient depend on the band gap? How can that coefficient be calculated theoretically? (Preferably with an example)"}
{"id":"60251","title":"Help me to understand this conversion (4-vectors)","text":"$u^{\\mu}$ - 4-velocity $b^{\\mu}$ - 4-vector of magnetic field $ u_{\\mu}u^{\\mu}=-1, \\qquad u_{\\mu}b^{\\mu}=0 $ $$ u_{\\beta}u^{\\alpha}\\nabla_{\\alpha}b^{\\beta}-u_{\\beta}b^{\\alpha}\\nabla_{\\alpha}u^{\\beta}+\\nabla_{\\alpha}b^{\\alpha}=0 $$ I don't understand why this equation gives this $$ u^{\\alpha}u^{\\beta}\\nabla_{\\alpha}b^{\\beta}+\\nabla_{\\alpha}b^{\\alpha}=0 $$ Help me please!"}
{"id":"19588","title":"How to prove that a motion is Simple Harmonic Motion (SHM)?","text":"I would like to know how one could show and prove that a given motion is simple harmonic motion. Once given an answer, I'll apply that technique to an example I am trying to figure out. Thank you in advance! I believe a motion can be proved simple harmonic, if the relation between its is as such: $$ a_x = - \\omega^2\\cdot x $$ And as such the period time is: $$ T =\\frac{2\\pi}{\\omega} $$ Question-so-far: How do you prove such for a given force $F = \\frac{G\\cdot m_e \\cdot M}{R_E} \\cdot r$ ? Or any force that has non-trivial constants?"}
{"id":"19586","title":"Hydrostatic equilibrium of a star derivation","text":"I've been told to revise the derivation that proves $\\frac{\\mathrm{d}P}{\\mathrm{d}r} =\\frac{GM(r)p(r)}{r^2}$ where brackets indicate a function of, P is pressure and p is density. Rather helpfully he hasn't given us it to revise, so if anyone knows it I'd be really grateful. Thanks."}
{"id":"52699","title":"How can one determine at which distance the Lennard-Jones potential reaches a given value?","text":"My question is fairly simple, but I do need clarification on how to get the inverse of the Lennard-Jones potential V(x). I am working with the following expression: $$ V(x) = e\\times[(R\/x)^{12} -2\\times(R\/x)^6] $$ So given a value $V$, how can I find $x(V)$ ?"}
{"id":"69036","title":"The unitary time-evolution in the interation picture","text":"I'm currently consuming a course on QFT where we need to define the unitary time-evolution to get the time evolution of the wave function in the interaction picture: $\\hat{U}(t_1,t_0) = \\exp\\left(\\frac{i}{\\hbar}\\hat{H}_0t_1\\right)\\exp\\left(-\\frac{i}{\\hbar}\\hat{H}(t_1-t_0)\\right)\\exp\\left(-\\frac{i}{\\hbar}\\hat{H}_0t_0\\right)$ . Now one can show that this operator follows a Schrödinger equation by simply taking the derivative to time: $i\\hbar\\frac{d}{dt}\\hat{U}(t,t_0) = \\hat{H}_1^I(t)\\hat{U}(t,t_0)$ . Where $\\hat{H}_1^I$ is the perturbation to our free-field Hamiltonian $\\hat{H}_0$. Now I started wondering whether $\\hat{U}(t_1,t_0)$ shouldn't also follow a Heisenberg equation since it's an operator. I believe it shouldn't since $\\hat{U}(t_1,t_0)$ gives a unitairy time- evolution which is a transformation, while the Heisenberg equation applies to observables. I was woundering if someone could confirm my reasoning or disprove it?"}
{"id":"64002","title":"Why should SUSY be expected naturally?","text":"In the last 40 years (approximately) people have been \"discovering\", \"rediscovering\" and \"studying\" SUSY as a powerful tool and \"symmetry principle\". Question: What if SUSY is not realized in Nature at the end? Is SUSY the only path to \"relate\" fermions and bosons or what else? Remark: SUSY has not been discovered yet, so keep you totally conservative. What if there is no SUSY? Bonus: What are the merits of SUSY? What are its main issues? I do know some answers to this, but I think it could very enlightening if we \"listed\" pros and contras of current supersymmetric theories to see where we are NOW."}
{"id":"52693","title":"Irreversible process","text":"I have this problem. I have an ideal gas that goes through an irreversible adiabatic decompression. I have the initial state (P,T,V), and the final pressure, and I have to calculate the entropy difference of the proccess. So, what I know is that I can make up any reversible proces bewteen the initial and final state because entropy is a state function, and integrate the heat over that process, but I can't solve it. I'm aplying $P_1V_1^\\lambda=P_2V_2^\\lambda$ to get the final state, but as I am creating the final point of the adiabatic, I get entropy difference of $0$ for an invented reversible path. That way of getting the final state is not correct, right? I have $PV=nRT$ but I only have the final $P$ so I need another equation. EDIT: I have tried also to use $\\Delta U=W\\Rightarrow C_v\\Delta T=P_{ext}\\Delta V$, get $T_{final}$, and go on, but I don't get the correct result either. I thought this was general: $dU=C_vdT$, when can I apply that equation?"}
{"id":"99520","title":"Why can't single LEDs produce white light directly?","text":"Why does production of white light using a LED require combining a short wavelength LED such as blue or UV, and a yellow phosphor coating? Why can't a single LED produce pure white light?"}
{"id":"45633","title":"The end of theoretical physics?","text":"About the meta questions 1193 and 2609, I've heard parallelly, that the complete branch theoretical physics is already done and that there isn't any thing else to do in this field, how true is it?"}
{"id":"51923","title":"Complex coordinates in CFT","text":"**The Setup** : Let's say we want to study a Euclidean $\\mathrm{CFT}_2$ on $\\mathbb R^2$ with coordinates $\\sigma^1$ and $\\sigma^2$ and metric $ds^2 = (d\\sigma^1)^2+(d\\sigma^2)^2$. It seems to me that in the usual discussion (e.g. di Francesco, Ginsparg, Polchinski), one proceeds to consider an analytic continuation of the CFT to $\\mathbb C^2$ with coordinates $z^1, z^2$ and complex metric $ds^2 = (dz^1)^2+(dz^2)^2$ and then, one performs the coordinate transformation $z = z^1+iz^2$ and $\\bar z = z^1-iz^2$. In this way the coordinates $z$ and $\\bar z$ can be considered \"independent\" because they are coordinates on a complex two-dimensional manifold. Also, in these coordinates the metric becomes $ds^2 = dz\\,d\\bar z$ and it becomes clear that conformal mappings consist of mappings: $(z, \\bar z)\\to (f(z), g(\\bar z))$. **My confusion is this** : Since our original theory was on $\\mathbb R^2$, books say that when we do calculations, we should consider the physical theory as living on the copy of $\\mathbb R^2$ embedded in $\\mathbb C^2$ given by the condition $\\bar z = z^*$. But consider the mapping $(z, \\bar z)\\to (z^2, \\bar z)$. This is a conformal mapping on $\\mathbb C^2$, but it does not map the surface $\\bar z = z^*$ to itself; for example the point $(z, \\bar z)=(2,2)$ gets mapped to the point $(z^2, \\bar z) =(4,2)$ and $2$ is clearly not equal to $4^*$. In particular, it seems to me that analytic continuation to a CFT on $\\mathbb C^2$ enlarges the set of mappings one can have, so what relevance does it really have to the original CFT on $\\mathbb R^2$? (my apologies for the long post)"}
{"id":"107241","title":"The Physical Meaning behind a Commutator","text":"I've just been introduced to the idea of commutators and I'm aware that it's not a trivial thing if two operators $A$ and $B$ commute, i.e. if two Hermitian operators commute then the eigenvalues of the two operators can be measured with certainty simultaneously. But what is the physical significance when two operators do not commute such as to give a certain value? For example the position and momentum operator do not commute and give a value of $i\\hbar$. What is the significance of the $i\\hbar$?"}
{"id":"113825","title":"Momentum of Light Question","text":"Since light beams carry momentum, why would a person holding a flashlight not feel a recoil similar to that of a rifle being fired?"}
{"id":"16910","title":"What's the difference between dim and bright light?","text":"When comparing two light sources, for example, a light bulb at 20W and a light bulb at 100W, what is it about the incoming light that makes the latter look brighter than the former? Are there different reasons why different light sources looks different in brightness (High five for cramming three instances of \"different\" in the same sentence)? For example, in this thread, it is stated that the human eye is most sensitive around 555nm, something that I guess translates to meaning that given a light of the same intensity (whatever that means, hence my question), it is going to be perceived as most bright when hitting 555nm. Does this question have different answers depending on if you're seeing light as a particle vs a wave?"}
{"id":"16913","title":"If a car appears in horison and within 2 seconds passes you by, whats the speed it's doing?","text":"While watching the first 4 seconds of driving at 745 km\/h is ludicrous from any angle wondered 1)If we knew the curvature of the earth in a \"flat\" desert, what would be the speed of the car? 2)Assuming we don't know the curvature of the earth, how long would it take for the car to go around the earth ( assuming a \"flat\" desert all the way travelling on a great arc)? Using my ninja pause skills it seemed it took 2 seconds from the time the car appeared ( as a dot ) in horizon to the time it passed by the camera, although 2 second window of observation seems to lead to a great error in overall estimates, using the 745 as the value for 1 and 2 the error of human observation could be calculated (?)"}
{"id":"12208","title":"How come an anti-reflective coating makes glass *more* transparent?","text":"The book I'm reading about optics says that an anti-reflective film applied on glass* makes the glass _more_ transparent, because the air→film and film→glass reflected waves (originated from a paraxial incoming wave) interfere destructively with each other, resulting on virtually no reflected light; therefore the \"extra\" light that would normally get reflected, gets transmitted instead (to honor the principle of conservation of energy, I suppose?). However, this answer states that _\"Superposition is the principle that the amplitudes due to two waves incident on the same point in space at the same time can be naively added together, but the waves do not affect each other.\"_ So, how does this fit into this picture? If the reflected waves actually continue happily travelling back, where does the extra transmitted light come from? * the film is described as (1) having an intermediate index of refraction between those of air and glass, so that both the air-film and film-glass reflections are \"hard\", i.e., produce a 180º inversion in the phase of the incoming wave, and (2) having a depth of 1\/4 of the wavelength of the wave in the film, so that the film-glass reflection travels half its wavelength back and meets the air-film reflection in the opposite phase, thus cancelling it."}
{"id":"12206","title":"Total power consumption of electric device","text":"For some electric devices, like a fan or air conditioner, I read about their power consumption in watts on their specification guide. Does it tell about the power at normal or full speed? or Does the speed even affect the power consumption? Can I find the consumption at different speeds?"}
{"id":"90120","title":"How do I determine $\\phi$ in a hollow circular cross section using torsion equivalence?","text":"I know that the maximum shear stress $$\\tau = \\frac{T}{J}\\rho$$ where $\\rho$ is the radial distance from the center of the cross section. I have also determined the torsional constant $J$, which is equal to $\\frac{\\pi}{2}(R_o^4-R_i^4)$ for this particular cross section. The following boundary condition applies: $$\\nabla^2\\phi = F$$ I know that $\\phi$ would be zero for a solid cross section, but I do not understand how I should calculate it with torsion equivalence. Any help is appreciated."}
{"id":"90129","title":"Photon propagator in terms of creation\/annihilation?","text":"As far as I understand it the photon propagator, $P(A\\rightarrow B)$, described in Feynman's QED book, gives the amplitude that a photon moves from spacetime point A to spacetime point B. I was wondering if in quantum field theory terms $P(A\\rightarrow B)$ is made up of the product of the following two amplitudes:   1. the amplitude that a photon is created at $B$ given that there is a photon at $A$.   2. the amplitude that a photon is annihilated at $A$ given that there is a photon at $B$. Is this the correct approach to describing a photon moving from $A$ to $B$ using creation and annihilation operators?"}
{"id":"46573","title":"What are the strings in string theory made of?","text":"This is a follow-up to an intriguing question last year about tension in string theory. What are the strings in string theory composed of? I am serious. Strings made of matter are complex objects that require a highly specific form of long-chain inter-atomic bonding (mostly carbon based) that would be difficult to implement if the physics parameters of our universe were tweaked even a tiny bit. That bonding gets even more complicated when you add in elasticity. The vibration modes of a real string are the non-obvious emergent outcome of a complex interplay of mass, angular momentum, various conservation laws, and convenient linearities inherent in of our form of spacetime. In short, a matter-based vibrating real string is the _outcome_ of the interplay of most of the more important physics rules of our universe. Its composition -- what is is made of -- is particularly complex. Real strings are composed out of a statistically unlikely form of long-chain bonding, which in turn depend on the rather unlikely properties that emerge from highly complex multiparticle entities called atoms. So how does string theory handle all of this? What are the strings in string theory made of, and what is it about this substance that makes string-theories simple in comparison to the emergent and non-obvious complexities required to produce string-like vibrations in real, matter-based strings? * * * **Addendum 2012-12-28 (all new as of 2012-12-29):** OK, I'm trying to go back to my original question after some apt complaints that my addendum yesterday had morphed it into an entirely new question. But I don't want to trash the great responses that addendum produced, so I'm trying to walk the razor's edge by creating an entirely new addendum that I hope expands on the intent of my question without changing it in any fundamental way. Here goes: The simplest answer to my question is that strings are pure mathematical abstractions, and so need no further explanation. All of the initial answers were variants of that answer. I truly did not expect that to happen! While such answers are sincere and certainly well-intended, I suspect that most people reading my original question will find them a bit disappointing and almost certainly not terribly insightful. They will be hoping for more, and here's why. While most of modern mathematical physics arguably is derived from materials analogies, early wave analogies tended towards placing waves within homogeneous and isotropic \"water like\" or \"air like\" media, e.g. the aether of the late 1800s. Over time and with no small amount of insight, these early analogies were transformed into sets of equations that increasingly removed the need for physical media analogies. The history of Maxwell's equations and then SR is a gorgeous example. That one nicely demonstrates the remarkable progress of the associated physics theories _away_ from using physical media, and _towards_ more universal mathematical constructs. In those cases I understand immediately why the outcomes are considered \"fundamental.\" After all, they started out with clunky material-science analogies, and then managed over time to strip away the encumbering analogies, leaving for us shiny little nuggets of pure math that to this day are gorgeous to behold. Now in the more recent case of string theory, here's where I think the rub is for most of us who are not immersed in it on a daily basis: The very word \"string\" invokes the image of a vibrating entity that is a good deal more complicated and specific than some isotropic wave medium. For one thing the word string invokes (perhaps incorrectly) an image of an object localized in space. That is, the vibrations are taking place not within some isotropic field located throughout space, but within some _entity_ located in some very specific region of space. Strings in string theory also seem to possess a rather complicated and certainly non-trivial suite materials-like properties such as length, rigidity, tension, and I'm sure others (e.g. some analog of angular momentum?). So, again trying to keep to my original question: Can someone explain what a string in string theory is made of in a way that provides some insight into why such an unusually object-like \"medium of vibration\" was selected as the basis for building all of the surrounding mathematics of string theory? From one excellent comment (you know who you are!), I can even give an example of the kind of answer I was hoping for. Paraphrasing, the comment was this: > \"Strings vibrate in ways that are immediately reminiscent of the harmonic > oscillators that have proven so useful analytically in wave and quantum > theory.\" Now I like that style of answer a lot! For one thing, anyone who has read Feynman's section on such oscillators in his lectures will immediately get the idea. Based on that, my own understanding of the origins of strings has now shifted to something far more specific and \"connectable\" to historical physics, which is this: > Making tuning forks smaller and smaller has been been shown repeatedly in > the history of physics to provide an exceptionally powerful analytical > method for analyzing how various types of vibrations propagate and interact. > So, why not take this idea to the logical limit and make space itself into > what amounts to a huge field of very small, tuning-fork-like harmonic > oscillators? Now _that_ I can at least understand as an argument for why strings \"resonated\" well with a lot of physicists as an interesting approach to unifying physics."}
{"id":"29359","title":"How to determine if an emergent gauge theory is deconfined or not?","text":"2+1D lattice gauge theory can emerge in a spin system through fractionalization. Usually if the gauge structure is broken down to $\\mathbb{Z}_N$, it is believed that the fractionalized spinons are deconfined. However in general, $\\mathbb{Z}_N$ gauge theory also have a confined phase. The question is how to determine if the discrete emergent gauge theory is really deconfined or not? For example, I am considering a $\\mathbb{Z}_3$ gauge-Higgs model defined on the Kagome lattice with the Hamiltonian $H=J\\sum_{\\langle i j\\rangle}\\cos(\\theta_i-\\theta_j-A_{ij})$, where $\\theta_i=0,\\pm2\\pi\/3$ is the matter field and $A_{ij}=0,\\pm2\\pi\/3$ is the gauge field. If the matter field is in a ferromagnetic phase, then I can understand that the gauge field will be Higgs out. But the matter field here is a Kagome antiferromagnet, which is strongly frustrated and may not order at low temperature. So in this case, I would suspect that the effective $\\mathbb{Z}_3$ gauge theory will be driven into a confined phase. Is my conjecture right? How to prove or disprove that? Thanks in advance."}
{"id":"63253","title":"Proof for commutator relation $[\\hat{H},\\hat{a}] = - \\hbar \\omega \\hat{a}$","text":"I know how to derive below equations found on wikipedia and have done it myselt too: \\begin{align} \\hat{H} &= \\hbar \\omega \\left(\\hat{a}^\\dagger\\hat{a} + \\frac{1}{2}\\right)\\\\\\ \\hat{H} &= \\hbar \\omega \\left(\\hat{a}\\hat{a}^\\dagger - \\frac{1}{2}\\right)\\\\\\ \\end{align} where $\\hat{a}=\\tfrac{1}{\\sqrt{2}} \\left(\\hat{P} - i \\hat{X}\\right)$ is a annihilation operator and $\\hat{a}^\\dagger=\\tfrac{1}{\\sqrt{2}} \\left(\\hat{P} + i \\hat{X}\\right)$ a creation operator. Let me write also that: \\begin{align} \\hat{P}&= \\frac{1}{p_0}\\hat{p} = -\\frac{i\\hbar}{\\sqrt{\\hbar m \\omega}} \\frac{d}{dx}\\\\\\ \\hat{X}&=\\frac{1}{x_0} \\hat{x}=\\sqrt{\\frac{m\\omega}{\\hbar}}x \\end{align} In order to continue i need a proof that operators $\\hat{a}$ and $\\hat{a}^\\dagger$ give a following commutator with hamiltonian $\\hat{H}$: \\begin{align} \\left[\\hat{H},\\hat{a} \\right] &= -\\hbar\\omega \\, \\hat{a}\\\\\\ \\left[\\hat{H},\\hat{a}^\\dagger \\right] &= +\\hbar\\omega \\, \\hat{a}^\\dagger \\end{align} These statements can be found on wikipedia as well as here, but nowhere it is proven that the above relations for commutator really hold. I tried to derive $\\left[\\hat{H},\\hat{a} \\right]$ and my result was: $$ \\left[\\hat{H},\\hat{a} \\right] \\psi = -i \\sqrt{\\frac{\\omega \\hbar^3}{4m}}\\psi $$ You should know that this this is 3rd commutator that i have ever calculated so it probably is wrong, but here is a photo of my attempt on paper. I would appreciate if anyone has any link to a proof of the commutator relations (one will do) or could post a proof here."}
{"id":"29353","title":"How is antenna gain correlated to beam width?","text":"Let's say you have two dipole type antennas. Antenna A has a gain of 2.15 dBi, a horizontal beam width of 360 deg and a vertical beam width of 45 deg. Antenna B is similar to antenna A, but has a horizontal beam width of 360 deg and a vertical beam width of 42 deg. Can you use the ratio of the vertical beam widths to predict the gain of antenna B? Note: In the application I'm asking about I'm not sure what method they used to calculate the beam widths. Maybe someone else knows which methods are most commonly used for dipole antennas."}
{"id":"53566","title":"What is the information content of a human being?","text":"How much memory would we need to represent a human? How would each atom be stored as? Bytes? Something more complex?"}
{"id":"113410","title":"Feynman Lectures on Physics","text":"I want to develop a good intuition of concepts in Physics, and I've heard that the Feynman Lectures on Physics are very good for this. However, I've also heard that the Mathematics is very complicated. I was just wondering, just how much Math do I need to know? I love Physics, and I'm willing to put in the work. How much time do you guys reckon this will take? Thank you :)"}
{"id":"5109","title":"Colder surface radiates to warmer surface","text":"When radiation from a colder source arrives at a warmer surface there is some debate about what happens next. To make the question more concrete lets say that the colder source is at temperature 288K. The warmer surface is at 888K and has emissivity of 1. 3 possibilities   1. We ignore such radiation because it cannot happen.   2. The radiation is subtracted from the much larger radiation of every wavelength leaving the hotter surface.   3. The radiation is fully absorbed and its effect is to be re radiated at characteristic temperature of 888K (plus infinitesimally small T increase due to radiation absorption). I would have thought that 2. and 3 are more plausible than 1. Both 2 and 3 satisfy the Stephan Boltzmann equation. 3 however seems to imply that the radiation from colder object is transformed into much higher quality radiation and a possible second law of thermodynamics infringement."}
{"id":"29357","title":"How does the grid on the microwave oven window prevent microwave radiation from coming out?","text":"If I look through the microwave window I can see through, which means visible radiation can get out. We know also that there is a mesh on the microwave window which prevents microwave from coming out. My question is how does this work? how come making stripes or mesh of metals can attenuate microwave radiation yet allow visible radiation? Looks like an electrodynamics problem to me with periodic boundary conditions (because of the partitions on the microwave oven window). Is it discussed in any textbook?"}
{"id":"54692","title":"Why is the free energy minimized by the Boltzmann distribution?","text":"Can someone show me, without glossing over anything, why $F = E - TS$ is minimized when $p_i = e^{-U_i\/k_bT}\/\\sum_ie^{-U_i\/k_bT}$? I understand it conceptually, but am having difficulty showing it formally."}
{"id":"21404","title":"Strict general mathematical definition of drag","text":"Is there a formal definition of drag, say, as some surface integral of normal and shear forces? There seem to be a lot of formulas for specific cases, but is there a general one? I need to accurately calculate the drag of three cylinders placed between two plates."}
{"id":"21405","title":"How much of the energy from 1 megaton H Bomb explosion could we capture to do useful work?","text":"The world is full of nuclear warheads being stockpiled. Controlled fusion power seems a long way away. Could we put these warheads to better use by exploding them in a controlled way and capturing the energy they produce? By useful work I mean the power is then available for the national grid to boil kettles or have a shower! Extra points for looking at the practicalities of building a facility to do this (although I guess that would get the question closed :-(... )"}
{"id":"37629","title":"Space expansion looking like time dilation","text":"Space looks like time depending on the motion of the observer so I was going to ask if space expansion was the same as the unfolding of time, but this was asked on physics.stackexchange before and the answer was that in GR time does not flow - there is no more a flow of time than there is a flow of space. So instead I'll ask: is space expansion the same as time dilation ?"}
{"id":"91345","title":"3-point correlation function for a massive scalar field","text":"I am a little bit perplexed as to how to compute the three-point correlation function for a massive scalar field, I know that it should be equal to zero. I need to show that: $\\lim_{T\\rightarrow \\infty (1-i\\epsilon)} \\int D\\phi \\phi(x_1) \\phi(x_2) \\phi(x_3) = 0$ How to show this? Edit: I forgot to mention without using the generating founction, $Z[J]$."}
{"id":"106626","title":"Could the universe be a series of Big Bangs?","text":"Imagine an eruption of energy\/mass $E$ from a singularity $O$, as in a Big Bang. After the energy\/mass $E$ is all at more than a distance $d$ from $O$, is it for some value of $d$ possible that there could be a new eruption of energy\/mass from $O$, i.e. a new Big Bang? If yes, is there an upper limit to the number of succeeding Big Bangs?"}
{"id":"24574","title":"2D - Kinematics - Linkage System using Vector Algebra","text":"I have this question that I dont know how to solve correctly : ![enter image description here](http:\/\/i.stack.imgur.com\/HsYdc.jpg) My question is, how do I find $V_B$ ? I will find the angular velocities myself, but I want to know the method to get $V_B$ ? I know I can start by using $V_C=V_B+V_{C\/B}$, but then I'm not sure what to do next...is the direction of $V_B$ and $V_{C\/B}$ the same ? I'm using vector algebra (cross product with i and j etc). More specifically : $$\\vec{V}_C=\\vec{V}_B+\\vec{V}_{C\/B}$$ $$\\vec{V}_C = -1 \\hat{j}$$ $$\\vec{V}_B= ?$$ $$\\vec{V}_{C\/B}= \\omega_{CB} \\hat{k} \\times \\vec{r}_{CB}$$ Shouldn't $\\vec{V}_B = \\omega_{CB} \\hat{k} \\times \\vec{r}_CB$ ??? If not, then what should it be ? If you need more info please let me know."}
{"id":"79721","title":"What are the values of effective mass approximation","text":"For my research work i am trying to calculate band gap of zinc oxide theoratically and found this paper (Determination of the Particle Size Distribution of Quantum Nanocrystals from Absorbance Spectr). What i am doing is I am making a small matlab programm which will reporduce the results using the same values from the above paper. I used the values given in the paper to make the program ![enter image description here](http:\/\/i.stack.imgur.com\/v8Fbp.jpg) **Absorbtion onset** $\\lambda ^{onset}=\\frac{c*h}{E^{*}}$                  Values used for calculating Band gap     8.85418782 ×  10^-12          permittivity of free space     1.60217657 ×  10^-19          charge of an electron     9.11 ×  10^-31                free electron mass     1.05457173 × 10^-34               hcut     0.59                      effective mass of holes     0.26                     effective mass of electrons     8.5                      relative permittivity     3.2                     zno bandgap     c                       velocity of light      But after running the programm i got these results               Size    Absorbtion Onset     1   6.21E-26     1.5 6.21E-26     2   6.21E-26     2.5 6.21E-26     3   6.21E-26     3.5 6.21E-26     4   6.21E-26     4.5 6.21E-26     5   6.21E-26     5.5 6.21E-26     6   6.21E-26     6.5 6.21E-26     7   6.21E-26     7.5 6.21E-26     8   6.21E-26     8.5 6.21E-26     9   6.21E-26     9.5 6.21E-26     10  6.21E-26      The results i got doesnt match with tha paper and I belive that my coding in matlab is correct but i am doubtful about the values i have taken for calculating band gap. Can you help me with this"}
{"id":"62120","title":"Is it physically meaningful to talk about the 'total temperature' of an object?","text":"If I had a semi infinite, 1-D object and a finite 1-D object, both heated at the same constant rate at one end each for the same time period and both begin at the same initial temperature, is it physically meaningful for me to integrate along the length of the object and consider this integral as a function of time and a measure of the 'amount' of heat on the object?"}
{"id":"28402","title":"Does classical physics predict the effects of shining a laser at a hair?","text":"The discussion on this webpage mentions that shining a laser beam at a hair produces an effect like that of the double-slit experiment. Does classical physics predict the effect you observe when you do this (since light is a wave)?"}
{"id":"115073","title":"Is it true that the sun will cause very high temperatures on Earth long before the red-giant-phase?","text":"I heard at least three claims about the development of the heat of the sun. In an old book, I read, that nothing dramatically will happen in the next few billion years. Wikipedia states, that the average temperature will hit 30° in a billion years, 100° a billion years later, and in an internet forum, someone claimed that in 500 million years, the oceans would cook. Which of this claims is true? How reliable are the claims about events in such far future?"}
{"id":"115070","title":"Does Light Experience Length Contraction?","text":"Lorentz length contractions states that the length of any moving object gets divided by the Lorentz factor equal to the Lorentz factor for that object (always $\\geq 1$), equal to $$ \\gamma=\\frac{1}{\\sqrt { 1-\\frac { { v }^{ 2 } }{ { c }^{ 2 } } } } $$ However, in massless particles $v=c$, so the Lorentz factor becomes $\\infty$, meaning that an object traveling at $c$ will have $0$ length. However, photons and obviously all forms of electromagnetic waves move at c when traveling through a vacuum, such as from a space shuttle to a space station or back to Earth. Does this mean that photons have no length? How does this affect wavelength?"}
{"id":"35058","title":"Aharonov-Bohm vs de Witt","text":"dewitt claimed in his paper > Bryce S. DeWitt. Quantum theory without electromagnetic potentials, _Phys. > Rev._ **125** no. 6 (1962), pp. 2189-2191, DOI: 10.1103\/PhysRev.125.2189, that the discovery of the Aharonov and Bohm that electromagnetic potentials play primary role in quantum mechanical theory is false. Who won? What are the errors in the argument of the losing side in this battle?"}
{"id":"35054","title":"Help on unit conversion problem","text":"This is a problem from school. I will show my attempt. The question: \"The gas constant for dry air R is 287 $\\frac{m^2}{s^2*K}$. Assuming the temperature is 330 K and the pressure is 1050 hPa, what is the atmospheric density.\" The professor said DO NOT produce an answer by finding a formula, but to use the magic of unit conversion to try to solve things. I know density is measured in kg\/m^3 or thereabouts so I tried the following: 1050 hPA = 105, 000 Pa 1 Pa = 1 kg\/m*s^2 105,000 $\\frac{kg}{m*s^2}$ * 330 K * 287 $\\frac{m^2}{s^2*K}$. This cancels some units... but not enough...in fact it cancels just K, so far as I understand, far from what I need for my density unit. Any ideas on what Im doing foolishly here?"}
{"id":"35052","title":"What makes Poynting's theorem consistent for a charge moving in a static electric field?","text":"Poynting's theorem is given by $$\\frac{\\partial}{\\partial t}\\int_{v}Udv \\+ \\oint_{A}\\vec S\\cdot \\vec {dA} +\\int_{v}\\vec E\\cdot\\vec J dv =0 $$ Where,   * the total electromagnetic energy inside the volume v is $U = \\frac 1 2 (\\vec E\\cdot\\vec D+\\vec B\\cdot\\vec H)$   * the Poynting vector $\\vec S=\\vec E\\times \\vec H$ This equation is interpreted as the conservation of electromagnetic and mechanical energy for a volume of space with each term representing respectively the rate at which   1. electromagnetic energy changes inside the volume   2. electromagnetic energy crosses the boundary of the enclosing surface   3. mechanical work is done on charges inside the volume  Now take the case of a charge accelerating from rest by a static electric field, and initially at the centre of a spherical volume fixed in space with radius cT where c is the speed of light and T the time taken for electromagnetic fields to propagate from the center to the spherical boundary. For 0 < t < T both the magnetic and mechanical energy inside the volume increases without electromagnetic energy crossing the boundary. So during this time, where does the negative term come from to maintain the RHS = 0 in Poynting's theorem?"}
{"id":"91987","title":"Physics of batteries (volts vs charge)","text":"Car batteries are usually 12 V. What is the difference between buying a car battery and hooking up a bunch of cheap household batteries in series? Both would register at 12 V. I assume that cars need much more current to start an engine then regular household things like lamps and toys. Does that mean that a car battery holds more charge within it? If we think of a simple cell battery, where there are two electrodes and an electrolyte. One electrode eventually, through chemical reaction, becomes positive and the other becomes negative. Thus in a car battery, does that mean the electrodes (if we can reduce a car battery to a primitive cell), have more charge separated on each electrode? But if that was the case, wouldn't a greater charge separation mean that the voltage would be greater between the terminals as well? In general, what is the relation between charge and voltage? I know the equation V = U\/q, just like E = F\/q (similar form, in the limit that the test charge is small as to not effect the PE or electric field. V = U\/q is sorta useless too, since differences matter, but nevertheless that is how we defined it). If charges are separated further, does that mean greater voltage? If more charge is separated, does that mean greater voltage? Lastly, what is the difference b\/w the charge in a 12 V car battery, and the charge in a 12 V makeshift, series strung battery from home?"}
{"id":"2838","title":"Total energy of the Universe","text":"In popular science books and articles, I keep running into the claim that the total energy of the Universe is zero, \"because the positive energy of matter is cancelled out by the negative energy of the gravitational field\". But I can't find anything concrete to substantiate this claim. As a first check, I did a calculation to compute the gravitational potential energy of a sphere of uniform density of radius $R$ using Newton's Laws and threw in $E=mc^2$ for energy of the sphere, and it was by no means obvious that the answer is zero! So, my questions:   1. What is the basis for the claim - does one require General Relativity, or can one get it from Newtonian gravity?   2. What conditions do you require in the model, in order for this to work?   3. Could someone please refer me to a good paper about this?"}
{"id":"133856","title":"Why is gravitational potential energy negative?","text":"Why is gravitational potential energy negative? How is it different from other forms of energy? I recently saw a video by Dr. Michio Kaku, he said that the total energy content of the universe is zero, since gravitational potential energy is negative, thus balancing all the positive energy. Can anyone help me out here?"}
{"id":"48120","title":"Total Energy of the Universe?","text":"I've heard the total energy is zero, but I've also heard it cannot be said to be zero since there's so much unknown stuff in the universe. Is that true?"}
{"id":"65695","title":"The Univere's mass-energy and uncertainty","text":"A virtual creation with total mass-energy = $E$ is allowed as long as that virtual creation doesn’t last longer than $E\/h$. Can the uncertainty principle also be used to estimate the mass-energy in the spontaneous creation of a universe - a spontaneous creation that has now lasted $13.6$ billion years? If so, the principle would require that universe to have a mass-energy less than $1.54\\times 10^{-51}$ Joules. Is there a flaw in this?"}
{"id":"17082","title":"Why is gravitational potential energy negative, and what does that mean?","text":"I usually think of gravitational potential energy as representing just what it sounds like: the energy that we could potentially gain, using gravity. However, the equation for it (derived by integrating Newton's law of gravitational force)... $PE_1 = -\\frac{GMm}{r}$ ..has me thrown for a loop, especially after this answer.   * If potential energy really meant what I thought it did, then it would always have to be non-negative... but this equation is _always_ negative. So what does \"negative potential energy\" mean!?   * If $KE + PE$ is always a constant, but PE is not only negative but becomes **more** negative as the particles attract, doesn't that mean the kinetic energy will become arbitrarily large? Shouldn't this mean all particles increase to infinite KE before a collision?   * If we are near the surface of the earth, we can estimate PE as $PE_2 = mgh$ by treating Earth as a flat gravitational plane. However, `h` in this equation plays exactly the same role as `r` in the first equation, doesn't it?      * So why is $PE_1$ negative while $PE_2$ is positive? Why does one increase with `h` while the other increases inversely with `r`?     * Do they both represent the same \"form\" of energy? Since $PE_2$ is just an approximation of $PE_1$, we should get nearly the same answer using either equation, if we were near Earth's surface and knew our distance to its center-of-mass. However, the two equations give _completely_ different answers! What gives!? Can anyone help clear up my confusion?"}
{"id":"134904","title":"What is the total observational energy of the universe?","text":"Based on what is measurable, how much energy is in the universe if the initial time is the big bang and the final time is the present?"}
{"id":"77322","title":"Is the total mass of the universe constant in time?","text":"Is the total mass of the universe constant in time?"}
{"id":"107918","title":"total amount of energy in the universe","text":"What is the total amount of energy in the universe? is there the same amount of energy as negative energy, cancelling each other out? Or is it something different?"}
{"id":"93588","title":"Why does Coulomb's constant have units?","text":"I think of Coulomb's constant as a conversion factor (not sure if this is correct). Kind of like how you would do calculations in kg and then times it by the conversion constant to convert your answer to pounds. The conversion factor would be $2.2\\: \\mathrm{lbs\/kg}$. Since the units for Coulomb's constant is $\\mathrm{N \\cdot m^2\/C^2}$, would it make sense to define the Newton as: $1\\:\\text{Newton} = \\frac{1}{1\/1\\: \\mathrm{meter^2} \\cdot 1\\: \\mathrm{Coulomb^2}}$ Would the above definition be valid? **EDIT:** So if $k$ is not a conversion factor since the above definition for a Newton is invalid and $k$ is not just a scaling factor, since it has units, then what is it? If its just a proportionality constant to adjust the magnitude then why does it have units? Shouldn't it be a unit less constant? **EDIT:** So $k$ is not just a scaling factor (since it has units) and its not a conversion factor since a Newton can't be expressed as the other units. So if its unit just exists so that things cancel out \"nicely\" doesn't this make dimensional analysis useless since you can add in random constants and units to cancel out whatever you want? **My question is not about the meaning of $k$. Its about its units.**"}
{"id":"23717","title":"Program for radiation and toxic hazards","text":"I worked in my masters thesis with $^{87}Rb$ and $^{40}K$, really small beta emitters. But there are so many other things around in the lab, that I want to keep track on all the things I might get in contact with. Is there any computer program to calculate the dose of the whole decay chain to get a picture of the artificial radiation and supports logging. I don't want to look up all the individual numbers and calculate it manually. Also in my apparatus various clusters (Cr, Ni, Co, Cu, Ag, Pd, Ca....) are produced. E.g. I know chromium(VI) is carcinogen, but the pure metal is not. But in the nanoworld things may change. So is there a database around where I can lookup the toxity of various substances, with an emphasize on nanoparticles? I use nitrile rubber gloves and try to do not inhale something if I clean the apparatus. But this might be not enough precaution. The laser dyes are not healthy too."}
{"id":"93585","title":"Objects made up of electrons?","text":"Say you have a neutral rod, and you bring a positively charged rod beside it (call the side the charged rod is brought near side A and the other side side B). The electrons from the side B will start moving towards side A and the positively charged nuclei in side A will start moving to side B. After a large part of side A consists of only of electrons, the electrons would start repelling each other and the movement of charges will stop. But at this point a very large portion of side A consists of electrons and a large part of side B consists only of positively charged nuclei. The system is at equilibrium. But the rod has now been mostly split into a electron side and a positively charged nuclei side, however it doesn't look different at all? Why? Shouldn't a rod made of atoms look and be completely different than a rod that is mostly made up of electrons on one side and nuclei on the other? I know that electrons are very small, and since side A mostly consists of electrons, shouldn't it be almost invisible?"}
{"id":"23711","title":"Interpretation of Stiffness Matrix and Mass Matrix in Finite Element Method","text":"I would like to have a general interpretation of the coefficients of the stiffness matrix that appears in FEM. For instance if we are solving a linear elasticity problem and we modelize the relation between a node $i$ and a node $j$ as a spring system, then $K_{i,j}$ (where $K$ is the stiffness matrix of the system) can be seen as the stiffness constant of the virtual spring between the two nodes. But does there exist a more general interpretation? Perhaps in terms of internal work? Another similar question is: What could be an interpretation of the coefficients $M_{i,j}$ of the mass matrix $M$?"}
{"id":"27001","title":"What is the state-of-the-art on spacelike singularities in string theory?","text":"What lessons do we have from string theory regarding the fate of singularities in general relativity? What happens to black hole singularities? What happens to cosmological singularities? Which points of view on string theory yielded results in this respect? String field theory? AdS\/CFT? Matrix theory? I suppose perturbative string theory is not applicable in the vicinity of singularities."}
{"id":"73605","title":"Why does nonlinearity in quantum mechanics lead to superluminal signaling?","text":"I recently came across two nice papers on the foundations of quantum mechancis, Aaronson 2004 and Hardy 2001. Aaronson makes the statement, which was new to me, that nonlinearity in QM leads to superluminal signaling (as well as the solvability of hard problems in computer science by a nonlinear quantum computer). Can anyone offer an argument with crayons for why this should be so? It seems strange to me that a principle so fundamental and important can be violated simply by having some nonlinearity. When it comes to mechanical waves, we're used to thinking of a linear wave equation as an approximation that is _always_ violated at some level. Does even the teensiest bit of nonlinearity in QM bring causality to its knees, or can the damage be limited in some sense? Does all of this have any implications for quantum gravity -- e.g., does it help to explain why it's hard to make a theory of quantum gravity, since it's not obvious that quantum gravity can be unitary and linear? > S. Aaronson, \"Is Quantum Mechanics An Island In Theoryspace?,\" 2004, > arXiv:quant-ph\/0401062. > > L. Hardy, \"Quantum theory from five reasonable axioms,\" 2001, arXiv:quant- > ph\/0101012."}
{"id":"74053","title":"How mirror equation can explain farsightedness correction?","text":"I have a friend who has just show me his medical prescription for hyperopia (farsightedness) correction and he needs glasses with 4,25 diopters for that, which seemed to be weird for me because I had learned, from the mirror equation, that the maximum correction possible for hyperopia is 4 diopters: $$ \\frac{1}{f} = \\frac{1}{p} + \\frac{1}{p'} $$ If we have $0.25m$ for the normal eye distant point and more than $0.25m$ for the farsighted eye distant point (negative sign, because it's a virtual image), then we would have: $$ \\frac{1}{f} = \\frac{1}{0.25} + \\frac{1}{p'} = 4 - \\frac{1}{|p'|} \\in\\quad ]0,4[, \\quad\\text{since}\\quad |p'| \\geq 0.25m \\quad\\text{and}\\quad p'<0 $$ I did some google search and find out that, indeed, hyperopia can reach values even greater, such as 20 diopters, but I can't find pages where doctors explain that with equations or physics teachers explain how things really work in ophthalmology. Either I am doing some terrible mistake, or doctors are doing some terrible mistake, or this equation just don't apply to hyperopia at all... Which one is true?"}
{"id":"55785","title":"Is there a formal definition of a macroscopic variable in statistical mechanics?","text":"Intuitively it's easy to accept that the usual variables like temperature, internal energy, etc. are 'macroscopic', but does there exist a formal definition of a macroscopic variable? In other words, is there a clear way to separate the set of all observables (and functions of observables) on a system into ones we would describe as 'macroscopic' and ones we would not? EDIT: Since apparently the answer is not completely straightforward, I'm interested in hearing any definitions which have appeared in literature, even if they are only conventions. I'm also interested in any necessary or sufficient conditions."}
{"id":"78921","title":"Entropy of the Sun","text":"* Is it possible to measure or calculate the total entropy of the Sun?   * Assuming it changes over time, what are its current first and second derivatives w.r.t. time?    * What is our prediction on its asymptotic behavior (barring possible collisions with other bodies)?"}
{"id":"98601","title":"Feynman Diagrams in 2 component notation","text":"When using two component notation people often prefer to refrain from using arrows in Feynman diagrams to denote charge flow as is done in four-component notation. Instead, if understand correctly, they use arrows to denote chirality. I'd like to know what is the prescription to draw out the diagrams. I have read here (pg. 39) that > arrows indicate the spinor index structure, with fields of undotted indices > flowing into any vertex and field of dotted indices flowing out of any > vertices (see the reference above for many examples). However, trying this out on Majorana and Dirac mass terms, this doesn't seem to be correct. A Majorana mass term, $\\psi ^\\alpha \\psi_\\alpha +h.c.$, is thus composed only of undotted indices. With the reasoning above, it should have two arrows pointing into the vertex, ![enter image description here](http:\/\/i.stack.imgur.com\/8B0z7.png) However, I'm pretty sure that this is a Dirac mass, and not a Majorana mass. What am I missing?"}
{"id":"35584","title":"How seriously do string theorists take the \"landscape\"","text":"The string theory landscape seems to this outside observer to be an intermediate step in the intellectual progress toward a more robust theory that explains why our one universe has the particular properties that it has. is this the majority opinion or do most string theorists view the landscape as a plausibly being included in the final form of the theory?"}
{"id":"9945","title":"Are all superalgebra's clifford algebra's","text":"I believe the answer to be yes, but I realize that sometimes physicists place additional constraints that might not be obvious. If superalgebras are clifford algebras, why make a literary distinction?"}
{"id":"56473","title":"Does the collapse of the wave function increase entropy of the atomic system itself?","text":"Does wave-function collapse cause the entropy of the atom (ie. the sub-atomic particle system that makes up the atom) to increase?"}
{"id":"28563","title":"Hours of light per day based on latitude\/longitude formula","text":"I'm looking for a formula that will return the number of hours per day given a specific location. I was thinking that can be calculated as a difference of sunrise and sunset, but I see that there are some other ways, like in this topic. What is the best, fast and correct way to calculate this?"}
{"id":"57789","title":"Why are electrons consider waves?","text":"I know the wave nature of electrons was evoked to explain why atoms are stable but I thought waves could be put in the same state like photons yet electrons can not exist in the same state."}
{"id":"57781","title":"Deriving equations of motion of polymer chain with Hamilton's equations","text":"This is related to a question about a simple model of a polymer chain that I have asked yesterday. I have a Hamiltonian that is given as: $H = \\sum\\limits_{i=1}^N \\frac{p_{\\alpha_i}^2}{2m} + \\frac{1}{2}\\sum\\limits_{i=1}^{N-1} m \\omega^2(\\alpha_i - \\alpha_{i+1})^2 $ where $\\alpha_i$ are generalized coordinates and the $p_{\\alpha_i}$ are the corresponding conjugate momenta. I want to find the equations of motion. From Hamilton's equations I get $\\frac{\\partial H}{\\partial p_{\\alpha_i}} = \\dot{\\alpha_i} = \\frac{p_{\\alpha_i}}{m} \\tag{1}$ $- \\frac{\\partial H}{\\partial {\\alpha_i}} = \\dot{p_{\\alpha_i}} = -m \\omega^2 (\\alpha_i - \\alpha_{i+1} ) \\tag{2}$ , for $i = 2,...,N-1$. Comparing this to my book, (1) is correct, but (2) is wrong. (2) should really be $- \\frac{\\partial H}{\\partial {\\alpha_i}} = \\dot{p_{\\alpha_i}} = -m \\omega^2 (2\\alpha_i - \\alpha_{i+1} - \\alpha_{i-1}) \\tag{$2_{correct}$}$ Clearly, I am doing something wrong. I suspect that I'm not chain-ruling correctly. But I also don't get, where the $\\alpha_{i-1}$ is coming from. Can anybody clarify?"}
{"id":"94549","title":"Spin state of electron after measurement","text":"I have a system of two spin 1\/2 particles in a superposition of spin states in the z-direction given by: $\\psi = \\frac{1}{2} |+ +\\rangle + \\frac{1}{2} |+ -\\rangle + \\frac{1}{\\sqrt{2}} |- -\\rangle$ where $+$ designates spin up, $-$ designates spin down and the first particle's state is the first term in each ket and the second particles' state is the second term in each ket. If I measure the spin on the first particle and get a value of $-\\hbar \/ 2$ (corresponding to a spin down state) is the new state of the particles simply $\\psi = | - - \\rangle$ meaning that the first particle is now \"set\" to being spin down? And if I determine the spin on the first particle to be spin up, would the subsequent state be $\\psi = \\frac{1}{\\sqrt{2}} |+ +\\rangle + \\frac{1}{\\sqrt{2}} |+ - \\rangle$ ? Basically, my question is once I make a measurement of a spin of a particle, does the wavefunction stay collapsed on the spin determined? And does having a second particle affect this in any way?"}
{"id":"52785","title":"Does special relativity unify the two phenomena at the base of Faraday's flux law (was Feynman wrong in this case)?","text":"Consider Faraday's flux law for the EMF generated in a conductor loop: $$ \\varepsilon = - \\frac{d \\phi}{dt},$$ where $\\varepsilon$ is the EMF, and $\\phi$ is the magnetic flux through the loop. There are two possible causes for the flux to variate over time: variations in the magnetic field (\"transformer EMF\") and variations in the area enclosed by the loop (\"motional EMF\").   Feynman has noted that this is a unique case where a single rule is explained by two different phenomena: > We know of no other place in physics where such a simple and accurate > general principle requires for its real understanding an analysis in terms > of _two different phenomena_. Usually such a beautiful generalization is > found to stem from a single deep underlying principle. Nevertheless, in this > case there does not appear to be any such profound implication.   >  —Richard P. Feynman, _The Feynman Lectures on Physics_ (Volume II, 17-2). **Is it really true that there is no way to view these two phenomena as one?**   For example, it says in this Wikipedia article that this apparent dichotomy was part of what led Einstein to develop special relativity.   **Does special relativity give us a unifying principle to derive Faraday's law from?**"}
{"id":"25716","title":"Pulsar beam radiant intensity distribution","text":"I'm curious about the radiant intensity distribution of pulsars: what's the general dependence of intensity on angle, and what are typical angular beam widths? How much does the beam width vary between pulsars? (Presumably this is tied to magnetic field strength.) Even something as simple as a very sketchy plot of intensity vs. angle would be great. It's easy enough to find plots of observed intensity vs. time for individual pulsars, but it takes a bit to get from those to the distribution at the source."}
{"id":"21877","title":"General integral to find resistance","text":"My question is: **is there a simple and truly general equation for the resistance between two electrical equipotential surfaces?**. Obviously, if so, what is it, and if not, why? It would be very difficult to solve, granted, but I just want to see a calculus equation that is fully descriptive. I have two frameworks under which this could be entertained, I'll write those out and then explain the motivation. To start with, we need propose that the volume separating the two surfaces has a volumetric resistivity, $\\rho$ in units of $(\\Omega m)$. # Single Volume Framework We can limit the discussion to a defined volume, then the surfaces reside in that volume or on the surface of it. This volume may have a constant resistivity $\\rho$ while everywhere outside the volume is completely electrically insulating. # Infinite Volume Framework An alternative to the above approach that might make the task more or less difficult would be to replace a constant resistivity with a spatial dependence $\\rho(\\vec{r})$ and no longer require a boundary condition. In that case we only have 3 mathematical inputs to the problem, which is the resistivity defined for all $\\vec{r}$ and a definition of the two surfaces, $S_1$ and $S_2$. # Known Algebraic Analogs The basic algebraic formulation that I find insufficient is: $$R = \\rho \\frac{\\ell}{A}$$ Where $l$ is the length of the restive material that is any shape which has translation symmetry over that length, and $A$ is the cross-sectional area. Obviously, this is a rather simple equation that won't apply to more complicated geometry. Even more sophisticated academic sources seem to give equations that fall short of what I'm asking. For example: $$R = \\rho \\int_0^l \\frac{1}{A(x)} dx$$ I think it's obvious that an equation such as this is built upon a myriad of assumptions. For a thought experiment, imagine that the area starts out as very small and then pans out to very large quickly. Well, accounting for the larger area in the above sense underestimates the resistance, because the charge has to diffuse out perpendicular to the average direction of flow as well as parallel to it. I have some reasons to suspect this might actually be rather difficult. A big reason is that all the approaches I'm familiar with require the flow paths to be established beforehand, which can't be done for what I'm asking. So maybe this will result in two interconnected calculus equations. # Motivation I had an interest in Squishy Circuits, and it occurred to me that I can't quickly and simply write down the equation for resistance between two points. The unique thing about Squishy Circuits is that it calls for two types of dough, one that conducts and one that is mostly insulating. However, the recipes aren't perfect and because of that, the young children who play with these circuits regularly encounter the limits of conductor and insulator definitions. If you make your conductor dough too long and\/or too thin, you will encounter dimming of the light you connect with it. Similarly, a thin insulator layer will lead to a lot of leakage current which also dims the light."}
{"id":"21873","title":"Moment of inertia of a sector about its center point?","text":"What is the moment of inertia of a pizza slice that has a radius r, an angle (radians) of theta, and a height of h about the center point perpendicular to the cheese plane?"}
{"id":"26068","title":"Recommended first accessories for starblast 4.5","text":"My son got an orion starblast 4.5 for Christmas. It comes with orion explorer II 17mm and 6mm eyepieces. We are looking at some additional accessories and wondering what you would recommend as \"first accessories\" to get the most out of the telescope. Our initial inclination is towards:   1. A barlow lens. Probably the orion shorty 2x barlow. Is the shorty-plus twice as good? (It is twice the price.)   2. A 25mm eye piece. From what we've read, a reflector with these specs is best for wide field views.   3. A solar filter. We think it would be cool to look at the sun."}
{"id":"64949","title":"How can I determine whether the mass of an object is evenly distributed?","text":"How can I determine whether the mass of an object is evenly distributed without doing any permanent damage? Suppose I got all the typical lab equipment. I guess I can calculate its center of mass and compare with experiment result or measure its moment of inertia among other things, but is there a way to be 99.9% sure?"}
{"id":"99956","title":"In Which of the following situations would force be exerted on on object and no work be done on the object?","text":"Consider the following question: > In which of the following situations would a force be exerted on an object > and no work be done on the object? > > I. a centripetal force is exerted on a moving object > > II. a force in the opposite direction as the object is moving > > III. a force is exerted on an object that remains at rest > > A.) I only > > B.) I and II > > C.) I and III > > D.) II and III > > E.) I, II, and III My response to this question is only II, but this is not a choice. I do not understand how I or III could have work. My logic is that in I, the force is centripetal and therefore is not parallel to the object's path, in any case it would be perpendicular because centripetal is towards the center. Work must be parallel. In III, if an object remains at rest, $d = 0$. And $W = fd$, so $W = 0$. I can't find the flaw in my logic. Any help is appreciated! Thanks!"}
{"id":"99386","title":"Eigenvalue problem for differential equations in QM","text":"I have a very simple question with regard to numerical methods in physics. I want to solve the eigenvalue problem for a particle moving in an arbitrary potential. Let's take 1D to be concrete. I.e. I want to find $(E,\\psi(x))$ satisfying \\begin{align} \\left[-\\frac{1}{2}\\partial_x^2 + V(x) \\right]\\psi(x)=E \\psi(x). \\end{align} Now how do I do it exactly? Naively I would implement the following algorithm: 1) Pick some $E$. 2) I want to find $\\psi(x)$ which is normalizable. So I could pick a large $L > 0 $, set $\\psi(-L) = \\epsilon > 0$ and $\\psi'(-L) = \\epsilon'>0$ and numerically integrate from there using the Schrodinger equation. 3) If I encounter a solution which is exponentially small far to the right of the origin, then I say the solution is normalizable (since it is decaying at $|x|\\to\\infty$), and I accept the pair $(E,\\psi(x))$. 4) I increment $E \\to E + dE$ and I repeat the process. In doing so, I should get the spectrum around my starting value of $E$. Does this algorithm actually work?? It also seems to me like a very uncontrolled way of doing it; I have no idea how accurate the spectrum is going to be. For example, would changing $L, \\epsilon, \\epsilon'$ make a difference? The thing is, I know from Sturm-Liouville theory that the spectrum $E$ is going to be discrete (given $V(x)$ satisfying some nice properties). So the spectrum is going to be a set of measure 0 amongst the entire real line that $E$ lives in. This means that I'm almost surely (i.e. with probability 1) never going to get a solution that is normalizable, and whatever solution I try to numerically integrate from my starting point is always going to blow up having integrated far enough to the right. So, what algorithm do people use to numerically obtain the spectrum and the eigenvalues? How do I also control accuracy of the spectrum generated?"}
{"id":"57431","title":"Electric field of a negative charge","text":"How was it discovered that the electric field of a negative charge points towards the charge itself? Is it true? ![field of a negative point charge](http:\/\/i.stack.imgur.com\/bh9uO.png) (Courtesy of wikipedia)"}
{"id":"53957","title":"Frequency Of Light","text":"I am confused on few topics... What is meant by \"Frequency of Light\"? Does the Photon(s) vibrate, that is known as its frequency? If the Photons vibrate, then they have a specific frequency, then What is meant by \"Higher frequency light\" as used in Photo- electric Effect? In which directions\/axis do they vibrate to have a specific frequency. Why is it that nothing can go faster than the Speed of light. Does these two things have any relation to each other? -Thanks."}
{"id":"57436","title":"Doubt in Kinematics","text":"I know that this isn't the place for such basic questions, but I didn't find the answer to this anywhere else. It's pretty simple: some particle moves in straight line under constant acceleration from one point $x_0$ to another point $x_1$ during the time interval $\\Delta t_1$. When the particle reaches the point $x_1$ it reverses it's movement and goes to another point $x_2$ during another time interval $\\Delta t_2$. I want to determine $x_2$, however I don't understand how to do it. My try was: Let $\\Delta x_1 =x_1 -x_0$ be the first displacement and let $\\Delta x_2 = x_2 - x_1$ be the second displacement. Then I can calculate two velocities: $$v_1 = \\frac{\\Delta x_1}{\\Delta t_1}$$ $$v_2 = \\frac{\\Delta x_2}{\\Delta t_2}$$ My thought is then to find the acceleration as: $$a = \\frac{v_2-v_1}{\\Delta t_2 + \\Delta t_1}$$ But I'm not sure it'll work, since the movement reverses at $x_1$ and since I'm assuming the velocities constant on the intervals. Can someone help me how to think with this problem, and how to solve it?"}
{"id":"101980","title":"Smaller mass in gravity well?","text":"When sitting in a gravity well, as we do on earth, does our effective mass become smaller than our rest mass due to having negative potential energy? Correspondingly, does a free falling mass (from infinity, radial path) have an effective mass equal to its rest mass, as here potential and kinetic energies are ballanced? The question is based on the concept of equivalence of energy and mass."}
{"id":"18567","title":"Can the field generated by a magnet domain extend to infinity?","text":"As a thought experiment let us assume that we have isolated a magnetic domain. This domain is of finite size and we know its dimensions. Assuming that we can measure an infinitesimal field, will there be a certain region beyond which the field won't be applicable? The instinctive answer to this question is no, but if you think about it we see the magnet's influence on the space around it as the result of equipotential regions then the contention is that only so many discrete equipotential regions are possible (the fact that something is not countable doesn't automatically mean it's infinite). Hence, that line of thought goes, there should be a limit theoretically and practically until which a field can exist. Can you please clarify this sticking point for me? Am I pushing the analogies we use to understand fields too far? What conceptual mistake am I making over here?"}
{"id":"38601","title":"How can we test if something is a wave?","text":"More specifically, I want to understand why a wave is a wave but a wave packet is not considered a wave (as discussed in this question). I would think that if something have these characteristics: 1. Wavelength; 2. Frequency; 3. Period; 4. Amplitude; and 5. Wave velocity; then it must be a wave. Does a wave packet have these characteristics? Do you have a better set of rules to apply to test if something is a wave?"}
{"id":"38603","title":"Breakdown voltage of a dielectric","text":"I know that a capacitor with a dielectric can operate normally up till a certain voltage (AFAIK called breakdown voltage) which depends on the strength of the dielectric placed between the plates. After this voltage, the circuit becomes short and current flows between the plates and thus the capacitor breaks down. But i want to know what is exactly happening when we say a dielectric \"breaks down\" ? What I know about a dielectric is that due to the electric field (because of the plates of the capacitor) the molecules of the dielectric align themselves accordingly and set up an electric field in the opposite direction, thus decreasing the net electric field. So, please can anyone tell me what happens during breakdown?"}
{"id":"83435","title":"Is information propagated across a medium in any other way than waves?","text":"Is information propagated in any other way than waves? Please distinguish \"propagation across a medium\" from information \"storage within stable states of matter\", which might difuse or interact chemically. Information might be stored in stable configurations of matter, which might diffuse, or interact chemically (odor,DNA), but these might be orders of magnitude weaker, in range and dissipation. Is there a domain in physics comparing wave vs non-wave propagation. The two most known are sensory: sound and electromagnetic propagation. I think gravity probes are still searching for waves in this medium. Why does nature prefer waves for long distance calls? Perhaps because it involves a minimum dissipation of energy?"}
{"id":"133300","title":"How to model an apartment's airconditioning?","text":"**Background:**   This reminds situation reminds be of an episode from the _The Big Bang Theory_ , although it is quite different from it. My roommate and I have similar temperature preferences and also share the objective of saving money on electricity. The optimal solution towards this objective is what we have differing opinions about. **The problem:**   In my opinion, during the daytime when no-one's around, the temperature setting should be at least 4 °F more than the normal setting (given an ongoing summer 90+ °F). I would go far as to argue that during the daytime the cooling should be completely switched off. My roommate's counter-argument is that assuming a PID controlled model, the overshoot caused by the cooling in the evening, in an effort to get the temperature down from the day's high, will more than nullify the electricity saved during the day thereby resulting in higher costs. The controller is a programmable digital thermostat with 4 separate settings for `sleep`, `wake`, `leave` and `return` for two regimens namely `weekday` and `weekend`. **Specifically:**   Instead of hand-waving I was looking for a mathematical model to simulate the heating-cooling of a typical apartment. A quick Google search yielded no useful models (most I could get my hands on are macro-models related to multiple commercial units). I am looking for a model — with elements like the `heat exchange` (i.e. the ac), the `heat sink` (i.e. the apartment) and the `environment` — a general model which can be used for a _first-order_ approximation. Assume that the cost of electricity is constant throughout the period under consideration (even if it isn't, a cost vs time graph can be easily incorporated in the model)."}
{"id":"791","title":"Question about moment of inertia and velocity","text":"First off, I swear this is not homework. I'm doing some practice problems because I got an exam coming up. I'm stuck on this one: ![alt text](http:\/\/i.stack.imgur.com\/w7KMz.png) I figured I would use energy conservation for this problem. So since the thing is not moving initially, I tried doing $mgh=1\/2 I\\omega^2+1\/2 mv^2$, but that doesnt give me the right answer. Any ideas?"}
{"id":"83384","title":"Will the Hubble constant reach zero asymptotically in the far future?","text":"In the current accelerated expansion universe model will the Hubble constant reach zero asymptotically in the far future?"}
{"id":"83382","title":"Is mass of a particle changed when it is charged?","text":"If a particle of mass $M$ is given an electric charge $Q$, will its mass change?"}
{"id":"15134","title":"Surface normal on the earth to the sun at a given point in time","text":"How complicated is it to calculate a surface normal on the spherical approximation of the earths surface pointing towards the sun at a given point in time? What I try do is to highlight a small area on a world map where the radiation from the sun tangential hits a imaginary sphere around the earth. This is just to get an idea where this area was at the given time, so this information has not to be very accurate. As a mater of fact I will just incorporated this information in the map if it is easy enough to calculate."}
{"id":"77939","title":"Scale invariance plus unitarity implies conformal invariance?","text":"What has the reaction been towards the recent paper claiming to have a proof that scale invariance plus unitarity implies conformal invariance in 4d?"}
{"id":"107137","title":"Gauss' Law for Magnetism Derivative Form: With or without volume integral?","text":"I've been reading through FLP Vol. II, and he has proven that as the flux through a closed surface is: $\\ \\int_{surface} \\mathbf{F} \\space \\mathrm{d}\\mathbf{a} $, according to the divergence theorem, the flux through a surface can be defined as: $\\ \\int_{volume} \\nabla \\cdot \\mathbf{F} \\space \\mathrm{d}V $, where $\\ \\mathbf{F} $ is any vector field, and the volume is that which is enclosed by the surface. Previously he had stated as a word equation that: $\\ \\text{Flux of } \\mathbf{B} \\text{ through any closed surface}=0. $ I would therefore assume that $\\ \\int_{volume} \\nabla \\cdot \\mathbf{B} \\space \\mathrm{d}V = 0$, however Gauss' law for magnetism states that: $\\ \\nabla \\cdot \\mathbf{B} = 0$. Does that mean that $\\ \\nabla \\cdot \\mathbf{B} = 0$ and $\\ \\int_{volume} \\nabla \\cdot \\mathbf{B} \\space \\mathrm{d}V = 0$ are equivalent statements, or am I making a fundamental error somewhere?"}
{"id":"77932","title":"When did we learn that stars die?","text":"As we all know, the stars we see in the night sky might already be dead. I was wondering though, when was this fact or conclusion commonly established? Today, most people (let's assume with an above average education) would probably be aware of this fact. When is the earliest time when the same could be said? I am particularly interested if the same could be said for the time period revolving around the period 1850 - 1900. I know that the speed of light was approximated fairly accurately in the 17th century. Knowing this (finite) speed, it's not hard for me to draw the conclusion that the source of the light I see may not be there anymore. Would this be an easy conclusion to draw a hundred years ago however? Maybe they thought stars don't die?"}
{"id":"77931","title":"Ambiguity in Beta Functions (2-loop)","text":"Beyond one-loop, the beta function of a QFT is scheme dependent. I would like to understand better this ambiguity. The easiest thing to say is that you haven't calculated something physical, so of course it does not need to be scheme independent. However, the anomalous dimension of operators is I think an observable quantity since we may measure critical exponents in the lab, and the anomalous dimension results from the same sort of calculation. Moreover, I can relate the beta functions to the trace anomaly. Schematically, $\\langle \\partial_\\mu j_{dilation}^\\mu \\rangle=\\langle T^\\mu_\\mu \\rangle \\approx \\beta$ (See Peskin 19.5 for the case of QED). If i couple some field to the trace of $T$ I think I should be able to turn this anomaly into a cross-section for some process which would be measurable (think of ABJ anomaly and $\\pi^0 \\to \\gamma \\gamma$ for instance). So the questions is: 1) Is it known how the terms in the beta function may differ between regularization schemes? If I try to calculate the couplings at the fixed point using different schemes, will I get the same answer (I am aware the location of the fixed point is not physical, but if I use the same field variables I could imagine this being scheme independent)? How may I see that although the beta function and location of the fixed point are ambiguous, the anomalous dimensions are not? 2) How would this ambiguity cancel out if I have a theory where I can turn the trace anomaly into a prediction of a scattering amplitude? Or can this simply not be done? Any clarification or suggestion for references is appreciated."}
{"id":"33890","title":"Using Lorentz Invariance of Charge To Calculate Current Density","text":"I'm attempting a problem from Zwiebach: A First Course in String Theory and am completely stuck. Could anyone give me a hint? The problem is as follows. Consider $S$, $S'$ two Lorentz frames with $S'$ boosted along the $+x$ axis. In frame $S$ we have a cubic box with sides of length $L$ at rest. The box is filled by a material, also at rest, of uniform charge density $\\rho$. In $S$ we assume that the charge density $\\underline{j}=0$. Use the Lorentz invariance of charge to calculate the charge density $\\rho'$ and current density $\\underline{j}'$ in $S'$. Verify that $(c\\rho,\\underline{j})$ a 4-vector. The charge density is easy. Indeed $L^3\\rho = Q = Q' = L'^3\\rho'=\\frac{L^3}{\\gamma}\\rho'$ so $\\rho' = \\gamma \\rho$. I know I'm right here because this agrees with what we'd expect from a 4-vector under Lorentz boost. To do the current density I tried to use $0=\\frac{\\textrm{d}Q}{\\textrm{d}t}=\\frac{\\textrm{d}Q'}{\\textrm{d}t'}=\\int_S\\underline{j}'.\\textrm{d}\\underline{a}=j'_xL^2$ so $\\underline{j}'=0$ since $j'_y=j'_z=0$ clearly must be zero. I know this is wrong though, because it doesn't agree with what I'd expect from a 4-vector! What am I doing wrong? And is this the right way to go about this question? Many thanks in advance!"}
{"id":"33892","title":"Is dimensional analysis used outside fluid mechanics and transport phenomena?","text":"Most dimensionless numbers (at least the ones easily found) used for dimensional analysis are about fluid dynamics, or transport phenomena, convection and heat transfer - arguably also sort of fluid mechanics. My understanding of dimensional analysis is the following: Derive dimensionless numbers from the description of a system, find the ones physically meaningful, and use them to compare different situations or to scale experiments. Is this possible in other fields, like classical mechanics, and their engineering applications? Example: describe a horizontal beam by: $$ X=\\frac {\\text{forces acting on the beam}} {\\text{forces beam can withstand without plastic deformation}} $$ Both parts of the ratio being functions of shape, density, gravity, material constants etc. My assumption is yes, it's possible, but most fields outside the sort-of fluid mechanics described above are easy enough to calculate without dimensional analysis."}
{"id":"103398","title":"Measurements for thermal diffusivity of graphene?","text":"We have known for a long time that graphene has in-plane thermal conductivity ranging between 2000 and 4000 $W m^{-1} K^{-1}$. But in order to model heat transport on a sheet of graphene, we need more than the conductivity: we also need specific heat in order to obtain the thermal diffusivity that is used in the equation. I couldn't find any measurement results online for this quantity. I've only seen some crude estimates based on phonon transport, but even so, no specific figures."}
{"id":"34046","title":"design of the heat exchanger...in chimney","text":"![heat exchanger diagram](http:\/\/i.stack.imgur.com\/w04j8.jpg) I want to design a heat exchanger in a chimney in order to utilize heat from chimney. I have done several experiments, but I could not determine the exact length of tube (carrying water), such that its inlet temp is ambient temperature and outlet temperature is expected to be 100 degree Celsius. I would be very thankful if some one would help me determine the length of the tube. tube diameter is 10mm (almunium), chimney diameter is(120 mm). Can anyone help me with formulas involved in calculating the heat transfer and length of the tube... I already calculated the lenght experimentally but I could not do it mathematically."}
{"id":"72212","title":"Tighten rope around cylindricaly shaped space","text":"Imagine we live on cylinder(we are 2d creature), put a rope around that cylinder and start pulling both ends of the rope against each another. Will the space get deformed? I guess it will, I have to put some energy to the rope and energy deforms the space. Is it possible to calculate shape of the space based on the force I put in the rope? In order to calculate this I guess you have to make 3d analogy and use laws of general relativity."}
{"id":"34049","title":"What really goes on in a vacuum?","text":"I've been told that a vacuum isn't actually empty space, rather that it consists of antiparticle pairs spontaneously materialising then quickly annihilating, which leads me to a few questions. Firstly, is this true? And secondly, if so, where do these particles come from?... (do the particles even have to come from anywhere?)"}
{"id":"31498","title":"Neutron decay and electron anti neutrino $n\\to p + e + \\bar{\\nu}_e$","text":"Why do we need neutrino to explain neutron decay? Is there any evidence regarding existence neutrinos in the context of $n\\to p + e + \\bar{\\nu}_e$?"}
{"id":"47253","title":"Why does the nature always prefer low energy and maximum entropy?","text":"Why does the nature always prefer low energy and maximum entropy? I've just learned electrostatics and I still have no idea why like charges repel each other. http:\/\/in.answers.yahoo.com\/question\/index?qid=20061106060503AAkbIfa I don't quite understand why U has to be minimum. Can someone explain?"}
{"id":"47252","title":"Simple explanation of Quantum Zeno Effect","text":"I'm a student and I had to give a talk on seminar about Quantum Zeno effect and Anti-Zeno effect to my colleagues (all listeners have had a course in quantum physics, but not a heavy one with all the bra and ket stuff). My first idea to give a simple explanation of Zeno's effect was this: Let's take a look at exponential decay where the chance for particle or state to survive some time $t$ is $P_S=e^{-t\/\\tau}$. If I measure it after time $\\tau$ I have chance $P_S=1\/e$ that it will still be intact. If I instead allow it to do it's things only for time $\\tau\/N$ and then measure it, the survival chance will be $P_S=e^{-1\/N}$ which approaches $1$ as $N$ increases. To achieve the same total time, I have to repeat this procedure $N$ times and the total survival probability is... $P_S=(e^{-1\/N})^N=1\/e$. So it obviously doesn't work, I get no Zeno's effect in this way. It's interesting that after I gave the talk professor rose and said \"Well, this can be easily understood if we look at the exponential decay\". Then he started drawing exponent and another exponent that's repeatedly interrupted and reset to initial state after small intervals. Later we agreed that this won't actually work, but the question is - why? Why doesn't this intuitively obvious way doesn't work and what would be the correct law out of which one could see the Zeno's effect? Is there any elegant way to explain this effect without heavy math and angles of state vectors? ADDITIONAL QUESTION (related): Is it correct to use name \"Quantum Zeno Effect\" for turning of polarization by series of inclined polarizers or the thing that is done in this article?"}
{"id":"88299","title":"An alternative definition of the creation and annihilation operators?","text":"Suppose we have a system of bosons represented by their occupation numbers $$\\tag{1} | n_1, n_2, ..., n_\\alpha, ... \\rangle$$ Then we can define creation and annihilation operators $$\\tag{2} a_\\alpha^\\dagger| n_1, n_2, ..., n_\\alpha, ... \\rangle = \\sqrt{n_\\alpha+1} | n_1, n_2, ..., n_\\alpha+1, ... \\rangle$$ $$\\tag{3} a_\\alpha| n_1, n_2, ..., n_\\alpha, ... \\rangle = \\sqrt{n_\\alpha} | n_1, n_2, ..., n_\\alpha-1, ... \\rangle$$ This is nice because the number operator is just $a_\\alpha^\\dagger a_\\alpha$. However, would it be sensible to define an alternate set of operators to work with? $$\\tag{4} b_\\alpha| n_1, n_2, ..., n_\\alpha, ... \\rangle = | n_1, n_2, ..., n_\\alpha+1, ... \\rangle$$ $$\\tag{5} c_\\alpha| n_1, n_2, ..., n_\\alpha, ... \\rangle = \\begin{cases} | n_1, n_2, ..., n_\\alpha-1, ... \\rangle & n_\\alpha>0 \\\\\\ 0 & n_\\alpha=0 \\end{cases}$$ $$\\tag{6} N_\\alpha| n_1, n_2, ..., n_\\alpha, ... \\rangle = n_\\alpha| n_1, n_2, ..., n_\\alpha, ... \\rangle $$ Why don't we work with these operators? The bosonic creation and annihilation operators $a_\\alpha^\\dagger$ and $a_\\alpha$ were defined to mimic the harmonic oscillator's raising and lowering operators ($x \\pm i p$), but is there any compelling reason to keep the $\\sqrt{n_\\alpha+1}$ and $\\sqrt{n_\\alpha}$ factors? I suppose $a_\\alpha^\\dagger$ and $a_\\alpha$ obey nice properties such as $[a_\\alpha,a_\\alpha^\\dagger]=1$ and the fact that they are Hermitian adjoints of each other. What are the analogous relationships that $b_\\alpha$ and $c_\\alpha$ would obey?"}
{"id":"131432","title":"Is there an efficiency-optimizing layout of items in a refrigerator?","text":"Is there some optimal layout of items, or types of items (characterized by size, shape, density, solid\/liquid, etc) that will allow a common household refrigerator to function with optimal energy efficiency? I know this is a very broad and any answer will have to make several assumptions, not the least of which include how often the refrigerator is opened and for how long. I'd be perfectly happy with a \"steady-state\" answer that assumes, e.g. the refrigerator is never opened. But more is always better. If nothing else, I'm curious about what considerations might go into conducting a more thorough analysis."}
{"id":"4761","title":"Why do white dwarfs shine white?","text":"The hotter something is glowing the more white\/blue it appears. A dying medium sized star expands, cools and becomes a red giant for a while, but eventually it is going to gravitationally collapse (once enough Iron (Fe) is accumulated in the core). Then it blows the outer layers away and what is left collapses into a white dwarf. What makes the dwarf shine? and why is it white? Does the luminosity decreases as the object cools down, or is there some other reaction that keeps it glowing for a long time? Can a white dwarf turn brown or black never to be seen again? Do all white dwarfs turn into Neutron stars eventually?"}
{"id":"4765","title":"Matter in superconductive state","text":"We distinguish between the states of matter: gas, liquid and solid. Possibly we could add the plasma state and\/or the superconductive state as new states of matter. Phase transistions at certain temperature perhaps with some other conditions should have to exist. What do you think, does it make sense?"}
{"id":"86268","title":"What happens to entangled particles when momentum is measured?","text":"In Wikipedia it is mentioned that position and momentum can be entangled as well as spin and polarization etc. I assume etc. is charge etc. I understand how if you measure spin up on one of a pair you get spin down on the second of the pair. What happens to the other particle in an entangled pair if I measure the momentum, position or charge of one of the particles? Is there a momentum up and down or charge up and down analog? http:\/\/en.wikipedia.org\/wiki\/Quantum_entanglement"}
{"id":"77061","title":"Can physics (ever) explain intrinsic properties of nature?","text":"I may be totally off with this quite abstract (?) question(s). But still, here are some closely related sub-questions:   * Is there a list of currently \"known\" intrinsic properties of nature?   * How exactly is an intrinsic property of nature defined? Is it defined as \" _it is so, because it is so_ \" or \" _it is so, because it cannot be otherwise (in our Universe)_ \"? For example, Wikipedia about mass as intrinsic property: > _An intrinsic property is a property that an object or a thing has of > itself, independently of other things, including its context. An extrinsic > (or relational) property is a property that depends on a thing's > relationship with other things. For example, **mass is an intrinsic property > of any physical object**._ Is this a correct explanation of an intrinsic property?"}
{"id":"52148","title":"Does light change color on its way through a window?","text":"Looking at the refractive index of glass, it's around $1.6$. Then the speed of light $x$ through light should be given by $$ 1.6 = \\frac{3.0\\times10^8}{x}, $$ so $x$ is about $2\\times10^8~\\mathrm{m}~\\mathrm{s}^{-1}$ The frequency is kept constant, so the wavelength must adapt to suit the slower speed, giving a wavelength of $2\/3$ the original. Does this mean that when passing through glass, say red light (wavelength $650~\\mathrm{nm}$) changes to indigo ($445~\\mathrm{nm}$), as $650 \\times 2\/3 = 433~\\mathrm{nm}$, or is my logic flawed somewhere?"}
{"id":"21336","title":"What determines color -- wavelength or frequency?","text":"What determines the color of light -- is it the wavelength of the light or the frequency? (i.e. If you put light through a medium other than air, in order to keep its color the same, which one would you need to keep constant: the wavelength or the frequency?)"}
{"id":"86843","title":"Pn junction voltage drop?","text":"This image from wikipedia, explains that there occurs a potential drop across a pn semiconductor junction, and an electric field confined to the depletion region.![enter image description here](http:\/\/i.stack.imgur.com\/7VaN3.png) I already know the reason for the existence of this drop and the calculation of the difference but I have two questions regarding this drop. **1) If then and p doped regions are externally connected using a perfectly conducting wire, why will not any current flow?**   Connecting the two regions should equalize their potentials (since wire is a resistance less) conductor, and therefore the gradient at the junction is destroyed resulting in a diffusion current which is obviously against conservation of energy as the semiconductor has non-zero resistivity. Where will extra potential drops will be created so that Kirchoff's voltage rule holds without any current and the built-in potential difference $V_o$ of the juction persists? **2)** (probably naive) If an external Bias is applied, of say $|V|<|V_o|$, then the pn potential difference across the junction will just reduce by that amount ($|V|$). Assuming that the external voltage source is ideal without any resistance, **what will be the potential drops which would sum to zero in this case? Will drop due to the resistance of semiconductor play a par in it? If yes, then there cannot be any ideal resistance-less semiconductor junction, can there?**"}
{"id":"86841","title":"Some questions on the Wilson loop in the projective construction?","text":"Based on the previous question and the comment in it, imagine two different mean-field Hamiltonians $H=\\sum(\\psi_i^\\dagger\\chi_{ij}\\psi_j+H.c.)$ and $H'=\\sum(\\psi_i^\\dagger\\chi_{ij}'\\psi_j+H.c.)$, we say that _$H$ and $H'$ are gauge equivalent if they have the same eigenvalues and the same projected eigenspaces_. And the Wilson loop $W(C)$ can be defined as the trace of matrix-product $P(C)$(see the notations here ). Now my questions are: (1)\"$H$ and $H'$ are gauge equivalent\" if and only if \"$W(C)=W'(C)$ for all loops $C$ on the 2D lattice \". Is this true? How to prove or disprove it? (2)If the system is on a 2D torus, is $W(L)$ always a _positive real number_ ? Which means that the 'total flux'(the phase of $W(L)$) through the torus is quantized as $2\\pi\\times integer$, where $L$ is the boundary of the 2D lattice. (3)If the Hamiltonian contains extra terms, say $H=\\sum(\\psi_i^\\dagger\\chi_{ij}\\psi_j+\\psi_i^T\\eta_{ij}\\psi_j+H.c.+\\psi_i^\\dagger h_i\\psi_i)$, is the Wilson loop still defined as $W(C)=tr(P(C))$? Thanks a lot."}
{"id":"86846","title":"Problem with Velocity of efflux","text":"I am stuck in this problem- ![The Problem](http:\/\/i.stack.imgur.com\/CBC4d.png) I need to find the velocity of efflux at the hole of the container. [We can assume that the area of the hole is negligible in comparison with the base area of the container]. Here's my approach Velocity of liquid at the upper-surface = $v_2$ Velocity of efflux (velocity of water at the hole, right?) = $v_1$ Using Bernoulli's equation for the surface and the hole - $$ P_{atm} + \\rho_2 g (h_1 + h _2) + \\frac{1}{2}\\rho_2 v_2^2 = P_{atm} + \\rho_1 g h_1 + \\rho_2 g h_2 + \\frac{1}{2}\\rho_1 v_1^2 \\\\\\ \\implies \\rho_2 g h_1 + \\rho_2 g h_2 - \\rho_1 g h_1 - \\rho_2 g h_2 = \\frac{1}{2}(\\rho_1 v_1^2 - \\rho_2 v_2^2) \\\\\\ \\implies \\frac{1}{2}(\\rho_1 v_1^2 - \\rho_2 v_2^2) = g h_1 (\\rho_2 - \\rho_1) $$ Now, let area of the base be $A_2$ and that of the hole be $A_1$ then, using equation of continuity, $$ A_1 v_1 = A_2 v_2 \\\\\\ \\implies v_2 = \\frac{A_1}{A_2} v_1 \\\\\\ \\implies v_2 \\approx 0 (\\because {A_1 << A_2}) $$ Using this value in the previous Bernoulli's relation $$ \\frac{1}{2}(\\rho_1 v_1^2) = g h_1 (\\rho_2 - \\rho_1) \\\\\\ \\implies \\frac{1}{2} \\rho_1 v_1^2 = g h_1 (\\rho_2 - \\rho_1) \\\\\\ \\implies v_1 = \\sqrt {\\frac {2gh_1(\\rho_2 - \\rho_1)}{\\rho_1}} $$ Which is not the correct answer. I did get a correct answer in the chat room, but it was using a different method. What's wrong with my method?"}
{"id":"86848","title":"Ramsey Interactions","text":"What are Ramsey interactions? I am researching atomic clocks and am not sure why the atoms need to be exposed twice to an electromagnetic field in order to cause excitation."}
{"id":"80141","title":"What is \"Symmetric Fission\"?","text":"Dose anyone has a clue what Symmetric Fission is? I couldn't find any explanation on what is it on internet."}
{"id":"80144","title":"Bose-Einstein condensate and nonlinear waves","text":"Can Bose-Einstein condensate be written as non-linear wave equation (in terms of mean field approximation theory)? the equation is: ![enter image description here](http:\/\/i.stack.imgur.com\/X8kEs.png) source: http:\/\/xxx.tau.ac.il\/abs\/1308.2288 > What I do understand by the Bose-Einstein condensate is, it is the state of > atoms at very low temperatues and at the moment the atoms forget their > previous identity. Therefore all the atoms stay in the same quantum state. Am I right?"}
{"id":"10356","title":"future light cones and light paths","text":"I understand that an event, in a four dimensional space-time, produces a light cone. As time increases the cones gets larger on either side of the event (past and future). For example the if the sun where to \"go out\" it would take 8 minuets for the earth to be affected by it simply because it takes approximately 8 minuets for light from the sun to reach the earth due to its location in the future light cone of the event (the sun being the event). Einstein made a suggestion that space-time is warped (vs flat) by the distribution of mass and energy and that bodies (like earth, jupiter, etc.) are meant to follow straight paths but cannot in a warped space (or appear not to because space is warped). However light supposedly follows these warped paths (called geodesics) as well even tho light is energy. How can space bend light? and why does light have to follow a specific path which is warped by space? Things with mass (like water) have to go around things (like rocks), but light can go through certain things or expand, but it doesn't move to the side like water does to a rock. Does it?"}
{"id":"122932","title":"Electromagnetic radiation and black body radiation","text":"I was taught today that the Electromagnetic wave Theory is unable to explain black body radiation. The example that was given to me: When a metal is heated, it emits different frequencies of light as it gets hotter. If electromagnetic wave theory was correct, it would not be so, the frequency (color) of light would remain the same, but only the intensity will change. I don't understand why this is so. My logic: Electromagnetic waves occur when a charged body oscillates in a electric and magnetic field. If the metal is provided with more energy (in the form of heat) won't the charged body vibrate faster, thus changing the frequency of the light emitted?"}
{"id":"830","title":"What is the direction of the friction force on a rolling ball?","text":"Suppose you have a solid ball on a horizontal table.   1. What is the direction of friction force when the ball I pushed horizontally and **starts rolling**?   2. Why is the direction of friction as it is?    3. Which forces acts at the contact point between delta time t0 to t1? (If we divide friction force in sub forces) V=1Vx m\/s Fx=?"}
{"id":"116853","title":"Are diffeomorphisms a proper subgroup of conformal transformations?","text":"The title sums it pretty much. Are all diffeomorphism transformations also conformal transformations? If the answer is that they are not, what are called the set of diffeomorphisms that are not conformal? General Relativity is invariant under diffeomorphisms, but it certainly is not invariant under conformal transformations, if conformal transformations where a subgroup of diff, you would have a contradiction. Or I am overlooking something important?"}
{"id":"134116","title":"Induced EMF in axle of car","text":"A car moves on a plane road in east-west direction.At what latitude of earth should it move so that induced EMF in the axle connecting it's wheels is maximum? A. At the poles B. At the equator C. At a latitude inclined at 45 degree to the equator"}
{"id":"134119","title":"How does a knife cut things at the atomic level?","text":"As the title says. It is common sense that sharp things cut, but how do they work at the atomical level?"}
{"id":"80858","title":"Why is velocity of outermost point on a rotating wheel double the velocity of centre of mass?","text":"'In the answers to one of the questions based on rotation of a disc in my physics book the answer includes the statement 'As we know that the velocity of outermost point on a rotating disc is double the velocity of center of mass'. But, I didn't know that and why is it like that? My thinking: I know that as we go away from the center of mass the tangential velocity of particles of the disc increases according to $v=wr$. But, how does that result in the above result?"}
{"id":"80856","title":"Does the 4\/3 problem of classical electromagnetism remain in quantum mechanics?","text":"In Volume II Chapter 28 of the Feymann Lectures on Physics, Feynman discusses the infamous 4\/3 problem of classical electromagnetism. Suppose you have a charged particle of radius $a$ and charge $q$ (uniformly distributed on the surface). If you integrate the energy density of the electromagnetic field over all space outside the particle, you'll get the total electromagnetic energy, which is an expression proportional to $c^2$. The energy divided by $c^2$ is what we usually call the mass, so if we calculate the \"electromagnetic mass\" in this manner we'll get $m = \\frac{1}{2}\\frac{1}{4\\pi\\epsilon_0}\\frac{q^2}{ac^2}$. If, on the other hand, you took the momentum density of the electromagnetic field and integrated it over all space outside the particle, you'd get the total electromagnetic momentum, which turns out (for $v<<c$) to be proportional to the velocity of the particle. The constant of proportionality of momentum and velocity is what we call mass, so if we calculated the electromagnetic mass in this way we would get $m = \\frac{2}{3}\\frac{1}{4\\pi\\epsilon_0}\\frac{q^2}{ac^2}$, which is $\\frac{4}{3}$ times the value we got before! That is the 4\/3 problem. Feynman claims that this fundamental issue remains when we move to quantum electrodynamics. Was he right, and if so has the situation changed since the 1960's when he was writing? I've seen claims on the Internet (I don't have the links) that the 4\/3 problem is still there in QED, but instead of $\\frac{4}{3}$ the coefficient is something closer to 1. Is that true, and if so what's the coefficient? All of this is of course related to issues of self- energy and renormalization. Any help would be greatly appreciated."}
{"id":"80854","title":"Calculating Length Scales from Passive Scalar Field","text":"I have a set of PLIF images of a passive scalar advected in a turbulent flow. I'm wondering if it's possible to estimate the integral length-scale based on the images of the passive scalar, and if so, how would I go about it. Any references to journal articles about this are welcome. It seems that a 2D cross-correlation of the images would help, with defined peaks at various length-scales, but then you'd have to filter out a lot of noise."}
{"id":"119044","title":"Universality of Quantum Operations","text":"Does an analog of the Solovay-Kitaev theorem exist for quantum operations, a generalization of quantum gates that also includes all completely positive maps?"}
{"id":"14301","title":"Degeneracies of the first excited state","text":"I have a box with $x,y,z$ all ranging from 0 to $l$. It has $V(x)$=0 inside and =$\\infty$ outside. By extending the 1D Schrodinger equation, I have that the allowed energy eigenvalues are $\\hbar^2\\pi^2\\over2ml^2$$(n_1^2+n_2^2+n_3^2)$. What is the degeneracy of the 1st excited energy level? By \"1st excited energy level\" does that mean 1 of the $n_k$'s, say $n_1$, =2 while $n_2=n_3=1$? Or does it mean all 3 dimensions are in their 1st excited state -- $n_k=2$ $ \\forall k\\in${$1,2,3$}? Also, how does one find the degeneracy? I am guessing that it is 3? Thanks for any help."}
{"id":"94912","title":"What is use of inertial frames","text":"In class of Newton's laws of motion, it was explained that Newton's second law is valid only in inertial frames. Teacher give us a example by considering a lift which is going downwards with acceleration$=a$ then a man watching the lift from outside will write $mg-t=ma$ (where $m$ is mass of the lift and $t$ is upward force on the lift) while a man inside the lift will write $mg-t=0$ (for him there is no acceleration of lift) both equations will give us different value of $t$ so we have to have a frame specified for work ie inertial frame but nearly after 5 months I have a problem. In the observation of the man inside lift acceleration due to gravity is stated as $g$ but I think that it should have value $g-a$ (if a stone fall on Earth with acceleration $g$ then inside the lift it should fall with acceleration $g-a$) and then by substituting this value for acceleration due to gravity in equation of man inside lift we will get the same value of $t$ as we got in equations of man outside the lift. So if both frames give same value of $t$ then why we prefer one frame after other."}
{"id":"94911","title":"Is terahertz radiation new?","text":"Regarding the future applications of terahertz technology, I thought we already knew the entire electromagnetic spectrum. In fact we do not, terahertz lies between microwaves and infrared radiation. The future applications of this is astonishing. For example, faster communications and next generation scanning and imaging technologies. The TSA will pretty much be out of a job, because T-Ray(terahertz) scanners will automatically detect chemical weapons and bombs. I know this because they leave a trace on the terahertz spectrum. The downside with communications is terahertz is absorbed easily by the Earth's atmosphere, so the communications cannot be too high in the sky. What I am asking is, are there any other frequencies on the electromagnetic spectrum we might discover? I was taught in school, that there are radio waves, microwaves, infrared, visible, ultra violent, x-rays, and gamma rays. Now the books have to be rewritten because of the existence of terahertz or t-rays. Remember it lies right between microwaves and infrared radiation. Another question is what about sound? So far there is infra sound, sound, and ultrasound. Do you think in the future we will discover other types of sound? Sorry for the length of my question, as you can see I know a lot about science and technology."}
{"id":"17966","title":"What are bootstraps?","text":"I've heard occasional mentions of the term \"bootstraps\" in connection with the S Matrix. I believe it applies to an old approach that was tried in the 1960s, whereby - well I'm not sure - but it sounds like they tried to compute the S Matrix without the interaction picture\/perturbation theory approach that we currently use. I'm aware that the approach was abandoned, but my question is: how was it envisaged to work ? What was the input to the calculation supposed to be and how did the calculation proceed ? I know it's got something to do with analyticity properties in terms of the momenta, but that's all I know.... For example, from the wikipedia article: > Chew and followers believed that it would be possible to use crossing > symmetry and Regge behavior to formulate a consistent S-matrix for > infinitely many particle types. The Regge hypothesis would determine the > spectrum, crossing and analyticity would determine the scattering > amplitude--- the forces, while unitarity would determine the self-consistent > quantum corrections in a way analogous to including loops. For example - I can't really understand how you would hope to compute the scattering amplitude just given crossing symmetry and assuming analyticity"}
{"id":"94919","title":"Capacitance of a free conductor","text":"Consider a ideal conductor in free space.For all purposes here,the zero of the potential is taken at infinity. Suppose I give a charge $Q$ to the conductor.As a result,the conductor will have a potential $V$. The question is can we say that $$Q=CV$$ where $C$ is a constant which depends only on **the shape and size on the conductor and not on the charge Q**?   1. If yes, how shall we prove it mathematically.   2. Also then how can we find the proportionality constant explicitly given the shape and size of the conductor?"}
{"id":"122201","title":"what about doing the laser beam in a moving reference frame but with a ball","text":"I realize the situation where a laser beam moves vertically in a moving vehicle![time dilation proof](http:\/\/i.stack.imgur.com\/qoDgm.gif) but what if the laser beam was a normal ball If we do the same steps of the proof considering that the velocity of the ball is not absolute and will have different velocities in different reference frames there will be no time dilation what is wrong in my understanding because according to special relativity there should be time dilation whether the event is a laser beam bouncing or a ball."}
{"id":"92884","title":"Equilibrium Condition","text":"In classical thermodynamics, equilibrium conditions means maximum entropy for a closed state. However, people always talk about equilibrium for open systems as well. How can one say that an open system has reached equilibrium with out negating the definition."}
{"id":"111006","title":"How does light bend around my finger tip?","text":"When I close one eye and put the tip of my finger near my open eye, it seems as if the light from the background image bends around my finger slightly, warping the image near the edges of my blurry finger tip. What causes this? Is it the heat from my finger that bends the light? Or the minuscule gravity that the mass in my finger exerts? (I don't think so.) Is this some kind of diffraction? ![Light bending around my finger](http:\/\/i.stack.imgur.com\/e32lY.jpg) To reproduce: put your finger about 5 cm from your open eye, look through the fuzzy edge of your finger and focus at something farther away. Move your finger gradually through your view and you'll see the background image shift as your finger moves. * * * For all the people asking, I made another photo. This time the backdrop is a grid I have on my screen (due to lack of grid paper). You see the grid deform ever so slightly near the top of my finger. Here's the setup: ![Setup](http:\/\/i.stack.imgur.com\/t2ioO.png)![Finger on a grid](http:\/\/i.stack.imgur.com\/biR3w.jpg) Note that these distances are arbitrary. It worked just as well with my finger closer to the camera, but this happens to be the situation that I measured. * * * Here are some photos of the side of a 2 mm thick flat opaque plastic object, at different aperture sizes. Especially notice hoe the grid fails to line up in the bottom two photos. ![Object photographed from the side](http:\/\/i.stack.imgur.com\/33jVJ.jpg)"}
{"id":"57015","title":"Energy-momentum conservation without translation symmetry?","text":"As I checked, the energy-momentum tensor defined as ${T^\\mu}_\\nu=\\frac{\\partial {\\cal L}}{\\partial(\\partial_\\mu \\phi)}\\partial_\\nu \\phi-{\\cal L}{\\delta^\\mu}_\\nu$ at the solution $\\phi$ of equation of motion(Euler-Lagrange equation) satisfies automatically the conservation law: $\\partial_\\mu{T^\\mu}_\\nu=0$, without any reference to the translation symmetry under $x^\\mu\\rightarrow x^\\mu-a^\\mu$.   So, what is the need of this symmetry?   Or, could there be something wrong with my calculation or conceptual issues?"}
{"id":"118560","title":"Speed of light affecting appearance of galaxies viewed not face on?","text":"So this has been really bugging me over the past few days (and forgive me if the answer is so simple). Let's say we're observing the Sombrero galaxy. It is about 29 million light years away and 50 thousand light yyears in diameter. So we should be observing the \"front\" of it at what it looked like 29 million years ago, and the \"back\" of it 29.05 million years ago. Why doesn't this extra distance change the galaxy's shape? If, for example, the galaxy was moving directly away from us in a straight line (not that it is), wouldn't the galaxy be compressed? Hope this makes sense."}
{"id":"9452","title":"How radio waves penetrate through buildings?","text":"For example how radio signals of a base transceiver station (BTS) penetrate through buildings?"}
{"id":"78589","title":"Why don't charges move transverse to an EM wave?","text":"Image we have an ultra-high intensity, ultra low frequency laser, with wattage on the order of terawatts and a wavelength on the order of a lightsecond. We rotate it that the electric field component is oriented on the $\\hat z$ axis, then fire it at a macroscopic block with a positive electric charge. Because of the low frequency the block will experience an electric that doesn't immediately change direction, and because of the high intensity the field will be very strong. So from this naive understanding of classical physics, the block will briefly levitate. Except this blatantly contradicts both QM (Compton scattering) and multiple macroscopic experiments (like solar sails), which both say that the block will be pushed in the direction of the laser. What assumptions in the original problem are missing\/wrong?"}
{"id":"78587","title":"when we rub objects together, what determines which material will pick up electrons?","text":"For example We know glass when rubbed by silk will become positively charged while the silk will be charged negative. What exactly makes glass appropriate for losing electrons in that experiment? ("}
{"id":"129603","title":"Unitary transformation behind gauge transformation","text":"It is very well-known that for bosonic operators a Gauge transformation can always be associated with it $$a\\rightarrow e^{i\\phi}a.$$ Obviously this is a Unitary transformation. Something like $$a^{\\prime}=\\mathcal{U}^{\\dagger}a\\mathcal{U}$$ I want to know what is $\\mathcal{U}$?"}
{"id":"56620","title":"Weightlessness for astronauts","text":"Well, this question has been puzzling me for kinda long time, many people believe that orbiting astronauts feel weightless because they are \"beyond the pull of Earth's gravity\"...How far from the Earth would a spacecraft have to travel to be truly beyond the Earth's gravitational influence? If a spacecraft were really unaffected by Earth's gravity would it remain n orbit? If so, what is the real reason for weightlessness in orbit?"}
{"id":"12461","title":"argument about fallacy of diff(M) being a gauge group for general relativity","text":"I want to outline a solid argument (or bulletpoints) to show how weak is the idea of diff(M) being the gauge group of general relativity. basically i have these points that in my view are very solid but i want to understand if there are misconceptions on my part that i'm simply not getting and if its so, i ask for help to make the case more solid, or understanding why it doesn't apply (to gravity):   * gauge groups are not the same as a symmetry group (thanks to Raymond Streater for making that point completely clear)   * gauge invariance in electrodynamics is an observation that physical observables are unchanged after a gauge transformation **without changing coordinate frame** ( we are ask to believe that in gravity someone did the same? that is, someone made the observation that physical observables are unchanged after a diffeomorphism-gauge-transformation, only to later argue that because of this, that there are no physical observables to begin with, that doesn't make a lot of sense, to not say that its just plain stupid circular argument )   * classic electrodynamics is also invariant (as in symmetry invariant, not as gauge-invariant) under Diff(M). The invariance is of course broken when the theory is quantized and $\\hbar$ makes an appearance, because it assumes a preferred scale for certain energies. The key point here is: **classical gravity is not special regarding having diff(M) as a symmetry group**   * from bulletpoints 2 and 3, if i cannot infer that Diff(M) is a gauge-invariance of electrodynamics, the same should apply to gravity For this question, i would say that a valid answer would either disprove any of the arguments as fallacies themselves (hence showing a solid argument why gravity is special and diff(M) is without a doubt its gauge group), or improve the argument for making it bullet-proof (sorry for the pun)"}
{"id":"12435","title":"Einstein's postulates $\\leftrightarrow$ Minkowski space for a Layman","text":"What's the cleanest\/quickest way to go between Einstein's postulates [1] of   1. Relativity: Physical laws are the same in all inertial reference frames.   2. Constant speed of light: \"... light is always propagated in empty space with a definite speed $c$ which is independent of the state of motion of the emitting body.\" to Minkowski's idea [2] that space and time are united into a 4D spacetime with the indefinite metric $ds^2 = \\vec{dx}^2 - c^2 dt^2$. Related to the question of what is the best derivation of the correspondence are:   Is the correspondence 1:1? (Does the correspondence go both ways?)   and are there any hidden\/extra assumptions? * * * ## Edit Marek's answer is really good (I suggest you read it and its references now!), but not quite what I was thinking of. I'm looking for an answer (or a reference) that shows the correspondence using only\/mainly simple algebra and geometry. An argument that a smart high school graduate would be able to understand."}
{"id":"76716","title":"Is it possible to derive the invariant spacetime interval from Einstein's two postulates for SR?","text":"In many textbooks, the interval $$ I = -(c\\Delta t)^2 + (\\Delta x)^2 + (\\Delta y)^2 + (\\Delta z)^2 $$ is taken for granted as the same for two events in any reference frame. Is it possible to derive this just from the two postulates,   1. That the laws of physics are the same in all reference frames   2. That the speed of light _c_ is constant for all reference frames?"}
{"id":"88887","title":"2D. Force applied at angle to body, where translational vector will be directed?","text":"I'm not a physicist and just making some research by the way of creating simple physics simulator, because of that, sorry if this is very dumb question, but I really need help with it. Let's assume that some body (rectangle, square, N-polygon, etc.), exists in 2D world in rest (no friction, gravity, etc.) and can be freely moved in any direction. If some pushing \/ pulling force will be applied to center of mass, then only translational force will exist, this case is very clear to me. But what if force will be applied on the edge of body? What will be if force will be applied at some angle to edge? I understand that this will involve a rotational forces. But how can I calculate the resulting translational vector in this case? Here is image, demonstrating the problem. Force F2 will not involve any rotational forces, I can calculate net force (= F2) and get acceleration vector. All this question is about F and finding resulting translational vector after apply of F. ![example of forces-related question](http:\/\/i.stack.imgur.com\/v8UwH.png)"}
{"id":"82282","title":"Expression for kinetic energy of gas per molecule","text":"The average kinetic energy (KE) per molecule of a gas is $\\frac{3}{2}kT$. While finding this we do $$ \\text{ Average KE} =\\frac{1}{2} M \\frac{1}{N}\\sum v^2=\\frac{3}{2}kT$$ But why do we not add rotational kinetic energy here?"}
{"id":"88882","title":"Will the \"Vacuum Wine Saver\" suck the bubbles out of Champagne?","text":"The \"Vacuum Wine Saver\" comes with the following \"warning\": > Not for sparkling wines Intuitively and naively, I imagine that the bubbles (or the \"bubble- potential\"—my made-up terminology) will be sucked out of the wine by the pump and that this is also the reason for the \"warning\". What is the better-formulated physical\/chemical description and explanation? In other words: What happens and why? * * * Extra: Is this (your answer) also a reason for not semi-compressing half-full flexible plastic cola bottles before closing them?"}
{"id":"28547","title":"What happens to water level when ice XII melts?","text":"There exists this famous idea that _if all floating icebergs melt, water level will stay the same_ (because the water replaced by ice is the volume of the melted ice). Now,   1. Is this always so, if you allow more exotic forms of ice (take e.g. ice XII with density 1.29) ?   2. Is this always so **on\/in earth** ? **EDIT:** I was looking a possibility that you would have water as a liquid that contains pieces of water as a solid. No land is assumed to exist here. \"Ice\" is always floating or sunken(if possible). To reformulate, can Ice be sunken in water in any circumstance (pressure, temperature)? **EDIT 2:** Wikipedia: \"Ice VII has a density of about 1.65 g cm-3 (at 2.5 GPa and 25°C)\". And water as an incompressible liquid has a density of 1 g cm-3. This means that ice VII is at the bottom of the \"lake\" and will expand when it melts. Is this true? And are there environments in earth where this (or similar) can happen?"}
{"id":"93955","title":"Should I learn Classical Physics if I want to learn Quantum Physics?","text":"I don't remember anything from school, so please what do you say about the title?"}
{"id":"53237","title":"What is the mathematical background needed for quantum physics?","text":"I'm a computer scientist with a huge interest in mathematics. I have also recently started to develop some interest about quantum mechanics and quantum field theory. Assuming some knowledge in the areas of topology, abstract algebra, linear algebra, real\/complex analysis, and probability\/statistics, what should I start to read to understand the math behind quantum mechanics and quantum field theory. If you could offer me books, and list them according to the order that I should read them, I would be glad."}
{"id":"22409","title":"A book on quantum mechanics supported by the high-level mathematics","text":"I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators etc, certainly the modern mathematics). If there isn't something similar please give me a reference to the book that is strictly supported by mathematics (given a set of mathematically descripted axioms author develops the theory using mathematics as a main tool)."}
{"id":"114415","title":"Basic maths theories for good understanding of the standard model","text":"I want to know what mathematical theories I should be aware of for a deep understanding of the standard particles model."}
{"id":"5014","title":"Mathematical background for Quantum Mechanics","text":"What are some good sources to learn the mathematical background of Quantum Mechanics? I am talking functional analysis, operator theory etc etc..."}
{"id":"16814","title":"What is the math knowledge necessary for starting Quantum Mechanics?","text":"Could someone experienced in the field tell me what the minimal **math knowledge** one must obtain in order to grasp the introductory Quantum Mechanics book\/course? I do have math knowledge but I must say, currently, kind of a poor one. I did a basic introductory course in Calculus, Linear algebra and Probability Theory. Perhaps you could suggest some books I have to go through before I can start with QM? Thx."}
{"id":"129479","title":"Study Basic Quantum Mechanics","text":"What is the appropriate mathematical background someone must attain in order to enroll in a quantum physics course for beginners?"}
{"id":"38735","title":"Mathematics for Quantum Mechanics","text":"What math should I study if I want to get a basic understanding of quantum mechanics and especially to be able to use the Schrodinger's equation."}
{"id":"19260","title":"Will a boiled egg or a raw egg stop rolling first?","text":"If we roll a normal egg and a boiled egg at the same time on a floor 1) with friction 2) without friction which one will come to stop first (if they will stop at all) and why? Can anyone tell me reason for this?"}
{"id":"105705","title":"Integrating for velocity","text":"Trying to determine velocity of a falling body with respect to traveled distance and initial speed. I've been provided with the following equation for acceleration as a function of distance and the grav. parameter(constant) of the attracting body : $a= GM\/r^2$ Where: a - acceleration.   GM - gravitational parameter(constant).   r - distance to the attracting body. I have entered inputs for GM and r and integrated this equation with respect to $r$. This obviously yielded total acceleration per traveled distance, in other words $m^2\/s^2$ at the given altitude. How do I proceed to determine velocity at this altitude?"}
{"id":"119773","title":"Total number of electric field lines coming out of a proton?","text":"I have to calculate the total number of electric field lines through a proton. I tried using Gauss' Law, i.e, $$\\phi = \\oint\\boldsymbol E.d\\boldsymbol s = {\\frac{q}{\\epsilon_0}} $$ $$So, \\phi = {\\frac{q}{\\epsilon_0}} $$ $$ \\textrm{Therefore}, \\phi = \\frac{1.6\\times10^{-19}}{8.854\\times10^{-12}}$$, as $q=1.6\\times10^{-19}C$ and $\\epsilon_0 = 8.854\\times10^{-12}C^2N^{-1}m^{-2}$. But the answer is $0.18\\times10^{-7}$ or something like that. How can that be the number of field lines? How can that be the number of anything? I mean it's a fractional number. Any help is appreciated. Thanks."}
{"id":"81484","title":"How does flux tubes between quarks bind them together?","text":"If you have, say, a proton it has gluon field fluctuations around it. Those flux tubes between the quarks suppresses the gluon field fluctuations and create a true vacuum between them(correct me if I'm wrong), but how does that bind the quarks together? I've read that it costs energy to clear the vacuum out, but I still don't quite get it. Thank you!"}
{"id":"83908","title":"Special Relativity and current in wire","text":"If I am a stationary observer and the electrons are moving relative to me,then shouldn't its density increase according to special relativity and thereby create an altogether negative net charge."}
{"id":"82828","title":"Explain what happens to object in lift","text":"In the following image,three cases have been mentioned. $N$ is the normal force acting on the object inside the lift and $mg$ is the force of attraction due to gravity.   In case 1, $N = mg$.   In case 2, $N = m(g+a)$ and   in case 3, $N = m(g-a)$. Why is it so in 2nd and 3rd case? ![enter image description here](http:\/\/i.stack.imgur.com\/xdA80.png)"}
{"id":"2552","title":"How to calculate cord tension in a vertical circle?","text":"Mass m is connected to the end of a cord at length R above its rotational axis (the axis is parallel to the horizon, the position of the mass is perpendicular to the horizon). It is given an initial velocity, V0, at a direction parallel to the horizon. The initial state is depicted at position A in the image. The forces working on the mass are MG from the earth and T the tension of the cord. How can I calculate the tension of the cord when the mass is at some angle $\\theta$ from its initial position (position B in the image)? ![image](http:\/\/img535.imageshack.us\/img535\/8167\/76300324.png). Here's what I thought: Since the mass is moving in a circle then the total force in the radial direction is T - MG*$\\cos\\theta$ = M*(V^2)\/R and so T = MG*$\\cos\\theta$+M*(V^2)\/R but since MG applies acceleration in the tangential direction then V should also be a function of $\\theta$ and that is where I kind of got lost. I tried to express V as the integration of MG*$\\sin\\theta$, but I wasn't sure if that's the right approach."}
{"id":"2558","title":"The final death of a black hole","text":"What are the different death scenarios for a black hole? I know they can evaporate through Hawking radiation - but is there any other way? What if you just kept shoveling more and more mass and energy into the black hole?"}
{"id":"94281","title":"Potential energy curve for intermolecular distance","text":"![potential energy vs intermolecular distance r](http:\/\/www.a-levelphysicstutor.com\/images\/matter\/E-r-graph.jpg) I'm trying to understand this curve better, but I can't quite figure out what \"negative potential energy\" means. The graph should describe a molecule oscillating between $A$ and $B$, however where I'm stuck in reasoning this is that the PE is equal in $A$ and $B$, but then why does this mean $r$ will increase in $A$ (repel) and decrease in $B$?"}
{"id":"11475","title":"Doubt concerning centripetal acceleration","text":"What is the centripetal acceleration and angular velocity of a child located 8.2 m the center of a carousel? The speed (size of the tangential velocity) of the child is 2.1 m \/ s A train moves in a straight path north until it turns to west. If the road segment used to change direction is shaped like a quarter circle of radius 30 m and the train takes 30 seconds to traverse that part of the road, What is it the speed (size of the velocity vector) and the centripetal acceleration acts on the train as it traverses the curve. I am reviewing some concepts like centripet force, `ar = ( v^2 ) \/ r` also this: > The direction of the centripital acceleration is always inwards along the > radius vector of the circular motion. The magnitude of the centripetal > acceleration is related to the tangential speed and angular velocity as > follows: ![enter image description here](http:\/\/i.stack.imgur.com\/XGojw.png) Can you please guide me to solve the 2 problems above? for the fisrt one is it only : `(2.1 m \/ s)^2 \/ 8.2 m` ?"}
{"id":"57237","title":"Working of a p-n junction diode when forward biased","text":"If p-type semiconductor and n-type semiconductor of a diode are equally doped, and if the diode is forward biased, then holes will move toward the n-type semiconductor and electrons will move toward the p-type semiconductor and they will diffuse with each other. Then will there be any electron that will go to the positive terminal of the battery if all of them have diffused with each other? I can't understand, please help me!"}
{"id":"98881","title":"Where did $\\mathcal{M}^\\mu(k) = \\int \\mathrm{d^4}x \\; \\exp(\\mathrm{i}k \\cdot x)\\langle f | j^\\mu(x) | i\\rangle$, in Peskin and Schroeder, come from?","text":"On page 160 Peskin & Schroeder, they say: > Therefore we expect $\\mathcal{M}^\\mu(k)$ to be given by a matrix element of > the Heisenberg field $j^\\mu$: $$\\mathcal{M}^\\mu(k) = \\int \\mathrm{d^4}x \\; > \\exp(\\mathrm{i}k \\cdot x)\\langle f | j^\\mu(x) | i\\rangle.$$ Why do they expect that? Where does this come from? I've seen similar expressions with e.g. two Lorentz indices and so on. I've been taught (obviously not enough) qft through path integrals and have a hard time to make connections between the different formalisms."}
{"id":"107144","title":"Is there an analogue of a geodesic for the evolution of the electromagnetic field?","text":"For a charged particle moving in free space, we can say from the homogeneity of space-time, that it moves along a geodesic. Is there an analogous principle for the evolution of the electromagnetic field in space-time?"}
{"id":"94135","title":"Minimum seperation between two Airy disks as a function of the distance between two point sources of coherent light passing through the same objective","text":"I have two coherent point sources of light, $A$ and $B$, separated by a distance $L$, which I focus down to the diffraction limit using a high-powered objective (e.g. a $\\approx 100x$ objective). If I turn on $A$ and turn off $B$, I have an Airy disk at position $c_1$, and I turn off $A$ and I turn on $B$, I have an Airy disk at position $c_2$. Given that both light sources are sent through the same objective, what is the minimum distance between $c_1$ and $c_2$? Is it simply $L$ scaled down by the objective (i.e. $\\frac{L}{100}$)? Or does something odd happen because of e.g. curvature of the lens in the objective? EDIT: A restatement of this question would be the following - Assuming all of the optics are perfect, if I shine a laser at a point (x,y) on an objective, and then shine the laser at a point (x2,y2), will the peak of the Airy disk move the same distance?"}
{"id":"78678","title":"Law of reversibility of light and total internal reflection","text":"When a light passes from a denser to a rarer medium at critical angle of incidence the light rays graces through the surface of the denser medium.According to the law of reversibility of light same thing should happen when we reverse the direction of light.But, how can that be? How does the light know when to go into the denser medium?"}
{"id":"17141","title":"Does a nonzero Poynting vector mean that there is propagation of energy?","text":"I don't know how this \"paradox\" can be solved. I'm given the following system: A permanent magnet with a magnetic field given by ($\\hat{a}$ are unit vectors in the x and y directions) $$\\vec{H}=H_0\\hat{a}_y$$ and a parallel plane capacitor with an electric field $$\\vec{E}=E_0\\hat{a}_x$$ Poynting's vector is given by: $$\\vec{S}=\\vec{E}\\times\\vec{H}=H_0E_0\\hat{a}_z \\neq 0$$ The funny part comes when the professor told that in a system like that \"clearly\" there is not propagation (wich I know will imply some short of energy flux) in the $z$ direction, hence the \"paradox\". Is there or is there not propagation of energy? Any hint will be appreciated, thank you for your time."}
{"id":"102513","title":"Using the boundary states, is there a precise way to write down a planar open string multi-loop amplitude as a closed string tree amplitude?","text":"The only explicit computation I have seen is the planar 1-loop one, but there should be a way to write the multi-loop case in terms of boundary states as well."}
{"id":"100498","title":"What is the \"discrete\" analogue to \"continuum\" mechanics?","text":"If I wanted to explore a discrete mathematics approach to continuum mechanics, what textbooks should I look into? I suppose a ready answer to the question might be: \"computational continuum mechanics\", but usually textbooks that discuss such a subject are usually focused upon applying numerical analysis to continuous theories (i.e. the base is continuous), whereas I would like to know if there is a treatment of the subject that builds up from a base that is discrete."}
{"id":"113519","title":"Electric field generated by a point charge moving at the speed of light","text":"![enter image description here](http:\/\/i.stack.imgur.com\/sUvHk.png) As you see, this is the electric field generated by a point charge moving at constant speed v. I know that when $v$ -> 0, $E$ is just the Coloumb Law. But how do you interpret $E$ when $v$ -> $c$ ? Can I just interpret it as the field of electromagnetic wave, because it moves at the speed of light? ![enter image description here](http:\/\/i.stack.imgur.com\/LvWkF.png)"}
{"id":"916","title":"Imagine a long bar floating in space. What force does it exert on itself in the middle due to gravity?","text":"# Problem If you had a long bar floating in space, what would be the compressive force at the centre of the bar, due to the self-weight of both ends? Diagram - what is the force at point X in the middle of the bar?:               <----------------------L--------------------->, total mass M     =======================X======================   <- the bar                      F---> X <---F      # Summary You should be able to simplify by cutting the bar into pieces, but that gives a different answer depending on how many pieces you use (see below). So the simplification must be wrong - but why? # My approach ### Split bar in two So, one approximation would be to cut the bar in half - two pieces of length L\/2, mass M\/2:                        (M\/2)<-------L\/2------->(M\/2)                #1          X           #2   <- bar approximated as blobs #1 and #2      Force at X is G(M1.M2)\/(R^2) = G (M\/2)^2 \/ (L\/2)^2 = G M^2 \/ L^2 Or Fx \/ (G. M^2 \/ L^2) = **1** But is that really valid? If so, shouldn't you get the same answer if you split the bar into four pieces? ### Split bar into four                  (M\/4)<-L\/4->(M\/4)<-L\/4->(M\/4)<-L\/4->(M\/4)          #1          #2    X     #3          #4      My assumption is that the force at X is the sum of the attractions of each blob on the left to every blob on the right. Force at X = #1<>#3 + #1<>#4 + #2<>#3 + #2<>#4 ('<>' being force between blobs #x and #y). Fx \/ (G.M^2 \/ L^2) = (2\/4)^-2 + (3\/4)^-2 + (1\/4)^-2 + (2\/4)^-2 = **1.61** This is bigger than the previous result (1.61 vs 1). ### Split bar into six Similarly, if you split into 6 blobs, the total force comes out as: Fx \/ (G.M^2 \/ L^2) = (3\/6)^-2+(4\/6)^-2+(5\/6)^-2 + (2\/6)^-2+(3\/6)^-2+(4\/6)^-2 + (1\/6)^-2+(2\/6)^-2+(3\/6)^-2 Fx \/ (G.M^2 \/ L^2) = **2.00** ### So what's wrong with my approach? And what is the real answer? So it seems the more pieces we split the bar into, the larger the result gets. There's clearly something wrong with my assumptions! - but what? I'd be very glad if someone here could explain this. Thanks! **EDIT** As Peter Shor pointed out, my calculations had some dodgy algebra and I'd calculated $$L^2\/M^2$$ values rather than $$M^2\/L^2$$. I've now corrected that - the value still increases as you divide into more masses. I'll do a bit more work with more divisions and see if this leads to convergence or not."}
{"id":"113515","title":"If gravity bends space time, could gravity be manipulated to freeze time?","text":"You age at a different rate depending on the force of gravity. Astronauts age fractions of fractions of fractions of a second less than earthlings. If you took a sphere of equal masses, separated by space, then found the exact center of the gravitational pulls of all masses. How would time react?"}
{"id":"104828","title":"Mario Livio's book on symmetry and the relationship between gravity and acceleration","text":"In his book: _The Equation That Couldn't be Solved_ Mario Livio explains the equivalence principle in laymen's terms. I took the statement on page 209: _The force of gravity and the force resulting from acceleration are in fact the same._ to mean that since our universe is accelerating, gravity is the result. In other words, we have gravity because we have acceleration. This made a lot of sense to me and seemed to explain things (like warps in space- time due to gravity of very large objects. They 'have' gravity because they are accelerating masses. In my beginner's head I extended that conclusion to mean if the acceleration of the universe was zero we would have mass but no gravity. Based on what I've read since, this does not seem to be true...is it?"}
{"id":"104822","title":"How to show the invariant nature of some value by the group theory representations?","text":"Let's have Dirac spinor $\\Psi (x)$. It transforms as $\\left( \\frac{1}{2}, 0 \\right) \\oplus \\left( 0, \\frac{1}{2} \\right)$ representation of the Lorentz group: $$ \\Psi = \\begin{pmatrix} \\psi_{a} \\\\\\ \\kappa^{\\dot {a}}\\end{pmatrix}, \\quad \\Psi {'} = \\hat {S}\\Psi . $$ Let's have spinor $\\bar {\\Psi} (x)$, which transforms also as $\\left( \\frac{1}{2}, 0 \\right) \\oplus \\left( 0, \\frac{1}{2} \\right)$, but as cospinor: $$ \\bar {\\Psi} = \\begin{pmatrix} \\kappa^{a} & \\psi_{\\dot {a}}\\end{pmatrix}, \\quad \\bar {\\Psi}{'} = \\bar {\\Psi} \\hat {S}^{-1}. $$ How to show formally that $$ \\bar {\\Psi}\\Psi = inv? $$ I mean that if $\\Psi \\bar {\\Psi}$ refers to the direct product (correct it please, if I have done the mistake) $$ \\left[\\left( \\frac{1}{2}, 0 \\right) \\oplus \\left( 0, \\frac{1}{2} \\right) \\right]\\otimes \\left[\\left( \\frac{1}{2}, 0 \\right) \\oplus \\left( 0, \\frac{1}{2} \\right) \\right], $$ what group operation corresponds to $\\bar {\\Psi} \\Psi$? This question is strongly connected with this one."}
{"id":"112646","title":"Potentiometer at null pointer!","text":"Consider the situation when a cell of an unknown emf is being measured using a potentiometer. We slide the jockey so as to obtain the null point. Now, is there any current in the potentiometer wire at the null point? Since we know that there is no current in the arm containing the unknown cell, its terminals have acquired equal potentials,how is it possible that there is any current in the potentiometer wire when that is in parallel to that cell. Potential difference across AB= Potential difference across CD? ![enter image description here](http:\/\/i.stack.imgur.com\/YcUOl.gif)"}
{"id":"64218","title":"Donors\/Acceptors in Metal Oxides","text":"Can anyone explain to me why most articles describe chromium as an acceptor in titanium dioxide? In TiO2, titanium has the charge state Ti$^{4+}$ and oxygen has the charge state O$^{2-}$. When Cr substitutes for Ti, it does so as Cr$^{3+}$. Now, at first glance, Cr has atomic number 24 and Ti 22. Cr therefore has two more valence electrons and is a donor. In TiO2, Cr$^{3+}$ actually has _three more_ valence electrons than the Ti$^{4+}$ ([Ar]$3d^34s^0$ vs [Ar]$3d^04s^0$). It should therefore be a donor, right? The thing is, it forms a deep impurity level near the valence band. TiO2 has an energy gap of around 3.2 eV, and the impurity state is about 1.0 eV from the valence band maximum. To me, that makes it a **deep donor**. For some reason, journals almost always describe it as an acceptor. Can someone help me make sense of this? My understanding has always been simply this: more electrons than host $\\Rightarrow$ donor, fewer electrons than host (more holes) $\\Rightarrow$ acceptor. The position of the impurity level, to my (perhaps incorrect) knowledge, does not determine whether or not the impurity is actually a donor or acceptor, but rather whether it is a recombination center or trap. We can have localized states near the middle of the bandgap that are technically donors\/acceptors but function as recombination centers, so I'm not sure what I'm missing here."}
{"id":"125932","title":"Special relativity and electromagnets","text":"This Veritasium video explains how electromagnets can be explained by special relativity, and how the magnetic field surrounding a current-carrying wire can also be viewed as an electric field, if your frame of reference is moving with respect to the wire. The example they use is a positively-charged cat, moving along a current- carrying wire in the same direction as the electron drift: ![positively charged cat next to a current-carrying wire](http:\/\/i.stack.imgur.com\/HHUiP.png) If you view this from the rest frame of the cat, then the electron's drift velocity is zero, while the protons are moving to the left. Because the protons are moving, length contraction makes it look like (to the cat) there are more of them, giving the wire a net _positive_ charge, _repelling_ the cat. This makes sense and is all kinds of elegant and intuitive. It explains electromagnets in a way that depends on only three simple concepts:   1. motion is relative   2. things contract in their direction of apparent motion   3. opposite charges attract, like charges repel Groovy. Now back up in the video. Derek says: > Now the number of protons is equal to the number of negative electrons, so > overall the wire is neutral. So if there were a positively-charged cat > nearby, it would experience no force from the wire at all. And even if there > were a current in the wire, the electrons would just be drifting in one > direction, but the density of positive and negative charges would still be > the same, and so the wire would be neutral, so no force on the kitty. ![Derek standing next to a current-carrying wire](http:\/\/i.stack.imgur.com\/Gb796.png) Wait... _what_? Why is it that in the cat's frame, the protons are moving, are contracted, and the wire is charged, but in Derek's frame, the electrons are moving, _but are not contracted_ , and the wire is still neutral? How can you say \"well, length contraction creates charge imbalances, allowing magnetic forces to be explained as electrical ones if you choose the right reference frame\", but simultaneously say \"but length contraction doesn't happen sometimes\"? That's not elegant at all. Is there an elegant, intuitive1 explanation? 1: meaning, I've seen the math on Wikipedia and it's over my head. There is also current in wire + special relativity = magnetism where the answer to my question seems to be \"the Lorentz force\". OK, but that negates the elegance of the explanation above, with only three simple axioms. Are they not sufficient? If so, why?"}
{"id":"168","title":"Intuitively, why is a reversible process one in which the system is always at equilibrium?","text":"A process is reversible if and only if it's always at equilibrium during the process. Why? I have heard several specific example of this, such as adding weight gradually to a piston to compress the air inside reversibly, by why should it be true in general? EDIT: Here is something that would firmly convince me of this: Suppose I have a reversible process that is not always in equilibrium. Describe a mechanism for exploiting this process to create a perpetual motion machine."}
{"id":"27590","title":"An entropy of the Wigner function","text":"Is there an entropy that one can use for the Wigner quasi-probability distribution? (In the sense of a phase-space probability distribution, not - just von Neumann entropy.) One cannot simply use $\\int - W(q,p) \\ln\\left[ W(q,p) \\right] dq dp$, as the Wigner function is not positively defined. The motivation behind the question is the following: A paper I. Białynicki-Birula, J. Mycielski, Uncertainty relations for information entropy in wave mechanics (Comm. Math. Phys. 1975) (or here) contains a derivation of an uncertainty principle based on an information entropy: $$-\\int |\\psi(q)|^2 \\ln\\left[|\\psi(q)|^2\\right]dq-\\int |\\tilde{\\psi}(p)|^2 \\ln\\left[|\\tilde{\\psi}(p)|^2\\right]dp\\geq1+\\ln\\pi.$$ One of the consequences of the above relation is the Heisenberg's uncertainty principle. However, the entropic version works also in more general settings (e.g. a ring and the relation of position - angular momentum uncertainty). As $|\\psi(q)|^2=\\int W(q,p)dp$ and $|\\tilde{\\psi}(p)|^2=\\int W(q,p)dq$ and in the separable case (i.e. a gaussian wave function) the Winger function is just a product of the probabilities in position an in momentum, it is tempting to search for an entropy-like functional fulfilling the following relation: $$1+\\ln\\pi\\leq\\text{some_entropy}\\left[ W\\right]\\leq -\\int |\\psi(q)|^2 \\ln\\left[|\\psi(q)|^2\\right]dq-\\int |\\tilde{\\psi}(p)|^2 \\ln\\left[|\\tilde{\\psi}(p)|^2\\right]dp.$$"}
{"id":"27598","title":"Massive excitations in Conformal Quantum Field Theory","text":"Single particle states in quantum field theory appear as discrete components in the spectrum of the Poincare group's action on the state space (i.e. in the decomposition of the Hilbert space of quantum states into irreducible representations of the Poincare group). Classification of irreducible unitary representations of the Poincare group leads to the notions of mass and spin. Now, suppose we have a conformal QFT and are doing the same trick with the conformal group. Which irreducible representations do we have? We still have the massless particles (at least I'm pretty sure we do although I don't immediately see the action of special conformal transformations). However, all representations for a given spin s and any mass m > 0 combine into a single irreducible representation. > What sort of physical object corresponds to this representation? Is it > possible to construct a scattering theory for such objects? Is it possible > to define unstable objects of this sort?"}
{"id":"99759","title":"range of the difference of two-qubit density matrix determinants","text":"The determinant of a two-qubit (4 x 4) density matrix lies between 0 and (1\/2)^8. (A pure state has determinant zero, and the fully mixed [classical] state, determinant (1\/2)^8.) The determinant of the partial transpose (transpose in place the four 2 x 2 blocks) of such a matrix (nonnegative values indicating separability) lies between -(1\/2)^4 and (1\/2)^8. (The minimum is achieved by a Bell state and the maximum, again by the fully mixed state.) What is the range (upper and lower limits) of the difference of these two determinants?"}
{"id":"29257","title":"Is the opening of the NOVA program a Calabi-Yau space?","text":"Is the opening of the NOVA program on PBS a Calabi-Yau space?"}
{"id":"57902","title":"Phase shift in electromagnetic potential","text":"In Aharonov-Bohm effect, how to derive that the wave function of a electric charge $q$ acquires a phase shift $\\phi=\\frac{q}{\\hbar}\\int \\mathbf{A} \\cdot d\\mathbf{x}$ after travelling in the non-zero magnetic vector potential $\\mathbf{A}$?"}
{"id":"127739","title":"How non-abelian gauge coupling runs below confinement or QCD scale?","text":"I know the famous beta function of asymptotic free, but that seems describe the running coupling beyond confinement\/QCD scale so that a perturbative analysis can apply. But how coupling runs below that scale? Any comment or references are greatly appreciated."}
{"id":"34286","title":"What would be likely to completely stop a subatomic particle assuming it was possible?","text":"Suppose that completely stopping a subatomic particle, such as an electron, could happen under certain conditions. What would be likely ways to get an electron to be perfectly still, or even just stop rotating the nucleus and collapse into it by electromagnetic forces? What would likely be required, below absolute-zero temperatures? Negative energy? Or could a 0 energy rest state not exist in any form, of any possible universe imaginable? Let's say there was a magnetic field of a certain shape that we could postulate that is so intensely strong that if we put an electron in the center of it, it could not move at all in any direction. Would the energy requirement of the field be infinite? What would be the particle's recourse under this condition? Further, suppose it were possible and one could trap an electron and stop all motion completely. What would this do to Heisenberg's Uncertainty Principle and\/or Quantum Mechanics, because its position and momentum (0) would both be known? If it can be done, is Quantum Mechanics no longer an accurate model of reality under these conditions? Could we say QM is an accurate model under most conditions, except where it is possible to measure both position and momentum of a particle with zero uncertainty? # Clarification: Please assume, confined in a thought experiment, that it IS possible to stop a particle so that is has 0 fixed energy. This may mean Quantum Mechanics is false, and it may also mean that under certain conditions the uncertainty commutation is 0. ASSUMING that it could physically be done, what would be likely to do it, and what would be the implications on the rest of physics? # Bonus Points Now here's the step I'm really after - can anyone tell me why a model in which particles can be stopped is so obviously not the reality we live in? Consider the 'corrected' model is QM everywhere else (so all it's predictions hold in the 'normal' regions of the universe), but particles can be COMPLETELY stopped {{inside black holes, between supermagnetic fields, or insert other extremely difficult\/rare conditions here}}. How do we know it's the case that because the uncertainty principle has lived up to testing on earth-accessable conditions, that it holds up under ALL conditions, everywhere, for all times?"}
{"id":"75788","title":"How to learn physics?","text":"I am an engineering student (CSE) in India..But recently I have developed a strong love to physics..I want to learn physics and understand it in deep..I know physics is the search of deep fundamental laws of nature.That means I need to start from first principles and extrapolate from it?How should i learn physics?"}
{"id":"116271","title":"Is it more efficient to stack two Peltier modules or to set them side by side?","text":"Is it more efficient to stack two Peltier modules or to set them side by side? And why? I have a small box that I want to cool down about 20 K below ambient -- cold, but not below freezing. (I want to keep my camera cool, so I'm putting in this cool box. The camera looks through a flat glass window on one side of the box). The heatsink I have on hand is about twice as wide as the widest Peltier module I originally planned on using. So there's room to put 2 Peltier modules side-by-side under the heatsink. Or I could center a stack of 2 Peltier modules under the heatsink. Which arrangement is more efficient? I have to cut a bigger hole in the insulation for the side-by-side arrangement, so the unwanted heat that \"back-flows\" through the side-by-side arrangement is worse. On the other hand, other effects are worse for the stacked arrangement. (Is http:\/\/electronics.stackexchange.com\/ a better place to post questions about Peltier coolers?)"}
{"id":"59501","title":"Identical fermions in the same quantum state","text":"If we are to take two Hydrogen atoms and subject them to the same potential, then wouldn't both Hydrogen atoms be in the same exact quantum state? This bother me because no two identical fermions can be in the same quantum state! This seems to contradict the principle. This applies to any two elements or molecules that are subjected to the same potential. Say these two Hydrogen atoms are located 1m from each other, then would the only way to distinguish them would be their spatial location? What is the technical term for two seemingly identical things to be distinguishable by their location?"}
{"id":"78682","title":"Is the firewall paradox really a paradox?","text":"The firewall paradox is a very hot topic at the moment (1207.3123v4). Everyone who is anybody in theoretical physics seems to be jumping into the action (Maldacena, Polchinski, Susskind to name a few). However, I am unable to see the paradox. To me Hawking's resolution of the information paradox (hep-th\/0507171) also resolves the so called firewall paradox. Hawking never says that the information is not lost on a fixed black hole background. In fact, he says the opposite. He says (page 3): \"So in the end everyone was right in a way. Information is lost in topologically non- trivial metrics like black holes. This corresponds to dissipation in which one loses sight of the exact state. On the other hand, information about the exact state is preserved in topologically trivial metrics. The confusion and paradox arose because people thought classically in terms of a single topology for spacetime.\" To surmise, in quantum gravity you don't know if you actually have a black hole or not, so you have to include the trivial topologies, including those when there isn't a black hole there, in the amplitude. Only then do recover unitarity. It seems to me that the error of the AMPS guys is that they use a fixed black hole background and assume conservation of information (i.e., late time radiation is maximally entangled with early time radiation). It is no wonder they are lead to a contradiction. They are simply doing the information paradox yet again. They give a menu of implications in the abstract: (i) Hawking radiation is in a pure state, (ii) the information carried by the radiation is emitted from the region near the horizon, with low energy effective field theory valid beyond some microscopic distance from the horizon, and (iii) the infalling observer encounters nothing unusual at the horizon. But the obvious solution is that (i) is wrong. The radiation, within the semi- classical calculation in which they calculate it (i.e., not quantum gravity), is non-unitary. So my question is, why is this a paradox? Something so obvious surely can not be overlooked by the ``masters'' of physics. Therefore I'd like to hear your opinions."}
{"id":"105882","title":"Why do we require manifolds to be a topological space?","text":"Roughly speaking, we define a manifold $M$ to be covered by a set of charts $\\\\{(U_i , \\varphi_i)\\\\}$ such that locally the $n$-dimensional manifolds looks like $\\mathbb{R}^n$. One of the conditions is that all the $U_i$ are open sets of the topology of the manifold. Why do we require the manifold to be a topological space? And why do we want $U_i$ to be open sets? What are the implications of these requirements on physics? (It appears to me that without these conditions the manifold still looks locally like $\\mathbb{R}^n$.) Edit to make my question more concrete: are there any physical theories that use manifolds that are not topological spaces? For instance, what would happen to general relativity if the spaces are diffeomorphic to each other, instead of homeomorphic (see answer below by Robin Ekman)?"}
{"id":"131631","title":"Current Density","text":"I'm trying to understand the definition given on my electromagnetism course for the current density. More specifically, I want to know why, as defined below, the current density is given the name \"current density.\" On my course, the current density is $\\vec{j}(t,\\vec{x}):=\\rho (t,\\vec{x}) \\vec{v} (t,\\vec{x})$ where $\\vec{v} (t,\\vec{x})$ is the velocity field governing how the charged particles move. I'm trying to get some intuition for what this quantity is. To give an example of what I'm talking about, in classical mechanics where you have momentum equal to mass multiplied by velocity, the definition makes sense intuitively because momentum is the oomph you will feel if an object hits you, and you feel that oomph more if either the mass or velocity of the object increases. So I have a really tangible idea of what momentum is. Wikipedia describing current density: \"In electromagnetism, and related fields in solid state physics, condensed matter physics etc. current density is the electric current per unit area of cross section.\" This justifies calling it a density (as it's an area density by defn.). I'm trying to understand what a current density could be, and in my head I've got an idea of a cross-sectional area with some fluid flowing through it (it's the same scenario in which I picutre Gauss' Law). I'm not sure what current at a point is, so I don't really understand what a current density could be! Then I need to relate this to the definition I've been given on my course somehow. Thanks for any help!"}
{"id":"105887","title":"Resistances connected in series","text":"When resistances connected in series then why through each resistances the different potential difference occurs?"}
{"id":"111990","title":"If we connect a source of electricity in a large water body ,will it be dangerous?","text":"I was wondering whether a person will be electrocuted if he is in large pool or a sea which is connected through a source of electricity . As sea water is saline it will conduct electricity so will it be dangerous !?"}
{"id":"26643","title":"Why Aren't Saturn's Rings Clumping into Moons?","text":"While reading with my son about how a Mars-like planet collided with the early Earth that resulted in our current moon, it said the initial debris also formed a ring, but that ring ended up getting absorbed by the Earth and the Moon. I couldn't answer his question then why Saturn still has rings. Shouldn't Saturn's rings be clumping into Moons or getting absorbed by Saturn's gravity?"}
{"id":"92574","title":"Ghost fields in particle physics","text":"In my particle physics lecture, ghost fields were briefly mentioned. As far as I understand, these come up when computing cross sections by the path integral method, to compensate for equivalent contributions due to gauge freedom. I'm still not sure how these equivalent contributions come up in the calculations. Say you want to compute the cross section of an electron anti- electron annihilation into a photon near a massive nucleus. I assume you would then take the sum of the path integrals over all possible Feynman diagrams corresponding to that interaction. QED has charge symmetry (i.e. $U(1)$), so the physics stay the same if the two charges are swapped. Where concretely do we have to introduce these ghost fields, and why can we not simply combine the Feynman diagrams that are graph-isomorphisms of each other into a single equivalence class and count the contribution of each equivalence class only once into the final probability amplitude. Please keep the answers simple, as I'm not a physics major."}
{"id":"92579","title":"Classical Mechanics & Coordinates","text":"1. What is the meaning generalised coordinates in Classical Mechanics?   2. How is Lagrangian formalism different from Hamiltonian formalism?   3. How are they related to Hamilton's Principle?   4. How are they related to Euler-Lagrange equation?"}
{"id":"66217","title":"Quantum Mechanics, Uncertainty Principle-- help understanding notes","text":"There is a section of my notes which I do not understand, hopefully someone here will be able to explain this to me. The notes read (after introducing the uncertainty operator): > > If the state $\\chi_A$ is an eigenstate of $\\hat O_A$ then the uncertainty > is zero and we measure it with probability 1. However, if $\\hat O_B$ is > another observable which does not commute with $\\hat O_A$, then the > uncertainty in any simultaneous measurement of the two observables will be > infinite. I understand the first sentence, but I can't see how to justify\/prove the second one. Can someone tell me how the second sentence is justified, please?"}
{"id":"112518","title":"If we suddenly lost track of time, how would we know what time is now?","text":"If we hypothetically lost all watches and all devices that keep track of time, how would we say what is the current time? Or we actually don't and time is just a convention?"}
{"id":"68841","title":"Is my boss wrong about our mechanical advantage from our pulley system?","text":"I work on a drilling rig as a roughneck and we had a lecture today (at the office) about mechanical advantage in pulley systems. Now, I know that my boss is well educated in oil drilling, but my instincts tell me that he may have this one wrong. A drilling rig works sort of like a crane in that it has a tall structure supporting a pulley system. There is a large winch permanently installed on the base platform and then it goes over the top of the structure (the crown of the derrick) and down through a floating sheave--this has a few wraps to give us more mechanical advantage. I am including pictures to help describe the situation. ![Floating Sheave \\(called the \"blocks\"\\)](http:\/\/i.stack.imgur.com\/naok2.jpg) Here the picture shows the floating sheave (the blocks) which we use to do most all of our operations. Most importantly, we use it to pick up our string of pipe that is in the ground. ![The set up of a drilling rig](http:\/\/i.stack.imgur.com\/nW0mS.gif) As seen in this picture, the blocks hold the weight of the string of pipe. Now he told us that if the pipe get stuck in the hole (maybe it snags something or the hole caves in), that we lose all of our mechanical advantage. He said that is why the weight indicator will shoot up and go back down after it is freed. He said that because when the pipe is snagged in the hole then we are not dealing with a free floating sheave anymore and that is what is required to have a mechanical advantage. I disagree with this because even if it is not free, there is still a mechanical advantage such that (say the normal mechanical advantage is 6 to 1) our pulling force is multiplied by 6. I would like somebody to confirm this for me. First picture taken from www.worldoils.com on June 21, 2013 Second picture taken from www.PaysonPetro.com on June 21, 2013"}
{"id":"29596","title":"The sound of coffee","text":"> **Possible Duplicate:**   >  Why does the sound pitch increase on every consecutive tick at the bottom > of a filled cup of coffee? A colleague suggested this experiment this morning :   * Get a cup of coffee   * Spin the coffee in the cup using a spoon   * Tap the bottom of the cup with the spoon _Observations_ : As the coffee slows down, the sound produced by the tapping gets higher in frequency. I'm very curious about this, does anyone have an explanation ?"}
{"id":"131188","title":"How did Fizeau make his famous speed-of-light experiment?","text":"I heard once in a TED talk how Fizeau measured the speed of light in the 19th century. Here is the link https:\/\/www.youtube.com\/watch?v=F8UFGu2M2gM You can read about it here in Wikipedia: http:\/\/en.wikipedia.org\/wiki\/Fizeau%E2%80%93Foucault_apparatus A short summary: He placed a kind of rotating wheel in front of a beam of light, and a mirror far away from these two things. The beam of light passes between two teeth of the rotating wheel, reaches the mirror and goes back from the original source. As the wheel is spinning very fast, during the time that the light has been travelling, the wheel has rotated a tiny bit, but enough to impede the passage of time through the point where it entered. Knowing the distance from the mirror as well as the speed at which the wheel is rotating, the speed of light can be easily calculated. The experiment is better explained in wikipedia, here I wrote a simplified version of it. I loved the experiment, because it seemed fairly easy to reproduce, so I ordered a green laser pointer on Amazon, which can reach up to 10 km. As a proof of principle, I went with a friend in the night to a place where there is a good visibility. We began setting a mirror somewhere 500 metres away from the laser, but, even from that far, the light had scattered so much that it was impossible to collect the light with the mirror. The laser is a very powerful one, of those that you can see the whole beam (usually used in astronomy). If I can't repeat the experiment using a laser like this, how on earth could Fizeau do that in the 18th century employing a much more rudimentary source of light and placing the mirror much farther away? It says in Wikipedia that the distance between them was like 8 km."}
{"id":"22348","title":"Why the shape of rainbow is semicircular after rain why not the whole atmosphere is colorful?","text":"I have a very simple question. Everyone must have seen the rainbow after rain. According to the theory the rainbow is created due to the passing of sunlight from small drops of water in the atmosphere(means by dispersion of light). Now my what I want to know is that after rain the rain drops are present in the entire atmosphere. So the whole atmosphere should look colorful. Why only a semicircular shape is formed (or is colorful)."}
{"id":"131186","title":"Determining the Wave Function From Initial Conditions","text":"This is Problem 2.6 (b) in Griffiths, _Intro to QM:_ > A particle in an infinite square well has its initial wave function an even > mixture of the first two stationary states: > > $\\Psi(x,0) = A[\\psi_1(x) + \\psi_2(x)]$. Here is the part of the problem that I am having a little trouble with: > (b) Find $\\Psi(x,t)$ and $|\\Psi(x,t)|^2$. Express the latter as a sinusoidal > function of time, as in Example 2.1. To simplify the result, let $\\omega > \\equiv \\frac{\\pi^2 \\hbar}{2ma^2}$ According to the answer key, even after $t=0$, the wave function continues to be a mixture of the first two stationary states. Why is that? I am having a little difficulty understanding this. Why can't it be a new 'mixture?' Are my questions sufficiently clear?"}
{"id":"131180","title":"Heat transfer to a cube from one of the faces","text":"Heat Flux= 10 MW Inlet water temperature = 28 degree centigrade for cooling purpose through a tube passing through the centre of retangular block of length 50 mm,height 30 mm and width 30 mm inner diameter of tube=10mm and outer diameter=12 mm tube is made of copper. Find out the heat transfer coefficient of water at 42 bar and given temperature Find out Nusselt number and Reynold's number."}
{"id":"44543","title":"Colliding bubbles in hyperspace","text":"Assuming the following   1. A universe is the surface from a bubble in hyperspace. Inside a bubble there is nothing, only the surface represents a universe. The size of the bubble is time.   2. Dark matter is from bubbles different then our own bubble that only shares gravity.   3. The big bang are bubbles colliding, creating a other bubble. My problem is if bubbles do only share gravity why would they have a collision, I expect that they would just pass trough each other without creating a big bang?"}
{"id":"77265","title":"Renormalizability of the Polyakov Action","text":"I was told today that the Polyakov action for a $p$-brane is (superficially) re-normalizable iff $p\\leq 1$. Of course, when I went to check for myself, I screwed up my power-counting, and I'm having trouble seeing why. We work in units with $c=1=\\hbar$, so that $L=T=M^{-1}$. In these units, any action must have dimension $1$, so from looking at the Nambu-Goto action, $$ S_{\\text{NG}}:=-T_p\\int d\\sigma ^{1+p}\\sqrt{-g}, $$ we see that $[T_p]=L^{-(1+p)}=M^{1+p}$. From the Polyakov action, $$ S_{\\text{P}}:=-\\frac{T_p}{2}\\int d\\sigma ^{1+p}\\sqrt{-h}h^{\\alpha \\beta}\\partial _\\alpha X^\\mu \\partial _\\beta X^\\nu G_{\\mu \\nu}(X), $$ we see that the the coupling constant of the interaction of of the scalar fields $X^\\mu$ with $h_{\\alpha \\beta}$ is precisely $\\frac{T_p}{2}$, which has dimensions $M^{1+p}$, and so is going to be (superficially) re-normalizable iff $1+p\\geq 0$ . . . But this, of course, is not the result I was looking for . . . Where is my mistake?"}
{"id":"79556","title":"Does a universe experiencing \"heat death\" have a temperature?","text":"As defined by Wikipedia: > The heat death of the universe is a suggested ultimate fate of the universe > in which the universe has diminished to a state of no thermodynamic free > energy and therefore can no longer sustain processes that consume energy > (including computation and life). Heat death does not imply any particular > absolute temperature; it only requires that temperature differences or other > processes may no longer be exploited to perform work. Does it even make sense to describe temperature in the system described? If so, would it be a very cold system or a very hot system?"}
{"id":"79553","title":"The surface area to volume ratio of a sphere and the Bekenstein bound","text":"I am trying to relate the surface-area-to-volume-ratio of a sphere to the Bekenstein bound. Since the surface-area-to-volume-ratio decreases with increasing volume, one would surmise that, per unit of volume, a small space is richer in information than a large one. How can this be and how can this bound work for black holes of various sizes? Thank you very much Mr. Rennie. I appreciate and have investigated your answer. It turns out that I am familiar with the AdS\/CFT correspondence and have sufficient understanding of the math (just barely) to be intrigued with the conjecture and, of course, the holographic theory. If the correspondence only works for a certain diameter black hole, the conjecture seems, to me, weak because of the changing surface-area-to-volume-ratio of a sphere. For myself, it would appear to be, likely, a mathematical curiosity or fluke. However, if, through some aspect that I do not understand, the correspondence holds for varying diameters, in fact, all diameters of black holes, then it seems quite astonishing, indeed. After searching for some time, I have once seen the amount described as trivial and possibly in another instance, that it may have something to do with informational redundancy. I’m afraid I cannot site these references as they were far too brief to be of any help."}
{"id":"28610","title":"Conformal fields on compactified manifolds? An apparent paradox!","text":"I would appreciate it if someone tells me how a cft on a compactified manifold (e.g. by means of periodic boundary conditions) can be meaningful? The global conformal invariance is broken due to the scale over which the manifold is compactified (e.g. the period). The local conformal invariance of course still classically exists, but for the simple case of a field on a cylinder for instance, the trace of the stress-energy tensor becomes non-zero (in proportion with the central charge); hence apparently compactification of the space-time manifold is in contrast with having conformal symmetry."}
{"id":"92392","title":"Collision between a photon and a massive particle","text":"Just a small question regarding collisions. Imagine a head-on collision between a photon and a particle with mass that moves with a non-relativistic speed, the particle was on its ground state, completely absorbs the photon, and moves to its next energy level. Is it always the case that the particle ends up with a non-relativistic speed after the collision? Something more specific: > To study the properties of isolated atoms with a high degree of precision > they must be kept almost at rest for a length of time. A method has recently > been developed to do this. It is called “laser cooling” and is illustrated > by the problem below. In a vacuum chamber a well collimated beam of Na23 > atoms (coming from the evaporation of a sample at 103 K) is illuminated > head-on with a high intensity laser beam (fig. 3.1). The frequency of laser > is chosen so there will be resonant absorption of a photon by those atoms > whose velocity is v0. When the light is absorbed, these atoms are exited to > the first energy level, which has a mean value E above the ground state and > uncertainty of (gamma). Find the laser frequency needed ensure the resonant > absorption of the light by those atoms whose kinetic energy of the atoms > inside the region behind the collimator. Also find the reduction in the > velocity of these atoms, ∆v1, after the absorption process. Data E = > 3,36⋅10-19 J Γ = 7,0⋅10-27 J c = 3⋅108 ms-1 mp = 1,67⋅10-27 kg h = > 6,62⋅10-34 Js k = 1,38⋅10-23 JK-1"}
{"id":"114176","title":"Solar neutrino problem in 1968 and experimental verification of neutrino oscillation in 2001. Why the huge delay?","text":"Solar neutrino deficit was first observed in the late 1960's. And theory of neutrino oscillation was developed in 1967. But,in 2001, the first convincing evidence of solar neutrino oscillation came in SNO. Why did it take nearly 35 years to verify the neutrino oscillation? reference: http:\/\/en.wikipedia.org\/wiki\/Neutrino_oscillation"}
{"id":"95610","title":"Could sphaleron-induced proton decay also cause vacuum decay?","text":"I will say right away that I don't mean standard-model sphalerons, I mean the sphalerons of some extension of the standard model. The reason to even think about this is last year's paper by Frampton and Hung (discussion), which proposes that the Higgs mass might be 126 GeV \"because\" the timescales for vacuum decay and for instanton-induced proton decay are the same. More precisely: the timescale of vacuum decay is unknown, but if it were the same as the timescale for the spontaneous occurrence of a sphaleron in a proton, that would indicate a Higgs mass near the observed mass. (This is not as remarkable as it may sound: the Higgs mass is right on the edge of making the vacuum genuinely stable, so _any_ extremely long lifetime for the vacuum corresponds to a Higgs mass near the value that we see.) They don't have a coherent causal model. But one way for it to work, would be if proton decay _caused_ vacuum decay. This ought to make for problems with early-universe cosmology, but put that to one side for now, maybe there's a way around that. What I would like to know is whether the idea of proton decay causing vacuum decay even makes sense. What I envisage is a standard model extension with new scalars that appear in the sphaleron action _and_ in an extended Higgs-sector potential. Perhaps the new scalar has a VEV of its own, the proton-destroying sphaleron action is minimized when that VEV changes, but the new VEV will in turn destabilize the existing Higgs VEV and drive it into a new, true minimum. The effect of an extra scalar on electroweak vacuum stability is a current topic of research, and section 4 of this paper shows sneutrinos nominally contributing to the sphaleron action in an extension of the MSSM (though here their contribution is negligible)."}
{"id":"23697","title":"charge moves if you scuff the rag with your shoes","text":"Why is that when you scuff with your shoes on, charges move (since electrometer moves back and forth), but if you don't have your shoes on, the electrometer doesn't move. Here's the corresponding video detailing the event. http:\/\/www.youtube.com\/watch?v=cJSp8v0YJrA It is at about 47 min of video from beginning. I don't get the logic behind it."}
{"id":"23696","title":"Clarification on Wald's book","text":"I have a question concerning the Wald's book: General Relativity. In the appendix E, he derived the Einstein equation by considering the surface term (GHY). I do not understand what he said after the equation (E.1.38). Actually he considers that $h^{bc}\\nabla_c(\\delta g_{ab})=0$, because we fix $\\delta g_{ab}=0$ on the surface, but therefore why the other term in (E.1.38) is not null, the term $h^{bc}\\nabla_a(\\delta g_{bc})$. They look the same for me, and after some algebra, where we replace the covariant derivative by the one compatible with the metric on the surface we should have a total derivative term on the surface that we can integrate away. Thanks in advance"}
{"id":"93624","title":"Spherical harmonics","text":"Given the following potential: $$V(\\theta,\\phi)=\\frac{Q}{a}\\left(\\sin\\theta \\cos\\phi+\\frac{1}{2}\\cos^2\\theta\\right)$$ on the surface of a sphere of radius $a$ I am trying to solve Laplace's Equation outside the sphere (where there aren't any charges). I know the general solution to Laplace's Equation outside the sphere is given by: $$\\phi(r,\\theta,\\phi)=\\sum_{l=0}B_l r^{-l-1}P_l(\\cos\\theta).$$ I am not quite sure how to proceed as am very new to spherical harmonics. Does the next step involve expressing the given potential as a Legendre polynomial? I'd appreciate some guidance."}
{"id":"79604","title":"Damping Coefficient for a Hydraulic-Controlled Door","text":"I am not very good with physics terms, so please treat me as an ignorant. I am trying to calculate a damping coefficient dynamically for a hydraulic- controlled door that opens and closes due to hydraulic pressure (opening\/closing). The formula where I need my damping coefficient is: viscous_fric_mom = $C \\times \\omega \\times 2\/\\pi$; The $\\pi\/2$ division is because the maximum angle that the door can be opened at is $\\pi\/2$ (i.e. 90°), $\\omega$ is the angular velocity of the door which is in $\\rm rad\/s$. And $C$ is the damping coefficient, which I need to calculate dynamically. The system specifies that my damping coefficient unit is in $\\rm N m s$! I thought it would be $\\rm N s \/m$, because usually it is `Force*time\/distance`. Apparently I am wrong. Could someone suggest what I should consider for calculating this damping coefficient? I am really bad at math and do not know any better of doing it. **UPDATE** I am trying to create a software model of a hydraulic-operated door. The door will open given that the effective hydraulic jack pressure have been applied. Same goes true when the door is closing i.e. an effective hydraulic jack retraction pressure must be applied. I use two part integration (integration of acceleration and velocity) to get the current angular position of the door, i.e. from the locked position. My current operating assumption is that the door will have either $\\pi\/2$ or $-\\pi\/2$ acceleration (i.e. opening or closing). If I integrate that, I get the velocity and double integration will give me the position. If I take the angular position and feed it back to my damping coefficient calculator, that should work, right?"}
{"id":"56626","title":"Does anybody know of any good sources that explain (generically) how we form Lagrangians\/Actions\/Superpotentials for different field content?","text":"I regularly find that I'll understand where the field content in a particular physics paper comes from, but then a Lagrangian or action or superpotential is stated and I don't know how it's derived. Is there a set of general rules for building a Lagrangian\/action\/superpotential if you already know the field content of the theory? Any suggestions of sources that explain how to do so would be very welcome as I'm having little trouble finding much that helps."}
{"id":"128403","title":"Is quantum field operator $\\psi$ same as quantum field $\\psi$?","text":"So in QFT, quantum field operator $\\psi$ is there. $\\psi$ seems to take the role of wavefunction in QM, which now acts upon vacuum state. Then, in lagrangian of various quantum field theories, $\\psi$ appears, but now it is called quantum field. So is quantum field operator no different from quantum field here? If it is not different, then how can $\\psi$ can have scalar $|\\psi|$ in any assumption as it only operates upon vacuum state? Assuming that vacuum state is represented by some $n \\times 1$ vector (let us forget about infinite-dimension for now), operator should be of the form $n \\times n$."}
{"id":"23962","title":"Partially filled orbitals and strongly correlated electrons","text":"Interesting behavior of strong correlation between electrons occur in metals with partially filled d or f orbitals (transition metals). Why these strong correlations do not appear with elements with incomplete p or s orbital for example ?"}
{"id":"61299","title":"Proof for angar speed equation","text":"I'm struggling to find a solution to a project for college. I have looked through my textbook countless times and googled for days, but I just can't seem to figure it out. The question is that we need to prove the angular speed of a hanging rotating weight is given by the expression: W = root(g\/Lcos.theta) Please can you help me or push me in the right direction. Any help would be appreciated. Thanks in advance"}
{"id":"65979","title":"Calabi-Yau manifolds and compactification of extra dimensions in M-theory","text":"I just finished learning M(atrix) theory and the basics of the compactification of extra dimensions. The extra 6 dimensions of superstring theory can be compactified on 3 Calabi- Yau manifolds (because 6 real dimensions means 3 complex dimensions). However, when it comes to M-theory, one cannot compactify on 3.5 Calabi-Yau manifolds, so after compactifying 6 dimensions, where does the extra 1 dimension go? Is it just compactified on a circle, or something like that?"}
{"id":"27272","title":"Stabilizer formalism for symmetric spin-states?","text":"This question developed out of conversation between myself and Joe Fitzsimons. Is there a succinct stabilizer representation for symmetric states, on systems of _n_ spin-1\/2 or (more generally) _n_ higher spin particles? By a \"stabilizer representation\", I mean that:   * every symmetric state (or some notable, non-trivial family of them which contains more than just product states) is represented as the unique +1-eigenstate of some operator or the unique joint +1-eigenstate of a list of operators, where   * each element of this set of stabilizing operators can be succinctly described, as an operator on the larger Hilbert space ( _i.e._  not only as a transformation restricted to the symmetric subspace itself), and    * where the stabilizing operators transform in a nice way in the Heisenberg picture under symmetric local unitaries ( _i.e._  unitary transformations of the form _U ⊗n_). Ideally, one would be able to efficiently describe all sorts of transformations between various symmetric states; but one cannot have everything. The constraint of being a unique +1-eigenstate of the list of stabilizing operators could also be made subject to the constraint of being a symmetric state. (For instance, many states on _n_ spin-1\/2 particles are stabilized by a σz operator on a single spin, but exactly one symmetric state is stabilized by that operator. Not that I would expect such an operator necessarily to arise in the formalism...) Does a representation with the above properties (or one close to it) exist?"}
{"id":"27277","title":"Introduction to neutron star physics","text":"I enjoy thinking about theoretical astrophysics because I want to understand black holes. Given that no one understands black holes, I like to ponder the nearest thing to a black hole: a neutron star! I have searched around the web for pedagogical discussions of the structure of neutron stars such as this link from NASA: http:\/\/heasarc.nasa.gov\/docs\/objects\/binaries\/neutron_star_structure.html, but none seems to be at an advanced enough level for my liking. The problem is that I do not know what literature I should read in order to learn more. What is the current state of neutron star research? What are some good review articles? More specifically, I am curious about theoretical predictions for \"starquakes\" referenced in the link above, and how they would look to an observer on Earth. I would also be interested in understanding what happens to gas falling into a neutron star -- specifically, if Sol was spiralling to its death by neutron star."}
{"id":"25527","title":"How to get started in Astronomy (UK based)","text":"I have always been interested in space and astronomy (in my youth - I wanted to be an astronaut). However for various reasons, I never quite got started. I now want to get started - small but steadily. I live in a city, ergo: light pollution is a problem - however, I would like to get a telescope (maybe a second hand one but a good make), maybe join a local astronomy club ? In an ideal world, I will get a \"good\" telescope which has the following attributes   1. Can be extended to make progressively more powerful   2. Can use one of the opensource astronomy packages (I am a programmer!)   3. Allow me to take photographs  Can anyone provide me with a series of steps to help me finally turn my dream of observing the skies into reality? Note: I am aware that it is very likely that I will have to start with a small, hand held telescope - but I like the idea of a telescope that will \"grow with me\" - if that is at all possible."}
{"id":"2489","title":"Is there a [set of] rules\/patterns that apply to elements","text":"> **Possible Duplicate:**   >  Do current models of particle physics explain the chemical properties of > elements\/compounds? Is there a rule\/pattern, or multiple rules and\/or patterns, which describe the properties of an element given the number of protons and neutrons it has. For example a carbon atom which has 6 protons and 6 neutrons, howcan you work out what it will act like, reactivity, if there was a bar of carbon, how could they be arranged and what would its properties be like strength. But the rules would work for any number of protons and neutrons."}
{"id":"191","title":"What is the mechanism of magnetic core saturation?","text":"Why does a magnetic core saturate? What is its actual mechanism?"}
{"id":"74000","title":"Books for learning Mathematics in Physics?","text":"Currently I'm doing Advanced Classical Mechanics courses. I'm finding it hard to understand due to the lack of knowledge in linear algebra, multi variable calculus and other chapters. Can anyone suggest a mathematical book which is dedicated to teaching all the math that is used in physics?"}
{"id":"117195","title":"Book for multivariable calculus","text":"Hi I want to start learning multi variable calculus specifically for learning electrodynamics. What are some good text books?"}
{"id":"24479","title":"Books on representation theory","text":"> **Possible Duplicate:**   >  Best books for mathematical background? I'm looking for a textbook on the group\/representation theory for a student- physicist. The main questions of interest are representations of SO, SU and the Lorentz\/Poincare group in various dimensions (or maybe some other topics that i've missed but important for particle physics)."}
{"id":"99883","title":"Mathematics for physics resources","text":"Looking for book suggestions. I have a good basic understanding of many mathematical concepts but I want to improve to a physicist level. This is mainly because I want to study atmospheric physics (fluid dynamics, gases, quantum mechanics, optics, etc.)."}
{"id":"15495","title":"Do eye-glasses decrease the Depth of focus?","text":"I was reading up about Depth of Focus and wondered if glasses affect depth of focus. If yes, is it noticeable to the user?"}
{"id":"98104","title":"Why does a faraday cage protect you from high currents?","text":"In an electrostatic case it is clear that that in a space enclosed with a conductor (without charge in it) the electric field is zero. This is often demonstrated in physics shows like on the following image: ![enter image description here](http:\/\/i.stack.imgur.com\/mKkdX.jpg) However it you have the lighting a current is flowing through the air and through the cage. So wie are not in the electro- **static** situation anymore since we have currents, i.e. moving charges. How can one account in the explanation properly that we have moving charges? Some people say that the fact that the man inside the cage is safe doesn't have to do anything with faradays cage, it's simply because the the cage is a better conductor. Sometimes also the skin-effect is mentioned. So what's true. It would be great do get a detailed and correct explanation of this. Do you have any good references?"}
{"id":"70548","title":"work done by tension","text":"## The problem statement, all variables and given\/known data Consider the following arrangement: ![enter image description here](http:\/\/i.stack.imgur.com\/Jge7p.jpg) Calculate the work done by tension on 2kg block during its motion on circular track from point $A$ to point $B$. ## The attempt at a solution We know that work done by a force is product of force and displacement. We know the displacement of point of application as 4m. How to find the work done by the tension as it is not constant it is variable! 2nd attempt(calculus approach) The 2 kg block moves along the circle, so its speed is Rdθ\/dt. It pulls the string, the length of the string between point O and the block can we obtained with simple geometry at any position θ (ignoring the size of the pulley). The total length of the string is unchanged, so the speed of the 1kg block is dL\/dt. http:\/\/imgur.com\/PbpBpOB Can u help after this"}
{"id":"57570","title":"How big is an inertial frame?","text":"How big is an inertial frame? Consider a huge rod which is rotating about a fixed point in a plane, its length is 1 light year. Thus light from its end closer to the fixed point to the end farther from the fixed point takes one year to reach. Now the angular velocity of the point closer to the fixed point is much slower than the angular velocity of the farther point. Thus the end closer to the fixed point has a relative velocity to the end farther away. At some point in time, the clocks at the two ends are synchronized by sending a light signal to both ends from exactly from the centre of the rod (half a light year from any point). My question is, are the two ends of the rod in the same inertial frame? Is it possible for 1 object to be in two inertial frames simultaneously? A simpler question. A rod is 1 light year long and travelling at a a velocity c\/2 along its length. There will be length contraction of the rod by gamma. Now imagine a light source at one end of the rod. The light from one end takes one year to reach the other end (as measured by either end), when the rod is stationary. Now since the distance is contracted when the rod is moving, so must the time be different (as compared to an observer looking at this rod) for the two ends of the rod as light must be measured with the same velocity on both ends. The time must be same at both ends by symmetry, so they are indeed in the same inertial frame. However, if there is length contraction, they will not measure one light year, they should each measure less than one light year. An external observer observing this rod would see the distance as less than one year. But to measure the same velocity of light, must also measure the same time as the ends of the rod, thus implying the observer is in the same frame as the rod? The second part of my question is this. In zero gravity, a person holds a ball in his hand. Thus the person and the ball are in the same inertial frame. Then the person \"throws\" the ball and as a result, both and ball and the person now acquire a relative velocity to each other (action and reaction). Are the two in the same inertial frame? The third part of my question is this. In zero gravity, can a ship and an observer on the ship be in the same inertial frame? That is, do they share acceleration? Assuming the ship is a plane, when the plane accelerates, the person is floating around, hence the person and the ship cannot share acceleration unless the person is strapped to the ship.Thus in zero gravity, every object must have its own inertial frame?"}
{"id":"104226","title":"How does quantum world affect us and why should I care about it?","text":"We live in a world which is much larger than quantum world. The laws of quantum physics are not valid. While I am pressing the keys on my laptop, I have 100% certainty that I am writing what I really want. Then how is such microscopic world going to affect me? The quantum world is entirely different, perhaps best suited for those Aliens. Then why should I worry about uncertainty and other stuffs happening at infinitesimally small scale?"}
{"id":"75968","title":"A simple question on $SU(2)$ gauge transformations in Wen's papers on projective symmetry group (PSG)?","text":"Recently I am studying the projective symmetry group (PSG) and the associated concept of quantum order first proposed by prof.Wen. In Wen's paper, see the last line of Eq.(8), the local SU(2) gauge transformation for spinor operators is defined as $\\psi_i\\rightarrow G_i\\psi_i$, where $\\psi_i=(\\psi_{1i},\\psi_{2i})^T$ are fermionic operators and $G_i\\in SU(2)$. Why we define it like this? Since as we know, the Shcwinger fermion representation for spin-1\/2 can be written as $\\mathbf{S}_i=\\frac{1}{4}tr(\\Psi_i^\\dagger\\mathbf{\\sigma}\\Psi_i)$, where $\\Psi_i=\\begin{pmatrix} \\psi_{1i} & -\\psi_{2i}^\\dagger \\\\\\ \\psi_{2i} & \\psi_{1i}^\\dagger \\end{pmatrix}$, and $G_i\\Psi_i$ which is the same as the above transformation $\\psi_i\\rightarrow G_i\\psi_i$ is in fact a spin rotation of $\\mathbf{S}_i$, while $\\Psi_iG_i$ does _not_ change spin $\\mathbf{S}_i$ at all. So in Eq.(8), why we define the SU(2) gauge transformation as $G_i\\Psi_i$ rather than $\\Psi_iG_i$?"}
{"id":"111704","title":"Literature on the time reversal operator","text":"Time reversal symmetry seems to be a very useful concept and is mentioned in a good number of papers I recently came across. Most of the time people claim that a certain system or Hamiltonian is time reversal invariant and deduce certain properties without going further into details. I have a hard time grasping the entire time reversal symmetry thing and would like to read more on the subject in general. So I was wondering whether anybody had good literature references? Preferably with a lot of applications and examples. I know the mathematical definition, but its applications usually elude me."}
{"id":"134468","title":"Determining the decay constant for a damped wave","text":"My question is short and simple. If a damped, travelling wave (say on a string) could be described as $y\\left( x,t\\right) =Ae^{-\\gamma x}\\sin \\left( kx-\\omega t\\right)$ how could\/would one determine the decay\/dampening constant $\\gamma$ mathematically?"}
{"id":"41333","title":"Same momentum, different mass","text":"The question is: if   * A bowling ball and ping pong ball    * are moving at **same momentum**   * and you exert **same force** to stop each one   * which will take a longer time? or some?    * which will have a longer stopping distance?  * * * So I think I can think of this as: $$F = \\frac{dp}{dt} = m \\cdot \\frac{v_i - 0}{\\Delta t} = \\frac{p_i}{\\Delta t}$$ Since both have same momentum, given same force and momentum, time will be the same? Is this right? * * * Then how do I do the stopping distance one?"}
{"id":"134461","title":"Applications of the Linearized Einstein Field Equations (EFE)","text":"Look up _linearized Einstein field equations_ anywhere and the first thing you'll see will be a discussion of gravitational waves. Using the linearized EFE's is pretty handy when studying gravitational waves, but it doesn't seem like they are used anywhere else! Is this true? If not, what are the other applications?"}
{"id":"86445","title":"Expanding Universe Balloon Analogy - Anything Similar for Time?","text":"It is difficult to imagine the infiniteness of space and how it itself is expanding rather than the universe expanding into something else. A helpful analogy is that of drawing little dots (representing galaxies or some other sub-universal structure) onto a deflated balloon and then blowing it up. The surface expands in all directions, with each dot moving away from every other dot. Although the analogous surface (the outside of the balloon) is effectively 2 dimensional, it's possible to imagine its translation into 3 dimensions. As for time, though, I have a hard time picturing its \"before \/ during \/ after\" states, and I realize those words aren't even accurate. Time supposedly began at the Big Bang and may end at the Big Crunch. But I'm wondering if anyone knows of an analogy for time, similar to the balloon analogy that applies to space. Is there a way to imagine time in some comprehensible way?"}
{"id":"133112","title":"Who foots the (magnetic) energy bill?","text":"Gravitational attraction and electrostatic attraction\/repulsion are intrinsic properties of matter, any particle (electron, proton) for some unknown reason can produce KE at a distance. But magnetic attraction\/force is not an intrinsic property of matter, a charged particle generates a magnetic field\/flux and a magnetic force only when it is moving: higher velocity = much higher force. The definition of KE says that it is _'work done to accelerate an object'_ , energy spent only to make it move, and does not mention the generation of other forces\/energy or doing 'extramural' work. _[Electrons moving in a current produce magnetic induction that makes an electric appliance do work, who\/what is spending the necessary energy to produce such work? Voltage (the difference of potential) coming from the mains provides the energy to accelerate the electrons, not the energy to blend your fruit. Same voltage would produce equivalent v\/Ke if the electron is accelerated in a vacuum (such as in a synchrotron, where there is no fruit). But this is difficult to prove.]_ **Edit** : I will not reply to any comment. They are attributing to me statements I never made. Now, to avoid technical complications ( _voltage_ , wattage etc,) that could trick me, forget the previous example, **let's consider another, simpler case:** An electron is travelling at high speed (say,0.9 C). If is moving near another electron, (proton, positron or a live wire) _it can make anything move, acquire KE_ , it **can do work**. When v approaches c, the attractive magnetic force gets so great that it _equals the huge electrostatic repulsion._ All the issues you have raised (intrinsic spin, magnetic field, electromagnetism etc., some comments have been deleted) are irrelevant here. If those properties exist they exist even when the electron is at rest. When the electron is at rest, it still has spin, nevertheless it is **not able any more** to do work. If this is an irrefutable fact, then, how come it can do work when it is moving, where is the necessary energy coming from?"}
{"id":"3081","title":"Why can't a piece of paper (of non-zero thickness) be folded more than $N$ times?","text":"Updated: > _In order to fold anything in half, it must be $\\pi$ times longer than its > thickness, and that depending on how something is folded, the amount its > length decreases with each fold differs._   >  – Britney Gallivan, the person who determined that the maximum number of > times a paper or other finite thickness materials can be folded = 12. Mathematics of paper folding explains the mathematical aspect of this.   I would like to know the physical explanation of this. Why is it not possible to fold a paper more than $N$ (=12) times?"}
{"id":"15818","title":"Why does the speed of light within a solid depend on frequency?","text":"Different frequencies of light travel at different speeds through solids, which along with Snell's law allows for rainbows. Has this phenomenon of variable speeds been predicted through derivations? What does it tell us about the interactions that occur when light travels through a solid?"}
{"id":"70896","title":"How \"things\" radiate electromagnetic radiation?","text":"How things radiate electromagnetic radiation? I don't ask why they radiate (higher temperature than 0K) but how they radiate this electromagnetic waves?"}
{"id":"93762","title":"Double-slit experiment formulated in a quantum circuit","text":"What would the double-slit experiment or something analogous to it look like implemented as a quantum circuit\/program? Also, what about the delayed choice quantum eraser, how would that look in a quantum circuit? For example, if implemented in this quantum computing simulator: http:\/\/www.davyw.com\/quantum\/?example=Toffoli . I couldn't find any direct answer via Google, so even good search terms would be useful."}
{"id":"103165","title":"Space time curvature due to electric charge or magnetic charges","text":"since we know that gravitational force is nothing but a curvature in space- time. I have a similar analogous for the electric or magnetic charges. Similarity is that both electromagnetic and gravitational forces follow the inverse square law. consequence of this can be that we can have a another interpretation of electron orbital motion causes by curvature in space due to electric charge (as mass of electron is very small so STR have no significance here). But problem is that we have two different type of charges (+ve and -ve) which gives repulsion and attraction, whereas in gravitation force mass always gives the attraction forces. Please correct me if I am wrong."}
{"id":"38035","title":"ratio between work and heat","text":"I am really stuck on a problem in my textbook: Water is heated in an open pan where the air pressure is one atmosphere. The water remains a liquid, which expands by a small amount as it is heated. Determine the ratio of the work done by the water to the heat absorbed by the water. MY ATTEMPT: We are given that: $P = 1.013 \\cdot 10^5 Pa$ We then have: $$\\frac{W}{Q} = \\frac{P \\Delta V}{cm \\Delta T} = \\frac{P \\beta V_0 \\Delta T}{cm \\Delta T} = \\frac{P \\beta m \\Delta T}{ cm \\rho \\Delta T} = \\frac{P \\beta}{c \\rho} = \\frac{1.013 \\cdot 10^5 \\cdot 207 \\cdot 10^{-6}}{4186 \\cdot 1} = 5 \\cdot 10^{-3}$$ But according to the textbook, the solution should be $4.99 \\cdot 10^{-6}$. If anyone can help me by pointing out what I'm doing wrong here, I would be extremely grateful!"}
{"id":"38034","title":"Explanation for self-rupture glass is needed","text":"I witnessed a phenomenon that I couldn't conclude its cause. Please bear with me for the length of the recall, for I merely want to include any details that might help us to investigate. I had a **cooking glass lid** sat on a wooden shelf that is **away from** the stove and oven and other heating objects. The shelf is nailed on the wall and is situated just above my eye level, and a counter top is also on the same side of the wall where the shelf is installed. Now here comes the surprise. In a winter afternoon 2011, my room had almost the same temperature as an autumn morning, and while I was cutting my lettuce on that counter top which I pointed out in above passage, **a pounding sound** , as if a heavy car door slam or a tree trump falling on top of the roof, knocked its introduction from the shelf that was just above my eye level. First, I thought I may had knocked something around me off(which I didn't believe that for there wasn't anything around me to knock off); then I thought it may be my neighbor next door dropping a heavy box; last, I suspected somewhere my roof top collapsed. But it was my third suspicion directed me to meet that glass lid I mentioned above, and **I found it had ruptured completely like glacier creaked BUT still having all broken pieces bounded without any pieces scattering toward random direction! Only the nob of the lid popped out partially.** Before this happened, I hadn't used that lid for cooking for years, and I didn't removed it from any heating object nor there was something on top of the lid that day, and I believe what the lid had maybe just an invisible layer of dust. I was glad my face hadn't been stung by any glass residues, but ponder what really happen to that glass lid and why it ruptured without collapsed. Below, I attached 2 pictures of the scene from that day. If you have any similar experience or know the theory behind it, may you please drop me an explanation to this incidence? Thank you in advance. I'm new and need reputation to post pictures. I definitely post that 2 pictures once I earn enough reputation point."}
{"id":"103016","title":"Need to use lasers,mirrors and prisms for a project.How to make the light visible and presentable at the same time?","text":"Alright so I really want to put some art into science for my project.project. I'm thinking of creating a beautiful image,preferably in air.I read about the visible lime-green lasers but i still want to be able to use other colors,and I need to make it visible.How can i make \"fog\" for it so that a pretty image is seen yet at the same time the inspectors won't be suffocating.Any ideas on how should I make it and what other things I can use to create pretty images?Please explain in details as I might not know some terms in optics.Thanks in advance!"}
{"id":"87364","title":"phase transformation of liquid under motion","text":"I was testing in my lab with water and found that it starts to solidify when it is stationery at 0 degree centigrade.but when I move the liquid with some velocity it dosent change its phase.my interpretation of this happening because when I move liquid it has kinetic energy which dosent allow it to crystallize.but at certain velocity and further lowered temperature it should start crystallising again.is there any graph which shows at what velocity and temperature it crystallize with atmospheric pressure?as in phase transition of water under temperature and pressure?"}
{"id":"87471","title":"How are atomic bonds created?","text":"From what I have learned in my chemistry course, Electrons with similar quantum numbers but with opposite spin are attracted to each other. What does this mean when there is a covalent bond being formed between lets say hydrogen and fluorine? I can think of four different results: A) A bond is not formed until an unpaired fluorine electron meets an unpaired hydrogen electron with the same energy shift (in this case, 0). B) The fluorine or hydrogen electron forces the other electron to have the same energy shift. C) The electrons don't change, but pair up regardless of quantum numbers. D) The electrons from both atoms enter into a different state that I have not learned yet. Thanks in advance for any help received!"}
{"id":"87472","title":"Gravastars: Are they observationally distinguishable from Black-Holes?","text":"Are observations of Hawking radiation at the acoustic event horizon in Bose- Einstein condensates consistent with Gravastars? To reconcile the second law of thermodynamics with the existence of a black hole event horizon, black-holes are necessarily said to contain high entropy while Gravastars not at all. An Event-Horizon forming out of a collapsing star's intense gravity sufficient enough to force the matter to phase change transforming into Bose-Einstein condensate would be such that nearby matter would be re-emitted as another form of energy, and all matter coming into contact with the Event-Horizon itself would become incorporated. So, it seems reasonable to wonder if Black-Holes are distinguishable from Gravastars since Gravastars appear to be better emitters, and Black-holes better entropy sinks. What do observations of Hawking radiation from acoustic black holes from Bose-Einstein condensate seem to suggest?"}
{"id":"65457","title":"Hilbert space of a free particle: Countable or Uncountable?","text":"This is obviously a follow on question to the Phys.SE post Hilbert space of harmonic oscillator: Countable vs uncountable? So I thought that the Hilbert space of a bound electron is countable, but the Hilbert space of a free electron is uncountable. But the arguments about smoothness and delta functions in the answers to the previous question convince me otherwise. Why is the Hilbert space of a free particle not also countable?"}
{"id":"98723","title":"Ambiguity in number of basis vectors","text":"The dimension of the Hilbert space is determined by the number of independent basis vectors. There is a infinite discrete energy eigenbasis $\\\\{|n\\rangle\\\\}$ in the problem of particle in a box which can be used to expand a general state $|\\psi\\rangle$ as: $$|\\psi\\rangle=\\sum\\limits_{n=0}^{\\infty} C_n |n\\rangle$$ Here, the basis implies that the Hilbert space has **countably infinite** number of basis vectors. But one could well expand $|\\psi\\rangle$ in any other basis as well, say the continuous position basis. Then that superposition, $$|\\psi\\rangle=\\int dx \\psi(x)|x\\rangle$$ is also possible. But this basis $\\\\{|x\\rangle\\\\}$ is continuous and **uncountably infinite**. Such sets cannot be equal because they have different cardinality. Then my questions is how many independent basis vectors are really there in this Hilbert space?"}
{"id":"22120","title":"Is the universe a quantum computer - is light speed barrier a computational constraint","text":"There is currently a debate ongoing on leading maths blog Gödel’s Lost Letter, between Gil Kalai and Aram Harrow, with the former arguing that building a quantum computer may not be possible due to noise propagation, and the latter arguing to the contrary. I am wondering if there is any argument to show that building a quantum computer is possible, by virtue of showing that quantum computation is evident in the physical world. So the question is: (A) Are there any known examples of physical interactions where macro level state transitions could be determined to only be in correspondence with an underlying quantum computation? I.e. similarly to Shor's algorithm being exponentially faster than any known classical factoring algorithm, are there any examples of known physical processes, for example perturbation stabilization in a very large particle cluster, that could be shown, assuming P<>NP, to only be efficiently solved by a quantum computation. Some, I admit highly speculative, additional questions would then be: (B) Is the speed of light barrier possibly a natural computational limit of our particular universe, so that for the computational complexity class of quantum mechanics, working on an underlying relational network-like spacetime structure, this is the maximum speed that the computational rules can move a particle\/wave representation through a network region of the lowest energy\/complexity (i.e. a vacuum)? (C) Is quantum mechanics an actual necessity for the universe to follow classical physical laws at the macro level? The informal argument being that in many-to-many particle quantum level interactions, only the capability of each particle to compute in parallel an infinite or quantum-quasi-infinite number of paths is what allows the universe to resolve a real-time solution at the macro level. Requesting references to research along these lines, or any arguments to support or contradict these speculations."}
{"id":"31516","title":"How would a semi-classical particle react to the double slit experiment?","text":"As far as I know, there is a smooth transition between quantum and classical regimes, so that even classical particle like a massive object has a wavefunction associated with it. However, the double slit experiment can either show quantum character, where the particle supposedly passes through both slits and interferes with itself, or classical character, where the particle passes through one slit with no interference. What would a semi- classical particle do when faced with this situation? In addition, I have a problem imagining a semi-classical particle in general. What is a good example of one?"}
{"id":"19464","title":"Does an interaction of entangled particles with each-other cause decoherence?","text":"I'll apologize in advance if this is not an appropriate place for my question. My background is not in physics, and my understanding of quantum mechanics is extremely rudimentary at best, so I hope you'll be forgiving of my newbish question. Given a system of entangled particles (eg, 2 or more electrons), possibly in a superposition state: if the particles interact with each-other, what effect does this have on their quantum state? Is their state now determined (but perhaps unknown until observed)?"}
{"id":"19466","title":"Derivation of Brillouin-Wigner theory for coupled subpaces","text":"I recall faintly from my quantum theory lecture that there was a really neat way to derive Brillouin-Wigner perturbation theory for the special case of two coupled subspaces that involved a geometric series in reverse. I know that the beginning was to have the Hilbert space split into subspaces 0 and 1 so that the Hamiltonian reads $$H = H_{00} + H_{01} + H_{01}^\\dagger + H_{11}$$ where the two indices indicate what subspace goes \"in\" and what subspaces comes \"out\" of each of the components. If $P$ is a projector onto subspace $0$ and $Q$ a projector onto subspace $1$, this means, for example, that $$H_{00} = PH_{00}P, H_{01} = PH_{01}Q$$ and so on. In matrix notation $$\\begin{pmatrix} H_{00} & H_{01} \\\\\\ H_{01}^\\dagger & H_{11}\\end{pmatrix} \\begin{pmatrix} \\psi_{0} \\\\\\ \\psi_1\\end{pmatrix} = E \\begin{pmatrix} \\psi_0 \\\\\\ \\psi_1\\end{pmatrix}$$ Now we can formally solve the Schrödinger equation for $\\psi_1$ only: $$|\\psi_1\\rangle = \\frac{1}{E - H_{11}} H_{01}^\\dagger |\\psi_0\\rangle$$ and insert that back into the SG for $\\psi_0$ to obtain $$H_{00} |\\psi_0\\rangle + H_{01} \\frac{1}{E-H_{11}} H_{01}^\\dagger |\\psi_0\\rangle = E|\\psi_0\\rangle$$ Now I know that some cool trick with geometric series and an inverted matrix was going on to immediately write down the expansion of $|\\psi\\rangle$. Usually in the literature one first shows the iterative formula and THEN notes that this is a geometric series, but here it was done in the reverse, somehow an operator of the type $1\/(A-B)$ with $A$ and $B$ matrices, was found and then expanded in the geometric series, but no matter how I massage the equations, I cannot seem to make it work, because there are so many different points where one could substitute one form into the other etc. Any ideas on how this works? I'm also relatively sure that the work was done using operators, not for \"individual\" matrix elements. Personally, I find this more elegant because it makes it clearer what's going on on a more abstract level instead of dealing with the low-level matrix elements. It could also be that at this point already the approximation $E \\approx E_0$ was made where $E_0$ is a typical energy of subspace $0$, assuming that this subspace is near-degenerate and well-separated from subspace 1."}
{"id":"88319","title":"What distance has the soldier travelled?","text":"> A troop $5$ meters long starts marching. A soldier at the end of the file > steps out and starts marching forward at a higher speed. On reaching the > head of the column, he immediately turns around and marches back at the same > speed. As soon as he reaches the end of the file, the troop stops marching, > and it is found that the troop has moved by exactly $5$ meters. What > distance has the soldier travelled? I thought that some info is lacking until my friend showed it in a book. How to get the answer?"}
{"id":"122973","title":"What would be walking speed in low gravity?","text":"In $1g$ the average adult human walks 4-5 km in an hour. How fast would such a human walk in a low gravity environment such as on the Moon $(0.17g)$ or Titan $(0.14g)$? Let's ignore the effects of uneven terrain (regolith or ice\/snow\/sooth); suppose our human walks on hardened pavement."}
{"id":"73973","title":"Do we move at the speed of light relative to light?","text":"My understanding of relativity isn't very sophisticated, but it seems to me that relative to a photon moving at the speed of light, we are moving at the speed of light. Is this the case?"}
{"id":"103132","title":"Explain Heat Transfer","text":"I would like to know what are these formulas used for. There is no intro about it in my book at all, and I am reading Heat Transfer book. ![enter image description here](http:\/\/i.stack.imgur.com\/nHffh.jpg) If needed Q. can be edited."}
{"id":"87036","title":"Comparing effect of electric and magnetic dipoles on their fields","text":"So at the end of one of my prof's lectures he gives us something to think about: > Both electric and magnetic dipoles tend to line up with their respective > fields. > > Materials made out of electric dipoles cause the electric field that turns > the dipoles to be **reduced**. > > Materials made out of magnetic dipoles cause the magnetic field that turns > the dipoles to be **increased**. > > Why are the two types of dipoles different in this regard? Now, this isn't a homework assignment or anything. It is just something to ponder on. I'm curious why this happens."}
{"id":"4047","title":"Energy conservation and quantum measurement","text":"Consider a particle in a potential well. Let’s assume it’s a simple harmonic oscillator potential and the particle is in its ground state with energy E0 = (1\/2) ℏω0. We **measure** its position (measurement-1) with a high degree of accuracy which localises the particle, corresponding to a superposition of momentum (and therefore energy) states. Now we **measure** the particle’s energy (measurement-2) and happen to find that it’s E10 = (21\/2) ℏω0. Where did the extra energy come from? In the textbooks it’s claimed that the extra energy comes from the act of observation but I wonder how that could work. Measurement-1 which probed the position of the particle can’t have delivered to it a precise amount of energy, while measurement-2 might just have been passive. No doubt there is entanglement here between the particle state and the measuring device but where, and which measurement?"}
{"id":"4040","title":"Weak measurement and Hardy's paradox","text":"How the notion of _weak measurement_ resolves Hardy's paradox?"}
{"id":"4048","title":"Why is it hard to extend the Feynman Checkerboard to more than 1+1 dimensions?","text":"The Feynman Checkerboard Wikipedia article states: \"There has been no consensus on an optimal extension of the Chessboard model to a fully four- dimensional space-time.\" Why is it hard to extend it to more than 1+1 dimensions?"}
{"id":"60991","title":"Mysterious spectra?","text":"In my blog post _Why riemannium?_ , I introduced the following idea. The infinite potential well in quantum mechanics, the harmonic oscillator and the Kepler (hygrogen-like) problem have energy spectra, respectively, equal to 1) $$ E\\sim n^2$$ 2) $$ E\\sim n$$ 3) $$ E\\sim \\dfrac{1}{n^2}$$ Do you know quantum systems with general spectra\/eigenvalues given by $$ E(n;s)\\sim n^{-s}$$ and energy splitting $$ \\Delta E(n,m;s)\\sim \\left( \\dfrac{1}{n^s}-\\dfrac{1}{m^s}\\right)$$ for all $s\\neq -2,-1,2$?"}
{"id":"46715","title":"Why does the milk frother on my coffee machine make so much noise?","text":"I have a Sunbeam home espresso machine with a steam wand. The steam roars out straight from the end of the wand. When it's first placed in the cold milk it really screams! Once the milk has a bit of a whirlpool action going it's much quieter, so I guess that the noise is because of fast-moving steam hitting stationary cold milk. Something something fluid dynamics?"}
{"id":"90425","title":"Why Landau Level quantization is observed only in low temperature and strong magnetic field in real experiment?","text":"I know that Quantum Hall Effect and Fractional Quantum Hall Effect origin from Landau Level quantization. In magnetic field, the energy of in-plane(plane perpendicular to magnetic field) degree of motion is quantized, which is $E=(n+1\/2)\\hbar\\omega$, $n$ is integer. In experiment, both QHE and FQHE are observed in low temperature and strong magnetic field, suggesting that landau level quantization is observable under the condition $k_B T<<\\hbar\\omega$, where $k_B$ is Boltzman constant. I am not sure why this condition is important in experiment."}
{"id":"13329","title":"How to prepare a desired quantum state?","text":"Given a quantum state function, we can Fourier expand it in terms of stationary states of the Hamiltonian. So if we want to build that same quantum state approximately all we need to do is to superpose stationary states with proper amplitudes. Assuming that we can prepare such stationary states individually, how is their superposition done experimentally? As a specific example, how to prepare experimentally, an ensemble described by the quantum state that is a superposition of 4 stationary states of the particle in a box (or the hydrogen atom) that corresponds to n = 1, 2, 3, and 4 say ?"}
{"id":"76532","title":"Pulley problem - unable to understand the concept","text":"Given:   * The pulley is moving towards the right.   * All blocks have different masses. (The pulley and the strings are massless.) ![Pulley](http:\/\/i.stack.imgur.com\/unSZF.png) What I don't understand:   * Is the tension the same for both A and B?   * If the tension is the same, then both blocks should have different accelerations, but this is not true?"}
{"id":"101321","title":"What do we mean when we talk about Gibbs Free Energy?","text":"Before I start, I'm aware that this question may be better suited on the Chemistry or Biology site, but it's my belief that physicists are more likely to have a clear understanding on what certain terms mean, so by all means move the question if you feel like it will get a better response elsewhere. Okay. In Chemistry we learn about this thing called the Gibbs Free Energy (which I understand is borrowed from Thermodynamics). It's pretty simple. $\\Delta G < 0$, and the reaction is spontaneous. $\\Delta G > 0$, and the reaction is not spontaneous. Other terms in the equation for Gibbs Free Energy are the total enthalpy change, which I interpret as the amount of energy that the system either takes in or releases, and also the temperature and total change in entropy. Observe these graphs of an ambiguous 'Energy' plotted against the progress of the reaction: http:\/\/upload.wikimedia.org\/wikibooks\/en\/a\/a6\/Gibbs_free_energy.JPG http:\/\/www.citruscollege.edu\/lc\/archive\/biology\/PublishingImages\/c05_10.jpg http:\/\/images.tutorvista.com\/cms\/images\/101\/exothermic-and-endothermic- reaction.png The idea is the same. Some reactions take in 'Energy,' and the curve ends higher than where it began. Some reaction release 'Energy,' and the curve ends lower than it began. All reactions seem to require an 'Activation Energy' which prevents the reaction from occurring spontaneously. Notice how the Y-axis has different names, such as Gibbs Free Energy, PE of molecules, and PE. Is Gibbs Free Energy the same or different from PE? I'm not sure anymore. Also, in one graph, the change in Energy is portrayed as $\\Delta G$, so a decrease implies spontaniety, and increase implies nonspontaniety. Yet both require an activation energy to proceed. One more thing to notice is the change in terms. In a Biology context, the terms are Endergonic and Exergonic. In Chemistry, it is Endothermic and Exothermic. Why different terms for the same idea? I would very greatly appreciate an explanation for this, which has been bugging me for a while."}
{"id":"62375","title":"Wald problem 11.4","text":"Consider a stationary solution with stress-energy $T_{ab}$ in the context of linearized gravity. Choose a global inertial coordinate system for the flat metric $\\eta_{ab}$ so that the \"time direction\" $(\\frac{\\partial }{\\partial t})^{a}$ of this coordinate system agrees with the time-like killing vector field $\\xi^{a}$ to zeroth order. (a) Show that the conservation equation, $\\partial^{a}T_{ab} = 0$, implies $\\int _{\\Sigma}T_{i\\nu} d^{3}x = 0$ where $i = 1,2,3$, $\\nu = 0,1,2,3$, and $\\Sigma$ is a $t = \\text{constant}$ hypersurface (therefore it has unit future-pointing normal $n^{\\mu} = \\delta ^{\\mu}_{t}$). (there is also a part b but it is trivial given the result of part a so I don't think there is any need to list it here) I am very lost as to where to start for this question. Usually for these kinds of problems, you would take the local conservation equation $\\partial^{a}T_{ab} = 0$ and use the divergence theorem in some way but that doesn't seem to be of any use here given the form of $\\int _{\\Sigma}T_{i\\nu} d^{3}x = 0$ (it isn't the surface integral of a vector field over the boundary of something nor is it the volume integral of the divergence of a vector field over something - it's just the integral over $\\Sigma$ of a scalar field $T_{i\\nu}$ for each fixed $i,\\nu$). The only thing I've been able to write down that might be of use is that since the linearized field equations are $\\partial^{\\alpha}\\partial_{\\alpha}\\gamma_{\\mu\\nu} = -16\\pi T_{\\mu\\nu}$, we have that $\\partial^{t}\\partial^{\\alpha}\\partial_{\\alpha}\\gamma_{\\mu\\nu} = \\partial^{\\alpha}\\partial_{\\alpha}\\partial^{t}\\gamma_{\\mu\\nu} = 0 = \\partial^{t}T_{\\mu\\nu}$ where I have used the fact that in this global inertial coordinate system with stationary killing field $\\xi^{a} = (\\frac{\\partial }{\\partial t})^{a}$, the perturbation cannot have any time dependence. This then reduces the conservation equation to $\\partial^{\\mu}T_{\\mu\\nu} = \\partial^{i}T_{i\\nu} = 0$ where again $i=1,2,3$. I really haven't been able to make much progress from here though. I would really appreciate any and all help, thanks."}
{"id":"44629","title":"Elastic Collision And Momentum","text":"The question I am working on is, \"Two blocks are free to slide along the friction-less wooden track shown below. The block of mass $m_1 = 4.98~kg$ is released from the position shown, at height $h = 5.00~m$ above the flat part of the track. Protruding from its front end is the north pole of a strong magnet, which repels the north pole of an identical magnet embedded in the back end of the block of mass $m_2 = 9.40~kg$, initially at rest. The two blocks never touch. Calculate the maximum height to which $m_1$ rises after the elastic collision.\" ![enter image description here](http:\/\/i.stack.imgur.com\/Uz4To.gif) This question comes from webassign. On webassign they have a feature called \"Watch It;\" this feature allows you to see a person solve a problem nearly similar to this one. I feel as though there is an error in what the person says in the video. The person says that the mechanical energy of the block- earth system is conserved, but that wouldn't be true; if we were looking at the block-block system--that is, $m_1$ and $m_2$--the mechanical energy would be conserved. By looking at just the block-earth system, there would be a loss in kinetic energy, because when the two blocks \"collide,\" they apply a force over a distance(work), causing a change in kinetic energy of each block, because the kinetic energy of the moving block is transferred into the other block, which is why the first block doesn't return to its initial height. Is this correct? If not, what am I misunderstanding? As a result of this contention with what the person said in the video, I am not very certain on how to solve this problem. EDIT (attempt to solve): Energy analysis: $m_1:$ $PE_i=mgh=KE_f$, where $h$ is the height from which it drops. $KE_i=0~J$; $PE_f=mgh_0$, where $h_0$ is the height is rises to after the collision $m_2:$ $KE_i=PE_i=PE_f=0~j$; $KE_f=\\frac{1}{2}m_2v^2_{f,2}$ Momentum Analysis: $m_1:$ $\\vec{p}_{i,1}=m_1\\vec{v}_{i,1}$; $\\vec{p}_{f,1}=m_1\\vec{v}_{f,1}$ $m_2:$ $\\vec{p}_{i,2}=m_2\\vec{v}_{i,1}$; $\\vec{p}_{f,2}=m_2\\vec{v}_{f,2}$ When I set up an equation for change in mechanical energy, and an equation for conservation of momentum, I get two equations, with a lot of unknowns. What did I do wrong?"}
{"id":"118886","title":"Deborah Number for harmonic excitation","text":"I think I do not understand well the concept of Deborah number. It is presented in the sources available to me as the ratio between the relaxation time of a fluid and a characteristic time scale of the flow. In other sources the denominator of the ratio is occupied by the \"observation time scale\". Firstly, I struggle in front of the definition of a \"relaxation time\" for a \"real world\" fluid, characterised by a continuous spectrum of relaxation times. But the worst is to come, so let us assume a fluid with a clearly defined relaxation time is at hand. If an harmonic excitation with, say, frequency $\\omega$, is applied to the viscoelastic and the flow observed for a duration of time $t$, how to define Deborah number? The relaxation time is fixed as a material property, but what to use in the denominator? The time scale linked to the frequency applied, or the observational time scale?"}
{"id":"118880","title":"Constructive Interference of Electromagnetic Waves","text":"So I was wondering if Electromagnetic wave has the same property of interference as normal waves. I understand that both the electric and magnetic parts of the wave would have to be in the same position at the same time. To negotiate the fact that only one part of the wave would match up at one time due to the fact that light can't go faster than it's self I had the idea for three waves all intersecting at the same point. All of these waves would be of different wavelengths so that they do not interfering with each other before the main point. If I knew of a good easy way off making a fbd I would but i am rather new to really doing physics Ioutside of a high school classroom. So I was wondering if constructive interference worked on electromagnetic waves and if it does whether or not is decrease's the wavelength (increasing the energy)."}
{"id":"134704","title":"Should I abandon my thought experiment about time?","text":"I'm trying to think about special relativity without \"spoiling\" it by looking up the answer; I hope someone can offer some insight - or at least tell me I'm wrong. Suppose I have an ordinary clock in front of me and I push it back with my hands. The force applied to the clock causes it to retreat away from me and after the push, it will travel away with uniform velocity. Suppose further, I can always see the clock clearly no matter how far away it is. Since the speed of light is constant, the light coming from the clock must travel a longer distance to reach my eye as it moves away. This would make time appear to slow down? If, on the other hand, the clock is moving towards me, the distance the light must travel to reach my eye becomes shorter and shorter, thus time would appear to speed up?"}
{"id":"134700","title":"Can a number of gravitational slingshot stop a planet?","text":"The answers to Where does the extra kinetic energy come from in a gravitational slingshot? state that in a gravitational slingshot the object being accelerated \"steal\" speed from the planet (or moon). Does that mean that an excessive number of g-slingshot could stop a planet?"}
{"id":"59748","title":"Which fraction of light is refracted from a source of light under a lake?","text":"I was trying to solve this problem: \"A punctiform source of light is standing inside a lake, at a height h of the surface. f is the fraction of the total of energy emitted that escapes directly from the lake, ignoring the light being absorbed in the water. Given n, the refractive index of water, determine f.\" I understand that, since the maximum refraction angle is 90°, there is a maximum incident angle. The next image explains the principle: http:\/\/img441.imageshack.us\/img441\/9573\/lake01b.png In it, the yellow incident light rays diverge (since the air is less dense than the water), until, at one point, the refractive angle is 90°. Then, the rays stop refracting. At this point, I applied Snell Law:               n1 * sin i = n2 * sin r     n * sin i = 1 (it's air) * sin 90     sin i = 1\/n      Now let's analyze the following triangle: http:\/\/img690.imageshack.us\/img690\/8421\/lake02.png As you can see, the triangle is formed by: _90 - i_ , _90 - i_ and _a_.               a + 90 - i + 90 - i = 180     a = 2i      The fraction of light that made out of the lake is _a_ over the total circle, that is, 360°. So:               f = 2i\/360 = i\/180     i = arc sin (1\/n)     f = (arc sin (1\/n))\/180      However, the answer I have for this exercise (and it does seem to be right, because it is from a University*) is $f = \\tfrac12 - \\tfrac{1}{2n} \\sqrt{n^2 - 1}$. And I don't know what I did wrong. It is very important for me to solve this exercise, and I hope someone would have a hint of what I am doing wrong. *It is a very old test (1969), and there is no resolution anywhere (just the final answer). * * * Second try, using Solid Angles: At is the total Area of the light sphere of radius h:               At = 4 * pi * rt²     rt = h     At = 4 * pi * h²      Ap is the partial area of the circle of light that gets out of the water:               rp = h \/ (tg(90 - i))          tg (90 - i) = sen (90 - i)\/ cos (90 - i)     sen (90 - i) = sen 90*cos i - sen i*cos90 = cos i     cos (90 - i) = cos 90*cos i + sen 90*seni = sen i     tg (90 - i) = cos(i)\/sen(i) = 1\/tg(i)          rp = h*tg(i)     Ap = pi * rp²      So f must be Ap\/At:               Ap\/At = (pi * rp²) \/ (4 * pi * rt²)     f = h² * tg²(i) \/ (4 * h²)     f = tg²(i)\/4      Still not there."}
{"id":"109574","title":"Coupled Oscillation Simulation","text":"I'm looking for an online coupled oscillation simulation. The best I have got so far is this --- https:\/\/phet.colorado.edu\/sims\/normal-modes\/normal-modes_en.html But I'm looking for something which has more options like changing the mass of the objects, changing the spring constants, cutting off the springs. Please suggest if you have come across better simulations. PS : I apologize if this question does not belong here. I need a coupled oscillation simulator for my work."}
{"id":"134155","title":"Scalar Particles, Flavor Changing Processes and Gauge Symmetries","text":"Let's consider an extended version of the Standard Model (SM) with a new Yukawa operator of the form $$ \\sum_\\ell g_\\ell\\bar{\\ell}\\ell \\phi ,$$ where $\\ell$ is any lepton of the SM and $\\phi$ is a new real spin-0 particle, which is assumed to be a singlet of $SU(2)_L$. This new term breaks the $SU(2)_L$ symmetry, but I'll not try to justify its existence. * * * Now, my question:   * I want to compute the loop correction to the vertex $\\mu e\\phi$, which does not exist in the original theory. One possible contribution for this term is show in the figure below (where I also suppose that neutrinos are massive). Does something guarantee that this loop computation will give a finite result in the framework presented here?  $\\hspace{6.5cm}$![enter image description here](http:\/\/i.stack.imgur.com\/rdP80.png),   * If this is not the case, what conditions must be imposed to the lagrangian in order to have finite contributions? Is it enough to have a Hamiltonian with dimension $d\\leq4$ operators? Or is it essential to have a perfectly defined gauge theory?"}
{"id":"47417","title":"How does rest mass become energy?","text":"I know that there's a difference between relativistic rest mass. Relativistic mass is \"acquired\" when an object is moving at speeds comparable to the speed of light.Rest mass is the inherent mass that something has regardless of the speed its moving at. When fusion happens, a certain percentage of the rest mass of two hydrogen atoms is converted into energy. I understand that $E = m_{rest}c^2$. **How does the rest mass of an object turn into energy?** Does this involve particle physics? Doesn't this suggest that rest mass can be viewed as some sort of potential energy? It seems to me that it's a sort of locked energy that has to undergo a certain process to expel its energy to surrounding particles. Also, **Why does only a small percentage of rest mass turn into energy? Why not all of it. What dictates how much rest mass gets converted into energy?**"}
{"id":"78119","title":"What are good examples to demonstrate Einstein's mass-energy relation","text":"According to Einstein's mass-energy relation mass and energy are interchangeable. Can you provide some examples where:   1. Mass gets converted into energy.   2. Energy gets converted into mass."}
{"id":"67647","title":"How does energy convert to matter?","text":"To my understanding, matter and energy are one and the same. Shifting from $E$ to $M$ in Einstein's famous equation requires only a large negative acceleration. If $M$ really is $E\/c^2$, does that make matter the solid state of energy? I've read a lot about positron-electron collisions at high energies creating larger particles, and there is obvious matter conversion in fusion and fission reactions, but I can't find anything describing the physics of the conversion from energy to matter, rather than the interactions of what is already matter. Specifically, the thing I'm getting hung up on is the reason energy would take on a solid state in the first place. If energy is represented by waves, how does it become particles? If gravity is determined by mass, and mass is nothing more than static energy, does that make gravity a static- electromagnetic force?"}
{"id":"103820","title":"What is the role of the speed of light in mass-energy equivalency?","text":"Where does $c$ squared come into play in the equation $E=mc^2$. Multiplication obviously but how does energy equal mass times the speed of light?"}
{"id":"14800","title":"Why are color values stored as Red, Green, Blue?","text":"I learned in elementary school that you could get `green` by mixing `blue` with `yellow`. However with LEDs, TFTs, etc. you always have RGB (red, green, blue) values? Why is that? From what you learned in elementary `yellow` would be the 'natural' choice instead of `green`."}
{"id":"14808","title":"Uses of Vectors In Real Life","text":"I always wonder how vectors are used in real life.Vectors and decomposition of vectors,dot and cross products are taught in the early stage in every undergraduate physics course and in every university.My question is how and where are vectors used? **Do physicists really use vectors in every day life? If so where?** I'am Looking for Motivation for learning Vectors."}
{"id":"104364","title":"A roadmap for learning standard model of particle physics","text":"Assuming that a person has understanding of theory of Lie groups, Lie algebras and basic quantum mechanics, what is the simplest route to gain a basic understanding of the SM of particle physics? Are there any particular books suited for people with this background?"}
{"id":"77622","title":"Book to read before \"introduction to gauge field theory\" by Bailin and Love","text":"A teacher has recommended me to read this book in order to prepare for a project I am doing. Anyway, I feel that I should need a book in order to prepare for this one. Any suggestions?"}
{"id":"20085","title":"How much energy from extreme coldness?","text":"Let's say I have: 1: one mole of extremely cold ideal gas   2: unlimited amount of ideal gas at temperature 300 K   3: one ideal heat engine Can I generate for example 1 MWh of mechanical energy using those three things? Alternative formulation: When temperature of the cold gas approaches 0 K, what does the amount of generated energy approach? (there is no other heat sink than one mole of cold ideal gas)"}
{"id":"48550","title":"Why does a capacitor discharge a percentage of the original energy in the same time?","text":"If I charge a capacitor ($220\\mu{F}$) using a 6V battery, and then measure the time it takes to discharge 90% of the initial energy over a resistor (${100k}\\Omega$), and then charge the same capacitor using a 12V battery and measure the time it takes to discharge 90% of its initial energy again (over the same resistor). Why are both times the same? Especially given that the second time there is 4 times more starting energy that the first time. ($E=\\frac{1}{2}CV^2$.)"}
{"id":"1680","title":"How long does it take a object captured by a star falling to the center?","text":"If the captured object do not have tangential velocity, it's just the free- fall time. But when it has, it may take longer time to fall in, right ? The function should be $\\ddot{r} = -GM\/r^2 + (v_0r_0\/r)^2 \/ r = -GM\/r^2 + v_0^2r_0^2 \/ r^3$ , where v_0 is the initial tangential velocity . After one integration, it becomes $\\dot{r}^2\/2=GM(1\/r-1\/r_0)-v_0^2r_0^2(1\/r^2-1\/r_0^2)\/2 $ . I don't know how to deal with it. But I guess there is a analytic solution. Anyone knows something about it ?"}
{"id":"70766","title":"Momentum paradox","text":"A cistern rail car is standing on infinitely slippery ice. The cistern is filled with water and it has an outlet in the form of a thin vertical pipe (spout) at the left end, so when the valve is open the water can escape vertically downward (in the car frame of reference). Initially the system is at rest, the valve closed. Then we open the valve and the cistern starts moving (presumably to the right). However the water dropping out must be moving (slipping on the ice) in the same direction as the cistern, according to the spout geometry. So we end up with everything moving in the same direction (say, to the right) in spite of initially having zero momentum. How to resolve this paradox? What will be the motion of the cistern in the process of water leaking out? How will the water on the ground and the cistern be moving after all water leaks out? ![enter image description here](http:\/\/i.stack.imgur.com\/lJYsB.jpg)"}
{"id":"111635","title":"How do you integrate an expression over a variable in the limit of an integral?","text":"I am trying to follow the steps to solve the integro-differential equation that arises from a plasma sheath problem given in this paper. This is the step I can't follow: > $$\\epsilon_o\\frac{d}{d\\varphi}\\biggl(\\frac{E^2}{2}\\biggr) = > \\sqrt{\\frac{m_e}{2e}}\\frac{j_{eo}}{\\sqrt{\\varphi}} - > \\sqrt{\\frac{m_i}{2e}}\\frac{j_{eo}}{\\lambda_I}\\int_\\varphi^{\\varphi_w}\\frac{\\varphi'\/\\varphi_I > - 1}{\\sqrt{\\varphi' - \\varphi}}\\frac{d\\varphi'}{E'},\\tag{4}$$ > > where $\\lambda_I =1\/\\sigma_o n_a$ is the ionization mean free path for the > electron energy $E_I$. The integration of Eq. (4) over $\\varphi$ leads to an > integral equation for the electric field $E(\\varphi)$, > > $$\\begin{multline}\\frac{\\epsilon_o}{4j_{eo}}\\sqrt{\\frac{e}{2m_e}}(E_w^2 - > E^2) = \\bigl(\\sqrt{\\varphi_w} - \\sqrt{\\varphi}\\bigr)\\\\\\\\- > \\frac{1}{\\lambda_I}\\sqrt{\\frac{m_i}{m_e}}\\int_\\varphi^{\\varphi_w}\\biggl(\\frac{\\varphi'}{\\varphi_I} > - 1\\biggr)\\sqrt{\\varphi(z') - > \\varphi(z)}\\frac{d\\varphi'}{E(\\varphi')},\\tag{5}\\end{multline}$$ The paper claims that instead of solving the integro-differential equation numerically from the form above, both sides of the equation can be integrated with respect to $\\varphi$. I am not sure if it is valid to integrate both sides of this equation with respect to this variable since it appears in the lower limit of the integral. Can someone explain how you would handle integrating both sides when $\\varphi$ is in the the limit of the integral? Thanks! Or if anyone has an argument for why a mistake might have been made in this step that would be helpful too."}
{"id":"95270","title":"Could each non-dependent physical contant represent dimentions, and our universe be a point on this n-dimentional structure?","text":"For example say the gravitation constant instead of equaling G, was actually a range bounded between 0 and infinity. Our Universe would be at a point on this range (equal to our G value) where things could physically exist and produce stars, galaxies and life. Yet in the gravitation constant's dimension it is able to be lower and higher then our G value. All other physical constants being equal, this would allow the sort of full universe we recognise to exist on only a very tiny part of the full range of the gravitation constant's dimension. The rest of the dimension would be full of non-universes that would not be able to support anything. The other non-dependent physical constants perhaps also represent dimensions. And there might even be a set of useful life supporting universe's on a continuous 'line' inside this n-dimensional structure."}
{"id":"29983","title":"What is the optical power level of common fiber optics lasers?","text":"In the book Nonlinear Optics, it is stated that the nonlinear effects start to become a problem in WDM systems (around 1550 nm) after about 1 mW of optical power. However, I measured the optical power at transmission of a transmitting laser of a 10GBASE-ZR on a short, 2 km link, and found that it was 2.5 mW. What is the operating optical power of telecommunications lasers? Does it differ for inter-continental links compared to short links, i.e. between buildings on a campus? If they are above 1 mW, are nonlinear effects a concern?"}
{"id":"29986","title":"Why do we stop using optics for photons above a certain energy?","text":"I'm reading about how the soon-to-be-launched NuSTAR is on the cutting edge of focusing x-rays, which captures 5 to 80 keV radiation by focusing them with optics that have a 10.15 meter focal length onto 2 sets of 4 32×32 pixel detector arrays. These are particular \"hard\" (high energy) x-rays, which is a part of what makes the task difficult and the NuSTAR telescope novel. If I understand correctly, imaging gets particularly difficult with electromagnetic radiation beyond a certain energy, as true gamma rays (above 100 keV) are detected with a family of radiation detectors that sense the Compton scatter or photoelectric absorption with an electrical pulse that is (in a naive sense) insensitive to the originating direction or location within the detector. It should be obvious that imaging can still be done with the use of an array of detectors, each constituting a single pixel, and these capabilities may improve with time as semiconductor detector technology evolves. So the critical distinction I'm trying to establish is between x-rays and gamma rays. It would seem that we focus x-rays and do not focus gamma rays. For a very good example of researchers _not_ focusing gamma rays, consider Dr. Zhong He's Radiation Measurement Group at UM, who do actual imaging of a gamma ray environment (the UM Polaris detector). They use a grid of room temperature semiconductors laid out bare in a room and use back-processing of the signals to triangulate a sequence of scatter-scatter-absorption reactions in 3D space. This is a lot of work that would be completely unnecessary if you could focus the gamma rays like we do for a large portion of the EM spectrum. Both of the technologies I reference, the NuSTAR telescope and the UM Polaris detector, use CdZnTe detectors. Functionally they are very very different in that the telescope uses optics to capture light from just a few arc-seconds of the sky. My question is what is the specific limitation that prevents us from focusing photons above a certain energy? It seems this cutoff point is also suspiciously close to the cutoff between the definition of x-rays and gamma rays. Was this intended? Could future technology start using optics to resolve low-energy gamma rays?"}
{"id":"126177","title":"Why are the experiments performed by Felix Ehrenhaft on magnetic monopole and magnetic current so much ignored?","text":"_It's really shocking that the following question was voted down twice although the question is yet to be answered by anyone._ So please read the following and give an answer why physics community has taken a double standard in the following case. We know about Michelson's experiment. We also know that despite of the null result how many times the experiment was revisited with the hope to get a positive result. Unfortunately, this is not the case for Ehrenhaft's experiments (http:\/\/www.rexresearch.com\/ehrenhaf\/ehrenhaf.htm). We have already performed a lot of experiments to find a real or artificial magnetic monopole but no monopole is found yet through any experiment. Furthermore, while studying literature on magnetic monopole, I have found that very few researchers even recognized the work of Ehrenhaft who published more than 60 papers on this topic. I don't understand why his work is so much ignored in the mainstream physics. **Is there a large number of references** (I don't expect this number to be comparable to the number of experimental attempts made in case of Michelson's experiment. I only expect a significant number of repetitions which can firmly conclude about the experimental outcome.) **which show that his experiments were revisited thoroughly, carefully and honestly but his claimed outcome could not be reproduced?** Please give your opinion about the following possible answers:   1. Michelson's NULL result was favorable to Einstein's Relativity, while Ehrenhaft's result was not.   2. Ehrenhaft's magnetic monopole was not in agreement with Dirac's theory of magnetic monopole."}
{"id":"94897","title":"Quantum mechanics book that emphasizes momentum space approach?","text":"I have a strong background in Fourier analysis, and I'm looking for QM resources that can build on that. Is there a book around or a little above the level of Griffiths that has that kind of emphasis? Any other resources are good too."}
{"id":"114370","title":"Time travel outside of light cone without causality violation","text":"If one is able to travel into the past but at a spatial distance that puts him outside of his own past light cone would this be considered a causality violating trip? Looking at a Minkoski diagram, it would seem that one ought to be able to travel to a spatially displaced past without producing causality violations. In fact, I'm not sure you could even say whether this was the past of not for sure. ![Generic Minkowski diagram](http:\/\/i.stack.imgur.com\/U34I5.png) For instance: If you found a one way wormhole, that took you outside of your own lightcone but into the past, could you say for sure that you traveled through time? Would this be a form of timetravel that did not violate causality? Note; I'm not necessarily suggesting that it is an FTL trip, it may be considered to be instantaneous teleportation, but in the reverse direction of x'."}
{"id":"113677","title":"Force as change in momentum vs. change in velocity","text":"Is there ever a situation where the distinction between $F = m \\frac{dv}{dt}$ and $F = \\frac{dp}{dt}$ is important? I can't think of a situation where one is true and not the other (assuming only conservation of momentum). * * * Edit: Obviously it is important to take a changing mass into account (e.g. for a rocket) when you're considering a full time evolution, i.e. $F(t) = m(t) \\frac{dv}{dt}$ (or in the relativistic case, perhaps something like $F(t) = m(t) \\frac{d}{dt} \\left( \\frac{p}{m} \\right)$ with $m$ the rest-mass). And perhaps there is a nontrivial relationship between the rate of change of mass, and the forces being exerted (again, e.g. with a rocket --- where the mass loss is tied to the propulsion). What is not clear is that there should ever be a $F = v\\frac{dm}{dt}$ term. * * * Edit 2: **My understanding of the solution:**   There should _not_ be a $dm\/dt$ term, as pointed out by @garyp. The change in momentum expression is, however, more accurate because $p \\neq mv$ in general (e.g. in relativistic cases, or when considering massless systems). It would seem that either one must take the caveat that $dp\/dt$ _cannot be used for mass-varying systems_ , or take the much less conceptual or aesthetically pleasing expression that $F = m \\frac{d (\\gamma v)}{dt}$ ( _which still only applies to classical systems_ )."}
{"id":"127227","title":"Could anti matter collisions be or make dark matter?","text":"I've recently seen that space is or could be a quantum vacuum full of particles like matter and anti matter appearing and possibly colliding causing in theory the same effects that dark energy has. My question is could Dark energy or dark matter be a left over waste product of matter\/anti matter collisions?"}
{"id":"110272","title":"$M^{+}_4$ Randall-Sundrum Brane Calculation","text":"The basic Randall-Sundrum model is given by the metric, $$\\mathrm{d}s^2 = e^{-2|\\sigma|}\\left[ \\mathrm{d}t^2 -\\mathrm{d}x^2-\\mathrm{d}y^2 - \\mathrm{d}z^2 \\right]-\\mathrm{d}\\sigma^2$$ where $\\sigma$ denotes the additional fifth dimension. Notice the brane is localized at $\\sigma=0$; this 'slice' is precisely Minkowski spacetime. To compute the stress-energy tensor, I define a vielbien, $$\\omega^\\mu = e^{-|\\sigma|}\\mathrm{d}x^\\mu \\qquad \\omega^\\sigma = \\mathrm{d}\\sigma$$ where $\\mu=0,..,3.$ Taking exterior derivatives and expressing in the orthonormal basis yields, $$\\mathrm{d}\\omega^\\mu = \\epsilon(\\sigma) \\, \\omega^\\mu \\wedge \\omega^\\sigma$$ where we have defined, $$\\epsilon(\\sigma)=\\theta(\\sigma)-\\theta(-\\sigma)$$ which arises because of the absolute value function in the exponent, and $\\theta(\\sigma)$ is the Heaviside step function. By Cartan's first equation, the non-vanishing spin connections $\\gamma^a_b$ are, $$\\gamma^{\\mu}_{\\sigma}= \\epsilon(\\sigma)e^{-|\\sigma|} \\mathrm{d}\\omega^\\mu$$ Taking exterior derivatives once again, and expressing in terms of the basis yields, $$\\mathrm{d}\\gamma^\\mu_\\sigma = \\left[ \\epsilon^2 (\\sigma)-2\\delta(\\sigma)\\right]\\, \\omega^\\mu \\wedge \\omega^\\sigma$$ which arises by applying the product rule, and noting that, $$\\frac{\\mathrm{d}\\epsilon(\\sigma)}{\\mathrm{d}\\sigma} = 2 \\delta(\\sigma)$$ because the delta function is the first derivative of the step function. From Cartan's second equation, $$R^a_b=\\mathrm{d}\\gamma^a_b + \\gamma^a_c \\wedge \\gamma^c_b$$ the components of the Ricci tensor are, $$R^\\mu_\\sigma = \\left[ \\epsilon^2 (\\sigma)-2\\delta(\\sigma)\\right]\\, \\omega^\\mu \\wedge \\omega^\\sigma$$ as the second term vanishes. By the relation, $$R^a_b = \\frac{1}{2}R^a_{bcd} \\omega^c \\wedge \\omega^d$$ we may deduce the Riemann tensor components, $$R^\\mu_{\\sigma \\mu \\sigma} = 2\\epsilon^2 (\\sigma)-4\\delta(\\sigma)$$ I believe, in this case, both tensors in the coordinate basis and orthonormal basis are identical. Therefore we obtain the rank $(0,2)$ Ricci tensor, $$R_{\\sigma \\sigma}=8\\epsilon^2 (\\sigma)-16\\delta(\\sigma)$$ As the only diagonal component, the Ricci scalar is identical to the Ricci tensor at $(\\sigma,\\sigma)$. Using the Einstein field equations, the stress- energy tensor is given by, $$T_{55} = \\frac{1}{8\\pi G_5}\\left[ 4\\epsilon^2 (\\sigma)-8\\delta(\\sigma) + \\Lambda\\right]$$ where $\\Lambda$ is the cosmological constant, and $G_5$ is the five- dimensional gravitational constant. The function $\\epsilon^2(\\sigma)$ is given by, $$\\epsilon^2(\\sigma)=\\theta^2(\\sigma)+\\theta^2(-\\sigma)-2\\theta(\\sigma)\\theta(-\\sigma)$$ The last term appears to be the delta function, as it is zero everywhere, but singular at zero. The first terms are unity everywhere, but undefined at zero, therefore, $$T_{55}= \\frac{1}{8\\pi G_5}\\left[ 4\\theta^2(\\sigma)+4\\theta^2(-\\sigma) -16\\delta(\\sigma) + \\Lambda\\right]$$ However, this disagrees with Mannheim's _Brane-Localized Gravity_ which states, $$T_{ab}=-\\lambda \\delta^\\mu_a \\delta^\\nu_b \\eta_{\\mu\\nu}\\delta(\\sigma)$$ where $\\lambda = 12\/\\kappa^2_5$. In his text, $T_{\\mu\\nu}\\propto -\\eta_{\\mu\\nu}$, but in my calculation the entire purely 4D stress-energy vanishes. I can only assume I've done something wrong."}
{"id":"25324","title":"How can I stabilize an unstable telescope?","text":"I have an 80 mm refractor telescope on a tripod, but it shakes on every touch. It's very hard to see via 6 mm (x120) ocular. Even a little wind causes the image to become too unsteady. How can I make my tripod more steady?"}
{"id":"25328","title":"Universe is expanding at enormous speed","text":"I'm not an expert but I've come to understand that the universe is expanding at enormous speed. That means that all of the visible galaxies are moving away from us at great speed. I also came to understand that, eventually (in many many years), all the galaxies and all of the rest of the objects outside our own galaxy will move so far away from us that it will be impossible for us to look (or measure) them. This means that eventually our entire observable universe will be our own galaxy and it will be impossible for us to measure anything else, and scientific measurements at that point in time will not correspond to the actual reality... because data will show that other galaxies don't exist. But I have two questions regarding this phenomena:   1. How is it that the light won't reach us anymore? Is the expansion happening at greater speed than light itself, making it impossible for it to ever reach anything?   2. If scientific measurements, at that point in time, will prove to be wrong, because they will show that galaxies don't exist (while they actually do exist), doesn't that mean that something similar could be happening right now as well? We could be measuring something about the universe that we're _dead_ sure about, but it won't be the actual reality."}
{"id":"110279","title":"What are the anomalies with General Relativity?","text":"If general relativity is the newest model of Gravity which is so far been proven. Does it still have any anomalies such as the problem of Mercury's orbit during Newtonian gravity period? If so are there other types of physics to be discovered?"}
{"id":"82320","title":"Food kept in a container in my pressure cooker cooks slower","text":"The pressure cooker is heated at the base to cook the food. The pressure cooker releases steam when the force exerted by pressure exceeds the counterweight, at which point the weight rises up allowing steam to escape. The time to cook is measured by the number of times steam is released. When food(dry lentils) is kept in a container inside the pressure cooker it does not cook as well as if it were dumped in there directly. The same amount of water is added to the food in both cases, though in the former case there is also some water in the base of the cooker outside the container. In both cases the cooker was allowed to release steam 3 times. I thought that based on Pascal's law the same pressure would be exerted on the water in the container as it would if kept there directly. Since the temperature is the same, in both cases the food should be completely cooked. However practical results differ. Please suggest a theory which explains this."}
{"id":"14056","title":"How does gravitational lensing account for Einstein's Cross?","text":"Einstein's Cross has been attributed to gravitational lensing. However, most examples of gravitational lensing are crescents known as Einstein's rings. I can easily understand the rings and crescents, but I struggle to comprehend the explanation that gravitational lensing accounts for Einstein's cross. I found this explanation, but it was not satisfactory. ![enter image description here](http:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/c8\/Einstein_cross.jpg\/300px- Einstein_cross.jpg) Image from http:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/c\/c8\/Einstein_cross.jpg\/300px- Einstein_cross.jpg Source of the second image. ![enter image description here](http:\/\/i.stack.imgur.com\/32yvT.png)"}
{"id":"51844","title":"Was Einstein's Cross Predicted by Einstein's Theories?","text":"> **Possible Duplicate:**   >  How does gravitational lensing account for Einstein’s Cross? Einstein's Cross is a fascinating phenomena for which I have asked explanation here. However, I'm also interested to know if Einstein's cross was a specific prediction that people had obtained from Einstein's theories such as General Relativity or if the phenomena was merely named after Einstein. The wiki page on Einstein's Ring explains how the gravitational lensing phenomena of rings was predicted by general relativity, however I have yet to discover evidence that Einstein explained why there are 4 dots instead of a ring\/crescent."}
{"id":"105143","title":"N=4 SYM in terms of N=1- SO(6) in the yukawa term","text":"I'm trying to write N=4 SYM in terms of N=1 superfields. I have the lagrangian $$\\mathcal{L}=\\frac{1}{16 k} \\int d^2 \\sigma \\text{Tr} \\big[W^a W_a\\big]+c.c+\\int d^4\\theta \\text{Tr}\\big[\\bar{\\Phi}^i e^V \\Phi^i e^{-V}\\big]+\\frac{\\sqrt{2}}{3}\\int d^2\\theta \\text{Tr}\\big[\\phi^i [\\phi^j,\\phi^k]\\big]\\epsilon_{ijk}+c.c $$ Where the $\\Phi^i$ are chiral superfields and V is a vector superfield. In components, this is all fine except for the Yukawa terms $$\\mathcal{L} \\supset i\\sqrt{2} f^{ABC} Z^{i\\dagger}_A \\psi^i_B \\lambda_C - \\sqrt{2}\\epsilon_{ijk} Z^i_A \\psi^j_B \\psi^k_C+c.c $$ Where $A,B,C$, are $SU(N)$ gauge group indices, $i,j,k$ number my 3 chiral superfields, which have an explicit $SU(3)$ symmetry, the $Z^i_A$ are the complex scalars from my chiral superfields, $\\psi^i_A$ are the fermions from my chiral superfields, and the $\\lambda_A$ is the fermion from my vector superfield. The fermions combine into an fundamental $SU(4)$ multiplet $\\chi^I=(\\psi^i, \\lambda)$, and I decompose my complex scalars into real ones in a fundamental $SO(6)$ (isomorphic to $SU(4)$) multiplet, $Z^i=X^a+iX^{a+3}$. I should be able to write the Yukawa terms as $$\\mathcal{L} \\supset f^{ABC} X^a_A C^a_{IJ}\\chi^I \\chi^J +c.c $$ essentially putting the scalars into the antisymmetric matrix representation of $SU(4)$, $X_{IJ}=X_{[IJ]}=X^a_{IJ}X^a$. So I need to show that the $C^a_{IJ}$ that I have are an invariant symbol of $SO(6)=SU(4)$, and thus my Lagrangian has that R symmetry. Not sure how to do that... a reference I found says they should be related to the $SO(6)$ gamma matrices (http:\/\/arxiv.org\/abs\/hep-th\/0201253, below equation 3.1), but that hasn't been very helpful."}
{"id":"70342","title":"Traditional Transfer Matrix on the Potts model -- how it grows for strip lattices?","text":"What is the transfer matrix size for a strip lattice of width $n$ vertices, with arbitrary $q$??   I am not sure if it is $q^n$ x $q^n$ or something else. Any reference is also welcome."}
{"id":"2481","title":"Would you be weightless at the center of the Earth?","text":"If you could travel to the center of the Earth (or any planet), would you be weightless there?"}
{"id":"119217","title":"gravitational force","text":"why does gravitational force of the earth decrease when we move towards the center of the earth whereas it depends on the radius of the earth.The radius of the earth also decreases when we move towards the center"}
{"id":"66383","title":"Does gravity change the closer you are to the Earth's core?","text":"Very simple question that has bugged me for some time, does the acceleration produced by gravitation (e.g. $9.8\\: \\mathrm{m\/s^2}$) would change if we placed an observer closer to the planet's core? I mean, if I were to sit on the center of the Earth, would I still feel the Earth's gravitational field?"}
{"id":"18446","title":"How does gravity work underground?","text":"Would the effect of gravity on me change if I were to dig a very deep hole and stand in it? If so, how would it change? Am I more likely to be pulled downwards, or pulled towards the edges of the hole? If there would be no change, why not?"}
{"id":"99117","title":"Why gravity decreases as we go down into the earth?","text":"We all know that gravity decreases as the distance b\/w the two increases. Hence $$ g = G \\frac{Mm}{r^2} $$ hence acceleration due to gravity is inversely proportional to the distance b\/w the 2 bodies. Then why the gravity decreases as we go deep into the earth?"}
{"id":"119067","title":"Gravitational force of the Earth","text":"Why does gravitational force of the Earth decrease as we move towards the centre of the Earth? Where as inverse square rule says that distance is less than gravitational force is more."}
{"id":"41686","title":"What is the gravity at the center of the Earth?","text":"> **Possible Duplicate:**   >  Would you be weightless at the center of the Earth? Supposing there is a cavity at the center of the Earth, what is the gravity there? What will be its direction and intensity? Will a body be attracted toward the center of mass of the Earth?"}
{"id":"69636","title":"The metric Tensor inside a massive shell","text":"Given a fixed shell with the mass of $M$ and a radius $R$ , what would be the metric tensor for $r<R$? I do know that using Birkhoff Theorem the metric for $r>R$ should be schwarzschild. I'm not sure how to solve $G_{\\mu\\nu}=0$ for the inner part, and I'm not sure if I can demand continuity at $r=R$."}
{"id":"52044","title":"What the effect of gravity in the center of a planet feels like","text":"> **Possible Duplicate:**   >  Would you be weightless at the center of the Earth? The issue of weightlessness at the center of the Earth has obviously already been discussed, however I am curious as to know what will it feel like as a human. Will it be absolute weightlessness as experienced by being in space, or will the mass of the Earth surrounding you pull you apart in all directions in a free floating expansion with your eyeballs popping out, etc."}
{"id":"43626","title":"Is spacetime flat inside a spherical shell?","text":"In a perfectly symmetrical spherical hollow shell, there is a null net gravitational force according to Newton, since in his theory the force is exactly inversely proportional to the square of the distance. What is the result of general theory of relativity? Is the spacetime flat inside (given the fact that orbit of Mercury rotates I don't think so)? How is signal from the cavity redshifted to an observer at infinity?"}
{"id":"69812","title":"Gravity in the center of a hollow neutron star","text":"Imagine a hollow 100 metre diameter (for example) sphere made of incredible dense material (ie neutron star dust etc) but is self supporting (ie the central cavity). Assuming that the sphere skin is reletively thick so that the whole object is exceptionally heavy (eg the mass of planet earth. What would someone who was stationed dead center of the sphere feel? IE would they be crushed by the potential gravity or ripped apart and smeared on the inside other sphere cavity (due to the gravity of the surrounding material?"}
{"id":"81324","title":"Why gravity decreases as we go under ground?","text":"We all know that gravity decreases as we go upward, we also know that gravity decreases as we go inside the earth? I don't know why gravity decreases as we go downward or inside the Earth? Please explain?"}
{"id":"126723","title":"What is the value of $g$ at centre?","text":"What is the value of $g$ at the centre of the Earth? Is it zero or infinity? My attempt: I know it's zero but applying law made me on fused."}
{"id":"132519","title":"Gravitational acceleration at half Earth's radius","text":"What would be the gravitational acceleration at half at Earth's radius? Something tells me it should be proportional to the mass distributed in that part, but I am not sure. Of course, we assume we know what's the acceleration at Earth's surface. Also, for the sake of it let's assume Earth has uniform density and is a perfect sphere."}
{"id":"116840","title":"quarks annihilation process","text":"I am appreciating particle physics and I read about mesons. In quark's model, mesons are pairs of quark-antiquark. Now I think that in general matter- antimatter annihilate and so I don't understand how meson could be possible. Why don't a couple up-antiup annihilates? Thank you!"}
{"id":"57005","title":"Besides transmutation, is there any way to \"speed up\" the decay rate of radioactive material?","text":"I'm well aware of transmutation as a way to effectively make radioactive material decay faster, however that isn't really what I mean. Doing a quick Google search I found references to several theoretical treatises on the subject, and a few references to experiments with, at best, inconclusive results. Among them, there were a few on the quantum anti-zeno effect, which I already knew about, as well as a couple on something called \"field enhanced beta decay\" and various methods utilizing lasers. Nevertheless, much to my frustration, there was essentially no example I could find with any sort of empirical testing of these effects. So, I suppose I have two questions:   1. How \"good\" is the physics behind these claims?   2. Have any of these effects ever been _unequivocally_ observed experimentally?"}
{"id":"5150","title":"Did spacetime start with the Big bang?","text":"Did spacetime start with the Big Bang? I mean, was there any presence of this spacetime we are experiencing now before big bang? And could there be a presence\/existence of any other space-time before the big bang?"}
{"id":"41501","title":"Before the Big Bang","text":"I've heard this saying before I don't know about anyone else. It says, \"What ever was before the big bang is _something physics can't explain..!_ Is this saying true (accurate)?"}
{"id":"2355","title":"Did time exist before the creation of matter in the universe?","text":"Does time stretch all the way back for infinity or was there a point when time appears to start in the universe? I remember reading long ago somewhere that according to one theory time began shortly before the creation of the universe. Does time have a starting point of note?"}
{"id":"28984","title":"Big Bang Anybody?","text":"> **Possible Duplicate:**   >  Did time exist before the creation of matter in the universe?   >  on causality and The Big Bang Theory I was recently watching a Discovery special on the Big Bang theory, and after a couple of drinks and a few detailed discussions with others, I have found that I do not believe the Big Bang ever happened. My reasoning is thus: no matter the precision in digressing the causation of material currently in our universe, a point arrives where the scientists give up (as far as I'm concerned) and say that the Big Bang happened out of nothing. So . . . these scientists say that there was nothing, then the Big Bang happened, and then the scientists go on to explain the glories of the universe in detail without discussing how something came from nothing! Here's my critique: if nothing is nothing, then there is no possibility of something coming from it, because there is nothing. I don't know how we came into existence, other than the possibility that the universe never had a beginning and has always been and is infinite, but that's why I'm asking the Mises community. Any ideas?"}
{"id":"55039","title":"Is this a great flaw in big bang theory?","text":"Einstein said that, Time & Space cannot exists without one another. Big bang says, time didn't exist before the big bang. So the Primordial ball referred in the Big Bang theory didn't had any space inside it ?? If primordial ball had space in it, then where did the time go ?? Is this a flaw in big bang theory ?? Or Am I missing something ??"}
{"id":"12807","title":"Does the Big Bang need a cause?","text":"> **Possible Duplicate:**   >  on causality and The Big Bang Theory Asking here in layman's terms.. When theoretical physicsists discuss the origin of our Universe, the wider consensus appears to be that it originates from a singularity; a position that rests on observations about the apparent expansion of our Universe. However, the question why singularity itself came into being, still remains. But this question takes a different angle: Does the Universe as a whole need a cause to exist at all? If the law of conservation of energy is universally valid, then questions about a beginning become irrelevant since there can't be any by definition and everything boils down to dynamics of interaction. Thoughts?$$\\mbox{ }$$"}
{"id":"7838","title":"on causality and The Big Bang Theory","text":"With the notion of causality, firmly fixed by GR, we derived the concept of a singular point from where space-time begun. Causality alone gives us the possibility to talk about a known past (i.e. every effect has a cause) and by this we can trace back in time every event. To reach the moment of the Big Bang we only have to extrapolate backwards the Hubble Law. (I see this like a reversal of effect with cause in every point of space-time.) With all this said, my question is about the moment of Big Bang; if even time started _there_ , we have an effect without a cause and causality breaks. In this perspective, when did causality started to play a role in the evolution of the universe? Does this path allow for a pre-Big Bang theory? Like a universe, where all its mass has been turned into energy (and in this case, with no mass present, the ideas of small and large are equivalent. And an infinite universe would be equivalent with a point sized one. Hence, the possibility of point like starting place for the Big Bang, but with preexisting _time_ ). From the point of view of causality we reach a logical nonsense at the moment of the BB. How can we modify the BBT to avoid this problem? (Or is has been done already?)"}
{"id":"46014","title":"What's a compact scientific answer to question \"(Why there is) \/ (what is before) the Big Bang?\"","text":"> **Possible Duplicate:**   >  Did spacetime start with the Big bang?   >  on causality and The Big Bang Theory   >  Before the Big Bang What's a compact scientific answer to question \"(Why there is) \/ (what is before) the Big Bang?\" Some fellow physics \/ science students ask question \"what is before the Big Bang?\" or alternatively \"Why there is a Big Bang\" (assuming the time dimension and hence 'before' is ill-defined at the event of Big Bang). We couldn't get any qualitative answer we universally agree on. Is there some compact way to answer this?"}
{"id":"102559","title":"Initial conditions of the origin of the universe","text":"I'm not quite sure this question fits the format of this site but I try to word it the best I can to comply the rules. The question is simple: How far can we go talking about the origin of the universe before admitting that the initial conditions cannot be explained without postulating some kind of god-like, physics-unexplainable, force\/whatever? I'm interested to know if there's a mainstream physics theory that aims to prove this."}
{"id":"61942","title":"How can a circuit function with two negative battery terminals facing each other?","text":"Here is a drawing of the circuit that is confusing me: ![enter image description here](http:\/\/i.stack.imgur.com\/Kag3r.png) I don't quite understand how batteries work in this diagram. If a battery has a negative and positive terminal, there must be a barrier preventing them from neutralizing one another, so how can the potential from either negative terminal ever make it through the top half of the circuit without passing through a battery?"}
{"id":"64536","title":"Why doesn't one-photon-irreducible function have any pole at $q^2=0$?","text":"I'm reading the QFT textbook by Weinberg. In volume one chapter 10 page 451, at the lower part of the page he says, > Now, because $\\Pi^*_{\\mu\\nu}(q)$ receives contributions only from one- > photon-irreducible graphs, it is expected not to have any pole at $q^2=0$. $\\Pi^*_{\\mu\\nu}(q)$ is the sum of all one-photon-irreducible graphs, with the two external photon propagators omitted. Weinberg states it within one sentence as if it's self-explanatory, but I cannot understand why it is true. Is there something simple I missed? **Update:** I think what Weinberg had in mind was Luboš Motl's answer, that why he's so brief. In addition Peskin & Schroeder used the same reasoning in page 245: > ...the only obvious source of such a pole would be a single-massless- > particle intermediate state, which cannot occur in any 1PI diagram However P&S also put a footnote immediately after: > One can prove that there is no such pole, but the proof is nontrivial. > Schwinger has shown that, in two spacetime dimensions, the singularity in > $\\Pi$ due to a pair of massless fermion is a pole rather than a cut; this is > a famous counterexample to our argument. There is no such problem in four > dimensions. Thus my original question stands justified. I'd be grateful if one can give a reference that elaborates P&S's footnote. Of course explanations by any SE user himself\/herself are even more welcomed."}
{"id":"56245","title":"Do mankind and manmade activities\/constructions have any effect on the rotation of the Earth?","text":"We walk or ride on our vehicles to our destinations daily. Does our movement have any effect on the rotation of the earth according to Newton's law? What will be the effect if we move all the peoples along with their vehicles at their maximum velocity in one line in one direction along the equator? How much effect will it make?"}
{"id":"64530","title":"Why can't supersonic planes \"just fly higher\" to go faster while maintaining cost?","text":"First post to this site, and I've got at most a high school background in physics - I really appreciate any answer, but I may not be able to follow you if you're too advanced. I suppose this goes for regular planes too, but I'm especially interested in supersonic planes. I read some reports in the news about various people working on commercial supersonic travel, but there were a lot of comments attached to these news posts listing essentially what were physics constraints that would make such travel severely cost ineffective:   1. \"skin friction,\" causing high heat and stress, leading to different metaled (thus more expensively researched\/manufactured) and heavier (thus less efficient) airplanes.   2. increased drag, requiring more fuel to overcome.   3. sonic booms. I'll leave alone sonic booms - I understand as well as I can why this could be hard to engineer around, and why various countries have made generating them over land illegal. The other two I don't get. After spending some time on wikipedia this evening, if I've got this right, it seems that, holding the shape of the airplane constant, skin friction, lift, and drag are each equal to a scalar times density times velocity squared. Density drops as the altitude raises, which seems to mean to me that you could keep drag, lift, and skin friction constant when increasing speed by merely increasing altitude. I assume this is right, so I guessed that the \"gas mileage\" issue had to do with needing to burn too much more gas to achieve thrust required for the higher velocity. And yet, the wikipedia article on jet engines states that the concorde was actually more fuel efficient than some conventional subsonic turbofan engines used in 747's. Given all this, what did I get wrong? Why can't supersonic planes just fly higher to be as cost effective, or more, than conventional subsonic commercial jetliners, using the same construction materials? Relatedly, why do current jets have service ceilings and max speeds (assuming it's not just about the high stress of breaking through the sound barrier)? Thanks!"}
{"id":"52184","title":"Does light induce an electric current in a conductor?","text":"I know that electromagnetic waves induce electric currents in conductors and that's the basis for radio, wi-fi etc. I also know that light is also an electromagnetic wave. So, can light induce a current in a conductor (like a metal wire? or a coil?). And, if the answer is yes, is the same visible for other high-frequency waves (X-rays, gamma)? I heard about the photoelectric effect, but it seems related to the particle theory of light (photons transferring their energy to electrons). So, do high-frequency electromagnetic waves generate electric currents? Is it possible to measure them? Is the skin effect relevant here?"}
{"id":"53278","title":"3D movie glasses making white light look red and blue","text":"While waiting for a 3D movie to start, I was playing with the glasses they give you. I understand each lens has different polarized filters, so the left and right superimposed images on the screen go to the correct eyes. The first thing that tripped me up was that rotating the glasses didn't affect the light that passes through it. After searching I bit I discovered about _circular_ polarization, which ignores the angle of the filter and seems to be the standard for cinema glasses. The second thing that tripped me up were the wall lamps. When looking through one lens, the light seemed to have a bluish color. From the other, a reddish\/orange color. It was subtle, but other people confirmed seeing it. I figured the lamp's white light had blue and red components, which are of different wavelengths (almost opposite in the visible spectrum, if I remember correctly), but what does the wavelength have to do with circularly polarized filters? And if this line of thought is correct, why does it seems to divide the visible spectrum?"}
{"id":"63544","title":"Consequences of Compactness in Physics","text":"If we understand spacetime as a $4$-dimensional manifold $M$, from the point of view of physics what are the consquences of a subset of it being compact? My point here is simple: in math we usually think of compactness as some analogue of finiteness because it shares many properties with finite sets, but what are the consequences of this when we deal with physics? Of course, we need not to go into relativity, we can even think about the usual three space $\\mathbb{R}^3$. What are again the consequences of a set $A \\subset \\mathbb{R}^3$ being compact? Are there any cool things we can get out from this, or we simply use compactness in physics to grant the mathematical properties desired without having any direct impact in the way we understand and interpret those sets? Thanks very much in advance."}
{"id":"63036","title":"Gravitational time delay and contraction of matter","text":"How can any matter contract to its Schwarzschild radius if gravitational time dilation clearly states that all clocks stop at that point. So any contraction any movement would stop. If that is so why all this talk about objects which can never form in the first place?"}
{"id":"47669","title":"Black hole formation as seen by a distant observer","text":"> **Possible Duplicate:**   >  How can anything ever fall into a black hole as seen from an outside > observer? Is black hole formation observable for a distant observer in finite amount of time? Specifically, let's take uniform non rotating distribution of 10 solar masses within solar radius, assume no outward pressure, and calculate its gravitational collapse. I assume distant observer would never witness complete formation of black hole with event horizon corresponding to entire 10 solar masses. What would be observable in finite amount of time, then? For example, would there be a black hole with smaller radius, while the rest of mass is still falling into its event horizon?"}
{"id":"48086","title":"Would dense matter around a black hole event horizon eventually form a secondary black hole?","text":"> **Possible Duplicate:**   >  Black hole formation as seen by a distant observer Given that matter can never cross the event horizon of a black hole (from an external observer point of view), if a black hole is \"fed\" with a large amount of matter then the new matter will eventually become extremely compressed, and presumably would be compressed below its Schwarzchild radius. Would secondary black holes eventually form near the original black hole? As an alternative one could also imagine that the combined mass of the original black hole and the new mass around the event horizon becomes contained within the Schwarzchild radius of both masses, and so a new event horizon forms, \"swallowing\" the new mass around the edge of the original black hole. This mechanism would allow black holes to swallow mass in a finite time.   Would this contradict GR predictions?"}
{"id":"48074","title":"Can a black hole actually grow, from the point of view of a distant observer?","text":"> **Possible Duplicate:**   >  Black hole formation as seen by a distant observer I've read in several places that from the PoV of a distant observer it will take an infinite amount of time for new matter to fall past the event horizon into a black hole, due to time dilation effects. This seems to imply that once a black hole is formed, its mass cannot actually grow any further as considered by most observers in the universe. Is that right?"}
{"id":"94074","title":"Contour for Klein-Gordon field transition amplitude","text":"In calculating transition amplitude for Klein-Gordon real-scalar field, I encountered the integral, $$ \\frac{-i}{2(2\\pi)^2\\Delta x} \\int^{\\infty}_{-\\infty} \\,dk \\frac{ke^{ik\\Delta x}}{\\sqrt{k^2+m^2}} $$ I can see here the integrand has branch cuts at $ k= \\pm im $ However, later they do a change of variables $ z= -ik $ and then the integral becomes, $$ \\frac{1}{2(2\\pi)^2\\Delta x} \\int^{\\infty}_{m} \\,dz \\frac{ze^{-z\\Delta x}}{\\sqrt{z^2-m^2}} $$ And it is said that they can wrap the contour around the upper branch cut for $ \\Delta x > 0 $ ![enter image description here](http:\/\/i.stack.imgur.com\/9qmWP.jpg) I am not able to see how this transformation happens and how the contour can be wrapped around the upper branch. Thanks for your inputs."}
{"id":"56974","title":"What limits the maximum attainable Fermi Energy for a material experimentally?","text":"Either through doping or gating. What are some good terms to search for if I'm looking for some experimentally obtained values for particular materials? I'm particularly interested in what the limit is for graphene, if anyone knows. For example, the DOS for regular graphene has states between -3t and +3t, but what is the maximum fermi energy we can examine experimentally? Is there a breakdown of the sample at some limit of doping or voltage?"}
{"id":"94071","title":"A confusion in the Derivation of Lorentz Transformation","text":"![http:\/\/thecuriousastronomer.wordpress.com\/](http:\/\/i.stack.imgur.com\/7xqyI.png) My doubt is in the equation (1) and (2). Aren't x,y and z also the radiuses? **EDIT** Thank you guys for trying to give a wonderful explanation but I figured out the answer myself and it was just my silly interpretation. I thought the the value of ct on x axis would be equal to ct and forgot that the value is given by a perpendicular and not an arc (of the sphere). I was thinking that where the sphere touches x axis, that is the value of ct but it isn't. The value of ct for x is given by a point on the axis that lies perpendicular to the axis."}
{"id":"127602","title":"why are the anthropometric units (which are about as big as we are) as large as they are relative to their corresponding Planck units?","text":"so this might have some duplicated inquiry that this question or this question had, and while i think i have some of my own opinion about it, i would like to ask the community here for more opinions. so referring to Duff or Tong, one might still beg the question: why is the speed of light 299792458 m\/s? don't just say \"because it's defined that way by definition of the metre.\" before it was defined, it was measured against the then-current definition of the metre. why is $c$ in the ballpark of $10^8$ m\/s and not in the order of $10^4$ or $10^{12}$ m\/s? similar questions can be asked of $G$ and $\\hbar$ and $\\epsilon_0$. EDIT: to clarify a little regarding $c$. i recognize that the reason that $c\\approx10^9$ m\/s is that a meter is, by no accident of history, _about_ as big as we are and a second represents a measure of how fast we think (i.e. we don't notice the flashes of black between frames of a movie and we can get pretty bored in a minute). so light appears pretty fast to us because it moves about $10^9$ lengths about as big as us in the time it takes to think a thought. so the reason that $c\\approx10^9$ m\/s is that there are about $10^{35}$ Planck lengths across a being like us ($10^{25}$ Planck lengths across an atom $10^5$ atoms across a biological cell and $10^5$ biological cells across a being like us). why? and there are about $10^{44}$ Planck times in the time it takes us to think something. why? answer those two questions, and i think we have an answer for why $c\\approx 10^9$ in anthropometric units. EDIT #2: the other two questions referred do **not** address this question. Luboš Motl gets closest to the issue (regarding $c$) but does not answer it. i think in the previous EDIT and in the comments, i made it (the question) pretty clear. i was **not** asking so much about the **exact** values which can be attributed to historical accident. but there's a reason that $c \\approx 10^9$ m\/s, not $10^4$ or $10^{12}$. reworded, i s'pose the question could be \" _why are the anthropometric units (which are about as big as we are) as large as they are relative to their corresponding Planck units?_ \" (which **is** asking a question about dimensionless values.) if we answer those questions, we have an answer for not just why $c$ is what it is, but also why $\\hbar$ or $G$ are what they are."}
{"id":"3644","title":"The origin of the value of speed of light","text":"Meaning, why is it the exact number that it is? Why not 2x10^8 \/mps instead of 3? Does it have something to do with the mass, size or behavior of a photon? To be clear, I'm not asking \"how we determined the speed of light\". I know there isn't a clear answer, I'm really looking for the prevailing theories."}
{"id":"53153","title":"What causes precession or nutation in a spinning object?","text":"1. What causes precession in a spinning object?   2. What causes nutation in a spinning object?    3. What causes a top, gyroscope, and the earth to wobble?  Just because it's a simple question, I'm not expecting a simple answer, but please do summarize whatever you say in laymen terms, thanks."}
{"id":"2745","title":"QCD phase diagram in the large N limit","text":"I am sending a couple of questions which seem a bit more specific than others on this site, partially to probe if there is a point in doing so. Not sure what is the range of expertise here, and no way to find out without trying, so here goes: I am wondering what is known about QCD, or other field theories, in the regime of large density and low temperatures, specifically studied in the large N limit. I know of the qualitative picture at finite N, but lots of the instabilities (e.g. the superconducting ones) are suppressed at large N and replaced by other interesting phenomena. I am only aware at the moment of the DGR instability to form chiral density waves, and I am wondering what else exists in the vast and possibly quite old literature. Any pointers or entry points to that literature will be appreciated"}
{"id":"102498","title":"How does Cooper pairing work?","text":"Cooper pairs are one of the models how superconductivity is explained. What still baffles me is how a vibration of the crystal lattice (the so-called phonon) can interact with the electron (an actual particle), in such a way that it then creates a coupled pair with an other electron... What is the explanation for this behaviour? What is the maths behind it?"}
{"id":"104579","title":"Small object in air","text":"What happens with very small spherical objects ($d=1\\mu m$, e.g. a bacterium) in air? Do they fall? How quickly? Does it depend on their mass? We often see objects of little mass e.g. leaves falling from trees, but these object ususally have a very large surface, so their behaviour is very different to spherical objects. _Disclaimer: I'm a biologist._"}
{"id":"95530","title":"Why do these equations result an incorrect unit for acceleration?","text":"![](http:\/\/i.stack.imgur.com\/BB7m9.jpg) Hello everyone. Imagine an object moving around a certain point on a circular orbit. Magnitude of the velocity is constant during the motion ($|v|$). The orbit radius is $r$. (I'd better notice that we're just talking about **kinematic** view of this motion.) According to the image I've uploaded, we'll have: $\\large v_x(\\theta)=|v|\\cdot \\cos\\theta$ $\\large v_y(\\theta)=|v|\\cdot \\sin\\theta$ Since perimeter of the circular path is $2\\pi r$, and magnitude of the velocity is constant, we'll have: $\\large\\theta (t)=\\frac{|v|\\cdot t}{2\\pi r} \\times 2\\pi =\\frac{|v|\\cdot t}{ r}$ Now we can combine these equations: $\\large v_x(\\theta)=|v|\\cdot \\cos(\\frac{|v|\\cdot t}{ r})$ $\\large v_y(\\theta)=|v|\\cdot \\sin(\\frac{|v|\\cdot t}{ r})$ By this point, everything is okay. But the problem occurs here, where we try to get derivative of $v_x(t)$ and $v_y(t)$ in order to find $a_x(t)$ and $a_y(t)$. As we know by differentiation we have: $\\cos^{\\prime}(x)=-\\sin(x)$ $\\sin^{\\prime}(x)=\\cos(x)$ And we know that acceleration(time) function is derivative of velocity(time). So: $\\large a_x(t)=(v_x(\\theta))'=|v|\\cdot -\\sin(\\frac{|v|\\cdot t}{ r})$ $\\large a_y(t)=(v_y(\\theta))'=|v|\\cdot \\cos(\\frac{|v|\\cdot t}{ r})$ Well, now something is wrong: These two equations result a $m\/s$ unit (or something like that) for acceleration, but that's wrong. Acceleration unit must be $m\/s^2$, (or something like that). The question is that: **Where does this problem come from?** I couldn't figure it out at all. I don't know, maybe some kind of misunderstanding about derivative concepts cause that. So please try to answer simple, clean as much as possible."}
{"id":"54720","title":"Restriction on vector fields","text":"The 2D vector field (x,-y) does not transform like a vector under rotation(Arfken Vol. 1)! Does this mean we cannot have such a vector field physically?"}
{"id":"126796","title":"Gravity on flat object","text":"I was wondering how gravity would behave on object of different shapes.   1. If the Earth was squeezed into a thin disk what would the gravitional acceleration be at the center of the flat surface? Would it be really low because the amount of matter beneath me would be small? If I stood on the edge would the gravitational acceleration be enormous because the amount of matter beneath me was huge?   2. If an object is lowered into the Mariana trench will the effect of gravity increase because it gets closer to the center of the Earth?"}
{"id":"62638","title":"How does this paper relate to standard QED?","text":"This paper proposes a microscopic mechanism for generating the values of $c, \\epsilon_0, \\mu_0$. They state that their vacuum is assumed to contain ephemeral (meaning existing within the limits of the HUP) fermion\/antifermion pairs. This affects the mechanism of photon propagation as follows: > When a real photon propagates in vacuum, it interacts with and is > temporarily captured by an ephemeral pair. As soon as the pair disappears, > it releases the photon to its initial energy and momentum state. The photon > continues to propagate with an infinite bare velocity. Then the photon > interacts again with another ephemeral pair and so on. The delay on the > photon propagation produced by these successive interactions implies a > renormalisation of this bare velocity to a finite value Now this description of what happens to the photon sounds awfully like a heuristic description you might give to the photon self energy contribution in standard QED perturbation theory, where, although it's used in the renormalization procedure, it most certainly doesn't make any changes to the velocity of light. Can someone explain how what's being proposed here relates to standard QED. I don't see how they fit together."}
{"id":"67046","title":"Can one define an acceleration operator in quantum mechanics?","text":"It seems most books about QM only talk about position and momentum operators. But isn't it also possible to define a acceleration operator? I thought about doing it in the following way, starting from the definition of the momentum operator: $\\hat{p} = -i\\hbar \\frac{\\partial }{\\partial x}$ Then we define a velocity operator in analogy to classical mechanics by dividing momentum by the mass $m$ $\\hat{v} = \\frac{-i\\hbar}{m} \\frac{\\partial }{\\partial x}$ In classical mechanics acceleration is defined as the time derivative of the velocity, so my guess for an acceleration operator in QM would be $\\hat{a} = \\frac{-i\\hbar}{m} \\frac{\\partial }{\\partial t} \\frac{\\partial }{\\partial x}$ Is that the general correct definition of the acceleration operator in QM? How about relativistic quantum mechanics?"}
{"id":"67040","title":"Vectors and motion in a plane","text":"> A particle travels with speed $50 m\/s$ from the point $(3,-7)$ in the > direction $7i-24j$ . Find its positional vector after 3 seconds. **My approach** : It has travelled a distance of 150m in the direction given by the unit vector $\\frac{7\\hat i - 24 \\hat j}{25} $ . So, its position is now $210\\hat i -720 \\hat j$. But it started from $(3, - 7)$, so I have to subtract that to get $197\\hat i - 713\\hat j$. But I think this answer is awkward, and I may have gone wrong. So, please tell me if I am right, and if yes, can you suggest any shorter way of doing it?"}
{"id":"132556","title":"Polchinski equation 4.3.16","text":"I am trying to obtain the polchinski's equation 4.3.16 which is following $ Q_B^2 = \\frac{1}{2}\\\\{ Q_B, Q_B \\\\} = -\\frac{1}{2}g^{K}_{IJ}g^{M}_{KL}c^Ic^Jc^L b_M =0$ Where $ Q_B = C^I(G_I^m +\\frac{1}{2}G_I^g)$ and $C^I$, $b^J$ are anticommuting(ghosts) and $[G_I, G_J]=ig^K_{IJ} G_K$,   $G_I^g = -ig^K_{IJ} C^Jb_K $ are ghost parts and $G_I^m$ are matter part and they satisfy above commutation relations What I have done are $\\\\{ Q_B, Q_B \\\\} = \\\\{ C^I(G_I^m +\\frac{1}{2}G_I^g), C^J(G_J^m +\\frac{1}{2}G_J^g)\\\\} =\\\\{C^IG_I^m, C^J G_J^m\\\\} +\\frac{1}{2} \\\\{C^IG_I^m, C^JG_J^g\\\\} +\\frac{1}{2} \\\\{C^I G_I^g, C^JG_J^m\\\\} +\\frac{1}{4} \\\\{C^IG_I^g, C^JG_J^g \\\\} = C^IC^J [G_I^m, G_J^m ] +\\frac{1}{2} C^IC^J [G_I^m, G_J^g] +\\frac{1}{2}C^IC^J[G_I^g,G_J^m]+\\frac{1}{4}C^IC^J[G_I^g, G_J^g] =C^IC^J [G_I^m, G_J^m ]+\\frac{1}{4}C^IC^J[G_I^g, G_J^g] = C^IC^J ig^K_{IJ}G_K^m+\\frac{1}{4}C^IC^J ig^{K}_{IJ}G_K^g =C^IC^J ig^K_{IJ}G_K^m+\\frac{1}{4}C^IC^J g_{IJ}^K g^M_{KL}C^Lb_M $ compare with the textbook $\\\\{ Q_B, Q_B \\\\} = -g^{K}_{IJ}g^{M}_{KL}c^Ic^Jc^L b_M$ My calculation is something wrong. How can I fix it?"}
{"id":"66395","title":"Addes mass forces: can a force depend on acceleration?","text":"My friend and I had a little discussion about added mass forces. I always interpreted $F=ma$ as a cause-effect relationship, so I find rather uneasy to accept that the cause can instantaneously depend on the effect. Is it fine to have a force which depends on an acceleration, in classical mechanics? I came up with some possible solutions to this:   1. It's perfectly fine for **F** =m **a** to be an implicit equation with respect to **a**.   2. The time derivative of the velocity appears as the result of an approximation of a time-delay.   3. It arises due to assumptions made on the nature of the fluid (i.e. incompressible).   4. None of them"}
{"id":"109583","title":"Photons emitted at the event horizon?","text":"While looking through the questions, a came across a section about black holes. I immediately though; what would happen if an atom is orbiting a black hole and emitted a photon perpendicular to the event horizon, going away from the black hole. How would light going away from a black hole react to the gravity? Photon \"stuck\" on the event horizon of a black hole actually talks about a photon stopping at the event horizon, but my question is about a photon just outside of the event horizon and if the photon is slowed down."}
{"id":"91291","title":"Why does centre of mass of ice-container system shift in absence of any net external force?","text":"Consider a cube of ice in a flat based container(the base is very broad).The temperature of the system is at first fixed at a minus Celsius temperature, but then the system is left on a table with the top open to atmosphere.   The ice starts melting, and finally there is only water, spread over the container's base(water doesn't touch walls of the container,so it looks like water spilled on the floor)   Here, the centre of mass of the $H_2O$-container system moves downward because of the lowering of height of water molecules.   But,this system experiences no net external force on it,all the time.Hence, the centre of mass of this system shouldn't accelerate at all!   How is this contradiction sorted out?I feel that since melting is very slow, the centre of mass might move only very slowly, but still that doesn't explain things.For example if the room was very hot, melting wouldn't have been slow, right? Edit:Many people are probably getting confused regarding what perspective I'm taking.Let the container and water(solid\/liquid) be a single system.We can think of them together as a point mass.This point mass is at equilibrium,and at rest,situated at the position of the centre of mass of ice+container system(let's say,at a height 'h' above the table.   There may be internal forces happening inside the point mass,but the net external force(resultant) is zero,all the time.Hence, according to Newton's first law,the point mass must remain at equilibrium and hence,at rest.But,when ice melts,the position of center of mass of ice+container system has moved down!   Hence the point mass has to move down, in the absence of any resultant external force.The gravitational force on ice+container is cancelled by normal reaction of table on container."}
{"id":"106556","title":"Shape Created by a Pile of Granular Objects Dropped Uniformly","text":"I've observed many times that if you drop a lot of a 'granular' substance in one place and keep the nozzle out of which the substance flows, that the shape of the pile created very much resembles a bell curve. The situation is a bit hard to explain so for example image somebody holding a small pipe vertically above the ground and dropping a large amount of sand through that small pipe. The shape created by the sand on the ground largely resembles a bell curve wrapped around the vertical axis. I come from a math background and am really unsure of how to even start proving or disproving the proposition that the shape formed by the grains is a bell curve. Any help and input would be appreciated."}
{"id":"106558","title":"Calculating the Probability Current of a Travelling Wave","text":"Calculate the probability current density vector $\\vec{j}$ for the wave function : $$\\psi = Ae^{-i(wt-kx)}.$$ From my very poor and beginner's understanding of probability density current it is : $$\\frac{d(\\psi \\psi^{*})}{dt}=\\frac{i\\hbar}{2m}[\\frac{d\\psi}{dx}\\psi^{*}-\\frac{d\\psi^{*}}{dx}\\psi]$$ By applying the RHS of the above equation : $$\\frac{i\\hbar}{2m}[-A^{2}ikxe^{-i(ωt-kx)}e^{i(ωt-kx)}-A^{2}ikxe^{i(ωt- kx)}e^{-i(ωt-kx)}]$$ This gives : $$\\frac{-2iA^{2}ik\\hbar}{2m}=\\frac{k \\hbar A^{2}}{m}$$ This is not the correct answer. :( What have I done wrong ? In the model workings instead of A in the complex conjugate of the wave function they have written $A^{*}$. Why is this necessary since $A$ is likely to be a real number anyways ?"}
{"id":"109589","title":"Is it possible to extract energy from mass rotating in space?","text":"Consider a single isolated rotating mass (for example planet), is it possible to extract energy out of its rotation? If yes, how could that theoretically be achieved?"}
{"id":"29047","title":"Physical Explanation of Being Able to \"Think\"","text":"This may look like a philosophical question, but I'm looking for physical explanations (if there's any), that's is why I'm asking it here. What is the ability of thinking? We are all creatures consist of flesh and bones. Our brains are also nothing but flesh and water? Why are we thinking? What happens to this thinking power when we die? How and why does it disappear? Does it really disappear, or go to somewhere else in a sense we don't know and understand yet? We observe that, different creatures have different capabilities of thinking. The creature which has the most advanced thinking ability is human. Besides human cats can think up to some degree; they know how to hunt, they decide where to hide and where to find food from. Ants think too, even if it is weaker then cats can do; they find food and quickly run away when you try to pick one of them from floor with your hand, because they sense danger and decide to escape from there. Cells can also think in some sense, but it is much less capable that we don't even call it \"thinking\". Most living creature sense themselves as separate being. What happens if we stop the time, we compile and exact verbatim copy of a normal human being, then start the flow of time once again? Would the second copy think too as the original one? Or would the second copy start his life in vegetative state? Or would it just be a stack of dead flesh? How does physics explain this? Is there any particle that causes us to think? One century ago, we didn't even dream that this many sub atomic particles existed. Can the ability of thinking be a cause of some unknown physical particle of a flow of vector field (like electromagnetic field) which hasn't been discovered yet? Do you expect that in the future some scientist(s) would discover this mystery? Is there any research going on this? What do we know about \"thinking\" today? (Note: There wasn't appropriate tags for my question. I would appreciate if another user with higher reputation could add some tags for this question and remove this note.)"}
{"id":"115185","title":"A brief explanation needed regarding newton's law in particle","text":"So based on the topic, this is the question. \"A particle with mass $m$ is moving along x-axis with $v_0$ at $t=0$ and $x=0$. The particle is acted by an opposing force with magnitude proportional to the square of velocity. Find out the a) velocity b) position and c) acceleration of particle at any time ($t>0$)\" I have the answers and solution for this but hardly understand them (I dont understand at all actually. Plus where does β come from?) Basically I just need a clarification on question a) as b) and c) are related to the first question. This is a self study for final exam. Answer provided by my friend but she also copied it from a source. So without understand it we do it correctly. But it seems to be useless. So I hope you guys can help me. This is the answer. a) velocity ![Part a](http:\/\/i.stack.imgur.com\/nU9kp.jpg) b) position ![Part b](http:\/\/i.stack.imgur.com\/PknJE.jpg) c) acceleration (t>0) ![Part c](http:\/\/i.stack.imgur.com\/cY0hx.jpg)"}
{"id":"86296","title":"Not so simple problem using momentum, energy and angular velocity...?","text":"I have an object in free space (no gravity) with angular momentum $ = \\omega_i $, and some velocity vector $=\\vec{V_i}$. To simplify we will say it has a mass-less rigid rod length $ = \\ell $, connecting two small masses both of mass $ = M $. The masses are small in the sense of a radius equal to the rod radius both much smaller than $ \\ell $. Of course for simplicity keep this in the 2D plane. I know precession can play a role, but accounting for it can make the problem easier, I've already done that. Now, we want to change the velocity of the object by $ \\Delta V $. We must do this by taking a bit of the matter of mass $ = m $ off the object and get it moving away at a velocity $ = \\vec{V_m} $. While keeping $M>>m$ What will be the most energy efficient mechanism to do this? What will be the most momentum efficient mechanism to do this? The final answer, like with the precession related solution, gives some $ \\vec{V_m} $ value parallel or perpendicular to $\\vec{V_i}$, and some mass \"$ m $\" proportional to \"$ M $\". The goal will be to obtain maximum $ \\Delta V $ with minimum energy or momentum. While keeping $M>2m$ Right now I am getting different solutions based on minimizing energy vs. momentum, is this logical? Why? This has helped me get to my solution thus far. Thanks JCooper and Maksim Zholudev. **EDIT:** The answer should be at least partially derived using formulas. Some relation between $ \\Delta V $ and the input energy or momentum must be shown."}
{"id":"91837","title":"Friction on an object moving with momentum over a surface","text":"I'm familiar with the equations for friction for a static object and an object moving at steady speed over a surface from high school physics. But we never learned how an object moving only due to momentum experiences friction. This is something I've modeled several times while building simple 2D games, but I have no idea if what I have made matches reality in any way. Is the force of friction dependent on the speed of the object across the surface? Or is the force constant as the object decelerates? Is there a simple equation giving the acceleration (deceleration) of the object depending on it's mass and velocity?"}
{"id":"91830","title":"Photon number conservation during scattering","text":"I was reading this writeup on the Kompaneets equation and the Sunyaev- Zel'dovich effect. On page 3, section 2 the author states > There is no way to increase the mean energy of a planckian distribution > without changing the particle number. But as far as I understand, photon number is conserved during a scattering process, and therefore isn't it possible that for a given Planckian distribution all the photons gain the same energy through scattering and thus increase the mean energy of the Planckian distribution? Or am I missing something here?"}
{"id":"91831","title":"A question about polarization in quantum mechanics","text":"We start our question we a definition A subbundle $P\\subset TM^{\\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if \\   1. $P$ is Lagrangian   2. P involutive   3. dim$P\\cap\\bar P \\cap TM$ is constant Now, Introduce an hermitian form on $P$ defined by  $$b(X,Y)=i\\omega(X,\\bar Y)$$ . Note that when $P$ is real then $b$ is vanishes identically on $P$. Cinsequently, $b$ projects onto a non-degenerate form on the quotient $P\/{(P\\cap \\bar P)}$ and we denote it by $\\bar b$. $P$ is said to be of type $(r,s)$ if and only if $\\bar b$ has signature $(r,s)$ i.e. its matrix is $$diag(\\underbrace{1,1...,1}_{r},\\underbrace{-1,-1...,-1}_{s} )$$ for $0\\le r+s=n-dim_{\\mathbf C }P\\cap \\bar P$ . Then, $P$ is said to be positive if $s=0$. In the case if $r=s=0$ then $P$ is real Philosophically the main goal in defining polarizations in quantum mechanics is to find the wave functions that are covariantly constant along its directions. When the form $b$ is not positive show that there are no globally defined wave functions"}
{"id":"100254","title":"Gravitational compression \/ compression in general","text":"How exactly does gravitational compression, or compression in general, increase temperature? It seems counter-intuitive seen as temperature usually increases from the solid to the gas phase."}
{"id":"107207","title":"Photons and Black holes","text":"How many photons in one Planck volume would it take to form a tiny black hole? A photon doesn't have mass but it does have energy, $1.0101 \\times 10^{-37}$ Joule for red $650$ nm wavelength light if I'm correct. A photon is a point- like boson so an infinite number of photons can fit into any given area. So more photons is equal to more energy which would bend space-time even more till they themselves wouldn't be able to escape from their own bend in space- time."}
{"id":"55547","title":"Can modern twistor methods to calculate scattering amplitudes be applied to renormalization group calculations?","text":"As explained for example in this article by Prof. Strassler, modern twistor methods to calculate scattering amplitudes have already been proven immensely helpful to calculate the standard model background in searches for \"new physics\". If I understand this correct, the \"practical\" power of these methods lies in their ability to greatly simplify the calculation of scattering processes, which are due to limited computer power for example, not feasable applying conventional Feynman diagrams. Depending on the system considered, a renormalization group transformation involves the calculation or summation of complicated Feynman diagrams too, which usually has to be simplified to obtain renormalization group equations which are numerically solvable in a finite amount of time. So my question is: Could the new twistor methods to calculate scattering amplitudes be applied to simplify investigations of the renormalization group flow, in particular investigations of the whole renormalization group flow field beyond a single fixed point, too? Are such things already going on at present?"}
{"id":"22135","title":"Can Mirror box simulate long light travel?","text":"Imagine that there is a cube box that has mirrors all 6 faces in . If we use a strong laser and enter in the box from a small hole on the box. The laser light travels in the box long time that we can detect the laser via a detector on other hole of the box. 1) Is it possible to simulate the long light travel in it (for example a day or week)? 2) Is it possible to proof that there is no ether via that box? If we move the box in a fixed velocity what we can observe about receive time on detector. I ask a question first time in this website If it is asked question or not appropriate for your format sorry for that."}
{"id":"77438","title":"Uniform distribution of charge on a plane and on a bar: conditions to have no motion of the bar","text":"On the plane z=0 there is a superficial charge distribution such that $\\sigma$ is constant. Near to the plane, there is a bar, charged uniform with total charge q. At the extremities the bar has two constraints, so it can't turn. If I want to find the constraints force and the force momentum needed to block the bar, can I consider the charge q as a single point charge and put it in the midpoint of the bar?"}
{"id":"77432","title":"Is spin just due to uncertainty in angular momentum?","text":"I can't seem to think of any way to envision electron spin. Can it be thought of as the uncertainty in angular momentum?"}
{"id":"73224","title":"Conservation of Angular Momentum, as related to a flywheel","text":"Trying to work out some pesky flywheel dynamics for a project I'm working on, would love some for your assistance to better understand the underlying concepts. For a given flywheel (thin-walled cylinder, assume a spoked bicycle wheel) rotating in the x-y plane, I'm trying to calculate the force generated in either direction along the z-axis. It seems to me, in line with Newton's first law of motion extended to rotational dynamics, what forces are physically being generated that prevent a rolling wheel from falling over?"}
{"id":"93475","title":"How to calculate fluid(oil \/ hydrocarbon) loss under pressure","text":"I'm trying to calculate the amount of fluid that would flow through an area dependant on the amount of pressure that there is. I'd also like to know the rate at which it would flow. Essentially I have a very basic model of a well drilling system. At the moment the variables \/ parameters for each of the objects in question are defined by the user but may be set to constants to enable testing. The assumptions I'm working on are: The viscosity of the oil \/ hydrocarbons is set to constant. Once drilled the area of the hole will not change (there will be no collapse etc). The well is never ending so constant pressure. If someone could point me in the right direction I'd be very thankful. Some of the assumptions may be way off so guidance in general is also appreciated, many thanks."}
{"id":"101126","title":"Representation of a Signal and its evolution via Quantum System","text":"I am a total beginner in the field of Quantum Mechanics. So, the question I am asking may be a silly one. So kindly give me possible answers or advice for modifications. Recently I am learning the concept of qubit. The quantum theory tells that a $n$-qubit system is represented by a unit vector in $(\\mathbb{C^2})^{\\bigotimes{n}}$ with some basis set. Now, we can also express a $2^n$ dimensional complex signal as a vector at any point of time, and with suitable normalization it is nothing but a qubit. Now, as far as I know, the evolution of a qubit is always dictated by a unitary operator, whereas, the evolution of a signal can be dictated by any arbitrary operator. So, is there any way, any sort of transformation that allows,given a signal and its evolution, to create its equivalent representation as an evolution of qubit, so that we can solve problems of signals using methods of Quantum Mechanics."}
{"id":"32749","title":"Why does the water-pressure of my shower fall if I hold the shower head high?","text":"I live in a very old house - build 1902 - in the 4th floor in the city of Karlsruhe (Germany). I have a shower and the gas-heater for the water is in it: ![enter image description here](http:\/\/i.stack.imgur.com\/tpiBK.jpg) If I hold the shower head over some height (see image), the water pressure is suddenly reduced. Whats the reason for that? (I guess the way the pump works might be the reason.) ![enter image description here](http:\/\/i.stack.imgur.com\/rvn6k.jpg)"}
{"id":"62398","title":"What were Feynman's objection(s) to a cubic lattice universe?","text":"In this video of Feynman discussing the scientific method, starting at around eight minutes and 30 seconds, Feynman describes the proposition that space consists of a cubic lattice of points (as opposed to space being continuous). Feynman says \"then we can prove that [the proposition] immediately is wrong,\" but he doesn't elaborate on the proof. > My question is: What are the objection(s) to a cubic lattice universe that > Feynman refers to in this video? I'm not looking for every possible objection to a cubic lattice universe. I'm wondering what in particular the (apparently immediate and obvious) objections are that made Feynman so dismissive of the idea."}
{"id":"4459","title":"Conserved quantities in generalized n-body problem","text":"Given a collection of point-particles, interacting through an attractive force $\\sim \\frac{1}{r^2}$. Knowing only $m_1a=\\sum_i \\frac{Gm_1m_i}{r^2}$ and initial conditions we can deduce the motion of the system. Consequently we can observe that three quantities remains constant A) center of mass of the system B) total energy C) angular momentum How can we derive these 3 facts directly from $m_1a=\\frac{Gm_1m_2}{r^2}$ ? Are these quantities conserved for any attractive force $\\sim\\frac{1}{r^n}$ ? Given any monotonically decreasing force for $r$ in $(0,\\infty)$, which are the conserved quantities?"}
{"id":"74166","title":"X-ray and neutron powder diffraction: peak broadening due to crystal size","text":"When analysing powder diffraction patterns, the broadening of peaks can be used to estimate crystal sizes. Smaller crystal size gives larger broadening according to the Scherrer equation: $$ \\beta = {{K\\cdot\\lambda}\\over{D\\cdot\\cos \\theta}}.$$ What is the physical origin of this effect? Edit: Added homework tag as this is related to my thesis on soil studies using diffraction methods."}
{"id":"130047","title":"Torque definition and right hand rule not arbitrary","text":"I have read the following: http:\/\/www.feynmanlectures.caltech.edu\/I_20.html#Ch20-S1 The formula for $\\tau_{xy}$ is derived in this chapter: http:\/\/www.feynmanlectures.caltech.edu\/I_18.html#Ch18-S2. In this derivation until equation 18.11 ($\\Delta W=(xF_y-yF_x)\\Delta\\theta$) the term $xF_y-yF_x$ does not seem to be arbitrary. It would not make any sense the other way around like: $yF_x-xF_y$, because of the equations 18.6 ($\\Delta x=-PQ\\sin\\theta=-r\\,\\Delta\\theta\\cdot(y\/r)=-y\\,\\Delta\\theta$) and 18.7 ($\\Delta y=+x\\,\\Delta\\theta$). Then $\\tau_{yz}$ and $\\tau_{zx}$ are derived (in the first link) by symmetry. $$\\begin{alignedat}{6} &\\tau_{xy}~&&=x&&F_y&&-y&&F_x&&,\\\\\\\\[.5ex] &\\tau_{yz}~&&=y&&F_z&&-z&&F_y&&,\\\\\\\\[.5ex] &\\tau_{zx}~&&=z&&F_x&&-x&&F_z&&. \\end{alignedat}$$ * * * I have some ideas and wonder whether or not they are true: I have got the feeling that they are wrong but I have no idea why they should be wrong.   1. _So, since the formula for the torque in the $xy$-plane ($\\tau_{xy}$) is not arbitrarily-derived, there is to torque $\\tau_{yx}$, since $yF_x-xF_y$ would not be correct in equation 18.11._   2. In my first link, below equation 20.1 there are 2 pictures which shall demonstrate how the letters $x$, $y$ and $z$ can be interchanged. _How would that make sense?_ Since equations 18.6 and 18.7 are true, we \"live\" in a right-handed coordinate system and $\\tau_{yx}$ would not make any sense.   3. Since $\\tau_{yx}$ would not make any sense, the arguments in my first link down to equation 20.9 show that the right-hand rule is not arbitrary. What is wrong with my arguments? What does $\\tau_{yx}$ mean?"}
{"id":"41271","title":"1D Acoustical Relations beyond nearest neighbor couplings","text":"Consider some 1D Lattice of atoms with nth neighbor coupling of strength k_{n}. I'm looking for the dispersion relation for acoustical phonons under these conditions. I start with the Lagrangian, $$L = K- V$$ $$L = \\sum^{\\infty}_{n} \\frac{1}{2}m \\dot{x}_{n}^{2} - \\sum^{\\infty}_{p=1} \\frac{1}{2}k_{p} \\\\{(x_n-x_{n+p})^2 + (x_n - x_{n-p})^2\\\\}$$ Mass is the same for each atom. The Lagrange equation should be $$m \\ddot{x}_{n}=\\sum_{p=1} k_p(x_{n-p}+x_{x+p}-2x_n)$$ Now, if I use a travelling wave solution as an ansatz, I should get my dispersion relation as some infinite series. Is this correct? If so, help me out because I can't make it work. Thanks!"}
{"id":"123299","title":"Absorption cross section and absorption coefficient","text":"What is the absorption cross section, how is it measured? How to convert it to the absorption coefficient (measured in cm$^{-1}$)?"}
{"id":"71262","title":"characterization of potential","text":"So i have a force field $F(x,y)$ and i have to find out wether it is a potential or not. My first idea was to calculate : $dU=Fdr$ (where $r$ is the radius vector) , to integrate on both sides and hence to see if $U$ is path dependent or not. That turned out to be right. I was thinking about another approach: we know that , if a potential exists then $F=-\\nabla U$ So if $F$ is a potential it must be possible to find such a $U$ , wich in my case wasn't possible, hence $F$ is not a potential. My question: (1) Is the second approach right? (2) are there othere ways for solving this problem? Thanks in advance."}
{"id":"18813","title":"What are local electrons in a crystal?","text":"I am reading Pekar's \"Research in Electron Theory of Crystals\" and I came across a passage I find a bit unclear: > The theory developed below takes into account the dielectric polarization of > a an ionic crystal by the electric field of the conduction electron. The > local polarization that results from this is related with the displacement > of the ions and consequently is inertial. It cannot follow the relatively > rapidly moving electron and therefore forms a potential well for the > electron. The depth of this potential well turns out to be sufficient for > discrete energy levels of the electron to exist in it. The electron, being > in a local state on one of these levels, can maintain with it sown field the > aforementioned local polarization of the crystal. Because of their inertia, > the ions are sensitive not to the instantaneous value of the electron field, > but to the average field. The latter can be calculated as the static field > of the $|\\psi|^2$ cloud of the electron; it produces a static polarization > potential well, which in turn maintains the electron stationarily in a local > state. Such states of the crystal with the polarization potential well, > which in turn maintains the electron stationarily in the local state. Such > states of the crystal with a polarization potential well, in which the > electron is localized, were called by the author polarons Now, what exactly does he mean by local vs. conduction electrons? Are local electrons those that are not moving and are in the crystal? That doesn't seem right. What does it mean for an electron to be in a local state? (Also what does he mean by \"inertial\"?) IN fact, it would be nice if one explains this passage in understandable terms so that I can have some intuitive picture in mind."}
{"id":"33035","title":"Zigzag flow of water along a vertical glass window","text":"I've observed this behavior many times. When it rains, the rainwater will form vertical channels along a glass window. The flow of water is mostly confined within these vertical channels and the channels are (more or less) stable. But sometimes - and I suspect this happens when the flow intensity in one of the channels increases - the channel will switch from a vertical configuration into a zig-zag configuration. The zig-zag is composed of short segments running horizontally that are connected by semicircular (vertical) segments. The zig-zag is unstable and lasts only for 0.1 second or so. Then the channel reverts to its vertical configuration. I have made photographs of this behavior but I cannot find them now. I have seen similar patterns in the book \"The self made tapestry\" by Philip Ball, page 145. This shows growth instabilities in glass cracks. Is says \"at higher speeds the crack becomes oscilatorry with a constant wavelength\". This is what I see in the water flow. It feels counterintuitive. There must be a good explanation for this behavior. Can you point me to it? EDIT Here is a video ."}
{"id":"123329","title":"What's the most fundamental definition of temperature?","text":"What's the most fundamental definition of temperature? Is it the definition concern about average energy, number of micro states, or what? By \"fundamental\", I mean \"to be applied\" in such general cases as Black Hole's Temperature, Accelerated Frame's Radiation,..."}
{"id":"123328","title":"Determine stationary angular velocity of wheel with circuit in magnetic field","text":"I have a wheel (free to spin around the $z-$axis) with four spokes that is connected by sliding contacts to a circuit with $U_0 = 0,72V$. Also, there is a B-Field parallel to the $z-$axis ![Imgur](http:\/\/i.imgur.com\/aLF3N7S.jpg) For the induced electric potential I have: $$U_i = - \\frac{1}{2}(R_{outer}^2 - R_{inner}^2)\\omega B$$ (with $\\omega$ = angular velocity) I'm now asked to find out the constant $\\omega_0$ after the system is in a stationary state (moves with constant speed). The assignment points out to look at one mesh (with one spoke) on the wheel and to determine if Kirchhoffs second rule applies and if there is a current flowing in a mesh. The most obvious way I can think of would be (since $U_0$ should be equal in any spoke): $$U_0 - U_i = 0$$ and to solve for $\\omega$. But elsewhere I was told that Kirchhoffs rules don't apply in systems with changing magnetic fields. Also I'm not sure if there still would be an emf induced in stationary conditions since the flow wouldn't change anymore then."}
{"id":"106203","title":"Instantaneous angular momentum of a disc","text":"> _Suppose we have a disk of radius $r$ and mass $m$ travelling at velocity > $v$. I want to calculate the instantaneous angular momentum with axis > through the edge of the disc (on the circumference)._ Angular momentum $= I \\omega$. $I = \\frac{1}{2}mr^2 + mr^2 = \\frac{3}{2}mr^2$ by the parallel axis theorem. $\\omega = \\frac{v}{r}$. Therefore, angular momentum $= \\frac{3mrv}{2}$. Alternatively, angular momentum $=p\\times r= m r \\times v = mrv$. Why do these two methods differ? Which, if any, are correct?"}
{"id":"2110","title":"Why does space expansion not expand matter?","text":"REFORMULATED: I have looked at the other questions (ie \"why does space expansion affect matter\") but can't find the answer I am looking for. My question: There is always mention of space expanding when we talk about the speed of galaxies relative to ours. Why, if space is expanding, does matter not also expand? If a circle is drawn on balloon (2d plane), and the balloon expands, then the circle also expands. If matter is an object with 3 spatial dimensions, then when those 3 dimensions expand, so should the object. If that was the case, we wouldn't see the universe as expanding at all, because we would be expanding (spatially) with it. I have a few potential answers for this, which raise their own problems:   1. Fundamental particles are 'point sized' objects. They cannot expand because they do not have spatial dimension to begin with. The problem with this is that while the particles would not expand, the space between them would, leading to a point where the 3 non-gravity forces would no longer hold matter together due to distance   2. Fundamental particles are curled up in additional dimensions ala string theory. These dimensions are not expanding. Same problems as 1, with the added problem of being a bit unsatisfying.   3. The answer seems to be (from Marek in the previous question) that the gravitational force is so much weaker than the other forces that large (macro) objects move apart, but small (micro) objects stay together. However, this simple explanation seems to imply that expansion of space is a 'force' that can be overcome by a greater one. That doesn't sound right to me. I think some of the problems in this question verge into metaphysics, but I think from his (?) previous answer, Marek can probably explain the physical side of things a bit more thoroughly. I will leave it there cos anything else I write sounds rambling!"}
{"id":"3518","title":"Expansion of the space-time metric","text":"If the space-time metric is expanding with the expansion of the universe, if I could travel back in time, would I be less dense than the matter in that previous era?"}
{"id":"32953","title":"Expansion of space","text":"> **Possible Duplicate:**   >  spacetime expansion and universe expansion? So I've heard that space is expanding very quickly and that the redshift we see when we look at other galaxies is evidence of this. But why doesn't this expansion affect orbits of the planets in our solar system, for instance? I would think that the expansion of space is uniform through the universe...why doesn't it tear our bodies apart?"}
{"id":"111456","title":"does space-time expansion affect on fundamental particle?","text":"does space-time expansion affect on fundamental particle or point particle?"}
{"id":"110684","title":"Expanding Universe and more","text":"We all are pretty familiar with the Friedmann models of the expanding universe, but at present I would like to concentrate on the model where the universe goes on expanding with the lapse of time. We know that if we mark spots on a balloon and inflate it, distance between these spots increases gradually(with time, of course). We may draw an analogy that these spots represent the galaxies. Now imagine a space within one of these spots, whose dimensions is indeed less than that of the spot concerned. With time, the dimensions of this space goes on increasing with time. If we compare this space to the interstellar(as well as interplanetary) space, and by the process stated above, go on imagining spaces within the one imagined before, we arrive at a limit where the rate of dimension variation of the space with time becomes very small in magnitude. Here I ask my question: could it happen so that objects in the universe(even solids) are expanding at a rate so very feeble(much much smaller than even the rate at which the universe expands), because all objects, be they solid or any other physical form, constitute some space within this universe, on account of its existence, and can be designated as a space within the spot considered in our balloon model. It is indeed a mere whim of mine to compare a universe model with that of a balloon-spot model! Nonetheless, please remove my doubts upon this trifling matter, that has perplexed me a good deal."}
{"id":"3660","title":"Effect of Cosmological Expansion at the Atomic Level","text":"> **Possible Duplicate:**   >  Why does space expansion not expand matter? Does cosmological expansion have any effects at the atomic level?"}
{"id":"131800","title":"How does the gravity well change as space expands?","text":"How does the gravity well change as space expands? If we assume that the Earth's gravitational field curves flat space to create a gravity well then how does the gravity well change as space expands also is the change in gravity well measurable."}
{"id":"1801","title":"Why space expansion affects matter?","text":"If space itself is expanding, then why would it have any effect on matter (separates distant galaxies)?   * Space is \"nothing\", and if \"nothing\" becomes bigger \"nothing\" it's still a \"nothing\" that shouldn't interact with matter in any way (it doesn't have mass, energy, etc).   * Gravity doesn't have a cutoff distance afaik, so even the most distant galaxies should be attracted to each other. Gravity force would be very very tiny, but it would still dominate \"nothing\" from space expansion.   * Lets take inertia from Big Bang into account. Inertia would be the primary force that moves galaxies away comparing to their tiny gravity and even more tiny, if any, force of our \"nothing\" that is still expanding inbetween. Wouldn't expansion decelerate if driven mostly by inertia?"}
{"id":"123061","title":"If the universe is expanding then why does the distance between Sun and the Earth is not expanding?","text":"It has been proved that universe is expanding in the accelerating way. If that is true than the space between the Sun and the Earth must also be expanding. But it is not so, why?"}
{"id":"56651","title":"How do we know space is expanding when we are part of space?","text":"From what I understand space itself is expanding, and the Big Bang attempts to describe this expansion at the very early stages of the universe. This is usually described in a visual way as 2 dots on the surface of a balloon as the balloon is being inflated. We exist in space, so as space expands we are expanding too aren't we? If this is correct how do we know space is expanding at all? I'll attempt to explain further... 2 dots on a balloon with a ruler drawn between them measuring 1 cm. As the balloon expands, so do the dots, and the ruler, and it always says that the distance is 1 cm. From an observers point of view (somewhere away from the balloon) the dots, ruler, and ruler are all getting larger - but from the point of view on the balloons surface nothing has changed. Another way to describe this would be a man in a room, the room and everything in it (including the man) are getting bigger at the same rate. From the man's perspective nothing is changing but from outside the room - you can see everything is getting bigger. I guess what I am trying to say is that we are in the universe, and expanding at the same rate as it (aren't we?) so how do we know it's getting bigger?"}
{"id":"70832","title":"What exactly is expanding when they say universe is expanding?","text":"We know that universe is expanding and galaxies are moving away from each other. Does this mean the galaxies are also expanding in itself and therefore I guess growing larger in volume? Depending on the answer of that, does the solar system also expands?"}
{"id":"99528","title":"Why doesn't everything expand when the universe expands?","text":"Everybody has been taught at one point, \"oh the universe expands, but that doesn't mean that everything is expanding uniformly, since that means we can't detect the expansion, but only that huge galaxies are moving away from each other\". But I'm rather confused. Can't the expanding universe simply be thought as a coordinate axis that expands? The definition of \"1\" on the coordinate axis constantly expands? After all, spacetime itself is expanding, and not only \"average distance of galaxies\". So our Planck length would expand, light wavelengths expand, we expand, etc. But clearly this isn't happening. Of course, the handwavy explanation is \"electromagnetism\/gravity overcomes expansion at small scales\", but the electromagnetic and gravity forces are defined in terms of constants relative to our \"coordinate intervals\", so if spacetime itself is expanding \"under the feet\" of gravity, shouldn't gravity not be able to do anything about it? We would also be unable to measure the expansion of spacetime since all of our measurement benchmarks and tools are also expanding. Clearly, this isn't happening. What is the _precise_ reason?"}
{"id":"95005","title":"Are atoms getting weaker?","text":"If the universe is expanding, it would make sense that the spaces between particles are getting bigger. If this is so, then the particles which make up atoms are also affected. Does that imply the spaces between the components of an atom will become large for the subatomic forces to hold? Are atoms getting weaker?"}
{"id":"127848","title":"Cosmic Expansion - Why aren't we ripped off yet?","text":"According to Hubble's law, the universe expands exponentially ever since the big bang.   1. If the space-time expands, what effects does it have upon us, (1.Earth, 2.solar system and 3. Milky Way).    2. What would be the fate of Gravitational force and other forces of nature if the expansion continues.    3. What happens to the dark energy and dark matter then?    4. Won't all the dark energy be used up in the acceleration process?"}
{"id":"29498","title":"Expansion of Universe","text":"> **Possible Duplicate:**   >  Why space expansion affects matter? If the Expansion is prevalent, i.e. it is observable and true then shouldn't that result in the expansion of the Milky Way galaxy, eventually our solar system would expand resulting in increase in the planetary distances (from the sun). Could the escape of Pluto be an example to this phenomena ? And is the Expansion resulting in increase in the sizes of the galactic objects ?"}
{"id":"131224","title":"Spacial curvature and expanding space","text":"If we take the analogy that in an empty space the space is just a flat sheet then if there is a single planet or a star then the flat sheet will curve below the planet leaving a curvature shaped like a hemisphere below the object, my question is, Does this curvature change as space expands?"}
{"id":"2982","title":"Why doesn't Brooklyn expand? (Or \"Is the expansion of the universe kinematic?\")","text":"> **Possible Duplicate:**   >  Why space expansion affects matter? Imagine two tiny spacecrafts that are moving with the Hubble flow and so are moving away from each other. Let's assume that they've been that way since the very early universe, never firing their engines, just drifting along their Hubble flow geodesics in a homogenous isotropic universe. They then momentarily fire their engines so that they \"cancel\" the Hubble flow and have a fixed proper separation. Will they now start drifting apart again (presumably due to expansion)? Or will they stay at fixed proper separation, and maybe very slowly move towards each other due to their mutual gravitational attraction? I guess this is another way of asking whether expansion is kinematic and hence can be forgotten (so that \"Brooklyn isn't expanding\" because gravitational collapse and structure formation have erased memory of the expansion). Or maybe someone will help me refine this question and make me realize I just haven't thought things through completely?"}
{"id":"130842","title":"Space within galaxies. Is it stuck by the gravity of the galaxy or expanding and \"slipping past\"?","text":"I understand that the inter-galactic space is expanding but galaxies themselves are not. What is happening to the space within a galaxy? Is it fixed by the gravity of the galaxy or is it expanding and moving out around it? Using the coins on a balloon analogy, is the balloon surface under the coins (galaxies) stuck to the coin and constrained or is it free to expand out, slipping underneath the coins leaving them in the same place (and the same size)?"}
{"id":"24324","title":"spacetime expansion and universe expansion?","text":"First of all, does the expansion of spacetime solely cause the expansion of universe? Secondly, if spacetime is the sole cause, do objects(matter with mass) themselves expand? Thirdly, by spacetime expansion, does time also expand? (at least in general relativity) Thanks."}
{"id":"31117","title":"How is the universe expanding?","text":"> **Possible Duplicate:**   >  spacetime expansion and universe expansion? Is the space between planets is growing or the space between stars is growing or the space between galaxies is growing?"}
{"id":"134149","title":"Expanding metre sticks","text":"Given the Universe is expanding. Therefore everything within the U is expanding, in all dimensions, subatomic to cosmic. Then all metre sticks are expanding. The question: How can we measure U expansion with metre sticks that are also expanding? Alice only knew she had grown ten feet tall by comparing herself to her surroundings."}
{"id":"37569","title":"Do atoms expand with universe?","text":"> **Possible Duplicate:**   >  Why space expansion affects matter?   >  Why does space expansion not expand matter? As we know, the universe is expanding, galaxies are away from each other. But what about atoms? Do they also in expanding? What's more, Bohr radius is $$a_0=\\frac{\\hbar}{m_e c \\alpha}$$, if it is increasing, does it means $m_e$ is decreasing due to the density of Higgs field is getting thinner. or $c$ is decreasing or $\\hbar$ is increasing?"}
{"id":"19012","title":"How is it possible to measure expansion of universe","text":"> **Possible Duplicate:**   >  Why does space expansion not expand matter? If according to Hubble's law space the expanding then shouldn't everything else along with the space expand and we should not be able to measure this expansion as our tools\/measuring sticks would also expand in same proportion"}
{"id":"129304","title":"What is the difference between matter & spacetime?","text":"If the universe is expanding why doesn't the matter in it expand proportionally making it seem as if the universe is static? Alternatively, as spacetime expands why does it not just slide past matter leaving matter unmoved? What anchors the matter to a particular point in spacetime?"}
{"id":"37625","title":"How is tritium illumination possible without negative health effects?","text":"Turns out there's tritium illumination \\- a tiny very strong plastic tube will be covered in phosphor and filled with tritium. Tritium will undergo beta decay and a flow of electrons will cause the phosphor to glow. This gives enough light for illuminating hours marks on a wristwatch dial and the hands of the wristwatch for many years and is claimed to not pose health hazard. Now how it is possible to have energetic enough radioactive decay and no health hazard at the same time?"}
{"id":"81470","title":"What do subatomic particles look like for a 'speed of light - observer'?","text":"Let's say that an observer is moving with the speed of light relatively to an atom that he wants to look into. He has equipment that precise that he can observe the atom and what is inside. From Einstein's theory we know that for light particles, everything else that moves with velocity smaller than the speed of light, 'looks like frozen, no move'. How would the elements inside the atom look?"}
{"id":"594","title":"Does the friction force change directions with a change in reference frame?","text":"In a basic friction problem with Block A sliding on top of Block B, the direction of the friction force is usually explained as being simply the opposite of the direction of motion. So if Block A is sliding to the right, the friction force is pointing to the left. But this reasoning implicitly assumes that we are calculating friction force from the reference frame of Block B. What if we instead look at the problem from the reference frame of Block A? To Block A, it looks as if Block B is sliding to the left, so an observer on Block A would say that there is a friction force which, to oppose the direction of motion, points to the right. It seems counterintuitive and probably wrong for the direction of friction force to depend on reference frame like this. Where is the flaw in the reasoning above? Are the two reference frames described above not exactly equivalent in a way that leads to the force changing directions?"}
{"id":"598","title":"Books for linear operator and spectral theory","text":"I need some books to learn the basis of linear operator theory and the spectral theory with, if it's possible, physics application to quantum mechanics. Can somebody help me?"}
{"id":"44999","title":"What's the motivation behind the action principle?","text":"1. What's the motivation behind the action principle?   2. Why does the action principle lead to Newtonian law?    3. If Newton's law of motion is more fundamental so why doesn't one derive Lagrangians and Hamilton principle from it?    4. Also does all Lagrangians obey $L=T-V$?    5. I think that it's related to the fact that the kinetic energy of the particle at all points on the path or it's travel time is as small as possible?    6. If so, How can we derive the principle of least action from this fact in detail?"}
{"id":"44998","title":"Charge of an electrolytic capacitors","text":"I can't understand the electrolytic capacitors, when a capacitor has a capacitance of 100 microfarads, does that mean that when it is charged with 100 volts will the charge of the plate be 0.01 coulomb? If there is a part of the plate with no isolation, then I touch it, I will be shocked with a charge of 0.01 coulomb and 100 volts?"}
{"id":"98595","title":"Does the Universe have finite number of particles?","text":"I read that the number of atoms in the entire observable universe is estimated to be within the range of $10^{78}$ to $10^{82}$. Does the Universe have finite number of particles? If so, how could it be determined?"}
{"id":"13907","title":"Positive Mass Theorem and Geodesic Deviation","text":"This is a thought I had a while ago, and I was wondering if it was satisfactory as a physicist's proof of the positive mass theorem. The positive mass theorem was proven by Schoen and Yau using complicated methods that don't work in 8 dimensions or more, and by Witten using other complicated methods that don't work for non-spin manifolds. Recently Choquet- Bruhat published a proof for all dimensions, which I did not read in detail. To see that you can't get zero mass or negative mass, view the space-time in the ADM rest frame, and consider viewing the spacetime from a slowly accelerated frame going to the right. This introduces a Rindler horizon somewhere far to the left. As you continue accelerating, the whole thing falls into your horizon. If you like, you can imagine that the horizon is an enormous black hole far, far away from everything else. The horizon starts out flat and far away before the thing falls in, and ends up flat and far away after. If the total mass is negative, it is easy to see that the total geodesic flow on the outer boundary brings area in, meaning that the horizon scrunched up a little bit. This is even easier to see if you have a black hole far away, it just gets smaller because it absorbed the negative mass. But this contradicts the area theorem. There is an argument for the positive mass theorem in a recent paper by Penrose which is similar. Questions:   1. Does this argument prove positive mass?   2. Does this mean that the positive mass theorem holds assuming only the weak energy condition?"}
{"id":"13901","title":"Normalizing the free particle wave function","text":"One way to normalize the free particle wave function > \"is to replace the the boundary condition $\\psi(\\pm{\\frac{a}{2}}) = 0$ [for > the infinite well] by periodic boundary conditions expressed in the form > $\\psi(x)=\\psi(x+a)$\" \\-- _Quantum Physics_ , S. Gasiorowicz How does this work? What does this mean physically? Or more precisely, why does this approximation suffice? I understand that this makes the wavefunction square-integrable (when integrated from $x=0$ to $x=a$) hence normalizable. Thanks."}
{"id":"18105","title":"Hydrostatics: Log floating in water near a dam","text":"So I am studying for a final and can't seem to solve this. There is a log floating in water and I need to find its weight. The question I have is what parts of the volume of the log count when summing the forces in the Y axis. ![enter image description here](http:\/\/i.stack.imgur.com\/108N2.png) What i have now for AREA alone is $r^2 - .25\\pi r^2$ pushes down and $.5\\pi r^2 + r^2$ pushes up. however this does not take into account the air above the log in the upper right quadrant. is the force pushing up taking into account the half circle below the water and also the $2r^2$ above the horizontal of the log? The actual question for this image says: A log is stuck against a dam as shown in the diagram. given the radius of the log of 1.4 m and the length of the log into the page, 10 m find the weight of the log in kN."}
{"id":"30074","title":"How to solve Poisson equation in electrodynamics?","text":"I learned electrodynamics. According to the vector potential determination, $$ \\mathbf B = [\\nabla \\times \\mathbf A ], $$ Coulomb gauge, $$ \\nabla \\mathbf A = 0, $$ and one of Maxwell's equations, $$ [\\nabla \\times \\mathbf B ] = \\frac{1}{c}4\\pi \\mathbf j, $$ I can assume, that $$ [\\nabla \\times \\mathbf B ] = \\nabla (\\nabla \\mathbf A) - \\Delta \\mathbf A = -\\Delta \\mathbf A = \\frac{1}{c}4 \\pi \\mathbf j. $$ How to prove that the one of the solutions of this equation is solution like newtonian potential, $$ \\mathbf A = \\frac{1}{c}\\int \\limits_{V} \\frac{\\mathbf j (r) d^{3}\\mathbf r}{|\\mathbf r - \\mathbf r_{0}|}? $$"}
{"id":"72900","title":"Sliding force less or equal?","text":"Why is the force required to slide a magnet off a steel plate A LOT less than the force required to directly pull it off? The force required to pull the magnet can be: 20lb While the force required to slide the magnet can be: 1lb more\/less. Why is that?"}
{"id":"72905","title":"Does the termination point(level) influence water flow from a pipe","text":"I get water to my home from a nearby Tank A at a certain height above ground level. I have a 1\" pipe through which I get this water to my home.. I leave this water into my well by connecting a 1\" tube to this pipe. Reason for question: I have seen water pressure vary(lower- more flow) depending upon the height I hold my pipe(on the outlet side) when I empty my fish tank, keeping how deep the pipe is immersed on the other side a constant. Now, if I elongate and leave the pipe from Tank A at deeper level below ground level should the flow rate increase?(will i get more water in the same time) as apposed to a shorter pipe which still goes into my well. or simply: Will I empty my overhead tank on 2nd floor quicker if I use the tap on the ground floor instead of the tap on the second floor. The taps being same size."}
{"id":"72906","title":"Reflection, transmission, absorption...how to calculate them?","text":"I was wondering whether there is an equation that enables me to calculate the reflection, transmission, absorption and polarization, when the electric field everywhere is given? Consider this: You have solved the full Mie scattering process, so incident field, the field in the sphere and the scattered field are known. How can one calculate those quantities then?"}
{"id":"31306","title":"Why does the car clock dim down so much when the headlight is turned on?","text":"Even on a current new 2012 car, when the green LED clock inside the car is quite bright, but when the car's headlight is turned on, the clock dims down to only about 1\/4 of its brightness, which makes the time hard to see. I thought the clock requires merely a watt or even less for its brightness, and the car stereo which output at least 30 watt of music (if comparing a 30 watt speaker for the PC to the car stereo), won't be less loud when the headlight is turned on. Why does the clock dim down so much? Is it explained by Ohm's law? Also, can't it be \"parallel\" instead of \"in series\" with the headlight, so that the brightness is not affected?"}
{"id":"127139","title":"Inclined conveyor belt mystery","text":"**The situation** An inclined conveyor belt with topmost point $h$ height above the ground receives sand at a constant rate $X$ from a container at a negligible height above the lowermost point of the conveyor belt. There is sufficient friction on the conveyor belt so that the sand stops almost immediately after coming in contact with the belt. **The question** On trying to find the minimum force required to maintain this situation, miraculously (after making some appropriate assumptions) the force needed at the bottom most point for the sand to start moving and the force required to maintain the motion of the sand on the conveyor belt above that point have to be same for the total force exerted by the conveyor belt to be minimum. Mathematically it is easy to reach to this conclusion but I can't understand why this happens physically."}
{"id":"98787","title":"What is the relation between General Relativity and Newtonian Mechanics?","text":"What is the relationship of General Relativity and Newtonian Mechanics? Namely, which laws does GR replace of Newtonian Mechanics, and which laws of Newtonian Mechanics are incorporated into it. Or is GR a complete replacement and overhaul?"}
{"id":"103226","title":"Step in a proof that $\\textrm{div} \\ \\mathbf{B} = 0$ from Biot-Savart's law","text":"Notation: The magnetic field $\\mathbf{B}$ generated by a point charge $e$ moving with velocity $\\mathbf{v}$ is given by Biot-Savart's law $$\\mathbf{B} = \\frac{\\mu_0 e\\ \\mathbf{v} \\wedge \\mathbf{r}}{4\\pi r^3}$$ where $\\mathbf{r}$ is the vector from the charge to the point at which the field is measured, $r = \\left| \\mathbf{r} \\right|$, and $\\wedge$ denotes vector product. Question: According to my book: Since $\\mathbf{r}\\, \/ \\, r^3 = - \\,\\textrm{grad} \\left(1\\, \/\\, r \\right)$, we have $$\\textrm{div} \\left( \\mathbf{v} \\wedge \\frac{\\mathbf{r}}{r}\\right) = \\mathbf{v} \\wedge \\textrm{curl} \\left( \\textrm{grad} \\frac{1}{r}\\right) = 0.$$ What I have is $$\\textrm{div} \\left( \\mathbf{v} \\wedge \\frac{\\mathbf{r}}{r}\\right) = \\mathbf{v} \\, . \\, \\textrm{curl} \\left( \\textrm{grad} \\frac{1}{r}\\right) - \\left( \\textrm{grad} \\frac{1}{r} \\right) \\, . \\, \\textrm{curl} \\ \\mathbf{v},$$ so I think there is a typo in the book ($ \\, \\wedge$ should be $\\, . \\,)$. However, I still don't know how to go from my equation to the correct one. Is it because $\\textrm{curl} \\ \\mathbf{v} = 0$? If so, why? I understand that the curl of a gradient vanishes identically."}
{"id":"103227","title":"If a planet were cut in half but stayed hemispherical, how hot would the ocean on the flat side be?","text":"Some time ago I asked a question about gravity on a hemispherical planet. What would gravity be like on a hemispherical planet? Would the water all boil away at first, quickly cooling the core of the planet? Would the ocean boil for centuries?"}
{"id":"103228","title":"Electric Field: distributed uniformly in one infinity tape of length","text":"> _One charge density surface is distributed uniformly in one infinity tape of > length with $2a$ width from distance $d$. Determine the Electric Field in > the point perpendicular from the distance $d$ of the centre of tape._ Answer: > $\\frac{\\rho_s}{\\pi\\epsilon_0}\\tan^{-1}(a\/d)$ PS.: I tried every ways (Coumlomb Law and Gauss Theorem) Gauss Theorem (I guess closer) Consider cylindrical surface: $Q = \\int_v \\rho_v dv = \\rho_s.2a.L$ (Call L as infinity length) $Q = \\oint_s \\vec{D}.d\\vec{S} = D_r . 2\\pi.d.L$ $\\Rightarrow E = \\large\\frac{\\rho_s}{\\pi\\epsilon_0}\\frac{a}{d}$ Coulomb http:\/\/i.stack.imgur.com\/Ks4gC.jpg The image woth more than thousand words. Please I'd like some advise to solve this problem. http:\/\/i.stack.imgur.com\/Ks4gC.jpg"}
{"id":"19055","title":"Is there really no meaning in potential energy and potential?","text":"I have been told all my physics life that potential energy between two mass\/charge has no meaning and only their difference has meaning. The same goes for electric potential, only the difference matter. Perhaps I am not understanding it correctly, but before I talk about masses, let's talk about potential energy\/potential associated with two charges. I am not sure where had I seen it, but a long time ago I was presented with a problem like this. Let's say I have a +Q and a -Q. What is the potential energy between them? The change in potential energy is the negative work done by the conservative force namely (vector sign and dot product got rid of, since the cosine 1) $\\Delta U = -\\int_{a}^{b} k(Q)(-Q) \\frac{1}{r^2} = -k(Q)(-Q) \\left. \\frac{-1}{r} \\right |_{a}^{b} = -k(Q)(-Q) \\left (\\frac{1}{b} - \\frac{1}{a} \\right) $ Now I assume that I brought it from infinitely far, so that 1\/a = 0 In that case I am left with $\\delta U = \\frac{kQQ}{b}$ Here are my questions 1) A long time ago, I saw a formula looking EXACTLY like what I just did there, but it doesn't concern with the _change_ , it's just gives me the potential energy. Now the formula I remember was $U = -\\frac{kQq}{r}$ where there is a negative. I thought this formula already takes care of the signs? Or am I wrong? 2) Kinda the same concept. If I tell you some charge (not telling you the sign) has a electric field and you have another charge (which is the test particle, not telling you the sign again even though it is conventional to use + charge ) somewhere in that field. I tell you that the electric potential at that point (not the difference) is K (where K is positive number). What can you conclude, if anything? What if it were negative? Suppose I tell you suddenly that the charges are the same signs, and I give you a location in which the electric potential is positive (I think it has to be). What does it mean? EDIT: Let me also just clarify a bit that I was taught that electric potential (not difference, I stress again) is the work that someone does to bring a charge from infinity to some point whereever. In my book however, it's defined as $V = \\int_{R}^{\\infty} \\vec{E} \\cdot \\vec{ds}$"}
{"id":"119831","title":"What is the intuition behind the direction of torque?","text":"I've seen the other posts on this question, but unfortunately I'm still having difficulty understanding the meaning of the direction of the torque vector. It makes intuitive sense that the magnitude of the torque exerted on an object is dependent on the length of the lever arm and the angle between the lever arm and the applied force. What I don't understand is what the direction of the torque vector is describing. How can I relate the direction of the torque vector to a real world example (that is not an example involving a wrench + screw and the screw \"unscrewing\" in the direction of the torque...which doesn't give me any more of an intuition on what torque means). Thanks!"}
{"id":"30824","title":"Can a photon be made to orbit a known (or undiscovered theoretical) body?","text":"Can a photon through some process be made to orbit a celestial or any other object? Two follow-up questions.   1. Can this orbit be described as the photon crossing its own path.   2. Will this wave-function be effected by positive interference. To the effect of increasing frequency?"}
{"id":"94273","title":"What force particle mediates electric fields and magnetic fields?","text":"The force carrier for magnetic fields and electric fields are supposedly photons. I don't get it: 1) Wouldn't that mean that a charged particle (e.g. an electron or even a polarized H2O molecule) would constantly be losing endergy from sending out photons? 2) Wouldn't that mean that an electric field is inseparable from a magnetic field, as photons have both - and that one can't have one without the other? 3) Would it be possible, then, to determine the wavelength of magnetic-field- mediating photons? If so, what is the wavelength - is it random or constant? 4) How can a photon (which has momentum) from one electrically charged particle to an oppositely charged particle cause these particles to _be pulled toward each other_ \\- or how can a magnetic field cause an electrically charged moving particle to experience a force _perpendicular to_ the source of the magnetic field if a particle with a non-zero mass moving between the two is the mediator of that force? If \"virtual photons\" are involved, please explain why they work differently from regular photons."}
{"id":"80651","title":"How does charge flowing between emf terminals reduce voltage difference?","text":"I'm currently learning what electromotive force is and while reading my book's description of an ideal source of emf, I had difficulty understanding what these sentences mean: > The nonelectrostatic force maintains the potential difference between the > terminals. If it were not present, charge would flow between the terminals > until the potential difference was zero. I don't quite get what it means by this. How does the nonelectrostatic force \"maintain\" the potential difference between the terminals? If it wasn't present and only the electric force remained, why would the potential difference decrease to zero? The only thing I could think of was that the electric force from the E field inside the source of emf would still be exerting forces on charges from high potential to low potential, turning its electric potential energy into kinetic energy by doing work on the charges. I still don't know how that would reduce the voltage difference across both of its terminals to zero though. Regarding the ideal source of emf, the book assumes there is a positive and a negative terminal with the positive terminal at a higher potential and the nonelectrostatic force going from low to high potential while the electric force goes from high to low, just in case that helps clear any ambiguity."}
{"id":"97743","title":"Did relativity make Newtonian mechanics obsolete?","text":"Did Einstein completely prove Newton wrong? If so, why we apply Newtonian mechanics even today? Because Newton said that time is absolute and Einstein suggested it relative? So, if fundamentals are conflicting, how can both of them be true at a time?"}
{"id":"88109","title":"Relative velocities of boats and a stone thrown by the boat","text":"> **Question:** A police boat is chasing a boat with criminals along a > straight river by moving against the stream. The speed of the river stream > is 3 miles per hour, the speed of the boat with criminals relative to the > river is 30 miles per hour, and the police boat is 4 miles per hour faster > than the boat with criminals. > > Currently the criminals are ahead of the police, and horizontally throw a > stone at the police boat at a speed 16 miles per hour relative to their boat > (i.e. relative to the boat of criminals). > > What is the horizontal velocity of the stone relative to the police boat and > to the river bank? You need to state what the origin and the positive > direction of motion are. Choose the direction of the river stream to be the positive direction and choose the origin to be in front of criminals' boat. Denote the velocity of the river stream by $\\dot{x}_R$ Denote the velocity of the police boat by $\\dot{x}_P$ Denote the velocity of the criminal boat by $\\dot{x}_C$. Denote the velocity of the stone thrown by $\\dot{x}_S$. From the question we have that $\\dot{x}_R = 3$, and we also have that the velocity of the criminal boat relative to the river stream is \\begin{align*} \\dot{x}_C - \\dot{x}_R &= -30 \\\\\\ \\implies \\dot{x}_C - 3 &= -30 \\\\\\ \\implies \\dot{x}_C &= -27 \\end{align*} Note that the minus 30 is because from the point of view of the river, the criminal boat is travelling in the negative direction. Now from $\\dot{x}_C$ I can calculate $\\dot{x}_P$, since \\begin{align*} \\dot{x}_P &= \\dot{x}_C + (-4) \\\\\\ &= -27 - 4 \\\\\\ &= -31 \\end{align*} Note that the minus 4 is because the police boat is 4 mph faster, but in the negative direction. Also from $\\dot{x}_C$ I can calculate $\\dot{x}_S$. From the question we have that \\begin{align*} \\dot{x}_S - \\dot{x}_C &= 16 \\\\\\ \\implies \\dot{x}_S - (-27) &= 16 \\\\\\ \\implies \\dot{x}_S &= -11 \\end{align*} I was wondering if my solution was right, even though $\\dot{x}_S = -11$ is negative, even though it moves in the positive direction."}
{"id":"128932","title":"Birds inside of a closed box","text":"Suppose there's a closed box with birds inside, on top of a scale. If the birds started flying inside, would the box get lighter?"}
{"id":"81955","title":"Weighing a flying bird","text":"Let thing of a bird standing still in a box on top of a weighing machine that shows a mass $m_0$. Now, imagine that the bird is flying, still in the same box and the same weighing machine shows a mass $m_1$. As the bird when flying, is applying a force towards the weighing machine, could we deduce that $m_0 = m_1$? I'm asking this question because saying that these masses are equal makes me as uncomfortable as saying that they are not equal and can't figure the right answer. So does $m_0 = m_1$ and why?"}
{"id":"77626","title":"Bird flying in a cage","text":"Assume that you are holding a cage containing a bird. Do you have to make less effort if the bird flies from its position in the cage and manages to stay in the middle without touching the walls of the cage?   Does it make a difference whether the cage is completely closed or it has rods to let air pass?"}
{"id":"35409","title":"Why is a nucleus isotropic?","text":"I believe in Neutron Scattering the neutrons after hitting a nucleus can bounce in any of 360*3 dimensions -> 1080 degrees? Why is this so? Shouldn't it only bounce \"off\" the neutron in approximately the same \"direction\" that it came in such as when a particle bounces off a mirror -> because of the cross-section ..."}
{"id":"88106","title":"Proof that QCD is the theory describing strong interactions?","text":"I would like to ask what are the experimental evidences that led to the conclusion that QCD is the right theory to describe strong interactions. I know that some of the key point are the decay of $\\pi_{0}$ and the measurement of Jets but I'd love to see a full answer to this question. Is there a still a chance to Regge theory nowadays?"}
{"id":"100752","title":"Electric Potential of a Cube Made of Point Charges","text":"I'm trying to find the potential energy of multiple geometric shapes made entirely out of point charges. This particular shape is a cube made out of two different point charges, A and B, each separated by a constant distance. I'm not entirely sure how to go about this problem but I'm pretty sure that the equations V=Ed and E=kQ\/r^2. I'm pretty sure you just add up the charges but I have no idea how to do this in a cube form.."}
{"id":"120008","title":"Why does lightning cause sound?","text":"If I understand correctly, lightning is the discharge of electricity from the atmosphere into the planet. However, if I switch on a lamp, the wires are not causing thunder (or any audible sound). I've also heard that the thunder comes from lightning breaking the sound barrier. This sounds weird to me since I would assume that lightning would be traveling at, well, light speed, so I'm not sure how the threshold could be crossed. How does lightning cause thunder?"}
{"id":"63706","title":"The gauge-invariance of the probability current","text":"It is simple to show that under the gauge transformation $$\\begin{cases}\\vec A\\to\\vec A+\\nabla\\chi\\\\\\ \\phi\\to\\phi-\\frac{\\partial \\chi}{\\partial t}\\\\\\ \\psi\\to \\psi \\exp\\left(\\frac{iq\\chi}{\\hbar}\\right)\\end{cases}$$ The Schrodinger equation $$\\left[-\\frac{\\hbar^2}{2m}\\left(\\nabla-\\frac{iq\\vec A}{\\hbar}\\right)^2+q\\phi \\right]\\psi=i\\hbar\\frac{\\partial}{\\partial t}\\psi$$ gives back the same equation. How does it follow that the probability current is gauge invariant?"}
{"id":"9113","title":"How sound intensity (dB) and sound pressure level (dB) are related?","text":"Source: http:\/\/www.engineeringtoolbox.com\/sound-power-intensity-pressure- d_57.html Both sound intesity and pressure level are measured in dB. Given a specific sound, are these two dB values the same?"}
{"id":"16998","title":"Duality and Fourier Transforms","text":"I read that $(FF(f))(x)=2\\pi f(-x)$, where $F$ is the Fourier transform _and_ $F(f(x-a))(k)=\\exp(-ika) X(k)$ where $X(k)=F(f(x))$ _implies_ $F(\\exp(iax)f(x))(k)=X(k-a)$. But I don't see how that is done... I am quite happy with getting $F^{-1}X(k-a)=\\exp(iax)f(x)$ by brute force calculation. I would like to see how to use duality though."}
{"id":"12284","title":"What's the definition of the time ordering operator for more than two particles?","text":"For two particles, $\\langle {\\mathcal T} a(t_1) a^\\dagger (t_2) \\rangle = \\langle a(t_1) a^\\dagger (t_2)\\rangle \\theta (t_1-t_2) + \\xi \\langle a^\\dagger (t_2)a(t_1) \\rangle \\theta (t_2-t_1)$ with $\\xi$ is a plus sign for bosons and a minus sign for fermions. How would I write, for example, $\\langle {\\mathcal T} a(t_1) a^\\dagger (t_2) a(t_3) a^\\dagger (t_4) \\rangle$ ?"}
{"id":"81300","title":"Velocity field induced by vortex points along ellipse","text":"I'm investigating the velocity field induced by a continuous distribution of 2D vortex points distributed along an ellipse $\\\\{a\\cos\\theta,b\\sin\\theta\\\\}$. I'm interested in the field inside the ellipse, and I need some help to prove whether this field is zero or not. The intensity of each vortex point is proportional to $d\\theta$ and not to the length along the ellipse. A vortex point located at a point $\\boldsymbol{x'}$ induces a velocity field $\\boldsymbol{u}(\\boldsymbol{x})=\\frac{d\\theta}{2\\pi |\\boldsymbol{x}-\\boldsymbol{x'}|} \\boldsymbol{e}_\\perp$ where ${e}_\\perp$ is the unitary vector orthogonal to $(\\boldsymbol{x}-\\boldsymbol{x'})$ which is in 2D: $\\boldsymbol{e}_z\\times(\\boldsymbol{x}-\\boldsymbol{x'})\/|\\boldsymbol{x}-\\boldsymbol{x'}|$. The total velocity field at a point $(x,y)$ inside the ellipse is obtained by integration over $\\theta$. Numerical experiments seem to show that the field is zero inside the ellipse, but I cannot prove it. Dropping the factor $2\\pi$, the field is in cartesian coordinates: $$\\boldsymbol{u}=\\int_0^{2\\pi} \\left\\\\{\\frac{-y + b \\sin\\theta}{(x - a \\cos\\theta)^2 + (y - b \\sin\\theta)^2}, \\frac{x - a \\cos\\theta}{(x - a \\cos\\theta)^2 + (y - b \\sin\\theta)^2}\\right\\\\} d\\theta \\stackrel{?}{=}\\\\{0,0\\\\}$$ Is the field really zero? Maybe there is no need for the integrals to prove it. Maybe complex analysis is of help? I should mention that the following property is true in this problem: For any closed contour inside the ellipse the circulation is zero: $$\\oint \\boldsymbol{u}\\cdot\\boldsymbol{dl} =0$$ Since we are in a simply connected region, the velocity is the gradient of a potential which is single valued. But then is this potential constant?... I'm able to prove that the velocity is zero on both the $x$ and the $y$ axis. Also $u_x$ is an even function of $x$ and an odd function of $y$. The opposite applies for $u_y$. I'm able to prove the result for a circle $a=b$. Can somebody help me for the ellipse? Any idea?"}
{"id":"21484","title":"Why was PACER abandoned?","text":"The PACER project is described in this question: How much of the energy from 1 megaton H Bomb explosion could we capture to do useful work? Why was it abandoned? It seems that it is the only readily economical and engineeringwise useful path to fusion power, and it seems that its breeder possibilities can easily let it pay for itself for generating fissile elements and helium (which is getting to be rare too nowadays!) Was it political or technical limitations that killed it? Is there hope for a renewed interest in this in todays energy conscious politics?"}
{"id":"21486","title":"Dielectric in Parallel Plate Capacitor","text":"Given a parallel plate capacitor of width $w$, length $l$, with a dielectric moving along the length $l$. Let the dielectric be from $x$ onwards. The capacitance will be $\\frac{w \\epsilon_0}{d} (\\epsilon_r l - \\chi_e x)$. Griffiths (p. 195) says that the total charge $Q$ in the $C=\\frac{Q}{V}$ expression is constant as the dielectric moves. But $Q$ here refers to the free charge, and the free charge definitely increases as you move the dielectric in increasing $x$. What am I misunderstanding?"}
{"id":"16484","title":"Classical car collision","text":"I have a very confusing discussion with a friend of mine. 2 cars ($car_a$ and $car_b$) of the same mass $m$ are on a collision course. Both cars travel at $50_\\frac{km}{h}$ towards each other. They collide. Ignoring any shreds and collateral damage, what is the speed of collision that the driver of $car_a$ felt? What I mean is, if $car_a$ were to be driven into an infinite mass wall, what would the velocity be to replicate the damage caused by the initial collision?"}
{"id":"82143","title":"Is a collision with an immovable object equal to a head-on collision?","text":"**Scenario 1** Two identical and bilaterally symmetrical cars, driven by identical drivers in the exact center of the car and in the same body position, each traveling the same speed, collide exactly head on. (Essentially, eliminate all variables that would cause rotation or interpenetration during the collision). **Scenario 2** One of the above cars hits an immovable wall instead of an oncoming car. **Question** Will each car in these scenarios experience the same forces? For clarity, the cars in scenario 1 have twice the closing speed of the car and wall in scenario 2."}
{"id":"45578","title":"Is two cars colliding at 50mph the same as one car colliding into a wall at 100 mph?","text":"I was watching a youtube video the other day where an economist said that he challenged his physics professor on this question back when he was in school. His professor said each scenario is the same, while he said that they are different, and he said he supplied a proof showing otherwise. He didn't say whether or not the cars are the same mass, but I assumed they were. To state it more clearly, in the first instance each car is traveling at 50mph in the opposite direction and they collide with each other. In the second scenario, a car travels at 100 mph and crashes into a brick wall. Which one is \"worse\"? When I first heard it, I thought, \"of course they're the same!\" But then I took a step back and thought about it again. It seems like in the first scenario the total energy of the system is the KE of the two cars, or $\\frac{1}{2}mv^2 + \\frac{1}{2}mv^2 = mv^2$. In the second scenario, it's the KE of the car plus wall, which is $\\frac{1}{2}m(2v)^2 + 0 = 2mv^2$. So the car crashing into the wall has to absorb (and dissipate via heat) twice as much energy, so crashing into the wall is in fact worse. Is this correct? To clarify, I'm not concerned with the difference between a wall and a car, and I don't think that's what the question is getting at. Imagine instead that in the second scenario, a car is crashing at 100mph into the same car sitting there at 0mph (with it's brakes on of course). First scenario is the same, two of the same cars going 50mph in opposite directions collide. Are those two situations identical?"}
{"id":"108502","title":"If two objects collide, will the collision force be twice as big as the original force?","text":"Say, if we have two trucks travelling towards eachother at 60m\/s, and those two trucks carry the same payload, and are the same model (aka they're exactly the same), will the collision force be as if just one truck crashed into a wall at 120m\/s? Theoretically the forces will balance each other out, but it won't always be this way."}
{"id":"90216","title":"Where does the loss in gravitational energy of the load go when a spring is pulled?","text":"A mass spring system is in equilibrium. If I pull on the load by $x$ meters, the energy stored in the spring is (this is what is given in my book): $$E=\\frac12kx^2 $$ However, doesn't the load lose gravitational potential energy as it moves down? Where would this energy go? By conservation law, shouldn't the energy equation be: $$E_{stored}= \\frac12kx^2 + mgx$$ In short, where does the loss of gravitational potential energy (mgx) of the load get transferred to if it is not stored in the spring? (Referring to a vertical mass spring system) The picture in my mind: ![enter image description here](http:\/\/i.stack.imgur.com\/OdVee.png)"}
{"id":"13155","title":"How to know the time a disc takes to stop from rotating and the numbers of revolutions","text":"Assuming I rotate a disk, I want to know how long it takes to completely stop, and the number of revolutions it made since I removed my fingers off the disk. Lets say a DVD I rotate with my fingers. I only know the radians per second (velocity) of the last moment I touched the disc. Can you guys tell me where to start? Im trying to implement this on an iPhone app. So it would be nice if you mention **equations**. It should not be exact."}
{"id":"10591","title":"Simulation of physics of chains\/ropes in force fields resources?","text":"I'm thinking about a project to tackle, and I'd like to make a simulation that allows the user to define a rope or chain of length L, pin it at arbitrary points r1, r2.... etc. and draw the resulting curve in real time. Also, I'd like the user to be able to alter the field that the rope exists in, for example more complex vector fields than just a straight gravitational field. This is a bonus, however, and I'd like to get the basic example working. Could you recommend some resources, preferably free\/online, for me to learn the physics involved (I'm not exactly sure what to CALL this area of classical mechanics)? I learn through concepts and then math so a resource that is concept-heavy would be nice. Thanks!"}
{"id":"94463","title":"how apparent weight varies due to the rotational motion of earth","text":"I learned that as the earth rotates about its axis, the bodies on the earth also follow a circular path. In most books I read, they give the example of a person standing on a weight balance at the equator... and I did understand that. However, by doing the following calculation, I am seeing that the apparent weight at other points on the earth (apart from the poles) is the same This is the picture on my mind: ![enter image description here](http:\/\/i.stack.imgur.com\/4C73k.png) At B, $$W-N=m{\\omega}^2 R$$ At A, a component of weight will provides the centripetal force to rotate around the circle with the radius $r$, $$Wcos{\\theta} - Ncos\\theta =m{\\omega}^2r $$ as $r=Rcos\\theta$, $$Wcos{\\theta} - Ncos\\theta =m{\\omega}^2R\\cos\\theta$$ The equation eventually ends up as... $$W - N =m{\\omega}^2R $$ So from this, I think that the normal reaction force which is the apparent weight remains the same as to the apparent weight at the equator. However the book states that the apparent weight varies along $A$ and $B$ Also, we assume that the earth is spherical. I am really sorry for making the question so long. Could someone please tell me which part of my concept is wrong."}
{"id":"20353","title":"Calculating lagrangian density from first principle","text":"In most of the field theory text they will start with lagrangian density for spin 1 and spin 1\/2 particles. But i could find any text where this lagrangian density is derived from first principle."}
{"id":"94469","title":"Can a bullet leave a gun and tumble to the ground?","text":"This question seems to have been asked a few times in different configurations, but none of them answer my variation. I've struggled to understand this for nearly 15 years and had conflicting answers from my school physics teachers and more recently friends who are physicists. So a round leaves my gun barrel at $40$ m\/s. Its initial Airspeed is $40$ m\/s. In another question they asked about firing down the length of a train traveling the same speed. I understand that firing toward the front of the train would result in a ground speed of $80$ m\/s and an airspeed of $40$ m\/s. Also, firing toward the back of the train results in a ground speed of $0$ m\/s and an airspeed of $40$ m\/s. My question... if you were to walk to the back of the train, open the door and fire directly out the back, would you end up with a ground **and** airspeed of $0$ m\/s? Meaning the projectile would literally just tumble in a straight trajectory down to the ground? Plenty of people have thrown spanners into this one over the years - like talking about the way an explosive force will 'hang' in a certain space if not pushed, pushing the projectile away from itself as well as the firing pin of the gun, giving it extra forward momentum in that instance. (unlike firing from the front of the train would always equal $80$ m\/s for air and ground. The explosion cannot be 'left behind'). I personally don't buy this one... But i don't know enough to judge."}
{"id":"82729","title":"Momentum of light in medium","text":"Maybe this has been asked before, but I didn't find anything about it. I am wondering about the momentum of light in media with refractive index n>1 (so to say, not in vacuum). There are two approaches to this problem I think, but they lead to different results. First, one can take the equation $p = \\frac{h}{\\lambda}$ Because it holds $\\lambda = \\frac{\\lambda_0}{n}$, one gets $ p = n \\frac{h}{\\lambda_0} = n\\cdot p_0$. So momentum is larger than in vacuum. Second, one can take the eqation $p = m \\cdot v$. Because $v= c$ for phonons and in medium it holds $c=\\frac{c_0}{n}$, one arrives at $p = \\frac{m c_0}{n} = \\frac{p_0}{n}$, which is clearly smaller than $p_0$. These two results dont fit together. So which result is right and why?"}
{"id":"132584","title":"A statics problem to find minimal friction","text":"consider the following: ![enter image description here](http:\/\/i.stack.imgur.com\/tpCcw.jpg) I need to find the minimal coeeficient of friction $\\mu _{min}$ so that both recatngle boxes would remain static. The lower angle in the triangle is $2\\alpha$ as indicated. I ended up with this expression: $$\\mu _{min}=\\frac{m_2}{2\\cdot m_1 \\cdot \\tan \\alpha}$$ but the answer in the book is rather this one: $$\\frac{m_2}{(2m_1+m_2)\\tan\\alpha}$$ Who is correct?I know that the normal force that that the triangle is exerting on both is like this: ![enter image description here](http:\/\/i.stack.imgur.com\/qYGQr.jpg) (on both sides of course) so with geometry and soome FBD that's what I came out with. Would like to hear your thoughts!"}
{"id":"29533","title":"top quark and Z,W bosons?","text":"The masses of the Z and W particle sum almost exactly to the mass of the Top quark,within the errors: Z + W = 80.385±0.015 + 91.1876±0.0021 = 171.57 ±0.0171 GeV Top quark 172.9± 1.5 GeV A: Is this one of those simple coincidences? B: The Z,W particles are decays of the T? C: Someone has a not too cranky idea connecting them? EDIT: After consideration of dmckee and Lubos posts. How about instead of a decay from a t quark, collide a $W^\\pm$ and a $Z$ to produce a red top and anti-red bottom. $$W^+ + Z^0 \\to t(r) + \\bar{b}(\\bar{r})$$ this conserves charge, spin, color, confinement and energy - provided the excess energy of the bottom quark comes from the kinetic term of the collision. It immediately decays as lubos and dmckee pointed out in an early question EDIT 2: Also note decay time of t-quark is $4.2\\ 10^{-25} s$, nearly matching the W,Z decays times of $3.0\\ 10^{-25} s$ , although I'm yet to find an uncertainty for these. And with incredible hubris I'm calling this the Metzgeer Momentary Meson $t\\bar{b} $   :) joke"}
{"id":"34844","title":"Can entanglement swapping be performed on already-entangled photons, and if so, can it preserve this entanglement over the swap?","text":"Consider 2 uncorrelated photon pairs (a1,a2), (b1,b2) such that (a1,a2) are entangled, and separately (b1,b2) are entangled. We wish to entangle-swap so as to end up with a new entanglement (a1,b1) by using the ancillary photon pair (a3,b3), such that (a1,a3) are entangled and (b1,b3) are entangled. This suggests that we do a multipartite (3-way in this case) entanglement preparation prior to the swap, such that (a1,a2,a3) are entangled, and separately (b1,b2,b3) are entangled. The swap proceeds in the usual way by Bell state measurement on a3 and b3, and we end up having entangled (a1,b1). The question is whether, after this operation, the entanglements (a1,a2) and (b1,b2) remain intact? If not, can we modify the procedure to guarantee this? If yes, will (a2,b2) now be entangled as a result of the swap? (they were uncorrelated before it). If not, how can we arrange that they are entangled, by only manipulating the other 4 photons?"}
{"id":"81085","title":"Why does dust stick so well to fan blades?","text":"After reading and understanding the reasons why dust stick to rotating fan propeller, I am interested to find out why the dust particles stick so well. Spraying powerful jets of water does not effectively remove the dirt. Some scrubbing is still required, whether it is metallic blades or plastic ones. Why is that so?"}
{"id":"59285","title":"Application of non maximally entangled state","text":"In quantum information and quantum computation, we generally use Bell type states which are maximally entangled. I find that the set of entangled states as interesting objects from a mathematical point of view and one can ask many questions regarding their structure and so on. But my question is, what are the practical uses of such states and why are they physically interesting. The only case I can find out is, when due to some environmental interaction such states are created. However I can only see the use of maximally entangled states in literature. Caveat: In multipartite cases, maximal entanglement can be tricky. I am not sure, whether I have have phrased the question correctly. Feel free to ask and edit. Advanced thanks for any help, suggestion etc."}
{"id":"92103","title":"Is the resistance 0 in the ballistic regime?","text":"Given that in the ballistic regime a particle (electron) can move freely without scattering (there are no impurities ), is the resistance through a ballistic sample zero?"}
{"id":"66624","title":"Perfectly focusing refractive surface","text":"On reading Feynman's lecture on physics, in the geometrical optics section he said that a curve which focuses all the rays coming from a point to another fixed point beyond the refracting surface perfectly is a complicated fourth degree curve which is actually the locus of all point with its distance $Op$ from one point, $O$ plus the distance $O_1p$, from another point $o_1$ times a number $n$ (refractive index of the material beyond the refractive surface) is a constant independent of point $p$ on the curve. But I don't know how to derive this curve, or as a matter of fact, how to even begin to. Any derivation would be greatly appreciated (simple preferred).given below is the excerpt from feynman's lectures which is what i reffered to in my question.![enter image description here](http:\/\/i.stack.imgur.com\/QnY7n.png) ![enter image description here](http:\/\/i.stack.imgur.com\/1pFTy.png)"}
{"id":"2392","title":"Have the Rowan University \"hydrino\" findings been replicated elsewhere?","text":"In 2009, Rowan University released a paper claiming to replicate Blacklight Power's results on energy generation using hydrino states of the hydrogen atom. The paper appears to describe the procedure in every detail as far as my untrained eye can tell. Has anyone else attempted to replicate these results and did they succeed?"}
{"id":"104050","title":"Quantum entanglement: does it necessarily imply superluminal information transfer?","text":"From what I understand, information is communicated _instantly_ between two quantum-entangled particles regardless of the spatial distance between them. However, does this necessarily imply superluminal data transfer? If the \"distance\" between the particles is simply assumed to be measured in our common 3D euclidean space then that would seem to imply superluminal communication. However, could there be other paths between the two particles that lie beyond our observable 3D euclidean space, other paths that involve additional dimensions in which the particles remain very close or some sort of folding of space on other dimensions that provide an information pathway? Anybody aware of any such research\/findings? Thnx."}
{"id":"131025","title":"Why can't classical information be transmitted through quantum entanglement by measuring time between change events?","text":"If the effect of measurement in quantum entanglement occurs instantaneously why can't it be used to transmit classical information faster than light? I'm based my question off this seemingly well cited Wikipedia article: http:\/\/en.wikipedia.org\/wiki\/Quantum_entanglement#cite_note-10 Edit: It would seem that my question should be more focused on why we can't use change as a method of data transmission."}
{"id":"34507","title":"Why do we think that quantum entanglement implies action at a distance?","text":"> **Possible Duplicate:**   >  Why quantum entanglement is considered to be active link between particles? I am a layman trying to read into quantum mechanics. As I understand it quantum entanglement is an experimentally proven mechanism by which the quantum state of two particles become 100% correlated, that is, measuring the quantum state of one object causes you to know the quantum state of its entangled particle with 100% certainty (they're opposite). Where I don't understand is why this is used to imply that information is \"traveling\", faster than light or not. Where is the intuition coming from that the correlation between two separated particles has anything to do with one particle effecting the other? For me, a much stronger explanation for why these two particles are correlated is that the entanglement itself effects the particles, that is, upon becoming entangled their quantum states are individually governed by some formula (as a function of time alone) where those formulas produce exact opposite results for each particle. Separating the particles doesn't have to change the formula, so if you measure one you know the other just because they must be opposite. Yet from what I've read it's taken for granted that quantum entanglement = information is traveling from one to the other. Do we have a reason for thinking that? Have we shown that we can definitely modify the quantum state of one particle and therefore change the quantum state of its entangled partner, or anything else to show that information is traveling? It's the old question of \"causation vs. correlation.\" In principle, the concept of measurement itself modifying the Universe itself (beyond any observer effects) strikes me as somewhat silly."}
{"id":"3158","title":"Why quantum entanglement is considered to be active link between particles?","text":"From everything I've read about quantum mechanics and quantum entanglement phenomena it's unobvious for me, why quantum entanglement is considered to be active link. I.e. it's stated every time that measurement of one particle _affects_ another. While in my head there is less magic explanation: the entangling measurement affect both particles in the way which makes their states identical, though unknown. In this case measuring one particle will reveal information about state of the other, but without magic _instant_ modification of remote entangled particle. Obviously, I'm not the only one who had this idea. What are the problems associated with this view and why _magic_ view is preferred?"}
{"id":"110591","title":"Suggest why electrical power input P1 differs from power used P2","text":"We have done an experiment heating water with a wire element and have determined it takes 21.5964 watts to heat 200ml of water to a specific temperature in 10 minutes. However the power of the electrical input is 29.9859 watts. It asks for an explaination of why they are different but I'm a bit lost"}
{"id":"110599","title":"How does $p_x$ commute with $p_y$, i.e. $[p_x,p_y]=0$?","text":"I know it's a simple and basic question but would someone show me how to evaluate $[\\hat{p}_x,\\hat{p}_y]$?"}
{"id":"22449","title":"What is a general definition of the spin of a particle?","text":"In quantum field theory, one defines a particle as a unitary irreducible representations of the Poincaré group. The study of these representations allows to define the mass and the spin of the particle. However, the spin is not defined the same way for massive particles (where the eigenvalue of the Pauli-Lubanski vector squared are $-m^2 s(s+1)$ where $$s = -S, -S +1, \\cdots, S,$$ and $S$ is the spin of the particle (and $m$ the mass)) and massless particles (where the helicity has eigenvalues $\\pm \\lambda$ and $S=\\left|\\lambda\\right|$ - not to mention continuous spin representations). With this definition, the \"spin\" $S$ that appears in both cases doesn't seem to be exactly the same thing. The eigenvalue that labels the irreducible representations are not from the same operator in the massive and the massless case ... however, it's tempting (for me) to see the maximum value of these eigenvalues as the same physical quantity if both cases. So I wonder if there is a more general definition that would embrace both massive and massless particles (even less practicable)?"}
{"id":"26354","title":"What would the universe look like if it started out without any irregularities?","text":"_(Warning! Newbie question coming up!)_ ## Background As seen on this picture of the cosmic microwave background (take from the Wikipedia entry on the very same topic) there exists irregularities in the distribution of matter: ![Cosmic microwave background. Boom baby!](http:\/\/i.stack.imgur.com\/KjAdE.png) To my knowledge, this is because the Universe wasn't formed with all the matter (or whatever you should call whatever existed right after the big bang) in a completely regular pattern (or if it was the case that anti-matter had this big fight with ordinary matter after the Big Bang, it's really not that important I guess), and I guess this is the reason why we have a highly diverse universe today, inhabited by galaxies, black holes, and Justin Bieber. ## Question What would the Universe look like if there were no irregularities in it from the get-go? Would we have a big massive black hole in the middle (if one even can talk about a middle), would the Big Bang never happen, or have I just misunderstood the whole thing, making my question completely nonsensical?"}
{"id":"32505","title":"How do I adjust the kinematic equations to avoid reaching speeds faster than light?","text":"I like some 'science' in my 'science fiction', so I started crunching out the kinematic equations for some of the scenarios my characters are getting involved in, and ran smack dab into an issue. (Please excuse my formatting, I can't figure out sub\/superscript notations) $v_f = \\sqrt{2ad}$, with $v_i$ assumed to be 'close enough' to zero. That's easy enough to crunch the math on. Except... With sufficiently high values for $a$ or $d$, you start crowding, or even violating, the light speed limit. And I can't find the equations to help handle relativistic distortion! I know they exist because I remember working with relativity when I took my physics class, years and years ago."}
{"id":"98029","title":"A question about spontaneous symmetry breaking at macroscopic limit","text":"The question relates to this post. Spontaneous symmetry breaking is somehow a volume effect, that in-principle only happens at infinity large system. Weinberg in the second volume of his QFT used a chair demostrated the lost of rotational symmetry due to large scale of the chair. p. 163 > We do not have to look far for examples of spontaneous symmetry breaking. > Consider a chair. The equations governing the atoms of the chair are > rotationally symmetric, but a solution of these equations, the actual chair, > has a definite orientation in space. Here we will be concerned not so much > with the breaking of symmetries by objects like chairs, but rather with the > symmetry breaking in the ground state of any realistic quantum field theory, > the vacuum. p.164-165 > Spontaneous symmetry breaking actually occurs only for idealized systems > that are infinitely large. The appearance of broken symmetry for a chair > arises because it has a macroscopic moment of inertia $I$, so that its > ground state is part of a tower of rotationally excited states whose > energies are separated by only tiny amounts, of order $\\frac{{\\hbar}^2}{I}$. > This gives the state vector of the chair an exquisite sensitivity to > external perturbations; even very weak external fields will shift the energy > by much more than the energy difference of these rotational levels. In > consequence, any rotationally asymmetric external field will cause the > ground state or any other state of the chair with definite angular momentum > numbers rapidly to develop components with other angular momentum quantum > numbers. The states of the chair that are relatively stable with respect to > small external perturbations are not those with definite angular momentum > quantum numbers, but rather those with a definite orientation, in which the > rotational symmetry of the underlying theory is broken. For the vacuum also, To a very good approximation, the chair is described by quantum electrodynamics $$L= -\\frac{1}{4} F_{\\mu\\nu}F^{\\mu\\nu} + i \\bar{\\psi}(\\gamma^{\\mu}D_{\\mu}-m) \\psi $$ The rotation of chair corresponds to $SO(3)$ group as a subgroup of the Lorentz group $SO(1,3)$. If vacuum is similar with the chair, the rotational symmetry could also break together with global rigid $O(N)$ symmetry. The standard argument that Higgs boson being a scalar, is to keep Lorentz invariance. If the $SO(3)$ rotational symmetry is lost, then this argument is not necessarily valid. Is there any other rationalization for the Higgs boson being scalar? If vacuum is _not_ similar with the chair, why only global rigid $O(N)$ symmetry is broken but not $SO(3)$? Is that a pure guess then confirmed experimentally?"}
{"id":"7359","title":"If the universe is expanding, what is it expanding into?","text":"1. If the universe is expanding, what is it expanding into?   2. When the big bang happened where did it occur?    3. When the big bang happened how did it occur?    4. Where did the energy come from? Energy can not be created or destroyed does that mean, energy has existed before the universe was here?"}
{"id":"126168","title":"Is the universe infinite?","text":"Is the universe infinite? Or is it finite? If it is infinite,it's very difficult to imagine an endless space(though not impossible). But if it's finite, the idea that we can't go beyond a certain space just creeps me out. So, what is it actually?"}
{"id":"25271","title":"Can we prove that the universe is finite or infinite?","text":"If I say that the universe is finite, how can you say with certain that I am wrong?"}
{"id":"77614","title":"What is the universe expanding into?","text":"Due to curiosity, it made me wonder what is outside of the universe, is it a new chemical or just empty space? But empty space have protons appearing and disappearing in it, it basically still has something in it. (It might have something to do with string theory.)"}
{"id":"24017","title":"Is the universe finite or infinite?","text":"I thought the universe was finite, but then I read this: How can something finite become infinite? And they seem to assume it is infinite. So which is it?"}
{"id":"8115","title":"If the Big Bang theory suggests that the Universe is \"expanding\" then what is it expanding INTO?","text":"I am a software engineer and not an astrophysicist but I want to know if anyone is working on sorting out what exists outside the universe? So what about outside spacetime? what is there? or shall I assume there are infinite galaxies? and even that is not a satisfying answer. I know this question might get closed soon ... but guys please tell me what is there outside the space as we know it? since childhood I have wondered about this. Its really a simple question. If we say that we live \"inside\" a Universe ... then we are objectively saying that there IS something \"outside\" the universe as well? Am I crazy? What exists there? Is it just empty space? Or what?"}
{"id":"100429","title":"Space and the size of infinity","text":"If you could build a spaceship and keep travelling in one direction, what would finally happen? One answer is that you would never ever reach the end. But this sounds purely platonic space and comes into conflict with the big bang theory (or the very first big bang, if you claim there have been many big bangs). That is, space was created in the big bang and is expanding and there is no space outside that realm. So what happens if the spaceship doesn't continue travelling forever? This brings us to the second possible answer: the spaceship would eventually stop travelling farther, change its direction and may return. To me, both of these answers are equally scary and unconvincing and contradicting. What would be the ultimate fate of the spaceship considering the big bang as the origin of space? UPDATE: The attempt here is to also make a _distinction_ between the space produced by the big bang and the dimensions outside the universe or whatever existed before the big bang, and come up with an answer as less contradictory as possible. Let us, only for the sake of discussion, imagine a virtual spaceship that can overtake the accelerating expansion of the universe and let it travel in a particular direction. When it reaches the edges of the universe, if that is not very inconvenient, it would surpass the universe itself, in case it doesn't go in circle. Even though this particular thought experiment wouldn't be fundamentally different than the expansion of the universe itself, it may help understand a bit about beyond the universe. Now if the spaceship continues travelling alone outwards, would it be _extending_ the space itself in that particular direction?"}
{"id":"98587","title":"What is actually universe? only matter or both matter and space?","text":"I have googled for the \"meaning of universe\" where I found the following: \"all existing **matter and space** considered as a whole; the cosmos. The universe is believed to be at least 10 billion light years in diameter and contains a vast number of galaxies; it has been expanding since its creation in the Big Bang about 13 billion years ago.\" Now,I have heard about big bang theory which tells that universe was initially a point of infinite density .But in the above definition I see space is also included in the universe. so if the universe was initially a point then **what was outside that point?** And **how this point is expanding if there is nothing outside it?** That is why I don't understand the balloon analogy also which is expanding in space. But here the universe itself contains the space according the google definition. Probably I have this question in my mind because I couldn't visualize how the space meets on itself in 3 dimension like earth's surface does on 2 dimension."}
{"id":"104424","title":"Capacitor with dielectric between the plates","text":"Let's assume we have a capacitor of capacitance $C$ and potential difference $U$. After charging it we disconnect it. Then we put a dielectric between the plates. I know that capacitance will increase by $C * k$, however what happens with the charge and potential difference on it. Let's say $k = 2$. Will $q$ double or will $ U $decrease to a half ?"}
{"id":"104421","title":"Frames, Tetrads and GR","text":"Given a general metric, $g_{ab}$ I can select an orthonormal basis $\\omega^{a}$ such that, $$g_{ab} = \\eta_{ab}\\omega^a \\otimes \\omega^b$$ where $\\eta_{ab}$ = $\\mathrm{diag}(1,-1,-1,-1).$ We may conveniently compute the spin connection and curvature form by employing Cartan's equations. The problem I have lies in getting back to the coordinate basis. I know the general formula, $$R^{\\mu}_{\\nu \\lambda \\tau} = (\\omega^{-1})^{\\mu}_a \\, \\omega^b_\\nu \\, \\omega^c_\\lambda \\, \\omega^d_\\tau \\, R^{a}_{bcd}$$ where the l.h.s. $R^{\\mu}_{\\nu \\lambda \\tau}$ is in the coordinate basis. The objects $\\omega^b$ are familiar, they're just the orthonormal basis, so what is the object $\\omega^b_\\nu$ (with the extra index)? The $(\\omega^{-1})^{\\mu}_a$ are the inverse vielbeins? How are these obtained? I've visited several sources, including Wikipedia, but it's still not 100% clear. I'd appreciate any clarification, especially a small explicit example if possible."}
{"id":"114657","title":"What does \"causally connected\" or \"causes\" really mean?","text":"In a different thread, a user stated the following in respect of events preceding or following other events: > However, if the two events are causally connected (\"event A causes event > B\"), the causal order is preserved (i.e. \"event A precedes event B\") in all > frames of reference. My question is what does \"causally connected\" really mean? What does \"causes\" mean? Further, given that we know that we can have instantaneous effects in typical quantum processes (e.g. flip the polarizer, effect on another a reading light years away, even though we cannot transmit useful information with this), does that not constitute \"causing\" for the purpose of this statement?"}
{"id":"116632","title":"Normal reaction - force without acceleration","text":"When a body lies on the surface of the Earth it is under the influence of gravity. The force on the body due to gravity causes it to exert a force on the ground and the normal reaction acts in the opposite direction causing the resultant force on the body to be zero. However, how can the body exert a force on the ground when it does not have any acceleration? Since force equals mass times acceleration how does a body without acceleration experience a force?"}
{"id":"106376","title":"Can entropy of Universe be constant?","text":"If I understand entropy correctly, then for example two objects orbiting a centre of mass have lower entropy than when said objects eventually crash into each other and form a new one. So let's say that a typical galaxy spirals around its centre of mass and eventually objects within it will fall into the center thus increasing its entropy. But if the entropy of the Universe was somehow to be constant, then maybe that's why space is expanding? As each galaxy becomes more chaotic while objects are going closer and closer together these galaxies are at the same time becoming more and more spread apart thanks to expansion of space. I don't know the exact calculations of entropy, but is it possible that there is a mechanism triggering space expansion as a reaction to gravity increasing local entropy? Also gravity is increasing in intesity when the distance between objects is shorter, so the longer two masses are gravitating, the shorter the distance between them and then the more intensive gravity becomes over time. This might correspond then to the increasing speed of expansion of Universe, as it has to compensate faster to keep entropy constant. Forgive me if what I said above is completely incorrect; I've been thinking about entropy and came with this idea, but I'm no professional physicist and would like to know if any of the above make sense."}
{"id":"48543","title":"Do some half-lives change over time?","text":"I was recently doing some physics tuition on radioactivity and the student claimed her chemistry teacher had said that radioactive substances can be grouped into two divisions: those whose half-life is constant and those whose half-life changes over time. I had never heard of this before and can't think of any reason why a half-life should change, so does anyone else know anything about this? (I know some half-lives can be altered under certain conditions, but I'm talking about a natural change over time)."}
{"id":"60115","title":"How does a thermal temperature gun work?","text":"I once worked as a kitchen porter over a winter season.   We had fun with thermal temperature guns (like these) which I learned can be used for measuring the temperature of something a reasonable distance away (aside from the obvious use of laser tag), which to my mind is pretty impressive. How do they work?"}
{"id":"107554","title":"Applying Ehrenfest's theorem to Hamiltonian","text":"It maybe a stupid question, but from the Ehrenfest's theorem, we have \\begin{eqnarray*} \\frac{d\\langle A\\rangle}{dt} &=& \\left\\langle\\frac{\\partial A}{\\partial t}\\right\\rangle + \\frac{1}{i\\hbar}\\left\\langle[A,H]\\right\\rangle \\end{eqnarray*} The if we apply it to the Hamiltonian, \\begin{eqnarray*} \\frac{d\\langle H\\rangle}{dt} &=& \\left\\langle\\frac{\\partial H}{\\partial t}\\right\\rangle + \\frac{1}{i\\hbar}\\left\\langle[H,H]\\right\\rangle \\end{eqnarray*} But since the last term vanishes \\begin{eqnarray*} \\frac{d\\langle H\\rangle}{dt} &=& \\left\\langle\\frac{\\partial H}{\\partial t}\\right\\rangle \\end{eqnarray*} But in general cases, the expectation value of the time derivative of the Hamiltonian is not zero, i.e. in the infinite potential well. $$ \\left\\langle\\frac{\\partial H}{\\partial t}\\right\\rangle=\\int\\Psi^*\\frac{\\partial H}{\\partial t}\\Psi dx=\\int\\sum_n c_n^* \\psi_n^*e^{iE_n t\/\\hbar}\\frac{\\partial H}{\\partial t}\\sum_m c_m \\psi_m e^{-iE_m t\/\\hbar}dx$$ $$ =\\int\\sum_n c_n^* \\psi_n^*e^{iE_n t\/\\hbar}\\sum_m c_m (H \\psi_m) \\frac{\\partial }{\\partial t}e^{-iE_m t\/\\hbar}dx$$ $$=\\int\\sum_n c_n^* \\psi_n^*e^{iE_n t\/\\hbar}\\sum_m c_m{1\\over{i\\hbar}}E_m^2\\psi_m e^{-iE_m t\/\\hbar}dx$$ $$={1\\over{i\\hbar}}\\sum_n\\sum_m e^{i(E_n -E_m) t\/\\hbar}c_n^*c_m\\int E_m^2 \\psi_n^*\\psi_m dx$$ $$={1\\over{i\\hbar}}\\sum_n |c_n|^2E_n^2 $$ But since the expectation value of the Hamiltonian in the infinite well is a constant, it is obviously a contradiction. Is it impossible to apply the Ehrenfest's theorem to the Hamiltonian, or is there any mistake in my calculation?"}
{"id":"82519","title":"Anharmonic oscillator solution function","text":"I am solving a CLASSICAL an-harmonic oscillator problem with Hamiltonian given by $H= (1\/2)\\dot{x}^2+(1\/2)x^2-(1\/2)x^4$ with all the constants (k's) and mass being taken as 1 (one). I find that $x= \\tanh(t\/\\sqrt{2})$ is satisfying the equation of motion. But my question is how to incorporate the Hamiltonian, $H$ in to this solution so that by providing $H$ we can control the initial conditions of this problem. Or any other solution function that can have $H$ in it. thanks in advance. PS= In SHO (m=k=1) lets say $x=A\\sin(t)$ then $A= \\sqrt{2H}$, where $H$ is the total energy or the Hamiltonian. So $x=\\sqrt{H}\\sin(t)$. I need a solution function like this."}
{"id":"118665","title":"Sun revolves around the Earth","text":"What if somebody says that **Sun revolves around the Earth** and not Earth revolving around the Sun? What would be the _consequences_?"}
{"id":"10933","title":"Why do we say that the earth moves around the sun?","text":"In history we are taught that the Catholic Church was wrong, because the Sun does not move around the Earth, instead the Earth moves around the Sun. But then in physics we learn that movement is relative, and it depends on the reference point that we choose. Wouldn't the Sun (and the whole universe) move around the Earth if I place my reference point on Earth? Was movement considered absolute in physics back then?"}
{"id":"102856","title":"Relative Motion and orbiting planets","text":"if all motion is relative to the frame of reference determined by an observer, why would the view that planet earth revolves around the Sun more correct than the view that the Sun revolved around planet Earth and all the other planets revolve around the Sun? Is there a mathematical proof?"}
{"id":"55444","title":"What made us think that Earth moves around the Sun?","text":"Trying to observe the night sky for a few weeks, the motion of the Sun and the stars pretty much fits into the Geocentric Theory i.e. All of them move around the Earth. **What then, which particular observation, made us think that it could be the other way around, that all the planets move around the Sun?**"}
{"id":"74007","title":"Why was Galilei \"right\"?","text":"Considering that all frames are equivalent, isn't it up to the observer to say \"The earth turns around the sun\" or \"The sun turns around the earth\"? Isn't this more or less like arguing about which meridian should be considered the null-meridian, or whether time should be counted according to years before\/after Jesus Christ or the conquest of Mecca?"}
{"id":"11185","title":"heliocentricity and the theory of relativity","text":"> **Possible Duplicate:**   >  Why do we say that the earth moves around the sun? I should preface by stating that I'm not a physicist and my knowledge of the theory of relativity is limited to what Encarta told me in the 90's (does everyone know what Encarta was?) and I don't even understand the Wikipedia entry! Encarta had a video on the theory of relativity that basically explained in part that (i suppose in vacuum) there's no way to tell which body is in motion relative to another (without observing from outside the space in which they are moving i suppose). so my question is whether this warrants a reassessment of our reliance on a heliocentristic model. is it possible the planets do not revolve the way we've modeled them? i'm not necessarily talking geocentric, just different from our current model. should we be exploring this? ps -- while researching this question i came across geocentricity.com, the first sentence of which starts \"Of all the sciences, the Holy Bible...\" I'm really not interested in starting a religious debate so please keep answers strictly physics related (though i hope i don't have to even say that on **_physics_**.stackexchange.com!)"}
{"id":"70150","title":"Revolution of Earth","text":"If all motion is relative, how do we know that the Earth revolves around the sun? Or we are just making the above statement from the frame of reference in which Sun is at the origin?"}
{"id":"68953","title":"Does the earth really move round the sun?","text":"Is there really any empirical proof that the earth moves round the sun? I've read simple explanations using mass-gravity relation. But then some questions remain, is gravity a function of mass alone? Is mass a function of size alone? if the earth is revolving round itself, and the sun is revolving round and moving towards the black hole in the centre of the Milky Way, how can we be sure that these two movements alone do not suffice to explain night and day or change in the position of the stars with time? Thanks."}
{"id":"86109","title":"How do we know the Earth orbits around the Sun and not the other way around?","text":"I know that describing the trajectory of all planets in the solar system around the Earth is much more complicated than if we take the Sun as the reference point. But besides this, what is the simplest experiment that can prove that the Sun is a more \"inertial\" point of reference than the Earth? Do we have to look at the stars in the background or something like that?"}
{"id":"99572","title":"Heliocentric Worldview","text":"Isn't the whole historic Discussion of Heliocentric vs. Geocentric Worldview just about a Calculation-Technique. I mean I could also choose my coordinate- center to be in the middle of Earth and setup my differential Equations and starting points and it would still describe all Movements correctly, right? Now if I want to solve this it might be pretty wise to choose the Sun as my coordinate origin. But nevertheless I could transform the resulting curve so that the origin is within earth and I can claim that the Sun is (in a fancy possibly self-intersecting way) rotating around Earth, and even the whole Universe is ( in a very fancy way) rotating around the Earth. What's the Problem here? Isn't this the General Principle of Relativity in Physics? How do we have to define Rotation, so that we get the Heliocentric Worldview? If we don't allow self-intersections, we would get a Problem with double-star- systems and Planets which rotate around both stars. It seems that the current Definition relies on the Masses of the involved Objects. But this seems to be a very naive View neglecting the Fact that Forces aren't imposed by the \"stronger\" Object ( = Human Intuition) but that they are just caused by \"Physics itself\"."}
{"id":"117303","title":"motion of an electron near a conducting wire","text":"An electron is ejected from the surface of a long thick straight conductor carying a current, initially in direction perpendicular to the conductor. The electron will: a) ultimately return to the conductor b) move in a circular path around the conductor c) gradually move away from the conductor along a spiral d) move in helical path with the conductor as the axis According to me, $$mv^2\/r=qvB\\implies r=mv\/qB$$ since B is the magnetic field due to the conductor, $B=\\mu i\/2\\pi R$ where R is the distance from a conductor. So$$r=2\\pi mvR\/\\mu qi$$ Since the force will always be perpendicular to the velocity, the electron moves in a circle. As the electron moves, $R$ increases since the electron has to move outwards first to complete the circular motion. As $R$ increases, $r$ also increases. So I think the electron should move away from the conductor along a spiral. What is the correct answer?"}
{"id":"9650","title":"Sterile Neutrinos as Dark Matter","text":"There has been recent activity by astrophysicists to determine whether a fourth flavor of neutrino, a sterile neutrino, exists. It would likely be more massive than electron, muon or tau neutrinos. However, it wouldn't be affected by the weak force, only by gravity. It would therefore have similar characteristics to dark matter. The LHC is looking for evidence of supersymmetry, and the existence of the stable neutralino, another dark matter candidate. Can the LHC be of any assistance in helping determine the existence of the hypothetical sterile neutrino?"}
{"id":"9657","title":"cm^3\/g as a unit of adsorption","text":"I recently saw cm^3\/g as a unit for amount adsorbed. Usually, you see either kg adsorbate\/kg adsorbent or mole adsorbate\/kg adsorbent. Does anyone know the meaning of this unit?"}
{"id":"107421","title":"Geodesic devation on a two sphere","text":"So the geodesic deviation equation gives the relative acceleration between two geodesics in motion. But given a pair of geodesic (let's say on the two sphere) that start at the equator, separated by some distance. Is there a way to compute their separation as a function of time without using the geodesic equation? Let's say they're moving at northward toward the pole along a line of constant longitude at unit velocity."}
{"id":"32840","title":"What isotope has the shortest half life?","text":"**Question:** What isotope has the shortest half life?"}
{"id":"102131","title":"Quantization of electrostatic $\\vec E$ field?","text":"Can a electrostatic field $\\vec E=\\vec E(x,y,z)$ (time-independent) or electrostatic potential $\\phi=\\phi(x,y,z)$ be quantized? If yes, will these quanta be photons again? But we don't have an electromagnetic field here."}
{"id":"113979","title":"What would cause a spinning fluid to stop spinning?","text":"I once saw a demonstration where an electric current caused a drop of mercury to spin. The drop contained bits of iron, which could be seen flowing around in a circular pattern. As soon as the current was turned off, the spinning slowed fairly quickly. What caused the circular motion within the drop to slow? It seems to me that there would be very little friction within the drop, and that the motion should be similar to that of a gyroscope. Why was this not the case?"}
{"id":"91501","title":"What keeps mass from turning into energy?","text":"I understand the energy and mass can change back and forth according to Einstein. It is fluid; it can go from one to the other. So, what keeps mass from just turning into energy? Is there some force holding a subatomic particle together? What keeps mass in it's state? I hope this is not a silly question but I am clueless. Thanks"}
{"id":"126154","title":"Can matter be created from energy?","text":"> _The small, hot, dense early universe the size of an atom was made up > entirely of energy, it wasn't until after the expansion began and the > universe cooled down some of that energy began converting into the first > atomic nuclei._ This quote seems a little dubious but I think this is worth asking anyway: Can someone explain the atomic process, if it even exists, of how this would work to convert energy in to matter, and what form of energy was initially present, and what is required to cause this change?"}
{"id":"95462","title":"Application of $E = mc^2$","text":"We very well know that mass-energy equivalence is given by $E = mc^2$. My question however is how would we actually convert an object or some mass into its pure energy state and then if possible even back to its original state. The answer needs to state the requirements for conversion into pure energy. We all know its possible theoretically but how would it be done in a practical world. Just assume that the mechanism required for the process is possible. In short I want the process\/mechanism for how to go about it."}
{"id":"112089","title":"Understanding emergent phenomena in the block universe. (Reworded question)","text":"Each person exists as an unchanging 4D worldtube in the block universe. At each slice of the worldtube there is a present, past and future. However, there is a black box* which appears to exist in only one slice of the block universe at a time. How could this be explained? I am not trying to attack the block universe concept, I am not trying to get into metaphysics and I am not trying to come up with my own crack-pot theory. I would just like to know how a world tube can contain something, a black box, that exists throughout the whole worldtube, and yet creates the illusion of only existing in one slice of the universe at a time. Is it an imaginary force like the centrifugal force? Or an emergent phenomena such as temperature? Something else? *The black box is consciousness, but I don't want to use that word as it seems to make physicists run away screaming."}
{"id":"106903","title":"velocity in inertial and nontial frames","text":"![enter image description here](http:\/\/i.stack.imgur.com\/Pik0X.jpg)![enter image description here](http:\/\/i.stack.imgur.com\/LL2TD.jpg) I got confused about the difference between the last term of both pictures. In the first one, we have w x r, but in the second we have w x r underlined. Does anyone have a better explanation? They should be the same."}
{"id":"3362","title":"Relativistic object impacts the earth","text":"A familiar trope within science-fiction is that of a large relativistic object hitting a planet such as the earth. This is normally an interstellar spacecraft or a kinetic weapon with a mass in the range 103 \\- 106 kg. But what would actually happen? The two scenarios seem to be: (a) the object creates a kind of conical tunnel through the earth with most of the material exiting on the far side; (b) the object dumps all of its kinetic energy within a few tens or hundreds of kilometres of the impact point and we have the equivalent of a conventional asteroid impact. Light transit time through an earth diameter of 12,800 km is just over 40 milliseconds. There’s not much time for lateral effects as the object barrels in. So what would happen if a 1 tonne object hit the earth, square on, at 0.99995c (γ = 100)?"}
{"id":"110220","title":"The velocity of sound in terms of Fermi velocity","text":"Assume this question: In a simple model of a monovalent metal consisting of point positive ion cores embedded in a uniform jellium of electrons, a value for the average energy for electron is: $$E = \\frac{9}{10} \\frac{e^{2}}{r} + \\frac{3\\hbar^{2}}{10mr^{2}}\\left(\\frac{9}{4\\pi}\\right)^{2\/3}$$ where r is the radius of a sphere containing one electron. Find the velocity of sound in terms of Fermi velocity. I'm attempting to solve this question, but I only can find the equilibrium vaule of $r$. I cannot understand the relation between the velocity of sound and this question."}
{"id":"30277","title":"Euler's buckling formula applicable for impact calculations?","text":"$$F = \\frac{\\pi^2 EI}{(KL)^2}$$ Is Euler's buckling formula applicable for impact calculations, considering speeds relevant for a car or aircraft crash? If there is a level where the formula becomes inapplicable or inappropriate in impact calculations, what determines this, and what behavior (and hence other formula) will then be relevant?"}
{"id":"76041","title":"What is physics behind States of matter?","text":"States of matter in physics are the distinct forms that different phases of matter take on. Four states of matter are observable in everyday life: solid, liquid, gas, and plasma. What is physics behind States of matter? **For Example:** I found a Trick to freeze water in about half a second, in this trick, When I open caps of a plastic bottle of cold water, cold Water freezes in a chain process, You can see the freezing process. What is physics of The process of freezing water?"}
{"id":"76042","title":"What sense can be made of the natural logarithm in physics?","text":"What sense can be made of the natural logarithm, When appearing in a physical process? For example, This integral in the thermodynamic $\\int_i^f \\frac {dV}{V}=Ln\\frac {V_f}{V_i}$ when $V$ denotes Volume. in general $Ln\\frac {Q_f}{Q_i}$ when the $Q$ denotes Physical quantities. or this one $S=k_BLn\\Omega$ Why sometimes natural logarithm can be interpreted as a physical process? What are the odds of that happening?"}
{"id":"109953","title":"Compact manifold taken as an Einstein Manifold","text":"In Kaluza-Klein theories I often see that the compact space is assumed to be an Einstein manifold, that is, its Ricci tensor is proportional to its metric. So, why is this done?"}
{"id":"4278","title":"What kind of invariants are proper time and proper length?","text":"Under the Lorentz transformations, quantities are classed as four-vectors, Lorentz scalars etc depending upon how their measurement in one coordinate system transforms as a measurement in another coordinate system. The proper length and proper time measured in one coordinate system will be a calculated, but not measured, invariant for all other coordinate systems. So what kind of invariants are proper time and proper length?"}
{"id":"108394","title":"R-symmetry commutator","text":"I've seen the claim made several placed; Terning's \"Modern Supersymmetry\" p. 5 on N=1 SUSY algebra states it as well as anyone: > The SUSY algebra is invariant under a multiplication of $Q_\\alpha$ by a > phase, so in general there is one linear combination of $U(1)$ charges, > called the $R$-charge, that does not commute with $Q$ and $Q^\\dagger$: > >> $[Q_\\alpha,R] = Q_\\alpha, \\;\\;\\;[Q^\\dagger_\\dot{\\alpha},R]=-Q^\\dagger_\\dot{\\alpha}$ The first statement is straightforward to see. But (1) Why is there is one linear combination of charges that does not commute? (2) How do we arrive at these commutators? (I imagine that the generators can be rescaled to give the coefficient $\\pm1$, but I would like a clearer explanation.) I spoke to a peer who said that the commutation relations could be found in a very general, mathematically heavy treatment of the most general possible SUSY algebra. Is there some easier way to understand?"}
{"id":"72292","title":"Slowing down light in an opaque crystal for a whole minute","text":"I just read about a team of physicists at the University of Darmstadt, Germany, that managed to completely slow down a beam of light that traveled through an opaque crystal (article here). How is it possible for a beam of light come to a complete stop? In the article they mentioned that they fired a laser at the crystal causing the atoms to go into a quantum superposition. How does this affect the stopping of the light? Also if the uncertainty principle applies to photons (which I do not know if it does), how does this not violate the uncertainty principle if the photons aren't moving?"}
{"id":"72295","title":"What is the difference between baryonic and gravitational mass?","text":"I was reading a webpage on neutron stars, and it mentioned that a neutron star's gravitational mass is about 20% lower than its baryonic mass due to gravitational redshift. I understand the basics of what the terms mean, but I do not see why gravitational redshifting would cause the gravitational mass to be reduced."}
{"id":"132292","title":"Will this achieve linear movement of a closed container?","text":"Can I create a device that moves in one direction with a design like Figure 1? I suspect that the forces cancel each other by the recirculation. It is a simple exercise of action and reaction, but I can not figure out the math to explain. Maybe my problem is of type XY problem. My intention is to create a device that produces movement from a **closed container** , something like a black box. (which would be added energy) Figure 1. ![enter image description here](http:\/\/i.stack.imgur.com\/dt1NI.png)"}
{"id":"131920","title":"Asymmetry in magnetic field direction of an electric wire","text":"The simplest magnetic field is that of an infinitely long wire with uniform current. It does enjoy radial symmetry about the wire and has the variation as 1\/r. To find the direction of the resulting magnetic field you use the right hand grip rule (for conventional current). This rule repeats the experimental fact. But it is a asymmetry. We get a bit less asymmetry calculating the magnetic field for a \"anti\"wire with positrons. Now we have to use the left hand grip rule. Where this asymmetry comes from?"}
{"id":"131925","title":"Proof that total derivative is the only function that can be added to Lagrangian without changing the eom","text":"So I was reading this: Invariance of Lagrange on addition of total time derivative of a function of coordiantes and time and while the answers for the first question are good, nobody gave much attention to the second one. In fact, people only said that it can be proved without giving any proof or any. So, if I have a Lagrangian and ADD an arbitrary function of q', q and t in such a way that the equations of motion are the same, does this extra function MUST be a total time derivative? **EDIT** Ok, I feel really dumb now. I guess the most voted answer of the question I posted was kinda wrong. So, I changed my question a little bit: If I have a function that obeys the Euler-Lagrange equation off-shell, this implies that my function is a time derivative? This was used in the most voted answer of this other question: Deriving the Lagrangian for a free particle , equation 7. Also, why people only talk about things that change the lagrangian only by a total derivative? If this is not always the case that keeps the equation of motion the same, so why is it so important? And why in the two questions I posted about the same statement on Landau's mechanics book only consider this kind of change in the lagrangian?"}
{"id":"1639","title":"What's the difference between running up a hill and running up an inclined treadmill?","text":"Clearly there will be differences like air resistance; I'm not interested in that. It seems like you're working against gravity when you're actually running in a way that you're not if you're on a treadmill, but on the other hand it seems like one should be able to take a piece of the treadmill's belt as an inertial reference point. What's going on here?"}
{"id":"132742","title":"kitaev-honeycomb : can't get wilson loop squared to yield +1","text":"I'm new here, loving this website and I'm having some difficulty with the wilson-loop operator in kitaev's honeycomb model. **problem statement** The Kitaev model (Kitaev, 2006 is the original paper) consists of spins residing at the lattice sites of a honeycomb lattice with separate nn couplings for the three directions that are identified for the bonds. The wilson-loop operator is $w_p=\\sigma^x_1 \\sigma^y_2 \\sigma^z_3 \\sigma^x_4 \\sigma^y_5 \\sigma^z_6$, where the indices $i \\in\\\\{1,...,6\\\\} $ indicate the $6$ lattice sites involved in the hexagonal loop (see picture). ![Kitaev honeycomb loop](http:\/\/i.stack.imgur.com\/aiT0X.png) In Jiannis K. Pachos' book (Introduction to topological quantum computation, 2012) the author states that $(w_p)^2=1$, which I'm trying to find myself. Actually this should be not at all hard, but I'm stuck unfortunately. **attempt at solution** I've tried the following $$ (w_p)^2 = \\sigma^x_1 \\sigma^y_2 \\sigma^z_3 \\sigma^x_4 \\sigma^y_5 \\sigma^z_6 \\sigma^x_1 \\sigma^y_2 \\sigma^z_3 \\sigma^x_4 \\sigma^y_5 \\sigma^z_6 \\\\\\ = -\\sigma^x_1 \\sigma^x_1 \\sigma^y_2 \\sigma^z_3 \\sigma^x_4 \\sigma^y_5 \\sigma^z_6 \\sigma^y_2 \\sigma^z_3 \\sigma^x_4 \\sigma^y_5 \\sigma^z_6 \\\\\\ = -\\sigma^y_2 \\sigma^y_2 \\sigma^z_3 \\sigma^x_4 \\sigma^y_5 \\sigma^z_6 \\sigma^z_3 \\sigma^x_4 \\sigma^y_5 \\sigma^z_6 \\\\\\ = + \\sigma^z_3 \\sigma^z_3 \\sigma^x_4 \\sigma^y_5 \\sigma^z_6 \\sigma^x_4 \\sigma^y_5 \\sigma^z_6 \\\\\\ = + \\sigma^x_4 \\sigma^x_4 \\sigma^y_5 \\sigma^z_6 \\sigma^y_5 \\sigma^z_6 \\\\\\ = - \\sigma^y_5 \\sigma^y_5 \\sigma^z_6 \\sigma^z_6 \\\\\\ =-1 $$ Where I've pulled $\\sigma_1$ through first, next the $\\sigma_2$, etc. And I've used $\\\\{\\sigma^\\alpha_i , \\sigma^\\beta_j \\\\}= 2 \\delta_{i,j}\\delta_{\\alpha,\\beta} I_2 $ (so that every swapping of unequal $\\sigma$'s gives a minus sign and $\\sigma^\\alpha_i\\sigma^\\alpha_i =I_2$ ). So I get $(w_p)^2=-1$, which is not what I wanted to find. All text on the subject state that $w_p$ acting on a lattice configuration yield $w_p=\\pm1 $ which can be easily concluded from $(w_p)^2=1$ (the expression I didn't get). My guess is that my commutation relations are not correct, but I'm unsure. Who can help me out? A big thanks in advance! Best, L"}
{"id":"87821","title":"Symmetries of a Uniform Magnetic Field","text":"Simple question. A system with a uniform electric field everywhere in space has translational invariance in the directions perpendicular to the electric field but no translational invariance parallel to it. This system also has rotational invariance in the plane perpendicular to the electric field. What about a uniform magnetic field? Judging from the Hamiltonian $(\\vec{p}-q\\vec{A})^2\/2m$, it would seem that the symmetries depend on our choice of gauge. Choosing a different $\\vec{A}$ breaks different symmetries. Is there a most symmetric choice for $\\vec{A}$?"}
{"id":"87826","title":"'Hole in the clouds' in the Sarychev Peak eruption","text":"One of my favourite ever pictures taken from space is a picture of the 2009 eruption of Sarychev Peak (Ostrov Matua island, Japan) taken by an ISS astronaut during a lucky fly-over. ![enter image description here](http:\/\/i.stack.imgur.com\/O1FfD.jpg) Image Source: Earth Observatory Image of the Day. I heartily recommend clicking through and seeing the animation. I always assumed that the hole in the cloud cover around the island was caused by some sort of shockwave originating in the eruption plume, most probably as a result of increased temperatures evaporating the cloud, or some such. However, upon revisiting the IOTD page, I was surprised to learn that this is not necessarily the case and that the origin of this hole is a matter of some controversy: > **Editor’s note: Following the publication of this photograph, the > atmospheric and volcanic features it captured generated debate among > meteorologists, geoscientists, and volcanologists who viewed it. Post- > publication, scientists have proposed—and disagreed about—three possible > explanations for the hole in the cloud deck above the volcano. > > One explanation is that the hole in the clouds has nothing to do with the > eruption at all. In places where islands are surrounded by oceans with cool > surface temperatures, it is common for a sheet of clouds to form and drift > with the low-level winds. When the cloud layer encounters an island, the > moist air closer to the surface is forced upward. Because the air above the > marine layer is dry, the clouds evaporate, leaving a hole in the cloud deck. > These openings, or wakes, in the clouds can extend far downwind of the > island, sometimes wrapping into swirling eddies called von Karman vortices. > > The other two possibilities that scientists have offered appeared in the > original caption. One is that the shockwave from the eruption shoved up the > overlying atmosphere and disturbed the cloud deck, either making a hole or > widening an existing opening. The final possibility is that as the plume > rises, air flows down around the sides like water flowing off the back of a > surfacing dolphin. As air sinks, it tends to warm; clouds in the air > evaporate. **Has this controversy been settled?** Is there a convincing, accepted explanation for the origin of the hole in the cloud cover?"}
{"id":"24010","title":"The equivalent electric field of a magnetic field","text":"I know that Lorentz force for a charge $q$, with velocity $\\vec{v}$ in magnetic field $\\vec{B}$ is given by $$\\vec{F} =q \\vec{v} \\times \\vec{B}$$ but there will exist a frame of reference where observer move at same velocity with that of charge $q$, so according to him $v=0$. hence he will see no magnetic force is exerted on charge $q$. I have work on this problem for a while and found that the special relativity predicts equivalent electric force will acting upon charge instead. I want to know the relationship between this equivalent electric force and magnetic force. Thanks in advance"}
{"id":"101730","title":"How does beta+ decay actually occur?","text":"I'm having difficulty in understanding beta plus decay. How can a proton which has slightly less mass than neutron transform into a neutron, positron and neutrino? Form where does the extra mass for neutron,electron and neutrino come? Is it that during the decay some of the binding energy gets converted into mass?"}
{"id":"25924","title":"Why is a new moon not the same as a solar eclipse?","text":"Forgive the elementary nature of this question: Because a new moon occurs when the moon is positioned between the earth and sun, doesn't this also mean that somewhere on the Earth, a solar eclipse (or partial eclipse) is happening? What, then, is the difference between a solar eclipse and a new moon?"}
{"id":"72435","title":"Why does a dielectric have a frequency dependent resistivity?","text":"This question has come about because of my discussion with Steve B in the link below. Related: Why is glass much more transparent than water? For conductors, I can clearly see how resistivity $\\rho\\,\\,(=1\/\\sigma)$ can depend on frequency from Ohm’s law, $\\mathbf{J}=\\sigma\\mathbf{E}$. So if the E-field is an electromagnetic wave impinging on a conductor, clearly the resistivity is frequency dependent. In a similar fashion, the frequency dependence of the electric permittivity $\\epsilon=\\epsilon_0n^2(\\omega)$ can be derived through the frequency dependence of the electric polarization and impinging electromagnetic wave (see How Does $\\epsilon$ Relate to the Dampened Harmonic Motion of Electrons?).   1. What does it mean physically for a dielectric to have a frequency dependent resistivity from (i) classical and (ii) quantum viewpoints? I am especially interested in the optical frequency range.   2. Can a simple mathematical relationship be derived similar to the frequency dependent resistivity (for conductors) and electric permittivity (for dielectrics)? Thank you in advance for any help on this question"}
{"id":"72432","title":"Would connecting p-type and n-type semiconductors work as a diode?","text":"If we placed `p-type` and `n-type` semiconductors close enough to be touching (see fig. 1), would this arrangement work as a diode? Please explain. ![connecting p-type and n-type semiconductors](http:\/\/i.stack.imgur.com\/n5UzB.jpg) Fig. 1 - Connecting p-type and n-type semiconductors"}
{"id":"109042","title":"Off-diagonal terms of the Husimi $Q$ function?","text":"The Husimi $Q$ function of a quantum state $\\rho $ is defined as $ Q (\\alpha)=\\langle \\alpha \\vert \\rho \\vert \\alpha \\rangle $, where $\\alpha = (x, p) $ is a phase space coordinate and $\\vert \\alpha \\rangle$ is a coherent state. Is the off-diagonal generalization $ Q (\\alpha, \\beta)=\\langle \\alpha \\vert \\rho \\vert \\beta \\rangle $ used for anything? Does it have a name? This is an interesting object because it essentially measures coherence (or decoherence) in the overcomplete basis of wavepackets ."}
{"id":"76950","title":"Do image charges radiate?","text":"Suppose I have a charge moving back and forth above an infinite, grounded, conducting plane. Can I calculate the total radiated power by using image charges? That is, are the scalar and vector potentials the same in the upper- half space for all time for both the image charge \"picture\" and the standard picture?"}
{"id":"32967","title":"A nonintegrable quantum system whose classical limit is integrable?","text":"In this discussion: http:\/\/chat.stackexchange.com\/rooms\/4243\/discussion- between-arnold-neumaier-and-ron-maimon Arnold Neumaier suggested that there might be a close link between classical and quantum integrability, while I think there are many more classically integrable systems than quantum integrable ones. The reason is that classically integrable systems are easy to make up--- you make up an infinite number of action and angle variables, and change canonical coordinates in some complicated way to x,p pairs, and say this x-p version is your system of interest. But quantum systems don't admit the same canonical transformation structure as classical systems, so there might be systems which have an integrable classical limit, but no real sign of integrability outside of the classical limit. But I don't know any examples! Most of the 1+1d integrable stuff is for cases where the classical and quantum integrability are linked up, for the obvious reason that people are interested in finding integrable systems, not examples where they are not. The reason I think finding an example is not trivial is because the classical integrability guarantees that the motion is not classically chaotic, and that the asymptotic quantum energy states are pretty regular. So I don't think one can look for a counterexample in finite dimensions, where all high enough energy states are permanently semi- classical. But consider a field theory on a lattice in 2+1 dimensions (continuous time). The lattice is so that the dynamics can be arbitrary, no continuum limit, no renormalization. Even if you have an integrable classical dynamics for the field theory, the energy can still dissipate over larger volumes (this isn't 1+1 d), and eventually the classical field will be weak enough that the classical limit is no longer valid, and you see the quanta. This allows the possibility that every finite energy state to eventually leave the semi- classical domain, and turns quantum, and then the integrability is lost. So is there a 2+1 (or 3+1) dimensional lattice scalar field theory where the classical dynamics is integrable, but the quantum mechanical system is not? By saying that the quantum system is not integrable, I mean:   * the many particle S-matrix doesn't factorize or simplify in any significant way (aside from the weak asymptotic relations implied by having a classical integrable limit)   * there are only a finite number of quantum conserved currents (but an infinite number of conserved currents in the classical limit)."}
{"id":"109044","title":"Is there any significance to the negative sign on this speed?","text":"A girl is riding a bicycle along a straight road at constant speed, and passes a friend standing at a bus stop (event #$1$). At a time of $60$ s later the friend catches a bus (event #$2$) If the distance separating the events is 126 m in the frame of the girl on the bicycle, what is the bicycle's speed? $$u = u' + v$$ can be written as: $$Δx\/t = Δx'\/t + v$$ $$v = Δx\/t - Δx'\/t$$ $$v = 0m\/60s - 126m \/60s$$ $$v = -126 m\/ 60s$$ $$v = -2.1 m\/s$$ Just wondering if the negative holds any significance? I know we're talking about speed which is scalar but how come the calculation gives a negative? Sorry I am just beginning to learn about relativity."}
{"id":"48731","title":"How can you test what color different people perceive?","text":"If I would show someone a yellow object and ask them, \"is this object yellow?\"   That person would say \"yes\". But I could never know if my perception of the color yellow is the same as that other person's.   Because he or she could actually be seeing, what I know to be the color green.   But then tells me that its the color yellow because that has been taught to him or her from young age. So how can you test if people are really seeing the same color? Question closed and can now be found @ http:\/\/biology.stackexchange.com\/questions\/5728\/how-can-you-test-what-color- different-people-perceive"}
{"id":"131210","title":"M-Theory and computer simulation","text":"I have a question. Is it possible to perform computer simulations based on the M-Theory? I was looking for such simulators or source codes but I have not found anything. However, M-theory must have a mathematical model that can be saved as a code to be executed by the computer. Something like this might work? Leaving aside the problem of computational complexity."}
{"id":"96620","title":"Calculate work done in an inclined plane","text":"How can you calculate the work done by a force (of unknown quantity) exerted on a 10kg block on an inclined plane. The force is pointing upwards and parallel to the incline (which is inclined 30 degrees with respect to the horizontal). a. frctionless plane b. coefficient of friction = 0.12 So the forces acting the block are the normal force, its weight, the friction force (for letter (b)), and the force exerted upwards the incline. All are given or can be solved almost instantly except for the force upwards denoted by F. How do I solve this problem? I am not sure what value of acceleration to use in the axis of the incline for F=ma. Sorry I could not provide a diagram for this."}
{"id":"103790","title":"Physical significance of negative temperature","text":"I read some answers regarding negative temperatures but I think my question is new. I want to know that what is the physical significance of negative temperature. Suppose I say a body has temperature -2 K. Can I interpret it physically?"}
{"id":"130877","title":"How does a fixed amount of transmitted radio energy supply an unknown number of destinations?","text":"I did some maths and physics up to the age of 18, and hold an amateur radio licence. This thing has puzzled me for a while - does reception of an electromagnetic wave imply an interaction with the transmitter? Does it drain some of the transmitter's energy?"}
{"id":"103208","title":"Heisenberg's uncertainty principle - Planck's (reduced) constant divided by two or not?","text":"The most common form of Heisenberg's uncertainty principle I've seen online is $$ \\Delta x \\Delta p ~\\geq~ \\dfrac{\\hbar}{2}.$$ However, I also regularly see $$\\Delta x \\Delta p ~\\geq~ \\hbar. $$ Sadly, I used the latter one in a project recently and I'm afraid it's incorrect. Obviously the upper one one is true if the latter one is, but why are these two versions used instead of only one of them?"}
{"id":"92080","title":"Proof of Gauss' Law","text":"How would you prove Gauss' law for an asymmetrical closed surface? I can find it for symmetrical surface but couldn't for Asymmetrical surfaces."}
{"id":"71389","title":"Understanding a paper: What is the meaning of $b_0$?","text":"I am looking at this paper (Multicoated gratings, J. Opt. Soc. Am., 1981) and I am getting confused around equation 22. I do not completely understand where he comes up with the equation $$\\xi^j_q=b_q^j(R^{-1}V_q^j)\\qquad \\text{(22)}$$ And then what is the meaning of the $b^{j+1}$. I initially thought they were the eigenvalues of the T matrix he defines, but all he says is that it is a vector of components $b^q_j$. This includes the $\\bf b^0$ and $\\bf b^{Q+1}$ which I do not see how they are vectors?"}
{"id":"105027","title":"band gaps in tight binding model","text":"What happens at the zone boundaries of the brillouin zones in the tight binding model? How does the band gap originate in the TB model?"}
{"id":"9857","title":"What is an observer in quantum mechanics?","text":"My question is not about (pseudo) philosophical debate; it concerns mathematical operations and experimental facts. **What is an observer?** What are the conditions required to be qualified of **observer** , both mathematically and experimentally?"}
{"id":"51374","title":"What is an \"observer\" really?","text":"> **Possible Duplicate:**   >  What is an observer in quantum mechanics? I'm sick of quantum physics explanations which term experiments where the outcome depends on \"if you observe it or not\". For example, the two-slit photon experiment. I figure that something physical happens, irrespective of if a \"being\" of some kind is there to consciously watch it. Is it really to do with absorption and emission? Like the only way to know for sure where a photon is, you need to have it absorbed and re-emitted by an atom\/electrons so that you can detect it (on one level, seeing with your eyes is about absorbing photons on the atoms in your eye and having that create a potential on your nerves which can travel through your brain). So in a sense, if a tree fell in the forest and nobody heard, it still would have fallen because the atoms in the tree would have absorbed thermal energy, and the atoms of the tree would have caused a vibration in the tree and the ground, which vibrated the air, and so this energy would have dissipated into the environment, and that counts as observation without a human ever getting involved? Do I understand things or am I very wrong?"}
{"id":"128335","title":"Quantum observation in the double slit experiment?","text":"As far as quantum observation, when a human observes electrons going through the double whole experiment, have scientist tried having a blind person observe the experiment to see if the electrons actually can tell if a person is able to view it or not? This will tell us a lot as far as how intelligent the electrons are and if a persons eyes gives off a certain radio signal to warn the electrons of its observation.. This may seem crazy but i had to ask."}
{"id":"111307","title":"What does it mean 'the observer' in Quantum Physics?","text":"Is it only necessary a human consciousness? A measurement device? Can it be said that any of them cause the wave function collapse?"}
{"id":"33203","title":"Some very basic questions on the Higgs Boson","text":"What exactly is a boson? Is the Higgs boson the cause of gravity or a result of it? Does the collision of particles at the LHC create a gravity field or waves or somehow interact with the gravity field of the earth? The Higgs Boson is supposed to be quite massive and equivalent to a large number of protons. Were many particles needed to create it or only a few travelling at high speeds? Was the high energy converted into the large mass? Why is the particle so short lived and what does it decay into?"}
{"id":"133923","title":"Space curvature based on net energy = 0","text":"In Neil DeGrass Tyson's epic video, at 2:26:50 https:\/\/www.youtube.com\/watch?v=AdHlVY8pEk0&list=PL1L9zQimONkUOxXJyM0xcJja9ItQu1P8l&src_vid=AdHlVY8pEk0&feature=iv&annotation_id=annotation_4179310099#t=12s He mentions how if net energy is negative, the spacetime curvature is spherical, and if it's net positive, saddle-shaped. He uses the terms \"flat\" and \"saddle-shaped\" which are 2d, but he's actually just using that as an analogy for 3d curvature, which we can't really visualize, correct? If so, what does it mean for 3d space to be \"saddle- shaped\" or \"spherical\"? 1) II'm guessing spherical implies that if we keep traveling in one direction in space, it would be possible to end up back where you started? (kind of like how you can end up in the same position traversing the surface of a planet if the 2d plane is curved? 2) If net energy is 0. But they also say there is far more dark energy and dark matter than energy and matter. So isn't this a contradiction? How does the accounting work out?"}
{"id":"118652","title":"Does gravity act in the centre of the Earth?","text":"If we we dig a hole from north pole to the south pole of the Earth and we throw a ball in this hole where would it stop? Will it come back on the surface or it will stop at the centre?"}
{"id":"133925","title":"What did recombination look like?","text":"I recently remembered that someone worked out what the big bang sounded like and that got me thinking... About 377,000 years after the Big Bang, electrons became bound to nuclei to form neutral atoms. Because of (?) this, the mean free path of photons became effectively infinite, i.e. the universe became transparent to radiation. **What would this have _looked_ like?** More precisely I could ask: did the sky suddenly become dark, or was the amount of radiation basically same after as before? What would the distribution and timescale have been like? In a perfectly homogeneous universe it would happen at the same rate everywhere. Did the universe have any structure at this point? Would you have been able to see blotches of lighter and darker patches of the sky (assuming you were within one of the more transparent patches), these being proto-shapes of galactic filaments perhaps, or the noisy grit of the CMB? Or would it just be a vague cloud at any scale, and in any part of the spectrum?"}
{"id":"7089","title":"Comparing Energy Technologies in terms of: Cost per Power Capacity  and Research Spendings","text":"### Cost per Power Capacity For a number of energy technologies I'd like to know what the minimum costs are to install a given power capacity. Are there any such comparisons available? ### Research Spendings Technologies often own much to publicly funded research efforts so I'm also interested in estimations regarding development cost of the technologies themselves. Of course foundations to a given technology may have been readily available, so let's limit this to spendings on larger research projects _dedicated_ to developing a technology and motivated by the need of energy. I'm aware it's unlikely any exhaustive comparison in full detail exists but maybe something related? (Maybe someone should tag this question 'energy-economics' -- I'm lacking the reputation)"}
{"id":"123486","title":"Q factor of parallel RLC circuit in series with a capacitor and resistor","text":"I know that for parallel RLC circuits, the $Q$ factor is given by: $$ Q = R \\sqrt {\\frac{L}{C}} $$ But now suppose it is connected in series to a resistor $R_2$ and capacitor $C_2$. Would the $Q$ factor be changed? ![enter image description here](http:\/\/i.stack.imgur.com\/dpLzs.png)"}
{"id":"71032","title":"why is mass of air bubble in material medium considered to be negative?","text":"The mass of air bubble in any medium is considered as negative. Is the air bubble is massless. I m in confusion. can we not neglect the mass of air bubble in material medium. But i have found in many books the mass of air bubble in material medium as negative. please discuss....."}
{"id":"8176","title":"Are Newton's gravity waves detectable by a laser interferometer?","text":"Newton's theory of gravity supports \"gravity waves\" in that moving objects cause changing gravitational fields. For example, two bodies rotating around their center of mass will have a stronger gravitational field when they are longitudinally oriented than when they are transverse oriented. Given two masses of mass $M$ orbiting on a circle of radius $r$, at a distance $d$ from an observer, the strength of the attraction is: $$\\begin{matrix} F_{min} &=& \\frac{2GM}{d^2},\\\\\\ F_{max} &=& \\frac{2GM}{d^2}\\frac{1+r^2\/d^2}{(1-r^2\/d^2)^2} \\end{matrix}$$ ![enter image description here](http:\/\/i.stack.imgur.com\/dIjHy.jpg) This should be detectable at long distance. My question is this: With that sort of gravity, would the laser interferometer based gravity wave detectors be able to detect the gravity wave? An example is the LIGO, Laster Interferometer Gravitational-wave Observatory."}
{"id":"39194","title":"Is there a published upper limit on the electron's electric quadrupole moment?","text":"I understand an electric quadrupole moment is forbidden in the standard electron theory. In this paper considering general relativistic corrections (Kerr-Newman metric around the electron), however, there is a claim that it could be on the order of $Q=-124 \\, \\mathrm{eb}$. That seems crazy large to me, but I can't find any published upper limits to refute it. Surely someone has tested this? Maybe it's hidden in some dipole moment data? If not, is anyone planning to measure it soon?"}
{"id":"86752","title":"Electron Wave Interference","text":"In the double slit interference pattern for the wave of an electron, what will happen if I make the slits to be smaller than the size of an electron ? Will I still observe an interference pattern on the opposite side of the screen or no electron will be able to cross the slit? If no, then how can a quantum prisoner escape ?"}
{"id":"133008","title":"Why is everything colder nowadays?","text":"I know that when the universe began it was incredibly hot. Ever since, it is been cooling, and nowadays the average temperature of the universe is quite close to 0 K. Is this a consequence of having the same energy spread in a wider space, or is there something else going on?"}
{"id":"88800","title":"Exciting Surface Plasmon-Polaritons with Grating Coupling","text":"I'm very new the topic of SPPs and have been trying to understand this particular method of exciting surface plasmons using a 1D periodic grating of grooves, with distance $a$ between each groove. If the light incident on the grating is at an angle $\\theta$ from the normal and has wavevector ${\\bf k}$, then apparently if this condition is met: $\\beta = k \\sin\\theta \\pm\\nu g$ where $\\beta$ is corresponding SPP wavevector, $g$ is the lattice constant $2 \\pi\/a$ and $\\nu={1,2,3,...}$ then SPP excitation is possible. I haven't ever really had a formal course in optics, so my question is where this condition comes from. It seems like Fraunhofer diffraction, but only for the light being diffracted at a $90^\\circ$ angle to the normal. Most books don't state how they get this result, they just say it's because of the grating \"roughness\" which really confuses me. Any help would be greatly appreciated."}
{"id":"127672","title":"Jumping sewer lid - WHY?","text":"**Intro:** Few hours ago, there was a storm. We heard some constant banging which couldn't be explained by thundering. Then we found out, it was a sewer lid jumping. Maybe it's normal in other parts of the world, but for me it was like the first time in my life. I've captured the video. **The question** is what was causing this to happen. It's kind of clear that it was air pressure so strong that it was capable of lifting this metal lid. But where did this air pressure appear?   * Wind blowing into the sewer? If so, would it be so strong to lift the lid? And shouldn't be the sewers protected against wind somehow?   * Water filling the sewer so quickly, that it made the air pressure this strong? Would this really be the easiest way for the air to leave the sewer?   * Something else (such as bored sewer worked :) )?"}
{"id":"47754","title":"Minimum size of an asteroid to actually impact earth","text":"From what I understand, an object entering the atmosphere will start to burn up from the tremendous resistance of the atmosphere. Presumably, for asteroids under a certain size, they will burn up completely and never impact the surface of the earth. Do we have a way of determining the minimum size needed for actual impact? If so, roughly what is the size and how does it compare to the average size of asteroids that pass by us regularly?"}
{"id":"47756","title":"Yang-Mills Coulomb Gauge","text":"My Question is how to explicitly move into the \"Coulomb gauge\" in Yang-Mills theory. Using the answer provided by QMechanic, one can move into the \"temporal gauge\" for Yang-Mills fields: Gauge fixing choice for the gauge field $A_0$ . Once in this gauge, to move into the \"Coulomb gauge,\" does one take an explicitly _time independent_ $g$, and demand that it satisfy: $$ i\\partial^\\mu (g^{-1}(\\partial_\\mu - i A_{\\mu})g) = 0 $$ Thus giving $\\partial ^{\\mu} A'_{\\mu} = 0$? Are we guaranteed that such a $g$ exists? Thanks."}
{"id":"81402","title":"How to formally write down the Boltzmann equation?","text":"Can someone write down the Boltzmann equation, not neglecting any of the variables of the involved functions and integrals? Specifically, how to concisely capture the \"primed\" variables in a sensible manner?"}
{"id":"69954","title":"Photons-Wave\/particle duality","text":"I know that photons and electrons and such are said to have a wave particle duality, but what does that mean for a photon? When light strikes an object, are many photons emitted, enough to draw infinitely many rays, is only one emitted, or something in between? In particular, I'm having trouble with thin film interference: ![enter image description here](http:\/\/i.stack.imgur.com\/E53nS.gif) The two resulting rays are said to constructively interfere, which is confusing to me. The two rays are clearly parallel, but not coinciding, so how do the two interfere at all? I think my problem is that I imagine light to be a single ray, with a linear oscillating magnetic field- what is the proper way to address these rays? Are they photons? Or are they small instances of a wave front? I've heard Huygens' Principle, but in this case we present single rays at the end, so I'm led to believe they really ARE rays, in which case they would be photons, and the interference problem would be a result of the wave\/particle duality. The only other thought I've had with regards to the interference is that, as opposed to looking at the rays as one dimensional rays, they could be some kind of representation of a wave 'centered' around that vector, but that doesn't make sense either. I know it's a heavy question, but it's really confusing me."}
{"id":"13739","title":"pressure exerted by fluid","text":"If I had a flexible tube sealed at both ends and I submerged it in water (held vertical) Would the bottom half of the tube compress and the top half expand? What would the pressure in the tube be? Say its a 2\" length of tube with the bottom being in 12\" of water"}
{"id":"13738","title":"Propagation of light in transparent media: absorption and reemission or scattering?","text":"In the two Phys.SE questions What is the mechanism behind the slowdown of light\/photons in a transparent medium? and Why glass is transparent? transparent media were discussed. But I'd like to clarify one detail: is a photon absorbed (and delayed) by the medium and then reemitted, or scattered instantly? Is e.g. a laser beam still coherent after passing glass? As medium molecules are disordered, this should distort the phase of photons taking different paths."}
{"id":"11386","title":"Electrons in CRT","text":"In a CRT, where do the ejected electrons go after they cause fluorescence on the screen, have they lost most of their energy, or do they actually go through the glass?"}
{"id":"89178","title":"Angular momentum of particle rolling around inside of sphere","text":"I have a hemispherical bowl in which I roll a small particle around the edge, starting from the top at point _A_ with a velocity $v_o$. It travels halfway around the sphere and reaches point _B_ , which is a vertical distance _h_ below _A_ , with a velocity $v_f$. Point _A_ is a radial distance of $r_o$ from the vertical centerline and point _B_ is a radial distance of $r$ from the vertical centerline. There is no friction. The goal is to solve for the angle, $\\theta$, between the horizontal and the velocity $v_f$. Here is a diagram of the problem scenario: ![diagram of sphere](http:\/\/i.imgur.com\/57qgEHI.png) My solution relies on the assumption that angular momentum only relies on the velocities in the plane perpendicular to the vertical centerline. Is that a safe assumption? Also, when dealing with energies, is rotational KE and linear KE the same? Should I be taking RKE into account? * * * $$ L_o=L_f $$ $$ mr_ov_o=mrv_f\\cos \\theta $$ $$ \\theta = \\arccos(\\dfrac {mr_ov_o}{mrv_f}) = \\arccos(\\dfrac {r_ov_o}{rv_f}) $$ * * * $$ KE_o + PE_o = KE_f $$ $$ \\frac 12 mv_o^2 + mgh = \\frac 12 mv_f^2 $$ $$ v_o^2 + 2gh = v_f^2 $$ $$ \\sqrt {v_o^2 + 2gh} = v_f $$ * * * $$ \\theta = \\arccos(\\dfrac {r_ov_o}{rv_f}) = \\arccos(\\dfrac {r_ov_o}{r\\sqrt {v_o^2 + 2gh}}) $$"}
{"id":"119299","title":"Question regarding FTL information transfer between two inertial frames","text":"Why can’t information travel faster than the speed of light, if the two endpoints to and from which the information is being sent are moving relatively to each other, as long as the information travels slower than the inverse of the velocity which the planets are moving apart at. According to the velocity addition formula: w=(u−v)\/(1−uv\/c2), as long as u < 1\/v, then w is positive and the laws of causality are not violated. Is there a reason then that information cannot travel faster than the speed of light between two points moving relatively to each other. Also, couldn’t information travel an infinite number times the speed of light, if two points were not moving relatively to each other."}
{"id":"19157","title":"What force is acting on the charge in the dielectric?","text":"For example I have a dielectric solid with a small charged ball in it. And I have external electric field $E$. So what force is acting on this ball? The field in dielectric is $\\frac{E}{\\epsilon}$, so the force should be $\\frac{Eq}{\\epsilon}$. On the other hand. If I remove a small piece of dielectric then the field in the hole will be $E$. Now I put charge in this hole and the force is $Eq$. The hole is just for charged ball and there is no free space. So what is the force?"}
{"id":"52269","title":"Does sound propagate further in freezing weather?","text":"A few days ago I went for a walk in the evening. We're having winter with a little snow and freezing temperatures. We're in a quiet, shallow valley with a train station about 1km from us. I heard a train coming so I wanted to wait for it to watch it arriving to the station. To my surprise, although I was hearing the sound coming from behind a hill, the train wasn't appearing. After several minutes, I gave up, and went back, and finally I saw the train arriving after another a few minutes. The train must have been several km away when I first heard it. I watched this phenomenon later and I realized that also I could hear and understand people talking on much larger distances than usual. This has not happened before, and my only idea is that it's because of the cold weather. I have two ideas how to explain it:   1. Cold air propagates sound better for some reason.   2. We have a few cm of snow covered by ice crust, as we had freezing rain a few days ago. I guess this can mean sounds aren't absorbed by earth and are reflected instead, which makes them propagate further. (I'd say this is more probable than 1.) Is any of this reasonable, or is there another explanation? (I'm not a native speaker so please feel free to correct any language errors.)"}
{"id":"52267","title":"Why does a salt solution conduct electrical current?","text":"How does e.g. sodium chloride (aq) conduct electricity? By accepting electrons (unlikely since they already have a full outer shell)? But they can't be hopping around themselves, can they? I mean, if I have two poles made of metals inserted into a beaker with this solution, and I try to let a current go through from the pole through the solution to the other pole, the ion itself moving can't be conducting electricity, right? The metal pole won't accept such an ion instead of an electron..."}
{"id":"52260","title":"Did force of gravity cause evolution of large scale structures?","text":"Did big bang create gravity? What role gravity is assumed to have played in the formation (starting from the big bang) of large structures of our universe and what other important physical mechanisms and processes probably led to the structure we observe today?"}
{"id":"43184","title":"So do I use this Lorentz's law or which law do I use?","text":"I have difficulty understanding exercise 24 in this document: > Two parallel wires I and II that are near each other carry currents i and 3i > both in the same direction. Compare the forces that the two wires exert on > each other. > > (a) Wire I exerts a stronger force on wire II than II exerts on I. > > (b) Wire II exerts a stronger force on wire I than I exerts on II. > > (c) The wires exert equal magnitude attractive forces on each other. > > (d) The wires exert equal magnitude repulsive forces on each other. > > (e) The wires exert no forces on each other. I think - if you use $F_m=IlB\\sin \\alpha$, which is Force on electric wire in uniform magnetic field - that $F_{II}=IlB\\sin \\alpha = 3IlB\\sin \\alpha > IlB\\sin \\alpha =F_{I}$ So answer would be b), but how is it possible because you have Newton's third law( the forces should be equal, but does it apply here) and there is not any magnetic field here. So do I use this Lorentz's law or which law do I use?"}
{"id":"60356","title":"How quark electric charge directly have been measured?","text":"How quarks electric charge directly have been measured when quarks never directly observed in isolation? (Due to a phenomenon known as color confinement.)"}
{"id":"65651","title":"Hexadecapole potential using point particles?","text":"We can get monopole $1\/r$, dipole $1\/r^2$, quadrupole $1\/r^3$ and octupole $1\/r^4$ potential falloff by placing opposite point charges at the corners of a point, line, square and cube, respectively. My book cryptically says \"and so on\", how does one get a $1\/r^5$ and higher potential falloff with a finite number of point-charges in 3 dimensions?"}
{"id":"65652","title":"How to derive the expression for Bose-Einstein distribution variance?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/Oo2mp.png) Can anyone point me to a derivation of this expression? $n_s$ is the number of bosons in a state."}
{"id":"87588","title":"I am looking for 'Programming+Electronics+ Physics' field","text":"I wish to pursue a career that somehow involves programming, electronics, and physics. What are such careers? Also, I have heard of some 1 year post-graduate diplomas\/courses for specialization in Physics. Which universities\/ colleges offer them?"}
{"id":"69376","title":"Finding out Energy value","text":"A Lagrangian is given by, $$L= \\left(\\frac{\\pi}{2}\\right)^2 R^d \\left[\\frac{1}{2}\\dot A^2 - V(A_{max})\\right]$$ $$E=\\left(\\frac{\\pi}{2}\\right)^2R^d V(A_{max}) $$ where V (A) now includes nonlinear terms and E is the energy which is found by taking the appropriate Legendre transform of the Lagrangian and evaluating it at the upper turning point of an oscillation, $A_{max}$. Now using the potential $V= \\phi^2-\\phi^3+\\frac{\\phi^4}{4}$, and $\\phi=A(t)e^\\frac{-r^2}{R^2}$we can write, $$V(A)= (1+\\frac{d}{2R^2})A^2-\\left(\\frac{2}{3}\\right)^\\frac{d}{2} A^3+ \\frac{A^4}{2^\\frac{d+4}{2}}$$ $$V''(A)= (2+\\frac{d}{R^2})-6\\left(\\frac{2}{3}\\right)^\\frac{d}{2} A+ 3\\frac{A^2}{2^\\frac{d}{2}}$$ > For $d=2$, they got $E_{\\infty}=4.44$ and $d=3$ they found the value > $E_{\\infty}=39.69$, but how? **Why do we write here $E_{\\infty}$?** For more > information please check equations 13 and 14 in the link"}
{"id":"119991","title":"Isotropic neutrino-lepton scattering","text":"I'm a physics student and I'm attending an introductory course of particle physics. My professor stated that, in center of mass frame, the $\\nu_\\mu e^- \\to \\nu_\\mu e^-$ elastic scattering has an isotropic angular distribution, while the $\\bar{\\nu}_\\mu e^- \\to \\bar{\\nu}_\\mu e^-$ scattering has not. I can't figure why this should be true. Any help would be appreciated."}
{"id":"46007","title":"Not working brakes: just another energy conservation problem","text":"> A car is driving down a mountain ($v=90 km\/h=25 m\/s$, when the driver > realizes that brakes aren't working. He try to lose velocity going up an > inclined ($20°$) plane, with a friction coefficient of $k=0.60$. How many > meters will it take to halt? I've tried as following ($s$ is the request): $$K=\\frac{mv^2}{2}$$ At the end, the potential energy gained is: $$U=mgh=mg\\cdot s\\cdot sin \\alpha$$ In the mainwhile the energy lost due to the friction is: $$L_f=F \\cdot s=mg \\cdot cos(\\alpha) \\cdot s$$ But the work done by non conservative forces (friction) is also: $$L_f=U-K$$ And I have: $$mg \\cdot cos(\\alpha) \\cdot s=mg\\cdot s\\cdot sin \\alpha-\\frac{mv^2}{2}$$ $$g \\cdot cos(\\alpha) \\cdot s=g\\cdot s\\cdot sin \\alpha-\\frac{v^2}{2}$$ $$9.22s=3.35s-312.5$$ But I get a negative time. What's wrong? I'm sure that there is a stupid error, but I can't find it. The correct result (reported on the textbook) is 120 m."}
{"id":"5277","title":"What happens when we connect a metal wire between the 2 poles of a battery?","text":"As I remembered, at the 2 poles of a battery, positive or negative electric charges are gathered. So there'll be electric field existing within the battery. This filed is neutralized by the chemical power of the battery so the electric charges will stay at the poles. Since there are electric charges at both poles, there must also be electric fields outside the battery. What happens when we connect a metal wire between the 2 poles of a battery? I vaguely remembered that the wire has the ability to restrain and reshape the electric field and make it within the wire, maybe like a electric field tube. But is that true?"}
{"id":"71757","title":"What is the mechanism of conduction of electric current in a conductor?","text":"I am just very much confused about what mechanism is followed when we connect the - terminal of battery to the + terminal through a conductor? > What happens inside the conductors that makes the electrons flow? I proposed a conduction mechanism that when we connect a wire to the - terminal, we are actually touching a neutral thing and a charged thing, electrons flow in the section in conductor just next to the - terminal and that section gets charged to equal potential but simultaneously there appears a potential difference between this and the next section in the conductor because the next section is still neutral so now to equalize the potential the electrons flow to this next section and this part goes on till the end of conductor and current flows. I just want to know whether I am thinking in the right direction."}
{"id":"99203","title":"Bystander effect: last works?","text":"Do you know the last works about Bystander Effect on cells caused by radiations? I have found research papers until 2004."}
{"id":"64296","title":"Some Dirac notation explanations","text":"Equation for an expectation value $\\langle x \\rangle$ is known to me: \\begin{align} \\langle x \\rangle = \\int\\limits_{-\\infty}^{\\infty} \\overline{\\psi}x\\psi\\, d x \\end{align} By the definition we say that expectation value is a sandwich: $\\langle \\psi|\\hat{x}|\\psi\\rangle$. So: \\begin{align} \\langle \\psi|\\hat{x}|\\psi \\rangle = \\int\\limits_{-\\infty}^{\\infty} \\overline{\\psi}x\\psi\\, d x \\end{align} * * * Can you first confirm that these three lines are correct (I am not sure if I understand Dirac's bra-ket notation right). If they are wrong please explain: \\begin{align} \\text{1st:}& & \\langle \\psi | \\hat{x} | \\psi \\rangle &= | \\psi\\rangle \\cdot \\hat{x}|\\psi \\rangle\\\\\\ \\text{2nd:}& & \\langle \\psi | \\hat{x} | \\psi \\rangle &= {\\langle \\psi|}^\\dagger \\cdot \\hat{x}|\\psi \\rangle\\\\\\ \\text{3rd:}& & \\langle \\psi | \\hat{x} | \\psi \\rangle &= {\\langle \\psi|}^\\dagger \\cdot \\hat{x} \\langle\\psi |^\\dagger\\\\\\ \\end{align} How do i derive relations $\\langle\\psi|\\hat{x}|\\psi\\rangle = \\langle \\psi |\\hat{x}\\psi\\rangle$ and $\\langle\\psi|\\hat{x}|\\psi\\rangle = \\langle \\hat{x}^\\dagger\\psi |\\psi\\rangle$?"}
{"id":"7865","title":"Intrinsic angular momentum in classical mechanics","text":"Please note, I am only interested in classical mechanics discussion on this. Please do not involve quantum mechanics. Inspired by this question: Is Angular Momentum truly fundamental? My question is: Can there be a concept of angular momentum separate from \"orbital\" angular momentum in classical mechanics? For example, can there be a thing such as \"intrinsic\" angular momentum in a classical theory that could be distinguished from the limit of shrinking a spinning ball to size zero?    It seemed obvious to me that the answer should be no. -- No such distinction could be made in a classical theory. However searching on related topics and reading more brought me to this article in wikipedia:   http:\/\/en.wikipedia.org\/wiki\/Einstein%E2%80%93Cartan_theory   While I don't follow the math details, the overview comments seem to claim there is a classical notion of intrinsic spin, but GR cannot handle intrinsic angular momentum. And a different classical theory is presented which does, and differs from GR. For instance, some important tensors can be non-symmetric now, which can't happen in GR no matter the size we make a spinning ball. Therefore there seems to be a real distinction between \"intrinsic\" angular momentum, and the limit of shrinking a spinning ball to size zero, even in classical mechanics. This blows my mind. So if the answer to the above questions are YES!, can someone help explain what this classical \"intrinsic\" angular momentum is?"}
{"id":"16819","title":"Is there an intuitive reason the brachistochrone and the tautochrone are the same curve?","text":"The brachistochrone problem asks what shape a hill should be so a ball slides down in the least time. The tautochrone problem asks what shape yields an oscillation frequency that is independent of amplitude. The answer to both problems is a cycloid. Is there an intuitive reason why these problems have the same answer? Proposed operational definition of \"intuitive\": Imagine modifying the problem slightly, either to the brachistochrone tunnel problem (tunnel through Earth), or by taking account of the finite radius of the ball. If the answer to the original question is intuitive, we should easily be able to tell whether these modified situations continue to have the same brachistochrone and tautochrone curves."}
{"id":"16818","title":"Transparent boundary condition","text":"I am interested in the finite-difference beam propagation method and its applications. I try to solve the Helmholtz equation. At first, i would like to solve numerically it for the easiest case, without nonlinearities. Just to make sure I'm on the right way. But i really don't understand how to wright the boundary condition. I chose the transparent boundary condition and i need to write it properly to solve numerically the equation. So, for a linear, homogeneous and instantaneous medium the Helmholtz equation is writen (in 3D case, z is the propagation direction) $$ \\frac{\\partial^{2} E(x,y,z)}{\\partial x^{2}} + \\frac{\\partial^{2} E(x,y,z)}{\\partial y^{2}} + \\frac{\\partial^{2} E(x,y,z)}{\\partial z^{2}} = - (k_{0} n )^{2} E(x,y,z) $$ It can be solved if the initial condition is known, $E(x,y,0)$. Introducing operator $\\hat{S}$ $$ \\hat{S} = \\frac{\\partial^{2}}{\\partial x^{2}} + \\frac{\\partial^{2}}{\\partial y^{2}} + (k_{0} n )^{2} $$ The equation can be written in the following form $$ \\frac{\\partial^{2} E(x,y,z)}{\\partial z^{2}} = -\\hat{S} \\ E(x,y,z) $$ The solution of this equation is $$ E(x,y,z) = \\exp \\left [ - i \\sqrt{\\hat{S}} z \\right ] E^{+}(x,y,0) + \\exp \\left [ i \\sqrt{\\hat{S}} z \\right ] E^{-}(x,y,0) $$ Considering only the forward propagating component and introducing the propagation operator $\\hat{P}^{+}$ the electric field at $z=\\Delta z$ can be written through the value of the field at $z=0$ (initial condition written earlier) and so on. $$ E(x,y,\\Delta z) = \\hat{P}^{+}(\\Delta z) \\ E(x,y,0) $$ where $$ \\hat{P}^{+}(\\Delta z) = \\sum \\limits_{n=0}^{\\infty} \\frac{1}{n!}\\left[- i \\sqrt{\\hat{S}} \\right]^{n} \\Delta z^{n} $$ Obtained expression can be adopted to the Crank-Nicholson scheme. But it is also necessary to write the boundary condition. How to write the boundary condition if the medium is confined in the transparent walls ?"}
{"id":"16817","title":"Penetration of armor plate","text":"Is there a simple mathematical expression for the stopping power of a given thickness of armor, given the thickness of armor plate, the radius of a cannon ball, the density of the cannonball and the armor, the tensile strength and\/or toughness of the armor, and the speed of the cannonball? For simplicity assume the cannonball is a solid metal sphere and that the armor plate is homogeneous. I realize that in modern warfare the projectiles are pointed and armor plate isn't a homogeneous slab, but I want to understand the simple case. (My question is inspired in part from reading about Civil War ironclads, but I also saw the question about chain mail and thought that if that question was legitimate this one should be more so.) In case my question isn't clear, what I'm asking is something like the following. Suppose it took 10 cm of iron armor to stop a 20 cm diameter cannonball moving at 300 meters\/sec. How thick would the armor have to be to stop a 40 cm cannonball moving at the same speed? Or what if you doubled the speed? Or what if you doubled the tensile strength of the armor? Etc..."}
{"id":"12129","title":"Speed of light, observed speed while travelling at the speed of light","text":"I was watching Discovery channel the other night, they were telling that time slows down when you travel at a higher speed. This means there is a difference between the actual speed you travel at, and the perceived speed. Does anybody know what the perceived speed is, the speed it seems you're travelling at? What does this imply? Does it have any meaning, in some theories perhaps, that the perceived speed is lower? Does this mean that you can never actually travel at the speed of light? (All this is if we don't account for the fact that's impossible to reach the speed of light.) I'm also not into physics, so try to keep it simple..."}
{"id":"12122","title":"Deriving Newton's Third Law from homogeneity of Space","text":"I am following the first volume of the course of theoretical physics by Landau. So, whatever I say below mainly talks regarding the first 2 chapters of Landau and the approach of deriving Newton's laws from Lagrangian principle supposing Hamilton's principle of extremum action. Please keep this view in mind while reading and answering my queries and kindly neglect the systems to which Action Principle is not applicable: If we use homogeneity of space in Euler-Lagrange equations, we obtain a remarkable result i.e. the conservation of momentum for a closed system. Now, this result, using the form of Lagrange for a closed system of particles, transforms into $ \\Sigma F = 0 $ . Now, how from this can we conclude that the internal forces that particles exert come in equal and opposite pairs? Is it because for 2 particles this comes out as $ F_{1} + F_{2} = 0 $ and we take the forces exerted by particles on one other to be independent of other particles (i.e. Superposition Principle) as an experimental fact? I doubt it as whole of Newtonian Mechanics is derivable from Lagrangian Mechanics and supposed Symmetries. So, according to me, a fact like Newton's Third Law should be derivable from it without using an additional experimental fact. I have an idea to prove it rigorously. Consider two particles $i$ and $j$. Let the force on $i$ by $j$ be $F_{ij}$ and on $j$ by $i$ be $k_{ij}F_{ij}$. Now the condition becomes $\\Sigma (1+k_{ij})F_{ij}=0$ where the terms to be included and rejected in summation understood. As this must be true for any value of $F_{ij}$, we get $k_{ij}=-1$. I don't know if this argument or refinement of such an argument holds or not. I can see many questions arising in this argument and it's not very convincing to me. I would like to hear from you people as to if it is an experimental result used or not? If not, then is the method given above right or wrong? If wrong, how can we prove it? **Addendum** My method of proof uses the fact of superposition of forces itself, so it is flawed. I have assumed that the coefficients $k_{ij}$ are constants and don't change in the influence of all other particles which is exactly what superposition principle says. As the superposition of Forces can be derived by superposition of potential energies at a point in space and potential energy being more fundamental in Lagrangian Mechanics, I restate my question as follows: _Is the principle of superposition of potential energies by different sources at a point in space derivable from the inside of Lagrangian Mechanics or is it an experimental fact used in Lagrangian Mechanics?_ I, now, doubt this to be derivable as the fundamental assumption about potential energy is only that it is a function of coordinates of particles and this function may or may not respect superposition."}
{"id":"57985","title":"The relation between Hamiltonian and Energy","text":"I know Hamiltonian can be energy and be a constant of motion if and only if:   1. Lagrangian be time-independent,    2. potential be independent of velocity,    3. coordinate be time independent. Otherwise $$H\\neq E\\neq {\\rm const},$$ or $$H=E\\neq {\\rm const},$$ or $$H\\neq E={\\rm const}.$$ I am looking for examples of these three situation."}
{"id":"90005","title":"Definition of Lorentz transformations as transformations of the universe?","text":"Following Arnold's [1] definition of the universe as an affine space $ A ^4$ with the group $\\mathbb R ^4$ acting on it, we may define a galilean transformation as _an affine map $g:A^4 \\to A^4$ which preserves the galilean structure_ , i.e. which preserves time intervals and spatial distance beetween simultaneous events. “Time” is an application $t:\\mathbb R ^4 \\to \\mathbb R $ and $P,Q\\in A^4$ are simultaneous if $t(P-Q)=0$? Furthermore, to say that a coordinate system $\\varphi _1$ (read: a bijection $\\varphi _1 :A^4 \\to \\mathbb R ^4$) is in uniform motion with respect to another $\\varphi _2$ means that $\\varphi_2 ^{-1}\\circ \\varphi _1 $ is a galilean transformation of $\\mathbb R \\times \\mathbb R ^3 $ (to give a galilean structure to this space we take $t$ to be the projection on the first coordinate). On the other hand, a Lorentz transformation beetween two coordinate systems in relative motion is often defined as a _linear_ transformation _of the space of coordinates_ : $\\Lambda :\\mathbb R \\times \\mathbb R ^3 \\to \\mathbb R \\times \\mathbb R ^3 $. Question is: is it possible in special relativity to characterize a) the universe as an affine space $A^4$ (with the group $\\mathbb R \\times \\mathbb R ^3$ acting on it) with some additional structure in a way similar to the galilean universe? I think the answer is yes, the structure is given by the pseudo metric on $\\mathbb R \\times \\mathbb R ^3$, $|x|^2 =c^2t^2 - x_1 ^2 -x _2 ^2 -x_3 ^2$. Is this sufficient to fully characterize the universe $A^4$. b) Lorentz transformation as transformations of $A^4$ that preserve its structure. Again, is it sufficient to say that Lorentz transformations are the linear ones which preserve the space-time distance given by $|.|^2$? And why linear and not affine?"}
{"id":"10419","title":"Fine tuning and parametric modelings","text":"When I perform parametric modeling, if there is significant multicollinearity between variables I think should be independent, but in fact are not, I run into the case where one or more of the coefficients becomes exceeding small (or large) relative to the others. How is that different than what occurs in fine tuning problems of the standard model?"}
{"id":"63114","title":"Negative temperature and Absolute hot","text":"This video explains that heat at negative temperatures flows from the negative object to the normal object. If the temperature of the normal object is absolute hot, what happens with the heat? The heat can't be transferred to the absolute hot object, and it apparently does not flow in the other direction, so what happens to it?"}
{"id":"127921","title":"Which experiment would be able to detect change in the speed of light?","text":"1. Since the unit of distance is defined in terms of speed of light in vacuum, if the speed of light in vacuum were to change by \"magic\", what experiment would be able to detect that?   2. Is there a theory which says that the speed of light is not a function of time (in the same medium)? edit. people voting for close, please see my comment"}
{"id":"54958","title":"Can infrared light be sent to long distance?","text":"I need to send infrared light from its emitter to a distance of about 10-12 feet. Is this possible?"}
{"id":"48361","title":"Equivilence of One Flux Quantum and Zero Flux","text":"In Ady Stern's review of the Quantum Hall effect, he says of a quantum hall system \"The spectrum at $\\Phi = \\Phi_0$ is the same as the spectrum at $\\Phi = 0$...\" Can someone explain why this is? It seems like the applied magnetic field certainly changes the hamiltonian, and thus the spectrum, but apparently not when the flux is a single quantum. Also I apologize for any newbie mistakes or if this is answered elsewhere, I'm pretty new to stackexchange. Thanks."}
{"id":"26972","title":"What are some ways to (approximately) symbolically diagonalize Hamiltonian operator?","text":"Specifically the Hamiltonian takes the form of $$\\hat H = \\frac{\\Delta }{2}{\\hat \\sigma _z} + {\\omega _1}\\hat a_1^\\dagger {\\hat a_1} + {\\omega _2}\\hat a_2^\\dagger {\\hat a_2} + {g_1}\\left( {{{\\hat a}_1}{{\\hat \\sigma }_ + } + \\hat a_1^\\dagger {{\\hat \\sigma }_ - }} \\right) + {g_2}\\left( {{{\\hat a}_2}{{\\hat \\sigma }_ + } + \\hat a_2^\\dagger {{\\hat \\sigma }_ - }} \\right),$$ a three body version of Jaynes-Cummings model. I'm currently trying to diagonalize this Hamiltonian, a first step in our application of quantum Zeno effect to a three-body system. I guess this Hamiltonian simply has no close-form diagonalization, just like in classical physics there is no closed-form general solution for a three-body system. So my question is: what are several symbolic approximation techniques to diagonalize an Hermitian operator? Better if that techniques particularly suits this Hamiltonian. The values of $\\Delta, \\omega_1, \\omega_2, g_1, g_2$ need not be general; they can be set, say, all equal in order to simplify calculation."}
{"id":"26977","title":"Global symmetry in string theory","text":"It is often stated that in quantum gravity only charges coupled to gauge fields can be conserved. This is because of the no hair theorem. If a charge is coupled to a gauge field then when it falls into a black hole the black hole acquires a corresponding field. However if it is not then the black hole doesn't \"remember\" it. Apparently it implies we can't have exact global symmetries, only gauge symmetries. How is this expectation realized in string theory? Is it true string vacuum sectors cannot posses global symmetry? Can we prove it?"}
{"id":"26979","title":"Poincare Symmetry in QFT","text":"Given that spacetime is not affine Minkowskispace, it does of course not possess Poincare symmetry. It is still sensible to speak of rotations and translations (parallel transport), but instead of $$[P_\\mu, P_\\nu] = 0$$ translations along a small parallelogram will differ by the curvature. I have not thought carefully about rotations and translations, but basically you could look at the induced connection on the frame bundle, to figure out what happens. This is all to say that spacetime has obviously not _exact_ Poincare symmetry, although the corrections are ordinarily very small. Most QFT textbooks seem to ignore this. Of course it is possible to formulate lagrangians of the standard theories in curved space and develop perturbation theory, too. But since there is no translation invariance, one can not invoke fourier transform. My questions are:   * Why is it save to ignore that there is no exact poincare symmetry? Especially the rampant use of fourier transforms bothers me, since they do require exact translation invariance.    * How does one treat energy momentum conservation? Presumably one has to (at least) demonstrate that the covariant derivative of the energy momentum tensor is zero. Any references that discuss those issues in more detail are of course appreciated."}
{"id":"114433","title":"Space between particles","text":"I am a high school student, and I am just wondering what is the space between each particle, like what is the gap around each atom? I have found no text book cover this topic. Is it a vacuum?"}
{"id":"7615","title":"What is in the space between a nucleus of an atom and its electrons?","text":"There is a common analogy about the structure of an atom, such as the nucleus is a fly in the centre of a sports stadium and the electrons are tiny tiny gnats circling the stadium (tip of the hat to 'The Greatest Show on Earth') but what is in the space between the 'fly' and circling 'gnats'?"}
{"id":"106770","title":"If empty space is not really empty, what does the space between an atomic nucleus and its electrons consist of?","text":"I'm currently reading Brian Greene's **The Elegant Universe** and have also picked up his other book - **The Fabric of the Cosmos**. In **The elegant universe** Brian makes up an interesting point that empty space is not really empty. This is according to string theory. Now I may be completely wrong, but lets consider the **assumption** that empty space is not really empty. Now there are several analogies to explain the composition of an atom. One favorite analogy says that the atomic nucleus can be considered to be the size of a football placed in the center of a football field, with the electrons moving just outside this field. _That's a lot of space!_ **Assuming** that empty space is not really empty, what does this atomic space consist of?"}
{"id":"6464","title":"Does vacuum (empty space) exist?","text":"Added: **5 times down vote for now! Down voter is this religion or physics, please try to explain your decision.** I'm confused about this. In physics we know for a vacuum, but I think that there is a contradiction in this term. The quantum fluctuations are the phenomena that contradicts the vacuum existence because, according to them, the vacuum isn't the empty space. In vacuum the creation of particle-antiparticle pairs is allowed for small times and this is also proven in practice. From the other side, the relativistic theory says about space-time that interactions like gravitation bend the vacuum (empty space). There seems to be a contradiction to me. If vacuum is an empty space then we can't bend it, because we can't bend something nonexistent. In other words: we can't bend 'nothing'. **Can empty space really exist in physics?** **EDIT1:** quote: Luboš Motl > \"By definition, the space without energy is the space whose total value of > energy is equal to 0.\" But this space is nonexistent, so it is abstract and should exist only in our mind... Why is so? My second statement of this post says that in physics we cant bend nothing because such a bending is only thinking and not physics! Another possibility is that space is unknown kind of energy, but this is contradiction in modern physics! **EDIT2:** Can any physics believe that nothing exists? By mathematical logic no! And mathematic is elementary tool in physics. Anything other than that is religion! **Nothing (empty space without energy) is only a logical state!** **EDIT3:** quote: Roy Simpson: > The General Relativity Vacuum is a space-time model region without matter. and Luboš Motl says: \"By definition, the space without energy is the space whose total value of energy is equal to 0.\" Agree... But this is only mathematical Euclidean space + time so this is only a mathematic and not physics! With other words: this is only a method of mathematical mapping. But, in real (not theoretic) physics we can't mapping empty things. Empty is only a logical state! **EDIT4:** Roy Simpsons argumentation seems to me acceptable. quote: Roy Simpson: > Einstein struggled with this too, and the problem has come to be known as > the \"Hole argument\" within GR. You have to decide whether you are just > interested in GR's vacuum (empty space) or the full physical vacuum which > includes quantum aspects as well. Thanks"}
{"id":"58185","title":"What would be the effects if Jupiter was to blown up?","text":"So, in one science fiction story, that tries to be realistic as possible apart from few space magics, humanity has contingency plan to blow up Jupiter. As in, totally destroy it in one massive nuclear explosion. I'd like to know the effects of such event. Would it totally wreck the solar system or would the whole plan be non-issue?"}
{"id":"59851","title":"A simple example of symmetry setting the properties of a Physical System","text":"Does anybody know of an example were one could derive some important properties of a physical system from a symmetry of said system. I´m specially looking for simple classical examples, which could serve to illustrate the importance of finding symmetries of a system to non-physicists (high school students or first year undergrads)"}
{"id":"45068","title":"How far can water rise above the edge of a glass?","text":"When you fill a glass with water, water forms a concave meniscus with constant contact angle $\\theta$ (typically $\\theta=20^\\circ$ for tap water): ![enter image description here](http:\/\/i.stack.imgur.com\/VV6ik.png) Once you reach the top of the glass, the water-air interface becomes convex and water rises up to a height $\\Delta h$ above the edge of the glass, allowing you to fill the glass beyond the naive capacity $\\pi r^2 h$: ![enter image description here](http:\/\/i.stack.imgur.com\/C44BZ.png) So when getting myself a glass of water, I came to wonder exactly how much this increases the capacity of a glass, and what physical constants are involved. My intuition would be that for a very large glass, $\\Delta h$ converges to a constant so that the effective water capacity of the glass grows like $\\pi r^2 (h+\\Delta h)$ (to make things simple I'm assuming that the glass is very thin: $\\Delta r\\ll r$). Perhaps such a constant depends on the precise shape of the rim of the glass. But if not, perhaps it is a constant multiple of the capillary length? So, what can we say about $\\Delta h$, the \"rim contact angle\" $\\alpha$, or the shape of the water-air interface when the glass is filled at maximum capacity?"}
{"id":"74190","title":"How do electrons move?","text":"Are they spinning around the nucleus? If so what makes them move? I just do not understand how they move or if they are \"attracted\" to something is it another particle that makes it move."}
{"id":"44331","title":"How to detect radiation on the metal (coin)?","text":"I've got metal coin : http:\/\/www.worldpeacecoin.org\/ Ruble\/dollar, a coin of disarmament with certificate. But, I am very spleeny person, I fear of it's radiance level and I don't know if I can trust it or check it somehow. That could look weird but my fears feels realistic. I only know that Soviet R-12 (SS-4) nuclear missile is nuclear missile and I'm not sure how do they extract that metal from it? I think that any metal that could be extracted from nuclear missile should be dangerous for my health because of radiance level. Please tell me if there is a method to check the radiance level of it or why should not I fear it?"}
{"id":"44335","title":"Poincare Patch covers half of the hyperboloid of AdS","text":"We start with the general case of $AdS_{p+2}$ i.e AdS space in $p+2$ dimension. \\begin{equation} X_{0}^{2}+X_{p+2}^{2}-\\sum_{i=1}^{p+1}X_{i}^{2} = R^2 \\end{equation} This space has an isometry $SO(2,p+1)$ and is homogeneous and isotropic. The Poincare Patch is given by \\begin{equation} ds^2 = R^{2}\\left(\\frac{du^2}{u^2}+u^2(-dt^2 +d\\mathbf{x}^{2})\\right) \\end{equation} According to Equation (2.27) of the article http:\/\/arxiv.org\/abs\/hep- th\/9905111, The second metric covers only half of the hyperboloid. Firstly, how do I show this. Secondly, when I go to the asymptotic limit (small radial distance), should the topology of the two spaces be different?"}
{"id":"65082","title":"Can the Earth's magnetic be used to generate electricity?","text":"Since the Earth has a magnetic field, can it, in theory, be run through a conductive metal coil to create electricity?"}
{"id":"66161","title":"Faraday Effect, Satellites, and Electromagnetic Atmosphere","text":"If I launched a Satellite into orbit and dropped a silver wire caged in carbon fiber with aerodynamic wings (for support), would the Faraday effect of the wire traveling through earth's electromagnetic atmosphere (as the satellite orbits earth) generate a lot of electricity? Also, would the wire being in the atmosphere eventually pull the satellite back down to earth?"}
{"id":"127928","title":"can we get electrical energy from gravitational energy?","text":"Now I'm trying to give an answer of the question. Satellite rotates about earth by the gravitational force of earth. Now if we place the satellite in such an orbit so that it rotate in opposite to earth rotation and having a greater angular rotational speed. Now there is an existence of earth magnetic field. Though it is very poor but strength of magnetic field is not the major factor of energy conversion. So as in electrical machine rotor rotates and stator gives magnetic field and electricity generates is it not possible to extract gravitational energy by using satellite as rotor and geomagnetic field as stator? how much it is effective? Edited: OK eventually this question becomes duplicate. Now I am asking how can we store this energy. Can we store this energy? or not."}
{"id":"1121","title":"Satellite Power","text":"Electromagnetic induction is the production of voltage across a conductor moving through a magnetic field. Since an orbiting satellite is passing through the Earth's magnetic field would a voltage be induced in a conductor inside the satellite? I would expect this voltage to be incredibly small but if true where is this energy \"coming from\"?"}
{"id":"113374","title":"Could we obtain free electrical energy from the Earth's magnetic field?","text":"If there exists a natural magnetic field due to the metallic core of the earth... could we then extract and get free energy from the earth core ??"}
{"id":"44339","title":"Right topology for infinite dimensional \"Hilbert\" spaces with indefinite or semidefinite norm","text":"For positive definite infinite dimensional Hilbert spaces, there is the standard Cauchy norm topology. What if this state space has an indefinite norm or a positive semidefinite one, as in gauge theories or Faddeev-Popov ghosts? Which infinite sums are valid, and which aren't? Similarly, for the algebra of operators, which norm topology do we choose? Not the W*-one? The C* one?"}
{"id":"14968","title":"Superluminal neutrinos","text":"![XKCD](http:\/\/imgs.xkcd.com\/comics\/neutrinos.png) I was quite surprised to read this all over the news today: > Elusive, nearly massive subatomic particles called neutrinos appear to > travel just faster than light, a team of physicists in Europe reports. If > so, the observation would wreck Einstein's theory of special relativity, > which demands that nothing can travel faster than light. --source Apparently a CERN\/Gran Sasso team measured a faster-than-light speed for neutrinos.   * Is this even remotely possible?   * If so, would it be a real violation of Lorentz invariance or an \"almost, but not quite\" effect? The paper is on arXiv; a webcast is\/was planned here. News conference video here"}
{"id":"131442","title":"Can neutrinos really travel faster than the speed of light?","text":"A few years ago scientists found that neutrinos were going faster than the speed of light. Then they found some problem with the calculations. I read an article that was made this year about faster than light neutrinos. I'm not sure if the journalist is ignorant or if they found that the neutrinos really were going faster."}
{"id":"17197","title":"What are the implications of the speed of light broken?","text":"> **Possible Duplicate:**   >  What would be the effects on theoretical physics if neutrinos go faster > than light? I don't know if it's been asked before, but I couldn't find a thread about it. I guess the news have spread to your ears already, but the speed of light has been broken with neutrinos and I can't come up with any scenario on the implications of this breakthrough. I suppose the 4 constants of the universe don't hold anymore, but I'm no physicist and would like the see your thoughts on the subject. What would Einstein think?? Sorry if this has already been answered."}
{"id":"14996","title":"What are the implications of superliminal neutrinos?","text":"> **Possible Duplicates:**   >  What would be the immediate effects if light does not go at the maximum > speed possible?   >  Superluminal neutrinos Do we re-write physics? Or can it be fitted in current theories? I always hear about Lorentz violation in Plank scale or somewhere there. But what about this?"}
{"id":"29340","title":"Neutrinos and the Cosmic Speed Limit","text":"> **Possible Duplicate:**   >  Superluminal neutrinos An article in the newspaper a few months ago said that Neutrinos travel faster than Light. Another article then said that this observation was flawed due to faults in the electrical systems. Is this true? And if not, will this lead to redefining the Cosmic Speed Limit ( _c_ ) as proposed by Einstein?"}
{"id":"107848","title":"Concerning a previous possible tachyon observation?","text":"A few years ago there was a story about the Large Hadron Collider where a possible tachyon was supposedly observed. It was later shown it didn't occur yet the incident made me think. If a large experiment using extreme amounts of energy trying to duplicate some processes or events of the 'distant' past , would the intense amount of energy in this 'early' Universe experiment create a situation where the speed of light was different?? If a particle was observed emanating from this experiment and it seemed superluminal could this be due to a different speed of light measurement due to this phenomenon."}
{"id":"14973","title":"What would be the effects on theoretical physics if neutrinos go faster than light?","text":"Earlier today, I saw this link on Facebook about neutrinos going faster than the speed of light, and of course, re-posted. Since then, a couple of my friends have gotten into a discussion about what this means (mostly about time-travel), but I don't really know what this really implies. This made me wonder... What are the biggest and most immediate implications of this potential discovery? **Related:** Superluminal neutrinos"}
{"id":"15010","title":"Can neutrinos travel faster than the speed of light?","text":"> **Possible Duplicate:**   >  Superluminal neutrinos   >  What would be the immediate effects if light does not go at the maximum > speed possible? This is a hot topic right now, so I thought we should get a question going on it and hopefully keep it up to date with the latest evidence for or against this discovery. Sources: [1] (Associated Press), [2] (Guardian.co.uk), [3] (Original Publication - Cornell University) > Scientists around the world reacted with cautious shock on Friday to results > from an Italian laboratory that seemed to show that certain subatomic > particles can travel faster than light. > > The journey would take a beam of light around 2.4 milliseconds to complete, > but after running the Opera experiment for three years and timing the > arrival of 15,000 neutrinos, the scientists have calculated that the > particles arrived at Gran Sasso 60 billionths of a second earlier, with an > error margin of plus or minus 10 billionths of a second. The speed of light > in a vacuum is 299,792,458 metres per second, so the neutrinos were > apparently travelling at 299,798,454 metres per second. Ignoring the boilerplate media hype about the possibilities of time travel and alternate dimensions - I'm looking for academic sources that might suggest how this could be true, or alternatively, how this discrepancy could be accounted for. * * * I read the article published with their findings. It looks like they took an insane amount of care with their measurement of distance and time. One of the most common skepticism of people who know nothing about the experiment is stuff like: > You might worry about[...] have they correctly accounted for the time delay > of actually reading out the signals? Whatever you are using as a timing > signal, that has to travel down the cables to your computer and when you are > talking about nanoseconds, you have to know exactly how quickly the current > travels, and it is not instantaneous. [2] This experiment doesn't use that sort of 'stopwatch' timing mechanism though. There is no 'T=0', and no single firing of neutrinos. What is detected is watermark patterns in the steady stream of particles. The streams at the input and output are time stamped using the same satellites and any position along each stream has a precise time associated with it. By identifying identical patterns at input and output streams, they can identify how long it took particles to travel between the points. [1] ![Time](http:\/\/i.stack.imgur.com\/MYJhd.png) As for distance, they use GPS readings to get the east, north, and altitude position along the path travelled to great precision. So much so that they even detect slow earth crust migration and millimetres of changes in distance between source and destination when something like an earthquake occurs. When your particles are travelling on the scale (730534.61 ± 0.20) metres, this is more than enough precision: ![Distance](http:\/\/i.stack.imgur.com\/o4gUG.png) It's going to take a lot more than grassroots skepticism to think of what could have caused this discrepancy. I've seen suggestions such as the gravity of the earth being different along the path of the neutrinos, which warps space\/time unevenly. The neutrino might not actually be travelling as far as they think if space\/time is contracted at one or more points along the path where gravity varies. Anyways, I'll be interested in seeing how it pans out. Like most scientists, my guess is an unaccounted for systematic error (because they definitely have statistical significance and precision and on their side) that has yet to be pointed out, but it probably won't take too long with all the theoretical physicists that will be pouring through this experiment."}
{"id":"14979","title":"What would the impact be on the physics world if neutrinos DO travel faster than light?","text":"> **Possible Duplicates:**   >  What would be the immediate effects if light does not go at the maximum > speed possible?   >  Superluminal neutrinos I was reading this article about a group of scientist thinking that they might have surpassed the speed of light with neutrinos, and are wanting other groups to check their work. If their research is proven true, what impact would this finding have on the world of physics?"}
{"id":"15053","title":"Do the particles that were found to break the speed of light really break Einstein's theory of relativity?","text":"> **Possible Duplicate:**   >  What would be the effects on theoretical physics if neutrinos go faster > than light? Update: Loose cable caused faulty results Apparently, researchers at CERN have found definitive evidence of particles traveling faster than the speed of light. The article writes: > If confirmed, the discovery would undermine Albert Einstein's 1905 theory of > special relativity, which says that the speed of light is a \"cosmic > constant\" and that nothing in the universe can travel faster. Is this just a publicity stunt, or is it possible that this is legit? (I would like to see the paper for these findings, I love how they conveniently leave that out). In other words, if this is found to be true, **would Einstein's theory of relativity _really_ be challenged?** There's some debate in the comments which suggest this doesn't do anything to Einstein's theory. E.g.: > The reporting here is incorrect. Einstein’s theory _DOES NOT HOLD_ that > nothing can travel faster than light. That is a very common misconception. > Einstein’s theory starts with one axiom (a self evident truth) and one > postulate (a statement deemed to be true without further argument). The > axiom from Einstein’s theory can be stated: the laws of physics should not > depend on the frame of reference of the observer. This is a self evident > truth. The postulate can be stated: light will be measured to travel at the > same speed by all observers regardless of reference frame. This postulate > was based on experimental evidence available in 1905 and still available > today. When the axiom and postulate are applied to observers traveling at > constant speed relative to one another, you get the special theory of > relativity, published as part of Einsteins 1905 paper. One of the > conclusions is that “the speed of light is constant and absolute in free > space”. It falls out of the mathematics. This conclusion is part of the > special theory of relativity. It has to date not been dis-proven. If it is, > then the postulate must be incorrect. It would mean that light _can_ be > measured to travel at a different speed depending on the frame of reference > of the observer. I don’t get from the article that this is what has > occurred. When the axiom and postulate are applied to gravitational and > accelerating frames of reference, you the general theory of relativity, > published in 1916. The general relativity mathematics bring forth strange > things like black holes, worm holes, time warps, time travel etc. in the so > called “fabric of space and time.” The stuff science fiction authors and > buff’s are so enamored with. Including me. Most of it has not been > practically realized. Only the more mundane stuff like gravity lenses, time > dilation, length contraction have been observed. If any experiment can be > performed that is in contradiction to the conclusions, then we would merely > say, as would Einstein, that the postulate of the constancy of the speed of > light, regardless of reference frame, must be incorrect. That’s not such a > big deal, really. It would change a lot of physics. It would be very > exciting. But it would just mean that the one postulate, one that none of us > have ever been able to intuit anyway, is incorrect. > > This is not the first time that experiments have been performed that have > particles traveling faster than light _in a medium other than free space_. > In this case neutrinos travel through, air, water and apparently rock faster > than light does. That does not violate the fundamental postulate that the > speed of light is constant regardless of the frame of reference of the > observer. > > Get it right. > > Dr. Karl Hudnut, UCAR – COSMIC. ## Who is correct?"}
{"id":"94785","title":"Dirac adjoint of a matrix","text":"The Dirac adjoint for Dirac spinors is defined as, $$ \\bar{u} = u^{\\dagger} \\gamma^{0} \\, . $$ However I have come across this, $$ \\overline{\\gamma^{\\mu}} = \\gamma^{\\mu} \\, , \\tag{1} $$ (where $\\gamma^{\\mu}$ are the $4\\times4$ gamma matrices). Naively applying the same rules as for the Dirac spinor clearly does not get us anywhere, $$ \\overline{\\gamma^{\\mu}} = \\gamma^{\\mu \\dagger} \\gamma^{0} = \\gamma^{0} \\gamma^{\\mu} \\gamma^{0} \\gamma^{0} = \\gamma^{0} \\gamma^{\\mu} \\neq \\gamma^{\\mu} \\, . $$ So it seems that the Dirac adjoint for a matrix is defined differently, so in trying to figure this out I make the following reasoning, let $A$ be a $4 \\times 4$ matrix and $u$ a Dirac spinor so that $Au$ is again a Dirac spinor. Taking the Dirac conjugate (which is defined) gives, $$ \\overline{A u} = (A u)^{\\dagger} \\gamma^{0} = u^{\\dagger} A^{\\dagger} \\gamma^{0} = u^{\\dagger} \\gamma^{0} \\gamma^{0} A^{\\dagger} \\gamma^{0} = \\bar{u} \\;\\underbrace{\\gamma^{0} A^{\\dagger} \\gamma^{0}}_{ = \\bar{A} ? } \\, . $$ So my guess is that $\\bar{A} = \\gamma^{0} A^{\\dagger} \\gamma^{0}$. If this is the case it is straightforward to show that $\\overline{\\gamma^{\\mu}} = \\gamma^{\\mu} $. **My question** is the following, is the above statement correct? Is it so that the Dirac adjoint is actually only defined for Dirac spinors but it can be sort of extended to $4 \\times 4$ matrices as above (allowing one to write $\\overline{A u} = \\bar{u} \\bar{A}$)? * * * Link where I found eq. (1) (page 93, eq. 3.249) Link where I found eq. (1) and the claim $ \\overline{X} = \\gamma^{0} X \\gamma^{0} $ which appears to be missing a \"$^{\\dagger}$\"? (page 9, eq. 5.54)"}
{"id":"94789","title":"Triangular lattice arrangement of vortices in a superfluid","text":"In a simply connected container containing a superfluid and rotating, there is a net circulation of superfluid. This is found due to the vortices formed, around which the superfluid rotates. These vortices have been found to generally form a triangular lattice arrangement. Why is the triangular lattice arrangement preferred by the vortices formed ?"}
{"id":"59589","title":"Use of Principle of Equivalence","text":"_Let $x^\\mu$ be the coordinates of a reference frame, $K$, where all bodies feel the same constant and uniform acceleration $\\textbf{a}=\\textbf{g}=-\\nabla\\varphi$; let $\\xi^\\mu$ be the coordinates of a Locally Inertial Frame, $LIF$. Using the **Principle of Equivalence** , show that the linear part in $\\textbf{g}$ of the interval for $K$ is_ $$ ds^2=(1+2\\varphi\/c^2)c^2dt^2-d\\textbf{x}^2+o(1\/c^2). $$ My attempt: P.E states the metric tensor, $g_{\\mu\\nu}$, respect to a generic system of coordinates its related to metric tensor of a flat Minkowskian $\\eta_{\\alpha\\beta}=\\text{diag(1,-1,-1,-1)}$ via the Jacobian of the diffeomorphism $x\\mapsto\\xi$. Obviously such a diffeo can be of the form $$\\xi^0=ct,\\xi^i=x^i-1\/2g^it^2,$$ and the P.E states that $$ g_{\\mu\\nu}=\\frac{\\partial\\xi^\\alpha}{\\partial x^\\mu}\\frac{\\partial\\xi^\\beta}{\\partial x^\\nu}\\eta_{\\alpha\\beta}, $$ so the 00 component is $$ g_{00}=\\frac{\\partial\\xi^\\alpha}{\\partial x^0}\\frac{\\partial\\xi^\\beta}{\\partial x^0}\\eta_{\\alpha\\beta}=\\frac{\\partial\\xi^0}{\\partial x^0}\\frac{\\partial\\xi^0}{\\partial x^0}-\\frac{\\partial\\xi^i}{\\partial x^0}\\frac{\\partial\\xi^j}{\\partial x^0}\\delta_{ij}=1-\\Big(\\frac{v^i-g^it}{c}\\Big)^2\\simeq1-2\\varphi\/c^2++o(1\/c^2,\\textbf{g}^2), $$ here i take $2v^itg^i\/c^2=2x^ig^i\/c^2=-2\\varphi\/c^2.$ Other problem arises becouse my $g_{0j}\\neq0$. What i wrong?"}
{"id":"34204","title":"relativistic acceleration equation","text":"A Starship is going to accelerate from 0 to some final four-velocity, but it cannot accelerate faster than $g_M$, otherwise it will crush the astronauts. what is the appropiate equation to constraint the movement so the astronauts never feel a gravity higher than $g_M$? for a moment i thought the appropiate relationship was $$ \\left\\lvert \\frac{d u}{d \\tau}\\right\\rvert \\le g_M $$ where the absolute value is of the spatial component of the four-acceleration But going down this route i get the following: $$ \\lvert u_F \\rvert = \\int_0^{\\tau_F} \\left\\lvert \\frac{d u}{d \\tau} \\right\\rvert\\,d \\tau \\le g_M \\int_0^{\\tau_F} d \\tau = g_M \\tau_F $$ where $u_F$ is the spatial component of the final velocity, and $\\tau_F$ is the proper time it takes to reach the final velocity. The above gives me: $$ \\tau_F = \\frac{ \\lvert u_F \\rvert }{ g_M } $$ i'm doing some silly mistake, because there are no gamma factors, and i'm getting a finite proper time to reach $\\lvert u_F \\rvert = c$"}
{"id":"134090","title":"How much has the Milky Way moved since it's forming?","text":"What i really want to ask how much has the Milky Way moved, relative to where it was \"at the big bang\" or the soonest time that makes sense (since i doubt \"at the big bang\" makes much sense in this question). I suppose the galaxies have non-zero impulse, otherwise we wouldn't see things like galaxy collisions. So, relative to where our galaxy or whatever was there (dust cloud?) \"in the beginning\", how much did we move?"}
{"id":"35153","title":"Dependance of temperature on color of metals","text":"I asked this question because I supposedly did last year, Stanfor Klein which belongs to the Solar Energy Laboratory of the University of Wisconsin says that \"the color of a car does not affect its internal temperature\". I wonder why Metals with different colors perhaps do not absorb different doses of temperatures? and as a consequence, when different metals of different colors are exposed to strong radiation it is not so differently warmed?"}
{"id":"35156","title":"the sounds of an exploding star","text":"We know that space cannot spread a sound wave as there is no \"air\" or a medium that would support the spread of a sound wave. However if we put ourselves in the vicinity of an exploding star, would it be possible to hear something? The question arises from the idea that within the explosion of a star (first few seconds or less) you may hear a noise due to the explosion of the star..."}
{"id":"28698","title":"Residual Resistivity in alloys and metals","text":"## Residual Resistivity * * * I saw that the graph of resistivity to temperature of alloys like nichrome is like so![enter image description here](http:\/\/i.stack.imgur.com\/KkiDU.png) Meaning that even at 0 K it has some resistivity just like copper : ![enter image description here](http:\/\/i.stack.imgur.com\/GnQ1v.png) I read some where \"It is the residual resistivity due to defect scattering\" Is this related to the defects that i studied in solid state chemistry about lattice defects.Can some body elaborate?   * An alloy is a mixture of metals and a temperature coefficient of resistivity comparable to metals then why is its graph more linear than metals.    * Is this because of the lattice structure of an alloy?"}
{"id":"91669","title":"Discrete movement vs wave function collapse","text":"I remember once, as a child, thinking that objects do not really \"move,\" but that at a very small scale they would have to \"disappear\" and then \"appear\" again at their newly shifted position, just the way computers render moving particles based on refresh rates. This relates to Zeno's paradox which is solved by infinite sums. Then I heard about quantum wave function collapse and the double slit experiment, and then thought: oh, maybe nature solved the problem by turning anything that wants to move into a wave instead of making a single particle \"appear\" and \"disappear\" in new positions as it moves. Waves is by the way a very elegant solution in comparison. My question is: was my thinking correct? are waves (and wave collapse) nature's way to make particles move around?"}
{"id":"91666","title":"Can a wormhole be created in space?","text":"Is wormhole a practical concept? If not what did the scientists do, to theorize it? Does it have any limitations pertaining to speed of light?"}
{"id":"91662","title":"Layman explanation and demonstration of positive lightning","text":"Inspired by the thunderstorm overhead, and after reading the question and answers Voltage and current of positive lightning \\- what is an effective means to explain this phenomenon to a layman? Also, as demonstrations are more often than not a better way of explanation, how could positive lightning be safely and meaningfully demonstrated?"}
{"id":"91660","title":"What is the reason behind band gap narrowing in semiconductors","text":"I want to know why some semiconductors band gap decreases after doping with elements. **_Burstein-Moss band-filling effect_** can be useful to explain band gap widing in a semiconductor materials but i was unable to find any logical explanation for band narrowing effect. Can you please explain the mechanism of band gap narrowing. This below quotes were taken from a research article. > There is general agreement that two competing phenomena are dominant in > affecting the absorption edge in heavily doped semiconductors. First, the > well-known Burstein-Moss band-filling effect which shifts positively the > measured band-edge energy with increasing carrier concentration. In this > case the measured optical gap $E_{m}$ is the sum of the optical gap of the > lightly doped material $E_{0}$, plus that due to filling of the conduction > band due to Is $\\Delta E_{BM}$, I. E. $E_{m}=E_{0}+E_{BM}$. Thc second > phenomenon which affects the optical absorption edge with increasing donor > density is due to a change in the nature and strength of the interaction > potentials between donors and the host crystal. This latter effect gives > rise to a band-gap shrinkage and to some increased tailing of the absorption > edge. In this case the measured optical gap is $E_{m}=E_{0}+E_{BM}-\\Delta > E_{g}$., where $\\Delta E_{g}$ is the gap shrinkage. I didnt understand the explanation given in second phenomenon. What is author meant by due **_to a change in the nature and strength of the interaction potentials between donors and the host crystal_**. Advance thanks for your help"}
{"id":"67745","title":"Triple-right triangle experiment: what's the minimum distance?","text":"Among the other ways, one way to prove the Earth is round is the **triple- right triangle**. The idea is simple:   1. Starting from point A you move in a straight line for a certain distance.   2. At point B, turn right 90° degrees, move along the line for the same distance.   3. At point C, turn again to the right and do the same.   4. We'll eventually get back at the starting point: point A and C are the same location, thus we just created a triangle with 90° degrees. This proves that that Earth has a spherical shape (not a perfect sphere), since these movements would only create a square with three sides if we were to do it on a flat surface. However, the \"problem\" of this experiment is that it's not really doable on a small scale. The distance must be so much that the curve of our planet can be taken into consideration. Walking 1 meter, then one meter and then another meter won't create a triangle, since the curve of the planet is not that strong. So my question is: what's the minimum distance we'd need to travel for this experiment to work?"}
{"id":"11321","title":"Why do two bodies of different masses fall at the same rate (in the absence of air resistance)?","text":"I'm far from being a physics expert and figured this would be a good place to ask a beginner question that has been confusing me for some time. According to Galileo, two bodies of different masses, dropped from the same height, will touch the floor at the same time in the absence of air resistance. BUT Newton's second law states that $a = F\/m$, with $a$ the acceleration of a particle, $m$ its mass and $F$ the sum of forces applied to it. I understand that acceleration represents a variation of velocity and velocity represents a variation of position. I don't comprehend why the mass, which is seemingly affecting the acceleration, does not affect the \"time of impact\". Can someone explain this to me? I feel pretty dumb right now :)"}
{"id":"17039","title":"why do all objects of varying mass acelerate the same?","text":"> **Possible Duplicate:**   >  Confused about the role of mass why is it that two object of varying mass will fall at the same speed in a frictionless enviorment like the moon? Is it because the object needs to overcome more momentum or what?"}
{"id":"36422","title":"Why do objects with different masses fall at the same rate?","text":"> **Possible Duplicate:**   >  Confused about the role of mass Today we were in our Literature class talking about the Renaissance and the Enlightement and our teacher also said that scientific experiments were being conducted, and she gave as an example the experiment in which they dropped objects with different mass from a tower to see which object would land first. She then said that she herself didn't know the outcome and since I'm known in my class as the #1, they asked me and I said that they landed at the same time (to my shock many classmates even disagreed with me about this fact). The teacher asked me to explain why and we haven't had anything about inertia in our physics class, so I was forced to use my own self-thought knowledge of physics: I said gravity does pull harder on heavier objects, but heavier objects have more resistance to move, in other words, they have more inertia. The entire class except for the teacher disagreed with me even though they usually don't. One fairly annoying kid asked me to explain why a feather falls way slower than a bowling ball, and I explained why and I also made the claim that in a vacuum they would fall at the same rate. Was I right or my class? Of course I know my explanation is lacking, but it has truth to it, right?"}
{"id":"13338","title":"Acceleration of two masses by gravitational force","text":"> **Possible Duplicate:**   >  Confused about the role of mass I know that two different masses fall at the same rate in the same gravitational field because the greater gravitational force of the heavier one is exactly offset by its greater inertial resistance. What I don't understand is why the larger mass wouldn't fall more slowly at first and then once the inertial resistance is overcome, it would then accelerate faster than the less massive one. It seems that the rate of acceleration (not velocity, which obviously does) would vary according to the distance of the fall"}
{"id":"1690","title":"Theoretical Physics - How to?","text":"Although I doubt somewhat whether this question is really appropriate for this site, I hope it gets answered anyways. I guess, what I'm wondering is:   1. How does one get to work as a theoretical physicist and - probably more importantly - what do theoretical physicist actually do all day long?   2. How are theoretical physicists distinguishable from mathematicians? Does a physicists day look very different from that of a mathematician?   3. I have a great interest in physics, but I'm not really much interested in doing experiments: Would it be advisable to do my bachelor in mathematics and try to get into theoretical physics later on?   4. Is there a real chance of getting into research afterwards? (not that any kind of answer to this question would ever stop me from trying...) Well, I hope this question is acceptable. I think 1) might for example be answered by giving a link to a blog of a working theoretical physicist, who gives some insight into his or her everyday life, or some kind of an essay on the topic. Of course any other kind of answer is greatly appreciated. Thanks in advance! Kind regards, Sam **Edit after several answers:** Thanks a lot for all the responses! I found it very interesting and helpful to to get some input from you guys. Although the opinions seem to differ a bit, one can definitely see many overlaps, too. I do still have some time do decide, and will definitely look at some books suggested here, visit some lectures and try to get a feel for what it would be like to do either physics or maths. Thanks again for your effort! :-)"}
{"id":"97847","title":"Non-Locality of Space - QFT (Srednicki's book)","text":"I was going through Mark Srednicki's book on QFT. It says in the relativistic limit the Schrodinger equation becomes something like : $$ i\\hbar\\frac{\\partial}{\\partial t} \\psi(\\vec x,t) = \\sqrt{-\\hbar^2c^2\\nabla^2+m^2c^4}\\psi(\\vec x,t) $$ Now he says that if I expand the square root (say binomially) it will have infinite no. of spatial derivatives acting on $\\psi(x,t)$; this implies that equation is not local in space. What exactly does it mean to say the equation is not local in space?"}
{"id":"97849","title":"Discrete Values for Observables vs Average Values (Quantum Mechanics)","text":"When considering observables and their corresponding operators, would it be correct to believe that discerning discrete values for an observable is possible ONLY when $\\psi$ is an eigenfunction of the operator? Alternatively, would it also be correct to believe that the average value of an observable is ALWAYS obtainable regardless if $\\psi$ is an eigenfunction of the operator? Thanks for your help."}
{"id":"129529","title":"Derivation of $a_{j}$ coefficients in the quantum harmonic oscillator","text":"In Griffiths' book page 53, when we derive the solution of the quantum harmonic oscillator by using the power series way, we have: $$a_{j+2} = \\frac{2j+1-K}{(j+1)(j+2)}\\, a_{j} .$$ And for large $j$, we have: $$a_{j+2}\\approx\\frac{2}{j}\\,a_j.$$ Up to this point I totally agree (one just takes the limit). However, the subsequent derivation of solution $a_{j}$ and $h(\\xi)$ I attached from the Griffiths' textbook are very confusing.   * How did it go from $a_{j+2}\\approx\\frac{2}{j}a_j$ to the solution of $a_{j}$?    * Also, how do the second and third approximations work in $h(\\xi)$? My questions are mainly mathematical. I very much hope someone can provide a derivation or refer a link where these questions may already be answered. ![enter image description here](http:\/\/i.stack.imgur.com\/qklld.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/wBTkg.jpg)"}
{"id":"73094","title":"Linearity of Quantum Mechanics?","text":"The proof of the No-Cloning Theorem states \"By the linearity of quantum mechanics, ...\" -- Could someone please give me a rough sketch\/outline of what this means. Does it have to do with the Hilbert Space that wave functions live in? I apologize if this question isn't specific enough, I just wanted to fully understand this concept."}
{"id":"32310","title":"What's the meaning of the general solution and the particular solution in differential equations?","text":"Can anybody cast some physical insight into this? I've been studying differential equations on my own and don't understand how you can have a whole host of general solutions. It seems like a rather curious situation which we don't come across in other areas of mathematics. Is there anything more to the discussion that I'm missing?"}
{"id":"14377","title":"Learn QM algebraic formulations and interpretations","text":"I have a good undergrad knowledge of quantum mechanics, and I'm interesting in reading up more about interpretation and in particular things related to how QM emerges algebraically from some reasonable real world assumptions. However I want to avoid the meticulous maths style and rather read something more meant for physicists (where rigorous proofs aren't needed and things are well- behaved ;) ) I.e. I'd prefer more intuitive resources as opposed to the rigorous texts. Can you recommend some reading to get started?"}
{"id":"129140","title":"Books on foundations of QM","text":"I am seeking for books on foundations of Quantum Mechanics with subjects like the EPR experiment, Bell's theorem, the problem of measurement, entanglement, decoherence, nonlocality, interpretations, QM without observer, different formalisms etc. I prefer modern books, post graduate oriented, with history, formalisms and different approaches for a young researcher specialize himself on the subject. EDIT: the other question marked as \"duplicate\" do not fit my needs. I prefer books with higher modern mathematics, deep arguments and well-defined positions within the philosophy of science, post graduate oriented, if those exists. I need a book to get plenty knowledge as to do research in that academic area, not an intuitive approach."}
{"id":"21100","title":"Good quantum physics textbooks","text":"> **Possible Duplicate:**   >  Learn algebra and interpretation of QM I'm a physics student it's my last year and I find my self struggling with usage of quantum mechanics. Now I'm starting to learn nuclear physics, solid state physics and soon quantum field theory and I see that I have holes in my knowledge. It's not that I'm completely ignorant, I've passed the exam in quantum physics I understand the concepts, but it doesn't feel right, I don't have the intuition like I do for example, classical mechanics or electrodynamics.   For example, of course, there are problems that I can't solve in classical mechanics or electrodynamics but I always know where to start I have an intuition about the problem and in most scenarios I have an ideas what solutions should look like. But in quantum physics, usually even for simple problems I can get stuck or don't know where to start.   I want to rebuild my knowledge from a new source.   The literature I've used so far is:   Introduction to Quantum Mechanics by David J.Griffiths,   Quantum physics by Leonard Schiff,   Modern Quantum Mechanics (Revised Edition) by J.J.Sakurai.      First one is an easy read, good introduction, but it doesn't use bra\/ket notation.   Second on I just didn't like, it was hard to read, short on examples.   Third one I really like, but I'd like to see more examples and I can't check solutions for problems at the end of the chapters anywhere.      Could you please recommend me couple of more choices? I prefer books with lot of examples of solving problems. Also, while it's fine that book starts with wave functions, solving Schrodinger equations etc, I'd like a book that uses bra\/ket notation."}
{"id":"104864","title":"How to calculate error of parallax and sextant based navigation?","text":"1. First of all, why wasn't the sextant ever used for land navigation? The horizon is easier to see at sea, but land based sextants could be used in conjunction with artificial horizons (as at sea when horizon is hidden by fog).   2. Parallax has been used by both the US army and navy to measure distance to targets. The devices that used this principle were called coincidence rangefinders. It seems this system was still used after the introduction of radar. Why was this system eventually phased out?   3. Finally my main question: How can one calculate the error for measurements made by sextants and parallax devices?"}
{"id":"114775","title":"Help explore a self-feedback camera-monitor chaotic system","text":"We are trying to emulate the chaotic system Jim Al-Khalili demonstrate (3 min video). In our chaos lab, we are trying to research the chaotic system shown in the video. We are using just a webcam and a regular PC monitor. Our goal is to build a bifurcation tree for this system and to show how by changing the parameters of the system (location and distance of the camera from the monitor, time delay (between capturing and showing what is captured on the screen)) we can see a transition from \"order\" to chaos. The problem is we are not sure what exactly is the mathematical representation of the system, and what is the thing that is \"doubling\" and going to chaos (the y axis of a bifurcation tree - like voltage peaks on the diode in a chaotic RLD system). How do we approach this subject (if it's even possible)?"}
{"id":"52929","title":"What is evidence for an irreversible change?","text":"Knowing some about thermodynamics and reactions, I do understand how it can be shown that a change is reversible. But irreversible? Why can't it be that a change that was deemed irreversible thousands of years ago via new changes perhaps developed by physicists or changes, processes and reactions from some other part of the world or space, a change that was deemed irreversible in the future can be shown to be reversible? I think that for instance diseases that were deemed irreversible as science progressed, we could make changes that were priorly said to be irreversible, in fact reversible."}
{"id":"78137","title":"Should entropy have units and temperature in terms of energy?","text":"I've been thinking about entropy for a while and why it is a confusing concept and many references are filled with varying descriptions of something that is a statistical probability (arrows of time, disorder, etc.). Could this confusion be in the nature of it's units and how it is scattered around in different equations? When Maxwell published his paper on the molecular distribution of velocities in 1859. This has led to the identification of temperature with the mean kinetic energy of atoms or molecules in the gas. At that point we could have redefined temperature to units with energy which make sense instead of K. This would make the new $T_a=kT$ where $T_a$ is the new absolute temperature in the units with energy. Once temperature is in the units of energy this makes entropy unitless (makes more sense to me): $S = ln(W)$ In addition this makes other equations more clear: Maxwell's gas law identity has the form (for atomic particles of mass m) $\\frac{3kT}{2} = \\frac{m<ν^2>}{2}$ where T is the absolute temperature and $<ν^2>$ , the average of the squared velocity of the atoms, and k Boltzmann’s constant. But with $T_a$, the relation $\\frac{3kT}{2} = \\frac{m<ν^2>}{2}$ will become simpler $\\frac{3T_a}{2} = \\frac{m<ν^2>}{2}$ And the gas constant R in the equation of state for ideal gases would be changed into Avogadro number $N_{AV} = 6.022 × 10^{23}$ and the equation state of one mole of an ideal gas will read: $PV = N_{AV}T_a$, instead of $PV = RT$ **Why bother?** This would make entropy identical conceptually and formally to information by redefining temperature in terms of units of energy. This creates a strong association between entropy and probability and makes the second law (which isn't absolute anyway) less mysterious. Would it make sense to have temperature in units of energy at other levels of physics and entropy unitless say in the case of black holes entropy?"}
{"id":"106975","title":"Correspondence principle and quantum computers","text":"I just read this article at https:\/\/medium.com\/the-physics-arxiv- blog\/7ef5eea6fd7a about the work of a physicist called Bolotin, that states that P!=NP (from computer science) implies that large quantum mechanical objects are not possible. The author starts by explaining the Schrodingers cat thought experiment. He then says \"Nobody knows why we don’t observe these kinds of strange superpositions in the macroscopic world\", which I find strange because you cannot observe a superposition, you either see that the cat is alive or dead. But then he writes \"For some reason, quantum mechanics just doesn’t work on that scale. And therein lies the mystery, one of the greatest in science.\" I thought that the correspondence principle actually explains nicely why quantum mechanics works very well on a large scale? The main point of the article is that if you can have a large quantum mechanical system, you would get P=NP, which is believed to be not true by most computer scientists. But isn't that exactly what they try to do with quantum computers?"}
{"id":"75441","title":"Why does string theory have such a huge landscape?","text":"I was browsing through _Foundations of Space and Time,_ a compilation of essays on various theories of quantum gravity. The following passage in the introduction intrigued me: > _Each compactification leads to a different vacuum state.... at least one > state should describe our Universe in its entirety.... the enormous number > (~10^500 at last count) of solutions, with no perturbative mechanism to > select mechanism to select among them, leads some critics to question the > predictive power of the theory..Even more worrying is that, while the theory > is perturbatively finite order by order, the perturbation series does not > seem to converge._ I don't know anything about string theory and so I could not make head or tails this. All I know is that ~$10^{500}$ is a very large number.   1. What exactly is a 'solution' in string theory? Is it a spacetime metric of some sort or the terms of a S-matrix of some sort?    2. Why are there so many 'solutions'?    3. I thought string theory was supposed to be finite, why do perturbative series still diverge?   4. Is there any experimental technique to limit the number of 'solutions'?    5. Will experimental techniques be able to pinpoint a solution within present day string theorists' lifetimes too? If not, how long will it take before we can experimentally probe these things?    6. Are string theorists completely relaxed about these issues? Or are they in anguish?"}
{"id":"26143","title":"Phases of the moon video","text":"I am an educator, and I am looking for a specific video. In the video, they ask some middle school students and some college graduates about why the moon has phases. Most of the students in both the groups get the answer wrong, saying that the phases of the moon happen because of the shadow of the Earth. I remember that they also interviewed the teacher of the middle school students, and she was really perplexed that her students didn't know the right answer. Another thing I remember from the video was that the middle school kids were not sure of their answers, but the college graduates (who are at their graduation ceremony) were really sure. I want to use that video for educational purposes, but I can't find it. Can you please help?"}
{"id":"26145","title":"Need help buying binoculars online for astronomy (10x50)","text":"After reading much online I've decided to by a 10x50 porro prism Binoculars. The one I have in mind http:\/\/www.letsbuy.com\/celestron-upclose-10x50-p-34652 has BK7 prisms. The one I would like to buy is this: Olympus 10x50 DPS I but it is never available and a tad bit more expensive. Is buying binoculars or telescopes online a bad thing as I've read you need to inspect them carefully before buying? Also will the prism difference (BK7 vs BAK4) make much of a difference for the 10x50 range?"}
{"id":"79824","title":"detailed balance in the context of the ising model","text":"I am having a very basic problem understanding the idea of detailed balance, particularly in the context of the Ising model. Most references I have found contain the following phrase: \"In equilibrium, each elementary process must be equilibrated by its reverse process\". What does it mean for one process to equilibrate another? In particular, I am trying to understand the section here: http:\/\/en.wikipedia.org\/wiki\/Ising_model#Algorithm_Specification in which they state that the given form of selection probabilities is required by detailed balance, but I suspect that that will follow once I understand what detailed balance actually means. Thanks!"}
{"id":"116750","title":"Grounded conductor inside a uniform electric field","text":"I am working on a textbook problem of a grounded conductor inside a uniform electric field. The textbook states that \"grounded\" means potential = 0. In my opinion, \"grounded\" should mean \"same potential as infinity\". But in this case we can't set potential at infinity equal zero. So my question is, what is actually the meaning of \"grounded\". Am I right that it means \"equal potential with infinity\"? What does it mean by \"grounded\" in this question when the potential at infinity can't be set to zero?"}
{"id":"75195","title":"If the velocity of particle $A$ exceeds that of $B$, is the acceleration of $A$ greater than $B$?","text":"Two particles $A, B$ are travelling along parallel straight paths. At some point, the velocity of $A$ exceeds that of $B$. Does this _necessarily_ mean that the acceleration of $A$ is greater than the acceleration of $B$? If you look at the $v - t$ graph of the two particles, the lines would intersect. Probably, starting off, the velocity of $B$ would be greater, but since the slope of the velocity of $A$ would be greater it would intersect with the graph of $B$ and exceed it. I couldn't think of any other situation. So, my conclusion was that the acceleration has to be greater. But my textbook says otherwise. How come? EDIT: This is question 13 from chapter 2 in Resnick halliday physics. To clarify: the problem does NOT assume that initally A's velocity was lower than B's. (See comments)"}
{"id":"79828","title":"Superpositions with two observers","text":"This is a bit of an odd question. I'm not a physicist, so bear with me if I say something wrong. Lets say you have some sort of quantum event where matter is in a superposition. Standing next to you is another scientist waiting to observe the results (and, in theory, collapse the suposition). You go to get a cup of coffee while your fellow scientist stays in the room. Your fellow scientist observes the result of the experiment while you are out of the room but does not tell you what the result was. There are three possible options here for when you return to the room:   1. The matter is still in a superposition for both you and the other scientist.   2. The matter is no longer in a superposition for either of you (even though you have not observed the event and have no knowledge of what happened).   3. The matter is in a superposition only to you. So which option is it? If the answer is option 2, would that mean that we as humans have some kind of \"superpower\" forcing quantum superpositions to collapse as soon as we see them?"}
{"id":"68294","title":"Why aren't there more than three generations of the leptons and quarks?","text":"There are three generations of electrons, neutrinos, and quarks. The second and third generations of electrons and quarks are unstable and decay into lighter particles. Why are there exactly three generations? Is it possible that there are more generations of increasingly massive and unstable particles that we aren't able to discover yet (for instance, at higher energy levels that we are capable of), or is there some known reason why it has to be exactly (and can't be more than) three? edit: in addition to the link above, I also found this question helpful: What Do We Get From Having Higher Generations of Particles?"}
{"id":"2023","title":"Is it safe to use any wireless device during a lightning storm?","text":"I need \"educated\" reasons whether it is safe to use any wireless device during a lightning storm. Most people said don't use it but they cannot explain why."}
{"id":"4820","title":"Radio waves and frequency of photon","text":"Is 89MHZ station emitting photons of 89MHZ frequency? (I mean $\\nu$ in $E=h\\nu$)."}
{"id":"104656","title":"Matter-antimatter asymmetry problem","text":"As we know that matter-antimatter asymmetry is one of unsolved problems in physics. One possible solution to this problem is given as baryogenesis which produce asymmetry in rate of creation between matter and antimatter particles. But doesn't alternate solution like \"different regions of space with different type of particles \" holds more ground.. Means in one region matter particle dominates (the region we live ) and another region anti matter particle will dominate. When particle-antiparticle pair created from energy (at that energy is so much high that particle- antiparticle pair could be created from energy according to $E=mc^2$) before they meet and annihilate with each other they also has equal chance of meeting with same type of particle in neighborhood, because after some time of big bang the four force were united and gravity was as powerful as remaining three forces. So now in 50% of all pairs destroyed by annihilation and other 50% clumps together with same type of particle means matter with matter and antimatter with antimatter and then inflationary epoch throw these clumps from each other at very large distance so that they can not meet each other and have chance of annihilation. And we now can not see these regions because of accelerated expansion of universe because of dark energy. Doesn't this hypothesis is more or at least equally valid with baryogeneis hypothesis."}
{"id":"72919","title":"Equation to estimate droplet volume falling from a needle of a known orifice diameter at a given flow rate","text":"I am doing some research to see if there is a simple equation (or an equation that can be simplified) to estimate the volume of a drop of liquid falling due to gravity based upon the approximate flow rate, and perhaps the viscosity\/surface tension of the fluid. Basically this is a drip chamber fluid set for delivering medications. Clinicians routinely calculate fluid flow based upon the number of drops per minute assuming a fixed drop size. However since drop size varies with flow rate. Using a fixed volume size has some amount of inaccuracy. I would like to see if I can be more accurate in determining actual flow rate."}
{"id":"105527","title":"The source of gravitation in a spacetime without matter","text":"In a discussion concerning: Physical meaning of non-trivial solutions of vacuum Einstein's field equations there were a number of answers claiming that the flatness of the Ricci space (Rµv=0) does not necessarily entail the flatness of the Riemannian spacetime. This is explained by the fact that besides the Ricci tensor, the Riemann curvature tensor depends also on the Weyl tensor said to govern the propagation of gravitational radiation. And although Rµv=0, the Weyl tensor can still be non-zero. That being so, it seems that there are some gravitational forces not included in Einstein's field equations, as the Ricci tensor (Rµv) and the Einstein's tensor (Gµv) are said to be trace-reversed (meaning that when one of them vanishes, the other does too). So, if Rµv=0 does not exclude the presence of gravitation - gravitational radiation can still be found through Weyl tensor - then (since Gµv=0) Einstein's gravitational field equations must be incomplete? Yet there immediately arises another - and in my opinion, more important - question: **What is the source of this gravitational radiation, since Rµv=0 means the whole universe is void of matter?**"}
{"id":"47193","title":"Intuitive explanation of the inverse square power $\\frac{1}{r^2}$ in Newton's law of gravity","text":"Is there an _intuitive_ explanation why it is _plausible_ that the gravitational force which acts between two point masses is proportional to the _inverse square_ of the distance $r$ between the masses (and not only to the inverse of $r$)?"}
{"id":"48447","title":"Gravity force strength in 1D, 2D, 3D and higher spatial dimensions","text":"Let's say that we want to measure the gravity force in 1D, 2D, 3D and higher spatial dimensions. Will we get the same force strength in the first 3 dimensions and then it will go up? How about if we do this with Electromagnetic force? I've include the Electromagnetic force just to see if I can find an analogy to the gravity force behavior."}
{"id":"78975","title":"Newtonian gravity equation in a 2 dimensional world","text":"I am wondering if my line of thought is correct - and thus the resulting answer to the problem above would be correct. As we know the gravitational force (of two point masses) is given by $$F = G\\frac{m_1m_2}{r^2}.$$ So the gravitational force\/vector field reduces with the distance squared. Now this is the formula in 3 spatial dimensions - and I always picture it as a point with gravitational field lines moving outward. Then the \"strength\" of the field would be the density of the lines. And hence the density drops with the distance squared (as it is inversely proportional to the area of the sphere at that distance). Now taking this line of thought to other situations we can think of course about a hypothetical 2 dimensional world. Here gravity would also be. And here we can also see the density of the \"gravitational field lines\". However as they propagate only in 2 spatial dimensions the density would be inversely proportional to the circumference of the circle at a distance $r$. And hence the formula would lose the square and become like: $$F = G\\frac{m_1m_2}{r}$$ (With change $G$, and obviously we can't talk about mass in 2d). Is this line of thought correct?"}
{"id":"134972","title":"Asymptotics of the Wigner 6j Symbol","text":"So, in doing some numerical computations in QFT, I've run into the following Wigner 6j-Symbol: $ \\left\\\\{ \\begin{array}{ccc} x & J_1 & J_2 \\\\\\ \\frac{N}{2} & \\frac{N}{2} & \\frac{N}{2} \\\\\\ \\end{array} \\right\\\\} $ In the regime where $x \\ll J_1,J_2,N$ and $J_1 \\approx J_2 \\approx N$, and $N$ is large. I would like to know if there is an asymptotic formula for such a symbol, or if one can be derived. Using symmetries we can get $ \\left\\\\{ \\begin{array}{ccc} x & \\frac{1}{2} \\left(J_1+J_2\\right) & \\frac{1}{2} \\left(J_1+J_2\\right) \\\\\\ \\frac{N}{2} & \\frac{1}{2} \\left(N+J_1-J_2\\right) & \\frac{1}{2} \\left(N-J_1+J_2\\right) \\\\\\ \\end{array} \\right\\\\} $ Perhaps this could help, I'm really not sure."}
{"id":"63875","title":"Solving the equation of relativistic motion","text":"How does one solve the tensor differential equation for the relativistic motion of a partilcle of charge $e$ and mass $m$, with 4-momentum $p^a$ and electromagnetic field tensor $F_{ab}$ of a constant magetic field $\\vec B$ perpendicular to the plane of motion. $$\\frac{dp^a}{d\\tau}=\\frac{e}{m}F^a{}_bp^b$$ ? Let the the initial condition be $$p^a=(E_0 ,\\vec 0)$$ I can see that the differential equation resembles that of a SHM equation or a cosh, sinh one if it's a scalar equation. However, I don't know how to deal with a tensor equation. Could anyone please explain? Thank you."}
{"id":"63873","title":"Redshifted from what?","text":"We need to know two of the following three to calculate the third: redshifted color, baseline color, and velocity. The velocity is related to the difference between the redshifted color and the objects baseline color. How do we know the baseline color of distant objects to know the amount of redshifting?"}
{"id":"123005","title":"can gapped systems have gravitational anomalies?","text":"The question is in the title. If it is possible, what are some examples of gapped systems--either quantum field theories or condensed matter systems--which exhibit some kind of anomaly when coupled to a metric with curvature or placed on a spacetime with non- trivial topology?"}
{"id":"112128","title":"Neutral shell(with a charge inside) in an electric field","text":"> A positive point charge $Q$ is kept eccentrically inside a neutral > conducting shell. An external uniform field E is applied. Then: > > a) Force on Q due to E is zero > > b) Net force on Q is zero > > c) Net force acting on Q and conducting shell considered as a system is zero > > d) Net force on the shell due to E is zero My try: Since the shell is neutral, thus net force on it by E must also be zero. Also since Q is electrostatically shielded(by placing it inside a shell), thus net force on it must also be zero. But the answer I'm getting(as you might have guessed) is wrong(that's why I've asked this question). Please point out the flaw in my reasoning."}
{"id":"112122","title":"Is a Perfect\/Lossless Mirror possible?","text":"In traditional mirrors, some of the input light is absorbed by atoms in the mirrors surface and are 'lost' as heat, degrading the quality of the reflected image. Could this loss be compensated by an array of \"powered elements\" arranged to reflect nearby photons repelling them with some sort of electo-magnetic force, to completely prevent loss? _This would realize theperfect mirror, reflecting all light back._ ![enter image description here](http:\/\/i.stack.imgur.com\/OD5FI.png)"}
{"id":"112127","title":"What is the Levi--Civita connection of a Wick rotated metric?","text":"A Wick rotation is a transformation that allows to change from a Lorentzian manifold to a Riemaniann manifold. In the cases when this is possible, is the Levi-Civita connection of the Riemaniann manifold the same connection as the initial Lorentzian manifold?"}
{"id":"34945","title":"Angular momentum of a rotating black hole","text":"Is there an upper limit to the angular momentum of a rotating (Kerr) black hole?"}
{"id":"10301","title":"How does reflection work?","text":"In Newton's model of light as being composed of particles, it's easy to imagine reflection as being the rebounding of individual corpuscles off a surface. However, since light can also behave like a wave, it poses a challenge in visualizing reflection. How does a wave reflect off of a surface, whether it be specular reflection or diffuse reflection? Must the wave be first absorbed, and then re-emitted? Or is there a different mechanism?"}
{"id":"32483","title":"How do mirrors work?","text":"Apparently, light is just a certain wavelength, or \"the visible spectrum\" of electromagnetic waves. If I recall correctly, my physics teacher explained to me that electromagnetic waves are basically consisted of two, interchanging parts. The \"electric\" and the \"magnetic\" parts of the wave are somehow manifesting each other in synchronized intervals, or they are (more probably) causing each other and canceling themselves out in the process. I checked wikipedia and the transition seems to be somehow interpolated (but then again, many things in nature are). So, which part of the mirror actually reflects the wave? Which of these two parts? Both? How come the wave doesn't get heavily distorted in the process? I guess the actual electrons of atoms of silver play a role, but why isn't every material reflective, then? Because is isn't \"perfectly\" flat? If I lined up atoms of a non-metal element in a perfect plane (maybe several rows, actually), would it reflect light just as mirrors do?"}
{"id":"83267","title":"What exactly is the connection between the Jacobi and Bianchi identities","text":"While reviewing some basic field theory, I once again encountered the Bianchi identity (in the context of electromagnetism). It can be written as $$\\partial_{[\\lambda}\\partial_{[\\mu}A_{\\nu]]}=0$$ Here, $A_\\nu$ is of course the electromagnetic potential. This formula is immediately reminiscent of the Jacobi identity: $$[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0$$ This is even clearer in general relativity, where we have $$\\nabla_{[\\lambda}R_{\\rho\\sigma]\\mu\\nu}$$ which we can rewrite, remembering the definition of the Riemann tensor in terms of the commutator of covariant derivatives, as $$[[\\nabla_\\lambda,\\nabla_\\rho],\\nabla_\\sigma]+[[\\nabla_\\rho,\\nabla_\\sigma],\\nabla_\\lambda]+[[\\nabla_\\sigma,\\nabla_\\lambda],\\nabla_\\rho]=0 $$ This all looks like there should be some profound connection here, but I'm incapable of pinpointing it. Maybe one of the experts here can make this more precise? I'd love to get to know more about this. Any comments are much appreciated. I'd also be grateful if someone could suggest (more) appropriate tags to use."}
{"id":"108921","title":"Heat flow in a hollow cylinder","text":"Consider a hollow cylinder of different outer radius and inner radius and two different temperatures are maintained at the outer and inner surfaces such that inner temperature is higher. Because of this heat will flow radially outward. Now I don't understand why we need to use integration to find rate of heat flow that is, why isn't the temperature gradient constant? Why can't we directly apply heat conduction formula?"}
{"id":"108928","title":"Constant of motion","text":"An exercise from Goldstein (9.31-3rd Ed) asks to show that for a one- dimensional harmonic oscillator $u(q,p,t)$ is a constant of motion where $$ u(q,p,t)=\\ln(p+im\\omega q)-i\\omega t $$ and $\\omega=(k\/m)^{1\/2}$. The demonstration is easy but the physical significance of the constant of motion is not so clear to me. Indeed I can show that $u$ can be rewritten like: $$ u(q,p,t)=i\\phi+\\ln(m\\omega A) $$ where $\\phi$ is the phase and $A$ the amplitude of the vibration of the oscillator. I can also demonstrate that $m\\omega A=\\sqrt{2mE}$, where $E$ is the total energy of the oscillator. But there is any further significance of $u$ that I'm missing?"}
{"id":"43327","title":"How to accurately explain evaporative cooling?","text":"I am trying to clearly express in one or two sentences how increased evapotranspiration could cool a region. The audience is educated but non- scientific. Is it accurate to say that the water vapor has removed latent heat? Is there a more clear explanation?"}
{"id":"9731","title":"What is the pure energy in matter antimatter annihilation made of?","text":"I used to read the term \"pure energy\" in the context of matter antimatter annihilation. Is the \"pure energy\" spoken of photons? Is it some form of heat? Some kind of particles with mass ? Basically, what does \"pure energy\" in the context of matter-antimatter annihilation refer to?"}
{"id":"95512","title":"Why is the charge on protons == to charge on electrons?","text":"I am not a expert on physics, just another high schooler, so sorry if the question is obvious. This is something I've been wondering about for a while. Why is the charge on a proton equal but opposite to the charge on the electron? A proton is much larger than a electron, and apparently a lot more heavier too. Why, then, is it's charge equal to that on a electron? Just what is charge, and what defines it? What factors decide the charge on a particle? Also while we're at it, why does the atom in it's default configuration have the same number of protons and electrons? Edit: To expand on this a bit, from what I know the attraction weakens as distance increases. So if theoretically a huge amount of protons were to be somehow brought together despite the repulsions constantly increasing, would a atom with a extremely high atomic number defy the proton = electron rule? Note: This is not a duplicate. I read through the Phys.SE post Why do electron and proton have the same but opposite electric charge? but I did not find a satisfactory answer (Or even understand many of the professional terms :s)"}
{"id":"78017","title":"Does the weight of a car frame has any effect on its horsepower?","text":"I want to know whether decreasing the weight of the frame of a car will increase its horsepower. From what I understand horsepower is a measure of the car's ability to transport load, and decreasing the frame's weight will increase this ability. I tried to search this on internet and all I got was that it depends on the horsepower of the engine not the car itself. It is very confusing. If my assumption that the car's weight has an effect on horsepower is right, then what is the possible way for me to calculate it?"}
{"id":"132020","title":"Is there an equivalent to wetness for air?","text":"I was wondering if there was something equivalent to the property of being wet with water, but with air instead. For example, if I drop water on my shirt, I'll notice by its appearance and feel that it is wet, so in a sense its properties were changed by being exposed to water. So I'm wondering if similarly, by being exposed to ambient air, my shirt is somehow being changed, i.e. if it was in a vacuum would it feel or appear different than when it's exposed to air?"}
{"id":"6068","title":"Recommendations for good Newtonian mechanics and kinematics books","text":"What are some good books for learning the concepts of Kinematics, Newton laws, 2D Motion of Object etc.?"}
{"id":"109478","title":"Any recommendations for self-studying physics?","text":"I'm new to this forum so I hope this post falls under the guidelines of what's acceptable. I'm currently finishing my grade 11 year of high school and I have hopes of attending one of the prestigious schools in the US such as MIT, Harvard, Stanford, Princeton, etc. I would love to be able to study physics in university and, since my school's physics courses are lacking any difficulty, I've been taking it upon myself to learn physics. I have a very strong ability to do the necessary math and have taken university calculus courses to prepare. However, the field is brimming with a wealth of resources and I'm just not sure what to choose and was hoping you would be able to help me out a little. I currently own Stephen Hawking's books: The Grand Design, The Universe in a Nutshell, and A Brief History of Time. Now I know these are written without the technical aspects, but I'm sure they'd make great reads when I find time. However, I'm more looking into resources which actually teach the math behind the ideas. I've found countless courses on MIT OpenCourseware and other online course resources including Leonard Susskind's The Theoretical Minimum courses. As far as books go, I've been interested in Leonard Susskind's two main books. I've also taken a great interest in Richard Feynman's work and so I'm interested in purchasing the Feynman Lecture on Physics boxed set with the accompanying exercise book being published this summer. I would like to listen to any recommendations any of you may have as to which resources I should use to gain a good understanding of physics. I would really like to learn using Feynman's lectures although I hope it isn't considered to be too old to be of much value. Any advice would be very much appreciated!!"}
{"id":"38687","title":"Reference request: Classical Mechanics as an Application to Smooth Manifolds","text":"> **Possible Duplicate:**   >  Classical Mechanics for Mathematician Last time I asked a question, but it does not sound specific. I am currently taking graduate topology class (using Lee's Introduction to Smooth Manifolds), but I am in math program (not in physics), so I might lack physical intuition. Is it better to learn Symplectic geometry first (such as da Silva's Lectures on Symplectic Geometry) then learn mechanics with symplectic geometry emphasis (such as Abraham & Marsden, etc), or is Spivak's Physics book self-contained in terms of both motivation and mechanics in manifold? How about Arnold? Is his physical explanation requires prerequisite in university level physics?"}
{"id":"128998","title":"Suggested reading for classical field theory","text":"I am reading a marvelous book _Classical Field Theory_ by E Soper, but it is mathematically too compact and sometimes I am unable to follow the equations. Can anyone suggest a side book for solution of my problem regarding reading the book? I also have _Classical Theory of Fields_ by Landau but its content and way of expression is not relevant with my book. The question which you people are saying is a possible duplicate of was asked by myself and in that question I tried to find a good book on Classical Field Theory.But here I need help to read a particular book on the same subject.So in this way I am trying to justify my attempt as to make it a new question."}
{"id":"31917","title":"Prerequisites for QFT?","text":"> **Possible Duplicate:**   >  Book recommendations Is there a book that covers everything you need to know (and possibly more) before starting a course on QFT. Alternatively, I need a list of what you need to know. It would have to contain Lagrangians and Poisson brackets, and quantisation. I've had a course in QM, and one in EM. I know SR too. I have an IQ of 160 and have had to give up up on my third book of QFT."}
{"id":"106337","title":"Book suggestion for Theoretical Physics with easy maths","text":"I am a Computer Scientist with literature interest in theoretical physics. I have already read books such as A Brief History of Time and Physics of the Impossible, and I am looking for suggestion for books with a slightly more scientific approach; where for example theorems are also explained in simplified equations. (like Eddington's Space Time and Gravitation) Can you suggest me some?"}
{"id":"108334","title":"Good math books for physicists","text":"In his first lesson (transcripted in \"Tips on Physics\"), Feynman talks about math for physicists in a very cool and practical way. And at the end of the section he talks something like \"so the first thing to do is to learn to learn derivative, integral and algebra\" (I don't know how much precise I'm being because I've read it in Portuguese). I would to know if there is some book that deals with math as Feynman did it in this lesson (respecting formalities, but teaching how to use the practical rules)? Also, someone have any recommendations for algebra book (college level)?"}
{"id":"96531","title":"Reference request for motion & related topics","text":"I'm starting a Robotics course at grad school and I could use some help to relearn and revise my concepts in the following topics :   1. Kinematics    2. Rigid Body Dynamics   3. Rotational Motion and Linear Momentum   4. Force Could you good people please let me know of 1) the best (subjectively speaking) intuitive sources that helped you learn the concepts? 2) some sources that showed examples instead of verbose explanations ? Any help\/suggestions much appreciated! Thanks. (if there are any more specifics required, please let me know, I will edit the post)"}
{"id":"74605","title":"What kind of physics topics exist?","text":"The question says everything I want to know, but for more details: I enjoy studying physics but the problem is that I can't find any information with a summary of all physics topics, collected together. I also Googled this and took a look at other websites and searched this website but without success. So if someone knows most of the topics, then please let me know them. The topics I am looking for are the ones from basics, to the university, and beyond university limit. Any comprehensive information you can provide would be useful. ps: I did some research on math topics and I did find a book called _Princeton Companion to Mathematics._ It is a really good book and I was curious if there is a book for Physics too."}
{"id":"123854","title":"Suggest me specific book for studying scattering theory and special functions","text":"I am doing msc physics. And we are studying major part of scattering theory. I used _Quantum Mechanics_ by Davydov, Griffiths, etc, to study scattering theory. But I am not understanding it properly, hence I want a specific book which will explain scattering theory very clearly."}
{"id":"121637","title":"A crash course in quantum mechanics","text":"I am shortly due to begin a summer project lasting roughly 6 weeks with the aim of performing some relatively basic calculations in the theory of open quantum systems. However, at this stage in my undergraduate physics degree, I am yet to have studied _any_ quantum mechanics at all. As such, the first part of my project will consist of self-studying quantum mechanics to gain sufficient working knowledge to be able to perform the required calculations. Which books, or other resources, would be suitable to gain such a working knowledge, given that I have already taken courses in Linear Algebra and some basic Lagrangian and Hamiltonian dynamics? My supervisor mentioned that it might be better to learn quantum mechanics using the matrix mechanics formalism, rather than the wave function formalism. (Although, given my situation, I can't really yet explain exactly what calculations I will be performing, it might be helpful to see this page of my supervisor's publications to see what sort of field I will be working in: http:\/\/arxiv.org\/find\/all\/1\/all:+AND+nazir+quantum\/0\/1\/0\/all\/0\/1)"}
{"id":"75129","title":"General physics landau","text":"I've found Landau 's book for a first course in physics (not theoretical physics, its title is general physics),the book is freely and legally avaible on archive.org ,but it's from 1967,is that okey,or this material in it outdated? What's the most recommended first course in physics? Which such books\/resources provide the best intuition, rigour, etc. ?"}
{"id":"63767","title":"Studying QM without math and physics background","text":"I rode all posted answers about this topic but i need to ask you another information. I have done a semester course called \"Principle of Physics\" (i am studying Biotechnology) and one called \"Principle of Analysis\" (integral and derivation basically). I would like to study QM so i want to ask you if it is better Griffiths or Dirac books considering my little background."}
{"id":"22696","title":"Best physics olympiad resources","text":"> **Possible Duplicate:**   >  Book recommendations I am looking for good book recommendations at the level of high-school physics. I am having in mind to find out as to what are the typical books that say IPhO competitors (say from US) would be studying."}
{"id":"79340","title":"Book recommendation","text":"I already learned the Physics' basics like Newton Laws, Electricity, and Optics. Can you recommend me a good book for advancing with my learning of physics(with some math)?"}
{"id":"92156","title":"Trouble with classical mechanics self-learning (How to avoid going down the Physics rabbit hole?)","text":"I'm a retired police officer trying to learn classical mechanics on my own. I have gone through many links on the Internet including the classical mechanics quick reference textbooks from Physics Stack Exchange. But, I always have the same problem just as anyone trying to learn classical mechanics _on his\/her own_ has had the experience of \"going down the Classical Mechanics Rabbit Hole\". > It turns out that only classical mechanics is the most difficult part of > physics to learn on ones own. I had a friend who confirmed this by comparing > how difficult it is to learn classical mechanics (including Lagrangian and > Hamiltonian formulation) on his own with electrodynamics and general > relativity. (Who are much much more difficult that all the field of CM) For example, suppose you come across the novel term vector space, and want to learn more about it. You look up various definitions, and they all refer to something called a field. So now you're off to learn what a field is, but it's the same story all over again: all the definitions you find refer to something called a group. Off to learn about what a group is. Ad infinitum. That's what I'm calling here \"to go down the Math Rabbit Hole.\" For example, I had lot of difficulties with the book \"An Introduction to Mechanics\" by _Daniel Kleppner_ , _Robert J. Kolenkow,_ which seemed according to many views to be an easy approach toward Newtonian and relativistic mechanics. The authors in general only and quickly pushes equations in my front without giving any reason for why a certain procedure is correct, and give no explanation on most of the things. I had then one choice: search on the net. But when I do, to search for a term X, I get to wikipedia page X, who give a definition that contains another term Y, where I click to understand the full meaning of term X, but who then contain another term Z, who redirects to... which leaves me with no understanding. Another thing is that when I go here on Physics Stack Exchange, and when I see answers like:   * http:\/\/physics.stackexchange.com\/a\/14752\/   * http:\/\/physics.stackexchange.com\/a\/67705\/   * http:\/\/physics.stackexchange.com\/a\/71093\/   * http:\/\/physics.stackexchange.com\/a\/64976\/   * and many many others... (like an answer by David Z for a question that kinda looks like: 'Would a heavier object fall faster because they attract earth stronger', I have no idea where he found the equations he wrote down. Also in many applied physics questions and answers by Lubos Motl. And in some questions: like: 'Why Newton's third law apply to all inertial frame?' I have no idea about that even if I already learn a lot from Daniel's book.) I don't know where those guys got all that stuff. I feel like: Mechanics is not well organized. For example, in relativity we first learn about Galilean relativity, then special relativity then general relativity. Everything is in order and it makes of the understanding a lot smoother. (according to my friend) But in classical mechanics I don't know where to start or what to pick. In Lagrangian and Hamiltonian mechanics book, it is even worse. Result? I fail to correctly answer some basic questions like: what happens when a cup of water starts to melt? or even more easy physics questions. **So I'm searching for a clear textbook that explains Newtonian mechanics well, then goes to special relativity, then to Lagrangian and Hamiltonian mechanics.** My dream for the next years of my life is to understand mechanics: Newtonian, SR, Lagrangian and Hamiltonian. And to start writing a web page about explanations of different phenomena like John Baez this week on mathematical physics. And maybe to do research on problems in classical physics which would make of me the most happy man in the world. Regards. Thanks for your understanding and time. My situation is similar to this guy My background: I'm very old, so I forgot almost all the math\/physics I've got in school, however, I've taken courses on Algebra, trigonometry and single variable calculus using KhanAcademy and some MIT videos. I've taken an MIT test on CalcI (just downloading the test online and verifying the solutions) and I scored 90%."}
{"id":"23107","title":"Classical Mechanics for Mathematician","text":"> **Possible Duplicate:**   >  Which Mechanics book is the best for beginner in math major? I am looking for suitable ways to learn mechanics in mathematician's perspective. I went through:   * multivariable calculus from Spivak,      * real analysis from Pugh,      * differential equations from Hirsh\/Smale\/Devaney (mostly focusing on linear system, existence & uniqueness, nonlinear dynamical system, bifurcation, and brief touch on chaos) (so no application covered)      * differential geometry from Pressley (but I hate pressley, so I am going to review through doCarmo)   * topology from Willard (but not all of them) The problem is I did not take freshman physics coures (because of annoying labs;;) My goal is to be able to read Abraham\/Marsden's Foundations of Mechanics or something of that level. I was thinking of reading differential equations book's applications section first and... idk. What books do you think is suitable for me to start learning classical mechanics? P.S. Some people mentioned Arnold's Mathematical Methods of Classical Mechanics, but do you think it is self-contained in terms of physical intuition required?"}
{"id":"88858","title":"Math required for learning Lagrangian mechanics","text":"How much knowledge of maths is required for learning Lagrangian mechanics? Also from where can I learn this math?"}
{"id":"79602","title":"Studying physics at undergraduate level: Prerequisites","text":"I have been really interested in physics for some time and also take active interest in quantum mechanics. I would like to pursue physics as my undergrad major. What kind of prerequisites does one need to major in physics? (I am going to study physics in Japan and right now is not so good at maths cause I am learning Japanese only for the past 6 months and has been totally out of it. )"}
{"id":"47621","title":"Introduction to quantum mechanics","text":"> **Possible Duplicate:**   >  Book recommendations   >  What is a good introductory book on quantum mechanics? I intend to learn quantum mechanics . But I don't have any suggestions about good books to start up with quantum mechanics. I will remain grateful if I get to know about books which start up the subject with the original physical insight as well as with mathematical rigor. So, I hope the learned people here will recommend few books that form their experience are going to be the best to start up with."}
{"id":"20149","title":"Physics book for 15 year old boy","text":"> **Possible Duplicate:**   >  List of good classical physics books my name is Bruno Alano. As stated in the title, **I'm 15 years old** (I'll do 16 on 7 of Feb) and much love **Computer Science** (C, C++), **Mathematics** and **Physics**. Some information may have been unnecessary, but my question is: **What is the suggestion of a good physics book for a teenager of my age?** I know basic things (speed, shoveller these issues and basic primary and secondary). A good reason for this is my Awe in mathematics and physics. Besides that maybe one day be useful in what I really want a career (science or computer engineering). **And another question: It is interesting physics in the area I want to go? I'm at an age that would be good to learn beyond what is taught in common schools?**"}
{"id":"132075","title":"Good explanation of phase transitions, second law of thermodynamics","text":"I am looking for a good explanation (website, book, video,...) of phase transitions. I am interested in atmospheric applications (so basically condensation, etc. of water in air). I would like an in depth explanation (so with derivations), which explains things from scratch (Second law of thermodynamics and so on). Maybe a video would be good, if anyone can point one out."}
{"id":"2219","title":"Beginner Physics Resources?","text":"I'm interested in learning physics. I do realize that the subject is large and that it would be easier if I had a specific area of interest. However, I do not. I suppose I want to learn about the fundamentals of it all; the axioms that combine all physics fields. Or, in other words, a high school physics class. Specifically, a book or series of videos would be helpful. I looked over MIT and unfortunately the material wasn't for me. I don't mean to be \"picky\" so I am not completely ruling out any resource just yet. Thanks in advance."}
{"id":"106344","title":"Link for resource on cosmology","text":"I have recently just begun studying cosmology. I have a background in physics. I would like a link to download video resources on Stephan Hawking's theories on physics and\/or cosmology that have not been proved wrong."}
{"id":"110298","title":"Can someone recommend good resources to self study physics please?","text":"I have a goal to educate myself up to the current level of knowledge we possess about the universe. I've tried textbooks, wikipedia, lectures, but i find each of them fundamentally flawed in different ways. Textbooks tend to be incredibly bloated, and I become unable to \"see the forest for the trees\" Wikipedia seems to be too technical\/mathematically rigorous for my level of understanding Lectures feel very time consuming and i dont really feel as though i get a structured sense of how everything fits together. I feel as though all mathematical and physical concepts can be explained in simple terms, yet i cant find resources which present concepts like this. I don't want explanations that are completely dumbed down or necessarily lacking in math, but i would prefer explanations which give math intuition and take time to explain what the variables and symbols mean in laymen's terms. If anyone knows any resources similar to what i described, or has any advice it would be greatly appreciated! Edit: I should also mention that i had been referred to Hooft's outline of learning, as seen here: http:\/\/www.staff.science.uu.nl\/~hooft101\/theorist.html#ssphysics Which i find to be a trememndous help, but I'm still struggling as to the ideal places to find clear resources in regards to these subjects."}
{"id":"363","title":"Getting started general relativity","text":"What are some good books, videos, websites for getting started with general relativity? Mathematically rigorous preferred!"}
{"id":"79477","title":"What is a good target to aim for when teaching myself quantum mechanics?","text":"I'm interested in teaching myself quantum mechanics, and I'm looking for a good goal to aim for. I've got an undergrad maths degree and a graduate degree in probability theory and stochastic processes. I've worked fairly extensively with diffusion processes, so I understand how the Fokker-Planck equation works, for example. As far as I understand, the Fokker planck equation is related to the Schrodinger equation via something called a Wick rotation. I'm a bit hazy on the details, but I think this may have been Feynman's approach to QM. If anyone could recommend a text for someone with my background (linear algebra, group theory, ODE and PDE, probability), but with relatively little experience in physics, that would be very helpful. The maths shouldn't be a problem, but there's a chance I'll just get bored unless I have a specific goal. Maybe deriving the emission spectra of various atoms, or understanding quantum computing. Any other suggestions?"}
{"id":"14951","title":"Introduction to relativity books for an engineer","text":"> **Possible Duplicate:**   >  Getting started general relativity I am an engineer who loves to read science fiction books especially when there's more science than fiction but usually I see that I lack the knowledge behind many of the relativity concepts in the novels. I've always felt curiosity in relativity so I decided that it was the time to buy an introductory book. I went to amazon and checked that there are dozens of \"Introduction to genereal relativity\" books in there. So here's my question: What book would you recommend to someone not matematician nor physicist but with (some) mathematical background as an introduction to relativity? Thank you so much!"}
{"id":"105021","title":"Good First year physics lecture notes","text":"My course textbook is Halliday fundamental of physics, this book is huge and since each week, they cover a lot of material in lectures (something about 6 chapters of the textbook), I find it hard to read this textbook because there are a lot of unnecessary details that reading them is not needed. I was wondering if you know of some good physics lecture notes that covers basically what is in Halliday fundamental of physics and it is short?"}
{"id":"16593","title":"Physics textbooks reference request","text":"> **Possible Duplicate:**   >  Book recommendations I am currently in the 11th grade (I am an Indian student) and I am looking forward to studying pure mathematics and theoretical physics in the future. I therefore request you to tell me books\/textbooks to study physics. I know some integration and differentiation and I plan to study Richard Courant's book \"Introduction To Calculus And Analysis: Volume 1\" from Tuesday onwards.Besides, I am studying Halliday-Resnick Walker for the moment."}
{"id":"129543","title":"Exam on Physics for Graduate","text":"Is there any book you recommend for graduation exam\/higher studies? I have an exam on physics this week so I need some books to brush up my knowledge. Is there any book you recommend for that purpose? Other than Nelsons Parker Physics. What I ment here is a book which has all in one **_like_** **\"Advanced Level Physics by Michael Nelkon (Author), Philip Parker (Author)\"** http:\/\/www.amazon.com\/Advanced-Level-Physics-Michael- Nelkon\/dp\/043592303X A book with all exercises alog with the lesson. For Advanced Level Examination."}
{"id":"33215","title":"What is a good introductory book on quantum mechanics?","text":"I'm really interested in quantum theory and would like to learn all that I can about it. I've followed a few tutorials and read a few books but none satisfied me completely. I'm looking for introductions for beginners which do not depend heavily on linear algebra or calculus, or which provide a soft introduction for the requisite mathematics as they go along. What are good introductory guides to QM along these lines?"}
{"id":"45172","title":"Quantum Mechanics Text for Electrical Engineers","text":"> **Possible Duplicate:**   >  What is a good introductory book on quantum mechanics? What is a good introductory text on quantum mechanics that could be used to train electrical engineers in device physics and band structure in a single semester?"}
{"id":"24621","title":"Beginning Physics, Advise on Books","text":"> **Possible Duplicate:**   >  Beginner Physics Resources?   >  Book recommendations I'm wanting to learn physics. In fact i'm a software engineer and i find physics very fascinating. But everything I read things concerning physics, I don't understand many things. What are the books that can be helpful for someone who don't know physics but have a good foundation of mathematics??"}
{"id":"106186","title":"Best books for high-school?","text":"Please don't close this thread, I now it might sound like a dozen others but I think that every case is different... I go to the first class of high school (that's how it's called in Greece, not sure how it's called in English standards) and I love physics. Recently I participated in a competition and I think I did well. Thing is, I can't find any really good textbooks. I want something inspiring with great problems and clear explanations. If I could find it as an ebook, it would be even greater. Also, it must also use simple maths or at least explain some of them, 'cause I'm still learning, naturally."}
{"id":"37865","title":"Undergraduate Math Major Wanting to Learn Physics","text":"> **Possible Duplicate:**   >  Book recommendations So I'm a Junior level math major. I've seen some abstract algebra, some differential geometry, and some lie theory. I'm currently working through some standard topology. I don't at all claim to be an \"expert\" in any of the above, but I'll attest that I have some decent mathematical maturity. I've always had a soft spot for physics, and I wanted to develop my knowledge of it much more formally. Particularly things like general relativity and quantum mechanics seem very cool, but I'm lamentably pretty ignorant of them. Given my somewhat okay math background, is there a good starting point for someone like me?"}
{"id":"88480","title":"Physics books for mechanics","text":"What are the best physics books for learning mechanics? I am in grade 12 and would love to learn in depth about Newtonian mechanics and also maybe get started on Lagrangian mechanics?"}
{"id":"119386","title":"Physics books covering classic mechanics","text":"I am going to be a high school freshman next year and I have acquired a strong interest in physics. I have a mathematical background, upto, but not including, Calculus. I am looking for in depth resources covering classic mechanics enough to move onto more in depth texts on relativity as well as quantum theory. Again, I have a strong math background to all the work leading up to Calculus, and I will be taking Calculus next school year."}
{"id":"110296","title":"Could you recommend some good books on physics?","text":"Could you recommend some good books on physics? I found out that some books on physics are not good, especially about Einstein's Relativity and quantum mechanics. Here is an example which makes me think so: While Space-man lives for 1 day, then how long does Earth-man live ? 1000 years or 1 second? A bad book makes my head spin for hours , it may be a good thing(I have some joy in it！）, but maybe it's a bad thing too sometimes. Someone says that \"Michael Berry, Principles of Cosmology and Gravitation\" is a good book. Any other good books ?"}
{"id":"12175","title":"Book recommendations","text":"Every once in a while, we get a question asking for a book or other educational reference on a particular topic at a particular level. This is a meta-question that collects all those links together. If you're looking for book recommendations, this is probably the place to start. All the questions linked below, as well as others which deal with more specialized books, can be found under the tag resource-recommendation (formerly books). If you find a question that should be added to the list, edit it into the existing list answer rather than creating your own answer for it. See our policy on resource recommendations for details. **Related Meta:** Do we need\/want an overarching books question?"}
{"id":"107048","title":"What resources may be helpful to learn physics from ground up (very basic level)?","text":"I want to learn physics and the math required, but I never got the chance to learn properly in school due to mental illness. But, now, as a 22 year old, I wish I can start learning. Attending a college is not an option as I do not think I would find one that is willing to provide the psychological support a person suffering from psychotic depression\/schizoaffective disorder needs in my city in India. Anyway, so I think I should start learning from resources online and by buying some (not too expensive) books. The problem is I'm not very good at math, as my knowledge of basics and foundation is very weak, probably worse than middle school level. Though I can probably do basic algebra, I'm not even sure I know what set theory is. My trigonometry and geometry is poor also and I doubt I would be able to do even simple problems. Physics is the same. So I really want something that will help me build my knowledge from ground up, as I now have an genuine interest in learning the subject. What got me interested was reading Stephen Hawking's Brief History of Time and some topics on quantum mechanics. So I want to learn now to be able to comprehend such stuff someday. So please recommend good resources\/books I can use to start from VERY basic level, as most similar questions and discussions elsewhere that I found were asking for resources for the high school level and up. Also, it would be nice if the books were very interesting and not dry, maybe like Bill Bryson or how Cecil Adams would do it. Edit with regards to Qmechanic's setting the question as duplicate: The resources mentioned in the other question do not seem to me to be aimed at a very basic level of math and physics (starting from middle school level, so to speak.) and they appear to be aimed at those who possibly have some well-to-do knowledge of at least the basics or the foundation, unlike what I asked for. However, I could be wrong, but their question does mention 'high-school level'."}
{"id":"62556","title":"Quantum harmonic oscilator - book that does it all right","text":"I am dealing with quantum harmonic oscillator. In every single book or video i have checked out i can read how the mathematical technique for solving this Schrödinger equation: $$ W\\psi = - \\frac{\\hbar^2}{2m} \\frac{d^2 \\psi}{dx^2} + \\frac{1}{2}m \\omega^2 x^2 $$ is beyond the level of the book\/video... I really want to learn this stuff and noone seems to be able to explain it. I have dealt with tunelling, particle in a box, potential steps... **so why do authors think i can't handle this?** **I need a good book recommendations**."}
{"id":"119957","title":"Would love to learn quantum physics","text":"I would love to learn quantum physics and am extremely keen on it. Lets get to points. I was extremely weak at math and physics while I was in school, but I guess it's just the way things were taught. I have read some theoretical (the literal word) on quantum physics and am really keen to do the math involved. I love reading books and that's how I've spent most of my life. It would be the greatest thing ever if you geniuses could recommend a step by step guide (books only) to becoming a quantum physicist. I intend to learn everything involved and\/or required. It would really help me and many other physics enthusiasts who currently see no other option than the conventional university education route. I am fully and completely prepared for this journey."}
{"id":"57514","title":"From Freshman Mechanics to String Theory: A Comprehensive Textbook Sequence in Physics","text":"If a student with no background in physics and an understanding of only single variable calculus wanted to learn string theory, what sequence of textbooks would most succinctly, clearly, and comprehensively fill in the gaps, assuming a 100% retention rate of the textbooks' material?"}
{"id":"105041","title":"An easy source to understand classical dynamics --- Rigid body Rotation","text":"I've been having an extremely hard time at understanding rigid body rotation. The source that I'm currently studying from has been suggested by 't Hooft on his webpage. It's by Richard Fitzpatrick. Here's the link --- http:\/\/farside.ph.utexas.edu\/teaching\/336k\/lectures\/node61.html I have spent countless hours trying to understand rigid body rotation in the fixed frame and in the body frame, i.e. Euler's Equation and the Eulerian angles, but I just don't get it. Can someone please suggest an alternative source which could be easier to understand? It's really important for my work that I understand rigid body rotation properly!"}
{"id":"103274","title":"Learning about group velocity, phase velocity and particle velocity","text":"I am studying quantum physics and I would like to know a bit more in detail about group velocity, particle velocity and phase velocity. Can you guys suggest some books\/online resources where I can learn the same."}
{"id":"60882","title":"Book suggestion : geometric approach to electromagnetism","text":"I´m looking for a book on electromagnetism that is introducing the topic from a geometric point of view, focusing more on the theoretical structure than on the application."}
{"id":"20641","title":"Book for learning about constructing correct physical modeling","text":"> **Possible Duplicate:**   >  List of good classical physics books I know physics. At least I thought so until I tried to make a model of the system of scales by myself. It was confusing. For example for a simple spring model everything is ok, but when you have 3 and more springs connected in some way, it is hard to determine how the forces and at which directions are applied. And even in simple one-spring model I do not understand how the direction is choosed (the direction of mx'') while drawing the system on the paper with forces applied on it. In general case, not only in case of springs I want to learn how to correctly determine the forces and direction in order to construct the correct differential equation (or system of diff.equations). Can you suggest me the books needed to learn for this purpose? I know some physics but no experience in constructing the models by myself. EDIT 1: For example I want to make a model of this system: ![enter image description here](http:\/\/i.stack.imgur.com\/h6Xz2.jpg) There will be 5 variables. The problem is that I can't decide the directions of the forces considering the influence on each other."}
{"id":"106607","title":"Green theory and Laplace theory","text":"I can not understand classical electrodynamics so well, maybe there is some book, video, lecture note can help me."}
{"id":"119913","title":"Informal book on Classical Mechanics","text":"I just want a book on classical mechanics that covers the same ground as Goldstein's book but is more on the line of DJ Griffiths's Classical Electrodynamics. I mean less formal and more conversational."}
{"id":"94249","title":"How can I answer the critical questions of mechanics?","text":"I have passed my 1st year of undergraduate study life somehow I could have managed. But recently I have decided to fill up the emptiness of knowledge over mechanics. Besides I have my studies of 2nd year. I am following \"PHYSICS\", volume 1 by Resnick, Halliday, Krane for mechanics. The questions of that book are so hard to answer. For example, a question - How would you criticize this statement : \"Once you have picked a standard, by the very meaning of 'standard' it is invariable\"? Being unable to answer so many questions does not making me happy. For classical mechanics I am following the book Herbert Goldstein. Actually what I want it is that I want to learn to answer those types of questions."}
{"id":"109698","title":"Books for a fresh physics undergraduate","text":"I'm going to start a degree course in physics next year. So far in high school I covered some physics without calculus. I know that I will start everything from the beginning at university, but I would like to prepare in some ways. Then the question is: what should I do? Should I revise what I studied? Should I go head? Can you suggest (from your experience) what should I do before starting a physics degree course? PS In case you recommend to study some material, can you suggest a book?"}
{"id":"133067","title":"Reference books for classical mechanics with good number of solved examples","text":"I am looking for some classical mechanics reference book with a good number of solved problems."}
{"id":"71096","title":"Material to read before taking College Physics 1?","text":"I'm transferring to a 4-year university from a 2-year college. I earned a Computer Science A.S., but must now combine Computer Science with Advanced Physics for my B.S. I'm actually **_very_** excited for Physics, but I am coming in from knowing nothing about the field. I did not take Physics in high school, and I will be taking Calc II while taking Physics I and Physics I Lab. This is a topic that I want to understand to my fullest extent, because I would like to work on Quantum Computers when I get out of college. Are there any \"pre-review\" or \"course-summary\" materials that I can read roughly within a month of starting classes? I would like to get a head start and ease in some topics that I will be learning in class, so the lectures are more of a review than a rush to take notes. Even something like Physics I For Dummies would be fine to purchase and read, I just want to see what Stack Exchange users would recommend. Thank you in advance! **EDIT:** I should be more specific - and say that I am taking Calculus based physics (which is what I was implying by mentioning Calc II), not algebra based."}
{"id":"57309","title":"Where to read about Minkowski space","text":"When I learned Special Relativity, it was taught in terms of basic linear algebra, without any mention of the Minkowski space, proper time as integration on the metric, etc. However, when I am trying to learn General Relativity from several books (Wald, Carroll, Weinberg), it seems they all assume I am already familiar with SR and Minkowski space, and only briefly review it. So, I'm afraid I'll be missing some material here. What is a good source for SR from a Riemannian geometry point of view?"}
{"id":"32533","title":"How to make strong base for fundamental physics?","text":"> **Possible Duplicate:**   >  Book recommendations How to make strong base for fundamental physics? I want to learn the fundamentals of physics. But in a simple manners with very general examples. Please suggest me any URL or Books for it."}
{"id":"83933","title":"Math and Theoretical Physics Topics & Textbook for Self-Study","text":"I am from Singapore, a civil engineering graduate and I've graduated from university in 2009. Throughout my school days I've been interested in Physics, unfortunately I was not accepted into the local university as my grades did not match up. And I couldn't afford to study abroad. Hence I took up whatever was offered to me at that point; civil engineering. Right now, I'm on a mission to self-study theoretical physics so that I can fully appreciate the beauty of it; I mean right down to the deep mathematical level of it. I am particularly interested in the subjects: General Relativity, Quantum Theory, M-theory. I acknowledge that first of all I'd have to master the math before I can delve deeper into the theoretical physics subjects. I would like to ask forumers the following questions: (1) - May I know what are the sequence of math & physics topics I have to master? I mean in a step-by-step way starting from high school level knowledge. (2) - Could you recommend texts both from the math & the theoretical physics side? Based on my research I have shortlisted a few and they are (a) Introduction to Mathematical Physics: Methods and Concepts by Chun Wa Wong. (b) A course in theoretical physics by P. john Shepherd (c) Introduction to Modern Physics: Theoretical Foundations by John Dirk Walecka & (d) A Unified Grand Tour of Theoretical Physics, Third Edition by Ian D. Lawrie. Are these books suitable and good for my cause? What other textbooks do you all recommend? I am willing to self-study patiently even if it takes me 10 years."}
{"id":"21576","title":"Good theoretical physics introduction for 6 year old very advanced in math?","text":"I think now is a good time to introduce my son to theoretical physics. He asks so many questions about the universe, black holes, gravity, atoms, molecules, light, etc. He's borderline obsessed with the idea of more than 3 space dimensions. And he tries to apply math to everything. So I'd love to find a great introductory book, lecture, online class or something of that nature to feed his curiosity. The issue is he's only 6 years old (his birthday was only a few months ago). So his life experience is obviously very limited. His vocabulary and general knowledge is probably not a whole lot more than the average 1st grader. However, his math skills are extraordinary. He's working on calculus now, having mastered algebra and everything that's come before. Is there any introductory physics material that is heavy on the math and light on the vocabulary and assumed general knowledge? This would also be beneficial to non-native English speakers."}
{"id":"92540","title":"Quantum Physics Books","text":"I`m very newbie in physics and want to study on amateur level. I`m interested in Quantum Physics, can anybody advice some books?"}
{"id":"74079","title":"Helping\/explanatory notes for Landau&Lifschitz Physics Course","text":"I've recently restored my interest on theoretical physics (I have a master degree in Electrical Engineering) and began my study with first volume of the Physics Course by Landau and Lifshitz. This is a very good and interesting book, and it just blows my mind (both in a positive and negative ways) - you spend really a lot of time trying to understand everything out of this textbook. Maybe someone can suggest somewhat sort of helping sidenotes for this book? Thank you for your help."}
{"id":"52143","title":"A Mathematician Who Wants to Learn Particle Physics","text":"> **Possible Duplicate:**   >  Book recommendations I'm a grad student in pure math, wrapping up a thesis in Lie theory. After years of talking to mathematicians and physicists, I've decided that it's time to get some formal training in physics, particularly in particle physics. I have no formal training in physics. We're talking only a childish understanding of basic mechanics and bits and pieces of relativity and CFT . . . I've seen Maxwell's equations once or twice . . . etc. But I know Riemannian\/pseudo-Riemannian geometry, the representation theory of compact and semi-simple Lie groups, some PDE, and so forth. So hopefully the math shouldn't be an issue. With so little physics training (and hence little physical intuition!) what kind of books do you guys recommend?"}
{"id":"102002","title":"help me.. please","text":"I'm a total novice to physics so please forgive me but I'm looking for a starting point on sound, frequencies and everything to do with electricity. Preferably books because my internet usage is limited, would love to here back so please.. help me understand the world"}
{"id":"119790","title":"What is a good mechanics book without calculus","text":"I am very interested in physics and I want to improve my basics especially in mechanics. I have only started high school (junior) so I don't know much about calculus, which is used in most of the books. I am only looking for a conceptual books explained basic concepts nicely."}
{"id":"78233","title":"Orbital motion (Mechanics)","text":"I am a student in Mechanical department and I took today a lesson with a title of _Orbital motion._ I need a good mechanics reference that discusses this topic."}
{"id":"88650","title":"best fundamental physics book","text":"Good evening. I'd like to know, in your opinion, what would be the best fundamental physics book for a freshman? I want to start all over again. Thanks in advance."}
{"id":"9165","title":"Which Mechanics book is the best for beginner in math major?","text":"I'm a bachelor student majoring in math, and pretty interested in physics. I would like a book to study for classical mechanics, that will prepare me to work through Goldstein's Classical Mechanics. What books would be good for a beginner?"}
{"id":"121430","title":"Books to independently study physics","text":"Can anyone give me a list of books to study so that I can \"master\" (whatever the precise definition of that may be) all the branches of physics? I want to start from the most elementary to the most advanced."}
{"id":"86546","title":"Recommended book for beginners on advanced science topics","text":"I have a background in engineering so I have some familiarity with basic math and science. I've recently been reading about other topics such as Einstein's relativity and have become interested in learning more about this and other similar topics in science. Are there any books you can recommend that explains this and similar topics (also stuff like astrophysics, spacetime, etc), but aimed at a more amateur audience? Thanks!"}
{"id":"105247","title":"Self-teaching physics recommendations?","text":"I've been very interested in mathematics for a while now and, after being inspired by reading Feynman's biography, would also like to learn about physics. My mathematics background is fairly extensive. I've taken courses through calc BC, the basics of multivariable calculus and a fair amount of linear algebra, with plans to take statistics next year. My physics background is okay, in that I know what many of the individual symbols mean, but have never seen them put together in such a way that they yield all the wonderful results we see today. Could anyone recommend a good place to start that makes use of this math background without being incomprehensible to my fledgeling physics knowledge and a freely available online textbook to accompany the former? Thank you very much!"}
{"id":"39188","title":"Books to study quantum thermodynamics and quantum decoherence","text":"> **Possible Duplicate:**   >  Book recommendations My friend is having a hard time finding books to self-study quantum thermodynamics and quantum decoherence. (search on amaxon would bring almost no book... so what books should he choose?)"}
{"id":"45941","title":"Need for a side book for E.Soper`s Classical Theory Of Fields.","text":"I am reading now E Soper Classical Theory Of Fields now and sometimes it is very hard to follow the equations.So I need a side book to read it comfortably.Landau`s book is not helping as its content and topics are very much different."}
{"id":"72088","title":"Where to begin with Physics beyond high school?","text":"So I'm in grade 10 but I finished all my physics classes, maths and further maths (Calculus). I would like to start with some harder stuff (university physics). What books do you recommend for both maths and physics? And if possible what steps should I take?"}
{"id":"11231","title":"Electric potential energy in curved space-time","text":"In flat space-time the electric potential energy between two charges is $\\frac{k Q_1 Q_2}{r_{12}}$, where $Q$'s are charges and $r_{12}$ is the distance between them. What would happen if the two charges are placed in a strongly curved space-time, which makes \"distance\" and \"duration\" measures different from place to place? Will the two feel the same strength of electric force\/potential energy?"}
{"id":"31598","title":"Magnetic field inside a charged stream","text":"Outside a narrow charged stream (say, a beam of ions or electrons) is the same as observing a current through a conducting wire - there is a circular magnetic field around it. What would happen inside a charged stream (for example, inside a conducting wire or inside a solar flare)? I have a feeling that symmetry will rule that there is no magnetic field, but I am not sure."}
{"id":"47511","title":"Are We Living in a Simulated Universe?","text":"If the universe is just a Matrix- like simulation, how could we ever know? Physicist Silas Beane of the University of Bonn, Germany, thinks he has the answer!. His paper “Constraints on the Universe as a Numerical Simulation” has been submitted to the journal Physical Review D !. Are We Living in a Simulated Universe? **Remark:** This person claimed that the whole world is a great simulation, and the whole world started with a simulated big bang. and every things are results of spontaneous program self-organization and automorphisms. The big bang happened in a kind of supercomputer And now we're inside of it. the Big Bang occurred approximately 13.75 billion years to our eyes, But perhaps within less than a second for simulator.   * If the universe is just a Matrix- like simulation, how could we ever know?"}
{"id":"47519","title":"When you apply the spin operator, what exactly is does it tell you?","text":"The example I'm trying to understand is: $ \\hat{S}_{x} \\begin{pmatrix} \\frac{1}{\\sqrt{2}}\\\\\\ \\frac{1}{\\sqrt{2}} \\end{pmatrix} = 1\/2 \\begin{pmatrix} \\frac{1}{\\sqrt{2}}\\\\\\ \\frac{1}{\\sqrt{2}} \\end{pmatrix} $ My interpretation of this is that the vector shows you the probabilities of a particle being spin up or spin down if you square them. And I've been told that $ \\hat{S}_{x} $ gives you the spin as an eigenvalue, but how? Since its 50:50 of getting -1\/2 and 1\/2. $ \\hat{S}_{x} $ has only given you one of them. Is it that $ \\hat{S}_{x} $ only measures the magnitude of spin in the x direction?"}
{"id":"1603","title":"Applications of Algebraic Topology to physics","text":"I have always wondered about applications of Algebraic Topology to Physics, seeing as am I studying algebraic topology and physics is cool and pretty. My initial thoughts would be that since most invariants and constructions in algebraic topology can not tell the difference between a line and a point and $\\mathbb{R}^4$ so how could we get anything physically useful? Of course we know this is wrong. Or at least I am told it is wrong since several people tell me that both are used. I would love to see some examples of applications of topology or algebraic topology to getting actual results or concepts clarified in physics. One example I always here is \"K-theory is the proper receptacle for charge\" and maybe someone could start by elaborating on that. I am sure there are other common examples I am missing."}
{"id":"48050","title":"Determination of auxiliary scale in dimensional regularization","text":"My questions are in italics. In the article [1] a dimensional regularization is presented on an electrostatic example of an infinite wire with constant linear charge density $\\lambda$. It is shown that the direct computation of the scalar potential gives infinity: $$ \\phi({\\bf x}) = {\\lambda\\over 4\\pi\\epsilon_0}\\int_{-\\infty}^\\infty { d l \\over |{\\bf x} - {\\bf l}| } = {\\lambda\\over 4\\pi\\epsilon_0}\\int_{-\\infty}^\\infty { d l \\over (x^2 + y^2 + (z-l)^2)^{1\\over 2} } = $$ $$ = {\\lambda\\over 4\\pi\\epsilon_0}\\int_{-\\infty}^\\infty { d u \\over \\sqrt{x^2 + y^2 + u^2} } = \\infty $$ But with dimensional regularization in the modified minimal subtraction scheme we get eventually: $$ \\phi_{\\overline{\\rm MS}}({\\bf x}) = {\\lambda\\over 4\\pi\\epsilon_0} \\log{\\Lambda^2\\over x^2 + y^2} $$ where $\\Lambda$ is the auxiliary scale parameter. One can then calculate the electric field (let's set $y=0$ from now on) as follows: $$ E_x = -{\\partial \\over \\partial x} \\phi_{\\overline{\\mathrm{MS}}}(x) = -{\\partial \\over \\partial x} {\\lambda\\over 4\\pi\\epsilon_0} \\log{\\Lambda^2\\over x^2}= $$ $$ = - {\\lambda\\over 4\\pi\\epsilon_0} {x^2\\over\\Lambda^2} \\Lambda^2 \\left(-{2\\over x^3}\\right) = {\\lambda\\over 2\\pi\\epsilon_0} {1\\over x} $$ The article claims that the original scalar potential is scale invariant: $\\phi(kx) = \\phi(x)$. But since both $\\phi(kx)$ and $\\phi(x)$ are infinite, _I don't understand the argument._ The article claims that the dimensional regularization preservers translational symmetry. However, the only way to make $\\phi_{\\overline{\\mathrm{MS}}}(kx)=\\phi_{\\overline{\\mathrm{MS}}}(x)$ is to choose different $\\Lambda$ for each side. _Are we allowed to do that?_ I thought that we have to set $\\Lambda$ once and for all and then just keep calculating with it and it must cancel at the end. Update: based on Michael's comment below I realized that the article claims _translational_ invariance of the original problem, i.e. $\\phi_{\\overline{\\rm MS}}(x, y, z+h)=\\phi_{\\overline{\\rm MS}}(x, y, z)$ and that is obviously true, because $\\phi_{\\overline{\\rm MS}}({\\bf x})$ does not depend on $z$. So I think that answers this particular question. Still a clarification from an expert would be nice. [1] Olness, F., & Scalise, R. (2011). Regularization, renormalization, and dimensional analysis: Dimensional regularization meets freshman E&M. American Journal of Physics, 79(3), 306. doi:10.1119\/1.3535586, available online here."}
{"id":"132835","title":"Gravity doesn't seem to work the way it is supposed to","text":"This has been a bit of an awkward question that's been plaguing me ever since I started watching space documentaries on discovery about 10 years ago. I was saving this for the day I would ever meet Professor Brian Cox where I can point and say \"HA!\" see you can't explain that one (he probably can and so could someone here). It's a simple question: when you get on a merry-go-round or something that rotates fast, then the force pushes you outward, but when they talk about things in space that rotate they refer to the force that pulls them inwards. So which is it. If I spun something like a big beach ball really really fast then I'd still be thrown off it. If I took that into space should that suddenly keep me glued to it? I apologise for my ignorance but I've never had this explained to me."}
{"id":"110305","title":"How can they prove the superposition of particle states prior to measurement","text":"If every time a particle's spin or momentum is measured, it gives a discrete answer (collapse of possibility states), how can they ever prove that prior to measurement it was in fact in a super-position of states? Is this solely a logical extrapolation from the wave-like interference patterns seen in the slit experiment? Clearly I don't understand something fundamental here."}
{"id":"132464","title":"Mode-dependent Andreev reflection","text":"Consider the following Hamiltonian which describes massless Dirac fermion on the surface of a topological insulator nanowire, $$H = -i\\hbar v_{F}\\left[ \\partial_{x}\\sigma_{x} + \\frac{\\sigma_{y}}{R}\\left( \\partial_{\\phi} + i\\eta\\right)\\right]\\ \\text{.}$$ The nanowire has translational symmetry in $x$-direction and $\\phi$ is the angle around perimeter. It has eigenenergies which can be written in the form, $$E_{k,n} = \\pm \\hbar v_{F} \\sqrt{k^{2} + \\frac{1}{R^2}\\left(l_{n} + \\eta \\right)^2}\\ \\text{,}$$ where $l_{n} = n-1\/2$ is quantized in integers $n$. The wavefunction can be written like this, $$\\Psi^{\\pm} = \\frac{1}{\\sqrt{2\\pi L}}e^{\\pm i k x}e^{i l_{n} \\phi}\\ \\text{.}$$ Now, if You do the same thing but with the Bogoliubov-de Gennes equation You will get electron-like and hole-like solutions where the wavenumber $k$ woould depend on the total energy $\\mathcal{E}$, transverse momentum $l_{n}$ and fraction of the magnetic flux quanta $\\eta$. I am wondering if the Andreev refelction at the boundary between normal-region and superconductor can take place between different modes $l_{n}$. Is there something which prevents from electron-electron, electron-hole etc. scattering processes between different modes at the N-S boundary?"}
{"id":"20622","title":"How to compute drag coefficient given initial position, initial velocity and final resting position?","text":"The equations I'm using are:               x = x + (DT * vx)     vx = vx * C      My DT is always 0.01 and the coefficient `C` (related to a linear drag coefficient, as mentioned in the comments) is greater than 0 and less than 1. The above will keep happening until x naturally reaches its limit. What I want is an equation to find C given the other three inputs: initial x, initial velocity x and final resting x(this is where the object has come to a stop and vx = 0). Right now it's tedious because I have to plug in ix, ivx and C and run my simulation to see where the object stops. Then I have to do further tweaking to get exactly what I'm looking for. Here are some samples:               initial x = 3.5     velocity x = -12.0     acceleration = 0.92     final resting x = 2.0          initial x = 3.5     velocity x = -14.0     acceleration = 0.92     final resting x = 1.75 For example(from the first dataset), we start at 3.5 we want to end at 2.0 and our velocity is -12.0 .. what do we need to use for C?"}
{"id":"74456","title":"Why does the electric field perpendicular to every point on the surface of a conductor?","text":"I am reading Berkeley Physics Course, Volume 2 (Electricity and Magnetism by Edward M. Purcell). I am in chapter $3$ pg $92$, and the book discusses conductors. The following is from the book: Because the surface of a conductor [in Fig $3.2$] is necessarily a surface of constant potential, the electric field, which is $-\\nabla \\varphi$ , must be perpendicular to the surface at every point on the surface I have omitted the picture because it is not relevant. Can someone please explain this reasoning ? I understand that the potential $\\varphi$ is a continuous function, and since $E=0$ inside the conductor and since $E=-\\nabla\\varphi$ I get that $\\varphi=0$ inside and on the surface (from continuity) of the conductor. However, I don't understand the reason the book gives for explaining why the field is perpendicular to every point on the surface"}
{"id":"135162","title":"What exactly do we see on the famous neutrino image of the sun?","text":"An answer to the question If we could build a neutrino telescope, what would we see? contains a link to a neutrino image of the sun  by the Super- Kamiokande neutrino detector. ![neutrino image of the sun](http:\/\/i.stack.imgur.com\/dSed3.jpg) There it says that the image actually covers a large part of the sky of about 90x90 degrees. As the diameter of the sun from earth is around one half of a degree, it must be that many of the neutrinos didn't come straight at us. This seems surprising (to me), as neutrinos should hardly interact with the atmosphere. Maybe the central few pixels of the image are extremely much brighter than the others, but this image doesn't show the difference between those and the surrounding pixels? Or is something else going on?"}
{"id":"102232","title":"Optical Simulation Tool","text":"I am a student and trying to design an aperture from which a light will pass will project on a sensor. I tried to find some software simulation tool for this purpose where i can draw the aperture (entrance side) and put diffuser or lens and then see the response of the light for some specific wavelength. I found one tool \"Synopsys Code V\" but it is very paid solution. So can anyone tell me some substitute for this. I am looking for Free simulation tools."}
{"id":"38865","title":"Free Optics Simulation Programs","text":"I'm having an extremely difficult time finding an optics program that is easy to use and offers accurate physics simulations. I'm not asking for much, I just want to be able to simulate a laser going through a beam splitter and then be able to drag and drop mirrors and angle them to be able to see where the laser beams end up. I want to intersect two laser beams that underwent beam splitting and redirect the beams... Does anyone know of any free software that can do this?"}
{"id":"6686","title":"Are we crystals?","text":"Can we say that we are crystals because just like crystals we are made up of very small unit (cell) making up almost the same shape (our body) everywhere."}
{"id":"130287","title":"How to transfer propane from a SMALL cylinder to a SMALL metal cylinder","text":"I made a metal cylinder that is the **SAME** size as my 400g propane cylinder (picture: here). I want to transfer **ALL** (or almost all) of the propane from the propane cylinder to my metal cylinder, preferably without using an air compressor or something like that. The reason why I'm doing this is because I'm building a rocket and I can't get the propane to flow out of the propane cylinder fast enough."}
{"id":"121383","title":"Thin Lens Methods and Object Perceived vs Literal Size","text":"When using the ray trace methodology to solve a given thin lens question, does the arrow (commonly used as the example object when tracing) represent the literal height of the object or the perceived height of the object relative to the lens? Refresher image below: ![a generic ray trace](http:\/\/boson.physics.sc.edu\/~rjones\/phys153\/raytracelens.GIF) Same for the equation methods. When using below equation for magnification, derived from the thin lens equation, does $h_0$ represent the the literal height of the object or the perceived height of the object relative to the lens? $$ \\frac{i}{o}=\\frac{h_i}{h_o} $$ Note: in the event that these aren't the standard notations, $i$ is the image's distance away from the lens, $o$ is the object's distance away from the lens, $h_i$ is the height of the image, and $h_o$ is the perceived or literal height of the object."}
{"id":"121381","title":"Which direction does a galaxy move in respect to its axis of rotation (Black Hole)","text":"As Galaxies travel through the universe, how do they orient? And, does this orientation apply to stars and their satellites? that is to ask if the movement of a galaxy or star is perpendicular to its satellites and its rotation.. One might even compare this proposed action to that of a tornado in that the planets or stars would 'follow' in the 'wake' of the star or blackhole like debris."}
{"id":"121380","title":"Special relativity and imaginary coefficient of the time coordinate","text":"I read somewhere that part of Minkowski's inspiration for his formulation of Minkowski space was Poincare's observation that time could be understood as a fourth spatial dimension with an imaginary coefficient. Clearly, taking the Euclidean norm of the vector $$(i \\Delta t, \\Delta x, \\Delta y, \\Delta z)$$ gives the correct spacetime interval (assuming appropriate units), but I don't really know where it goes from there (possibly something to do with Moebius transforms?) I think this is mentioned in Taylor and Wheeler's book, but I may have read it elsewhere. After the historical note, the author (whoever it was) said it was \"preferable\" to use Minkowski geometry straight off, rather than mucking about with time as an imaginary space coordinate. Could anyone elaborate on Poincare's formulation? Why is Minkowski's methodology better?"}
{"id":"6350","title":"Double slit experiment  and perforated walls' properties","text":"I have a doubt about how double slit experiment is made. Let's think about the perforated wall, what are the requirement for it? Can a photographic plate could be used as a wall ? I see a problem here, as a photographic plate serve also as a detector, then the single photon experiment could end up with a single point in _the wall that contains the slits_ and not in the plate located _behind the slit's wall._ Of course the \"one photon count\", is knowing the emition before the slits (if this count were made after, it will be a detector making a measurement and would simply reflect the particle effect) The wall with the slits could absorb the photon(removing electrons from the material) before something happen on the other side, if that happened, there would be no experiment, the photon won't pass through the slits, just because its energy was already used (supposing all energy is used and the photon doesn't split). There must be some kind of specification about the material of the wall, need to be special, in the sense of needing a higher energy (than the test photon) to pickup electrons, then the light won't stop there, for that, I think the wall can't be made of a photosensible material, but I don't know, and here I am asking.. thanks for any answer"}
{"id":"18500","title":"Gravitational waves detection, any news?","text":"Is the detection of gravitational waves a reality with nowadays technology? Are there recent news?"}
{"id":"72094","title":"What cause scientists to study Black Body Radiation?","text":"After spending hours understanding what exactly Black Body radiation and Ultraviolet catastrophe is, I cannot help myself asking what was the reason that make scientists such as Wilhelm Wien and Max Planck to study Black Body Radiation at the first place? What intrigue them to study a hypothetical situation? What they were looking for exactly that make them in studying this phenomenon."}
{"id":"112719","title":"MHD Flow in a channel with and external B field and circuitry determining E. Different circuits and Ohm's law","text":"I was wondering if someone can explain why E is the way it is in cases 2,3,4 in page 9 of these notes ? In case 2 \"Short Circuit\", do I just have to assume that for a perfect conductor E = 0 for short circuit ? I don't really get why current density is negative in this case (or the 3rd case for that matter) and what it means physically, for current density to be negative... Any extra comments on what is going on in there would also be appreciated."}
{"id":"68809","title":"How do I calculate the work of a gas showed up on a graph?","text":"I have this graph of a gas: ![gas graph](http:\/\/i.stack.imgur.com\/DMQlN.jpg) Now, I need to calculate work of it, based on ABCD cycle of changes (that's a bit confusing to me, do I calculate AB, BC, CD separately?). How do I do it? Also, how can I calculate heat of this gas needed to exchange with enviroment, in order to maintain it's internal energy?"}
{"id":"80955","title":"Can all theories of theoretical physics be generalized to any (arbitrary) number of dimensions?","text":"Please explain from: mathematical point of view \"laws of mathematics\", and, physical point of view \"laws of physics\"? Or is there any bound on number of dimensions?"}
{"id":"133783","title":"How to get the pressure?","text":"A system of solar panels are fixed on top of a large water tank of height $40m$ and area $10m^2$. Atmospheric pressure is $100kPa$. What is the pressure at the bottom of the tank? The density of water is $\\rho =1000kg\/m^3$ and $g=10m\/s^2$.               a 10kPa     b 1000kPa     c 2000kPa     d 200kPa     e 1kPa      My doubt: why is the answer not $100kPa+\\rho gh$? explanation of concept will be appreciated. What I did was simply atmospheric pa +pressure due to water. The confusion lies in the fact that will we only consider the pressure due to water as the solar panel roof will balance the atmospheric pressure?"}
{"id":"80959","title":"$a^3B_2$, $b^3A_2$ and $A^1A_2$ states in electron spectroscopy","text":"I am new to electron spectroscopy. I have a basic question regarding the molecular transition. I have learned about the states $\\Sigma_g^+$ $\\Pi_g$.. etc electronic states. But I could never find in any books the states like $a^3B_2$, $b^3A_2$ and $A^1A_2$ states! For example, I am attaching a part from from the page 770 of the article (Analyst, 2003,128,765-772). Can anyone please explain me what are these states mean? Or any books\/reference suggestions to learn will also be good. ![enter image description here](http:\/\/i.stack.imgur.com\/s4u2M.jpg)"}
{"id":"52628","title":"Could there be a star orbiting around a planet?","text":"I wonder if there ever could be a star (really small) which may orbit around a planet (really big)?"}
{"id":"111514","title":"Rocky Planet in the center of System","text":"We all know that mostly stars are at the center of planetary systems, but is it possible that instead of star there was a rocky planet in the center with stars (and other planets and moons) orbiting it? To be more concrete: Is it possible for a star to have the same mass and radius as e.g. the Moon and orbit a planet like Earth at the same distance (at which Moon orbits Earth in actuality)? To further distinguish this from similar questions, I want to further ask whether the star will still be able to shine and fuse hydrogen if its mass and radius would be same as the mass and radius of the Moon. Or is there a lower limit to the size or a star that can shine and fuse hydrogen?"}
{"id":"94813","title":"Cayley-Klein Parameters","text":"I have a very simple question(I guess )to ask $$\\frac{d\\mathbf{m}}{dt}= \\mathbf{C} \\times \\mathbf{m}$$ where $\\mathbf{m}$ and $\\mathbf{C}$ are vectors. Assume that $\\mathbf{C}$ is constant over a certain period of time $[0,T]$. Then would someone please explain me how can we find a rotation matrix using Cayley-Klein parameter so that, for $t\\in [0,T]$, we can express $\\mathbf{m}(t)=R(t) \\mathbf{m}_0$? Here $R(t)$ is a rotation matrix and $\\mathbf{m}_0$ is the initial vector. I know that, in $[0,T]$, it can be solved analytically as $\\mathbf{m}(t)=exp(At) \\mathbf{m}_0$. Moreover would anyone please explain the relation between this two solution?"}
{"id":"2166","title":"Is time travel possible? Is it possible to go back in time?","text":"I read somewhere that according to relativity, black holes and other space related stuff it is possible to jump into past.   Is it possible for anything to go back in time either continuously or by jumping? I doesn't fit my mind and seems totally illogical, unreasonable and even stupid thought because if it was possible, we must had someone who came from future. **UPDATE:** Time Travel Impossible, Say Scientists"}
{"id":"7823","title":"Is time travel possible?","text":"Time travel -- often featured in movies, books, or facetiously in conversation. There are also theories treating time as simply another dimension, which to the layperson might imply forward and backward movement is possible at will. But what do we know scientifically with respect to the _possibility_ or _impossibility_ of controlled time travel? **Are there any testable theories on the horizon that may support or eliminate controlled time travel as a possibility?** By \"controlled time travel\" I mean the ability to willingly transport a complex object or being through time (perhaps only to the past, or a copy of the past, which would be an answer too)."}
{"id":"29487","title":"Can we develop in the future a technology that can send a message to the past?","text":"> **Possible Duplicate:**   >  Is it possible to go back in time? I believe that everything is possible in this world. If that can be done then why aren't we receiving any message from the future."}
{"id":"56135","title":"Possibility of Time Travel?","text":"Is time travel still considered as a possibility? What does the newer theories like string theory, M theory, etc. say about it."}
{"id":"41507","title":"Does Stephen Hawking not believe in Negative movement?","text":"> **Possible Duplicate:**   >  Is it possible to go back in time?   >  Is time travel possible? According to Stephen Hawking, time travel to the future is not only possible but proven, however time travel to the past is impossible, however this doesn't make sense. It seems to me that an extension to Newtons laws of motion could be that there is something opposite of motion which exists in the universe. The concept is that the universe (or a universe) is ending, and sort of crunches itself up, this would result in negative motion, which in theory would result in backwards time as well. He believes it's not possible because of the crazy scientist theory, where a scientist uses his vortex to kill himself in the past, so who shot him? Well the answer to this is very simple, the scientist would require a dieing universe to perform the time travel, and he himself would have to be in it for him to travel through time, therefore, he wouldn't survive, as he would perish inside of the dieing universe, and never get the chance to shoot himself. Also, he had a party, where future scientists were invited. He wrote up a invitation which would be preserved through time for his future guests. However, his experiment isn't all that swift, because, a number of things can go wrong with his invitation over time, which would mean his \"party people\" would never arrive, because they never received the invitation. It's kind of like those that say the infinite universe is impossible, because someone in this infinite universe would have blown it up, it just simply does not preclude the existence of the infinite universe, but that's another topic."}
{"id":"86111","title":"Did physicists solve the grandfather paradox?","text":"Now, physicists are trying to send information backward in time. But, why are physicists almost sure that this would happen and why are they so confident about it? Did physicists solve the grandfather paradox? In that case it would mean that result would occur before the cause. That is very weird. Of course the universe is very weird(and hence very interesting) but is that the case? Can information be sent backward in time?"}
{"id":"20599","title":"Time travel to future","text":"> **Possible Duplicate:**   >  Is time travel possible? Is time travel possible? According to my friend, it is possible to go to the future but not the past. In Physics, particles move faster than the speed of light,$c$. However, consider a train moving close to the speed of light along the Earth's equator, and a particle moves in that train in the opposite direction of the train's motion. Thus, the speed of the particle with respect to Earth now becomes the products of the two velocities. Since the particle cannot travel faster than light, what happens? Is it theoretically possible to time travel into the future using this concept?"}
{"id":"74181","title":"Time travel, is it possible?","text":"I see there are many questions with this title, but this one is different, so please give it due thought: Is time travel possible? If so, Can we go both forwards and backwards in time? Paradox: I am studying for a very competitive exam, wherein I must put in a LOT of effort to hope for any success. Now, I am lazy. And assume that time travel is possible, and I have access to it. I decide to check my future, and (god-permitting) I see that I have passed with flying colors. Now, I \"come back in time\" and enjoy life, and don't study. So, it is just Impossible that I succeed. So, where is the flaw?? Does this prove that time travel must not exist, and will not exist?"}
{"id":"41576","title":"Why do bigger tires = more friction?","text":"> **Possible Duplicate:**   >  Why do people recommend wider tyres in car for better road grip? Static\/kinetic friction says that the friction force is proportional to the normal force, via the coefficient of static\/kinetic friction. Great. Imagine that you have a sports car. You'd like to improve your lap times at a nearby race track. So, you buy larger (width) tires. Bigger contact patch, so your car sticks to the road better. But wait, the normal force on the car didn't increase, and the coefficients of friction of the tires didn't change. Just the contact patch. What explains the increase in grip from larger tires?"}
{"id":"129727","title":"How to calculate the horizontal acceleration?","text":"I am trying to measure the acceleration and deceleration of a car by using an 3 axis accelerometer which is build in an iPhone 5s. Placing the iPhone flat inside the car with the y axis to the top of the car this works pretty ok. But now I want to be able to place the iPhone basically arbitrary inside the car. (like in the picture below) ![iPhone inside a car](http:\/\/forums.everythingicafe.com\/data\/MetaMirrorCache\/farm3.static.flickr.com_2355_2206921565_5fd67cf1b6.jpg_a4f0a23c50a330cf1bf56cae6c33a893.jpg) Is there a way to calculate only the horizontal acceleration of the device in such a placement? (I am aware that you won't be able to differ a directional acceleration but that's sufficent for my project. If I am wrong, I would be glad to know if it would work) To do the calculations the following data is available   * User Acceleration (uX,uY,uZ) - only the acceleration the user imparts to the device   * the total acceleration of the device (tX,tY,tZ) - user acceleration plus gravity   * Gyro Data (gX,gY,gZ) - the device’s rate of rotation around it's axes   * the attitude of the device (quaternion, rotation matrix, (pitch,roll,yaw)) Many thanks in advance!"}
{"id":"129723","title":"Macroscopic Bose condensate in Special Relativity","text":"I remember from an experiment about the Josephson effect the state of each of the super conductors is fully described by a phase factor. From there I assume that is true for any Bose-Einstein condensate. So... Let there be a huge blob of Bose condensate in vacuum. In its reference frame, you can describe the wave function as a complex phase number, which is the same over the whole condensate. It will change over time, but at all times, it is the same everywhere. Enters Special Relativity: there is no simultaneousness. If someone in one reference frame \"sees\" the condensate in one phase constant over the whole condensate, but changing with time, another observer with relative velocity will \"see\" a phase gradient. Does that mean, that to the moving observer it does not seem like a Bose- Einstein condensate? Or does it just seem to be an excited version of it? I am confused."}
{"id":"47659","title":"What's the difference between photoelastic constant, photoelastic coefficient and the acousto-optic coefficient","text":"I'm reading a few papers about how the optical properties of materials change when a under stress or a force acts upon them. I seem to be encountering the following three terms:   1. Photoelastic constant    2. Photoelastic coefficient    3. Acousto-optic coefficient Is there a difference between these three terms, as they seem to be used in context of describing very similar phenomena? Also, does Acousto-optic constant exist?"}
{"id":"29128","title":"What causes a Phase-Transition","text":"A phase transition occurs when for example, heat is applied continuously to a liquid and after a certain time it converts into a gas. How does this process work in detail? Is their a chain reaction that causes to liquid to reach a 'critical' point? Does the liquid syncronises in some specific vector, facilitating the phase transition? Finally could it be that the liquid theromodynamically self organisises into a state that causes the transition? To paraphrase, what exactly is a 'phase-transition' what occurs before, during and after one? Any additional comments you think would help explain this phenomenom to me would be great."}
{"id":"129893","title":"Does Newton's law and Quantum mechanics also apply for the matter which is not dead?","text":"The following quote is extracted from the book \"The Field-The quest for the secret force\": > ...There was other, quite practical, unfinished business with quantum > theory. Bohr and his colleagues only got so far in their experiments and > understanding. The experiments they’d conducted demonstrating these quantum > effects had occurred in the laboratory, with non-living subatomic particles. > _From there, scientists in their wake naturally assumed that this strange > quantum world only existed in the world of dead matter. Anything alive still > operated according to the laws of Newton and Descartes, a view that has > informed all of modern medicine and biology._ Even biochemistry depends upon > Newtonian force and collision to work. Are the above statements correct? I always get stuck here, according to Newton's laws of motion, everybody continues in the state of motion or rest (w.r.t to an inertial frame), unless and until a force is applied on it. But, a rat, a dog, a girl!, always pass before me and fluctuate to rest and motion, rest and motion. Are they acted upon by any force to set them in motion or to get them into rest? If muscles help them move (Pratyay gosh has noticed this significant point), which force make the muscles move. According to Newton's law, they must be acted upon by a force, right? So, does Newton's laws of motion also apply for the matter which is not dead? Is Quantum mechanics applicable only for dead matter?"}
{"id":"129892","title":"Light has a wave particle duality, how do we know?","text":"I've been told my whole life that light is either a wave or a particle. When it's traveling through space, it's a wave. When it hits a wall, or a photo- sensitive chemical strip or something similar, it's a particle. However, upon looking back all of the examples I've seen I can only recall instances in which we observe light as a particle. Are there in fact ways we can measure it as a wave?"}
{"id":"129898","title":"The n-point Green functions and Heisenberg picture","text":"Let's have the S-matrix: $$ S_{\\beta \\alpha} = \\langle \\beta | \\hat{S} | \\alpha\\rangle . $$ Here $|\\alpha \\rangle , | \\beta \\rangle$ are $t \\to \\mp \\infty$ limit of the free states, $\\hat {S} = \\hat{T}e^{-i\\int \\hat{L}_{\\int}d^{4}x}$, $\\hat{L}_{\\int}$ refers to the operator in the interaction picture. When we decide to get the matrix element of some process we will get $$ \\int d^{4}x_{1}...d^{4}x_{n}\\langle \\beta |\\hat{T}(\\hat{\\varphi}_{1_{int}}(x_{1})...\\hat{\\varphi}_{m_{int}}(x_{n})) | \\alpha \\rangle . $$ So it's convenient to introduce n-point Green function, $$ \\tag 1 G_{n}(x_{1},...x_{n}) = \\langle 0| \\hat{T}(\\hat {\\varphi}_{1_{int}}(x_{1})...\\hat{\\varphi}_{n_{int}}(x_{n}))| 0\\rangle $$ and generation functionals for it. But recently I have read anywhere that as n-point Green function people use expression $$ \\tag 2 G^{H}_{n}(x_{1},...x_{n}) = \\langle 0| \\hat{T}(\\hat {\\varphi}_{1}(x_{1})...\\hat{\\varphi}_{n}(x_{n}))| 0\\rangle , $$ where the operators of fields are in the Heisenberg picture. So they need to rewrite the operators into interaction picture: $$ \\tag 3 G^{H}_{n}(x_{1},...x_{n}) = \\langle 0| \\hat{T}\\left( \\hat {\\varphi}_{1_{int}}(x_{1})...\\hat{\\varphi}_{n_{int}}(x_{n})\\hat{S}\\right)|0\\rangle . $$ I don't understand why we need the Green function $(2)$ where the fields operators are in the Heisenberg picture if $S$-matrix \"generates\" rather Green functions with operators in interaction picture. Can you explain it why we don't use $(1)$ when talk about the Green functions?"}
{"id":"7540","title":"How would I go about detecting monopoles?","text":"A question needed for a \"solid\" sci-fi author: How to detect a strong magnetic monopole? (yes, I know no such thing is to be found on Earth). Think of basic construction details, principles of operation and necessary components of a device capable of detecting\/recognizing a macroscopic object emitting magnetic field of equivalent of order ~0.1-10 Tesla near its surface, but with only one pole, reliably distinguishing it from normal (2-pole) magnets, preferably at a distance. Preferably a robust method, not involving extremely advanced technology. Detect the presence, possibly distance (or field strength) and direction. I know of SQUIDs, but these concentrate on extreme sensitivity. I'm thinking of something less sensitive but more robust (like, no need for the monopole to fall through the loop) and still able to recognize a monopole against a magnet. Also, how would such a macroscopic object behave practically? Such a \"one-pole magnet\" about the size and strength of a refrigerator magnets - how would it behave around ferromagnetics, normal magnets and so on?"}
{"id":"110148","title":"Coset space and transitiviy","text":"I have a question regarding coset space or homogeneous space $SO(n+1)\/SO(n)$ which is simply $S^n$. I need some intuition regarding this result. As everyone knows that for a simple case of $SO(3)\/SO(2)$, one can have $SO(3)$ as a group acting on $\\mathbb{R}^3$ and $SO(2)$ as an isotropy group of $x\\in\\mathbb{R}^3$, then the group $SO(3)$ acts transitively on $S^2$ and we get $S^2$ as the coset. Since the result is just 2-sphere or $n$-sphere, is there an intuitive way of seeing it?"}
{"id":"65191","title":"What is the electric field in a parallel plate capacitor?","text":"When we find the electric field between the plates of a parallel plate capacitor we assume that the electric field from both plates is $${\\bf E}=\\frac{\\sigma}{2\\epsilon_0}\\hat{n.}$$ The factor of two in the denominator comes from the fact that there is a surface charge density on both sides of the (very thin) plates. This result can be obtained easily for each plate. Therefore when we put them together the net field between the plates is $${\\bf E}=\\frac{\\sigma}{\\epsilon_0}\\hat{n}$$ and zero everywhere else. Here, $\\sigma$ is the surface charge density on a single side of the plate, or $Q\/2A$, since half the charge will be on each side. But in a real capacitor the plates are conducting, and the surface charge density will change on each plate when the _other_ plate is brought closer to it. That is, in the limit that the two plates get brought closer together, _all_ of the charge of each plate must be on a single side. If we let $d$ denote the distance between the plates, then we must have $$\\lim_{d \\rightarrow 0}{\\bf E}=\\frac{2\\sigma}{\\epsilon_0}\\hat{n}$$ which disagrees with the above equation. Where is the mistake in this reasoning? Or more likely, do our textbook authors commonly assume that we are in this limit, and that this is why the conductor behaves like a perfectly thin charged sheet?"}
{"id":"32325","title":"Is it possible to control a treadmill's tread speed such that a plane on the treadmill will be prevented from moving?","text":"I've posed the question in this particular way to avoid the ambiguity usually found in the posing of the \"airplane on a treadmill\" puzzle, e.g. I'm not specifying how the treadmill is controlled but asking _if_ it can be controlled in such a way that the thrust of the plane's engine is countered with an equal and opposite force. Assume the wheel bearings are frictionless and the wheels rotate freely. Please justify your answer. [EDIT] Idealize the problem such that we can ignore rolling resistance."}
{"id":"65447","title":"Why doesn't this equation for orbital motion change with position in the orbit?","text":"The question and answer are on pg.8-10 of this PDF: At first, I went through it, thinking nothing of it. But then, I wondered: \"What if we picked a final state in which the space junk was NOT at closest approach, but an arbitrary distance away from the center of the moon?\" The equation (eq.11) would be exactly the same! What does that mean? Since obviously the distance from the space junk to the moon changes continuously, yet the form of the equation remains the same. Using conservation of energy: $$\\frac{1}{2}mv_i^2-\\frac{GMm}{R}=\\frac{1}{2}mv_f^2-\\frac{GMm}{r}$$ And conservation of angular momentum: $$mRv_i=mrv_f$$ For any $v_f$ and $r$. Now look. We have two equations and two unknowns, $v_f$ and $r$. This suggests that there is a unique solution for both. If we solve for one and plug that into the other equation, we'll get a unique result (or perhaps end up with a quadratic equation, which doesn't fix the problem). How do we reconcile this?"}
{"id":"119548","title":"Does quantum entanglement imply the existence of a non-causal structure connecting space-time together?","text":"In contrast to a \"time-like\" or \"causal\" structure connecting space-time together, Does quantum entanglement imply the existence of a \"space-like\" or \"non-causal\" structure holding space-time together as well. A more general question; is there even any relevance to the _discussion_ of the existence of a non-causal structure connecting space-time together? The reason I ask is because it initially seems too assuming to suggest that causal structure is the only meaningful structure just because it's intuitive; Consider the fact that two space-like separated events are even allowed to exist in a definable space (space-time diagram). Is there nothing physical _in principle_ between the two events which can be defined?"}
{"id":"116598","title":"Why more than one Higgs?","text":"Since by introducing _one_ Higgs Boson we can give a mass to the leptons and gauge bosons of the weak interaction: Why should we consider more than one Higgs (doublet) once we go beyond the standard model?"}
{"id":"25254","title":"Why does the moon sometimes appear giant and a orange red color near the horizon?","text":"I've read various ideas about why the moon looks larger on the horizon. The most reasonable one in my opinion is that it is due to how our brain calculates (perceives) distance, with objects high above the horizon being generally further away than objects closer to the horizon. But every once in a while, the moon looks absolutely huge and has a orange red color to it. Booth the size and color diminish as it moves further above the horizon. This does not seem to fit in with the regular perceived size changes that I already mentioned. So what is the name of this giant orange red effect and what causes it?"}
{"id":"20844","title":"Why does the moon appear bigger  close to the horizon,  rising or setting?","text":"> **Possible Duplicate:**   >  Why does the moon sometimes appear giant and a orange red color near the > horizon? I made a little research about this and found this article http:\/\/en.wikipedia.org\/wiki\/Moon_illusion that states the explanation of the moon being bigger at the horizon is still debated. I found this which looks pretty 'big' http:\/\/www.psychohistorian.org\/img\/astronomy\/deep- sky\/photos\/ayiomamitis\/20090804-moonrise.jpg but however this kind of repetitive shots http:\/\/apod.nasa.gov\/apod\/image\/0706\/UludagMoonrise_tezel.jpg show no decrease in the aparent moon radius. What I am trying to understand is if it is a real illusion, if the atmosphere makes the image bigger or what other explanation could be possible and of course plausible."}
{"id":"109900","title":"Does dark matter get drawn into black holes?","text":"Would dark matter get drawn into a black hole? Also, could enough dark matter be concentrated to create a \"dark black hole\"?"}
{"id":"90903","title":"I need help understanding a step in the derivation of the Schwarzschild solution","text":"I am looking at Wikipedia's article on deriving the Schwarzschild solution. In the section \"Simplifying the components\", it says, > On the hypersurfaces of constant $t$ and constant $r$, it is required that > the metric be that of a 2-sphere: > > $$dl^2=r^2(d\\theta^2+\\sin^2\\theta d\\phi^2)$$ My question is why does the metric have to be this particular 2-sphere with a coefficient of $r^2$? We are not necessarily dealing with Euclidean space here."}
{"id":"9421","title":"strange modulation of radiactive decay rates with solar activity","text":"Recently i found out this uber strange article about nuclear decay rates being somehow showing seasonal variations with a high correlation with sun activity. Two very precise questions: 1) **has this been experimentally confirmed\/disproved?** an experiment using neutrinos from a fission reactor would be awesome, although probably a couple orders of magnitude below the required luminosity (at least to be comparable with solar sources) 2) **could the standard model possibly allow neutrinos to modulate decay rates in this way? or do we need new physics?** link to the public version of the paper **EDIT** brief explanation why i tag this question as `cavity-qed`; because _it is the only other known mechanism we are aware that you can use to shift decay rates of energy levels_ , it might be interesting to see if there are deeper relationships between both mechanisms involved **EDIT 27\/01\/2013** Another paper about this, but now the SuperKamiokande data is compared with data from Brookhaven: http:\/\/arxiv.org\/abs\/1301.3754 They also propose a model of the effect called neutrino \"resonant spin-flavor precesion\", and i'll be damned if i knew what that is."}
{"id":"9423","title":"What is 656 Beagle?","text":"What kind of object is 656 Beagle (1908BU)? I know it's a minor planet, but that includes a large array of different stuff. Specifically, I am looking at the general chemistry\/geology of the object."}
{"id":"11645","title":"Some questions about chapter I.1 (by Minahan) of the \"Review of AdS\/CFT Integrability\"","text":"These questions are in reference to this beautiful review article by Minahan - http:\/\/arxiv.org\/pdf\/1012.3983v2 I gained a lot by reading some of its sections but not everything is clear to me. I would like to ask a few questions to clarify some of the things in it.   * On page 5 between equation 3.7 and 3.8 it says that the R-charge representation of the $S$ are \"reversed\" compared to that of the $Q$. What does it exactly mean ?    * There it does not explicitly specify the commutation relationship between $M_{\\mu \\nu}$ and $S^a_\\alpha$ and $\\bar{S}_{\\dot{\\alpha}a}$. Should I assume that its similar to that with $Q_{\\alpha a}$ and $\\bar{Q}^a_\\dot{\\alpha}$ ? Like if I may think -  $[M^{\\mu \\nu},S^a_\\alpha] = i \\gamma ^{\\mu \\nu}_{\\alpha \\beta} \\epsilon ^{\\beta \\gamma} S^a_{\\gamma }$ $[M^{\\mu \\nu},\\bar{S}_{\\dot{\\alpha} a} ] = i \\gamma ^{\\mu \\nu}_{\\dot{\\alpha} \\dot{\\beta}} \\epsilon ^{\\dot{\\beta} \\dot{\\gamma}} \\bar{S}_{\\dot{\\gamma} a}$ ?   * Comparing equation 3.12 to 3.9 I see some possible discrepancies. Is there a factor of $\\frac{1}{2}$ missing with the term containing $D$ on the RHS of equation 3.12 ? In the same RHS of equation 3.12 in the $M_{\\mu \\nu}$ term why has the $\\gamma ^{\\mu \\nu}$ of equation 3.9 become $\\sigma ^{\\mu \\nu}$ ?    * In the statement just below equation 3.14 it says \"Hence a primary operator with R-charges $(J_1,0,0)$ is annihilated by $Q_{\\alpha 1}$ and $Q_{\\alpha 2}$ if $\\Delta = J_1$\"...Is this a consistency statement?  From this how does it follow that thesame operator is also annihilated by $\\bar{Q}^3_{\\dot{\\alpha}}$ and $\\bar{Q}^4_{\\dot{\\alpha}}$ In the same strain can one also say that $\\Delta = -J_1$ is consistent with the primary operator being annihilated by $Q^3_\\alpha$ and $Q^4 _ \\alpha$ ?   * I guess the above conclusions follow from taking different values of $a$ and $b$ in the equation 3.13. But one would get an extra factor of $\\frac{1}{2}$ on the RHS of 3.13 if one puts in the factor of $\\frac{1}{2}$ with the term containing $D$ on the RHS of equation 3.12 (..as I think it should be ..)   * On page 8 one creates bispinors $F_{+\\alpha \\beta}$ and $F_{-\\dot{\\alpha} \\dot{\\beta}}$ out of $F_{\\mu \\nu}$. I would like to know what is the intuition\/motivation\/reason for doing this ? Is the bispinor version of $F$ still have the meaning of a field strength ? If so then how does it relate to the bispinor version (?) $D_{\\alpha \\dot{\\beta}}$ of the covariant derivative ? I think I have already put in too many questions for one question. May be I will ask some more about this review in a separate question."}
{"id":"11647","title":"how does human brain compare to a modern CPU in energy per bit?","text":"Can someone compare the energy efficiency of human brain as a computer ? What is the energy in joules \/ flop ? may be some reasonable assumptions on the computational load of common tasks such as pattern recognition or speech synthesis can be used."}
{"id":"52765","title":"How does the voltage between two charged sheets change if change their distance","text":"Suppose I have two charged capacitor plates that both are isolated and carry a charge density $D = \\frac QA$. According to textbook physics the electric field between them is given by $E=\\frac D {\\epsilon\\epsilon_0}$ and the voltage by $U = Ed = \\frac {Dd}{\\epsilon\\epsilon_0}$ with $d$ the distance between the plates. According to the formula for the voltage from above I could set any voltage between the plates if I just separate them far enough from each other and also the electric field would be constant no matter how far the plates are apart which is also quite counter-intuitive. As far as I remember this is true as long as $d$ is small compared to the size of the charged plates. But what if this condition no longer holds? What is happening then? Is there another formula for this case that is comparably simple? I would suppose that for very large $d$ the whole thing can be seen as two point charges which would give a $\\frac1r$ dependency of the voltage. But what is happening in between?"}
{"id":"27941","title":"What is the time correlation function in the Green-Kubo formulation of ionic current?","text":"I am reading a paper, and I came across the Green-Kubo formulation, where the conductivity $\\sigma$ of charged particles is related to the time correlation function of the $z$-component of the collective ionic current $J_z(t)$: $$\\sigma_{GK} = \\frac{1}{V k_B T} \\int_0^{\\infty} dt \\; C_{JJ}(t)$$ where $C_{JJ}(t) = \\langle J_z(0) J_z(t) \\rangle$ and the collective current along the $z$ direction is $J_z(t) = \\sum_{i=1}^N q_i v_{z, i}(t)$. $V$ is the volume of the system and $q_i$ and $v_{z, i}$ are the charge and $z$-component of the velocity of the $i$th charged particle. $\\langle ... \\rangle$ is an equilibrium ensemble average. My question is, what is the time correlation function? Is the time correlation function $C_{JJ}(t) = \\langle J_z(0) J_z(t) \\rangle$? Or is the time correlation function the integral: $\\int_0^{\\infty} dt \\; C_{JJ}(t)$?"}
{"id":"21395","title":"In the Niagara Falls, which factors prevent rise of T H2O falling a certain height?","text":"I need some ideas on a problem. The first part says: Whats the posible rise in the temperature of the water falling 49.4 m in the Niagara Falls? That one was easy, with answer 0.112 Kelvin. ($\\Delta T = \\frac{g*h}{c_{H_2 O}}$) The second part asks what factors tend to prevent that rise in temperature? Im thinking kinetic energy, preassure, air conductivity but im not sure. Thanks in advance"}
{"id":"80545","title":"naive question on Boltzmann equation and conservation laws","text":"The Boltzmann equation in absence of external force reads: $\\frac{\\partial f}{\\partial t} + \\vec{v} \\cdot \\frac{\\partial f}{\\partial \\vec{r}} = \\left( \\frac{\\partial f}{\\partial t}\\right)_{coll}$ Where the r.h.s. stands for the change in the distribution function of the velocities owing to collisions. I won't specify here a particular collision kernel as it is not necessary for my question...I think. It is common to introduce   * The particle density $n(\\vec{r},t) \\equiv \\int d^3v \\:f(\\vec{r}, \\vec{v},t)$   * The mean velocity field $\\vec{u}(\\vec{r},t) \\equiv \\frac{1}{n(\\vec{r},t)}\\int d^3 v \\:\\vec{v} f(\\vec{r}, \\vec{v},t)$ It is then easy to integrate over the velocities the force-free Boltzmann equation above and find the famous conservation law: $\\frac{\\partial n(\\vec{r},t)}{\\partial t} + \\nabla_{\\vec{r}}\\cdot(n(\\vec{r},t)\\vec{u}(\\vec{r},t)) = 0$ That is because, if the collisions conserve the number of particles, then there should not be any 'source' or 'sink' on the r.h.s. My naive question is the following: The above conservation law for matter can be recast as $\\frac{\\partial n(\\vec{r},t)}{\\partial t}+ \\nabla_{\\vec{r}}\\cdot \\vec{J}(\\vec{r},t) = 0$ with the particle flux $\\vec{J}(\\vec{r},t) = n(\\vec{r},t)\\vec{u}(\\vec{r},t)$. This is fine but it does not seem very general. For instance, it does not seem to encompass a diffusive flux does it? Hence, although this equation is very general, it does not seem to lead to Fick's laws of diffusion and yet I kind of remember that one can get the latter from some kind of expansion of at least a linearized Boltzmann equation... Could somene unlighten me on this apparent paradox?"}
{"id":"48116","title":"Pictures of nuclear explosions some milli\/nano seconds after detonation","text":"Where I can find photos of nuclear explosions just after detonation (before 5-10 ms, the shorter the better)?"}
{"id":"95744","title":"Curvature of Spacetime","text":"I have been exploring for some time both the Special and General Relativity, hoping to glean at least a conceptual grasp of their basic tenets. In reading the book \"Gravitation\" by Misner, Thorne and Wheeler, the authors stress that Riemann came very close to make a decisive connection between gravitation and curvature of space, but he failed to do so, they say, precisely because he thought of SPACE and curvature of SPACE instead of curvature of SPACETIME and this makes the whole difference! Can anybody explain in conceptual terms, as far as possible, why spacetime, unlike SPACE alone, can be seen and understood as curved? I firmly believe that mathematics is only a language, albeit a complex one, which facilitates our understanding of reality, but that the same reality is not hopelessly beyond reach without maths. We should remember after all that Einstein's mathematical formulations of the Special and General Relativity are rooted in thought experiments and in a basic conceptual grasp, which preceded its mathematical formulation."}
{"id":"104606","title":"Ampère's Law for an infinitely long straight wire","text":"In order to find the magnetic field generate by an infinitely long straight wire of radius R, I have to use local Ampère's Law : $\\boxed{ \\oint_L B\\cdot dl = \\mu_0 \\sum_j I_j}$ If $r > R$ then $ \\oint_L B\\cdot dl = B(r)\\cdot 2\\pi r = \\mu_0 I \\Rightarrow B(r) = \\dfrac{\\mu_0 I}{2\\pi r} $ (I'm okay with this expression) But if $r<R $ I'm supposed to find $ \\oint_L B\\cdot dl = B(r)\\cdot 2\\pi r = \\mu_0 I \\cdot \\dfrac{\\pi r^2}{\\pi R^2} \\Rightarrow B(r) = \\dfrac{\\mu_0 I r}{2\\pi R^2} $ I don't understand where come from the $\\dfrac{\\pi r^2}{\\pi R^2}$ factor. Thanks for your help (and sorry for my bad english)"}
{"id":"78253","title":"If the net work on a particle is zero, can the speed change?","text":"The following question was on a quiz in physics class: > If the net work done on a particle is zero, which of the following > statements must be true? > > a) The velocity is zero   >  b) The velocity is decreased   >  c) The velocity is unchanged   >  d) The speed is unchanged   >  **e) There is no displacement for the object** The correct answer was e. In what scenario would the speed change? There was an explanation beside the question: > Since the work done is zero, it indicates that the applied force is zero. > Since Force = Mass X Acceleration, and the mass is not zero, this implies > that the acceleration is zero. When asked about this question, the teacher responded: > If there is no displacement, then only work done is zero. If the speed is > unchanged, then there is no acceleration. This will lead to an absence of a > force. Hope it explains the situation."}
{"id":"109501","title":"BPS state and annihilation of SUSY charges","text":"As we all know that anticommutator of one set of supercharges in massive extended supersymmetry is something like $$\\\\{b_\\alpha, b_\\beta^\\dagger \\\\} = \\delta_{\\alpha \\beta} (M-\\sqrt{2} Z).$$ My question is that everyone says that it is obvious that BPS states annihilate half of the supersymmetric charges. Why is this so? It may be trivial but I don't know how. May be it is just because the annihilation operator acts on the lowest energy state and annihilate BPS States? But why only half? Can anyone clear that up."}
{"id":"109500","title":"Does centrifugal force exist?","text":"Currently in my last year of high school, and I have always been told that centrifugal force does not exist by my physics teachers. Today my girlfriend in the year below asked me what centrifugal force was, I told her it didn't exist, and then she told me her textbook said it did, and defined it as \"The apparent force experienced towards the outside of a circle is the centrifugal force and is due to the mass of the object resisting the inward centripetal acceleration that the object is experiencing\". I was pretty shocked to hear this after a few years of being told that it does not exist. I did some reading and found out all sorts of things about pseudo forces and reference frames. I was wondering if someone could please explain to me what the hell is going on (in high school student terms), is it wrong to say that centrifugal force does not exist? This has always nagged me a bit as I often wonder that if every force has a reaction force then a centripetal force must have a reaction centrifugal force, but when I asked my teachers about this they told me that centrifugal force does not exist. Now I'm just confused. Help."}
{"id":"234","title":"How should a physics student study mathematics?","text":"**Note: I will expand this question with more specific points when I have my own internet connection and more time (we're moving in, so I'm at a friend's house).** This question is broad, involved, and to some degree subjective. (I started out as a physics-only student, but eventually decided to add a mathematics major. I am greatly interested in mathematics; the typical curriculum required for physics students is not deep or thorough enough; mathematics is more general (that means work!); and it only requires a few more classes. Naturally, I enjoy mathematics immensely.) This question asks mainly of undergraduate-level study, but feel free to discuss graduate-level study if you like. Please _do not_ rush your answer or try to be comprehensive. I realize the StackOverflow model rewards quick answers, but I would rather wait for a thoughtful, thorough (on a point) answer than get a fast, cluttered one. (As you probably know, revision produces clear, useful writing; and a properly- done comprehensive answer would take more than a reasonable amount of time and effort.) If you think an overview is necessary, that is fine. For a question this large, I think the best thing to do is focus on a specific area in each answer. * * * **Update:** To Sklivvz, Cedric, Noldorin and everyone else: I had to run off before I could finish, but I wanted to say I knew I would regret this; I was cranky and not thinking clearly, mainly from not eating enough during the day. I am sorry for my sharp responses and for not waiting for my reaction to pass. I apologize. **Re: Curricula:** Please note that I am not asking about choosing your own curriculum in college or university. I did not explicitly say that, but several people believed that was my meaning. I will ask more specific questions later, but the main idea is how a physics student should study mathematics (on his or her own, but also by choosing courses if available) to be a competent mathematician with a view to studying physics. _I merely mentioned adding a mathematics major to illustrate my conclusion that physics student need a deeper mathematical grounding than they typically receive._ And now I have to run off again."}
{"id":"102547","title":"To all experienced theoretical physicists out there, what is the step by step process in your math education?","text":"I am not doing a physics degree but an engineering degree but i am planning using my free time to self study all the math in preparing myself to self study subjects in theoretical physics. (I've always wanted to do theoretical physics but there are no scholarships offered in my country for the subject and my family is not rich so I had no choice but to self study) By \"self studying\" I mean buying books and reading online materials on my own and try to understand everything I can. So far this is my plan, Analytic geometry -> Single Variable Calculus -> Advanced Calculus (basic Linear algebra + multivariable and vector calculus) -> Linear Algebra -> ODEs -> but here I am stuck as to where do I go from here? Well first of all is this plan okay?   If it is : how do I continue? Should I continue to real and complex analysis and do more things from pure math? Or should I just stick to the more computational part of math and learn more advanced techniques in solving differential equations or learn more algebraic techniques? How about geometry? Topology? Group theory and more things from pure Math? If its not : what is the better way of doing it? Should I learn all the basic math first before reading Goldstein's classical mechanics ? Or should I just start with the physics and just pick up the math along the way? And I also have the time issues to worry about how I wish I can just do a physics degree but unfortunately things don't always go your way. Due to imminent time constriants (engineering has lots of projects im sure) i would like my self study and learning to be as efficient as possible. Hence good advice from all experienced physicists would be very much helpful. Thanks in advance."}
{"id":"14074","title":"What are the prerequisites to studying general relativity?","text":"This question recently appeared on Slashdot: > Slashdot posts a fair number of physics stories. Many of us, myself > included, don't have the background to understand them. So I'd like to ask > the Slashdot math\/physics community to construct a curriculum that gets me, > an average college grad with two semesters of chemistry, one of calculus, > and maybe 2-3 applied statistics courses, all the way to understanding the > mathematics of general relativity. What would I need to learn, in what > order, and what texts should I use? Before I get killed here, I know this > isn't a weekend project, but it seems like it could be fun to do in my spare > time for the next ... decade. It seems like something that would be a good addition to this site: I think it's specific enough to be answerable but still generally useful. The textbook aspect is covered pretty well by Book recommendations, but beyond that: **What college-level subjects in physics and math are prerequisites to studying general relativity in mathematical detail?**"}
{"id":"45912","title":"How should a theoretical physicist study maths?","text":"> **Possible Duplicate:**   >  How should a physics student study mathematics? If some-one wants to do research in string theory for example, Would the Nakahara _Topology, geometry and physics_ book and other geometry and topology books geared at physicists be sufficient for that purpose, or should one read abstract math textbooks e.g. Spivak _Differential geometry._ What about real analysis and functional analysis (not just the introductory functional analysis chapter that's present in quantum mechanics textbooks)?"}
{"id":"37961","title":"Mechanics Energy (Calculus)","text":"A particle moves with force $$F(x) = -kx +\\frac{kx^3}{A^2}$$ Where k and A are positive constants. if $KE_o$ at x = 0 is $T_0$ what is the total energy of the system? $$ \\Delta\\ KE(x) + \\Delta\\ U(x) = 0$$ $$F(x) = -\\frac{dU}{dx} = m\\frac{dv}{dt} = m v\\frac{dv}{dx}$$ Integrating to get U(x) and 1\/2mv^2 I get $$\\Delta\\ U(x) = \\frac{kx^2}{2} - \\frac{kx^4}{4A^2}$$ $$\\Delta\\ KE(x) = -\\frac{kx^2}{2} + \\frac{kx^4}{4A^2}$$ Which Makes sense. But how do I find the function KE(x) where KE(0) = $T_0$? Do I Even need to? The total energy in the system is $T_0$ Correct? Also a kind of side note. What is really confusing me, is when should I add limits of integration and under what circumstances should I just use an indefinite Integral?"}
{"id":"30505","title":"Someting almost faster than light traveling on something else almost faster than light","text":"> **Possible Duplicate:**   >  Travelling faster than the speed of light   >  Double light speed Lets say that   * an airplane can fly at 4\/5 of the speed of light, and   * I can run at 2\/5 of the speed of light, and   * I'm in the airplane. Suddenly I start running towards the cockpit down the middle lane. Now what? From what I know, the speed of light is not the speed of light just because it is. The speed of light is instead the maximum speed limit because light moves as fast as possible and therefore equals the max speed.   But when it's the maximum speed limit, nothing can ever move faster. So, I'm running in the airplane... now what?"}
{"id":"122335","title":"Driving car with (almost) the speed of the light and switch the headlights on","text":"I'm curious what will happen if we 'drive' a car with (almost) the speed of light, and all of sudden we switch the car headlights on? Will the car headlights ray have double speed (speed of 600.000km\/h) or that ray will never be seen?"}
{"id":"102063","title":"Can we travel faster than light?","text":"Consider two cars going in opposite direction one at speed $c\/2$ and other just greater than $c\/2$, then one bus will appear to other moving at speed more than $c$. How can an object travel at speed more than light?"}
{"id":"116367","title":"Relative speed when approaching the speed of light","text":"According to this chart of the Lorentz factor as a function of speed: ![Lorentz factor](http:\/\/i.stack.imgur.com\/jgzti.gif) If a spacecraft neared (roughly) 0.85c, would it appear to be traveling at 1.7x the speed of light from the perspective of those on board - i.e. covering 0.85c distance, but in half the time amount of time?"}
{"id":"16807","title":"What happens to body chemistry at the speed of light?","text":"Assume that I'm traveling at the speed of light in one direction. My brain is also traveling at the speed of light in that direction. Presumably there is at least one receptor site in my brain that is oriented in such a way that particles would need to travel faster than the speed of light to reach their destination receptor. What would happen to those particles? Would my brain cease to function because these particles couldn't reach their destination?"}
{"id":"26723","title":"Is there more mass in stars or interstellar medium?","text":"From what I gather the interstellar medium has about about 1 atom per cubic centimeter. But on the other hand, as they say, \"Space is big, really really big\" So if it is known (or at least theorized about) what percent of the universes matter is in the following: black holes, stars, and interstellar medium?"}
{"id":"92436","title":"cause of maximum kinetic energy and elastic potential energy at the same point of a transverse wave in a string","text":"Suppose a string element oscillating transversely in _SHM_ having both kinetic energy and elastic potential energy travels with velocity $u$. Let $y(m)$ be the amplitude. When it reaches $y(0)$, it has max. $K.E.$ and elastic $P.E.$ together at the same time. But in _SHM_ , when the kinetic energy is max. the potential energy is min. but in the string it is contradicted. Does transverse wave violate _SHM_ and if not why at the same time it has both K.E. and P.E. maximum unlike _SHM_?"}
{"id":"92433","title":"Does diamond dust conduct heat as well as diamond?","text":"Diamond is one of the best thermal conductors you can get. If the diamond is crushed into dust and spread out over a flat surface, but still held fairly compact (for instance in a small petri dish), would it still conduct as well as before?"}
{"id":"114859","title":"Single photons: Is there a 90° offset of the electric to the magnetic component in the direction of propagation?","text":"Single photons: Is there a 90° offset of the electric to the magnetic component in the direction of propagation? ![Electromagnetic wave](http:\/\/i.stack.imgur.com\/ksHYn.png)"}
{"id":"414","title":"Number theory in Physics","text":"As a Graduate Mathematics student, my interest lies in Number theory. I am curious to know if Number theory has any connections or applications to physics. I have never even heard of any applications of Number theory to physics. I have heard Applications of linear algebra and analysis to many branches of physics, but not number theory. Waiting forward in receiving interesting answers!"}
{"id":"114851","title":"Time-dependent Schrodinger equation from variational principle","text":"In the paper, \"Density-functional theory for time-dependent systems\" Physical Review Letters 52 (12): 997 the authors mentioned that the action $$ A= \\int_{t_0}^{t_1} dt \\langle \\Phi(t) | i \\hbar\\partial \/ \\partial t - \\hat{H}(t) | \\Phi(t) \\rangle \\tag{1} $$ provides the solution of time-dependent Schrodinger equation at its stationary point. Wikipedia  called (1) as the Dirac action without further reference. If I do a variation, indeed the stationary point of action (1) gives $$ i \\hbar\\partial \/ \\partial t | \\Phi(t) \\rangle = \\hat{H}(t) | \\Phi(t) \\rangle $$ However, from path-integral point of view, the least action principle is only a limiting case when $\\hbar \\rightarrow 0$. In general, there is no least action principle in quantum mechanics. My question is, how to reconcile these two aspects? What does vary of action (1) mean?"}
{"id":"68927","title":"What does a curved natural log graph suggest?","text":"Sorry if this is rather simple, but I've only just started learning about using logarithms in experimental physics. I did an experiment to test the amount of time it would take for an amount of water to leave a burette. I used the starting volume of water in the burette as a control variable, $50cm^3$. I recorded the time it took for the a given volume of water to be left in the burette. For example, $10cm^3$ left took a time of roughly $71\\mbox{s}$; $45\\mbox{cm}^3$ left took roughly $6\\mbox{s}$, and then many values in between. I would expect this to represent exponential decay, seen as different concentrations and masses of water in the burette would have different effects on the speed of the water leaving the burette. (Correct me if I'm wrong.) So I plotted a graph of volume against time and it showed exponential decay, but it was only very slightly curved, but curved nonetheless. So I decided then to plot a graph of $\\ln\\left(V\/\\operatorname{cm}^3\\right)$ against time\/s. However, this did not produce a straight line. If I were to follow the plotted points with a curve, the gradient of the line would have been negative and increased in negative 'magnitude'. I'm meant to analyse the extent of whether or not my experiment shows exponential decay. I'm quite stuck, because my original graph shows very slight decay, whereas my log graph isn't a straight line. Does the fact that the log graph doesn't produce a straight line show that there isn't exponential decay? Does it not matter? Would it have been straight had there been very few experimental errors\/uncertainties (there would have been a lot)? So I guess, fundamentally, my question is: What does the curved line on my natural log graph suggest?"}
{"id":"77346","title":"When is the Fermi surface a surface of constant mean curvature?","text":"Fermi surfaces are surfaces of constant energy in reciprocal space. They provide information about the properties of a material in solid state physics. Constant mean curvature surfaces are a superset of minimal surfaces, which minimize area and have zero mean curvature. Soap bubbles are surfaces of constant mean curvature. The two certainly coincide in the case of a sphere, but must they always coincide? If so, why? Is there a sharper constraint that Fermi surfaces must obey as well? I found a few papers that mention the two as connected, but they never establish the exact connection between the two. Preliminary literature review: Mackay, Alan L. Periodic minimal surfaces, 1985. > Minimal surfaces are found, as mentioned above, in soap films . . . The > surfaces of constant energy in reciprocal space, used in solid state physics > for finding the Fermi surface, are very similar. Mackay, Alan L. Periodic minimal surfaces from finite element methods, 1994. > Fermi surfaces, which are surfaces in reciprocal space, are closely related > to nodal surfaces."}
{"id":"114958","title":"Why is the space-time interval squared?","text":"The space-time interval equation is this: $$\\Delta s^2=\\Delta x^2+\\Delta y^2+\\Delta z^2-(c\\Delta t)^2$$ Where, $\\Delta x, \\Delta y, \\Delta z$ and $\\Delta t$ represent the distances along various coordinates according to an observer, and $\\Delta s$ is the space-time interval. All observers agree on the space-time interval, it is constant. My question is **why is it squared?** If we had in equation like this: $$\\Delta s'=\\Delta x^2+\\Delta y^2+\\Delta z^2-(c\\Delta t)^2$$ $\\Delta s'$ would be constant as well. It would also never be imaginary. It would have units of $[length]^2$ instead of $[length]$ though. Is there a theoretical or practical reason that we define the space-time interval based on squaring, or is it just to make it look similar to Pythagoras' theorem\/give it simpler units or something else entirely?"}
{"id":"94594","title":"Regarding state of Klein-Gordon field","text":"In regular quantum mechanics of particles, I have the Schrodinger evolution picture for a **general state** $$ i\\hbar \\frac{d}{dt} \\left|\\psi(t)\\right> = \\hat H \\left|\\psi(t)\\right> $$ then we take the inner product with respect to $ \\left< x\\right| $ to obtain the equation in the position representation. $$ i\\hbar \\frac{d}{dt} \\psi(x,t) = -\\frac{\\hbar^2}{2m}\\nabla^2\\psi(x,t) - V(x)\\psi(x,t) $$ where $$ \\psi(x,t) = \\left< x |\\psi(t)\\right> $$ and $$ \\left< x |\\hat H|\\psi(t)\\right> = -\\frac{\\hbar^2}{2m}\\nabla^2\\psi(x,t) - V(x)\\psi(x,t) $$ In the case of quantum fields, the position and momentum and demoted from operator status to simply paramaters and the field operator (here for KG real field) is given by $$ \\hat \\psi(\\vec x) = \\int \\frac{d^3p}{(2\\pi)^3} \\frac{1}{\\sqrt{2\\omega_p}} \\Bigl[\\hat a(\\vec p) e^{i\\vec k. \\vec x} + \\hat a^\\dagger(\\vec p) e^{-i\\vec k. \\vec x} \\Bigr] $$ in the Schrodinger picture. The states are generated from vacuum by the operator $$ \\hat \\psi(x) \\left|0\\right> = \\int \\frac{d^3p}{(2\\pi)^3}\\frac{1}{2E_p}e^{-i\\vec p.\\vec x} \\left|\\vec p\\right>$$ Now what exactly are these states, are they eigenstates of some operator (seems like the field operator) or quite arbitrary ? In what representation is the field operator given here ? EDIT : I think I understood the first part of the question, these states are the eigen-states of Hamiltonian of the field. But the question about the choice of basis of the operator still remains."}
{"id":"79614","title":"Why does the echo of the last chord appear to rise in pitch","text":"I - and others - observe that when a musical performance ends, the echo of the last chord appears to rise in pitch by up to a quarter tone while the echo decays. This effect appears to be independent of the type of performance - orchestral or choral. I am tempted to assume that this is related to what is known in electro-acoustic circles as Space Echo, but I cannot see why this should occur."}
{"id":"106885","title":"Small question about accuracy and precision","text":"Let's say I have a law like this, $$D=\\frac{c}{r}$$ where $c$ is a constant, $r$ a distance in meter. my measures of $r$ are [$0.02m$, $0.01m$], then $<r>=0.015m$ and $\\delta r = \\pm 0.005m$. So now if I want to calculate $D+\\delta D$ should I use $+\\delta r$ or $- \\delta r$ in my equation? because if I use $+\\delta r$ I get a smaller value than if I use $-\\delta r$ since $r$ divide $c$ edit: in my real problem I have a lot of data, all is fine when I use the minus delta. I just want to be sure..."}
{"id":"433","title":"In interferometry, what is the origin of the name \"Airy function\"?","text":"In interferometry (specifically, in the domain of Fabry-Perot cavities), the function $$f(\\phi) = \\frac{1}{1 + F \\sin^2 \\phi}$$ , which describes the shape of the resonant structure of the cavity, is often called the \"Airy function\" (for instance, in Wolfram Mathworld). However, it is obviously quite different from the special functions Ai(x) that usually go by that name. This function resembles probability density function of the wrapped Cauchy distribution. **How did it get the name \"Airy function\"?** I've heard that Fabry and Perot gave it this name in one of their original papers (maybe this one? PDF, in French, which I can't read), in honor of (the same) George Biddell Airy who had earlier considered similar interferometers. It would be great if someone could help ferret out the first reference to that function by this name."}
{"id":"94599","title":"Vanishing of Weyl Tensor Contraction","text":"Within the context of Einstein space-times, we know that the contraction of the Weyl tensor across a set of indices always vanishes, like so : $$C{^{\\alpha }}_{\\mu \\alpha \\nu }=0$$ From a purely mathematical standpoint this should be straightforward ( but perhaps tedious ) enough to prove from the definition of the conformal tensor in terms of the Riemann tensor and its contractions. However, I am wondering what the physical and\/or geometric meaning and significance - if any - of this vanishing contraction really is ? I am a very visual person and learner, so an intuitive geometric understanding of this would be very helpful to me."}
{"id":"90822","title":"Will accelerating a massive particle generates a blackhole?","text":"I have a naive question about blackhole. If I accelerate a massive particle very close to the speed of light, the particle will have large energy-momentum tensor. Will it become a blackhole?"}
{"id":"90482","title":"Relativistic Black Hole?","text":"So recently, looking at high energy particles through the lens of General and Special Relativity has peaked my interest. One thing I was considering, using the electron as the first example, is as follows: If gravity is the result of mass (stress-energy tensor), and as particles approach the speed of light, their mass is increased as measured by an outside observer (A) by a factor, $ \\frac{1}{\\sqrt{(1-(v\/c)^{2})}} $. The mass of the electron, $ m_{e} $, is $ 9.109 \\times 10^{-31} kg $, and the radius is $ r_{e} = 2.818 \\times 10^{-15} m $. Given the Schwarzschild Condition is: $$ r_{s} = \\frac{2Gm}{c^{2}} $$. Observer A, due to the Lorentz dilation, will measure a dilated mass term that is increasing monotonically as a function of the electron's velocity. The mass that observer A will then measure is: $$ m_{A} = \\frac{m_{e}}{\\sqrt{(1-(v\/c)^{2})}} $$. Combining the Swarzschild Condition with the mass as measured by A, we attain a relation between the mass that observer A measures with the particle's velocity, accounting for relativistic effects. This allows me to now pose the question that given the Schwarzschild radius is the electron radius, at what velocity does the mass increase to the point where the electron, as seen by observer A, become a black hole? This also leads to the more conceptual question of what are the implications of simultaneity in this situation? If the electron's mass is still $m_{e}$ in it's own frame, then shouldn't the electron not really turn into a black hole? Of course, this whole argument falls apart if the derived speed is greater than $c$. I went own to calculate that. I found that speed to be: $$ v = \\sqrt{c^{2}-\\frac{4G^{2}m^{2}_{e}}{r^{2}_{e}c^{2}}}$$ The velocity at which an object of a given radius and mass would become a black hole. To my disappointment, this came out to be $8.988 \\times 10^{16} m\/s $. Over twice the speed of light. I have yet to calculate the velocity for more massive objects, such as stars ( which could theoretically reach relativistic velocities as the result of being flung by a galactic collision ). Either way, if this velocity is attainable for anything, what would simultaneity say about this?"}
{"id":"127198","title":"Gravitational atraction of fast object","text":"Let's imagine a asteroid that travels with 0.99999999999999999c. (I know it's impossible). Anyway... Relativistic mass of such object would be almost equal to earth's stationary mass. Now let's imagine that such object passes closely to me (assuming I'm immortal etc.). According to $F_g=GMm\/r^2$ (Where M is relativistic mass of this asteroid and m is my mass) such an object would attract me. So would I gravitate after it? Would I gain it's speed constantly falling on it? Or maybe it would just pulled me for a moment, but then I would return to my ol' good non-relativistic speed in \"everyday reference system\"? Or maybe something else would happen? What happens with gravity of relativistic, NON ACCELERATINNG objects? (Let's assume that there's something that makes this asteroid speed constant)."}
{"id":"28422","title":"Can a black hole form due to Lorentz contraction?","text":"> **Possible Duplicate:**   >  If a 1kg mass was accelerated close to the speed of light would it turn > into a black hole? Imagine, a rod of length **L** is moving with velocity approaching the speed of light with respect to a human observer on Earth. Due to Lorentz contraction, the rod will observed to be very short. And since all laws of physic hold true in every frame of references, the gravitation law which state the force acting is inversely proportional to square of distance between them, will be acting between various part of the rod. Now, as velocity approach **c** , **L** will approach 0. This should cause an enormous gravitational force enough to form a black hole. Isn't this suggesting a black hole can be formed when the object velocity comes near the speed of light? According to the rod's frame of reference, it will see itself as stationary and a man who observe it is moving, so according to the rod, the human must be a black hole. Isn't this a paradox?"}
{"id":"34874","title":"Has the speed of light changed over time?","text":"Could someone judge my (stoner) hypothesis that the speed of light has changed over time -- ie. as the universe has expanded in volume light has slowed down, perhaps going so far as back to the big bang when it was infinitely fast and there was no time because everything happened at once etc. Thinking that the speed at which information can propagate through the universe is linked to the size of it seems intuitive to me. My question -- is there an easy disproof of this? Would Einstein have to be wrong? Does it violate anything supposedly more fundamental such as quantum or string theories? Do any current experiments invalidate it? If not can you show me in any case why you think its unlikely. # Edit 8\/24 I'm accepting Mark M.'s answer but will post this here because there is a character limit on comments @Mark M thanks, good answer but. as someone whos only read some popular physics, and should leave this to the experts, im still muddled in my personal theory. i dont see why you should need two units to measure the speed of light. the thing i have a hard time wrapping my head around is the relation of time and distance. they seem like they could fundamentally be the same thing. if you say time is measured fundamentally by the vibration of so and so quantum object in space....why cant we just measure that vibration distance as the constant... ill repeat myself to try to be clear... there is a certain minimum distance that particles have to go to interact with each other ....... if it wasnt vibrating there wouldnt be time,its what creates the illusion of time...so instead of talking about speed or c as distance\/time.... cant we simply talk about that distance a quantum object vibates....I'll lead up to my point.....perhaps there can need be only one constant here, and that is the physical size of the universe. a tiny metal tuning fork doesnt appear to be vibrating at all but if you blew it up to the size of the empire state building the metal rods would move from window to window. perhaps as our universe expanded in size the length of that minimum vibration (perhaps infinite at point zero) would have expanded, and therefore created the illusion of time and the speed of light, which, as the universe expands, will continue slowing down. perhaps we are like a big balloon and we have been blown up and all the fields\/particles-without-size are vibrating more and more in that space.. am i missing something obvious here?"}
{"id":"113540","title":"Curl of a vector field with two different systems of coordinates","text":"Let $$\\mathbf{H} = H_x \\mathbf{u}_x + H_y \\mathbf{u}_y + H_z \\mathbf{u}_z$$ be a vector field whose components are defined with respect to the unit vectors $\\mathbf{u}_x$, $\\mathbf{u}_y$ and $\\mathbf{u}_z$, so in the $(x,y,z)$ system of coordinates. **Question 1** : If we computed its curl in a _new_ system of coordinates, that is $(x' = x, y' = y, z' = -z)$, how would we do it? $$\\nabla \\times \\mathbf{H} = \\mathrm{det} \\begin{vmatrix} \\mathbf{u}_{x'} & \\mathbf{u}_{y'} & \\mathbf{u}_{z'}\\\\\\ H_x? & H_y? & H_z?\\\\\\ \\displaystyle \\frac{\\partial}{\\partial x'} & \\frac{\\partial}{\\partial y'} & \\frac{\\partial}{\\partial z'} \\end{vmatrix} $$ In other words, which quantities should we put in the second line? $H_x$, $H_y$ and $H_z$ are along $\\mathbf{u}_x$, $\\mathbf{u}_y$ and $\\mathbf{u}_z$ and _not_ $\\mathbf{u}_{x'}$, $\\mathbf{u}_{y'}$ and $\\mathbf{u}_{z'}$. **Question 2** : And if we would like to keep $H_x$, $H_y$ and $H_z$ in the second line, which would be their meaning in this computation? These questions are strictly relative (but not equal) to my previous one."}
{"id":"100832","title":"SR time dilation","text":"Two clocks are located at either end of a two light-hour long pole and motionless relative to the pole. Each clock transmits its time and notes that the other clock shows a reading two hours behind its own. That is, the clocks can be considered synchronised with each other. There is a flashbulb at the midpoint of both clocks. It goes off, and when each clock sees it (one hour later), it starts accelerating toward the other and each at the same rate (applying the same amount of thrust for the same local time). They do this for a short time until reaching a steady speed of 0.4c, relative to the flashbulb. Now according to the most recent generation of relativity experts, each clock should observe the other running more slowly. This needs to be the case because time dilation is based on the square of velocity so direction of travel is unimportant.   As they approach each other, the observed time difference will reduce because it takes less time for the transmitted signal to arrive. But the observed clock rate (after adjusting for doppler shift) will be slower at all times. By logical extension then, when they finally pass alongside or stop adjacent to each other, each clock should observe the timestamp of the other clock to be less than its own. Now obviously that outcome can’t be acceptable. Therefore we must conclude that any time dilation observed during their passage is nothing other than an illusion, and certainly not ‘real’ according to any experimental measurement, since only the final side-by-side comparison counts. It goes without saying then that if the clocks were instead moving apart from each other then any observed time dilation must also be an illusion. Am I correct?"}
{"id":"100833","title":"Rotatory motion of uniform disk","text":"Consider a uniform disk rolling without slipping with a certain constant angular velocity.Firstly it is moving in sufficiently rough surface.What will happen if it crosses the rough surface and just enters the smooth frictionless surface in its way?Will it be in the state of pure rotation or attains translatory motion or remains in pure rolling state.Please explain."}
{"id":"74696","title":"How is Special Relativity (SR) shown to NOT be an artifact of perception or measurement","text":"Special Relativity (SR) paradoxes are old-hat. But as I read explanations, they tend to resolve issues of simultaneity by applying the appropriate math... but that seems to me to be proving the theory by positing the theory. Just because the math is valid doesn't mean an argument is sound. All I have to do is believe warping space and time against curves and yes... it all works out. For instance... A space ship is traveling toward an asteroid and away from a stationary body at 60 percent the speed of light while at the same time an asteroid is traveling directly toward the space ship and toward the stationary body at 60 percent the speed of light. So the typical question is, \"when does the space ship meet its demise?... distance \/ (1.2*c)?\" _My question is distinctly different._ Why is it we are confident that the conclusion matches reality... i.e. how do we know the space ship doesn't just blow up before it thought it should based on only the perception and measurements available to it which seem to be governed by SR?"}
{"id":"74694","title":"Study of Black-body Radiation","text":"Why did scientists study black body radiations from something as complicated as a hollow container rather than the radiation from something simple like a thin solid cylinder?"}
{"id":"130330","title":"Water pump with a tap connected at its opening not pumping water when water is drawn from the pump before starting the pump","text":"Please give your explanation in layman terms, without throwing any complex equations at me. We have a 5000 liter tank dug in ground and a water pump connected to this tank that draws water from it and fills a overhead tank. We also have a tap at the opening of the pump. (A T joint is connected to the output of the pump, one goes to overhead tank and the other has a tap attached to it). Now, people draw water from this tap without turning on the pump (We try a lot to prevent this, yet people stubbornly do it). Nowadays when we turn on the pump, unless we pour water into the tap (About a liter of water) after turning on the pump we are not getting water output from the pump. A plumber suggested to us that it is because of Air lock and said to turn the tap upside down, like its a fountain :| Because of some physical constraints we cannot do this and it will be freaking expensive to do it. I am somehow not convinced that it is a solution to the problem. Could someone tell me what is happening here and how to solve this problem ?"}
{"id":"81314","title":"Reference Frame and Angular Speed Related?","text":"I am given the following problem: > If an airplane propeller rotates at 2000 rev\/min while the airplane flies at > a speed of 480 km\/h relative to the ground, what is the linear speed of a > point on the tip of the propeller, at radius 1.5m, as seen by (a) the pilot > and (b) an observer on the ground? The plane’s velocity is parallel to the > propeller’s axis of rotation. I was able to solve part a pretty easily by just using the formulas that relate linear speed with angular speed, but I didn’t get part b correct. I thought that the answers would be the same from both perspectives because the speed of the plane does not contribute to the rotation of the propeller, but I’m assuming that is the wrong way to think about it (because that produces the wrong answer). Can someone explain why this is the wrong approach? The solution to this problem involved noting that \"The plane’s velocity $v_p$ and the velocity of the tip $v_t$ (found in the plane’s frame of reference), in any of the tip’s positions, must be perpendicular to each other.\" How is this relevant to the problem?"}
{"id":"27138","title":"Kähler potential vs full effective potential","text":"In evaluating the vacuum structure of quantum field theories you need to find the minima of the effective potential including perturbative and nonperturbative corrections where possible. In supersymmetric theories, you often see the claim that the Kähler potential is the suitable quantity of interest (as the superpotential does not receive quantum corrections). For simplicity, let's consider just the case of a single chiral superfield: $\\Phi(x,\\theta)=\\phi(x)+\\theta^\\alpha\\psi_\\alpha(x) + \\theta^2 f(x)$ and its complex conjugate. The low-energy action functional that includes the Kähler and superpotential is $$ S[\\bar\\Phi,\\Phi] = \\int\\\\!\\\\!\\\\!\\mathrm{d}^8z\\;K(\\bar\\Phi,\\Phi) \\+ \\int\\\\!\\\\!\\\\!\\mathrm{d}^6z\\;W(\\Phi) + \\int\\\\!\\\\!\\\\!\\mathrm{d}^6\\bar{z}\\;\\bar{W}(\\bar\\Phi) $$ Keeping only the scalar fields and no spacetime derivatives, the components are $$\\begin{align} S[\\bar\\Phi,\\Phi]\\big|_{\\text{eff.pot.}} = &\\int\\\\!\\\\!\\\\!\\mathrm{d}^4x\\Big(\\bar{f}f\\,\\frac{\\partial^2K(\\bar\\phi,\\phi)}{\\partial\\phi\\partial{\\bar\\phi}} + f\\,W'(\\phi) + \\bar{f}\\, W(\\phi)\\Big) \\\\\\ \\xrightarrow{f\\to f(\\phi)} -\\\\!&\\int\\\\!\\\\!\\\\!\\mathrm{d}^4x\\Big(\\frac{\\partial^2K(\\bar\\phi,\\phi)}{\\partial\\phi\\partial{\\bar\\phi}}\\Big)^{-1}|W'(\\phi)|^2 =: -\\\\!\\int\\\\!\\\\!\\\\!\\mathrm{d}^4x \\ V(\\bar\\phi,\\phi) \\end{align}$$ where in the second line we solve the (simple) equations of motion for the auxiliary field. The vacua are then the minuma of the effective potential $V(\\bar\\phi,\\phi)$. **However** , if you read the old (up to mid 80s) literature on supersymmetry they calculate the effective potential using all of the scalars in the theory, i.e. the Coleman-Weinberg type effective potential using the background\/external fields $\\Phi(x,\\theta)=\\phi(x) + \\theta^2 f(x)$. This leads to an effective potential $U(\\bar\\phi,\\phi,\\bar{f},f)$ which is more than quadratic in the auxiliary fields, so clearly not equivalent to calculating just the Kähler potential. The equivalent superfield object is the _Kähler potential + auxiliary fields' potential_ , as defined in \"Supersymmetric effective potential: Superfield approach\" (or here). It can be written as $$ S[\\bar\\Phi,\\Phi] = \\int\\\\!\\\\!\\\\!\\mathrm{d}^8z\\;\\big(K(\\bar\\Phi,\\Phi) + F(\\bar\\Phi,\\Phi,D^2\\Phi,\\bar{D}^2\\bar{\\Phi})\\big) \\+ \\int\\\\!\\\\!\\\\!\\mathrm{d}^6z\\;W(\\Phi) + \\int\\\\!\\\\!\\\\!\\mathrm{d}^6\\bar{z}\\;\\bar{W}(\\bar\\Phi) $$ where $F(\\bar\\Phi,\\Phi,D^2\\Phi,\\bar{D}^2\\bar{\\Phi})$ is at least cubic in $D^2\\Phi,\\bar{D}^2\\bar{\\Phi}$. The projection to low-energy scalar components of the above gives the effective potential $U(\\bar\\phi,\\phi,\\bar{f},f)$ that is in general non-polynomial in the auxiliary fields and so clearly harder to calculate and work with than the quadratic result given above. * * * **So my question is** : when did this shift to calculating only the Kähler potential happen and is there a good reason you can ignore the corrections of higher order in the auxiliary fields?"}
{"id":"32294","title":"Areas of computer science required for quantum computing","text":"What knowledge of computer science should I have, to be able to pursue research in quantum computing. I am a Physics undergrad and would take three core courses in QM, before the completion of my degree. SO I guess necessary QM would be done. What about computer science?"}
{"id":"32296","title":"Introduction to differential forms in thermodynamics","text":"I've studied differential geometry just enough to be confident with differential forms. Now I want to see application of this formalism in thermodynamics. I'm looking for a small reference, to learn familiar concepts of (equilibrium ?) thermodynamics formulated through differential forms. Once again, it shouldn't be a complete book, a chapter at max, or an article. **UPD** Although I've accepted David's answer, have a look at the Nick's one and my comment on it."}
{"id":"5508","title":"How to learn physics effectively and efficiently","text":"How do you effectively study physics? How does one read a physics book instead or just staring at it for hours? (Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd certainly appreciate any suggestions for its modification if need be.)"}
{"id":"101105","title":"How to study physics as a first year student?","text":"I firstly apologise if this question is off the topic. I am a first year undergraduate student and the required textbook for the course Advanced Physics is Fundemental of physics by Halliday. This textbook is huge and to be honest I was scared when I saw the size of the textbook. There are too much text to read and there are many exercises at the end of each chapter and I do not think that I would have time to do every single problem at the end of each chapter. So with this information, what is the effective way of studying physics? what to read and what not to read? what exercises should I do? should I try to do all exercises? any other advice that I should do to be successful in physics is also appreciated. If you need also some information about my background: I am a first year combined bachelor of electrical engineering and bachelor of mathematics and I have good knowledge of math but did not do any physics in highschool."}
{"id":"102569","title":"What is an effective and efficient way to read research papers?","text":"I will be a grad student in condensed matter theory starting this fall. As an undergrad, I did the basic physics and math courses as well as a few grad classes (qft, analysis, solid state physics etc.) When I start reading research papers, I often feel overwhelmed because there is so much that I don't know and I find it hard to decide which points to gloss over and which points to spend time on and understand more thoroughly (which in my case, would probably require supplementary reading of textbooks or related papers) What are some things to keep in mind while reading a paper so that:   1. I get a general overview of the paper and I more useful insights into parts   2. I can do the above reasonably fast (say, finish reading at least 1 paper a week for a start) You don't have to be specific to condensed matter theory papers when you answer."}
{"id":"102970","title":"Good ways for learning and cramming formulas?","text":"I got an exam coming up and its numerical based (its pre-university level exam) but really tough. I want to know about various ways and methods to learn formulas. I know how can I derive them but its time consuming. I want to know how you people learnt these formulas?"}
{"id":"61679","title":"Effects of a very large magnetic field on the human body","text":"Ever since reading about the NHMFL I have always wondered about this and asked several people without getting a good satisfactory answer. My question is, considering the simplest case let's say a uniform magnetic field with a very high magnitude constant both in time and space permeating a large room, what would happen to my body in such a field like if I was to just walk through it? How large would the magnitude need to be before I \"feel\" anything? One tesla is fairly large, would I feel anything? I imagine as we crank up the magnitude, I would feel queasy and sick. What if I crank it up to fifty teslas or a hundred teslas? Would it hurt? Would it mess up my synaptic potentials? Would I go crazy with a dysfunctional brain? Would I pass out or go into a coma? At what point will the damage become irreversible? What magnitude will cause death? With a high enough magnitude would molecules in my body start falling apart? What happens at hundreds of teslas? Thousands of teslas or millions of teslas? The closest thing here I found was here in the accepted answer \"a magnetar that would be 1000 miles away would kill us due to diamagnetism of water in our cells\". This is the kind of stuff I am looking for, the magnitude of the B-field and then its effect on a human body. Like \"at 100T, your body would **___ _** because **_ _**and at 1000T, your body would **___ ___ __** because **___ ___**\" and so on. ~~If we allow changes in the magnetic field in time, does the induced electric field \"hasten\" the effects somehow? Would a strong enough E-field cause a shock within us burning our organs and killing us even if the average B-field is small but the db\/dt is \"large\" for example?~~ I don't know if any experiments on biological samples have been done because artificially creating magnetic fields beyond a 100T or so hasn't been quite done yet much less sustaining them and observing the effects on a biological tissue. But if there are any cool references even if they are on theoretical grounds, that would be interesting. Thanks. * * * Edit: Just to make the question more answerable, I'll focus only on static fields. Anyone know of any references\/experiments regarding the effects of large magnetic fields on biological tissues?"}
{"id":"131986","title":"Why is there a $\\pi$ phase difference when light goes from a rarer medium to a denser medium?","text":"I want the basic reason. I want the description of answer graphically also."}
{"id":"78261","title":"Why is a $\\pi$ phase added on reflection and why do things change with absorption?","text":"1. Mathematically, how does a $\\pi$ phase shift appear upon reflection of light off a optically denser medium?   2. Why is it always $\\pi$?   3. If the medium is absorptive it is no longer $\\pi$?"}
{"id":"46599","title":"Why does the echo for soundwaves hitting a vacuum come back out of phase?","text":"> **Possible Duplicate:**   >  Phase shift of 180 degrees on reflection from optically denser medium I've read in a physics book for musicians that, when a soundwave hits a near- solid object, it generates an almost completely in-phase echo. However, when the wave hits a near-vacuum, the echo generated is almost completely out of phase echo. Can someone explain, in detail, why this happens (or point me at a detailed explanation)? (My math-fu is fairly strong, but my physics-fu is not)."}
{"id":"71722","title":"why does the phase change when wave reflects from rigid boundary","text":"when the wave is reflected from open end there is no phase change but when it reflects from the rigid surface its phase changes. in open surface its phase change is 0 and in rigid surface its phase change is pi. How is it possible. Is is possible to reflect from open end."}
{"id":"66783","title":"Bouncing ball simulation computer science","text":"In my Computer science class I was given a problem where I have to simulate a bouncing ball using \"real physics\". I have been trying to find a equation that will simulate the height of the bounce given a gravity and an arbitrary mass. And I will need to calculate the next bounce and it's height. A lot of the equations I've found require a time. But I don't really care about a time, all I want is to get the next height after the previous bounce until it finally hits height of 0 or close to it. I haven't taken a physics class since high school( 5 years ago) and that was basic physics."}
{"id":"26266","title":"Do the stars imaged by a telescope even exist at present?","text":"I know that we now have telescopes which can capture the images of the stars and galaxies millions of light-years away from us. Does the telescope capture the past image of the star, i.e. the light which it emitted centuries ago? What guarantee is there that the star is still alive? What basis do organizations like NASA plan missions for evading such stars?"}
{"id":"56097","title":"Why is it important that the equation of state parameter of dark energy is measured?","text":"The equation of state for a perfect fluid is that $p=\\omega \\rho c^{2}$, where $p$ is the pressure, $\\rho$ is the density, $c$ is the vacuum speed of light, and $\\omega$ is called the equation of state parameter. $\\omega$ may be constant or varying in time. I'm looking for broad answers (and references if possible) to the following:   * Why is it important that the equation of state parameter of dark energy is measured?   * What will it tell us?    * What are the implications?"}
{"id":"101870","title":"Does the photon-phonon interaction always rotates the photon polarization of 90°?","text":"I'm reading about the acousto-optic effect and on the Acousto-Optical Tunable Filters on particular and wanted to understand the physics under its working. I found this paper http:\/\/dx.doi.org\/10.1016\/S0079-6727(03)00083-1 where they talk about the interactions of polarized light within a crystal. They say > \"For instance, if the incident radiation is an ordinary ray (o-ray), it will > be converted into an extraordinary ray (e-ray) upon interaction with the > acoustic wave.\" and a few paragraphs later > \"Considering the entering o-ray [...], photons interact with phonons [...] > and are converted to e-rays by a 90° polarization rotation.\" They do not explain the origins of this polarization rotation (and I imagine that it's beyond my knowledge) but is this rotation of 90° always true and why of 90° particularly?"}
{"id":"113197","title":"Question on Newton's law","text":"I have heard my friends referring to quantity ma as the \"force of acceleration\" whereas my teacher told us it can't be referred as so. Is it correct to refer that quantity a force? If not, how can we better describe this quantity?"}
{"id":"109666","title":"How does a wave packet acts rigidly?","text":"Particles can be represented as wave packet. So how do particles get scattered? Waves superimpose on one another, they don't bounce off of on one another."}
{"id":"53009","title":"Is time dilation an illusion?","text":"It is said that we can verify time dilation by flying a very accurate clock on a fast jet or spaceship and prove that it registers less time than the clocks on earth. However, the clocks on earth would be moving relative to the clock on the spaceship, and since time always dilates and never goes faster regardless of the direction of relative motion, the clocks on earth should register less time than the clock on the spaceship. Is this true? Whenever there is a fast-moving object such as a rocket do all clocks on earth really become slow? If the rocket with the clock landed after moving at relativistic speed, would its clock and the earth's clock again show the same time since during its travel both appeared slow to each other? Or is all this just an illusion, ie. the clocks just appear to be slow to each other but in actually run at normal speed, and neither is behind when the rocket actually lands?"}
{"id":"98832","title":"Calculate time dilation with Lorentz transformation","text":"It is possible to calculate the time dilation with Lorentz transformation? So with the equation $$t'=\\gamma(t-\\frac{x v}{c^2}) \\tag{1}$$ in which $\\gamma$ is the Lorentz-Factor? In an exercise I have succeeded. So why can be calculated with the equation $(1)$ the time dilation? What is the reason?"}
{"id":"100186","title":"Is time dilation real?","text":"If a body with an accurate clock is moving away from A which is stationary, then the time in B would be slower than that in A. Since relative to B A would have an equal velocity time in A would be slower than that in B. This results in a contradiction. This is the argument given in this website. Can someone say how this argument is wrong from a deeper study of SR.?"}
{"id":"95587","title":"Is aging independent of time progression when relativistic effects are taken into account?","text":"I have been trying to understand Einstein's principle of relativity for quite some time. I have at least I believe, understood the notion that time is relative and can actually 'change' relative to some frame of reference close to c. My question is, time is actually a variable that we invented based on how we thought about it; it's the progression of events, life and existence itself. So when, we say in Twin Paradox, one of the twin when he comes back to Earth is younger, how can that be possible? Again, it is true time would slow for him during his travel in space but that's because of the way we defined time and consequently its limitations close to c. How can that affect his physical structure (aging process)? Time should be independent of his personal physique."}
{"id":"102208","title":"Is time dilation an illusion? Variation on the twins paradox","text":"Consider the twins paradox with a slight variation: Twins A and B are in separate space ships both capable of going at the speed of light instantly (i.e. without any acceleration). Both ships are stationary relative to each other in intergalactic space facing in opposite directions. They synchronize their clocks. Then Twin A sees ship B zooms off to the \"right\" at the speed of light, and ship B travels a round trip of 8 years. Twin A sees ship B recede away from him at the speed of light (in fact, ship A just disappears). When ship B returns, Twin B's clock will show he has been gone 8 years. But from Twin B's perspective, it's ship A that zooms off to the \"left\" at the speed of light, does an 8 light year round trip, and according to Twin A's clock, he has also been gone 8 years. So they both would agree that one has been away from the other 8 years, and both have aged the same amount of time. It would seem clocks do not actually run slower as they move closer to the speed of light. So is time dilatation just an illusion?"}
{"id":"53232","title":"The real meaning of time dilation","text":"Is this true or false: If A and B have clocks and are traveling at relative velocity to each other, then to B it APPEARS that A's clock moving slower, but A sees his own clock moving at normal speed. Similarly, to A it APPEARS that B's clock is moving slower, but B sees his own clock moving at normal speed. If the above is true, then both A seeing his own clock moving at normal speed and B seeing his own clock moving at normal speed means that in reality both clocks are moving at normal speed, and neither has slowed down, whereas the other person's clock APPEARING to move slowly is merely an illusion. Now is this true or false: If A and B have clocks and are traveling at relative velocity to each other, then A sees his own clock move slowly (compared to the speed of the clock when A was at rest with respect to B) and B sees his own clock move slowly, so that in reality both clocks are moving slowly, but they still remain synchronized (since both are slow by the same amount) If the above statement is not true, then why do muons decay slowly when moving fast? [it could only be possible if the muon saw its own clock as moving slowly. If we saw the muon's clock moving slowly, but the muon saw its own clock moving at the normal rate, then the muon would decay at the normal rate, and not slowly] Can anyone please explain where I went wrong?"}
{"id":"51919","title":"How does a snowflake \"know\" to form symmetrically?","text":"> **Possible Duplicate:**   >  Why are snowflakes symmetrical? Under ideal situations, a snowflake forms into near perfect hexagonal symmetry. How? For instance, when a water molecule moves towards the edge of the snowflake, how does it \"know\" where to place itself so that the crystal formed matches the other five points?"}
{"id":"4986","title":"Could ball lightening be a form of plasma?","text":"With regard to the recent arXiv article: J. D. Shelton, _Eddy Current Model of Ball Lightening_   http:\/\/arxiv.org\/abs\/1102.1224 I wonder if this is a reasonable explanation of ball lightening, or if there is such an explanation. The paper is somewhat technical and E&M is one of my worst subjects. Please feel free to edit this question to one better suited, or if you don't have the rep, add a comment suggesting changes."}
{"id":"2158","title":"Why does measured pressure change over time in closed hose with temperature gradient","text":"I have a 4' hose that is closed at one end and connected to a Airdata Test Set (precise control of pressure) and a high accuracy pressure monitor on the other end with a T and valve. The valve allows the Airdata Test Set connection to be closed off resulting in a hose connected to the pressure monitor and closed at the other end. The valve is a high quality needle valve. The pressure monitor is a a Druck DPI 142. Half the length of the hose is in a temperature chamber controlled to 70 C. The Druck connected end is outside the chamber at roughly 22 C. When the airdata test set is commanded to a pressure of 1300 mb and allowed to settle for 30 seconds or so, then the valve closed, the pressure reported by the Druck drops over 20 minutes or so with a decreasing rate of change. The airdata test set draws air from the room when operating. The hose is ~0.190\" ID neoprene, Saint-Gobain P\/N 06404-15. The temp chamber, hose, etc, are given 1 hour to thermally stabilize prior to commanding the pressure to 1300 mb. The difference between initial pressure and stable pressure is ~18 mb. Why does the pressure take 20 minutes to stabilize?"}
{"id":"134544","title":"Value of G, gravitational constant","text":"if the universe is a 0 energy universe, could the value of G be worked out through summing up the strong, weak and electromagnetic force strengths for an elementary particle up to an infinite distance, and take the value of the negative energy gravity provides as that value in reverse? Or at the very least the value of the sum of the force of gravity and the cosmological constant?"}
{"id":"77275","title":"Why do we use operators in quantum mechanics?","text":"In classical mechanics, physical quantities, such as, e.g. the coordinates of position, velocity, momentum, energy, etc, are real numbers, but in quantum mechanics they become operators. Why is this so?"}
{"id":"113441","title":"Quantum Field Theory without LSZ, how is it possible?","text":"Most modern texts spend some time deriving the LSZ reduction formula that connects S matrix elements to time ordered field correlation functions. It seems essential, and really helps clear up what you are calculating. Yet some earlier texts and even some modern texts (e.g. \"Student Friendly Quantum Field Theory\" by R. Klauber) seem to skip right past this, working everything out in the \"interaction\" picture. It seems there must be something going wrong with this latter procedure, but I am not quite able to put it together."}
{"id":"113443","title":"$\\mathcal N=2$ Weyl multiplet and chiral superfields in 4d","text":"It is more or less known that a given antisymmetric tensor $F$ in two indices can be written in terms of spinorial indices, splitting into self-dual and anti-self-dual parts $$ F_{\\mu\\nu} = F_{\\alpha\\beta} (\\sigma_\\mu)^{\\alpha\\dot\\alpha}(\\sigma_\\nu)^{\\beta\\dot\\beta} \\varepsilon_{\\dot\\alpha\\dot\\beta} + F_{\\dot\\alpha\\dot\\beta} (\\sigma_\\mu)^{\\alpha\\dot\\alpha}(\\sigma_\\nu)^{\\beta\\dot\\beta} \\varepsilon_{\\alpha\\beta} .$$ My question is the following: start from (equations 5.3 and 5.4) $$ W_{\\mu\\nu}^{ij} = T_{\\mu\\nu}^{ij} - R_{\\mu\\nu\\lambda\\rho}\\theta^i\\sigma_{\\lambda\\rho}\\theta^j + \\cdots $$ which is self-dual in Lorentz indices $\\mu\\nu$ and antisymmetric in $\\mathrm{SU}(2)$ indices $ij$. Then it is squared to $$W^2= \\varepsilon_{ij}\\varepsilon_{kl} W_{\\mu\\nu}^{ij}W_{\\mu\\nu}^{kl}.$$ How do I see (possibly using the first equation above) that this is the same as a tensor (formula 4.1 and below) $W_{\\alpha\\beta}$ with self-dual $T$ as top component, where $\\alpha$, $\\beta$ denote symmetric spinor indices and $$W^2=W_{\\alpha\\beta}W_{\\alpha'\\beta'} \\varepsilon^{\\alpha\\alpha'}\\varepsilon^{\\beta\\beta'} ?$$"}
{"id":"86526","title":"AC through a pure inductor","text":"_I've studied the AC circuit for an ideal inductor in many physics books. After deriving the final equation for current the integration constant $C$ is assumed to be $0$ by giving inadequate reasons. In this question I seek for an adequate reasons._ Suppose an ideal AC voltage source is connected across a pure inductor as shown: ![i](http:\/\/i.imgur.com\/aevwKxN.jpg) The voltage source is$$V=V_0\\sin(\\omega t).$$ From Kirchhoff’s loop rule, a pure inductor obeys $$V_0 \\sin(\\omega t)=L\\frac{di}{dt},$$ so $$\\frac{di}{dt}=\\frac{V_0}{L} \\sin(\\omega t)$$ whose solution is $$i=\\frac{-V_0}{\\omega L}\\cos(\\omega t)+C$$ Consider the (hypothetical) case where the voltage source has zero resistance and zero impedance. > In most of elementary physics books $C$ is taken to be $0$ for the case of > an ideal inductor. $$\\text{Can we assume that } C \\neq 0?$$ >   1. (To me this is one of the inadequate reasons). This integration > constant has dimensions of current and is independent of time. Since source > has an emf which oscillates symmetrically about zero, the current it sustain > also oscillates symmetrically about zero, so there is no time independent > component of current that exists. Thus constant $C=0$. > >   2. (Boundary condition) there might exist a finite DC current through a > loop of wire having $0$ resistance without any Electric field. Hence a DC > current is assumed to flow through the closed circuit having ideal Voltage > source and ideal inductor in series if the voltage source is acting like a > short circuit like a AC generator which is not rotating to produce any > voltage. When the generator starts, it causes the current through the > circuit to oscillate around $C$ in accordance with the above written > equations. > >"}
{"id":"20919","title":"Relativistic centripetal force","text":"The thought randomly occurred to me that a circular particle accelerator would have to exert a lot of force in order to _maintain the curvature of the trajectory_. Many accelerators move particles at fully relativistic speeds, and I want to ask how that affects things. Why does this matter? Well, if I understand correctly, a particle in the LHC moving at $0.999 c$ would be _dramatically_ more difficult to keep moving in the circle than a particle moving at $0.99 c$. Reading Wikipedia, I was delighted to find a fully specified problem within a single paragraph. http:\/\/en.wikipedia.org\/wiki\/Large_Hadron_Collider > The LHC lies in a tunnel **27 kilometres** (17 mi) in circumference, as deep > as 175 metres (574 ft) beneath the Franco-Swiss border near Geneva, > Switzerland. Its synchrotron is designed to collide opposing particle beams > of either **protons at up to 7 teraelectronvolts** (7 TeV or 1.12 > microjoules) per nucleon, or lead nuclei at an energy of 574 TeV (92.0 µJ) > per nucleus (2.76 TeV per nucleon). I can (or Google can) calculate the proton case to come to $0.999999991 c$. In order to correctly calculate the force that the LHC must exert on it, do I need to start from $F=dp\/dt$, or can I get by with $v^2\/r$ times the relativistic mass? I'm not quite sure how to do the former. **Question:** Say I have 2 protons, both moving so fast they're going almost the speed of light but one has twice the energy of the other. They move side- by-side and enter either an electric or magnetic field that causes them to accelerate perpendicular to the velocity vector. Is the radius of curvature mostly the same for the two, or is it different by a major factor (like 0.5x, 1x, or 2x)? I want to read some comments from people who understand these physics well. The particles beyond a certain energy are all going _almost exactly_ the same speed. Is it still okay for me to say they have the same \"acceleration\" too? They just happen to have absurdly huge inertia compared to the same particle at a more modest fraction of the speed of light. Is this the correct perspective?"}
{"id":"106066","title":"Is the existence of electromagnetic standing waves dependent on the observers reference frame?","text":"If I take two plane EM waves travelling in opposite direction e.g. $E = E_0 \\sin(kx-\\omega t)$ and $E=E_o \\sin (kx + \\omega t)$, they sum to give a standing wave with a time-averaged Poynting vector of zero. If I use the appropriate special relativistic transformations to derive how these fields appear to an observer travelling at $v$ along the x-axis, I find that one E-field is diminished, one is boosted, whilst at the same time one wave is blue-shifted and the other red-shifted. These waves do not sum to give a standing wave in the moving frame of reference and have a non-zero time- averaged Poynting vector. So, is the phenomenon of a standing wave dependent on the frame of reference of the observer?"}
{"id":"106061","title":"Viability of Dysons bladeless fans?","text":"Do Dyson's bladeless fans produce enough air pressure to cool say an air conditioning unit and do they produce a vortex pressure like a bladed fan?"}
{"id":"4310","title":"Does boundedness of observables in the Haag-Kastler axiomatization rule out interactions?","text":"In an interacting theory I expect there to be caustics, resonances, and other situations in which _some_ observables would give an infinite experimental result. Of course, these are idealized states and observables -- if a real device's measurement results are quite accurately modeled by such an observable, and we created a state in which its expected value is large enough, the device would be destroyed. An idealized world in which models never predict infinite results seems somehow different from the world we live in. Even though experiments never return the measurement result \"infinity\", they do return the measurement result \"I'm so sorry, I'm breaking now, it's bigger than you thought it could be\". The Wightman axioms do not require operators to be bounded, and to me seem much better for it. QFT as it's used in practice isn't constructed within the Wightman axioms, but it's much less constructed within the Haag-Kastler axioms, and, it seems to me, particularly for this reason. I take boundedness to mean that all the eigenvalues of an operator in its action on a Hilbert space of idealized Physical states are required to be finite. This is much stronger than requiring the expected values of an operator to be finite for a dense subset of the Hilbert space (or other, relatively weaker, requirements). It's certainly mathematically more convenient to use bounded operators (because we don't have to keep track of for which states we get a finite result for a given measurement, because they all do), but is that enough? At least, is this acceptable as part of a major attempt to axiomatize theoretical Physics? Is it obvious enough to be an axiom? I'm prompted to ask this question in this way by a comment in Doplicher's \"The principle of locality: Effectiveness, fate, and challenges\" J.Math.Phys. 51, 015218 (2010), http:\/\/arxiv.org\/abs\/0911.5136, which I'm reading this morning, where he sets out on the first page (2nd page in the arXiv) that \"In quantum mechanics the observables are given as bounded operators on a fixed Hilbert space\", which seems a specially sanitized version of QM, insofar as it rules out position, momentum, and energy observables. Finally, this question is asking, as always, have I got something (very) wrong?"}
{"id":"71406","title":"In a large city how much hotter on average is it outside due to the air conditioning of all the buildings?","text":"Title pretty much states the question. How much hotter do air conditioning units make it outside in a large city like NYC, Chicago, etc?"}
{"id":"113626","title":"Proof that 4-potential exists from Gauss-Faraday field equation","text":"This is a problem concerning covariant formulation of electromagnetism. Given $$\\partial^{[\\alpha} F^{\\beta\\gamma]}= 0 $$ how does one prove that $F$ can be obtained from a 4-potential $A$ such that $$F^{\\alpha \\beta}=\\partial^{\\alpha} A^{\\beta} - \\partial^{\\beta} A^{\\alpha} $$"}
{"id":"83549","title":"What is the linearized form of relativistic hydrodynamics?","text":"**What I'm looking for:** Let $\\vec{W}$ be the vector of conserved variables for a 1-dimensional, adiabatic, (special) relativistic, electrically neutral fluid. (Yes, something that simple!) I'm looking for a paper that derives the form of the matrix $A$ that linearizes the evolution equation. That is, $$ \\partial_t \\vec{W} \\approx A \\partial_x \\vec{W}. $$ Alternatively, the matrix $B$ that does the same for the conserved variables will work: $$ \\partial_t \\vec{U} \\approx B \\partial_x \\vec{U}. $$ These matrices are useful in fluid computations for several reasons. I need more than just the eigenvalues (used in certain solvers) - I also need the eigenfunctions. I've found plenty of papers that treat the nonrelativistic case (for one of many, many examples, see the appendices of Stone et al. 2008, ApJS 178 137) both with and without magnetism. I've found some papers that just quote a few eigenfunctions for relativistic MHD, but these are often the ones that are interesting only with magnetism in play. I'm looking for whatever paper derives these matrices in the rather simple case I'm dealing with. An answer that gives the derivation would be nice, but I'm also trying to locate the relevant literature. In particular, I would like a paper addresses the physical reliability\/usefulness of the linear approximation. **Background:** There are three primitive variables defining my fluid: rest-mass density $\\rho$, velocity $v$, and pressure $p$. By convention, these variables are combined into a vector $\\vec{W} = (\\rho, v, p)^\\mathrm{T}$. Many approaches to evolving fluids deal with the equations in flux- conservative form. Here, the conserved variables are \\begin{align} D & = \\gamma\\rho && \\text{(lab-frame density),} \\\\\\ M & = Dh\\gamma v && \\text{(relativistic momentum),} \\\\\\ E & = Dh\\gamma - p && \\text{(relativistic energy).} \\end{align} Here I define \\begin{align} \\gamma & = \\frac{1}{\\sqrt{1-v^2}} && \\text{(standard Lorentz factor),} \\\\\\ h & = 1 + \\frac{\\Gamma}{\\Gamma-1} \\left(\\frac{p}{\\rho}\\right) && \\text{(enthalpy),} \\end{align} where the ratio of specific heats $\\Gamma$ is assumed to be constant. These are often combined as $\\vec{U} = (D, M, E)^\\mathrm{T}$. Along with the vector of conserved quantities, we can define the vector of fluxes $$ \\vec{F} = \\begin{pmatrix} Dv \\\\\\ Mv + p \\\\\\ M \\end{pmatrix}. $$ Then we have the relation $$ \\partial_t \\vec{U} + \\partial_x \\vec{F} = 0, $$ which is used as the basis for most Riemann solvers and many fluid codes in general."}
{"id":"10615","title":"What would be the effective resistance of the ladder of resistors having n steps","text":"I'm a tutor. This is a high school level problem. In high school, every one have might have solved a problem of effective resistance of a ladder of resistors having infinite steps. Now the problem is little different. what if it has `n` steps instead of infinite steps. How to calculate effective resistance in that case?![enter image description here](http:\/\/i.stack.imgur.com\/n1eIX.jpg)"}
{"id":"112717","title":"Any liquid metallic alloys which are safe to handle with bare hands?","text":"http:\/\/en.wikipedia.org\/wiki\/Liquid_metal describes some alloys which are liquid at room temperature containing gallium, and sodium-potassium. We are advised not to handle them, and mercury, with unprotected skin. Are there any which are safe (as safe as, say, alcohol or glycerine)?"}
{"id":"108898","title":"How does ultrasonic horn produce a convection current in the water?","text":"When I was using ultrasonic horn in a beaker, I notice that there are convection currents in the beaker and stir up my substance. I don't understand why it produce water current, I thought that it will just vibrate like the ultrasonic bath. So why does ultrasonic horn create water currents? Thank you."}
{"id":"89723","title":"Lasers can demagnetize ferromagnets?","text":"How is it possible to demagnetize a magnet with a laser? Source: http:\/\/www.helmholtz- berlin.de\/pubbin\/news_seite?nid=13657&sprache=en&typoid And the paper: http:\/\/prb.aps.org\/abstract\/PRB\/v88\/i21\/e214404 How does this work?"}
{"id":"103113","title":"Effect of waters changing specific gravity on objects apparent weight placed in liquid","text":"My goal is to monitor the change in specific gravity of a liquid over a period of time. My question is: What are the appropriate formula for determining expected apparent weight of an object immersed in a liquid where the liquids specific gravity g\/ml is expected to change? EG. If I were to take an object who's density is 2.6 (average for glass) weighing 100 grams and plunk it into distilled water I believe I should expect an apparent weight should be roughly 61.53 grams. Please let me know if I am just horridly wrong. So then if that distilled waters density\/specific gravity were to change say to 1.010, would my new apparent weight of the object be 61.15 grams? My math is not solid in this. I'm basically using ratios in order to produce these answers. Please for the sake of simplicity if you are to choose to answer leave out extenuating circumstances such as temperature of the liquid\/object and possible compression of the object due to pressure. If you do chose to add extenuating circumstances I would ask to add those concepts as tertiary answers. I'm sure that my question is probably very basic, but grasping the concepts has proven perplexing to me. I am probably not using the correct search. Your help in this simple question is greatly appreciated."}
{"id":"71055","title":"Why do nearsighted people see better with their glasses *rotated*?","text":"If you are nearsighted (like me), you may have noticed that if you _tilt_ your glasses, you can see distant objects more clear than with normally-positioned glasses. **If you already see completely clear, you can distance your glasses a little more from your eyes and then do it.** To do so, rotate the temples while keeping the nosepads fixed on your nose, as is shown in the figures. As I said, starting with your glasses farther than normal from your eyes, you can observe the effect for near objects too. (By _distant_ , I mean more than 10 meters and by _near_ I mean where you can't see clear _without glasses_ ) **Note that if you rotate more than enough, it will distort the light completely.** Start from a small $\\theta$ and increase it until you see blurry, distant objects more clear. (You should be able to observe this at $\\theta\\approx20^\\circ $ or maybe a little more) When looking at distant objects, light rays that encounter lenses are parallel, and it seems the effect happens because of oblique incidence of light with lenses: ![enter image description here](http:\/\/i.stack.imgur.com\/QpPys.png) The optical effect of oblique incidence **for convex lenses** is called coma, and is shown here (from Wikipedia): ![enter image description here](http:\/\/i.stack.imgur.com\/lIUw6.png) I am looking for an explanation of how this effect **for concave lenses** (that are used for nearsightedness) causes to _see_ better. One last point: It seems they use plano-concave or convexo-concave lenses (yellowed lenses below) for glasses instead of biconcave ones. ![enter image description here](http:\/\/i.stack.imgur.com\/s3HdJ.png)"}
{"id":"90202","title":"Black Hole evaporation through virtual couples at the horizon","text":"I read about vacuum energy. It explains the Hawking radiation, the black hole necessary radiation: > Physical insight into the process may be gained by imagining that particle- > antiparticle radiation is emitted from just beyond the event horizon. Vacuum > fluctuations are always created as particle–antiparticle pairs. The creation > of these virtual particles near the event horizon of a black hole has been > hypothesized by physicist Stephen Hawking to be a mechanism for the eventual > \"evaporation\" of black holes. Is this \"insight\" supposed to make us to think that   1. the virtual pair created is a part of black hole so that when half flies away then the BH mass is reduced? The article on vacuum energy says that the vacuum energy is an underlying background energy that exists in space throughout the entire Universe. So, it is not related to the black holes, though one particle of the couple may create the salute that you may consider as BH evaporation, it actually covers up the fact that the **second particle flies into the BH, increasing its mass**.   2. Why do we believe that the receding half of the couple does not fall back to the Black Hole? Yes, it has escaped the horizon but the (pretty strong) BH gravitation is still in action, and it is only few neutons less than $\\infty$ because we are still almost at the horizon initially. This means that to actually escape and not to fall back into BH, the speed of the receding particle must be virtually infinite. I doubt that it is likely that you will have many such virtual particles right at the event horizon. It is much much (Almost surely) more likely that **\"escaped\" half of the couple will also eventually fall onto the black hole, increasing its mass**. So, considering the virtual particles of vacuum energy, we find two ways to add mass to the BH. Why do they call it BH (mass) evaporation?"}
{"id":"108896","title":"Is there a general term for the situation where an improperly chosen measurement range results in a bias?","text":"For example, consider the following measurement: A sensor can measure a specific physical quantity, and has a range of $0$ to $100$. All values above $100$ will be shown as 100. We now take the following measurements: $78$, $100$, $82$, $94$, $100$, and we conclude that the average is $90.8$. However, the true values were $78$, $112$, $92$, $94$, $105$, resulting in the true average of $94.2$. Both end results are in the valid measurement rage, but individual measurements are not. Can we call it saturation? It has a similar meaning in photography, but in mathematics it seems to have a completely different meaning."}
{"id":"132053","title":"The thermodynamics of heating water in a rigid container","text":"When the pressure on the liquid surface is less than the vapor pressure of the liquid at a given temperature, the liquid will start to evaporate. This is common sense. The problem is more difficult when the liquid and its vapor are heated inside a rigid container, with the specific volume of the mixture less than the critical specific volume: ![enter image description here](http:\/\/i.stack.imgur.com\/ESHNv.png) According to my professor's notes, the level of the liquid in the container would fall. Why? What is the physics (or the thermodynamics) behind this? If a mixture is heated in a constant volume what happens to it? I know that it will not boil completely because it would attain equilibrium pressure with its vapor soon enough, but how do we fix the state when it would stop boiling? Why is it that if the specific volume of the mixture is less than the critical specific volume, the level of the liquid will fall when heated?"}
{"id":"132052","title":"Centre of mass, integral","text":"I was answering a question on proving the parallel axis thereom for angular momentum and came across this: $$\\int Yy'dm=Y\\int y' dm=0$$ Where the position of the center of mass of an object is given by $(X,Y,Z)$, $(x',y',z')$ is a position relative to the centre of mass and m is the mass of the object. My text book (Introduction to classical mechanics) says that this is due to the definition of the centre of mass. There are two things that I don't understand firstly why is $Y$ independent of mass whilst $y'$ is not? and secondly please can you explain what definition of the centre of mass they are using to get the reslut above? I really have no idea to the answer for either of these questions?"}
{"id":"2774","title":"Rotate a long bar in space and get close to (or even beyond) the speed of light $c$","text":"Imagine a bar spinning like a helicopter propeller, At $\\omega$ rad\/s because the extremes of the bar goes at speed $$V = \\omega * r$$ then we can reach near $c$ (speed of light) applying some **finite** amount of energy just doing $$\\omega = V \/ r$$ The bar should be long, low density, strong to minimize the amount of energy needed For example a $2000\\,\\mathrm{m}$ bar $$\\omega = 300 000 \\frac{\\mathrm{rad}}{\\mathrm{s}} = 2864789\\,\\mathrm{rpm}$$ (a dental drill can commonly rotate at $400000\\,\\mathrm{rpm}$) $V$ (with dental drill) = 14% of speed of light. Then I say this experiment can be really made and bar extremes could approach $c$. What do you say? **EDIT** : Our planet is orbiting at sun and it's orbiting milky way, and who knows what else, then any Earth point have a speed of 500 km\/s or more agains CMB. I wonder if we are orbiting something at that speed then there would be detectable relativist effect in different direction of measurements, simply extending a long bar or any directional mass in different galactic directions we should measure mass change due to relativity, simply because $V = \\omega * r$ What do you think?"}
{"id":"3119","title":"Reaching speed of light","text":"> **Possible Duplicate:**   >  Rotate a long bar in space and reach c Sorry this is very naive, but it's bugging me. If you had a straight solid stick attached on one end and rotating around that attachment at a certain rpm, there would be a length at which the end of the stick would theoretically reach, with that rpm, the speed of light. Well, doesn't seem possible - what specifically would be the limitations that would prevent the end of the stick to reach the speed of light? What would happen?"}
{"id":"63974","title":"Why is this thought experiment flawed: A vast lever rotating faster than the speed of light","text":"If there were a vast lever floating in free space, a rigid body with length greater than the width of a galaxy, made of a hypothetical material that could endure unlimited internal stress, and this lever began to rotate about its middle like a propeller so that a person looking at the universe would simply see it spinning at a gentle pace like a windmill, would not its ends be moving many, many times the speed of light? Or am I making so many errors in my thought process that the whole question is absurd? I'm trying to establish why, in essence, a sufficiently large mechanical device (large beyond reason) could not exceed the speed of light."}
{"id":"107097","title":"What prevents an orbiting object from getting a speed which is greater than $c$?","text":"Consider an object orbiting around a point with radius $r$ and angular velocity $\\omega$. Here its linear velocity is $v=\\omega r$. If we choose a large enough $r$ and reasonable $\\omega$, $v$ might be greater than $c$. If this fact is impossible, what prevents it from happening?"}
{"id":"6537","title":"Knotted token-ring network","text":"Suppose we have a rigid token-ring network. An observer at any node can seemingly determine the angular momentum of the network by measuring the time it takes for a packet to travel around the ring in each of the two directions. Is it possible by any means for an observer to determine whether the network is knotted?"}
{"id":"25151","title":"What allows the modified Urca process to work at lower density than direct Urca in neutron star cooling?","text":"> **Possible Duplicate:**   >  What allows the modified Urca process to work at lower density than direct > Urca in neutron star cooling? This comes from an unanswered question over at physics.se: The dominant method of neutron star cooling is neutrino emission. There are two regimes usually presented, the \"direct Urca\" and \"modified Urca\" processes, each of which are sequences of neutron decay and inverse reactions. The direct Urca looks like this: $$n\\rightarrow p+l+\\overline{\\nu_l},\\quad p + l \\rightarrow n + \\nu_l$$ where $l$ is a lepton - either an electron or a muon. These processes cause continuous emission of neutrinos which cools a neutron star relatively quickly. But below a density of $\\rho\\approx 10^{15}\\mathrm{\\,g\\,cm^{-3}}$ (about three times the nuclear density) this process is suppressed, which means that the direct Urca process only occurs in the core. This is the reason according to a review of neutron star cooling from Pethick and Yakovlev (2004): > The process can occur only if the proton concentration is sufficiently high. > The reason for this is that, in degenerate matter, only particles with > energies within ~$k_BT$ of the Fermi surface can participate in reactions, > since other processes are blocked by the Pauli exclusion principle. If the > proton and electron Fermi momenta are too small compared with the neutron > Fermi momenta, the process is forbidden because it is impossible to satisfy > conservation of momentum. Under typical conditions one finds that the ratio > of the number density of protons to that of nucleons must exceed about 0.1 > for the process to be allowed. This makes some sense. But what surprises me is that this process can still work with a slight modification at lower densities. The _modified_ Urca process can cool the star $$n+N\\rightarrow p+N+l+\\overline{\\nu_l},\\quad p + N + l \\rightarrow n + N + \\nu_l$$ where $N$ is a nucleon - a proton or a neutron. This process, I'm told, can work at much lower densities, but produces 7 orders of magnitude less emissivity. As a result, it's the dominant process in the superfluid outer core. My question is why does the additional nucleon permit lower densities? How does an additional neutron or proton get us out of the conservation of momentum problem with the direct Urca process?"}
{"id":"54874","title":"Where do electrons get the energy to remain in orbit?","text":"As we know electrons continuously revolve around the nuclus without falling in it at a high velocity beating it's force of attraction. My question is where do electrons get energy to revolve around the nucleus and withstand its force of attraction."}
{"id":"55976","title":"What is the source for magnetic energy?","text":"_\"The electron around the nucleus is in a quantized energy level and can change it only if an external interaction intervenes.\"_ That is OK but when there is a magnet, it has energy of attracting iron particles for months and years (natural magnet) just by right order of atoms. Where this energy come from? In nature we have to spend one sort of energy to gain another form, but in a magnet what is spending? Thanks"}
{"id":"88459","title":"Commutator evolution operator and position operator","text":"Let $H= \\frac{p^2}{2m}$, then I am supposed to calculate $[x,e^{-iHt}]$. My idea was to use $[x,p^n]=i \\hbar n p^{n-1}$ and so I ended up by using the series for the exponential function with $-\\frac{t \\hbar}{m} e^{-iHt}$. Could anybody tell me, whether this result is correct?"}
{"id":"88452","title":"What will happen to matter if there is a Higgs metastability decay?","text":"Previously, on Save us from swallowing baby universes, please!, I pointed out the dangers associated with false vacuum decay. I wish to be more specific here. Suppose the Standard Model remains a good description of our universe even at higher energies. The observed value of the Higgs mass predicts that our current phase is metastable. If our phase decays in a bubble, its interior will have a different much higher value for the Higgs field. What happens to matter as it encounters the expanding bubble wall? Coleman-de Luccia assumed the exterior is initially in the vacuum state. They did not take the presence of matter into account. A changing value for the Higgs field is unlikely to affect the electromagnetic field much. But we expect the electromagnetic coupling strength to change inside the bubble due to a different running of the coupling constants under the renormalization group in the new phase. How much will this affect photons, and what is its frequency dependence? The leptonic and quark masses due to the Yukawa Higgs coupling will jump up dramatically in the new phase. Will this lead to a partial reflection and a partial transmission of leptons as the bubble passes through? What about confined bound states of quarks in hadrons? How will hadrons be affected? Leptons and quarks also happen to be dressed states, and the dressing will differ between the two phases. Might this cause a particle shower if these particles are transmitted? The masses of the W and Z bosons will also jump up. What will happen to them? Will neutrinos be unaffected? What if neutrino oscillations are also taken into account? This leads us to the ultimate question. Suppose we have some lump of matter, like a cyborg, which is a complicated bound state of leptons, hadrons and photons. What will happen to this cyborg as the bubble expands and hits it?"}
{"id":"122310","title":"Does light bend in a vacuum?","text":"I'm familiar with gravitational lensing but still I'm wondering if there is experiments (conducted here on Earth) which show that light bends due to gravity. For example mirrors setup to hold the light or something like that. My question is inspired by this bounty question."}
{"id":"113709","title":"Mach-Zehnder with PBS = Bit-Flip?","text":"Is it true that a Mach-Zehnder interferometer with two polarizing beam splitters (PBS) is nothing but a bit flip for the polarisation degree of freedom? Say the PBSs reflect vertical polarized light and transmit horizontal polarized light. If we send in a state like $|+\\rangle = \\frac{1}{\\sqrt 2}(|H\\rangle+|V\\rangle)$ the horizontal part gets reflected only once (by the mirror in the interferometer) but the vertical part gets reflected three times (first PBS, mirror, second PBS). At the end we have a relative Phase of $i^2=-1$ between $|H\\rangle$ and $|V\\rangle$ which means that we end up with the state $|-\\rangle = \\frac{1}{\\sqrt 2}(|H\\rangle-|V\\rangle)$. Thus it is a bit flip of the polarisation. Right?"}
{"id":"133904","title":"Neutron to antiproton decay","text":"Would it be possible for a neutron to lose a positron and become an antiproton? Or would would it need to be the decay of a antineutron to antiproton instead?"}
{"id":"133908","title":"What does \"P-wave\" mean when referring to a particle?","text":"In scattering theory, P wave means $l=1$, where $l$ is the azimuthal quantum number. However, what does P wave mean when referring to particle states? For example, in this paper (arXiv link), the authors are talking about _P-wave charmonia states_. What does that mean? More specifically, I understand that in some sort of potential model, solved using Schroedinger equation for example, there will be states that may be labeled by $n$=something, $l=1$. But here, the article says P-''wave'' charmonia! What is this wave?"}
{"id":"92091","title":"What are the ways of finding the spin of a particle","text":"How many ways are there for this mission? Any article about that on the internet. I heard about TODAI made a reprogrammable quantum computer. Any more ideas are welcome"}
{"id":"93722","title":"How do you measure proton's spin?","text":"I've probably read it somewhere in Sakurai but I cannot recall it at the moment. So how does one really measure the proton's spin? I mean the proton's spin and not its constituents. Do you measure it using a Stern-Gerlach type of setup? How about atoms and other stuff, how is their spin measured?"}
{"id":"35805","title":"Complex part of the solution for physical values","text":"What's a physical meaning of, for example, complex part of the solution for coordinate change of the anharmonic oscillator? Why after substitute (for diff. equation solve) for real x we can earn $x = Re(z) + iIm(z)$? It's because of substitute?"}
{"id":"98439","title":"Some questions about the large-N Gross-Neveu-Yukawa model","text":"Consider the following action with a fermionic field $\\psi$ and a scalar field $\\sigma$, $S = \\int d^dx \\\\{ -\\bar{\\psi}(\\gamma^\\mu \\partial_\\mu +\\sigma )\\psi + \\Lambda^{d-4}[ \\frac{(\\partial_\\mu \\sigma )^2 + m^2\\sigma^2 }{2g^2 } + \\frac{\\lambda \\sigma^4 }{4!g^4 } ] \\\\} - (N'-1)Trln(\\gamma^\\mu \\partial_\\mu + \\sigma )$ Assuming that this has a large-N saddle with uniform $\\sigma$ one gets the large-N free energy density as, $E(\\sigma) = \\Lambda^{d-4}(\\frac{m^2\\sigma^2}{2g^2} + \\frac{\\lambda \\sigma^4 }{4!g^4 } ) - \\frac{N}{2}\\int^\\Lambda \\frac{d^dq}{(2\\pi)^d}ln [\\frac{q^2 + \\sigma^2 }{q^2 } ]$ And the large-N saddle value of $\\sigma$ is determined by the large-N gap equation, $E'(\\sigma)=0$ Now from here how do the following conclusions come?   * Firstly that a non-trivial solution to the gap equation exists only when, $\\frac{m^2}{g^2} < N\\Lambda^{4-d}(\\frac{1}{(2\\pi)^d} \\int^\\Lambda \\frac{d^dk }{ k^2 } ) $  How does this one come?   * Secondly from this apparently follows that the inverse $\\sigma$ propagator in the massive phase is,  $\\Delta_\\sigma^{-1}(p) = \\Lambda^{d-4}(\\frac{p^2}{g^2} + \\frac{\\lambda \\sigma^2}{3g^4} ) + \\frac{N(p^2+4\\sigma^2) } {2(2\\pi)^d}\\int^\\Lambda \\frac{d^dq }{(q^2+\\sigma^2)((p+q)^2 + \\sigma^2)} $ How does this equation come?   * Now from this one can show that $\\Delta_\\sigma \\sim \\frac{2}{N b(d) p^{d-2} }$ From the above it follows that the canonical dimension of $\\sigma$ is 1. How does one understand that the mass dimension of the field $\\sigma$ does not depend on the space-time dimension?   * Now I don't understand this argument which says that now since $[\\sigma] =1$, both the terms $(\\partial_\\mu \\sigma)^2$ and $\\sigma^4$ are of dimension $4$ and hence for $2\\leq d \\leq 4$ these terms vanish in the IR critical theory? For this argument to work was it necessary that the IR theory was critical?"}
{"id":"116350","title":"Inertial navigation system: am I doing it wrong?","text":"I'm trying to develop an inertial navigation system. I can access data from an accelerometer sensor (acceleration on three axes) and gyroscope sensor (angular velocity on three axes). First of all, I integrate my angular velocity data with respect of time, and get angles on all three axes at every moment ($x \\to \\phi, y \\to \\theta, z \\to \\psi$). Then I feed the angles to this rotation matrix ![Rotation matrix](http:\/\/i.stack.imgur.com\/USq1f.png) and I use it to rotate my acceleration vector at that time, thus taking all my acceleration values on the same reference frame. Finally, I integrate acceleration to get space travelled, using the simple formula $$s(t) = \\frac12 a(t) t^2 + v(t - 1)t + s(t - 1)$$ My method seems to work fine with fake data, but performs really bad when I plug in the real data, the output is almost meaningless. Am I doing something wrong with the math or I have to search the problem in my implementation?"}
{"id":"94362","title":"What causes different decays?","text":"Nuclei spontaneously decay according to a certain decay rate. There are however different kinds of decay, alpha, beta, gamma... What causes then the nuclei, when they decay, to do so in one way of another? Is there a different decay rate for each kind of decay?"}
{"id":"1557","title":"Accelerating particles to speeds infinitesimally close to the speed of light?","text":"I'm in a freshmen level physics class now, so I don't know much, but something I heard today intrigued me. My TA was talking about how at the research facility he worked at, they were able to accelerate some certain particle to \"99.99% the speed of light\". I said why not 100%, and I didn't quite understand his explanation, but he said it wasn't possible. This confused me. Since the speed of light is a finite number, why can we go so close to its speed but not quite? Edit: I read all the answers, and I _think_ I'm sort of understanding it. Another silly question though: If we are getting this particle to 99.99% the speed of light by giving it some sort of finite acceleration, and increasing it more and more, why cant we increase it just a _little more_? Sorry I know this is a silly question. I totally accept the fact we cant reach 100%, but I'm just trying to break it down. If we've gotten so close by giving it larger and larger acceleration every time, why cant we just supply it with more acceleration? And how much of a difference is there between 99.99% the speed of light, and _the_ speed of light? (I'm not really sure if \"difference\" is a good word to use, but hopefully you get what I'm asking)."}
{"id":"80365","title":"What is so special about speed of light?","text":"I will try to be as explanatory as possible with my question. Please also note that I have done my share of googling and I am looking for simple language preferable with some example so that I can get some insight in this subject. My question is what is so special about $c$? Why only $c$. Its like chicken and egg puzzle for me. Does Einstein reached to $c$ observing light or does he got to light using some number which turned out equal to $c$. Why is $c$ not relative. If something has zero rest mass like a photon why they only travel at $c$ in vacuum and not with $c+1$ or $c-1$?"}
{"id":"81243","title":"The LHC - Proton Speed Limit","text":"I'm originally from stack overflow, so this is my first foray into this particular community. I've read in numerous places that the Large Hadron Collider is capable of accelerating protons at 0.999999991 _c_ , which mathematically works out to being 3 metres per second slower than the speed of light. That seems so incredibly close to the speed of light, that it's hard for me to understand why we can't quite get all the way there. **My Main Question :** What is preventing the LHC from achieving 1 _c_ , in terms of how fast it can propel these protons? **Side Question :** If we were able to accelerate the protons to the speed of light, what would the results of their collisions be?"}
{"id":"34134","title":"Is it possible to build an instrument which can travel faster than light?","text":"> **Possible Duplicate:**   >  Accelerating particles to the speed of light I have heard about atomic rockets in novels which have the capability to travel faster than light. I have also heard about fictional stories where objects capable to travel faster than light. My question is: > Is it scientifically possible to build an instrument which can travel faster > than light? And atlast i like to clarify that whether this question suitable to your site or maybe asked in anyother stackexchange site's,usually downvote will be awarded when there is lack of research ,i am seeking answer where the research be-ginned for it,am i wrong ? if my question is off the topic here please let me to know where can i ask this question?"}
{"id":"108575","title":"About accelerating particles","text":"Can a particle moving below the speed of light be accelerated more and more until it is travelling at c? IF so does it behave like electro-magnetic radiation?"}
{"id":"91442","title":"Why does light travel at finite speed?","text":"We have known for quite some time now that light travels at a finite speed of 3x10^8m\/s (approx) through vacuum. But why is that? Why can't the speed be infinite? Or at least higher than the current value? What is limiting its speed? I mean, the speed of sound is limited by the density of the medium. Then, what limits the speed of light in vacuum? Does finiteness of speed of light mean the possibility of existence of a omnipresent medium like aether, which, may be is intangible (or doesn't interact with the particles we know of currently)? * * * I do know that light is propagated by photons. That only makes me more curious as to why the speed is fixed to the current value? Especially as to WHAT is limiting the speed of a particle with no inertia?"}
{"id":"120067","title":"Why is the speed of light arbitrarily the limit?","text":"I know Einstein was great and all. Why is it that exactly at the speed of light is where infinite energy is required to accelerate any object with mass? Is it simply because the math of relativity checks out and explains most of everything? Are there any physicists who disagree with Einstein's theory?"}
{"id":"69528","title":"radiation thermodynamics paradox","text":"This question is concerned with a thermodynamic paradox for radiating bodies and radiation in a cavity of a specific shape. Consider two nested shells that are axisymmetric ellipsoids with the same two foci, A and B, as shown in the figure (line AB is the axis of symmetry). Cut the system along the vertical plane of symmetry and remove the right side of the outer shell, and remove the left half of the inner shell. Then connect the two halves with vertical surface, as shown in the figure, to make it a continuous enclosure. The result is a figure of rotation shown by the thick black line in the figure. Next, make the inner surface of it a perfect mirror. The property of such a cavity is that each ray emitted from point B comes to point A; but not each ray emitted from point A comes to point B - some rays emitted from A (shown in blue) come back to A. Now, put two small black bodies (say, two spheres of some small radius) at points A and B. Thermodynamic equilibrium requires that eventually the temperatures of the two spheres equilibrate. However, according to the geometric properties of this cavity, all energy emitted from B comes to A but only a fraction of energy emitted from A comes to B; so the equality of temperatures is not consistent with balance of emitted and absorbed power. How to resolve this paradox? ![enter image description here](http:\/\/i.stack.imgur.com\/igGAT.jpg)"}
{"id":"69525","title":"physical difference between A and B","text":"We all know that when we say A it sounds different than when we say B. I was wondering what exactly can be the difference between saying A and B in terms of physics. I first thought that it may due to difference in the combination of frequencies. Then I realized I can say any alphabet in many different tones. So what exactly in terms of physics is the difference between saying A and B ?"}
{"id":"113823","title":"Numerical Computation of Linbald Equation","text":"Can anybody suggest me a good algorithm for the time evolution of the reduced density matrix using Linbald equation. My Hamiltonian is time dependent. I am aware about Qotoolbox and Qutip. I have checked both things but I don't have any clues about the algorithms they are using. I have to find the entanglement dynamics in a bipartite system under dissipation and forcing."}
{"id":"135080","title":"How is free energy built into a Metropolis Monte Carlo simulation of an Ising model?","text":"In the Metropolis algorithm, the change in the energy given by the hamiltonian is compared for flipping a spin. This is not the free energy, but for systems above absolute zero you are trying to minimize the free energy, not the energy. So how is free energy built into this kind of simulation? How is entropy?"}
{"id":"43550","title":"how do you destroy magnetic field - demagnetize?","text":"And what happens with the magnetic field of a star that goes supernova? The magnetic radiation is scattered through the cosmos? Each particle will go away with its own magnetic radiation?"}
{"id":"74861","title":"A question about BRST current in bosonic string theory","text":"I have a question about Eq. (4.3.3) in Polchinski's string theory book volume I, p. 131. It is said > Replacing the $X^{\\mu}$ with a general matter CFT, the BRST transformation > of the matter fields is a conformal transformation with $v(z)=c(z)$, while > $T^m$ replaces $T^X$ in the transformation of $b$. Noether's theorem gives > the BRST current $$ j_B = c T^m + \\frac{1}{2} : cT^g : + \\frac{3}{2} > \\partial^2 c, $$ $$ = c T^m + : bc \\partial c : + \\frac{3}{2} \\partial^2 c, > \\tag{4.3.3}$$ My question is, what is the explicit expression of $T^m$? According to this thesis, p 29, $$-\\frac{1}{\\alpha'}: c \\partial X \\cdot \\partial X = :c T_X:$$ Suppose this expression is correct, I cannot use it to vertify Eq. (4.3.11) $$T(z) j_B(0) \\sim \\frac{ c^m - 26}{2z^4} c(0) + \\frac{1}{z^2} j_B(0) + \\frac{1}{z} \\partial j_B(0) \\tag{4.3.11}$$ if in (4.3.11), $T(z)= -\\frac{1}{\\alpha'} : \\partial X^{\\mu} \\partial X_{\\mu} : \\tag{2.4.4}$ and I applied contraction Eq. (2.2.11)."}
{"id":"41680","title":"Why is light called an 'electromagnetic wave' if it's neither electric nor magnetic?","text":"How can light be called _electromagnetic_ if it doesn't appear to be _electric_ nor _magnetic?_ If I go out to the sunlight, magnets aren't affected (or don't seem to be). And there is no transfer of electric charge\/electrons (as there is in AC\/DC current in space). In particular, the photons (which light is supposed to be composed of) have no electric charge (nor do they have magnetic charge). I'm looking for an explanation that can be appreciated by the average non- physicist Joe."}
{"id":"120165","title":"Bandpass of diffraction grating - exit slit width?","text":"I'm supposed to derive a relation for the range of wavelengths that's being transmitted by a spectrometer (bandpass) in terms of the dispersion, focal length and exit slit width. Given grating is $100mm$ wide, slit separation $d=5.6\\times 10^{-7} m$, focal length $f=1\\space m$, exit slit width $w=100\\mu m$ and spectrometer working in first order regime. Then I'm supposed to find value of slit width that corresponds to a bandpass of theoretical resolving power of $\\lambda \\approx 500 nm$ **Attempt** I found dispersion $\\frac{d\\theta}{d\\lambda} = \\frac{p}{d cos \\theta} $ and resolving power $Np$. Thus The amount of wavelengths contained in the dispersion is simply $\\delta \\theta \\times \\frac{d\\lambda}{d\\theta} $ But what is the angle of spread? Is it simply $\\delta \\theta = \\frac{w}{f}$. If so, then I find the bandpass $\\Delta \\lambda = \\frac{w}{f} \\times \\frac{d}{p} = 5.6 \\times 10^{-11} $. Last part of the question doesn't make sense, as resolving power is only dependent on $N$."}
{"id":"101402","title":"Exponential of a differential operator","text":"I have a differential operator $L$, $\\displaystyle L = i (t\\frac{\\partial}{\\partial z} - z\\frac{\\partial}{\\partial t})$ I can trivially hit this operator to $x,y,z$ and $t$ as $L x$, $L t$, $L y$, $L z$. But I have a problem with _exponential of that operator_. I want to hit this operator to $x,y,z$ and $t$ as well. $\\exp(i\\eta L)\\,\\,x$ ($\\eta$ is rapidity in this case) The first thing that comes to my mind is to use definition of exponential of an operator: $\\displaystyle \\exp(A) = I + A + \\frac{A^2}{2!} + \\frac{A^3}{3!} ...$ But, I don't know why, I don't want to use this infinite sum. There should be a smart way of doing this.. Do you have any suggestions for me?"}
{"id":"101403","title":"Conservation of stress-energy and the fluid equation in cosmology","text":"I'm trying to derive the equation for the cosmological fluid: $$\\dot \\rho + 3 \\frac{\\dot a}{a}(\\rho +P)=0$$ by starting from the conservation of the stress-energy tensor: $$\\nabla^\\mu T_{\\mu \\nu} = 0$$ with the stress-energy for a perfect fluid in its own frame being: $$ T_{\\mu \\nu} = \\text{diag} (\\rho, a(t)^2 P,a(t)^2 P,a(t)^2 P) $$ in a spatially flat FLRW metric: $$g_{\\mu \\nu} = \\text{diag}(1,-a(t)^2,-a(t)^2,-a(t)^2)$$ But I keep getting a bogus answer! Consider the equation you get from $ \\nabla^\\mu T_{\\mu \\nu} = 0$ when $\\nu =0$: $$ \\begin{align*} \\nabla^\\mu T_{\\mu 0} &= 0 \\\\\\ g^{\\mu \\alpha}\\nabla_\\alpha T_{\\mu 0} &= 0 \\end{align*} $$ $T$ is diagonal, so $\\mu$ must be zero, but $g$ is diagonal as well, so if $\\mu$ is zero, then so is $\\alpha$. This gives: $$ \\begin{align*} g^{0 0}\\nabla_0 T_{0 0} &= 0 \\\\\\ \\nabla_0 \\rho &= 0 \\\\\\ \\dot \\rho &= 0 \\end{align*} $$ Because $\\rho$ is just a scalar, so the covariant derivative is the partial derivative. Except this answer is wrong."}
{"id":"102598","title":"Positive Mass Theorem","text":"I'm a third year maths undergrad doing a project on minimal surfaces. However I'm really struggling to understand what the PMT is trying to explain? Could anyone help explain this (as simply as possible)"}
{"id":"45180","title":"About Efimov States and Halo-Nuclei","text":"I read that > _Halo nuclei could be seen as special Efimov states, depending on the subtle > definitions._ (The last sentence in the second to last paragraph of this > Wikipedia article.) This does not seem trivial to me in the least. Can someone shed some light on this for me?"}
{"id":"104037","title":"Why can't we have a wave of particles?","text":"I understand the nature of light can be complex and has extensive theories\/experimental data. We hear light can be both a wave and particle, so why can't it be both, a wave of particles?"}
{"id":"81923","title":"Identity in quantum operator tutorial","text":"I'm reading this tutorial by Ben Simons entitled _Operator methods in quantum mechanics_ in connection with his course in advanced QM, and I'm a bit puzzled by an identity in page 25, a bit above relation (3.3): With the momentum operator $\\textbf{p}=-i\\nabla$ and the vector $\\textbf{a}$ we have $e^{-i\\textbf{a}\\cdot\\textbf{p}}=e^{\\ \\textbf{a}\\cdot\\nabla}=\\displaystyle\\sum_{n=0}^{\\infty}\\frac{1}{n!}a_{i_1}\\cdots a_{i_n}\\nabla_{i_1}\\cdots\\nabla_{i_n}\\ ,$ where repeated indices are summed over. What confuses me here is that (in 3D) $\\textbf{a}$ and $\\nabla$ have 3 components while this expression seems to refer to component $i_n$ where $n\\rightarrow\\infty$, and I don't recognize the usual expansion of the exponential. What am I missing here?"}
{"id":"135337","title":"First variation of the action in relativistic notation - Landau & Lifshitz \"Classical theory of fields\"","text":"In Landau & Lifshitz's book, _Classical theory of fields,_ the action for a free particle is defined as: $$\\tag{8.1} S= \\int ^b _a {-mc \\ \\text d s}=0,$$ where $$\\text d s=c\\,\\text d t\\sqrt{1-\\frac {v^2}{c^2}}$$ is the the invariant space-time interval beetween points along the particles worldline. From the latter expression of the Lagrangian, it's easy to obtain the expressions of the momentum and the energy of the particle. However, after a \"classical\" derivation, the author repeats the calculations with a different notation (I'll post the derivation from the \"particle in a electromagnetic field\" case omitting the 4-potential terms, because I think that my book contains an error in the free particle case): > Since $\\text d s= \\sqrt {\\text dx^i \\text dx_i}$: $$\\delta S = -mc \\int ^b > _a\\dfrac{\\text dx_i \\text d\\delta x_i}{\\text ds}.$$ Integrating by part, > introducing the 4-velocity $u_i =dx_i\/ds$, we get: $$\\tag{9.10}\\delta S =-mcu_i\\delta x^i|^b _a+\\int_a ^bmc\\dfrac{\\text d u_i}{\\text d s} \\delta x^i \\,\\text ds.$$ I'm very puzzled about the meaning of these two lines; I have three questions: **1)** What does the notation $\\text d \\delta x_i$ mean? **2)** How does he obtain the $\\delta S$ expression? **3)** How does he pass from the first integral to the second? If someone could explain in detail I'd be very grateful. Note, I have no problem in getting the result, I can obtain it by replacing $\\text d s =c\\text d t \\sqrt {...}$ and doing the variation on $v$."}
{"id":"135334","title":"Is energy since the big-bang still here in the present?","text":"Based on the concept that energy can never be destroyed and is only transferred. Does it mean that energy since the formation of literally everything still here today?"}
{"id":"46988","title":"Operator Ordering Ambiguities","text":"I have been told that $$[\\hat x^2,\\hat p^2]=2i\\hbar (\\hat x\\hat p+\\hat p\\hat x)$$ illustrates **_operator ordering ambiguity_**. What does that mean? I tried googling but to no avail."}
{"id":"135338","title":"Would the charges on a superconductor near an another charged conductor move and produce magnetic field?","text":"When there is a charged conducting object near an another charged conducting object the charges on these objects accumulate to the sides where the closest points between these objects are. However, since these objects are conductors the charges also have to move on these objects. For a perfect conductor(~superconductors)would the charges still move on the object because to my mind it should accumulate on the sides and be static. I am pretty confused at this point so I hope someone could explain it to me: Would the charges on a superconductor near an another charged conductor move and produce magnetic field?"}
{"id":"134060","title":"Finding the least capacitance?","text":"A circuit has a self-induction of 1 H and carries a current of 2 A. To prevent sparking when the circuit is witched off, a capacitor of which can withstand a voltage of 400 V is used. The least capacitance of the capacitor connected across the switch must be? I have no idea how to go about this question."}
{"id":"134064","title":"Does the speed at which sound travel depend on the volume (amplitude) of the sound?","text":"Lets say you have a plank is you hit it once and get t time if you hit is 2x as hard will it travel t\/2? will it be the same or will it travel only slightly faster?"}
{"id":"52252","title":"Intuitive meaning of the Hilbert Space formalism","text":"> **Possible Duplicate:**   >  Intuitive meaning of Hilbert Space formalism I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points:   1. the observables are given by self-adjoint operators on the Hilbert Space.   2. Gefland-Naimark Theorem implies a duality between states and observables   3. what's the significance of spectral decomposition theorem in this context?   4. What do the Hilbert Space itself corresponds to and why are states given as functionals on the hilbert space. I need a real picture of this."}
{"id":"48460","title":"Measuring the quantum state of light","text":"**A clarification please** The following scheme to measure **linear** polarization states (a single polarizing beam splitter and two photo counters) orientation (as $arctan \\sqrt{\\frac{v}{h}}$) of **coherent** light pulses cannot discriminate two states which make the same angle with the measuring bases. $\\langle n \\rangle$ is the average number os photons per pulse. ![a](http:\/\/s14.postimage.org\/evo2hbpzl\/g4322.png) In the example, the $+45^o$ and $-45^o$ polarized pulses will result on the same output. I'd like to know if it is OK to do the following setup and what would be change in precision of the polarization angle, if any: ![b](http:\/\/s2.postimage.org\/xsyztz1g9\/g4361.png)"}
{"id":"130236","title":"Rectifying incomplete popular notions in cosmology","text":"In looking at the answers to this question regarding light from distant galaxies ever being visible to us: Expansion of the Universe, will light from some galaxies never reach us? I came across a few concepts that were quite surprising to me. In particular:   * Movement faster than the speed of light   * The big bang was not an explosion outwards from a single point. Granted I am just a rank beginner and self-studier, yet I did study a QM course from Oxford, have read several sets of notes on SR, and readily went through the first hundred pages of \"Student Friendly QFT.\" Yet I have never encountered these notions. My question is where does one acquire this type of information. Not necessarily the technicalities (of, e.g., GR); but just a correct awareness."}
{"id":"44584","title":"How exactly are the different motions of only one kind of fundamental string assumed to give rise to the spectrum of elementary particles we observe?","text":"In string theory, it is assumed that all particles can be described as quanta corresponding to the excitations of only one kind of fundamental string. How can in principle the different motion patterns of one kind of string give rise to the whole particle zoo in the standard model? How are the different properrties, that characterize an elementary particle such as their   * mass   * spin   * charges encoded in the allowed motion pattterns of the string? (I know that the vacuum corresponding to our standard model can not be uniquely identified at present, but I am interested in the general concepts and ideas that should in principle give rise to the different characteristics of the particles we observe.) **Note** _This is just an attempt to extract and reask the useful part ofthis now a bit too overloaded original question._"}
{"id":"44587","title":"What is the electric current $I_{12}$ and voltage $U_{12}$ in following electric circuit?","text":"What is the electric current $I_{12}$ and voltage $U_{12}$ in following electric circuit? $I_{12}$ and $U_{12}$ are between points 1 and 2. ![picture of electric circuit of the question](http:\/\/i.stack.imgur.com\/FldEO.jpg). I have used this in constructing this electric circuit and as you can see simulator shows wrong electric current($I_{12}=0,34 A$) between points 1 and 2, but why? I mean, because both bulbs has resistivity of $10 \\Omega$ and Cell has voltage as 10 V, then should be $I_{12}=0 A$. ![Another configuration showing another value of $I_{12}$](http:\/\/i.stack.imgur.com\/nm86W.jpg) Now $ I_{12} \\approx 0$ so I think something is wrong with this simulator."}
{"id":"54528","title":"Is it really possible to walk on water?","text":"Is it really possible to walk on water or levitate in air. If not then how do some magician,s like dynemo do it simply while walking on street? And i have asked for possibility of walking on water so please exclude the high velocity concept."}
{"id":"70413","title":"Calculating the amount of dissolved gases in a liquid?","text":"The most basic situation, water at room temperature, has dissolved oxygen, CO2, and more. Why is this? How would one calculate how much gas _should_ be dissolved in any given liquid?"}
{"id":"103653","title":"How do I properly write Newton's second law for a particle with drag?","text":"> A heavy particle is projected at speed $U$ at an angle $\\alpha$ to the > horizontal. The particle is subject to air resistance which is > experimentally found to vary proportionally to the square of the speed. Show > that > > $$\\vec{\\dot{v}} = -\\frac{g}{V^2}\\lvert\\vec{v}\\rvert\\vec{v} - g\\vec{j},$$ > > where $V$ is the terminal velocity of the particle. If $\\alpha = > \\frac{\\pi}{2}$ (so that the particle is projected directly upwards), find > the maxium height reached and the time taken to reach it. What is the speed > of the particle when it returns to the horizontal? I'm having troubles solving this question, firstly I have some fundamental mis-understandings,   * if the particle is subject to air resistance which is proportional to the square of the speed then do we model newton second law as: $$\\vec{\\ddot r} = \\vec{g} - k|\\vec{v}|^2\\dfrac{\\vec{v}}{|\\vec{v}|}$$ or $$\\vec{\\ddot r} = \\vec{g} - k|\\vec{v}|^2$$? I'm assuming the former by looking as what's required but _why_? The question says it's just proportional to speed, why in the direction of the velocity?   * Secondly, why do we have to artificially put a negative sign? Does this mean that $k>0$ from now on? As for my attempt of the question I done it as follows: $$\\vec{\\ddot r} = \\vec{g} - k|\\vec{v}|^2\\dfrac{\\vec{v}}{|\\vec{v}|} = -g\\vec{j} - k|\\vec{v}|\\vec{v} $$ so we have, $\\vec{\\dot v} = -g\\vec{j} - k|\\vec{v}|\\vec{v}$ solving this I get the solution of $$ \\vec{v} = \\dfrac{-g}{k|\\vec{v}|} + \\vec{c}e^{-kt|\\vec{v}|} $$ where $\\vec{c}$ is a constant vector, now from here I get terminal velocity as $\\dfrac{-g}{k|\\vec{v}|} \\vec{j}$ but I am unsure how to get $V^2$ from here. on second thoughts if $|\\underline{v}| = V$ then I get the required result, but _why_ would the speed be the same as the terminal speed"}
{"id":"121648","title":"Question about two vehicles moving toward each other, but the answer doesn't seem correct, am I missing a key concept?","text":"**The problem statement** Vehicle A and Vehicle B are moving in opposite directions on the NJTP. Vehicle A is heading south toward atlantic city while vehicle B is heading north towards Hoboken. In situation 1 and 2 described below, at t = 0s both vehicles are at a distance of separation of 400m and are moving towards each other. Situation 1: When vehicle A is moving at a constant velocity of 30 m\/s and travels a distance of 120 m, the vehicles pass each other on the turnpike. Situation 2: When vehicle A is moving at a constant velocity of 80 m\/s and after a time of 3s, the vehicles pass each other on the turnpike. For vehicle A label variables as: Via, Vfa, aa, da,ta For vehicle B label variables as: Vib, Vfb, ab,db, tb In the ORDER INDICATED: (a) Calculate the initial velocity of vehicle B or Vib; and (b) Then calculate the acceleration of vehicle B or ab Show all work in DETAIL and keep all numerical values to the nearest hundredth value. Draw and label all pictures Note: You will receive NO CREDIT if you first calculate part (b) and then part (a) For situations 1 and 2 vib = constant ; vfb ≠ a constant value since its value changes w\/ time ab = constant **My Work** Situation 1: Da + Db = 400 Via = 30 m\/s Vfa = 30 m\/s a = 0 m\/s2 da = 120 m ta = ? Vave = d\/ t = vf + vi \/2 Vave = da\/ta ta = 4s db = 280 ta = 4s Situation 2: Via = 80 m\/s aa = 0 m\/s2 vfa = 80 m\/s ta = 3s = tb da = ? Using same equation as in situation 2, da = 240 db=160 tb = 3s Situation 1 db = vibtb + 1\/2abtb^2 280 = vib(4) + 1\/2 (ab)(4)^2 Situation 2 db = vibtb + 1\/2abtb2 160 = vib(3) + 1\/2 (ab)(3)^2 Solving for ab I get ab = 11.85 \/ vib Plugging that back into the equation I get Vib = .34 m\/s and plugging that into the 280 = (.34)(4) + 1\/2 (ab)(4)^2 = ab = 34.82 m\/s^2. I'm sorry if I'm new and breaking the rules, but this problem has really been bothering me for the last week. My friend says it's correct, but it just seems too low of an initial velocity."}
{"id":"121642","title":"Induction on a circuit with switch","text":"I have the following circuit: ![enter image description here](http:\/\/i.stack.imgur.com\/Ss7Iv.jpg) It is subject to a steady, time-invariant magnetic field which points out of the paper. At $t = 0$, the switch closes. I thought that the magnetic flux would decrease at the moment the switch closes and, by Faraday's law, cause the voltmeter to change its reading. However, my textbook says that the voltmeter won't change. How does one arrive at this conclusion?"}
{"id":"91567","title":"How can I integrate in $\\mathrm{d}t$ the cube of the harmonic oscillator propagator?","text":"I'm redoing the calculations of \"Point Canonical Transformations in the path integral\", by Gervais and Jevicki; while doing so I stumbled in integrals like $$ \\int \\mathrm{d}t \\, \\Delta_F^3(t) = -\\frac{1}{12} \\frac{i}{\\omega^4}, \\\\\\ \\int \\mathrm{d}t \\, \\dot{\\Delta}_F^2(t) \\Delta_F(t) = \\frac{1}{12} \\frac{i}{\\omega^2} $$ where $$ \\Delta_F(t) = \\int \\frac{\\mathrm{d}\\nu}{2\\pi} \\, e^{i\\nu t} \\frac{i}{\\nu^2-\\omega^2+i \\varepsilon}. $$ I tried various methods, without success. For instance, I integrated explicitly $\\Delta_F(t)$ and after some calculations I found $\\Delta_F(t) = \\frac{1}{2\\omega} \\cos(\\omega t)$; but this expression, inserted in the previous ones, does not make the integrals converge. Maybe in these integrals it's important to integrate in $\\mathrm{d}t$ before doing the $\\mathrm{d}\\nu$ integration; but when trying to do so what I obtain is terribly complicated."}
{"id":"122591","title":"Computational package to find the ground state of a particle in 3D domain","text":"I am developing a numerical algorithm to find the ground state of a Hermitian matrix. Obvious applications are quantum many-body systems and particles in various potentials. I am a little stuck with the comparison for the latter example. Could anyone recommend a numerical package to find a ground state of a particle in a simple 3D domain, say, L-shape? UPDATE: Having read the rules of phys.SE more accurately, I want to emphasise that the question is not about numerical algorithms, but about tools of the trade that people in physics community use and recommend."}
{"id":"123644","title":"Electrostatics - Inserting a brass plate between two charges","text":"The question is: if I were to insert a brass plate between two charges, what will happen to the force between the charges? Would it increase, decrease or stay the same?      Does the brass plate increase the value of permittivity of the medium and therefore the force decreases?      The correct answer is that it will increase. But I do not understand how."}
{"id":"119369","title":"Have I calculated Angular Acceleration correctly?","text":"I am teaching myself basic mechanics from a standing start. I am trying to understand Angular Acceleration and have set myself a problem to solve. My answer 'feels' wrong, so I'd like some help to understand if I've misunderstood, or miscalculated anything. I've taken many liberties with rounding, please ignore, this is more about the basic process\/theory than accuracy. Thanks in advance!! **Problem** > _An object is travelling around a circle with a radius of 40m. It's speed at > (A) is calculated as 50mph. 5 seconds later, it's speed at (B) is calculated > as 40mph. Determine the Angular Acceleration._ **Basic conversions** Circumference = $2\\pi r = 251\\text{ m}$ Velocity (A) = $22\\text{ m\/s}$ Velocity (B) = $18\\text{ m\/s}$ **Angular Velocity at (A)** 251 \/ 22 = 11.4. Therefore one full revolution would take 11.4 seconds. $$\\omega = \\theta\/t$$ $$\\omega = 2\\pi \/t$$ $$\\omega = 2\\pi\/11.4$$ $$\\omega = 0.55 \\text{ rad\/s}$$ **Angular Velocity at (B)** $251\/18 = 13.9$. Therefore one full revolution would take $13.9$ seconds. $$\\omega = \\theta\/t$$ $$\\omega = 2\\pi \/t$$ $$\\omega = 2\\pi\/13.9$$ $$\\omega = 0.45\\text{ rad\/s}$$ **Angular Acceleration** $$\\alpha = \\frac{d\\omega}{dt} $$ $$\\alpha = \\frac{0.45 - 0.55}{5 - 0}$$ $$\\alpha = -0.1 \/ 5$$ **Answer to Problem** $\\alpha = -0.02\\text{ }\\mathrm{rad\/s^2}$"}
{"id":"69037","title":"What do mathematicians mean by Navier Stokes existence and smoothness problem?","text":"I still don't know what mathematicians mean by Navier-Stokes existence and smoothness. Since there is a reward for proving it, it seems important to them. (in past several months I've read online articles on this topic). Physically, what do we get from such a proof? Is it possible that we _don't_ have existence and uniqueness mathematically, but that our physics still \"works\" somehow? Also, is there a consensus on whether, based on physical intuition (we are modelling real things after all), there _must_ be existence and uniqueness to the pure math problem?"}
{"id":"129918","title":"Unknown Function in the Tolman-Bondi-de Sitter Metric","text":"I've been working with some dust solutions in General Relativity, practicing calculating the Riemann curvature tensor, and I came across an odd metric: the Tolman-Bondi-de Sitter metric. A quick internet search (to supplement the book I'm reading) can tell you that it describes spherical dust, while accounting for a cosmological constant. It's a pretty simple solution, with a line element of the form $$ds^2=dt^2-e^{-2\\Psi(t,r) } dr^2-R^2 (t,r)d\\theta^2-R^2 (t,r) \\sin^2⁡\\theta d\\phi^2$$ There's one term in there that has me a bit befuddled, and that I can't find an explanation for in a book or on the Internet: $\\Psi(t,r)$ At first, I thought it had to be a simple wavefunction, but after looking at it more, I'm not quite sure. What is it, and what is its significance in the metric?"}
{"id":"11063","title":"Can quantum annealing be used for factorization?","text":"It is known that there is a famous quantum factorization algorithm by Peter Shor. The algorithm is thought to be suitable only for quantum gate computer. But can a an adiabatic quantum computer especially that which is capable of quantum annealing be used for factorization? I am asking this because it seems that Geordie Rose claims in his blog that they have a quantum factorization algorithm that is somehow \"better than Shor\". But the details are unavailable as of now."}
{"id":"36362","title":"Relativistic contraction for a wave packet and uncertainty on momentum","text":"Consider an electron described by a wave packet of extension $\\Delta x$ for experimentalist A in the lab. Now assume experimentalist B is flying at a very high speed with regard to A and observes the same electron. The extension of the wave packet will appear contracted, and the uncertainty on momentum will increase. What happens when the later become larger than the electron's rest mass?"}
{"id":"114525","title":"How does hot water clean better than cold water?","text":"I had a left over coffee cup this morning, and tried to wash it out. I realized I always instinctively use hot water to clean things as it seems to work better. A google search showed other people with similar results, but this yahoo answer is a bit confusing in terms of hot water \"exciting\" dirt. What is the physical interaction between hot water and oil or a material burnt onto another vs the cold water interaction?"}
{"id":"126697","title":"Stability Group of the Poincare Group","text":"The stability group $G_\\Sigma$ is a subgroup of the Poincare group $P(1;3)$. Its generators $X$ in the **front form** leave the hypersurface $\\Sigma: x^+ = 0$ invariant. Phrased differently they satisfy the condition: $x'^{+} = x^+ + [x^+, X] = x^+ = 0$. So, for the generators $X$ of the stability group we have $$ [x^+, X]=0 . $$ How can I relate the matter of leaving the hypersurface invariant with the condition: $x'^{+} = x^+ + [x^+, X] = x^+ = 0$ ?"}
{"id":"61317","title":"To which real densities do carrier densities in the semi-classical model of a crystal correspond?","text":"In the semi-classical model of a crystal in solid state physics, electrons and holes are assigned effective masses that account for their different mobilities. E.g. in silicon, holes have a bigger mass than electrons. This results in different electron\/hole densities as the temperature increases. Why do those different densities not violate conservation of charge? In what way do these imaginary electrons\/holes correspond to real particles, in what way do they not?"}
{"id":"28848","title":"Is a 125 GeV Higgs large (or small) for the MSSM?","text":"As far I as I know, and from naturalness considerations, a 125 GeV Higgs mass is rather large for the MSSM. This is because in the MSSM $$m_h^2 \\lesssim M_z^2 \\cos^22\\beta + \\Delta$$ where $\\Delta$ represents top\/stop loop corrections to the Higgs. It takes the form $$\\Delta \\sim \\ln(\\frac{m_\\text{stop}}{m_\\text{top}}) + \\text{mixing}$$ Moreover, the Z mass is determined by the Higgs mass parameter, $m_{H_u},$ and $\\mu$ (in the large $\\tan\\beta$ limit. i.e. $\\tan\\beta \\geq 10$). And $m_{H_u} \\sim - m_\\text{stop}$, so the larger $m_\\text{stop}$ the larger the fine- tuning in the MSSM. Now, only by large $m_\\text{stop}$ (and mixing) one can reach a Higgs mass greater than LEP\/LHC bounds. But this is already associated with large fine-tuning as I said. So, in this view, a 125 GeV Higgs mass is large for the MSSM. (At least for the cMSSM.) However, I realised that some people think 125 GeV is much less than what the MSSM predicts! Now this I don't understand at all. So could someone please explain what this view is based on?"}
{"id":"82597","title":"Why is momentum (instead of something else) the canonical conjugate of position?","text":"Why did nature decide to make conjugate of position to be momentum? Since energy and position do not commute, why not energy? What determines the pairing of time with energy and momentum with position?"}
{"id":"82595","title":"How does time relate to mass and velocity","text":"I understand that the larger the mass the greater gravity is and the slower time is, as well the faster an object is traveling the slower time passes. My question is that since the faster an object travels the more mass it has, is the increase in mass the reason for the change in time, or is it the velocity?"}
{"id":"82596","title":"A satellite in orbit fires it's engines for a short interval. Is the new orbit closer or further away?","text":"A satellite is in a circular orbit when its engines turn on to exert a small force in the direction of the velocity for a short time interval. Is the new orbit further or closer to the Earth? The solution is that the new orbit is further away (which is also intuitive) and is justified by stating that there is a positive increase in the total energy which is given by the formula: $$\\:E_{T}=-\\frac{GMm}{2r}$$ And that states that an increase in the total energy would result in a larger radius. Thus, the new orbit is further away. However, the problem arises when I look at this equation which relates the speed with the radius: $$\\:v^2=\\frac{GM}{r}$$ Since the small force was in the direction of the velocity, an increase in veloctiy should result in a DECREASE in the radius meaning that the new orbit is closer in according to this equation. Why is it that the first equation is correct to use, while the second one is wrong? Why is the second equation not working here? Thanks!"}
{"id":"10811","title":"Do apparent event horizons have Hawking radiation?","text":"As I understand it, black holes have an absolute event horizon and an apparent horizon specific an observer. In addition to black holes, an apparent horizon can come from any sustained acceleration. Firstly, there is the \"particle horizon\" of the universe, which is the furthest co-moving distance from which light can reach us before the acceleration of the expansion of the universe prohibits it, then there is the case of a constantly accelerating observer in just regular flat space. My zeroth question is if the Hawking radiation from a black hole could be said to come from the apparent or absolute horizon - which is it? Obviously the former would be specific to each observer. For the case of the universal acceleration, dark energy, cosmological constant (please help me with terminology as appropriate): One could ask \"do we observe radiation from the horizon?\", for which we already have an available answer in the form of the cosmic microwave background (CMB). But I am curious if this would match the temperature and intensity you would expect if you treated the edge of the universe as a black hole horizon. Are these two be equivocal in some sense? Finally, I'm most interested to ask what would occur for the case of a constantly accelerating observer. An apparent horizon exists behind the observer beyond which light may never reach. Could it be possible the Hawking radiation is observed as coming from behind? ![Accelerating Observer](http:\/\/i.stack.imgur.com\/IJRj1.png) **Could P possibly observe Hawking radiation coming from behind?** Granted, if you took the simple version of the temperature of the Hawking radiation ($T = \\frac{\\hbar c^3}{8 \\pi G M k_B}$) and plugged in some super large mass, you will correspondingly get some super small temperature which will not agree with the 2.7 K of the CMB. Of course, the two cases I'm discussing do not have any definable mass that can be used. I would imagine that in the case of a simply accelerating particle, a greater acceleration would correspond with greater Hawking radiation, which is why it's not an issue for normal stuff, but if we accelerated something small with great force, could it observe radiation from that (false) event horizon?"}
{"id":"116962","title":"Dipole and multipole bound state anions: Do these bound electrons behave exactly like conventional electrons in the molecular orbitals?","text":"Recently, I read about dipole and multipole bound anions. Dipole bound anions are those, if I understood correctly, when an electron is attached electrostatically on a neutral molecule which is polar. If the molecule is having a quadrupole moment, the anion is said to be a quadrupole bound anion. These bound electrons will have less binding energy compared to the electrons in molecular orbital. My question is, these dipole, quadrupole and other multipole bound electrons behave like normal electrons in orbitals except that it has less binding energy or different? To be more clear, do these electrons have specific orbitals with orbital angular momentum or not? If so, are these orbitals like normal molecular orbitals (S P D F)? Can we assign electronic states to these electrons? Also, is it possible to excite any electron in the molecule (say valance electron) to this bound state (dipole or any multipole bound state)?"}
{"id":"116967","title":"Minkowski metric and definition of coordinate differentials?","text":"This is probably a really silly confusion I have about the definition of “coordinate differentials”, which I thought were things like $dx,dy,dz$ etc. The Minkowski line element $$ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}$$ defines the Minkowski metric $$\\left[\\eta_{\\mu\\nu}\\right]=\\left(\\begin{array}{cccc} c^2 & 0 & 0 & 0\\\\\\ 0 & -1 & 0 & 0\\\\\\ 0 & 0 & -1 & 0\\\\\\ 0 & 0 & 0 & -1 \\end{array}\\right).$$ Using index notation, the line element can be written as $ds^{2}=\\eta_{\\mu\\nu}dx^{\\mu}dx^{\\nu}$. In textbooks I have seen the terms $dx^{\\mu},dx^{\\nu}$ called “coordinate differentials”, which seems OK except $dx^{0}=cdt$. I realise this is trivial, but is it correct to call $cdt$ a “coordinate differential”? To me it looks like a coordinate differential $dt$ multiplied by $c$."}
{"id":"57128","title":"reversible cellular automata","text":"Let's suppose a cellular automaton has a value $b(r,t)$ belongs to $Q$ at site $r$ and time $t$, where $Q$ is the set of possible states at each site. Let $N(r, t)$ be the values of the states of all the sites in some (e.g., Von Neumann or Moore) neighborhood of $r$ at time $t$, taken in some canonical order. The cellular automaton rule is then $$b(r, t + 1) = F(N(r, t))$$ where $F$ is the update function. Let us suppose that $Q = \\\\{0, 1\\\\}$ so that we have a bit at each site. One way to construct reversible cellular automata (RCA) is to look at rules of the form $b(r, t + 1) = F(N(r, t)) \\text{ xor } b(r, t − 1)$ To see why this is reversible, how to solve for $b(r, t −1)$ in terms of $b(r, t)$ and $b(r +1, t)$. Does it mean that this obeys a condition even stronger than reversibility? One apparent disadvantage of this approach seems to be that computation of $b(r, t+1)$ requires knowledge of the state at time $t−1$ in addition to that at time $t$. Is there way to fix this problem by adding extra state at a site so that the state at time $t + 1$ depends only on that at time $t$? Are there other functions besides $\\text{xor}$ that could be used to create RCA in the above fashion for $Q = \\\\{0, 1\\\\}$?"}
{"id":"65761","title":"If $SU(2)_{L} \\times U(1)_{Y}$ breaks to $U(1)_{em}$ when a non-zero mass for the Higgs boson is chosen, why do we still have weak interactions?","text":"As I understand it, when we say that the $SU(2)_{L} \\times U(1)_{Y}$ is broken via the Higgs mechanism, this is because the symmetry acts on the Higgs mass in a way that would change it's value. If we want to pick a particular model we need to pick a fixed value of the Higgs mass, and this is only possible if we say $SU(2)_{L} \\times U(1)_{Y}$ is broken to $U(1)_{em}$. The Lagrangian is always invariant to $SU(2)_{L} \\times U(1)_{Y}$ (even if the Higgs mass changes under $SU(2)_{L} \\times U(1)_{Y}$ the Lagrangian is such that it remains invariant). The symmetry breaking is just neccesary in choosing a theory with one particular value of the Higgs vev. Real life does appear to have a fixed Higgs mass so we require that $SU(2)_{L} \\times U(1)_{Y}$ breaks to $U(1)_{em}$. But then aren't we saying that observable physics is described by $SU(3)_{C} \\times U(1)_{em}$, why does the weak interaction still work in our universe that contains a fixed Higgs mass?"}
{"id":"10670","title":"What nonlinear deformations will a fast rotating planet exhibit?","text":"It is common knowledge among the educated that the Earth is not exactly spherical, and some of this comes from tidal forces and inhomogeneities but some of it comes from the rotation of the planet itself. The deformation from the rotational effect makes it more oblate spheroid-like, or as I would prefer, \"like a pancake\". Here is one site illustrating the behavior, and image: ![Earth's shape from rotational distortion from Mathematical Imagery](http:\/\/i.stack.imgur.com\/gV7Ay.jpg) Literature exists detailing the mathematical expectations for a rotating planet using just hydrostatic forces, for example, see Hydrostatic theory of the earth and its mechanical implications. I like to imagine a ball of water in space held together by its own gravity. I also don't want to deviate from consideration of only hydrostatic (and gravitational) forces because I think it is sufficient for this discussion. It would seem that the solution of the described problem is in terms of a small change in radius as a function of the azimuth angle, or z-coordinate if you take the axis of rotation to be the z-axis. This is using rotational symmetry. In other words, Earth's deformation due to rotation does not depend on longitude. I want to ask about the extreme case. Imagine a planet rotating so fast that it is a very thin pancake. What will occur in this case? I am curious:   * Will the center hollow out, creating a donut shape?    * Will it break up into a multi-body system? It seems to me that it would be logical for the high-rotation case to break up into 2 or more separate bodies. The reason is that a 2 body system is stable an can host a very large angular momentum. But would it be an instability that leads to this case? When would such an instability occur and could a rotating planetary body deform in a different kind of shape from the beginning, such as a dumbbell-like shape, which would transition into a 2-body system more logically than the pancake shape? ![Would THIS happen?](http:\/\/i.stack.imgur.com\/aHZ9N.jpg) image link To sum up, how would a pancake shape transition into a dumbbell shape? Or would it? What are the possibilities for the described system?"}
{"id":"61865","title":"Tunneling and transmission","text":"Lets say we have a tunelling problem in the picture, where $W_p$ is a finite potential step: ![enter image description here](http:\/\/i.stack.imgur.com\/aFgQ4.png) If particle is comming from the left a general solutions to the Schrödinger equations for sepparate intervals I, II and II are: \\begin{align} \\text{I:}& & \\psi_1 &= \\overbrace{A e^{i\\mathcal L x}}^{\\psi_{in}} + \\overbrace{Be^{-i \\mathcal L x}}^{\\psi_{re}}& \\mathcal L &= \\sqrt{\\tfrac{2mW}{\\hbar^2}}\\\\\\ \\text{II:}& & \\psi_2 &= C e^{\\mathcal K x} + De^{-\\mathcal K x}& \\mathcal K &= \\sqrt{-\\tfrac{2m(W-W_p)}{\\hbar^2}}\\\\\\ \\text{III:}& & \\psi_3 &= \\underbrace{E e^{i \\mathcal L x}}_{\\psi_{tr}}& &\\\\\\ \\end{align} Where $\\psi_{in}$ is an incomming wave, $\\psi_{re}$ is a reflected wave and $\\psi_{tr}$ is transmitted wave. I used the boundary conditions and got a system of 4 equations: \\begin{align} {\\tiny\\text{boundary}}&{\\tiny\\text{conditions at x=0:}} & {\\tiny\\text{boundary conditions}}&{\\tiny\\text{at x=d:}}\\\\\\ A + B &= C + D & Ce^{\\mathcal K d} + De^{-\\mathcal K d} &= E e^{i \\mathcal L d}\\\\\\ i \\mathcal L A - i \\mathcal L B &= \\mathcal KC - \\mathcal K D & \\mathcal K C e^{\\mathcal K d} - \\mathcal K D e^{-\\mathcal K d}&= i \\mathcal L E e^{i \\mathcal L d} \\end{align} So now i decided to calculate coefficient of transmission $T$: \\begin{align} T &= \\dfrac{|j_{tr}|}{|j_{in}|} \\\\!=\\\\! \\Bigg|\\dfrac{\\dfrac{\\hbar }{2mi}\\\\! \\left( \\dfrac{d\\overline{\\psi}_{tr}}{dx}\\, \\psi_{tr} - \\dfrac{d \\psi_{tr}}{dx}\\, \\overline{\\psi}_{tr} \\right)}{\\dfrac{\\hbar}{2mi} \\\\!\\left( \\dfrac{d\\overline{\\psi}_{in}}{dx}\\, \\psi_{in} - \\dfrac{d\\psi_{in}}{dx}\\, \\overline{\\psi}_{in} \\right) }\\Bigg| \\\\!=\\\\! \\Bigg|\\dfrac{\\frac{d}{dx}\\big(\\overbrace{Ee^{-i\\mathcal L x}}^{\\text{konjug.}}\\big) Ee^{i\\mathcal L x} - \\frac{d}{dx} \\left( Ee^{i\\mathcal L x}\\right)\\\\! \\overbrace{Ee^{-i\\mathcal L x}}^{\\text{konjug.}}}{ \\frac{d}{dx}\\big(\\underbrace{Ae^{-i\\mathcal L x}}_{\\text{konjug.}}\\big) Ae^{i\\mathcal L x} - \\frac{d}{dx} \\left( Ae^{i\\mathcal L x}\\right)\\\\! \\underbrace{Ae^{-i\\mathcal L x}}_{\\text{konjug.}}}\\Bigg|\\\\! = \\nonumber\\\\\\ &=\\Bigg|\\dfrac{-i\\mathcal L Ee^{-i\\mathcal L x} E e^{i \\mathcal L x} - i\\mathcal L E e^{i \\mathcal L x} Ee^{-i \\mathcal L x}}{-i \\mathcal L A e^{-i\\mathcal L x} Ae^{i \\mathcal L x} - i \\mathcal L A e^{i \\mathcal L x}Ae^{-i \\mathcal L x} }\\Bigg|=\\Bigg|\\dfrac{-i\\mathcal L E^2 - i\\mathcal L E^2}{-i \\mathcal L A^2 - i \\mathcal L A^2}\\Bigg|=\\Bigg|\\dfrac{-2 i \\mathcal L E^2}{-2i\\mathcal L A^2}\\Bigg| = \\frac{|E|^2}{|A|^2} \\end{align} It accured to me that if out of 4 system equations i can get amplitude ratio $E\/A$, i can calculate $T$ quite easy. Could anyone show me how do i get this ratio?"}
{"id":"60197","title":"calculating work done by friction","text":"I want to calculate the work done by friction if the length $L$ of uniform rope on the table slides off. There is friction between the cord and the table with coefficient of kinetic friction $\\mu_k$. $$ W = \\int F \\cdot d \\vec{s}$$ I think it would be: $$ W_{fr} = \\frac M L g \\int_{0}^{L} dx$$ But the solutions (which could be mistaken) say: $$ dW_{fr} = \\mu_k \\frac M L g \\, x \\, dx$$ which is then integrated. Should there be an $x$ in the integral? I don't think there should be because you are summing up over an infinitesimal displacement $dx$ and the force of friction is not proportional to the displacement at any instant (I think). ![enter image description here](http:\/\/i.stack.imgur.com\/lKzuO.png)"}
{"id":"53611","title":"Is the photon energy required to cause an atomic transition $\\Delta E+\\Delta KE$, where $\\Delta E$ is the \"transition energy\"?","text":"An atom \"at rest\" can absorb a photon, and while some of this energy goes into increasing the energy level of the electron, momentum must be conserved, and so some energy must also increase the kinetic energy of the atom. The amount of $KE$ delivered to the atom is easy to calculate. Therefore, is the energy to cause a transition $\\Delta E+\\Delta KE$, where $\\Delta E$ is the \"transition energy\"? So that no commenters cover old ground, please see this question."}
{"id":"63425","title":"What is the real-world significance of the Bekenstein bound?","text":"The Bekenstein bound sets the maximum amount of information that can be contained in a region of space\/energy, and is usually referred to in the same way as computer storage density: > For example, a single hydrogen atom, if it were to code as much information > as permitted by the Bekenstein Bound, would code about 4×106 bits of > information, since the hydrogen atom is about one Ångström in radius, and > has a mass of about 1.67×10−27 kilograms. (source) > Nature permits a surprising amount of information to be encoded before the > Bekenstein bound is reached. For example, a hydrogen atom can encode about 1 > Mb of information — most of a floppy disk. (source) > The \"Bekenstein bound\" leaves room for a million bits in a hydrogen atom > (source) But what does this really mean? How could _any_ information be stored in a hydrogen atom?"}
{"id":"63428","title":"In solar cells, do photons break apart electron-hole pairs, or create them?","text":"Some sources say that when a photon hits the PV cell, it breaks apart electron-hole pairs. Other sources say that photons _create_ electron-hole pairs. Can anyone explain which one is right? I've read several explanations of what goes on in the solar cell, but they don't seem very clear. To me, who has little prior knowledge, a few aspects of the many different explanations seem to contradict each other. I don't know if it's because some of the explanations said things a certain way for simplicity's sake, but this part seems too significant to ignore. For example, http:\/\/www.solarenergyexperts.co.uk\/buyersguides\/photovoltaic- glass-how-does-it-work\/: \"When photons (light particles) from the sun hit the cell, the energy breaks up the paired particles. The freed electrons go into the n-type layer, while the holes go down into the p-type layer.\" science.howstuffworks.com\/environmental\/energy\/solar-cell3.htm : \"When light, in the form of photons, hits our solar cell, its energy breaks apart electron- hole pairs.\" solarjourneyusa.com\/bandgaps.php has a hole section called \" _Generation_ of electron-hole pairs\" This one basically seems to be talking about the same thing, but doesn't really mention electron-hole pairs, and the explanation seems more confusing to me, but anyway, http:\/\/www2.pv.unsw.edu.au\/nsite- files\/pdfs\/UNSW_Understanding_the_p-n_Junction.pdf: \"...broken bonds created by the light act as holes ... and these holes are also free to move throughout the material. Electrons and holes created in this way are physically near each other: for every electron excited by the light there is a corresponding hole generated. These electrons and holes can remain excited only for a short period of time. In a process called recombination, excited electrons stray too close to holes and the two fall back into bonded positions.\" This one seems to be saying that photons break apart the bonds between the electron and its atom, which creates a free electron and a hole, i.e. another one saying that an electron-hole pair is created. solarcellcentral.com\/junction_page.html : \"When photons hit the solar cell, free electrons attempt to unite with holes in the p-type layer.\" www.solarenergy.net\/Articles\/how-photovoltaic-cells-work.aspx : \"When enough photons are absorbed by the negative layer of the photovoltaic cell, electrons are freed from the negative semiconductor material.\" So what actually happens when light hits a solar cell?"}
{"id":"90298","title":"Can sand falling in a floating hourglass cause it to sink? (Follow up to hourglass question)","text":"This is related to this question:Does the weight of an hourglass change when sands are falling inside? At Brigham Young University, there is a display consisting of a sealed off liter bottle with a sunken sealed off hourglass. When one turns over the bottle, the hourglass floats as the sands flow down. Eventually, the hourglass reaches some transition, and precipitately changes from floating to sinking. It has a sign next to it saying that it's for a fishing lure,and that you should not reveal the answer if you get it. What causes it to change from floating to sinking so quickly?"}
{"id":"68067","title":"How general relativity gets to an inverse-square law","text":"I understand that a general interpretation of the $1\/r^2$ interactions is that virtual particles are exchanged, and to conserve their flux through spheres of different radii, one must assume the inverse-square law. This fundamentally relies on the 3D nature of space. General relativity does not suppose that zero-mass particles exchanged. What is the interpretation, in GR, of the $1\/r^2$ law for gravity? Is it come sort of flux that is conserved as well? Is it a postulate? Note that I am not really interested in a complete derivation (I don't know GR enough). A physical interpretation would be better. Related question: Is Newton's Law of Gravity consistent with General Relativity?"}
{"id":"60736","title":"Potential step and its transmission \/ reflection","text":"Lets say we have a potential step with regions 1 with zero potential $W_p\\\\!=\\\\!0$ (this is a free particle) and region 2 with potential $W_p$. Wave functions in this case are: \\begin{align} \\psi_1&=Ae^{i\\mathcal L x} + B e^{-i\\mathcal L x} & \\mathcal L &\\equiv \\sqrt{\\frac{2mW}{\\hbar^2}}\\\\\\ \\psi_2&=De^{-\\mathcal K x} & \\mathcal K &\\equiv \\sqrt{\\frac{2m(W_p-W)}{\\hbar^2}} \\end{align} Where $A$ is an amplitude of an incomming wave, $B$ is an amplitude of an reflected wave and $D$ is an amplitude of an transmitted wave. I have sucessfuly derived a relations between amplitudes in potential step: \\begin{align} \\frac{A}{D} &= \\frac{i\\mathcal L-\\mathcal K}{2i\\mathcal L} & \\frac{A}{B}&=-\\frac{i \\mathcal L - \\mathcal K}{i \\mathcal L + \\mathcal K} \\end{align} I know that if i want to calculate transmittivity coefficient $T$ or reflexifity coefficient $R$ i will have to use these two relations that i know from wave physics: \\begin{align} T &= \\frac{j_{trans.}}{j_{incom.}} & R &= \\frac{j_{trans.}}{j_{incom.}} \\end{align} * * * **Question 1:** I know that $j = \\frac{dm}{dt} = \\frac{d}{dt}\\rho V \\propto \\rho v \\propto \\rho k$ But what is a density $\\rho$ equal to? **Question 2:** I noticed that $\\mathcal L$ and $\\mathcal K$ are somehow (i dont know how) connected to the wavevector $k$ from the equation in 1st question but how? How can i make it obvious?"}
{"id":"94628","title":"equation of motion for the scalar field via variational principle in general relativity","text":"I would like to find the equation of motion for the scalar field $\\phi$ by varying the following action in General Relativity. Special Relativity: $$ S = -\\tfrac{1}{2}\\int d^4\\xi\\, \\eta^{ab} \\partial_a \\phi\\partial_b\\phi $$ General Relativity: $$ S = -\\tfrac{1}{2}\\int d^4x \\sqrt{g}\\, g^{\\mu\\nu} \\partial_{\\mu} \\phi\\partial_\\nu \\phi $$ I was able to get the correct equation using the covariant derivatives. Since they are constant with respect to the metric partial integration works and one obtains $ \\Box\\phi = 0$. But since for the scalar field the covariant and the partial derivates are the same I wanted to vary the action with the partial derivatives. Note: $\\phi \\rightarrow \\phi + \\delta\\phi$ $$ \\begin{align} \\delta S &= -\\tfrac{1}{2}\\int d^4x \\sqrt{g}\\, (g^{\\mu\\nu} \\partial_{\\mu} \\delta\\phi\\partial_\\nu\\phi + g^{\\mu\\nu} \\partial_{\\mu} \\phi\\partial_{\\nu} \\delta\\phi ) \\\\\\ % &=-\\int d^4x \\sqrt{g}\\, g^{\\mu\\nu} \\partial_{\\mu} \\delta\\phi\\partial_\\nu\\delta\\phi \\\\\\ % &=\\int d^4x \\, \\partial_\\nu(\\sqrt{g}\\,g^{\\mu\\nu}\\partial_\\mu\\phi)\\delta\\phi \\end{align} $$ The third line one obtains after partial integration. But at this point I'm stuck. I know the following definition: $$ \\Box\\phi = \\tfrac{1}{\\sqrt{g}}\\partial_\\mu(\\sqrt{g}g^{\\mu\\nu}\\partial_\\nu\\phi) $$ But I'm not able to convert my obtained solution into this. Thanks for your help!"}
{"id":"60733","title":"Why does a temperature increase on a fixed volume increase entropy?","text":"I heard that this statement is correct. However, it seems odd to me. The number of possible microstates is still the same, so isn't the entropy constant?"}
{"id":"60738","title":"Towing of asteroid","text":"I recently studied that NASA has planned to tow and place it in the orbit of the moon. My doubt is when asteroid is placed in the orbit near moon.since the gravitational field of earth is very high.what will it revolve around the moon or the earth. Can anyone clarify my doubt ??"}
{"id":"113605","title":"4-velocity and 4-acceleration in instantaneous rest frames","text":"I am trying to solve this problem: > _Consider a rocket moving relative to an inertial frame $\\mathcal{F}$ , such > that its worldline is given by > $$x^{\\mu}=c^2\/g(\\sinh(g\\tau\/c),\\cosh(g\\tau\/c)-1,0,0).$$ What are the > components of four acceleration relative to the instantaneous rest frame of > the rocket, $\\mathcal{F}'$?_ I (think I) understand how to do this using Lorentz transformation: $dt\/d\\tau=(1\/c) \\cdot dx^0\/d\\tau=\\cosh(g\\tau\/c)$. This is equal to $\\gamma$ and it is then straightforward to compute 3-velocity and then use the Lorentz matrix to get $$dx'^\\mu(\\tau)\/d\\tau=(c,0,0,0) \\quad ; \\quad d^2 x'^\\mu(\\tau)\/d\\tau^2=(0,g,0,0)$$ However, when I first saw this I immediately thought that by definition the 4-velocity in the instantaneous rest frame of the rocket would be (c,0,0,0) because in $\\mathcal{F}'$ the 3-velocity is zero and $\\gamma$ would be 1 and I was wondering if this is a valid reasoning. Even if it is, **why is the following wrong**? If $v'^\\mu=dx'^\\mu(\\tau)\/d\\tau=(c,0,0,0)$, then $a'^\\mu=dv'^\\mu\/d\\tau=(0,0,0,0)$ since $c$ is a constant. But this contradicts the calculations given by the Lorentz transformation and I don't understand why and given this I have no idea of **how to interpret 4-acceleration**."}
{"id":"53334","title":"Why do clocks measure arc-length?","text":"Apologies in advance for the long question. My understanding is that in GR, massive observers move along timelike curves $x^\\mu(\\lambda)$, and if an observer moves from point $x^\\mu(\\lambda_a)$ to $x^\\mu(\\lambda_b)$, then his clock will measure that an amount of time $t_{ba}$ given by the curve's arc length; $$ t_{ba} = \\int_{\\lambda_a}^{\\lambda_b}d\\lambda \\sqrt{-g_{\\mu\\nu}(x(\\lambda))\\dot x^\\mu(\\lambda)\\dot x^\\nu(\\lambda)} $$ will have elapsed where $g_{\\mu\\nu}$ is a metric on spacetime with signature $(-,+,+,+)$. **Why is this so?** * * * Here is how I would attempt to justify this fact in special relativity with $g_{\\mu\\nu} = \\eta_{\\mu\\nu}$. Consider an inertial observer $O$ in $\\mathbb R^{3,1}$, and suppose that this observer sees a clock, which I'll call observer $O'$ moving around on a curve $x^\\mu(\\lambda)$. If $O'$ were also an inertial observer, then given any event with coordinates $x^\\mu$ as measured by $O$, observer $O'$ would measure the coordinates of the event to be $x'^\\mu = \\Lambda^\\mu_{\\phantom\\mu\\nu} x^\\nu + x_0^\\mu$ for some Lorentz transformation $\\Lambda$. If $O'$ is not inertial, then this is no longer true, and there is some more complicated family of transformations, say $T_\\lambda$ between events as seen by both observers. I would argue, however, that if we were to partition the interval $[\\lambda_a, \\lambda_b]$ into a large number $N$ of intervals $I_1=[\\lambda_a, \\lambda_i], \\dots, I_N=[\\lambda_{N-1}, \\lambda_b]$ with $\\lambda_n = \\lambda_a+n\\epsilon_N$ and $\\epsilon_N=(\\lambda_b-\\lambda_a)\/N$, then on each interval $I_n$, $O'$ is approximately an inertial observer in the sense that $$ T_{\\lambda_n} = P_n + \\mathcal O(\\epsilon_N), \\qquad (\\star) $$ for some Poincare transformation $P_n$. Then we would note that since $O'$ is stationary in his own reference frame, he measures his worldline to have the property $\\dot x'^\\mu(\\lambda) = (\\dot t(\\lambda), \\mathbf 0)$ so that $$ I_{ba}=\\int_{\\lambda_a}^{\\lambda_b}d\\lambda \\,\\sqrt{-\\eta_{\\mu\\nu}\\dot x'^\\mu\\dot x'^\\mu} = \\int_{\\lambda_a}^{\\lambda_b} d\\lambda \\, \\sqrt{\\dot t^2} = t(\\lambda_b) - t(\\lambda_a) = t_{ba} $$ On the other hand the integral on the left can be written as a Riemann sum using the partition above, and we can invoke ($\\star$) above to get \\begin{align} I_{ba} &= \\lim_{N\\to\\infty}\\left[\\sum_{n=1}^N \\epsilon_N\\sqrt{-\\eta_{\\mu\\nu}\\dot x'^\\mu(\\lambda_n)\\dot x'^\\nu(\\lambda_n)}\\right] \\notag\\\\\\ &= \\lim_{N\\to\\infty}\\left[\\sum_{n=1}^N \\epsilon_N\\sqrt{-\\eta_{\\mu\\nu}\\dot x^\\mu(\\lambda_n)\\dot x^\\nu(\\lambda_n)} + \\mathcal{O}(\\epsilon_N^2)\\right] \\notag\\\\\\ &= \\int_{\\lambda_a}^{\\lambda_b}d\\lambda \\,\\sqrt{-\\eta_{\\mu\\nu}\\dot x^\\mu\\dot x^\\mu} \\end{align} Combining these two computations gives the desired result. **How do others feel about this argument?** I'm not completely comfortable with it because of the assumption $(\\star)$ I made on $T_\\lambda$. I imagine that in GR a similar argument could be made by invoking local flatness of the metric."}
{"id":"427","title":"Nature of gravity: gravitons, curvature of space-time or both?","text":"General relativity tells us that what we perceive as gravity is curvature of space-time. On the other hand (as I understand it) gravity can be understood as a force between objects which are exchanging (hypothetical) virtual particles called gravitons, similar to the way electromagnetic forces are due to objects exchanging virtual photons? At least at first glance, the two concepts seem mutually exclusive. Is there a description of gravity which includes both, or is this contradiction one of the problems in combining GR with quantum mechanics?"}
{"id":"75457","title":"How do gravitons impact on general relativity?","text":"As I'm reading about GR a lot lately, I was wondering: how do gravitons (if they exist ofc) impact the general relativity? Since in GR, when we look at particles moving in space-time, we are only looking from geometric point of view so to say. Since gravity is represented with curvature of space-time via Einstein equations, we don't say gravity is a force that influences on bodies, gravity is just curvature affecting the bodies. So if there is a graviton (gravitons) which would be mediators of gravity as a force within or not within the Standard Model, how would this be reconciled with the view of gravity as a curvature of space and time? I would guess that this kinda question was asked by some scientists and answered, but I never really read anything on it. I don't even remember seeing gravitons mentioned in standard books about GR. Are there any explanations about it?"}
{"id":"102206","title":"How do I calculate stopping power?","text":"Malcolm Gladwell made a claim in a recent talk that a sling with a stone going at 30m\/s has the same stopping power as a .45 calibre handgun. How would I verify whether or not this claim is true - even given some assumptions, like the stone has a weight of 1lb or 2 lbs or whatever. Not quite sure how to work this out. Assume that the bullet is a 185grain 0.45 calibre bullet."}
{"id":"102203","title":"How does Newton's law apply to a person throwing snow onto their snowbank at the side of their driveway?","text":"We know Newton's three laws:   1. A object at rest will remain at rest, and an object in motion will remain in motion unless a external force acts upon it.   2. If an unbalanced force acts on a object, the object will accelerate in the direction of the net force.   3. If an object $A$ exerts a force on object $B$, then object $B$ will exert a equal force to object $A$ in the opposite direction. $F_{a\\text{ on }b} = -F_{b\\text{ on }a}$ What I think is that as you throw the snow, it begins moving at a constant velocity. However, when the shovel stops moving, the snow will remain in motion, causing it to accelerate\/fly toward the snow bank, according to Newton's first law. But is the snow **accelerating** when it leaves the shovel? How does this apply to Newton's second law? Thanks!"}
{"id":"104916","title":"Self-adjointness","text":"I know I have posted this question before some time ago. But no one could help so I decided to put my problem in another background. The Schrödinger equation of a free scalar field is given by $i\\partial_{t}\\Psi[\\Phi,t]~=~\\underset{A}{\\underbrace{\\frac{1}{2}\\int d^{3}x\\left(-\\frac{\\delta^{2}}{\\delta\\Phi^{2}(\\vec{x})}+|\\nabla\\Phi|^{2}+m^{2}\\Phi^{2}\\right)}}\\Psi[\\Phi,t]$. This is the Schrödinger representation of QFT. Now I want to know, whether the operator $A$ on the r.h.s is essentially self- adjoint? Any idea or advise? My problem is how to handle the functional derivative here."}
{"id":"54643","title":"Strong interaction and the Lagrangian for electromagnetic interaction","text":"The Lagrangian for electromagnetic field has the following expression: $$ L = -\\frac{1}{c^{2}}A_{\\alpha}j^{\\alpha} - \\frac{1}{8 \\pi c}(\\partial_{\\alpha} A_{\\beta})(\\partial^{\\alpha}A^{\\beta}) $$ (I used Lorentz calibration $\\partial_{\\alpha} A^{\\alpha} = 0 $). If I add the summand $\\frac{\\mu^{2}}{8 \\pi c}A_{\\alpha}A^{\\alpha}$, I'll get an equations for field (which characterized by some 4-vector $A^{\\alpha}$ (not electromagnetic (!!!))) of strong interaction and (for static case) the expression for Yukawa potential. So what is the physical meaning of summand written above? This term is somehow characteristic of the mass of the interaction carriers, but I don't understand the physical meaning of $A_{\\alpha}A^{\\alpha}$."}
{"id":"62772","title":"A question to the theory of multiverse","text":"Guys I couldn't catch a point of multiverse theory.. Theory: If space-time goes on forever, then it must start repeating at some point, because there are a finite number of ways particles can be arranged in space and time.. Question: It seems like as if multiverse theory stands on a optical argument. We call there is a next universe because we are not seeing that since there is a speed limit of light and our optical universe is 14 billions light yrs in diameter(or whatever it be..since we can not assume our space time a sphere or 2dimensional plane).Doesn't it sound weird? If there is a space between two universe then what is the value of calling it a system of two universe, Isn't it just a single universe? Question:Is it a good idea to call there is a next universe because it is exactly alike? Question: Another question is How does the multiverse theory provides solution to the grandfather paradox? Isn't it like we fold the two dimensional spacetime graph where universe exactly alike to our universe exits??If my above argument is correct then how is it valid?? Note that I am not questioning multiverse theory this time..So please Don't give answer to this question proving my above arguments wrong..Let this be independent question.. I know I am wrong But please Make me clear."}
{"id":"62770","title":"Acceleration, velocity and speed","text":"It is given that acceleration is constant, so can we infer that average speed and velocity are the same?? Moreover, circular motion is out of the question, as the function of x(t) where x=displacement, suggests, that for any t>=0, displacement can not be zero... This is the conceptual problem I am facing in a question: My teacher was reading out the question, and it was asked only to find the avg velocity from the acceleration. She, on her own, added a part to it, asking us to also find avg speed, and then, while discussing solutions, said that a graph must be made in order to solve this...so do you think that it is absolutely necessary? Moreover, if my premise is flawed, then how can the graph even help?? Thanks in advance!"}
{"id":"62779","title":"Phys.org Spectral geometry to unite relativity and quantum mechanics, restate in laymens terms?","text":"Lingua Franca links relativity and quantum theories with spectral geometry Could someone give me a short synopsis of this article in laymens terms? What implications does this have in the physics community? Is this work ground breaking or just the start of something that might be? From what I can understand this physicist related two types of maths that we use to model the world around us. Which has been hard to do, because the types of maths are incompatible with each other. But what is this spectral geometry? How does it relate the two types of maths in relativity and quantum theory? What might those maths be? What about the two has made it hard to unite them?"}
{"id":"62778","title":"What is the difference between Cramer and Vaidman?","text":"Two very interesting new papers on arXiv last night by Lev Vaidman and friends lead me to ask about the differences between Cramer's transactional interpretation of quantum mechanics (TIQM) and the two state vector formalism (TSVF) advanced by Aharonov, Vaidman and others. At a first look, they both seem very similar to me.   The two papers:   http:\/\/arxiv.org\/abs\/1304.7474 “The past of a quantum particle”, and   http:\/\/arxiv.org\/abs\/1304.7469 “Asking a particle where it has been”.   From these two papers, you can find further references for TSVF, including its origin.   For TIQM, you can start with Wikipedia, and also trace back to the origin. http:\/\/en.wikipedia.org\/wiki\/Transactional_interpretation"}
{"id":"99590","title":"Are atoms unique?","text":"Do atoms have any uniquely identifying characteristic besides their history? For example, if we had detailed information about a specific carbon atom from one of Planck's fingerprints, and could time-travel to the cosmic event in which the atom formed, would it contain information with which we could positively identify that they two are the same?"}
{"id":"128741","title":"What is it that makes an electron maintain a distance from the positively charged nucleus?","text":"What is it that makes an electron maintain a distance from the positively charged nucleus? Why aren't electrons merely pulled into and absorbed by the nucleus ?"}
{"id":"9415","title":"Why do electrons occupy the space around nuclei, and not collide with them?","text":"We all learn in grade school that electrons are negatively-charged particles that inhabit the space around the nucleus of an atom, that protons are positively-charged and are embedded within the nucleus along with neutrons, which have no charge. I have read a little about electron orbitals and some of the quantum mechanics behind why electrons only occupy certain energy levels. However... How does the electromagnetic force work in maintaining the positions of the electrons? Since positive and negative charges attract each other, why is it that the electrons don't collide with the protons in the nucleus? Are there ever instances where electrons and protons _do_ collide, and, if so, what occurs?"}
{"id":"132216","title":"What was a second in the early universe?","text":"I have read some popsci articles and documentaries about the early universe and they often explain how various features of the universe came about and at what time. For example hydrogen atoms came about after hundreds of thousands of years. Now, the official SI time unit, the second, is based on the caesium atom's properties. How can we talk about time lengths in seconds or years if there are no atoms yet in the universe? (Or going back further where there aren't even electrons.) Clearly, there must be a way to do it. Then the next question is: why don't we use those field properties or whatever that existed before atoms to define the SI second? If atoms are not needed to get \"the beat of time\" in the early universe because something else can provide the tick then we could also use those things today instead of relying on the caesium atom for the definition of the second. I hope it's clear what I mean. Thanks!"}
{"id":"91397","title":"Quantum Quench Problem","text":"I read about the quantum quench problem in condensed matter physics. But what does really mean? Has anybody a good explanation about the origin of quantum quench problem?"}
{"id":"106450","title":"Four-current, Induced Charge and Magnetic Flux","text":"I'm studying Jackiw's \"Fractional Charge and Zero Modes for Planar Systems in a Magnetic Field\" DOI: 10.1103\/PhysRevD.33.2500 but I have difficulties at some points. One of the problems is $$\\langle j^0\\rangle=\\pm\\frac{e}{4\\pi}B$$ where $j$ is four-current, $B$ is magnetic field. How can I derive this result (and why we used only $j^0$ component)?."}
{"id":"106453","title":"Can a magnet or a magnetic field push gravity?","text":"I have been asking around at my school and at the high school and at EWU but no one can answer this question: can a magnet or a magnetic field push gravity?"}
{"id":"99625","title":"What's wrong with Schwarzschild equations?","text":"I don't know much about black holes physics and so I find the Schwarzschild equations with a few contradictions. In particular I am trying to understand this little puzzle. The Schwarzschild Newtonian gravitational field equation is expressed as follows (see http:\/\/en.wikipedia.org\/wiki\/Schwarzschild_radius): $\\frac { r^2 }{r_s} \\frac {g}{c^2} = \\frac {1}{2}$ So once a particle is close to the event horizon such that $r\\to r_s$ the equation becomes: $ r \\frac {g}{c^2} = \\frac {1}{2}$ But then at that point light cannot escape so I would think that $g\\to c\/t$ where $t=1 sec$. So the equation would now approximate to: $r = \\frac {c\/t}{2}$ But this implies that $r$ is generalized for all black holes regardless the mass. If $r$ is indeed $r\\equiv r_s$ then that contradicts the other Schwarzschild formula where $r$ depends on $m$: $r_s = \\frac {2 G M} {c^2}$ Also based on what we know about $r_s$ in massive black holes, a radius of 150,000 km is quite small. What's wrong with this picture?"}
{"id":"79794","title":"How is black hole affected by gravity of nearby bodies?","text":"I run into wikipedia articles about ergosphere of rotating black holes. What if some massive body passes nearby some black hole: is something like ergosphere produced, or is the event horizon distorted by the gravity of the passing body?"}
{"id":"23101","title":"The most challenging physical phenomena","text":"What are examples of endeavors, in the history of mankind, to understand physical phenomena with models which were proved to be incorrect later, reformed significantly, or are still under development?"}
{"id":"3534","title":"Don't heavier objects actually fall faster because they exert their own gravity?","text":"The common understanding is that, setting air resistance aside, all objects dropped to Earth fall at the same rate. This is often demonstrated through the thought experiment of cutting a large object in half, the halves of which clearly can't then fall more slowly just by being sliced in two. However, I believe the answer is that when two objects fall together, attached or not, they do \"fall\" faster than an object of less mass alone does. This is because not only does the Earth accelerate the objects toward itself but the objects also accelerate the Earth toward themselves. Considering the formula: $F_g = G m_1 m_2\/d^2$ We can see that the force of gravity is dependent on **both** the masses, not just that of the more massive object. Of course in everyday situations, we can for all practical purposes treat objects as falling at the same speed. But I'm hoping not for a discussion of practicality or what's measurable or observable, but what we think is actually happening. Am I right or wrong? What really clinched this for me was considering dropping a small Moon-massed object close to the Earth and a small Earth-massed object close to the Earth. This made me realize that falling isn't one object moving toward some fixed frame of reference, but that the Earth is just another object, and falling consists of _multiple objects mutually attracting in space_."}
{"id":"19388","title":"Radial fall in a Newtonian gravitational field","text":"Suppose an object of mass $m$ starts at rest at a radial distance $ r_0$ from a perfectly spherical mass $M$ (where $m << M$), $r_0 > R =$ radius of $M$. Can we analytically determine when $m$ will hit the surface of the $M$? In other words, can we analytically solve this initial value problem: $$ \\frac{d^2r}{dt^2} ~=~ - \\frac{GM}{r^2} ,$$ $$ \\dot{r}(0) ~=~ 0 ,$$ $$ r(0) ~=~ r_0? $$"}
{"id":"88528","title":"Do heavier objects fall faster?","text":"This question has been asked multiple times here and all over the internet yet I can't find a conclusive answer:   * Some claim that heavier objects do fall faster: Don't heavier objects actually fall faster because they exert their own gravity?   * Others claim that all objects fall at the same speed regardless of their mass: Free falling of object with no air resistance Which one is the right one?"}
{"id":"51279","title":"Gravity question","text":"> **Possible Duplicate:**   >  Don’t heavier objects actually fall faster because they exert their own > gravity? When true: The force pulling the \"heavy\" object down is greater BUT it also takes more force to accelerate a heavy object. These two effects cancel out. As I heard, the bigger mass\/size an object the bigger attraction force between each other, then, why the big mass object did not accelerate faster due to the attraction force is greater than light object."}
{"id":"114362","title":"Falling from Earth's sphere of influence","text":"How can I make a graph of an object falling from say, the Earth's sphere of influence, where acceleration(,a) is the real force of gravity at the objects radius? ie, using $$s=ut+\\frac{1}{2}at^2$$ I can find the position s after a time t due to acceleration, a. This does not account for any gravity gradient, which is what I want to account for. Basically, $$ s=ut+\\frac{1}{2}(\\frac{GM}{r^2})t^2 $$ But there is a problem, firstly, I don't think that would work as it still only uses a constant acceleration, also there is a circular dependancy with r and s basically having the same meaning. (s + r would be the original radius at t=0) I'm not sure how helpful this will be, but recall (Hopefully accurately) that the area under an acceleration time graph is the distance travelled. Currently I have a function for the force at any radius [f(x)=GM\/r^2], and a function for the position with the acceleration for a point r in it. In my case, it is [s(x)=ux+1\/2 f(r) x^2] **Edit:** What I have realised is that when I calculate the position it returns s=ut+1\/2 at^2 (duh...) So every point on the graph is the distance travelled had the object been accelerating for time,x all the way from the starting distance, rather than accelerating from the last point. Hope that makes it clearer, is there a way to get around this? P.S. In the graph, the actual force is not what I want and I am assuming that the object has a mass of 1kg, thus the force is the acceleration. Thanks in advance, Leo"}
{"id":"55278","title":"Distance traveled in a simple two body problem","text":"I'm trying to program an $N$-body simulation and I'd like to be able to test it with a known solution to a simple, two-body problem. I've looked at multiple sources, but I just don't know how to apply it to my simple test case. Two objects at rest placed 10 meters apart with mass of 1. The force between them is a modified gravitational force of F = 10 * m1 * m2 \/ r^2. How long will it take for each object to travel 4 meters?"}
{"id":"83346","title":"If an object fell from the moon","text":"Ignoring the moons gravity, if an object sitting still (relative to the Earth, i.e. not in orbit) was dropped from the moon. How long would it take to hit the Earth?"}
{"id":"45078","title":"One dimensional motion with changing acceleration. Falling to a large body from a great distance","text":"> **Possible Duplicate:**   >  Radial fall in a Newtonian gravitational field My math and physics are rusty. I am trying to calculate the time an object takes to fall to a large body. Before you answer $1\/2at^2$, the conditions where $1\/2at^2$ apply is where $a$ is a constant. Constant acceleration is a fine approximation when the distance the object falls is $<<$ than the distance to the center of the large body. i.e. 100 meters on the surface of the Earth. I'm looking at problems where distance fallen is 50% or more of the distance between objects. Gravity varies with the inverse square of distance. Given two bodies, $m_s$ and $m_l$. $$ F = m_sa = m_la $$ $$ F = Gm_sm_l\/r^2 $$ $$ m_sa = Gm_sm_l\/r^2 $$ $$ a = Gm_l\/r^2 $$ Thus acceleration of $m_s$ varies with the inverse square of the distance to the center of $m_l$, and versa visa. Given R is the radius of the $m_l$, defining the surface of $m_l$. Given $h$ is the distance $m_s$ is above $m_l$'s surface, subsitute $R+h$ for $r$. $$ a = Gm_l\/(R+h)^2 $$ This is as discussed at stackexchange 35878. Given the initial velocity of $m_s$ is zero, what is the time, $t$, it takes $m_s$ to fall distance $h$? An alternative question is give a $t$, how far with $m_s$ fall? This is where I'm rusty. I don't believe one can't simply substitute $Gm_l\/(R+h)^2$ for $a$ in $1\/2at^2$, but I don't know what else to do. The stackexchange articles, 15587 and 41741, discuss $a(x)$ where $a$ changes linearly with distance. Here $a$ changes with the inverse square of the distance. I did a lot of web searching and did not find this topic discussed. The discussions are with $a$ being constant or changing in a linear manner. Thanks for the help."}
{"id":"64185","title":"What's an equation for two astronomical entities both of 4000 tonnes in weight, colliding?","text":"I have next to no knowledge of any physics, but would be happy if you could answer my question... I want to know an equation for two astronomical entities such as the star Sirius (2.02 solar mass) colliding."}
{"id":"19869","title":"Basic question about law of gravitation","text":"> **Possible Duplicate:**   >  Radial fall in a Newtonian gravitational field This is how Wikipedia defines Newton's law of Gravitation: > Every point mass attracts every single other point mass by a force pointing > along the line intersecting both points. The force is proportional to the > product of the two masses and inversely proportional to the square of the > distance between them: > > $F=G\\frac{m1m2}{r^2}$ > > where > >   * F is the force between the masses, >   * G is the gravitational constant, >   * m1 is the first mass, >   * m2 is the second mass, and >   * r is the distance between the centers of the masses. > Now, say 2 spheres, one the size of the earth and the other the size of a ping pong ball are placed say 10 km apart. There are no other forces acting on the system other than gravitation. If I've understood rightly, then _both_ the ball & the earth sized sphere will be pulled towards each other with the same huge force. My question is- how can we calculate when the 2 spheres will meet? And while calculating that shouldn't we consider the fact that the force is changing every moment because the distance between them $r$ changes every moment? How can I include this factor into my calculations? Further, since the force is changing every moment, are the spheres undergoing acceleration or _accelerated_ acceleration?? (Forgive me if my terminology is improper. I'm a beginner.) Is there any name for such forces & accelerations which change at a predictable rate as in this question?"}
{"id":"99583","title":"Determining impact velocity over long distances","text":"Everything I have found concerning impact velocity from a given height deals with constant acceleration due to gravity. I would like to know how to work in varying acceleration into the equation. So, time for the question... How does one go about determining the velocity at which an object with negligible mass $m$ and radius $a$ will hit the surface of a much larger object with mass $M$ and radius $r$ from a initial distance $R$? To be a bit more specific, let's say we have a closed system with two objects, one comparable to the moon and another with a small mass, let's say a basketball. The problem I have with this situation is the acceleration due to gravity is not constant at any given distance. Newton's equation for this is: $$g = GM \/ R^2.$$ Ideas?"}
{"id":"77572","title":"Is there any problem a quantum finite state machine can do faster than a classical finite state machine?","text":"All of the quantum algorithms I've seen so far require a turing-complete quantum computer, at least as far as I can tell. Are there any quantum algorithms that require only a quantum finite automaton? If so, how does their asymptotic complexity compare to the classical versions of those algorithms?"}
{"id":"60541","title":"How do you calculate heat flux (Kw\/m2) at the focal point of a mirror?","text":"can anyone help me to determine the heat flux (Kw\/m2) on a focal point of a parabolic dish having a diameter of 1.5 meter and a focal length 60 cm ??? please awaiting your soonest reply for my senior project :( Regards"}
{"id":"73832","title":"Generalized Coulomb's Law","text":"This question is about the generalization of Coulomb's law to continuous bodies of charge. The basic statement of Coulomb's Law involves two discrete charges $q_1$ an $q_2$: $$\\vec{F}_i = \\frac{1}{4 \\pi \\epsilon_0} \\frac{q_1 q_2}{r_{12}} \\hat{r}_i $$ Here $i$ represents the charge on which the force is exerted, and $\\hat{r}_i$ represents the unit displacement vector between the other charge and the charge $i$. Many treatment of electrostatics extend this law to the case that one charge is not discrete, but rather a continuous body. The force on the discrete charge $Q$ is then: $$\\vec{F} = \\frac{Q}{4 \\pi \\epsilon_0} \\int \\frac{dq}{r^2} \\hat{r} $$ Here $dq$ is the infinitesimal charge element of the continuous body, while $r$ and $\\hat{r}$ represent the distance and displacement vectors between $dq$ and $Q$. Continuing this way, we could probably propose an expression for force between two continuous bodies of charge, like so: $$\\vec{F} = \\frac{1}{4 \\pi \\epsilon_0} \\int \\int \\frac{dq_1 dq_2}{r^2} \\hat{r}$$ However, I have not really seen this expression in the literature\/treatments of electrostatics. Does anyone know why this is the case? Is the expression not useful, or are there no applications demanding the above expression?"}
{"id":"26887","title":"Stability of the vacuum state of interacting quantum fields","text":"\"Stability\" is generally taken to be the justification for requiring that the spectrum of the Hamiltonian should be bounded below. The spectrum of the Hamiltonian is _not_ bounded below for thermal sectors, however, but thermal states are nonetheless taken to be stable because they satisfy thermodynamic constraints. In classical Physics, we would say that the thermal state has lowest _free energy_ , which is a thermodynamic concept distinct from Hamiltonian operators that generate time-like translations. The entropy component of free energy, meanwhile, is a _nonlinear_ functional of the quantum state (presuming that the definition of entropy in quantum field theory would be at least this much like von Neumann's definition in terms of density operators), so we can reasonably expect the sum of the energy and entropy components to have a minimum in the state of greatest symmetry, as we see for thermal states. [It seems particularly notable in this context that the entropy is not an observable in the usual quantum mechanical sense of a linear functional of the quantum state.] The presence of irreducible randomness in quantum mechanics presumably puts quantum field theory as much in the conceptual space of thermodynamics as in the conceptual space of classical mechanics, despite the quasi-functorial relationship of \"quantization\", so _perhaps_ we should expect there to be some relevance of thermodynamic concepts. Given this background (assuming, indeed, that no part of it is _too_ tendentious), **why should we think that requiring the Hamiltonian to have a spectrum that is bounded below should have anything to do with stability in the case of an interacting field?** The fact that we can construct a vacuum sector for free fields in which the spectrum of the Hamiltonian is bounded below does not seem enough justification for interacting fields that introduce nontrivial biases towards statistically more complex states. This question is partly motivated by John Baez' discussion of \"quantropy\" on Azimuth. I am also interested in the idea that if we release ourselves from the requirement that the Hamiltonian of interacting fields must have a spectrum that is bounded below, then we will have to look for analogues of the KMS condition for thermal states that restore some kind of analytic structure for interacting fields. I asked a related Question here, almost a year ago. I don't see an answer to the present Question in the citations given in Tim van Beek's Answer there."}
{"id":"78776","title":"Relation between decay probability and the energy of particle","text":"Is there any way to find the energy of a particle through its decay probability?"}
{"id":"7738","title":"Why there's a whirl when you drain the bathtub?","text":"At first I thought it's because of Coriolis, but then someone told me that at the bathtub scale that's not the predominant force in this phenomenon."}
{"id":"72242","title":"What causes a whirlpool to form in a bathtub?","text":"I mean, when you have a full bathtub and then let it empty, under what conditions does a whirlpool form? Could we devise an experiment and measure this effect? When it comes to fluids, I don't know why I can't think of anything to test a hypothesis."}
{"id":"34661","title":"Why can glass absorb ultraviolet light?","text":"Please see the following photos. (I cannot post them...) http:\/\/i1163.photobucket.com\/albums\/q554\/startanewww\/CIMG4545.jpg http:\/\/i1163.photobucket.com\/albums\/q554\/startanewww\/CIMG4546.jpg From the first photo, the book mentioned _Don't cover the paper with glass because glass absorbs enough ultraviolet light to slow the damage process._ (line 8) My questions:   * Why can glass absorb ultraviolet light?   * Will papers with dark colours fade fastest? Why is that the case? (see the second last paragraph)"}
{"id":"66106","title":"Slowdown rate of rotating body due to friction force","text":"This isn't a homework question, but it might as well be. The problem I have been pondering is: > If a disc (or children's roundabout if you like), of radius r, mass m, is > spun around it's center with an initial force F, and thereafter there is the > friction force (of either the axle or air resistance or both) of f, then how > long will it take to come to a stop? I have thought about it and have come up with not much. My first way is thus: $F = ma$, so $a = F\/m$, the initial acceleration ( or should that be $(F-f)\/m$ ?). And then the deceleration is $a = -f\/m$. I'm not sure how to calculate the initial linear velocity, but assuming I have it, $u$, say, then I could say that after time $t$ the velocity is $v = u -at = u -ft\/m$, where f is the friction force. So then the disc would stop spinning when $ t= um\/f$. I am aware that this is wrong (well, it might work if we were dealing with linear motion). Straight away it seems wrong because it doesn't take into account the radius of the disc and also the slow down seems linear, when from observation it seems rotating discs slow down and taper off to a standstill. But that is as far as I got. I have tried to use angular motion equations (well $\\omega r = v$) but I am stuck at this point, and of course, finding the initial velocity. Any help is appreciated."}
{"id":"90943","title":"Do Photons interfere when it passes through a slit (one)?","text":"When a light (photons) goes through two slits it creates interference patterns. if the light goes through a \"single\" slit, does it create interference patterns or does it behave like particles (photons)? is the pattern same for both particles and light waves if it is a \"single slit\"? In the double slit experiment, if you close one slit (or observe) it is said that light behave as particles (bullets) which means that through one slit light exactly behaves as stream of particles??"}
{"id":"24653","title":"Is this really how a capacitor works? Why doesn't it behave like a resistor?","text":"My book says a capacitor is two conducts being connected by an insulator. Now let's take a parallel plate capacitor to simplify the problem I have. Suppose I got two parallel plate capacitor in series and I hook the circuit up with a battery. ![enter image description here](http:\/\/i.stack.imgur.com\/YWnG8.jpg) As soon as I hook it up, electrons flow (forget conventional current for now) into on the right plate and builds up on that capacitor (and spreads on the surface of conducting plate) and remains stuck there because there is an insulator that blocks the electrons from going anywhere. Now here is my confusion, how does the left plate of $\\ C_1$ even build the positive charges and how do the current even run through the circuit if there an insulator blocking the electrons from moving? Is there even current through the circuit?"}
{"id":"24650","title":"Gravitational field with cavity removed","text":"I'm just struggling a little with this question: A uniform sphere, of radius $R$, contains a spherical cavity of radius $R\/4$, whose centre is $3R\/8$ from the surface. The diameter passing through the centres of the sphere and cavity meets the surface at points $X$ and $Y$. Find the ratio of the gravitational field at $X$ and $Y$. My attempt at the solution goes something like this: Using the superposition principle, the gravitation field due to the whole mass is equal to the sum of the gravitational fields due to the remaining mass and the removed mass. The gravitational field due to a uniform solid sphere is zero at its centre. Therefore, the gravitational field due to the removed mass is zero at its centre. The gravitational field due to the solid sphere is equal to the gravitational field due to the remaining mass. Now we know g acts towards the centre of the sphere. As such, both the gravitational field of the combination of the sphere and removed mass and the gravitational field of the sphere only act in the same direction, so we can use the scalar form of the equation. Therefore the gravitational field is given by $g=GMr\/R^2$. Then insert $r=-R$ and $R$ for the gravitational field at $X$ and $Y$. But this doesn't seem to be correct as it is just the same as if the removed mass wasn't there..... have I gone wrong in my logic somewhere?"}
{"id":"35535","title":"Optical laser pumping and reflectors","text":"Let's say we are building Nd:YAG laser. It is optically pumped by some linear xenon flash lamps, it absorbs light around 750nm and 800nm, and emitted light is at 1064nm. The question is why doesn't 1064nm emission from the flash lamps interfere with laser operation? Why doesn't 1064nm photons emitted in the Nd:YAG at 'wrong' directions (not coaxially to the resonator path) and reflected back and forth from the cylindrical\/oval reflector (for the flashlights) interfere with the laser emission? As far as I see it, both these factors should consume precious atoms in excited state, and probably require some 1064nm filter around the Nd:YAG rod..."}
{"id":"73307","title":"what is the radial extent of the last scattering \"shell\"?","text":"At CMB recombination (z=1090), what is the radial extent of the last scattering \"shell\"? a) Delta(z) = .... b) Delta(comoving angular distance)= ....Mpc The WMAP first-year parameters give Delta(z) = 195. Is this still correct?"}
{"id":"31852","title":"Flow of liquid among branches","text":"If water is flowing through big pipe is branched into 4 branches of small pipe. Lets say the flow is around 4 m\/sec. I have the following questions:     1. What will be the flow rate in each of the pipes? I would be knowing diameter, height of the pipes. Considering all pipes are rigid.     2. What will be the flow rate in each of the pipes, if I close one pipe of the 4 branches? Will water flow speed increase in other 3 pipes? If yes, how we can calculate the gain of speed of flow?     3. What if the pipes are non-rigid, will that have any effect?"}
{"id":"55226","title":"Practical method to weigh human limbs with common household items?","text":"What methods could be used to determine ( _or_ estimate within a _reasonable_ margin of error) the mass of a living human's limbs, short of cutting them off? And more interestingly, how can this be done without any high tech equipment, just with the means commonly found in households? A scale for example is allowed. An MRI isn't ;)"}
{"id":"32627","title":"Has anyone else thought about gravity in this way?","text":"Picture yourself standing on a ball that is expanding at such a rate that it makes you stick to the ball. Everything in the universe is expanding at this same rate. To escape the earths gravitational pull we would need to jet upward faster than the expansion of the earth. Each object expands at a different rate on its surface according to its size. Thus different gravity affects for different size planets. When in space we are subject to being affected by the most distant body if we stand in its way. I just can not explain the reaction of our tides with our moon. Have any scientists seriously considered an idea like this? Follow up July 23 I am no scientist, but I think someone with more knowledge might explore this idea a little further. At the very lease the idea that every thing in the total universe is expanding, including all parts of the atom can be used as a simple way to see formulas and the same results to the effects of gravity of anything on the surface of a sphere planet or a donut shaped planet. The area of mass will grow but the density will remain the same. The idea can be cross referenced by light shifting etc, to see if it falls in line with the known action of planet gravity and the known expansion affect of the whole universe. Maybe the gravity affect of a planet on its surface dweller is a completely different force than is the force that maintains the orbits of planets. keplers law I believe. What happens when we have an eclipse of the moon?, does the earths orbit around the sun change for time of this eclipse? My summary is that if all scientists can not explain gravity totally, then maybe the common thought for all these years is not completely a correct one."}
{"id":"18255","title":"Define Pressure at A point. Why is it a Scalar?","text":"I have a final exam tomorrow for fluid mechanics and I was just looking over the practice exam questions. They do not provide solutions. But pretty much I have to define pressure at a point and also say why pressure is scalar instead of a vector. I am thinking pressure at a point is $P=\\lim_{\\delta A \\to 0} \\frac{\\delta F}{\\delta A}$. Please let me know if I am wrong. But I do not know at all why pressure is a scalar instead of a vector. I know it has something to do with $d \\mathbf{F}=-Pd \\mathbf{A}$"}
{"id":"38486","title":"Intercept a moving object","text":"Object A can move at 50km\/h, wants to intercept object B (currently $15^{\\circ}$, east of north from A) moving at 26km\/h, $40^{\\circ}$ east of north. What angle should A take to intercept B? AB is 20km apart ![enter image description here](http:\/\/i.stack.imgur.com\/Kcm8I.png) The provided answer looks like: * * * Choose x axis along 20km distance. $26t \\sin{(40-15)} = 50t \\sin{\\theta}$ $\\theta = \\sin^{-1}{\\frac{11}{50}} = 12.7$ $15 + 12.7 = 27.7$ * * * I took a different approach and used $\\cos$ and got a different answer ... why is that? $26t \\cos{(40-15)} = 50t \\cos{\\theta}$"}
{"id":"56751","title":"Buoyancy fluxes in a stratififed fluid and units","text":"I am calculating the buoyancy flux ($B$) for a stratified fluid as follows: $$ B=\\frac{g\\alpha S}{C_{pw}\\rho_0} $$ where $g = 9.81$ $m\/s$; $\\alpha = 1.6 t\\times10^{-5} + 9.6\\times10^{-6} \\times (20 \\text{ degC})$; $S = 100\\text{ }Wm^{-2}$, $\\rho_0 = 1000$ $kg\/m^3$, and $C_{pw}$ is the specific heat of water. The question I have is: Is $C{pw} = 4200$ or $4.2$ I've seen it used both ways, and I am unsure which I would use with the units of the other terms in the equation."}
{"id":"4558","title":"Treatment of boundary terms when applying the variational principle","text":"One of the main sources of subtlety in the AdS\/CFT correspondence is the role played by boundary terms in the action. For example, for a scalar field in AdS there is range of masses just above the Breitenlohner-Freedman bound where there are two possible quantizations and which one you get depends on what boundary terms you add to the action. Boundary terms are also essential in the treatment of first-order Lagrangians for fermions and self-dual tensor fields. These all involve the \"UV\" boundary as $z \\rightarrow 0$ in Poincare coordinates. Then there are dual models of QCD like the hard-wall model where one imposes an IR cutoff and imposes boundary conditions at the IR boundary and\/or adds IR boundary terms to the action. My question is a bit vague, but basically I would like references to reviews, books or papers that give a good general treatment of the variational principle when one has to be careful about boundary terms. It would help if they clearly distinguish the requirements that follow from mathematical consistency from those that are imposed because of a desire to model the physics in a certain way."}
{"id":"16322","title":"Confront Order Of magnitudes","text":"Is it correct to say that 9.0 is one order of magnitude smaller than 10.0? Has anyone a link\/source about confronting order of magnitudes, apart from wikipedia?"}
{"id":"24068","title":"Isn't the uncertainty principle just non-fundamental limitations in our current technology that could be removed in a more advanced civilization?","text":"From what I understand, the uncertainty principle states that there is a fundamental natural limit to how accurately we can measure velocity and momentum at the same time. It's not a limit on equipment but just a natural phenomenon. However, isn't this just an observational limit? There is a definite velocity and momentum, we just don't know it. As in, we can only know so much about the universe, but the universe still has definite characteristics. Considering this, how do a wide range of quantum mechanical phenomena work? For example, quantum tunneling - its based on the fact that the position of the object is indefinite. But the position is definite, we just don't know it definitely. Or the famous light slot experiment? The creation of more light slots due to uncertainty of the photon's positions? What I am basically asking is why is a limit on the observer, affecting the phenomenon he is observing? Isn't that equivalent to saying because we haven't seen Star X, it doesn't exist? It's limiting the definition of the universe to the limits of our observation!"}
{"id":"114133","title":"Is the uncertainty principle a property of elementary particles or a result of our measurement tools?","text":"In many physics divulgation books I've read, this seems to be a commonly accepted point of view (I'm making this quote up, as I don't remember the exact words, but this should give you an idea): > Heisenberg's uncertainty principle is not a result of our lack of proper > measurement tools. The fact that we can't precisely know both the position > and momentum of an elementary particle is, indeed, a property of the > particle itself. It is an intrinsic property of the Universe we live in. Then this video came out: Heisenberg's Microscope - Sixty Symbols (skip to 2:38, if you're already familiar with the uncertainty principle). So, correct me if I'm wrong, what we may claim according to the video is: > the only way to measure an elementary particle is to make it interact with > another elementary particle: it is therefore **incorrect** to say that an > elementary particle **doesn't have** a well defined momentum\/position before > we make our measurement. We cannot access this data (momentum\/position) > without changing it, therefore it is **correct** to say that our ignorance > about this data is not an intrinsic property of the Universe (but, rather, > an important limit of how we can measure it). Please tell me **how** can both of the highlighted paragraphs be true or how they should be corrected."}
{"id":"127253","title":"Uncertainty principle implies the non-deterministic universe?","text":"Does the uncertainty principle imply the non-deterministic universe, or just the fact that our model of the universe, the one based on observation, can be at most non-deterministic, since we will not be able to measure with perfect accuracy, even if such would exist (and therefore we will have to provide probability distributions rather than an event which should happen with overwhelming probability (term used by Terence Tao))?"}
{"id":"59960","title":"About Heisenberg uncertainty principle","text":"What would happen if someone invented a way to measure both position and momentum precisely? If it is impossible why?"}
{"id":"54184","title":"Is the uncertainty principle just saying something about what an observer can know or is it a fundamental property of nature?","text":"I ask this question because I have read two different quotes on the uncertainty principle that don't seem to match very well. There are similar questions around here but I would like an explanation that reconciles these two interpretations specifically:   1. Feynman talks about the uncertainty principle in one of his lectures and mentions it as the reason why electrons don't crash into the atom's nucleus: If they did they would have an exact location and momentum which is not allowed by the uncertainty principle. In saying this it is clear that the uncertainty principle is a fundamental property of nature because it has an effect on where an electron can reside.   2. Recently I read - somewhere else but I forgot where exactly - an account of the uncertainty principle where there was explained how we can measure position of a particle by firing another particle into it, the collision disturbs the velocity of the observed particle therefore we can not know its momentum anymore. Now, 2) very much seems like a limitation of what the observer can know, while 1) attributes a fundamental property of nature to it (electrons don't crash into the nucleus). What is the correct way to think about this?"}
{"id":"387","title":"A list of inconveniences between quantum mechanics and (general) relativity?","text":"It is well known that quantum mechanics and (general) relativity do not fit well. I am wondering whether it is possible to make a list of contradictions or problems between them? E.g. relativity theory uses a space-time _continuum_ , while quantum theory uses _discrete_ states. I am not merely looking for a solution or rebuttal of such opposites, more for a survey of the field out of interest."}
{"id":"71568","title":"What are the Conflicting Predictions of General Relativity & Quantum Mechanics?","text":"I see a lot of questions in various sites about why the 2 theories are or aren't incompatible, I'm satisfied as to why that's the case. However it has been mentioned that both theories make predictions about phenomena that contradict or are incompatible, and I've been unable to find any examples. What are the conflicting\/contrary\/incompatible predictions made by General Relativity versus Quantum Mechanics? Or are the claims false?"}
{"id":"66265","title":"Incompatibility of GR and QM","text":"I am told that the theories of General Relativity and Quantum Mechanics are fundamentally incompatible... Why is that? Someone explained that it had to do with the fact that quantum particles such As quarks and photons are points without volume... Leading to singularities. Do people really believe that ^ What data do we have to suggest that quarks are really zero dimensional points?"}
{"id":"61731","title":"What is an example of a situation where Quantum Mechanics and Relativity do not work together?","text":"I've learned special relativity in school last semester, and this semester we began learning about Quantum Mechanics, and my teacher told us that there was a Relativistic Schrodinger equation. I was under the impression that you couldn't use Quantum Mechanics and Relativity together, so why is it appropriate to have a Relativistic Schrodinger equation, and what is an example of a situation where QM and Relativity cannot be used together?"}
{"id":"94833","title":"Quantum Mechanics and General Relativity in Macroscopic Level","text":"Hi I read a book yesterday.The book was Brian Greene's The Elegant Universe. I learned that uncertainty principle affects space-time very microscopic levels and this affection makes conflict in General Relativity and Quantum Mechanics. I wanted to know this is the only reason that this two theoris can not be conbined or there are more reasons ?"}
{"id":"92259","title":"General relativity && quantum mechanics \"incompatibillity\"","text":"Now this may be utterly weird layman-physics-question, but anyways... I have read recently following: \"The fundamental universe laws are everywhere the same. It's just that the manifestation (!) of (these) laws is different (!) on a specific \"scale\".\" Which brings me to the question: Could it be that GR and QM are one and the same thing on a \"different scale\"? If yes (LOL), that would \"explain\" why it is not possible to \"merge\" GR and QM. Was question like this ever considered?"}
{"id":"104140","title":"Sound as a use to separate molecular structures","text":"Sound can be a destructive force. However, could it be used to separate say the Hydrogen atom from the Oxygen atoms?"}
{"id":"74081","title":"Zero modes ~ zero eigenvalue modes ~ zero energy modes?","text":"There have been several Phys.SE questions on the topic of zero modes. Such as, e.g.,   * **zero-modes** (What are zero modes?, Can massive fermions have zero modes?),    * **majorana-zero-modes** (Majorana zero mode in quantum field theory),   * **path-integral-with-zero-energy-modes** (Path integral with zero energy modes), etc. Here I would like to understand further **whether \"Zero Modes\" may have physically different interpretations** and **what their consequences are** , or **how these issues really are the same, related or different.** There at least 3 relevant issues I can come up with: ## (1) **Zero eigenvalue modes** By definition, **Zero Modes** means zero eigenvalue modes, which are modes $\\Psi_j$ with zero eigenvalue for some operator $O$. Say, $$O \\Psi_j = \\lambda_j \\Psi_j,$$ with some $\\lambda_a=0$ for some $a$. This can be Dirac operator of some fermion fields, such as $$(i\\gamma^\\mu D^\\mu(A,\\phi)-m)\\Psi_j = \\lambda_j \\Psi_j$$ here there may be nontrivial gauge profile $A$ and soliton profile $\\phi$ in spacetime. If zero mode exists then with $\\lambda_a=0$ for some $a$. **In this case, however, as far as I understand, the energy of the zero modes may not be zero.** This zero mode contributes nontrivially to the path integral as $$\\int [D\\Psi][D\\bar{\\Psi}] e^{iS[\\Psi]}=\\int [D\\Psi][D\\bar{\\Psi}] e^{i\\bar{\\Psi}(i\\gamma^\\mu D^\\mu(A,\\phi)-m)\\Psi } =\\det(i\\gamma^\\mu D^\\mu(A,\\phi)-m)=\\prod_j \\lambda_j$$ In this case, if there exists $\\lambda_a=0$, then we need to be very careful about the possible long range correlation of $\\Psi_a$, seen from the path integral partition function ( **any comments at this point?** ). ## (2) **Zero energy modes** If said the operator $O$ is precisely the hamiltonian $H$, i.e. the $\\lambda_j$ become energy eigenvalues, then the zero modes becomes zero energy modes: $$ H \\Psi_j= \\lambda_j \\Psi_j $$ if there exists some $\\lambda_a=0$. ## (3) **Zero modes $\\phi_0$ and conjugate momentum winding modes $P_{\\phi}$** In the chiral boson theory or heterotic string theory, the bosonic field $\\Phi(x)$ $$ \\Phi(x) ={\\phi_{0}}+ P_{\\phi} \\frac{2\\pi}{L}x+i \\sum_{n\\neq 0} \\frac{1}{n} \\alpha_{n} e^{-in x \\frac{2\\pi}{L}} $$ contains zero mode $\\phi_0$. * * * Thus: **Are the issues (1),(2) and (3) the same, related or different physical issues?** If they are the same, why there are the same? If they're different, how they are different? I also like to know when people consider various context, which issues they are really dealing with: such as the **Jackiw-Rebbi** model, the **Jackiw- Rossi** model and **Goldstone-Wilczek** current computing induced quantum number under soliton profile, **Majorana zero energy modes**, such as the Fu- Kane model (arXiv:0707.1692), Ivanov half-quantum vortices in p-wave superconductors (arXiv:cond-mat\/0005069), or the issue with **fermion zero modes under QCD instanton** as discussed in Sidney Coleman's book ``Aspects of symmetry''. ps. since this question may be a bit too broad, it is totally welcomed that anyone attempts to firstly answer the question partly and add more thoughts later."}
{"id":"27303","title":"Negative probabilities in quantum physics","text":"Negative probabilities are naturally found in the Wigner function (both the original one and its discrete variants), the Klein paradox (where it is an artifact of using a one-particle theory) and the Klein-Gordon equation. The question is if there is a general treatment of quasi-probability distributions, besides naively using 'legit' probabilistic formulas? For example, is there a theory saying which measurements are allowed, so to screen negative probabilities? _Additionally_ , is there an intuition behind negative probabilities? (Providing other examples than ones mentioned in the question can illuminate the issue.)"}
{"id":"25233","title":"Smaller free remote control telescopes?","text":"There are several online services that let you control a large telescope (eg, lightbuckets.com and slooh.com), even some that are free (eg, telescope.org). Unfortunately, the pay services are expensive, and you get very little reserved time on the scopes. The free services are painfully slow: I had to wait several months for a picture of Jupiter I wanted. Has anyone set up automated (ideally free) remote access to a smaller telescope? I'd much rather play with a 10\" telescope in real-time than a larger one in limited\/delayed time. In fact, even a CCD or camera would be nice for wide-angle shots."}
{"id":"17651","title":"Is it possible to recover Classical Mechanics from Schrödinger's equation?","text":"Let me explain in details. Let $\\Psi=\\Psi(x,t)$ be the wave function of a particle moving in a unidimensional space. Is there a way of writing $\\Psi(x,t)$ so that $|\\Psi(x,t)|^2$ represents the probability density of finding a particle in classical mechanics (using a Dirac delta function, perhaps)?"}
{"id":"32112","title":"Classical limit of quantum mechanics","text":"I have heard that one can recover classical mechanics from quantum mechanics in the limit the $\\hbar$ goes to zero. How can this be done? (Ideally, I would love to see something like: as $\\hbar$ goes to zero, the position wavefunction reduces to a delta function and that the Schrodinger equation\/Feynman path integral reduces to the Newtonian\/Lagrangian\/Hamiltonian equations of motion.)"}
{"id":"77294","title":"How to prove Newton's second law with quantum mechanics?","text":"Newton's second law claims that $F=ma$. In terms of quantum mechanics, the equality can be written as $ \\frac{d\\langle p \\rangle}{dt} = -\\langle \\nabla V(x) \\rangle$. How can I prove this with non-relativistic Schrodinger Equation? I've calculated the expectation of $p=mv$, and tried to take the derivative under the integral. And there I'm stuck."}
{"id":"32237","title":"Classical Limit of the Feynman Path Integral","text":"I understand that in the limit that h_bar goes to zero, the Feynman path integral is dominated by the classical path, and then using the stationary phase approximation we can derive an approximation for the propagator which is a function of the classical trajectory (see http:\/\/www.blau.itp.unibe.ch\/lecturesPI.pdf pg 46). I am under the impression that this further implies that the particle follows the classical trajectory but I don't understand how the above mentioned fact implies this. The propagator describes the time-evolution of the wavefunction, so I would think that this classical limit form of the propagator should give a time- evolution in which the wavefunction follows the classical trajectory, but I have not been able to find such work. Moreover, even this statement itself is problematic since the wavefunction describes a probability distribution and not a single trajectory. $\\textbf{New Edit:}$ In section 7 of Feynman's paper introducing the path integral (see http:\/\/imotiro.org\/repositorio\/howto\/artigoshistoricosordemcronologica\/1948c%20-FEYNMAN%201948C%20Invention%20of%20the%20path%20integral%20formalism%20for%20quantum%20mechanics.pdf) he discusses the classical limit. It appears that the key to understanding why the fact that the classical path dominates the path integral further implies that the particle follows the classical trajectory may be found in Feynman's remark on pg 21: \"Now we ask, as $\\hbar → 0$ what values of the intermediate coordinates $x_i$ contribute most strongly to the integral? These will be the values most likely to be found by experiment and therefore will determine, in the limit, the classical path.\" However, I don't understand why \"These will be the values most likely to be found by experiment\" ?"}
{"id":"65964","title":"Can Newton's laws be explained by Quantum Physics?","text":"I have only basic knowledge of physics. Could you please explain to me if a \"Quantum\" laws can theoretically (perhaps in the future?) be used to explain everything in macro levels? I'm having problems to understand how we can have \"two physics\" at the same time. are Newton's laws only a simplification?"}
{"id":"32110","title":"Classical limit of the path integral formulation of quantum mechanics","text":"It is well-known that if $S \\gg \\hbar$, then the classical path dominates the Feynman path integral. But is there some to show that if $S\\gg\\hbar$, then the particle's trajectory will approach the classical path?"}
{"id":"108222","title":"From Quantum Mechanics to Classical Mechanics","text":"Is it possible, and has it been attempted, to use quantum mechanics to deduce Newtonian, macroscopic level mechanics laws as was the case of statistical mechanics deriving thermodynamic relations?"}
{"id":"109510","title":"How are the Lagrange equation and Feynmann path integral related?","text":"My question is, where could I get some more info on how the Euler-Lagrange equations are related $$ \\delta S [y(x)] =0 $$ with the Feynmann path integral formulation $ \\int D[y(X)]e^{iS[y(x)]\/\\hbar} $ I believe that when $ \\hbar \\to 0$ (semiclassical), only the points that satisfy (1) contribute to the path integral. If someone can give me more info about it :D thanks"}
{"id":"44926","title":"How to go from Quantum World to Classical World?","text":"> **Possible Duplicate:**   >  Is it possible to recover Classical Mechanics from Schrodinger’s equation?   >  Classical Limit of the Feynman Path Integral In the quantum world we don't have specific trajectories, the particle so to speak goes through all possible paths. In the classical and macroscopic world we have definite paths, and usually one specific trajectory is assigned to a body's motion. How would you go from a trajectoryless world to trajectoried world? Are there any theories about this bridge between the two worlds? I guess there should be such a theory, cause one world is the building block of the other."}
{"id":"127919","title":"How do inflationary models predict the generation of gravitational waves during the inflationary period?","text":"Recent results from the BICEP2 experiment have produced a lot of talk about the primordial gravitational waves produced during the inflationary period. I would like to have some explanation about how inflationary models predict the generation of these gravitational waves. Have these gravitational waves been described as metric perturbations around a de Sitter spacetime? Are they predicted using a semiclassical gravity formalism so as to take into account the quantization of a scalar field (the inflaton) in such \"perturbed\" spacetime?"}
{"id":"71303","title":"What is the speed of electrical current in salt water?","text":"I am wondering about a specific question regarding the speed at which an electrical current traverses through salt-water \/ saline. By this I do _not_ mean the electron drift speed - I mean, at what speed would a current travelling from an anode to a cathode immersed in salt water be? A ball park figure will do. Thank you."}
{"id":"71300","title":"Is Electromagnetic Mass Possible?","text":"If the sinusoidal electric component of a light wave were off-set to one side of the magnetic component and then the smaller \"lobe\" were to cancel out with much of the larger side, then where would the energy go? Would it not form a closed loop much like a mass-bearing string? Could the electrical energy not be converted into a gravitational field to bend space-time over the length of the wave to form a 1-D \"string\" along the junction of the perpendicular E and B fields? This would be much like folding a sheet of paper in half so that one edge protrudes past the other. The protruding edge being the electric component and the rest being electrical energy converted into mass\/gravity (comparable to a mass-bearing string). The magnetic component could then arise from Lorentz symmetry as described by Lubos: > If you only start with the $E_z$ electric field, the component $F_{03}$ is > nonzero. However, when you boost the system in the $x$-direction, you mix > the time coordinate $0$ with the spatial $x$-coordinate $1$. Consequently, a > part of the $F_{03}$ field is transformed into the component $F_{13}$ which > is interpreted as the magnetic field $B_y$, up to a sign. **Is the described \"string wave\" model possible?** I realize that I haven't provided a mechanism to explain the proposed model. However \"I do not reject food because I do not understand digestion.\" -Oliver Heavenside From here on the rest of this post is merely a compilation of the \"evidence\" that circumstantially supports the proposed idea. Please do not feel required to address these topics. They are here because I think electromagnetic mass is real. I realize this borders on \"promoting unaccepted theories,\" but it is merely my way of assessing the possibility. I promise not to bring this up again on SE if it is refuted. **The concept seems compatible with the standard model:** a.) an origin of charge has been proposed. b.) eliminating one lobe has reduced the \"spin\" or magnetic field by 1\/2 relative to a light wave. c.) Lepton number 1 applies due to the \"strong interaction\" of the remaining electric component d.) a mechanism of mass has been proposed. **Maxwells equations seem to fit:** The Laws of Electromagnetism have geometric relations incorporated into them that naturally arise from the proposed electron-as-an-EM-wave model: the dot product and the cross product specifically. Ampere's Law Ampere’s Law describes the magnetic field produced by the flow of electrons along a wire. The negative components of electrons flowing along the wire should repel each other, which would mean that the negative components should protrude from the wire like the dorsal fins of sharks swimming parallel and breaking the surface. Since the electrons are travelling in the same direction along the wire the magnetic components (in the direction of the side fins of the shark analogy) should be tangent to the surface of the wire, which results in a circular magnetic field around the wire just as Ampere’s Law and the Biot-Savart Law predict. Faraday's Law Faraday’s Law describes the electric voltage produced in a coil of wire as a magnetic field through it changes. The voltage is proportional to the number of loops of wire, which is counter-intuitive\/non-conservative. Why should the voltage depend on the number of loops? An explanation naturally arises from the proposed model. Electrons inherently have velocity in the form of a \"Poynting-like\" vector. When the loop of wire encounters a changing magnetic field the \"poyinting vectors\" align and under the right orientations they align with the wire loops and thereby form electric current. ![enter image description here](http:\/\/i.stack.imgur.com\/eyCkq.jpg) In the bottom orientation the magnet produces no EMF along the wire. In the top orientation the EMF is along the wire as seen in generators. Schrodinger's statistical model won because Maxwells equations had already divorced physics from first causes. Saying that the surface covered area of a loop was the cause of EMF instead of the poynting vectors of electrons adding over the distance of the circumference. Gauss's Laws Gauss’ laws of electricity and magnetism are easily integrated with the proposed model as the net magnetic flux is zero and the net charge is unchanged. However Coulombs Law does not hold true at the subatomic level. The force between two charged electrons is modified by the presence of other electrons if electrons are not point charges, but electric components of an EM-wave. Coulomb's law Coulomb's law for point charges: $$F=k\\frac{q_1 q_2}{r^2}=\\frac{1}{4\\pi\\epsilon_0}\\frac{qq_0}{r^2}\\hat{u}$$ Does not hold at the subatomic level for all directions if a blip of negative charge sticks off one side of a mass-bearing electromagnetic wave. This naturally provides a classical explanation for \"quantum tunnelling:\" certain orientations of subatomic particles behave unlike point charges. **Atomic Orbitals** Take one electron and one proton and place them near each-other. The proposed model suggests an intrinsic and intuative reason why electrons don’t fall into the nucleus, which current theories lack. The velocity of an imbalanced EM wave is perpendicular to both the electric and magnetic components, which means that the radial acceleration of attraction towards the nucleus experienced by the electric component of an electron is always perpendicular to its velocity or the “Poynting vector” of an electron. All that is left for the electron to do is to set up the lowest energy standing wave possible. This also suggests that the proton rotates with the electron's orbit so that the \"fins\" constantly point toward eachother. Spherical harmonics should arise naturally from this arrangement and approximate to the schrodinger equation. **Relativistic Explanation of the Lorentz Force** Using relativistic tensors: \"If you only start with the $E_z$ electric field, the component $F_{03}$ is nonzero. However, when you boost the system in the $x$-direction, you mix the time coordinate $0$ with the spatial $x$-coordinate $1$. Consequently, a part of the $F_{03}$ field is transformed into the component $F_{13}$ which is interpreted as the magnetic field $B_y$, up to a sign\\cite{Lubos}.\" If the electron is an E-M wave as proposed and the magnetic components align with the extern magnetic field, then performing the reverse of the above transformation should convert the external magnetic field into an electric component in the $E_z$ direction so that the electron feels the equivalent to a charge perpendicular to its direction of motion, which explains the Lorentz force. **Numerous Other Explanations** There are many other phenomena that seem to fit with the proposed model. I'm just running out of steam! **A Little History** Electrons were viewed as \"matter waves\" in DeBrogli's model. \"Matter waves\" is a polite way of saying: something is waving but we don't know what. Heisenberg came along and said that despite not knowing what is waving we can assume that the wave doesn't have any undiscovered properties that would allow for knowing both position and momentum (an arrogant assertion!). Schrodinger then came along and noticed that tweaking spherical harmonics provided a reasonable model of atomic orbitals (statistically). That all led to the Copenhagen interpretation, which Einstein called the \"Born-Heisenberg tranquilizing philosophy, or religion.\" Electrons have continued to be treated as point charges or matter waves as is convenient for interpreting experimental results ever since. QED and QCD have introduced \"virtual photons\" in explaining the interactions of \"point charges\" and light, etc. Surely \"virtual photons\" would be unnecessary if the point charges were instead modelled as EM waves themselves. Particle physicists have invented a \"higgs field\" which they propose vibrates to create mass. The higgs field seems like adding epicycles: sorta unnecessary when there is a simpler and better alternative (that I have not really understood yet!). It has been obvious for decades that physics has stalled while trying to model hadrons as point charges. String theory has been a lonely success story waiting to happen. Electromagnetic string-waves are the future! (unless someone refutes me :-)."}
{"id":"25759","title":"How exactly does time slow down near a black hole?","text":"How exactly does time slow down near a black hole? I have heard this as a possible way of time traveling, and I do understand that it is due in some way to the massive gravity around a black hole, but how exactly does that massive gravity slow down time?"}
{"id":"22876","title":"Does a photon exert a gravitational pull?","text":"I know a photon has zero rest mass, but it does have plenty of energy. Since energy and mass are equivalent does this mean that a photon (or more practically, a light beam) exerts a gravitational pull on other objects? If so, does it depend on the frequency of the photon?"}
{"id":"65361","title":"How does gravity effects both time and light if they have no mass","text":"I've been reading about how black holes can effect both time and light with gravity. So I was wondering, doesn't something have to have mass to be effected by gravity? And if so, does this mean both light and time have mass? And if not, how can gravity effect something that has no mass?"}
{"id":"54701","title":"Is light affected by gravity? Why?","text":"I would like to know if light is affected by gravity, also, I would like to know what is the correct definition of gravity: \"A force that attracts bodies with mass\" or \"a force that attracts bodies with energy, such as light\"? Is light massless after all?"}
{"id":"132971","title":"What causes light to travel?","text":"What is the force that causes it to move and why does it maintain the speed for so long? If it has no mass, why is it effected by mass?"}
{"id":"72819","title":"Can electromagnetic radiation (i.e. photons) produce gravity?","text":"I don't want to play with physical laws in a frivolous way. Assuming that the nature of matter and energy is the same, can a high density of highly energetic photons produce a gravity force? We do know that radiation is affected by space-time distortions, or in another way \"feels gravity\". Why do photons can (or cannot) produce a gravity field?"}
{"id":"107930","title":"Why can light (photons) bends in a curve through space without mass?","text":"I've heard that light can form a curve if they travel near high-mass stars or even a black hole with strong gravity. Which is according to this Newtonian formula $$\\large F_{g}=\\dfrac{Gm_1m_2}{r^2}.$$ But I've also heard that photons do not have (rest) mass! So it doesn't fit that equation anymore! But why can photons be pulled by gravities without (rest) mass? Could someone explain that?"}
{"id":"10612","title":"Explain how (or if) a box full of photons would weigh more due to massless photons","text":"I understand that mass-energy equivalence is often misinterpreted as saying that mass can be converted into energy and vice versa. The reality is that energy is always manifested as mass in some form, but I struggle with some cases: **Understood Nuclear Decay Example** In the case of a simple nuclear reaction, for instance, the total system mass remains the same since the mass deficit (in rest masses) is accounted for in the greater relativistic masses of the products per $E=\\Delta m c^2$. When a neutron decays and you are left with a fast proton and a relativistic electron. If you could weigh those two without slowing them down, you would find it weighed as much as the original neutron. **Light in a Box** This becomes more difficult for me when moving to massless particles like photons. Photons can transmit energy from one heavy particle to another. When a photon is absorbed the relativistic mass (not the rest mass) of the (previously stationary) particle that absorbs it increases. But if my understanding is correct, the energy must still be manifested as mass _somehow_ while the photon is in-flight, in spite of the fact that the photon does not have mass. So let's consider a box with the interior entirely lined with perfect mirrors. I have the tare weight of the box with no photons in it. When photons are present the box has an additional quantifiable amount of energy (quantified below) due to the in-flight photons. Say there are $N$ photons... obviously assume $N$ is large. $$\\Delta m = \\frac{ E }{ c^2 } = \\frac{ N h }{ \\lambda c}$$ Interactions are limited to reflections with the wall, which manifest as a constant pressure on the walls. If I hold this box in a constant gravitational field (like the surface of Earth) then there will be a gradient in the pressure that pushes down slightly. Is this correct? Wouldn't there still technically be mass as the photons are in-flight, which would cause its own gravitational field just as all matter does? How is this all consistent with the assertion that photons are massless? Is it really correct to say that photons don't have mass? It seems to be a big stretch. Please offer a more complete and physically accurate account of this mirror- box."}
{"id":"18900","title":"If photons are deflected by a strong gravitational field, then how come photons do not have mass?","text":"> **Possible Duplicate:**   >  Explain how (or if) a box full of photons would weigh more due to massless > photons It has been proved and showed through experiments that light can be bent by the Sun or any other body with considerable mass. Also light is nothing but photons. So can these photons be attracted by massive bodies if they have no mass?"}
{"id":"62411","title":"What is the cause the light is affected by gravity?","text":"I know that photons have no mass and that a photons exist only moving at the speed of light. So what is the cause that a massive astronomical object can bend a ray of light? I have two thoughts, but I am confused which of these, if any is correct:   1. A gravitational field of a massive astronomical object curves spacetime and affects the light traveling near this object **indirectly**. That is, for external viewers the light bends, but for the light itself it is still traveling in a straight line. So I mean, that the coordinates in that place are curved, but only for the external viewer.   2. Energy of a photon is equivalent to mass, so the gravity field interacts with photons **directly** the same way, as if they would have mass (equivalent to the energy)."}
{"id":"107808","title":"Does non-matter energy curve spacetime?","text":"I know that matter (mass) curves spacetime, but do other forms of energy do the same? I.e. is matter the only form of energy that curves spacetime?"}
{"id":"103918","title":"What are the factors affecting the spacetime curvature?","text":"Large masses in space as stars and planets cause a curvature in the spacetime fabric. What are the factors that affect this curvature? Is it only mass? And can we conclude these factors using Tensors?"}
{"id":"122003","title":"Light and Gravity - bending of light around a massive body","text":"Well, as I have read, a massive body can cause light to bend around itself due to its gravitational attraction. What I don't understand is how, as the Newtonian formula for the force of gravitational attraction is $$F = \\frac{Gm_1m_2}{r^2}.$$ As photons do not have (rest) mass, shouldn't there be no attraction between light and the celestial bodies, and therefore no bending? Surely I am missing a key point. Please point out my logical flaw?"}
{"id":"30489","title":"How to find the Green's Functions for time-dependent inhomogeneous Klein-Gordon equation?","text":"I'm trying to find the Green's functions for time-dependent inhomogeneous Klein-Gordon equation which is : \\begin{align*}‎‎ \\left[ -‎ ‎\\nabla ‎^2 + ‎‎‎‎\\frac{1}{c^2} ‎‎\\dfrac{\\partial ^2}{\\partial t^2} +‎ ‎‎‎\\kappa ‎^2 ‎‎\\right] ‎‎\\psi(‎{‎\\mathbf{r},t )}‎ = ‎‎‎‎\\rho‎(‎\\mathbf{r},t‎)‎ \\end{align*} It has been mentioned in the question that I can find the Green's functions : \\begin{align*}‎ ‎‎&G_R(‎\\mathbf{r} , t , ‎\\mathbf{r'} , t') = ‎\\dfrac{c}{8 \\pi ^2 ‎\\mathbf{R} i }‎\\dfrac{d}{d‎\\mathbf{R}} ‎\\int_{- ‎\\infty‎}^{+‎\\infty‎} ‎‎\\dfrac{e^{ i ‎\\frac{R}{c} ‎‎\\sqrt{q^2 - k^2 c^2}‎‎}}{‎\\sqrt{q^2 - k^2 c^2}‎} ‎e^{ - iq (t - t')} ‎dq‎ \\\\\\‎ &G_A(‎\\mathbf{r} , t , ‎\\mathbf{r'} , t') = ‎-\\dfrac{c}{8 \\pi ^2 ‎\\mathbf{R} i }‎\\dfrac{d}{d‎\\mathbf{R}} ‎\\int_{- ‎\\infty‎}^{+‎\\infty‎} ‎‎\\dfrac{e^{- i ‎\\frac{R}{c} ‎‎\\sqrt{q^2 - k^2 c^2}‎‎}}{‎\\sqrt{q^2 - k^2 c^2}‎} ‎e^{ - iq (t - t')} ‎dq‎ \\end{align*}‎ using the fourier transform, but when I use the fourier transform I don't gain the proper answer. The fourier transform which I use is the one which is generally given as : ‎\\begin{align*}‎ ‎f(r) = ‎\\dfrac{1}{‎\\sqrt{2 \\pi}‎} ‎\\int_{- \\infty}^{\\infty} ‎e^{ik.r}‎\\hat{f}‎(k) ‎dk ‎‎ \\end{align*} but from this transform I cannot find $G_A$ and $G_R$. Is there another transform which I should use to find the Green's functions? **Edit** The Green's function which I wind up with is : ‎\\begin{align*}‎‎ G_A(‎\\mathbf{r} , t , ‎\\mathbf{r'} , t')‎ = ‎‎‎‎\\dfrac{1}{(2\\pi)^4} ‎\\int ‎d^3\\mathbf{k} ‎dk' ‎‎\\frac{1}{k^2} ‎e^{i\\mathbf{k}.(‎\\mathbf{r} - ‎\\mathbf{r'}‎‎)}e^{ik'(t-t')}‎ \\end{align*} which is not even similar to the answer given here!"}
{"id":"30482","title":"Speed of light, breaking the barrier","text":"when the sound barrier is broken, a series of concentric waves of sound is produced.Does it mean when the speed of light barrier is broken, a ripple of photons are created in the space-time fabric?"}
{"id":"30485","title":"Electric Force is to Magnetic Force as Gravitational Force is to ...?","text":"One can no nothing about the magnetic force and yet arrive at it by taking the relativistic effects of a current and a moving charge system into account. I ask whether there exists such an inherent force in case of gravity."}
{"id":"7003","title":"Physics of tsunami: the relationship between wavelength, sea depth and the height of the water","text":"If I understand correctly, when an earthquake occurs, energy will be transferred to the water, resulting in water waves. As the waves reach seashore, because the sea depth is getting shallower and wavelength is getting shorter, the height of the wave gets push up, resulting in tsunami. In other words in deep sea, water won't get pushed up as high as the water in shallow seashore. Is my understanding correct? Is there a quantitative way to express the physics behind all this?"}
{"id":"13611","title":"Planck's Law in terms of wavelength","text":"I am drawing a blank when it comes to equation transformation. Wikipedia gives two equations for the spectral radiance of black body:   * First as a function of frequency $\\nu$: $$I(\\nu, T) = \\frac{2 h \\nu^3}{c^2}\\cdot\\frac{1}{e^\\frac{h \\nu}{k T} - 1}$$   * Then as a function of wavelength $\\lambda$ : $$I'(\\lambda, T) = \\frac{2hc^2}{\\lambda^5}\\cdot\\frac{1}{e^\\frac{h c}{\\lambda k T}-1}$$ And I don't see how they get $\\lambda^5$ term. I'm assuming that the transformation is just $\\nu \\rightarrow c\/\\lambda$, but that gives $$ \\frac{2 h \\nu^3}{c^2} \\Rightarrow \\frac{2 hc}{\\lambda^3} \\neq \\frac{2hc^2}{\\lambda^5} $$ Similar transformation happens at other parts in the article also. I'm obviously missing something, likely completely trivial."}
{"id":"78960","title":"Wien's displacement law in frequency domain","text":"When I tried to derive the Wien's displacement law I used Planck's law for blackbody radiation: $I_\\nu = \\frac{8 \\pi \\nu^2}{c^3} \\frac{h \\nu}{e^{h\\nu\/k_bT}-1}$ Asking for maximum: $\\frac{dI_\\nu}{d \\nu}=0:~0= \\frac{\\partial}{\\partial\\nu}(\\frac{\\nu^3}{e^{h \\nu\/k_bT}-1}) = \\frac{3\\nu^2(e^{h \\nu\/k_bT}-1) - \\nu ^3h\/k_bT \\cdot e^{h\\nu\/k_bT}}{(e^{h \\nu\/k_bT}-1)^2}$ It follows that numerator has to be $0$ and looking for $\\nu>0$: $3(e^{h \\nu\/k_bT}-1) - h \\nu\/k_bT \\cdot e^{h\\nu\/k_bT}=0$ Solving for $\\gamma=h\\nu\/k_bT$: $3 (e^\\gamma-1) - \\gamma e^\\gamma=0 \\rightarrow \\gamma=2.824$ Now I look at the wavelength domain: $\\lambda = c\/\\nu:~ \\lambda =\\frac{h c}{\\gamma k_b} \\frac{1}{T}$ but from Wien's law $\\lambda T = b$ I expect that $hc\/\\gamma k_b$ is equal to $b$ which is not: $\\frac{h c}{\\gamma k_b}= 0.005099$, where $b = 0.002897$ Why the derivation from frequency domain does not correspond the maximum in wavelength domain? I tried to justify it with chain rule: $\\frac{dI}{d\\lambda} = \\frac{dI}{d\\nu} \\frac{d \\nu}{d \\lambda} = \\frac{c}{\\nu^2} \\frac{dI}{d \\nu}$ where I see that $c\/\\nu^2$ does not influence where $dI_\\lambda\/d \\lambda$ is zero."}
{"id":"55856","title":"Different versions of Planck's law","text":"For a presentation in physics I am going to talk about black body radiation since our book just mentions Planck's law, Wien's displacement law and Stefan- Boltzmann's law. I want to derive Wien's law and Stefan-Boltzmann's law from Planck's law. I have managed to derive Stefan-Boltzmann's law from: $$u(\\nu)=\\frac{2\\pi h \\nu^3}{c^2}\\frac{1}{e^\\frac{h\\nu}{kT}-1}$$ by integrating over all $\\nu$ from $0$ to $\\infty$. However, my textbook says: $$u(\\lambda,T)=\\frac{2\\pi hc^2 }{\\lambda^5}\\frac{1}{e^\\frac{hc}{\\lambda kT}-1}$$ If I can get from the second formula to the first, I will be happy alltough, a simple and informal derivation of Plack's law would be very satisfying. I also managed to derive Wien's law by derivation and setting equal to zero by using: $$u_\\lambda=\\frac{8\\pi hc }{\\lambda^5}\\frac{1}{e^\\frac{hc}{\\lambda kT}-1}$$ What are the differences? And please, since I am Norwegian and do not know very advanced physics, do not use any advanced expression without explaining them. Thanks!"}
{"id":"86282","title":"What exactly means \"is a singlet under $SU(N)$\"","text":"I don't get a grip of what that exactly means. What IS an abstract singlet, doublet,... under $SU(N)$ or other groups?"}
{"id":"110253","title":"What does this notation mean?","text":"Terminology question. Reading this, in the middle of the second page, when it says > Left-handed quarks form 3 (3; 2; + 1 6 ) multiplets Qn (n = 1; 2; 3); What does this (3;2;+1\/6) mean?"}
{"id":"68484","title":"Notation for Standard Model Charges?","text":"Does anybody know what these following numbers describing an electron $(1, 1, -1)$ represent in $SU(3) \\times SU(2) \\times U(1)$? Or, these numbers that describe an up quark: $(3, 1, 2\/3)$? I'm really confused!"}
{"id":"41423","title":"What happens to the Lagrangian of the Dirac theory under charge conjugation?","text":"Consider a charge conjugation operator which acts on the Dirac field($\\psi$) as   $$\\psi_{C} \\equiv \\mathcal{C}\\psi\\mathcal{C}^{-1} = C\\gamma_{0}^{T}\\psi^{*}$$ Just as we can operate the parity operator on the Lagrangian, and we say that a theory has a symmetry if $$\\mathcal{P}\\mathcal{L}(t,x^{i})\\mathcal{P}^{-1} = \\mathcal{L}(t,-x^{i})$$ Suppose we operate $\\mathcal{C}$ on the Dirac Lagrangian what should we get? $$\\mathcal{C}\\mathcal{L}_{Dirac}(x^{\\mu})\\mathcal{C}^{-1} = \\mathcal{L}^{*}_{Dirac}(x^{\\mu}) ?$$ in analogy to the transformation of the scalar field $\\phi$ under charge conjugation. On a same note one can ask what equation should $\\psi_{C}$ satisfy? Should it satisfy the conjugated Dirac equation as $$(i\\gamma^{\\mu}\\partial_{\\mu} + m)\\psi_{C} = 0 ?$$ If so can someone give me the physical interpretation for it. I am asking this question as I want to explicitly use $\\psi_{C}$ and check whether it keeps the Dirac Lagrangian invariant. I have done a calculation by substituting $\\psi_{C}$ in the Dirac equation and have found it is not satisfying as shown below. $$(i\\gamma_{\\mu}\\partial_{\\mu} - m)\\psi_{C} = i(\\gamma_{\\mu}C\\gamma_{0}^{T})\\partial^{\\mu}\\psi^{*} - (C\\gamma_{0}^{T})m\\psi^{*}$$ We will use $C^{-1}\\gamma_{\\mu}C = - \\gamma_{\\mu}^{T}$ and $\\\\{\\gamma_{\\mu}^{T},\\gamma_{\\nu}^{T}\\\\} = 2g_{\\mu\\nu}$.   Consider \\begin{align} \\gamma_{\\mu}C\\gamma_{0}^{T} &= CC^{-1}\\gamma_{\\mu}C\\gamma_{0}^{T} \\\\\\ &= -C\\gamma_{\\mu}^{T}\\gamma_{0}^{T} \\\\\\ &= C\\gamma_{0}^{T}\\gamma_{\\mu}^{T} \\end{align} Hence substituting back we will get   \\begin{align} (i\\gamma_{\\mu}\\partial_{\\mu} - m)\\psi_{C} &= C\\gamma_{0}^{T}(i\\gamma_{\\mu}^{T}\\partial^{\\mu}\\psi^* - m\\psi^*) \\\\\\ &= C\\gamma_{0}^{T}[(i\\gamma_{\\mu}^{T}\\partial^{\\mu}\\psi^* - m\\psi^*)^{T}]^{T} \\\\\\ &= C\\gamma_{0}^{T}(i\\psi^{\\dagger}\\gamma_{\\mu}\\partial^{\\mu} - m\\psi^{\\dagger})^{T} \\\\\\ &= C\\gamma_{0}^{T}[(i\\bar{\\psi}\\gamma_{0}\\gamma_{\\mu}\\partial^{\\mu} - m\\bar{\\psi}\\gamma_{0})]^{T} \\\\\\ \\end{align} Now \\begin{align} \\gamma_{0}\\gamma_{\\mu}\\partial^{\\mu} &= (\\gamma_{0}\\partial^{t} + \\gamma_{i}\\partial^{i})\\gamma_{0} \\end{align} If we substitute back we will get \\begin{align} (i\\gamma_{\\mu}\\partial_{\\mu} - m)\\psi_{C} &= C\\gamma_{0}^{T}[\\\\{i\\bar{\\psi}(\\gamma_{0}\\partial^{t} + \\gamma_{i}\\partial^{i}) - m\\bar{\\psi}\\\\}\\gamma_{0}]^T \\\\\\ &\\neq 0 \\end{align}"}
{"id":"83357","title":"String tension in vertical circular motion","text":"Suppose that I have a point mass attached to a massless string and I am rotating it vertically. That means The mass is in uniform circular motion and the path of its motion is vertical circle. How does the tension change with respect to the position of the mass. More specifically is the tension in the string is only due to circular motion ($mv^2\/R$) or gravity plays a part in it ($mv^2\/R$ + something due to weight)?"}
{"id":"2378","title":"What is our location relative to the Big Bang?","text":"Given what we know about space, time and the movement of galaxies, have we or can we determine what our position is in relation to the projected location of the Big Bang? I've read some introductory papers on the superstructure and galaxy cluster movements, but none of them specifically mentioned space in terms of relative or absolute positions relating to the original position of the Big Bang. So my question is, does our current understanding of the structure and workings of the Universe give us a good enough estimate to determine our location relative to the Big Bang or can we never guess at it, since every viewpoint in our universe looks the same in every direction?"}
{"id":"33823","title":"The ideal trampoline","text":"Suppose we have a mass attached to the top of an ideal (linear and massless) spring oriented vertically in a uniform gravitational field, and on top of that mass there is another mass resting on it. The two masses are not attached at all, so they will lose contact with each other as the normal force is about to become negative. Also suppose that once the two masses separate and collide again, they undergo perfectly elastic collisions. First of all, is there a name for systems like this? It seems like an \"ideal trampoline\" to me but searching for that doesn't yield much. Has anyone ever discussed it in a book? Second of all, is this system chaotic? For sufficiently small oscillations, of course, the masses remain in contact the whole time and you get simple harmonic oscillation, but above some threshold the free mass will keep bouncing off the spring-attached mass and it's quite nontrivial to figure out what eventually happens. Do you get interesting things like period doubling?"}
{"id":"679","title":"Learning physics online?","text":"I'm thinking of following some kind of education in physics online. I have a master degree in Computer Science and have reasonable good knowledge in physics. I would like a program of 1-2 years and I'm more interested in particle physics. Is there any good online program that offer something similar?"}
{"id":"14568","title":"How would a newtons cradle act in zero gravity?","text":"I imagine that the first ball would strike the rest as normal, but what would the last ball do, without gravity to swing it back?"}
{"id":"72825","title":"Can an Incandescent Bulb be considered as a heater?","text":"Due to an experiment, I need a small heater (around 70 to 100 watts). I intend to use an incandescent bulb so it can act as a heater. What I wonder here is will a 70 watts Incandescent Bulb be equal to a 70 watts heater?"}
{"id":"31463","title":"Intensity of unpolarized light","text":"If the electromagnetic field of an unpolarized plane wave is written as $$\\bar{E}(t,\\bar{x})=(\\bar{E}_{0x}+\\bar{E}_{0y}e^{i\\delta(t)})e^{i(\\bar{k}\\bar{x}-\\omega t)}$$ $$\\bar{B}(t,\\bar{x})=\\frac{1}{\\omega}\\bar{k}\\times\\bar{E}(t,\\bar{x})$$ where $\\delta(t)$ is a random phase shift, then the intensity of this light is given by the time-average of the norm of the Poynting vector $$I(\\bar{x})=\\left<\\|\\bar{P}(t,\\bar{x})\\|\\right>_{t}$$ $$\\begin{split}\\bar{P}(t,\\bar{x})=&\\frac{1}{\\mu_{0}}\\mathcal{R}e(\\bar{E}(t,\\bar{x}))\\times\\mathcal{R}e(\\bar{B}(t,\\bar{x}))\\\\\\ =&\\frac{1}{\\omega}\\bar{E}_{0x}^{2}\\cos^{2}(\\bar{k}\\bar{x}-\\omega t)\\bar{k}+\\frac{1}{\\omega}\\bar{E}_{0y}^{2}\\cos^{2}(\\bar{k}\\bar{x}-\\omega t+\\delta(t))\\bar{k} \\end{split}$$ $$\\Leftrightarrow I(\\bar{x})=\\frac{1}{2}c\\epsilon_{0}\\bar{E}_{0x}^{2}+c\\epsilon_{0}\\bar{E}_{0y}^{2}\\left<\\cos^{2}(\\bar{k}\\bar{x}-\\omega t+\\delta(t))\\right>_{t}$$ Can we simplify this further? Is the remaining average also $1\/2$?"}
{"id":"31467","title":"Speaker cabinet to improve sound of mobile phone loudspeaker (music)","text":"I want to experiment with an enclosure for my phone so the frequency response has a little more punch at the bottom end. I understand that something can't be created from nothing, but enclosures work for drivers so I can't see why not for the phone? What sort of cabinet design would do the job? It would be nice if it preserved a natural mid and treble as well. Even if this design just dampens medium and high frequencies that would be fine too. The response without any enclosure seems to have started to tail off at about 260Hz and is almost gone by 130Hz. I can just about hear 60Hz if I put the volume on full and my ear against the speaker. (Don't try this on your hifi at home kids) Thanks"}
{"id":"31466","title":"Deviation from power law distribution of earthquakes","text":"One of the most accepted frameworks for the relationship between the magnitude and frequency of an earthquake is that of the critical phenomena. In this framework, the magnitude of events must be distributed following a power law. However the Gutenberg-Richter law clearly shows a deviation at low magnitudes from the power law called a _roll-off_. How can the roll-off deviation of the GR prediction be explained?"}
{"id":"31464","title":"Hilbert of quantum gravity: bulk $\\otimes$ horizon","text":"I was reading a paper dealing with the Hilbert of quantum gravity (or more precisely what should it look like considering what we know from QM and GR) ref: http:\/\/arxiv.org\/abs\/1205.2675 and the author writes the following: $${\\cal{H}_M} = {\\cal{H_{M,\\,\\textrm{bulk}}}}\\otimes{\\cal{H_{M,\\,\\textrm{horizon}}}}$$ for a specific manifold $\\cal{M}$. I know very little about the holographic principle and the AdS-CFT correspondence but isn't it a redundant description? If there is a duality between the gravitational theory in the bulk and the CFT on the boundary, knowing one means knowing the other, so why can't we restrict ourselves to one of the Hilbert spaces? Moreover, the author writes, a couple of lines after this first element, that the two Hilbert space have same dimension ( $\\textrm{exp}({\\frac{{\\cal{Area}}}{4}})$ ) so they are totally equivalent, as a complex Hilbert space is only defined by its dimension."}
{"id":"47266","title":"Rolling (without slipping) ball on a moving surface","text":"I've been looking at examples of a ball rolling without slipping down an inclined surface. What happens if the incline angle changes as the ball is rolling? More precisely I've been trying to find equations for programming a simulated (2D) ball rolling inside a swinging bowl\/arc. I thought I could still use the same equations for just having a ball rolling inside a still bowl (see below) and the changes in the incline angle (tangent at the contact point of the ball and the bowl) would take care of itself: For a ball rolling inside a bowl: The only torque acting on the ball is the frictional force: $τ=Iα=fr$, using the rolling without slipping condition $a=rα$ and the moment of inertia for a solid sphere, $I = \\frac{2}5 mR^2$, we get $f=\\frac{2}5ma$. The net force acting on the system is gravity and the force of friction, $F=ma=mgsinθ−f$ and therefore, $a=\\frac{5}7gsinθ$ I am speculating that due to the surface itself moving (swinging on a circular path), it's the relative motion that contributes to the friction? But, I don't know how to include that. Can someone please help me?"}
{"id":"47139","title":"Who used the concept of symmetries first?","text":"Who \"invented\" the concept of symmetries? This article is quite extensive, but it blurs the history with the modern understanding. http:\/\/plato.stanford.edu\/entries\/symmetry-breaking\/ Some of the concepts can be traced to Galileo and Newton, but I'm quite certain the modern notion is incompatible with their view of the world. Does the notion come from group theory specifically? Can the first mention be traced accurately? > Although the spatial and temporal invariance of mechanical laws was known > and used for a long time in physics, and the group of the global spacetime > symmetries for electrodynamics was completely derived by H. Poincaré [7] > before Einstein's famous 1905 paper setting out his special theory of > relativity, it was not until this work by Einstein that the status of > symmetries with respect to the laws was reversed."}
{"id":"45802","title":"Stiffness tensor","text":"Let's have a stiffness tensor: $$ a^{ijkl}: a^{ijkl} = a^{jikl} = a^{klij} = a^{ijlk}. $$ It has a 21 independent components for an anisotropic body. How does body symmetry (cubic, hexagonal etc.) change the number of independent components of the tensor? For example, for cubiс symmetry it has three components. How to explain it? Update. Is the explanation a simple realization of idea $$ a_{ijkl}' = \\beta_{im}a^{m}\\beta_{jt}a^{t}\\beta_{k f}a^{f}\\beta_{ld}a^{d} = a_{ijkl}, $$ where $\\beta_{\\alpha \\beta}$ is a components of a matrix $\\beta$ for rotation around z-, x-, y-axis at the same time?"}
{"id":"87502","title":"Why do we must know the Weyl tensor for 4-dimensional space-time?","text":"I heard that we must know the Weyl tensor for fully describing the curvature of the 4-dimensional space-time (in space-time with less dimensions it vanishes, so I don't interesting in cases of less dimensions). So I have the question: what is physical (or geometrical) sense of the Weyl tensor and why don't we need only Riemann tensor for describing the curvature? Does it connected with gravitational waves directly?"}
{"id":"39277","title":"How can we have massive states of strings and CFT on the string worldsheet at the same time?","text":"Ok, so we can have conformal invariance on a string world sheet. However, it is well known that to preserve conformal symmetry we require states to be massless. So how is it that string theories incorporate CFT but allows massive states? Is it because the CFT is on the worldsheet and therefore applies to the worldsheet coordinate X (X is treated as the field) - however the physical states arise from the the creation\/annihilation operators that we get from X? Therefore the CFT doesn't actually act on the states (massive or massless) but instead it acts on the field X."}
{"id":"84971","title":"Rotation axis of a rigid body","text":"I am confused about a trivial concept. Let the rotation of a rigid body, say with one point fixed, be described by the equation $\\vec{x}(t)=R(t)\\vec{x}(0)$, with $R(0)=I$. Then, at each instant there is only one real eigenvector of $R(t)$ with eigenvalue 1 that we may call $\\vec{v}(t)$ and which we may take to be normalized. That vector $\\vec{v}(t)$ is what geometrically we would call the (instantaneous) axis of the rotation. Kinematically, however, the instantaneous axis of rotation is the line of points with vanishing instantaneous velocity $\\dot{\\vec{x}}(t)=\\vec{\\omega}(t)\\times\\vec{x}(t)=0$. That is the direction of $\\vec{\\omega}(t)$. As is obvious (for example from the Rodrigues formula), in general $\\vec{v}(t)$ and $\\vec{\\omega}(t)$ are not parallel. So, why are there two axes of rotation, and does $\\vec{v}(t)$ play any role in the kinematics\/dynamics of the motion?"}
{"id":"84972","title":"Diffraction Grating in spectrophotometer","text":"In terms of spectrophotometer, we need to make the light monochromatic before passing it to sample. For that, we use diffraction grating. From the research, I found out that the grating reflects light into different wavelengths. Also, the grating moves in such a way that each wavelength has an opportunity to pass through slit to sample. According to one journal, the monochromatic light is needed to make sure every photon has same energy and wavelength, which means everyone of them, has an equal opportunity to hit the particle. Is there better explanation in this theory? Also, Am I right in thinking that all wavelengths scattered from grating will hit the particles but particles will only absorb certain wavelength?"}
{"id":"12650","title":"units of measure","text":"This question is rather historical one. kilometres can be defined in metres, metres in centimetres, centimetres in millimetres. There must be some elementary unit (like millimetre or smth.) which cannot be defined in smaller units. The question is : How does this elementary unit came into being?? e.g. How did scientists decide about exact distance between point A and point B which is considered millimetre??"}
{"id":"88227","title":"What's the importance of conformal transformations in general relativity?","text":"I tried to understand the importance of conformal transformations in general relativity, but I failed. I didn't see that conformal transformations help to simplify the metrics, and also I didn't see that some physical metric (i.e., metric which describes geometry of some physical system) with conformal scale factor. Can you give some examples when conformal transformations are physically useful?"}
{"id":"21057","title":"Work and Area under a Curve relating to Hooke's Law","text":"> If it takes work W to stretch a Hooke’s-law spring (F = kx) a distance d > from its unstressed length, determine the extra work required to stretch it > an additional distance d (Hint: draw a graph and give answer in terms of > W!). I don't understand why the answer is not 2W since Force is proportional to x, or even how to begin using a graph to disprove why the answer is not 2W. Any help would be greatly appreciated."}
{"id":"110940","title":"Nuclear waste reclamation","text":"Is it possible to reclaim nuclear waste from commercial reactors for useful purposes, if not necessarily energy production?"}
{"id":"45573","title":"Will I recieve a shock if I charged a capacitor then i touch the 2 poles in same time?","text":"What happens if I charged a capacitor then i touch the 2 poles in same time?"}
{"id":"43632","title":"Definition of Fine-Tuning","text":"I've looked in and out the forum, and found no precise definition of the meaning of fine-tuning in physics. **QUESTION** Is it possible to give a precise definition of fine-tuning? Of course, I guess most of us understand the empirical meaning of the phrase... but it seem so ethereal, that's the reason behind my question."}
{"id":"53137","title":"How is multiplicity given by 2S+1?","text":"Suppose there are two electrons in an atom with $s_1 = \\frac{1}{2}$, $l_1 = 1$ and $s_2 = \\frac{1}{2}$, $l_2 = 1$. Hence the total $S$ (of the atom) may be +1 or 0. And total $L$ is either $+2$, $+1$ or $0$. Now If we consider $$\\begin{align} S=1,L=2 &\\to 2S+1=3; J=3,2,1\\\\\\ S=0,L=2 &\\to 2S+1=1; J=2\\\\\\ S=1,L=1 &\\to 2S+1=3; J=2,1,0\\\\\\ S=0,L=1 &\\to 2S+1=1; J=1\\\\\\ S=1,L=0 &\\to 2S+1=3; J=1\\end{align}$$ But the last one does not show that $2S+1$ is multiplicity as it has only one $J$ value. Where am I making a mistake?"}
{"id":"90555","title":"Help in understand Magnetostatic Energy","text":"$$E_{\\mathrm{ms}} = \\frac{1}{2}\\mu_0 \\int_V \\mathbf{M} \\cdot \\mathbf{H}_{\\mathrm{ms}} d^3 r$$ I can't understand this formula, what is the magnetostatic stored potential energy?! What does it show? Does it explain anything relative to magnetization?"}
{"id":"90552","title":"Constraints of massive relativistic point particle in hamiltonian mechanics","text":"I try to understand constructing of Hamiltonian mechanics with constraints. I decided to start with the simple case: free relativistic particle. I've constructed hamiltonian with constraint: $$S=-m\\int d\\tau \\sqrt{\\dot x_{\\nu}\\dot x^{\\nu}}$$ $\\phi=p_{\\mu}p^{\\mu}-m^2=0$ $-$ first class constraint. Then $$H=H_{0}+\\lambda \\phi=\\lambda \\phi.$$ **So, I want to show that I can obtain from this Hamiltonian the same equation of motion, as obtained from Lagrangian.** But the problem is that I'm not sure what to do with $\\lambda=\\lambda(q,p)$. I tried the following thing: $\\dot x_{\\mu}=\\\\{x_{\\mu},\\lambda \\phi\\\\}=\\\\{x_{\\mu},\\lambda p^2\\\\}-m^2\\\\{x_{\\mu},\\lambda\\\\}=\\lambda\\\\{x_{\\mu},p^2\\\\}+p^2\\\\{x_{\\mu},\\lambda\\\\}-m^2\\\\{x_{\\mu},\\lambda\\\\}=2\\lambda \\eta_{\\mu b} p^b+p^2\\\\{x_{\\mu},\\lambda\\\\}-m^2\\\\{x_{\\mu},\\lambda\\\\}=2\\lambda \\eta_{\\mu b} p^b+p^2\\frac{\\partial \\lambda}{\\partial p^{\\mu}}-m^2\\frac{\\partial \\lambda}{\\partial p^{\\mu}}$ $\\dot \\lambda=\\\\{\\lambda, \\lambda \\phi \\\\}=\\\\{\\lambda,\\lambda p^2\\\\}-m^2\\\\{\\lambda,\\lambda\\\\}=\\lambda\\\\{\\lambda,p^2\\\\}+p^2\\\\{\\lambda,p^2\\\\}=2\\lambda\\eta_{ak}p^{a}\\frac{\\partial \\lambda}{\\partial x^{k}}$ $\\dot p_{\\mu}=\\\\{p_{\\mu},\\lambda p^{2}-m^2\\lambda \\\\}=p^{2}\\\\{p_{\\mu},\\lambda\\\\}-m^2\\\\{p_{\\mu},\\lambda\\\\}=-p^{2}\\frac{\\partial \\lambda}{\\partial x^{\\mu}}+m^2\\frac{\\partial \\lambda}{\\partial x^{\\mu}}$ If we recall that $p^2-m^2=0$, then we get from the third equation: $\\dot p=0$, and from the first: $\\dot x_{\\mu}=2\\lambda\\eta_{ak}p^{a}$. So we have 1) $\\dot x_{\\mu}=2\\lambda\\eta_{\\mu b}p^{b}$ 2) $\\dot \\lambda=2\\lambda\\eta_{ak}p^{a}\\frac{\\partial \\lambda}{\\partial x^{k}}$ 3) $\\dot p=0$ _But I dont know what to do next._ Can you help me?"}
{"id":"7737","title":"Introductory texts for functionals and calculus of variation","text":"I am going to learn some math about functionALs (like functional derivative, functional integration, functional Fourier transform) and calculus of variation. Just looking forward to any good introductory text for this topic. Any idea will be appreciated."}
{"id":"58102","title":"microcanonical distribution","text":"We know that in an isolated system, the density matrix is the microcanonical distribution matrix. That this the possibility for all the states with energy in a certain interval is a constant? But how can I deduce this from the postulate of equal probability?"}
{"id":"37729","title":"Is there a finite amount of mass in the universe?","text":"So, I'm not too physics savvy but I am curious to ask. Is there a finite amount of mass in the universe? or is there more and more being created from somewhere or something? If the universe is infinite, and there's a finite amount of mass, that just seems kinda weird I guess. Hopefully this isn't too dumb a question..."}
{"id":"90087","title":"Adiabatic proccess and Carnot cycle in a photon gas","text":"I am making a comparation between the photon gas and the ideal classic gas for my Thermodynamics class. The photon gas is defined by the equations: $$U=aVT^4 $$ $$P=\\dfrac{1}{3}aT^4$$ I found this document: http:\/\/www.csupomona.edu\/~hsleff\/PhotonGasAJP.pdf which explain how to find some basic things, like enthalpy and entropy. It says that a great exercise is to compare the Carnot cycle of the photon gas with the Carnot cycle of the ideal gas. According to it, the efficiency is $\\eta=1-\\frac{T_2}{T_1}$ the same as the ideal gas. I think that this is really interesting for my comparation, so I'm trying to calculate the Carnot cycle efficiency for this gas. I have no problem with the isothermal process, which is solved in that document: $$W_{ab}=-\\dfrac{1}{3}aT^4\\Delta V$$ However, I'm not sure if my result of the adiabatic process is correct. Work is $W=\\int PdV$. Now, I can use the photon gas adiabatic equation (see the document) $PV^{4\/3}=k$, where $k$ is a constant, to substitute $P$ in work equation, and integrate to obtain: $$W_{bc} =\\dfrac{3}{4}k\\left( \\dfrac{1}{V_b^3} - \\dfrac{1}{V_c^3} \\right)$$ I'm not sure if this result is correct. When I try to calculate the efficiency of the cycle, I have: $$\\eta=\\dfrac{|W_T|}{|Q_{ab}|}=\\dfrac{|W_{ab}+W_{bc}+W_{cd}+W_{da}|}{|Q_{ab}|}$$ where $W_{ab}$,$W_{cd}$ are isothermal and $W_{bc}$,$W_{da}$ are adiabatic. The heat is also defined in the document as: $$Q_{ab}=\\dfrac{4}{3}aT^4\\Delta V$$ But with these values I can't obtain the correct expression for the efficiency, or I don't know how to reduce the efficiency expression to obtain what I want. Which is the correct way to calculate adiabatic work in a photon gas? And the Carnot cycle efficiency? Thank you all for your answers :D"}
{"id":"13217","title":"Shaping a wire such that a bead sliding on it has exactly isochronous oscillations","text":"Let a wire be shaped according to some even function $y=f(x)$, with $f'(0)=0$ and $f''(0)>0$, and let a bead of negligible size slide frictionlessly on the wire. Let the bead oscillate under the influence of gravity about $x=0$ with amplitude $A$ (i.e., between $x=-A$ and $x=+A$) and frequency $\\omega(A)$. Clearly $\\omega$ is nearly constant for small $A$; it differs from the frequency $\\omega_o$ of simple harmonic motion by at most $O(A^2)$. By choosing $f$ to be a fourth-order polynomial, we could presumably adjust the wire's shape so as to eliminate the errors of order $A^2$ and make $\\omega(A)$ constant up to $O(A^4)$. Possibly we could continue this process of approximation and make all the derivatives $d^n\\omega\/dA^n$ vanish up to some finite $n$, or maybe for all $n$. If the derivatives can be made to vanish for all $n$, then I think $\\omega$ would have to be nonanalytic at $x=0$. It seems impossible that there is any $f$ such that isochrony holds for arbitrarily large $A$. No matter how steep you make the sides, the bead can't do any better than accelerating downward with acceleration $g$. Therefore I think the best $f$ you can find is probably one that blows up to infinity at $|x|$ equal to some $x_{max}$. On dimensional grounds, we would have to have $x_{max}=cL$, where $c$ is a unitless constant and $L=g\/\\omega_0^2$. So my multipart question is: (1) Is there a function $f$ that gives $d^n\\omega\/dA^n=0$ for all $n$? If so, ... (2) How is $f$ characterized, and what is $c$? (3) Is $\\omega(A)$ analytic at $x=0$, and if so, what is its radius of convergence to its Taylor series in units of $L$?"}
{"id":"33957","title":"What are the properties of the partially polarized light on refraction?","text":"When a ray of ordinary light is passed on the surface of the water the reflected light will be completely polarized( vibrations in one plane). My question is what will be **plane of vibration in the partially polarized light** that undergoes refraction? **How many planes of vibration** will be there? Deep explanation focusing on the **planes of vibration** of the partially polarized light would be appreciated."}
{"id":"123167","title":"Partial derivatives in Lagrangian formalism","text":"Suppose I have a function $f = xy$. A partial derivative of $f$ with respect to $x$ implies holding $y$ constant: $$ \\frac{\\partial f}{\\partial x} = y $$ Does this mean that in order to evaluate this derivative, $y$ cannot depend on $x$? For example, if $y = x$ then $$ \\frac{\\partial f}{\\partial x} = \\frac{\\partial (x^2)}{\\partial x} = 2x $$ Which is inconsistent with the first calculation of $\\frac{\\partial f}{\\partial x}$. If $y$ indeed cannot depend on $x$, then how does the Lagrangian formalism of classical mechanics make sense? The Lagrangian is a function of $q$ and $\\dot{q}$, and when we evaluate $\\frac{\\partial L}{\\partial \\dot{q}}$ we 'ignore' all of the $q$ dependence, even though $\\dot{q}$ is a function of $q$."}
{"id":"107061","title":"Why does a cup with 100 g water float when placed on another cup with 50 g of water?","text":"Imagine we have cup A with 50 g of water and cup B (smaller in width than A) with 100 g of water. Now put cup B into cup A. If the width of both cups are of comparable size then the cup with 100 g of water floats. It does not touch the bottom of cup B. Now think about Archimedes law of flotation. It says that the weight of displaced liquid = weight of the floating object. However in this case the bottom cup has only 50 g of water. How can an object float without displacing water equal to its own weight? Am I not applying Archimedes principle correctly or because of both things beings of comparable size Archimedes principle does not apply?"}
{"id":"20011","title":"why is there no ninth gluon?","text":"A teacher of mine told me once that there were no ninth gluon because such a one should be white and interact infinitely far, and no one has been observed. Is there also a theoretical reason?"}
{"id":"45391","title":"Can we transfer the charge on a plate of a capacitor elsewhere?","text":"I know it is hard , but can we transfer the charge on a capacitor plate elsewhere?"}
{"id":"34178","title":"How can an object's instantaneous speed be zero and it's instantaneous acceleration be nonzero?","text":"I'm studying for my upcoming physics course and ran across this concept - I'd love an explanation."}
{"id":"94702","title":"Calculating axle load in cargo container (pics are included)","text":"I'm current developing cargo loader software, but i have a little challenge with calculating load for each axle when a cargo is placed on container. I know that all axles will be affected (the nearest ones are the most affected ones of course), and I want to find approximately the value of each load. Here's example scenario in pictures: ![enter image description here](http:\/\/i.stack.imgur.com\/LCqjF.jpg) I tried to find loads using balance rules taking each point as a pivot and calculating equation based on that position as below (EDIT: on the following figure in case 4 the 5th axle is pivot, I forgot to show it on figure): ![enter image description here](http:\/\/i.stack.imgur.com\/3AP9g.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/v0D88.jpg) But I think I have problem with above solution as I could not find F3, F4, F5 in special examples (putting \"d\" distance values from pivot - P.S in this last picture all \"d\"s are different distances from pivot in that case - so d3 in the first equation is not equal to d3 in second equation as these both have different distances to different pivots) - for easy calculation you can use Cramer's rule online calculator here. I tried to solve this problem with moment but I could not done as I'm not good in physics :( I could not find relations between different forces in that case, I had only F1*d1 + F2*d2 + F3*d3 = F4*d4 + F5*d5 and F1 + F2 + F3 + F4 + F5 = mg ===> i got 5 unknowns but only 2 equations :( So, i need your help! Any suggestions will be appreciated!"}
{"id":"55774","title":"Static Friction work and Energy","text":"I have this problem: A 4.0 kg block is given an initial speed of 8.0 m\/s at the bottom of a 20° incline. The frictional force that retards its motion is 15.0 N. (a) If the block is directed up the incline, how far does it move before stopping? (b)Will it slide back down the incline? I've managed to get part (a) which was 4.51m but I'm not sure how to start part (b).. The coefficient of Kinetic Friction was 0.407 Any ideas?"}
{"id":"91456","title":"How does a transverse wave propogate in a medium?","text":"I have been told that trasverse wave propogates by the oscillation of medium particles in direction perpendicular to propogation. Consider a wave on a taught string (x-y plane). What is the mechanism of movement of the same sinosudial function along the string? Suppose we start moving the particle on one end sinosudialy the how come the next particle also moves sinosudialy? How does elastic force between them does this work? If we displace the particle, there is some elastic force developed which is at some angle with horizontal so the next particle should move at some angle and its vertical displacement should be less than the previous one. So how come in a propogating wave all particle rise to the same height?"}
{"id":"93242","title":"Can I make a cold air trap (thermal equilibrium)","text":"Thermodynamic buoyancy. I have an air intake for combustion air entering into a basement furnace room. During cold weather, the air enters unchecked (no damper on pipe is allowed). I have this 5 foot vertical pipe, suspended downwards within a 2 foot high box, terminating 12\" from the bottom. The cold air spills over the top of the box (combustion appliances are off at this point). How do I create a \"cold air trap\" like a p-trap (thermal equilibrium)? Would raising the height of the box help?"}
{"id":"29471","title":"How is spacetime depicted in quantum field theory?","text":"How is spacetime depicted in quantum field theory? Is space and time completely separate, and time is just nature of law as in Newtonian mechanics?"}
{"id":"29475","title":"What happened to David John Candlin?","text":"This is an ultra-soft question about relatively recent history. While reading some of Mandelstam's papers, I noticed that he cites David John Candlin consistenly whenever he does anything with Grassman path-integral. Everyone else cites Berezin. So I read Candlin's 1956 paper, and I was stunned to find a complete and correct description of anticommuting variables, presented more lucidly than anywhere else, with a clear definition of Grassman integration, and a proof that it reproduces the Fermionic quantum field. This is clearly the original source of all the Grassman methods. I was stunned that the inventor of this method is quietly buried away. I wrote the Wikipedia page on the guy, but I couldn't find out anything beyond the sketchy stuff I found on an old Princeton staff listing. The fellow doesn't google very well at all. Here are the questions:   * Is he still alive? (Hello? Are you there?)   * Did he become the experimental physicist David John Candlin in the late 1970s\/early 1980s? Or is this someone else with the same name?   * Did he get any credit for his discovery? I mean, this is one of the central tools of modern physics, it is used every day by every theorist, and the inventor is never mentioned. It's 50% of the path integral. Why the silence?"}
{"id":"29805","title":"Richtmyer Meshkov instability in MHD","text":"In magnetohydrodynamics, the Richtmyer Meshkov instability is found to get suppressed by application of longitudinal magnetic field. Exactly what happens at the interface? Why instability gets suppressed? (How one can get the physical intuition of what is happening?)"}
{"id":"129037","title":"Differentiation operator with respect to observable acting as a function of the observable?","text":"In his _Principles of Quantum Mechanics_ Dirac writes: $$\\int \\langle \\phi \\frac{d}{dq}|q'\\rangle dq' \\psi(q')=\\int \\phi(q') dq' \\frac{d\\psi(q')}{dq'}.$$ To me it is rather strange, and it seems as if he was treating the operator $\\frac{d}{dq}$ as a function of the observable canonical coordinate $q$, beacuse for functions of observables he gave the definition: $$f(\\xi)|\\xi'\\rangle=f(\\xi')|\\xi'\\rangle,$$ where $|\\xi'\\rangle$ is an eigenket of the observable $\\xi$. Using this analogy, taking $f(q)=\\frac{d}{dq}$ one could write $$\\frac{d}{dq}|q'\\rangle=\\frac{d}{dq'}|q'\\rangle$$ but then we would also need to differentiate $|q'\\rangle$, since it is a function in $q'$. So what happens here? Please explain! Also, as a side question: why does $$\\int \\langle \\phi \\frac{d}{dq}|q'\\rangle dq' \\psi(q')=-\\int \\frac{d\\phi(q')}{dq'} dq' \\psi(q')$$ imply $$\\langle \\phi \\frac{d}{dq}|q'\\rangle=-\\frac{d\\phi(q')}{dq'}~?$$ Simply because the results of the integrations equal, that doesn't mean that their arguments also equal. The book can be accessed here. The formulas are on page 90, using the books original numbering."}
{"id":"95023","title":"Does a moving object curve space-time as its velocity increases?","text":"We always hear how gravity bends space-time; why shouldn't velocity? Consider a spaceship traveling through space at a reasonable fraction of the speed of light. If this spaceship, according to special relativity, gains mass as a factor of _y_ as it approaches _c_ , then its gravitational field should increase in strength as well. Hence, space-time should warp. Note: Changes in space-time, gravity and mass should only be measureable by an outside observer with a different velocity. Those inside of the ship moving with it would not be able to measure the change in these properties."}
{"id":"134105","title":"Does a moving star have a larger gravitational pull?","text":"A moving star's relativistic mass is larger than its rest mass. Is its gravitational pull larger? What about its inertial mass? Does it have larger inertial mass, keeping in mind that inertial mass should equal gravitational mass according to the equivalence principle. I suppose this quote from wikipedia is relavant: > However, it turns out that it is impossible to find an objective general > definition for the concept of invariant mass in general relativity. At the > core of the problem is the non-linearity of the Einstein field equations, > which makes it impossible to write the gravitational field energy as part of > the Stress–energy tensor in a way that is invariant for all observers. For a > given observer, this can be achieved by the Stress–energy–momentum > pseudotensor.[21]"}
{"id":"8782","title":"How to determine viscous dampening coefficient of spring?","text":"I'm trying to determine the viscous dampening coefficient of a spring $c$. Read about it on Wikipedia here. The two equations which I have are: $f=-cv$ and $ma+cv = -kx$ I know the spring constant $k=5$, the mass is $50\\text{ }\\mathrm{g}$ and the initial amplitude of the spring is $10\\text{ }\\mathrm{cm}$."}
{"id":"556","title":"What is the fallacy in this infinite motion machine?","text":"![alt text](http:\/\/i.stack.imgur.com\/IKFiu.jpg) I realize this isn't possible, but I can't see why not, especially if you change the model a little bit so that the balls simply travel through a tube of water on the way up, rather than exactly this model. Please be clear and detailed. I've heard explanations like \"the balls wouldn't move\" but that doesn't do it for me - I really do not see why the balls on the right would not be pulled\/pushed up, and the rest of the chain wouldn't continue on."}
{"id":"82934","title":"Why does this perpetuum mobile not work?(Gases and Densities)","text":"I recently came up with the following concept. It is very simple, and may have been thought of before. A picture says more than a thousand words, so here is it explained in a picture: ![Conveyor belt using densities of gases to keep it in motion](http:\/\/i.stack.imgur.com\/DQhdL.png) _Note that water was used to make the example easier to understand. Another gas (denser than Gas A and B) could be used instead, resulting in less friction than when using water._    At first glance, it seems that this machine could run for an indefinite amount of time. But I do not deem it possible to break the law of energy conservation. However, I have a hard time finding out what kind of force would cause this machine to slow down and stop."}
{"id":"95077","title":"Force and Energy of interaction of conducting sphere and point charge","text":"> _A completely isolated neutral conducting sphere of radius $R$ is kept such > that its center is at a distance of $r\\left( >R\\right)$ from a point charge > $+Q$._ How can I find the force of interaction of the induced charges and the point charge, or at least the energy? I can't use \"method of images\" because the sphere is not grounded. **Note:** The actual question has a sphere already charged with $+Q$ charge and it asks for the $r$ at which the point charge is in equilibrium. I thought of breaking the force down into a superposition of two forces, one from the $+Q$ of sphere and the other from the induced charges. If there is some other way to solve this, I would like to know that too."}
{"id":"68041","title":"Best EM\/Photon rocket using avalable tech?","text":"Given 10-1000 Watts of electrical power (and no other consumables), what is the current best way to turn it into thrust? Just running it through a heater on an insulating pad would result in an IR thruster, but has bad focus. A laser has good focus but only for a small percentage of the energy."}
{"id":"34984","title":"What does it mean to increase volume by X decibels?","text":"I am trying to decipher what decibels are: http:\/\/en.wikipedia.org\/wiki\/Decibel It seems to be a log ratio of audio amplitude multiplied by a constant. I am confused by what this means though. If my original volume is X, what does say increasing the volume by Y decibels mean? Does it mean `New Volume = 10 log ( Y \/ X )`?"}
{"id":"119139","title":"Significance of an operator with a negative variance","text":"Suppose we have a quantum state, well described by its time-independent wave function $\\Psi$. And we have a well-defined Hermitian (self-adjoint) operator $\\hat{A}$. We successfully evaluate the expectation value of the operator $\\hat{A}$. Next we derive the general formula for the higher moments of $\\hat{A}$ (i.e. the expectation value of $\\hat{A^n}$ for $n=2,3,4…$). Finally we scale the operator $\\hat{A}$ appropriately, in order to make the result dimensionless and to remove a possible growth factor (of type $C^n$) in the moments. We obtain: $$ <\\hat{A^n}> = Cn + D $$ for $n=1,2,3,...$ and where $C$ and $D$ are constants. Let us now define a new operator $\\hat{B}$ as follows: $$ \\hat{B} = \\hat{A^{n+1}} - \\hat{A^n} $$ We can easily verify that the first and second moment of B are given by: $$ < \\hat{B} > = C $$ $$ < \\hat{B^2} > = 0 $$ Therefore the variance of operator $\\hat{B}$ is negative! In violation of statistical laws. Should we conclude from this example that the results derived for the moments of $\\hat{A}$ must be flawed? Or should we conclude that the new operator $\\hat{B}$ is not a proper operator after all and therefore its strange properties are insignificant with respect to questions about the validity of $\\hat{A}$? $$\\begin{align} <\\hat{B}> &= <\\hat{A^{n+1}}> \\- <\\hat{A^n}> \\\\\\ &= C(n+1) + D - Cn - D = C \\end{align}$$ $$\\begin{align} <\\hat{B^2}> &= <\\hat{A^{2n+2}}> \\- 2<\\hat{A^{2n+1}}> \\+ <\\hat{A^{2n}}> \\\\\\ &= C(2n+2) + D - 2C(2n+1) - 2D + C(2n) + D \\\\\\ &= 0 \\end{align}$$ $$\\begin{align} \\text{Variance of }\\hat{B} &= <\\hat{B^2}> \\- \\left(<\\hat{B}>\\right)^2 \\\\\\ &= -C^2 \\end{align}$$"}
{"id":"134517","title":"Newtonian physics","text":"According to the book I am referring while working with non inertial frame of reference we have to apply a pseudo force equal to mass x acceleration in the opposite direction of the acceleration of the non inertial frame of reference with respect to the inertial frame of reference. But which frame is to considered to be absolutely inertial? Please don't say earth because Earth is not perfectly inertial its just inertial to good approximation. This can be justified while calculating the apparent weight of object due to earth's rotation."}
{"id":"3193","title":"What determines which frames are inertial frames?","text":"I understand that you can (in principle) measure whether \"free particles\" (no forces) experience accelerations in order to tell whether a frame is inertial. But fundamentally, what determines which frames are inertial (i.e. what principle selects in which frames free particles will not appear to accelerate)? I've been told that the cosmic microwaves determine the ultimate rest frame of the universe, but that doesn't make sense to me, since one can still ask why that frame is an inertial frame. Also, I understand that there are no real inertial frames in general relativity, but it seems like there certainly exists approximate inertial frames and we can ask why those frames are approximately inertial and not others. For example, in the frame of a person riding a merry go round, free particles appear to greatly accelerate; while in the frame of someone standing next to the merry go round there are no such great accelerations. Why does the guy (or gal) on the merry go round see free particles accelerating while the other guy doesn't. And if you're gonna tell me that it's \"the rest of the stuff in the universe\" that determines whether the person on the merry go round sees free particles accelerate, I'll ask how you know that all that stuff is not spinning. I hope this question sort of makes sense, it's been bothering me for a while and my study of relativity (most of special relativity and just the outline of general relativity) hasn't really clarified it for me much."}
{"id":"8133","title":"Unit conversion help (school)","text":"Can someone please help me explaining how to convert units in problems like the one below? I've done many things, but I am very unsure of all my methods. Could you tell me what steps should I take? $$5.66\\ \\frac{\\mathrm{hm}}{\\mathrm{min.}} \\frac{\\hspace{2cm}}{}\\frac{\\mathrm{km}}{\\mathrm{hr.}}$$ Thank you."}
{"id":"61123","title":"Heating and Recooling of an Object","text":"Consider a piece of metal of length $L$ and linear thermal expansion coefficient $\\alpha$. We eat the metal $\\Delta T$ degrees, causing the metal to increase to length $$ L' = L + L \\alpha \\Delta T$$ Now, cool the object back to the original temperature. This causes the metal to decrease in length to $$L'' = L' - L'\\alpha\\Delta T \\\\\\ = L + L \\alpha \\Delta T - (L + L \\alpha \\Delta T)\\alpha \\Delta T \\\\\\ =L (1 - \\alpha^2 \\Delta T^2)$$ My intuition would lead me to believe that heating up and then recooling an object would cause it to return to the same size it began at (if it did not, then bridges which repeatedly warmed and cooled would continually shrink) but this is not true according to my mathematics, which indicates that warming and recooling an object leaves it slightly smaller than it was before. **Do objects really not return to their original size when recooled? If so, then why do objects which warm and cool on a daily basis not slowly shrink?** I suspect that this may be related to $\\alpha$ varying over the range of temperatures, but I didn't think that was a significant effect for solids."}
{"id":"22569","title":"How is contact resistivity defined for a Schottky contact, or the Schottky barrier height for an ohmic contact?","text":"Based on the transfer length method (TLM), one can accurately calculate the contact resistivity for an ohmic contact, by evaluating the absolute resistance measured through the test structure and plotting it as a function of the gap spacing between the two ohmic contacts. By extrapolation, the contact resistance and transfer length (and thus, the contact resistivity) can be calculated. However, what if a measurement of the contact resistivity of a Schottky contact was desired? In this case, the forward biased current is non-linear (does not follow Ohm's law), and thus the absolute resistance measured is a function of voltage. Is there another way to measure the contact resistivity in this case? On the flip side of the coin, I have only seen the Schottky barrier height calculated for Schottky contacts. However, some ohmic contacts (e.g. tunneling ohmic contacts) still have a positive Schottky barrier height. How is the height measured in this case?"}
{"id":"114281","title":"Tensor algebra doubt","text":"Is it possible to take a tensor to the other side of the equation, and the tensor becomes its inverse(i.e contravariant becomes covariant and vice versa)? It is a stupid question, but It confuses me. For example, if $A_{ij} = B_{ij},$ Can I write $A_{ij}B^{ij} = \\delta_{j}^{i}$, (though I think it should be $A^2$) Or is it only valid for the metric tensor? Also, is there a difference in the matrix representations of a tensor's contravariant and covariant form, or only the transformation rules differ?"}
{"id":"32422","title":"QM without complex numbers","text":"I am trying to understand how complex numbers made their way into QM. Can we have a theory of the same physics without complex numbers? If so, is the theory using complex numbers easier?"}
{"id":"110075","title":"'Pseudo-Relativistic' behavior in Graphene","text":"I've read that electrons in Graphene behave 'pseudo-relativistically'; what does this mean? how do they behave differently from electrons in other materials?"}
{"id":"114750","title":"Hawking radiation at the mouth of Schwarzschild wormholes","text":"I was researching a question for another post and it occurred to me that you might expect to see Hawking radiation at the mouth of wormholes. * * * Given the mechanism of Hawking radiation at the event horizon of black holes: virtual partial pairs forming at the edge and being separated by the event horizon; would virtual particles at the edge of wormholes likewise be divided? As I understand it, the trip is one way, which should cause the spontaneous formation of electrons and separation of positrons. * * *   1. Am I on the right track or is this a ridiculous assumption?   2. I don't see this in the literature on Hawking radiation or wormholes. Can you point me to any resources?   3. If this wouldn't occur, why not?"}
{"id":"110079","title":"Why a mono-atomic crystal layer (2D) can't be stable?","text":"According to Peierls and Landau, 2D crystals were thermodynamically unstable. They can't exist! Of course, this theory was disapproved in 2004 (example: graphene). What is the general definition of stability of a general system? What is the thermodynamics' stability?"}
{"id":"7041","title":"\"Speed\" of Gravity and Speed of Light","text":"Some threads here touching speed of gravity made me think about that. This lead to some questions.   * The speed of gravity was not measured until today (at least there are no undebated papers to that effect).   * It seems common knowledge\/belief among physicists that the speed of gravity is the same as the speed of light.  And this is my question: Is that speed of light = speed of gravity a result of GR or is that fed into the theory? Or is there some evidence from other sources than GR for this? Does the same speed imply some deep-lying relation between gravity and electromagnetism?"}
{"id":"23760","title":"Is it sure that gravitons are massless bosons?","text":"> **Possible Duplicate:**   >  \"Speed\" of Gravity and Speed of Light I'm wondering if gravitational waves have the same speed of light? They must if gravity is mediated by a graviton spin 2 massless boson. One can argue that since gravity is a long range force, then its boson must have zero mass. But neutrinos travel through the universe too and despite their tiny mass yet have non zero mass. So, I'm wondering if there is a deeper justification for believing that gravity's speed =c?"}
{"id":"118668","title":"Self-adjoint extensions with 'teletransporting' boundary conditions","text":"When choosing a self-adjoint extension of a Hamiltonian, in general one can obtain domains in which (i) the probabilities teleport* between points on the boundary and (ii) boundary conditions locally conserve probabilities. The ones which locally conserve probability currents somehow seems nicer to me. But this is not at all an argument especially since tunneling is allowed in quantum mechanics. Is there any fundamental physical reasoning one can use to discard teleporting boundary conditions? Thanks in advance for any useful inputs. *I have used terminology from discussion about a related question : Physical interpretation of different selfadjoint extensions"}
{"id":"51215","title":"Where to find the current positions and velocities of the planets?","text":"I've written a program which simulates the motions of planets and other bodies. I'd like to run it on our own solar system, but to do so I need to know the current positions (preferably in heliocentric coordinates) of the planets as well as their current velocities. Is there a website where I can find this? I've found all the positions of the planets here, and I can find their average orbital speed fairly easily, but for some planets (e.g. Mercury) the orbital speed varies a fair amount."}
{"id":"51211","title":"why is orbital moment quenched while atoms forming solid","text":"atom has well defined spin(up and down) and orbital(s,p,d,etc) momentum, but when forming crystals, why the spin degree continues to be good quantum number while orbital momentum is quenched?"}
{"id":"106234","title":"Motion of two gravitational bodies?","text":"I'm doing a basic realtime simulation of two bodies, but the orbits are unstable for some reason. This code is run at every timestep:               rsq = (a.x - b.x)*(a.x - b.x) + (a.y - b.y)*(a.y - b.y)          a.vx = a.vx - (a.x - b.x) * g * b.m * (1\/rsq) * dt      a.vy = a.vy - (a.y - b.y) * g * b.m * (1\/rsq) * dt          b.vx = b.vx - (b.x - a.x) * g * a.m * (1\/rsq) * dt     b.vy = b.vy - (b.y - a.y) * g * a.m * (1\/rsq) * dt          a.x = a.x + a.vx * dt     a.y = a.y + a.vy * dt          b.x = b.x + b.vx * dt     b.y = b.y + b.vy * dt      Where m is mass, x&y are position, vx&vy are velocity, g is the gravitational constant, and dt is the elapsed time. My bodies orbit eachother but the orbit is unstable and acts as though R is to an exponent other than 2. Have I missed something in my understanding of this problem or somewhere in my code?"}
{"id":"106236","title":"Does mass affect velocity when travelling through frictionless medium?","text":"I found the following question on an standardized test, and was debating with some friends what the answer would be: A car of mass **M** is travelling with a constant velocity through a plane in which friction is non-existent. An object of mass **m** (`m = M\/3)` that is falling perpendicularly to the car lands inside of it. How will the velocity of the car be affected? This illustration can help explain the problem. ![Problem Illustration](http:\/\/i.stack.imgur.com\/ZYLtp.jpg) My initial thought was that the velocity would be the same, given that friction is non-existent and that the momentum of the falling object is perpendicular to that of the car. However, some friends suggested that, since the mass of the car increases, the velocity should decrease."}
{"id":"25038","title":"What does the sky look like to human eyes from orbit?","text":"There are numerous pictures, obviously, of the blackness of space from the shuttle, the space station, and even the moon. But they all suffer from being from the perspective of a camera, which is not sensitive enough to pick up the stars in the background when compared to the bright foreground objects (the limb of the Earth, the station, moon, etc). I've seen some photos that show a few of the brightest stars, but nothing special. Are there any photos or eye witness accounts from astronauts of what it looks like to a human with night-adjusted vision? If I were in an orbit similar to the space station and looked away from the Earth, would I be able to see more stars than I ever could on Earth, or would it only be marginally better than the best terrestrial night viewing?"}
{"id":"104974","title":"SU(N) Yang-Mills $gg \\to ggg$ scattering at tree level","text":"When talking about the spinor-helicity formalism in his new textbook on quantum field theory, Matthew D. Schwartz claims as a highly nontrivial example, it is quite easy to use the Parke-Taylor formula to calculate the $gg\\to ggg$ scattering cross section at tree level by hand, which is also one of the problems in the book. (Problem is found on page 560, problem 27.6) Can anyone tell me how to do this exactly? It looks like we cannot avoid summing $24^2$ terms, and in the paper, http:\/\/www.sciencedirect.com\/science\/article\/pii\/0550321386902300, the cross section of $gg\\rightarrow gggg$ spans several pages! This makes me expect the cross section for $gg\\rightarrow ggg$ is also complicated. Is there any smart way to do it?"}
{"id":"65359","title":"What is the reason that relativistic corrections for hydrogen atom work?","text":"Here I cite part from Sidney Coleman's lectures on Quantum Field Theory: > It is a phenomenal fluke that relativistic kinematic corrections for the > Hydrogen atom work. If the Dirac equation is used, without considering > multi-particle intermediate states, corrections of $O \\big(\\frac{v}{c}\\big)$ > can be obtained. This is a fluke caused by some unusually low electrodynamic > matrix elements. What is the fluke about? Also, how can one justify the usage of Pauli- Schrodinger type equations that comes from first quantization of Dirac's equation? Schrodinger's equation is universal postulate valid for any quantum theory, and is equation for wave functionals in field theory. Could one go from non-relativistic QED field theory and then justify the usage of Pauli equation in which $\\psi$ is interpreted as \"wave function\" in certain kinematical conditions (approximation)?"}
{"id":"24927","title":"How do astronomers measure the distance to a star or other celestial object?","text":"How do scientists measure the distance between objects in space? For example, Alpha Centauri is 4.3 light years away."}
{"id":"2585","title":"Pauli exclusion principle and Entangled pairs","text":"It is true for fermions in the same potential that the total wavefunction of two particles must be antisymmetric with respect to exchange of electrons. Which means the spin wavefunction is given by $\\chi=\\frac{1}{\\sqrt{2}}[\\chi_+ (1)\\chi_- (2)-\\chi_+ (2)\\chi_- (1)] $ which looks very much like the bell state, $\\beta_{11}=\\frac{1}{\\sqrt{2}}[ |01\\rangle - |10 \\rangle]$. So, can we call those fermions, entangled states, as long as they are within the potential or there is something fundamentally special about entangled states (e.g. difference in measurement statistics) which makes them more unique? Apologies if the question is too simple for the level of this website. However, apparently it has made a lot of confusion for many people!"}
{"id":"20289","title":"Do photons have acceleration?","text":"Photons travels with the largest speed in our universe, the speed of light. Do photons have acceleration?"}
{"id":"94049","title":"Does a photon instantaneously gain $c$ speed when emitted from an electron?","text":"An excited electron looses energy in the form of radiations. The radiation constitutes photons which move at a speed $c$. But, is the process of conversion of the energy of the electron into the kinetic energy of the photon instantaneous. Is there a a simple way to visualize this process rather than math?"}
{"id":"130359","title":"Is there any acceleration of light?","text":"I find it hard to believe that photons always travel with 3 x 10^8 m\/s just from the start. But there must be some acceleration of light. Maybe huge or taking place in picoseconds. So what maybe the acceleration of light or photons?"}
{"id":"79738","title":"Does light initially accelerate?","text":"Light travels with a speed of $3\\times10^8{m\\over s}$. My question is that was the light initially accelerating or it archived the speed in an instance? If it was accelerating then why it did not accelerate beyond that?"}
{"id":"111189","title":"When photons are emitted, do they accelerate to reach the speed of light?","text":"Photons are considered mass-less particle with a specific velocity but according to the electromagnetic theory, a photon is considered to have both energy and momentum. So what happen when they are emitted, do they accelerate to reach the speed of light or is it somehow instantaneous?"}
{"id":"30658","title":"Does light photons have jerk?","text":"While searching in web regarding whether rate of change of acceleration is possible or not; I came across the concept of jerk. I want to know whether photons which can be accelerated can also have jerk or not?"}
{"id":"117228","title":"Deflection of magnetic compass sorrounded by alternating poles","text":"What happens to the deflection of the magnetic compass if it is surrounded by south poles and north poles in alternating direction of a magnet around it in a circular pattern. Will it be deflected in any particular direction ? ![enter image description here](http:\/\/i.stack.imgur.com\/MHqqH.jpg)"}
{"id":"99125","title":"rectlinear motion with constant acceleration","text":"Friends, this is a numerical homework problem. I tried my best to solve it but my answer is not matching with the one given at the back of the text book. Please help me out: A motor car moving at a speed of 72 km\/h can come to a stop in 3 seconds, while a truck can come to a stop in 5 seconds. On a highway, the car is positioned behind the truck, both moving at 72 km\/h. The truck gives a signal that it is going to stop at emergency. At what distance the car should be from the truck so that it doesn't collide with the truck. The typical human response time is 0.5 sec. My logic and answer: since car can decelerate to a stop much faster than the truck, it only need to worry about human response time which is 0.5sec. car would cover 10m in 0.5seconds at a speed of 72 km\/h. so it just need to be 10m behind the truck minimum. but the answer in the book is 1.25 m How is this possible?"}
{"id":"23866","title":"Why the pressure of atmosphere doesn't crush you when you e.g. walk outside?","text":"Why the pressure of atmosphere doesn't crush you when you e.g. walk outside? I mean the density of air is $1.26 kg\/m^3$, so with $100 km$ above us, it exerts much pressure on you when you walk outside."}
{"id":"107543","title":"Kicking a soccer ball","text":"I wonder which type of kicking may cause the ball to go a larger distance? One way is kicking the ball when it is at rest and another is kicking a moving ball in the opposite direction. If the ball goes farther in the second way,then what is the reason despite the fact that a moving ball will exert extra force to the player's leg?"}
{"id":"62717","title":"How to calculate Riemann and Ricci tensors for a sphere?","text":"Let's have the metric for a sphere: $$ dl^{2} = R^{2}\\left(d\\psi ^{2} + sin^{2}(\\psi )(d \\theta ^{2} + sin^{2}(\\theta ) d \\varphi^{2})\\right). $$ I tried to calculate Riemann or Ricci tensor's components, but I got problems with it. The Ricci curvature must be $$ R_{ij}=\\frac{2}{R^{2}}g_{ij}. $$ But when I use definition of Ricci tensor, I can't turn the expression into the expression for the metric tensor Maybe, there are siome hints, which can help?"}
{"id":"113835","title":"How to increase grain size in iron","text":"Cut, polished and etched iron meteorites have grain size about 1000 times greater than air quenched iron. The reason for the large grain size is the very slow cooling of the core of a planet or protoplanet. Check web for cross section picture Widmanstätten pattern What methods can be used to grow large iron crystals on earth. What is the largest iron grain size achieved on earth? How about zone refining?"}
{"id":"106347","title":"How can a clock work if the uncertainty principle is true?","text":"If the uncertainty principle and Copenhagen Interpretation are true, then how can a clock tick? Supposedly particles can do all sorts of things when not measured, then how can they be formed into gears that make the clock tick when not measured?"}
{"id":"134044","title":"Uncertain if invoking uncertainty principle for wave function is handwaving","text":"Why doesn't the electron collapse onto the proton in a hydrogen atom? One explanation seems to be given by the Heisenberg uncertainty principle, which follows from the purely physical assertion that the wave function in the position representation is the Fourier transform of the wave function in the momentum (or more precisely the wave number) representation. A different explanation is obtained by solving (or inspecting) the Schrӧdinger equation. To what extent are these explanations independent? To what extent is the latter more fundamental? Is the latter explanation more correct, or more complete? The hydrogen atom is just an example, a more general form of the question would be to what extent it is correct to use the Heisenberg uncertainty principle to answer this kind of questions, as is often done in popular science texts? Elaborating a bit on what I mean to ask, the fact that the Fourier transform of the position wave function is the momentum wave function seems to allow us to draw conclusions about the behaviour of a system from this principle alone. However, when loosely drawing these conclusions, we tend to invoke our classical idea of momentum, whereas the dynamics of the system are really governed by the Schrӧdinger equation."}
{"id":"78090","title":"Friction problem having two blocks on separate inclined planes connected by a string","text":"This is a question I found in my old high-school textbook(I'm revising the topics for a course).Two blocks A and B of masses $1kg$ and $2kg$ are placed on a fixed triangular wedge by a massless in-extensible string as shown. ![enter image description here](http:\/\/i.stack.imgur.com\/vkP06.png) The pulley is massless and friction-less. The coefficient of friction between block A and wedge is $\\mu _1 = 2\/3$ and that between B and wedge is $\\mu _2 = 1\/3$ (let's neglect the difference between static and kinetic friction).The inclines are at $45 ^\\circ $ with the horizontal.We need to find the frictional force exerted on each block,and the tension in the string. I worked out the problem,and I thoroughly checked my steps.Since the maximum resistive(friction) force is higher than the components of weights of the bodies along the inclines,there should be a state of equilibrium.I got the equations: $$T= \\frac {10}{\\sqrt2}+f_A \\space .........(1)$$ $$T+f_B=\\frac {20}{\\sqrt 2} \\space .........(2)$$ $$=> f_A +f_B = \\frac {10}{\\sqrt 2} \\space ....(3); f_A-f_B=2T-\\frac{30}{\\sqrt 2} \\space ....(4)$$ $$max(f_A)=max(f_B)=\\frac{10\\sqrt2}{3}\\space .........(5)$$ (Considering $g=10 m\/s^2$) Where $f_A$ and $f_B$ are frictional forces experienced by blocks $ A$ and $B$. There seems to be no way to get exact values of $f_A$ and $f_B$.Still,the book says(only numerical answers are given in the end,no complete solution) that $$f_A=\\frac {10}{3\\sqrt2} , f_B=\\frac{10\\sqrt2}{3} and \\space T= \\frac{40}{3\\sqrt2} $$ All I can see by looking at the book's answers is that the author simply guessed that let there exist limiting(max) friction between $B$ and the wedge,and he might have put the value $f_B= max(f_B)$ and solved.In that case,why not the friction between $A$ and the wedge be limiting and the answers be $$f_A=\\frac{10\\sqrt2}{3} , f_B=\\frac {10}{3\\sqrt2} and \\space T= \\frac{50}{3\\sqrt2} $$ But when we look at eq$(3)$,we see that neither of the frictions needs to be limiting,their sum needs to satisfy $\\frac{10}{\\sqrt2}$. Any way to justify the book's answers?"}
{"id":"134915","title":"If gravitation causes constant acceleration why moon does not fall into earth?","text":"If moon travels with constant speed in one direction and earth gravitation causes constant acceleration in perpendicular direction why moon does not eventually fall into earth? I mean if gravitation causes moon to fall faster each second (10m\/s2) shouldn't after time velocity toward earth be big enough to cause it to fall ?"}
{"id":"1193","title":"Why does the atmosphere rotate along with the earth?","text":"![Now im isolating Atmosphere](http:\/\/i.stack.imgur.com\/FkZ4H.jpg) I was reading somewhere about a really cheap way of travelling: using balloons to get ourselves away from the surface of the earth. The idea held that because the earth rotates, we should be able to land in a different place after some time. As we all know, this doesn't happen. Someone said that the reason why this doesn't happen is because the atmosphere (air, clouds etc.) also revolves around the earth (with the same angular velocity as the earth's rotation). Since we are also part of the atmosphere, our position doesn't change relatively. Well, I'm not convinced with that answer. Why does the atmosphere rotate along with earth? Gravitational force is towards the centre of the earth, but I don't see how it's making the atmosphere rotate."}
{"id":"16390","title":"Does the rotation of the earth dramatically affect airplane flight time?","text":"Say I'm flying from Sydney, to Los Angeles (S2LA), back to Sydney (LA2S). During S2LA, travelling with the rotation of the earth, would the flight time be longer than LA2S on account of Los Angeles turning\/moving away from our position? Or, in the opposite direction, would the flight to Sydney be faster since the Earth turns underneath us and moves Sydney closer? ===   * Please ignore jet stream effects and all other variables; this is a control case in an ideal environment.   * By \"dramatically\" I suppose I mean a delay of 1 hour or more."}
{"id":"112349","title":"Flight time Toronto to Moscow the same","text":"I have a question. How come that the flight from Toronto to Moscow takes the same time as the flight from Moscow to Toronto. Should it take much longer because of the earth rotation? One direction we travel in direction or an orbit and traveling back we are going against the Earth orbit... From Toronto 9hours From Moscow 9:45 hours. If flying from Moscow we are traveling against the Earth rotation so we should add half of the travel to our current travel. It means that the travel from Moscow should take about 13 hours... Thanks,"}
{"id":"58154","title":"Can a hovering helicopter travel half the globe in 12 hours?","text":"Suppose we have a helicopter that is able to stay stationary in flight for extended periods of time. If such a helicopter stayed at point A in the sky for 12 hours straight, would it reach the other side of the globe?"}
{"id":"112653","title":"Relation between Earth's rotation-gravity and moving object on Earth","text":"This might be silly question. It's about the rotation of the Earth and the objects moving on it. If I jump into the air, why did I also move with the Earth though I am in the air? I am supposed to be land in backward\/forward location, as it keeps rotating. But this doesn't happen. What is this phenomenon called?"}
{"id":"133996","title":"How come the atmosphere moves with Earth?","text":"now, I have read a lot of explanations on that but still can't really understand why it would happen so if you can give some examples for a such a thing happening. I mean lets say gravity attracts the air and due to air viscosity all of the air gets carried with the Earth but why will the speed be same ?"}
{"id":"80090","title":"If I jump will I land in the same spot?","text":"If I were to jump one meter in the air and hang for one second, would I fall back down in the same spot or would the earth rotate ever so slightly under me, causing me to land a short distance away from my original point of departure? I am conflicted on this. If I look at the equation for angular velocity, I see that $w = v\/r$ where $w$ is the angular velocity, $v$ is the linear velocity, and $r$ is the radius of the object I am on (in this case the Earth). There is another version of this stating that $w$ = $2\\pi\/t_{rev}$ where $t_{rev}$ is the time it takes to complete one revolution. Being on the Earth, I have a certain linear velocity, $v_{G}$. When I jump in the air 1 meter, I am not applying any force except for vertically so I do not believe my linear velocity would change. However, I am increasing the distance I am from the center of the earth so now my angular velocity would be $w_{A} = v_{G}\/(r+1)$. Therefore, it seems that my angular velocity in the air ($w_{A})$ would be slightly less than my angular velocity on the ground ($w_{G}$). If I were to recalculate the linear velocity given this discrepancy in $w$, I would get $v_{G}=w_{G}r$ and $v_{A}=w_{A}r$. This shows I get a small difference unless the radius in $v_{A}$ must account for the jump ($r+1$), in which case I end up with $v_{A} = v_{G}$. I've gone around in circles trying to decide if one would actually move. This top answer seems to think you would: Earth moves how much under my feet when I jump? Any insight would be appreciated. Thank you!"}
{"id":"48287","title":"Earth moves how much under my feet when I jump?","text":"If I'm standing at the equator, jump, and land 1 second later, the Earth does NOT move 1000mph (or .28 miles per second) relative to me, since my velocity while jumping is also 1000mph. However, the Earth is moving in a circle (albeit a very large one), while I, while jumping, am moving in a straight line. How much do I move relative to my starting point because of this? I realize it will be a miniscule amount, and not noticeable in practise, but I'd be interested in the theoretical answer."}
{"id":"44669","title":"Earth is rotating","text":"> **Possible Duplicate:**   >  Why does the atmosphere rotate along with the earth? If i take off from land on a helicopter straight above the earth surface to a certain height and stay there for few mins\/hours and come down. Why am i coming down to the same place where i took off? If the earth is rotating i should land on a different place right because i have not moved i am just coming down straight.I moved only vertically. I got this thought because i was thinking why are we spending so many hours on flights to reach a country on west if i started from a eastern country. May be there are a lot of scientific reasons behind this which i am not aware excuse me if it sounds silly.I thought this would be the best place to ask."}
{"id":"54278","title":"Sideways motion between a vertical launch from a planet and landing","text":"I saw a video some days ago (Hello Kitty in Space) of a schoolgirl successfully launching a balloon into space which later popped and landed ~47 km from launch site. If I vertically launch an object (say via balloon or rotors) from the surface of a spinning planet with an atmosphere like ours, then as it rises, will the planet's surface move away below it, or will the object rotate with the earth? (Assume that there are no local strong winds relative to the atmosphere, and it is completely calm.) The latter would mean the atmosphere, being itself a collection of such objects (air molecules), also spins with the planet. Related SE question: Why does the atmosphere rotate along with the earth? It seems from common sense that since at launch the object shares the motion of the surface it would keep rotating with the planet, and hence would later land at the launch site. However, as it ascends, the lateral speed required to maintain the same angular velocity with the planet rises linearly. Where would it get this extra momentum\/energy from? From the rotating atmosphere? How did the atmosphere get moving in the first place then? It might seem a lot of questions for a single SE post, but they're all connected, so a single explanatory answer should suffice."}
{"id":"100564","title":"Does the ground beneath a levitating object change due to the rotation of the earth?","text":"I don't know whether this question pertains to physics. But anyways, here goes... If I were to hover 1m above the ground in a helicopter, would the ground below me change after 1 hr due to the rotation of the earth?"}
{"id":"55042","title":"Would Oscars made of pure gold bend?","text":"Since gold is often considered \"soft\": If the Oscars were made of pure gold (instead of gold-plated britannium), would they bend (deform) over time?"}
{"id":"39512","title":"Stacking gold Bars","text":"I once heard that because gold is so malleable, while also being very heavy; that if one were to stack several layers of bars on top of on another (like is so familiarly depicted in movies & TV), the shape of the ones on the bottom would be smushed down because of the weight of the ones on top and gold's high malleability. Is there any truth to this?"}
{"id":"20976","title":"How can I estimate the power between the guide and the sky diver?","text":"I would like to estimate the Maximum power\/ tension that can be, between the sky diving guider and the \"tourist\". In this picture what the max tenstion will be, I think it is maximum when the parachute is opened."}
{"id":"68186","title":"Tension in parallel springs","text":"> _A block of mass $200$kg is connected to a horizontal ceiling by four > identical light elastic ropes, each having natural length $7$m and stiffness > $180$N\/m. It is also connected to the floor by a single light elastic rope > having stiffness $80$N\/m. All five ropes are stretched and vertical, and air > resistance is neglected._ > > _Find the tension in one of the upper ropes._ I know it's just $kx$, but I don't understand why it is not $\\frac{kx}{4}$ because tension is shared between four ropes."}
{"id":"68185","title":"Magnitude of force to keep stick in equilibrium","text":"**Problem statement** A straight and homogenous stick with mass m is pressed against a wall with the force F. The stick is horizontal perpendicular against the wall. Given that the friction between the wall and the stick is μ, determine the horizontal component of F in order for the stick to not fall down. **My thoughts** Forces involved: We have: The gravitational force **mg** in the negative y-direction. The normal force from the wall, **N** (negative x-dreiction). The friction force in the positive y-direction which is **f=μN** and the force **F** which acts in the positive x-direction. Thus: $$\\sum F_{x}: F-N=0 \\Rightarrow F=N \\\\\\ \\sum F_{y}: mg-f=mg-\\mu N=0 \\Rightarrow N=F=\\frac{mg}{\\mu }$$ $$\\boxed{F=\\frac{mg}{\\mu}}$$ ![picture](http:\/\/i.imgur.com\/QFJq7wT.png) * _Correct answer is $\\boxed{F=\\frac{mg}{2\\mu}}$ *_"}
{"id":"68232","title":"Is there an equation that tells you more about the amplitude of an object which is in resonance?","text":"I'm a high school senior and I have to write a paper about resonance and differential equations. I've been searching the Internet for a long time, but I haven't found an equation that is properly explained. Could anyone explain whether there is an equation to describe the amplitude and if so, whether you could explain to me how you can obtain this formula?"}
{"id":"17756","title":"Understanding the Seebeck effect","text":"Thermoelectricity is, as I understand it, the difference in voltage between the hot and cold ends of two dissimilar materials. If two materials are connected at two different junctions, the hot junction will effectively liberate electrons which will flow to the cold junction, along with the heat. All the diagrams of this I could find showed exactly two materials, connected at both the hot and cold ends. In order to have the hot side remain hot, and the cold remain cold, while transmitting the electricity, ideally a substance is desired that conducts electricity well, but heat poorly. This is the measure of quality. A good overview paper is: New Directions for Low- Dimensional Thermoelectric Materials by Dresselhaus et al. \"ZT= S2rT\/j, where S, r, T, and j are, respectively, the Seebeck coefficient, electrical conductivity, temperature, and thermal conductivity\" Maximizing efficiency of a thermocouple is, in fact, a matter of finding materials that conduct heat poorly, while conducting electricity well. In practice, that's difficult, so in constructing nanomaterials to do so, they are trying to scatter phonons while not scattering electrons. Questions:   1. How do you make a series circuit with lots of thermocouples? Surely this must be done, yet when I draw a diagram, I can't come up with a way that doesn't look like two identical thermocouples are simply operating in opposing directions. This is answered below (and thanks), but note the answerer is incorrect about the need for thermal conductivity. Ideally, you want to insulate these things, and have as little heat as possible produce as much electricity as possible.   2. In the diagram drawn by the answer, it shows a thermopile as a zig-zag connection of two materials between hot and cold. This makes perfect sense now that I see it, but then, does material A transport electrons from hot to cold, while material B transports them from cold to hot? I thought I intuitively understood the Seebeck effect as a liberation of electrons from the hot end, effectively by \"shaking them loose\" but it appears one material must be doing the reverse? I'm guessing this is similar to galvanic action, where one material wants an electron to complete a valence shell, and the other wants to give one away?   3. I have now drawn a diagram showing a single thermocouple, a thermopile per the answer below, and my question, which is whether you can introduce a third material connecting the other two. I'm guessing you can, and that whatever its electropotential, since it's symmetric it should have no effect on the voltage of the thermopile other than its resistance. ![diagrams of thermopiles](http:\/\/i.stack.imgur.com\/W714r.png)"}
{"id":"104895","title":"Adiabatic Expansion with expanding control volume","text":"I have a tube with a length \"L\" and diameter \"d\" that is open on 1 side . At a certain point ( say \"x\" ) from the closed end of the tube, I have a gas with a high pressure. At this point, \"x\", there is also a weight of mass \"m\". It is also known what the velocity of the weight is at point \"L\". Initially : ![initial](http:\/\/i.imgur.com\/DPXFndD.png) Finally : ![final](http:\/\/i.imgur.com\/xu5Qmmr.png) Assuming that the tube is held stationary, what force is imparted on the closed section of tubing ( left side )? Basically, think of it as a rifle barrel with a bullet inside. I wish to find a expression for the force on the closed end of the tube as it varies with time. What I propose : Determine the Work by the gas. And since work is defined as $W = Fx $ and then $ \\frac{dW}{dt} = \\frac{d(Fx)}{dt}$ perhaps this could lead to an expression for force?"}
{"id":"129730","title":"Size of an elementary particle","text":"Do we have a well defined mathematical expression denoting the size of a fundamental particle with no internal structure (electron for example) ? If we do, how does it fit in with the uncertainty principle ? And if we don't then what exactly do the experiments reveal claiming radius of electron is of the order $10^{-18} m$ or something like that ? Also in this particular case, does this mean that the probability of the electron being within that radius is 1 and zero outside ? A proper clarification would be very helpful."}
{"id":"31269","title":"Is the Higgs a quantum field or a particle?","text":"The Higgs is not detected in the asymptotic data, so it is possible that there is no particle interpretation for the Higgs quantum field. Indeed, the Higgs potential is only positive definite if the quartic term is included --- the quadratic term corresponds to a negative mass term. It would seem, therefore, that the Higgs field does not have an on-shell mass spectrum, so that there is no straightforward particle interpretation. One can say that there is an effective field theory in which there is a resonance near a given mass that we will call the Higgs _resonance_ , but in the absence of a pure mass shell spectrum (that is, if there is a continuous mass spectrum), it is generally taken in QFT that there is no particle interpretation. The resonance is clearly not a $\\delta$-function, so is there some other precise way in which we can call the Higgs a particle? Of course this doesn't call into question the empirical effectiveness of the Standard Model of Particle Physics, it only asks about its interpretation and about how we put the Mathematics into words. I was somewhat struck by Rolf Heuer's observation (this morning) that this is the first observation of a scalar particle. Indeed, according to the SM, there are no quantum fields that have non-zero mass terms in the absence of interactions. In the absence of interactions, the Higgs field is a massless scalar field. Should we say that it is the Higgs _interaction_ that gives mass to the standard model? (EDIT: Is it better to say that every term that is not quadratic in the fields _contributes_ towards the effective masses of each of the asymptotically observed fields? Or what alternative phrasing is closer to the Mathematics of the interacting fields?) EDIT(2, $\\scriptstyle\\mathsf{see\\ below\\ for\\ the\\ comment\\ that\\ prompted\\ this\\ possible\\ rephrasing}$): Is there any part of the definition of \"particle\" that is not a matter of convention? Does the Higgs cross that bar?"}
{"id":"31261","title":"What is the difference between pole and running mass?","text":"For example, when we meassure Higgs boson mass to be 125 GeV, do we think about renormalized or pole mass? Should the mass of the Higgs change if it is produced at higher energies?"}
{"id":"3134","title":"Is the force carrier of the magnetism in a common household magnet a photon?","text":"As I have understood it, the Standard Model includes particles that carry the different forces, e.g. the electromagnetic (EM) force, the gravitational (G) force. When talking about EM fields such as visible light or microwaves, the associated particle is said to be the photon. But what about a static EM field without any electricity, like a common household magnet? How does that magnet communicate its force? Via photons?"}
{"id":"3133","title":"Why does frozen water burst a pipe?","text":"When water freezes in a pipe it can crack the pipe open. I assume this takes quite a lot of energy as when I try to crack a pipe it can be hard work! I think water freezing is a result of energy (heat) being lost from the water and out of the pipe into the freezing environment around it. So what energy is cracking the pipe and how? When warm and not frozen there is more energy in the pipe than when frozen? My secondary question might be - is this a particular phenomenon of water or would other matter crack open a pipe when it freezes solid from liquid? Andrew"}
{"id":"108296","title":"Loop-the-loop question","text":"> Problem: The marble rolls down the track and around a loop-the-loop of > radius R. The marble has mass $m$ and radius $r$. What minimum height $h$ > must the track have for the marble to make it around the loop-the-loop > without falling off?   >  Express your answer in terms of the variables $R$ and $r$. I found this solution to be very reasonable: $$mg = m a_c = m \\frac{V^2}R $$ which leads to $$V = \\sqrt{g R} $$ The energy at the top of the loop KE = Delta PE $$\\frac12 m V^2 + m g (2R) = m g h \\\\\\ \\frac12 (g R) + g (2R) = g h \\\\\\ \\left(\\frac12+2\\right) R = h $$ so $h = 5\/2 R $ However the correct answer is actually $\\frac52(R-r)$, I think it's because the radius of the loop is measured from the center of the ball rolling on it, so the they subtracted $R$ from $r$, but how would you derive that? I tried the same steps just using $R-r$ instead of $R$ but I got a different answer."}
{"id":"82507","title":"Practical example of stabilizer codes","text":"Given the Steane code $$ \\left|0\\right\\rangle_L \\equiv \\frac{1}{\\sqrt{8}}(\\left|0000000\\right\\rangle + \\left|1010101\\right\\rangle + \\left|0110011\\right\\rangle + \\left|1100110\\right\\rangle + \\left|0001111\\right\\rangle + \\left|1011010\\right\\rangle + \\left|0111100\\right\\rangle + \\left|1101001\\right\\rangle) $$ $$ \\left|1\\right\\rangle_L \\equiv \\frac{1}{\\sqrt{8}}(\\left|1111111\\right\\rangle + \\left|0101010\\right\\rangle + \\left|1001100\\right\\rangle + \\left|0011001\\right\\rangle + \\left|1110000\\right\\rangle + \\left|0100101\\right\\rangle + \\left|1000011\\right\\rangle + \\left|0010110\\right\\rangle) $$ and its relative stabilizers: $$ K^1 = IIIXXXX $$ $$ K^2 = XIXIXIX $$ $$ K^3 = IXXIIXX $$ $$ K^4 = IIIZZZZ $$ $$ K^5 = ZIZIZIZ $$ $$ K^6 = IZZIIZZ $$ The stabilizer set establishes valid codewords for a state if the equation $$s\\left|\\psi\\right\\rangle=\\left|\\psi\\right\\rangle,\\;\\;\\;\\forall s \\in S \\;\\;\\;\\;\\; (1)$$ is satisfied. That means $\\left|\\psi\\right\\rangle$ is a +1 eigenstate of $s$. We then consider a practical example of the usage of these stabilizers ![enter image description here](http:\/\/i.stack.imgur.com\/kg9F6.png) The state of the system is represented by: $$\\left|\\psi\\right\\rangle_F={1\\over 2}(\\left|\\psi\\right\\rangle_I+U\\left|\\psi\\right\\rangle_I)\\left|0\\right\\rangle + {1\\over 2}(\\left|\\psi\\right\\rangle_I-U\\left|\\psi\\right\\rangle_I)\\left|1\\right\\rangle$$ where $U \\in \\left\\lbrace K^1,K^2,K^3\\right\\rbrace$. We apply $U$ to the input state and we measure the ancilla qubits (syndrome measurement) to verify the integrity of the input (if $\\left|\\psi\\right\\rangle_I$ is +1 eigenstate of $K^1,K^2,K^3$). If the equation $(1)$ is not satisfied, then the corrupted qubit is corrected with a $Z$ gate addressed by the syndrome measurement. This is how does the system work?"}
{"id":"102202","title":"How long does it take for a black hole to form?","text":"The well-known fable of an astronaut sending signals out to an external observer while falling toward an event horizon states that the time lapse between such signals becomes greater even if in the astronaut sends them out periodically (as judged in his inertial frame). When viewed from earth and weighing time-dilation due to the gravitational field of the collapsing star, how is it possible that a black hole can form in finite time (for any external observer) if it takes an infinite amount of time to \"see\" events occurring at the event horizon?"}
{"id":"91583","title":"What is the direction of static friction?","text":"**Note** : My question is duplicate of the following   1. Direction of friction when a car turns   2. Why does friction cause a car to turn? * * * I've gone through many related questions especially the first. As I understand the static friction is always opposite to the force applied on the object as shown:   ![image](http:\/\/www.school-for-champions.com\/science\/images\/friction- slide_kinetic.gif)   But in the case when front wheels of a vehichal are turned the force of static friction is not opposite to the applied force. For example consider a car accelerating forward. The net force on the car is in forward direction which is provided from the rear tyres, if eventually break is pressed static friction(assuming tyres aren't skidding) comes into picture. This friction **should be and is** opposite in direction to the direction of force applied by the rear tyres. When the front tyres are turned the direction of static friction is changed(radially inward) means the direction of static friction is not opposite to the direction of applied force as shown:   ![image 2](http:\/\/img.gawkerassets.com\/img\/18zoykoy35nhfpng\/ku-xlarge.png) **Question:** Is the force of static friction is always opposite to the applied force ? If not then what determines its direction?"}
{"id":"103675","title":"Magnetic Shielding?","text":"Can magnetic fields be \"blocked\"? For example, in the game, TitanFall, a robot stop bullets with ( presumably ) a magnetic shield. I wish to calculate the magnetic force required to stop a bullet within a few microseconds. But the problem I have is that my entire electronic system or the vehicle ( most likely even a building ) will be subjected to massive fields most likely destroying it as well. Does any know of a way to \"block\" or divert the field around my vehicle? Is it even possible? My knowledge in magnetic fields are quite limited at this time."}
{"id":"67464","title":"Can a light be bent by a magnetic field?","text":"I'm struck with two competing ideas on the question in the title. Listing #1: http:\/\/van.physics.illinois.edu\/qa\/listing.php?id=2009 Q: \"How far can a magnetic field bend light?\" A: \"Unfortunately, the path light takes is not affected by the presence of a magnetic field. Light itself is composed of an oscillating electric and magnetic field, and one very important property of electric and magnetic fields is what we call \"linearity.\" That is, if you have two sources of electric and\/or magnetic fields, you can predict what the combined field is just by adding the two source fields together. The two fields don’t change each other at all. \" Listing #2(Answer #1): Does electric charge affect space time fabric? Q: \"Does electric charge affect the space time fabric? If so, why?\" A: [See link. Rather, see both links if you must.] I'm more inclined to consider the latter question and answer as the correct interpretation. Anyway, if anyone could help me out with this conceptualization that would be great, thanks."}
{"id":"67467","title":"Aharonov-Bohm Effect in Torus","text":"I had a very brief introduction to the Aharonov-Bohm effect in class. The lecturer introduced the notion that $H(\\Phi=\\Phi_0)$ and $H(\\Phi=0)$ gives identical energy spectrum and that the Hamiltonians are related by a the large gauge unitary transformation. I did a quick Google on the large gauge transformation, but I did not quite understand much about it aside from the fact that it is a topological related gauge transformation. Can someone explain a bit more about what that gauge is about and how it is performed? Also, in a many-body system with the following Hamiltonian (1), in a 3D Torus with flux, $\\Phi$ piercing through the hole on the torus, how do show that the energy eigenvalues of $H(\\Phi=\\Phi_0)$ and $H(\\Phi=0)$ are indeed identical? $\\Phi_0 = \\frac{h}{e}$ in this case. $$H(\\Phi) = \\Sigma_{j} \\frac{1}{2M}\\biggl(\\vec{p_j}+e\\frac{\\Phi}{L}\\hat{x}\\biggr)^2 +\\Sigma_j \\ U(\\vec{r_j}) + \\Sigma_{j<k} \\ V\\Bigl(\\vec{r_j}-\\vec{r_k}\\Bigr)\\tag{1}$$ The general idea of how I will go about solving this is by perhaps acting the hamiltonian on a wavefunction to determine the energy eigenvalue though I am not sure how to do it explicitly."}
{"id":"25804","title":"What telescope would I need to observe Mars's polar caps?","text":"What telescope refractor aperture size would I need to observe Mars's polar caps? I have a Levenuhk Strike NG 80mm, and I can see red disk of planet but cannot see any details on the surface."}
{"id":"130829","title":"Can a wave propagate in any substance? Aren't there any prerequisites?","text":"We see waves propagate in air, water, through the cristal of a metal and along a rope. Isn't a wave a wonder of Nature, or is it just a simple phenomenon? Are homogeneity and isotropy necessary properties for the correct propagation of waves? Update are a rope, water and space\/EM field elastic in the same way?"}
{"id":"74983","title":"Why don't FCC metals have a brittle-to-ductile temperature transition?","text":"I initially thought that it had something to do with the number of slip systems in FCC vs. BCC, but they're both the same."}
{"id":"66738","title":"Is Brownian motion a deterministic system?","text":"Is Brownian motion a deterministic system? I.e the motion of all particles are completely determined or is there an innate indeterminism like quantum systems?"}
{"id":"15324","title":"Is the Sun-set and the Sun-rise Symmetrical for the Observer?","text":"Is there the effect of sun rising and sun setting, in terms of Rayleigh scattering and visual spectrum and other factors completely similar and symmetric? I mean can one recognise them from a picture taken from the sky?"}
{"id":"105854","title":"Differential equation for evolution of probability density in Quantum Mechanics?","text":"I have come up with this differential equation for the evolution of $\\vert \\Psi \\vert^2$, the probability density in quantum mechanics. Is there a name for this equation? Is the logic sound? So I start from the conservation of the total probability: $$ \\frac{d}{dt} \\int _{all space} \\vert \\Psi \\vert^2 d^3\\textbf{r} = 0$$ so that $$\\int _{all space} \\vert \\Psi \\vert^2 d^3\\textbf{r} = {\\rm const}$$ This means that $$\\int _{all space} \\vert \\Psi (\\textbf{r},t) \\vert^2 d^3\\textbf{r} = \\int _{all space} \\vert \\Psi (\\textbf{r}+d\\textbf{r},t+dt) \\vert^2 d^3\\textbf{r}={\\rm same\\,const}$$ So since the integration is over the same volume I can equate the intergrands? This gives us: $$\\vert \\Psi (\\textbf{r},t) \\vert^2 = \\vert \\Psi (\\textbf{r}+d\\textbf{r},t+dt) \\vert^2$$ or $$\\Phi (\\textbf{r},t) = \\Phi (\\textbf{r}+d\\textbf{r},t+dt) $$ if we call $\\vert \\Psi (\\textbf{r},t) \\vert^2 = \\Phi (\\textbf{r},t)$. A Taylor expansion to first order gives us: $$ \\Phi (\\textbf{r}+d\\textbf{r},t+dt) =\\Phi (\\textbf{r},t) + \\nabla\\Phi\\cdot d\\textbf{r}+\\frac{\\partial \\Phi}{\\partial t} dt$$ Plugging this into the previous equation gives us: $$\\boxed{ \\nabla\\vert \\Psi \\vert^2\\cdot \\frac{d\\textbf{r}}{dt} = -\\frac{\\partial \\vert \\Psi \\vert^2}{\\partial t} }$$ What is $\\frac{d\\textbf{r}}{dt} $ in the context of quantum mechanics? Is this equation correct? What is the physical meaning of this equation?"}
{"id":"77182","title":"How does the gravity of a massive non-spherical object act on things around it?","text":"Firstly, not sure if this question ought to be in the space SE site. Please let me know if it should. (Posted in both for now) Secondly, I don't know a whole lot about physics (I'm just inquisitive). So please try to keep answers simple (or at least dummed down to laymans terms). Inspired by \"A Hitchhikers Guide to the Galaxy\" movie (haven't read the book yet) My question is, for a non-spherical object (lets say a cube for now) of a size and mass like that of Earth, how would objects act in it's gravity? Aspects of this question lead to the following questions •Would something weigh more\/less in certain areas over other areas? •I'm aware that it's possible to 'slingshot' around massive spherical objects to change their directory. Would it be possible to slingshot objects around something non-spherical to change their velocity? •If this massive object was orbiting our sun, would it's orbit be as uniform as ours? if not what would that look like and why?"}
{"id":"92587","title":"Pipe in a flowing river problem","text":"I'm working on a certain problem in fluid mechanics, which isn't really my strongest area. The problem is as follows: Curved pipe is partially submerged in a flowing river so that one end is pointing in the same direction as the velocity of river. Level of water inside of the pipe is 7cm higher than the level of river. Determine the speed of river. Basically what I argued is that this reduces to Torricelli's law: level of water in pipe is constant so the velocity of the surface is zero (or very nearly so, I assume it would oscillate in real life), therefore water should move on the other end with $v_{2}=\\sqrt{2gH}$, but it isn't since $v_{2}=v_{river}$ counteracts the movement. I've tried to be more rigorous so I took pressures at the submerged end of pipe: $$p_{\\text{water in pipe}}=p_{\\text{atmosphere}}+\\rho g h_{\\text{depth}} + \\rho g H_{\\text{above water}}$$ $$p_{\\text{river}}=p_{\\text{atmosphere}}+\\rho g h_{\\text{depth}} + \\frac{1}{2} \\rho v^{2}$$ Since they have to be in equilibrium, pressures are equal and you get Torricelli's expression above. Is my reasoning correct? It's somewhat counter-intuitive to me because the water is moving _away_ from the pipe."}
{"id":"87437","title":"Can the phrase \"Terminal Velocity\" be used to describe non-gravity situations?","text":"According to Wikipedia: > [Terminal Velocity] is the velocity of the object when the sum of the drag > force (Fd) and buoyancy equals the downward force of gravity (FG) acting on > the object. Since the net force on the object is zero, the object has zero > acceleration. I was wondering if this can be more generalized. Assume that we replace the force of gravity with another type of force (for example, the force of a theoretical engine on a rocket whose fuel never loses mass). In this case, the force is not one of gravity (along the y-axis) but of the engine (along the x-axis). However, drag would still play a part in causing the acceleration due to the rocket's engine to reach 0. As far as I'm concerned, these concepts are the same, but whenever one talks about \"Terminal Velocity\" they always discuss it in context of gravity. Can the term \"Terminal Velocity\" be used in the \"rocket\" case as described above, where the force is NOT that of gravity?"}
{"id":"131155","title":"Measuring position and momentum at the same time?","text":"In a non-relativistic quantum mechanical system in an infinite potential well. I try to measure the energy and the position of the system simultaneously. Since, the respective operators do commute according to Heisenberg's uncertainty relation I should be able to measure them both with infinite precision. Now, since I know that there is no potential energy in the well I can use $ E=\\frac{p^2}{2m} $ since the potential energy is 0 and determine it's momentum provided I know it's mass. But I shouldn't be able know the momentum and position simultaneously with infinite precision! So where am I going wrong?"}
{"id":"110911","title":"Exhaustive list of assumptions for the Clauser-Horne-Shimony-Holt inequality","text":"I am trying to create an exhaustive list of all assumptions which work as the base of the CHSH inequality.   1. Locality - this means an object can be influenced only by its surroundings. So, the events taken place at Alice and Bob's ends cannot influence each other.   2. Realism - the value of the observed quantity is independent of observation.   3. The quantity being observed is a discrete random variable.   4. Repeated rounds of experiments are independent of each other and evenly distributed.   5. A measurement will always produce a result (a photon will always be detected).   6. No enhancement assumption - it means when a measuring setup is placed between the source of the entangled particle and a detector the probability of measurement doesn't increase. Am I missing anything?"}
{"id":"106712","title":"Loss of kinetic energy in inelastic collision","text":"I know that momentum and energy are always conserved in collisions, but if we have a perfectly inelastic collision in which an object sticks to another object $m_1 v_1 + m_2 v_2 = (m_1+m_2)v_{12}$, the kinetic energy is not conserved. I know that kinetic energy converts into thermal or sound energy, but I don't see how this would account for the whole of the lost kinetic energy. Does the kinetic energy transform into some sort of potential energy between the bonding of the two objects? For example, if the two objects stick together with a small magnet (assuming the initial attraction between the objects is negligible) does the kinetic energy transfer to some sort of magnetic potential?"}
{"id":"100324","title":"Detailed balance formulation in solar cells?","text":"Hello I wanted to know where does the integral of following picture come from and what are the alternatives in it? How and where can i find information i need to know to understand this text?   thank you in advance! ![Detailed balance formulation](http:\/\/i.stack.imgur.com\/LBBlE.png)"}
{"id":"121308","title":"How to show $ \\epsilon_{iab}\\epsilon_{jcd}(x_ap_d\\lbrace x_c,p_b \\rbrace+x_cp_b\\lbrace x_a,p_d \\rbrace) = x_ip_j-x_jp_i$","text":"If $ \\lbrace f,g \\rbrace $ is Poisson bracket and $\\epsilon_{ijk}$ is Levi- Civita symbol, how to show that $$ \\epsilon_{iab}\\epsilon_{jcd}(x_ap_d\\lbrace x_c,p_b \\rbrace+x_cp_b\\lbrace x_a,p_d \\rbrace) = x_ip_j-x_jp_i$$ where $x_i$ are generalized coordinates and $p_i$ are momentums ?"}
{"id":"103712","title":"Introduction and overview of Condensed Matter Physics","text":"Is there any book that provides an overview of Condensed Matter Physics? I have had a course in QM and statistical physics and some. I dont know anything about this field, so is there a readable introduction to this large field? One that provides an overview, instead of too much detail? Edit: I am looking for something of a lower level than this Phys.SE question. Can we reopen?"}
{"id":"54675","title":"Supplements for Kittel's Solid State Physics?","text":"I think by supplement I really mean replace. I spent a lot of time agonizing over the first chapter of Kittel as he introduces a bunch of concepts such as Bravais lattice and he doesn't clearly define them. It's frustrating and infuriating considering that I have no background in crystallography nor solid state physics, and as an introduction to the subject I haven't found the book to be very helpful."}
{"id":"43137","title":"Could someone introduce books or reviews on electron-electron interaction to me?","text":"> **Possible Duplicate:**   >  Books for Condensed Matter Physics Could someone introduce books or reviews on electron-electron interaction to me? Especially its effects on screening and transport?"}
{"id":"86926","title":"Doesn't the use of a thermometer alter the temperature of the system?","text":"If I place a mercury thermometer in hot water, heat energy will transfer from the water to the mercury inside the thermometer. Will this continue until thermal equilibrium is reached and thus the mercury will show the temperature of the water? However, if this is so, will the thermometer show the right temperature as some of the heat energy is transferred to the thermometer and this in turn will cause original temperature of water to fall? Please correct me if I am wrong."}
{"id":"86920","title":"Can we \"trivialize\" the equivalence between canonical quantization of fields and second quantization of particles?","text":"As Weinberg exposited in his QFT Vol1, there are two equivalent ways of arriving at the same quantum field theories: **(1).** Start with single-particle representations of Poincare group, and then make a multiparticle theory out of it, while preserving principles of causality etc. I would call this approach the second quantization of particles, since second quantization is usually used to emphasize the many- body nature of a theory. **(2).** Start with field representations of Poincare group, canonically quantize it, while preserving principles of causality, positive definiteness of energies etc. I would call this approach the quantization of fields, just as everyone else would call it. Weinberg showed the proof of the equivalence between the above two approaches using some, though not hard, but let's say nontrivial, mathematics. The equivalence seems like a sheer miracle to me, or a complete coincidence. I do not feel that I understand the equivalence with the current state of mind. Is there a way to trivialize the equivalence? Or putting it another way, is there an a priori reasoning to argue, given the two sets of starting points of (1)(2), we have to get the same theory in the end? Just as a side remark, many have suggested the term \"second quantization\" should be totally dumped, because it is really just the first quantization of fields. To me however, it still serves some purposes since the equivalence is not transparent."}
{"id":"77728","title":"Horizontal magnetic attraction?","text":"I came across and interesting effect today, I have a dozen of Neodymium magnets around my house. And, they are very strong. Anyhow, one of them got attracted to my large steel plate table(use for cutting and building). It was almost impossible to take it apart, till I contacted the manufactured of those magnets and they proposed to slid it off. Amazingly, it worked. But what shocked me is the level of huge force decrease. It's like I'm going against friction alone... Only at the edge of the plate I felt a force that is strong, but half way through... The magnet as almost \"off\" only the edge of the magnet was attracted strongly to the edge of the plate, what explains this?"}
{"id":"43920","title":"Did the universe always have 4 space time dimensions?","text":"Is the dimensionality of spacetime in all usual models constant?"}
{"id":"89621","title":"What is the universality class of transition in one-dimensional XXZ Heisenberg at $\\Delta$ = -1?","text":"In the one-dimensional spin-$\\frac12$ XXZ Heisenberg model, $$H=J\\sum_i{S_i^x S_{i+1}^x + S_i^y S_{i+1}^y+\\Delta S_i^z S_{i+1}^z},$$ with $J>0$. There are two transition points:   * $\\Delta=1$   * $\\Delta=-1$ The transition at $\\Delta=1$ is of BKT type. What about the transition at $\\Delta=-1$? Could anyone provide some reference?"}
{"id":"122657","title":"How are the angles equal?","text":"At the back of my mind I know they should be equal, but mathematically, how are the two $\\Delta \\phi$ angles equal? ![Angles Image](http:\/\/i.stack.imgur.com\/qIxBs.png) The only explanation present in the text is that, \"both velocities are perpendicular to the radii vectors,\" but I don't see how that makes them equal. Also how will you make those two triangles in the 2 diagrams similar? Any images to support the explanation would be appreciated."}
{"id":"76466","title":"The lab have a constant electric field and a constant magnetic field, what is the electric and magnetic field inside a conductor and far from it","text":"The following is an old question from an exam in a Physics $2$ course I am taking, I have tried to solve the question and after I thought I got the answer I looked at the solution and saw it isn't correct. The solution explains what the answer is, but doesn't really explains the facts used. > In the lab frame there is an electric field $E=E_{y}$ and a magnetic field > $B_{z}=\\beta E$. > > We place a cube made of a conducting material, point $a$ is at the center of > the cube while point $b$ is far from it. > > Calculate the following: $$ E_{y}(a),E_{y}(b),B_{z}(a),B_{z}(b) $$ and $$ > E'_{y}(a),E'_{y}(b),B'_{z}(a),B'_{z}(b) $$ as they are seen in a frame of > reference moving at a speed $c\\beta\\hat{x}$. **My efforts:** I wrote that $$ E_{y}(a)=E_{y}(b)=E_{y}=E $$ because the electric field is constant (in the lab frame). Similarly, I wrote that $$ B_{y}(a)=B_{y}(b)=B_{z}=B $$ because the magnetic field is constant (in the lab frame). Then I used field transformation $$ E'_{\\perp}=\\gamma(E_{\\perp}+\\beta\\times B) $$ $$ B'_{\\perp}=\\gamma(B_{\\perp}+-\\beta\\times E) $$ to calculate the other four values requested. I then looked at the solution, the solution claims that $$ E_{y}(a)=0,\\, E_{y}(b)=E $$ $$ B_{z}(a)=\\beta E,\\, B_{z}(b)=\\beta E $$ and the other field were calculated using the above transformations. I then remembered that in a conductor the electric field is $0$, this explains why $E_{y}(a)=0$. I don't understand this situation completely, Please help me understand it by answering the following questions (that goes a bit beyond the question, but I find interesting and am having difficulties answering myself): > 1) Why does $b$ have to be far from the conductor so we can say that the > electric is $E$ there ? what happens near the conductor ? (I wonder if we > can say something about the charge distribution on it boundary, $\\sigma$) > > 2) Why does the magnetic field does not change in space, inside or outside > the conductor ? (I know that the electric field does change, at least inside > the conductor, why shouldn't the magnetic field change as well ?) > > 3) What can we say about the work done while moving the conductor in the lab > ?"}
{"id":"33326","title":"Can human hand move at a speed rate like this baseball pitch or is it just the speed of ball?","text":"Fastest baseball pitch ever recorded was in 1974 at a speed of 100.9 miles per hour. Does this mean that the pitcher's hand was also traveling at that speed or just the ball? Is it physically possible to move hand\/leg at that speed? I'm asking this because basically pitcher's hand was moving 17.7 inches per 10 millisecond which is really a super(unbelievable) speed for human."}
{"id":"45809","title":"Will Randall-Sundrum extra dimension scenario become defunct if not supported by LHC?","text":"The Randall-Sundrum extra dimension scenario had been one of the most extensively studied class of theories. This offered a solution to the hierarchy problem. However, if this picture is not supported by the LHC, will it become completely defunct? What about theories like little higgs, composite higgs, technicolor, higgsless models (perhaps already practically abandoned?)?"}
{"id":"87323","title":"On motivation for the definition of ADM mass","text":"The **ADM mass** is expressed in terms of the initial data as a surface integral over a surface $S$ at spatial infinity: $$M:=-\\frac{1}{8\\pi}\\lim_{r\\to \\infty}\\int_S(k-k_0)\\sqrt{\\sigma}dS$$ where $\\sigma_{ij}$ is the induced metric on $S$, $k=\\sigma^{ij}k_{ij}$ is the trace of the extrinsic curvature of $S$ embedded in $\\Sigma$ ($\\Sigma$ is a hypersurface in spacetime containing $S$). and $k_0$ is the trace of extrinsic curvature of $S$ embedded in flat space. Can someone explain to me why ADM mass is defined so. Why is integral of difference of traces of extrinsic curvatures important?"}
{"id":"10309","title":"Conservation law of energy and Big Bang?","text":"Did the law of conservation of energy apply to the earliest moments of the Big Bang? If so, what theoretical physics supports this? I hear that Einstein's theory of relativity disputes the law of conservation of energy- so does that mean the law is false, or only some aspect of it?"}
{"id":"107013","title":"Dark energy and conservation of energy","text":"With accelerated expansion of universe which is same in all direction we know that dark energy increase with time because space between any two point in space time increases with time. So after some finite time we can not see nearby galaxy cluster which we can see now. So doesn't that violate conservation of energy which says energy neither can created nor can destroyed. Because with expanding universe energy in the form of dark energy increases with time so if we consider whole universe (visible + invisible) as isolated system then energy of whole universe increase means energy is created from nothing. Am I missing something over here?"}
{"id":"83837","title":"Is false vacuum bubble nucleation possible in our universe?","text":"Is it possible that a false vacuum bubble to nucleate into our universe rather than a true vacuum one ? If yes,it will expand at speed of light within our spacetime or what ?"}
{"id":"83832","title":"Ampere's law on a long wire with varying current density","text":"On a question from my book: > A long straight wire with a circular cross section of radius $R$ carries a > current $I$. Assume the current density is not constant over the cross > section of the wire, but rather varies as $J=\\alpha r$ where $\\alpha$ is a > constant. Given $I, R$ > > Find $\\alpha$ > > Find the magnetic field as a function of r both inside and outside the wire I think it's just the calculus parts confuses me. My attempt: $$J=\\alpha r' = \\frac{dI}{dA}$$ $$dI = 2 \\pi r'^2 dr' \\alpha$$ $$I = 2 \\pi \\alpha \\int_0^R r'^2 dr'$$ $$I = 2 \\pi R^3 \\alpha \/3$$ $$\\alpha = \\frac{3I}{2 \\pi R^3}$$ from here you just use ampere's and I believe there's no variance issues? $$\\oint \\vec{B} \\cdot \\vec{dl} = \\mu_0 I_{in}$$ apply J=I\/A $$B 2 \\pi r = \\mu_0 \\alpha r A$$ $$B = \\frac{\\mu_0 \\alpha A}{2 \\pi}$$] $$B = \\frac{3 \\mu_0 I r^2}{2 R^3}$$ Is this right? The units seem to line up so I'm hopeful. Outside the wire is treated as the general uniform wire case I assume and am not too worried about that."}
{"id":"134372","title":"How can you calculate how fast a spinning ring\/cylinder will accelerate a mass via gravitomagnetism?","text":"Lets say you have a cylinder of length L, radius R, and mass M. How fast will it accelerate a mass of mass M2 that is entering the \"throat\" of the cylinder, considering the effects of gravitoelectromagnetism? By Gravitoelectromagnetism, I'm specifically talking about the effect where a spinning body will pull an object through the \"throat\" of its spin. Also, I'm wondering what the practical limit of this is: considering the strongest known materials, how fast could a cylinder be spun?"}
{"id":"70845","title":"Discretization of Hamiltonian using finite difference always justified?","text":"I have this continuum version $$ H_{R}=\\int dx\\psi^{\\dagger}(x)(\\frac{p^{2}}{2}+V)\\psi(x) $$ with $V$ as constant potential. Is it always justified to go from this to $$ \\sum_{i}c_{i}^{ \\dagger }\\left[c_{i+1}+c_{i-1}-2c_{i}\\right] +V \\sum_{i}c_{i}^{ \\dagger }c_{i} $$ using the finite difference form of the one-dimensional second derivative? Ignore the factor $-1\/2 $ and assume lattice constant $a= 1 $ and $\\hbar =1 $. Actually I am thinking whether it is justified to use this even when the derivative of eigenfunction is discontinuous at some of the points in real space like for the delta-function barrier. Will that affect the second derivative of the field operators ?"}
{"id":"72143","title":"Can relativistic energy transformation be explained by time dilation and E=h f?","text":"Can one explain the relativistic energy transformation formula: $$E = \\gamma\\ E',$$ where the primed frame has a velocity $v$ relative to the unprimed frame, in terms of relativistic time dilation and the quantum relation $E=h\\nu$? I imagine a pair of observers, A and B, initially at rest, each with an identical quantum system with oscillation period $T$. Now A stays at rest whereas B is boosted to velocity $v$. Just as in the \"twin paradox\" the two observers are no longer identical: B has experienced a boost whereas A has not. Both observers should agree on the fact that B has more energy than A. From A's perspective B has extra kinetic energy by virtue of his velocity $v$. Relativistically A should use the energy transformation formula above. But we should also be able to argue that B has more energy from B's perspective as well. From B's perspective he is stationary and A has velocity $-v$. Therefore, due to relativistic time dilation, B sees A's oscillation period $T$ increased to $\\gamma\\ T$. Thus B finds that his quantum oscillator will perform a factor of $\\gamma\\ T\/T=\\gamma$ more oscillations in the same period as A's quantum system. Thus B sees that the frequency of his quantum system has increased by a factor of $\\gamma$ over the frequency of A's system. As we have the quantum relation, $E=h\\nu$, this implies that B observes that the energy of his quantum system is a factor of $\\gamma$ larger than the energy of A's stationary system. Thus observer B too, using his frame of reference, can confirm that his system has more energy than observer A's system. Is this reasoning correct?"}
{"id":"83789","title":"Intuitively Re-Deriving Equations of Mathematical Physics","text":"Using the intuitive interpretation of the Laplacian $\\vec{\\nabla}^2$ as the difference between the average value of a field in the neighbourhood of a point & the value of the field at that point, one can pretty easily & quickly derive the form of the heat equation, Poisson's equation & the wave equation (as is done in that link if anyone's interested: Davis - Fourier Series & Orthogonal Functions P196). I quite honestly cannot remember those equations, I re-derive them using the intuition that the Laplacian affords me mixed with physical reasoning drawn from the situation (drawn from the field we're using whether it's temperature, concentration, electric potential or displacement). I'm wondering if similar intuition can be used to derive the form of the Helmholtz equation, the Schrodinger equation, the Dirac equation, & really any other nice equation from mathematical physics that people have nice intuition for in their head & wouldn't mind sharing. It doesn't have to be in any way rigorous or even necessarily completely logical so long as you get the right result, though it should really be quick & to the point, thanks."}
{"id":"30675","title":"What exactly are super WIMPs?","text":"I recently got confused (and slightly annoyed by the lack of technical details) when reading a popular article (authored by Jonathan Feng and Mark Trodden) introducing the concept of super WIMPs. The article characterized super WIMPs (without giving more detailed explanations) as follows:   * WIMPs could probably decay to so-called super WIMPs, which would only gravitationally interact with visible matter    * different kinds of super WIMP particles could interact via additional newly postulated weak \"dark forces\" ( = gauge bosons ?) with each other   * this kind of dark matter particles can probably interact with dark energy ( how? What is dark energy in this particular scenarios suposed to be? )   * the authors vaguely stated the super WIMP models are some kind of extensions of supersymmetric models that lead to the \"ordinary\" WIMPs From this characterization I really dont get what super WIMPs are suposed to be so my question is: What are the underlying theoretical ideas behind these phenomenological models? Are they derived in some \"top down\" approach from high energy theories or is some \"buttom up\" extension of something like the MSSM for example applied ? And I would appreciate a technically more accurate description of the super WIMP particles and their interactions."}
{"id":"12385","title":"Are there books on Regularization and Renormalization in QFT at an Introductory level?","text":"Are there books on Regularization and Renormalization, in the context of quantum field theory at an Introductory level? Could you suggest one? _Added_ : I posted at math.SE the question _Reference request: Introduction to mathematical theory of Regularization_ and accepted this answer by Willie Wong."}
{"id":"123217","title":"Schrödinger evolution for a Klein-Gordon equation","text":"I have a problem with the transition from quantum relativistic wave equations (specifically Klein-Gordon equation) to QFT, since a lot of assumptions seem implicit. For example I have a problem with the time evolution operator, which is crucial on deriving the perturbative expansion $-$ the main tool in QFT I believe. c So here's what I have a problem with: when we make the leap from Schrödinger equation to a Klein-Gordon equation, we get a second order time derivative, and hence loose the simple concepts from nonrelativistic QM like: the Hamiltonian, time evolution operator etc. But for a scalar quantum field we can make a Lagrangian density: $$ \\mathcal{L}(x) = \\hbar^2 c^2 g^{\\mu \\nu} \\partial_\\mu \\phi \\partial_\\nu \\phi^* - m^2c^4 \\phi \\phi^* $$ and perform the \"second quantization\", from which we get a Hamiltonian, canonical commutation relations and the ability to use pictures (Schrödinger's, Heisenberg's...). So how does this work? Before there was no Hamiltonian in principle, and now there is. Is this the Hamiltonian we pluck into the perturbative expansions' formulas? What changed, when compared to the single solution wave equation in the beginning?"}
{"id":"4669","title":"What are the most important discoveries\/breakthroughs in physics recently?","text":"Can you provide a list of the most important discoveries\/breakthroughs in physics recently? By recent, I mean the past decade or so. All branches of physics are welcome. Basically, I am interested in major physics breakthroughs\/discoveries which haven't become well-known yet outside their narrow specialties. Most breakthroughs in the 90s like string dualities and the accelerating universe have already become common knowledge."}
{"id":"123218","title":"Seeing one's back on the event horizon","text":"If we would hypothetically be exactly on the event horizon, we should see our own back, because of the circular motion of photons on the event horizon, right? But what would be the image size, or $-$ asking differently $-$ how far away, would our back seem to be? Would it be magnified or minified, when compared to the image of a person $2 \\pi R_{Schwarzschild}$ away?"}
{"id":"104094","title":"Magnetic field of a wire with over 1000 amperes?","text":"Does a large thick wire that has over 1000 amperes of current flowing through it generate powerful magnetic fields? What formula is best here to predict $B$?"}
{"id":"86491","title":"Calculating the power of a lightbulb","text":"How do I calculate the power of a lightbulb? I have values but I don't know the equation to use."}
{"id":"88925","title":"$p\\ dq$ is the \"tautological\" one-form?","text":"The one-form $$\\theta=\\sum_i p_i\\, \\text dq^i$$ is a central object in hamiltonian mechanics. It has a bunch of applications: $\\omega=\\text d\\theta$ is the symplectic structure on phase space, $S=\\int\\theta$ is the classical action, and so on and so forth. It is associated with the names Liouville one- form, Poincaré one-form, canonical one-form, and symplectic potential, none of which surprises me, but its Wikipedia entry informs me that the preferred[by whom?] name for it is actually \"tautological\" one-form, on the grounds that 'canonical' (which would be my natural choice) is 'already heavily loaded', and because of the risk of confusion with some algebraic thingammy. This name completely mystifies me. **Why was the name \"tautological\" chosen for this object?** When, where, and by whom? Or was this name chosen because that's its name?"}
{"id":"88921","title":"Velocity after inelastic collision between bullet and block","text":"The following is the problem that I am working on. > A bullet of $.01\\: \\mathrm{kg}$ is shot into a block of mass $.89\\: > \\mathrm{kg}$ that is hanging from the ceiling. After the bullet has been > caught in the block, it swings and rises $.40\\: \\mathrm{m}$ from its initial > height. Find the initial velocity of the bullet. I though that $\\frac{1}{2}m{v_0}^2 = (m+M)gh$ would give me the solution $25.6\\: \\mathrm{m\/s}$, but the answer is supposedly $2.5 \\times 10^2\\: \\mathrm{m\/s}$. What am I missing ?"}
{"id":"40763","title":"Why does light of high frequency appear violet?","text":"When people are asked to match monchromatic violet light with an additive mix of basic colours, they (paradoxically) mix in red. In fact, the CIE 1931 color space chromaticity diagram shows this effect begins at about 510nm (greenish- cyan), where people mix in no red. From that point on, the higher the frequency of the light source, the more red they mix in. This effect is reflected by the red curve of the CIE standard observer color matching functions, which has an additional bump in the area of blue light. However, that curve does not match the actual spectral sensitivity of red cones. So where does this additional perception of red at higher frequencies come from?"}
{"id":"75710","title":"Is the color wheel just an optical illusion?","text":"As a kid, I was taught that that blue and yellow make green, yellow and red make orange, and red and blue make purple - forming the subtractive color wheel. As an adolescent I was taught that blue and green make cyan, green and red make yellow, and red and blue make magenta - forming the additive color wheel. Somewhere in that time I was also taught that the colors of the rainbow are ROY G. BIV. That last one is the only one that makes _much_ sense to me in physics because perceivable light is not somehow a cycle, but rather a tiny segment of the EM spectrum. Why do the colors at the top and bottom of the spectrum seem to be mixable? Is there anything in nature that indicates this should be the case? Is this just a psychological or physiological phenomenon?"}
{"id":"73003","title":"Why does the visible light spectrum appear to be circular","text":"I'm not sure if this is an acceptable place to ask this question, as it may have more to do with the biological workings of human eyes than with the physical properties of light, but I'd rather hear a physical explanation than a biological one (if one exists). Essentially, we often describe the visible spectrum with a color wheel. In this wheel, Red appears next to Purple (violet), but Red and Violet are at opposite ends of the visible light spectrum. What accounts for this? That is, why do we perceive the Visible Light Spectrum as circular, instead of linear?"}
{"id":"107439","title":"Quantum Fourier Transform and Entropy","text":"QFT is a nonlocal unitary transformation and so can generate entanglement in a system. It means a separable pure state can be converted into an entangled pure state. Now since the presence of entanglement can be witnessed via an increase in the entropy of the subsystems. Since all the subsystems witness a positive entropy change ,does the entropy of the complete system also increase (it seems to increase since entropy is additive) ? Now if it does increase , It seems to violate reversible nature of Quantum algorithms. I am very confused."}
{"id":"65237","title":"If light rays obey to the wave equation, why can they be thought as straight lines?","text":"I'm a newbie with physics but I'm wondering how a ray of light can essentially be represented. I have always known that a ray of light proceeds in a straight line until it encounters another object (or material) that refracts\/reflects it. But a light ray should be part of an electro-magnetic wave, is this correct? If it is so, it should obey to the wave equation and this doesn't seem to me to describe a straight line ray. I'm having problems visualizing how light is emitted and how it relates with the wave equation. Can someone with a clear understanding of the problem explain it to me in simple terms?"}
{"id":"69859","title":"Why and how maximum force is $\\frac{dF}{dx}=0$","text":"In an certain question my teacher asked to find the maximum force. She said that the maximum force in electrostatics means $\\frac{dF}{dx}=0$. Why is it like that?"}
{"id":"107434","title":"Difference between the increase in optical path due to refraction and that due to reflection","text":"If we place two glass plates of refractive index n and each having thickness t,on the way of a light ray the increase in optical path becomes ![enter image description here](http:\/\/i.stack.imgur.com\/7p0h5.jpg) (S2P-S1P)=2(n-1)t due to refraction through them,and the path difference(or extra path traversed by light) due to reflection at the second surface of one glass plate is ![enter image description here](http:\/\/i.stack.imgur.com\/NNxzf.gif) 2ntcos(alpha).Without seeing the expression if we look at the phenomenon directly,aren't they basically same? as both are going through a glass media of same length!!"}
{"id":"33273","title":"Is spacetime discrete or continuous?","text":"Is the spacetime continuous or discrete? Or better, is the 4-dimensional spacetime of general-relativity discrete or continuous? What if we consider additional dimensions like string theory hypothesizes? Are those compact additional dimensions discrete or continuous? Are there experimental evidences of continuity\/discreteness? When particles move inside space do they occupy spacetime by little chunks? What would imply if spacetime is discrete on continuous theories? I've found little information on the web and books. Probably my question is ill-posed and I apologize for this."}
{"id":"121408","title":"The 6-j symbol and intersecting Wilson loops, redux","text":"This is a quite specific question continuing the problems I have with computing the expectation value of intersecting Wilson loops I laid out here. Using the tools from the answer there, I quite quickly arrive at the following expression for the local factor associated to a vertex, where two Wilson loops with reps $\\alpha_1$ and $\\alpha_2$ meet, and where the four surrounding regions have reps $\\beta_1$ to $\\beta_4$: $$ G(\\alpha_1,\\alpha_2,\\beta_{1,2,3,4})_{\\mu\\nu}^{\\sigma\\rho} := \\epsilon_\\mu^{ijk}(\\alpha_1,\\beta_1,\\beta_4)\\epsilon_\\nu^{lmn}(\\alpha_2,\\beta_1,\\beta_2){\\epsilon^*}^\\sigma_{ijk}(\\alpha_1,\\beta_2,\\beta_3){\\epsilon^*}^\\rho_{lmn}(\\alpha_2,\\beta_3,\\beta_4)$$ The Greek indices are the indices incurred from decomposing tensor products as $a^i \\otimes b^j \\otimes b^k = \\epsilon_\\mu^{ijk}e^\\mu$, leading to integral results like $$\\int \\alpha_i(V_b)^i_{i'} \\beta_c(V_b)^j_{j'} \\beta_{c'}(V_b)^k_{k'} \\mathrm{d} V_b = {\\epsilon^*}^\\mu_{i'j'k'}\\epsilon^{ijk}_\\mu$$ (see previous answer). Since the $\\epsilon^*$ that has the $\\mu$ this is summed with lives on the opposite end of the (part of) the Wilson line, the Greek indices must necessarily remain open at the vertices. I am totally fine with this being the result of the computation, but I am still puzzled why the relation to the 6j symbol is so casually tossed about. Let me first remark that the above equation is already suspiciously similar to the very first equation in the definition of $6j$ symbols, but the free indices are irritating me. If $\\epsilon_\\mu^{ijk}(\\alpha_l,\\beta_m,\\beta_n)$ is the $3jm$ symbol (with $i,j,k$ playing the role of the $m$ and the reps corresponding to the $j$), what is the additional index $\\mu$ doing here? If it is not the $3jm$ symbol (which I am currently thinking), then why would the $G$ defined above be the $6j$ symbol (and why has it free indices)? (If these are neither $3jm$ nor $6j$ symbols, then why do Witten, Ramgoolam, Moore, etc. insist they are?) Note that the $6j$ symbol cannot arise after summing the Greek indices, since the $G$s the second index belongs to are, in general, at other vertices, and so have not exactly the same 6 reps as arguments. Furthermore, the $3jm$ symbols are, if I understand them correctly, essentially the Clebsch-Gordan coefficients for expanding a tensor product of _two_ irreducible reps in a third, and the $\\epsilon$ above expand the tensor product of _three_ irreducible reps in all possible fourths (which are then summed over in form of the Greek indices). Something does not add up here, and I heavily suspect it is only in my understanding of the symbols, so I would really appreciate someone clearing up my confusion. **EDIT** : Ok, I think I have found something, but I am still a far shot from solving this riddle, and it requires to think more carefully about the coefficients $\\epsilon^{ijk}_\\mu$: Let $\\alpha,\\beta,\\gamma$ be reps with basis elements $a^i,b^j,c^k$as before. Then, we can decompose the tensor product stepwise instead of at once as: $$ a^i \\otimes b^j \\otimes c^k = \\sum_{\\rho \\subset \\alpha \\otimes \\beta} C(\\alpha,\\beta,\\rho)^{ij}_\\zeta e(\\rho)^\\zeta \\otimes c^k = \\sum_{\\rho \\subset \\alpha \\otimes \\beta} \\sum_{\\sigma \\subset \\rho \\otimes \\gamma} C(\\alpha,\\beta,\\rho)^{ij}_\\zeta C(\\rho,\\gamma,\\sigma)^{\\zeta k}_\\mu e(\\sigma)^\\mu$$ (I apologize for the abundance of symbols, but it really becomes clearer what's happening that way.) Here, the $C(j_1,j_2,j_3)$ are now manifestly Clebsch-Gordan coefficients for $j_1,j_2$ in $j_3$, and thus essentially $3jm$ symbols, and the notation $\\rho \\subset \\alpha \\otimes \\beta$ means that the irreducible rep $\\rho$ occurs as a subrep in $\\alpha \\otimes \\beta$. Since Clebsch-Gordan coefficients for reps not appearing in a given tensor product are zero, we can drop the constraint on the sums and sum over all irreducible reps. Thus, $\\epsilon^{ijk}_\\mu = \\sum_\\rho C(\\alpha,\\beta,\\rho)^{ij}_\\zeta C(\\rho,\\gamma,\\sigma)^{\\zeta k}_\\mu$. Now, in the integral result above, by the Peter-Weyl theorem (see also previous answer), the $\\epsilon$ are only summed over the $\\mu$ belonging to trivial subreps of $\\alpha \\otimes \\beta \\otimes \\gamma$, i.e $\\sigma = 0$, if we denote the trivial rep by $0$ in analogy to $j = 0$ in the spin case. Therefore, we have that the result of the integral is $$I := \\int \\alpha(g)^i_{i'}\\beta(g)^j_{j'}\\gamma(g)^k_{k'} = \\left(\\sum_\\rho C(\\alpha,\\beta,\\rho)^{ij}_\\zeta C(\\rho,\\gamma,0)^{\\zeta k}_\\mu C^*(\\alpha,\\beta,\\rho)_{i'j'}^\\eta C^*(\\rho,\\gamma,0)_{\\eta k'}^\\mu\\right) $$ But the trivial irreducible rep has only one dimension, so the sum over the $\\mu$ is just the multiplicity $n(\\rho,\\gamma,0)$ of $0$ in $\\rho \\otimes \\gamma = \\bigoplus_\\sigma n(\\rho,\\gamma,\\sigma)\\sigma$, i.e. $$ I = \\sum_\\rho n(\\rho,\\gamma,0)C(\\alpha,\\beta,\\rho)^{ij}_\\zeta C(\\rho,\\gamma,0)^{\\zeta k} C^*(\\alpha,\\beta,\\rho)_{i'j'}^\\eta C^*(\\rho,\\gamma,0)_{\\eta k'}$$ This gets rid of the annoying $\\mu$, would lead to the $G$ from the beginning of the question to be comprised of the sum over a product of 8 $3jm$ symbols (the $C(\\alpha,\\beta,\\gamma)$), of which 4 each are summed over their $m$ indices, yielding the product of two $6j$ symbols summed over one of their $j$s (the $\\rho$). Also, the index structure in $e^{ijk}e^*_{ijk}$ in $G$ would translate to an index structure of the $C$ exactly matching that of the $3jm$ in a $6j$ symbol. But before I work that out, can anybody tell me if this is the right track or if I butchered something along the way (I am not comfortable enough with my skills yet to fully trust my reasoning when it leads me to an answer whose shape I already know)? Or should I perhaps take this to the mathematicians, since the answer seems to be purely group theoretic so far?"}
{"id":"17741","title":"How does electricity propagate in a conductor?","text":"On a systems level, I understand that as electrons are pushed into a wire, there is a net field and a net electron velocity. And I've read that the net electron drift is slow. But electricity travels through the wire, essentially at c, and I want to understand that mechanism. My apologies if my question is poorly stated, I know bare bones undergraduate quantum theory circa 1990s but it doesn't explain the motion of electricity in detail. Here's my conception, I'm hoping someone will fill in the holes, ha ha. An electron moves into the wire. It's got a kinetic energy. After travelling a short distance, it spontaneously emits a photon, which hits another electron in a valence shell. That electron then presumably does the same. If this conception is simply wrong, please enlighten me. The questions that arise:   1. Presumably at this level, electrons are acting more like waves and less like particles, but is there any classical component in the picture, ie are electrons coming in imparting other electrons with kinetic energy through repulsion, or does it not work that way?   2. If electrons momentarily have energy, then pass it on by a photon, what determines when that photon is emitted, and what frequency it will be? I assume that electrons in this cloud are not limited by any kind of exclusion principal, and that any frequencies are possible?   3. Why should a photon emitted by an electron be in the direction of travel? Conservation of momentum tells me that if an electron is moving, the photon should be emitted in that direction, slowing the electron, but could an electron emit a photon in the opposite direction? If it did, I assume it would somehow have had to absorb energy from elsewhere? That sounds possible by analogy with quantum tunneling.   4. What is the mechanism by which electrons propagating increase the temperature of the material? Are they transmitting energy to the electrons in the valence shell, which tug at the nucleus, do some photons hit the nuclei directly, or is there some other way?    5. Presumably, electricity travels slower than light, because there is some time in each exchange, and some time when electrons are moving at sublight speeds before emitting a photon. By how much is this slower than light, and what is the speed of each interaction?"}
{"id":"102566","title":"Conduction and propagation","text":"What is the difference between conduction of electric wave in conductor and propagation of electromagnetic wave in dielectric? Why propagation term is used for dielectric and conduction for conductor?. Somehow why propagation of electromagnetic wave (is it energy wave) is not possible in conductor, but in dielectrics, and conduction (power signal) not possible in dielectric."}
{"id":"31355","title":"what exactly do electric waves transfer from one point of the wire to other..?","text":"> **Possible Duplicate:**   >  how does electricity propagate in a conductor? so basically i was considering the speed of the charges inside wire defined by the drift velocity and speed of electric waves that's equal to speed of light..so my point is if electric waves don't carry any charge (as mostly light that we encounter from sun's radiation is em and since they don't carry any charge we don't get shock)so what do they actually carry..i was trying to figure out its resemblance to mechanical waves ..as mechanical wave don't itself carry any matter with it they only transfer energy..then so do the electric waves they can't carry charge but they must transfer energy in the form of you know electric waves..or could i say now that these electric waves are source of electric field..then thing that's confusing me is that if electric field propagates at speed of light(speed of electric waves)..then do the voltage that we define as eL or voltage or potential at a point is=electric field intensity * L(distance) then should the voltage drop or potential also varies accordingly or i could say that the potential also following the electric field intensity at the speed of light..but this is contradictory to our general observation in which we define particular time period and phasors which define variation of voltage that..so not at the speed of light.."}
{"id":"116519","title":"Does the average momentum vanish for an eigenstate of the simple harmonic oscillator?","text":"Suppose we have a simple harmonic oscillator, let's consider the ground state, $|0\\rangle$ and the first excited state $|1\\rangle$. $\\langle 0|\\hat p|0 \\rangle$ is zero right? Since the particle can either be travelling to the left or right, where $\\hat p$ is the momentum operator. Similarly, I think $\\langle 1|\\hat p|1 \\rangle = 0$ But, $\\langle 0 | \\hat p | 1 \\rangle$ is non-zero, right? Since they are different states. Also, since $\\hat p$ is Hermitian, $\\langle 0 | \\hat p | 1 \\rangle = \\langle 1 | \\hat p | 0 \\rangle $, right?"}
{"id":"116510","title":"Surface Tension - Lung Alveoli","text":"So, the way I understand this is as follows : The alveoli (pretend they're bubbles) have diameters of the order of microns implying a massive pressure required to inflate them by the Young-Laplace equation. $p_{in}-p_{out}=\\frac{2\\gamma}{r}$ However, the presence of pulmonary surfactant molecules (lets just pretend they're like detergents molecules in washing liquid) can effectively reduce the surface tension at the unexpanded alveoli and hence allow easy inflation. Now this bit I don't understand : As the alveoli expand the **distance between the individual surfactant molecules on the alveoli increases and hence the surface tension rises again** therefore decreasing the rate of expansion. What is the mathematical connection between surface tension and separation between surfactant molecules ? How can I rationalise the statement in bold ?"}
{"id":"114971","title":"Why are the bounds to the permittivity $\\epsilon$ a circular arc in the complex plane?","text":"I'm reading this paper which is essentially about connecting the complex permittivity $\\epsilon$ with the microstructure of a thin film. They talk about how you can place limits on the possible values of $\\epsilon$ depending on how much information you have on the microstructure, and say: > We use the fact that all bounds are circular arcs in the complex $\\epsilon$ > plane to give... Why is this the case? My intuition tells me it has something to do with the Kramers-Kronig relations or energy conservation, but I don't know why."}
{"id":"5950","title":"Justification of ignoring large set of entanglements","text":"If we can think about the universe as a wave function then many particles should be entangled with many other particles in the universe. The obvious question arises why we don't see those entanglements in everyday circumstances. One standard explanation given is those entanglements average out and cancel so we can ignore those. However, hardly any mathematical justification is given for them to cancel. My question is how much trust one should have on that particular assertion? Is there any mathematical arguments already put forward by anyone?"}
{"id":"5955","title":"Is holomorphy the real reason for nonrenormalization in supersymmetry?","text":"Seiberg traced the nonrenormalization of supersymmetric theories to holomorphy of the superpotential in chiral superspace. However, this overlooks the fact that with a different number of supersymmetric generators, supersymmetry can be real or symplectic, instead of complex. But yet, even for those cases, we still have nonrenormalization. If holomorphy isn't the reason, what is? If the reason for nonrenormalizability is totally different for different dimensions and number of supersymmetry generators and choice of superfields, is it then _ad hoc_ that nonrenormalizability always holds in each case?"}
{"id":"127442","title":"If light was able to pass through a wall, would the wall be invisible to the human eye?","text":"In addition, to get light to other side of the wall, could it be converted to radio waves and then back to light waves? Edit: My idea was if there was a special material that was painted on both sides of a wall that converted radio waves to light waves and light waves to radio waves would the wall appear invisible? (My thought being that the electromagnetic radiation would travel through the wall since radio waves can travel through walls) Also, thanks for all the responses!"}
{"id":"64644","title":"Currents and the Speed of Light","text":"Why is it that currents don't flow at the speed of light, but rather significant ratios of the speed of light. I don't have any formal reasoning as to why they would flow at the speed of light-I just feel as if it would make sense. That being said, the fact that they do move at near the speed of light is also peculiar to me. Lastly, if you have current and area determined, can you figure out the velocity of the charges? I suppose you would also need to know how big the charges are. I just have no intuition as to how to go about analyzing the speed of current in some given wire."}
{"id":"72072","title":"Windflow in a car","text":"When driving on a highway, if the front two windows of a car are open , how does the air flow in and out of the windows? Clearly, air cannot keep accumulating inside the car, so how\/where does the air flow out?"}
{"id":"64642","title":"How large of a solar sail would be needed to travel to mars in under a year?","text":"I'm attempting to approach this using the identity $$F\/A = I\/c$$ I can solve for Area easily enough $$A = F(c\/I)$$ and I know the distance $d$ is $$d=1\/2(at^2)$$ But I'm having difficulty trying to relate this to the time it would take to reach mars. Any ideas where to go from here?"}
{"id":"12059","title":"what is an inverse femto barn?","text":"I came across the use of the unit barn and inverse barn while reading about the operation of LHC. What is an inverse femtobarn ? What does it tell about the experiment being described ?"}
{"id":"46166","title":"Relatively how much energy do fluorescent light tubes take to turn on?","text":"Fluorescent lights are already efficient when they’re running but I’ve heard that it takes a lot of energy to turn a fluorescent light tube on. So is it more efficient to turn off a fluorescent tube immediately when you’ve finished using it or is it better to leave it on and then wait until you’re more likely to not use it again for a while?"}
{"id":"121845","title":"Why does angular momentum shorten the Schwarzschild Radius of a black hole?","text":"Angular momentum causes the event horizon of a black hole to recede. At maximum angular momentum, $J=GM^2\/c$, the Schwarzschild radius is half of what it would be if the black hole wasn't spinning. Can someone explain why angular momentum reduces the Schwarzschild radius?"}
{"id":"121840","title":"Confused about Newton's 3rd law","text":"I am confused about Newton's 3rd Law. If a person jumps off the ground a force is applied both to the person and to the ground. However, as $F=ma$ acceleration experienced by the Earth is much less than that experienced by the person. But: I press with gravitation force on the ground so it should press with the same force on me, so if my mass is less than earth mass my acceleration should be greater as well but I am not moving (flying)? Second: If small rocket(with small mass) pressure against bigger rock with greater mass the rocket should have greater acceleration towards direction opposite to its flying path so how the rocket can actually move the rock(towards left at picture where rock acceleration is small) and not the opposite(rocket acceleration at picture towards right is bigger)?   _Edited_   If a rocket with mass $m_{rocket}$ pushes against a rock with mass $m_{rock}$ with force $F_{thrust}$ the rock will push back with equal force (Newton's 3rd Law). The rocket will experience an acceleration $a_{rocket}$ in the opposite direction of $F_{thrust}$ and the rock will experience $a_{rock}$ in the _same_ direction as the thrust. However, in this example $m_{rocket} < m_{rock} \\therefore a_{rocket} < a_{rock}$   Why doesn't the rocket move in the opposite direction of $F_{thrust}$ since the rock has a greater mass? ![enter image description here](http:\/\/i.stack.imgur.com\/9kKKg.jpg) Not sure if it helps but added picture to second question Edit:To put second question simple how a small(low mass) object can push big(great mass) object if according to newton 3rd law a big mass object causes greater acceleration on small mass object"}
{"id":"99325","title":"Ideal gas law problems","text":"I got really confused about real gases volume and ideal gas volume. Ideal gas molecules take up no space, if we put gas into a 2.4L water bottle, we know that all the gas will expand all over the bottle and we say at this moment the gas has volume of 2.4L. So what is this volume?"}
{"id":"69141","title":"How to justify matter-field interaction for non-gauge-invariant Hamiltonian?","text":"I'm wondering how can one formally justify the electromagnetic response of a system which does not verify local U(1) gauge invariance. A good example of what I would like to consider is given by the two-body interaction term discussed in relation with superconductivity as I'll elaborate below, but many examples can be found and the question is rather general. Most of the people starts with a BCS Hamiltonian having generically the following form   $$H_{\\text{BCS}}=\\sum_{k,k'}\\hat{c}_{k\\alpha}^{\\dagger}\\left(\\mathbf{i}\\sigma_{y}\\right)_{\\alpha\\beta}^{\\dagger}\\hat{c}_{-k\\beta}^{\\dagger}U\\left(k,k'\\right)\\hat{c}_{k'\\alpha}\\left(\\mathbf{i}\\sigma_{y}\\right)_{\\alpha\\beta}\\hat{c}_{-k'\\beta}$$ _i.e._ describing singlet Cooper pairing of electron with fermionic operators $\\hat{c}_{k}$ in mode $k$, the greek indices being the spin ones. I think this Hamiltonian is manifestly not U$(1)$ local gauge invariant, due to the $k$ and $k'$ on different operators. **I'm wondering** whether it makes sense to talk about the electrodynamic response of a superconductor when one starts with a non gauge invariant Hamiltonian. More generally, **does it makes sense to discuss non-gauge invariant Hamiltonian in the context of condensed matter ?** How should we understand such non-gauge-invariant Hamiltonians, $H_{\\text{BCS}}$ being a simple example ? More details:   * The original BCS Hamiltonian has $k=k'$ and $U\\left(k,k'\\right) \\rightarrow -g$ a constant, and so the $s$-wave interaction _is_ both local and global U(1) invariant. $H_{\\text{BCS}}$ is only U(1) symmetry global invariant, as far as I can see.   * The Hamiltonian $H_{\\text{BCS}}$ given above is particularly useful to describe some non-conventional effects ($d$-wave pairing for instance) and can be further generalise. I have not doubt about the validity of the results obtained using this Hamiltonian (some of them are even justified experimentally).   * _I'm wondering about the possibility to formally define an electromagnetic response in a non-U(1) gauge invariant theory._ It is for instance clear that one can add some gauge invariant part of the above Hamiltonian, such that the constitutive Maxwell equations are preserved (no magnetic monopole and Faraday's law). But it seems also clear for me that one intrinsically imposes from the beginning some different matter-field interaction, isn't it ? Or at least that the covariant substitution is no more a correct prescription..."}
{"id":"79444","title":"projectile that splits into two fragments of equal mass","text":"I am studying for an exam, and this is part of a problem in my book. A projectile is launch from level ground and is intended to hit a target 100m away. Instead, it explodes into two fragments of equal mass, one of which lands 100m beyond the target. If the fragments don't take the same amount of time to land, must the second one land 100m short of the target? The back of the book tells me that this isn't true, but I'm having a hard time figuring out why. I know that the trajectory of the center of mass for the system is going to follow the trajectory of the projectile before it exploded. Thus, the CM of the system is at a distance of 100m from the origin. However, the equation for the center of mass would be $100 = mx_{cm} = \\frac{1}{2}mx_1 + \\frac{1}{2}mx_2$. Now $x_1 = 200$ by the conditions of the problem, so $ \\implies x_2 = 0$. So clearly there must be some reason why this doesn't hold, but I can't figure it out. Could anyone explain to me why this won't work?"}
{"id":"91917","title":"Triple points for other substances","text":"Can substances other than H2O have a triple point, where the three usual phases of matter (solid\/liquid\/gas) can exist?"}
{"id":"46292","title":"Rough, easy DIY method of measuring magnetic field strength","text":"How to easily, using standard DIY equipment measure the strength of magnetic field generated by a permanent magnet? Narrowing down the \"loose language\" of the above: strength of magnetic field: either flux density **B** at given point relative to the magnet or magnetic flux ΦB over area enclosed by a loop made of wire - whichever will be easier to measure, either of those is fine. standard DIY equipment: commonly found household items, rudimentary tinkering tools. Soldering tools, multimeter, simple electronic parts, or maybe an easy to make spring-based dynamometer - anything of this class of complexity. The distance of measurement is such that the field is easily noticeable through simplest methods e.g. another magnet held in hand exerts perceptible force - distance of maybe 5cm away at most. The measurement doesn't need to be very accurate - error of order of 50% is quite acceptable. Simplicity is preferred over accuracy. Rationale: trying to estimate what coil I need to generate sufficient amount of power to light a LED with a frictionless generator based on that magnet (knowing speed of movement of the magnet and location of the coil relative to the path of the magnet). If you know other simple methods of doing that (without need for measuring the field), they are most welcome them too."}
{"id":"57874","title":"Do protons exchange photons with electrons?","text":"I'm sorry for this question but, I just don't get it. According to the electromagnetic field theory, electrons repel each other by exchanging photons. How do protons attract electrons, by photon exchange?"}
{"id":"9204","title":"Buoyancy: helium vs hydrogen balloons","text":"Given I have two identical balloons on earth, how will the buoyancy compare between the one filled with helium and another filled with hydrogen? How can I calculate the ratio of buoyancy given two different substances and identical balloons? I am also interested in the equations relating to this problem."}
{"id":"9201","title":"What is meant by positive and negative gravity\/energy\/spactimecurvature?","text":"I have recently come across some cosmological assertions (based on empirical data) about the universe being self contained in the sense that it is entirely capable of coming into existence from a zero-energy initial state . This is based on the observation that at grand scale the positive and negative gravity\/energy etc. cancel out each other. What do the terms positive and negative actually mean in this context ?"}
{"id":"63018","title":"How to determine a reaction force?","text":"An object sits on an inclined plane. The weight of the object will have a normal and parallel component. I always thought that the reaction of the plane was simply the negative of the normal component of the weight. Similarly, an object swings on a pendulum. The weight of the object can be decomposed into a radial and tangential component. I assumed that the reaction (tension) of the string must be the negative of the radial component. But several examples have made me doubt my assumptions. One exercice in my textbook involves determining the reaction on a mass as it slides down a parabolic surface, as a function of $\\theta$, the angle the vertical makes with the surface at any given point. Under my assumptions about reaction force, the answer is so trivial as to make the exerice pointless (since they give you $\\theta$), but instead the exerice launches into a lot of complicated reasoning involving the Frenet-Serret base and comes out with a very different answer to just $-mg\\ sin(\\theta)$. Also, consider the second example I gave. If the object is swinging in a circle (to make it even simpler, say it has uniform circular motion), then it must have radial acceleration (centripetal, specifically). But that's impossible if the reaction is exactly the negative of the radial weight. But then... how can I determine a reaction force? In the absence of the simple rule I gave in the first paragraph, what else is there? Can it be done without making assumptions about the motion (eg. if you suppose the motion of an object is uniform circular, you have a formula all ready for the centripetal force and may thus be able to deduce the reaction force)."}
{"id":"123808","title":"Double slit experiment from first principles of QM","text":"I have read many descriptions of electron double slit experiment but I could not find the description from the first principles of quantum mechanics. Most of the descriptions makes comparison with light waves or water waves and after some arguments from optics explain why the interference happens. Light waves and water wave description are not fundamental, they are phenomenological models. Could somebody explain the electron double slit interference only from the first principles of quantum mechanics? I know the answer from the Feynman path integral approach but I would like to understand it from the pre-Feynman Integral approach."}
{"id":"54856","title":"Lagrangian definition of stress energy tensor","text":"Can anyone explain why $T_{\\mu \\nu} = \\frac{2}{\\sqrt{-g}} \\frac{\\delta \\mathcal{L}_M}{\\delta g^{\\mu \\nu}} $, other than justifying it from the Einstein field equations?"}
{"id":"62945","title":"Error calculation with linear regression","text":"I am trying to determine the boltzmann constant by using a bipolar junction transistor. In my circuit (apparently I don't have enough point to join a image sorry), the Ebers and Moll model gives the relation $ i_c = I_s\\exp\\left(\\frac{V_{BE}}{V_T}\\right)$ where $ V_T= \\frac{k_b\\cdot T}{q}$ and $V_{BE}$ is the difference of potential between the emiter and the base. I'm plotting the $\\log(i_c)$ as a function of $V_{BE}$ and use the slope of the line ($\\alpha$) to deduce boltzmann constant. So $ log(i_c)= \\alpha\\cdot V_{BE} + b $ from which I deduce $k_b = \\frac{q}{\\alpha\\cdot T\\cdot \\ln(10)}$ I have done that for 4 different temperatures. I average the different values of $k_b$ to find $k_b = 1.57\\cdot 10^{-23}$ How can I calculate the error margin for each of the $\\alpha$ coefficient and how do I calculate the error for the mean coefficient $\\bar\\alpha$?"}
{"id":"53498","title":"What is the importance of the Higgs-strahlung process in the Higgs search?","text":"I would particularly like to know why this process is considered the main search mode for Tevatron but useless for search at LHC."}
{"id":"118855","title":"average speed and velocity","text":"A distance on a straight line from point $a$ to $b$ is $2 km$. A student walks from this line with a speed of 4km\/h and another student walks with a speed of 6km\/hour. what is the average velocity and average speed of student?"}
{"id":"10283","title":"Is there a \"Size\" Cutoff to Quantum Behaviour?","text":"We all know that subatomic particles exhibit quantum behavior. I was wondering if there's a cutoff in size where we stop exhibiting such behavior. From what I have read, it seems to me that we still see quantum effects up to the nanometer level."}
{"id":"21573","title":"Quantum perpetual motion","text":"> Perpetual motion describes hypothetical machines that operate or produce > useful work indefinitely and, more generally, hypothetical machines that > produce more work or energy than they consume, whether they might operate > indefinitely or not. (Source:Wikipedia) With this definition in mind, particularly the \"operates indefinitely\" (I don't care about producing work), won't quantum mechanics allow perpetual motion due to energy quantization? For example, an electron in hydrogen can be thought of as perpetual motion. It's indefinite(I think so); unlike gravitational orbits (which slowly release energy). This is due to the quantization of energy. Without it, the electron would have fallen into the nucleus. More generally, if we energy is quantized in a system, dissipative forces of lesser magnitude cannot act on it, due to quantization. For example, if a block can have only an integer value of energy in Joules, then frictional forces of power $P<\\frac{1 J}{\\text{planck time}}$ cannot act. Or something like that. So does quantum mechanics permit an infinitely advanced civilization to build a machine which operated indefinitely without doing work? I'm not well versed in quantum mechanics, so I may be making a mistake here, or I may just be confused. Refer to equations if you want, but try not to use them too heavily unless the answer depends on it. It's OK if they're explained a bit."}
{"id":"80807","title":"Why do same\/opposite electric charges repel\/attract each other, respectively?","text":"I know plus pushes another plus away, but why, really, do they do that? On the other hand, molecules of the same type are attracted to each other. I find that weird. I do know some stuff about four universal forces. But why in general the general \"rule\" is that opposite charges pull each other? Yes, I do realize this could be connected to very basic stuff that science is still trying to figure out, and can be traced to the Higgs, but still, there must be something to tell. Please don't answer like my chemistry teacher: \"The reason this is not a metal because it's is non metallic element\", or similar explanations."}
{"id":"105196","title":"Attraction and repulsion of charge?","text":"Why do like charges on identical bodies cause a repulsion and unlike charges cause an attraction?"}
{"id":"129249","title":"Why does proton and electron attract each other?","text":"Not that their charges are opposite! \"+\" & \"-\" are the signs we named them. By nature Why do they attract each other? And Why do they repel each other?"}
{"id":"6108","title":"Comprehensive book on group theory for physicists?","text":"I am looking for a good source on group theory aimed at physicists. I'd prefer one with a good general introduction to group theory, not just focusing on Lie groups or crystal groups but one that covers \"all\" the basics, and then, in addition, talks about the specific subjects of group theory relevant to physicists, i.e. also some stuff on representations etc. Is Wigner's text a good way to start? I guess it's a \"classic\", but I fear that its notation might be a bit outdated?"}
{"id":"131972","title":"Lie theory and particle physics","text":"I have recently been reading _Intro to Lie algebras and representation theory_ by Humphreys, and when I am finished I am interested in reading about Lie groups and Lie algebras and their applications to particle physics. > Is there a book that assumes basic knowledge of Lie algebras, and no > knowledge of lie groups and particle physics\/quantum mechanics? I have seen Howard Georgi's book, but it assumes good knowledge of particle physics. In addition, I do not know differential geometry."}
{"id":"103169","title":"Would anyone suggest me usefull web resources on lie groups and lie algebra and a good book to start with?","text":"Would anyone suggest me useful web resources on lie groups and lie algebra and a good book to start with?"}
{"id":"44475","title":"The formula of the force exerted on an electric dipole by non-uniform electric field","text":"When an electric dipole of moment $\\mathbf{P}$ is located in a non-uniform electric field $\\mathbf{E}$, there is an net force exerted on it. However, the formula of the force in some books is read $\\mathbf{F}=\\nabla(\\mathbf{P}·\\mathbf{E})$, while in other books, it is $\\mathbf{F}=(\\mathbf{P}·\\nabla)\\mathbf{E}$. Obviously, the two formula are not the same. So, which one is true?"}
{"id":"11887","title":"What is the conserved canonical momentum for a relativistically moving charge in a static Coulomb electric field?","text":"The canonical momentum is a fundamental conserved quantity from Noether's theorem for translational invariance of the Lagrangian. Yet I'm finding it very difficult to see its derivation, or even a statement of what it is for something as fundamental as a relativistically moving charge in a static Coulomb electric field. Can anyone state what it is, or even give a derivation if it's not too much trouble?"}
{"id":"112984","title":"Liouville-von Neumann equation can be directly derived from Heisenberg picture?","text":"The Liouville-von Neumann equation for the density matrix is: $$ i\\hbar\\frac{\\partial\\rho}{\\partial t}=[H,\\rho],$$ while in the Heisenberg picture: $$ \\frac{d}{dt}A(t)=\\frac{i}{\\hbar}[H,A(t)] +\\frac{\\partial A(t)}{\\partial t}$$ if we adopt some kind of conservation law (like in classical theory), i.e. $$\\frac{d}{dt}A(t)=0. $$ Then we have $$\\frac{i}{\\hbar}[H,A(t)] +\\frac{\\partial A(t)}{\\partial t}=0 $$ by replacing $\\rho=A(t)$, we can directly reach the von Neumann equation. Is this derivation correct? if it is, what is the physical meaning behind $\\frac{d}{dt}A(t)=0$ then?"}
{"id":"111879","title":"Can a vacuum cleaner be used to purify the air in a small room?","text":"A hepa vacuum cleaner will pick up fine dust from the floor, filter it and send the clean air out through the exhaust. However with movement in the room fine dust will also be goinng up in the air and so the vacuum will not take it in and this fine dust will settle hours later. As far as i can see, a vacuum cleaner is very similar to an air scrubber, takes air in, filters it and sends it out. 1) is it not possible to close windows and leave the vacuum on in the middle of the room and expect it to filter fine dust in the air\/room?   2) what if i maneuvered around and tried to vacuum the air aswell as the floor for several hours, would this do the job? 3)is a air scrubber\/filter necessary? The room in question is about $18\\,\\mathrm{m}^2$ and the vacuum cleaner i intend to use is a sealed hepa unit which is about 500 air watts and says it can do $58\\,\\mathrm{L\/s}$ which i think is litres per second. Please give general answers to the questions I have asked aswell as specific to the vacuum and room in question. Thanks."}
{"id":"54145","title":"What can be the lightest possible moon launch vehicle?","text":"I tried calculating this, but it gets too complicated. Assume, we have a Moon orbit station and ISS on Earth orbit. We have a Moon base. We want to send a tourist for a week on the Moon and back. We need to launch only the oxygen\/fuel cells and the fuel for trans-lunar and trans-earth injection. The lunar lander lands and takes off in one piece (we don't need a pile of used stages on the lunar base). It can be lighter than 10 metric tons of Apollo LEM. Fuel cells (200 kg each) may go, because now we have solar panels. So, how low can we get with newer technologies? And the main question, how small can the launch weight be? Is this doable with conventional Soyuz or Proton rocket? Assume, we might have a coilgun to do trans-Earth injection right from the lunar surface. Delta-v of 2.7 km\/s means 90 km of rail and accelerating @ 5g, if I remember my calculations correctly. Some data from Wikipedia:   * Apollo Lunar Module      * Ascent stage 4,547 kg of which 2,353 kg is propellant     * Descent stage: 10,149 kg (8,200 kg of propellant for 2,500 m\/s delta-v)   * Command and Service module      * command module 5,809 kg     * service module 24,523 kg all together 46,980 kg   * Saturn V      * 1st stage with fuel: 2,300,000 kg     * 2nd stage: 480,000 kg     * 3rd stage: 120,800 kg The third stage fired only partially (165 + 335 seconds) to get to LEO, and then was used for TLI. So, 120 tons were sent to LEO, and only 47 tons left after TLI. How low can the latter get? The lunar module will be orbiting the Moon and reused. This means several tonnes less for TLI and Lunar orbit injection. But the fuel for it has to fly from Earth. If CM gets smaller, this can also make the vehicle lighter. Service module can be reduced by using inflatable materials, and it can be reused and stored at ISS. So we save 25 tons x 9 km\/s (launch from Earth), but add 3km\/s of delta-v to park it after the way back. So, all together, we need to launch   * the Earth landing module (CM analog)   * fuel for      * TLI     * LOI     * lunar landing and takeoff     * TEI     * fuel to park SM in LEO If coilgun is used, we don't need takeoff and TEI fuel. (Hm... we need to launch the SM back too :) How much does this weigh?"}
{"id":"79698","title":"Integration of 3-momentum","text":"During a lecture that I missed, I was trapped when the lecturer uses the relation $$dp_x~ dp_y ~dp_z ~=~d^3\\mathbf{p} ~=~ 4\\pi p^2 dp.$$ Can I know how is this relation derived please?"}
{"id":"91741","title":"S-Matrix, String theory, Matrix mechanics and Quantum Mechanics","text":"I'm trying to learn theoretical physics up to string theory. I know linear algebra, calculus 1+2, complex analysis. I know the basics of homology, homotopy, group theory and differential geometry. Now I'm starting to read a first introduction to QM, which is this PDF: http:\/\/arxiv.org\/abs\/1007.4184 . And I read a lot on physics exchange on these topics. My main goal is to get a good sense of string theory and maybe that I can solve some basic problems. Now I did read that the historical background of string theory is the S-Matrix(which I guess has to do with the matrix formulation of QM). Should I first learn this approach to QM and then switch to S-Matrix and finally string theory? Or just completely skip these topics(S-Matrix and Matrix Mechanics) and learn string theory in an ordinary style?(like in the usual literature, which I assume is not taught with the S-Matrix model). Edit\/ I know usual mechanics, theoretical mechanics(Lagrangian and Hamiltonian too) and a bit electromagnetism, but didn't do much problems on these topics, but I know the concepts."}
{"id":"62050","title":"Negative emf in AC generator","text":"At a certain instant in AC generator, when the normal of the plane (rectangular coil) makes an angle of 270 degrees with with the magnetic induction B, the value of emf is: $E = -NAB\\omega$ My teachers would usually say that this is the minimum value of emf that a generator produces. Does it really mean that? Or does the negative sign only mean that emf is at its peak value but the current is flowing in opposite direction?"}
{"id":"79695","title":"Punching - Force or Momentum?","text":"If I want to punch a person inflicting maximum damage, what do I need to care about? My force of punching, i.e, do I need more acceleration? Or do I need momentum, i.e my velocity for punching?"}
{"id":"126181","title":"Calculate amount of energy based on height of object","text":"How can you calculate the amount of energy an object produces that falls from a particular height? Or water. In an eg i would like to know how can you calculate the amount of electricity water can produce at a certain height?"}
{"id":"23550","title":"Why does your car lurch toward an oncoming truck as it passes you?","text":"I notice that the larger the truck the greater the magnitude of the lurch. Can anyone give a physical explanation to this?"}
{"id":"5031","title":"Can black holes form in a finite amount of time?","text":"One thing I know about black holes is that an object gets closer to the event horizon, gravitation time dilation make it move more slower from an outside perspective, so that it looks like it take an infinite amount of time for the object to reach the event horizon. It seems like a similar process should slow the formation of the black hole itself: As the star collapses, its gravitational time dilation make itself collapse more slowly. This make me wonder, are what astronomers claim to be black holes really black holes, or are they stars that progressively make themselves more similar to one without actually reaching the stage of having an event horizon? EDIT: Contemplating one answer, I realize the question is ambiguous. What does finite time mean in general relativity. Here is a less ambiguous question: Is there a connected solution of 3+1 dimensional general relativity with one space-like slice not have a singularity, and another space-like slice having one."}
{"id":"66535","title":"Specific energy and specific angular momentum of photon","text":"In this PDF [1], is made reference to specific energy and angular momentum of a particle. If the particle has no mass, like a photon, how should I define these terms in the equations further down for the path of the particles? [1] Lecture XIX, Christopher M. Hirata, Caltech M\/C 350-17"}
{"id":"27647","title":"Relativistic corrections to quantum mechanics of Coloumb potential","text":"Systems of charged particles (such as atomic nuclei and electrons) can be described by nonrelativistic quantum mechanics with the Coloumb interaction potential. A fully relativistic description is given by quantum electrodynamics which is much more complex. Is it possible to expand various quantities in QED as power series in 1\/c i.e. around the nonrelativistic approximation? Examples of relevant quantities are:   * Ground state energy of a given set of charged particles   * Excited state energies   * Scattering cross sections of charged particles & their bound states (assuming we trace over the photons in the final state)"}
{"id":"38930","title":"Using quantum entanglement to send messages back to the past","text":"> **Possible Duplicate:**   >  Entanglement in time I heard that there is an experiment that uses quantum entanglement to try to send messages back to the past. I am having a hard time understanding how such experiments would work theoretically. Can anyone offer me some insights toward these types of experiment?"}
{"id":"97965","title":"How do I combine three resistors which are in parallel with each other?","text":"For this circuit (a and b are connected by a battery), ![circuit diagram](http:\/\/i.stack.imgur.com\/CVtE3.png) Will I be able to find the total resistance of the circuit by adding resistors that are in series and combining resistors that are in parallel without using the method of Kirchhoff's Voltage and Current Law? I tried adding R3+R4 and this forms one resistor, which is in parallel with R2 and R1. And R2 is in parallel with R1 and R3, and R1 is in parallel with R2 and R3. This can be verified by Kirchhoff's Voltage laws. The problem is, I get stuck here. I know that if R1,R2 and R3+R4 can be combined into one resistor, then parallel-series would solve the problem neatly. But I can't see how these resistors be combined."}
{"id":"74611","title":"weak interaction coupling constant","text":"in wikipedia, the weak interaction coupling constant is said to be 10^-13 times weaker than that of the strong interaction but hyperphysics (http:\/\/hyperphysics.phy-astr.gsu.edu\/hbase\/forces\/couple.html) says it's 10^-6. It seems wikipedia is right but why many other resources also provide a value similar to that of hyperphysics?"}
{"id":"8570","title":"Determine Charge With Electroscope?","text":"You have three separate glass rods, and you know one is positive, one is negative and one is neutral. You also have an electroscope that is positively charged. How can you determine which rod is positive, negative and neutral? I think you will be able to determine which rod is negatively charged, as that rod will cause the electroscope leaf to rise. I'm not sure how to differentiate between the positive+neutral though, as both will cause the leaf to collapse. Could you place it next to the negative rod and see which repels? Essentially it would be great if somebody could explain the actual physics behind the situation. Why will the negative rod cause the leaf to rise and vice versa? Thanks!"}
{"id":"119071","title":"What is being deprived when a photon is being watched in double slit experiment?","text":"How are photons being watched in the double slit experiment? What exactly does being observed mean, as it is obviously changes the state of the photon somehow - it must be depriving the photon of something or emitting something that interacts with the photon."}
{"id":"51395","title":"Photons in a gravitational field","text":"I have been really staring for a while in a MP-Beiser book and I totally disagree with a statement he does there. On a page 85 he states that photons act as they have a mass $m$. He derives this by stating that: $$ \\begin{split} p &= m v\\\\\\ \\frac{h\\nu}{c} &= m c\\\\\\ m&= \\frac{h \\nu}{c^2} \\end{split} $$ But I totally disagree with this. We have learned and derived that momentum of a particle is: $$ \\begin{split} p &= m v \\gamma (v)\\\\\\ \\frac{h \\nu}{c} &= m c \\gamma{(c)}\\\\\\ m &= \\frac{h \\nu}{c^2 \\underbrace{\\gamma(c)}_{=0}}\\\\\\ m &= 0 \\end{split} $$ Something here is totally wrong, but what? How can an author state what he does? I know that on Harvard they did an experiment resulting in different $\\nu$ of a photons falling in a gravitational field, but they must have been wrong or something... Please someone explain."}
{"id":"32210","title":"Using the covariant derivative to find force between 't Hooft-Polyakov magnetic monopoles","text":"I am reading this research paper authored by NS Manton on the Force between 't Hooft-Polyakov monopoles. I have a doubt in equation 3.6 and 3.7. We assume the gauge field for a slowly accelerating monopole to be $A_0 = \\epsilon^2 a_i t A_1$, where $\\epsilon^2$ is an infinitesimal. Also, we write $\\partial_0 \\phi = -\\epsilon^2 a_i t \\partial_i \\phi$. Using this he writes $D_0\\phi=-\\epsilon^2 a_i t D_i \\phi$, where $D_i\\phi=\\partial_i \\phi + [A_i,\\phi]$. Isnt the sign of the second term wrong? Secondly, he says differentiation wrt t gives us, $D^0 D_0 \\phi = \\epsilon^2 a_i D_i \\phi$. Shouldnt it be $\\partial^0D_0 \\phi$? Cause we are taking the actual derivative wrt t rather than the covariant derivative, WE should get some extra terms, do they cancel out? How does the minus sign disappear? Does the covariant derivative behave like a normal derivative in any case?"}
{"id":"61089","title":"English translation of Helmholtz' paper: “On the Physical Significance of the Principle of Least Action”","text":"I am asking about an English translation of a Helmholtz paper: > Ueber die physikalische Bedeutung des Princips der kleinsten Wirkung. > _Journal für die reine und angewandte Mathematik (Crelle's Journal), Volume > 100, Issue 2, 1887, Pages 137-166, and Volume 100, Issue 3, 1887, Pages > 213-222._ http:\/\/www.degruyter.com\/view\/j\/crll.1887.issue-100\/crll.1887.100.137\/crll.1887.100.137.xml?format=INT. (Also see link and link.) The title in English: _On the Physical Significance of the Principle of Least Action._ Has it ever been translated (to English)?"}
{"id":"26593","title":"For the long-term evolution of atmosphere\/orbit, when is perihelion more important than mean distance?","text":"When we want to figure out the long-term evolution of a planet's atmosphere\/orbit, when is perihelion more important than mean distance? E.g. some processes (like Jeans Escape and escape of atmospheres) are disproportionately affected during perihelion (point of closest approach) rather than during aphelion."}
{"id":"36469","title":"Are the protons and neutrons in the nucleus arranged in any particular way?","text":"I was wondering this: suppose you have two oxygen atoms. They will both have 8 protons and 8 neutrons in the nucleus (at least if they are the most common isotope). Now, will all those particles be arranged in the same way in both atoms? If they are, why would that be, and if not, does that affect the element's properties in any way? But then I also thought that maybe the uncertainty principle doesn't let us even ask this question. Maybe you can't tell the particles' positions so accurately, so all you can say is that you have 8 protons and 8 neutrons all together in a small space. So, which one is it? Can we even tell where all the particles are, and if we can, does it matter exactly how they are arranged?"}
{"id":"90501","title":"Geometric structure of atomic nucleus?","text":"The electrons in an atom have a certain configuration and we can predict in which space the electrons can be present of a certain energy level (s-orbitals have a sphere-shape,....). Recently I read that there are magic numbers for nuclei: special configurations of the nuclei that makes them more stable (the magic numbers are 2, 8, 20, 28, 34, 50, 82, and 126) - so there are definitely shells in a nucleus. I was wondering : do the neutrons and protons form certain geometric structures in these sub-shells of the nucleus? Or are they all stacked closely on each other?"}
{"id":"52842","title":"The difference between the Wannier function and atomic orbit in a tight binding model","text":"In a tight binding model, we usually start from the atomic orbits and linearly combine them to get the wave function of the crystal energy band. My questions are:   1. Since this kind of tight binding is an approximate method due to using atomic orbits, is it exact to use the Wannier function formalism? If so, how do I get the Wannier function systematically?   2. What is the use of maximally localized Wannier functions?   3. Why can't we get the maximum localized Wannier function when the Berry phase is not zero?   4. Also, in tight binding, formalism, taking atomic orbits or Wannier functions as the basis function, respectively, what does position operator (diagonal or not), velocity operator and angular momentum operator look like?"}
{"id":"60970","title":"Does gravity affects temperature reading of a mercury thermometer?","text":"I remember when I was in primary school, the science teacher put me in charge of a mercury thermometer. I do not quite understand the mechanics behind except that mercury expands when it is hot and contracts when it is cold, and that this could be read off a temperature scale along the stem of the thermometer. At the back of my mind, there is this doubt on how gravity affects the movement. Having seen most thermometers being hanged vertically on a wall, I did likewise, placing the thermometer on the floor, upright, leaning against the window sill beside my desk. This is fine until one day when a strong gust of wind blew and cause the thermometer to fall flat on the floor, smashing the glass casing and causing the mercury to leak out resulting in the evacuation of the whole class. After that incident, I learnt my lesson and place the new thermometer lying flat instead of upright. But, the lesson that I didn't learn is how gravity affects the behaviour of such thermometer. Hope to learn something here."}
{"id":"65256","title":"Solar sail area going to Proxima Centari","text":"I have a physics question that I need some help with: \"Proxima Centauri is a star in the Alpha Centauri solar system, it’s the nearest star to our sun (4.24light−years) http:\/\/en.wikipedia.org\/wiki\/Listofneareststars. How large of a solar sail would be needed to accelerate a solar sail of mass m to a velocity which will get the ship to Proxima Centauri in two lifetimes?\" I'm assuming that the time is 160 years for 80 years each lifetime. Can anyone help?"}
{"id":"21909","title":"Electric Flux Density - Ring Charge","text":"A ring placed along $y^{2}$ + $z^{2}$ = 4, x = 0 carries a uniform charge of 5 $\\mu$C\/m. Find D at P(3,0,0) Should I be using Gauss's Law to solve this problem? I was considering using a spherical Gaussian Surface, and then using the formula D = $\\epsilon_0 $E to find D, but I'm not sure how to set up my integral."}
{"id":"79909","title":"Some ambiguous points on Spontaneous Symmetry Breaking (SSB)?","text":"Almost in every textbook of condensed matter physics, the standard description of SSB could be formulated as follows: Consider the lattice Heisenberg model in an external magnetic field $H=\\sum_{ij}J_{ij}\\mathbf{S}_i\\cdot\\mathbf{S}_j+hS_z$, where $h$ is the magnitude of magnetic field and $S_z=\\sum_iS_i^z$. Now the average magnetization per site is a function of both magnetic field $h$ and number of lattice sites $N$, say $m\\equiv \\sum_i\\left \\langle S_i^z \\right \\rangle\/N=m(N,h)$, where $\\left \\langle S_i^z \\right \\rangle\\equiv tr(\\hat{\\rho }S_i^z)$ with $\\hat{\\rho }=e^{-\\beta H}\/tr(e^{-\\beta H})$ the density operator. Then if $$\\lim_{h\\rightarrow 0}\\lim_{N\\rightarrow \\infty }m(N,h)\\neq 0$$, we say the system has SSB at temperature $T$. Now I get some questions: (1)We know at finite $N$ and zero $h$, $m(N,h=0)=0$ due to spin-rotation symmetry. But **there is no reason** for that $$\\lim_{h\\rightarrow 0}m(N,h)=m(N,h=0)—[1]$$, right? Since the function $m(N,h)$ may _not be continuous_ at $h=0$, from the math viewpoint. (2)If Eq.[1] is correct, and hence $\\lim_{h\\rightarrow 0}m(N,h)=0$, then $\\lim_{N\\rightarrow \\infty }\\lim_{h\\rightarrow 0}m(N,h)=0$, right? (3)If Eq.[1] is wrong, say $\\lim_{h\\rightarrow 0}m(N,h)\\neq m(N,h=0)$ and hence $\\lim_{h\\rightarrow 0}m(N,h)\\neq0$, then what about $$\\lim_{N\\rightarrow \\infty }\\lim_{h\\rightarrow 0}m(N,h)?$$ And why don't we use this identity to define SSB? Thank you very much."}
{"id":"4909","title":"How do alpha and beta particles ionise surrounding particles?","text":"I've been wondering about this question for a while. If you have alpha and beta particles released from a radioactive core, how do they ionise surrounding particles?"}
{"id":"115098","title":"Wikipedia's derivation of torque related to angular acceleration","text":"Wikipedia derivation of the relationship between a torque and an angular acceleration is given here. Could someone help me to see how the following: $$\\vec{\\tau} = \\left(-\\sum^n_{i=1}m_i [\\Delta r_i]^2\\right) \\vec{\\alpha} + \\vec{\\omega} \\times \\left(-\\sum^n_{i=1}m_i [\\Delta r_i]^2\\right) \\vec{\\omega}$$ is obtained from the following: $$\\vec{\\tau} = \\sum^n_{i=n} (\\vec{r}_i - \\vec{R}) \\times (m_i \\vec{a}_i)$$ by using Jacobi identity, please? My attempt at the derivation is as follows: $$\\vec{\\tau} = \\sum^n_{i=n} (\\vec{r}_i - \\vec{R}) \\times (m_i \\vec{a}_i) \\\\\\ \\hphantom{\\vec{\\tau}} = \\sum^n_{i=n} \\vec{\\Delta r_i} \\times (m_i \\vec{a}_i) \\\\\\ \\hphantom{\\vec{\\tau}} = \\sum^n_{i=n} m_i (\\vec{\\Delta r_i} \\times \\vec{a}_i) \\\\\\ \\hphantom{\\vec{\\tau}} = \\sum^n_{i=n} m_i (\\vec{\\Delta r_i} \\times (\\vec{\\alpha} \\times \\vec{\\Delta r_i} + \\vec{\\omega} \\times (\\vec{\\omega} \\times \\vec{\\Delta r_i})))\\;\\ldots\\;\\vec{R}\\text{ is centroid} \\\\\\ \\hphantom{\\vec{\\tau}} = \\sum^n_{i=n} m_i (\\vec{\\Delta r_i} \\times (\\vec{\\alpha} \\times \\vec{\\Delta r_i}) + (\\vec{\\Delta r_i} \\times (\\vec{\\omega} \\times (\\vec{\\omega} \\times \\vec{\\Delta r_i})))\\;\\ldots\\text{ cross-product distributivity over addition} \\\\\\ $$ Then, I try the Jacobi identity on the second term as follows: $$\\vec{\\Delta r_i} \\times (\\vec{\\omega} \\times (\\vec{\\omega} \\times \\vec{\\Delta r_i})) + \\vec{\\omega} \\times ((\\vec{\\omega} \\times \\vec{\\Delta r_i}) \\times \\vec{\\Delta r_i}) \\+ (\\vec{\\omega} \\times \\vec{\\Delta r_i}) \\times (\\vec{\\Delta r_i} \\times \\vec{\\omega}) = \\vec{0}$$ The last term on the LHS is $\\vec{0}$ because $$(\\vec{\\omega} \\times \\vec{\\Delta r_i}) \\times (\\vec{\\Delta r_i} \\times \\vec{\\omega}) \\\\\\ = (\\vec{\\omega} \\times \\vec{\\Delta r_i}) \\times -(\\vec{\\omega} \\times \\vec{\\Delta r_i}) \\\\\\ = -[(\\vec{\\omega} \\times \\vec{\\Delta r_i}) \\times (\\vec{\\omega} \\times \\vec{\\Delta r_i})] \\\\\\ = \\vec{0}$$ So, $$\\vec{\\Delta r_i} \\times (\\vec{\\omega} \\times (\\vec{\\omega} \\times \\vec{\\Delta r_i})) + \\vec{\\omega} \\times ((\\vec{\\omega} \\times \\vec{\\Delta r_i}) \\times \\vec{\\Delta r_i}) = \\vec{0}$$ That is different from the one stated in the Wikipedia. However, I try to continue with my own finding as follows: $$\\vec{\\Delta r_i} \\times (\\vec{\\omega} \\times (\\vec{\\omega} \\times \\vec{\\Delta r_i})) + \\vec{\\omega} \\times ((\\vec{\\omega} \\times \\vec{\\Delta r_i}) \\times \\vec{\\Delta r_i}) = \\vec{0} \\\\\\ \\vec{\\Delta r_i} \\times (\\vec{\\omega} \\times (\\vec{\\omega} \\times \\vec{\\Delta r_i})) = -[\\vec{\\omega} \\times ((\\vec{\\omega} \\times \\vec{\\Delta r_i}) \\times \\vec{\\Delta r_i})] \\\\\\ \\vec{\\Delta r_i} \\times (\\vec{\\omega} \\times (\\vec{\\omega} \\times \\vec{\\Delta r_i})) = \\vec{\\omega} \\times -((\\vec{\\omega} \\times \\vec{\\Delta r_i}) \\times \\vec{\\Delta r_i}) \\\\\\ \\vec{\\Delta r_i} \\times (\\vec{\\omega} \\times (\\vec{\\omega} \\times \\vec{\\Delta r_i})) = \\vec{\\omega} \\times (\\vec{\\Delta r_i} \\times (\\vec{\\omega} \\times \\vec{\\Delta r_i}))$$ to have the following: $$\\vec{\\tau} = \\sum^n_{i=n} m_i (\\vec{\\Delta r_i} \\times (\\vec{\\alpha} \\times \\vec{\\Delta r_i}) + (\\vec{\\Delta r_i} \\times (\\vec{\\omega} \\times (\\vec{\\omega} \\times \\vec{\\Delta r_i}))) \\\\\\ \\hphantom{\\vec{\\tau}} = \\sum^n_{i=n} m_i (\\vec{\\Delta r_i} \\times (\\vec{\\alpha} \\times \\vec{\\Delta r_i}) + \\vec{\\omega} \\times (\\vec{\\Delta r_i} \\times (\\vec{\\omega} \\times \\vec{\\Delta r_i}))) \\\\\\ \\hphantom{\\vec{\\tau}} = \\sum^n_{i=n} m_i ([\\Delta r_i]^2 \\vec{\\alpha} - (\\vec{\\Delta r_i} \\cdot \\vec{\\alpha}) \\vec{\\Delta r_i} + \\vec{\\omega} \\times ([\\Delta r_i]^2 \\vec{\\omega} - (\\vec{\\Delta r_i} \\cdot \\vec{\\omega}) \\vec{\\Delta r_i})) \\\\\\ \\hphantom{\\vec{\\tau}} = \\sum^n_{i=n} m_i ([\\Delta r_i]^2 \\vec{\\alpha} - (\\vec{\\Delta r_i} \\cdot \\vec{\\alpha}) \\vec{\\Delta r_i} + \\vec{\\omega} \\times [\\Delta r_i]^2 \\vec{\\omega} - \\vec{\\omega} \\times (\\vec{\\Delta r_i} \\cdot \\vec{\\omega}) \\vec{\\Delta r_i}) \\\\\\ \\hphantom{\\vec{\\tau}} = \\sum^n_{i=n} m_i ([\\Delta r_i]^2 \\vec{\\alpha} - (\\vec{\\Delta r_i} \\cdot \\vec{\\alpha}) \\vec{\\Delta r_i} + (0) - \\vec{\\omega} \\times (\\vec{\\Delta r_i} \\cdot \\vec{\\omega}) \\vec{\\Delta r_i}) \\\\\\ \\hphantom{\\vec{\\tau}} = \\sum^n_{i=n} m_i ([\\Delta r_i]^2 \\vec{\\alpha} - (\\vec{\\Delta r_i} \\cdot \\vec{\\alpha}) \\vec{\\Delta r_i} - \\vec{\\omega} \\times (\\vec{\\Delta r_i} \\cdot \\vec{\\omega}) \\vec{\\Delta r_i})$$ Then, I get stuck there. If you want to help, please show the complete derivation instead of just hinting here and there unless you have the complete derivation already in your mind."}
{"id":"57632","title":"Are Neutron stars transparent?","text":"Neutrons have no charge so they would not, I think, interact with photons. Would a neutron star be transparent?"}
{"id":"60028","title":"How do you actually define an orbit?","text":"How do you actually define an orbit? I believe, Newtonian Mechanics describes an orbit as one object in free fall around another where projectile paths become elliptical. I think, Einstein describes an orbit as an object taking the shortest distance through curved space. And in Quantum Mechanics, orbits are quantized orbitals or states. Is there a definition for orbits, where all these characteristics are true?"}
{"id":"36462","title":"Where does the \"g\" force that pilots experience come from?","text":"I understand that it has to do with acceleration. Say a pilot does a quick maneuver and experiences a force of 5g. What exactly is happening here? And what is this force relative to? If someone can show an example with some calculations that would be really helpful. Thank you"}
{"id":"113113","title":"Is the solar energy Infinite?","text":"Is the solar energy coming from the sun infinite and will continue to be radiated to our earth forever? (discarding any outer factors) what's the sun's fuel?"}
{"id":"102911","title":"How does time dilation work in this situation?","text":"It is my understanding that when moving near the speed of light, time slows down relative to other things not moving so fast. Based on this principle, would it be theoretically possible to travel a thousand light years in a year, with the thousand years only having passed on the place you're moving relative to? If something is wrong with my premises or question, that knowledge is welcomed, too. (I am still a bit shaky on the whole subject of time-dilation because I don't fully understand how it's possible without a privileged reference frame.) Understanding is the prime goal, here."}
{"id":"96254","title":"Oil drop experiment and quantization of charge","text":"How to systematically show that the resulting charges in oil drop experiment are integers multiplied by $e$ in other word how to extract $e$ from the data?"}
{"id":"99698","title":"What are qubits made of in Wen's string-net theory?","text":"In Prof. Xiaogang Wen's theory, photons and electrons are described as quasi- particles appeared as a result of the existence of the string-net liquid, which is the topological order of the qubits that form the space. This is a little confusing to me. Consider when we talk about other quasi-particles, for example, phonons, we say they are excitons appear from the oscillation of the atoms that form the solid, which means that 'atoms' are 'real' while phonons are 'quasi-particles'. But when we make an analogy between photon and phonon, (topological order with lattice model, qubit with atoms), what are those qubit in essence? Are they some kind of ultimate thing that build up our world?"}
{"id":"99699","title":"Why Can We See Stars In The Sky","text":"Why is it when we look up into the night sky we can see stars. but when you see pictures taken from the ISS you don't see any stars. Why is this?"}
{"id":"22784","title":"Wick Order and Radial Ordering in CFT","text":"I am not so much familiar with the computations tools of conformal field theory, and I just run into an exercise asking to demonstrate the following formula (related to the bosonic field case): $$\\cal{R}j(z_1)j(z_2)~=~\\frac{1}{(z_1-z_2)^2}~+~:j(z_1)j(z_2):$$ with $j$ defined as $$j(z)~=~\\sum_k \\alpha_k z^{-k-1}.$$ My question is should I start the calculation form the Wick ordered term and make the two others appear, because starting from the left side, I don't see how could I develop some calculus?"}
{"id":"6824","title":"How good is current tsunami prediction?","text":"We all know that predicting tsunami and earthquake is difficult, with too many variables involved. But with the advent in data collection and computing power and better models, one should be able to predict tsunami better than in the past. How accurate is current tsunami prediction?"}
{"id":"18277","title":"Solar wind and the Earth's magnetic field","text":"I have again an old question from a comprehensive exam I took a couple of months ago. Lucky for me one could pick 5 out of 8 questions, because on some of the problems I didn't even know how to start. Now that classes are over I've now the time to revisit those problems I was dumbfounded by, such as this one: (Abriged version) > Life on earth would be impossible if we were constantly exposed to charged > solar particles. Luckily, earth's magnetic field protects us from them. The > solar particles have a typical energy spectrum of $d\\Phi \/ dE \\propto > E^{-3}$ particles\/$m^2\/s\/J$. What is the _minimal_ field strength of the > earth's magnetic field based on the anthropic principle, i.e., it couldn't > be weaker or else we wouldn't live to observe it. Well, this quantity $d\\Phi\/dE$ looks like a flux, so I guess the general setting is that of a scattering problem. But first, $d\\Phi \/ dE$ isn't given completely, only a rough form of its energy dependence. And second, I'm not sure what a reasonably simple model for this entire process would be. Easiest in terms of calculation would be to assume some sort of homogeneous magnetic field aligned with the earth's magnetic axis, because I guess it's a pain to calculate the path of a particle in a dipole field... Maybe the idea of this is to calculate the total cross section of earth's magnetic field and then demand that it should \"cover\" the earth? Or they want me to solve an equation of motion for incoming solar particles and show that all of them are deflected? My problem right now is that I don't even know how to interpret the $d\\Phi\/dE$ quantity whose energy dependence I'm given. I guess it makes more sense to someone with a background in elementary particle physics? Right now I'm trying to write a vector potential $\\vec{A} = \\mu_o\/(4\\pi r^2) \\vec{m} \\cdot \\vec{e}_r$ where $\\vec{e}_r$ is the unit vector in $r$-direction in spherical coordinates, and then try to get equations of motion from the Hamiltonian $$H = \\frac{(\\vec{p} + q\\vec{A})^2}{2m}$$ but I am not sure if I'll be able to solve whatever comes out of that, or if I'm completely on the wrong track with this. * * * EDIT * * * ![Schematic of Deflection of Solar Particles](http:\/\/i.stack.imgur.com\/jbZDA.png) Image taken from here Maybe it helps trying to understand this schematic, but I cannot easily see how the Lorentz force would create such a trajectory. * * * ANOTHER EDIT * * * From further searching, I know suspect that this has something to do with how a plasma current (the charged particles) interact with a magnetic field. That would mean that I have to calculate the radius of the ensuing magnetosphere and then demand, via the anthropic principle, that it should be at least of the same size (or larger) as the radius of earth. So the Lorentz force would probably not directly have anything to do with it. But I also have no training in plasma physics. (Some of the problems in the exam were specifically geared towards Astronomy students, so I guess they'd find it a breeze)."}
{"id":"30309","title":"What is your prefered toolkit \/ program for 3D visualisations of data?","text":"If you recorded data which represents some physical value in space (e.g. electron density) and you need to explore this dynamically in 3d (say you have a isosurface and you change the value it represents) what toolkit\/program would you use? Personally I used MayaVi and it has all the all the features I could need. Unfortunately it is sometimes unstable and often slow. So in principle I am looking for equally powerfull alternatives. Edit: I know this should probably not be in the physics SE but I think the answer to my question could be relevant for many physicists."}
{"id":"30308","title":"Changing the Half-Life of Radioactive Substances","text":"Is there a way to extend or reduce the half-life of a radioactive object? Perhaps by subjecting it to more radiation or some other method."}
{"id":"15081","title":"What causes a gyroscope to eventually rotate\/fall over?","text":"Hey so I've just learned about angular velocity and momentum and how torque changes it. Looking at a wheel spinning around an axis, with one end being held up by a rope, what causes the wheel to rotate downwards over time, and eventually fall?"}
{"id":"109827","title":"Why use lasers for intense, localized heat instead of some other light source?","text":"Lasers are used in various industrial processes that need intense, localised, heat (3d printers and laser cutters come to mind). My question is: why use lasers? There are many other (cheaper, brighter) light sources. There are even other monochromatic and coherent light sources (LEDs and mercury vapour lamps respectively), and this video shows someone sintering desert sand using a Fresnel lens and sunlight, which is of course neither monochromatic nor coherent. So, what is it about lasers that make them so better than a conventional light source combined with appropriate focusing elements?"}
{"id":"112798","title":"Neutral quantum particle in inhomegeneous magnetic field","text":"I'm trying to understand Stern-Gerlach experiment on a computational level. Suppose we have a neutral particle with magnetic moment (e.g. a neutron), and apply an inhomogeneous magnetic field to it (let it change linearly with coordinate). As I understand, its Hamiltonian would look like: $$\\hat H=-\\frac{\\hbar^2}{2m}\\nabla^2+\\left(\\frac e{mc}\\right)\\hat{\\vec s}\\vec B$$ Now the spin operator is $$\\hat s_i=\\frac{\\hbar}2\\sigma_i,$$ where $\\sigma_i$ is $i$th Pauli matrix. So, for magnetic field $\\vec B=\\vec e_x B_0 x$ we'd have Schrödinger 1D (Y and Z directions can be separated due to translation symmetry) equation: $$-\\frac{\\hbar^2}{2m}\\frac{\\partial^2\\psi}{\\partial x^2}+\\left(\\frac {\\hbar e}{2mc}\\right)\\sigma_x B_0 x\\psi=i\\hbar \\frac{\\partial\\psi}{\\partial t}.$$ I now try to solve this equation numerically, taking initial wave function in the following form: $$\\psi(x,t=0)=\\begin{pmatrix}\\psi_0(x)\\\\\\ \\psi_0(x)\\end{pmatrix},$$ where $\\psi_0(x)$ is a gaussian wave packet with zero average momentum. The problems start when I select $\\sigma_x$ as is usually given: $$\\sigma_x=\\begin{pmatrix}0&1\\\\\\1&0\\end{pmatrix}.$$ The solution appears to look like showed below. I.e. both wave function components accelerate left! ![enter image description here](http:\/\/i.stack.imgur.com\/Rrbsk.gif) I thought, what if I choose another axis as $x$, so I tried doing the same with $\\sigma_y$: $$\\sigma_y=\\begin{pmatrix}0&-i\\\\\\i&0\\end{pmatrix}.$$ The result in the animation below. Now it's a bit better: the wavefunction at least splits into two parts, one going left, another right. But still, both parts are composed of a mix of spin-up and spin-down states, so not really what one would expect from Stern-Gerlach experiment. ![enter image description here](http:\/\/i.stack.imgur.com\/8hyNA.gif) Finally, I tried the last option — using $\\sigma_z$: $$\\sigma_z=\\begin{pmatrix}1&0\\\\\\0&-1\\end{pmatrix}.$$ The result is again showed below. Finally, I get the splitting into \"independent\" spin parts, i.e. one spin part goes left, another one goes right. ![enter image description here](http:\/\/i.stack.imgur.com\/KPe1s.gif) **Now, the question** : how to interpret these results? Why does choice of active axis result in such drastic differences in results? How should I have done instead to get meaningful results? Shouldn't permutation of Pauli matrices not affect results?"}
{"id":"101982","title":"Integrating the gauge covariant derivative by parts","text":"I was watching a set of lectures on effective field theory and the lecturer said that you can always integrate the covariant derivative by parts due to gauge symmetry. For example, if I understand correctly, we can write: \\begin{equation} \\int {\\cal D} \\phi \\exp \\left\\\\{ i \\int d ^4 x D _\\mu \\phi D ^\\mu \\phi + \\, ...\\right\\\\} = \\int {\\cal D} \\phi \\exp \\left\\\\{ i \\int d ^4 x - \\phi D ^2 \\phi + \\,...\\right\\\\} \\end{equation} where $D_\\mu = \\partial_\\mu + i g T ^a G_{a, \\mu} $. This would be obvious if we didn't have the gauge boson contribution but why does integration by parts hold for the covariant derivative?"}
{"id":"10745","title":"Can heat be transfered via magnetic field in a vacuum?","text":"Say you want to store hot coffee in a container surrounded by a vacuum. To remove all sources of conductive energy loss the container is suspended in the vacuum by a magnetic field and does not have a physical connection to the sides of the vacuum chamber, My question is would the magnetic field be a path for energy to be conducted out of the suspended container? Another way to look at this question would be two magnets are suspended in a vacuum with their poles aligned. A heat source is attached to one of the magnets. Would the second magnet show a corresponding increase in temperature, excluding radiated heat transfer?"}
{"id":"108152","title":"Relativistic Transverse Doppler Effect","text":"In Minkowski spacetime, two observers, A and B, are moving at uniform speeds u and v, respectively, along different trajectories, each parallel to the y-axis of some inertial frame S. Observer A emits a photon with frequency $\\nu_{A}$ that travels in the x-direction in S and is received by observer B with frequency $\\nu_B$. Show that the Doppler shift $\\nu_B\/\\nu_A$ in the photon frequency is independent of whether A and B are travelling in the same direction or opposite directions. Relevant equations: $$\\lambda\/\\lambda' = \\nu_B\/\\nu_A = \\gamma(1-\\beta\\cos\\theta)$$ Aberration formula: $\\cos\\theta' = (\\cos\\theta - \\beta)\/(1-\\beta\\cos\\theta) = -\\beta$   (for transverse case) The answer is apparently that the Doppler shift is independent of the relative direction of motion. I have tried to transform to the frame S' where B is stationary, finding the velocity of A using the addition of velocities formula - to then get gamma. I have used the abberation formula to insert $\\cos\\theta'$ into the Doppler shift formula above to get $\\nu_B\/\\nu_A = \\gamma(1+\\beta^2)$. Plugging in the velocity of the emitter A in frame S' doesn't seem to get the required result."}
{"id":"131588","title":"Sequence of E and B field in radio waves and in single photons","text":"In antenna technology we distinguish between nearfield and widefield. In the nearfield the electric and the magnetic fields are shifted by 90°. If you look closer you can see that there are two possibilities of this shift, 90° and -90°. To explain it you have to remember what this 90° means. Let (in vacuum) the coordinate systems X axis be parallel to the E field, let the Y axis be parallel to the B field and the Z axis is parallel to c * t . In Z equal zero let the E field be maximum and directed in the X direction. The B field is zero. 90° later (in terms of E = E(max) * cos α and B = B(max) * sin α) and this is a quarter of the wavelength the B field can be directed to the left or to the right. And this is natural because B = B(max) * - cos α is the second possible state of the nearfield radio waves. Regardless of the approach to see radio waves as one electromagnetic wave (statistical method) it is obvious that all radio waves are made from photons which are emitted during the electrons acceleration in the antenna rod. My question is, do these photons all have the same sequence of the E and B fields? The same question appears for quantum dots which produce single photons. Edit: There has to be a right or left hand rule because if half the photons have B-field with 90° to E and half have -90° then there wouldn't be magnetic field at all. Update: I get it. It's the right hand grip rule (conventional direction of current) because there is no principal difference to a straight wire."}
{"id":"131589","title":"Does this count as moving faster than light?","text":"I'm not familiar with any complicated physics equation, however I do understand some basics. Suppose there is two objects, both of them are moving away from each other in a 3-dimensional space, which they both have the speed of half the speed of light ($c\/2$). Relative to the reference frame of object A, object B would be moving away from itself equal to the speed of light.                  A          B     <-----      ----->       -c\/2       c\/2   to a 3rd observer        0          c    to A       -c          0    to B      This seems to break the law of \"nothing can move faster than the speed of light\". I believe that there is some other explanation for this case. I have done some research, and there are quite a few questions on this topic. Take this one and this one for example. However, they are about adding velocities together, which isn't quite the same as in the case I was describing. This question is also similar to my case, however the two objects described are both moving toward the same direction, and I'm not shooting any beam or laser toward the other object. Most probably in reality there are some extremely complex laws and equations which makes this question more complicated. However, the two objects are not interacting with each other and I'm not trying to \"detect\" the speed of the other object in the reference frame of A or B. Also none of the objects are moving at a significant fraction of the speed of light relative to the third observer, so special relativity doesn't apply(?) I just could not think of any reason why wouldn't B be moving away from A in the speed faster than light. A simple explanation would be great."}
{"id":"33726","title":"Would Einstein have accepted the presumptions that lead to the Bell inequality?","text":"To check the correlation between Hidden Variable Theory and Quantum Mechanics, Bell calculated the expectation value $<\\sigma_{e}(\\vec a,\\vec V) \\sigma_{p}(\\vec b,\\vec V)> = \\int d^n V \\rho(\\vec V) \\sigma_{e}(\\vec a,\\vec V) \\sigma_{p}(\\vec b,\\vec V)$ Here I am assuming that \"Alice\" is measuring the spin of an electron e along $\\vec a$ and \"Bob\" is measuring the spin of the positron p along $\\vec b$. Then $\\sigma_{e}(\\vec a,\\vec V)$ and $\\sigma_{p}(\\vec b,\\vec V)$ are the resulting spin values ($\\pm \\frac{1}{2}$) of the electron and positron, respectively. The vector $\\vec V$ is an n-dimensional vector containing the hidden variables and $\\rho(\\vec V)$ is a probability distribution for the hidden variables. But does this not assume QM is probabilistic? I thought Einstein disagreed with the probabilistc nature of Quantum Mechanics (God does not throw dice)."}
{"id":"91265","title":"Motion of charge in magnetic field with drag force","text":"Say you have a charged particle in a region that contains a fluid that will produce a drag force that goes as $F=-kv$ where $v$ is the speed and $k$ is some constant. The region also contains a uniform magnetic field. Suppose you give the particle some initial velocity $v_0$ in the plane perpendicular to the magnetic field. What will be the particle's subsequent motion? Please provide semi-quantitative answers. Note that this is not a homework question."}
{"id":"35225","title":"How close can spaceship get to the Sun","text":"If you want to fly a spaceship with human passengers as close to the Sun as possible, then what effects would the spaceship have to be designed to counteract in order to keep the passengers alive and how close to the Sun could you get before there would be no way to counteract the effects ?"}
{"id":"131038","title":"If you run faster than speed of light what happens with your shadow?","text":"If you are running faster than the speed of light, and behind you have an a object projecting a burst of light what Happens with your shadow? If you are running slower than the light, when the light touch you, its generate a shadow of the opposite side, right? So, when you are running faster than the light, your shadow runs counter the light \"clashing\" with her? **EDIT:** I know that is impossible to move's faster than the light in vacuum, it is logically a assumption. Sorry my bad English, please correct the errors."}
{"id":"101779","title":"Equation of a flying kite","text":"My question is the following: _What is the shape of the rope which holds a kite flying?_ (Steady state.) I am not a physicist (I am a mathematician), so I can not work the physics part of the question. My guess is that it should be the same as catenary (hyperbolic cosine)"}
{"id":"132639","title":"Computationally solving bodies that push and pull","text":"Is there a way to find a solution for the positions (as a function of time) of multiple free bodies that push and pull on each other? Say for instance I have a collection of cells which can individually contract or expand, and are joined to one another. For simplicity, I'm assuming two dimensions; no gravity, friction, or other external forces; and the cells are circular, and therefore connect at a point. When a cell contracts or expands, it simply reduces or increases its radius, respectively. When it contracts, it pulls on any cells that are attached to it. When it expands, it pushes on any attached cells, and in both cases it does so with a particular force. The mass of each cell is known. I suspect this may be related to an N-body problem (perhaps even more difficult) and therefore cannot be feasibly solved exactly. If that's the case, can anyone suggest any approach for reasonably approximating this system?"}
{"id":"101770","title":"Inward Pressure of Gravity","text":"We are looking at White Dwarfs in Quantum and specifically how they are stabilized between gravitation pressure inwards balanced by the Fermi pressure due to electrons. > To start the problem off we are asked to calculate the inward pressure due > to gravity in terms of the total gravitational energy. We are examining a > white dwarf with the mass of our Sun (ignoring where this is > probable\/possible). If we take the total gravitation energy to be $U_g$ then I figured $U_g=PV$ and that the inward pressure due to gravity would be $P=\\frac{U_g}{V}$ where $V={4 \\over 3}\\pi r_{sun}^3$. However when I use this later on to calculate the radius of a white dwarf with mass equal to our sun I get an obscenely small number. While I understand the densities of white dwarfs are enormous I think I made a mistake in the inward pressure due to gravity part. Should it be that $V={4\\over 3}\\pi K f^3$ and then...? Not sure where to go or what I did wrong. Ideas?"}
{"id":"79257","title":"Gravitational acceleration on the Moon and Mars","text":"There are plenty of formulas that use gravity acceleration of Earth. This is represented with the symbol $g$. In my school work (I am a high school student) we usually take it as $g= 9,8 \\,\\text m\/\\text s^2$. This thing is obviously a number that is only usable on Earth. What I want to know is that, what if I want to make my calculations according to another planet? How the number is going to change?"}
{"id":"132633","title":"At the instant of release of an object from rest. Is the only force that can act its weight?","text":"Q3 from a mechanics exam past paper:![enter image description here](http:\/\/i.stack.imgur.com\/95nD6.png) I can do parts i) and ii) but for iii) in finding the angular acceleration, i used $C=I\\alpha$, where $C$ is the applied couple or torque, $I$ is the moment of inertia for the lamina about A and $\\alpha$ is the angular acceleration. At the instant the object is released the only force acting on it is its own weight. Hence, $6g*0.8=9\\alpha$ which yields $\\alpha=5.227$. However the mark-scheme says the answer is $\\alpha=1.65$, as they have taken into account the frictional couple. Here is the mark-scheme:![enter image description here](http:\/\/i.stack.imgur.com\/l1s2i.png) **It was my understanding that at the moment of release no friction _(and hence no frictional couple)_ can act as there is no movement (yet).** So could someone please kindly explain what the mark-scheme is talking about?"}
{"id":"132630","title":"Potential due to charge over infinite grounded plane conductor using the method of images","text":"I am reading section 3.2.1 of Griffiths 3ed which explains how to calculate potential using first uniqueness theorem. Griffiths\/3.2.1 ![](https:\/\/i.imgur.com\/hTmCHFX.png) Griffiths\/First Uniqeness theorem (Its corollary actually) ## ![](https:\/\/i.imgur.com\/JlGMUEl.png)   1. As I understand we are taking volume $V = R^3$ , $\\Phi (\\vec{r})=0\\ $on the boundary of $V$. But what does it mean by \"boundary of $R^3$\" ?   2. Since $\\ \\rho_1(\\vec{r}) \\ne \\rho_2(\\vec{r})$ which does not satisfy the prequisites of the theorem, we cannot use the theorem. Moreover no other distribution can be used which makes this theorem useless which it is not. So what am I missing here ?   3. Does not single charge distribution also satisfies the same boundary condititons ? So should not $\\Phi$ also be $\\frac{1}{4\\pi \\epsilon_0}\\frac{q}{R}$ ?"}
{"id":"87958","title":"Integration with Grassmann variables","text":"How to show that $$ \\int d\\Psi d\\bar {\\Psi}e^{i \\int d^{4}x\\bar {\\Psi} \\hat {A} \\Psi} = det (\\hat {A})? $$ $\\Psi , \\bar {\\Psi}$ refers to Dirac spinors (the second is $\\bar {\\Psi} = \\Psi^{\\dagger}\\gamma^{0}$)."}
{"id":"103883","title":"What do the latest FERMI results say about dark matter?","text":"There was an announcement at a recent UCLA symposium on dark matter by the FERMI collaboration which hints at some evidence of dark matter. The results aren't yet published, but the symposium news release is here. Most of the news release is just general dark matter fluff, but it says that > One search technique involves using the vast amount of dark matter in our > galaxy. The NASA Fermi Satellite Telescope, an international collaboration > involving NASA, the Goddard Space Flight Center and the SLAC National > Accelerator Laboratory, searches for gamma rays — very high-energy light > particles — from this dark matter. > > There are models of dark matter that would allow a signal in the galactic > dark matter consistent with the claims at the meeting and provide a small > interaction consistent with the \"null results\" in the direct dark matter > searches all over the world. This obviously doesn't contain much information. Can anyone closer to the field expound on the findings?"}
{"id":"43886","title":"Does the Photino have mass or is it mass-less like the photon","text":"Does the photino in super-symmetry have a mass, Or is this different in different super symmetric models?"}
{"id":"101482","title":"Realistic interpretations of quantum entanglement: Are there alternatives to the \"Transactional Interpretation\"?","text":"Quantum entanglement bridges space and time: entangled particles show correlations independently of where and when they are measured. This is most evident in \"delayed choice quantum eraser\" experiments, where the outcome of single measurments _now_ depends on which measurements are done _in the future_ (though there is no retro-causality in the strict sense). I am looking for realistic interpretations which directly address this topic, i.e. discussions of underlying physical mechanisms or basic structures of the universe on which this correlations may be built. So far I found John Cramers \"Transactional Interpretation\" of 1986; Bohmian mechanics may also be considered. Are there other physical interpretations or philosophical papers on this topic of correlations over spacetime?"}
{"id":"101481","title":"Solar panel maximum efficiency at specific frequency and temperature?","text":"I talked to a professor about solar panels and their efficiency. It seemed that the main reason solar panels aren't that efficient is because it can only accept a single energy input size. Anything below is ignored and anything above only gives the energy of that single limit. That is, say the limit is 1W. A photon with 0.999W energy wouldn't give any voltage. A photon with 10W energy would give the same voltage as a 1W photon. Does this mean I can get very high efficiency from a light source with a specific frequency? For example a solar-powered calculator in a room with only sodium lights. How high? How does the efficiency change depending on the amount of light or the temperature of the solar cell? How about an extremely small amount of light, like 1nW, close to 0K? This could be terribly wrong as there was quite a language barrier between us. I want to learn, so please point out any misstakes."}
{"id":"128801","title":"scattering matrix superccurent density of states","text":"while studying scattering formula for the supercurrent (Beenakker) I encountered that the density of states for discrete and continuous spectrum can be simultaneously described by taking $\\epsilon\\ ->\\ \\epsilon + i\\delta $ in the following formula: $\\rho(\\epsilon) = -\\frac{1}{\\pi}Im\\frac{d}{d\\epsilon}\\left\\\\{lnDet\\left[1 - S_{A}(\\epsilon)S_{N}(\\epsilon) \\right] - \\frac{1}{2}lnDet\\left[S_{A}(\\epsilon)S_{N}(\\epsilon)\\right]\\right\\\\}$. I am wondering if one could argument this by referring to scattering theory and Green's function - eg. Laszlo Szunyogh."}
{"id":"130674","title":"Moving wedge and pulley system","text":"A wedge of mass $M$ rests on a rough horizontal surface with coefficient of static friction $\\mu$. The face of the wedge is a smooth plane inclined at an angle $ \\alpha$ to the horizontal. A mass $m_1$ hangs from a light string which passes over a smooth peg at the upper end of the wedge and attaches to a mass $m_2$ which slides without friction on the face of the wedge. A) Find the accelerations of $ m_1, m_2 $ and the tension in the string when $ \\mu $ is very large. B)Find the smallest coefficient of friction such that the wedge will remain at rest. Attempts: No specification on whether or not any of the masses $m_i$ are bigger than the others, so let us assume that $m_1$ goes down and $m_2$ goes up the slope. Then A) is fine. (a minus error in the final answer for A) is introduced if I assume the other orientation) As for B) if we consider the wedge+masses as a system, then to conserve linear momentum if the masses move as above, then the wedge must move rightwards. So the frictional force acts leftwards until the wedge slips. The only other force acting on the wedge is via the normal contact force from $m_2$. A component of this force acts to accelerate the wedge horizontally. Let's write this relative to some fixed inertial frame of reference with $x$ rightwards. Then the above recast into symbols gives $$-\\mu (M+m_1+m_2)g - m_2 g \\cos \\alpha \\sin \\alpha = 0 $$ at the point of slip of the wedge. However, solving this for $\\mu$ gives a negative result. Where did I go wrong? ![enter image description here](http:\/\/i.stack.imgur.com\/IQBDT.png)"}
{"id":"109535","title":"What is charge?","text":"I know this isn't the right place for asking this question, but in other places the answers are so awfull.. I'm studying eletricity, so, I start seeing things like \"charges\", \"electrons has negative charges\",etc. But I didn't quite understand what charge is. I search a little bit on the internet and found it related to electromagnetic fields, then I thought \"negative and positive may be associeted with the behaviour of the particle in the field, great!\", but the articles about e.m. fields already presuppose \"negative\" and \"positive\" charges. In other places, I see answers relating charges to the amount of electrons\/protons in an atom, but if that's right, the \"negative\" electron is an atom without any protons? What about the neutron? So, my questions are (1) What are charges; and (2) How a particle can \"be\" electrically charged. What does that really mean? Thanks for your time."}
{"id":"57199","title":"Origin of electric charge","text":"Baryons have charges that are the result of a polynomial calculation of their building blocks (quarks)'s fractional charges. But what gives these quarks electric charges? What interactions do they have with photons? And what about leptons, like muons and electrons? How do they get their -1 charge? And then, how do positrons and anti-muons get +1 charge? Same goes for quarks: how does up-antiquarks get -2\/3 charge? Or I may want to formulate my question this way: **How does any elementary particle interact with the EM-Force to get an electric charge?**"}
{"id":"106605","title":"What is charge actually? How to define it?","text":"Is charge of something for (e.g.) an electron related to electromagnetic space if it exists due to energy, due to which it may have mass? I don't know about quantum mechanics or advanced particle models. Can anyone just simply give an intuitive idea? EDIT I want to mean what actually gives electron charge if it is not assumed fundamental but result of some other physical phenomenon or it is just the quantity defined to explain physical interactions?I think now it is clear"}
{"id":"75693","title":"Particles and their charges","text":"It is always known that electrons and protons have opposite charges but what gives electrons or protons the charges they have?"}
{"id":"118756","title":"What exactly is charge?","text":"If gravity is really the bending of space\/time causing objects with mass to experience acceleration, is there a similar physical meaning to 'charge' besides 'a property of matter which causes it to experience a force when placed in an electric field.' What exactly is charge?"}
{"id":"72847","title":"Is there any theory for origination of charge?","text":"We have a theory of a Higgs field that describes how a particle gets mass. Since mass and charge both are intrinsic properties of a particle, is there any similar theory for how particles get electric charge?"}
{"id":"123337","title":"What gives an elementary particle a charge?","text":"We know that proton is positive, and electron is negative. But where does come notion of negativity and positivity? Does charge come from some specific particles, or they specific order?"}
{"id":"26877","title":"Regularization of the Casimir effect","text":"For starters, let me say that although the Casimir effect is standard textbook stuff, the only QFT textbook I have in reach is Weinberg and he doesn't discuss it. So the only source I currently have on the subject is Wikipedia. Nevertheless I suspect this question is appropriate since I don't remember it being addressed in other textbooks Naively, computation of the Casimir pressure leads to infinite sums and therefore requires regularization. Several regulators can be used that yield the same answer: zeta-function, heat kernel, Gaussian, probably other too. The question is: > What is the mathematical reason all regulators yield the same answer? In physical terms it means the effect is insensitive to the detailed physics of the UV cutoff, which in realistic situation is related to the properties of the conductors used. The Wikipedia mentions that for some more complicated geometries the effect _is_ sensitive to the cutoff, so why for the classic parallel planes example it isn't? EDIT: Aaron provided a wonderful Terry Tao ref relevant to this issue. From this text is clear that the divergent sum for vacuum energy can be decomposed into a finite and an infinite part, and that the finite part doesn't depend on the choice of regulator. However, the _infinite_ part does depend on the choice of regulator (see eq 15 in Tao's text). Now, we have another parameter in the problem: the separation between the conductor planes L. What we need to show is that the infinite part doesn't depend on L. This still seams like a miracle since it should happen for all regulators. Moreover, unless I'm confused it doesn't work for the toy example of a massless scalar in 2D. For this example, all terms in the vacuum energy sum are proportional to 1\/L hence the infinite part of the sum asymptotics is also proportional to 1\/L. So we have a \"miracle\" that happens only for specific geometries and dimensions"}
{"id":"56991","title":"Going through a ring of black holes","text":"Mathematician here with a speculative physical question -- feel free to boot me if the level isn't right. Suppose one finds, or builds, a constellation of several black holes arranged in a circle. (To get a stable arrangement, presumably one could put a bunch of small positively charged black holes at equal angles around a central positively charged object, or alternatively suspend uncharged black holes gravitationally in a circle surrounded by a quickly rotating ring of heavy matter). Suppose moreover that the event horizons of neighboring holes overlap -- or if there's some reason this is theoretically impossible, suppose they're really, really close to being tangent, while leaving room for light and matter to pass through the middle of this ring. (Is this possible?) My question is whether a trajectory from point A to point B that goes through the middle of the ring is causally related to one that does not. For example say a space explorer measures the spin of an electron, writes down the result and puts it in a box which he leaves at some fixed point in space (we've chosen a reference frame), then goes in a loop through the middle of the ring of black holes and comes back. When he comes back and opens the box again, will the recorded result be the same? I got this question after someone told me about quantum decoherence, and I was curious whether the world can change significantly if you go around a loop that either cannot be contracted or cannot be contracted without at some point becoming ridiculously long (where it's up to you to decide the value of \"ridiculously\")."}
{"id":"126488","title":"Is the expansion accelerating or decelerating?","text":"I have already asked this on Astronomy.SE but I couldn't understand the answer there. > According to Hubble's Law, the farther a galaxy is, the farther it is moving > away. But do we take into account the fact that we are actually looking in > the past? > > For example, there are two galaxies A and B at distance of 5 and 10 billion > light years respectively. Now, when we observe A we are looking at how it > was moving 5 billion years ago. The same applies for B. So, now we conclude > that 5 billion years ago space was expanding at a slower rate while it was > expanding comparatively faster 10 billion years ago. What's wrong with this > conclusion?"}
{"id":"6010","title":"What is the name and value of the constant that relates to electrons and that coincidentally has the same exact value as the speed of light?","text":"There's some constant relating to electrons that also has the same value as the speed of light. What is it, what is the value, and how are they related? EDIT: Is it the fine-structure constant?? Are there any other similar constants? If you posted that answer before, you shoulda left it (to the person who deleted their answer)!"}
{"id":"130147","title":"Is a vacuum needed in photoelectric effect?","text":"This question was asked to me. My first thought was that electrons may ionise the air and potential difference that was applied may increase or decrease the current which should have been observed. I'd like to know whether this is the right reason or if there is any other reason which I should take into account. $$hv-w.f=hv'=eV$$ where $V$ is stopping potential , my second question is: if there isn't a vacuum will that stop potential change?"}
{"id":"10049","title":"What is the ram-facing side of spacecraft?","text":"what is the ram-facing side of a satellite? What does it mean and why is it called \"ram\"-facing? Thanks."}
{"id":"7584","title":"How do we know that some radioactive materials have a half life of millions or even billions of years?","text":"If a radioactive material takes a very long time to decay, how is its half life measured or calculated? Do we have to actually observe the radioactive material for a very long time to extrapolate its half life?"}
{"id":"45102","title":"Poisson brackets: prove that they are canonical invariants","text":"**EDIT** : I haven't forgotten to accept answer, the question is still open.. * * * I need a clarification about Poisson brackets. I'm studying on Goldstein's Classical Mechanics (1 ed.). Goldstein proves that Poisson brackets are canonical invariants for any functions F and G. But there is a step that I can't understand. After some steps, he says that: $$ \\tag{1} [F, G]_q,_p = \\sum_k ( \\frac { \\partial G}{\\partial Q_k} [F,Q_k]_q, _p +\\frac {\\partial G}{\\partial P_k}[F, P_k]_q, _p)$$ After other steps, he writes: $$ \\tag{2}[F,Q_k]= - \\frac {\\partial F}{\\partial P_k}$$. and $$ \\tag{3}[P_k, F]_q, _p = \\sum_j \\frac {\\partial F}{\\partial Q_j} [P_k, Q_j] + \\sum_j \\frac {\\partial F}{\\partial P_j}[P_k, P_j]$$ -> $$ \\tag {4} [F,P_k]=\\frac {\\partial F}{\\partial Q_k}$$ and now he replaces these relations in the first expression I have written, obtaining: $$\\tag {5}[F, G]_q, _p=[F, G]_Q, _P$$ Why does he obtain in the second last step $\\frac {\\partial F}{\\partial Q_k}$ and not $-\\frac {\\partial F}{\\partial Q_k}$? $[P_k, F]=-[F, P_k]$ isn't it? **EDIT** : Golstein starts from (1) and substituites $Q_k$ to $F$ and $F$ to $G$ and so he obtains (2). Then he substitutes $P_k$ to $F$ and $F$ to $G$ and obtains (3). Immediately after he writes (4), that according to me is opposite to (3). And so I have thought to a printing error. I have tried to substitute $-P_k$ to $F$ and I have obtained $$[-P_k, F]=\\frac {\\partial F}{\\partial Q_k}$$ Then, as $[-P_k, F]=[F,P_k]$, I can say that $$ [F,P_k]=\\frac {\\partial F}{\\partial Q_k}$$. And so I can obtain (5). Could you confirm that my argumentation is correct? Many thanks"}
{"id":"128117","title":"Thermal Velocity","text":"What is thermal velocity? What is it's physical significance? Wikipedia says: > The thermal velocity or thermal speed is a typical velocity of the thermal > motion of particles which make up a gas, liquid, etc. Thus, indirectly, > thermal velocity is a measure of temperature. But I cannot understand and comprehend it."}
{"id":"44278","title":"Can exhaust gases be diverted to other cylinders during engine operation?","text":"Here's a description of new combustion engine improvement by Mazda, called SkyActive-G. They claim that in a \"generic\" engine... > when the exhaust manifold is short, the high pressure wave from the gas > emerging immediately after cylinder No. 3’s exhaust valves open, for > example, arrives at cylinder No.1 as it finishes its exhaust stroke and > enters its intake stroke. As a result, exhaust gas which has just moved out > of the cylinder is forced back inside the combustion chamber, increasing the > amount of hot residual gas I always though, that exhaust manifold and intake manifold are separated, so exhaust gases just can't possibly enter other cylinders combustion chambers as described in the above quoted text. Can exhaust gases be diverted into other cylinders as claimed in that text?"}
{"id":"44272","title":"Is the structural similarity between atoms ( smallest) and universe (biggest) a conincidence. Or there can a reason for this beyond imaginations","text":"Is the structural similarity between atoms ( smallest) and universe (biggest) a coincidence? Or there can a reason for this beyond imaginations? It seems like, if one starts travelling out from atoms... and grows bigger and bigger, one ends in a similar structure somewhere in the universe. Kind of a circular ring. PS: By structural similarity I mean : single nucleus and electrons revolving around in an atom, is structurally similar to planets revolving around the sun."}
{"id":"44270","title":"Computational Science involve programming?","text":"I read what is computational science in Wikipedia but the explanation and understanding are not very clear. So, I could you please give a simple example computational science project and what all basic skills a person should have? Also,   1. Does computational science involves programming?   2. How different are computational science and computational materials science?   3. I am from Electrical and Computer Science (basically programming) background. I was assigned a computational materials science project. So, is it in my scope? Probably, the prof assigned based on what individual subjects I studied (Engineering Math, Engineering Physics, Engineering Chemistry, Probability, Programming)."}
{"id":"36141","title":"Acceleration of table tennis racket","text":"I can't find any numbers about table tennis. What accelerations occur during a forehand smash for example? Thanks in advance & kind regards, Hans"}
{"id":"36143","title":"picture of newtonian gravity and black hole","text":"![Picture of gravity](http:\/\/i.stack.imgur.com\/Vl3a6.jpg) We assume that the earth planet is a black hole. When a light beam is fired. The only way that even light can move is A and B which means this is impossible that light move on other ways C, D, E Isn't it true?"}
{"id":"70494","title":"Force-Weight Pinewood Derby Car","text":"Force-Weight Pinewood Derby Car. I'm trying to make a fast derby car. I was wondering if I could make a car that can drop\/eject weight once it comes off of the incline. Would that increase speed or would it make it slower?"}
{"id":"70493","title":"Nonlinear waves superposition","text":"Non-linear waves do not superimpose to each other, but why? What characteristics give this property?"}
{"id":"108830","title":"2 Nucleon Potential","text":"I am looking at a 2 Nucleon potential of the form $$V(r)=V_0(r)[a+bI_1\\cdot I_2]$$ Where a and b are constants. $I_1,I_2$ are isospins. $V_0(r)$ is of the square well form. My goal is to find an equality for a and b, given that deuteron exists, and that diproton and dineutrons do not. My approach has been the following: 2 nucleons can either be in an isosinglet or an isotriplet. The Isosinglet has I=0 and is the following: $$|00\\rangle=\\frac{1}{\\sqrt2}(pn-np),I_1\\cdot I_2=-\\frac34$$ The isotriplet has I=1 and $$|11\\rangle=pp,I_1\\cdot I_2=\\frac12$$ $$|10\\rangle=\\frac{1}{\\sqrt2}(pn+np),I_1\\cdot I_2=\\frac12$$ $$|1-1\\rangle=nn,I_1\\cdot I_2=\\frac12$$ So I get the following equations based on V $$\\text{Isosinglet: }\\frac{V(r)}{V_0(r)}=a-\\frac34 b$$ $$\\text{Isotriplet: }\\frac{V(r)}{V_0(r)}=a+\\frac12 b$$ Now I am supposed to use the fact that $V_0(r)$ is of the square well form to create inequalities, but I am unsure of where the inequalities come from."}
{"id":"71727","title":"how to determine direction of gyroscopic couple on car?","text":"How can I determine the direction of the force acting due to gyroscopic couple on a car's wheels when it is taking a turn to it's left side?"}
{"id":"71721","title":"Temperature of rod if end points are different temperature","text":"How does the temperature vary along the length of the rod if its both ends are at different temperature. As an example, consider the problem: > 20 cm long rod has rod at one end 100 ºC and another end at 0 ºC. Find the > temperature at the center of the rod when it's in thermal steady state."}
{"id":"18738","title":"What causes light to move through a vacuum","text":"I was looking at rockets and stuff and thought about how they move through a vacuum using newtons 3rd law, and then I started thinking of any other ways you could move through a vacuum without using this and then I thought about the photon. I then thought how does the photon move through a vacuum? So I searched it online and couldn't find an answer. So my question is simply how does a photon move through a vacuum? Is it because its massless? Is it because it is thrown from an electron or whatever like a rock from a slingshot? Or is it something completely different? I'm very curious!"}
{"id":"17816","title":"non-exotic traversable wormholes and string theory","text":"In this article it is claimed that certain type of string theory called EGBd allows for traversable wormhole solutions that do not require exotic matter. What is this EGBd model and how it fits in the grand scheme of things of string theory? is this fringe science or is serious? (i know that there might be fringe authors in ArXiv). But my main concern is how general or realistic is this result?"}
{"id":"17076","title":"Is the spring constant k changed when you divide a spring into parts?","text":"I've always been taught that the spring constant $k$ is a _constant_ -- that is, for a given spring, $k$ will always be the same, regardless of what you do to the spring. My friend's physics professor gave a practice problem in which a spring of length $L$ was cut into four parts of length $L\/4$. He claimed that the spring constant in each of the new springs cut from the old spring ($k_\\text{new}$) was therefore equal to $k_\\text{orig}\/4$. Is this true? Every person I've asked seems to think that this is false, and that $k$ will be the same even if you cut the spring into parts. Is there a good explanation of whether $k$ will be the same after cutting the spring or not? It seems like if it's an inherent property of the spring it shouldn't change, so if it does, why?"}
{"id":"17818","title":"Question on ladder operators","text":"Suppose we have a finite , discrete set of orthonormal states $|k\\rangle $ We can construct raising and lowering operators intuitively, for example $$a_+ =\\sum_{k=1}^nC_{k+1}|k+1\\rangle \\langle k|$$ However most textbooks begin by defining ladders in terms of the linear combinations of hermitian operators. How do we get from the above construction to showing that such ladder operators have the form $$a_\\pm = \\frac{\\alpha\\pm i\\beta}{\\sqrt{2}}$$ where $\\alpha,\\beta$ are hermitian?"}
{"id":"17819","title":"How does cooling scale with volume?","text":"What equation would give me the answer to the question, \"If i have a cup of water at a tempature of say boiling, how long would that cup of water take to cool off compared to say half that size of a cup of water.\" So the volume is in half. Its a general question I am just looking for where to start."}
{"id":"29827","title":"What's the amount of deviation of cellestial orbits from perfect ellipses","text":"It's well known that the planets don't orbit the sun in perfect circles and the characteristics of the elliptical orbits which serve as better approximations to their motion have been calculated fairly accurately. How accurately do elliptical orbits model their actual paths?"}
{"id":"29829","title":"Calculating the Uncertainty for an Average Value","text":"How would I calculate the uncertainty for the average of this set? $32.5 \\pm 0.1$ $32.0 \\pm 0.1$ $32.3 \\pm 0.1$"}
{"id":"29082","title":"Would time freeze if you can travel at the speed of light?","text":"I read with interest about Einstein's `Theory of Relativity` and his proposition about the speed of light being the speed limit for anything with mass. So, if I were to travel in a spacecraft at the speed of light, would I freeze and stop moving? Would the universe around me freeze and stop moving? Who would the time stop for?"}
{"id":"8142","title":"How does a particle of light reach the max speed of light?","text":"> **Possible Duplicate:**   >  How can a photon have no mass and still travel at the speed of light? First of all I am not a professional physicist. I was curious as to how a particle of light can reach the maximum speed of light c given that particles experience an increase in mass as they approach c ?"}
{"id":"2229","title":"If photons have no mass, how can they have momentum?","text":"As an explanation of why a large gravitational field (such as a black hole) can bend light, I have heard that light has momentum. This is given as a solution to the problem of only massive objects being affected by gravity. However, momentum is the product of mass and velocity, so, by this definition, massless photons cannot have momentum. How can photons have momentum? How is this momentum defined (equations)?"}
{"id":"3541","title":"How can a photon have no mass and still travel at the speed of light?","text":"I've read a number of the helpful Q&As on photons that mention the mass\/mass- less issue. Do I understand correctly that the idea of mass-less (a rest mass of 0) may be just a convention to make the equations work? From a layperson's view, it's difficult to understand how a particle of light (photon) can be mass-less. A physical object (everyday world-large or quantum- small) must have a mass. Yet, if my understanding is correct, the mass of a moving object\/particle increases in proportion to its speed\/velocity...so that at the speed of light, its mass would be infinite. A photon travels at the speed of light, but it obviously doesn't have infinite mass, right? Can someone formulate a practical explanation that can be understood by middle- school to high school kids? Much thanks for the help. * * * Wow--your answers to my original Q below clear up much of my confusion. I now have the daunting task of going over these nuggets and working up an equation- less (hopefully) explanation of the mass-less photon for non-physicist types. Yes, _from a layperson's view_ , it does seem remarkable that an existing piece of _matter_ \\-- which has to be made of _physical substance_ \\--could have zero mass at rest (though a photon is never at rest). It would be almost understandable if a piece of matter made of _nothing_ had zero mass, but that seems to be an oxymoron, and \"nothing\" would equate to nonexistent, right? In case you might find it interesting: I'm working on a writing project that posits we inhabit a universe that consists of matter (physical stuff) only, and that the NON-physical (aka supernatural) does not (and cannot) exist. For instance, if a purported supernatural phenomenon is found to actually exist, then by definition, its existence is proof that it is mundane\/natural. All it would take to disprove this premise is reliable proof that ONE supernatural event has occurred. Despite thousands of such claims, that's never yet happened. Who else better than physicists to confirm my premise? However, I do wish the TV physicists would explain the terms they throw about, some of which mislead\/confuse their lay viewers. Case in point: \"The universe is made up of matter and energy\" (without properly defining the term \"energy\" as a property of matter). The result is that laypersons are left with the impression that energy must therefore be something apart from or independent of matter (ie, nonphysical). Their use of the term \"pure energy\" without specifying exactly what that means adds to the confusion. (Thanks to your replies on this forum, I now understand that \"pure energy\" refers to photon particles.) However, \"psychics\" and other charlatans take advantage of such confusion by hijacking terms like energy (as in \"psychic energy\"), frequencies, vibrations, etc to give perceived scientific legitimacy to their claims that a supernatural spirit world, etc., exists. As you may realize, the majority of people in the US (per 2009 Harris Poll) and around the world believe in the existence of nonphysical\/supernatural stuff such as ghosts and spirits. My purpose is to give laypersons the information they need to distinguish what's real from what's not. Thanks so much for help...And, PLEASE, add any further comments you think might be helpful\/insightful to better inform laypersons."}
{"id":"57442","title":"How does the solar sailing concept work?","text":"Wikipedia describes solar sailing as > a form of spacecraft propulsion using a combination of light and high speed > ejected gasses from a star to push large ultra-thin mirrors to high speeds. I understand the part where ejected gasses bump into the sail pushing the spacecraft. On the other hand, I don't understand how light can do this, since light has no mass. How does that work? Does this mean that if I have a mirror balancing on a needle I would be able to push it over with my flashlight?"}
{"id":"89675","title":"Clocks tick steadily, so why is there no photon time?","text":"Consider a photon bouncing left and right between two mirrors in a photon clock. Seen from inside the clock, the photon bounces at a constant frequency. Time ticks regularly. No matter whether the clock begins moving relative to some outside observer, the clock continues to tick regularly. Light always moves at the same speed relative to the observer in the clock. If the clock is moving forwards relative to an outside observer, we know that _to that outside observer_ , the photon clock appears to tick more slowly. This happens because the speed of light relative to that observer is constant, and, since the photon is moving diagonally relative to the outside observer, the distance traveled by the photon in the clock between ticks is greater, so the photon must take longer to pass between the mirrors and the clock must tick more slowly. But notice how, if you are inside the clock, it continues to tick at the normal speed. This brings me to my question. It is widely reported (including on this very QA site) that a photon traveling uninhibited in a vacuum experiences no time. This seems to be because all the speed of the photon in the clock is now taken up keeping up with the forwards movement of the clock, so there is no speed left to deal with left\/right bouncing and ticking. The clock stops ticking. Well, this is the thing. Surely the clock only stops ticking _from the perspective of the outside observer_. If you're inside the light-speed clock, time continues to tick normally. The photon continues to bounce. So why do we say a photon experiences no time? Surely we mean it _appears to us_ as stationary observers to experience no time? If you're a photon, time is just the same as it is to the rest of us, right? Fair enough, your average photon might well feel that it got from A to B in no time at all, because distance was compressed to zero and all the other clocks appeared to have zipped along beyond the end of days in less than an instant, but its own clock, if the photon could observe it, is just ticking normally. This leaves my brain mushed, because now the photon has seen the entirety of an infinity of time in no time at all, yet its clock is still ticking. So what does that mean? What happens the moment after the infinity of external time has completely passed? Surely there is nothing left to happen?"}
{"id":"68812","title":"Does distance really exist at least for observer which moves at the speed of light?","text":"To our time a light year is the distance that light travels in a vacuum but if we consider time of observer which moves at speed of light we found for light time and distance doesn't really exist. Does distance really exist? ($d\\tau=dt\\gamma^{-1}$ where the at speed of light $\\gamma^{-1}=0$) isn't universe just a point for observer at light speed? * * * it is not duplicate guys!."}
{"id":"68600","title":"Frame of reference of the photon?","text":"In the frame of photon does time stop in the meaning that past future and present all happen together? If we have something with multiple outcomes which is realized viewed from such frame? Are all happening together or just one is possible? How the communication between two such frame s work meaning is there time delay for the information as $c$ is limited? If there is time delay does it mean that time does not stop? My question does not concern matter at that speed rather how it looks viewed from the photon reference. Thanks Alfred! I think I understand it now."}
{"id":"51933","title":"Speed of light and lorentzian factors","text":"> **Possible Duplicate:**   >  How can a photon have no mass and still travel at the speed of light? If light travels at the speed of light, and anything with rest mass will experience relativistic effects based on the Lorentzian equations, why doesn't light experience these kinds of effects? For example, relativistic mass and rest mass are related via $$m = \\frac{m_0}{\\sqrt{1-\\dfrac{v^2}{c^2}}}$$ Shouldn't light therefore have no rest mass (since the Lorentzian is $0$)?"}
{"id":"63969","title":"Why does Lorentz factor not hold for relativistic mass when we apply it to photons?","text":"We know that the photon itself is massless particle $m_0=0$. But we also know, that the mass of the objects does increase with their energy. And we know that under certain circumstances (gravity, collision with objects) the light does behave like a beam of particles that do have a mass. Now, this is the equation to get the (relativistic) mass of the object with certain speed: $$m=\\frac{m_0}{\\sqrt{1-v^2\/c^2}}$$ We already do know some things: $m_0 = 0$, $v = c$. $$m=\\frac{0}{\\sqrt{1-c^2\/c^2}}$$ $$m=\\frac{0}{0}$$ Expression $\\frac{0}{0}$ is **not** equal to 0. So what is this?   I'll remind you of those weird clock that are run by light - I failed to find any images, but the clock principle was similar to windmill. And it works.   Also there was an idea to make long-distance autonomic spaceships that are pushed by light rays from stars."}
{"id":"94877","title":"how photons are said to be in momentum if they are considered to have no mass and no charge?","text":"According to quantum theory,Einstein said that light waves have small packets called photons and they are said to have no mass and charge but having energy E=hv and they behave has both particle and wave character. And they travel with certain velocity. How photons have momentum and energy if they said to have no mass? Momentum = p=mv ,where m= mass v=velocity."}
{"id":"133556","title":"How can light travel at speed of light?","text":"I am well aware that the speed of light is a universal constant at which nothing but light can travel. But why? Why can even light travel at such a speed? Our maths tell us that anything with rest mass takes infinite energy to reach speed of light, so do we just mean that light is the lightest of all things and it can reach speed of light with negligible energy? Why isn't the speed constant which appears not only as speed of light, but various other places like GR etc something even more than speed of light? Why can light reach the speed of light? I think my question was highly misinterpreted, my question has nothing to do with mass of light. My question is why is the speed limit that appears universally equal to speed of light, why isn't even light allowed to reach the universal speed limit? Yes after that I did ask, the reason for light reaching the highest possible speed. Whether it is it's unusually less mass(mass-less) or something else. So to summarize my question is : 1\\. Why anything at all can reach the top speed limit, why is even light permitted to reach the top speed limit? 2\\. Why light can reach the top speed? (Only this part is related to the question whose duplicate my question is marked)"}
{"id":"30764","title":"Does $p=mc$ hold for photons?","text":"Known that $E=hf$, $p=hf\/c=h\/\\lambda$, then if $p=mc$, where $m$ is the (relativistic) mass, then $E=mc^2$ follows directly as an algebraic fact. Is this the case?"}
{"id":"14896","title":"Masses of all the particles in the Standard Theory","text":"> **Possible Duplicate:**   >  If photons have no mass, how can they have momentum? I'm sure this question has been asked here before but I wasn't able to find it clearly answered in one q\/a session. I'm a newb, yes. In the standard model (which I understand is well experimentally supported except for the absent Higgs Boson), all the elementary particles must carry some mass, correct? For E=MC2 to be true, you can't have energy without mass. And energy has been carried away in collisions of sub-atomic particles that can't be accounted for (the Higgs). So EVERY elementary particle in the Standard Model must have mass, correct?"}
{"id":"131645","title":"How does light travel at the speed of light?","text":"If it is impossible for matter to accelerate to $c$ (because doing so would take infinite energy), and if light can be deemed matter (because of wave- particle duality, photons are matter, right? And photons clearly have mass per solar radiation pressure), how does light manage to travel at $c$?"}
{"id":"74107","title":"What happens with time at the speed of light?","text":"Say that by some means you could become massless, and therefore travel at the speed of light. I understand that the closer you get to the speed of light, the slower time appears to the person traveling. Like say you were traveling a 99.999 % of the speed of light, ever year for you would be 200 or so years for people not traveling. So I suppose at the speed of light, time stops for the traveler. Would that mean you'd be frozen in time and an infinite amount of time would have passed for the rest of the world while you were travelling at the speed of light?"}
{"id":"99885","title":"If photons don't have mass, how can they accelerate objects?","text":"As far as I know photons don't have mass but they do have momentum ($p=mv$). Scientists say that if we put a shiny (reflective) shield of large radius in the vacuum of space, then light from sun will will start moving the shield. How can these two facts be commensurate?"}
{"id":"76917","title":"attaining the speed of light??","text":"WHY our time would run slow if we move with the speed close or equal to the speed of light, and is that time dilation restricted to only just '' time' ' or it affects our biological time also (does it stops or slows down our ageing process) ??"}
{"id":"133216","title":"Are photons travelling in future? Are they lifespan 0 (or close to 0)?","text":"We know that if we are going closer to speed of light, we are travelling in future. Since the photons are travelling at the speed of light, that means that they are travelling into future? Also, if we could possible attach a clock to a photon and measure how much time passed for 1 earth year(we put the clock to be 0, then we wait 1 year, then we look at the clock), would that clock show 0?"}
{"id":"62744","title":"Do photons have mass?","text":"As a student in a highschool physics class, my teacher has repeatedly told me that photons are massless. Yet, I have also heard from other sources that photons have momentum. If photons were to have momementum, that would mean that they have mass as according to p = mv. Do photons really have mass? Also, how would this mass be calculated?"}
{"id":"51733","title":"Does the increase of (relativistic) mass, while flying near speed of light, has any impact on astronauts?","text":"> **Possible Duplicate:**   >  Would travelling at relativistic speeds have any impact on human biology? I am asking myself this question for a few day. What is the answer on: Does the increase of (relativistic) mass, while flying near speed of light, has any impact on astronauts? I mean let's pretend a spaceship will be able to travel in space with nearly the speed of light. Does the astronaut on this ship, while its (relativistic) mass increases, feel any difference on gravity or anything else? Or is the increase simultaneous to his own?"}
{"id":"125962","title":"Why doesn't light travel instantly?","text":"I've read that the faster you travel in space, the slower you travel in time. And when you reach the speed of light (which we won't be able to) time will stand still. So when light travels at the speed of light, it doesn't move in time, so no time will have past when the light arrives at it's \"destination\". Right?"}
{"id":"61794","title":"What is the mass of a photon moving at the speed of light?","text":"1. What is the mass of a photon moving at the speed of light?    2. And if it does not have mass, how is it affected by gravity?   3. Also why does Einstein's general relativity support that a gravitational wave must travel at the speed of light? I'm just an A-level student. So, I would appreciate it if you could explain it as idiotically as possible."}
{"id":"116464","title":"Photons have no mass. So, why does $E = pc$ hold?","text":"It's a somewhat theoretical question. In special relativity, The energy of a photon is given by $E = pc$. But, my argument is that, since photons have no mass, how can they have a momentum $p$? The energy $E$ turns out to be 0 always. So, why does this equation hold?"}
{"id":"119490","title":"How to get the accurate relativistic momentum form for photons?","text":"I have studied from Griffiths, the relativistic form of momentum is $$p = \\frac{1}{\\sqrt{1-\\frac{v^2}{c^2}}} m_0v$$ Now when I evaluate the momentum for photon, I just insert $v=c$ and $m_0=0$ and I get $p= 0\/0$. How does it make sense? Can you tell me that where I am wrong?"}
{"id":"48502","title":"Would travelling at relativistic speeds have any impact on human biology?","text":"If a person was sitting on a craft that has accelerated to near light speed speed from Earth (e.g. 99.999% of light speed) would there be any impact on his or her human biology due to relativistic effects on mass, time and distance? For example, would the heart still be able to pump blood effectively if the relativistic mass of the blood has increased? Or would accelerating the blood a bit further towards light speed requires more energy than when stationary on earth?"}
{"id":"2095","title":"Why does the cart move?","text":"A while ago someone proposed the following thought experiment to me: A horse attached to a cart is resting on a horizontal road. If the horse attempts to move by pulling the cart, according to the 3rd Newton's Law, the cart will exert a force equal in magnitude and opposite in direction, cancelling each other out and thus the horse and the cart _should_ not move. And yet it moves (pun intended ;) Why? I never got a satisfactory answer, my guess is that the answer lies in the frames of reference involved: horse-cart and horse-road. Any ideas?"}
{"id":"118452","title":"Topolgical insulators order parameter","text":"_For topological insulators_ **Is there any way to defineorder parameter for topological phase transitions?**"}
{"id":"118450","title":"Why is this not a violation of parity invarance for EM","text":"I read that Wu's experiment illustrates that parity violation is possible for weak processes. In that experiment, when Co-60 undergoes beta decay, the emitted electrons come out opposite to the direction of nuclear spin. The mirror image process does not occur. I also read that parity is conserved for EM and strong processes. But consider the following from EM. Say I have a positively-charged particle moving to the right, and the magnetic field is into the page. Then force on the particle is up (i.e. the particle is accelerated upwards initially). And here, the mirror image also does not occur! The mirror image (the mirror I am thinking of lies in the plane that cuts your head between the eyes) would be moving left, magnetic field into page, acceleration still pwards - but EM would predict that the acceleration is supposed to be downwards!! Why isn't this an example of parity violation?"}
{"id":"123396","title":"How to convert physical ligth quantities (frequency, intencity, spectrum) to RGB and back?","text":"Suppose I would like to have maximally physical image format. Which quantities should I store in the pixel? Obviously, it should be frequency spectrum, i.e. table with frequency vs. WHAT? How to convert RGB values to any sort of physical quantities and vise versa?"}
{"id":"123395","title":"Is there Phase difference between voltages at end points of a long AC Power line","text":"I found this explanation somewhere: Since wavelength=c\/freq so in a AC power line of 50 Hz, wavelength=(3*10^8)\/50 = 6000 Km, so voltage phase reverses after 6000 Km in a ac Power line. Now this would have been correct if it was a 50 Hz E.M. Transmission line, but does the same occur in metal wire with A.C. ? Clearly the power transmission through the wire would not be through E.M. waves. I am also confused if a wave phenomenon does occur in a AC wire or not. If it occurs then how (in terms of flow of electrons) and at what speed\/wavelength ?"}
{"id":"129995","title":"Energy stored in a capacitor in an RC circuit","text":"Suppose we have a capacitor of capacitance $C$ and a cell of emf $E$, why is the maximum energy stored in a capacitor equal to $\\tfrac{1}{2}CE^2$? I feel confused because, the potential difference against the capacitor will be less than the emf of the cell because of the potential drop across the resistor. What is the flaw in my thinking?"}
{"id":"59882","title":"What is the observable Earth we can see?","text":"Yes I know we can see the whole earth, but how far can we see left to right waving out the limitations of sight. Because it's impossible to see the whole earth right? because it's spherical? So is there some kind of equation to work this out?"}
{"id":"71641","title":"What is the invariant associated with the symmetry of boosts?","text":"Noether's Theorem states that if a Lagrangian is symmetric for a certain transformation, this leads to an invariant: Symmetry of translation gives momentum conservation, Symmetry of time gives Energy conservation etc. The Galilean principle stating that all reference frames that move with constant speed relative to each other are equivalent is also a symmetry principle: Setting up a physical system that is identical to the original except for a constant velocity (boost) added will have the same behaviour. Shouldn't there be an invariant associated with this symmetry? If yes, what is that invariant?"}
{"id":"44253","title":"Noether's charge due to lorentz transformation","text":"> **Possible Duplicate:**   >  What conservation law corresponds to Lorentz boosts? For a relativistic free particle, What is the Noether charge generated due to Lorentz transformations? What is conserved quantity associated with boosts?"}
{"id":"16072","title":"What is a correct and simple definition of quantum physics?","text":"Is it correct to define Quantum Physics as the study of Physics in sub-atomic scale? Does Quantum Physics studies something else other than sub-atomic phenomena? This may be a very stupid question but realize that I am **!!!VERY NOOB!!!** at Physics."}
{"id":"16074","title":"Does (it make sense to say that ) the universe has a center?","text":"I was reading this page: http:\/\/www.guardian.co.uk\/science\/2011\/oct\/23\/brian-cox-jeff-forshaw-answers and I found this sentence by Brian Cox: > That seems to imply that everything is flying away from us and we're > therefore somehow in a privileged position; that isn't true. The way it's > often described is if you imagine some bread with raisins in it that you're > baking in the oven and as you heat it, it expands. On any particular raisin, > if you look, you can see all the other raisins receding from it. So it's > space that stretching, it's not that everything's flying away. I already heard this raisins analogy, but it never persuaded me: I understand that the \"big bang\" is more like a \"big stretch\", and I see how every 2 observers in the universe are being distanced farther and farther away (regardless of their position) Yet one of the Big Bang ideas is that the universe isn't anymore considered infinite and completely homogeneous But the fact that the universe is finite, while inflating to me implicates that it should have some kind of bounds (not that we can reach these \"bounds\", since our distance to them is getting bigger, but they should still exist) (And the fact that it's spreading inhomogeneous mass and energy over big distances, is thus making it more homogeneous, but this doesn't probably matter) So: the very idea of a big bang seems to me in contradiction to the assertion that there's no such thing as a \"center of the universe\": If it has a finite mass and some kind of bounds, then it should also have a barycenter. And if we consider the bread with raisins analogy: the bread has a center from which it's expanding Surely, the universe isn't homogeneous (like the distribution of the raisins), and so, in its hypothetical center, there may not be actually anything... but I think (even if it's really unlikely) it should still be theoretically possible to have a raisin in the exact centre of the bread"}
{"id":"88714","title":"Finding possible values of $L_x$ given $L^2$","text":"Here's a homework problem I'm working on. I am not asking for the answer, but any guidance or comments on the approach are appreciated. > Given that a measurement of $L^2$ for a free particle has resulted in the > value $6\\hbar^2$, what are the possible results for a measurement of $L_x$? I've been following the approach given on pp. 378-380 in Liboff (4th ed.). I've made it as far as expressing the eigenfunctions of $L_x$ as sums of the eigenfunctions of $L^2$... Here's what I've got: $$\\begin{align*} X_1 &= \\frac{1}{2}\\left(|2,2\\rangle - |2,1\\rangle + |2,-1\\rangle - |2,-2\\rangle\\right)\\\\\\ X_2 = -X1 &= -\\frac{1}{2}\\left(|2,2\\rangle - |2,1\\rangle + |2,-1\\rangle - |2,-2\\rangle\\right)\\\\\\ X_3 &= \\frac{1}{2}\\left(\\sqrt{\\frac{3}{2}}|2,2\\rangle - |2,0\\rangle + \\sqrt{\\frac{3}{2}}|2,-2\\rangle\\right)\\\\\\ X_4 = -X3 &= -\\frac{1}{2}\\left(\\sqrt{\\frac{3}{2}}|2,2\\rangle - |2,0\\rangle + \\sqrt{\\frac{3}{2}}|2,-2\\rangle\\right)\\\\\\ X_5 &= \\frac{1}{2}(|2,2\\rangle + |2,1\\rangle - |2,-1\\rangle - |2,-2\\rangle)\\\\\\ X_6 = -X_5 &= -\\frac{1}{2}\\left(|2,2\\rangle + |2,1\\rangle - |2,-1\\rangle - |2,-2\\rangle\\right) \\end{align*}$$ The next step appears to be to express $L^2$ as sums of the eigenfunctions of $L_x$, which does not seem possible. It was a lot of algebra, matrices, determinants, etc. to get to this point so it is certainly possible that I've miscalculated something accidentally, but does anyone see anything fundamentally wrong here with this approach? And\/or is there a simpler way?"}
{"id":"119462","title":"Mathematically possible vs physically probable outcomes","text":"A good buddy of mine and I have had a friendly debate about the origins of the current state of our universe (namely; Earth and life on Earth) and have fundamentally disagreed in our stances with respect to probabability, infinity in time and space and possible\/probable event outcomes. He maintains the position that given a set of possibilities and enough trials, each outcome _must_ have occured; which is his reasoning for why life must exist. I do not necessarily take issue with this particular concept, as given an infinite amount of time and states of matter life is bound to come from one of those states. In our discussions, however, we have been using a specific example in which I disagree vehemently with his stance. The example: > If you throw a handfull of sand in the air an infinite number of times, and > that sand lands on a flat surface, every configuration _will_ happen > (according to his position). For instance, the sand landing in a pattern > which spells your name out is a mathematically possible outcome, and will > therefore happen given enough trials. To counter his stance on this example, I took the position that there is a mathematical (but not physical) possibility that every grain of sand lands in the same one inch square of the surface; but I maintain that even though it is a mathematically possible outcome it will never happen because of the way the physical world works - that sand will be roughly evenly distributed for each throw, even if over an infinite number of trials, assuming consistent and fair trials (ie, no God or other being moving grains of sand). I submit that even though it is a mathematical possibility, you'll never see your name spelled out in block letter English anywhere in the universe without the influence of intelligence, even if you were able to attempt a verification for this - he disagrees. I held him liable for mathematical\/physical proof of reasoning for his stance and he has taken to dismissing me as ignorant of probability and infinity. Can anyone provide some good reasoning for either side of this argument? I realize that either is an impossible stance to prove, since we can't verify our positions, but any well-reasoned insight will be appreciated. A similar question, with an answer I found to be relatively useful: Infinite universe - Jumping to pointless conclusions **EDIT:** After reading some of the comments and answers here it has become apparent that I may have misrepresented my ultimate question. I realize that given a non-zero probability and an infinite number of trials, the mathematical probability of encountering the event described by said probability converges to 1. Some have taken the position that there is no disconnect between a mathematical probability and the likelihood (read: possibility) of a physical event happening. To simplify the argument, the surface can be thought of as a grid - in which case every single configuration has some mathematical probability associated with it. My stance regards certain configurations as physically impossible, however, which is the reasoning behind my one-inch-square analogy. Can anyone show clear reasoning (and sources!) for their belief that it is possible to toss a handful of sand into a one inch square?"}
{"id":"119464","title":"Losing mass in space","text":"So I came across a question while studying laws of motion. Roughly, this is how it goes: There are two astronauts in a space shuttle, who together have mass 200 kg. If by doing exercise, they manage to lose 80 kg, what will be the percentage increase in speed of the shuttle. The question is pretty straight forward, if thought about directly. However, my instant reaction was that by conservation of mass, the mass that the astronauts lose will still be contained within the space ship in the form of water, CO2, etc. So technically there won't be any change in mass, thus no change in speed. I would like to know if this assumption is correct and in what forms is the mass we lose released."}
{"id":"118687","title":"insufficient electrons in a conductor","text":"We all know that if an electric field is applied across an isolated conductor, then charges are induced at either surface such that the net electric field inside becomes zero. Now if the applied electric field is so strong that there are not sufficient free electrons in the metal to move and cause the required surface charge seperation. Then will the conductor now behave as a non- conductor(i suppose so) or will it ionise nearby gas and extract electrons from it ."}
{"id":"2317","title":"Including air resistance, what is the escape velocity from Earth?","text":"Including air resistance, what is the escape velocity from the surface of the earth for a free-flying trajectile?"}
{"id":"114109","title":"Conservation of energy (or lack thereof) in Doppler cooling","text":"I did not find my question answered elsewhere, so here it is. I'm doing a project in my first optics course, and we are reading a bit about Doppler cooling. I understand that a laser is tuned to a frequency where the photon energy $E_p = h \\nu$ corresponds to an energy slightly below the energy $E_{abs}$ required to excite the gas atoms, and that an atom moving towards the laser at a high speed (relative to rest of the gas) will experience a blueshift that allows the atom to absorb the photon, thus slowing down the atom. I also understand that the photon is re-emitted in a random direction, which of course means that the atom's velocity in the direction of the laser will be decreased. What I'm not sure I understand is how the re-emission of the photon doesn't just bring the atom back to the same kinetic energy (but with a different direction of motion). I have not been able to find a source explaining this, but here's my own attempt at an explanation (which I am not sure is correct, which is why I am asking here): The absorbed photon has an energy (in the frame of the laboratory) of $E_1 = h \\nu_1$ which is slightly less than the absorption energy $E_{abs}$. But after the absorption the atom's speed is reduced, thus the photon will be emitted with a different frequency $\\nu_2$ (in the frame of the laboratory). Since the speed of the atom (relative to the frame of the laboratory) has been reduced by the absorption, the energy $E_2$ of the emitted photon will be closer to $E_{abs}$, thus $|E_{abs} - E_2| > |E_{abs} - E_1|$ (the last step of the explanation only works if the atom's speed perpendicular to the laser is not greater than the speed parallel to the laser for most atoms, but since the laser is tuned to only hit the fast moving atoms, this seems improbable enough to not invalidate the explanation). Is this explanation correct or have I made mistake? I have taken a special relativity course but no quantum mechanics yet, so please keep the quantum mechanics to a minimum if possible."}
{"id":"76962","title":"Modeling of exact stick-slip-friction using Linear complementarity problems (LCP)","text":"Within my master's thesis, I try to model a machine elements and have to model the exact slipping and exact sticking between the bodies. So I noticed that such problems can be formulated as a LCP. However I have difficulties at finding any literature where the usage is described well with examples.. I would like to have something like a 1-DOF model such a box on a plane pulled with a spring and friction modeled with coulomb or so. I would be very grateful of mentioning any literature\/sites which can bring me further. I am really desperate at implementing the coulomb friction.."}
{"id":"56838","title":"Dalitz plot analysis","text":"I have seen a few Dalitz plots so far and tried to understand how they are useful. So one of the advantages of these plot is that the non-uniformity in the plots can tell something about the intermediate states that we cannot detect. My question is how do you extract the mass of these resonant particles from such plots? What additional information would you need?"}
{"id":"18527","title":"Does the Pauli exclusion principle instantaneously affect distant electrons?","text":"According to Brian Cox in his _A night with the Stars_ lecture$^1$, the Pauli exclusion principle means that no electron in the universe can have the same energy state as any other electron in the universe, and that if he does something to change the energy state of one group of electrons (rubbing a diamond to heat it up in his demo) then that must cause other electrons somewhere in the universe to change their energy states as the states of the electrons in the diamond change. But when does this change occur? Surely if the electrons are separated by a significant gap then the change cannot be instant because information can only travel at the speed of light. Wouldn't that mean that if you changed the energy state of one electron to be the same as another electron that was some distance away, then surely the two electrons would be in the same state until the information that one other electron is in the same state reaches the other electron. Or can information be transferred instantly from one place to another? If it can, then doesn't that mean it's not bound by the same laws as the rest of the universe? \\-- $^1$: The Youtube link keeps breaking, so here is a search on Youtube for Brian Cox' _A Night with the Stars_ lecture."}
{"id":"16382","title":"The shape of the earth$\\ldots$","text":"....is an oblate spheroid because centrifugal force stretches the tropical regions to a point farther from the center than they would be if the planet did not rotate. So we all learned in childhood, and it seems perfectly obvious. However... I am at $45^\\circ$ north latitude. Does that mean   * An angle with vertex at the center of the earth and one ray pointing toward the equator at the same longitude as mine, and one ray pointing toward me, is $45^\\circ$ (that would mean I'm closer to the north pole than to the equator, measured along the surface, as becomes obvious if you think about really extreme oblateness); or   * The normal to the ground where I stand makes a $45^\\circ$ angle with the normal to the ground at the equator at the same latitude (this puts me closer to the equator than to the north pole); or   * something else? If for the sake of simplicity we assume the earth is a fluid of uniform density, it seems one's potential energy relative to the center of the earth would be the same at all points on the surface.   * Would the force of gravity at my location, assuming no rotation, be directly toward the center? Would it be just as strong as if the whole mass of the earth were at the center and my location is just as far from the center as it is now?   * Would the sum of the force of gravity (toward the center or in whichever direction it is) and the centrifugal force (away from the axis) be normal to the surface at my location?   * Given all this, how does one find the exact shape?   * How well does that shape in this idealized problem match that of the actual earth?"}
{"id":"11763","title":"how does dynamic casimir effect generate correlated photons","text":"There is a recent paper on arxiv receiving lot of acclaim http:\/\/arxiv.org\/abs\/1105.4714 The authors experimentally show that moving a mirror of a cavity at high speeds produces light from high vacuum. The usual doubts about the experimental techniques seem to be very clearly addressed and reviewed (as per Fred Capasso's comments) http:\/\/www.nature.com\/news\/2011\/110603\/full\/news.2011.346.html) My question is : Can someone explain how correlated\/squeezed photons are generated in this process ? I can get a feel for how a moving mirror can generate real photons by imparting energy to the vacuum (correct me if this is not consistent with the detailed theory). But, I don't see how photons are generated in pairs. Could someone describe the parametric process happening here ?"}
{"id":"11760","title":"Landau's ambiguous statement about the existence of inertial frames","text":"Landau writes \"It is found, however, that a frame of reference can always be chosen in which space is homogeneous and isotropic and time is homogeneous.\" Does he mean that we can prove the existence of an inertial frame or does he want to say that it is assumed by doing enough number of experiments? Can we start with some axioms and definitions of properties of space and time and then deduce the existence of such a frame in which space is homogeneous and isotropic and time is homogeneous?"}
{"id":"69718","title":"Why is quantum mechanics based on probability theory?","text":"What makes us formulate quantum mechanics based on probability theory? Isn't the real quantum world based on unknown laws to us? Is it possible that results of an experiment will be measurable in another way but not expected value?"}
{"id":"90792","title":"Resource about Supercontinuum Generation in Fibers","text":"I would like to consult a nice reference that explains the theoretical background of SC generation in optical fibers in detail but more or less self- contained. I would also like to have your opinions on nonlinear optic books that present material at introductory level. Thanks in advance"}
{"id":"64053","title":"What is the difference between spin glass and spin liquid?","text":"What is the difference between spin glass and spin liquid? Do they both originate from frustration?"}
{"id":"46316","title":"Harmonic oscillator and Lorentz symmetry","text":"There is a analog between harmonic oscillator $x=\\frac{1}{\\sqrt{2\\omega}}(a+a^\\dagger)$ and quantum field $\\phi=\\int dp^3\\frac{1}{(2\\pi)^3}\\frac{1}{\\sqrt{2\\omega_p}}(a_p e^{ipx}+a^\\dagger e^{-ipx})$, which is used to quantize the field operator. However, one thing confuse me is about the coefficient $\\frac{1}{\\sqrt{2\\omega}}$. For field operator, this comes from Lorentz invariance, just because we have integrated time t. However, for harmonic oscillator, there seems no apparent Lorentz symmetry give me this. Is there any hidden symmetry behind the harmonic oscillator?"}
{"id":"46311","title":"Does the attached \"poster\" work as a hook into the arXiv paper cited, \"Nonlinear Wightman fields\"?","text":"\"Nonlinear Wightman fields\" are my current response to a wish to do interacting quantum field theory differently, no matter how successful what we currently do may be. The following image of a single page \"poster\" attempts to get across why I think the mathematics is interesting, with the intention that an ordinary physicist would find the hook curious enough to look at the 22 page arXiv paper. The mathematics of section IV of that paper is really as simple as \"use the Hadamard product of matrices\" to construct something nontrivial that might have more relevance to Physics than the Wightman axioms have had since they were created more than 50 years ago. This is at the same time familiar stuff and far up a different mountain, so there is some hope at the same time as there is concern that physicists generally will not understand the motivation or that I've made a mistake in the mathematics. [Note that the image itself is legible even if what is displayed inline is marginal.] ![Nonlinear Wightman fields -- a poster](http:\/\/i.stack.imgur.com\/hzGN8.jpg) The paper this cites is something of an outcome from the thinking that led to my question \"Does renormalization make quantum fields into (slightly) nonlinear functionals of test functions?\", 18 months ago, when I was using PhysicsSE quite actively as a way to help my research. Some of the content of the arXiv paper may also be found in a much rawer form as an entry in this year's FQXi essay competition (so raw that I can't recommend that you chase the link). People who've been on PhysicsSE long enough may remember my attempts, for over a year, to use PhysicsSE as a tool for research, which I eventually felt to be counterproductive because it led to too little long-term thinking, and also because I felt unable to do much to help people whose research directions were different from mine. I will submit the arXiv paper to a journal, probably JMathPhys, fairly soon, where it will hopefully generate a worthwhile critique either of the whole or of parts, but I would prefer to iron out kinks in the presentation or in the mathematics before doing so (or, if someone is kind enough to make a killing remark about the mathematics, you could save the editors or referees of JMathPhys or of some other journal some trouble). As an aside, anyone who takes the time to read sections II and III of the arXiv paper will find calculations and an unconventional perspective on interacting quantum field theory that I would expect anyone who has thought for a while about foundational issues in QFT to find curious and perhaps useful."}
{"id":"21298","title":"Keeping air in a giant gravitationally-bound space balloon","text":"Let's say a space-faring society wants to make a space station that has a large volume filled with air (or other gas), but no gravity. Using normal pressure tanks will require gathering an amount of material proportional to the volume, which is a ratio set by the tensile strength and desired pressure. It's likely that some other method for containing gas would be more economic for anything over a certain volume. Could you confine a large volume of air in space using self-gravitation to hold the container together? Well obviously you can. Consider: a spherical volume of air surrounded by a solid sphere of matter through which the air cannot leak or diffuse and the pressure balances the gravity of the walls. Think of a big balloon ball in space, or alternatively, injecting the middle of the moon with air until it starts expanding. A hollow planet, if you will. There will be little gravity within the air because the walls don't contribute to the gravity inside and air has a low density. So here is what I'd like to ask:   * For a given pressure, what surface mass density ($kg\/m^2$) would you need? What would the wall thickness be?   * Going by the amount of material required, at what size would this approach be more economic?   * How stable would this thing be?   * Hypothetically, could you use the same principles to drape a airtight tarp over Mars and keep atmosphere from leaking out? It would be the surface mass density that determines the altitude at which it rests, right? Disclosure type statements: I can do a lot of the calcs for this myself, but I don't want to because I have doubts about certain parts and I want to avoid influencing other people with my potentially incorrect thought process. If you look at my Physics SE activity, you might notice that I'm fascinated by self-gravitation problems. I came up with this question reading Keeping air in a well."}
{"id":"20525","title":"What happens to angular momentum when matter is converted to energy?","text":"Let's say a spinning star radiates mass-energy only from it's pole regions. How does the loss of mass-energy effect the angular momentum of the star?"}
{"id":"45660","title":"Meaning of negative frequency of sound wave","text":"Suppose that Alice and Bob are both holding speakers emitting sound at a frequency $f$. Alice is stationary while Bob is moving towards Alice at twice the speed of sound. In the case of Alice, if I forget for a minute that she is holding a speaker herself, and just think about the Doppler effect from Bob moving towards her, I get $$f_{\\text{effective}}=f\\frac{1}{1-\\frac{2v_a}{v_a}}=-f$$ I do not know what \"negative\" frequencies mean in the context of sound waves, and so I do not really know how to describe what Alice will hear as Bob becomes very close to her. I also do not know how the fact that she is emitting sound at a frequency $f$ will affect, if at all, what she hears. And, from Bob's perspective, I get $f_{\\text{effective}}=3f$, which is a bit more meaningful, but does not factor in the fact that Bob is emitting sound himself. Any clarification is appreciated."}
{"id":"89383","title":"Does String Theory explain wave-particle duality?","text":"Does string theory explain the weird things that happens at the quantum level, especially wave-particle duality?"}
{"id":"16925","title":"Alternate layman's metaphors for illustrating curved space-time","text":"The metaphor of a surface (typically a pool table or a trampoline) distorted by a massive object is commonly used as a metaphor for illustrating gravitationally induced space-time curvature. But as has been pointed out here and elsewhere, this explanation seems (to a layman like me, at least), to be \"hopelessly circular\", and in the end contributes little to an understanding of how modern theories of gravitation work. Are there other (or additional) metaphors that might be helpful in illustrating to lay readers (a) what motivates modern gravitational theory and (b) why it has greater explanatory power than Newtonian gravitation?"}
{"id":"80374","title":"Why closed bottle change its weight when I put effervescence tablet?","text":"I use several shape, several type of material (glass, metal, plastic), I use two different balances with 0.01g of accuracy. I put oil on gasket, and put upside down (like that I can see if water escape). But always it's the same result, when I put one, two or more effervescence tablets, the weight decrease: one tablet => -0.03g two tablets => -0.06g three tablets => -0.09g Sure the difference is not big. But if I put an object without effervescence tablet, the weight is always the same (move sometimes +0.01 or -0.01g but never more). The time for decreasing is about 2 minutes so it's not enough for change something from temperature I think especially with glass container. So, maybe someone has done this experimentation before and know where is my error ? Or maybe someone can test and try this experimentation ? @Luboš Motl: \"Recall that the air density is about 1.3 grams per cubic centimeters\", you're sure ? it's not 0.0013 ? Yes, I have the same values for glass (2 mm of thickness and one of 3 mm of glass) with a metal cover of 0.8 mm of metal. I done about more than 100 measures. After 2 minutes, the tablet is full dissolved in water and weight move very few after (-0.01g next 2 minutes) but never more. Maybe I found: when bubbles move up in water, they have gas in it (CO2), this gas move up with a speed, so there is a quantity of movement, the weight losses is the sum of mass of bubbles multiply by speed. Like quantity of movement is conserved, the weight don't change. It's possible to use the formula of Newton: force=2mv if top speed is 0.25 m\/s the weight losses is 2*0.3\/1000*0.25\/10 = 0.015 g with 0.3 g of CO2, image show speed that I found on Internet. If it's that, the weight must change with a ball full of air (ping-pong ball) in water. But the CO2 in a tablet is very powerfull, 0.3g give 0.187 liter of CO2, even the ball move faster the volume is not great in a bottle. The ball must be down before close the container. And the gas must relax when the ball reached surface. Is it possible it is the rotation of Earth that change the pressure in water due to the centripetal forces ? This give -0.03N for each kg of water. Good day"}
{"id":"16922","title":"Desperately Need Help with Grade 9 Static Electricity","text":"I am preparing to teach Grade 9 Static Electricity next week and am going crazy trying to figure out what is happening in one of my experiments. I have a short piece of PVC pipe, 4 inches diameter, and I rub it with wool to charge it negatively. I can observe excellent repulsion when I touch it with my foil bit (dangling from a thread). Here is the problem: I am holding down the PVC pipe down on a wooden base with two brass-plated wood screws, and these screws somehow collect an INSANE amount of positive charge, even when I am very careful not to touch them with the wool. The foil bit is strongly attracted to the screws, and when it touches them it bounces off more violently than anything I've seen in any of my other static electricity experiments. Can anybody explain what I am seeing?"}
{"id":"80379","title":"Correct way to calculate torque produced by axle","text":"For my electrical engineering course, we had to build a simple DC motor that can lift a coin. I have tested the motor, and here are the results:   * rotational speed (no load): 3630 RPM (380 rad\/sec)   * Current with load (9g): 3A (limited)   * Current with no load: 0.7A   * Maximum weight lifted: 0.1422 N   * Max voltage 12V I now want to plot a torque-speed curve and calculate its efficiency. My problem is that the weight it lifted is a force quantity. To convert it to a torque quantity, I multiply by the axle radius (6mm) which gives 0.00085 Nm. This seems wrong to me becausekjg when I calculate the efficiency $\\epsilon = \\frac{\\omega \\tau}{VI}$ I get $\\epsilon = 0.9 \\% $ which seems extremely low. Is this the correct way to calculate the torque produced by an axle when lifting a load?"}
{"id":"12237","title":"What is the fundamental reason of the fermion doubling?","text":"Recall that the fermion doubling is the problem in taking the $a \\to 0$ limit of a naively discretized fermionic theory (defined on a lattice with lattice spacing $a$). After such a limit one finds themselves with an additional amount (precisely $2^d$) of fermionic fields. One can fix this by considering different discretizations of the action that make unwanted fields decouple in the continuum limit. The downside is that the additional terms have to spoil some nice features of the theory (chiral symmetry, locality, lattice symmetry, ...). Now, I wonder what is the true reason for the appearance of new fields. Is it the fermionic nature of the theory? (In other words, is a similar problem ruled out for bosonic fields?) And do all (naive?) fermionic theories (that is, independent of the continuum form of the action) suffer from this problem? More generally, how can one tell _a priori_ what will the field content of a lattice theory in the continuum limit be? Or is the field content fundamentally a continuum limit property that has to be calculated?"}
{"id":"107393","title":"Nature of Microscopic space-time","text":"I am going through the introductory chapter's of Schwinger's Source theory. He writes, > It [Source Theory] is a phenomenological theory, designed to describe the > observed particles. No speculations about the inner structure of the > particles are introduced. No abstract definition of particle is devised. The > theory is thereby firmly grounded in space-time, where the experimenter > manipulates his tools, **but the question of ultimate limitation to > microscopic space-time description is left open, with the decision reserved > to experiments. Correspondingly, no Operator-fields are used.** Now in this regard, I would want to know how operator fields answer the question of ultimate limitations to microscopic space-time (If they are related to each other)? **EDIT 1 :** It just struck me that the limitation could be due to canonical commutation between field operators and their conjugates. However, I don't see how to formalize a restriction using this commutation."}
{"id":"81438","title":"does the current's direction on a loop with a magnetic field affect the torque?","text":"I have a question in my book ![http:\/\/i.imgur.com\/P5R55sR.png](http:\/\/i.stack.imgur.com\/7yZhT.png) The question is very easy, but then my solutions manual gave me unexpected answers ![http:\/\/i.imgur.com\/tSMmJEY.png](http:\/\/i.stack.imgur.com\/cam6h.png) I don't get how in d) they conclude the vectors go in the opposite direction, but in b) they go in the same direction. The only thing I could guess is that there's some assumed current direction but I can't find anything in my book about that. Thanks for any help"}
{"id":"29365","title":"If the temperature of 2 materials are the same does that mean the molecules are vibrating at the same speed?","text":"Pretty much what the title says. My base question is this. Assuming I take a piece of steel, and a piece of PVC plastic and I measure both their temperatures and find they are the same. I then take a look at the vibration speeds of the individual molecules would they be the same as well? Here's a rough example: I measure both the steel and PVC and find them both at 100F, and then I measure the vibrations of a molecule in the steel and find it to be moving at 10 miles an hour. Would the PVC molecules also be moving at 10 miles an hour? I'm sure I'm not using the correct units of measure to measure the vibration, but I didn't know what the correct unit of measure is for something like that. Hopefully it gets the point across."}
{"id":"29367","title":"Why is there a phase factor when the two composite angular momentum is exchanged in Clebsch–Gordan coefficients","text":"An identity exists for CG coefficients: $$\\langle j_1 m_1 j_2 m_2 |J M \\rangle = (-1)^{j_1+j_2-J} \\langle j_2 m_2 j_1 m_1|J M\\rangle,$$ But why is there a phase factor $(-1)^{j_1+j_2-J}$? It seems to me that $$|JM\\rangle =\\sum_{m_{1},m_{2}}|j_{1}m_{1}\\rangle\\otimes|j_{2}m_{2}\\rangle\\langle j_{1}m_{1}j_{2}m_{2}|JM\\rangle =\\sum_{m_{1},m_{2}}|j_{2}m_{1}\\rangle\\otimes|j_{1}m_{2}\\rangle\\langle j_{2}m_{2}j_{1}m_{1}|JM\\rangle $$ And since $|j_{1}m_{1}\\rangle\\otimes|j_{2}m_{2}\\rangle$ and $|j_{2}m_{1}\\rangle\\otimes|j_{1}m_{2}\\rangle$ are the same physical state, there should be no difference between $\\langle j_1 m_1 j_2 m_2 |J M \\rangle$ and $\\langle j_2 m_2 j_1 m_1|J M\\rangle$. What do I get wrong?"}
{"id":"6047","title":"Physics for mathematicians","text":"How and from where does a mathematician learn physics from a mathematical stand point? I am reading the book by Spivak Elementary Mechanics from a mathematicians view point. The first couple of pages of Lecture 1 of the book summarizes what I intend by physics from a mathematical stand point. I wanted to find out what are the other good sources for other branches of physics. Thanks"}
{"id":"91326","title":"Prereqs for The Geometry of Physics by Frankel","text":"I'm interested in giving The Geometry of Physics a read, and I was wondering what the mathematical and (more importantly) physical prerequisites are. My background is a bit stronger on the mathematical side, and I'm worried that if heavy (or even moderate) exposure to physics is required that I'll miss out on a lot. If someone has read it and has a good sense of this sort of thing, I'd very much appreciate the input! A link to the book."}
{"id":"78442","title":"What is a phonon?","text":"I am trying to understand intuitively what a phonon is, but for the moment I find it quite difficult (having a limited background in quantum mechanics, an undergraduate course in non-relativistic QM). In fact, I find it hard to formulate good questions, so I hope my questions below make some sense. I read that phonons are (the quantum mechanical analog of) normal modes of vibration in a crystalline system of atoms or molecules, so I guess a superposition, i.e. a general vibration should also be a phonon. Is that so? Why would they then be described as normal modes? Could we say that a phonon is a particle whose position wave function extends over the whole crystal? Are the quantum mechanical frequency and wave vector the same as the frequency and wave vector of the corresponding classical oscillation (vibration in the crystal)? In what sense is it (like) a particle? In that it is always observed or it always interacts at a specific location?"}
{"id":"21434","title":"What is \"charge discreteness\"?","text":"I assume it is some kind of quantity. Google only made things more confusing. I get that it has something to do with circuits. I also get what a discrete charge is. In fact, I thought charges were, by definition, discrete - because each individual proton\/electron\/hole contributed one unit of charge."}
{"id":"49780","title":"Speed astronauts measure moving at the speed of light","text":"Lets suppose a spaceship travels with v = 0.9c relative to the Earth. The time inside the spaceship would pass slower than on Earth. Would the astronauts measure a different speed (that means, a different one that the observer on Earth does) in relation to the same reference frame (Earth)?"}
{"id":"63204","title":"Is the speed of light related to the mass of the universe?","text":"If the mass of the universe were cut in half, would it affect the speed of light? Would it be twice as fast? Would it stay the same? Do we have instruments that are sensitive enough to measure the speed of light at different positions relative to high-mass objects to empirically answer this question? The speed of light is (something of) a universal constant, but is it really dependent on the universe or on something intrinsic to photons? **EDIT:** Related question: Since gravity is a relationship between one atom and every other atom in the entire universe, and it takes all the energy in the universe to travel at the speed of light, is there something about the energy\/gravity\/mass of the universe that \"slows\" light from going a faster speed?"}
{"id":"49789","title":"Examples of systems with energy as an intensive variable","text":"I need to consider a couple of examples of systems which have energies that are intensive variables - not extensive. I'be been thinking about this and I am not coming up with anything. My understanding is that extensive variables (at least wrt usual energies) scales with mass or length (system size). It also seems that some 'energies' depend upon the model used, such as how strong the interactions are in neighbors of atoms or dipoles, etc., or whether one is considering chemical potential or not, etc. Any good suggestions?"}
{"id":"67587","title":"Electrical properties of molten gallium arsenide","text":"I'm looking for the resistivity and magnetic permeability of molten gallium arsenide, but can only seem to find the values for the solid material at room temperature (e.g., Wikpedia). Not even temperature-dependency of these parameters seems to be available. Is this because the value just doesn't change with temperature? If it does, what's a good resource for such parameters?"}
{"id":"115026","title":"Is quantum uncertainty principle related to thermodynamics?","text":"Would like to ask a question, but first i would like to say **Hello Everybody** in a way that plays the system, since some geniouses decided that one should not be able to say hello in a question. The uncertainty principle in quantum mechanics is well known and considered one of most basic properties of natural reality. The 2nd Law of thermodynamics is also well known and also considered one of the most basic processes of natural reality. The uncertainty principle uses and is related to Planck's constant. Planck's constant has the dimensions of _action_ and in a statistical mechanics approach, also relates nicely with the partitioning of the phase-space providing the basic measure for the entropy functional (this answer provides a nice outline of this). Apart from that, there are relatively recent papers which relate the Heisenberg Uncertainty Principle in quantum mechanics directly and intuitively to the 2nd Law of Thermodynamics. Is this relation correct? And if so can we derive one from the other? Thank you PS. One can also check this question, which although not the same, is related in an interesting way. UPDATE: anna's answer is accepted since by mentioning the derivation of (part of) the 2nd law from unitary dynamics, answers the question at least in one way. Please consider this as still open so you can add another answer. There are more alternatives (and one of which is my stance, ie thermodynamics -> uncertainty)"}
{"id":"29582","title":"What kind of light is needed to light up Venus?","text":"Let's say I would want to light up Venus, such that we can see Venus all day long and not have to wait for a Venus Transit. What kind of light would I need for it? How powerful would it need to be?"}
{"id":"24526","title":"What is a fluid flux?","text":"I need a concise definition of a fluid flux and an accompanying example. I've never taken a single physics course before, but I'm required to understand this concept so I can do the calculations for a Complex Analysis class."}
{"id":"24520","title":"Neutrino Oscillation and their gravitational implications","text":"As I understand neutrinos, there are three different flavors, all with different masses. Although the masses of these neutrinos have not been directly measured, their mass differences have been. Current experiments, KATRIN and Project8 are going to measure neutrino masses and we shall know soon enough. Regardless, their mass states change as they travel through space. This leads to my question... Since an object's gravitational field is related to its mass and neutrinos have different mass states while they are traveling, it must mean that every point in space must be constantly altering in gravity intensity! Although every object alters the gravitational field intensity as it travels and passes a given point, neutrinos would do it differently because they keep changing mass states! Let's assume a constant stream of neutrinos pass by a point in space versus neutrinos that don't oscillate (This is hypothetical) doing the same thing. Wouldn't these extremely weak gravitational waves be different given oscillations than not?"}
{"id":"66187","title":"G(2) lattice and the M-theory landscape","text":"In a previous question (Calabi-Yau manifolds and compactification of extra dimensions in M-theory), I was told that the $G(2)$ lattice can be used to compactify the extra 7 dimensions of M-theory and preserve exactly $\\mathcal N=1$ supersymmetry. However, since there is only 1 $G(2)$ lattice, there should be only 1 4-dimensional M-theory. Then, why is there such a huge fuss about the M-theory landscape? Thanks!"}
{"id":"72122","title":"How does one show using QED that same\/opposite electric charges repel\/attract each other, respectively?","text":"Why do same charges repel each other and opposite charges attract each other (please explain the phenomenon using real laws of nature (QED) not with the approximation model)?"}
{"id":"88522","title":"Are all elementary particles of the same type exactly the same?","text":"Are all elementary particles of the same type EXACTLY the same? Is there some variation in what an electron is, for example, or are they all the same?"}
{"id":"55112","title":"Can low temperature plasma exist?","text":"Plasma is ionized gas which as far as I know only occurs at high temperatures. When plasma cools down it tends to recombine with the electrons present and turn back into gas. But what if the disassociated electrons in the plasma were removed and the plasma were allowed to cool down in a vacuum, while being held in place by a strong magnetic field. Would this substance still be plasma? Is this possible?"}
{"id":"111230","title":"Help calculating field of view for ball lens I just bought","text":"![Diagram](http:\/\/imgur.com\/KwER0d8) I have a 5 mm (diameter) solid ball lens that I intend to place in front of a 36mm X 24mm camera sensor (I have already made preparations for the image to be focused and for the sensor to not be flooded with light). I want to calculate the resulting horizontal and vertical field of view from the ball lens when I take a picture. Please see the link for a diagram of the setup if the image does not work. ![http:\/\/imgur.com\/KwER0d8](http:\/\/i.imgur.com\/KwER0d8.jpg) I have already calculated the back focal length of the ball lens to be 1.479 mm."}
{"id":"22271","title":"Which bonds are the cross-links and which are secondary bonds (in elastomers)?","text":"Elastomeres are \"defined\" as: \"linear-chain polymers with widely spaced cross-links attaching each molecule to its neighbours\" Now I found sentences (talking about glass transition): \"This means that at room temperature the secondary bonds are melted and the molecules can slide relative to each other with ease. Were it not for the cross-links, the material would be a viscous liquid, but the cross-links give it a degree of mechanical stability.\" So, what exactly are these bonds that melt? and which ones do not? An example would be nice. Chemistry is not my strongest field so be rigorous. The quotations are from cellular solids by gibson and ashby."}
{"id":"74009","title":"Has the Double-Slit Quantum Eraser Experiment ever been tried on a large scale?","text":"I was just reading about quantum entanglement and the example was the Double- Slit Quantum Eraser Experiment. Then this was used as a basis for saying that particles might be half a universe apart and still be just as connected. So I was just wondering if entanglement experiments have been done to show that this is really the case or if it is considered \"academic\" to conclude that there's no difference when the distances are huge? As a side query, quantum mechanical interference seems to be based on the existence of probability - does this mean probability should be thought of as something as tangible as matter\/energy?"}
{"id":"74001","title":"Frequency of earthquakes","text":"Are earthquakes getting less frequent across the centuries? I know that more seismic stations have register more earthquakes in the last century, but that doesn't imply there were more. I am interested on the geophysical side of it. The logic is that things settle down, so there is less stuff to shuttle. Add to it that the earth (at its core) get colder with the passage of time."}
{"id":"98672","title":"Work done by isothermal expansion from two different viewpoints","text":"Consider an adiabatic system as follows. It consists of a gas in a container and a piston. Initially, the system is at equilibrium and the gas inside it occupies a volume $V_i$ at a pressure $p_i$ which is equal to the outside pressure. Suddenly, the outside pressure changes and reduces to $p_{atm}$. The piston moves to equalize the pressure and the gas expands isothermally to obtain equilibrium. The gas now occupies a volume $V_f$ at a pressure of $p_f$ which is equal to $p_{atm}$ Now, two textbooks I have define the work done from two different viewpoints. **1: From the viewpoints of the surroundings** : The work done on the system by the surroundings equals $-p_{atm}\\Delta V$. Since $p_{atm}$ is pretty much constant for the whole of the whole of the process, we can say that the work done equals: $$W_1 = -p_{atm}(V_f - V_i)\\tag1$$ **2: From the viewpoint of the system:** We can write the internal pressure of the system as a function of its volume: $p_{in}(V)$. As during the expansion, the internal pressure changes, the work done by the system equals $$W_2 = \\int_{V_i}^{V_f}p_{in}(V)dV$$ Now, I don't know which definition to use. The work is done by the system (from definition 2) is done on the surroundings. But what about the negative work that the surroundings did on the system? Where did that energy go? Maybe, the two definitions express the same thing: the work done by the system. The negative sign in definition 1 signifies that the work is done by the system. But that would mean that $$W_1 = W_2$$ We can simplify $W_2$ as follows. Clearly, $$p_{in}(V) = \\frac{p_iV_i}{V}$$ $$W_2 = \\int_{V_i}^{V_f}\\frac{p_iV_i}{V}dV = p_iV_i \\ln{\\frac{V_f}{V_i}} \\tag2$$ The internal pressure when the volume is equal to $V_f$ is $p_{atm}$ $$\\implies p_{atm} = p_{in}(V_f) = \\frac{p_iV_i}{V_f}$$ $$\\implies V_f = \\frac{p_iV_i}{p_{atm}}$$ Putting this into $(1), (2)$ gives us, $$W_2 = p_iV_i\\ln{\\frac{p_{i}}{p_{atm}}}$$ $$W_1 = -p_{atm}\\left(\\frac{p_iV_i}{p_{atm}} - V_i\\right) = -V_i(p_{atm} - p_i)$$ Consider some values. Let $p_i = 5 \\ \\rm{Pa}, V_i = 1 \\ \\rm{m^3}, p_{atm} = 1 \\ \\rm{Pa}$. $$W_1 = -1\\cdot(1-5) = 4$$ $$W_2 = 1\\cdot1\\cdot\\ln{\\frac{5}{1}} = 1.609$$ Where am I making a mistake?"}
{"id":"98671","title":"Does there exist a single plate capacitor(conductor)?","text":"Does there exist a single plate capacitor(conductor)? _if yes_ How will you define the **capacitance** and potential(difference) of such conductor?"}
{"id":"123426","title":"Probability and the propagator","text":"Due to the Wiki article, \"...In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum...\". Let's have the expression for the propagator of some field $\\hat {\\Psi}_{l}$ in the free theory: $$ \\tag 1 D_{lm}(x - y) = \\langle |\\hat{T}\\left( \\hat {\\varphi}_{l}(x)\\hat {\\varphi}_{m}^{\\dagger}(y)\\right) |\\rangle = -\\frac{i}{(2 \\pi )^{4}}\\int \\frac{F_{lm}(p)e^{-ip(x - y)}d^{4}p}{p^{2} - m^{2} - i\\varepsilon}. $$ Let's use two examples: spinor and scalar fields: for them $(1)$ takes the form $$ \\tag 2 D_{lm}(x - y) = D(x - y) = -\\frac{i}{(2 \\pi )^{4}}\\int \\frac{e^{-ip(x - y)}d^{4}p}{p^{2} - m^{2} - i\\varepsilon}, $$ $$ \\tag 3 D_{lm}(x - y) = -\\frac{i}{(2 \\pi )^{4}}\\int \\frac{(\\gamma^{\\mu}p_{\\mu} + m)_{lm}e^{-ip(x - y)}d^{4}p}{p^{2} - m^{2} - i\\varepsilon} $$ respectively. How to convert $(2), (3)$ into the probability (which lies at interval $[0, 1]$)? Particularly i don't understand what to do with spinor indices in $(3)$ (the probability must be Lorentz scalar, so I need to sum over the indices (?)). And also, why doesn't probability depends on momentum (the propagator doesn't contain info about momentum)?"}
{"id":"98674","title":"Why is the potential energy of a particle $mgh$ regardless of force?","text":"Consider a particle raised to height $h$ by applying a constant force of $F$ vertically opposite to the direction of force of gravity on it. The potential energy of the particle is $$W = mgh$$ But, according to the definition of work, the work done on the particle is $$W' = Fh$$ According to conservation of energy, $W = W'$. But what if $F>mg$?"}
{"id":"99988","title":"Dead time in data acquisition","text":"I am creating a data acquisition software, based on Sparrow's Kmax. There I would like to add a feature that will show the system's dead time. I have already a text field that shows the real time and live. My question has to do, with the dead time. Is it right to say that the dead time is $\\text{dead time}=\\dfrac{\\text{total time of recording events}}{\\text{total time of acquisition}}=\\dfrac{\\text{live time}}{\\text{real time}}$"}
{"id":"104493","title":"Quantum Anomalies for Bosons","text":"We know that there is Adler and Bell-Jackiw(ABJ) type anomalies for fermions. In some case, the ABJ anomaly affecs particle physics pheonomelogy, such as pion decays or kaon decays(in the case of pion, we still have a calculation on left\/right chiral fermions running on the 1-loop triangle diagram). In some other case, there is 1+1D QED Schiwinger or axial anomaly for chiral fermions. The commutation of fermionic anomaly is usually done by, either a 1-loop Feynman diagram, or a Fujikawa path integral method. The above may be some examples of anomalies for fermions. **Is there any example of quantum anomalies for bosons (pure bosonic systems)?**"}
{"id":"104498","title":"Electric Field Change Freezing Temperature of Water?","text":"I assume that the reason water freezes is because as you decrease the temperature, the kinetic energy of the water molecules decreases and the dipole bonding potential eventually over comes the escape velocity of the molecules and they form a crystal structure. What would happen if you attempted to freeze water in the presence of a very powerful electric field? Will the water molecules align to the field and make it harder to form a solid or easier to form a solid?"}
{"id":"32008","title":"Could quarks and leptons mix if they carried flavor charges?","text":"If quarks and leptons carried flavor charges that differed across generations (as they do in some theories), then could mixing take place?"}
{"id":"55809","title":"Calculating Average Velocity","text":"I understand that the concept of an average of a data list means finding a certain value 'x', which ensures that the sum of the deviations of the numbers on the left of 'x' and on the right of 'x' yields the value 0. Assume we are given the values 2,6,9 and 12. I label these values from left to right in the sequence I have provided as: a,b,c and d respectively. The definition of finding these values' average (let's call this average value 'x') in mathematical form is:      $${(x - a) + (x - b) = -(x - c) - (x - d)}$$      So this equation can be reduced to:      $$\\Large x = \\frac{(a + b + c + d)}{4}$$      This will provide you with the value for the average of any number of values. Similarly there is a formula in physics that calculates the average velocity from a graph by dividing the sum of the initial and final velocities by 2.      $$\\Large\\frac{v_f + v_i}{2} = \\bar{v}$$      There is also another formula which calculates the average velocity from a graph by calculating the slope of the line through two points on the graph. This is done by dividing the change in position by the change in time:      $$\\Large \\frac{x_f - x_i}{t_f - t_i} = \\bar{v}$$      I can understand the third equation from the top of the page, as I can understand this calculation of average velocity in terms of what I explained in the first two equations from the top of the page. I am unable to reconcile the calculation of the average of velocities as seen in the last equation with the method as listed above in the first two equations. Can someone help me solve this dilemma?"}
{"id":"64874","title":"Time evolution of Gaussian wave packet","text":"I'm slightly confused as to answer this question, someone please help: Consider a free particle in one dimension, described by the initial wave function $$\\psi(x,0) = e^{ip_{0}x\/\\hbar}e^{-x^{2}\/2\\Delta^{2}}(\\pi\\Delta^2)^{-1\/4}.$$ Find the time-evolved wavefunctions $\\psi(x,t)$. Now I know that since it is a free particle we have the hamiltonian operator as $$H = -\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x^2},$$ which yields the energy eigenfunctions to be of the form $$\\psi_E(x,t) = C_1e^{ikx}+C_2e^{-ikx},$$ where $k=\\frac{\\sqrt{2mE}}{\\hbar}$, and the time evolution of the Schrödinger equation gives $$\\psi(x,t)=e^{-\\frac{i}{\\hbar}Ht}\\psi(x,0)$$ but the issue I face is what is the correct method to find the solution so that I can then calculate things such as the probability density $P(x,t)$ and the mean and the uncertainty (all which is straight forward once I know $\\psi(x,t)$. In short - how do I find the initial state in terms of the energy eigenfunctions $\\psi_E(x,t)$ so that I can find the time evolved state wavefunction."}
{"id":"32003","title":"Does the Lorentz transformation not apply to light?","text":"Since you would know that light always travels at the constant velocity with respect to all frame of reference ....according to relativity whenever we are traveling at speed of light our time with respect to relative rest observer would become stopped ..if. it means light travels with respect to all frame of reference at light speed, so it implies that light from the sun would never reach us ..but sadly it would reach us within 8 minutes ..How is that possible?"}
{"id":"98583","title":"Is the canonical momentum conserved when a particle moves in magnetic field?","text":"Here is a question about the canonical momentum that I had asked some days ago, but I still have one point that I am not understand. Considering a particle moves in a magnetic field with charge $q$ and mass $m$, its hamiltonian is $$H=\\frac{\\vec{P}^2}{2m}=\\frac{(\\vec{p}+q\\vec{A})^2}{2m}$$ where $\\vec{p}$ is the momentum of the particle, $\\vec{A}$ is the vector potential of the magnetic field and $\\vec{P}$ is the canonical momentum of the particle. I think, because of the expression of the hamiltonian, the canonical momentum $\\vec{P}$ is a conserved quantity. But by the answer in the previous link, it seems that the canonical momentum is not conserved even in a simple example that a particle moves in a homogeneous magnetic field. I am confused about this question. Is the canonical momentum conserved when a particle moves in magnetic field?"}
{"id":"128588","title":"Are O-planes not dynamical?","text":"In Polchinski's book \"String theory\", he is saying \"Unlike the case of D-branes, there are no string modes tied to the orientifold plane to represent fluctuations in its shape. Our heuristic argument that a gravitational wave forces a D-brane to oscillate does not apply to the orientifold plane. Essentially, the identifications (8.8.3) become boundary conditions at the fixed plane, such that the incident and reflected waves cancel. For the D-brane, the reflected wave is higher order in the string coupling.\" I want know how to check \"For O-plane,the incident and reflected waves cancel. For the D-brane, the reflected wave is higher order in the string coupling.\" Is there any papers or books or Web sites calculating it?"}
{"id":"57406","title":"How does a Fresnel rhomb work (half and quarter wave plate)?","text":"How does a Fresnel rhomb work (half and quarter wave plate)? I am aware of birefringence, which creates a phase shift of $\\Delta\\phi=\\dfrac{2\\pi\\Delta nL}{\\lambda_0}$. But this doesn't explain how a plane polarised light shifts it's polarization angle after a half-waveplate or how a linearly polarised light becomes elliptical after a quarter-waveplate. For the half one let's assume that wikipedia has an answer http:\/\/en.wikipedia.org\/wiki\/Waveplate but why this electric field $E(\\hat{f}+i\\hat{s})e^{i(kz-\\omega t})$ describes an elliptical polarised light? What I need is a mathematical description of how the retarders work..."}
{"id":"24892","title":"How do you clean a dusty lens?","text":"The lenses of my telescope and binoculars are dusty. What is the best way to clean them without damaging the optic coating?"}
{"id":"3698","title":"Flat space limit of the Schwarzschild metric and Hawking temperature","text":"The Schwarzschild metric reduces to the Minkowski metric in the limit of vanishing $M$, but the Hawking temperature which is proportional to $1\/M$ diverges in the same limit. This would imply that flat spacetime has infinite rather than zero temperature. What am I missing? EDIT: This question is back up on the front page because I've posted an answer of my own \\- let me know if I'm thinking about this right."}
{"id":"64933","title":"Interpreting the results","text":"I have preformed the muon lifetime experiment at my uni's lab, and got the data. It's text file with 8190 numbers. My TDC unit was set so that the time gates were at 10 $\\mu s$, and it has 8192 channels (first and last contained some kind of noise so they were not included in data analysis). Now, I made a program in python that will sum every n data, and make a new list that I can then plot in bar plots and from that calculate the mean muon lifetime, because it decays by exponential law. I also found that I can estimate the muon lifetime by using this formula: $$<t>=\\frac{\\sum\\limits_{i=n_l}^{n_u}N_i t_i}{\\sum\\limits_{i=n_l}^{n_u}N_i},$$ where $N_i$ is the number of counts in the bin, $t_i$ is the 'time bin', and the bins go from $n_l$ (lower) to the $n_u$ (upper) bin. Using that as a guide, I found that for my data I get the best estimate if I set the bin count at 7 (7 bins), and if I put 10, I get higher result. Now if I set the bin number to 20, I get even higher number. But that sounds wrong. I mean, there must be some kind of catch. If I divide the time I get for 20 bins by two I get the correct result (~2.5 $\\mu s$). The pictures are like this: ![enter image description here](http:\/\/i.stack.imgur.com\/hZ1Z4.png) ![enter image description here](http:\/\/i.stack.imgur.com\/QgNQJ.png) ![enter image description here](http:\/\/i.stack.imgur.com\/eS8a0.png) So what am I doing wrong with interpreting this?"}
{"id":"26037","title":"Is building your own reflector telescope worth it?","text":"I have always fancied building my own reflector telescope. I am wondering - For a certain budget, can you get better results with a home made rig? Or is this a folly, and really it would be better to buy a ready made unit?"}
{"id":"37788","title":"What is the fundamental reason for noise?","text":"I have read that noise is a result of there being \"no such thing as a perfect one-way valve\". That energy transfer is never perfectly one-way; there will always (at some level) a finite flow of energy from output to input. I understand this much, but what is the fundamental reason for this? Does it have its roots in thermodynamics? Edit: The idea of a \"one-way valve\" is merely a construction in the mind, it doesn't refer to an actual valve, like in a car engine or something. It's just a general way of describing 'what connects two systems'. Additionally, although I am looking for the _fundamental_ reason for noise, it may be easier to let people know that I am referring specifically to _measurement noise_."}
{"id":"3527","title":"Newton's cradle","text":"Why, when one releases 2 balls in Newton's cradle, two balls on the opposite side bounce out at approximately the same speed as the 1st pair, rather than one ball at higher speed, or 3 balls at lower speed?"}
{"id":"131462","title":"Information of things inside a black hole","text":"**Can we get the information of things that are gone in black hole?**"}
{"id":"37784","title":"Boundary conditions in AdS\/CFT","text":"This question is in reference to this very famous paper of Witten.   * In general through the whole paper why is the author able to just focus on the scalar field propagating in the bulk and not need to take into account all the other fields and the complicated Lagrangian in the bulk (a Type IIB superstrings?)    * To construct the example in equation 4.1 (middle of page 6) why did the author choose half-BPS operators and is there a simple way to see that an example of ${\\cal O}$ written down is a half-BPS operator? (..what are other such?..is there a classification?..)    * How generic is the argument in equation 4.8 (top of page 8) to get the RG flow equation? Or is this a special case which works here for some special reason?  With a change in the mass\/renormalization scale\/cut-off one usually asks for the connected n-point functions or the effective potential to be invariant - but here the author seems to want to have the scalar field's boundary asymptotics to be invariant - I found this renormalization condition very new and mysterious.   * I guess the most exciting analysis in this paper is the argument in the first paragraph on the top of page 9. Can someone help understand that?      * To start off how one know that the operators ${\\cal O}_1$ and ${\\cal O}_2'$ related to the boundary values of the two scalar fields are actually (super?)conformal primaries of the boundary (S?)CFT?      * I did not understand how one sees that the deformation as stated in equation 4.12 (and the line before it) preserves quantum conformal invariance.      * and the main point about the structure of equation 4.12 and the conformal invariance of the boundary being maintainable for $f \\neq 0$.."}
{"id":"37785","title":"Bicycle Wheel Drag in Slipstream","text":"I was recently driving behind a car that had a bicycle mounted on a carrier over the rear bumper. The bicycle wheels were not bound so they were rotating in the slip-stream of the car. I wonder, the fact that the wheels are turning; does this theoretically increase or decrease the drag on the car?"}
{"id":"75000","title":"Motivation behind studying the asymptotic structures","text":"I am trying to explain to myself the motivation behind studying the asymptotic structures at null, time-like and space-like infinities (For the purposes of this post, I will stick to four dimensional Minkowski spacetime). I think I have the motivation behind the first two of them down, but not the third.   1. Null Infinity is interesting since radiation and fluxes of massless particles can reach it. One must understand these in order to answer questions such as energy loss due to gravitational radiation, or charge fluxes. Further, information can travel from the bulk to null infinity only, and is therefore an important thing to study in that context as well. Geroch's paper notes that > At null infinity, one's description involves \"what the system is doing > through time\", i.e. the dynamics of the system. I agree with this.   1. Time-like infinity corresponds to the starting and ending points of massive particles travelling along geodesics. I would presume that these would be of importance when describing \"In\" and \"Out\" states of a quantum theory.   2. Space-like infinity is a little harder for me to motivate. In Geroch's paper, he notes > At spatial infinity, \"what the system does\" is never even recorded. Rather, > one sees only the state of the system\", once and for all. Here are my questions: >   1. Are the motivations I provided above for null and timelike infinity > correct? Is there anything more to add? > >   2. What does Geroch's comment mean? Surely, in general, a system will > evolve in time. What then describes the \"state of a system\"?? Is it the > initial conditions? What, precisely is the information about the system that > is stored at space-like infinity? > > I have often seen null and spacelike infinities being discussed thoroughly without much literature on time-like infinity. While the extensive study of null infinities does not surprise me, the study of spacelike infinity does. I therefore feel like I am missing something quite important here. EDIT: I just had a thought. It seems to me that both null and spatial infinity can be reached in a finite value of coordinate time (appropriately defined). On the other hand, time-like infinity can only be reached asymptotically. This implies that study of time-like infinity requires taking two limits $r \\to \\infty$ and $t \\to \\pm \\infty$, whereas, study of null and spacelike infinity involve single limits like $r \\to \\infty,~v$ fixed and $r \\to \\infty,~t$ fixed respectively. Maybe this has something to do with my answer, but I'm not quite sure how."}
{"id":"3543","title":"Understanding boundary conditions on slices of AdS5","text":"This is a thing Iïve seen on many papers dealing with Warped Extra Dimensions, specifically on slices of AdS5. But the one where it appears more clearly is a lecture by Tony Gherghetta: http:\/\/arxiv.org\/abs\/hep-ph\/0601213 Essentially what is done is that one builds a 5-dimensional theory in a slice of a Anti-De-Sitter space, the slice meaning that the fifth dimension has a small size and ends with a 4D brane at each extreme. The principle of least action applied to the 5D action leads us to two terms (eq. 7 in the paper above), one on the bulk (the vanishing of this one leads us to the bulk equation of motion) and one on the branes. The vanishing of this second term can be accomplished in two ways: 1) the variation of the field on the branes is zero 2) the term multiplying the variation is zero on the branes (let?s call it B) Now, all authors affirm that, in absence of extra terms on the brane, this two conditions lead to Neumann or Dirichlet conditions for the field on the branes (in the paper above this is said just after equation 10). My question is: why is that? For scalar fields it can be shown that B equals the derivative of the field, so I can see the Neumann condition there (am I wrong?). But the 1st condition says the variation of the field is zero not the field itself. Clearly I am misunderstanding something here..."}
{"id":"120139","title":"Why is charge not taken as a fundamental unit?","text":"According to the definition of electric current, it appears to be a derived quantity. Charge on the other hand seems more fundamental than electric current. Then why is current taken as fundamental quantity instead of charge? Is it arbitrary choice? Is it because we can measure current more efficiently than charge or some other reason?"}
{"id":"70650","title":"Is electron volt an alternate unit for electric potential?","text":"My question is: Can an electron volt be considered an alternate unit for electric potential?"}
{"id":"14650","title":"Function of air conditioner","text":"What i always thought that air conditioners blow the cool air without knowing that they actually take the warm air from indoor and from outdoor. But whats the point of taking air from indoor and outdoor, Shouldn't they just be blowing the cool air inside and out air outside? How does it really make the difference? Also, Does all Air conditioners function the same way as in shown in this figure, even split A.Cs? ![enter link description here](http:\/\/i.stack.imgur.com\/iXrvX.png) I've run out of **tags** for questions :\\"}
{"id":"14652","title":"Fluid Mechanics from a variational principle","text":"It is posible to define a good variational principle to describe Fluid Mechanics? if so, wath is the correct tratement of the issue. I guess something like: $I=\\int d^4x (\\frac{1}{2}\\rho v^2-P-\\rho g x)$"}
{"id":"14657","title":"How and why will the Milky way collide with the Andromeda?","text":"Hubble's law says that the universe is expanding.How come the milky way and the andromeda are on a collision course?How will they end up colliding with each other?"}
{"id":"21557","title":"How does the Milkovic Two-Stage Mechanical Oscillator Pendulum-Lever System work?","text":"See http:\/\/peswiki.com\/index.php\/Directory:Milkovic_Two- Stage_Mechanical_Oscillator The Two-Stage Mechanical Oscillator Pendulum-Lever System is very simple, yet very puzzling because it appears that more energy is going out (since it is able to pump water) than is put in (only lift one lift of the pendulum and an occasional push). Where is the apparent excess energy coming from? The reason I am asking this question is because all of the explanations I have seen claim that somehow Newton's Laws are violated, which I cannot believe."}
{"id":"38632","title":"Microphones, Loudspeaker and their analogies to spring mass system","text":"I have just started studying Microphones and Loudspeakers. I need a good text to refer which can explain their mechanical analogies with simplicity and basics too."}
{"id":"30402","title":"Why do we think of light as a wave?","text":"I've read that light travels in a straight line and has a wavelength of 400nm to 700nm. But I don't understand why does it have a wavelength and what creates its wavelength? I agree with the concept of sound which also has wavelength, thus called sound waves which are created by the vibrational movement in air. I'm not aware of calling light a wave. Does the light vibrate too? If so, then how?"}
{"id":"134886","title":"Nuclear fission difference in energy calculation?","text":"If you have a uranium atom of radius $10^{-14}m$ that undergoes fission into two fragments each with 46 protons and radius $8\\times10^{-15}m$. There are two ways you can calculate the electrostatic energy realised (according to my textbook):   1. Find the electric potential energy of the two fragments as if they where point charges at a distance of $2\\times8\\times10^{-15}m$ apart (i.e. find the potential energy one of these charges would have in the field of the other). This gives an answer of around 180 MeV.   2. Find the electrostatic energy of the original nucleus by imagining the nucleus was built layer by layer each of thickness $dr$ and then integrating over it to find the energy of the whole nucleus. If this is then repeated for the two fragments and the sum of the electrostatic energies of the fragments is taken from that of the original nucleus that is the electrostatic energy released. But this gives an energy of around 270MeV. Here is my question, why is there a difference between these two values (the 180MeV and the 270MeV)?"}
{"id":"133354","title":"Basics of osmosis. What about excluded volume?","text":"I may not understand osmosis very well. Let us suppose two compartments filled with water, separated by a semi-permeable membrane. At equilibrium, both levels are equals. Let us introduce now a given volume of solute in one of the compartments (say right). At first, the level of the right compartment will increase, to accommodate the extra volume of solute. Because of this, the concentration of water in the right compartment has decreased, and is no more at equilibrium with the left compartment. Thus a net flux of water from left to right occurs until equilibrium is reached. That's what I understand from what I read so far, but I have a problem with this. In this particular example, it is true to say that the water concentration on the right side decreased when the solute is introduced, but because of the excluded volume (from the solute), the water pressure should remain the same, and thus I would not expect a net flux. Unless there is other effect I don't consider? Thank you for your explanation."}
{"id":"113321","title":"Does the speed of medium affect the path of light?","text":"Let's say I shine a laser from a stationary medium into a moving medium (suppose the water is moving very quickly) perpendicular to the interface and back to a stationary medium like this: ![Scenarios](http:\/\/i.stack.imgur.com\/SiL6P.png) (Note: left and right sides of the image are stationary mediums, center is a medium moving in the direction indicated by the arrow) Which of the above scenarios (A, B, C, or \"I'm way off\") correctly reflects the path the light will take (even if the translation is incredibly small)? **Edit:** To answer some good questions (and things I left out of the original question):   * The center (moving) medium is water   * The left and right medium (stationary) is air   * The first angle of incidence (on the left) between the air and the water is perpendicular"}
{"id":"106059","title":"When one monkey accelerates up a rope, what happens to the other?","text":"The question is: Two equal weight monkeys each hang from the ends of a rope passing over a weightless, frictionless pulley. If one accelerates up the rope, what happens to the other? I'm going to assume here that if one monkey climbs, the other monkey goes up (I tried to imagine how it would look in real life). But in physical terms, why would this be? Because of the additional force exerted by the climbing monkey?"}
{"id":"83461","title":"Good book on deriving approximate solutions from first principles?","text":"I have always been excited by examples in which a few simple assumptions and first principles are used to characterize a system. For example, I did an exercise in which Crawford estimates a lake to be rectangular and then predicts the frequency of its waves through the harmonic motion of its center of mass. I'd love to come up with questions and simple answers about natural phenomena, but I am not very good at coming up with either the questions or the answers, and I'd like to see lots of examples. What are some of the books you like on the subject?"}
{"id":"27811","title":"Will a precessing spinning wheel fall down if there is no friction at all?","text":"If there where no friction at all, would a spinning wheel held up by one end of the axis spin precess forever without falling down? ![Spinning wheel](http:\/\/i.stack.imgur.com\/S3f2n.gif) I just asked another question about the same problem: Direction of torque precession of a spinning wheel Since it seems to be a good practice on stackexchange not to ask several questions in one post, I splitted them up into two questions. However if I am wrong, feel free to merge this questions."}
{"id":"40778","title":"Time of a ball going up and down with air resistance","text":"A ball is shot directly upward, and then it comes back to the place where it was shot. Suppose we have air resistance. Suppose $t_1$ is the time period from the moment that the ball was shot to the moment that it reached its highest altitude, and $t_2$ is the time period from the moment it reached its highest altitude to the moment it reached its original position. Is $t_1=t_2$? Why or why not?"}
{"id":"40777","title":"Why is there no oxidizer in space?","text":"Just as a thought experiment. One factor in the economies of space exploration is that of fuel. This may be split as   * MSL - Earth Orbit    * Earth Orbit to Inner Solar System    * Beyond. In each of these cases, the rocket demands an oxidizer to be loaded at the point of launch as neither oxygen, nor oxidizers exist in free space. Why does space not have free oxygen\/oxidizers? p.s. According to wikipedia Oxygen is the third most abundant element in the known Universe ..."}
{"id":"2964","title":"What is a resonating valence bond (RVB) state?","text":"There's something known as a \"resonating valence bond\" (RVB) state, which plays a role in at least some attempts to understand physics of high-$T_c$ superconductors. This, roughly, involves a state that's in a superposition (hence the \"resonating\" part of the name, if I understand correctly) of different ways to pair electrons into strongly-bonded spin singlets. My question is: what is a more precise definition of this sort of state? What's the underlying physics, when does it arise, and why is it interesting? Points an answer might address: is there a simple toy model for which this is the ground state, that sheds light on what sort of system it could arise in? Is there an interesting continuum limit, in which we can characterize this state in a more field-theoretic language? Are there particular kinds of instabilities such a state tends to be subject to? I think I know where I would start digging if I wanted to really understand this for myself, but mostly I'm asking it to probe the community and see what kind of expertise might be lurking here, since there haven't been so many condensed matter questions."}
{"id":"103383","title":"A question to clarify the use of divergent series in calculating the casimir effect","text":"Some time ago I posted a question here on this forum. I would like to ask some questions regarding the way the energy per unit area between metallic plates is calculated. The full calculation is on wikipedia. At some point in the calculation on the relevant wikipedia page (see the link above), we have the equation: $$\\frac{ \\langle E \\rangle }{ A} = - \\frac{ \\hbar c \\pi^2 }{6a^3}\\cdot\\zeta(-3) . $$ In the next step, it is written rather casually that $\\zeta(-3) = - \\frac{1}{120} \\qquad (*) $. This is true when considering the analytic continuation of the riemann zeta function or the Ramanujan Summation method. Therefore, it is concluded, that $$\\frac{ \\langle E \\rangle }{A} = - \\frac{ \\hbar c \\pi^2}{720 a^3} . $$ I am wondering under which circumstances people decided to assume the $(*)$-marked equation is 'true'. I can think of a couple of scenarios:   1. The formula for $\\frac{ \\langle E \\rangle }{A} $ was already derived by means of another method which did not require the use of (regularised) divergent sums. Therefore, physicists could infer that $\\zeta(-3)$ _had_ to be equal to $ - \\frac{1}{120} $, making the derivation of the formula by means of this method, which does use divergent series, correct.    2. The exact formula for $\\frac{ \\langle E \\rangle }{A} $ was not already known. Physicist did have some data points that roughly showed them how the formula should look. Therefore, they tried some different constants for $\\zeta(-3)$. At some point they guessed $\\zeta(-3) = - \\frac{1}{120} $, which yielded a formula that coincided with the known data points. They might have already known that $\\zeta(-3) = - \\frac{1}{120} $ by means of zeta function regularisation, making it easier to use this equation as a \"guess\" to find a suitable formula for $\\frac{ \\langle E \\rangle }{A} $ .    3. Some other scenario.  Which scenario roughly describes how the formula for $\\frac{ \\langle E \\rangle }{A} $ came into existence? If it was scenario 1, which other method did physicists formerly employ to derive the formula? If it was scenario 3, how _did_ this whole process unfold? Thanks a lot, Max"}
{"id":"109981","title":"Work done or not in this case?","text":"I have a very simple question. A motorboat directed upstream is seen to be at rest from the bank of a river. Is the engine doing any work? Is it right to say that since it is not causing any displacement, so it is not doing any work."}
{"id":"39584","title":"The Ultimate Hand Dryer","text":"I have come across many hand dryers that attempt to dry your hands really fast after you wash them. Here are two of them: XLERATOR http:\/\/www.exceldryer.com\/ Dyson Airblade http:\/\/www.dysonairblade.com\/homepage.asp So I guess I have a ridiculously high standard cause I think even these are too slow. Would it be possible to create a large static electric field to attract the water molecules off of your hand? Can someone offer some ideas that would remove the water off of your hand using some type of electric field generated by charged plates or something?"}
{"id":"43051","title":"Dropping an anchor from a boat","text":"> A yacht on a lake drops its anchor overboard. What happens to the water > level in the lake? > >   1. It rises very slightly. >   2. It falls very slightly. >   3. It stays exactly the same. >   4. It's impossible to say. > My understanding is that due to Archimedes principle, when the anchor is in the boat, it contributes to the mass of the boat, and thus the mass of water displaced. When it is thrown overboard, it is now the volume of the anchor which contributes to the amount of water displaced. Therefore, without knowing the density, mass and volume of the anchor, it is impossible to determine the effect on the water level, therefore the correct answer is option 4. Can anyone tell me if my reasoning is correct as there is no mark scheme for the test where this question came from. Thanks in advance!"}
{"id":"78126","title":"What really is the smallest \"mass\" or \"object\" in the universe?","text":"Look at this here. With respect to the sciences, the atom is obviously not the smallest piece of mass. Apparently, if people have already broken down the atom in to particles smaller than so, why haven't particles been understood yet? Old scholars reasoned that everything has smaller parts, so what's smaller than subatomic particles? Or is there a limit in the size of mass, only being able to be small to an extent? Because mass always seems to keep growing, but when scaling opposite in size we reach a limit. Therefore, big always gets bigger, but why does small have limits?"}
{"id":"88552","title":"explicit matrix elements for a representation decomposed into subgroup by branching rules","text":"I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not sure how to derive the explicit representation from the weight system deformed by extended Dynkin diagram? To decompose a irreducible representation under a maximal regular subalgebra, we look at the extended Dynkin diagram, which has one more node from the minus highest root added on the original Dynkin diagram, and we then eliminate another node to deform the extended Dynkin diagram into the Dynkin diagram for the maximal regular subalgebra. The key point is to replace the Dynkin coefficient corresponding to the eliminated node by the one corresponding to the extra node of the minus highest root. Thus, deforming the weight system of the given irrep with above replacement gives us the branching rules and the Cartan matrices for the subalgebra in the given irrep. For example, under the subgroup $SU(2)\\times SU(2)\\times SU(2) \\Subset SO(7)$, we have the 8 dimensional irrep of SO(7) deformed as $8 \\rightarrow (1,2,2) + (2,1,2)$, see P33-34 LieART. To construct the explicit matrix representation, we define the coordinates for each simple roots in an orthonormal basis (Cartan-Weyl basis) and therefore the coordinates for the fundamental weights as well. Thus, the generators corresponding to Cartan matrices have its matrix entries as the coordinate of every weights. Now, my question is what are the fundamental weights for the deformed Dynkin diagram? Should I keep the old fundamental weights except the replaced one or the new fundamental weights are resolved from the deformed Dynkin diagram? Most importantly, what's the new fundamental weight corresponding to the extended root? I would also need help to reveal how generators of the subalgebra are in the subset of generators of the mother algebra. When I replace the Dynkin coefficient with the extended root, how could the corresponding Cartan matrix belong to linear combination of Cartan matrices of the mother algebra? I'm a Physics student with poor knowledge in Lie algebra. Please reinterpret my description with proper language whenever you needed to."}
{"id":"109673","title":"How we can manipulate the momentum of a particle?","text":"Is there any way to affect a particle's momentum value?"}
{"id":"133824","title":"Eigenvalue of the adiabatic Hamiltonian of Farhi's three qubit 2-SAT problem","text":"I was trying to reproduce example 3.3 of Quantum Computation by Adiabatic Evolution by Edward Farhi et. al. This is an adiabatic algorithm to solve an instance of three qubits 2-SAT problem. I think I have created the initial Hamiltonian, $H_B$ correctly. $$H_B = \\left(\\frac{1}{2}\\left(I_2 - \\sigma_x \\right)\\right)\\otimes I_2\\otimes I_2+I_2\\otimes \\left(\\frac{1}{2}\\left(I_2 - \\sigma_x \\right)\\right)\\otimes I_2+\\left(\\frac{1}{2}\\left(I_2 - \\sigma_x \\right)\\right)\\otimes I_2\\otimes I_2+I_2\\otimes I_2\\otimes \\left(\\frac{1}{2}\\left(I_2 - \\sigma_x \\right)\\right)+I_2\\otimes \\left(\\frac{1}{2}\\left(I_2 - \\sigma_x \\right)\\right)\\otimes I_2+I_2\\otimes I_2\\otimes \\left(\\frac{1}{2}\\left(I_2 - \\sigma_x \\right)\\right)$$ When I evaluate it, the matrx is: $$H_B=\\left( \\begin{array}{cccccccc} 3 & -1 & -1 & 0 & -1 & 0 & 0 & 0 \\\\\\ -1 & 3 & 0 & -1 & 0 & -1 & 0 & 0 \\\\\\ -1 & 0 & 3 & -1 & 0 & 0 & -1 & 0 \\\\\\ 0 & -1 & -1 & 3 & 0 & 0 & 0 & -1 \\\\\\ -1 & 0 & 0 & 0 & 3 & -1 & -1 & 0 \\\\\\ 0 & -1 & 0 & 0 & -1 & 3 & 0 & -1 \\\\\\ 0 & 0 & -1 & 0 & -1 & 0 & 3 & -1 \\\\\\ 0 & 0 & 0 & -1 & 0 & -1 & -1 & 3 \\\\\\ \\end{array} \\right)$$ According to the example, the unique satisfying assignment is $011$. The problem Hamiltonian is a sum of three sub-Hamiltonians each correspond to a clause. $$H_P = H^{12}_{imply} + H^{13}_{disagree}+H^{23}_{agree}$$ Here are my results for sub-Hamiltonians. $H^{12}_{imply}$ clause can be satisfied with any of $00$, $01$ or $11$. So, $$H^{12}_{imply} = I_8 - \\left(\\left(\\frac{1}{\\sqrt{6}}\\right)\\left(|000\\rangle+|001\\rangle+|010\\rangle+|011\\rangle+|110\\rangle+|111\\rangle\\right)\\right) \\left(\\left(\\left(\\frac{1}{\\sqrt{6}}\\right)\\left(|000\\rangle+|001\\rangle+|010\\rangle+|011\\rangle+|110\\rangle+|111\\rangle\\right)\\right)\\right)^{\\dagger} $$ The ground state of this Hamiltonian satisfies the assignment constraint. $H^{13}_{disagree}$ clause can be satisfied with any of $01$ or $10$. So, $$H^{13}_{disagree} = I_8 -\\left(\\left(\\frac{1}{2}\\right)\\left(|001\\rangle+|011\\rangle+|100\\rangle+|110\\rangle\\right)\\right) \\left(\\left(\\left(\\frac{1}{2}\\right)\\left(|001\\rangle+|011\\rangle+|100\\rangle+|110\\rangle\\right)\\right)\\right)^{\\dagger}$$ The ground state of this Hamiltonian satisfies the assignment constraint. $H^{23}_{agree}$ clause can be satisfied with any of $00$ or $11$. So, $$H^{23}_{agree} = I_8 - \\left(\\left(\\frac{1}{2}\\right)\\left(|001\\rangle+|011\\rangle+|100\\rangle+|110\\rangle\\right)\\right) \\left(\\left(\\left(\\frac{1}{2}\\right)\\left(|001\\rangle+|011\\rangle+|100\\rangle+|110\\rangle\\right)\\right)\\right)^{\\dagger}$$ The ground state of this Hamiltonian satisfies the assignment constraint. So, $$H_P = \\left( \\begin{array}{cccccccc} \\frac{17}{6} & -\\frac{1}{6} & -\\frac{1}{6} & -\\frac{1}{6} & 0 & 0 & -\\frac{1}{6} & -\\frac{1}{6} \\\\\\ -\\frac{1}{6} & \\frac{7}{3} & -\\frac{1}{6} & -\\frac{2}{3} & -\\frac{1}{2} & 0 & -\\frac{2}{3} & -\\frac{1}{6} \\\\\\ -\\frac{1}{6} & -\\frac{1}{6} & \\frac{17}{6} & -\\frac{1}{6} & 0 & 0 & -\\frac{1}{6} & -\\frac{1}{6} \\\\\\ -\\frac{1}{6} & -\\frac{2}{3} & -\\frac{1}{6} & \\frac{7}{3} & -\\frac{1}{2} & 0 & -\\frac{2}{3} & -\\frac{1}{6} \\\\\\ 0 & -\\frac{1}{2} & 0 & -\\frac{1}{2} & \\frac{5}{2} & 0 & -\\frac{1}{2} & 0 \\\\\\ 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 \\\\\\ -\\frac{1}{6} & -\\frac{2}{3} & -\\frac{1}{6} & -\\frac{2}{3} & -\\frac{1}{2} & 0 & \\frac{7}{3} & -\\frac{1}{6} \\\\\\ -\\frac{1}{6} & -\\frac{1}{6} & -\\frac{1}{6} & -\\frac{1}{6} & 0 & 0 & -\\frac{1}{6} & \\frac{17}{6} \\\\\\ \\end{array} \\right)$$ This Hamiltonian, which represents the instance of the problem is expected to be satisfied only with the assignment 011. So, the ground state should be $|011\\rangle$. But in reality, the ground eigenvalue is greater than $0$ and the ground state is $\\left(1,4,1,4,3,0,4,1\\right)^{\\dagger}$. Here, $I_n$ is the $n \\times n$ identity matrix. **What was I doing wrong?** My Mathematica code is available here."}
{"id":"108561","title":"Energy in an electromagnetic wave","text":"A radio antenna creates EM waves through switching the polarization in the antenna at a certain frequency. I assume the the energy of the photons produced in this process amount to E=hf for each photon. So good so far, but classically, i read that the EM field is an oscillation of the electric field between positive and negative values, as would seem reasonable from what the antenna is doing. If individual photons simply have a single positive energy relating to the frequency of the wave, how is this positive-negative oscillation represented in that stream of photons? I guess to put it simply, how is a changing electric field represented in a stream of photons in general, and from a radio antenna specifically?"}
{"id":"133983","title":"doppler shift through two mediums","text":"When considering the Doppler shift, the 'canonical equation' is $$f=\\frac{c+vr}{c+vs}f_0$$ However, this equation seems to run into trouble in the following situation: A light source inside water is moving at a speed $v$ towards a receiver outside the water. Can we modify the above equation to deal with a transition between two media? Or is there a completely different formula that applies here?"}
{"id":"35928","title":"A Different Lasing Medium","text":"Powerful lasers are highly intense, diverge negligibly and are also coherent. These radiations are emitted through partially reflecting mirrors after simultaneous reflections within the lasing medium. Don't these EM radiations affect the lasing medium or reflecting mirrors? Also, How is plasma used as a lasing medium in certain lasers?"}
{"id":"35920","title":"Maximum Possible Information in the universe?","text":"I remember hearing about this in one of the programs in discovery science. The physicist claimed that the maximum possible information in the universe is (10)^(10^123) whereas the maximum possible information that can be known by man is (10)^(10^90). Can anyone explain to me how can we arrive at such a specific number, and also how can information be represented by only numbers?"}
{"id":"127815","title":"What is the advantage of segmented particle traps?","text":"I'm currently trying to familiarize myself with the physics of ion and particle traps, especially with linear Paul traps. Many scientific experiments I've come across use segmented electrodes (like in this image from a Nature article). My question is _what do you gain by segmenting the electrodes?_ I could imagine that you can move the trapped particles along the axis to some extent, or determine their position in the trap. But in both cases I believe there are more precise methods."}
{"id":"127814","title":"Compactification and off-diagonal terms of the metric tensor","text":"In standard 3+1 dimensional spacetime, the metric tensor is of order 4 and had ten independent coefficients, hence there are 6 terms off the diagonal in the corresponding $4\\times 4$ real symmetric matrix. On the other hand, superstring theory postulates that there are 6 additional spatial dimensions that have to be compactified in a Calabi-yau manifold. My question is thus: are these two facts somehow related or is it just mere coincidence? For example, would the observable dimensions correspond to eigenvalues of the metric tensor?"}
{"id":"103968","title":"Picture of supports","text":"This questions stems from Axiomatic Quantum Field Theory and is mathematical in nature. However, I feel that an answer from physicists is more in line with what I will be asking. Let $\\phi$ be a real quantum field, namely $\\phi$ is an operator-valued distribution. One of the requirements of $\\phi$ is that it is _local_. In case $f\\in C{^\\infty _0 }_{real}$, then the assumption of locality requires that $$\\phi(f)\\phi(g)=\\phi(g)\\phi(f)$$ when the supports of f and g cannot be connected by a light ray. One says that such supports are _space-like separated_. **Reference** : Page 7 http:\/\/www.arthurjaffe.com\/Assets\/pdf\/Quantum- Theory_Relativity.pdf I'd like to gather further insight into this statement. Namely, how does one picture the supports? Should I have light cones in mind?"}
{"id":"69400","title":"Does a magnet contain (and potentially produce) energy?","text":"Very quick question, does a magnet contain energy? The general consensus seems to be, it does not. And this is generally confirmed by the fact that it would break the first law of thermodynamics. Whatever the hell that is (joke:) The reason I ask is because a) I'm no genius and b) because I'm perplexed. So maybe some of you smart people could help me out please. Here's the scenario; Now, if I took, oh I dunno, say a metal ball and lifted it say six inches. I have converted some of my man boob calories into energy that is now stored in the ball. When I release it and it drops to original level, the energy is released. Makes sense. Now if I took the same metal ball and rolled it along the ground, it would continue to roll until the kinetic energy was depleted, through friction and stuff like that. Now if I take the same ball and roll it along the ground, but this time with a magnet suspended 6 inches from the ground and directly in the line of movement. The magnet is strong enough to attract the ball and is therefore lifted 6 inches and sticks to the magnet. Where has that energy come from? It can't have come from me putting the magnet there, as once I put the magnet back on the ground, I have released that energy. As I said, I'm not smart, nor educated, just been pondering this question for a couple of days. Would be great if you could allow my brain to get back to menial tasks. Thanks"}
{"id":"52395","title":"Help with the understanding of boundary conditions on $AdS_3$","text":"So I am trying to reproduce results in this article, precisely the 3rd chapter 'Virasoro algebra for AdS$_3$'. I have the metric in this form: $$ds^2=-\\left(1+\\frac{r^2}{l^2}\\right)dt^2+\\left(1+\\frac{r^2}{l^2}\\right)^{-1}dr^2+r^2d\\phi^2$$ And I have the boundary conditions. So if I'm correct, I should find the most general diffeomorphism, by solving $\\mathcal{L}_\\xi g_{\\mu\\nu}=\\mathcal{O}(h_{\\mu\\nu})$, where $h_{\\mu\\nu}$ are the boundary conditions (subleading terms). So, if I'm doing things right, I get 5 equations. Because the $t\\phi$ term of Lie derivative vanishes. Now, I should use the power expansion of $\\xi$, as given in the paper, and solve these 5 differential equations or? I'm not certain if I'm on a right path, so any advice is welcome..."}
{"id":"79144","title":"Entropy inequality","text":"Assume that you have two bipartite systems $\\rho_1^{AB},\\rho_2^{AB}$ then I would like to prove the following: $$S(\\frac{1}{2}( \\rho_1^{AB}+I^A\\otimes\\rho_2^B))+S(\\frac{1}{2}(\\rho_2^{AB}+I^A\\otimes\\rho_1^B)) \\geq S(\\frac{1}{2}(\\rho_1^{AB}+I^A\\otimes\\rho_1^B))+S(\\frac{1}{2}(\\rho_2^{AB}+I^A\\otimes\\rho_2^B))$$ where $S$ is the von Neumann entropy, $\\rho_1^B=tr_A(\\rho_1^{AB}),\\rho_2^B=tr_A(\\rho_2^{AB})$ and $I^A$ is the maximally mixed state on $A$. It looks like it should pass with some monotony property, any hints or counterexample are welcome."}
{"id":"77030","title":"What is Quantization?","text":"In classical mechanics you construct an action (involving a Lagrangian in arbitrary generalized coordinates, a Hamiltonian in canonical coordinates [to make your EOM more \"convenient & symmetric\"]), then extremizing it gives the equations of motion. Alternatively one can find a first order PDE for the action as a function of it's endpoints to obtain the Hamilton-Jacobi equation, & the Poisson bracket formulation is merely a means of changing variables in your PDE so as to ensure your new variables are still characteristics of the H-J PDE (i.e. solutions of the EOM - see No. 37). All that makes sense to me, we're extremizing a functional to get the EOM or solving a PDE which implicitly assumes we've already got the solution (path of the particle) inside of the action that leads to the PDE. However in quantum mechanics, at least in the canonical quantization I think, you apparently just take the Hamiltonian (the Lagrangian in canonical coordinates) & mish-mash this with ideas from changing variables in the Hamilton-Jacobi equation representation of your problem so that you ensure the coordinates are characteristics of your Hamilton-Jacobi equation (i.e. the solutions of the EOM), then you put these ideas in some new space for some reason (Hilbert space) & have a theory of QM. Based on what I've written you are literally doing the exact same thing you do in classical mechanics in the beginning, you're sneaking in classical ideas & for some reason you make things into an algebra - I don't see why this is necessary, or why you can't do exactly what you do in classical mechanics??? Furthermore I think my questions have some merit when you note that Schrodinger's original derivation involved an action functional using the Hamilton-Jacobi equation. Again we see Schrodinger doing a similar thing to the modern idea's, here he's mish-mashing the Hamilton-Jacobi equation with extremizing an action functional instead of just extremizing the original Lagrangian or Hamiltonian, analogous to modern QM mish-mashing the Hamiltonian with changes of variables in the H-J PDE (via Poisson brackets). What's going on in this big Jigsaw? Why do we need to start mixing up all our pieces, why can't we just copy classical mechanics exactly - we are on some level anyway, as far as I can see... I can understand doing these things if they are just convenient tricks, the way you could say that invoking the H-J PDE is just a trick for dealing with Lagrangians & Hamiltonians, but I'm pretty sure the claim is that the process of quantization simply must be done, one step is just absolutely necessary, you simply cannot follow the classical ideas, even though from what I've said we basically are just doing the classical thing - in a roundabout way. It probably has something to do with complex numbers, at least partially, as mentioned in the note on page 276 here, but I have no idea as to how to see that & Schrodinger's original derivation didn't assume them so I'm confused about this. To make my questions about quantization explicit if they aren't apparent from what I've written above: **a)** Why does one need to make an algebra out of mixing the Hamiltonian with Poisson brackets? (Where this question stresses the interpretation of Hamiltonian's as Lagrangian's just with different coordinates, & Poisson brackets as conditions on changing variables in the Hamilton-Jacobi equation, so that we make the relationship to CM explicit) **b)** Why can't quantum mechanics just be modelled by extremizing a Lagrangian, or solving a H-J PDE? (From my explanation above it seems quantization smuggles these idea's into it's formalism anyway, just mish-mashing them together in some vector space) **c)** How do complex numbers relate to this process? (Are they the reason quantum mechanics radically differs from classical mechanics. If so, how does this fall out of the procedure as inevitable?) Apologies if these weren't clear from what I've written, but I feel what I've written is absolutely essential to my question. **Edit:** Parts **b)** & **c)** have been nicely answered, thus part **a)** is all that remains, & it's solution seems to lie in this article, which derives the time dependent Schrodinger equation (TDSE) from the TISE. In other words, the TISE is apparently derived from classical mechanical principles, as Schrodinger did it, then at some point in the complicated derivation from page 12 on the authors reach a point at which quantum mechanical assumptions become absolutely necessary, & apparently this is the reason one assumes tons of axioms & feels comfortable constructing Hilbert spaces etc... Thus elucidating how this derivation incontravertibly results in quantum mechanical assumptions should justify why quantization is necessary, but I cannot figure this out from my poorly-understood reading of the derivation. Understanding this is the key to QM apparently, unless I'm mistaken (highly probable) thus if anyone can provide an answer in light of this articles contents that would be fantastic, thank you!"}
{"id":"86233","title":"Under which representation of U(1) transform electron and photon gauge field?","text":"I know that under $SU(2) \\times SU(2)$, the left-handed electron transforms under $ ( \\frac{1}{2},0 ) $ representation and the vector gauge field $A_\\mu$ under $ ( \\frac{1}{2},\\frac{1}{2}) $. Since the electron transforms under $U(1)$, there must be a represenation under which it transforms. What is this representation? Does it have a name? Apparenly $A_\\mu$ does not transform under the same representation, which would mean $e^{\\alpha(x) Q} A_\\mu$, but instead as $A_\\mu + i \\partial_\\mu \\alpha(x)$ ? What representation is this? Of course I realize that the transformation of $A_\\mu$ can't be different for the Lagrangian to be invariant, but that shouldn't be used to define the it."}
{"id":"52511","title":"Any suggestions for units conversion?","text":"> **Possible Duplicate:**   >  Photometer: measured Irradiance L converted to photon rate I am conducting a experiment where stimulus output of $470\\ nm$ is measured by a radiometer at $30\\ \\mu W\\ cm^{-2}$. The stimulus is $1$ inch from the detector. Any suggestion on how I might go about converting $\\mu W\\ cm^{-2}$ to log photon $cm^{-2} s^{-1}$?"}
{"id":"43457","title":"What does the * mean in spherical harmonics?","text":"In Jackson's book about classical electrodynamics, this formula comes up: $$q_{lm} = \\int \\mathrm d^3 x' \\, Y^*_{lm}\\left(\\theta', \\phi'\\right) r'^l \\rho\\left(\\vec x'\\right)$$ What does that $^*$ mean?"}
{"id":"43453","title":"Symmetry and overlapping of ground states","text":"In a quantum mechanics, there is the following formula to derive the zero energy $E_0$ of a perturbed Hamiltonian $$H = H_0 + V$$ knowing the zero energy $W_0$ of the free Hamiltonian $H_0$: $$E_0 = W_0 + i\\frac{d}{dt}\\text{ln}R(t)|_{t\\rightarrow\\infty(1-i\\eta)}$$ The exponential killing the excited states faster than the lowest energy one. However, one needs to suppose a non-vanishing overlap of the two ground states $|\\phi_0\\rangle$ for $H_0$ and $|\\psi_0\\rangle$ for $H$. I read that if two have different symmetry they must be orthogonal but I didn't manage to derive why. Let's suppose that $G$ is a symmetry of $|\\phi_0\\rangle$ then $G|\\phi_0\\rangle=0$ and $\\langle\\psi_0|G|\\phi_0\\rangle=0$ and if $|\\psi_0\\rangle$ is not invariant under $G$ I still need to have $$G|\\psi_0\\rangle \\propto |\\psi_0\\rangle$$ to derive $\\langle\\psi_0|\\phi_0\\rangle=0$. ($|\\psi_0\\rangle$ needs to be an eigenvector of $G$ with non-vanishing eigenvalue)"}
{"id":"19771","title":"Electrical flow in a simple parallel circuit","text":"I'm having trouble understanding something in one of my text books: > Let’s have a look at the implications of each circuit configuration. Figure > 3.13 shows the Conventional representation of a parallel circuit. If you > assume that the resistance of the wires can be neglected, then the voltage > drop across each bulb is equal to the e.m.f. of the source, the dynamo. The > Current flowing from the source is divided between each bulb depending on > its resistance (remember I = E\/R), Removing one of the bulbs would not > affect the voltage drop across the other bulb and would therefore not affect > the Current in it, although the overall Current from the dynamo would drop > as the demand has been reduced. Specifically, the line about removing one of the bulbs not affecting the voltage drop. My understanding is that the sum of voltage drops in a circuit must be equal to the output of the emf, but the way I'm reading the text suggests that if you have three bulbs in a parallel circuit and you remove one of them, the voltage drop remains the same - this is what I don't understand, though. If you a lamp is removed, then that's one less lamp consuming emf. Does it not get redistributed to the other two lamps? Likewise, the text mentions the overall current from the dynamo dropping because demand has been reduced. My initial understanding was that current from an emf is only affected by the emf - not the demands of components further along the circuit, as in this case. What have I misunderstood?"}
{"id":"19775","title":"Quantum Field Theory Variants","text":"I am a math guy, so sorry for the naivety. When I peruse the wikipedia I see many \"variants\" of quantum field theory...conformal quantum field theory, topological quantum field theory, axiomatic\/constructive quantum field theory, algebraic quantum field theory, etc. Whether or not these are actually variants of something is unclear to me. I don't really have a specific question, but I was wondering if you guys could help me understand what these different things are and\/or point me to somewhere to get a clearer picture."}
{"id":"17638","title":"Has anyone theorized a connection between entropy and quantum uncertainty?","text":"I apologize if this kind of idle theorizing is frowned upon here, but I was wondering if it is possible that the Second Law of Thermodynamics is a consequence of quantum uncertainty. I've heard entropy of a system defined as the number of micro-states that it can have to correspond to the macro-states it has. So that definition makes it sound like entropy is simply losing information. As we know, entropy increases as time goes on. Now this seems contradictory to me; we know more as time goes on, not less. Is it possible that, because you can gain more and more information about a system as time goes on as you can interact with it more that, some information needs to be \"hidden\" from you. And that this process of losing information is entropy? P.S. I know that what I \"know\" about any system does not approach the limits set out by the uncertainty principle. But as System A interacts with System B, over time System A's state is more influenced by System B and in that sense System A has gained knowledge of System B."}
{"id":"20723","title":"The quantum state can be interpreted statistically, again","text":"Now there are two papers **The quantum state cannot be interpreted statistically** http:\/\/arxiv.org\/abs\/1111.3328 (It was discussed here the consecuences of this \"no-go theorem\") And this one (two of the authors are the same as the previous paper): **The quantum state can be interpreted statistically** http:\/\/arxiv.org\/abs\/1201.6554 I would like to note this: **titles give only poor information about the content** , and they seem even maliciously chosen, but the mere existence of the two papers is funny anyway.. The question is : **Which is more general!?** From the paper: _\"Recently, a no-go theorem was proven [21] showing that a $\\psi$-epistemic interpretation is impossible. A key assumption of the argument in [21] is preparation independence situations where quantum theory assigns independent product states are presumed to be completely describable by independently combining the two purportedly deeper descriptions for each system. Here, we will show via explicit constructions that without this assumption, $\\psi$-epistemic models can be constructed with all quantum predictions retained\"_ **About being general** As I understand the second one just _seems_ more general (because of \"less assumptions\"), but by no means Newton's dynamics is more general than Einsein's relativity because \"it lacks of c=constant assumption\". It's weird anyway, because it would mean that if \" wavefunction is a real physical object\" (a funny phrase from www.nature.com article) would depend on assumptions!, then what kind of realism depend on assumptions? Perhaps a point to discuss (assuming the theorem is well proven) is whether those assumptions have sense, if they come from experiments, or if they are just limiting the scope (toy model), or if those are random assumtions that have no source."}
{"id":"24934","title":"Do all black holes have a singularity?","text":"If a large star goes supernova, but not enough mass collapses to form a black hole, it often forms a neutron star. My understanding is that this is the densest object that can exist because of the Pauli exclusion principle: It's made entirely of degenerate matter, each particle of which cannot occupy the same quantum state of any other. So these objects are so massive that they gravitationally lens light. If you make them more massive, they bend the light more. Keep going and going until they bend the light _so much_ that light passing near the surface can barely escape. It's still a neutron star. Add a bit more mass, just enough that light passing just over the surface cannot escape. Now it's a black hole with an event horizon (I think?). Does this mean the neutron star has become a singularity? Isn't it still just a neutron star just beneath the event horizon? Why are black holes treated as having a singularity instead of just an incredibly massive neutron star at its center? Does something happen when an event horizon is \"created?\""}
{"id":"122946","title":"E&M and geometry - a historical perspective","text":"Recently, I was contemplating the beautiful formulation of electromagnetism (specifically Maxwell's equations) in terms of differential forms: $$F=\\mathrm{d} A\\implies \\mathrm{d}F=0 \\hspace{1cm}\\text{and}\\hspace{1cm} \\mathrm{d}\\star\\mathrm{d}F=\\mu_0 J $$ I started thinking about the history of this way of looking at things, and realized that I don't know much about it at all. My first question was therefore: Was it known already at the time of Maxwell (or soon after) that electromagnetism could be cast in this geometric form? How was this first introduced and who did it? After consulting Maxwell's treatise, it became clear that at least Maxwell himself was not aware of this formulation. But maybe someone else immediately recognized the geometric formulation once Maxwell published his results... In modern times, one is - at least as a physicist - usually first introduced to the field strength tensor $F$ through the covariant formulation of Maxwell's equation using tensor calculus, where it is defined as $F_{\\mu\\nu}=\\partial_\\mu A_\\nu -\\partial_\\nu A_\\mu$. When one then learns about differential forms etc. it is then obvious that $F=\\mathrm{d}A$ and the geometric formulation follows quite naturally. However, was this also the case historically? Did 'they' come up with the tensor calculus formulation of $F$ first, and did they only then recognize the geometric description? Or was the geometric description discovered first? Another possibility is that it took the introduction of Einstein's general relativity for anyone to realize that fields can be interpreted in terms of geometry. In conclusion, I am interested in a chronological description of the development of the different formulations of electromagnetism, with emphasis on the following points:   1. Who first came up with the geometric formulation in terms of differential forms?   2. Is it known at all how this person arrived at this?    3. Was the geometric interpretation discovered _before_ tensor calculus became popular, or only after it was know that $F_{\\mu\\nu}=\\partial_\\mu A_\\nu -\\partial_\\nu A_\\mu$? Was this after the introduction of GR, and was it at all influenced by Einstein's work?"}
{"id":"13296","title":"Suns emission spectrum","text":"I learnt that in astrophysical spectroscopy, the emission spectrum of distant stars is used to determine what they're made of. So why is it that our own Sun is emitting the whole spectrum ? (or is that information incorrect)"}
{"id":"128636","title":"Why do my ice cubes stick together?","text":"When I put ice cubes in a glass of water, I find that sometimes they will stick together and form a sort of \"bridge\" between them as they melt. There is usually a visible line where one ends and the other begins, and they break apart if pushed (but for the most part they stick together if they aren't interfered with). I'm wondering how they form this bridge in the first place if they're melting, and why it stays together."}
{"id":"80177","title":"Follow up question on \"Wilson Loops as Raising Operators\"","text":"This is a follow-up question on the topic that I opened a few days ago, Wilson Loops as raising operators. The paper > Topological Degeneracy of Quantum Hall Fluids. X.G. Wen, A. Zee. _Phys. Rev. > B_ **58** no. 23 (1998), pp. 15717-15728. arXiv:cond-mat\/9711223. gives a nice derivation of the explicit ground states of the $U(1)$ Chern- Simons Theory on a torus in Section 2 on Abelian Quantum Hall States. In particular Eq. (12) gives the generic form of a ground state $\\psi(y) = \\sum_{n=-\\infty}^{\\infty} c_{n} \\ e^{i\\ 2\\pi ny}$. Due to the fact that the theory lives on a torus the ground state manifold is found to be $k$-fold degenerate. My question: Is it possible (by direct calculation) to obtain the relations \\begin{align} W(b)|n \\rangle &= |n + 1 \\text{ mod } |k| \\rangle, \\nonumber \\\\\\ W(a) |n \\rangle &= e^{2\\pi i n \/k} |n \\rangle. \\end{align} from the previous question? I don't have a particularly strong background in field theory so I am feeling somewhat uneasy when it comes to the explicit evaluation of the Wilson Loop (with its exponentiated gauge field and the path ordering) acting on the constructed state. I am looking forward to your responses."}
{"id":"131281","title":"Gravity of Light","text":"I'm reading Quantum field theory in a Nutshell and I find a very interesting calculation that leads to the gravitational interaction between 2 light beam. Is this kind of interaction permitted in general relativity? (not necessarily a quantum gravity theory or the like) Is there a energy tensor $T_{\\mu\\nu}$ asscociated with a photon so that it creates $R_{\\mu\\nu}\\ne 0$? It's awesome if someone can give a simple derivation in the framework of general relativity."}
{"id":"110359","title":"principle in a water Faucet","text":"If you try to block with your finger the flow of the water out of the faucet or a water hose, the water's speed increases. But while you're blocking the water flow out of the faucet by turning the handle, the speed doesn't increase. So what's the difference between the two ?"}
{"id":"93087","title":"Differences in the behaviour of pinching a garden hose and closing a tap","text":"Let's say you have a garden hose connected to an ordinary water tap which is opened fully. If you pinch the end of the hose, water leaves the hose at a higher speed (and this can be useful while watering plants, to reach pots which are further away). However when a tap (with no hose connected) is opened only slightly, water flows out at a low speed, possibly even in drops. The actions of pinching the end of a hose and of almost-closing an open tap seem similar, so why the difference in behaviour?"}
{"id":"134148","title":"Vacuum stability in quantum field theory","text":"What exactly do people mean when they talk about the scale dependence of the effective potential ($V$)? I explain the motivation for my question (and hence my confusion) below. Please correct me as appropriate. * * * If one defines the effective potential as the non-derivative part ($p^2 \\rightarrow 0$) of the 1-PI effective action, then something like the Callan- Symanzik equation will imply that $$\\frac{d}{d \\log \\mu} V = 0 \\implies \\left[ \\frac{\\partial}{\\partial \\log \\mu} + \\beta_i \\frac{\\partial}{\\partial \\lambda_i} + \\gamma \\frac{\\partial}{\\partial \\log \\phi} \\right] V = 0$$ If the value of the effective potential at any field value (and zero momentum) is scale independent, then shouldn't the vacuum be stable at all scales, if we know it's stable at some scale? * * * All that I've said above holds so long as perturbation theory has been used correctly. I've also seen some people \"RG-improve\" the effective potential by resumming the leading logs across all loop order. But then, the moment you do a partial resummation across loop orders, is there any reason for the thing you calculate to be scale independent? It's also not clear what the physical interpretation of such a scale-dependent quantity should be -- so why should one take the appearance of another vacuum seriously -- after all, you don't see any such thing when you do your calculations at some \"low\" scale -- and physical observables (at zero momentum) better not be RG-scale dependent?"}
{"id":"116235","title":"Optimal placement of support joist under shelf","text":"Assume an ideal board, supported by two joists. Where should those two joists be optimally placed? Instinctively, I'd say at somewhat less than 25% and somewhat more than 75% of the extend of the board. Presumably this also depends on whether the board can move freely with respect to the joists, or whether the board is attached to the joists. I'm unclear even where to start analyzing this problem ..."}
{"id":"10364","title":"Do quantum states contain exponentially more information than classical states?","text":"Do quantum states contain exponentially more information than classical states? It might seem so at first sight, but what about in light of this talk?"}
{"id":"100670","title":"Uncertainty principle and commutation relations","text":"What connection exists between the uncertainty principle and commutation relations amongst the operators representing observables in Quantum Mechanics?"}
{"id":"24116","title":"Heisenberg Uncertainty Principle scientific proof","text":"Heisenberg's uncertainty principle states that: _if the x-component of the momentum of a particle is measured with an uncertainty_ $$\\Delta \\vec p_x$$ _then its x-position cannot, at same time, be measured more accurately than_ $$\\Delta\\vec x=\\frac {\\hbar}{2\\Delta\\vec p_x},$$ $$\\Delta\\vec x\\Delta\\vec p_x \\ge \\frac {\\hbar}{2}.$$ What is the scientific proof of this principle? Operators Uncertainty"}
{"id":"7200","title":"Can T-duality resolve spacelike singularities?","text":"Schwarzschild singularities are described by the Kantowski-Sachs metric with a contracting S2. Of course, T-duality doesn't apply to S2. But what about a Kasner-type singularity with two contracting spatial dimensions compactified over a torus T2, and an expanding spatial dimension? The T-dual of the torus gives rise to a geometry which is expanding in all spatial directions."}
{"id":"6227","title":"Grip of the train wheels","text":"How do the wheels of a train have sufficient grip on a metal track? I mean both of the surfaces are smooth (and not flexible) and it is okay if there is no inclination, but how about on an inclined track?"}
{"id":"6483","title":"Why can you see virtual images?","text":"In optics it is widely mentioned real images are projectable onto screens whereas virtual ones can only be seen by a person. Isn't that contradictory? I mean in order to see the virtual image it has to be _projected_ onto the retina (ultimately acting as a screen). So, why can you see virtual images in the first place?"}
{"id":"83755","title":"conceptual meaning of \"virtual image\"","text":"I am trying to learn about optics and I am having a hard time understanding the meaning of \"real\" vs \"virtual\" image. My understanding is that for a concave mirror, the image focuses on the same side as the object so it is a real image. For a convex mirror, the image focuses on the opposite side of the object from the mirror so it is a virtual image. However, we can both see either real or virtual images, so how are we seeing something that focuses on the other side of the mirror in the convex case ? I am very confused."}
{"id":"93191","title":"What are virtual objects, Reflection of light?","text":"While studying reflection through a plane mirror, I have been told that when the object is real the image will be virtual and **the image will be real while the object is virtual**. What are virtual objects? Do they really exist in nature?"}
{"id":"109085","title":"What is the difference between virtual object and virtual image?","text":"What is the difference between virtual object and virtual image?"}
{"id":"99341","title":"Help me understand virtual images versus real images","text":"Simple question; Is a virtual image that is produced by a lens always - in front of or in back of the lens?"}
{"id":"2658","title":"Virtual vs Real image","text":"I'm doing magnification and lens in class currently, and I really don't get why virtual and real images are called what they are. A virtual image occurs the object is less than the focal length of the lens from the lens, and a real image occurs when an object is further than focal length. By why virtual and real? What's the difference? You can't touch an image no matter what it's called, because it's just light."}
{"id":"121763","title":"Why do I get readings from my radiation meter with magnetized pieces?","text":"What causes those radiation readings in my radiation meter (See the linked video)? Metal pieces were magnetized. Used radiation detector is RADEX RD1503. \"Radiation\" is not generated if the tube has been wrapped inside an aluminium foil. Should induction occur through thin aluminium foil? At least magnet holds through that foil and plastic tube."}
{"id":"121764","title":"If non-zero cosmological constant interpreted as a repulsive field, what would be the properties of this field's quanta?","text":"If non-zero cosmological constant interpreted as a repulsive field, what would be the properties of the excitation of such field, i.e. the particle which serves as the field's quantum? What would be its spin, mass, possible interactions and other properties?"}
{"id":"121767","title":"Protection of the electron mass by chiral symmetry","text":"In many textbooks it is said that mass renormalization of the electron mass is only logarithmic $\\delta m \\sim m\\, log(\\Lambda\/m)$ because it is protected by the chiral symmetry. I understand that in case of massless fermions to keep them massless in the renormalization procedure it must be like this. Or differently said, the renormalization procedure respects the axial current conservation. But is there a compulsory reason for the renormalization procedure to respect the axial current conversation ? Does every renormalization procedure respect that ? Apparently Pauli-Villars and dimensional renormalization do it, but what for other procedures ? I also know that in triangular Feynman diagrams anomalies occur which do break the axial current conversation. So why can't it happen for something simpler like the electron mass respectively self-energy ?"}
{"id":"122498","title":"When we get an electrical shock, the ciruit is incomplete but still the current flows, why?","text":"If we touch the live wire and ground at the same time, we will get a shock. But the current goes from live to ground and not to neutral i.e, circuit is open. Then how can we get a shock? ![Image](http:\/\/i.stack.imgur.com\/fLKbj.png) This is the same circuit: will the current flow?   ![Image](http:\/\/i.stack.imgur.com\/nryo4.png) I think current should not flow. So, what is the reason that we still get shocks when the circuit is open?"}
{"id":"119201","title":"thrust-bursting towards a destination in space","text":"<\\--burst of thrust------|ME in a ship>----->>--<<\\--laser radar--<<\\---->>---||destination In the above, I imagine myself in a ship, marking my speed to a far off destination with laser radar. In addition, someone at the destination is clocking me as well with a separate laser. So, moving toward my destination at speed S1, I'm at rest. Then I fire a burst of thrust and experience acceleration, a, for time, t, and arrive at a new speed S2, right? This is all just simple physics right? My question is, where does special relativity play into this and how would it be perceived by A) me in the ship and B) the person at the destination clocking the ship? Would it be reasonable to want to graph velocity and distance to destination over time, given regular bursts of thrust, for both me in the ship and the observer at the destination? If so, any tips\/equations to use? PS: Although this might sound like a homework problem (or maybe I'm giving myself too much credit), it's actually just a lunchtime conversation :)"}
{"id":"119207","title":"Tension on an object","text":"According to a website, $T_3 = T_2 + T_1$. Also, the rope and pulley device are mass less. But I don't really understand this. Let me try and explain how I see the problem. $M_2$ receives tension in the up direction. This tension is the result of the resultant force of $M_1$ on the rope. $M_1$ receives tension in the up direction. This tension is the result of the reactant force of tension in $T_2$ The forces on the rope connected to the masses are: $T_{1m}$, $T_{2m}$ and these are both resultant forces from said masses. But since they are technically in different directions, they cancel each other out. So the net force on the rope is zero. But why is $T_3 = T_2 + T_1$? I do not understand this at all. It just doesn't make sense to me. ![enter image description here](http:\/\/i.stack.imgur.com\/lRZc2.gif)"}
{"id":"119208","title":"Explanation of binding energy in decays","text":"Everyone knows that the mass of a system is less than the mass of its components, with the equation: $M = \\sum_i m_i - BE(M) $ Now, if we consider a general decay, lets say $A \\rightarrow \\sum_i B_i$ then, for the conservation of the first component of the four-vector momentum we obtain a necessary condition for the decay: $ M(A) \\ge \\sum_i M(B_i) $ Isn't this last condition in contradiction with the first one? The only answer I get is a particle that can decay is not an eigenstate of the system. For this reason it shouldn't have a defined binding energy. So, for example, Tritium, that can decay, hasn't a BE? This appears so illogical if we think about nuclei with an half-life of years or more. Moreover I know this energy is experimentally defined and is bigger that 3He's BE."}
{"id":"80826","title":"Christoffel symbols and Dirac matrices mathematical similarities?","text":"Maybe mine is a silly question, but are there **mathematical similarities** or **common roots** between the Christoffel symbols: $ \\nabla - \\partial = \\Gamma $ and the Dirac matrices $ ( \\gamma^\\mu \\gamma^\\nu + \\gamma^\\nu \\gamma^\\mu = 2g^{\\mu\\nu}I )$ obtained through: $ (\\gamma^\\nu \\partial_\\nu)(\\gamma^\\mu \\partial_\\mu) = \\partial^\\mu \\partial_\\mu $ **EDIT:** What I mean is that the Dirac matrices are obtained by trying to match two different derivatives, so I was wondering if that had some common ground with the Christoffel symbols that are defined as the difference between the connection and the coordinate derivative."}
{"id":"80824","title":"What is the difference between Quantum Physics, Quantum Theory, Quantum Mechanics, and Quantum Field Theory?","text":"What is the difference between Quantum Physics, Quantum Theory, Quantum Mechanics, and Quantum Field Theory? Are they the same subject? I believe that they are not the same subject! Maybe there is not big difference between those subjects but I need to know what is main difference between those subjects and what is main intersection? Also I need to know which one is big subject relative to another?"}
{"id":"41140","title":"Calibrating an electronic temperature sensor based on power consumption","text":"I'm working with an electronic temperature logger that is being affected by heat generated internally. How does one come up with a calibration equation to calculate a more accurate reading of ambient temperature based on what the temperature sensor reads, taking into account its own power consumption? details: After a few hours and in equilibrium, the sensor reports values that are actually 1 degree Celsius higher than the ambient room temperature (22C) measured by a calibrated device. The sensor is accurate to 0.1 degree C at reporting the temperature of the device itself (which due to heat generated by the electronics has gotten warmer) The device consumes ~0.1 watts of power, weighs about 200g and has an average specific heat capacity of 1.0 j\/g (weighted mix of glass, abs, fr-4, copper). Dimensions are 1\"x 3\"x 4\". What I've got so far is this heating calculation: 200g * 1c * 1.0j\/g \/ 0.1w \/ 60s = ~33 minutes to heat up 1 degree. I'm assuming what we need is to figure out Sensor value - Heat-generated + Heat-dissipated to arrive at actual temperature. Which will require measure the K in newton's law? then what? I'd really appreciate you help here."}
{"id":"41149","title":"Why does bad smell follow people (assuming they are not the source)?","text":"When you are sitting in a room where there is a source of bad smell, such as somebody smoking or some other source of bad smell, it is often a solution to simply move to another spot where bad smell is not present. Assuming you are not actually the source of the smell, this will work for a while until you notice the smell has somehow migrated to exactly the spot where you are now sitting. Frustrating. This got me thinking about the fluid mechanics of this problem. Treat bad smell as a gas that is (perhaps continuously) emitted at a certain fixed source. One explanation could be that human breathes and perhaps creates a pressure differential that causes the smell to move around. Is there any truth to this? Please provide a reasoned argument with reference to the relevant thermodynamic and\/or fluid quantities in answering the question. Theoretical explanation is desired, but extra kudos if you know of an experiment."}
{"id":"17994","title":"How to write the Fröhlich Hamiltonian in one dimension?","text":"I am currently working on a (functional) analysis problem refining Pekar's Ansatz (or adiabatic approximation, as it is called in his beautiful 1961 manuscript \"Research in Electron Theory of Crystals\"). Anyways, I have two related questions, which the members of this community may find simple. The Fröhlich Hamiltonian is given as follows in three dimensions $$H=\\mathbf{p^{2}}+\\sum_{k}a_{k}^{\\dagger}a_{k}-\\biggl(\\frac{4\\pi\\alpha}{V}\\biggr)^{\\frac{1}{2}}\\sum_{k}\\biggl[\\frac{a_{k}}{|\\mathbf{k}|}e^{i\\mathbf{k\\cdot x}}+\\frac{a_{k}^{\\dagger}}{|\\mathbf{k}|}e^{-i\\mathbf{k\\cdot x}}\\biggr]$$ The physical scenario here is an electron moving in a 3-dimensional crystal. Each $k$ signifies a (vibrational) mode of the crystal. If we restrict ourselves to just a 1-dimensional crystal, why is it that the Hamiltonian can be written as follows: $$H=\\mathbf{p^{2}}+\\sum_{k}a_{k}^{\\dagger}a_{k}-\\biggl(\\frac{4\\pi\\alpha}{V}\\biggr)^{\\frac{1}{2}}\\sum_{k}\\biggl[a_{k}e^{i\\mathbf{k\\cdot x}}+a_{k}^{\\dagger}e^{-i\\mathbf{k\\cdot x}}\\biggr]$$ Namely, why do we drop the $|\\mathbf{k}|$ factor in the third term? Furthermore, I see how the creation and annihilation operators work on the (bosonic) Fock space (referring to the crystal here), especially when we write the creation operator in the form $\\sum_{k=0}^{\\infty}\\frac{(a^{\\dagger})^{k}}{\\sqrt{k!}}\\left|0\\right\\rangle =\\left|k\\right\\rangle$. Namely, the creation operator is jumping from one tensored state in Fock Space to the next. However, I also see the form $a_{k}=\\frac{1}{\\sqrt{2}}\\bigl(k+\\frac{d}{dk}\\bigr)$. How are the two forms connected? How do you intuitively think of the latter form? For example, I thought of the former form as the creation operator jumping from one state in fock space to the next, but the latter form I am not quite sure."}
{"id":"51322","title":"Wrinkling paint - soluto\/thermocapillarity - is it due to the primer or solvent","text":"This is really a one-and-a-half part question. I know that when paint is mixed with a solvent or used with a primer, it sometimes wrinkles. As I understand, a key physical phenomena here is a non- uniform evaporation of the primer or solvent that gives rise to different evaporation rates at the paint surface. This can create a non-uniform temperature difference. This in turn can create a thermocapillarity (surface tension effects due to non-uniform temperature distribution) or solutocapillarity (surface tension effects due to non-uniform solvent\/solute concentration) effects that wrinkle the paint surface. So is this a solvent problem or a primer problem? Am I understanding the physics right but the nature of primer and solvent wrong?"}
{"id":"11104","title":"Constructing a maximally entangled qutrit state from $n$ Bell states","text":"I've read that maximally entangled qubit states are a good \"unit\" of bipartite entanglement since it is possible to create any other entangled state from them using local operations and classical communication (LOCC) provided sufficiently many copies are available. What would the protocol be to construct a maximally entangled qutrit ($\\vert \\psi \\rangle_{AB} = \\frac{1}{\\sqrt{3}}(\\vert 00 \\rangle + \\vert 11 \\rangle + \\vert 22 \\rangle)$) between two space separated parties from a set of $n$ Bell states ($\\vert \\Phi^+ \\rangle^{\\otimes n}_{AB} = 2^{-n\/2} (\\vert 00 \\rangle + \\vert 11 \\rangle)^{\\otimes n}$) initially shared by those parties using only LOCC? If you can, please include in your answer why local operations alone would not be sufficient."}
{"id":"8704","title":"Expansion of multi-particle state vector as a sum of n-entangled states","text":"Physically, quantum entanglement is ranged from full long-range entanglement (Bose-Einstein condensate), described by a basis of states that look like this: $$ |\\Psi\\rangle = |\\phi_{i_{0} i_{1} ... i_{N}}\\rangle $$ to full-decoherence (a Maxwell-Boltzmann ideal gas) which a basis of states that look like this: $$ |\\Psi\\rangle = \\prod_{i}{ |\\phi_{i}\\rangle } $$ And in the middle of this range we have 2-particle entanglement terms, 3-particle entanglement, etc. So it seems natural to arrange the wavefunction as a series looking like: $$ |\\Psi\\rangle = \\sum{ \\prod_{i}{ |\\phi_{i}\\rangle } } + \\sum{ \\lbrace \\prod_{i_{0} ,i_{1} > i_{0}}{ |\\phi_{i_{0} i_{1}}\\rangle } \\rbrace } + \\sum{ \\lbrace \\prod_{i_{0} ,i_{1} > i_{0} , i_{2} > i_{1}}{ |\\phi_{i_{0} i_{1} i_{2}}\\rangle } \\rbrace } + \\dotsb $$ **BEGIN EDIT** I feel that i need to put in a bit more clear footing the mathematics behind this separation. Let's take an arbitrary state vector $|\\Psi\\rangle$, we might write it like this: $$ |\\Psi\\rangle = \\prod_{i}{ c^{0}_{i} |\\phi_{i}\\rangle } + |\\Psi_{Remainder}\\rangle $$ That is, we write the state vector as a vector that is completely separable and a remainder that is not. _This decomposition is unique_. Proof: take another decomposition with $\\widehat{c^{0}_{i}}$, take the difference between both decompositions and verify that both remainders are completely separable, which is against the definition The idea is that one should be able to further this decomposition of the remainder state vector, for instance lets take a vector with zero completely separable part (that is, we are on the equivalence class of our remainder above) and attempt to write it like: $$ |\\Psi_{R_0}\\rangle = \\prod_{i_{0} ,i_{1} > i_{0}}{ c^{1}_{i_0 i_1} |\\phi_{i_{0} i_{1}}\\rangle } + |\\Psi_{R_1}\\rangle $$ Analogously, one can prove that the $c^{1}_{i_0 i_1}$ are unique and depend only on the $|\\Psi_{R_0}\\rangle$ vector. And we don't need to do any symmetrization operation to get this result, so this is an universal decomposition of the state vector (for bosons and fermions) (note: i'm aware that this leaves out a lot of products with mixed entanglement, i.e: some single-particle states multiplied by two-particle entangled states, but since they don't add anything to this particular argument i choosed to leave them aside) **END EDIT** **BEGIN 2ND EDIT** separability of states is a property that is invariant under unitary transformations, so a separable state is not equivalent to a separable one in any basis you choose. I've investigated a bit more and the problem in general of knowing if a state is separable or not is known as QSP (quantum separability problem). For a definition please look at this paper **END 2ND EDIT** Question: Where can i read more about this sort of expansion and do you know if there is a computational framework to estimate the relative magnitudes of each term (for instance, i would expect that for Bose-Einstein condensates you need to keep all the terms of the expansion, while for relatively high- temperature solids you would be able to get away with 3 or 4 terms)"}
{"id":"69700","title":"Can the mass of an orbiting object and the object being orbited be determined by the distance and orbit velocity alone?","text":"This question is the inverse of: \"Could an object orbit while moving at twice the speed, but at the same distance, if it had half the mass?\" I'm curious about the nature of orbits, but am not well enough versed in mathematics to understand Kepler's laws well. I have been wondering if the mass of a planet and a star it orbits could be determined based solely on the distance and speed of the orbit, or if the ability to orbit at a given speed\/distance was based _relatively_ on the mass of both objects (i.e., we could determine the ratio of the mass of the two objects, but not the actual mass)."}
{"id":"41020","title":"What is the relationship between mass, speed and distance of a planet orbiting the sun?","text":"After reading this fascinating story about a new exoplanet, I was wondering about how mass, speed and distance determine a circular orbit of a planet around a star. Given the mass of the sun and star, and the distance between them, is there only one possible orbital velocity? Given any 2 of the following, is it possible to calculate the 3rd?   * Relative mass of the planet to the sun   * Orbital velocity   * Distance between the 2 masses And, given 1 of the above, are there an infinite number of possible values for the other 2?"}
{"id":"110780","title":"Transformation Law for Covariant Derivative in $SU(2)$ Yang-Mills","text":"In page 488 of Peskin and Schroeder, it is stated (emphasis mine): > It is not difficult to check using (15.27) and (15.21) that, even for > _finite_ transformations, the covariant derivative has the same > transformation law as the field on which it acts. I was trying to indeed verify that. This is what I have tried:   1. $V\\left(x\\right)\\in SU\\left(2\\right)^{\\mathbb{R}^4}$   2. $\\psi\\left(x\\right)\\mapsto V\\left(x\\right)\\psi\\left(x\\right)$.   3. $D_\\mu\\left(x\\right)\\equiv\\partial_\\mu-igA_{\\mu,\\,j}\\left(x\\right)\\frac{\\sigma^j}{2}$.   4. $igA_{\\mu,\\,j}\\left(x\\right)\\frac{\\sigma^j}{2}\\mapsto V\\left(x\\right)igA_{\\mu,\\,j}\\left(x\\right)\\frac{\\sigma^j}{2}\\left[V\\left(x\\right)^\\dagger\\right]-V\\left(x\\right)\\left\\\\{\\partial_\\mu\\left[V\\left(x\\right)^\\dagger\\right]\\right\\\\}$ Thus: \\begin{align} D_\\mu\\left(x\\right)\\psi\\left(x\\right) \\mapsto & \\left\\\\{\\partial_\\mu -V\\left(x\\right)igA_{\\mu,\\,j}\\left(x\\right)\\frac{\\sigma^j}{2}\\left[V\\left(x\\right)^\\dagger\\right]+V\\left(x\\right)\\left\\\\{\\partial_\\mu\\left[V\\left(x\\right)^\\dagger\\right]\\right\\\\}\\right\\\\}V\\left(x\\right)\\psi\\left(x\\right) = \\\\\\\\\\ &= \\left[\\partial_\\mu V\\left(x\\right)\\right]\\psi\\left(x\\right)+V\\left(x\\right)\\partial_\\mu\\psi\\left(x\\right)-V\\left(x\\right)igA_{\\mu,\\,j}\\left(x\\right)\\frac{\\sigma^j}{2}\\psi\\left(x\\right)+V\\left(x\\right)\\left\\\\{\\partial_\\mu\\left[V\\left(x\\right)\\dagger\\right]\\right\\\\}V\\left(x\\right)\\psi\\left(x\\right) = & \\\\\\\\\\ &= V\\left(x\\right)D_\\mu\\left(x\\right)\\psi\\left(x\\right)+\\left\\\\{\\left[\\partial_\\mu V\\left(x\\right)\\right] + V\\left(x\\right)\\left\\\\{\\partial_\\mu\\left[V\\left(x\\right)\\dagger\\right]\\right\\\\}V\\left(x\\right)\\right\\\\}\\psi\\left(x\\right) \\end{align} So as far as I understand, $\\boxed{\\left[\\partial_\\mu V\\left(x\\right)\\right] + V\\left(x\\right)\\left\\\\{\\partial_\\mu\\left[V\\left(x\\right)\\dagger\\right]\\right\\\\}V\\left(x\\right)\\stackrel{?}{=}0}$ should be zero. To prove that I have used the fact that $VV\\dagger=1$: \\begin{align} \\partial_\\mu\\left[ V\\left(x\\right)\\right] + V\\left(x\\right)\\left\\\\{\\partial_\\mu\\left[V\\left(x\\right)\\dagger\\right]\\right\\\\}V\\left(x\\right) &= \\\\\\ \\partial_\\mu\\left[ V\\left(x\\right)1\\right] + V\\left(x\\right)\\left\\\\{\\partial_\\mu\\left[V\\left(x\\right)\\dagger\\right]\\right\\\\}V\\left(x\\right) &= \\\\\\ \\partial_\\mu\\left[ V\\left(x\\right)V\\left(x\\right)^\\dagger V\\left(x\\right)\\right] + V\\left(x\\right)\\left\\\\{\\partial_\\mu\\left[V\\left(x\\right)\\dagger\\right]\\right\\\\}V\\left(x\\right) &= \\\\\\ \\left[\\partial_\\mu V\\left(x\\right)\\right]V\\left(x\\right)^\\dagger V\\left(x\\right) +V\\left(x\\right)\\left[ \\partial_\\mu V\\left(x\\right)^\\dagger \\right]V\\left(x\\right) +V\\left(x\\right)V\\left(x\\right)^\\dagger\\left[ \\partial_\\mu V\\left(x\\right)\\right] + V\\left(x\\right)\\left\\\\{\\partial_\\mu\\left[V\\left(x\\right)\\dagger\\right]\\right\\\\}V\\left(x\\right) &= \\\\\\2\\left\\\\{\\partial_\\mu\\left[ V\\left(x\\right)\\right] + V\\left(x\\right)\\left\\\\{\\partial_\\mu\\left[V\\left(x\\right)\\dagger\\right]\\right\\\\}V\\left(x\\right)\\right\\\\}\\end{align} Is this all correct?"}
{"id":"110785","title":"Galilean Transform","text":"I tried to solve a problem using two different ways and I had some trouble, the problem is: We define a symmetry transform of the expected value of $\\vec{P}$ like this: $$\\langle \\psi|\\vec{P}|\\psi \\rangle \\rightarrow \\langle \\psi|\\vec{P}+m\\vec{V_0}|\\psi \\rangle$$ being $\\vec{V_0}$ a constant vector. If we define the generators $\\vec{K}$ like $$\\vec{K}=-m \\vec{T} $$ prove that we obtain the following relations. $$ [K_j,P_k]=-im\\delta_{jk} $$ $$ [K_j,K_k]=0 $$ **And my two solutions are** Using $$U^-1 \\vec{P} U = (1-i \\vec{\\epsilon} \\vec{T}) \\vec{P} (1+i \\vec{\\epsilon} \\vec{T}) = \\vec{P} + m\\vec{V_0} = \\vec{P} + \\vec{\\epsilon} $$ i end up with (where $\\vec{\\epsilon} \\equiv m\\vec{V_0} $) $$ [\\vec{K},\\vec{P}] = -im $$ My problem is if we expand those vectors like this $$ \\vec{K}\\vec{P}-\\vec{P}\\vec{K} = K_iP_i+K_jP_j+K_kP_k-K_iP_i-K_jP_j-K_kP_k = [K_i,P_i]+[K_j,P_j]+[K_k,P_k]=3[K_i,P_i] $$ and this gives $[K_i,P_i]=\\frac{-im}{3} $ wich is wrong But if I start with $$(1-i \\epsilon_i T_i) P_i (1+i \\epsilon_i T_i) = P_i + m V_{i0} $$ we obtain the good answer $$[K_i,P_i]=-im $$ so, my question is, whats wrong with the first method and why it doesn't give me the right answer."}
{"id":"123862","title":"Has a metric formulation of electromagnetism ever been attempted?","text":"I understand that electromagnetic fields carry energy, and this energy curves spacetime gravitationally. That's not my question. I'm asking if anyone has tried to formulate electromagnetism in such a way that EM charges impart an EM geometry onto spacetime that is only experienced by EM charges. That is, the EM geometry of spacetime would be a function of charges such that charges themselves produce EM curvature and the motion of charges produces EM torsion, and that charged objects move according to how their charge (electrical or magnetic, positive (N) or negative (S)) experiences that geometry. For example, a positive electric charge would appear as a \"hill\" to other pe charges, a \"valley\" to ne charges, and either a left or a right hand \"whirpool\" to either nm or sm charges (I'm not immediately sure which would match with which) if it had some velocity. This formulation would be directly analogous to how energy creates gravitational curvature in spacetime (and if one accepts Einstein-Cartan, gravitational torsion comes from intrinsic angular momentum), and then this resulting geometry is experienced by the energy in that region. If Einstein is correct about gravity, would it be too much of a stretch to suppose separate metric functions for each of the fundamental forces, considering that they were a single force moments after the big bang? I'm not currently concerned with the quantum mechanical approach to electromagnetism. I understand that to be truly fundamental, EM has to be formulated quantum mechanically, but right now I want to limit my question to macroscopic charged objects."}
{"id":"79994","title":"Magnification for a concave mirror","text":"Here is the question as given in my textbook: > Find the distance of the object from a concave mirror of focal length 10 cm > so that the image size is 4 times the size of the object. The solution in my textbook has the following data stated: $u=-x$ as it is assumed that the object is real. $v=-4x$ as it is assumed in case 1 that a real image will be formed and $|\\frac{v}{u}|=|m|=4$ So now I am not able to understand why the image distance $v$ is taken $-4x$. In the question it is given that the _object size_ is magnified 4 times, and not the distance of the of the object from the mirror. And the magnification formula is $\\frac{-v}{u}=m$, so why does the solution include the modulus of the formula? Am I missing something in analysing the solution?"}
{"id":"116803","title":"Does relativistic mass violate the conservation laws?","text":"When an object's speed increases, its (relativistic) mass increases. Are new atoms created inside the object by its increased speed? or is its \"gravitational charge\" increased by its increased speed, without more atoms?"}
{"id":"126807","title":"How to properly construct the electromagnetic tensor in curved space-time?","text":"How do I properly construct the electromagnetic tensor in curved space-time? I have my curved spacetime metric $(+,-,-,-)$ and my magnetic vector potential $A$. I tried two ways but not sure which is right (if there is one). First way:   1. Compute the magnetic field $B$ from the curl of the magnetic vector potential $A$: $$ \\mathbf{B} = \\nabla \\times \\mathbf{A}. $$   2. Place the resulting components directly in the contravariant electromagnetic tensor definition in cylindrical coordinates: $$ F^{\\mu\\nu} = \\begin{pmatrix} 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & B^z \\\\\\ 0 & 0 & 0 & -B^r \\\\\\ 0 & -B^z & B^r & 0 \\end{pmatrix}. $$ Second way:   1. Define the electromagnetic four-potential ($\\phi$ is zero in my problem): $$ A^\\alpha = (\\phi, \\mathbf{A}). $$   2. Lower the four-potential index by contracting it with my covariant metric tensor.   3. Compute the electromagnetic field components with the formula $$ F_{\\mu\\nu} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu. $$ I replaced the ordinary derivatives with covariant derivatives.   4. Raise the indexes of this covariant electromagnetic field tensor to compare with the first way. The problem is that I can't seem to have the same results with both methods, which says pretty clearly that I am doing something wrong. Is there something fundamentally wrong in taking these steps?"}
{"id":"12456","title":"What is the terminal velocity for a mobile phone","text":"You may have seen the story of the iPhone which was dropped from perhaps 13,500 feet by a skydiver - it survived. This made me wonder how to work out the terminal velocity for something like that. Obviously calculating terminal velocity for a sphere can be relatively straightforward, but with a flattened oblong, what factors come into play? End on will be fast, flat will be slow, but is there a stable configuration?"}
{"id":"80574","title":"Why is charge $q$ symmetrically distributed?","text":"Simple question: Why is charge $q$ outside symmetrically distributed?   The material is a conductor.   ![enter image description here](http:\/\/i.stack.imgur.com\/8Tutg.png)"}
{"id":"56119","title":"Are quantum mechanics and determinism actually irreconcilable?","text":"As a preface, I am not a physicist. I'm simply interested in abstract physics and fundamental principles of the universe and such. As such, if you can provide an answer for the layman (as non-academic and unjargonized as possible), it would be very, very appreciated so that I can actually understand it. Everything I ever learned about physics seemed to be built off of an assumption that the universe and everything in it behaved deterministically. So it should always be a (theoretical) possibility that, given perfect knowledge of every particle and force in the universe at a given moment, we can calculate with 100% accuracy what the state of the universe will be in the next moment. This of course assumes omniscience and unlimited computational capacity, which is why I said this is only a theoretical possibility. However, we can define our closed system to be much smaller -- say, a bottle full of nitrogen and helium -- and apply this principle more directly. And it seems like this assumption is absolutely necessary for scientific experiments to even take place or have any validity, since without this kind of determinism, the observations and the results inferred from them can't ever actually be trusted. I don't understand quantum mechanics very well, but it seems like this theory breaks this assumption completely. From what I understand, there is no way to predict what the state of the particle will be at the next moment. The most I can know is that, given that a particle is in state `A`, it will next be in state `B` or state `C`. There is absolutely no way to know for sure, and the only way to find out is to observe it actually change. Furthermore, observations of this kind don't yield any insight into what other particles in state `A` will do. So, in classical physics, laws used to look like this:               A -> B   [A implies B]      But with quantum physics, all of this is gone, and our laws can at best look something like this:               A -> ((B v C) v D) v E   [A implies B, or C, or D, or E, or ...]        How does this not break everything physics is built upon? The implications of this are seriously troubling to me, and I feel like it destroys everything I thought I knew. Can anyone explain how this works in slightly lower-level terms, or show how it's still possible for the theories and laws of classical physics to hold any weight?"}
{"id":"65682","title":"Given mean insolation, can one place an upper bound on the daily peak insolation?","text":"Photovoltaic panels are rated under standard conditions: eg a $100\\mathrm{W}$ panel if irradiated with $1000\\mathrm{W}\/\\mathrm{m}^2$ at $25^\\circ\\mathrm{C}$. When the cost effectiveness of photovoltaics is discussed one is often presented with daily irradiation maps averaged yearly or possibly monthly. This is fine for cost effectiveness calculations. But in designing the power conversion electronics or effects of grid injection one cannot work with average values - they must be able to sustain peak powers that are possibly larger than the nominal power of the panels, right? If so, by what factor are these electronics oversized? I cannot find any similar maps of the average daily peak irradiance. Given the daily solar energy incident ($\\mathrm{kW hr \/m^2\\cdot day}$) averaged either annually or monthly, can one place an upper bound on the daily peak irradiance ($\\mathrm{W \/ m^2}$)? I am particularly interested in Mediterranean countries."}
{"id":"65680","title":"Magnetic field due to a conducting sphere in unpolarized light","text":"From Jackson, problem 10.3: > A solid uniform sphere of radius $R$ and conductivity $\\sigma$ acts as a > scatterer of a plane-wave beam of unpolarized radiation of frequency > $\\omega$, with $\\omega R \/c \\ll 1$. The conductivity is large enough that > the skin depth $\\delta$ is small compared to $R$. (a) Justify and use a > magnetostatic scalar potential to determine the magnetic field around the > sphere, assuming the conductivity is infinite. (Remember that $\\omega \\neq > 0$.) We'd like to show $\\nabla \\times {\\bf B} = \\nabla \\cdot {\\bf B}=0.$ I have two questions. First, what does it mean for a plane wave to have a definite frequency and be unpolarized? For example, are there many sources out of phase, all at a given frequency, radiating with varying amplitudes and in all directions? If this is true, then is it possible for the superposition of all these waves to vary faster than the original frequency due to interference? Assuming that the above question is resolved and that we can make the long- wavelength approximation so the magnetic field is roughly constant over the sphere, why does $$\\nabla \\times {\\bf B} =0\\neq \\frac{1}{c^2}\\frac{\\partial {\\bf E}}{\\partial t}$$ (assuming that the electric field is just like an oscillating dipole and has a term free from powers of $\\omega$)?"}
{"id":"16227","title":"what is the kinematics of a particle with complex mass?","text":"* particles with real-mass have time-like kinematics ($ds^2 > 0$).   * particles with zero-mass have light-like kinematics ($ds^2 = 0$).   * particles with imaginary-mass have space-like kinematics ($ds^2 < 0$) (tachyons). So the question is pretty simple: **What would be the kinematics of a particle with both non-zero real and imaginary parts?**"}
{"id":"90685","title":"Allowed transformations in General Relativity","text":"So in Special Relativity we have: $$ \\Lambda \\eta \\Lambda^T=\\eta $$ Is there an analagous formula for the metric in General Relativity?"}
{"id":"105038","title":"Simultaneous Charging and Discharging Capacitor","text":"sorry if I sound little noobish. Though I have a fairly good understanding of physics, I sometimes don't understand the electrical aspects. Say there is a capacitor. This capacitor is expected to act as a storage buffer. By extension, the capacitor will have a \"charge\" interface and a \"discharge\" interface. There may or may not be an electrical circuit between these interfaces and the capacitor. The expected behavior of the system is, electrical energy may be input to the system via the charge interface, which will charge the capacitor, and energy may be simultaneously drawn also, via the discharge interface, which will draw energy from the capacitor, and this process can happen so long as the energy stored within the capacitor is within its maximum and zero. The actual path of the energy may be from the interface, through the circuit, to the capacitor, and back through the circuit to the other interface; or part of the energy may be routed from one interface to the other by the circuit, and the net energy difference between the two interfaces be actually sent to, or drawn from the capacitor. Though I may think that this is possible, I'm not aware if any such system exists currently. The home inverter seems to be doing quite the same thing, but both the cycles don't happen at the same time though. Edit: Yeah, a diagram will help me also to explain better what I have in mind. ![Energy Storage Buffer](http:\/\/i.stack.imgur.com\/cXRGC.jpg) The two interfaces are part of the circuit which shields the capacitor. This circuit may work in two possible ways, which I've mentioned as flow 1 and flow 2. In Flow 1, \"all\" of the energy which flows into\/out of the system, does so through the capacitor. In Flow 2, the circuit redirects part of the energy flow in one interface to the other interface, and only the net difference between the energy flows is actually transmitted to\/from the capacitor. Hope this makes it more clear. Now, let me restate my question. Is such a system possible, importantly, such a circuit possible. Are any systems available today, which do exactly the same thing. And your own views on this is really welcome."}
{"id":"122815","title":"Having trouble understanding how the centrifugal force works","text":"I thought that I understood the centrifugal force earlier, but I can't seem to grasp how it interacts when considering that everything is relative? Let's imagine that you are the only one in the entire universe, and that you are spinning with high angular velocity, with the rotation axis pointing in the same direction that your eyes are directed. Surely you would feel the centrifugal force pulling your feet and head apart, wouldn't you? A problem with this, though, is the following: Since you are the only object in the universe, there's no way to tell if you're rotating. Your angular velocity isn't even defined, since you aren't rotating in relation to anything else. How, then, can one know how what the centrifugal force is? Is it defined in relation to all the other mass in the universe, in such a way that it's negligible in classical mechanics problems?"}
{"id":"93724","title":"Quantum fluctuations in a classical domain?","text":"\"In the presence of chaos, even small fluctuations (including quantum fluctuations) can be amplified to produce large uncertainties in later behavior\"(http:\/\/arxiv.org\/pdf\/gr-qc\/9210010v2.pdf) Is there some experimental evidence for the amplification of the quantum fluctuations in a classic domain (typically m, s, kg)?"}
{"id":"64480","title":"Stark Effect on the 1st excited state of Hydrogen","text":"I know the ground state of hydrogen is unaffected by the Stark effect to first order. And I also know that the 1st excited state is split from 4 degenerate states to 2 distinct, and 1 degenerate state like this: ![Hydrogen splitting](http:\/\/i.stack.imgur.com\/zO0Zt.png) But I don't quite understand why. I imagine it is something to with (anti)symmetry of the wavefunctions and selection rules. Can anyone explain?"}
{"id":"2501","title":"Why are the orbits of planets in the Solar System nearly circular?","text":"Except for Mercury, the planets in the Solar System have very small eccentricities. Is this property special to the Solar System? Wikipedia states: > Most exoplanets with orbital periods of 20 days or less have near-circular > orbits of very low eccentricity. That is believed to be due to tidal > circularization, an effect in which the gravitational interaction between > two bodies gradually reduces their orbital eccentricity. By contrast, most > known exoplanets with longer orbital periods have quite eccentric orbits. > (As of July 2010, 55% of such exoplanets have eccentricities greater than > 0.2 while 17% have eccentricities greater than 0.5.1) This is not an > observational selection effect, since a planet can be detected about equally > well regardless of the eccentricity of its orbit. The prevalence of > elliptical orbits is a major puzzle, since current theories of planetary > formation strongly suggest planets should form with circular (that is, non- > eccentric) orbits. What is special about the Solar System that orbits of planets here are nearly circular, but elsewhere they are moderately or highly eccentric?"}
{"id":"46505","title":"Thermodynamic process when nebula is heated","text":"The basic thermodynamics problem is stated as follows. > The nebula contains a very tenuous gas of a given number density (atoms per > volume) that is being heated to a given temperature. What is the gas > pressure?   1. What are the basic assumptions that should be taken in solving this problem? There is no sealed container, obviously, but if nebulæ were to allowed to expand indefinitely then it would be an isobaric process, would it not? Seen how that would not allow us to determine the pressure (?), then some sort of constraint is to be placed. If we are to assume that it is, in fact, an isochoric process, would it be a simple matter of finding molar mass from given number density and plugging it in the formula of Ideal Gas Law ($p = nRT \/ V$) assuming the volume of 1 metre cubed?   2. Given the numbers ($1 × 10^{8}$ atoms per m$^3$, 7500K) what should be realistic order of magnitude for the answer (in $Pa$ or $atm$) for the purposes of assessment as many certainly would not have intuitive grasp of your average nebular pressures? In general, for questions like this (advanced question from introductory chapter) is it detrimental to overthink the problem, that is: do I look for more difficulty than I should?"}
{"id":"46502","title":"Is it possible to create an electrified silicon gas vortex?","text":"Is it possible to create, and sustain, an electrified silicon gas vortex?   If it is possible: would it produce an electromagnetic field?   And how would that field affect the vortex?"}
{"id":"103872","title":"Explicit solutions to simple one-dimensional fluid flow problems?","text":"In order to test and verify numerical simulation results, I'm looking for explicit (or approximate) solutions to simple fluid flow problems to recalculate (like e.g. the pressure development during 1-dimensional flow through a planar crack etc.). However, I've found it quite hard to find solutions like that online, because the sources mostly either refer to really old (pre-pdf) publications or expensive textbooks. Does anybody know a good online source, that might be able to help me? Any links or other suggestions? Edit: I'm looking for solutions in a Eulerian frame of reference, since that would be the easiest to compare with simulation results where I would typically get the pressure as a function of time at distinct coordinates."}
{"id":"94107","title":"Rigid bodies - the wheel","text":"As I've been taught lately in my mechanics course: > the wheel has a unique property: at every moment of motion, the touching > point between the wheel and the ground is not in movement and therefore no > work is done by the friction force. Now, many of those problems are solved by using the 2nd Newton law and its rotational analog. For instance consider having a wheel with a mass $m$ and a radius $R$ rolling on a slope that creates an angle of $\\theta$ and we want to calculate its acceleration then we can start by writing: $$ma=mg\\sinθ−F_f$$ and the analog equation for torque: $$F_f R=I\\alpha$$. where $F_f$ is the frictional force. Now, the first equation is the 2nd Newton law applied on the centre of mass of the wheel, and as we see, one of the forces is the external frictional force. Now, though the touching point is not in movement at the moment, the center of mass is, and in the equation we assume there is a friction force on the center of mass and therefore work is done indeed. Now, after thinking about this for a while, I've come to the conclusion that this makes sense, cause if we see the wheel as point of mass located in the center, then energy is not preserved because some of it is transfered to the spin and that's why we have the second equation. > The question I'm having trouble with is whether the \"work\" of the friction > force on the center of mass is equal to the energy transfered to the spin of > the wheel?"}
{"id":"9516","title":"How close can two extremal black holes with the same charge can get?","text":"Here's a puzzle I have been pondering over. If we have two extremal black holes with the same charge, the electrostatic repulsion between them ought to cancel the gravitational attraction between them. Without any net attraction or repulsion between them, how close can we bring these two black holes to each other without merging? A peculiarity of the metric seems to suggest the event horizon is always infinitely far away for extremal black holes $\\int_R^r dr' \\frac{1}{r'-R} = \\infty$. Does this give enough elbow room for both black holes to get arbitrarily close to each other without merging?"}
{"id":"67366","title":"Is there any potential associated with magnetism","text":"Can anybody please tell me if magnetism is a conservative force or if there is a field associated with it? How to reason? One thing I know is that the work done by a magnetic force is $0$."}
{"id":"118498","title":"Is magnetic force non-conservative?","text":"If magnetic field is conservative, then why not the magnetic force? My professor thinks it is non conservative but he couldn't explain to me why?"}
{"id":"16326","title":"Work done by the Magnetic Force","text":"The magnetic part of the Lorentz force acts perpendicular to the charge's velocity, and consequently does zero work on it. Can we extrapolate this statement to say that such a nature of the force essentially makes its corresponding work independent of the choice of path, and hence that the magnetic force is conservative?"}
{"id":"9512","title":"Why is it that the splash pattern for every drop is never the same?","text":"This was just a random thought that crossed my mind, and I'm sure there is a simple physics explanation. Say we have a leaky faucet 20 feet above the ground that lets out the exact same amount of water every few seconds or so. Why is it that the splash pattern for every drop is never the same as any other drop? Now it's not like I've sat there and observed every splash, but sometimes its obvious, like a certain splash's water reached a farther distance than another splash. If we are letting the same amount of water come out everytime, and nothing else changes, why do the splashes vary? (pretend the water disappears off the ground after every splash, that way every splash has the same environment.)"}
{"id":"90612","title":"Why is a vacuum cleaner not as good heater as an electric radiator?","text":"I've read this question and answer: How efficient is an electric heater? , but still don't understand. If I have an electric radiator it heats the room with 1000 Watts of power. And I feel the room's getting warmer. In contrast, if I turn on a vacuum cleaner which consumes 1000 Watts as well as the radiator, it doesn't seem to heat the room as well. Why? Won't all kind of energy transform into heat ultimately?"}
{"id":"104810","title":"Currents and magnetic fields produce forces that affect things they're not touching?","text":"I was doing this past paper and am a little confused by question 5) part c)ii) ![Diagram from past paper](http:\/\/i.stack.imgur.com\/3MBvP.png) ![force on current carrying wire from magnetic field](http:\/\/i.stack.imgur.com\/3p3C8.png) ![how this force affects the balance](http:\/\/i.stack.imgur.com\/PzDp1.png) I correctly calculated that the force acting on the rod due to the magnetic field is 0.016N but I can't see how this could affect the scales as they're not touching. I had a look at the mark scheme but I can't understand why Newton's 3rd law would apply when the two objects aren't touching? Here is a print screen of the mark scheme for part c): ![mark scheme](http:\/\/i.stack.imgur.com\/gfeWK.png) Could anyone explain this please? Thank you :)"}
{"id":"111399","title":"Contradiction of a scalar product","text":"Can anyone resolve this contradiction: $$\\vec{r}\\cdot\\dot{\\vec{r}}=\\frac{1}{2}\\frac{d}{dt}\\left(\\vec{r}^2\\right)=\\frac{1}{2}\\frac{d}{dt}\\left(\\left|\\vec{r}\\right|^2\\right)\\equiv\\frac{1}{2}\\frac{d}{dt}\\left(r^2\\right)=r\\dot{r}, \\qquad r=|\\vec{r}|.$$ But the velocity $\\vec{v}=\\dot{\\vec{r}}$ has not to be parallel to $\\vec{r}$, so actually: $$\\vec{r}\\cdot\\dot{\\vec{r}}=r \\dot{r} \\cos{\\angle\\left(\\vec{r},\\dot{\\vec{r}}\\right)}$$ What am I doing wrong? Has anyone an idea? P.S. I have this problem from the book \"Electromagnetic Theory\" from Ferraro (p. 543)."}
{"id":"64574","title":"Definition of the entropy","text":"In physics, the word entropy has important physical implications as the amount of \"disorder\" of a system. In mathematics, a more abstract definition is used. The (Shannon) entropy of a variable $X$ is defined as $$H(X)=-\\Sigma_x P(x)log_2 [P(x)]$$ bits, where $P(x)$ is the probability that $X$ is in the state $x$ , and is defined as $0$ if $P=0$. **Question:** Can anyone explain to me why \"disorderness\" of a system defined as this? especially, where has \"$log_2$\" come from in this formula?"}
{"id":"20412","title":"A simple question about the stationary wave and fundamental frequency","text":"Suppose we have 2 fixed end connected with a wire and now we insert a vibrator **in the middle** of the wire, and resonance occur. How would the fundamental frequency looks like? I know the case when the vibrator is at one ends and another ends are fixed while in this case, there are 2 fixed point and the vibrator is at the middle. ![](http:\/\/i.stack.imgur.com\/WMmgR.png) Is the fundamental frequency like this? I imagine half of the original wire acts like a wire with a vibrator at one end and get the result. Would the fundamental frequency be different if the vibrator which is on the string is vibrating with a very large amplitude?"}
{"id":"132316","title":"My model of Conductors in \"static\" condition - Please analyze","text":"My textbook presents an idealization of a conductor as made up of infinitesimal units of charge and derives results. I was not convinced, so I started thinking of how electric fields are in real metals. Here is what I think now: i) My description of a \"static\" situation - There are electric fields inside a conductor, but no net electric field that does anything meaningful. Electrons are flying around left, right, up and down, but no net current is present. If you take a reasonably macroscopic chunk of metal, no net current will flow. ii) The entire metal is at a lower potential energy than let's say, air, that surrounds it. Therefore, there exists a work function to pull out electrons. It's like a well. You need to pull hard enough to yank an electron out. Precisely, you have to do more work than the force that holds the electron in. iii) The interior of a metal cannot have excess charge in a static condition - If the interior does have charge, the charges will repel until they reach a point of equilibrium, which means the metal isn't static. This is almost like a definition. iv) Excess charge lies at the surface of a conductor, in static situation - Since the metal is static, all the inside of the metal should behave like a regular neutral metal. For a sphere for example, this means that excess charges must form a ring of equal charge density. This is similar to saying that electric field is zero inside a ring. The excess charges will not contribute any field v) The excess charges exert a field on each other. For a sphere for example, every charge exerts charge on every other charge, so that net electric field points outside. The existence of the work function means that any attempts to pull out charges will be countered. This countering force (essentially due to protons in atoms) will counter outside field. An equilibrium is reached and therefore, those charges are more or less still. vi) The electric field inside is more or less zero, and the excess charge on the outside experiences zero field too, so the metal is an equipotential surface. vii) Any external electric field is countered using by an arrangement of charge that yields zero field inside, and these charges do not fly out because they are held in. I know there are many approximations in this model. But I think it beats the unrealistic idealization given by my textbook isn't really connected to real life. Are there any flaws in it? I will learn better models when I do solid state physics, right?"}
{"id":"48068","title":"First and Second Moment of Mass","text":"I recently came across the definition of the Center of Mass of a system as the point about which the **first** _moment of mass_ is zero. Further, it defined Moment of Inertia as the **second** _moment of mass_. My question is, What is this ' **moment of mass** '?"}
{"id":"29281","title":"Exist some relationship between irradiance units and wavelenght of the incident sunlight?","text":"Exist some relationship between irradiance units and wavelength of the incident sunlight? What about irradiance? I want to establish a relationship between wavelength and irradiance, because I would try to model photosynthesis on Vensim."}
{"id":"72080","title":"Lasing in a 2-Level system?","text":"What exactly is the difference between 2-Level, 3-Level and 4-Level systems? Why can we not achieve stimulated emission in a two-level system using optical pumping?"}
{"id":"72081","title":"First principle calculation of boiling point of water","text":"How can we theoretically calculate the boiling point of water at given pressure (other subtle parameters as well, if any)? What is the most accurate (minimum discrepancy with experimental value) computation that can analytically predict the boiling point of water? Possibly we need to invoke quantum mechanics for this. I anticipate that many answer would say that EXAXCT prediction is computationally infeasible, but please give the outline in algorithmic form regardless of computational cost. My main goal is to learn how quantum mechanics can be applied to phenomena which can be observed by layman. Other example where quantum mechanics is used to predict physical properties of small molecules from first principle are also welcome."}
{"id":"109471","title":"Does two Neodymium magnets stick together twice the power?","text":"Assume I use a coil to create B-flux and put ONE Neodymium cylinder magnet (1\"diameter & 0.25\" thick) close to the flux, it would create 10 lbs force. Does it mean that I put TWO cylinder magnets (1\"diameter & 0.25\"+0.25\" = 0.5\" thick) would create 20lb force?"}
{"id":"95779","title":"Why is $U(\\Lambda)^{-1} = U(\\Lambda^{-1})$ for a unitary representation?","text":"This is from the beginning of Srednicki's QFT textbook, where he writes (approximately): In QM we associate a unitary operator $U(\\Lambda)$ to each proper orthochronous Lorentz transformation $\\Lambda$. These operators must obey the composition rule $$U(\\Lambda'\\Lambda) = U(\\Lambda')U(\\Lambda).$$ So far OK. But where does he get the following from? $$U(\\Lambda)^{-1} = U(\\Lambda^{-1})$$"}
{"id":"68817","title":"Battery and current confusion?","text":"How exactly does a battery produce a current in the circuit connected across its ends? I dont want to know the chemical reactions in the battery core, but just the essence of it. I believe it doesn't do this by creating an excess of electrons at the -ve terminal and a deficit at the positive terminal. Moreover, how is the voltage and the EMF different in their definitions and value. Electric field being a conservative field, can the work done in motion of electrons in the conducting wire and all the components be compared to the work done in any other path across the terminals of the battery? And on a side note, how can we theoretically derive an relation between the potential difference and the electric current?"}
{"id":"59573","title":"Why a person with a further near point experience a larger magnification with a magnifier","text":"> Two people, Micah and Lyra, with different near points are equally close to > an object. Both inspect the object through the same magnifier by holding the > lens close to the eye. Micah's near point is located farther away from his > eye than Lyra's near point is located relative to her eye. Micah will > experience a larger magnification for which of the following reasons? > > The answer is: > >   * When the object is located at his near point, the angular size of the > object is smaller for Micah than for Lyra. > >   * The angular size of the image relative to the angular size of the object > at the near point is greater for Micah than for Lyra. > > * * * My question is why the 1st point? Not really able to visualize whats happening ... Also, why is \"Micah can see the image clearly from a larger distance than Lyra can.\" not correct too? He has a larger near point so he can see the image from a larger distance?"}
{"id":"59572","title":"Water entering hole at a depth, surface tension","text":"The following is the question that very commonly appears in all HS textbooks.   A hollow sphere with a hole is taken to a depth of 40cm when the water starts entering the hole. if the surface tension of water is 70dyne per cm find the radius of the hole.   All the textbooks solve it in a similar manner as given below:- When water enters the sphere, radius of the bubble of air formed, will be equal to radius of the hole say r. Then excess pressure inside the bubble equals the external pressure at the depth of 40cm and thus on equating 2T\/r to h*d*g, they find the answer. but i dont find this explanation satisfying at all. why should a bubble form only at that depth and that too of the radius of hole. cant it form at any depth and the radius depend on the depth with maximum being equal to the radius of the hole.?"}
{"id":"25070","title":"Why can we see the cosmic microwave background (CMB)?","text":"I understand that we can never see much farther than the farthest galaxies we have observed. This is because, before the first galaxies formed, the universe was opaque--it was a soup of subatomic particles that scattered all light. But before the universe was opaque, the Big Bang happened, which is where the cosmic microwave background (CMB) comes from. If the opaque early universe scattered all light, and the first few galaxies are as far back as we can see, why is the CMB observable? Where is it coming from?"}
{"id":"25888","title":"All significant objects in the universe?","text":"Taking it to the bottom of layman's terms, what would be the shortlist of significant _things_ in the universe? A list I could think of myself would put   * **Energy** (at whatever wavelength travelling through space)   * **Gas** (Atoms and molecules floating interplanetary, interstellar and intergalactic space)   * **Rocks** (Anything from dust to planets)   * **Stars** (In more general terms, anything that is or were a star... would include also black holes, magnetars and novae)   * **Galaxies**.. Things that does not fit very well in this classifications however are:   * The super massive black hole of galaxies (although I could simply put it in galaxies)   * Dark matter (but I do not worry too much until it has been properly measured :) )   * Wormholes (if they even exist..)   * Anything else What else is to be put in either list? cheers!"}
{"id":"79509","title":"How to determine the positions of two points in a radial line by an intensity level dB?","text":"The following is the question from my school. **A source emits sound uniformly in all directions. A radial line is drawn from this source. On this line, determine the positions of two points, 1.00m apart, such that the intensity level at one point is 2.00dB greater than the intensity level at the other.** I have no idea what to do because I haven't met a question about determining the positions by dB. How can I deal with this question? _**Thank you for your attention._**"}
{"id":"937","title":"How does gravity escape a black hole?","text":"My understanding is that light can not escape from within a black hole (within the event horizon). I've also heard that information cannot propagate faster than the speed of light. It would seem to me that the gravitational attraction caused by a black hole carries information about the amount of mass within the black hole. So, how does this information escape? Looking at it from a particle point of view: do the gravitons (should they exist) travel faster than the photons?"}
{"id":"69095","title":"Graviton through the horizon and force felt outside a black hole","text":"Gravitational force is mediated by graviton exchange. If I am standing outside a black hole, I can of course feel the attracting force towards the black hole. This should correspond to gravitons mediated between the matter inside the horizon and myself; but then these gravitons should cross the horizon from the inside to the outside. My question is: how is this paradox precisely solved? I guess a starting point for an answer is that these gravitons are off-shell, much like in QED where photons exchanged between electrons that feel each other are virtual. But still, this confuses me a bit - is there some references explaining this in detail?"}
{"id":"63970","title":"Is there any proof that the speed of gravity is limited?","text":"I must warn that though I'm argumenting with black holes I'm **not asking how does gravity escape the black hole!**. I want to know if the absolute speed of gravity waves were **proven** bu an experiment. We know that anything with $v\\le c$ cannot escape the event horizon of a black hole. But the gravity itself seems quite unaffected by itself. There can be the explanation that the gravity does **not** interact with itself - but no matter what, in this case it directly violates the rule that no information can leave the black hole.   This brings me to he question itself - I'm having serious doubts that the speed limit was proven for the gravity. I've found nice explanation of the problem with the black hole though: > As I understand it, to the outside observer the black hole takes an infinite > time to form, although in its own frame, the black hole forms right away. In > any case, the gravity felt by the outside observer is that of the object > becoming a black hole."}
{"id":"130979","title":"Speed of light versus pull of gravity - Is $c$ really the limit?","text":"The understanding I have is that the speed of light is considered to be the highest attainable speed in physics. Of course there are theories of tachyons but since those haven't been proven we'll dismiss them for this statement. So, thinking mathematically, the speed of everything else is relative to the maximum value (using this logic I've always had an issue with believing there IS a max value instead of just infinity). Using this line of thinking, how is it possible for some force to be able to _overcome_ this universal constant? For example, the gravity of a black hole. Shouldn't that _at most_ be able to equal the amount of force carrying a photon (or other body traveling at the speed of light) in the opposite direction? By this logic the only place that gravity could possibly even match the speed of light is at the center of the singularity, so it wouldn't trap any photons at all that weren't generated at that exact point. I guess my point is that mathematically speaking, the largest \"number\" is infinity, and you can't have \"infinity minus one\" so why should the absolute largest possible speed be able to be influenced at all, let alone in a way that it can be completely negated and then some? Doesn't that completely invalidate $c$ as being the fastest possible speed? Perhaps it's just the fastest speed we've observed? People far smarter than me have developed these theories and built whole areas of physics off of them, so I don't expect to be right. But can someone please explain to me why that's wrong?"}
{"id":"41930","title":"If nothing in the universe can travel faster than light, how come light can't escape a black hole?","text":"> **Possible Duplicate:**   >  How does gravity escape a black hole? If nothing in the universe can travel faster than light, how come light can't escape a black hole? I mean, Einstein's relativity says nothing can travel faster than light, but yet, light can't escape a black hole. Does this mean that light really _isn't_ the fastest thing? That the pull of the black hole is really _faster_ than light? That Einstein was wrong, even though it's been backed up by scientific evidence? I'm very confused. If anyone would be able to answer my question, I would appreciate it: Why can't light escape a black hole if nothing can travel faster than light?"}
{"id":"130172","title":"Blackholes gravity, why it has effect outside the event horizon?","text":"If gravity is a curvature of space time, and the event horizon is the collapse of this curvature on itself, then if the blackhole gets more matter after it is formed we should see no effect at all (i.e. increased gravity field) outside the event horizon, apart from the effect of matter collapsing before entering the event horizon itself (energy relaeased), is this correct? Does this mean that the law of conservation of matter and energy is violated? Or does this mean all the matter entering the event horizon is trasformed in energy before entering? Viceversa, if we see an effect outside the event horizon, does this mean that the interaction of gravity (gravity waves) is not following the law of \"no information outside event horizon\"? Same question if gravity is modelled with gravitons: how can the effect of increased mass be seen outside the event horizon if the gravitons are like any other particle and cannot move >c ? thanks in advance"}
{"id":"129546","title":"If gravitational radiation (or anything) cannot escape a black hole, how can it produce redshift or curve spacetime?","text":"There is an apparent paradox in a Black hole. Keenan Pepper wrote: > _Electromagnetic radiation cannot escape a black hole, because it travels at > the speed of light. Similarly, gravitational radiation cannot escape a black > hole either, because it too travels at the speed of light._ If this is true, then evidently a BH cannot exert a pull at the centre of a Galaxy, neither it can suck off the energy out of light trying to excape from it , etc? EDIT: _should I ask a new question about this_? @John Rennie, saying that gravity is curvature, seems like **shifting the problem** (,like saying that electrostatic pull is the electric field). Relativity says that mass curves spacetime, but that supposes interaction. How can spacetime _know_ that a BH is there, if it cannot communicate? a planet modifies surrounding spacetime in some way, and that , in its turn, affects bodies and light. The question now becomes: **how does a BH acts on surrounding spacetime** to make it curve, since nothing can excape it, except Hawkings radiation?"}
{"id":"45999","title":"Nothing escapes BHs, gravitons mediate gravity, so why do BHs gravitate?","text":"> **Possible Duplicate:**   >  How does gravity escape a black hole? Nothing escapes black holes, gravitons mediate gravity, so why do black holes gravitate? My question is, \"where is the hole (no pun intended!) in the above understanding?\""}
{"id":"131665","title":"Why aren't train headlights brighter?","text":"I suspect a valid scientific, physics answer for this question, because I'd venture that train, insurance companies would've calculated and contemplated this question. Yet the train headlights at http:\/\/www.youtube.com\/watch?v=0TY9Kdj5PJI don't appear to brighten\/illuminate sufficiently far ahead. If an obstacle or hindrance were in front, then wouldn't the limited range of the headlights prevent a safe stop? Also, as can be seen at the 8:15 mark, the train's lights don't corner or turn together with the train. Why not? Wouldn't they befit and help such a train? Shouldn't such heavy, menacing trains be equipped with headlights of the 'lumens' resembling http:\/\/www.youtube.com\/watch?v=jPKy1KAz8OM and http:\/\/www.youtube.com\/watch?v=bWS30yHBuLQ _(same car and driver)_ , how gleaming so ever it is for a car?"}
{"id":"67799","title":"Fiber optics with broadband, incoherent light","text":"I wonder if someone could help clarify waveguiding with broadband, incoherent light please. If we take a telecomms fiber, which is single-moded above ~1.4 μm and couple a laser beam in, we assume this excites the only mode, which is the fundamental mode. The mode-field diameter of the transverse mode is determined by the core radius and wavelength of light. I assume laser light here is coherent. But what happens if you couple light in from a broadband incoherent source (e.g. an incandescent bulb with spectrum from 1.5-3 μm)?. Can you say that a single mode will be excited containing all wavelengths from 1.5 to 3 μm? And what about if the spectrum extends from 1 to 3 μm - will there be multiple modes excited, since 1.4 μm is the cut-off for single mode behavior? Or is this picture invalid since the light is incoherent?"}
{"id":"92502","title":"Why do high current conductors heat up a lot more than high voltage conductors?","text":"120 volts x 20 amps = 2,400 Watts However, if I increased the voltage and lowered the current, you can also use a smaller wire size (more inexpensive), also have less heat and achieve the same watt Power. 1,000 volts x 2.4 amps = 2,400 Watts   1. Why doesn't it heat up like current?   2. To me this approach seems more efficient and less costly because you don't use as much material, so why isn't this common?"}
{"id":"23066","title":"Experiment to find structure of water","text":"Who first determined the structure of water (two hydrogen atoms stuck to an oxygen atom at approx 105 degrees), and, more importantly, how was this done?"}
{"id":"23062","title":"How to calculate the evaporative cooling rate needed to protect a house from forest fire","text":"Recently in our area there has been a large forest fire and I've been looking into home defense from such things. I am not a physicist - but can do some basic math. I was wondering how I could calculate if a 'evaporative mister' type system could be used for such to reduce the ambient air temp to stop the house catching fire (dont care about cosmetic damage). The goal would probably be to reduce temperature of air\/surfaces of the house by approx 1000F to keep them below combustible temperatures. The area is very low humidity between 4% to maybe 15% during wildfire season. How can I calculate how much water\/mist I need to put into the air to reduce temperature below 400F. Very rough simplest equations are fine - I know its not an exact science when dealing with wildfires. I found this formula on wiki but I don't know how to adapt it to use it to calculate water need for a temp drop of               TLA = TDB – ((TDB – TWB) x E)     TLA = Leaving Air Temp     TDB = Dry Bulb Temp     TWB = Wet Bulb Temp     E = Efficiency of the evaporative media.       Anything I forgot\/missing - be appreciated to know. Some restrictions\/thoughts I had   1. The roof would need to stay generally wet light continuous layer of water   2. Misting\/irrigation could not use more then 12 gallons per minute (max well output)   3. Be nice to second use the misting system for outdoor aircon on off time (ie 20% capacity)   4. Windows\/glass would need some of IR shielding to stop ignition of furniture inside the house."}
{"id":"24283","title":"What would be the path of the earth seen by the astronaut in the lunar sky?","text":"An astronaut camps on the moon for a period of one month as per the earth’s calendar. What would be the path of the earth seen by the astronaut in the lunar sky? (A) The earth remains approximately at a fixed altitude and direction. (B) The earth completes one revolution parallel to the lunar horizon in one month. (C) The earth completes one revolution from east (direction of rising sun) to west (direction of setting sun) in one month. (D) The earth completes one circle around the Pole Star in one month but never goes below the horizon."}
{"id":"66226","title":"A different proof for 6 degrees of freedom","text":"I want a different proof of 6 degrees of freedom of a solid object made of $\\ N$ particles. I am thinking along these lines: Definition of rigid body is $\\ modulus[\\vec{r_i}-\\vec{r_j}]=constant \\ \\forall\\ i,j$ This gives me $\\ ^NC_2$ constraints. There exist in total $\\ 3N$ equations. So the number of free variables should be $\\ n= 3N- \\ \\ ^NC_2=\\frac{N(5-N)}{2}$ Which is clearly not the answer as $\\ n$ is $\\ N$ dependent, but it should be $\\ 6$. **What I want to do is show that :** $$\\ number\\ of\\ constraints \\ actually\\ required= 3N-6$$ which is the correct answer since I know $\\ n=6$ _I am aware of the proof given in Goldstein, Rana Joag etc. What I am asking is how to do it following this approach._"}
{"id":"67071","title":"Spin-Statistics Theorem - Proof","text":"Is this proof of spin-statistics theorem correct? http:\/\/bolvan.ph.utexas.edu\/~vadim\/classes\/2008f.homeworks\/spinstat.pdf This proof is probably a simplified version of Weinberg's proof. What is the difference? What is the physical meaning of $J^{+}$ and $J^{-}$ non-hermitian operators? I'm especially interested in the beginnig of proof of second lemma. How to get this: \\begin{eqnarray} F_{AB}(-p^{\\mu}) = F_{AB}(p^{\\mu})\\times (-1)^{2j_{A}^{+}} (-1)^{2j_{B}^{+}} \\\\\\ \\nonumber H_{AB}(-p^{\\mu}) = H_{AB}(p^{\\mu})\\times (-1)^{2j_{A}^{+}} (-1)^{2j_{B}^{+}} \\end{eqnarray} Also why under CPT field transform as \\begin{eqnarray} \\phi_{A}(x)\\rightarrow \\phi_{A}^{\\dagger}(-x) \\times (-1)^{2J_{A}^{-}} \\\\\\ \\nonumber \\phi_{A}^{\\dagger}(x) \\rightarrow \\phi_{A}(-x) \\times (-1)^{2J_{A}^{+}} \\end{eqnarray} conjugation is from charge reversal, - from space inversion and time reversal. What about $(-1)^{2J_{A}^{-}}$?"}
{"id":"61792","title":"What is the coefficient $\\mu_\\text{air}$?","text":"I am answering some exercises in thermodynamics, and I am still in trouble with some values and constants. In the exercise, it is given: $\\mu_\\text{air}=1.8\\cdot10^{-5}~\\text{kg\/m s}$. What for is this value used? In the case, I need to find some tensions in given points, and I know the speed profile of the air. Other question: As I need the tension in some points, and it is about air, is it the same thing as pressure?"}
{"id":"61809","title":"Two masses attached to a spring","text":"I'm trying to understand the solution of the following problem. > Two masses $m_{1}$ and $m_{2}$ slide freely in a horizontal frictionless > track and are connected by a spring whose force constant is $K$. Find the > the frequency of oscilatory motion for this system. In the solutions manual, it is considered that $x_{1}$ and $x_{2}$ are the coordinates of $m_{1}$ and $m_{2}$, respectively, and the length of the spring at equilibrium is $l$. Then, it is defined that, the equations of motion for each mass are: $$ m_{1}\\ddot x_{1} = -k(x_{1} - x_{2} + l), \\\\\\ m_{2}\\ddot x_{2} = -k(x_{2} - x_{1} - l).$$ And, with some algebraic manipulations, we arrive at the answer. My question is about the right hand side of these two equations. Why is the displacement in the restoring force equal to the difference between the positions plus the length of the spring in the first equation and minus on the second? What is the behavior of the system during the move? What changes in the analysis when I consider the frictional force? Could I consider that this motion has some relationship with the center of mass of the system?"}
{"id":"62216","title":"Period of oscilation","text":"Two masses $m_1$ and $m_2$ are connected by a spring of spring constant $k$ and slide freely without friction along horizontal track. What is period of oscillation? No force influence."}
{"id":"28660","title":"Is the electric field zero inside an ideal conductor carrying a current?","text":"By an ideal conductor, I mean one with zero resistance. Inside an ideal conductor with no current, the electric field is zero, but is the electric field still zero with the ideal conductor carrying a current?"}
{"id":"23396","title":"Maximize Magnetic Self-Inductance Through a Wire","text":"You are given a long length W of copper wire. How would you arrange it to obtain the maximum self-inductance? Why? I am trying to use the equation $$L=\\mu_o n^2 l A$$ I try to solve it using a fixed length wire of 10 units, width 1mm and winding it into a solenoid. I plug in values of circumference 10, 5, 2.5 and finding the inductance through number crunching. However, I am getting a larger values for multiple loops but the answer is a single loop (ie. a circle) rather then a solenoid. Here are the sample values I got: $n=1; C=10; r=1.59; L=0.079 \\mu_o$ $n=2; C=5; r=0.79; L=0.156\\mu_o$ $n=4; C=2.5; r=0.3978; L=0.318\\mu_o$ If anyone could enlighten me on the proper way of solving this, I'd appreciate it."}
{"id":"74906","title":"Blasius boundary layer solutions","text":"I'm trying to understand the Blasius boundary layer solution, but I'm having some difficulties. Using wikipedia, I wonder how they get the first formula: http:\/\/en.wikipedia.org\/wiki\/Blasius_boundary_layer. $$\\frac{U^2}{L}\\approx\\nu \\frac{U^2}{\\delta^2}$$ And how they get to the fourth formula: $$\\delta(x) \\approx \\sqrt{\\frac{\\nu x}{U}}$$ I feel that those formulas are correlated somehow, but I don't really see how they derive those. I hope someone can help."}
{"id":"74903","title":"container with low pressure inside of the high pressure container","text":"How will act low pressure container, or what would happen inside the low pressure container when it is placed inside the container with high pressure? And if it is high pressure container inside the high pressure container? Will the container which is inside stay in the centre?"}
{"id":"74902","title":"accelerated charged particles and interaction with magnetic field","text":"In high school we are taught that magnetic field perpendicular to velocity of an charged particle experience perpendicular force that causes it to move in circular path by relation $$qvB=\\frac{mv^2}{r}$$ but in drawbacks of Bohr's theory it was proposed that accelerated charged electron orbiting along nucleus will immediately loose energy in form of electromagnetic waves and collapse into nucleus.So My question basically is are we taught wrong about this relation that charged particle according to Lorenz's force will perform circular motion as far as required conditions in equations are provided but will it eventually loose energy in form of electromagnetic radiation and halt it's circular orbit ?"}
{"id":"61249","title":"Potential energy during vertical fall","text":"Suppose I have a weightless spring connected perpendicularly to the ground, and it has on top of it some weightless surface. Now, I release some sticky object from height $h$ above the system of light spring-surface. The object eventually hits the surface and the spring is starting to contract till all the kinetic energy of the object is transformed to elastic potential energy. And the system continues to oscillate harmonically. I want to find the maximal contraction of the spring, therefore, I did: (we assume that no energy is lost during the collision) $U_{GR}=E_{k,max}=U_{SP, max}$ $mgh=\\frac{1}{2}kx_{max}^2$ where $k$ is the stiffness coefficient. Therefore: $x_{max}=\\sqrt{\\frac{2mgh}{k}}$ I've chosen my 0 level for gravitational potential energy to be on the same level as 0 level of potential elastic energy. Suppose I want to choose another 0 level for GPE, e.g. at maximum contraction. Then my equation would be: $mg(h+x_{max})=\\frac{1}{2}kx_{max}^2$ It obviously leads us to a slightly different $x_{max}$ value. According to the numbers in the answer in my book it seems like the second equation is correct. My question is - how then correct equations should look like if I want to arbitrary choose my PE 0 level at any point? Or why the first one is wrong? Also, I'm wondering why the amplitude of the oscillations is not $x_{max}$ (according to the book it is slightly less the correct $x_{max}$ value, even though we assumed that no kinetic energy was lost. Interesting coincidence: the amplitude on the contrary, according to the book, equals to the $x_{max}$ value of my first equation)."}
{"id":"73028","title":"The relation between Gauss's law and Coulomb law and why is it important that the electric field decrease proportionally to $\\frac{1}{r^{2}}$?","text":"My question relates to the third MIT's video lecture about Electricity and Magnetism, specifically from $21:18-22:00$ : http:\/\/youtu.be\/XaaP1bWFjDA?t=21m18s I have watched the development of Gauss's law, but I still don't quite understand the link between Gauss's law and Coulomb law: How does Gauss's law change if Coulomb law would of been a different one. I also don't understand why is it so important for Gauss's law that the electric field decrease proportionally to $\\frac{1}{r^{2}}$ ? For example, what would of happened if the electric field decrease proportionally to $\\frac{1}{r}$ , or $\\frac{1}{r^{3}}$ ?"}
{"id":"31954","title":"How we can find the kinetic energy?","text":"A solid disk of mass m is rolling along a surface its center has velocity v what is the kinetic energy of disk?I cannot solve the problem,"}
{"id":"55307","title":"Why is there a potential difference?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/R70Fu.png) The question then asks for the potential difference between $X$ and $Y$, which is claimed to be $3.6\\text{ volts}%$. Why would there be a potential difference in this case? If I connect a lightbulb on $X$, and another on $Y$, it seems very obvious that the brightness would be the same. Or is the standard direction of current important? So the current \"encounters\" the 60 ohm \"first\" when flowing to $X$, and the 30 ohm \"second\" when flowing to $Y$? That seems a little crackpot to me. So if we change the resistor to the left of $X$ to 100 ohms, and the resistor to the right of $Y$ to be 0 ohms, the differing definitions of current flow would actually cause a difference in measurement of potential?!?! The correct answer is worded: ![enter image description here](http:\/\/i.stack.imgur.com\/kpT1i.png)"}
{"id":"100902","title":"Speed to run a loop","text":"So this guy was the first to run a loop and in this (german) article (and also in the video) a certain speed (13.8km\/h) is mentioned. Why must he run at this speed and not just \"as fast as possible\"? My intuition says it has to do with to much centrifugal force and him not being used to it $\\rightarrow$ trip hazard."}
{"id":"98424","title":"phospholipid bilayer","text":"I'm a high school student. The head is hydrophilic, the tail is fatty acid,in other words hydrophobic. Here is the thing I don't understand, all textbooks state that water is repelled by hydrophobic tail, why? The hydrophilic head is composed of polar molecule,and water is polar,so they will attract each other. But why hydrophobic tail will repel polar molecule?Isn't hydrophobic tail is just composed of nonpolar molecules?"}
{"id":"4619","title":"Water pressure in free fall","text":"The increasing water pressure as you go deeper is generally explained in terms of the weight of the water column above the observation point pressing down. The question, then, is what would happen if you had a big blob of water in free fall, say 100m in diameter-- not big enough to produce large gravitational forces on its own-- and swam to the center of it. I'm imagining some sort of contained environment, here-- a giant space station, or the planet-sized envelope of air in Karl Schroeder's Virga novels, that sort of thing-- so you don't worry about the water boiling off into vacuum or freezing solid, or whatever. Would there be any pressure differential between the surface of the blob and the center of the blob? At first glance, it would seem not, or at least not beyond whatever fairly trivial difference you would get from the self-gravity of the water. But it's conceivable that there might be some other fluid dynamics thing going on that would give you a difference. For that matter, what difference would you expect between the pressure inside the water and outside the water? We know from shots of astronauts goofing around that water in free fall tends to stay together in discrete blobs. This is presumably some sort of surface tension effect. Does that lead to a higher pressure inside the water than out? How much of a difference would that be? Or would you not particularly notice a change from sticking your head into a blob of free-falling water? (Other than, you know, being wet...) (This is just an idle question, brought on by thinking about the equivalence principle, and thus free-falling frames. It occurred to me that somebody here might know something about this kind of scenario, so why not post it?)"}
{"id":"96390","title":"Why does ebonite rod gets negatively charged when rubbed with fur","text":"What I think is : The protons are present at the centre of the atom with rotating electrons around it so when it is rubbed by fur the electrons get passed from the ebonite rod to the fur leaving the rod negatively charged. But in reality the rod is negatively charged Why ?"}
{"id":"103292","title":"electrons in elecrtostatic force(please read specification given below)","text":"when comb is rubbed with hair, and brought near small pieces of papers, small paper pieces gets a towards comb. Here the comb gets negatively charged and when brought near piece of paper where do electrons go because after sometime attraction of paper pieces towards comb stops and comb doesnt have negative charge then. Why does it happen so ? What is the reason for it ? If a glass rod is rubbed with silk, rodna and silk gets positive or negative charge respectively, an when they are seperated after charge gets nuetral. Why does it happens? What is the reason behind this ?"}
{"id":"44486","title":"How does rubbing cause the transfer of electrons from one object to the other?","text":"I have just learnt about electrostatics. Why would there be a transfer of electrons? Is it because of the difference of the materials (i.e. triboelectric series)? So in the case of two different materials **contacting** each other, will there still be transfer of electrons?"}
{"id":"23934","title":"understanding the oscillating part of the Gutzwiller trace","text":"given the density of states according to Gutzwiller's trace formula $ g(E)= g_{smooth}(E)+ g_{osc}(E) $ i know that the 'smooth' part comes from $ g_{smooth}(E)= \\iint dxdp \\delta(E-p^{2}-V(x)) $ for one dimensional system however how it is the oscillating part of the trace obtained ?? :D i mean the sum over lenghts of the orbit (in the phase space) also how does the condition for WKB energies appear ?? $ \\oint _{C} p.dq= 2\\pi \\hbar (n+ \\alpha) $ from the Gutzwiller trace ??"}
{"id":"118799","title":"can silence happens when 2 sound waves destroy each other","text":"Hi is there any possibility that you located between 2 sound sources and u hear nothing? as we know 2 wave in opposite direction will destroy each other..."}
{"id":"55318","title":"Where does energy go in destructive interference?","text":"I have read that when two light waves interfere destructively, the energy contained within is transferred to other parts of the wave which have interfered constructively. However, I am having some trouble grasping this. While in experiments such as Young's Double Slit experiment, there are visible bright bands of higher energy, I would imagine that it be possible to configure light waves to propagate linearly such that the waves interfere only destructively and not at all constructively. Is such an arrangement possible? And if so, to where is the energy in the wave transferred? Similarly, how does the energy transfer from one part of a wave which is interfering destructively to another part which is interfering constructively? These regions may be several meters apart for long wavelength light, and I find it strange that energy can travel between these potentially distant and non-interacting regions."}
{"id":"101056","title":"How accurately can we measure human electromagnetic fields?","text":"How accurately can our current technological tools measure the human bio- electromagnetic field emitted by a person? Or, to put it differently, does each person have a different electromagnetic field signature, and can we measure that with enough accuracy to tell two people apart by their frequencies?"}
{"id":"104028","title":"Numerical aperture of a lens","text":"Using a laser setup, I was asked to determine the aperture of a given lens and then use some geometrical arguments and compare the theoretical value from the manufacturer and the experimental value. However, when I did this, the values _differed by orders of magnitude_. The way I got the aperture was to let the laser light pass through the lens onto a white screen and then keep the screen at a fixed distance from the lens and measure the radius of the light spot obtained. Then I used geometry to determine the angle and thus its sine. Could diffraction be the reason of such a large error?"}
{"id":"65945","title":"How can the accurate value of electric field intensity be calculated?","text":"When we calculate electric field intensity for a point charge at any point inside electric field the field intensity is $E = F\/q$ where $F$ is the force acting on charge $q$. In this case, the charge $q$ should be very small. The practical value of $q$ cannot be so small as needed. In defining electric field by measuring value $F\/q$ is smaller than the actual value. **My question:** How can the accurate value of electric field intensity be calculated?   I'm quite confused. Can someone point me out?"}
{"id":"65947","title":"Deriving the Angular Momentum Commutator Relations by using $\\epsilon_{ijk}$ Identities","text":"I've been trying to derive the relation $$[\\hat L_i,\\hat L_j] = i\\hbar\\epsilon_{ijk} \\hat L_k $$ without doing each permutation of ${x,y,z}$ individually, but I'm not really getting anywhere. Can someone help me out please? I've tried expanding the $\\hat L_i = \\epsilon_{nmi} \\hat x_n \\hat p_m$ and using some identities for the $\\epsilon_{ijk} \\epsilon_{nmi}$ which gives me the LHS as something like $-\\hbar^2\\delta_{ij}$ but I've got no further than this."}
{"id":"65940","title":"How to design a deliberately biased coin?","text":"For demonstrating basic probability concepts, it would be nice to have a coin- like object that lands heads\/tails not in 50\/50% ratio, but biased in a way that can be revealed in a short experiment. What I'd like is to make an object satisfying:   * Thin disk shape, say thickness around 1\/10 to 1\/20 of diameter.   * Lands heads\/tails with some given lopsided ratio such as 60\/40.   * Feels evenly balanced to the students' hands.   * Can be made by the average machinist, hobbyist woodworker, or 3D printing designer.   * Size not important, but maybe 10-20cm diameter. It's meant to be a theatrical prop visible to a small audience, not an actual coin. Would a disk with an interior hollow zone closer to one side than the other do the job? I doubt it, since the coin will still rotate uniformly in the air, exposing both sides equally to any direction, including the floor. Making both sides \"heads\" is too obvious a cheat, and I don't want 100\/0% probabilities anyway. Note I'm not asking for any practical how-to workshop details, just the physical principle for designing such an object."}
{"id":"27220","title":"Why\/When can the gauge superfield and\/or chiral superfield kinetic term in $(2,2)$ SUSY be ignored?","text":"This is in reference to the argument given towards the end of page $61$ of this review paper. There for the path-integral argument to work the author clearly needed some argument to be able to ignore the kinetic term(Kahler potential). But I can't get the argument fully. Just above equation 5.5 in that review the author says, _\"..since the Kahler potential is irrelevant we may choose it at will,and in particular we may choose it very small. To a first approximation, in fact, we may ignore it...\"_ And on Page 62 in his point 1, he says, _\"..Nothing in our argument assures that the resulting kinetic term in the Calabi-Yau sigma model will be sufficiently “large” to ensure that sigma model perturbation theory will be valid..\"_   * It would be great if someone can help make precise as to what the above argument means (..espeically the second part about \"alrge\"..) and flesh out its contents in may be an equation form or something.. Conceptually I would think that trying to ignore the kinetic term of the chiral fields is _different_ than trying to ignore the kinetic term of the gauge fields. like say as written in equation 2.22 on page 11 of this paper.   * Now one can see that in the above paper (unlike in the initially linked review) the author is trying to ignore the gauge kinetic term by trying to take the gauge coupling $e \\rightarrow \\infty$ limit as in Page 30 of the above paper by Witten.  ..but naively one might think that large gauge coupling is precisely the region where perturbative arguments would have started to fail and the entire idea of trying to minimize the classical potential would have started to become bad..   * But unlike in the review as linked to earlier, Witten's argument crucially needs the kinetic terms of the scalar chiral superfields (..from where the $F^2$ and the Yukawa terms come..). And Witten's argument also crucially needs the gauge kinetic term (from there the $D^2$ terms come..) Then why is he trying to justify the $e \\mapsto \\infty$ limit? Like he comments towards the end of Page 60, _``..the effects of finite e, are believed to be “irrelevant” in the technical sense of the renormalization group..\"_ It would be great if someone could explain these two arguments about ignoring the kinetic term of the gauge fields and the chiral fields."}
{"id":"25573","title":"How to make a moondial?","text":"A question came up on Outdoors StackExchange, How to tell time at night. I wrote an answer to that question, and in my answer I said that a standard sundial wouldn't work without some fiddling, but that an astronomer would know what fiddling was required. (I invite\/encourage any qualified astronomers to go there and answer the question, BTW.) So I did some searching around, and some thinking about it. Here's the question. Suppose we have an ordinary horizontal sundial, at a fixed latitude (in the Northern hemisphere), which has been calibrated to correctly show solar time at that latitude. The sundial _works_. Now, suppose there's a Full moon. What adjustments would be necessary, what additional information is needed, what would it take to transform\/convert the ordinary sundial into a moondial? The definition of \"success\" would be that we could look at the shadow and get the \"correct\" time, where correct might mean up to the precision of an ordinary sundial which might neglect the Equation of Time (is a different equation needed?). One website I read said that no adjustment was necessary, but I'm not sure I agree. If that's not too bad, how would the answer change if we were some days _off_ of a full moon? (Obviously, if we are sufficiently far off then there will not be enough light to make a discernible shadow.) **EDIT:** I've been thinking about it a lot, and thinking about the E.o.T.. The EoT has two (dominant) parts: the Kepler eccentricity part, and the oblique axis part. At first I thought the Kepler part wouldn't matter, but then I learned that the Moon\/Earth eccentricity (0.055) is even bigger than the Earth\/Sun eccentricity (0.017). So it would seem any reliable moondial would need to be corrected for this problem. I'm still wrapping my head around the obliqueness part. I understand that the style of the gnomon must be angled from the horizontal (equal to the latitude), but that's to make sure the style points toward true north. In fact, even if a horizontal sundial is used at an incorrect latitude, we can fix it by angling the whole sundial so the style points north. Clearly the Moon is in another orbital plane (apparently by about 5 degrees, I'm guessing that's an average), but if we change the angle of the style to compensate for the Moon then it wouldn't be pointing north anymore. So, at the moment, I'm thinking to leave the sundial as-is and then figure (somehow) the sine-wave that describes the Moon's obliqueness. The further this goes the more it looks like I'm talking about figuring an Equation of Moon Time or similar."}
{"id":"128388","title":"What happens when Dark Matter comes in contact with the event horizon of a large Black Hole","text":"Josh Hill, 9, Oakdale Elementary has always talked Theory of Relativity and Astrophysics etc., I can answer most but lately he has stumped me and has been begging me to ask a pro, so here it is.... \"What happens when Dark Matter comes in contact with the event horizon of a large Black Hole?” Josh Hill & Mike Hill (DAD) P.S. we should probably anticipate “ What happens when Dark Energy comes in contact with a Black Hole?” as the follow up question , just so we kill 2 birds with one stone , Thanks."}
{"id":"70590","title":"periodic boundary conditions for vortex in a square lattice","text":"I am trying to follow this paper and track the dynamics of vortex motion on a discrete (square) lattice. The idea is to simulate the time evolution of the Gross-Pitaevskii (GP) equation, which reads (in rescaled units) $$i \\partial_t \\psi = - \\nabla^2 \\psi + |\\psi|^2 \\psi.$$ As initial condition one takes a wavefunction of the form of Eq. (10) in the above-mentioned paper, $$\\psi(z) = \\prod_{z_+ \\in [Z_+]} \\frac{(z-z_+)}{|z-z_+|} \\prod_{z_- \\in [Z_-]} \\frac{(z^*-z_-^*)}{|z-z_-|}$$ where $z=x+iy$ are complex coordinates. This wavefunction describes a set of vortices (V) with coordinates $z_+$ in $Z_+$ and a set of anti-vortices (AV) with coordinates $z_-$ in $Z_-$. This is equivalent to using a wavefunction of the type $e^{i \\phi}$ (polar coords.) for vortices and $e^{-i \\phi}$ for anti- vortices, with $\\phi$ the polar angle. One can calculate the velocity as gradient of the phase of the complex wavefunction. An example of the associated velocity field, for a V at (x,y)=(0,2.5) and AV at (0,-2.5) is shown below. ![velocity-lines](http:\/\/i.stack.imgur.com\/nLZqZ.png) This should correspond to Fig. 1 in the paper. However, there are several things I do not understand. First, why do the authors say that they cannot look at the dynamics of just one vortex, but must include an anti-vortex in the initial conditions to satisfy the periodic boundary conditions? Second, apart from the initial V-AV pair, they also seem to include \"a few mirrow images with respect to the boundary\". However, they do not specify how many nor where they are positioned. (I assume that the mirrow image of a V is and AV and vice-versa, please correct me if I'm wrong). Lastly, they mention \"evolving the system in imaginary time\" in order to \"cool it down\" (although no temperature is included in the model!) and then \"turning on a superflow\" and continuing the evolution in real-time. I though all was needed was the discretization of the GP equation and following its evolution given the initial wavefunction and periodic boundary conditions ($\\psi(\\text{left margin})=\\psi(\\text{right margin})$ and $\\psi(\\text{top margin})=\\psi(\\text{bottom margin})$). **Edit** Following the suggestion of @BebopButUnsteady, I repeated the 'unit' of my system, the V-AV pair, until obtaining the following tiling pattern (vortices are circles, AVs are squares) ![tiles](http:\/\/i.stack.imgur.com\/OQSXa.png) Let us first look at the real (left) and imaginary (right) parts of the wavefunctions for the initial V-AV pair: ![initial-boundaries](http:\/\/i.stack.imgur.com\/ECgmG.png) One can see that, especially in the imaginary part, the values on the boundaries are not equal. Now we start adding copies of the \"unit cell\" along the x axis, and plot a cut though the imaginary part of the wf: ![boundaries-tiled](http:\/\/i.stack.imgur.com\/VVDSm.png) It appears that, as we increase the number of copies, the values at the ends get closer and closer to 0, which would be the ideal periodic case. Of course this is just numerics, I have no formal proof yet that this procedure actually converges."}
{"id":"70591","title":"Intensity of particles with zenith angle dependence","text":"The intensity of particles come to sea level depend to the zenith angle $I=Iv (\\cos(z))^n$, where $z$ is the zenith angle and $n$ depend to the particle. I know the vertical intensity depend to the energy or momentum of the particle, for exemple, to muons, $n$ is about 2 and vertical intensity is $0.0094\\, \\mathrm{cm}^{-1}\\,\\mathrm{s}^{-1}\\,\\mathrm{sr}^{-1}$ to $>0.35 \\mathrm{GeV\/c}$. I would like to know what is the vertical intensity and the $n$ for electrons, positrons, protons and hadrons."}
{"id":"7073","title":"Why isn't the wave equation $\\nabla^2 \\psi - 1\/c^2 \\partial_{tt} \\psi = (\\frac{mc}{\\hbar})^2\\psi$","text":"Special relativity was well established by the time the schrodinger equation came out. Using the correspondence of classical energy with frequency and momentum with wave number, this is the equation that comes out, and looks sensible because this is of the form of a wave equation, like the one for sound etc, except with an inhomogeneous term $$\\nabla^2 \\psi - 1\/c^2 \\partial_{tt} \\psi = (\\frac{mc}{\\hbar})^2\\psi$$ Instead, we have schrodinger's equation which reminds of the heat equation as it is first order in time. **EDIT** Some thoughts after seeing the first posted answer What is wrong with negative energy solutions? When we do freshman physics and solve some equation quadratic in, say, time... we reject the negative time solution as unphysical for our case. We do have a reason why we get them, like, for the same initial conditions and acceleration, this equation tells usthat given these iniial velocity and acceleration, the particle would have been at the place before we started our clock, since we're interested only in what happens _after_ the negative times don't concern us. Another example is if we have areflected wave on a rope, we get these solutions of plane progressive waves travelling at opposite velocities unbounded by our wall and rope extent. We say there is a wall, an infinite energy barrier and whatever progressive wave is travelling beyond that is unphysical, not to be considered, etc. Now the same thing could be said about $E=\\pm\\sqrt{p^2c^2+(mc^2)^2}$ that negative solution can't be true because a particle can't have an energy lower than its rest mass energy $(mc^2)$ and reject that. If we have a divergent series, and because answers must be finite, we say.. ahh! These are not real numbers in the ordinary sense, they are _p-adics_! And in this interpretation we get rid of divergence. IIRC Casimirt effect was the relevant phenomenon here. **My question boils down to this**. I guess the general perception is that mathematics is only as convenient as long as it gives us the answers for physical phenomenon. I feel this is sensible because nature can possibly be more absolute than any formal framework we can construct to analyze it. How and when is it OK to sidestep maths and not miss out a crucial mystery in physics."}
{"id":"105370","title":"What parity has an electron?","text":"I couldn't find anything about the parity of an electron. Neither in the german, nor in the spanish and nor in the english version of Wikipedia. I only found one sentence in the parity article of Wiki: > One way to fix a standard parity operator is to assign the parities of three > particles with linearly independent charges B, L and Q. In general one > assigns the parity of the most common massive particles, the proton, the > neutron and the electron, to be +1. - Wiki But I cannot find other sources to confirm it. I do not need it for a special exercise, it's just to understand the whole thing..."}
{"id":"123522","title":"A simple way of calculating Euler Angles from Rotation Matrix --- help!","text":"This is a follow up of this question : I have the rotation matrix $$ \\left( \\begin{matrix} a_{11} & a_{12} & a_{13}\\\\\\ a_{21} & a_{22} & a_{23}\\\\\\ a_{31} & a_{32} & a_{33}\\\\\\ \\end{matrix}\\right) $$ I'm using pre-multiplying rotation matrix (that operates on column vectors) for intrinsic rotations (i.e. I make rotations about the axes of the plane that rotates). And since the fixed frame is my reference frame --- $$ \\left( \\begin{matrix} 1 & 0 & 0\\\\\\ 0 & 1 & 0\\\\\\ 0 & 0 & 1\\\\\\ \\end{matrix}\\right) $$ My rotation matrix is nothing but the column unit-vectors of the axes of the rotated frame, i.e. $$ \\left( \\begin{matrix} x_{1} & x_{2} & x_{3}\\\\\\ y_{1} & y_{2} & y_{3}\\\\\\ z_{1} & z_{2} & z_{3}\\\\\\ \\end{matrix}\\right) $$ So therefore I have the values of a11,a12,a13,a21,a22,a23,a31,a32,a33 as x1, x2, x3, y1, y2, y3, z1, z2, z3. Now if I consider a particular set of rotation (say X first, then Y , then Z), with the corresponding Tait-Bryan angles \\--- a,b and c. My rotation matrix will be the following. Rx(a)*Ry(b)*Rz(c) \\--- $$ \\left(\\small{ \\begin{matrix} \\cos(b)\\cos(c) & -\\cos(b)\\sin(c) & \\sin(b)\\\\\\ \\cos(a)\\sin(c) + \\cos(c)\\sin(a)\\sin(b) & \\cos(a)\\cos(c) - \\sin(a)\\sin(b)\\sin(c) & -\\cos(b)\\sin(a)\\\\\\ \\sin(a)\\sin(c) - \\cos(a)\\cos(c)\\sin(b) & \\cos(c)\\sin(a) + \\cos(a)\\sin(b)\\sin(c) & \\cos(a)\\cos(b)\\\\\\ \\end{matrix}} \\right) $$ Now if I have to solve for the above angles a, b & c (pitch, yaw and roll), I basically have nine equations but three unknowns. Following are the equations ---               a11 =  cos(b)∗cos(c)     a12 = −cos(b)∗sin(c)     a13 =  sin(b)     a21 = cos(a)∗sin(c)+cos(c)∗sin(a)∗sin(b)     a22 = cos(a)∗cos(c)−sin(a)∗sin(b)∗sin(c)     a23 = −cos(b)∗sin(a)     a31 = sin(a)∗sin(c)−cos(a)∗cos(c)∗sin(b)     a32 = cos(c)∗sin(a)+cos(a)∗sin(b)∗sin(c)     a33 = cos(a)∗cos(b)       This is where I learnt about it. Now I am using a non-linear least squares curve fitting method to solve the above set of over-determined equations. There are two major problems that I am encountering   1. The final values of a,b, c change as I change the initial values in the iterative algorithm. I get different results if I start from [50 50 50] and different results with [0 0 0].   2. Secondly I don't think that the values obtained are correct; since the angles of pitch, yaw & roll seem pretty much different in the video. I'm using the lsqcurvefit(click for the question I asked on stackoverflow) command in Matlab. (click for documentation) I have been on this problem of how to calculate pitch, yaw & roll for quite some time now. I think this post will give you all the details of what I have tried. I need your help to know if what I am doing is the best approach to tackle my problem. If so, please point out what is wrong in my method. If you think there are other simpler methods, please let me know about them. I'm sure there has to be a better method, since this seems like a pretty simple thing to do. Should I change my Matlab algorithm? Anyone know any special Matlab\/Mathematica toolbox that calculates the yaw, pitch, roll? Thanks!"}
{"id":"123526","title":"Kallen–Lehmann spectral representation for an arbitrary spin","text":"Let's have Kallen–Lehmann spectral representation for the scalar theory: $$ \\tag 1 D(p) = \\int \\limits_{0}^{\\infty} d(\\mu^{2})\\frac{\\rho (\\mu^{2})}{p^{2} - \\mu^{2} + i\\varepsilon}. $$ We can represent $(1)$ in a form $$ D (p) = D_{free}(p, m)Z + \\int \\limits_{m^{2}}^{\\infty}d(\\mu^{2})\\rho (\\mu^{2})D(p, \\mu), $$ where $\\mu$ refers to the mass of the multiparticle states and $m$ refers to the mass of one-particle free state. From here there is interesting result: it can be showed that $0 \\leqslant Z \\leqslant 1$. How to get analogous result for the arbitrary spin field (or at least for fermions with spin $\\frac{1}{2}$ and for photons)? I know the general structure of the propagator of the free theory (for simplicity I assume massive case): for field $\\Psi $ with spin $s$ $$ \\hat {\\Psi}_{l} (x)= \\sum_{\\sigma = -s}^{s}\\int \\frac{d^{3}\\mathbf p}{\\sqrt{(2 \\pi)^{3}2E_{\\mathbf p}}}\\left( u^{\\sigma}_{l}(\\mathbf p )e^{-ipx}\\hat {a}_{\\sigma}(\\mathbf p ) + v^{\\sigma}_{l}(\\mathbf p )e^{ipx}\\hat {b}^{\\dagger}_{\\sigma}(\\mathbf p ) \\right) $$ it will be $$ D_{lm}(p) = \\frac{F_{lm}(p)}{p^{2} - m^{2} + i\\varepsilon}, \\quad F_{lm} = \\sum_{\\sigma}u^{\\sigma}_{l}(\\mathbf p )\\left(u^{\\sigma}_{m}(\\mathbf p) \\right)^{\\dagger}. $$ For the derivation of spectral representation for the scalar field see the reference above. Here also you can see the problem with this derivation which appears for the case of field with nonzero spin: it is impossible to introduce the scalar density function $\\rho (\\mu^{2})$. Also there is bigger problem: there was using relation $\\langle |[\\hat {\\Psi}(\\mathbf x , t), \\frac{d}{dt}\\hat {\\Psi}^{\\dagger}(\\mathbf y, t)]|\\rangle = i\\delta (\\mathbf x - \\mathbf y)$ for getting relation $0 \\leqslant Z \\leqslant 1$ (look to the Weinberg's \"QFT\" (chapter \"Kallen–Lehmann spectral representation\" of the section \"Nonperturbative methods\") or to Greiner's \"Field Quantization\" (pp. 278-282)). But this relation is correct only if one field is canonical conjugated to another (one field is canonical coordinate while the another one is canonical momentum). But for the arbitrary spins the canonical momentums must be chosen \"individually\"."}
{"id":"41309","title":"Does entropy decrease through measurement?","text":"For an electron in its rest frame, we have an entropy $$ S = \\log 2, $$ which comes from the 2 possible spin directions along z-axis. If the measurement $S_z$ changes its state to $\\left| + \\right>$, the entropy goes to zero. Does this violate the second law of thermodynamics?"}
{"id":"88202","title":"How many years would it take the Pioneer space probe to travel to Proxima Centauri with its current speed?","text":"Pioneer space probe moving at a speed of $30km\/s$. Assuming its heading for Proxima Centauri, which is situated at $4.2ly$ from earth, calculate how long it would take to get there in years, to the nearest year? Could you please help."}
{"id":"86419","title":"What's the Noether charge associated with Kaehler invariance of SuGra?","text":"What is the Noether charge associated with Kahler invariance of supergravity (SUGRA)? As the question is rather tangential to what I need to do, I have not tried explicitly calculating it myself, but I'm sure that I'm not the first one to wonder."}
{"id":"15827","title":"How can a conductor be grounded yet there are induced charges on it?","text":"A classic example for the method of images is the following, quoted from Griffiths's Introduction to Electrodynamics, page 121: \"Suppose a point charge $q$ is held a distance $d$ above an infinite grounded conducting plane. Question: What is the potential in the region above the plane?\" Griffiths continued on solving the example using the method of images setting V=0 on the plane as one of the boundary conditions saying \"since the conducting plane is grounded\". Now, of course there will be an induced surface charge density. My question is, how can this be since the plane is grounded? Does the word grounded have different meanings? sometimes it means not charged and the others it means the potential there is 0?"}
{"id":"33947","title":"Does reactive force require the two force involved have to have two different medium for reactive force to occur?","text":"Does reactive force require the two force involved have to have two medium for reactive force to occur? I know the fuel-thruster is working on vacuum space, but we human could not use arm to swim in space? For example, two baseball hitting together have two medium"}
{"id":"34997","title":"Scattering problem: Converting the two-body lab frame problem into a one-body center-of-mass frame problem","text":"I'm reading the section on scattering in Goldstein's Classical Mechanics, and I have a rather basic question about this. It says that scattering in the laboratory is a two-body problem because of the recoil of the scatterer. Therefore, we convert the obtained data into center of mass coordinates in order to find the true scattering angle, the angle that we would get if the scatterer didn't recoil. However, I'm having a hard time understanding what exactly this center of mass system is. The figure (3.25) below shows the path of two particles moving towards each other, but it looks like it is simply the laboratory problem shown in the frame of reference where the center of mass is not moving. My question: It still looks like a two-body problem to me, especially since the scatterer is deviating from its path. How does this simply the situation? Figure 3.25 still doesn't show a situation where the scatterer's position is fixed. Neither does figure 3.24. ![enter image description here](http:\/\/i.stack.imgur.com\/9OSm9.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/Fl79e.jpg)"}
{"id":"13870","title":"Gauge symmetry is not a symmetry?","text":"I have read before in one of Seiberg's articles something like, that gauge symmetry is not a symmetry but a redundancy in our description, by introducing fake degrees of freedom to facilitate calculations. Regarding this I have a few questions:   1. Why is it called a symmetry if it is not a symmetry? what about Noether theorem in this case? and the gauge groups U(1)...etc?   2. Does that mean, in principle, that one can gauge any theory (just by introducing the proper fake degrees of freedom)?   3. Are there analogs or other examples to this idea, of introducing fake degrees of freedom to facilitate the calculations or to build interactions, in classical physics? Is it like introducing the fictitious force if one insists on using Newton's 2nd law in a noninertial frame of reference?"}
{"id":"13871","title":"How does the quantum path integral relate to the quantization of energy?","text":"So, the quantum path integral is a generalization of the classical principle of least action- but here we know that all paths contribute something finite to the probability density. What confuses me is that this doesn't seem to involve the quantization of energy (action) at all. How does this come into play? Is it a separate assumption, only coming into play when we worry about the non-commutativity of certain operators?"}
{"id":"89039","title":"Polarized sunglasses: should the axes in both lenses be parallel?","text":"See the pictures below. A pair of sunglasses I recently purchased has the polarization axis in one lens offset about 20 degrees (by eyeball estimation) from the other. I don't have much experience with other polarized sunglasses, but this seems very obviously wrong to me. And it's very noticeable while wearing the glasses. When facing into glare from certain directions, one eye filters our considerably more glare than the other, which is very distracting and uncomfortable. I can see arguments for making the polarization axis angle just about anything (though horizontal or vertical seem most logical), but what I cannot see is any argument for having it different in each lens on the same pair of glasses. Thoughts? ![enter image description here](http:\/\/i.stack.imgur.com\/TIu4z.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/TW0Fd.jpg)"}
{"id":"100363","title":"Recommendation request for a book explaining string theory to a common idiot (me)","text":"I find the idea of string theory fascinating, but I'm not a student of physics, mathematics or science in general. Are there any books which effectively break down the concepts in a way that laymen like me can understand without being overly simple but also overly technical?"}
{"id":"103023","title":"phase difference between incident plane wave incident on a dipole and radiation fields from dipole","text":"i have an incident plane wave and a dipole, consider that plane wave incident on dipole. at this moment what happen for dipole ? we know that after incident of plane wave on dipole, the radiation have occurred, and i want derive the **phase difference between radiation field and incident plane wave field** ? please introduce me a reference"}
{"id":"47078","title":"What is the Laughlin argument?","text":"The fundamental question is > Why is Hall conductance quantized? Let's start with the Hall bar, a 2D metal bar subject to a strong perpendicular magnetic field $B_0$. Let current $I$ flow in the x-direction, then the y-direction develops a voltage $V_H$. The Hall conductance is $\\sigma_H = I\/V_H$ ![Hall bar \\(http:\/\/members.home.nl\/skoric\/quantum\/setup.gif\\)](http:\/\/i.stack.imgur.com\/0bhnE.gif) > To make Laughlin's charge pump, how should we wrap the Hall bar? Identify > the left and right, or top and bottom sides? Based on my understanding, we should paste top and bottom side together. (correct? Figure 1. left of the Paper maybe a little confusing.) Laughlin assumes the Fermi level is in the middle of the gap, so that the ring is an insulator. But the changing flux will induce an current by taking \"adiabatic derivative\" of total energy w\/r flux $$I = c\\frac{\\partial U}{\\partial \\Phi}$$ which flows in y-direction and where $c$ is speed of light. Following Laughlin's calculations, as one threads one flux quantum, there will be $p$ (number of filled Landau levels) electrons transported. Then $$U=peV$$ where $V$ is the potential difference of two edges. From the current formula, we find the quantized Hall conductance. The heart of the problem is > What is an adiabatic derivative? Why is ${\\bf j} = \\partial {\\cal > H}\/\\partial {\\bf A}$ valid?"}
{"id":"47072","title":"Fermi's Golden Rule and Density of States","text":"I know Fermi's Golden Rule in the form $$\\Gamma_{fi} ~=~ \\sum_{f}\\frac{2\\pi}{\\hbar}\\delta (E_f - E_i)|M_{fi}|^2$$ where $\\Gamma_{fi}$ is the probability transition rate, $M_{fi}$ are the transition matrix elements. I'm struggling to do a derivation based on the density of states. I know that under certain circumstances it's a good approximation to replace $\\sum_f$ with $\\int_F \\rho(E_f) \\textrm{d}E_f$ to calculate the transition probability, for some energy range $F$. Doing this calculation I obtain $$\\Gamma_{fi} ~=~ \\int \\rho(E_f) \\frac{2\\pi}{\\hbar}\\delta (E_f - E_i) |M_{fi}|^2\\textrm{d}E_f.$$ Now assuming that the $M_{fi}$ are constant in the energy range under the integral we get $$\\Gamma_{fi} ~=~ \\rho(E_i) \\frac{2\\pi}{\\hbar} |M_{fi}|^2.$$ Now this is **absolutely not** what is written anywhere else. Other sources pull the $\\rho(E_f)$ out of the integral to obtain Fermi's Golden Rule of the form $$\\Gamma_{fi} ~=~ \\rho(E_f) \\frac{2\\pi}{\\hbar} |M_{fi}|^2$$ for any $f$ with $E_f$ in $F$ which makes much more physical sense. But why is what I've done wrong? If anything it should be more precise, because I have actually done the integral! Where have I missed something?"}
{"id":"15318","title":"Can sound be used to purify\/filter water?","text":"I remember reading about all chicken at a poultry farm being violently sick\/dying for apparently no reason. It turned out the culprit was machinery at a nearby factory that emitted sound at a frequency equivalent to the resonant frequency for brain cavity of the chicken.   * Could the story be true?   * Could sound be used to purify water of bacteria\/viruses (preferably filter, else cull)?"}
{"id":"118728","title":"Is this a photograph of an electron-positron annihilation?","text":"With degrees in Mechanical and Electrical engineering but no advanced education in physics, I submit a query based on ellipsometric macro photography of TEMS supplied by FDA\/NIH. In one TEM a triangle of RNA\/DNA indicates ionic activity.   PHOTO 1. ![100nm RNA\/DNA Traingle](http:\/\/i.stack.imgur.com\/mMdGD.jpg) This piqued my curiosity, so I photographed the 100nm size triangle with a macro lens in order to enlarge the 35-42,000x TEM further and obtained unexpected images upon which the question is based. I angled the macro lens camera to the light source hoping to enhance the images within the photographic gel. The black-gray shades in TEMS are not ink but silver atoms crystallized in the original film negative by electron beams then projected into a silver-halide photographic gel. Electrons convert silver halide over 10,000 times more efficiently than photons. The biological samples, dark stained with nano-gold, provide a plethora of particle targets and possibly emit surface plasmon resonances that generate latent photon induced images hidden due to the shortened photographic development time required to form primary images with electron precipitated silver atoms. Collisions between electrons and particles must have occurred as can be seen in the spiraling burst emanating from a point above the bottom angle of the triangle that impressed me as having ionic interactions. The triangle legs are about 2nm wide with 100nm between vertices. Microvilli sized ion channels appear to be in the picometer range and may be waves generated from electron collisions. PHOTO 2. ![Electron-particle collision erupts above lower angle](http:\/\/i.stack.imgur.com\/djUBK.jpg) An invisible clear cloud seemed to lie between the left two vertices and I took hundreds of photos in an attempt to illuminate it. ![A spiraling cloud illuminated](http:\/\/i.stack.imgur.com\/RFgEb.jpg) PHOTO 3 is the result of those efforts. I assume the cloud to be one of those hypothetical latent images of energy associated with electron-particle collision. Unfortunately the color forces in QED do not relate to colors. PHOTO 4 was taken in the same area as PHOTO 3 with a different lens-light configuration. ![Annihilation?](http:\/\/i.stack.imgur.com\/2407g.jpg) Might the simultaneous Lycurgus Cup-like red\/green~emission\/absorption seen in PHOTO 4 correspond to an electron travelling back in time (positron) colliding with an electron beamed or displaced in the TEM vacuum and depict an annihilation?"}
{"id":"90987","title":"Comparison of 1D and 3D wave functions","text":"When discussing the Schroedinger equation in spherical coordinates, it is standard practice in QM handbooks to point out that the radial part of the 3-dimensional wave equation bears a strong analogy to the corresponding 1-dimensional case. This is because the Laplace operator in spherical coordinates can be written in the form $$L = \\frac{1}{r}\\,\\frac{d^2}{dr^2}r.$$ Hence my making the substitution $\\psi\\to r\\psi$, the 3D radial wave function $r\\psi(r)$ satisfies exactly the same Schroedinger equation as the wave function $\\psi(x)$ in the 1D case. So far so good. However, at this stage some authors point out that this substitution is merely feasible, as long as the condition $r\\psi\\to0$ in the limit $r \\to0$ is met. If not, then the wave function diverges at the origin, and this is unacceptable on physical grounds. [One may counter this argument, by pointing out that in the calculation of expectation values the square of the absolute value of the wave function is always multiplied by the spherical shell $4\\pi r^2\\,dr$. The factor $r^2$ neutralizes the above mentioned divergence.] Indeed there appears to be a slight difference between the 3D and 1D case, when one observes the elementary case of a particle in a box. In both cases, outside of the box the wave function is exponentially decreasing. Inside, where the particle is \"free\", the solution is oscillatory. In the 1D case, the wave function can be written as $$\\psi(x) = A\\sin(kx) + B\\cos(kx).$$ In the 3D case, the wave function is given by the zeroth order Bessel function $$\\psi(r) = C\\sin(kr)\/r.$$ [Apparently the term $D\\cos(kr)\/r$ is omitted since it is considered unphysical, see previous paragraph.] Now if the wave function inside the box has only one free parameter $C$, all one can really do is determine its value by demanding continuity of the wave function on the boundary of the box. On the other hand, QM textbooks claim that one should always seek a solution in which not only the wave function itself, but also its first derivative is continuous on a boundary! All in all I find the omission of the $D\\cos(kr)\/r$ term on physical grounds confusing. First of all it creates a difference between the 3D and 1D case. This seems odd to me since the corresponding wave functions satisfy essentially the same Schroedinger equation. Secondly, by omitting this term one loses an adjustable parameter $D$, which plays an important role in meeting the full set of boundary conditions."}
{"id":"100599","title":"Is this picture correct from the point of view of physics?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/Hgytn.jpg) This picture violates some laws of physics. Am I right?"}
{"id":"115210","title":"how does different colour wavelength intefere?","text":"Say a red colour with 400nm mixes with blue colour with 700nm, but out of phase by a little. What would be the resultant colour? I would know how to represent the resulting wave mathematically, but what colour is it? Is the colour only determined by the resulting wavelength?"}
{"id":"45203","title":"Higgs boson in LHC","text":"Recently,the higgs bosons are discovered in LHC. My question is How did they come to know that the particle that are created are actually Higgs boson? On the basis of which properties,they confirmed them as Higgs boson?"}
{"id":"12048","title":"Could we enable ourselves to send messages to and receive messages from the future?","text":"Based on John Isaacks' question, \"If you view the Earth from far enough away can you observe its past?\" and the responses, it appears that we could use mirrors to see into the past. Using Vintage's example, a single mirror on the moon would only allow an observer to see 0.1μs into the past. Would it therefore be theoretically possible to align a massive series of mirrors that sent a single beam of light back and forth over a larger span of time (e.g., months, years, decades)? If so, would it be possible to send messages to ourselves in the past via such an arrangement? Or would we always be looking into the past and only able to send messages to the future?"}
{"id":"123197","title":"Physics of a Railgun","text":"I have a couple of questions about how railguns actually work, and the mathematics behind them. I understand that the projectile is driven forward by the Lorentz force caused by high currents traveling through the projectile. Because of these high currents, when the projectile initially comes in contact with the rails it can be welded into place instead of pushed forward, and thus needs an initial velocity. So...   1. How do you calculate what the initial velocity needs to be in order to counteract the high current that wants to hold the projectile in place?   2. Would there be any difference in the Lorentz force in a pure-translational velocity versus a rotational and translational velocity?   3. How high do the currents need to be compared to the mass of the projectile?    4. Where would I find these kinds of formulas for these calculations?  Thanks!"}
{"id":"119812","title":"Why does time dilation cause you to age slower? And is time considered relative to the observer?","text":"I understand that the higher your velocity the slower light will move. But how does time itself slow down while you are moving faster?"}
{"id":"43792","title":"Why is mass renormalization insufficient to explain electron mass?","text":"In the Standard Model, I understand that the mass of the electron is assume to arise from two effects:   1. A bare mass given by Yukawa interaction with the Higgs field, and   2. A mass correction from mass renormalization effects In this framework, why do we need to assume 1? Could mass renormalization explain the mass of fermions in general?"}
{"id":"100044","title":"Does specific heat change with pressure? If so, why?","text":"As pressure is increased, do we require more energy to increase the temperature from given temperature for the same mass. For example, if we heat water at 1 atm to raise its temperature by 20 degree Celsius, will the heat required be the same when pressure is 3 atm?"}
{"id":"21643","title":"How does smoke move in the air and how can I direct it so it will go to a place I want it to go?","text":"Let's assume a close room with 1-2 people who only one of them smoking cigarette. What is the equation describe the smoking spreading? is it diffusion? what are the parameters is so? Is there a way\/ device I can build or put that make the smoke go in some specific way, like a vacuum that will suck all the smoke even if it is far away?"}
{"id":"21646","title":"Can we determine the force an object exerts by its mass and acceleration?","text":"I understand that the objects acceleration is determined by the force exerted _on it_ , and that the force exerted _on it_ is determined by its acceleration. But, does an object's (named A) acceleration (and mass) tell us anything about how much force the object will exert on another object (named B)?"}
{"id":"131013","title":"Will the Universe eventually stop expanding","text":"Sorry if this is a naive question, not being even a part qualified physicist in any way shape or form. I've read that the universe is expanding and the rate of expansion is increasing. The assumption being that it will continue expanding indefinitely. However isn't there another possibility. Let me illustrate my question prior to posing it. If I throw a ball in the air.. it accelerates away from my hand and keeps accelerating, until momentum is lost and it starts to slow down, eventually falling back to my hand. Is it possible that the expanding universe is also in the throws (excuse the pun) of an initial acceleration phase, prior to that acceleration slowing and eventually the universe compressing? Again sorry if its a stupid observation!"}
{"id":"90453","title":"Applications of quarks or string theory?","text":"We wouldn't have computers if we didn't know about quantum physics. I understand understanding of general relativity is needed to make GPS work well. Has knowledge of quarks or string theory resulted in useful things? **EDIT:** I suspect quarks and string theory could be used by people who study traces of the Big Bang and similar topics. That aside have they resulted in technological advances that could be useful to people (besides improving our understanding of nature)? Perhaps advances in medical imaging have been possible. There might also have been improvements in nuclear fission, nuclear fusion. Other than that I will be surprised to hear of applications. I suspect many applications are possible, but I expect it will take years to get there."}
{"id":"90457","title":"Clarify formula in quantum perturbation theory","text":"I'm studying perturbation theory in the context of quantum mechanics. My lecture notes say that in order to calculate the first-order correction of eigenfunction $\\psi_n$, that is $\\psi_n^{(1)}$, I shall use the formula: \\begin{equation} \\psi_n^{(1)}=\\sum_{m,m\\ne n} \\frac{V_{mn}}{E_n^{(0)} - E_m^{(0)}}\\psi_m^{(0)} \\end{equation} where $E_n^{(0)}$ is the zeroth-order correction (i.e., the unperturbed), and $V_{mn}$ are given by: \\begin{equation} V_{mn} = \\left(\\psi_m^{(0)}, V\\psi_n^{(0)}\\right) \\end{equation} and V is the \"weak\" potential that represents the physical disturbance to our system. $(x,y)$ represents inner product. **My question is** : What values does $m$ take in the above summation ? Say I'd like to calculate the first-order corrected $\\psi_1$, that is to calculate $\\psi_1^{(1)}$. What values would $m$ assume ? Infinite values except for $1$ ? Values $0,\\ldots,n-1$ ?"}
{"id":"58023","title":"So gravity turns things round","text":"It makes sense, since gravity tends to push the surface of a body towards it's center. Unless I'm mistaken, everything with mass has it's own gravity, every atom and for instance, our own bodies should also have their own gravity. The question is: how strong is our own gravitational pull? I know it must be extremely weak, but is there actually anything at all that gets attracted to us, like maybe, bacteria or molecules? And finally (this will sound ridiculous, but I'd really want to get an answer or at least a way of calculating it myself): What size would a human body have to reach in order for it to collapse into a sphere?"}
{"id":"87434","title":"Derive non-linear $\\sigma$ model from a theory of SU(2) matirx","text":"It's said in Chapter VI.4 of A. Zee's book _Quantum Field Theory in a Nutshell_ , a theory defined as $L(U(x))=\\frac{f^2}{4}Tr(\\partial_{\\mu}U^{\\dagger}\\cdot\\partial^{\\mu}U)$, can be write in the form of a non-linear $\\sigma$ model (up to some order) $L=\\frac{1}{2}(\\partial\\vec{\\pi})^2+\\frac{1}{2f^2}(\\vec{\\pi}\\cdot\\partial\\vec{\\pi})^2+...$, where $U(x)=e^{\\frac{i}{f}\\vec{\\pi}\\cdot\\vec{\\tau}}$ is a matrix-valued field belonging to $SU(2)$, $\\vec{\\pi}$ is a three components vector, $\\vec{\\tau}$ are Pauli matrices. Maybe it's not hard but I meet some problems to derive it. I suppose the first step is the Taylor expansion of $U$, $U=1+\\frac{i}{f}\\vec{\\pi}\\cdot\\vec{\\tau}-\\frac{1}{2f^2}(\\vec{\\pi}\\cdot\\vec{\\tau})^2+...$, and then $\\partial^{\\mu}U=\\frac{i}{f}\\partial^{\\mu}(\\vec{\\pi}\\cdot\\vec{\\tau})-\\frac{1}{f^2}(\\vec{\\pi}\\cdot\\vec{\\tau})\\partial^{\\mu}(\\vec{\\pi}\\cdot\\vec{\\tau})$, then $(\\partial_{\\mu}U^{\\dagger})(\\partial^{\\mu}U)=\\frac{1}{f^2}[\\partial(\\vec{\\pi}\\cdot\\vec{\\tau})]^2+\\frac{1}{f^4}.[(\\vec{\\pi}\\cdot\\vec{\\tau})\\partial(\\vec{\\pi}\\cdot\\vec{\\tau})]^2$. Now there are my questions, (1) Can I write $\\partial^{\\mu}(\\vec{\\pi}\\cdot\\vec{\\tau})=\\partial^{\\mu}\\vec{\\pi}\\cdot\\vec{\\tau}$? Then by $\\vec{\\tau}^2=1$, I get $L=\\frac{1}{4}(\\partial\\vec{\\pi})^2+\\frac{1}{4f^2}(\\vec{\\pi}\\cdot\\partial\\vec{\\pi})^2$, which is almost correct but differ to the wished answer by a pre-factor $\\frac{1}{2}$. (2) Suppose $\\partial^{\\mu}(\\vec{\\pi}\\cdot\\vec{\\tau})=\\partial^{\\mu}\\vec{\\pi}\\cdot\\vec{\\tau}$ is correct, however, if I do $\\partial^{\\mu}U=\\frac{i}{f}U\\partial^{\\mu}(\\vec{\\pi}\\cdot\\vec{\\tau})=\\frac{i}{f}U\\partial^{\\mu}\\vec{\\pi}\\cdot\\vec{\\tau}$ first, it seems $\\partial^{\\mu}U^{\\dagger}\\cdot\\partial^{\\mu}U=|\\frac{i}{f}U\\partial^{\\mu}\\vec{\\pi}\\cdot\\vec{\\tau}|^2=\\frac{1}{f^2}(\\partial\\vec{\\pi})^2$, say, only the first term of the wished answer. I probably made something wrong somewhere, can anyone hit me?"}
{"id":"38800","title":"What does the term liquid mean in condensed matter physics?","text":"In condensed matter physics, people always say quantum liquid or spin liquid. What does liquid mean?"}
{"id":"17404","title":"$E=mc^2$ why is it $c^2$ and not just $c$?","text":"Why is constant for the conversion of mass to energy square of the ligths speed? is it bedside it's the fastest real matter?"}
{"id":"17406","title":"what this Lagrangian stands for?","text":"i saw this Lagrangian in notes i have printed: $$ L(x,dx\/dt) = (m^2(dx\/dt)^4)\/12 + m(dx\/dt)^2*V(x) -V^2(x) $$ what is it? is it physical? it seems like it doesn't have the right units of energy, thanks"}
{"id":"62321","title":"Application of plasma actuators for flow control of moving objects","text":"I've recently been to a converence on plasma physics were, to my surprise, a lot of presentations were concerned with plasma actuator. Could someone, preferably in the field, tell me how long people have been reseaching this now and, more importantly, give an honest opinion on how far (in years) this technique is from regular industrial use?"}
{"id":"11803","title":"How is it possible for astronomers to see something 13B light years away?","text":"In a NPR News story from a few years back: > \"A gamma-ray burst from about 13 billion light years away has become the > most distant object in the known universe.\" I'm a layman when it comes to physics, so cut me some slack if this is an ignorant question, but assuming the universe is around 14B years old, and has been expanding since the Big Bang, how is it that we can see events so far back in time? I understand how it would work if you had a static universe and the GRB happened 13B years ago at 13B Light years away and the light just arrived. However, at the time of the burst wouldn't we (or at least the matter that we are made from) have been much closer to the source of that burst, and wouldn't the light have blown by us eons ago? How is it we are seeing it now? If we were expanding away from it at close to light speeds it would seem to make sense for why it took so long for it to get here, except for that whole notion that light moves at the same speed relative the to the observer, which I think would blow that idea out of the water. Perhaps gamma rays travel at sub-light speeds? But, I'd still think the math would require that they travel MUCH slower than light for this scenario to play out. Another, possibility is that the light has wrapped around a finite universe a few times before reaching us. Of course if that were the leading theory, there wouldn't be any remaining controversy about the finite vs. infinite universe models. What am I missing here?"}
{"id":"48669","title":"\"Hard wall\"\/ \"soft wall\"","text":"I have encountered those terms in various places. As I understand it, \"soft wall\" can correspond to a smooth cutoff of some spacetime, while \"hard wall\" can be a sharp one, which can be described in terms of D-branes. Could somebody please explain the terminology, and in which context it can occur?"}
{"id":"29735","title":"On Bolte's semiclassical law","text":"i have seen on internet the following, for $ E >> 1 $ the Eigenvalue Staircase can be approximated by $ N(E)= \\frac{1}{\\pi}argZ(1\/2+i \\sqrt E ) $ http:\/\/books.google.es\/books?id=hLEJkA34GMQC&pg=PA35&lpg=PA35&dq=quantum+chaos+bolte%27s+formula&source=bl&ots=jqMMu5KK-L&sig=L9ZlQ_Wq7XQLDvEHyf1nvIc1TdE&hl=es&sa=X&ei=oqHMT4zdEoO_8APlk4UM&ved=0CGUQ6AEwBzgK#v=onepage&q=quantum%20chaos%20bolte%27s%20formula&f=false PAGE 34 section 1.28 (of course you may use $ \\lambda =1$ ) of course the situation is very familiar if we put $ Z(s)= \\zeta (s) $ Riemann zeta, but how could i prove Bolte's Law ?? also from the reconstruction of the potential could we deduce that $ V^{-1} (x)=A \\frac{d^{1\/2}}{dx^{1\/2}}argZ(1\/2+i \\sqrt E) $ where could i find some more info and a proof of Bolte's Law ?? thanks."}
{"id":"40919","title":"Is the electric field at the edge of a uniformly charged disk infinite?","text":"Consider a disk with a radius $R$ (I'll use $R=1$ at various points here) that has a constant surface charge density $\\sigma$. Unlike the similar problem of the field in the vicinity of a infinitely thin ring, the field directly above the disk is very well behaved. Like the case of an infinite sheet of constant surface charge, the field limits to a constant as you get closer to the surface, and a point on the surface has a defined electrical potential. This recent paper gives equations for the specific case of points on the plane of the disk. Since we have rotational symmetry, there is only one variable, $\\rho$, the distance from the axis of rotational symmetry. Graph from the paper: ![Potential](http:\/\/i.stack.imgur.com\/PvFoJ.jpg) As you can imagine, the point of the highest slope is the edge of the disk. Is the slope (field) at this point infinite? From what I can make out of their equations, signs seem to point to \"yes\". The Elliptic functions are tricky, however, and I don't trust my conclusions. Up to the point $\\rho=R$, the form of the above function is: $$ V(\\rho) = V(0) \\frac{2}{\\pi} E(\\rho)$$ Here, $E(\\rho)$ is an elliptic function, and is different from the paper, because it uses the \"Matlab\" convention while I use the \"Wikipedia\" convention. The slope of E() doesn't _look_ convincingly upright, but the derivative seems to indicate it should be. I'm also troubled by the fact that I can't seem to numerically observe a divergence at $\\rho=0.9999999999$. The physics at play are even more perplexing to me. I expected that a 2D surface, any 2D surface, would have a finite field on its face. Not so for a line, I understand that. But the ring that defines the edge of the disk only has a differential amount of charge on it. Consider another thought experiment: start at $\\rho=1$ and $z=0$, I'll denote it $(1,0)$. Now move upward, but keep $\\rho$ the same. If you're at $(1,0.001)$ you should have a finite field. But what happens as $z\\rightarrow 0$? Does it just jump from a defined finite value to infinity? What on Earth is going on with the electric field in the vicinity of this edge, or ANY edge of a 2D surface for that matter?"}
{"id":"129177","title":"dose physics affectthe biological like this","text":"According to neural sciences the brain process information and reacts in 1 teen of a seconds the numbers add up to be 10.596674 loest is 30 to 20 years the course of time in reality recorded by our time keeping devices. but whant to know if perhaps i estimated wrong or something or if it really is what it sound like and what i mean by this that like stars from far out galaxies we to are seeing what happen years. Back in the past"}
{"id":"129283","title":"Deflection of light by the Sun","text":"Can you give details of a recent experiment of deflection of light by the Sun? What is the distance from the surface of the Sun and what is the exact value of the angle of deflection?"}
{"id":"135258","title":"Does the sun \"drag the solar system through space\"?","text":"In this video: https:\/\/www.youtube.com\/watch?v=0jHsq36_NTU#t=55 at 0:55, it is claimed that \"the sun is [...] dragging the planets in its wake\". Is this true? My understanding is that the sun and the planets are all moving together through the solar system at the same velocity, and there is no friction involved, so there's no dragging or indeed motor force required. What's really going on?"}
{"id":"95249","title":"Different units in the equations of electrostatics in macroscopic media","text":"I have the usual equations of electrostatics in macroscopic media expressed in Gaussian units: $$\\nabla\\cdot\\vec{D}=4\\pi\\rho$$ $$\\nabla\\times\\vec{E}=0$$ $$\\vec{D}=\\vec{E}+4\\pi\\vec{P} \\tag1$$ My teacher wants me to convert these Equations to the SI system using the following transformation rules: > $$\\vec{E}\\rightarrow\\frac{\\vec{E}}{k_1}$$ > $$\\vec{D}\\rightarrow\\frac{\\vec{D}}{k_1k_4}$$ > > Where $k_1=\\frac{1}{4\\pi\\epsilon_0}$ and $k_4=\\epsilon_0$, which gives: $$\\vec{E}\\rightarrow 4\\pi\\epsilon_0\\vec{E} \\tag2$$ $$\\vec{D}\\rightarrow 4\\pi\\vec{D}. \\tag3$$ Applying this to my electrostatics equations I get: $$\\nabla\\cdot\\vec{D}=\\rho$$ $$\\nabla\\times\\vec{E}=0$$ $$\\vec{D}=\\epsilon_0\\vec{E}+\\vec{P},$$ which is consistent with what I find in books. If now I suppose that $\\vec{P}=\\chi\\vec{E}$, doing the appropriate manipulations to the equation (1) I obtain: $$\\vec{D}=\\epsilon\\vec{E} \\quad , \\,\\,\\, \\mbox{Where I define:} \\,\\,\\, \\epsilon=1+4\\pi\\chi$$ If now I apply the transformation rules (2) and (3) to this in order to express it in SI units I get: $$\\vec{D}=\\epsilon\\epsilon_0\\vec{E} \\,\\,\\ , \\,\\,\\, \\mbox{Where I define:} \\,\\,\\, \\epsilon=1+4\\pi\\chi$$ But this is not consistent with what I find in books, which is: $$\\vec{D}=\\epsilon\\epsilon_0\\vec{E} \\,\\,\\ , \\,\\,\\, \\mbox{Where I define:} \\,\\,\\, \\epsilon=1+\\chi$$ The only way to obtain the solution that appears in the books is to assume that $\\vec{P}=\\epsilon_0\\chi\\vec{E}$ (This is the accepted equation in SI units) and do the same I did before (Replace this in (1) and use the transformation rules).   1. My question is: is there any way of obtaining the equations in SI units without using the relation $\\vec{P}=\\epsilon_0\\chi\\vec{E}$? (using only the transformation rules)?   2. And if this is not possible, how do I convert between $\\vec{P}=\\epsilon_0\\chi\\vec{E}$ (SI units) and this $\\vec{P}=\\chi\\vec{E}$ (Gaussian units)?"}
{"id":"23295","title":"Why can't we know the speed, $\\vec{v}(t)$, and position, $\\vec{r}(t)$, of an electron (the two) at the same time $t$?","text":"I've read something about this and I conclude that it happens because of the uncertainty principle. But I don't understand very well the meaning of that. I mean, it's very abstract that the speed, $\\vec{v}(t)$, and position, $\\vec{r}(t)$, of a particle can't be known at the same time. I don't understand that statement."}
{"id":"23294","title":"How much is 1 electron-volt (eV)?","text":"I am interested in knowing how much is one eV of energy. Everywhere I found are the technical definitions. Can anybody please tell me how much is this much energy. I need something which I can feel. I mean how much work I can do with 1 eV? Can I drive a 1000cc car for 1hour? Any of example in context of real life usage would be interesting."}
{"id":"1106","title":"Wi-Fi in the presence of very strong magnets?","text":"A friend of mine claims to have been able to surf the Internet without fuss on a Wi-Fi connection while performing NMR on samples he was analyzing. I would have thought the strong magnets needed for this would have washed out Wi-Fi signals due to the radio waves emitted. How plausible is his claim? Or, to be more general, what kind of (electro?)magnet would it take to interfere with mobile devices?"}
{"id":"63243","title":"How can we detect a black hole?","text":"If black holes are phenomena of very high density (gravitational singularities) which don't emit radiation **how can we detect them** so far away from us where so much **other radiation can hide the black hole**? Aren't there many objects and things in front of black holes that would obstruct our view?"}
{"id":"28065","title":"$\\pi$ and the Curvature of Space","text":"If one draws a circle on a sphere and measures the ratio of the diameter to the circumference, that value varies depending on the diameter of the circle compared to the diameter of the sphere it is drawn on (for a circle much smaller than the sphere it's drawn on, the ratio will converge to $\\pi$, whereas for a circle the same size as the sphere, the ratio will be 2). Does the same or similar hold for curved 4-dimensional space? As I approach a massive body, will the ratio of the circumference of a circle to its diameter change? I found a number of sometimes conflicting statements online, including http:\/\/www.physicsforums.com\/archive\/index.php\/t-9869.html http:\/\/mathforum.org\/library\/drmath\/view\/55198.html http:\/\/www.last-word.com\/content_handling\/show_tree\/tree_id\/2339.html"}
{"id":"126102","title":"Difference between single mode and multi mode optical fibres?","text":"What is the difference between single mode and multi mode optical fibres? First off, I guess that by _modes_ we mean the spatial modes of the electric (or magnetic?) field right? Now: what makes a fibre able to support more than a single mode? I mean, what aspect of its structure corresponds to which mode(s) can be transmitted?"}
{"id":"28060","title":"Boiling when I plunge my french press?","text":"Occasionally when I make coffee in my french press I experience something odd. It happens pretty infrequently but certainly enough to be curious about. I have the grounds ready in the carafe. The water just heated in an electric kettle. I pour the water over the grounds and place the plunger on top. Then, maybe 1\/4 inch into plunging, the water \"explodes\" out the top like a busted sprinkler head. I'm curious what's going on here. Is the slight pressure increase (it can't be much) - or simply the plunging action - adding enough energy to the system to induce a flash boil? I've ruled out that the additional volume that the plunger itself adds is not enough to make the coffee overflow (this is a normal setup in every other respect where plunging would simply deliver a refreshing hot beverage)."}
{"id":"5246","title":"Energy levels in disordered organic semiconductors?","text":"Now in disordered organics, the band picture is thrown out the window, from what I can tell (due to lack of symmetry). But don't HOMO\/LUMO levels basically take the place of conduction\/valence bands in molecules? In a organic system (a lot of molecules with no order), then, I am correct to believe that the HOMO and LUMO levels broaden into a Guassian density of states? However, the LUMO 'band' does not act like a conduction band in that the states are still localized. From my reading, it appears that the Fermi Level is in the LUMO 'band'. My question is, how can the LUMO 'band' have filled states since a single molecule of course has no electrons in the LUMO. What am I missing?"}
{"id":"111597","title":"How to do calculation in relativity of simultaneity","text":"I have great trouble in understanding simultaneity in special relativity. Let me illustrate it with a concrete example. Assuming there is a train, its two end points are $A$ and $B$, the length of the train is $x$. The train moves at speed $v$. Assuming the train is moving in the direction from $A$ to $B$. For a ground observer observing the train movement, he notices that two lightnings strike simultaneously at $A$ and $B$ when the middle part of the train $O'$ passes through right in front of him. In other words, the ground observer is located at the middle part of the train($O'$) when the lightnings strike simultaneously at $A$ and $B$. Now there is another moving observer sitting inside the train and he sits right in the middle of the train ($O$, equidistant from $A$ and $B$). Does this moving observer think that the lighting happens at the same time? If no, how much time has passed before he notices lightning at $B$, after he had observed lightning at $A$?"}
{"id":"111590","title":"Light absorbs and Color","text":"I'm curious about how the material absorb the light and reflect the light back as colors in a sense of Quantum Mechanics (Quantum Electro Dynamics)   Does Hadron related to the absorbs of photon ? or are there any other factors ?"}
{"id":"26510","title":"Why is oxygen the third most abundant element?","text":"I was reading the article _Oxygen finally spotted in space_ today in which it stated > Oxygen is the third most abundant element in the cosmos, after hydrogen and > helium. Why would oxygen take the third spot when it is so heavy (relative to the five elements ignored for it)? It would seem logical to me that the third most abundant element would be lithium or beryllium as hydrogen and helium smash into each other."}
{"id":"103608","title":"Can an inhomogeneous magnetic field be built to slow down and catch a neutron?","text":"Neutrons have a measureable magnetic dipole momentum from their intrinsic spin. Is it possible to slow down and catch the neutron by imposing a force by an inhomogeneous magnetic field. I think the force the neutron experiences in a magnetic field is \\begin{align} \\bar{F} = \\bar{\\nabla}(\\bar{m}\\cdot\\bar{B}). \\end{align}"}
{"id":"131147","title":"What is entropy of the universe?","text":"What is entropy of the universe? What is the change in entropy of our universe? Is this change increases or decreases? If the entropy of universe keeps on increasing then what are the consequences? Is there any limiting factor to this increase? One practical application of entropy where it is applied in day to day life?"}
{"id":"95863","title":"Faster than light galaxies\/clusters?","text":"A few years ago in an astronomy course, we calculated some (transverse?) velocity of a moving object and got super luminal results. The answer was apparent and not physical velocity of the object. Hence no problem. But at the moment, I don't recall the solution to this apparent issue. Anyone?"}
{"id":"93165","title":"Is this configuration correct for this spherical capacitor?","text":"This capacitance contain 4 dielectric as shown in the figure dielectric 1 in half sphere and 2,3 in for 1\/4 of the sphere and the fourth one in the last 1\/4 of the sphere as shown and I want to find total capacitance. i think that 2nd dielectric parallel with potion of 1st and portion of 4th then this group series with (3rd dielectric and other portion of 1st and 4th one which are also parallel) is the correct answer as the voltage between a,b is the same at any dielectric as the electric field is the same from boundary condition ET1'=ET2=ET4' AND ET1''=ET3=ET4'' ![enter image description here](http:\/\/i.stack.imgur.com\/4tJ6n.png) is it correct"}
{"id":"118740","title":"The Helium mass fraction from the Big Bang Nucleosynthesis","text":"In Perkin's book Particle Astrophysics (page 144): I do not understand how one comes to the following expression (the second equality with $r$) for the Helium mass fraction due to the Big Bang Nucleosynthesis: $$Y= \\frac{4N_\\text{He}}{4N_\\text{He}+N_\\text{H}}= \\frac{2r}{1+r} $$ where $r=N_\\text{n}\/N_\\text{p}$. The first equality follows from the fact that He is (approximately) 4 times heavier than H: $$Y = \\frac{m_\\text{He}}{m_\\text{He}+m_\\text{H}}= \\frac{4N_\\text{He}}{4N_\\text{He}+N_\\text{H}}.$$ However I can't derive the second equality relating $Y$ to $r$: $N_\\text{He}= 2N_\\text{p} + 2N_\\text{n}$ and $N_\\text{H}= N_\\text{p} + N_\\text{n}$ $$Y= \\frac{4N_\\text{He}}{4N_\\text{He}+N_\\text{H}}= \\frac{8(N_\\text{n}+N_\\text{p})}{9(N_\\text{n}+N_\\text{p})}\\quad???$$"}
{"id":"55674","title":"Fourier transform between $x$ and $p$","text":"On this page right at the top they mention two sets of fourier transform. First set is connection between $x$ (position) and $k$ (wave vector) space: $$ \\begin{split} f(x) &= \\frac{1}{\\sqrt{2\\pi}} \\int\\limits_{-\\infty}^{\\infty} A(k) e^{ikx} dk\\\\\\ A(k) &= \\frac{1}{\\sqrt{2\\pi}} \\int\\limits_{-\\infty}^{\\infty} f(x) e^{ikx} dx \\end{split} $$ while the second set is connection between $x$ (position) and $p$ (momentum): $$ \\begin{split} \\psi(x) &= \\frac{1}{\\sqrt{2\\pi \\hbar}} \\int\\limits_{-\\infty}^{\\infty} \\phi(p) e^{i\\frac{p}{\\hbar}x} dp\\\\\\ \\phi(p) &= \\frac{1}{\\sqrt{2\\pi \\hbar}} \\int\\limits_{-\\infty}^{\\infty} \\psi(x) e^{-i\\frac{p}{\\hbar}x} dx\\\\\\ \\end{split} $$ * * * **Q1:** How do i derive the second set out of first one? I know De Broglie relation $p = k \\hbar$. Hence from $\\exp[\\pm ikx]$ in the first set we get $\\exp \\left[\\pm i \\frac{p}{\\hbar} x\\right]$ in the second set of equations. This is clear to me. What i dont know is how do we get from $1\/\\sqrt{2\\pi}$ in the first set to $1\/\\sqrt{2 \\pi \\hbar}$ in the second set. Where does a $\\hbar$ come from?"}
{"id":"56323","title":"Bernoulli's theorem: $\\frac{p}{\\rho}+\\frac{1}{2}u^2+\\phi$ is constant along a streamline","text":"I am trying to understand **the Bernoulli's theorem:**   * **$\\frac{p}{\\rho}+\\frac{1}{2}u^2+\\phi$ is a constant along a streamline** I got that: $\\frac{\\partial u}{\\partial t}$ + ($\\nabla \\times u)\\times u$ = $-\\nabla(\\frac{p}{\\rho}+\\frac{1}{2}u^2+\\phi)$ For a steady flow: $\\frac{\\partial}{\\partial t}$ = 0 and then: ($\\nabla \\times u)\\times u$ = -$\\nabla H$ with the scalar: $H$ = $\\frac{p}{\\rho}+\\frac{1}{2}u^2+\\phi$ **now, I didn't understand the next steps:** Taking the “dot product” of ($\\nabla \\times u)\\times u$ = -$\\nabla H$ the left hand side vanishes, as ($\\nabla \\times u)\\times u$ is perpendicular to $u$ and we get: $(u \\cdot \\nabla)H = 0$ This implies that $H$ is constant along a streamline **Can someone explain me this thing in other words please?**"}
{"id":"45554","title":"Why can we analyse force balance on a dislocation?","text":"Dislocation (like screw or edge dislocation) is not a 'real' thing, while Newton's laws only apply to a real object (no matter macroscopic, like stars, or microscopic, like atoms). In the derivation of Peach-Koehler force (stress acting on a dislocation), I understand that the force is actually acting on those atoms around the dislocation, which is equivalent to acting on the dislocation in the mathematical sense. However, based on the above point (treating dislocation as a 'real' thing), some books directly use force balance and other mechanical analysis on dislocation. Can we treat dislocation as a 'real' thing in all mechanical cases just as the void in a solid when analysing electrical properties? Is there some way to think about it easily other than a complicated mathematical argument?"}
{"id":"35443","title":"Why should space be empty?","text":"> **Possible Duplicate:**   >  How vacuous is intergalactic space? The _emptiness_ of space is explained in many articles... But, space does contain some matter due to these possible reasons:   * During the explosion of supernovae, some elements are created. Not all supernovae nucleosynthesis are successful. So, there should be some scattering of matter into outer space.   * Cosmic rays are coming from interstellar space (and even from sun) in all directions. Would all of the particles have enough efficiency to reach their destination (I mean, take earth or any other interstellar object)?. Some protons or neutrons could be scattered into space.   * Even the gaseous molecules in atmosphere of celestial bodies have possibility to reach escape velocity and go into space. Are my assumptions correct? If so, then space must contain elements up to some extent. Isn't it? Or is it due to the infiniteness of space, that these scattered particles are ignored?"}
{"id":"81436","title":"Thin lens formula workout method (Query)","text":"![enter image description here](http:\/\/i.stack.imgur.com\/0nuvB.jpg) In order to workout the method for establishing the formula of thin lens, my teacher says that the optical path is: $PA + AQ = PS_1 + nS_1S_2 + S_2Q$ ($n$ is the refractive index of the lens)   Why is she multiplying $S_1S_2$ by $n$? Also, she says that for spherical surface $S_1$, with centre of curvature $C_1$, we can write from geometry,   $h^2 = 2(S_1C_1 - OS_1) OS_1 $ How is she deriving this?"}
{"id":"25372","title":"Where does the dust on the moon come from?","text":"From the Apollo missions we know that the moon is covered with dust. Where does it come from? Is it from the erosion of the moon rock? By what? Or by accretion of dust from space? Which comes from where?"}
{"id":"104591","title":"When combining three spin $\\frac{1}{2}$ particles what are the corresponding states?","text":"I want to combine three spin half particles and this is what I have so far. I used the lowering operator $J_{-}$ on the top states and found the following states fine: $$|\\frac{3}{2},\\frac{3}{2}\\rangle , |\\frac{3}{2},\\frac{1}{2}\\rangle , |\\frac{3}{2},0\\rangle , |\\frac{3}{2},-\\frac{1}{2}\\rangle , |\\frac{3}{2},-\\frac{3}{2}\\rangle $$ So that is the combination of three spin $\\frac{1}{2}$ particles is equivalent to a spin $\\frac{3}{2}$ particle, right? In the case where I did it for combining two spin $\\frac{1}{2}$ particles I found this was equivalent to a spin 1 particle and an additional spin zero particle. So my question is, is there a singlet or anymore that accompany what I found for the case where I combined three spin $\\frac{1}{2}$?"}
{"id":"82353","title":"Streamlines tangent to velocity vector","text":"As from the title, I'm not too sure how they are related. Definition is that streamlines are instantaneously tangential to the velocity vector of the field. Why would a streamline that shows direction be a tangent to the velocity?"}
{"id":"82352","title":"LC Oscillator and relativity","text":"There are two identical LC oscillators with electronic counters attached indicating how many times they have oscillated (from the time they are turned on). They are turned on simultaneously and one is kept on earth and other one is hurled to outer space at very large speeds on a rocket and brought back to earth after traveling several million miles. Does relativity say they will show two different counts ? (One traveled will show less than the one on earth)"}
{"id":"127766","title":"Does the geomagnetic field rotate?","text":"The Earth rotates about it's own axis. Do the geomagnetic field lines rotate due to this rotation or not?"}
{"id":"82359","title":"How was Newton's third law discovered?","text":"How did Newton's third law came into being? Was it his original finding like the second law? Or was it more of a restatement of someone else like the 1st one coming from Galileo? In either case what initiated the thought of what is now known as Newton's 3rd law?"}
{"id":"96302","title":"Thermal equilibrium and kinetic energies","text":"Is temperature solely a function of a kinetic energy? If a solid and a gas are at thermal equilibrium at a temperature of 20 degrees Celsius, the solid has much less kinetic energy than the gas. How can the temperatures of both be the same? What is keeping the solid at the same temperature as the gas?"}
{"id":"21980","title":"Angular momentum equations","text":"I do not understand this because angular momentum is $L=I\\omega$ ($I$ is moment of inertia;$\\omega$ is angular velocity) but it I have also seen equations where $L= rmv\\sin(x)$. I do not understand how these are related, could someone please explain the connection?"}
{"id":"16408","title":"Conveyor scales modeling","text":"> **Possible Duplicate:**   >  Conveyor scales modeling Assume we have a conveyor scales. Which consists of scales, and motor with conveyor belt placed above, so that the boxes can be measured (weight) while moving above. What I want is to create the model of the oscillations of that scale. The equation of the oscillations of the spring is well known for everybody. How to include the influence of the motor, conveyor and other important effects to the system? Is there any papers (articles) about this thema? I think that the influence of the motor in simle case can be modelled as the force: $F(t) = A\\sin(\\Omega t)$ Where $A$ is a constant and $\\Omega$ is the parameter which depends on the motor (frequency), which is also constant (depending on the speed of the motor). How to add the ocsillations created by conveyor belt and may be some other oscillations such as box moving above can create additional vibrations?"}
{"id":"16416","title":"Finding two dimensional critical point","text":"I'm reading an article about bi layered membranes which state that for the free energy function   $f(\\theta) = \\theta \\ln \\theta + (1-\\theta)\\ln(1-\\theta) + \\chi \\theta (1-\\theta)$ Where $\\phi_i$ is the mole fraction of certain types of lipids in layer $i$ and $\\chi$ is the energy of interaction between two lipids molecules composing the layer. The article states that a critical point exist at $\\chi=2$ and $\\theta=0.5$. Reading in wikipedia, I concluded that removing the layers interaction part, I simply need to take the second and third derivative of the free energy of the membrane. How do I do this with the interactions? Where can I read about this?"}
{"id":"16412","title":"The gravitational potential of ellipsoid","text":"In the literature (Kirchhoff G. - Mechanic (1897), Lecture 18 or Lamb, H. - Hydrodynamics (1879)) one can find the following analytical closed form expression for the gravitational potential of homogeneous ellipsoid of unit density, whose surface is given by \\begin{equation} \\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1 \\;. \\end{equation} Gravitational point for internal points is \\begin{equation} \\Omega=\\pi abc\\int_0^\\infty\\left(1-\\frac{x^2}{a^2+\\lambda}-\\frac{y^2}{b^2+\\lambda}-\\frac{z^2}{c^2+\\lambda}\\right)\\frac{d\\lambda}{\\Delta} \\end{equation} and for external points \\begin{equation} \\Omega=\\pi abc\\int_u^\\infty\\left(1-\\frac{x^2}{a^2+\\lambda}-\\frac{y^2}{b^2+\\lambda}-\\frac{z^2}{c^2+\\lambda}\\right)\\frac{d\\lambda}{\\Delta} \\;, \\end{equation} where \\begin{equation} \\Delta=\\sqrt{(a^2+\\lambda)(b^2+\\lambda)(c^2+\\lambda)} \\end{equation} and $u$ is the positive root of equation \\begin{equation} \\frac{x^2}{a^2+u}+\\frac{y^2}{b^2+u}+\\frac{z^2}{c^2+u}=1 \\;. \\end{equation} The expressions in these formulas appear similar to confocal ellipsoidal coordinates. How can these formulas be derived? (perhaps something more readable than the original papers) Can they be derived in terms of ellipsoidal harmonics?"}
{"id":"93835","title":"A basic question: what is accelerating voltage?","text":"Or would it be acceleration voltage? Acceleration sounds like it makes more sense, but my paper says accelerating. What are possible ways you could go about calculating it?"}
{"id":"61912","title":"Composition of solar spectrum","text":"I read some where that there are three types of UV and infrared rays namely UV-A, UV-B, UV-C and near infrared, mid infrared and far infrared. Which is the most abundant among the the three in Ultraviolet and infrared radiation from sun? I mean there is a total of 1000 W per unit area illumination by sun among which more than 500 watts is infrared, around 450 watts in visible and the rest part is ultraviolet so in the infrared rays which are the most abundant means which contribute to most of the 500 watts? Also, what are the sources of infrared at night and what is the power density of infrared rays at night!."}
{"id":"61918","title":"Three polarizers, 45° apart","text":"If light is passed through two polarizing filters before arriving at a target, and both of the filters are oriented at 90° to each other, then no light will be received at the target. If a third filter is added between the first two, oriented at a 45° angle (as shown below), light will reach the target. Why is this the case? As I understand it, a polarized filter does nothing except filter out light--it does not alter the light passing through in any way. If two filters exist that will eliminate all of the light, why does the presence of a third, which should serve only to filter out additional light, actually act to allow light through? ![Image of three polarizers, target is at the right](http:\/\/i.stack.imgur.com\/OogSm.png)"}
{"id":"58127","title":"Strings and QFT: particles moving backward in time?","text":"New question: In string theory and QFT, do particles travel back in time? Not related to antimatter: Do they travel back and forth in time in reality or are these just interpretations of mathematical formulas used to make sense of calculations? This is a different question (from this one) as it concerns strings and not antimatter..."}
{"id":"53201","title":"Why is the turbulent energy cascade described as function of a wavenumber?","text":"In all the literature I've seen the turbulent energy spectrum described as $E(k)$ instead of $E(L)$, i.e. as a function of a wave number not eddy size. The connection via $k=2\\pi\/\\lambda$ is clear, but exactly what wave process is meant here. Is the idea that turbulent flow can be viewed as a superposition of waves? Waves of what? Or is this just a common notation used for energy spectra?"}
{"id":"90592","title":"Better explanation of the common general relativity illustration (stretched sheet of fabric)","text":"I've seen many science popularisation documentaries and read few books (obviously not being scientist myself). I am able to _process and understand_ basic ideas behind most of these. However for general relativity there is this one illustration, which is being used over and over (image from Wikipedia): ![Spacetime curvature \\(from Wikipedia\\)](http:\/\/upload.wikimedia.org\/wikipedia\/commons\/2\/22\/Spacetime_curvature.png) I always thought that general relativity gives another way how you can describe gravity. However for this illustration to work, there needs to be another force, pulling the object down (referring to a direction in the attached image). If I put two non-moving objects in the image, what force will pull them together? So where is my understanding incorrect? Or is general relativity not about explaining gravity and just describes how heavy objects bends spacetime (in that case the analogy is being used not correctly in my opinion)? * * * **UPDATE** Thank you for the answers and comments. Namely the XKCD comics is a spot on. I understand that the analogy with bent sheet of fabric pretty bad, but it seems that it can be fixed if you don't bent the fabric, but just distort the drawn grid. Would you be so kind and answer the second part of the question as well - whether general relativity is explaining gravitational force. To me it seems that it is not (bending of spacetime simply can not affect two non-moving objects). However most of the time it is being presented that it does."}
{"id":"9729","title":"potential energy of an object due to other two objects","text":"consider object A with mass $m_{A}$ and positional vector $\\overrightarrow{r_{A}}$ object B with mass $m_{B}$ and positional vector $\\overrightarrow{r_{B}}$ object C with mass $m_{C}$ and positional vector $\\overrightarrow{r_{C}}$ since reference frame is inertial (assumed) so $m_{A}\\frac{d^{2}}{dt^{2}}\\overrightarrow{r_{A}}=\\overrightarrow{F}{}_{AB}+\\overrightarrow{F}{}_{AC}$ $\\Rightarrow$$m_{A}\\frac{d^{2}}{dt^{2}}\\overrightarrow{r_{A}}=G\\frac{m_{A}m_{B}}{\\left|\\overrightarrow{r_{B}}-\\overrightarrow{r_{A}}\\right|^{3}}(\\overrightarrow{r_{B}}-\\overrightarrow{r_{A}})+G\\frac{m_{A}m_{C}}{\\left|\\overrightarrow{r_{C}}-\\overrightarrow{r_{A}}\\right|^{3}}(\\overrightarrow{r_{C}}-\\overrightarrow{r_{A}})$ ....[1] for potential energy equation is to be integrated with a positional vector and i am trying hard to figure out what that vector could be, but still no success. please help * * * Potential energy of a object due to another object $m_{A}\\frac{d^{2}}{dt^{2}}\\overrightarrow{r_{A}}=\\overrightarrow{F_{A}}=G\\frac{m_{A}m_{B}}{\\left|\\overrightarrow{r_{A}}-\\overrightarrow{r_{B}}\\right|^{3}}(\\overrightarrow{r_{B}}-\\overrightarrow{r_{A}})$ assuming $\\overrightarrow{r_{0}}=\\overrightarrow{r_{B}}-\\overrightarrow{r_{A}}$ $\\Rightarrow m_{A}\\frac{d^{2}}{dt^{2}}\\overrightarrow{r_{0}}=-(1+\\frac{m_{A}}{m_{B}})(G\\frac{m_{A}m_{B}}{r_{0}^{2}}\\hat{r_{0}})$ $\\Rightarrow\\intop_{\\overrightarrow{r_{0i}}}^{\\overrightarrow{r_{0f}}}\\left(m_{A}\\frac{d^{2}}{dt^{2}}\\overrightarrow{r_{0}}\\right).d\\overrightarrow{r_{0}}=-\\intop_{\\overrightarrow{r_{0i}}}^{\\overrightarrow{r_{0f}}}\\left((1+\\frac{m_{A}}{m_{B}})(G\\frac{m_{A}m_{B}}{r_{0}^{2}}\\hat{r_{0}})\\right).d\\overrightarrow{r_{0}}$ $\\Rightarrow\\left.\\frac{1}{2}m_{A}v_{0}^{2}\\right|_{v_{0}(\\overrightarrow{r_{0i}})}^{v_{0}(\\overrightarrow{r_{0f}})}=\\left.G\\frac{m_{A}(m_{A}+m_{B})}{r_{0}}\\right|_{\\overrightarrow{r_{0i}}}^{\\overrightarrow{r_{0f}}}$ So What positional vector should be integrated with equ [1] ? How it is decided ? (i mean what kind of property(s) that vector should possess ?) EDIT 1 $\\int_{\\overrightarrow{r_{i}}}^{\\overrightarrow{r_{f}}}\\left(m_{A}\\frac{d^{2}}{dt^{2}}\\overrightarrow{r_{A}}\\right).d\\overrightarrow{r}=\\int_{\\overrightarrow{r_{i}}}^{\\overrightarrow{r_{f}}}\\left(G\\frac{m_{A}m_{B}}{\\left|\\overrightarrow{r_{B}}-\\overrightarrow{r_{A}}\\right|^{3}}(\\overrightarrow{r_{B}}-\\overrightarrow{r_{A}})+G\\frac{m_{A}m_{C}}{\\left|\\overrightarrow{r_{C}}-\\overrightarrow{r_{A}}\\right|^{3}}(\\overrightarrow{r_{C}}-\\overrightarrow{r_{A}})\\right).d\\overrightarrow{r}$ .....[2] questions a) what is $\\overrightarrow{r}$ in terms of $\\overrightarrow{r_{A}}$,$\\overrightarrow{r_{B}}$or $\\overrightarrow{r_{C}}$ ? b) solve equ [2]."}
{"id":"24132","title":"How does gravitation propagate along curved spacetime?","text":"In this wikipedia article it is described how a beam of light, with its locally constant speed, can travel \"faster than light\". That is to say it travels a distance, which, from a special relativistic point of view, is surprisingly big. I wonder if a gravitational wave on such a curved spacetime (of which the wave is actually part of) behaves equally. > Does a gravitational wave also ride on expanding spacetime, just as light > does? Do the nonlinearities of gravitation-gravitation interaction influence > the propagation of a wave (like e.g. a plasma) such that light and gravity > are effectively not equally fast? > > If I want to send a fast signal in this expanding universe scenario, in what > fashion do I decide to I send it?"}
{"id":"112248","title":"What is light localisation?","text":"Reading about plasmonic nanoparticles I faced the term \"localised light\". How can one localise light? What are applications of it?"}
{"id":"45963","title":"Do black holes have infinite areas and volumes?","text":"How to calculate the area \/ volume of a black hole? Is there a corresponding mathematical function such as rotating $1\/x$ around the $x$-axis or likewise to find the volume?"}
{"id":"45969","title":"Does the kinetic theory of gases means gases mix almost instantaneously?","text":"This theory has bugged me ever since my first physics class on the subject. If this (http:\/\/en.wikipedia.org\/wiki\/Kinetic_theory) is true, it leads me to a few weird conclusions. Opening the rear window in a pickup truck at 10 m\/s doesn't empty the cabin of all its air. This means that gas molecules can catch up with us from behind so we can conclude that gases move much faster than 10 m\/s at normal temperatures. Then, shouldn't gases mix almost instantaneously? For example, shouldn't a fart's smell reach us faster than a car on an highway and then dissapear almost immediately into such a large area that the concentration is too low for our noses to detect?"}
{"id":"15587","title":"How to get distance when acceleration is not constant?","text":"I have a background in calculus but don't really know anything about physics. Forgive me if this is a really basic question. The equation for distance of an accelerating object with constant acceleration is: $$d=ut +\\frac{1}{2}at^2$$ which can also be expressed $$d=\\frac{\\mathrm{d}x}{\\mathrm{d}t}t+\\frac{\\mathrm{d^2}x}{\\mathrm{d}t^2}\\frac{t^2}{2}$$ (where x(t) is the position of the object at time t) That's fine for a canonball or something like that, but what about a car accelerating from 0 to cruising speed? The acceleration is obviously not constant, but what about the change in acceleration? Is it constant? I suspect not. And then what about the change in the change of acceleration, etc. etc.? In other words, how does one know how many additional terms to add in the series? $$d=\\frac{\\mathrm{d}x}{\\mathrm{d}t}t+\\frac{\\mathrm{d^2}x}{\\mathrm{d}t^2}\\frac{t^2}{2}+\\frac{\\mathrm{d^3}x}{\\mathrm{d}t^3}\\frac{t^3}{3}+\\frac{\\mathrm{d^4}x}{\\mathrm{d}t^4}\\frac{t^4}{4}\\cdot etc. \\cdot ?$$"}
{"id":"56595","title":"Getting pairs of angle and velocity for a projectile to a given destination","text":"I'm trying to calculate the initial velocity $v_0$ and angle $\\theta$ for a given destination $(x, y)$ with a launch height of $y_0$. Obviously there will be a set of pairs of velocity and angle that will pass through the destination point. This set is given by $$ \\left\\\\{(v_0,\\theta) \\middle|y = - \\frac{g}{2 v_0^2 \\cdot \\cos^2(\\theta)} \\cdot x^2 + \\tan(\\theta) \\cdot x +y_0 \\right\\\\} \\ .$$ I have already looked at this questions: Solving for initial velocity required to launch a projectile to a given destination at a different height How to get the angle needed for a projectile to pass through a given point for trajectory plotting So for example I could rewrite the set to $$\\left\\\\{ (v_0,\\theta) \\middle| v_0 = \\frac{1}{\\cos(\\theta)}\\sqrt{\\frac{\\frac{1}{2} g x^2}{x \\tan(\\theta)+y_0}} \\right\\\\}\\ .$$ Is there an easy way to compute this set for one given destination without iterating over all possible velocities or angles?"}
{"id":"94233","title":"TISE for a triangular potential","text":"I want to solve the TISE of a particle of charge $q$ and mass $m$ in a one dimensional triangular potential, with an infinitely high potential wall at $x = 0$, i.e. $$\\hat{H}=-\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x^2}+V(x),$$ with $$V\\left(x\\right)=\\begin{cases}qEx&{\\rm if}\\,x>0 \\\\\\ \\infty &{\\rm if}\\,x\\leq0\\end{cases}$$ For $x>0$, this gives the TISE $$-\\frac{\\hbar^2}{2m}\\frac{\\partial^2\\Psi_n}{\\partial x^2}+qEx\\cdot\\Psi_n = E_n\\Psi_n.$$ I know that the solutions of $$\\frac{d^2y}{dx^2} - xy = 0$$ are given by the Airy functions, however I don't know how I can transform the TISE to this form. I know it's possible, because I have the solution to the problem, but I don't know how I can myself come up with this, I don't see the aim which has to be reached in order for a substitution to be successful. Also, I don't know how the differential will transform under a substitution including $x$. Any help? The substitution used in the solution is $$u=\\left(\\frac{2mqE}{\\hbar^2}\\right)^\\frac{1}{3}\\left(x-\\frac{E_n}{qE}\\right).$$"}
{"id":"3076","title":"What would ACTUALLY happen to a person jettisoned into space?","text":"_[insert obligatory statement of my lack of knowledge in physics]_ Alright, so we have all seen the movies where someone gets blasted out of the airlock on their starship, or their suit decompresses while on a space walk. The poor schmoe usually either decompresses so violently that blood is oozing out of every orifice in their body, or they freeze instantly. From this I have two questions:   1. **Would the decompression really be that violent?**     1. Clearly the drastic difference in pressure from a normal \"earth\" like environment to space would be bad, but would it be _that_ devastating.     2. I vaguely remember that standard atmospheric pressure was something like 15 psi, which doesn't seem like enough to mess you up that bad.   2. **Would you _actually_ freeze instantly in space?**     1. Heat, or lack thereof is a measure of internal energy, but in a vacuum there wouldn't be anything to have internal energy, so does space even have a temperature?     2. Wouldn't some form of matter have to be present in order to cool off? If there were no matter besides yourself and a few stray particles here and there, it seems like it would take a very long time to cool off."}
{"id":"91155","title":"Traditional Kirchoff voltage law in AC circuit?","text":"The traditional (not taking into account phasor addition or complex addition) application of Kirchoff Voltage law, i.e. $\\Sigma\\Delta V=0$ along a loop, does not work for AC circuits. We can sum the voltage drops to zero if we take into account their phase differences. But at some particular time, no matter what the phase differences in the general equation for the voltages across various components, there will be only one numeric value of $\\Delta V$. > **Why doesn't this sum to zero over the loop in case of AC circuits with > non-zeroreactance?** Is this because there is some contribution to the $\\oint E.dl$ due to changing magnetic flux? If it is, even in absence of any inductance, the voltage drops still do not sum to zero. why?"}
{"id":"59990","title":"Exploiting the Heisenberg Uncertainty Principle as a means to communicate","text":"It seems as though I've come across a rather unusual conclusion that could either simply be a misinterpretation or a contradictory discovery. I seem to have found a way to utilize the Heisenberg Uncertainty Principle (HUP) to our benefit to communicate faster than the speed of light (FTL). I am aware of many proposals that try to utilize entangled spin particles and try to communicate, but this schemes fail because the measurement outcome can NOT be controlled. Therefore, even if two people share an entangled pair of particles, and Alice measures spin up, although she knows Bob has a spin down particle, Bob didn't measure yet and has no way of knowing that Alice has measured her particle. Hence, FTL is impossible based on the reason that the measurement outcome can not be controlled. This leads to my proposal of using position and momentum entangled pairs. Consider **Alice and Bob hold an ensemble of entangled particles in position and momenta** , where each particle is trapped in a separate harmonic potential. At some specified time agreed upon in the future, Alice measures all of her particles' position to high precision. **Alice and Bob have synchronized clocks** and Bob measures all of his entangled particle's momentum to whatever accuracy he chooses. Alice can calculate the average value and also the standard deviation of position. The standard deviation of position would be extremely narrow, i.e, the spread of her measurements would be very small. From the entanglement relation, **x1 = x2 and p1 = -p2** , we know that when Bob measures his particle's momentum, the spread of momentum will be very large. This must be true because position and momentum cannot be measured to arbitrary accuracy at the same time. This is very similar to Einstein's proposal to violate the HUP, however I am exploiting HUP. What this all means is that the if Bob measures a very large momentum spread, it must mean that Alice has made her measurements. If Bob measures a relatively moderate momentum spread, then he knows Alice did not measure her particles. **Since the position measurement can be made to arbitrary accuracy, we are \"controlling the spread or the standard deviation as the means to communicate.\"** (MAIN RATIONALE) Say Alice and Bob have multiple ensembles of entangled particles. Alice can relay a message by simultaneously measuring her first ensemble, meaning its a \"1\" and not touch her second ensemble meaning its a \"0\", and perhaps she chose to measure the third ensemble, \"1\", etc. Hence generating the series 101..., where **each ensemble of entangled particles represents one bit of information.** What is flawed in this proposal? Entanglement in position and momenta is well established. We can also choose to measure the position of a particle to arbitrary precision. The HUP must hold for Bob and everyone involved."}
{"id":"111924","title":"Would the inside of a black hole be like a giant mirror?","text":"As any light reflected or emitted from objects inside a black hole (if it is possible to be there) does not leave the event horizon and comes back inside, would it be like seeing yourself? What I mean is that would the light we might emit\/reflect return to our eyes and make us sort of look at ourselves?"}
{"id":"106521","title":"force on a moving charge in magnetic field","text":"Need help in understanding the direction of magnetic force in the magnetic field!Totally confused by directions. Why is it that magnetic force is perpendicular to the direction of magnetic field and velocity of charged particle. Why is it(force) not in the same direction as the magnetuc field"}
{"id":"106526","title":"Electron distribution around atom when moving","text":"I do not have much experience on this but if an atom has some electrons around nucleus and the atom itself it is moving at some speed does that affect the distribution of electrons around? I am presuming that the interaction between the nucleus and electrons has a constant speed $c$. Anything I found so far is a calculation for interactions that presume an infinite speed. As an argument I am thinking of relativistic Doppler effect that does not change proportional with $v\/c$. So I am thinking that maybe the speed does affect the distribution and so that is why the difference in the emission energy."}
{"id":"23212","title":"Shor's Algorithm: Why throw away the f(x)?","text":"I'm having a little trouble understanding Shor's algorithm - namely, why do we throw away the result f(x) that we get after applying the F gate? Isn't that the answer we need? My notation: $\\newcommand{\\ket}[1]{\\left|#1\\right>}$ $F(\\ket x \\otimes \\ket0) = \\ket x \\otimes \\ket{f(x)}$ , $f(x) = a^x \\mod r$"}
{"id":"109204","title":"S-operator lorentz invariance","text":"How to show that $\\hat {S}$-operator must be lorentz-invariant operator? $$ |\\Psi (t)\\rangle = \\hat {S} | \\Psi (0) \\rangle , \\quad \\hat {S} = \\hat {T}e^{-i\\int \\hat {H}_{I}d^{4}x}. $$ I have read that this result follows from the unitarity of the Poincare group operator $U_{0}(\\Lambda , a)$ and the covariance of S-matrix $S_{out, in} = \\langle out| \\hat {S}|in \\rangle$, but I don't understand how do we conclude that from this follows that $U_{0}(\\Lambda , a)\\hat {S}U^{-1}_{0}(\\Lambda , a) = \\hat {S}$. I'm not interesting in the derivation of lorentz-invariance of S-operator from causality principle at this moment."}
{"id":"105954","title":"How do the different types of energy apply to voltage in a circuit","text":"I was just wondering what happens in a circuit in terms of different types of energy transformations. If you apply a voltage to a circuit then electrons start moving (very slowly). Since the electrons want to flow to the positive terminal they will have electrical potential. But the electrons are also now moving once the voltage is applied as there is a current so would they have kinetic energy as well? Some information I have found online says that electrons collide with atoms in a bulb and this is why the filament heats up. But this would imply that kinetic energy is being transformed into heat and this can't be correct because surely any change in kinetic energy would alter the current flowing? Also, in a series circuit, if you measure the potential difference between any 2 points after all of the loads then you always get zero. I was just wondering why it is zero because don't the electrons keep moving even after passing through all the load in order to get back to the power source so surely they cannot do this without some form of energy?"}
{"id":"105956","title":"Active and passive transformations and the change in potential energy","text":"Under active transformation, the particle moves. On the other hand, for a passive one, the coordinate is just relabel. I've read that the passive one will not affect the potential energy and the active one will change it. This is the case for two massively different size objects, like earth and a ball. But basically there will be no difference between passive and active transformation if the two objects making the system are comparable in size. My question is how to prove that and which mathematical framework one should use?"}
{"id":"46376","title":"Superconducting nanowire in a parallel magnetic field","text":"Consider a nanoscopic wire (with radius $R$) of superconducting material. The wire lies along the $z$-axis and a magnetic field $\\mathbf{H}_a = H\\mathbf{e}_z$ is applied. The magnetic field is too weak to destroy superconductivity or to induce vortices and we assume that the superconducting parameter $\\psi$ can be put equal to the constant value $\\psi_{\\infty}$ in the material (and of course, zero outside). Now, I want to use the second Ginzburg-Landau equation to find the vector potential $\\mathbf{A}$ (and the magnetic induction field $\\mathbf{B} = \\nabla\\times\\mathbf{A}$ from this potential). With the assumptions mentioned above the second Ginzburg-Landau equation simplifies to   $\\nabla\\times\\left(\\nabla\\times\\mathbf{A}\\right) = \\dfrac{\\mu_0Q^2}{M}\\left|\\psi\\right|^2\\mathbf{A}$   where   $\\psi(\\mathbf{r}) = \\left\\\\{\\begin{array}{lr} \\psi_{\\infty} & \\mathrm{in\\,nanowire} \\\\\\ 0 & \\mathrm{outside} \\end{array} \\right.$   This expression is simplified even further by using the Coulomb gauge $\\nabla\\cdot\\mathbf{A} = 0$, leading to the following vector laplace equation in the case outside the wire:   $\\nabla^2\\mathbf{A} = 0$   Since the system has cylindrical symmetry, I'm working in cylindrical coordinates. The scalar equations associated with the general vector equation are then:   $\\left\\\\{\\begin{array}{rcl} \\dfrac{\\partial^2A_{\\rho}}{\\partial\\rho^2} + \\dfrac{1}{\\rho^2}\\dfrac{\\partial^2A_{\\rho}}{\\partial\\phi^2} + \\dfrac{\\partial^2A_{\\rho}}{\\partial z^2} + \\dfrac{1}{\\rho}\\dfrac{\\partial A_{\\rho}}{\\partial\\rho} - \\dfrac{2}{\\rho^2}\\dfrac{\\partial A_{\\phi}}{\\partial\\phi} - \\dfrac{A_{\\rho}}{\\rho^2} & = & \\dfrac{\\mu_0Q^2}{M}\\left|\\psi\\right|^2 A_{\\rho} \\\\\\ \\dfrac{\\partial^2A_{\\phi}}{\\partial\\rho^2} + \\dfrac{1}{\\rho^2}\\dfrac{\\partial^2A_{\\phi}}{\\partial\\phi^2} + \\dfrac{\\partial^2A_{\\phi}}{\\partial z^2} + \\dfrac{1}{\\rho}\\dfrac{\\partial A_{\\phi}}{\\partial\\rho} + \\dfrac{2}{\\rho^2}\\dfrac{\\partial A_{\\rho}}{\\partial\\phi} - \\dfrac{A_{\\phi}}{\\rho^2} & = & \\dfrac{\\mu_0Q^2}{M}\\left|\\psi\\right|^2 A_{\\phi} \\\\\\ \\dfrac{\\partial^2A_z}{\\partial\\rho^2} + \\dfrac{1}{\\rho^2}\\dfrac{\\partial^2A_z}{\\partial\\phi^2} + \\dfrac{\\partial^2A_z}{\\partial z^2} + \\dfrac{1}{\\rho}\\dfrac{\\partial A_z}{\\partial\\rho} & = & \\dfrac{\\mu_0Q^2}{M}\\left|\\psi\\right|^2 A_z \\end{array} \\right.$   while for the case outside the wire the right hand sides are zero. These equations are coupled and I'm not sure how to efficiently solve them. So my question is: could anyone point me in the right direction as to how I should try to solve this vector laplace equation in cylindrical coordinates? Or are there perhaps any possible further assumptions I failed to make? I'm not looking for a complete answer by the way, just a push down the right path, a method. Note: I've asked for a method to solve the vector laplace equation in cylindrical coordinates with less context on math.se yesterday (Solving the vector Laplace equation in cylindrical coordinates) but I thought it might be better to put this here, so I can describe the full physical problem. Perhaps the question on math.se can\/should be deleted then?"}
{"id":"102469","title":"Calculating the height objects fall from","text":"* Imagine an object of a given mass.   * The object falls from a certain height.   * On contact with the ground, the object has a certain force. If the force of the object and the mass of the object are know, what formula would be used to find the height fallen from?"}
{"id":"92714","title":"Relation of fraction of binary stars with spectral class (mass)","text":"What is relation of fraction of binary stars with spectral class (mass)? For example, how many binary stars are among O,B,A,F,G,K,M stars separately?"}
{"id":"32715","title":"What Cat States of light have been experimentally produced?","text":"I'm specifically looking for Schrödinger's Cat states involving superpositions of two, or if it's been done more, coherent states, i.e. monomodal states of the form $$|\\psi\\rangle=a|\\alpha\\rangle+b|\\beta\\rangle.$$ What states of this form have been produced in experiment? How even can the weights be? What regions of the $\\hat{a}$-eigenvalue $\\alpha$ and $\\beta$ are accessible? If more than two coherent states can be superposed, how many? What phase-space geometries are possible so far? I'm also interested in what techniques are currently used to generate these states."}
{"id":"32713","title":"Confused about indices of the Ricci tensor","text":"In an intro to GR book the Ricci tensor is given as: $$R_{\\mu\\nu}=\\partial_{\\lambda}\\Gamma_{\\mu \\nu}^{\\lambda}-\\Gamma_{\\lambda \\sigma}^{\\lambda}\\Gamma_{\\mu \\nu}^{\\sigma}-[\\partial_{\\nu}\\Gamma_{\\mu \\lambda}^{\\lambda}+\\Gamma_{\\nu \\sigma}^{\\lambda}\\Gamma_{\\mu \\lambda}^{\\sigma}]$$ I have gotten to the point where I can work out a given Christoffel symbol, but I am still having trouble working out the above tensor as a whole (just algebraically speaking). If I'm not mistaken, $R_{\\mu\\nu}$ should end up a $\\mu$x$\\nu$ (i.e. 4x4) matrix just like the energy-momentum tensor on the other side of the field equations. In the above rendering $\\sigma$ is clearly a dummy index to be summed over, and I can see how $\\lambda$ is also a dummy index in the first term. But the $\\lambda$s in the other terms seem to be free indices, which would then introduce incompatible dimensions in the matrix operations. I appreciate it if someone can point out the error of my ways."}
{"id":"19285","title":"If microwave ovens and WiFi both operate on the same frequency, why doesn't WiFi cook things?","text":"If we ignore 5GHz WiFi, then both microwaves and WiFi create photons at ~2.4GHz but one of them will boil water in a few seconds but the other doesn't have any effect. So what's the difference? Is it simply the number of photons created? Is that what the wattage of a microwave measures? If so, what would be the wattage of a wireless router? Does the enclosed space have anything to do with it? If it all has to do with power output could I put enough WiFi routers together in a room to cook a turkey (from microwaves and not waste heat)?"}
{"id":"6098","title":"Reynolds number, turbulence regime, and drag force","text":"I am trying to model a system in which cubes of about 2 cm in size are floating in a circular water thank of about 30 cm in diameter. The cubes move around under the influence of the fluid flow induced by four inlets that point toward the center of the tank, and are located at the positions $0$, $\\pi\/2$, $\\pi$, and $3\\pi\/2$. The flow velocity ranges from 0 to 10 cm\/s, with an average velocity around 6 cm\/s. My questions are the following:   * What would be the Reynolds number of the system? In particular, should I take as characteristic length the size of the cubes, or that of the tank?   * For such a system, what is the limit Reynolds number for the turbulent regime?   * What would be the correct form of the drag force, and do you intuitively think that the orientation of the blocks is negligible from a drag coefficient point of view? Thanks for your help!"}
{"id":"131366","title":"Similarity in two events from Dynamics point of view","text":"What is similar from dynamic point of view between book lying stationary on horizontal table and rain drop falling down with constant speed? * * * I can find two similarities   1. Force acting on them is equal   2. Both have horizontal velocities zero Is there any other similarities possible? It is not mentioned how many similarities we have to give that means we should give all possible similarities. Answer should contain related basic physic like acceleration, velocity, displacement, force,etc. We can say stuff which can be understand by high school student."}
{"id":"22165","title":"Why does optical pumping of Rubidium require presence of magnetic field?","text":"The optical pumping experiment of Rubidium requires the presence of magnetic field, but I don't understand why. The basic principle of pumping is that the selection rule forbids transition from $m_F=2$ of the ground state of ${}^{87} \\mathrm{Rb}$ to excited states, but not the other way around ($\\vec{F}$ is the total angular momentum of electron and nucleus). After several round of absorption and spontaneous emission, all atoms will reach the state of $m_F=2$, hence the optical pumping effect. But what does the Zeeman splitting have anything to do with optical pumping? Granted, the ground state, even after fine structure and hyperfine structure considered, is degenerate without Zeeman splitting, but the states with different $m_F$ still exists. In addition, how is the strength of optical pumping related to the intensity of magnetic field applied?"}
{"id":"22163","title":"Differentiating inside an integral sign","text":"I'm reading John Taylor's Classical Mechanics book and I'm at the part where he's deriving the Euler-Lagrange equation. Here is the part of the derivation that I didn't follow: ![enter image description here](http:\/\/i.stack.imgur.com\/7B0wm.jpg) I don't get how he goes from 6.9 to 6.10 by partial-differentiating the term inside the integral. If this is allowed, I was probably missed my calculus class the day it was covered. Can someone tell me more about this? Which part of calculus is this from?"}
{"id":"77443","title":"Newton third law. Is there a limit on how much force you can apply to an object?","text":"Recently in class we went over Newton's Third Law. In the book they put an example of hitting a punching-bag with your fist and hitting a piece of paper, or an object with much less mass. It's clear that you cannot exert more force on the paper than what the paper can exert on you, otherwise it might stop your fist in the process, just like the punching-bag stops your fist. Is there a limit, however, on much force you can exert on a paper? This might sound silly but it got me thinking for the past couple of days. In order to make it a little more clear consider a Car with a paper in front of it. The car-system can continuously increase the acceleration, but this means that the paper will increase the acceleration as well. Ergo, by Newton 2nd Law, the force exerted on the paper has to increase. Right? That same force back to the car. Therefore, the car can continuously increase the force it exerts on the paper."}
{"id":"77447","title":"Expression of electrostatic field","text":"An electrostatic field is characterized by the fact that it depends only by $r$, isn't it? If it is true, I don't understand why this expression, given in cylindrical coordinates, $${\\bf E(r)}=\\frac{\\alpha}{z^2}{\\bf u_r}-2 \\frac{\\alpha r}{z^3}{\\bf u_z} $$ rapresents an electrostatic field."}
{"id":"111919","title":"Can we develop a new kind of principles?","text":"We have a basic equation $F=ma$. Now can we change this equation to another variable and start a new era in physics? If so, then how? If not, then why?"}
{"id":"134774","title":"Police laser speed detection guns","text":"### Situation Police shooting a laser device into his rearview mirror, bouncing off the oncoming car license plate. I want to know if this second surface mirror, and the symmetric system--laser gun to mirror to license plate of moving car to mirror to laser gun---has a built in error because of the wavelength shift due to the glass of the mirror. Velocity and wavelength change for the raypath inside the mirror's glass, but would revert to the original wavelength for the \"in air\" portion of the raypath. ### Question Is this symmetry perfect and creates no error in measuring the car's speed?"}
{"id":"130014","title":"How the Bloch sphere of a Hahn echo in NMR looks like? 90-t-90-t-echo","text":"I have tried to find in the literature a proper nice and beautiful Bloch sphere to describe the trajectory of a nuclear spin, starting in z-axis, using a pulse sequence of an initial 90º pulse with zero phase (which means that the pulse is launched in the x-axis), wait for a time $\\tau$, and then launch another 90º pulse with zero phase, and wait the same amount of time tau to measure an echo. Please notice that I am talking about Nuclear Magnetic resonance, and most of the books don't offer a Bloch sphere to explain this echo, which is supposed to be the first! echo, and only use a 90º and 180º pulse to explain that. I have read the original article of Hahn echo, but I saw that his diagram is not clear."}
{"id":"12140","title":"Why does everything spin?","text":"The origin of spin is some what a puzzle to me, everything spin from galaxies to planets to weather to electrons. Where has all the angular momentum come from? Why is it so natural? I was also thinking do photons spin? we always think of the wave as a standard 2d sin wave but could this rotate in 3d? What implications would this have? And what about spacetime how does all the spinning effect? This has always been avoided in all lectures and classes I ever went to."}
{"id":"24888","title":"Why does each celestial object spin on its own axis?","text":"AFAIK all the celestial objects have a spin motion around its axis. What is the reason for this? If it must rotate by some theory, what decides it's direction and speed of rotation? Is there any object that does not rotate about its axis?"}
{"id":"56864","title":"Why planets are rotating only in one plane?","text":"Since gravity is three dimensional why planets are rotating only in one plane around sun."}
{"id":"75489","title":"Why does earth spins?","text":"I understand the governing force causing the earth's cycle around the sun is gravity. It can be described by Keplar's law. But what causes the earth's spin? what is its governing law?"}
{"id":"7819","title":"Why does Venus rotate the opposite direction as other planets?","text":"Given: Law of Conservation of Angular Momentum.   * Reverse spinning with dense atmosphere (92 times > Earth & CO2 dominant sulphur based).   * Surface same degree of aging all over.   * Hypothetical large impact is not a sufficient answer. Assuming any object large enough to alter a planets rotation or even orbit would likely destroy most of its shape, yet Venus has retained a spherical property with a seemingly flat, even terrain indicating no volcanoes,and few if any visible meteor impacts. It would be fragmented and dispersed for billions of years. Even the question of what meteor, comet, asteroid composition could survive traveling that close to the sun's temperature, radiation, electromagnetic energy, solar flares, or gravity to equal a mass reactionary change as to alter it's spin."}
{"id":"105248","title":"Why do we have a galactic and a solar plane","text":"Why do all of the planets in our solar system move is (more or less) the same plane? It would seem that the planets should all have very different orbital planes, each at a different angel to each other. The fact that all of their orbits are so closely parallel suggests that a force is pulling and\/or maintaining the orbits of the planets in a flat plane. What is that force? A similar force also seems to be working at a galactic level too."}
{"id":"42004","title":"Why are the orbits of the planets in our solar system along the same basic plane?","text":"> **Possible Duplicate:**   >  Why are our planets in the solar system all on the same disc\/plane\/layer? After watching this video I realized that the orbits of the planets in our solar system basically reside along a similar 'flat' plane: ![enter image description here](http:\/\/i.stack.imgur.com\/q4O5U.gif) Meaning that the angles of orbit are no greater than 20 degrees (if you still count Pluto as a planetary object). Why are there none of the planets at a 45 or even 90 degree orbit (in relation to the earth)? Is this due specifically to the gravitational pull from the Sun?"}
{"id":"38713","title":"Single plane Ring system ","text":"> **Possible Duplicate:**   >  Why are our planets in the solar system all on the same disc\/plane\/layer? I've noticed this in many pictures, Planets are shown with a single ring around them (in some particular plane). Taking extreme case... As gravity should act in all the directions, such planets must be covered with asteroids all around them. Not just _single_ ring in some _single_ plane..! So, My question is: Why don't planets have many rings instead of just a single ring..?"}
{"id":"93830","title":"Why the galaxies forms 2D plane (or spiral-like) instead of 3D ball (or spherical-like)?","text":"> **Question** : As we know (1) the macroscopic spatial dimension of our > universe is 3 dimension, and (2) Gravity attracts massive objects together > and the gravitational force is isotropic without directional preferences. > **Why do we have the spiral 2D plane-like Galaxy(galaxies), (or more common > than), instead of spherical, 3D ball or elliptic like galaxies?** **Input** : Gravity is (at least, seems to be) **isotropic** from its equation of force law (Newtonian gravity). It should show **no directional preferences** from the form of force vector $\\vec{F}=\\frac{GM(r_1)m(r_2)}{(\\vec{r_1}-\\vec{r_2})^2} \\hat{r_{12}}$. The Einstein gravity also does not show directional dependence at least microscopically. If the gravity attracts massive objects together **isotropically** , and the macroscopic space dimension is **3 dimension** , it seems to be natural to have a spherical or ball like shape of massive objects gather together. Such as the Globular clusters, or **GC** , are roughly spherical groupings Star cluster, and its Wiki picture: ![Star cluster](http:\/\/i.stack.imgur.com\/KKEUZ.jpg) However, my impression is that, even if we have observed some more-spherical or more-ball-like Elliptical galaxy, it is more common to find more-planer Spiral galaxy? ( **Is this statement correct? Let me know if I am wrong.** ) Such as our Milky Way Also, such as this NGC 4414 galaxy: ![Galaxy](http:\/\/i.stack.imgur.com\/jrklh.jpg) **Is there some physics or math theory explains why theGalaxy turns out to be planer-like (or spiral-like) instead ball-like or spherical-like?**"}
{"id":"17325","title":"Why does the earth rotate?","text":"> **Possible Duplicate:**   >  Why does every thing spin? So why would the earth, or any planet for that matter, rotate along an axis? I know of no force which could come into play here, so i assume it just started off with an initial rotation. But where did this rotational energy come from? And are there any known planets which do not rotate? Assuming no force accounts for earth's rotation, would it be true that the angular velocity of the earth would decrease(by a very very small amount), when bombarded by meteors(Due to an increase in the moment of inertia by the increase in mass of the earth i.e. the additional mass of the meteors)?"}
{"id":"18502","title":"On the origin of the rotation of celestial bodies","text":"> **Possible Duplicate:**   >  Why does Venus rotate the opposite direction as other planets?   >  Why does every thing spin? As far as I can imagine, almost each celestial body, star, planet, solar- systems, galaxies do rotate on their center. Where this come from ? Is it the normal work of gravity? And why Venus has a retrograde rotation? (Or, why all the solar systems rotate in the same way - except Venus?)"}
{"id":"100076","title":"why galaxies look like discs rather than spheres?","text":"Black holes have much gravitation to hold all stars and nebulas but why they are aligned in disc type shapes rather than spheres because gravitation is everywhere around black holes. Even on upper and lower sides. Even if rotation is the cause why cant they move above black holes in smaller circles."}
{"id":"26083","title":"Why are our planets in the solar system all on the same disc\/plane\/layer?","text":"I always see pictures of the solar system where our sun is in the middle and the planets surround the sun. All these planets move on orbits on the same layer. Why?"}
{"id":"130019","title":"Rotating uniform rod","text":"> A uniform rod of mass 1.2 kg and length 1.8 m is pivoted in the horizontal > position as shown (black point).![enter image description > here](http:\/\/i.stack.imgur.com\/cGmYy.png) The rod is at rest and then > released. The acceleration due to gravity is $g = 9.8 m\/s^2$. 1) Find the > rotational inertia of the rod relative to the axis perpendicular to the > screen and passing through the pivot point. 2) What is the angular speed (in > rad\/s, but do not include units) of the rod as it passes through the > vertical position (when end marked B is at the bottom)? My attempt of solution: To find moment of inertia, I use the moment of inertia of a uniform rod respect to it's center of mass (middle in this case), which is $$I_{cm}=mL^{2}\/12$$ where $m$ is mass of the rod and $L$ its length. Then I use the parallel axis theorem to find the moment of inertia in this case: $$mL^{2}\/12+0.45^{2}m=0.567$$ For the 2º part, since the net torque on the rod is not constant I use potential energy to find first the speed of center of mass at the vertical position. The initial height of center of mass is 0.45 m taking as reference point position of the middle point of the rod at vertical position. So: $$mg0.45=1\/2mv_{cm}^{2}+1\/2I\\omega ^{2}=1\/2mv_{cm}^{2}+1\/2I(v_{cm}^2\/0.45^{2})$$ From here I get $v_{cm}$ and later the angular velocity. Where am I wrong?"}
{"id":"122041","title":"Why does the light intensity increase as I approach a distance light source?","text":"Analogy: assume that I have constant rain fall and I have a water bucket to collect this rain. If I am rest relative to the earth, I will catch a certain amount of rain. However, if I now move towards the rain, I will increase the amount of water I collect. Now I want to apply this idea to a source emitting photons instead of rain drops. I image a distance light source (distance star) that is emitting a constant number of photons such that I can ignore the $1\/r^2$ fall off of the intensity (that is, the intensity does not change very much over some appreciate distance). So instead of a water bucket, I now have a light collector that measures light intensity. Here is my question: **Does the light intensity measured by my light collector that is moving towards the distant star increase or stay the same when compared to a light collector at rest realtive to the distant star?** My first thought was that the light intensity would increase because the light is blue shifted and higher energy photons will therefore produce a higher intensity. However, I believe that there might be another contribution due to length contraction. Since an observer moving towards a light source has to account for length contraction, does this mean that there is an increase in photon density? If so, this higher photon density in the frame of the light collector will also contribute to increasing the light intensity. Can someone verify or correct my thinking."}
{"id":"122045","title":"Doubt in collision","text":"> _If a ball under the influence of gravity falls straight down from a height > $h$, collides elastically with the floor, at the instant of collision, what > forces does it experience?_ Shouldn't the floor exert a force equal to the weight of the ball, on the ball itself? Then won't the forces cancel out?(since the only to forces on the ball are it's weight and the force of the floor, which are equal and Opposite) I know that the better way to find resulting velocity, etc. Would be to use work-energy or kinematics, but I would like to reconcile those with the force treatment"}
{"id":"130094","title":"Has a theory of a small Big Bang in a much larger universe been proposed?","text":"I have a theory I think might explain why the universe has accelerated in a way that doesn't require dark energy. I'm wondering if someone has proposed this theory before (did some research and couldn't find anything). The theory is that the big bang was essentially a huge super-nova-like eruption inside an even larger universe. Just like the visible universe has super novae that happen inside much much larger galaxies and dust clouds, so could the Big Bang have been a huge explosion in a much huger universe. The acceleration we have seen evidence of could, then, have been caused if the big bang was off center in the huge amount of matter in the outer universe - ie if one side exerted more gravity than the other, mass closer to that side would accelerate away from mass closer to the center of the big bang. I imagine a huge black-hole sucking in a giant swath of matter from the greater universe, coalescing into a small area that leaves a large space around it rather empty. Then when it explodes (or perhaps it just emitted large amounts of matter out of it via something like hawking radiation), it fills this space again with what we now see as our universe. This theory doesn't seem to require any sort of bizarre unknown physics, like dark energy, singularities or anything else that causes divide by 0 errors in established physical equations. Has it been proposed before? Also, I'd be interested to know if anyone has any concrete reasons why this theory wouldn't work."}
{"id":"25433","title":"Stellar Viscosity in Galaxies","text":"Is there such as thing as the viscosity of stars in a galaxy, along the lines of gravitational attraction between stars changing the dynamics. If so, how is that put in terms of the Virial Theorem?"}
{"id":"133666","title":"Hamilton's characteristic and principle functions and separability","text":"Just hoping for some clarity regarding Hamilton's characteristic function (W). When we take a time independent Hamiltonian we can separate the Principle function (S) up into the characteristic function minus $ht$, yes I know its the Legendre transform but, \\begin{equation} W=S+ht \\end{equation} Meirovitch in his _Methods of Analytical Dynamics_ p 356 gives $h$ as being the Jacobi energy function $h$ as defined in earlier chapters both his and Goldstien's texts. This is the only time I have seen it being called $h$ rather than the Hamiltonian. I was just wondering if anyone had read through it and perhaps noticed something different in Meirovitch's definition that escapes me. Most authors define these integration constants as the Hamiltonian instead. I know the difference is subtle but it is intriguing as to why he chose $h$ not $H$! Is it just down to how you express the conjugate momenta?"}
{"id":"128198","title":"Do the standard cosmology models spontaneously break Lorentz symmetry?","text":"In standard cosmology models (Friedmann equations which your favorite choice of DM and DE), there exists a frame in which the total momenta of any sufficiently large sphere, centered at any point in space, will sum to 0 [1] (this is the reference frame in which the CMB anisotropies are minimal). Is this not a form of spontaneous Lorentz symmetry breaking ? While the underlying laws of nature remain Lorentz invariant, the actual physical system in study (in this case the whole universe) seems to have given special status to a certain frame. I can understand this sort of symmetry breaking for something like say the Higgs field. In that situation, the field rolls down to one specific position and \"settles\" in a minima of the Mexican hat potential. While the overall potential $V(\\phi)$ remains invariant under a $\\phi \\rightarrow \\phi e^{i \\theta}$ rotation, none of its solutions exhibit this invariance. Depending on the Higgs model of choice, one can write down this process of symmetry breaking quite rigorously. Does there exist such a formalism that would help elucidate how the universe can \"settle\" into one frame ? I have trouble imagining this, because in the case of the Higgs the minima exist along a finite path in $\\phi$ space, so the spontaneous symmetry breaking can be intuitively understood as $\\phi$ settling **randomly** into any value of $\\phi$ where $V(\\phi)$ is minimal. On the other hand, there seems to me to be no clear way of defining a formalism where the underlying physical system will randomly settle into some frame, as opposed to just some value of $\\phi$ in a rotationally symmetric potential. [1] The rigorous way of saying this is : There exists a reference frame S, such that for all points P that are immobile in S (i.e. $\\vec{r_P}(t_1) = \\vec{r_P}(t_2) \\forall (t_1, t_2)$ where $\\vec{r_P}(t)$ is the spatial position of P in S at a given time $t$), and any arbitrarily small $\\epsilon$, there will exist a sufficiently large radius R such that the sphere of radius R centered on P will have total momenta less than $\\epsilon c \/ E_k$ (where E is the total kinetic energy contained in the sphere). * * * Ben Crowell gave an interesting response that goes somewhat like this : Simply put then : Causally disconnected regions of space did not have this same \"momentumless frame\" (let's call it that unless you have a better idea), inflation brings them into contact, the boost differences result in violent collisions, the whole system eventually thermalises, and so today we have vast swaths of causally connected regions that share this momentumless frame. Now for my interpretation of what this means. In this view, this seems to indeed be a case of spontaneous symmetry breaking, but only locally speaking, because there should be no reason to expect that a distant causally disconnected volume have this same momentumless frame. In other words the symmetry is spontaneously broken by the random outcome of asking \"in what frame is the total momentum of these soon to be causally connected volumes 0?\". If I'm understanding you correctly, this answer will be unique to each causally connected volume, which certainly helps explain how volumes can arbitrarily \"settle\" into one such frame. I'm not sure what the global distribution of boosts would be in this scenario though, and if it would require some sort of fractal distributions to avoid running into the problem again at larger scales (otherwise there would still be some big enough V to satisfy some arbitrarily small total momentum)."}
{"id":"128190","title":"What is the relation between entropy and mass of black hole?","text":"1. What is the relation between entropy and mass of black hole?   2. And what is the relation between symmetry of physics operation and entropy?For instance,measuring or doing measure on state of quantum system especially pure state of a system has to lead to increase in entropy of the system,and the measuring is not symmetric operation,but it seems that the statement is tautologic,because that operations make processes symmetric,means nothing has been changed."}
{"id":"41512","title":"What makes a Feynman diagram real or virtual?","text":"Simple question: as the title says, what makes a real Feynman diagram real, and what makes a virtual diagram virtual? Or in other words, how do I tell whether any given diagram is real or virtual? I've never gotten a really satisfying explanation of this. I would imagine it has something to do with virtual particles, but all internal propagators are virtual particles and I know for a fact that having internal lines doesn't make a diagram virtual."}
{"id":"80748","title":"Small oscillations of heavy string","text":"I'm solving problem in classical field theory and I have some difficulties. I'm trying to study small oscilations of heavy string with fixed points. First of all I wrote down this Lagrangian: $$S=\\int dt ds \\left[\\frac{\\rho}{2}(\\dot{x}^2+\\dot{y}^2)-\\rho g y(s,t)+\\frac{\\lambda(s,t)}{2}\\left(\\left(\\frac{\\partial x}{\\partial s}\\right)^2+\\left(\\frac{\\partial y}{\\partial s}\\right)^2-1\\right)\\right]$$ This Lagrangian describes heavy string with fixed ends in gravitational field. Where $\\rho$ is density, $g$ is gravitational acceleration, $s$ is natural parameter. So I have 3 equations from Euler-Lagrange equations. $$\\rho\\ddot{x}+\\frac{d}{ds}\\left(\\lambda(s,t)\\frac{\\partial x}{\\partial s }\\right)=0$$ $$\\rho\\ddot{y}+\\frac{d}{ds}\\left(\\lambda(s,t)\\frac{\\partial y}{\\partial s }\\right)+\\rho g=0$$ $$\\left(\\frac{\\partial x}{\\partial s }\\right)^2+\\left(\\frac{\\partial y}{\\partial s }\\right)^2=1$$ After that I've found stationary solution ($\\frac{\\partial x}{\\partial t}=\\frac{\\partial y}{\\partial t}=\\frac{\\partial \\lambda}{\\partial t}=0$). (I just put $\\ddot{x}=\\ddot{y}=0$) $$y_0(x)=-\\frac{C_1}{\\rho g}\\cosh\\left(\\frac{\\rho g x}{C_1}+C_2\\right)$$ Where $C_1,C_2$ is integration constants (depends on positions of ends of string). And $\\cosh(x)$ is hyperbolic cosine. To study small oscillations I've tried to use pertrubation theory. So, I put $$y(s,t)=y_0(s)+\\bar{y}(s,t)$$ $$x(s,t)=x_0(s)+\\bar{x}(s,t)$$ $$\\lambda(s,t)=\\lambda_0(s)+\\bar{\\lambda}(s,t)$$ But after that I get difficult differential equations, which I can't solve. **Maybe someone know the more simplier aproach to solve this problem or know how to solve it in this way?**"}
{"id":"71236","title":"Saturation of the Cauchy-Schwarz Inequality","text":"Going to as little details as possible, here is a statement from Wald's text on QFT in curved spacetimes(I am not quoting the book) He considers two vector spaces ${\\cal S}$ and ${\\cal H}$. > Note - For more details about ${\\cal S}$ and ${\\cal H}$, read this box. I > believe for most part of the question, the following details are irrelevant, > but I will provide them nonetheless. Otherwise, skip below. > > He starts by considering the solution space of a classical system ${\\cal S}$ > with symplectic structure $\\Omega$. This has a natural vector space > structure. He complexifies it to ${\\cal S}^{\\mathbb C}$ and extends $\\Omega$ > to ${\\cal S}^{\\mathbb C}$ by complex linearity on each variable. He then > defines the map $(\\cdot, \\cdot): {\\cal S}^{\\mathbb C} \\times {\\cal > S}^{\\mathbb C} \\to {\\mathbb C}$ on ${\\cal S}^{\\mathbb C}$ as $$ (y_1, y_2) = > - i \\Omega( \\overline{y_1}, y_2) $$ This satisfies all the properties of an > inner product except positive-definiteness. He then considers the subspace > ${\\cal H}$ of ${\\cal S}^{\\mathbb C}$ on which the inner product above is > positive-definite. (There are of course many such choices of ${\\cal H}$. Any > one of them will do.) He then shows that there is a one-one onto map $K: {\\cal S} \\to {\\cal H}$. He shows that one can define a real inner product $\\mu: {\\cal S} \\times {\\cal S} \\to {\\mathbb R}$ on ${\\cal S}$. He then goes on to show that one can use this to define a complex inner product on $\\cal H$ as $$ \\left( K y_1, K y_2 \\right)_{\\cal H} = \\mu(y_1, y_2) - \\frac{i}{2} \\Omega(y_1, y_2)~\\forall~y_1, y_2 \\in {\\cal S} $$ where $\\Omega: {\\cal S} \\times {\\cal S} \\to {\\mathbb R}$ is an antisymmetric function on ${\\cal S}$, i.e. $\\Omega(y_1, y_2) = - \\Omega(y_2, y_1)$. He then uses the Cauchy-Schwarz Inequality for ${\\cal H}$. This reads $$ \\left( K y_1, K y_1 \\right)_{\\cal H} \\left( K y_2, K y_2 \\right)_{\\cal H} \\geq \\left| \\left( K y_1, K y_2 \\right)_{\\cal H} \\right|^2 \\geq \\left| \\text{Im} \\left( K y_1, K y_2 \\right)_{\\cal H} \\right|^2 $$ Expanding it out, he writes $$ \\mu(y_1, y_1) \\mu(y_2, y_2) \\geq \\mu(y_1, y_2)^2 + \\frac{1}{4} \\Omega(y_1, y_2)^2 \\geq \\frac{1}{4} \\Omega(y_1, y_2)^2 $$ More specifically $$ \\boxed{ \\mu(y_1, y_1) \\mu(y_2, y_2)\\geq \\frac{1}{4} \\Omega(y_1, y_2)^2 } $$ Now, here is the statement that confuses me > **Indeed, since $K$ is one-to-one and onto and since the Schwarz inequality > on ${\\cal H}$ always can be ``saturated\", we obtain the following stronger > version of the last inequality: For each $y_1 \\in {\\cal S}$ we have** $$ > \\mu(y_1, y_1) = \\frac{1}{4} \\max_{y_2 \\neq 0} \\frac{ \\Omega(y_1, > y_2)^2}{\\mu(y_2, y_2)} $$ Here's my question **Q. Where did he get the above > expression from?** He seems to be claiming that the boxed inequality is always saturated for some vector $y_2 \\in {\\cal S}$. Is that true? Why? PS - I will understand if some people think that this question is more of a math question than a physics one. But, I thought that it might be possible that the answer relies on some of the assumptions we make in physics, so I asked it here. Any comments will be helpful"}
{"id":"51332","title":"Relationship between current through a motor and it's load","text":"When a motor, connected to a battery that has a constant voltage, spins without a load it's speed is higher than with load. I'm told that because of back emfs the current is very small when there's no load because of the higher speed. And so when there is a load the back emf is less as the motor spins slower, and so the current is higher. What is a back emf and what is the relationship between that and the speed of the motor (well the coils inside the motor)? Is the output energy of the motor constant, whether it has a load or not? (ignoring friction and electrical resistance) because I don't understand how the current can be higher when the motor is (or seems to be) doing more work due to the load?"}
{"id":"7179","title":"Gravitation as the source of redshift of light beams","text":"According to Hubble's law, light and other kinds of electromagnetic radiation emitted from distant objects are redshifted. The more distant the source, the more intense is the redshift. Now, the expansion of the universe is expected to explain the redshift and its nearly linear dependence on distance between source and observer. But isn't there an other source influencing the redshift? We know that a light beam passing the Sun is deflected by the Sun's gravity in accordance with predictions made by Einstein's general theory of relativity. This deflection is dependant on a gravitational interaction between the Sun and the light beam. Thus, the position of the Sun is affected by the light beam, though by such a tiny amount that it is impossible to detect the disturbance of the Sun's position. Now, during its journey to the Earth a light beam, originating from a distant source in the Universe, is passing a certain amount of elementary particles and atoms. If the light beam interacts gravitationally with those elementary particles and atoms, affecting the microscopic mechanical properties of the individual elementary particles and atoms at issue, can this interaction be detected as a redshift of the light beam? If so, could we use this gravity redshift to measure the mean density of matter and energy in space? The beginning of the sentence \"If the light beam interacts gravitationally with those elementary particles and atoms...\" should be interpreted to say \"If the light beam _interacts gravitationally_ with those elementary particles and atoms by way of leaving them in a state of acceleration different from their initial state of acceleration....\" This clarification seems to necessitate the additional question: \"why would a gravitationally interacting object (a cluster of photons) passing another gravitationally interacting object (a mass) leave that mass in the same state as before the passage?\""}
{"id":"86593","title":"How to measure speed of a ceiling fan?","text":"I thought for fun I might measure how my ceiling fan slows down when turned off, and maybe find out what forces act on it when it slows down. My problem is that for that I need to measure its rotational speed when it's turned on. When it's in the lowest speeds it's easy because you can see it rotate, but for higher speeds that's not an option. I thought one way would be to use a strobe light and try to match up the frequency, but I don't have one of those. Can you think of any DIY way to measure the speed of a very fast ceiling fan?"}
{"id":"4068","title":"Formalizing Quantum Field Theory","text":"I'm wondering about current efforts to provide mathematical foundations and more solid definition for quantum field theories. I am aware of such efforts in the context of the simpler topological or conformal field theories, and of older approaches such as algebraic QFT, and the classic works of Wightman, Streater, etc. etc . I am more interested in more current approaches, in particular such approaches that incorporate the modern understanding of the subject, based on the renormalization group. I know such approaches exists and have had occasions to hear interesting things about them, I'd be interested in a brief overview of what's out there, and perhaps some references. Edit: Thanks for all the references and the answers, lots of food for thought! As followup: it seems to me that much of that is concerned with formalizing perturbative QFT, which inherits its structure from the free theory, and looking at various interesting patterns and structures which appear in perturbation theory. All of which is interesting, but in addition I am wondering about attempts to define QFT non-perturbatively, by formalizing the way physicists think about QFT (in which the RNG is the basic object, rather than a technical tool). I appreciate this is a vague question, thanks everyone for the help."}
{"id":"6530","title":"Rigor in quantum field theory","text":"Quantum field theory is a broad subject and has the reputation of using methods which are mathematically desiring. For example working with and subtracting infinities or the use of path integrals, which in general have no mathematical meaning (at least not yet) ect. My question is a little vague, but i am interested in hearing what is the status of rigor in QFT. What is known to be mathematically rigorous and consistent, what is known to be not rigorous? Any examples and references are welcome. **Added:** Just to clarify by rigorous I meant anything that a mathematician would find satisfactory. Also my question wasn't for books with rigorous (in some sense) approach, although that was welcomed. It was about specific examples of what is considered mathematically satisfactory and what not. For example the quantization of free fields satisfying the Klein-Gordon equation can be done rigorously. There is no mathematical definition in general of the Feynman path integral and so on."}
{"id":"33598","title":"Will a charged capacitor discharge if one lead is connected to ground?","text":"If I charge a capacitor and connect one lead to ground keeping the other lead floating, will the capacitor discharge ?                  G-------||------ open\/floating              +q -q      (G for ground)"}
{"id":"30020","title":"Does a complete theory of quantum gravity require anthropic post-selection?","text":"Does a complete theory of quantum gravity require anthropic post-selection? Certainly the black hole complimentarity and causal patch conjectures highlights the essential role of observers, at least in the asymptotic future of their future timelike trajectories. Does the measure problem in quantum gravity cosmology suggest that a \"global god's eye view\" of the universe might be an incoherent fiction? However, if observers are essential, don't we have to post-select to those states containing the observer of interest? It might be suggested the S-matrix of string theory in a superselection sector with only a finite energy difference from the vacuum over a BPS background provides a counterexample, but can an asymptotic future state of noninteracting Fock space particles really support the future trajectory of an observer?"}
{"id":"30021","title":"Are physics and philosophy compatible?","text":"Lately, there has been a war between physicists and philosophers. Some of the exchanges can be found at http:\/\/www.nytimes.com\/2012\/06\/10\/opinion\/sunday\/what-physics-learns-from- philosophy.html?bl, http:\/\/opinionator.blogs.nytimes.com\/2012\/05\/10\/can- physics-and-philosophy-get-along\/?ref=sunday and www.3ammagazine.com\/3am\/time- lord\/. are physics and philosophy compatible? Does physics need philosophy?"}
{"id":"44945","title":"Idea of precursors of the electro-magnetic waves","text":"The idea of the material Maxwell equation is almost clear. But I'm curious about the idea that except for material equation the pure Maxwell equation should work, but in harder sense: more currents and charges. There are plenty of vacuum in solid body, so, I think, that there should be some precursors of electro-magnetic waves that propagates through the body: $v = c\/n$ the speed of wave in body, but part of wave propagates in vacuum with the speed of $c$. And for **any** frequency some part of amplitude would be observed much earlier. Is there some theory behind this and does this phenomena is observed?"}
{"id":"13170","title":"Particle physics plots","text":"I'm having a hard time understanding what some of the plots that are presented by ATLAS\/CMS actually show. See for example: http:\/\/resonaances.blogspot.com\/2011\/07\/higgs-wont-come-out-of-closet.html What is the y-axis? How would the plot look like if a Higgs was in the given mass range? ![enter image description here](http:\/\/i.stack.imgur.com\/IbqWV.png)"}
{"id":"127185","title":"Torque due to Gravity","text":"I have learned that a torque on an object due to the gravity can be calculated as if the gravity acts on the center of mass of the object when the object is near the surface of the earth. Now I want to prove it... So, I have $$\\tau=\\int d\\tau=\\int r(dmg)sin\\theta=g\\int rsin\\theta dm.$$ Then, how do you obtain $$\\int rsin\\theta dm= mr_{cm}sin(\\theta_{cm}),$$ where $r_{cm}$ is the distance between an axis of rotation and the center of mass, and $\\theta_{cm}$ is the angle between the two vectors $\\vec{r_{cm}}$ and $m\\vec{g}$ at the center of mass??? Would $$\\int r dm=mr_{cm}$$ be helpful to prove this? Would the integral be dependent on the object I use, or would there be a general way to prove this?"}
{"id":"13937","title":"Question about Rayleigh scattering","text":"To quote from Wikipedia on elastic scattering, \"In this scattering process, the energy (and therefore the wavelength) of the incident photon is conserved and only its direction is changed.\" How does this work? The impacted particle will also start moving to conserve momentum. So it'll also carry some energy. How can the photon not lose any energy during the collision? Another possibly unrelated question: how exactly can the photon and the particle interact? The photon essentially carries information about a changing electromagnetic field. How can it influence a particle that carries neither charge nor magnetism?"}
{"id":"13931","title":"What does the Rayleigh Phase Function tell us","text":"I am working on some radiative transfer equations, and struggling as I'm fairly new to this field. I have read about the Rayleigh Phase Function which is: $P(\\theta) = \\frac{3}{4}(1 + cos^2 \\theta)$ I can plot this function, and generate values from this function in my computer program (which is what I need to do) - but what is it actually telling me? If I know I have a ray of incident light at a certain angle, will this function tell me what angle it will scatter at? Or is it more of a statistical generalisation where the areas where the function is higher are the angles at which the scattering is more likely to happen? Furthermore, when I see the function plotted it is always from 0-180, rather than 0-360 as I would expect for the scattering. This makes me think that the function may be giving the difference between the incoming angle and the outgoing angle rather than the absolute angle it is scattered at. Is that correct?"}
{"id":"76272","title":"Continuous vs. Discrete Spectra in various materials","text":"I read that the reason solids emit continuous spectra is that they don't have time to let their electrons decay-they are too close together. Given that electrons decay on the order of 100 nanoseconds I find this difficult to believe. Also, do electromagnetic waves move the electrons, or the atom, or both? If it is simply exciting the electrons, I don't know why is should also give way to the vibration of the atoms. If it does give way to vibration, then shouldn't gases _also_ give way to continuous spectra?"}
{"id":"43341","title":"Non-relativistic Kepler orbits","text":"Consider the Newtonian gravitational potential at a distance of Sun: $$\\varphi \\left ( r \\right )~=~-\\frac{GM}{r}.$$ I write the classical Lagrangian in spherical coordinates for a planet with mass $m$: $$L ~=~ \\frac{1}{2}m (\\dot{r}^{2} + r^{2}\\dot{\\theta ^{2}} + r^{2}\\dot{\\phi ^{2}}\\sin^{2}\\theta ) + \\frac{GM}{r},$$ and find that the canonical momentum $p_{\\phi }$ is a constant of motion, because: $$\\dot{p_{\\phi }}~=~ \\frac{\\partial L}{\\partial \\phi} ~=~ 0.$$   1. What is the physical interpretation of the canonical momentum?   2. How can we from the Lagrangian see that it is a constant of motion?"}
{"id":"40836","title":"Killing vector fields","text":"I am facing some problems in understanding what is the importance of a Killing vector field? I will be grateful if anybody provides an answer, or, refer me to some review or books."}
{"id":"96466","title":"Euler-Lagrange equations and friction forces","text":"We can derive Lagrange equations supposing that the virtual work of a system is zero. $$\\delta W=\\sum_i (\\mathbf{F}_i-\\dot {\\mathbf{p}_i})\\delta \\mathbf{r}_i=\\sum_i (\\mathbf{F}^{(a)}_i+\\mathbf{f}_i-\\dot {\\mathbf{p}_i})\\delta \\mathbf{r}_i=0$$ Where $\\mathbf{f}_i$ are the constrainded forces and are supposed to do no work, which it's true in most cases. Quoting Goldstein: > [The principle of virtual work] is no longer true if sliding friction forces > are present [in the tally of constraint forces], ... So I understand that we should **exclude friction forces** of our treatmeant. After some manipulations we arrive to: $$\\frac{d}{dt}\\frac {\\partial T}{\\partial \\dot q_i}-\\frac{\\partial T}{\\partial q_i}=Q_i$$ Further in the book, the Rayleigh dissipation function is introduced to **include friction forces**. So given that $Q_i=-\\frac {\\partial \\mathcal{F}}{\\partial \\dot q_i}$ and $L=T-U$, we get: $$\\frac{d}{dt}\\frac {\\partial L}{\\partial \\dot q_i}-\\frac{\\partial L}{\\partial q_i}+\\frac {\\partial \\mathcal{F}}{\\partial \\dot q_i}=0$$ **Question:** Isn't this an inconsistency of our proof, how do we know the equation holds? Or is it just an educated guess which turns out to be true?"}
{"id":"96467","title":"What advantages does action-at-a-distance description have over the field view of forces?","text":"It is written in Jackson (page 3) : > In fact, though there are **recurring attempts** to eliminate explicit > reference to the fields in favor of action-at-a-distance descriptions of the > interaction of charged particles, the concept of the electromagnetic field > is one of the most fruitful ideas of physics, both classically and quantum > mechanically. What does he mean? **What advantages can a action-at-a-distance description have over the field view of forces, that makes (at least some) people try toward that description?**"}
{"id":"127563","title":"Gas pressure and centrifugal force","text":"I think about a rotating torus (simplified tire) filled with ideal gas. Mass of gas is $m$ and molar mass is $M$. Pressure in non rotating torus is $p_0$. Temperature is constant $T$. Inner radius of torus is $r$ and outer radius is $R$. Then the cylinder begins to rotate with angular velocity $\\omega$ How can I derive difference of pressures near the wall between inner wall and outer wall? Do I need more data (for a simple model)? If someone knows how to make it numerically, It would be also interesting for me. I would also appreciate if someone could tell which book should I read to know more about solving such problems."}
{"id":"65018","title":"Physics of every-day life: rotating bag of tea","text":"Whilst studying for my physics courses, I like to drink tea. Today, I noticed that if you pull a bag of tea out of a hot cup of water, it gradually starts to rotate, picking up speed as time progresses and reaching a asymptotic speed after a while. My first question is: why does it start rotating? Of course it has something to do with the liquid water entering the gas phase just above the cup of tea, but how does that give the bag of tea a directed motion to one side? I presume that the momentum of the gaseous water is quite randomly directed vertically, so I don't see a reason for a horizontal rotation. Second question, how is the direction determined? I haven't tried, but might the direction of rotation be equally probable to be left-handed and right-handed? Third question: might the shape of a tea bag have an influence on what we observe? If we take a pyramid tea bag, for example, does the behavior change? I will perform some of these experiments while making my cup of teas and report back to you :)"}
{"id":"88799","title":"Numerical simulation of sound propagation in air","text":"How does one go about accurately modeling sound propagation in a room (with reflections, absorption, and diffusion characteristics) from the motion of a loud speaker? More specifically what are the governing equations that are needed? Obviously Naiver-Stokes but this is too general? Is there an easier way?"}
{"id":"120034","title":"30kHz solenoid and power amplifier design","text":"I have some difficulties designing my experimental apparatus: The goal is creating a Solenoid in air (or iron powdered core) able to produce at least 50G of peak B field on his axis at a distance of 1\". I would like to drive this solenoid in such a way that the B field varies sinusoidally from 0G to 50G (hopefully even higher) with a frequency ranging from DC to 30kHz. Probably the easiest way is to use a function generator as an input for a power amplifier (I have trouble choosing it, I don't know if I need a four quadrant amplifier or a traditional one) Since I need a fast varying B field I suppose that I have to try to keep inductance as low as possible -> so maybe I'll put not so many turns but more current..inductance will probably also create problem in the correct impedance matching with the power amplifier (I don't have a strong background on this topic.)"}
{"id":"121921","title":"What invariant counting process derives the Minkowski metric?","text":"Starting with Euclidean space, suppose I make a number of copies of a coordinate system all coincident at the origin, together with copies of a standard unit length. For any space interval common to all the coordinate systems, all the observers will count the same minimum number of times their unit length can be laid end to end to one another between this interval. And from this invariant counting process we derive the Euclidean metric in terms of its orthogonal components. What invariant counting process do we carry out to derive the Minkowski metric in terms of its orthogonal components?"}
{"id":"121920","title":"Integrating the generator of the infinitesimal special conformal transformation","text":"(c.f Di Francesco, Conformal Field Theory chapters 2 and 4). The expression for the full generator, $G_a$, of a transformation is $$iG_a \\Phi = \\frac{\\delta x^{\\mu}}{\\delta \\omega_{a}} \\partial_{\\mu} \\Phi - \\frac{\\delta F}{\\delta \\omega_a}$$ For an infinitesimal special conformal transformation (SCT), the coordinates transform like $$x'^{\\mu} = x^{\\mu} + 2(x \\cdot b)x^{\\mu} - b^{\\mu}x^2$$ If we now suppose the field transforms trivially under a SCT across the entire space, then $\\delta F\/\\delta \\omega_a = 0$. Geometrically, a SCT comprises of a inversion, translation and then a further inversion. An inversion of a point in space just looks like a translation of the point. So the constant vector $b^{\\mu}$ parametrises the SCT. Then $$\\frac{\\delta x^{\\mu}}{\\delta b^{\\nu}} = \\frac{\\delta x^{\\mu}}{\\delta (x^{\\rho}b_{\\rho})} \\frac{\\delta (x^{\\gamma}b_{\\gamma})}{\\delta b^{\\nu}} = 2 x^{\\mu}x_{\\nu} - x^2 \\delta_{\\nu}^{\\mu}.$$ Now moving on to my question: Di Francesco makes a point of not showing how the finite transformation of the SCT comes from but just states it. $$x'^{\\mu} = \\frac{x^{\\mu} - b^{\\mu}x^2}{1-2x\\cdot b + b^2 x^2}$$ I was wondering if somebody could point me to a link or explain the derivation. Is the reason for its non appearance due to complication or by being tedious? I am also wondering how, from either of the infinitesimal or finite forms, we may express the SCT as $$\\frac{x'^{\\mu}}{x'^2} = \\frac{x^{\\mu}}{x^2} - b^{\\mu},$$ which is to say the SCT is an inversion $(1\/x^2)$ a translation $-b^{\\mu}$ and then a further inversion $(1\/x'^2)$ which then gives $x'^{\\mu}$, i.e the transformed coordinate."}
{"id":"82621","title":"Strömgren Sphere of Sun","text":"I have a homework problem: > The Sun emits $ \\sim5 x 10^{23}$ photons per second with $hν > 13.6$ $eV$. > If the density of hydrogen atoms in interplanetary space is $n =$ $109 > m^{-3}$, what is the size of the Stromgren sphere? Assume a recombination > coefficient $α = 2.6 x 10^{-19} m^3s^{-1}$. From Wikipedia, I was able to get to $$ R_S = \\left( \\frac{3S_*}{4\\pi n^2 \\beta_2} \\right)^\\frac{1}{3} $$ And I know $\\beta_2 = \\frac{\\alpha}{T}$, but I have no idea what the $S_*$ is. Wikipedia describes it as a source of flux, which is obviously the sun, but I cannot figure out anything else about it. I'm actually quite sure that the professor would have given us a different equation, but he never went over it, so I don't know. Anyone know how to solve this?"}
{"id":"19685","title":"Clouds in closed hydrosphere","text":"Is possible estimate the needed size of an geodesic dome (like in the Eden project) for creating an real hydrosphere - especially clouds (and rain)? With other words, under what circumstances can happen clouds formation in an closed space -isolated from the outer atmosphere?"}
{"id":"19684","title":"Modification of de Donder gauge","text":"The de Donder gauge is often used to simplify the linearised equations of motion of general relativity. If the metric is linearised as $g_{ab} = \\bar g_{ab} + \\gamma_{ab}$, then the de Donder gauge reads   $\\nabla^a(\\gamma_{ab} - \\frac{1}{2}\\bar g_{ab}\\gamma) = 0$. The partial differential equation for the gauge transformation vector $v^a$ is $ \\nabla^b\\nabla_b v_a + R_a^b v_b = \\nabla^a(\\gamma_{ab} - \\frac{1}{2}\\bar g_{ab}\\gamma)$. In chapter 7.5 of Wald, I read that this equation can always be solved because it is of the form $g^{ab}\\nabla_a\\nabla_b \\phi_i + \\sum_j (A_{ij})^a\\nabla_a \\phi_j + \\sum_j B_{ij}\\phi_j + C_i$.   Theorem 10.1.2 of Wald says that in a globally hyperbolic spacetime this equation has a well posed initial value formulation on any spacelike Cauchy surface. In stead of de Donder gauge, I want to use a similar gauge:   $\\nabla^a(\\gamma_{ab} - n \\bar g_{ab}\\gamma) = 0$.   The partial differential equation changes to   $ \\nabla^b\\nabla_b v_a + (1 -2n)\\nabla_a\\nabla_b v^b + R_a^b v_b = \\nabla^a(\\gamma_{ab} - n\\bar g_{ab}\\gamma)$. This equation is not covered by theorem 10.1.2 of Wald. My question is: is the existence of a solution for this equation guaranteed in an AdS background when $n=1$?"}
{"id":"80668","title":"What is the relation between N=2 super Yang-Mills and its twist","text":"My question is what is the relation between N=2 super Yang-Mills and its twisted version topological field theory? After twisting N=2 super Yang-Mills, i.e. diagonally embedding $SU(2)'_R$ into $SU(2)_R \\times SU(2)_I$, we get a topological field theory. My question is since N=2 SYM and TQFT are different i.e. one is physical and the other is topological. Why can we use TQFT to calculate partition of N=2 SYM? What are the same for these two different theories? **Update** : From the second paper of Trimok, the authors claim that SYM under twist are just redefination. How to understand it?"}
{"id":"88132","title":"What is the sense of introducing generating functional to the summands of expansion of S-matrix?","text":"Let's have generating functional $Z(J)$: $$ Z(J) = \\langle 0|\\hat {T}e^{i \\int d^{4}x (L_{Int}(\\varphi (x)) + J(x) \\varphi (x))}|0 \\rangle , \\qquad (1) $$ where $J(x)$ is the functional argument (source), $\\hat {T}$ is the chronological operator, $\\varphi (x)$ - some field. I want to understand the reasons for its introduction for the summands of expansion of S-matrix. As I read in the books, it helps to consider only the vacuum expectation values​​, forgetting about in- and out-states. But in $(1)$ appear summands like $\\int \\frac{J(p)dp}{p^2 - m^2 + i0}$ instead of the contributions from external lines. It may refer to the internal lines. So what to do with them and are there some other reasons to introducing $(1)$ except written by me?"}
{"id":"46393","title":"How can two time theories be compactified to 3+1 without any Kaluza-Klein remnants","text":"I have recently been looking into the two-time theories and the implied concepts. For me this seems slightly hard to grasp. How can I see the basic concept in this theory in a fundamental way based on its implied interaction with normal 3+1 dimension? I am interested specifically in how gauge symmetries that effectively reduce 2T-physics in 4+2 dimensions to 1T-physics in 3+1 dimensions without any Kaluza-Klein remnants."}
{"id":"33639","title":"On Bell's Inequality (Classical Intuition) and Quantum Mechanical Counter Intuition","text":"This posting is directly related to the issue in The System and the Measuring Gadget. The QM expectation is given by: $$\\langle\\sigma_{1}.\\vec{a}{\\;}\\sigma_{2}.\\vec b\\rangle=-\\vec a.\\vec b$$ In the above relation we are considering _the measured value_ of spin which is the outcome between the value of some property of the system itself and the measuring gadget The \"classical\" formula for evaluating the expectation with the hidden variable is as follows: $$P(\\vec a.\\vec b)=\\int d\\lambda\\,\\rho (\\lambda) A(\\vec a)B(\\vec b)$$ Now some property of the system may depend on the value of $\\lambda$ and the probability distribution $\\rho (\\lambda)$. Is the effect of measurement being fully accounted for by the the hidden variable $\\lambda$ and the pdf $\\rho (\\lambda)$, especially in view of the fact that the process of measurement modifies the wave function itself. Would it be possible remove the contradiction between QM and commonsense intuition,expressed through Bell's Inequality, by considering the above factors? Reference for Bell's Original Paper."}
{"id":"17300","title":"Projectile motion equations @ very (!) high starting velocity","text":"I've already searched Physics StackExchange for some similar question but I didn't find anything about this. Assumptions:   * Earth is a perfect sphere with it's core (X,Y,Z) -> (0,0,0) as a reference-frame center   * Air resistance can be ignored   * Earth rotation can be ignored   * Moon gravity-effect can be ignored And if I know (projectile starting properties):   * Current Earth GPS-coordinates   * Starting angle   * Starting direction (relative to (0,0,0))   * Starting velocity (can be larger than Earth-escaping speed)   * (mass of the projectile is irrelevant (I guess) when we know starting velocity of the projectile) How could I calculate aprox. coordinates of a projectile landing somewhere around the Earth globe (OR detect that projectile will \"leave\" the Earth)? EDIT: I've found tons of links on the net about projectile trajectories when starting velocities are quite small (& Earth can be considered like a flat plane), but non about upper situation."}
{"id":"123793","title":"Why there is added a partial time derivative in formula for time derivative of potential energy?","text":"In proving the total energy in conservative field is constant we have this equation(picture) why it added partial derivative? Why? I mean where it did come from? ![enter image description here](http:\/\/i.stack.imgur.com\/xkrx5.png)"}
{"id":"69055","title":"What is difference between $\\frac {dr}{dt}$ and $\\frac {\\partial r}{\\partial t}$?","text":"What is difference in **physical meaning** of partial time derivative and ordinary derivative of $r$? $$\\frac {\\partial r}{\\partial t}\\quad\\text{and}\\quad \\frac {dr}{dt}.$$ I know that ordinary time derivative is velocity."}
{"id":"63220","title":"What is the common difference between partial time derivative and ordinary time derivative?","text":"What is difference between partial and ordinary time derivative? for example: what is difference between $\\frac {\\partial v}{\\partial t}$ and $\\frac {dv}{dt}$? where the $v$ is velocity."}
{"id":"53045","title":"Entropy exchange of a free fall","text":"I have a problem in which the tell me that you drop a bag of 50 kg of sand from 10 meters high, and you have to caltulate the entropy difference of the sand, asuming that the speific heat of the sand is so high that its temperature (298K) doesn't change. My result is 16.5 J\/K, and my problem is with the sign. The book doensn't give an explanation but just an answer and it's positive, but I'm thinking that in the impact the heat must flow out of the sand, as it's loosing its kinetic energy, so the entropy would decrease, and the surroundings entropy would increase; what am I missing, or what is really going on here?"}
{"id":"10656","title":"Misunderstanding of Special Relativity","text":"Person A in reference frame A watches person B travel from Star 1 to Star 2 (a distance of d). Of course, from person B's reference frame, he is at rest and is watching Star 2 traveling to him. Now we know from the principle of relativity, each one will measure the other one’s clock as running slower than his own. Let’s say that Person A measures Person B’s speed to be v, and that Person A measures 10 years for person B to make it to Star 2. Let’s also say that person B is moving at the speed so that the Gamma Factor is 2. This means person A observes person’s B’s clock to have elapsed a time of 5 years. Now let’s look at this from Person B’s perspective: Person B observes Star 2 approaching (and Star 1 receding from) him also at speed v. Since the two stars are moving, the distance between them is length contracted (after all, if there were a ruler in between the stars, the moving ruler would be contracted) by a factor of 2. Since person B measures the initial distance to Star 2 to be d\/2 and its speed v, he calculates the time to Star 2’s arrival to be 5 years. Since he observes person A’s clock as running slow (since Person A is moving also at speed v), when Star 2 arrives, he measures Person A’s clock to have elapsed a time of 2.5 years. Do you see why I’m confused? Person A measures Person B’s elapsed time to be the same as Person B measures Person B’s elapsed time (both 5 years), but Person B does not measure Person A’s elapsed time to be the same as Person A measures Person A’s elapsed time (Person B get’s a measurement of 2.5 years while Person A measured 10 years). This is asymmetrical, which probably means it is wrong. But I’m not sure what the error is. I suspect if I had done this correctly, each person should measure his own elapsed time to be 10 years and measure the other’s elapsed time to be 5 years. This would be symmetrical and would make the most sense, but again, I can’t seem to justify how person B wouldn’t measure his trip time to be 5 years. What's my mistake?"}
{"id":"7911","title":"Why can different batteries with the same  voltage send different currents through the same object?","text":"According to an answer in this thread on Skeptics: > If you take one of the little 12V garage door opener batteries and short out > (directly connect) the two terminals with a piece of wire or something else. > You'll get a light current flow through the wire or metal. It may get a > little warm. This battery is only capable of supplying a small amount of > current. > > If you take a 12V car battery and short out the two terminals (don't do it, > it's not fun), you will be met with a huge current arc that will likely > leave a burn mark on whatever was used to short it. This is because the car > battery is capable of discharging a large amount of current in a very short > period of time. I'm not sure how this could work given Ohm's law V=IR. If we assume the resistance of toucher is constant, then we'd expect the current to be the same as well.   1. Could it be that a car battery has less internal resistance than a garage battery?   2. Does it have anything to do with contact area? If it does, then how would you model it? Generally resistances add in series, but if I only half touch a contact, then neither the battery nor I change, so you'd expect our resistances to stay the same, but somehow our total resistance changes. So which objects resistance would change - mine or the batteries?"}
{"id":"7910","title":"Are black holes really that special?","text":"~~Science and science fiction alike~~ Science fiction describes black holes as these amazingly different entities in space that don't behave according to the same laws of physics that the rest of the universe is bound to. I've heard them described as wormholes to other universes, singularities, or tears in space-time. However, are they really that special? My thought is that all matter has event horizons. However, most of it isn't dense enough for that event horizon to be large enough to affect the way the matter interacts with the rest of the universe. So, a black hole is simply just a dense star that is simply dark. And, if black holes _must_ have a singularity at their center because of general relativity, can't we conclude from that that all matter creates singularities, since the gravitational field intensifies into infinity the closer you get to the point-mass? What is the difference being really close to a point-mass and being very close to a black hole?"}
{"id":"63753","title":"recommendation for a physics history\/non-fiction book","text":"I know that there are a lot theses being published on lives of physicists. Is there a history\/non-fiction book that tracks the development of a problem chronologically? Like pieces of a puzzle. I would like it to be mathematical and trying to get into the heads of people trying to solve that problem. Something like a case study."}
{"id":"7915","title":"Is it possible for wind to break the sound barrier?","text":"I understand that in nature wind would never get high enough, but I am just curious as to whether physics would allow this to occur or not."}
{"id":"111093","title":"Definition of Information in Information Theory","text":"I am not sure in which SE site I have to put this question. But since I have learnt Shannon Entropy in the context of Statistical Physics, I am putting this question here. In the case of Shannon Information theory, he defines information $I$ for an $i^{th}$ event as, $$ I_i = -\\ln P_i \\qquad \\qquad \\forall i=1,...n. $$ Based on this definition we further define Shannon Entropy as average information, $$ S_\\text{Shannon} =\\langle I\\rangle = -\\sum\\limits_{i=1}^n P_i\\log_2P_i .$$ My question is what is the motivation behind defining entropy as some function that is inversely related to probability? I was told by my professor that lesser the probability of an event more information it possesses, although am still not convinced about this fact. Secondly, what is the reason in choosing the logarithmic function in this definition? Are there places where this definition of information is forfeited?"}
{"id":"20836","title":"Are all classically impossible quantum possibilities entangled?","text":"Any entangled state represents a quantum possibility that is classically impossible.   Is the converse true?   That is, are all states that are quantum mechanically possible but classically impossible entangled in some way?   If so, can you give a proof, or a reference to a proof?   If not, can you give a counterexample?"}
{"id":"53593","title":"Resultant vector problem","text":"$\\newcommand{\\v}[1]{\\vec #1}\\newcommand{\\i}{\\hat i}\\newcommand{\\j}{\\hat j}$ Problem statement (1,2) > A shopper at the supermarket follows the path indicated by vectors $\\v A, \\v > B, \\v C, \\v D$ in the figure. Given that the vectors have magnitudes > $A=51\\:\\mathrm{ft}, B=45\\:\\mathrm{ft}, C=35\\:\\mathrm{ft}, > D=13\\:\\mathrm{ft}$, find the total displacement of the shopper using (a) the > graphical method and (b) the component method of vector addition. Give the > direction of displacement relative to $\\v A$ > > ![enter image description here](http:\/\/i.stack.imgur.com\/iJbOUm.jpg) My work: ![enter image description here](http:\/\/i.stack.imgur.com\/RCyp8m.jpg) $$\\begin{align}A &=&0&\\i + &51&\\j \\\\\\B &=&45&\\i + &51&\\j\\\\\\C &= &10&\\i + &-35 &\\j\\\\\\ D &= &10&\\i + &-13 &\\j \\\\\\\\\\text{Resultant} &=&54&\\i+&65&\\j\\end{align}$$ Essentially I am adding all the components to get the resultant vector but it is not leading me to the right answer. What am I doing wrong here?"}
{"id":"48251","title":"Does altitude affect sound pitch?","text":"Due to differences in air pressure, temperature, and other factors, the speed of sound varies with altitude on Earth. Does this affect the pitch of the sound in any meaningful way? For example, if I had a tuning fork that vibrates at around 262 Hz, would I hear the same \"Middle C\" from it while standing on the shores of the Dead Sea as I would while standing at the peak of Mount Everest - an altitude difference of 9,271 meters or 30,417 feet? Would there be any difference at all, and would it be perceptible to the human ear at close range? Would the altitude difference affect how the sound is heard more at a distance than it would nearby? Or, am I not quite understanding properly how sound works?"}
{"id":"112881","title":"What is the experimental uncertainty of an ensemble measurement?","text":"Let's say you measure the time it takes for 10 oscillations of a mass undergoing simple harmonic motion to within ± 0.01s, what is the uncertainty of the period of one oscillation?"}
{"id":"13103","title":"Cheat sheet of elementary particles","text":"I am trying to teach myself some particle physics. There are too many particles and its too much for me. I hated biology just because of this sort of stuff. Too many names and it was all Greek to me. Is there a good cheat sheet\/ reference sheet of elementary particles? It will also be very helpful if you share how you people manage to remember these things. This is the first time I have ever hated studying physics. :("}
{"id":"62156","title":"Rømer's determination of the speed of light","text":"I am trying to understand Rømer's determination of the speed of light ($c$). The geometry of the situation is shown in the image below. The determination involves measuring apparent fluctuations in the orbital period of Io. (Jupiter's moon) ![Geometry of the problem](http:\/\/i.stack.imgur.com\/J5dyj.png) The Earth starts from point A. $r(t)$ is the distance between the Earth and Jupiter. $r_e$ is the radius of the (assumed) circular orbit of the Earth around the Sun, while $r_0$ is the same for Jupiter. $T$ is the period of the Earth's orbit. Under the assumption that the Jupiter-Io system is stationary, $r(t)$ can be expressed as $$r(t) = \\sqrt{r_E^2 + r_0^2 -2r_0 r_E \\cos \\left(\\frac{2\\pi t}{T}\\right)}$$ If we further assume that the period of Io's orbit around Jupiter, $\\Delta t$ is much smaller that $T$, then it can be shown that the distance the Earth moves, $\\Delta r$ when Io completes one orbit is: $$\\Delta r = \\frac{2\\pi r_E \\Delta t}{T} \\sin\\left( \\frac{2\\pi t}{T} \\right)$$ The point I am stuck is about why is there an apparent fluctuation in Io's orbit as observed on the Earth? And how can we derive the observed delay using these expressions?"}
{"id":"62151","title":"Cancel out Earth's Magnetic field","text":"Is there any tools except helmholtz coil to cancel out earth's magnetic field to calibrate magnetometers in practice."}
{"id":"54064","title":"What's the exact gravitational force between spherically symmetric masses?","text":"Consider spherical symmetric$^1$ masses of radii $R_1$ and $R_2$, with spherical symmetric density distributions $\\rho_1(r_1)$ and $\\rho_2(r_2)$, and with a distance between the centers of the spheres $d$. What is the exact force between them? I know point masses are a good approximation, but I'm looking for an exact formula. This would be useful for a gravity-simulation toy software. \\-- $^1$ Assume for simplicity the idealization where tidal or centrifugal forces do not deform the spherical symmetric, i.e., the various mass parts are held in place by infinitely strong and rigid bonds."}
{"id":"20304","title":"Why did my windshield freeze instantaneously?","text":"It was very cold outside, this morning, when I took the car that slept in the snow, with a simple cloth on the windshield. I entered the vehicle, drove a kilometer or so. The air inside was so cold I could see my breath (or maybe I forgot to brush my teeth). Warm air was blowing on the windshield from the inside and suddenly… The glass in front of my became opaque, starting from the bottom (the place where the hot air was blowing) and freezing up, up, up until the whole screen was filled in about 5 seconds. It reminded my a little bit of how some baterias spread. When I tried to remove the mist, I realised it was ice **in** the car. My windshield instant froze. I know the theory behind supercooling: a very cold and still liquid can freeze when moved. I'm not sure what happened here, but there were some pretty big turns in the road before it froze. It really looked like the hot air froze the place and not the movement. Any idea about what happened ?"}
{"id":"7470","title":"Chemical potential","text":"This is something probably very basic but I was led back to this issue while listening to a recent seminar by Allan Adams on holographic superconductors. He seemed very worried to have a theory at hand where the chemical potential is negative. (why?)   * For fermions, isn't the sign of the chemical potential a matter of definition? The way we normally write our equations for the Fermi-Dirac distribution the chemical potential happens to that value of energy at which the corresponding state has a occupation probability of half. And within this definition the holes in a semiconductor have a negative chemical potential.   * It would be helpful if someone can help make a statement about the chemical potential which is independent of any convention. {Like one argues that negative temperature is a sign of instability of the system.}   * Also isn't it possible for fermions in an interacting theory to have a negative chemical potential?   * Also if there is a \"physical argument\" as to why bosons can't have a positive chemical potential? (Again, can an interacting theory of bosons make a difference to the scenario?)   * And how do these issues change when thinking in the framework of QFT? (No one draws the QCD phase diagram with the chemical potential on the negative X-axis!)   * In QFT does the chemical potential get some intrinsic meaning since relativistically there is a finite lower bound of the energy of any particle given by its rest mass?"}
{"id":"100217","title":"On the distinction of past and future: could one theoretically reverse direction of particles and cause time to appear to go backwards?","text":"Based on my understanding of physics after seeing _The Distinction of Past and Future_ on Project Tuva, there is no distinction between past and future on a fundamental level- all particle interactions can occur in reverse. So my question is whether or not one could theoretically reverse the direction of all particles in the observable universe relative to each other and have time essentially go backwards indefinitely. If you think about it, things could \"fall\" upwards because the air resistance would be much lower due to the way the air was moving when it fell, and the velocity from the gravity downwards would be reversed as well as air under the ball pushing up (again due to the way the air was moving previous to the switch). I don't see why this same logic couldn't be applied to a more complex system. Does this logic make sense? If not, where is the flaw? What other constraints would need to be added to make time essentially go backwards other than reversing direction, if it is possible at all, in theory?"}
{"id":"66678","title":"Universe with ZERO Fundamental Forces","text":"Warning: I am not a physicist so please excuse my naivety! As you all know, physicists think that there exist four fundamental forces. Would a universe with zero fundamental forces be possible, at least in principle? Would elementary particles be able to exist in such a universe? On a larger scale, what would such a universe look like?"}
{"id":"130437","title":"Does pulling or pushing take more effort?","text":"In which case is more effort needed: when you push or when you pull? We have a common experience that man tends to first push anything. How it is justified and what are the reasons behind this?"}
{"id":"23766","title":"Derivation of Electric Force between Parallel Plates","text":"So the electric field between two parallel plates is given by $E = V\/d.$ How do you derive this?"}
{"id":"13465","title":"Choice and identification of vacuums in AdS\/CFT","text":"> **Possible Duplicate:**   >  Choice and identification of vacuums in AdS\/CFT I think I know how do we define a vacuum in flat space QFT and also in a curved space QFT. But, can somebody tell me how do the choice of vacuum state in say the CFT side of AdS\/CFT changes the choice of vacuum state in gravity side? Let me ask the other way. I mean if we pick a vacuum (say in bulk side, because it may not be unique), how does it reflect on the CFT vacuum (and vice versa)? The CFT side is conformally flat. So, there is no reason in the CFT side that there will be a unique vacuum, is there? So, how does this choice reflect on both sides and how do we generally make the identification? Thanks."}
{"id":"92157","title":"Off-shell corrections to massive vector boson propagator in polarization form","text":"As an exercise for myself, I have been working on rewriting the massive vector boson propagator (unitary gauge). I have run into a problem interpreting some of the terms that stick around when the propagator is rewritten this way. Here's what I have: I've taken the unitary gauge vector propagator $$ D_{\\mu\\nu}(q) = \\frac{i}{q^2 - M_W^2 + i \\varepsilon} \\left( -g_{\\mu \\nu} + \\frac{q_\\mu q_\\nu}{M_W^2} \\right) $$ and projected it into its helicity components. The convention I am using is that $$ \\epsilon^1_\\mu(\\vec q) = (0,1,0,0) \\\\\\ \\epsilon^2_\\mu(\\vec q) = (0,0,1,0) \\\\\\ \\epsilon^0_\\mu(\\vec q) = \\frac{1}{M}(|q|,0,0,E_q) \\\\\\ \\epsilon^s_\\mu(\\vec q) = \\frac{1}{M}(E_q,0,0,|q|) $$ where $E_q = \\sqrt{M^2+|q|^2}$. These are an orthonormal set, where $$ \\epsilon^{\\lambda}_\\mu \\epsilon^{\\lambda' \\mu} = - \\eta_{\\lambda} \\delta_{\\lambda \\lambda'} $$ where $\\eta_\\lambda = 1$ for $\\lambda = \\pm,0$ and $-1$ for $\\lambda = s$, the scalar polarization. So, it's easy to show that $$ X_{\\mu \\nu} = \\sum_{\\lambda,\\lambda'} X_{\\lambda,\\lambda'} \\epsilon^{\\lambda}_\\mu \\epsilon^{\\lambda' }_{\\nu} \\Rightarrow X_{\\mu \\nu} \\epsilon^{\\lambda \\mu} \\epsilon^{\\lambda' \\nu} = \\eta_\\lambda \\eta_{\\lambda'} X_{\\lambda \\lambda'} $$ In particular, $$ \\- g_{\\mu \\nu} \\Rightarrow g_{\\lambda \\lambda'} = \\frac{\\eta_\\lambda' \\delta_{\\lambda \\lambda'}}{\\eta_{\\lambda} \\eta_{\\lambda'}} = \\frac{\\delta_{\\lambda \\lambda'}}{\\eta_{\\lambda}} = \\eta_{\\lambda}\\delta_{\\lambda \\lambda'} $$ Where I've run into difficulty is breaking up the transverse term. $$ q_\\mu \\epsilon^{1\\mu}(\\vec q) = 0 \\\\\\ q_\\mu \\epsilon^{2\\mu}(\\vec q) = 0 \\\\\\ q_\\mu \\epsilon^{0\\mu}(\\vec q) = \\frac{|q|}{M}(q_0-E_q) \\\\\\ q_\\mu \\epsilon^{s\\mu}(\\vec q) = \\frac{1}{M}(q_0 E_q - |q|^2) $$ The last of these can be rewritten $$ q_\\mu \\epsilon^{s\\mu}(\\vec q) = M + \\frac{E_q}{M}(q_0-E_q) $$ So, the helicity components of $$ T_{\\mu \\nu} = \\frac{q_\\mu q_\\nu}{M^2} $$ are $$ T_{1\\lambda} = T_{\\lambda1} = T_{2\\lambda} = T_{\\lambda2} = 0 $$ $$ T_{00} = \\frac{|q|^2}{M^2} \\left( \\frac{q_0-E_q}{M} \\right)^2 $$ $$ T_{ss} = 1 + \\frac{q_0^2-E_q^2}{M^2} + \\frac{|q|^2}{M^2} \\left( \\frac{q_0-E_q}{M} \\right)^2 $$ $$ T_{0s} = T_{s0} = - \\frac{|q|}{M} \\left( \\frac{q_0-E_q}{M} \\right) - \\frac{E_q|q|}{M^2}\\left(\\frac{q_0-E_q}{M}\\right)^2 $$ The term equal to $1$ in the scalar polarization term cancels out the corresponding scalar polarization term $g_{ss}$. Furthermore, all of the terms proportional to $(q_0-E_q)^2$ I understand. They cancel out the pole in the propagator, since $$ \\frac{1}{q^2 - M^2 + i \\epsilon} = \\frac{1}{q_0^2 - E_q^2 + i \\epsilon} \\sim \\frac{1}{q_0 - E_q + i \\epsilon} \\frac{1}{q_0 + E_q} $$ and after canceling out the pole they retain a factor $q_0 - E_q$ which forces them to be 0 while on-shell. These are explicitly off-shell corrections. However, I'm not sure how to interpret the terms $$ T_{ss} \\ni \\frac{q_0^2-E_q^2}{M^2} = \\frac{q_0+E_q}{M} \\frac{q_0-E_q}{M} $$ and $$ T_{0s} = T_{s0} \\ni - \\frac{|q|}{M} \\left( \\frac{q_0-E_q}{M} \\right) $$ Naively, it appears to me that these cancel out the pole at $q_0 = E_q$, but the remaining portion does not vanish at at $q_0 = E_q$ (or at least as at $q_0$ approaches $E_q$) I think that a careful analysis of the behavior of these terms the pole might shed light on this, or that maybe it is a gauge artifact, but I am stuck."}
{"id":"78973","title":"Trouble evaluating an integral arising from particle collision","text":"Assume we have two charged particles colliding. He have particle 1 with mass $m_1$, charge $Z_1 \\cdot e$ which travels in $x$-Direction passing by a STATIONARY particle 2 (mass $m_2$, charge $Z_2 \\cdot e$) at distance $b$ (impact parameter). I want to calculate the change in momentum in $y$-Direction (as the change in momentum in x-Direction vanishes). $\\Delta p_y = Z_1 e \\int_{-\\infty}^\\infty F_y dt = Z_1 e \\int_{-\\infty}^\\infty F_y \\frac{1}{v}dx $ (as $dx\/dt = v)$ in the script im trying to follow they now multiply by $1$ as follows: $\\Delta p_y = \\frac{Z_1 e}{2 \\pi b v} \\int_{-\\infty}^\\infty 2 \\pi b F_y dx$ They now immediately evaluate this theorem using Gauss-Theorem to: $\\frac{Z_1 e}{2 \\pi b v} \\frac{Z_2 e}{\\epsilon_0}$ Generally the Gauss-Theorem relates the integral over the surface to an integral over the space that surface encloses. I have a bit of a problem seeing over which surface\/space area im integrating here, since its just an integral along the x-axis. Thanks in advance for any help! Cheers"}
{"id":"78970","title":"plasma frequency in non-neutral plasma","text":"Is it possible to see the oscillations with plasma frequency in a gas of particles of the same charge (not mixture of positive and negative charges)?"}
{"id":"78977","title":"Calculating Phase Diagrams (Calphad)","text":"I am calculating the phase diagrams for Li-Mg binary alloy with reference to the following text: http:\/\/ieeexplore.ieee.org\/xpl\/articleDetails.jsp?arnumber=5600701 In the equation used in the above paper, i am confused about whether G0 for a component is a function of temperature? ![enter image description here](http:\/\/i.stack.imgur.com\/V78Oq.png)"}
{"id":"32556","title":"Question regarding the Bohm interpretation","text":"I tryed to understand the Bohm interpretation and this is what picture appeared to me. Please tell me if I understood something incorrectly.   * All particles have definite positions and follow deterministic rules of dynamics   * Any future configuration of an isolated subsystem is only dependent on initial conditions   * Even slight difference in initial conditions may result in huge differences in the result. The problem is that those initial conditions, are inherently unknown. This is fundamental: even if an observer manages to measure the whole Bohm state of the entire universe, he still would not know the Bohm state of himself. This is like making predictions bout future states of a three-body system based on Newtonian mechanics with initial coordinates known only with finite precision. Due to apparent chaoticity of the solution the possible results may be dramatically different. Correct me if I am wrong."}
{"id":"32554","title":"How do I extend the Lorentz transformation metric to dimensions>4?","text":"How do I extend the general Lorentz transformation matrix (not just a boost along an axis, but in directions where the dx1\/dt, dx2\/dt, dx3\/dt, components are all not zero. For eg. as on the Wikipedia page) to dimensions greater than 4? Thanks"}
{"id":"55736","title":"Gradient of the electric potential","text":"I was wondering if for a point-like charged object, does the gradient of the electric potential point in the direction of maximum increase or maximum decrease of the function $V$?"}
{"id":"32553","title":"What should I call an n>4 dimensional Minkowski metric?","text":"I am manipulating an $nxn$ metric where $n$ is often $> 4$, depending on the model. The $00$ component is always tau*constant, as in the Minkowski metric, but the signs on all components might be + or - , depending on the model. (I am not trying to describe physics with this metric). Can I call this metric a Minkowski metric? Or what should I call it?"}
{"id":"115201","title":"Langreth rules and Keldysh formalism","text":"I am trying to confirm the proof of Langreth's theorem \/ rules as seen in http:\/\/www.iue.tuwien.ac.at\/phd\/pourfath\/node52.html . My problem is equation 3.55. I would do it like this: $\\int_{C_{1}} \\mathrm{d}\\tau A(t,\\tau) B(\\tau,t') = \\int_{-\\infty}^{t} \\mathrm{d}t_1 A^{>}(t,t_1) B^{<}(t_1,t') + \\int_{t}^{-\\infty} \\mathrm{d}t_1 A^{<}(t, t_1) B^<(t_1, t')$ And now I flip the integration boundaries on the right and replace $A^r = A^>-A^<$: $ = \\int_{-\\infty}^{t} \\mathrm{d}t_1 (A^{>}(t,t_1) - A^<(t,t_1) ) B^{<}(t_1,t') = \\int_{-\\infty}^{t} \\mathrm{d}t_1 A^r(t,t_1) B^<(t_1,t)$ And thus my integration does not go from $-\\infty$ to $\\infty$ as shown in the source. What am I missing? * * * Alright, I got it. It works because of the definition of the retarded function with a theta function: $$ A_r(t,t') = \\theta(t-t') [ A^>(t,t') - A^<(t,t')] $$ So the retarded function is always zero when $t' > t$. One can therefore extend the integration to $\\infty$. I leave this here so it might help someone else later."}
{"id":"107945","title":"light entering a black hole's singularity","text":"I already understand that light cannot escape a black hole after passing the event horizon, so please do not explain that to me. What I would like to know is this: a well known fact about light (a photon specifically) is that it travels at the speed of light, and at no other speed, which means that it has no rest mass, as an example, as it does not stop. As a photon aproached a black hole, it would begin to spiral around it as it got ever closer to the singularity at the centre. The closer the photon got to the singularity, the shorter the amount of time it would take to go once around the singularity, as it remained at its constant speed. However, on reaching the perfect centre, it would stop moving completely relative to the blackhole, so would no longer be travelling at the speed of light. Can you explain why this happens or (more likely) where I have gone wrong?"}
{"id":"107949","title":"In what form does proton give energy to electron?","text":"According to Bohr, electron revolves around the nucleus because of force of attraction between electron and proton. This force of attraction gives energy to the electron. So my question is this that- In which form does this electron get the energy?"}
{"id":"100401","title":"'ting' 'ting' sound from a tube-light?","text":"Whenever I light up a tube-light it makes 'ting' 'ting' sound every-time it blinks. I am talking about this tube-light   ![enter image description here](http:\/\/i.stack.imgur.com\/j5j97.jpg) Why is it so? I think its because of sparking(inside glass tube) similar sound in the 6th box hover and listen here"}
{"id":"14212","title":"Collision of Phobos","text":"Mars has two moons: Phobos and Deimos. Both are irregular and are believed to have been captured from the nearby asteroid belt. Phobos always shows the same face to Mars because of tidal forces exerted by the planet on its satellite. These same forces causes Phobos to drift increasingly closer to Mars, a situation that will cause their collision in about 50 to 100 million years. How I can calculate, given appropriate data, the estimated time at which Phobos will collide with Mars?"}
{"id":"26304","title":"Is the moon a planet?","text":"Can our moon qualify as a planet? With regard or without regard to the exact definition of the planet, can the moon be considered as planet as Mercury, Venus and Earth etc. not as the satellite of the planet Earth."}
{"id":"54299","title":"Semi-conductor band-gap and deformation potential","text":"Submitting a semi-conductor to stress leads to a deformation in the energy- bands, roughly described by:$$H_{ij} = {\\cal{D}}_{ij}^{\\alpha\\beta}\\;\\epsilon_{\\alpha\\beta}$$ $\\epsilon$ being the strain (linked to the stress by Hooke's law), $H$ the perturbation Hamiltonian to the Hamiltonian describing a stress-free semiconductor, $i,j$ being indexing the energy level of the previous \"free\" Hamiltonian. Still, I lack intuition regarding the apparition of this term, it seems that compression enlarges the band-gap whereas dilation tightens it. Why do we have this behaviour? I have tried thinking on electrostatic arguments, the potential decreasing as $r^{-1}$ we do have an increase of energies in $r \\rightarrow \\alpha r$ for $\\alpha < 1$ or also seeing the dilatation as a renormalization group transformation, basically going to a coarser grain (although there probably other pertinent length scales in an atomic lattice (spreading of the electronic orbitals, ...) which would make this argument wobbly). Cutting to the point, what is your hand-waved way of seeing it? Books on the subject of deformation potential don't really seem to offer intuition on it, more numerical values for specific materials."}
{"id":"75351","title":"Are electrodynamics problems in the complex plane relevant to real life?","text":"This is a question I asked in Maths SE, and it was suggested I ask it here. This is a direct copy of that question. I have been reading Tristan Needham's excellent Visual Complex Analysis. The end of the book deals almost entirely with physics, using symmetries of conformal mappings to generalise the famous method of images technique in electrodynamics. The method of images is used in finding the electric field due to a charge when a grounded surface (such as a sphere or plane) is nearby. (See e.g. Wikipedia.) However, the problems seem to have very little \"real life\" applications to me, the main problem being that the complex plane is two dimensional, whereas we live in a 3 dimensional world. To see this problem concretely, the electrostatic force is goes like $F\\sim \\frac{1}{r^2}$ because the surface area of a ball of radius $r$ centred at the charge is proportional to $r^2$. However since the complex plane is 2 dimensional, a charge in the complex plane produces a field which goes like $\\frac1r$. So any solution we find to a problem of this kind in the complex plane isn't relevant in 3d. And this is my question, is there any physical application of this technique? Or is it completely irrelevant?"}
{"id":"3611","title":"Why does a ballerina speed up when she pulls in her arms?","text":"My friend thinks it's because she has less air resistance but I'm not sure."}
{"id":"59753","title":"Huge confusion with Fermions and Bosons and how they relate to total spin of atom","text":"I am supremely confused when something has spin or when it does not. For example, atomic Hydrogen has 4 fermions, three quarks to make a proton, and 1 electron. There is an even number of fermions, and each fermion has a 1\/2 spin. Since there are an even number of fermions, the total spin value is an integer. This spin number is the \"intrinsic\" spin number that cannot be changed, but its orientation \"up\" or \"down\" can be changed. For atomic Hydrogen, it is a Boson because it has integer spin, however it also has a single electron. I read on physics forums, http:\/\/www.physicsforums.com\/showthread.php?t=69992, that the spin of atom comes from the electrons and not its nucleus. I also read on here, How to find that a molecule has zero spin?, that the spin of atomic Hydrogen is 1\/2! The answer says atomic Hydrogen has spin 1\/2 because it ignores the nuclear spin. This is one thing that is confusing me. Shouldn't atomic Hydrogen have an integer spin because of the nuclear component? So does atomic Hydrogen have spin and is affected by a magnetic field? Nuclear spins are affected by magnetic fields, but they aren't as affected as electrons according to the discussion on physics forums. Why do we ignore nuclear spin sometimes? Also, can someone help me out here with all the possibilities? Is there a Boson with an half integer spin value? (Surely, there must not be) However atomic Hydrogen is one of those cases! (It seems...) (Why don't we cancel out the nuclear spin with the electron spin?) Say we have another atom that is a Boson, It has unpaired electrons in different orbitals, so what determines whether or not electrons fill in orbitals as spin up or down? Does spin down nuclear spin cancel out a electron up spin?"}
{"id":"75359","title":"How can systematic errors be calculated?","text":"Usually, it is said that systematic errors can not be handled in a well- defined way, unlike statistical errors. My question(s): A) How can systematical errors be calculated for any experimental device or experiment? Maybe does it involve a \"calibration\" of experimental devices? B) I have heard that bayesian methods provide THE tool to estimate and calculate systematical errors. How is it done?"}
{"id":"21827","title":"Addition of a neutral electrolyte to water-- how can it increase conductivity?","text":"> Sparked off by Is sea water more conductive than pure water because > \"electrical current is transported by the ions in solution\"? This question really belongs on chemistry.SE, which is still in area 51. While answering this question, I realised that there was a flaw in the standard logic for these situations. Let's take $NaNO_3$ solution in water, and compare it with pure water (same size cell). (I'm not taking $NaCl$ for a reason*). In both cases, we have nearly the same(~7) PH and pOH, right? So concentrations of $H^+$ and $OH^-$ are the same, and thus contribution of these ions to the overall conductance\/conductivity is the same. These ions both migrate to the electrodes, get reduces\/oxidised, and emit\/absorb electrons, facilitating passage of current. OK. Now, let's consider the $Na^+$ and $NO_3^-$ ions. Yes, they migrate as well. But, they don't get redoxed (the water ions are preferentially redoxed). So, all I see happening here is a buildup of charges on either electrode, which will stop once equilibrium is attained. This buildup of charges cannot migrate to the outside circuit like a capacitor, as it cannot translate to electrons. But, these ions still make a significant contribution to the net conductance by Kohlrausch law. **So how does a general good neutral electrolyte help water conduct electricity?** I feel that it should conduct electricity at the same rate as water; but by Kohlrausch law, it clearly doesn't. And anyways, I've always heard that impure water conducts electricity better. *$NaCl$ has the issue of overvoltage of $O_2$, causing oxidation to of $Cl_2$, which complicates the situation. The $Cl^-$, being able to get oxidised, facilitate current and thus don't serve as a good example here."}
{"id":"116992","title":"Decoupling of Holomorphic and Anti-holomorphic parts in 2D CFT","text":"This maybe a very naive question. I have just started studying CFT, and I am confused by why we have two separate parts of everything in CFT (operator algebras and hilbert space), the holomorphic and anti-holomorphic, which are decoupled from each other. We initially introduced $z$ and $\\bar{z}$ as two independent variables instead of say $t$ and $x$ in two dimensions. But now, we have got two isomorphic parts, the holomorphic and antiholorphic (they may given by $z \\to \\bar{z}$ and $h \\to \\bar{h}$), then what extra info. does the anti-holomorphic parts provide? And how is all the information contained in just one independent coordinate $z$? Also a physical theory should be the tensor product of the verma modules? Why do we need both the parts, and what is the physical significance of each."}
{"id":"32081","title":"Basic Concepts of Water Pressure for Plumbers","text":"I am learning how to install water pipes and I was told that to check if a pipe is leaking water I need to check the water pressure. But I don't understanding how water pressure works inside the pipe. For example, a water pump is connected to a valve and to a tap like this: pump ----> valve ---->tap the water pressure is caused by the pump. Now if I close the valve and blocking the water from the pump to the tap, what will be the water pressue in the segment of pipe between the valve and the tap? i.e. pump ----> X ----> tap My understanding is that the pressure should drop to zero as this segment is disconnected from any \"source of water pressure\" (the pump). But If I connect a meter to the tap before closing the valve: pump ----> X ----> meter The reading on the meter doesn't drop to zero. Instead, if there is a water pressure drop, it indicates a leakage. But what causes the pressure when the valve is closed? I know air pressure inside a sealed system is caused by the Brownian motion of the air molecule does the same thing applies to water?"}
{"id":"116667","title":"Critical point vs Gibbs phase rule","text":"Why is the critical point for the phase diagram of pure water degrees of freedom equal to 0? Maybe, you know what is the mathematical explanation for the fact that the number of degrees of freedom at the critical point is 0? What else is affected by the lack of degrees of freedom in addition to the Gibbs phase rule?"}
{"id":"102815","title":"Why is $\\left.\\frac{\\partial C_V}{\\partial V}\\right|_T$ different in these derivations?","text":"I want to show that $\\delta q$ is not an exact differential. Starting from $dE = \\delta q - pdV$ and because $E := E(V, T)$ is a state function, which allows to express the exact differential as $$ dE = \\left.\\frac{\\partial E}{\\partial V}\\right|_T dV +\\left.\\frac{\\partial E}{\\partial T}\\right|_V dT, $$ the two expressions can be set equal giving after rearrangement $$ \\delta q = \\left[ \\left.\\frac{\\partial E}{\\partial V}\\right|_T + p \\right]dV + \\left.\\frac{\\partial E}{\\partial T}\\right|_V dT $$ and therefore also $$ \\delta q = \\left[ \\left.\\frac{\\partial E}{\\partial V}\\right|_T + p \\right]dV + C_V dT. $$ Now, by definition $\\partial E\/\\partial T|_V = C_V$, and so $$ \\left.\\frac{\\partial C_V}{\\partial V}\\right|_T =\\left[\\frac{\\partial}{\\partial V}\\left.\\frac{\\partial E}{\\partial T}\\right|_V\\right]_T $$ and because $E$ is a state function, the sequence of partial derivatives can be exchanged (according to Schwarz' theorem, while I don't understand how it works), allowing to write $$ \\left.\\frac{\\partial C_V}{\\partial V}\\right|_T =\\left[\\frac{\\partial}{\\partial T}\\left.\\frac{\\partial E}{\\partial V}\\right|_T\\right]_V. (*) $$ Also, multiplying the above expression for $\\delta q$ by $1\/\\partial V$ at constant $T$, I obtain $$ \\left.\\frac{\\delta q}{\\partial V}\\right|_T = \\left[ \\left.\\frac{\\partial E}{\\partial V}\\right|_T + p \\right]\\left.\\frac{\\partial V}{\\partial V}\\right|_T + C_V \\left.\\frac{\\partial T}{\\partial V}\\right|_T $$ where $\\partial V\/\\partial V = 1$ and the second term on the right hand side equals $0$ because $\\partial T = 0$ at constant temperature. Multiplying the remaining equation by $\\partial \/ \\partial T$ at constant $V$ gives $$ \\left[\\frac{\\partial}{\\partial T}\\left.\\frac{\\delta q}{\\partial V}\\right|_T\\right]_V = \\left[ \\frac{\\partial}{\\partial T} \\left(\\left.\\frac{\\partial E}{\\partial V}\\right|_T + p \\right)\\right]_V. $$ Now _assuming_ $\\delta q$ were exact, again the sequence of partial derivatives would not matter and I could write $$ \\left[\\frac{\\partial}{\\partial V}\\left.\\frac{\\delta q}{\\partial T}\\right|_V\\right]_T = \\left[ \\frac{\\partial}{\\partial T} \\left(\\left.\\frac{\\partial E}{\\partial V}\\right|_T + p \\right)\\right]_V $$ and by using $q=E$ since the \"inner\" differential on the left side is evaluated at constant volume, $$ \\left[\\frac{\\partial}{\\partial V}\\left.\\frac{\\partial E}{\\partial T}\\right|_V\\right]_T = \\left.\\frac{\\partial}{\\partial V} C_V\\right|_T = \\left[ \\frac{\\partial}{\\partial T} \\left(\\left.\\frac{\\partial E}{\\partial V}\\right|_T + p \\right)\\right]_V. (**) $$ From this we find that $(*)$ and $(**)$ are different and thus the assumption must be wrong and therefore $\\delta q$ is not an exact differential. Does this make any sense? * * * Exact wording from book: > Starting with $dE = \\delta q - pV$, show that > > a) $\\delta q = C_V dT + [P+(\\partial E\/\\partial V)_T] dV$ > > b) $\\left(\\frac{\\partial C_V}{\\partial V}\\right)_T = > \\left[\\frac{\\partial}{\\partial T} \\left(\\frac{\\partial E}{\\partial > V}\\right)_T\\right]_V$ > > c) $\\delta q$ is not an exact differential. For c), the book states > If $\\delta q$ were an exact differential, then by solution to a), $(\\partial > C_V\/\\partial V)_T$ would have to be equal to $[\\partial \/\\partial > T(P+(\\partial E\/\\partial V)_T)]_V$ but it is not according to solution of > b), hence $\\delta q$ is not exact."}
{"id":"105450","title":"Ratio of position error of orbiter to the size of the orbit","text":"This is an assignment question for online AP physics. As if it isn't already tough. This question is killing me. I even asked my physics teacher from last year and my calculus teacher. It stumped them, too. I would show some kind of attempt at solving the question, but I've got nothing. I don't even understand how to begin solving this. The question is: > The space shuttle tracking system predicts the position of the shuttle > orbiter with an accuracy that varies between 30 m and 100 m. Its orbital > radius is slightly larger than the radius of the Earth (average orbit > altitude is 340 km). What, approximately, is the ratio of the position error > to the size of the shuttle’s orbit? The orbiter is approximately 37 m long > and 17 high. ($R_{Earth} = 6378$ km)` Any help with this would be greatly appreciated!"}
{"id":"105454","title":"Free-particle solution to Schrödinger Equation","text":"The free particle solution in stationary state (with definite energy) to the Schrödinger equation is $$\\psi(x,t) =Ae^{i(kx-\\omega t)} + Be^{-i(kx+\\omega t)}$$ Since the energy is definite, and hence the momentum is definite, the uncertainty in position must be infinite. How is this reflected by the probability distribution function: $$\\Psi = |\\psi(x,t)\\psi^*(x,t)| $$ The book that I am using just look at the first term of the solution, and derive that the probability distribution function is $A^2$. However, I do not understand why we can do that? Does it imply that if wave function is made up of n terms such that each individual term has a constant probability distribution function, the whole wave function also has a constant probability distribution function? If so, how can I prove it? I know my question might be very vague but that is precisely the problem I am facing now, I don't even know how to ask about the things that I don't understand."}
{"id":"106663","title":"Query into the cumulative velocity of mounted platforms","text":"Consider throwing a stone at an object from rest, it travels at Vms-1. Now throw that stone whilst running at Ums-1. It seems in the latter scenario the total speed of stone is V + U. Now imagine Running at Ums-1, throwing a stone at Vms-1 whilst on a moving train with speed Wms-1 - total stone speed would be V+U+W. Let's extrapolate this to the case where you have a stack of moving platforms, the bottom platform begins to accelerate, once reaching top speed, the platform on top begins to accelerate, and so on and so forth. In a vacuum could it be theoretically possible to reach near infinite projectile velocities using these cumulative platform velocities? ![enter image description here](http:\/\/i.stack.imgur.com\/G5wtQ.jpg)"}
{"id":"27898","title":"Curvature of spacetime in only required to explain tidal forces?","text":"I'm a bit confused about the equivalence principle in GR. I'm quoting from Wikipedia: > An observer in an accelerated reference frame must introduce what physicists > call fictitious forces to account for the acceleration experienced by > himself and objects around him. One example, the force pressing the driver > of an accelerating car into his or her seat, has already been mentioned; > another is the force you can feel pulling your arms up and out if you > attempt to spin around like a top. Einstein's master insight was that the > constant, familiar pull of the Earth's gravitational field is fundamentally > the same as these fictitious forces Later it is written: > The equivalence between gravitational and inertial effects does not > constitute a complete theory of gravity. When it comes to explaining gravity > near our own location on the Earth's surface, noting that our reference > frame is not in free fall, so that fictitious forces are to be expected, > provides a suitable explanation. But a freely falling reference frame on one > side of the Earth cannot explain why the people on the opposite side of the > Earth experience a gravitational pull in the opposite direction Here are some things I hope I understand correctly:   * A particle in free fall is in an inertial frame of reference   * Curvature of spacetime in only required in order to explain tidal forces, as long as you ignore tidal forces, you can explain gravity without curvature.   * Gravity is a fictious force experienced in a non-inertial reference frame My Questions (2 very related questions)   * 1) The statement that curvature of spacetime in only required to explain tidal forces seems weird to me. In the case that there is no curvature of spacetime, what explains gravity? I mean, if gravity is a \"fictitious-force\", what is the \"real cause\" of it? (Again this question stems from the statement that curvature is only needed to explain tidal forces, and not all of gravity). Last example from Wikipedia: > For gravitational fields, the absence or presence of tidal forces determines > whether or not the influence of gravity can be eliminated by choosing a > freely falling reference frame   * 2) If I'm in outer space and I'm freely falling towards earth, let's say I'm very small and I don't experience tidal forces, both me and earth are freely falling and thus in inertial reference frames, and yet I see the earth accelerating towards me, in my frame is it said that \"gravity is eliminated\"? just because I feel no tidal forces?"}
{"id":"27897","title":"Difference b\/w Kinetics & Kinematics w\/concrete example","text":"(I know whether I understand this or not doesn't matter much to my work & study but am just curious.) I still can't differentiate in my head kinetics and kinematics (similar thread is found but doesn't explicitly answer to my question yet What is the difference between \"kinematics\" and \"dynamics\"?). Some websites out there say (ex.) explain that force is only considered in kinematics. Does this mean for example Newton-Euler method is in kinetics and Lagrangian is in kinematics? I also prefer concrete examples in both category."}
{"id":"27896","title":"Can a force in an explicitly time dependent classical system be conservative?","text":"> If I consider equations of motion derived from the pinciple of least action > for an _explicilty time dependend_ Lagrangian > > $$\\delta S[L[q(\\text{t}),q'(\\text{t}),{\\bf t}]]=0,$$ > > under what circumstances (i.e. which explicit functional $t$-dependence) is > the force conservative? By force I understand here the term on the right hand side of the equation, if I shove everything to the right except the expression $mq''(t)$. * * * As a sidenote, besides the technical answer I'd be interested here in some words about the physical motivations involved. I'm somewhat unhappy with a formal $\\text{curl}[F]=0$ condition, since it seems to be to easy to fulfill (namely we have to consider closed circles only at single points in time, respecively). The physical motivation behind conservative forces is the conservation of energy on closed paths, where any parametrization $q(s)$ of curves can be considered. But practically, only loops tracked in finite time are physically realizable, i.e. we would move in a circle while t changes. I guess as soon as one computes the r.h.s. for the equations of motion, one would also be able to define a more physical alternaltive to the above stated idea of conservative forces in this case. I.e. a ask-if-the-forces-integrate- to-zero-on-a-closed-loop functional for a rout between two points in time $t_1$ and $t_2$. This would be an integral where the momentarily force along the point in the path I'm taking would be taken into account. It wouldn't be path independend of course. (We could then even construct another optimization problem on its own, by asking for path with the smallest energy difference, which really would be a sensible question if friction is involved.)"}
{"id":"1968","title":"Lunar twilight and sixth magnitude stars","text":"Summary: when the Moon is x degrees below the horizon, it interferes with stargazing the same as astronomical twilight would. What is x (as a function of the Moon's phase)? We define civil, nautical, and astronomical twilight as when the sun is 0-6, 6-12, and 12-18 degrees below the horizon respectively. This corresponds roughly to what most people call twilight, the ability to distinguish a horizon at sea, and the ability to see 6th magnitude stars at the zenith. However, even when below the horizon, the Moon shines brightly enough to interfere with stargazing. What are the equivalent twilight angles for the Moon? I realize that even the full Moon overhead isn't bright enough for civil twilight, so my real interest is in astronomical twilight. Of course, this will vary greatly with the Moon's phase, and slightly with Moon's distance."}
{"id":"30220","title":"Commutator of scalar fields","text":"So, in the calculation of $ D(t,r) = \\left[ \\phi(x) , \\phi(y) \\right] $, where $ t= x^0 - y^0,~ \\vec{r} = \\vec{x} - \\vec{y} $ you need to calculate the following integral $$ D(t,r) = \\frac{1}{2\\pi^2 r} \\int\\limits_0^\\infty dp \\frac{ p \\sin(p r) \\sin \\left[(p^2 + m^2)^{1\/2} t \\right]} { (p^2 + m^2 )^{1\/2}} $$ For $m=0$, the integral is simple. We get $$ D(t,r) = \\frac{1}{4\\pi r} \\left[ \\delta(t - r) - \\delta(t + r) \\right] $$ I even know what the answer for $ m \\neq 0 $. I have no idea how to calculate it though. Any help?"}
{"id":"107160","title":"Energy realized in fusion and electrons?","text":"I was doing this question about the energy released in a fusion reaction: ![enter image description here](http:\/\/i.stack.imgur.com\/Cogxc.jpg) In the mark scheme it included the mass of the electrons (for part cii) on the left and just used the mass of protons for the H. Do the H in this case not contain electrons and if so why do we not include them in the calculation but do the once on the left??"}
{"id":"107164","title":"How does an electron adjust itself to fit in an excited state that is completely filled?","text":"According to quantum mechanics each state has a specific shape. So, how does the electron get into that shape of the orbital?"}
{"id":"39504","title":"decoherence free subspace of a single photon","text":"Take the state vector for a single photon as $\\psi = \\int \\gamma_{\\omega} | \\omega \\rangle \\otimes (\\alpha |H \\rangle + \\beta | V \\rangle )d \\omega$ $H, V, \\omega$ are the horizontal polarization, vertical polarization and frequency components of the photon. We have a nice, big tensor product space of states for this photon. In particular, we have entangled single photon states! The entanglement of the single photon comes from entangling the polarization with its own frequency. Is it possible to use this effect to create decoherence free subspaces that would be useful in transmitting information within the polarization state of single photons?"}
{"id":"67266","title":"Non-associative operators in Physics","text":"Are non-associative operators (or other kind of elements) used in Physics? For example, in QM I'm looking for something like this: $A(BC)|\\psi\\rangle \\ne (AB)C|\\psi\\rangle$ NOTE: I think that this question does not make much sense, in that case I will close it."}
{"id":"45095","title":"Gauge-invariance of pole mass using Ward Identity","text":"I am able to explicitly verify to one-loop order that pole masses are independent of the choice of gauge paramter. But how do I use the Ward-Identity\/Taylor-Slavnov identity show that the position of the poles in Greens functions are gauge-independent to all orders? The difficulty I'm running into is in using the identities to make statements about the analytic structure of Green's functions."}
{"id":"80485","title":"Can we \"safely\" assume that quantum computing systems will be finite-dimensional?","text":"This is a common assumption in the study of quantum computation to assume that the quantum systems involved are finite-dimensional, since qubits lives in the two-dimensional Hilbert space. According to Roman Gielerak and Marek Sawerwain, the quantum registers corresponding to a quantum computing machine with coherent pulses of light must have an infinite-dimensional character. I am looking for arguments for and against the hypothesis that finite- dimensional structures are sufficient to study quantum computation."}
{"id":"76077","title":"B physics and CP violation","text":"So I am studying CP violation in SM. Experimentalists are trying to study B meson decays now a days. B meson systems involve quarks from the third generation and hence B physics gives more information than the Kaon system. Is there any other reason?"}
{"id":"57232","title":"Zeolite-based oxygen concentrators","text":"I wonder, what is content of output of zeolite-based pressure swing adsorption oxygen concentrators (both oxygen output, and exhaust output)? Yes, they can produce 95% oxygen. But what's the remaining 5%? Is it just argon and other noble gases, or some air contaminants could also be concentrated (like CO2, NO2, CO)? Any references to quantitative gas analysis results would be extremely useful (wasn't able to find any, probably bad google skills)."}
{"id":"34035","title":"Is this paragraph on probabilities of sub atomic partials accurate?","text":"I am working on a concept for something and i want to make sure i understand something clearly before i start on everything else. Note, my project is more about the interactions of elements of complex systems rather than physics. Im just using this paragraph as an example, not doing a project on quantum mechanics. Is this paragraph accurate: The universe is built on probabilities. In the whole scope of the universe, absolute mathematics and absolute certainty do not exist. Its incorrect to say 1 + 1 = 2, the more accurate way to say it is 1 + 1 probably equals 2, but not always. It is all based on the probability that what you expect to happen, will. Its quantum probabilities that dictate that a group of sub atomic particles, at this exact moment in time, line up precisely in a certain way that allow an atom to exist. That atom lives in an environment (dictated by other probabilities) that has a calculable probability to grab onto another atom and form an object with mass. That object was hammered into a shape by a craftsman trusting that the probabilities dictating the actions of the subatomic particles allow it to not crack, or disintegrate, or break the tool. That object is now sitting on your desk holding your soft drink in the form of a can. The subatomic particles that are almost insignificantly small form a thread of interaction that leads up to that can sitting on your desk. If that thread breaks at any point, that can ceases to exist as you know it and your soda is running all over your desk. Thanks everyone, im not sure if this falls in the scope of this site, but i can't think of any other place to ask this question."}
{"id":"79218","title":"Casimir effect as an entropic force","text":"When I first learned about the depletion interaction, my initial reaction was that it looks very similar to the Casimir effect. On making this remark to the professor, he replied somewhat mystically: \"It **is** the Casimir effect.\" No further detail was supplied, however. Nevertheless, it really looks like both problems can be understood in terms of degrees of freedom maximising their positional entropy by reducing the volume of some \"forbidden region\" between two objects. For the depletion interaction, the fluctuating degrees of freedom are small particles, while in the Casimir effect, these degrees of freedom are long-wavelength modes of the free radiation field. However, a key difference is that in the first case the fluctuations arise thermally, while in the second case they are unavoidable quantum fluctuations of the vacuum. > Is it possible to derive the Casimir force between two conducting plates > from entropic considerations alone?"}
{"id":"79217","title":"Relation between (super)integrability and closed orbits","text":"Inspired by this recent question, I would like to understand from a more general and mathematical perspective why closed orbits are only found for the Kepler ($V(r) \\sim 1\/r$) or harmonic ($V(r) \\sim r^2$) potential problems, as follows from Bertrand's theorem. There are two aspects that make these problems special, which I suspect may be related to the closed-orbit property. First, both problems are superintegrable. This property sits intuitively well with the idea that phase- space orbits should close \"as quickly as possible\", thus implying that real- space orbits close after a single revolution. Second, each problem possesses an additional \"unexpected\" conserved quantity, due to a larger symmetry of the problem than the obvious $O(3)$. For the Kepler problem, this is the Runge- Lenz vector, related to the $O(4)$ symmetry of the Hamiltonian. Meanwhile, the harmonic oscillator Hamiltonian conserves the Fradkin tensor: $$ F_{ij} = \\frac{p_i p_j}{m\\omega^2} + m\\omega^2 q_i q_j, $$ which is related to an $SU(3)$ symmetry. In fact these symmetries and corresponding conserved quantities exist for _any_ central field problem (D. M. Fradkin, Prog. Theor. Phys. **37** (1967), p.798). However the conserved quantities only take a \"nice\" form for the Kepler and harmonic problems, which also allows the corresponding quantum problems to be diagonalised exactly by symmetry arguments alone. These considerations motivate the following question: > What specific physical\/mathematical feature(s) do these two problems share > that gives them the property of closed orbits? Does this feature bear > relevance to the quantum counterpart?"}
{"id":"79213","title":"Scalar two loop diagram in $\\varphi^4$ theory","text":"Could someone explain how, or at least show me a link that explicitly shows the calculation of a two-loop corrections to scalar’s two-point function in $\\varphi^4$ theory in the massless limit."}
{"id":"35342","title":"What is the missing proportionality constant in the magnetic levitation formula?","text":"The formula for magnetic levitation is $$ B \\frac{dB}{dz} = \\frac{ \\rho g }{\\chi} $$ but as always, I have a hard time figuring the units in SI. The left hand side is $\\mathrm{T^2 \/m}$, while $\\chi$ has units of $\\mathrm{m^3\\,mol^{-1}}$, which makes the right hand side with units of.. $$\\mathrm{kg\\,m^{-5}\\,s^{-2}\\,(mol)}$$ Obviously there must be some constant, but I don't know what is it, as all texts use this natural unit system. Help? **Update:** using the permeability of free space as reported by Wikipedia, which seems to be Tesla-meter per Ampere, leaves the right hand side as $$\\mathrm{kg\\,T\\,m^{-4}\\,s^{-2}\\,Ampere\\,mol}$$ still far from being recognizable to the left side"}
{"id":"35343","title":"Is emission\/absorption of a photon lossy?","text":"I recall vaguely that energy is absorbed\/radiated in packets called quanta. Quanta were what are now known as photons. What I'm curious about - Is absorption\/radiation vis-a-vis photon lossy? Do the total number of photons exactly match the energy acquired\/released?"}
{"id":"28510","title":"X-Ray crystallography using Bragg's Law","text":"I was looking up X-Ray crystallography using Bragg's Law: $2d\\sin\\theta = n\\lambda$ and I can understand the values of everything except this integer value $n$. As far as my research got $n$ is used to describe the atom spacing in the crystal lattice, but I don't understand how you'd express $n$ or how it would describe it. Could someone please explain this to me please? Note: diagrams tend to be very useful in developing my understanding and if anyone has any reference to a video that might help as well. Thanks."}
{"id":"87853","title":"How to use The Schwarzchild Metric formula to get distribution representing \"free-fall\"","text":"Given formula: ![enter image description here](http:\/\/i.stack.imgur.com\/JsywY.png) How I can use to calculate distribution of points in space, so if i choose path which contains most of the points I get path that close to \"free-fall path\". As far as I know i should use square root of Determinant of given formula to do this, but i am not sure. Please, be gentle, I am computer scientist student that in process writing black hole simulation for academic purposes. NOTE: Simulation is in 2D. so some parameters may be irrelevant."}
{"id":"130736","title":"Variations of S-matrix functional and Feynman diagrams in Weinberg QFT","text":"Weinberg on p. 287 of his QFT vol. 1 introduces the extended interaction operator: $$ \\tag 1 \\hat{V}(t) \\to \\hat{V}(t) + \\sum_{a}\\int d^{3}\\mathbf x \\hat{o}_{a}(\\mathbf x ,t)\\varepsilon_{a}(x). $$   Here $$ \\hat{S} = \\hat{T}e^{i\\int \\hat{V}(t)dt}, \\quad \\hat{o}_{a}(\\mathbf x, t) = e^{i\\hat{H}_{0}t}\\hat{o}_{a}(\\mathbf x , 0)e^{-i\\hat{H}_{0}t}. $$ Then he says that S-matrix arbitrary element $S_{\\beta \\alpha} = \\langle \\beta | \\hat{S}| \\alpha \\rangle$ after extension $(1)$ becomes the $\\varepsilon $-functional, and after that he introduces generalized Feynman rules by adding new vertexes corresponding to $\\hat{o}_{a}$ with $n_{a}$ lines (the number $n_{a}$ coincides with the number of fields in $\\hat{o}_{a}$) and c-factor $\\varepsilon_{a}$. After that he introduces variational derivative $$ \\tag 2 \\left( \\frac{\\delta^{r}S_{\\beta \\alpha}}{\\delta \\varepsilon_{a_{1}}(x_{1})...\\delta \\varepsilon_{a_{r}}(x_{r})}\\right)_{\\varepsilon = 0} = (i)^{r}\\langle \\beta | \\hat{T}\\left(e^{i\\int \\hat{V}(t)dt} \\hat{o}_{a_{1}}(x_{1})...\\hat{o}_{a_{r}}(x_{r})\\right)|\\alpha\\rangle $$ and notices that all $n_{a_{1}}, ..., n_{a_{r}}$ lines correspond to $\\hat{o}_{a_{1}},...,\\hat{o}_{a_{r}}$ respectively are internal, i.e. in case when $n_{a_{1}} = ... = n_{a_{r}} = 1$ they are compared to the propagators. Finally, he says, that if we want to get Feynman diagram with $r$ external lines with types $a_{1},...,a_{r}$ in momentum representation we need to do following with $(2)$: 1) to throw out of propagators $D_{a_{1}a_{r}}(x_{1} - x_{r})$, 2) to apply the Fourier transformation, 3) to add corresponding coefficient functions $u_{a_{1}},...$. Here is the question: сould you make the sense of introduction of mechanism 1)-3) clearer for me? Why do we need additional $r$ external lines which with corresponding vertexes which aren't connected to other vertexes (so the diagram is non-connected), as I think? I need to rephrase the question. I know that n-point Green functions $\\langle \\beta | \\hat{T}\\left(\\hat{o}^{H}_{a_{1}}(x_{1})...\\right)|\\alpha \\rangle$ where the operators are in Heisenberg picture are equal to $(2)$. Also these functions are widely used for derivation of some non-perturbative results for scattering processes. But I don't understand how to apply these non- perturbative results for S-matrix, because we need to set $r$ to zero for obtaining the exact sets of Feynman amplitudes $S_{\\beta \\alpha}$, but I'm not sure that it is possible without loss of nonperturbative results. Here is some incorrect understanding of Weinberg method, so it will be very good if someone help me."}
{"id":"130730","title":"Longitudinal waves in a large (infinite) solid block","text":"Specifically, I am trying to roughly determine the sound produced by a ball when it hits the floor and bounces. If the ball exerts a pressure onto the floor, then certainly this pressure will go on to create a sound wave in the floor (infinite block) and the air. I was just wondering if there is any way to find the sound wave (or the energy of the sound wave) released by the ball, thus finding the energy lost and the ball's approximate rebound height. Is the velocity of the sound wave given as follows?: ![](http:\/\/i.stack.imgur.com\/r3bgq.gif) or is it more complicated than the basic equations provided in most physics textbooks?"}
{"id":"23408","title":"shouldn't we add the oscillating terms into Bohr-Sommerfeld quantization formula","text":"shouldn't be the quantization formula (in one dimension) equal to $ N_{smooth}(E)+N_{osc}(E) = \\oint_{C}p.dq $ ?? where the Oscillating term is just the correction from Gutzwiller trace formula or a sum over Orbits why is just the oscillating term ignored .."}
{"id":"76920","title":"How to find the angle of elevation and launch speed?","text":"A particle is projected from a point on level ground with a speed of $u$ meters per second and an angle of elevation $\\theta$. The maximum height reached by the particle is $42$ meters above the ground and the particle hits the ground $196$ meters from its point of projection. Find the values of $\\theta$ and $u$."}
{"id":"76928","title":"Four vector manipulation","text":"I have started doing _Relativistic Quantum Mechanics_ from _Greiner_. I'm having difficulty understanding the following derivation for commutation relations (page 4): $$[\\hat{p}^\\mu, x^\\nu] = i\\hbar[\\frac{\\partial}{\\partial{x_\\mu}},g^{\\nu \\sigma}x_\\sigma]=$$ $$= i\\hbar g^{\\nu \\sigma}\\frac{\\partial x_\\sigma}{\\partial x_\\mu}=$$ now this is the part I don't understand, how did we got from last equation to next one: $$=i\\hbar g^{\\nu \\sigma}\\delta^{\\mu}_\\sigma$$ Why is: $$ \\frac{\\partial x_\\sigma}{\\partial x_\\mu}= \\delta^{\\mu}_\\sigma$$ where $$\\delta^{\\mu}_\\nu = g^{\\mu \\sigma}g_{\\sigma \\nu}$$"}
{"id":"129562","title":"Relationship between Schrodinger equation and string\/membrane","text":"In Sakurai's _Modern Quantum Mechanics_ (2nd ed) p.99, he says > We know from the theory of partial differential equations that ( _time- > independent Schrodinger equation_ ) subject to boundary condition > ($\\psi(\\mathbf{x}')\\rightarrow 0$ as $|\\mathbf{x}'|\\rightarrow\\infty$) > allows nontrivial solutions only for a discrete set of values of $E$. It is > in this sense that the time-independent Schrodinger equation yields the > quantization of energy levels. Once the partial differential equation ( > _time-independent Schrodinger equation_ ) is written, the problem of finding > the energy levels of microscopic physical systems is as straightforward as > that of finding the characteristic frequencies of vibrating **strings or > membranes**. Although I've found a paper (http:\/\/www.scientificexploration.org\/journal\/jse_21_1_hocking.pdf) about this, the relationship is still not very explicit. So would you please explain briefly about the relationship between the Schrodinger equation and string\/membranes? Many thanks."}
{"id":"91243","title":"Festive physics: gold flake vodka","text":"I have a bottle of vodka that has a load of gold flakes suspended in it. It has been sat still for over 24 hours and the flakes are all still suspended within the liquid: they have not risen to the surface or sunk to the bottom. Any ideas as to the physics behind this?"}
{"id":"114439","title":"Why is the chiral symmetry $SU(2)_A$ not anomalous?","text":"Using Fujikawa's path integral treatment of the triangle diagram, one can show that $$\\mathrm{Tr} \\gamma^5 = \\int d^4 x\\ \\partial_{\\mu}j^{\\mu} $$ Where $j^{\\mu}$ is the Noether current of $U(1)_A$. Thus, the $U(1)_A$ anomaly can be traced to to fact that the trace of $\\mathrm{Tr} \\gamma^5 \\neq 0$ at loop order. My question is, why isn't $SU(2)_A$ anomalous too? I fail to understand why this applies only to $U(1)_A$."}
{"id":"81842","title":"Orthonormality of Radial Wave Function in the Hydrogen Atom","text":"`Question`: Show that the radial wave function for the hydrogen atom are orthogonal with no exceptions. `Attempt`: We know that the wave function for the hydrogen atom is orthonormal; that is, $$\\langle\\psi_{nml}\\mid \\psi_{n'm'l'}\\rangle=\\delta_{nn'}\\delta_{mm'}\\delta_{ll'}$$ Splitting up $\\psi$ into a radial component $R_{nl}$ and a angular component $Y_{lm}$, we have $$\\langle R_{nl}Y_{lm}\\mid R_{n'l'}Y_{l'm'}\\rangle=\\delta_{nn'}\\delta_{mm'}\\delta_{ll'}$$ We know that the angular component $Y_{lm}$ is orthonormal, and thus $$\\langle Y_{lm}\\mid Y_{m'l'}\\rangle=\\delta_{ll'}\\delta_{mm'} $$ I believe that, if we do not have orthogonality in the radial component, we will not have orthogonality in the wave function. For example, if we do not have radial orthogonality, then at $n=n'$, we might have the radial component be zero. Thus, the radial wavefunction must be orthonormal. Is this correct reasoning? i' worried I' missing something."}
{"id":"81841","title":"Large object is pushed through a small wormhole, what happens?","text":"A simple question about wormholes: What happens if a large object is pushed through a much smaller wormhole? For example, what happens if I push an elephant through a palm-sized wormhole? My intuition is that depending on the magnitude of the force applied, either a part of the elephant is ripped out, or the elephant sort of gets stuck in the wormhole, as if hitting a wall. Is this correct?"}
{"id":"86819","title":"Can Kirchhoff laws be applied to any circuit?","text":"Kirchhoff's loop\/current rule is just law of conservation of energy and Kirchhoff's junction rule is just law of conservation of charge.So, I think that these can be applied to any circuits unlike Ohm's law. Is there any exceptions?"}
{"id":"81848","title":"Infinite period in Simple Harmonic Motion","text":"I'm studying the Simple Harmonic Motion, and I am hesitant about, how to get mass values for infinite period?   * When mass is 0.   * When mass is infinite. With $\\tau=2\\pi\/\\sqrt{k\/m}$."}
{"id":"128018","title":"Is there some other name used for \"ping rigidity\"?","text":"In MTW, p. 398, \"Box 16.4 (continued)\", there's an interesting sketch (which can also be seen on   p. 15 of this excerpt (www.pma.caltech.edu\/~ph236\/yr2008\/readings\/MTW_Chapter16.pdf).   (It's not the only sketch of this kind in MTW; but it happens to be the only sketch on p. 398.)   Shown are (cmp. text of p. 397)   * two open-ended lines representing two \" _particles_ \" (They are supposed to satisfy additional relations which are not of immediate interest here, however.),    * several line segments criss-crossing between the two open-ended lines, representing \" _light rays bouncing back and forth between [the two particles]_ \", and   * two of the criss-crossing segments being extended such that they meet, with the coincidence point of these two segments denoted as $\\mathscr Q$. In order to describe what's represented by that third feature (which, unfortunately, MTW don't seem to spell out explicitly) it is convenient to refer to the two explicitly shown particles by explicit distinct names; say \"Paul\" for one and \"Eric\" for the other (i.e. by names which have been suggested through this sketch by M. Goossens). Accordingly, it may be said that the sketch of MTW p. 398 shows that `(a)`: Paul observed $1$ _light ray bounce to, and back from_ event $\\mathscr Q$ (or: $1$ \"ping wrt.\" event $\\mathscr Q$), and `(b)`: for his same initial\/signal indication, and his same final\/reception indication, Paul counted $3$ consecutive _back and fourth bounces_ (or: \"pings\") with respect to Eric. And likewise: `(c)`: Eric observed $1$ _light ray bounce to, and back from_ event $\\mathscr Q$ (or: $1$ \"ping wrt.\" event $\\mathscr Q$), and `(d)`: for his same initial\/signal indication, and his same final\/reception indication, Eric counted $2$ consecutive _back and fourth bounces_ (or: \"pings\") with respect to Paul. Now, rather than considering only one such event $\\mathscr Q$, we may ask about, and Paul and Eric may under certain conditions even actually find, a set $\\cal L$ with several **additional events** (and including event $\\mathscr Q$) such that the conditions `(a)` \\- `(d)` had been satisfied by **all events of** $\\cal L$. Taking again the suggestion from the already mentioned sketch by M. Goossens it may be found that there can be (at least) one particle identified who participated in all events of such a set $\\cal L$; let's refer to it by the name \"Lutz\". If \"Paul\", \"Eric\" and \"Lutz\" have been identified accordingly, as having satisfied conditions `(a)` \\- `(d)` throughout some sufficiently extended trial, then obviously `(1)`: Paul found any $3$ consecutive pings wrt. Eric same as the corresponding $1$ ping wrt. Lutz; symbolically: \" $\\overset{\\leftarrow ~ \\rightarrow}{{}^EP{~}^L} = 3$ \", `(2)`: Eric found any $2$ consecutive pings wrt. Paul same as the corresponding $1$ ping wrt. Lutz; symbolically: \" $\\overset{\\leftarrow ~ \\rightarrow}{{}^PE{~}^L} = 2$ \", and apparently (without further conditions or reservations) it follows that `(3)`: Lutz found any $3$ consecutive pings wrt. Eric same as the corresponding $2$ consecutive pings wrt. Paul; symbolically: \" $\\overset{\\leftarrow ~ \\rightarrow}{{}^EL{~}^P} = \\frac{3}{2}$\". This relation between such three participants together may be recognized as a general kind of \" _rigidity_ \"; which I'd call in default of any other terminology: \"(triplewise) ping rigidity\"; in the above example symbolically: \" $\\underset{\\\\{P, E, L\\\\}}{\\shortmid \\leftrightarrows \\shortmid} = \\\\{2, 3, 1\\\\}$ \". **My question:** Have such relations, of three (or more) participants \" _bouncing light rays back and forth between_ \" each other and counting **constant integer or rational ratios** for the numbers of (consecutive, or individual) \"corresponding pings\" been recognized and already named elsewhere?"}
{"id":"130002","title":"How can I ensure my scale is not affected my surface beneath it","text":"I have person weighing spring scale. I live in Japan where the house has wood floors which deflect significantly under weight. I weighed my self in the bathroom and in the tiled bath area. The difference was 2.5 kilograms greater in the bath area but it could still be affected by possibly an angled floor or deflection. So how you get as true a result as possible with that spring scale?"}
{"id":"38478","title":"If $L$ is a matrix that represents real physical quantity, why is $L^2$ non-negative real physical quantity?","text":"In my textbook, it says that when $L$ is a matrix that represents real($\\mathbb{R}$) physical quantity, $L^2$ represents non-negative real physical quantity. What would be the proof of this?"}
{"id":"130008","title":"My orbiting body is orbiting about the wrong focus of it's elliptical orbit… why?","text":"I am coding in c++ and am computing the position of an orbiting body as a function of time. Everything is almost working. I have a nice elliptical orbit. Except, my orbiting body speeds up as it moves away from the \"sun\" and slows down as it approaches it. The best way I can describe it is that it's like the sun is at the wrong focus of the ellipse. I'm hoping someone can point me to what could cause this to happen? I've gone through all my code and I don't see any mistakes. My steps are:   1. Compute the Mean Motion of a satellite.   2. Use this to compute the Mean Anomaly   3. Calculate the Eccentric Anomaly from this (that was a doozy)   4. Calculate the True Anomaly from the Eccentric Anomaly   5. Calculate the Heliocentric Distance   6. Finally I take the Polar Coordinates and convert to Cartesian Coordinates which I then position over the top of my \"sun\" Somewhere in here I'm screwing up. I don't believe it's the final step as the calculated coordinates are derived from the angle and radius, which should be based on the correct focus. EDIT for the equations I'm using: **Mean Motion** $$ n = \\sqrt{\\frac{G(M+m)}{4\\pi^2a^3}} $$ This is from Wikipedia. **Mean Anomaly** This is just $n \\times t$ elapsed. **Eccentric Anomaly** I'm not really sure how to write this up for a non programmer. Basically the way I did this was based very much on some code that I found on the internet that I can no longer find. There's some recursion involved to gradually refine the answer. I'm convinced the issue isn't in here because I can enter my values into something like http:\/\/www.jgiesen.de\/kepler\/kepler.html and I have similar results. **True Anomaly** $$ \\nu = 2\\ \\arg\\left(\\sqrt{1-e} \\ \\cos \\left(\\frac{E}{2}\\right), \\sqrt{1+e} \\ \\sin \\left(\\frac{E}{2} \\right)\\right) $$ $e$ is Eccentricity $E$ is Eccentric Anomaly from above Taken from here Wikipedia. **Heliocentric Distance** $$ r = a \\frac{1-e^2}{1+e \\ \\cos(\\nu)} $$ $a$ is my semi-major axis $\\nu$ is the True Anomaly from above Taken from Wikipedia."}
{"id":"71009","title":"IIA and IIB Compact on 8D","text":"How can compactifying IIA (non-Chiral) and IIB (Chiral) Superstring on $T^2$ (2-torus) gives rise to ($2$ dual descriptions of) the same $\\mathcal N = 2$ supergravity in $8$ dimensions? I don't see it. Could you please explain it to me or recommend me some literature to read about it? Thank you!"}
{"id":"18672","title":"Will perfect black hole apply force on matter?","text":"While standing on a planet its gravitation force is noticeable and manifests only through the normal force, e.g. if jupiter had rocky surface, standing on the surface will apply about 2.5 times more force than when standing on earth. However sky diving on earth and on jupiter (without atmosphere) would apply the same amount of \"destructive\" force, that is 0. Thus gravitation force cannot really destruct matter without something for the matter to interact with.   If a black hole with no matter falling into it exists, will a person with space suite die if he would \"jump\" into it? he will have nothing applying \"destructive\" force on it, just kinetic energy that cannot manifest without something to hit."}
{"id":"128378","title":"Finding particles in the classically forbidden regions","text":"Have particles ever been found in the classically forbidden regions of potentials? For example, in a square well: has an experiment been able to find an electron _outside_ the rectangular well (i.e. in the exponential fall-off regions) ? And more importantly, has anyone ever observed a particle **while** tunnelling?"}
{"id":"77585","title":"Quantum mechanics potential barrier problem","text":"While reviewing some quantum mechanics, I cam across a very interesting situation. For a potential barrier, if a particle has an energy $E$ less than the potential barrier $V_0$, it is possible to measure it inside the potential barrier or the _classically forbidden region_ quantum mechanically. But, if we calculate reflection and transmission coefficients for the wave, we find out $R=1$ and $T=0$ which means the wave will be fully reflected and no transmission will take place. But still there is probability of finding it inside the potential barrier. How?"}
{"id":"39616","title":"What is the physical interpretation of force times area?","text":"I know that $\\text{Force} \\times \\text{Distance = Work}$.   But, what would be the physical meaning of $\\text Force \\times \\text Area?$ Is such a quantity used in physics?"}
{"id":"87038","title":"Commutator of Momentum with a Position dependent function","text":"I heard from my GSI that the commutator of momentum with a position dependent quantity is always $-i\\hbar$ times the derivative of the position dependent quantity. Can someone point me towards a derivation, or provide one here?"}
{"id":"87037","title":"Classical dynamics with Schrodinger equation","text":"What are some interesting classical systems for which the dynamics can be reduced to a many-body Schrodinger equation, at least in some useful regions of phase space, and in particular, with many variables."}
{"id":"45935","title":"It seems to me that superpotentials can be defined in a theory with or without supersymmetry. Is this true?","text":"I recently read \"An Introduction to Supersymmetry in Quantum Mechanical Systems\" by T. Wellman (amongst other sources) in an effort to find out what a superpotential actually is and how it relates to the potentials of particles\/fields). Here's the link: http:\/\/www.google.co.uk\/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CDkQFjAA&url=http%3A%2F%2Fphysics.brown.edu%2Fphysics%2Fundergradpages%2Ftheses%2FSeniorThesis_Wellman.pdf&ei=ulm- UPSCLZOY1AWjwYHYDw&usg=AFQjCNGrg_2jv5NZ7b6k4Fs7er34jgtw3w&sig2=yjQYy1Lf_gZVS- RRefUCsQ It occurred to me that the Fermionic Hamiltonian and the Bosonic Hamiltonian can be formulated without supersymmetry. On page 13, Wellman expresses these Hamiltonians in terms of the superpotential W in a way that is purely algebraic and doesn't require supersymmetry. In other words we have a bunch of terms that give us the two Hamiltonians; these terms are then simply replaced by W's (see equations 3.2 to 3.9). We only actually get supersymmetry when the separate assertion is made that Q operators exist that transform between our fermionic and bosonic states. So would it be true to say that non-supersymmetric theories contain superpotentials, W, within their Hamiltonians in the same way that supersymmetric theories do? If this is the case the superpotential is just a useful function that is especially helpfully when we consider supersymmetric theories? I.e. superpotentials exist with or without supersymmetry."}
{"id":"44040","title":"Dirichlet's work on gravity in non-Euclidean space?","text":"In the book _The Norton History of Astronomy and Cosmology_ by the late John North I have found the following statement (page 514): \"The German mathematician Lejeune Dirichlet studied the law of gravitation in non-Euclidean space towards the end of 1850.\" Does anyone perhaps know more about that?"}
{"id":"133413","title":"Is it possible to build a huge spaceship like the one of of Star Trek or Prometheus?","text":"Ok, so theoretically, is it possible to build a spaceship like the ones we see off of Star Trek or Prometheus or will physics not allow it? What are some challenge faces such an task. Also, is there other ways we can use to launch heavier payload from earth or are we just stuck with using tons of fuel to launch small object?"}
{"id":"47894","title":"Dynamics of a Rocket","text":"I am interested in modelling the trajectory of a rocket from the Earth to the Moon by solving a differential equation numerically. Below are some key facts and assumptions I am using. I want to make sure that I have not made any serious mistakes, nor disregarded any necessary facts. We will consider the following equation, $$ \\vec{T} + \\vec{c}(\\vec{r})\\dot{\\vec{x}} + \\vec{G}(\\vec{r})= m(t) \\ddot{\\vec{x}}, $$ where $T$ is the _**constant**_ rocket thrust, $c$ denotes air resistance and is a function of radial distance from the earth, and the rocket has mass that drops at a rate that is _**constant**_ with respect to time (we are assuming that a constant amount of fuel is always used for constant rocket thrust -- is this a valid assumption?). Now a question:   * The trajectory of the rocket is not straight; how do we incorporate parabolic motion into the numerics?"}
{"id":"41725","title":"Causal and Global structure of Penrose Diagrams","text":"What kind of global and causal structures does a Penrose diagram reveal? How do I see (using a Penrose diagram) that two different spacetimes have a similar global and causal structure? Also, I have the following metric $$ds^2 ~=~ Tdv^2 + 2dTdv,$$ defined for $$(v,T)~\\in~ S^1\\times \\mathbb{R},$$ e.g. $v$ is periodic. This is the according Penrose diagram: ![](http:\/\/www.matheplanet.de\/matheplanet\/nuke\/html\/uploads\/8\/8403_misner_penrose_2.jpg) Is the Penrose diagram that I have drawn correct?"}
{"id":"82128","title":"Regarding formulation of a multipoint model of fluid dynamics","text":"Suppose I am trying to formulate a multipoint model of fluid dynamics. I have a procedure for doing so the details of which is not important to this question, but only that it is based on a series expansion, the higher order terms in which series being dependent on larger number of spacial points. The equations for the first two terms in a Burger formulation is something like shown below: \\begin{align} &\\frac{\\partial u^0}{\\partial t}(x,t)+\\frac{\\partial^2 u^0}{\\partial x^2}(x,t) \\+ u^0(x,t)\\frac{\\partial u^0}{\\partial x}(x,t)\\\\\\ &\\qquad\\qquad\\qquad\\quad\\;\\;+\\int_0^1 u^1(x,t;x_1)\\frac{\\partial u^1}{\\partial x}(x,t;x_1) dx=0\\\\\\ &\\frac{\\partial u^1}{\\partial t}(x,t;x_1)+\\frac{\\partial^2 u^1}{\\partial x^2}(x,t;x_1)+u^0(x,t)\\frac{\\partial u^1}{\\partial x}(x,t;x_1)+u^1(x,t;x_1)\\frac{\\partial u^0}{\\partial x}(x,t)=0 \\end{align} it is however an odd formulation, the point $x_1$ appears as a mere parameter in the second formulation while it appears only in the integrand of the first equation, so that apparently there is no enough restriction on the behavior of the function $u^1$ with respect to $x_1$, so it seems there s no unique solution for $u_1$. If the flow is a homogeneous flow but $u^1$ can be written as $u^1=u^1(x-x_1,t)$, so that derivation with respect to $x$ will also take into play the point $x_1$, so that $x_1$ will now behave like it is a more important variable. This will get more importance if we further consider the higher order terms like $u^2(x,t;x_1,x_2)$ which in a homogeneous flow will be writable as $u^2(x-x_1,x-x_2,t)$ and whose $x$-derivative becomes: $$\\frac{\\partial u^2}{\\partial x}=\\frac{\\partial u^2}{\\partial(x-x_1)}+\\frac{\\partial u^2}{\\partial(x-x_2)}$$ Such a treatment imports the variables $x_1$, $x_2$ and etc. in the second and higher order equations in a more proper manner, in such a way that hope to find unique solutions increase.   1. The problem is that how the first formulation which was more general is lame in giving such unique solutions as the second formulation is apparently capable of?   2. Or maybe there is a must in every multi-point modeling of fluid dynamics (actually turbulence) to consider one of the following two procedures? a- To expand the series in terms of functions of the form: $u^0(x,t)$, $u^1(x,x-x_1,t)$, $u^2(x,x-x_1,x-x_2,t)$ and etc. ? b- To write the equations once at the point $x$, then once at the point $x_1$ available also in the arguments of $u^1$ and higher order functions, then once at the point $x^2$ available also in the arguments of $u^2$ and higher order functions, and etc., then consider all those equations in one place, for example by adding them together. As the series is truncated somewhere at a function $u^n$ this will not contain infinitely of equations but I am not very hopeful that this is the right path to take.     * any idea about these or other methods for multipoint formulations?"}
{"id":"15402","title":"What is the difference between electric potential, potential difference (PD), voltage and electromotive force (EMF)?","text":"This is a confused part ever since I started learning electricity. What is the difference between electric potential, potential difference (PD), voltage and electromotive force (EMF)? All of them have the same SI unit of Volt, right? I would appreciate an answer."}
{"id":"6936","title":"A water drop in a falling lift","text":"Consider a lift, which is at rest in an homogeneous gravity field. There is a thin layer( **with thickness** $h$) of water on the floor of the lift. At some moment a single cable, supporting the lift, breaks and the lift begins free falling(forever). It is easy to describe qualitatively what happens with water, i think: the formation of a drop begins. During this process the drop jumps up from the floor and once the total kinetic energy of the drop is dissipated into the heat, the center of the drop stands at some height $H$ from the floor. (All the kinetic energy of the drop is coming from the difference in surface tension energy of water between initial and end moments.) Question: How to determine(approximately) $H$ ? For simplicity let's assume that the cross-section of the lift is a disc- shaped with radius $R$. $(h<<R)$ Remark: **A key point is to determine how the total kinetic energy distributes between internal kinetic energy and translational kinetic energy of the drop, i think.** With this the problem will be solved. So it is convenient at the early stage of formation of the drop ignore viscosity effects of water and air resistance. **Edit:** I would like to clarify that by internal kinetic energy i mean the kinetic energy of macroscopic motion of water inside the drop. At early stage there is no energy loss assumed, for simplicity."}
{"id":"15404","title":"What was so surprising about accelerating cosmological expansion?","text":"The 2011 Nobel Prize in Physics, as far as I understand, concerns the expanding universe -- galaxies moving away from each other at ever increasing speed (that's what I think I read in newspapers). Now, the idea of expanding universe is not new. What is the novelty in this study? What was unexpected in it?"}
{"id":"7707","title":"Energy conservation and interference","text":"I have a problem with energy conservation in case of interfering waves. Imagine two harmonic waves with amplitudes $A$. They both carry energy that is proportional to $A^2$, so the total energy is proportional to $2A^2$. When they interfere, the amplitude raises to $2A$, so energy is now proportional to $4A^2$ and bigger than before. The equivalent question is what happens to the energy with the superposition of two waves that interfere destructively. Also, if someone could comment on the statement about this problem in my physics book (Bykow, Butikow, Kondratiew): the sources of the waves work with increased power during the interference because they feel the wave from the other source."}
{"id":"7702","title":"Does quantum mechanics prohibit extraterrestrial life?","text":"There is a known thought experiment, connected to quantum immortality: a duel between physicist and a philosopher. Each turn physicist and philosopher fire at each other with a pistol. The quantum immortality predicts that each of the participants will find themselves alive and the opponent dead after a number of shots. We can modify the experiment in the following ways:   1. They do not fire at each other, but it is lightning that shots them. The outcome should not change.   2. There are not two but more participants.   3. They are located at separate islands in the ocean. It seems that even if there are hundreds of participants, each of them will eventually find all others killed by lightning. If he could not observe other isles directly, after he discovers and explores other isles, he will find them uninhabitable. Now pretend that the participants actually live on separate planets, and in each million years there happens disaster that kills all inhabitants on 1% of all planets. It seems that after years the physicist that lives on one of the planets should discover that all other planets are uninhabitable. Even more: there is even no chance that organic matter on each other planet could actually evolve into anything resembling actual life. That said the physicist will find out that there is no other inhabitable planet in the Uinverse. On the other hand, this does not mean that extraterrestrial life does not exist. It actually cannot not to exist. But it exists in parallel universes (in terms of MWI) and thus unobservable. Both physicist on planet A and philosopher on planet B exist, but will never meet in the same universe. Thus, QI it seems predicts one concrete observable consequence that extraterrestrial life can be never detected. What is your opinion?"}
{"id":"102256","title":"Could moving land mass alter Earths gravity?","text":"This is potentially a very stupid question but I'm going to ask it anyway. With all these huge buildings such as the Abu Dhabi Mosque, where an unbelievable amount of materials such as marble was moved from one side of the earth to the other, is it possible that if we 'shift enough stuff' that we could change the earths centre of gravity, and potentially alter its orbit?"}
{"id":"126558","title":"Does the weight of fluid in a conical container act entirely on the base?","text":"# Scenario description Lets assume both containers have a capacity of 300 litres. One is a vertical tube as shown in pic 1 Other one is more or less a V shaped containers as shown in pic 2 Both are at ground level # Question No doubt that whole weight of 300 litres will act on the base of the tube (pic 1). But will the weight of the whole 300 litres act on the base of the container as shown in pic 2? I.e if you keep both the containers on a weighing scale, for sure, it will show a weight of approx. 300 kg each. But what I'm asking is whether the weight of whole 300 litres acts on the **base** of container 2 (since the container's walls are slanting, as opposed to the walls of the tube in pic 1). If the whole weight will not act on the base of the container, can you state the reasoning and also the principle, so that I can further get to know about it? Pic 1                          |         |                        |         |                |         |                |         |                |         |                |         |                |_________|      Pic 2                        \\              \/               \\            \/                \\          \/                 \\________\/"}
{"id":"133222","title":"Was the firewall paradox resolved?","text":"For quite sometime there has been a claim that the firewall paradox has been resolved (via lasers). For instance, http:\/\/global.ofweek.com\/news\/Lasers-to- solve-the-black-hole-information-paradox-9867 . I was wondering what the physics community thought about it? I haven't heard much of a reaction of the physics community. Why is this? His paper: http:\/\/arxiv.org\/abs\/1310.7914"}
{"id":"126881","title":"Does the entropy of an adiabatically isolated system stays the same after we decrease\/increase the pressure?","text":"> Even if the system is isolated and there is no heat exchange with > surroundings, shouldn't the decrease\/increase of pressure result in > increase\/decrease of entropy? > > Does this property of an isolated system means that increase of pressure > means equivalent decrease in volume, so after the process the $pdV$ i.e. the > work is still the same? > > Can someone explain this in more detail? Quoted text above was original question. Please read on further, I clarified the question. To be more specific, situation is this. We have an adiabaticaly isolated system and (for example) a piston inside it. If there is piston, that means that volume work or $PdV$ can be done. **So if we do work on the system, by some force from surroundings which pushes piston(compresses the gas) we would get increase of the pressure inside the system... right?** **If we increase the pressure in the system, the entropy will decrease as well, altough we haven’t bring in the heat .. right?** So if we increased the pressure and mutually decreased the volume, the molecules of the gas are packed more tightly and they have lesser space to move in...This is followed by system having more internal energy now, then it had before process...Since the activity of molecules inside is higher, they are moving at higher speeds(cause of smaller volume)... **That means that system should have more internal energy after the process, right**? Taking that into consideration, and if we follow the rule that increase in pressure is also increase in temperature, that is molecules have higher activity on higher pressure and that kind of activity is measured with temperature... **that means that system also has more amount of heat?** _Correct me if I am wrong._ **And even if it is adiabatically isolated (dQ=0) shouldn't the internal energy increase and thus amount of heat in the system increase as well when we do the work on the system (move the piston to compress the gas )?** I am talking that higher activity leads to more heat inside the system even if it is prevented (isolated) to exchange heat with its surroundings. I hope you understand me now."}
{"id":"22700","title":"About binary stars and calculating velocity, period and radius of their orbit","text":"I saw somewhere about being able to measure the velocity, period and radius of a binary star orbit by looking at red shift and blue shift. I understand it but can someone give me an example of calculations etc done to calculate the velocity, period and radius of a binary stars in orbit?"}
{"id":"119704","title":"Could the collision of two pairs of quantum entangled protons cause a temporary \"wormhole\"?","text":"I recently read this article from MIT News. I then started thinking about how a particle accelerator creates a temporary microscopic black hole. My question is: > _If quantum entangled pair $A$, consisting of $A_1$ and $A_2$, and a second > pair $B$, consisting of $B_1$ and $B_2$, were collided, $A_1$ collides with > $B_1$, and $A_2$ collides with $B_2$, would this create a temporary > wormhole?_"}
{"id":"119705","title":"What is an \"Interaction Hamiltonian\"","text":"I'm an undergraduate reading up on some quantum physics so that I can help out more in the lab that I'm working in this summer. In the book I'm reading (Shankar's \"Principles of Quantum Mechanics\") I just came across the term interaction Hamiltonian in describing how orbiting electrons interact with a magnetic field. I have an idea of what it might mean, but I can't find a good explanation anywhere. What is an \"interaction Hamiltonian\", and how does it differ from a standard Hamiltonian?"}
{"id":"52295","title":"Dark energy before radiation or after?","text":"In the cosmic history of the universe, does the dark energy comes before the radiation epoch, or only now, in the 'matter' universe (matter dominated era)? Because, we now know that like 75% of our visible universe is made up of dark energy, about 20% is dark matter and the rest is the 'ordinary matter', but does that mean that this dark energy came after the radiation and matter domination of the universe or was it there in the beginning, at the BB, and then was just spread out via inflation, while radiation era and matter domination era followed?"}
{"id":"82875","title":"Basic Interpretation of Compostion of Observables and their Measurement","text":"Given two (or more) observables $A, B$ which commute one can construct a third observable $C= A \\circ B$. If $\\psi$ is a common eigenvector of $A, B$ with eigenvalues $\\lambda_1, \\lambda_2$ then it is clear that the measurement of $C$ of the state $\\psi$ gives the measurement result $\\lambda =\\lambda_1 \\lambda_2$, i.e. the result of the measurement of the observable $C$ is the result from the measurement of $A$ times the result from the measurement $B$. But what if $\\psi$ is an eigenvector of $C$, but not of $A$ and $B$? Is there any connection between the measurement results of $A$, $B$ and $C$? Example: Let there be three observers which measure a spin state with the corresponding observables $A = \\sigma_x \\otimes \\mathbb{I} \\otimes \\mathbb{I}$, $B=\\mathbb{I} \\otimes \\sigma_y \\otimes \\mathbb{I}$ and $C=\\mathbb{I} \\otimes \\mathbb{I} \\otimes \\sigma_y$. They commute and we can construct $D= A \\circ B \\circ C = \\sigma_x \\otimes \\sigma_y \\otimes \\sigma_y$. Now the GHZ-state $\\psi = \\frac{1}{\\sqrt{2}} ( | +z, +z, +z \\rangle - | -z, -z, -z\\rangle)$ is an eigenvector of $D$ with eigenvalue $\\lambda =-1$ but it is not an eigenstate of $A, B$ or $C$. Each of the observers will get a result $\\pm 1$. Is there any connection between this individual results and the eigenvalue of $\\psi$ (respectively the expectation value $\\langle \\psi | D | \\psi \\rangle = -1$)? Intuitively I would say that the product of the results should give the eigenvalue of $\\psi$ but I can't see how this should follow from any quantum mechanical postulate or mathematical reasoning like in the case of the common eigenvector."}
{"id":"81785","title":"Counting of brownian particles: Point Process","text":"Imagine a point process defined by the passage time of purely brownian particles through a given point (in 1D), line (2D) or plane (3D). I'm interested in the variance of the counts (number of particles passing the points) as a function of sampling time. Unlike brownian motion, I expect my signal to show non-vanishing correlation, because a particle that just crossed the point, has a high probability to cross it again. Thus the correlation should look like a dirac delta function (classic for discrete point process) plus a decreasing time function. However, I'm a bit confuse with the way to calculate it analytically. I search in areas of photon counting, geiger counter, and gaz dynamics, but I didn't find any explicit calculation of the variance. Two idea I had: 1\/ Studying the Spatio-Temporal correlation function of a field of Brownian particles (see Gardiner, Stochastic Methods, eq. 13.3.22 for instance): $G(x,t)=\\frac{1}{\\sqrt{4\\pi Dt}}\\exp\\left(-x^2\/(4Dt)\\right)$ in 1d. We should have: $Var(T)=\\lim_{x\\rightarrow 0} \\int_T\\int_T G(x,t) dt dt'$. However, the last expression is hard to compute. 2\/ Using the probability of return time to origin of Brownian motion (the correlation arise from successive passage of the same particle) Any idea or suggestions would be welcomed !"}
{"id":"116465","title":"Quantum Locking Superconductors on Earth's Magnetic Field","text":"Is the earth's magnetic field strong enough to hold a quantum locked superconductor, or the superconductor wouldn't be able hold its own weight? How strong should be the earth's magnetic field to hold quantum locked superconductors?"}
{"id":"11404","title":"Ascent rate and size of balloon","text":"I am part of a school project, Project Stratos to send a balloon to the edge of space (the closer side :P) and was wondering how you would work out the accent rate of a large balloon (roughly 1m^3 of helium with 100g of mass) and the size of it as it increases its Altitude. I am creating a live map (that will be based on predictions rather than its actual location) and want to know the speed it will float up into the atmosphere. Currently we are assuming the ascent rate will be about 5m\/s but I doubt that is very accurate and would this speed increase as it gets higher? Edit: I would also quite like to know the burst height of the balloon."}
{"id":"126258","title":"Understanding the cause of the big bang","text":"Ok, as I understand the expansion of the initial singularity was caused by quantum fluctuations like the ones predicted by the Heisenberg Uncertainty Principle. But how can these fluctuations occur within a singularity? And how can they cause the expansion of that singularity? Or do I have a terrible misunderstanding of the theory?"}
{"id":"19238","title":"Does a Lorentz-contracted object float or sink?","text":"Consider the following thought experiment: Imagine an object of a certain mass density which allows it to float in water. Now if this object is viewed from a moving frame with high speed, it will look Lorentz contracted hence its density will look greater than the rest density. So from the rest frame it floats but from the moving frame it sinks! What am I missing?"}
{"id":"575","title":"Is it possible\/correct to describe electromagnetism using curved space(-time)?","text":"Comparing the simples form of the forces of both phenomena: the law of Newton for gravitation $V\\propto \\frac{1}{r}$, and the Coulomb law for electrostatics $V\\propto \\frac{1}{r}$, one might think that if one can be extended relativistically to a curvature in space-time, that the other one would lend itself to a similar description. Is this so? Thanks for any useful thoughts and\/or suggestions!"}
{"id":"76126","title":"Is there any relationship between Gravity and Electromagnetism?","text":"We all know that the universe is governed by four Fundamental Forces which are The strong force , The weak force , The electromagnetic force and The gravitational force . Now, is there any relationship between Electromagnetism and gravity?"}
{"id":"9560","title":"Theories that Relate Gravity, Electricity, and Magnetism","text":"There are some people who (without having a stated theory that I know of) insist that Gravity, Electricity, and Magnetism are related. Some point to symmetry in Maxwell's Equations as a potential indication of this (intuitive) connection. Is it possible that there are \"Maxwell-like\" equations that relate Electricity and Magnetism directly to Gravity? All forms of electromagnetic energy do interact with gravity. Are there any physicists who are working on this sort of theory?"}
{"id":"80781","title":"Additive Trichromatic Color: Seeking Answer for a Fun Application","text":"I'm not the brightest person and I was wondering if it was possible, in the same way that it's possible to make ink that's visible under UV light, if it's possible to make ink that requires a distinct combination of colours of light to work. If there are different coloured UV lights would it be possible to make an ink (or buy one) that reacts only when all three are present and overlapping?"}
{"id":"16845","title":"Lorentz Transformation via Geometry","text":"Today, I was tutoring and explained the space-time. I explained how one can convert North-South into West-East by rotating, and how you can convert time into space with velocity. Below the Energy-Momentum stuff the book had some problems. One was the following: > Given an event at (1 Lightsecond, 2 s), give it's coordinates in a frame of > reference that is moving with 0.6 of speed of light to it. This the diagram I came up with, in blue there is the event. (Not really to scale.) ![](http:\/\/wstaw.org\/m\/2011\/11\/11\/m8.png) Then I projected the point onto the orange axes, and I get smaller values on each. This makes sense to me, as a moving observer would see thing smaller and slower, resulting in less values in either measurement. But how do I get the orange lines? I cannot see how I get this angle. If I got them, I could tell you the speed of the reference frame, but the speed is given here … And did I do the projection of the speed right?"}
{"id":"12179","title":"Stern-Gerlach-Experiment with j=1 atoms?","text":"Suppose you do a Stern-Gerlach experiment with atoms in a $j=1$ state. There would be three separate beams ($m_j = -1, 0, 1$) coming out of the apperatus. But what would be the relative distribution of atoms in these beams? On first thought I would say that the distribution should be $\\frac{1}{3}$, $\\frac{1}{3}$, $\\frac{1}{3}$. But considering the \"classical\" distribution of the projections of the angular momentum onto one axis I could imagine that there could be more atoms in the $m_j = 0$ state. This would still be compatible with symmetry considerations. To state the question differently: how does the density matrix of an unpolarized beam of $j=1$ atoms look like?"}
{"id":"91234","title":"Why is a coherent state an eigenfunction to the annihilation operator?","text":"In class when we talked about the harmonic oscillator in QM we noticed that the eigenfunctions to the annihilation operator are coherent states in the sense that they have minimum uncertainty in momentum and position. My question is: What is the physical meaning behind this? Why are coherent states eigenfunctions to the annihilation operator and: Was this only a lucky punch and a special property of the harmonic oscillator or is this true for more physical problems?"}
{"id":"35572","title":"Is carbon dioxide a greenhouse gas?","text":"> **Possible Duplicate:**   >  What experiments prove the greenhouse effect? I am seeking for a proof that CO2 is a greenhouse gas. I posted this on **Skeptic.SE** recently but found no help in seeking for proof: > I assisted to a physicist conference in my university a few years ago > against the case that carbon dioxide was a cause of global warming. The main > point was that CO2 is **not** a greenhouse gas. I did a research to find > evidence for either side and found absolutely nothing. > > So, is carbon dioxide a greenhouse gas? If yes, has it been demonstrated in > a scientific paper? > > Here are some discussion articles describing arguments against CO2 being a > greenhouse gas: Home experiments Ph.D. Tim Ball Evidence from temperature rise through history I understand that the sources do not prove that CO2 isn't a greenhouse gas...but I can't find evidence proving the contrary. Since a real physicist with many years of experience a demontrated to me that it is not possible by definition, I need to find proof to accept the other premise. I found this post on physics.SE, but the asker do not seek proofs and simply accept what the others are saying. Baically, to have a greenhouse effect, a gas needs to absorb only a portion of a wavelength and let pass another portion. Since CO2 absorbs some IR, a part of it is remitted to earth. The spectrum is absorbed by the earth (everything) and it is remitted on another wavelength which allows a different proportion to pass through the CO2 layer. The question is, why the heat would be remitted toward earth (it should go in the colder direction)? And also, if CO2 is all around the earth and blocking a certain wavelength, why would increasing it warm the earth? Am I missing something?"}
{"id":"10449","title":"What are the temperatures of objects in Low Earth Orbit (LEO)?","text":"What is the temperature of objects in Low Earth Orbit? Consider LEO to be 600km to 800km."}
{"id":"53655","title":"Optical refocusing efficiency","text":"What is the material for optical focusing that produces the less diffraction losses? Suppose one have a sequence of serial optical elements $R$ that keep refocusing a beam of collimated light, each a distance $D$. Now suppose that the wavelength $\\lambda$, the optical element and the distance are such that the $\\epsilon$ is the amount of radiation that falls outside the next optical element, which i'll take to be small. I'm concerned with other losses in this system, for instance, the optical material can have some intrinsic diffusion, or it absorbs a portion of the radiation What are the main losses that an optical element can have of this kind, and which are the best refocusing optical elements that minimise those losses?"}
{"id":"26464","title":"How will the super massive black hole affect our galaxy?","text":"I've recently learned that the general consensus is that several (if not, most) galaxies have super massive black holes in their center, in particular the Milky Way. This, at least to me, makes perfect sense seeing as we are in a spiral galaxy which means we need something to \"spiral\" around (a large body or a bunch of mass). But seeing as we're rotating around this massive black hole, won't we inevitably end up sucked in by it? Aren't we spinning towards the center of the galaxy, or are we staying steady where we are?"}
{"id":"63129","title":"Reachable area of cannonball given fixed initial speed","text":"The trajectory of a cannonball fired from the origin with initial speed $v_0$ at an angle $\\theta$ is given by $$y = x\\tan\\theta - \\frac{g}{2v_0^2 \\cos^2\\theta}x^2.$$ For fixed $v_0$, at what angle $\\theta$ should the canon ball be fired so that it will hit a target located at $(x_0,y_0)$? What is the area of the region in which the target can be located such that it is possible to hit the target?"}
{"id":"78327","title":"Suppose we fill a membrane with ferro fluid, and alter the magnetic field around it in specific ways, what dynamics can come into play?","text":"im trying to develop a flexible robotic arm filled with ferro fluid. Different flexible motions can be performed using this concept."}
{"id":"63127","title":"How does one subtract two light beams?","text":"From what I understand, it seems like you can only \"add\" beams together. You can use a beam combiner, basically using a beam splitter in reverse, to combine two beams. In homodyne detection, you use a local oscillator beam to mix with the signal beam so you can decipher the difference in phase change which would give you information about the system your signal beam is probing. Homodyne detection can give information regarding the position of an atom. Say I wanted to detect the relative distance of two atoms. Each atom\/ion is stuck inside an electric harmonic potential. The location of the harmonic potentials are known, (if this is possible) Now, let's say I turn off the harmonic potential trap and at the same time I use the homodyne measurement technique to determine the position of both of the atoms. This means shining a laser to detect the atom's position. (The light is absorbed and re-emitted?, or reflected?) In any case, light that comes out of the atom\/ion cavity is related to the atom's position\/momentum and it is combined with a local oscillator beam to determine the position and momentum of the atom. The light hits photodetectors which then integrate the signals together producing a photocurrent. The analysis of this photocurrent will allow one to figure out the atom's position relative to the center of the harmonic trap. To figure out the relative position of the atoms from each other, I'll need to \"subtract\" the position of one atom with the other. If one atom's position is labeled x1 and the other is labeled x2, than wouldn't we simply do x2 - x1? If so, how is one to do this?! How does one \"subtract\" the position of one from the other using light beams?"}
{"id":"62838","title":"Could the outer structure of the Universe be made from Antimatter?","text":"Cern recently stated that antimatter may be repelled by matter, much like the opposite effect of Gravity. So is it possible that antimatter is actually repelled to the edges of the Universe to create a sort of outer-shell, something that allows the expansion of the Universe into nothingness.."}
{"id":"54924","title":"Is restoring force a particular type of force?","text":"I have a question about the restoring force in elastic band or rope which confusing me for a long time. As I was told in high school physics, for an elastic band (or spring), if Hooke's law holds, we have $F = k\\Delta x$. What's confusing me is: should F be the total force acting on the object or the restoring force only? Or I ask this way: is there anything called \"restoring force\" existing independently, just like gravity, friction or tension? To my understanding, restoring force should be the total force which is pointing to the equilibrium point. For example, if we consider a bungee cord, we should always count the tension of the cord as well as the gravity so the restoring force at any time should be the total force of tension and gravity; hence, when we apply Hooke's law, we should always have $F$ being the total force, not just the tension. Is that correct? This is pretty confusing to me because there use many terms in the book. Sometimes they said it is the tension in Hooke's law, sometimes they say the restoring force and sometimes the total force...."}
{"id":"87746","title":"Does changing the electric \/ magnetic field cause self-reinforcing induction of the other?","text":"I understand that changing electric field produces magnetic field and changing magnetic field produces electric field. Are these produced magnetic and electric field produced due to one defined to be constant or variable? If these are defined to be variable then do they continue to produce one another? By this I mean if changing electric field produces changing magnetic field, does this changing magnetic field produce a new electric field or the same one again?"}
{"id":"68728","title":"Do radio waves go faster than the speed of light?","text":"My science teacher used to say a lot of weird stuff, but I'm just making sure on this one."}
{"id":"68721","title":"Does a car consume more fuel when it's raining?","text":"Yesterday my wife asked me that question, and I couldn't answer. Consider a car, in a sunny day, and that is consumes x gallons per mile. Considering that everything is equal, except that it's traveling in a rainy day, but at the same temperature as the sunny day so that air density is the same. Will the lower friction of the tires make it consume more or less fuel? And the fact that rain drops are falling over and in front of it ? I answered that it'll consume more fuel, since friction is what makes car move and that the rain will act against it... but I'm not sure ?"}
{"id":"26696","title":"How do we determine the mass of a black hole?","text":"Since by definition we cannot observe black holes directly, how do astronomers determine the mass of a black hole? What observational techniques are there that would allow us to determine a black hole's mass?"}
{"id":"35143","title":"Elements of a Planet reveals nearby supernova remnant?","text":"During a random reading through this site, I found this one: Origin of elements heavier than Iron (Fe)... The answer was \"The formation of many elements in earth was due to Supernova nucleosynthesis\" as told by some guy. Here, A question crosses my mind:   * If the elements were formed due to the explosion of a supernova, then there should be a **remnant** like a black-hole or a neutron star nearby... Were there any nearby? Or, the _famous_ **Big Bang** is responsible for this? ( _Down-voters - leave comments_ )"}
{"id":"88262","title":"Partial waves and the velocity expansion of a scattering cross section","text":"I'm confused about the relation between the velocity expansion of a scattering cross section and the angular momentum (partial wave) expansion. For example, for dark matter annihilation, we write $\\sigma v = \\sigma_s + \\sigma_p v^2 + \\cdots$, where $v$ is the velocity and $\\sigma_s$ corresponds to $s$-wave scattering (orbital angular momentum $\\ell =0$) and $\\sigma_p$ is the $p$-wave scattering ($\\ell =1$). **Question:** : why does the $v^{2\\ell}$ term correspond to initial states in an $\\ell$-wave configuration? * * * I understand that in these calculations we treat the scattering states as plane waves, and that these can be expanded as a series of partial waves ($\\ell$ eigenstates). And I can see that $\\langle \\mathbf x | E,\\ell,m\\rangle \\sim j_\\ell(kr) \\sim k^\\ell$. Is there more to this story?"}
{"id":"34231","title":"What is the particle residence time for given flow rates of gas mixture components?","text":"For two different chemical substances there are two open valves with flow rates $$Q_1=a\\frac{m^3}{h}\\ \\text{ and }\\ \\ Q_2=b\\frac{m^3}{h},$$ leading into seperate cables. Next, the cables join, the substances mix perfectly and they flow along together. Then during the route, there is an observational volume $V$ (e.g. a cylinder or length $d$ and cross section $A$). > How long is the residence time $\\Delta t$ of a particle of the mixture > inside the volume?"}
{"id":"34235","title":"A relative time dilation paradox.","text":"Let us assume that there are two astronauts A and B who are floating in space. A sees B passing by and vice versa. A sends signals to B every minute. According to A since B is moving his clock will be slower. So B will receive the signals prior to the appointed minute. The same argument can be applied for B who will conclude A's clock is running slow. Who is right?"}
{"id":"79057","title":"Does gravity limit the number of bosons that can occupy the same single-particle state?","text":"QFT says that an unlimited number of bosons can occupy the same \"state\" (what I mean by that is that the whole system's wavefunction is composed of a product of many identical wavefunctions). However, gravity increases monotonically with energy density. It seems that at some point, one additional boson would create a high enough energy density to create a black hole. Is this true? Could I calculate the number of bosons necessary to cause this?"}
{"id":"35181","title":"Ward Takahashi identities from Z invariance","text":"I'm trying to get Ward-Takahashi identities using the approach used in Ryder's book (pages 263-266). I like that he starts from demanding gauge invariance of Z in a explicit way and them explores the consequences of that to functional generators of vertex functions. But the actual calculation is bugging me out. The author seems oblivious to the fact that the fermionic fields and sources ($\\psi$ and $\\eta$) are Grassmann variables and keeps commuting them out with no regard for my sanity. For instance, equation (7.102) has a term: $$i e (\\bar{\\eta}\\psi-\\bar{\\psi}\\eta) $$ that promply becomes (exchanging the fields by ${1\\over i}$ times the derivatives on the sources, acting on Z to the right): $$ e (\\bar{\\eta}{\\delta \\over \\delta \\bar{\\eta} }-\\eta{\\delta \\over \\delta \\eta }) $$ In my opinion that should be: $$ e (\\bar{\\eta}{\\delta \\over \\delta \\bar{\\eta} }+\\eta{\\delta \\over \\delta \\eta }) $$ This one has no consequences because he commutes them again right after. But when I try to calculate it being careful with the Grassmann variables, I can never get the right signs in (7.111). I'm specially troubled by the derivatives below (this is what I'm getting, but one of them should have a different sign in order to get the right WT identities): $${\\delta \\over \\delta \\bar{\\psi}(x_1) } {\\delta \\over \\delta \\psi(y_1) } \\left[{\\delta \\Gamma \\over \\delta \\psi(x) }\\psi(x)\\right]_{\\psi=\\bar{\\psi}=0}=-\\delta^4(x-y_1){\\delta^2 \\Gamma \\over \\delta \\bar{\\psi}(x_1) \\delta\\psi(x) }$$ $${\\delta \\over \\delta \\bar{\\psi}(x_1) } {\\delta \\over \\delta \\psi(y_1) } \\left[\\bar{\\psi}(x) {\\delta \\Gamma \\over \\delta \\bar{\\psi}(x) } \\right]_{\\psi=\\bar{\\psi}=0} =\\\\\\=-\\delta^4(x-x_1){\\delta^2 \\Gamma \\over \\delta \\psi(y_1) \\delta\\bar{\\psi}(x) } = \\delta^4(x-x_1){\\delta^2 \\Gamma \\over \\delta \\bar{\\psi}(x) \\delta\\psi(y_1) }$$ Does anybody ever did this calculation in detail and has some pointers? Are there any other references that follow this same approach? EDIT: Just a shameless bump: I still looking for some light on this. Any reference on where this is done in detail would help."}
{"id":"57901","title":"Noether theorem, gauge symmetry and conservation of charge","text":"I'm trying to understand Noether's theorem, and it's application to gauge symmetry. Below what I've done so far. First, the global gauge symmetry. I'm starting with the Lagragian $$L_{1}=\\partial^{\\mu}\\Psi\\partial_{\\mu}\\Psi^{\\ast}-m^{2}\\left|\\Psi\\right|^{2}$$ with classical complex fields. This Lagragian is invariant with respect to the global gauge symmetry $\\Psi\\rightarrow\\tilde{\\Psi}=e^{\\mathbf{i}\\theta}\\Psi$, ... such that I end up with $$\\delta S=\\int dv\\left[\\dfrac{\\delta L_{1}}{\\delta\\Psi}\\delta\\Psi+\\dfrac{\\delta L_{1}}{\\delta\\Psi^{\\ast}}\\delta\\Psi^{\\ast}+\\mathbf{i}\\left(\\Psi\\partial^{\\mu}\\Psi^{\\ast}-\\Psi^{\\ast}\\partial^{\\mu}\\Psi\\right)\\partial_{\\mu}\\delta\\theta\\right]=\\int dv\\left[\\partial_{\\mu}j^{\\mu}\\right]\\delta\\theta$$ provided the equations of motion ($\\delta L \/ \\delta \\Psi = 0$, ...) are valid. All along I'm using that $$\\dfrac{\\delta L}{\\delta\\phi}=\\dfrac{\\partial L}{\\partial\\phi}-\\partial_{\\mu}\\dfrac{\\partial L}{\\partial\\left[\\partial_{\\mu}\\phi\\right]}$$ and that $\\int dv=\\int d^{3}xdt$ for short. The conserved current is of course $$j_{1}^{\\mu}=\\mathbf{i}\\left(\\Psi^{\\ast}\\partial^{\\mu}\\Psi-\\Psi\\partial^{\\mu}\\Psi^{\\ast}\\right)$$ since $\\delta S \/ \\delta \\theta =0 \\Rightarrow\\partial_{\\mu}j_{1}^{\\mu}=0$. **Here is my first question:** Is this really the demonstration for conservation of charge ? Up to now, it seems to me that I only demonstrated that the particle number is conserved, there is no charge for the moment... Then, I switch to the local gauge symmetry. I'm starting with the following Lagrangian $$L_{2}=\\left(\\partial^{\\mu}+\\mathbf{i}qA^{\\mu}\\right)\\Psi\\left(\\partial_{\\mu}-\\mathbf{i}qA_{\\mu}\\right)\\Psi^{\\ast} -m^{2}\\left|\\Psi\\right|^{2} -\\dfrac{F_{\\mu\\nu}F^{\\mu\\nu}}{4}$$ with $F^{\\mu\\nu}=\\partial^{\\mu}A^{\\nu}-\\partial^{\\nu}A^{\\mu}$. This Lagrangian is invariant with respect to the local gauge transformation $$L_{2}\\left[\\tilde{\\Psi}=e^{\\mathbf{i}q\\varphi\\left(x\\right)}\\Psi\\left(x\\right),\\tilde{\\Psi}^{\\ast}=e^{-\\mathbf{i}q\\varphi\\left(x\\right)}\\Psi^{\\ast},\\tilde{A}_{\\mu}=A_{\\mu}-\\partial_{\\mu}\\varphi\\right]=L_{2}\\left[\\Psi,\\Psi^{\\ast},A_{\\mu}\\right]$$ Then I have $$\\delta S=\\int dv\\left[\\dfrac{\\delta L_{2}}{\\delta\\Psi}\\delta\\Psi+\\dfrac{\\delta L_{2}}{\\delta\\Psi^{\\ast}}\\delta\\Psi^{\\ast}+\\dfrac{\\delta L_{2}}{\\delta A_{\\mu}}\\delta A_{\\mu}\\right]$$ with $\\delta\\Psi=\\mathbf{i}q\\Psi\\delta\\varphi$, $\\delta A_{\\mu}=-\\partial_{\\mu}\\delta\\varphi$, ... such that I end up with $$\\dfrac{\\delta S}{\\delta\\varphi}=\\int dv\\left[\\mathbf{i}q\\Psi\\dfrac{\\delta L_{2}}{\\delta\\Psi}+c.c.+\\partial_{\\mu}\\left[j_{2}^{\\mu}-\\partial_{\\nu}F^{\\nu\\mu}\\right]\\right]$$ with $j_{2}^{\\mu}=\\partial L_{2}\/\\partial A_{\\mu}$ and $F^{\\nu\\mu}=\\partial L_{2}\/\\partial\\left[\\partial_{\\nu}A_{\\mu}\\right]$ Then, by application of the equations of motion, I have $$\\partial_{\\mu}\\left[j_{2}^{\\mu}-\\partial_{\\nu}F^{\\nu\\mu}\\right]=0\\Rightarrow\\partial_{\\mu}j_{2}^{\\mu}=0$$ since $\\partial_{\\mu}\\partial_{\\nu}F^{\\nu\\mu}=0$ by construction. Of course the new current is $$j_{2}^{\\mu}=\\mathbf{i}q\\left(\\Psi^{\\ast}\\left(\\partial^{\\mu}+\\mathbf{i}qA^{\\mu}\\right)\\Psi-\\Psi\\left(\\partial^{\\mu}-\\mathbf{i}qA^{\\mu}\\right)\\Psi^{\\ast}\\right)$$ and is explicitly dependent on the charge. So it seems to me this one is a better candidate for the conservation of charge. NB: As remarked in http:\/\/arxiv.org\/abs\/hep-th\/0009058, Eq.(27) one can also suppose the Maxwell's equations to be valid ($j_{2}^{\\mu}-\\partial_{\\nu}F^{\\nu\\mu} = 0$, since they are also part of the equation of motion after all, I'll come later to this point, which sounds weird to me), and we end up with the same current, once again conserved. Nevertheless, I still have some troubles. Indeed, if I abruptly calculate the equations of motions from the Lagrangian, I end up with (for the $A_{\\mu}$ equation of motion) $$j_{2}^{\\mu}-\\partial_{\\nu}F^{\\nu\\mu}\\Rightarrow\\partial_{\\mu}j_{2}^{\\mu}=0$$ by definition of the $F^{\\mu \\nu}$ tensor. So, my **other questions** : Is there a better way to show the conservation of EM charge ? Is there something wrong with what I did so far ? Why the Noether theorem does not seem to give me something which are not in the equations of motions ? said differently: Why should I use the Noether machinery for something which is intrinsically implemented in the Lagrangian, and thus in the equations of motion for the independent fields ? (Is it because my Lagrangian is too simple ? Is it due to the multiple boundary terms I cancel ?) Thanks in advance. PS: I've the feeling that part of the answer would be in the difference between what high-energy physicists call \"on-shell\" and \"off-shell\" structure. So far, I never understood the difference. That's should be my last question today :-)"}
{"id":"91635","title":"Could fast vibrations cause us to travel forward in time","text":"Assuming it's possible to vibrate a human at near light speed without harming him, would a few minutes of this from his point of view be much longer from a stationary observer's point of view? In other words do vibrations work the same as normal movement with regards to time dilation? So a person could walk into such a machine, and walk out hundreds of years in the future, even though a much smaller amount of time would have passed from their perspective?"}
{"id":"79587","title":"What is bulk viscosity and how does it affect the flow?","text":"What is bulk viscosity and how does it affect the flow? Explain the idea of introducing such a term in the Navier-Stokes equation. What are the consequences if not taken into account?"}
{"id":"79580","title":"Vibrating system at angular frequency in a vibrating system","text":"Can anyone clarify what does mean the angular frequency of a system in case of the vibrating membrane. Angular frequency is measured in radians per second, what does this have with the vertical displacement in case of the wave equation."}
{"id":"91639","title":"Does turning sharply on a bicycle conserve more energy than a wide turn?","text":"I use a bike to commute, so I spend a lot of time thinking about how to get the most bang out of my momentum. Aside from the extra distance traveled in a wide turn, does making a sharp turn save you any energy? My guess is no, because these things tend to even out, but it definitely _feels_ like I'm going much faster. I've even considered that maybe taking a turn sharp is worse, because the extra pressure will cause more friction in the bearings and the tires. Either way it's more fun, though that probably doesn't get a term in the equations."}
{"id":"34588","title":"Shor's algorithm and Bohmian Mechanics","text":"Do quantum computer's tell us anything about the foundations of quantum theory? In particular Shor argued in the famous thread by 't Hooft Why do people categorically dismiss some simple quantum models? that quantum computation was at odds with 't Hooft's ideas. Does quantum computation tell us anything new about hidden variables like Bohmian mechanics (which, at least so far, is 100% in agreement with everything we know about physics, contrary to what some people (e.g. Motl) claim)?"}
{"id":"130533","title":"Name of battery voltage when load connected\/disconnected","text":"If I had a 3V battery, and when no load connected it reads 3.2V, and with a load 2.8V (just a hypothetical example), what is the name for these two terms, with a load or no load? I know the voltage drop occurs due to its internal resistance when a load is connected, however I still am not sure of the terms. Here's a few which may be of help:   * emf ($\\mathcal{E}$)   * Potential difference   * Terminal voltage Any help would be appreciated in what these terms are called, this is one of those questions where it almost seems too simple to find this answer online."}
{"id":"114126","title":"Scattering of two particles - phase factor","text":"I did see some posts on stackexchange on this matter, but I find them to be beyond my scope or not directly related to what I am looking for. I am reading Feynman Lectures III, chapter 4. It talks about scattering of two particles $a$ and $b$ and it defines $f(\\theta)$ to be the probability amplitude of particle $a$ scattering at angle $\\theta$. Then it says \"You might also think that the amplitude for the second process (where particle $b$ instead enters the detector placed at angle $\\theta$) is just $f(\\pi-\\theta)$. But that is not necessarily so, because there could be an arbitrary phase factor.\" I'm not questioning why things behave the way they are in quantum mechanics. But, isn't it **_by definition of $f$_** that the probability amplitude of particle $a$ scattering at angle $\\pi-\\theta$ is $f(\\pi-\\theta)$? I don't see why there is a phase factor. If there's a phase factor, it should already be part of $f$."}
{"id":"114634","title":"Gauge invariance of Rarita-Schwinger action in curved spacetime","text":"The Rarita-Schwinger action in curved $n$-dimensional spacetime is $$ \\int \\sqrt{g} \\overline{\\psi}_a \\gamma^{abc} D_b \\psi_c $$ Here $g = \\det(g_{\\mu \\nu})$, and the indices $a, b \\dots$ are 'internal' indices that transform under e.g. $\\mathrm{SO} (3,1) $ in $3+1$ dimensions. $\\gamma^{abc} = \\gamma^{[a} \\gamma^{b} \\gamma^{c]}$ with the gamma matrices obeying $\\gamma^a \\gamma^b + \\gamma^b \\gamma^a = 2 \\eta^{ab} $, and $\\eta^{ab}=\\mathrm{diag}(1,1 \\ldots 1,-1,-1 \\ldots -1)$ is the 'internal metric'. $\\psi_{\\mu} = \\psi_{c} e^{c}_{\\mu} $ is a spinor-valued one form. Spacetime indices $\\mu, \\nu$ can be 'converted' to internal indices using the frame field $e_a^{\\mu}$, and vice versa. The covariant derivative is $D_{\\mu} \\psi_{\\nu} =\\partial_{\\mu} \\psi_{\\nu} + \\frac{1}{4} \\omega_{\\mu}^{ab} \\gamma_{ab} \\psi_{\\nu} $. Here $\\omega$ is taken to be the torsion free spin connection, and $\\gamma^{ab} = \\gamma^{[a} \\gamma^{b]}$. In flat space, the covariant derivative becomes a normal derivative, and the action then has a symmetry $\\psi_c \\rightarrow \\psi_c + \\partial_c \\phi $, with $\\phi$ an arbitrary function. This freedom can be used to eliminate some of the degrees of freedom from the field $\\psi_c$ which correspond to lower spin. However, in curved space there is no corresponding symmetry under $\\psi_c \\rightarrow \\psi_c + D_c \\phi$. For this reason, it is said that the Rarita-Schwinger action in curved spacetime is inconsistent. My question is, what goes wrong when you don't have this extra symmetry? And do the problems manifest at the classical level or only at the quantum level?"}
{"id":"93671","title":"Why is the blue color of sky darker in high altitude regions?","text":"I read that as the air density is lower at high altitudes, the scattering is less, hence the darker blue but then other colors are scattering lesser than blue, why don't we see a darker shade of the other colors?"}
{"id":"61715","title":"Why does a rod rotate?","text":"I'm a physics tutor tutoring High School students. A question confused me a lot. Question is: > Suppose a mass less rod length $l$ has a particle of mass $m$ attached at > its end and the rod is hinged at the other end in vertical plane. Another > point object of mass $m$ is moving with velocity $v$ and hits the rod at its > end and continues in its path with velocity $v\/2$. The rod gets enough > velocity that it can complete a vertical circular path. What would be the > force due to hinge on the rod just after collision? Assume that time taken > for collision is very small. From conservation of momentum, the particle attached to rod gets velocity $V' = v\/2$ which is equal to $2\\sqrt{gl}$ and therefore $$T-mg=\\frac{mv'^2}{l}$$ The tension in the rod is the Normal reaction due to hinge on the other end of the rod. Hence, the force exerted by rod is $$\\frac{mv'^2}{l}+mg$$ But other teacher claims that I'm wrong. He says there will be horizontal force due to hinge on the rod. I say, there won't be any horizontal force. He gives an example that suppose there is a rod and you flick it at one end with finger then won't the other end move (or tend to move)? Well it does. **EDIT:** I've understood your answers mathematically.I still don't know how to counter my friend's argument. I'm convinced by it. Because it sounds intuitive. He says, suppose you have a rod (massless or with mass) in space, and you flicked at one of its end, then the other end surely will have velocity.Similarly, in the above problem, the collision will impart velocity to one end, so other end also must move but it is hinged and can't move because of hinge. So, there is a horizontal force due to hinge on the rod."}
{"id":"51409","title":"Idealized trajectory from sloped surface","text":"I am a GIS programmer implementing a visualization. I am modeling the idealized trajectory of a particle ejected from a volcanic vent using: $$\\text{distance} = \\frac{(v^2 \\times sin(2\\theta))}{g}.$$ Where $g = 1.62\\:\\mathrm{m\/s^2}$, $v$ is velocity, and $\\theta$ is ejection angle. $g$ is the lunar gravity constant I was supplied. How can I incorporate the slope of the underlying surface assuming a single point of ejection? $$$$$$$$ EDIT: My current workflow is to compute total travel distance, extract a topographic profile along the total theoretical travel distance and then check the height of the projectile to the height of the actual surface at 100m intervals. In this way I can compute the landing site for the projectile. EDIT 2: I updated the question with the correct formula. Apologies for the incorrect transposition. My implementation now assumes a completely flat surface. What happens when the ejection surface is sloped either uphill or downhill?"}
{"id":"55380","title":"Quantum Teleportation Fidelity","text":"I understand that quantum teleportation fidelity is the overlap of the initial quantum state with the teleported quantum state. If the teleportation is perfect, then the fidelity would equal 1 or 100% successful of retrieving the quantum state at the desired location. In all real experiments I've glanced at and read thoroughly, the fidelities are never 1. Doesn't this suggest that what the experimenters \"teleported\" was really just a different state? Doesn't the fidelity have to be exactly 1 in order to really remove a quantum state from one location and make it appear at another location? My concern is that either I've missed something, or that we aren't really teleporting at all because the final state achieved on the other side is not equal to the initial state!"}
{"id":"55387","title":"Can a divider \"laminarize\" turbulent flow and thus reduce friction?","text":"Looking at the Moody chart I think to myself, the _friction factor_ doesn't decrease much at all with Reynolds number after a certain point. I wonder if laminar flow is more efficient in a sense, and what sense would that be? ![Moody diagram](http:\/\/i.stack.imgur.com\/1bIeg.jpg) I understand that in laminar flow you have clear lines that the fluid doesn't cross, whereas that wouldn't be true in turbulent flow. We could imagine a pipe with a divider placed in the middle to keep the fluid from mixing in eddies, but that would just create more friction with the divider. There are thinkable cases, however, where you could introduce a divider that moves along with the fluid, in particular, the Taylor-Couette flow... http:\/\/en.wikipedia.org\/wiki\/Taylor%E2%80%93Couette_flow This setup describes basically one cylinder rotating within another cylinder with a fluid in-between them. ![Concentric rotating cylinders](http:\/\/i.stack.imgur.com\/zA1r3.png) Let's say that you kept the fluid the same, and the distance between the inner cylinder and outer cylinder the same. In that system, let's say you insert a divider at a radius in the middle of the annular area, and this divider was mostly buoyant in the fluid, so it's not experiencing friction on the edges, and it also is free to rotate with the fluid. Would doing so actually reduce the frictional torque on the rotating inner cylinder? If you could introduce an infinite number of infinitely thin dividers is there a theoretical limit to how much you could reduce the retarding torque? Would that just make it laminar, or laminar-ish?"}
{"id":"129512","title":"Why do liquids boil when their vapor pressure equals the ambient pressure?","text":"Given that the boiling point of a liquid is the temperature at which the vapor pressure is equal to the ambient (surrounding) pressure, what significance does a liquid's vapor pressure have in the formation of bubbles that happens at and above the boiling point? The definition of boiling point seems to imply that the pressure inside of the bubbles must be at least as great as the liquid's vapor pressure in order to balance the outside pressure, but is there any particular reason why the pressure inside of the bubbles is related to the vapor pressure? The vapor pressure seems to be a measurement describing the tendency of the molecules to escape from the surface of the liquid, but I don't see how that relates to bubble formation within the liquid. This question has bothered me for a while, so any help would be much appreciated."}
{"id":"22337","title":"How do I go from exponents to a formula?","text":"This is a continuation of this question. http:\/\/ocw.mit.edu\/courses\/physics\/8-01-physics-i-classical-mechanics- fall-1999\/video-lectures\/lecture-1\/ skip this lecture to around 25:50. After doing dimensional analysis on $t\\propto h^\\alpha m^\\beta g^\\gamma$ Lewin concludes that: $$\\alpha = \\frac{1}{2}, \\gamma = -1\/2, \\beta = 0$$ This is all fully understood, but he then goes to conclude from this that: $$t = C \\sqrt{\\frac{h}{g}}$$ How did he get to this? And why is he allowed to just assume that there is a constant, C, there when he doesn't even know its value or what it is? Keep things as simple as possible please, I'm 16."}
{"id":"22331","title":"Current formula and form factor","text":"I am currently struggling with the formula for an exact current in QFT, a fermion with an upcoming momentum $p$ and an outgoing momentum $p'$. My problem is to show whether or not a term of the form $\\bar{u}(p')F(q^2)i\\sigma^{\\mu\\nu}(p+p')^{\\nu}$ where $\\sigma^{\\mu\\nu}=\\frac{i}{2}[\\gamma^{\\mu},\\gamma^{\\nu}]$. I would like it to be forbidden by current conservation but I can't prove it, is my intuition wrong?"}
{"id":"130286","title":"Classical Viewpoint on Electromagnetism","text":"**Note:** This question may be difficult or impossible to answer within the rules of these forums due to its philosophical nature. I will delete the question if I am violating the rules. Onto the question! Recently I have been curious whether classical electromagnetism is fully solved (up to the divergences). Specifically, can we completely mathematically describe and then interpret the classical world of charges and currents and them the associated fields. Let us assume that our world is classical and that electromagnetism is a complete theory even though there are certain inconsistencies (self-interactions, infinite energy of point charges, etc). A common description of $\\textbf{E}(\\textbf{r}, t)$ and $\\textbf{B}(\\textbf{r}, t)$ among physics textbooks is that changing electric fields induce magnetic fields and vice-versa. This is assuming that there are no external fields or sources of field present. In symbols, $$ \\partial_t\\textbf{E} \\neq \\textbf{0} \\implies \\exists \\textbf{B} \\quad (1) $$ $$ \\partial_t\\textbf{B} \\neq \\textbf{0} \\implies \\exists \\textbf{E} \\quad (2) $$ Many physicists come to this conclusion from Maxwell's equations. Specifically, they argue that Faraday's law, $$ \\nabla \\times \\textbf{E}(\\textbf{r}, t) = -\\partial_t\\textbf{B}(\\textbf{r},t), $$ implies (1), and that Ampere's law (with Maxwell's correction term and no currents), $$ \\nabla \\times \\textbf{B}(\\textbf{r}, t) = \\partial_t \\textbf{E}(\\textbf{r},t), $$ implies (2). Note that we are using natural units with $c = 1$. However, these equations do not have any obvious causal connection. While we may like to pretend that _right_ implies _left_ , this is purely notational convention. Who is to say from these equations alone that one field having a nonzero curl doesn't produce a changing dual field? One attempt at reconciling this problem seems to be in Jefimenko's equations. I will state the equations without derivation, but the fields can be solved for completely in terms of the source charges and currents (I'm lazy and the following equations are in mks units from Wikipedia): $$ \\textbf{E}(\\textbf{r}, t) = \\frac{1}{4 \\pi \\epsilon_0}\\int [\\frac{\\rho(\\textbf{r}', t_r)}{|\\textbf{r} - \\textbf{r}'|^3} + \\frac{1}{c}\\frac{\\partial_t \\rho(\\textbf{r}', t_r)}{|\\textbf{r} - \\textbf{r}'|^2}] (\\textbf{r} - \\textbf{r}') - \\frac{1}{c^2}\\frac{\\partial_t \\textbf{J}(\\textbf{r}', t_r)}{|\\textbf{r} - \\textbf{r}'|^2} d^3\\textbf{r}', $$ $$ \\textbf{B}(\\textbf{r}, t) = \\frac{\\mu_0}{4 \\pi}\\int [\\frac{\\textbf{J}(\\textbf{r}', t_r)}{|\\textbf{r} - \\textbf{r}'|^3} + \\frac{1}{c}\\frac{\\partial_t \\textbf{J}(\\textbf{r}', t_r)}{|\\textbf{r} - \\textbf{r}'|^2}] \\times (\\textbf{r} - \\textbf{r}' )d^3\\textbf{r}' , $$ where $t_r = t - |\\textbf{r} - \\textbf{r}'|\/c$ is the retarded time. These equations seem to imply that neither of the fields \"causes\" the other. Instead, Jefimenko's equations imply that only the source charges and currents generate the fields (without the presence of external charges, currents, or fields). My question is related to this approach. Is it valid? What are the arguments for and against? Is the matter settled in the classical context of electromagnetism, or are there certain subtleties I've skipped over? As an extra question, is it instead better to consider $F_{\\mu \\nu}$, and treat it as one object arising solely from $J^\\mu = (\\rho, \\textbf{J})$, instead of looking at the individual fields? Thanks in advance for any and all answers!"}
{"id":"78873","title":"Does the term \"dark matter\" apply to nonluminescent bodies which still interact electromagnetically?","text":"On the new Astronomy.SE site, I was having a short discussion on one of my answers. The basic discrepancy was; can MACHOs like black holes\/brown dwarfs\/neutron stars be termed \"dark matter\"? My reasoning is that these objects do not radiate EM radiation on their own but they do gravitate, and thus constitute a small part of the total dark matter in the universe. I agree that there is a lot of dark matter which doesn't In other words, can the term \"dark matter\" be applied to nonradiating (or faintly radiating) bodies which still participate in the electromagnetic interaction (baryonic or otherwise)? Or is it necessary for all dark matter to not interact electromagnetically?"}
{"id":"9896","title":"Paper airplane physics","text":"I am working on a game involving flying and steering a paper airplane for WP7. I want the plane to fly just like how normal paper airplanes fly (see this game for an example http:\/\/armorgames.com\/play\/7598\/flight) but I can't seem to find an equation for how paper airplanes fly. Anyone have any experience with this? In my game now, it just follows the usual motion for an object in a vacuum, which makes for some flight, but it doesn't feel perfect, and traveling at a slight downward angle makes you lose speed, which isn't right. Thoughts?"}
{"id":"55961","title":"Quantum computers: are they possible or impossible?","text":"I know quantum computers are very complicated and my question is is there any way in \"Principle\" to create one? Are there already quantum computers being created?"}
{"id":"22596","title":"How do magnets work?","text":"I've read a classbook on the field theory (including EM): it perfectly describes quantitive patterns in EM-theory, but I have no luck understanding how and why it works. I mean, magnetic substances are described mainly by magnetic moments of electrons, but all explanations of the phenomenon I've found are rather of deep high-level focus on fields exclusively, than on explanation why it works and what underlying mechanisms bring all those ideas to life (including explanation of what field is, except that it is an abstraction). So, the question is: may anyone try to give (or point to) a popular and thorough explanation of magnets and on low-level mechanisms, which unifies and explains how this long-distance interaction really works (i.e., not only modelled and described mathematically)? p.s.: also, I'd like to see some papers on computing magnetic properties of bodies (iron ball, for example, finite plane, NeoCube's ball chains, etc). Thanks in advance!"}
{"id":"22594","title":"A question about definition of Fermi energy","text":"Wikipedia states the definition of Fermi energy as for \"a system of non- interacting fermions\". If we have to assume free electrons in a solid behave this way before we are able to calculate Fermi energy, how can Pauli exclusion be justified (because electrons are non-interacting)? Can Fermi energy be similarly defined for electrons confined to a single atom?"}
{"id":"100988","title":"interior surface of furnaces","text":"For industrial furnaces, in order to improve radiation heat transfer and save energy, some people say applying a high emissivity coating on to the interior surface of a furnace will do, while some people say applying a high reflectivity coating will do. we all know that at given wavelength and for an opaque object surface, emissivity + reflectivity =1, anyone can explain to me please? With best regards Sean"}
{"id":"57523","title":"Do you use the magnitude equation to get speed from an accelerometer?","text":"A guy suggested to me that getting speed from an accelerometer required the use of this equation: $\\text{speed} = \\sqrt{x^2 + y^2 + z^2}$ This does not make any sense to me, all that you would get from this equation would be the magnitude in $m\/s^2$ of the acceleration in the $x, y$ and $z$ axes. Am I correct? Or is my reasoning flawed?"}
{"id":"104604","title":"What is the meaning of inflection points in the dispersion relation inside the first Brillouin zone?","text":"I have a question regarding the $E$ vs $k$ curve in the first Brillouin zone. Why does the curve have an inflection point at some value of $k$ in the curve? How does it physically support it? ![E vs k](http:\/\/i.stack.imgur.com\/Y4DGl.jpg)"}
{"id":"57527","title":"How do you explain Kepler's third law in general terms without complex math?","text":"I understand the first law-elliptical orbits, and the second-equal area in same time, but I need help with the third one. Note that I am not in an AP course or taking calculus at the moment so simple quadratics\/cubics\/CP level explanations would suffice."}
{"id":"104600","title":"What's the meaning of the age of the universe?","text":"I'm not asking about how we worked backward from an expanding universe to the age of the big bang, but rather what is the meaning of time in a near infinitely dense point in the context of general relativity? Wouldn't time flow infinitely slowly for a theoretical (though physically impossible) observer?"}
{"id":"107564","title":"Are retrograde capture orbits \"easier\" than prograde capture orbits?","text":"After reading up on irregular moons in the solar system - moons that are thought to be captured, most seem to be in retrograde orbit around their parent body. That led me to wonder if retrograde orbits are easier to capture objects than prograde orbits - say prograde orbits are more likely to gravitationally slingshot the object away from the parent body before capture, whereas retrograde orbits would be more likely to capture before flinging the object away. When viewing the capture from the perspective of the parent body, an object that is moving retrograde past the body appears to slow down as it interacts with the gravity well of the body, whereas an object moving prograde past the body appears to accelerate in the same frame of reference. Is there any validity to this, or is that just a flaw in reasoning?"}
{"id":"26172","title":"How much detail can telescopes actually provide?","text":"For example, could the numbers \/ letters on a postage stamp in a randomly specified location be clearly visible from space. This is to settle a discussion with a friend that piqued my curiosity."}
{"id":"26175","title":"What determines a progenitor's fate as a spiral or elliptical?","text":"I was thinking about my answer to Are the inner planets on planar orbits because there was more dust in the inner solar system (early on in planetary accretion)? \\- when it occurred to me that maybe I was reversing cause and effect. Specifically, that perhaps spiral galaxies simply arise from high- angular-momentum progenitors, while elliptical arise from low angular-momentum progenitors. In this scenario, the spiral wave-pattern would merely be how that excess angular momentum organizes itself, and the matter can't help but stir itself. For a warm-up question and consistency checker, do spiral galaxies in fact have substantially greater total angular momentum than elliptical?"}
{"id":"79874","title":"Curve Fitting and Multiple Experiments","text":"Say I do an an experiment 5 times, each of which gives you a list of data points. Do I fit a curve to each one separately and then average the parameters and their uncertainties? Or do I take the average of all the experiments and then do fit a single curve to that?"}
{"id":"79875","title":"Light Apparent Brightness","text":"Let's say I have a triangular light source. From that light source I want to calculate a pyramidal frustum (tetrahedron with no apex). How would I calculate the maximum bottom area where light would be visible to the naked eye? And how would I calculate the apparent brightness?"}
{"id":"79870","title":"How deep is the region near an event horizon where Hawking radiation is generated?","text":"In other words, how strong does gravity have to be to cause Hawking radiation to occur?"}
{"id":"30538","title":"Why does salty water heat up quicker than pure water?","text":"> **Possible Duplicate:**   >  Why does adding solutes to pure water lower the the specific heat? Why do higher concentration salt solutions heat up more quickly?"}
{"id":"70047","title":"Can the Hubble constant be measured locally?","text":"The Hubble constant, which roughly gauges the extent to which space is being stretched, can be determined from astronomical measurements of galactic velocities (via redshifts) and positions (via standard candles) relative to us. Recently a value of 67.80 ± 0.77 (km\/s)\/Mpc was published. On the scale of 1 A.U. the value is small, but not infinitesimal by any means (I did the calculation a few months ago, and I think it came out to about 10 meters \/ year \/ A.U.). So, can you conceive of a measurement of the Hubble constant that does not rely on any extra-galactic observations? I ask because, whatever the nature of the expansion described by the Hubble constant, it seems to be completely absent from sub-galactic scales. It is as though the energy of gravitational binding (planets), or for that matter electromagnetic binding (atoms) makes matter completely immune from the expansion of space. The basis for this claim is that if space were also pulling atoms apart, I would naively assume we should be able to measure this effect through modern spectroscopy. Given that we are told the majority of the universe is dark energy, responsible for accelerating the expansion, I wonder, how does this expansion manifest itself locally? Any thoughts would be appreciated."}
{"id":"121512","title":"Terminologies for moment of inertia","text":"Perhaps someone can suggest the right terms for the following mathematical objects related to moment of inertia?   1. A _inertia tensor_ $I$. $$I \\equiv \\begin{bmatrix} I_{1,1} & I_{1,2} & I_{1,3} \\\\\\ I_{2,1} & I_{2,2} & I_{2,3} \\\\\\ I_{3,1} & I_{3,2} & I_{3,3} \\\\\\\\\\end{bmatrix}$$   2. A _product of inertia_ is an off-diagonal entry in the tensor: $I_{1,2} = I_{2,1}$, $I_{1,3} = I_{3,1}$, or $I_{2,3} = I_{3,2}$.   3. A _principal moment of inertia_ is a diagonal entry in the tensor: $I_{1,1}$, $I_{2,2}$, or $I_{3,3}$. This is the semantic of moment of inertia discussed in elementary treatment of Physics.   4. What is the term for $I_2$ and $I_3$ in the last line below? $$\\begin{align*} I_{1,1} &= \\sum_{j} m_j\\;\\left(r^2_{j,2} + r^2_{j,3}\\right) \\\\\\ &= \\sum_{j} m_j\\,r^2_{j,2} + \\sum_{j} m_j\\,r^2_{j,3} \\\\\\ I_{1,1} &= I_2 + I_3 \\end{align*}$$"}
{"id":"113209","title":"Speed of light as a universal speed limit","text":"It follows from special relativity that nothing can travel faster than light. Einstein believed this would have to hold so generally that he assumed the Einstein-Podolsky-Rosen paradox to indicate a contradiction in quantum physics. Nowadays we know we have to be a bit more specific, and maybe it would be safe to say that nothing that can carry information can travel faster than the speed of light. If information can be transferred faster than light, it would be possible to change the past. If I understand correctly it is the contradictions arising from the hypothetical possibility to influence the past that lead to the assertion that nothing can travel faster than light. My questions:   1. Is this indeed the reason why it is said to follow from special relativity that nothing can travel faster than the speed of light?   2. Aren't the contradictions associated to influencing the past more of a philosophical nature than of a purely logical (or at least physical) nature? **EDIT** I realize now that instantaneous transfer (action at a distance) is actually physically in contradiction with special relativity: what is instantaneous\/simultaneous in one inertial frame is not in the other, so the laws of physics would not be the same. I suspect that for finite velocities faster than that of light there must be a similar argument, I would appreciate if anyone could elaborate on that."}
{"id":"112153","title":"Can you clean the air in a small room using a vacuum cleaner?","text":"You can use a vacuum cleaner to vacuum the floor but if you tried to vacuum the the air in a small room would this work or will the laws of physics stop this from working in any way? In what way would attempting to vacuum clean the air be different from using an air scrubber(in terms of physics). Thanks"}
{"id":"112155","title":"Using Ampere's Law for a Solenoid","text":"![solenoid](http:\/\/hyperphysics.phy- astr.gsu.edu\/hbase\/magnetic\/imgmag\/sol.gif) To calculate the magnetic field, a rectangle amperian loop was drawn, and since the sides of the rectangle are perpendicular to the magnetic field, and the top is too far away to have any field lines cross it, only BL makes a contribution. ![equation](http:\/\/hyperphysics.phy- astr.gsu.edu\/hbase\/magnetic\/imgmag\/sol2.gif) I don't understand why this magnetic field is only contained within the solenoid. Shouldn't the magnetic field at any point on the amperian loop equal μnI, according to Ampere's law?"}
{"id":"112157","title":"Is this webcomic accurate?","text":"I was considering this xkcd comic from 5\/10\/14, with the alt-text \"Trains rotate the Earth around various axes while elevators shift its position in space.\" I'm wondering about its accuracy. ![http:\/\/xkcd.com\/1366\/](http:\/\/i.stack.imgur.com\/EvUsH.png) I first considered the elevator scenario. As the elevator accelerates (to start or stop its motion), the occupants feel a fictitious force, demonstrating conclusively that it is the elevator and not the planet which is in motion. By this argument, an elevator is fundamentally different from a device which physically displaces the planet. Similarly, I suspect that a train doesn't \"rotate the Earth,\" since such a rotation would be felt by occupants on Earth as an angular or centripetal acceleration. Any observer can conclude definitively that it is the train which is in motion around the Earth, not the Earth rotating under the train. For this reason, is the comic is not strictly correct? (Note that there is a minimal motion of the Earth resulting from the action- reaction force pair between the train and the planet, but it's safe to say that this motion is negligible relative to the distance traveled by the train, since $I_{earth} \\gg I_{train}$.)"}
{"id":"38068","title":"Is it possible to hear the past?","text":"From this Stack Exchange Physics Post, I am certain that it is possible to view the past. But then this interesting question came to me. Is it possible to hear the past? Ok, you might say, \"Well, we are hearing the past, aren't we?\" and go onto immense details about the speed of sound against the difference and stuff. But what if I wanted to hear the sound of a place extremely close by days, weeks, months or even years back? Would that be possible?"}
{"id":"34917","title":"Origin of Electromagnetic Interactions between Molecules","text":"What is the origin of electromagnetic interaction between molecules? Anyway, it should have some relation with atoms. Also, These electromagnetic interactions are playing a major role in different properties of matter including the transition between solid-liquid-gas. Hence, what would be the **source** of these interactions..? If these interactions originate from atoms, then some other questions come into **focus** :   * Are they related to the transition of electrons between various energy levels and emission of photons from an atom..?   * Are they related to cohesive forces between molecules in solids & liquids?"}
{"id":"83257","title":"Do really, two oppositely charged bodies (of equal charge in magnitude) attract each other?","text":"It is well known fundamental behaviour that, _oppositely charged bodies attract each other_ (I don't know whether it applies also for charges of equal magnitude or not), and identical charges repel each other. It is also well known that, a system of oppositely charged bodies with equal charge in magnitude, has zero net charge. If a system has oppositely charged bodies, with equal charge in magnitude, it would imply that, there will be zero net charge, and the vector field's had got cancelled each other, like around a current carrying conductor, there is no electric field because, charge on current carrying conductor is zero, as one electron enters the conductor, the other will be leaving the conductor. Now, if there is no field, how could the oppositely charged bodies (of equal charge in magnitude) attract each other, they should not feel any force, isn't? Taking into account all the above statements, would it imply that, two oppositely charged bodies (of equal charge in magnitude) attract each other? _[All statements made, are up to my view. Any correction advisory is welcome]_ _LINKS_   * What are the fields produced around a current carrying conductor? (Provides an idea of whether, there exists a electric field around a neutral body or not, like the current carrying conductor)"}
{"id":"133160","title":"equivalent spring constant for a drum floating in water","text":"If I have a drum floating in water, how do I find the equivalent spring constant? I know that the water has density $\\rho$, the drum has diameter $d$ and height $\\ell$, and the positive direction is in the downward direction. Therefore, the buoyancy pushing up on the barrel is my spring. I have been reading the section of this topic in Mechanical Vibrations by Rao but there isn't any information that helps me figure this out."}
{"id":"133163","title":"What measurement of time is so small that it qualifies as quantum?","text":"> Quantum mechanics (QM; also known as quantum physics, or quantum theory) is > a branch of physics which deals with physical phenomena at nanoscopic scales > where the action is on the order of the Planck constant. -wiki Time is much less finite than matter. Matter can be split to the point where it is so small that it gains the classification of nanoscopic and subatomic. However the Planck constant is a measurement of mass and time so I find it difficult to separate the measurements. What measurement of time is so small that it qualifies as quantum? Could someone clear up the misconception?"}
{"id":"129218","title":"How should semiconductors be understood?","text":"In the chemistry setting, a doped semiconductor is a crystal lattice with holes\/extra electrons in it. In the band\/quantum mechanical picture, these holes\/extra electrons can be seen as the amount of missing\/excess charges from a full band of energy states. Therefore, in a pure silicon crystal, with full covalent bonds and thus no holes or extra electrons, the valence band is completely full, and the conducting band is completely empty. Which means that Silicon is an insulator, since the Fermi level lies above the valence and below the conducting band. What are the holes in my reasoning?"}
{"id":"107646","title":"Mathematical Physics SUSY QM Resource Recommendation","text":"I want to study SUSY QM. I found some excellent physically motivated articles on Arxiv. Despite, I am especially interested in the mathematical structure behind SUSY QM. Does anybody know whether there is a particularly good reference to start studying SUSY QM from a rigorous mathematical point of view. The reference should cover Spectral analysis and a wide range of all the algebraic properties concerned with the topic."}
{"id":"107648","title":"What are the general solutions to a hard sphere collision?","text":"Surely someone has found the solutions to the hard sphere collisions (in $n$ dimensions) of two bodies of mass $m_1$ and $m_2$, respectively--that is the resultant velocities (or momenta) of the two bodies after a hard sphere collision given $\\vec{p}_1$ and $\\vec{p}_2$. I have a solution (both for elastic and inelastic collisions), but I wonder whether or not it's correct. FYI, here are my solutions for elastic, hard-sphere collisions: $$ \\vec{p}_{1f} = \\vec{p}_1 + 2\\frac{\\left(m_1\\vec{p}_2 - m_2\\vec{p}_1\\right)\\circ \\hat{r}}{m_1 + m_2}\\hat{r} \\\\\\ \\vec{p}_{2f} = \\vec{p}_2 - 2\\frac{\\left(m_1\\vec{p}_2 - m_2\\vec{p}_1\\right)\\circ \\hat{r}}{m_1 + m_2}\\hat{r} $$ wher $\\hat{r}$ is in the direction of the \"hard sphere\" (i.e. the unit vector connecting the two radii of the spheres). You can see that my formulation trivially conserved momentum--the question is whether or not it conserves energy as well."}
{"id":"78049","title":"Symmetry breaking in Bose-Hubbard model","text":"According to Landau's symmetry breaking theory, there is a symmetry breaking when phase transition occurs.   1. What is the symmetry breaking of superfluid-Mott insulator transition in Bose-Hubbard model?    2. Why metallic state to Mott insulator state transition in Fermi-Hubbard model is not a phase transition, but a crossover."}
{"id":"127912","title":"Why do physicists use LHC?","text":"1. My question is why are we colliding particles in LHC to produce new ones?    2. And these particles that they sometimes say live for a fraction of a second, how in space they exists then?    3. In space all these particle exist without smashing to each other, why do we need to smash them to produce them?"}
{"id":"127154","title":"If batteries are a source of energy, would not lower-valued resistors cause a violation of the conservation of energy?","text":"Here's the conundrum I have been facing. Yesterday I asked the question: how can batteries constantly motivate electrons to complete a circuit? (What maintains constant voltage in a battery?) After some thought I realized that the concept is that of an energy source. The batteries are a source of energy, and regardless of the mechanism, they will constantly 'push' for a goal. My solution finally came when I realized that speaking about someone's 'motivation' was identical to the constant push of the battery. Here's the issue: My sister wanted to get me coffee yesterday for my birthday. She was attempting to give it to me before my class started. Unfortunately, she had difficulty with traffic, and so was able to deliver only after my class started. I realized when she told me that she tried as hard as she could, and that what really mattered was her motivation. The push she gave was constant regardless of the resistance she encountered. The parallel here to the constant push of a battery was strong; voltage didn't matter in respect to resistance for a given battery. Now I have a different issue. The voltage drop is described as the per-capita potential energy difference of electric charges at points a and b right? The total energy would then be the amount of 'capitas' multiplied by this per capita potential energy difference. More capitas mean more total energy. Here's the problem. If the motivation of a battery is described by voltage, then if we look at a circuit with a resistor and compare it to another circuit with an equal-length resistor but with a wider surface area, the potential difference is supposed to stay the same in both cases. However, in taking the charge from one surface to the other, more energy is required for the resistor with larger surface area! The amount of energy supplied is different in either case, yet from the same 'constant' energy source. This seems so wrong to me right now. Yet I know that Power=Voltage*(Current)^2, so this actually makes sense in terms of the equation. I wonder what is lacking in my fundamental understanding of the potential difference in comparison to the idea of motivation\/constant energy source. I'm really fundamentally misunderstanding something complex here that I don't quite have my finger put on yet. Please help clear up my view of a battery as a constant pusher to line up with this changing energy ouput a battery can give."}
{"id":"86993","title":"How is it possible to move something without completely lifting it?","text":"For example, let's assume a chair here: It can be \"slid\" across the force if we use minimal upwards force, but not enough to actually \"lift\" the chair. Why should it move? Here's a better example: I have seen some strong people lift and slide the back of cars, however, the back tires of the car never elevate completely off the ground. With regards to friction, though, how is it possible to actually \"move\" something without lifting it completely? How does this work is a better question. This image should illustrate the car example: ![enter image description here](http:\/\/i.stack.imgur.com\/foRJg.jpg) Is there enough space within it to allow the angular force to make it \"slid\" a pinch, or what else would explain its ability to move without being completely suspended in the air that would, say, differ from this with a human?"}
{"id":"21121","title":"Are there known turbulent nonlinear equations where the cascade is a thermal gradient?","text":"In a recent answer (here: The equipartition theorem in momentum space ), I suggested that if you have an appropriate first order equation (in the answer I used a second order equation, but it is more likely to be true for a first order equation) of the form: $$ i\\partial_t \\phi = (\\nabla^2)^N \\phi + \\lambda \\phi^3 + f + D $$ (EDIT: I first had $|\\phi|^2\\phi$, which is the nonlinear Schrodinger equation, but this has the same frequency shifting property as the relativistic version--- the frequencies are not completely perturbatively decoupled even with a resonance condition. The only reason I went to first order was to fix this annoying problem, and there is no condition of being physical. Although this will not conserve the total $|\\phi|^2$, it has a conserved energy.) Where f is a driving force with only long-wavlength components, and D is a damping force which only affects short wavelength components. The field $\\phi$ is a complex scalar. Then if N is large and $\\lambda$ is sufficiently small (by scaling, $\\phi$'s magnitude can substitute for $\\lambda$, so the limit of small $\\lambda$ is really the limit of small $|\\phi|$), the cascade due to the nonlinear term is a thermal gradient draining energy from the hot low frequency modes to the cold high-frequency modes. I don't know if this model is in the literature, of if the thermal analysis is accurate (I am pretty sure it is correct, because the resonance matching does lead to local flow of energy in |k|, but I am not sure if there isn't a block to thermalization due to some problem with mixing in each |k| modes separately). In the original answer, I used a second order equation, which has frequencies of both signs. This allows some extra mixing between distant modes, which shifts the frequency of low modes based on the occupation of high modes. While I don't think this changes the cascade significantly, I wanted a clean example, so I used a first order equation here. (EDIT: The clean example does not use the absolute value, so that the frequencies have to strictly additively combine.) Is this model analyzed in the literature? Does it thermalize into a thermal gradient cascade?"}
{"id":"101396","title":"One more relation with spherical spinors","text":"Let's have the spherical spinors: $$ \\mathbf {Y}_{j, m, l = j \\pm \\frac{1}{2}} = \\frac{1}{\\sqrt{2l + 1}}\\begin{pmatrix} \\pm \\sqrt{l \\pm m +\\frac{1}{2}}Y_{l, m - \\frac{1}{2}} \\\\\\ \\sqrt{l \\mp m +\\frac{1}{2}}Y_{l, m + \\frac{1}{2}} \\end{pmatrix}. $$ How to prove the relation $$ (\\hat {\\sigma} \\cdot \\mathbf r )\\mathbf {Y}_{j, m, l = j \\pm \\frac{1}{2}} = -r\\mathbf {Y}_{j, m, l = j \\mp \\frac{1}{2}}? $$ I tried to rewrite the $(\\hat {\\sigma} \\cdot \\mathbf r )$ as $$ (\\hat {\\sigma} \\cdot \\mathbf r ) = r\\begin{pmatrix} \\cos(\\theta ) & \\sin(\\theta )\\,e^{-i\\varphi } \\\\\\ \\sin(\\theta )\\, e^{i \\varphi } & -\\cos(\\theta ) \\end{pmatrix}, $$ but it's hard to use recurrence relations for associated Legendre functions after that. Is there some hint which simplifies the proof? Maybe, I can first show that $(\\hat {\\sigma} \\cdot \\mathbf r )\\mathbf {Y}_{j, m, l = j \\pm \\frac{1}{2}} = -ra\\mathbf {Y}_{j, m, l = j \\mp \\frac{1}{2}}$, because the parity of the left side is opposite to the parity of $\\mathbf {Y}_{j, m, l = j \\pm \\frac{1}{2}}$, and $(\\hat {\\sigma} \\cdot \\mathbf r )$ commutes with rotation operator, because refers to the scalar pdoduct?"}
{"id":"101390","title":"The Holstein-Primakoff Representation (approximation)","text":"I have a question regarding the Holstein-Primakoff representation. In the HP-representation we define the spin operators in terms of bosonic creation and annihilation operators. $$ S_j^+ = \\sqrt{2S - n_j} a_j \\\\\\ S_j^- = a^\\dagger_j\\sqrt{2S - n_j} \\\\\\ S^z_j = S - n_j $$ Where $a_j$ and $n_j$ are operators. When we derive the magnon dispersion relation, we make the assumption that $$ \\frac{\\langle n_j \\rangle}{\\langle S^z \\rangle}\\ll 1 $$ which is fine. As far as I understand this only means that we assume that most of the spins are pointing along the z-direction. However, when we go further in the derivation, we do a series expansion in $n_j\/S$, so that i.e. $$ S_j^+ = \\sqrt{2S}\\sqrt{1- \\frac{n_j}{2S}+...} \\approx \\sqrt{2S}\\sqrt{1- \\frac{n_j}{2S}} $$ This is where I don't understand. Say that we are in a spin $1\/2$-system. In that case we can for one site have at most 1 magnon excitation and $S=\\pm 1\/2$, so for each individual site $n_j\/2S$ is not much smaller than one. My question is then, how can we justify that the series expansion makes sense in the operators at each individual site? Will the contributions from each site when summed up not contribute because $$ \\frac{\\langle n_j \\rangle}{\\langle S^z \\rangle}\\ll 1 $$ or have I misunderstood something? Any thoughts would be much appreciated."}
{"id":"47660","title":"Research on galaxy \"stiffness\"?","text":"> **Possible Duplicate:**   >  Are galactic stars spiraling inwards? Has there been any research on the effects of interstellar gravitational attraction that would make collective star motion in a galaxy behave more like a solid disk? I understand rotational velocity has been observed to be higher than expected for stars farther from galaxy centers. This doesn't agree with Newtonian physics in which Pluto goes around the Sun more slowly than Mercury. An explanation for the faster star motion observed in galaxies was created and called dark matter. This seems to not comply with Occam's Razor, where the most simple possibility is probably right. I visualize the attraction between stars as stiffening the collection, or increasing the viscosity of the rotational behavior which could explain faster rotation towards the outside of a galaxy. The rotation would behave somewhere between atoms locked together in a frisbee, and a fluid circling the drain. Has any research been done to prove this wrong?"}
{"id":"135130","title":"How would water drain out of a sealed pipe?","text":"While hiking through the Grand Canyon, I started wondering. Say we have a pipe for the purpose of transporting water across a canyon, with the bottom submerged in a pool of water. Like the blue line in this terribly drawn image: ![enter image description here](http:\/\/i.stack.imgur.com\/7iVLE.png) Now let's seal the two ends of the pipe, which is full of water. What would happen if the pipe broke at the bottom of the U? Note: It shouldn't matter whether the pipe is a U or a vertical line, I simply thought about this with a U-shaped pipe. * * * If the ends of the pipe weren't sealed, the water inside the pipe would leak out until the height of the water outside the pipe was the same as the height of the water inside the pipe. But the ends are sealed. A vacuum can support 10m of water, so perhaps the water will drain out of the pipe until it is 10m above the height of the water outside of the pipe, leaving a big near-vacuum behind. This would explain why the Grand Canyon's pipe has such a large number of valves. On the other hand, we learn from http:\/\/what-if.xkcd.com\/6\/ that water boils into a vacuum. Does this leave a significant amount of water vapour behind?"}
{"id":"10137","title":"Uniqueness of quasiclassical consistent histories","text":"The current zeitgeist here is on interpretations of quantum mechanics, so let me add my own two cents here. As you may know, consistent histories is an alternative interpretation proposed in a series of papers 1990-1994 by Gell- Mann and Hartle. In a shocking article, Dowker and Kent proved there are many distinct consistent histories for most quantum systems. Even if we constrain ourselves to quasiclassical histories, i.e. those histories which are close to classical, being quasiclassical now does not guarantee being quasiclassical in the future, or in the past. This is disturbing. Do histories which are quasiclassical for all times exist? If they do, are they unique? Is the nightmare scenario of two everywhere quasiclassical consistent histories which are mutually incompatible possible?"}
{"id":"70255","title":"Weyl Large Number Coincidence","text":"I've just learned of the Weyl Large Number Coincidence on the Wikipedia. It looks interesting. Here is my interpretation of it. What is the smallest \"quantum\" of energy in the Universe? $$E_{min} = \\hbar H$$ where $H$ is the Hubble parameter. This make sense as the Hubble parameter is roughly the inverse of the age of the Universe ($H=2\\times10^{-18}s^{-1}$). You can't define a non-zero frequency smaller than that. Now consider the gravitational self-energy of a particle, $E_g$, given by $$E_g = \\frac{Gm^2}{r}$$ where $m$, $r$ is the mass,size of the particle. The smallest possible particle must have a gravitational self-energy of the size of $E_{min}$ therefore we have $$\\frac{Gm^2}{r} = \\hbar H$$ We can link the mass $m$ and size $r$ of a particle by using the Compton wavelength quantum relation $$m c r = \\hbar$$ Putting the two equations together and eliminating $r$ we find $$m^3 = \\frac{\\hbar^2H}{Gc}$$ If we plug values into this expression we get a mass given by $$m \\approx 10^{-28}kg$$ This is nearish to the proton mass. Interesting or just a coincidence?"}
{"id":"10138","title":"D-branes wrapping divisors and\/or cycles","text":"What is the difference between a divisor and a homology cycle? What is the difference between a D-brane wrapped around a divisor and a D-brane wrapped around a cycle? Thanks."}
{"id":"121406","title":"How is it possible for objects to travel faster than the speed of sound when particles interact at the speed of sound","text":"First of all, I am sorry if this is a stupid question but: I've heard that atoms interact with each other at the speed of sound (when you for instance push a chair, the atoms collide with each other in a chain reaction at the speed of sound, making the chair move). How can then airplanes fly faster than the speed of sound without something going crazy?"}
{"id":"121400","title":"Ballistic Pendulum Demo Problem","text":"I have a question about the following problem: > ![enter image description here](http:\/\/i.stack.imgur.com\/nOtw4.png) I got the solution $v=\\frac{M+m}{m} \\sqrt{2gh}$. But my real question is in the following picture: > ![enter image description here](http:\/\/i.stack.imgur.com\/gVtk4.png) In the above slide, how can you derive the solution $v=(\\frac{M+m}{m})d\\sqrt{\\frac{g}{L}}$ and thus, $v=3d$? Also, it seems that $v=3d$ is not correct since their unit is not matching... Can somebody explain this???"}
{"id":"40996","title":"Schrödinger's equation, time reversal, negative energy and antimatter","text":"You know how there are no antiparticles for the Schrödinger equation, I've been pushing around the equation and have found a solution that seems to indicate there are - I've probably missed something obvious, so please read on and tell me the error of my ways... Schrödinger's equation from Princeton Guide to Advanced Physics p200, write $\\hbar$ = 1, then for free particle $$i \\psi \\frac{\\partial T}{\\partial t} = \\frac{1}{2m}\\frac{\\partial ^2\\psi }{\\partial x^2}T$$ rearrange $$i \\frac{1}{T} \\frac{\\partial T}{\\partial t} = \\frac{i^2}{2m}\\frac{1}{\\psi }\\frac{\\partial ^2\\psi }{\\partial x^2}$$ this is true iff both sides equal $\\alpha$ it can be shown there is a general solution (1) $$\\psi (x,t) \\text{:=} \\psi (x) e^{-i E t}$$ But if I break time into two sets, past -t and future +t and allow energy to have only negative values for -t, and positive values for +t, then the above general solution can be written as (2) $$\\psi (x,t) \\text{:=} \\psi (x) e^{-i (-E) (-t)}$$ and it can be seen that (2) is the same as (1), diagrammatically ![energy time diagram](http:\/\/i.stack.imgur.com\/0JBc2.jpg) And now if I describe the time as monotonically decreasing for t < 0, it appears as if matter(read antimatter) is moving backwards in time. Its as if matter and antimatter are created at time zero (read the rest frame) which matches an interpretation of the Dirac equation. This violates Hamilton's principle that energy can never be negative, however, I think I can get round that by suggesting we never see the negative states, only the consequences of antimatter scattering light which moves forward in time to our frame of reference. In other words the information from the four-vector of the antiparticle is rotated to our frame of reference. Now I've never seen this before, so I'm guessing I've missed something obvious - many apologies in advance, I'm not trying to prove something just confused."}
{"id":"40993","title":"Distribution of charge on a hollow metal sphere","text":"A hollow metal sphere is electrically neutral (no excess charge). A small amount of negative charge is suddenly placed at one point P on this metal sphere. If we check on this excess negative charge a few seconds later we will find one of the following possibilities: (a) All of the excess charge remains right around P. (b) The excess charge has distributed itself evenly over the outside surface of the sphere. (c) The excess charge is evenly distributed over the inside and outside surface. (d) Most of the charge is still at point P, but some will have spread over the sphere. (e) There will be no excess charge left. Which one is correct and why? I guess it is some kind of electrostatic induction - phenomena going on. Am I right? I understand that excess charge is distributed over hollow sphere and that negative and positive charges are distributed opposite sides, but don't know which one positive or negative go to inside surface."}
{"id":"123805","title":"On the shape of magnetic and electric fields in an electromagnetic wave","text":"Electromagnetic waves are generally depicted like this: ![enter image description here](http:\/\/i.stack.imgur.com\/Gn4wv.png) Where the electric fields and magnetic fields exist in the planes perpendicular to the direction of propagation. I also realize that as the electric field changes while the wave is propagating, a magnetic field is induced and vice versa (by faraday's and maxwell's laws of induction). But, those laws predict that the fields will be circular. So, won't the electric and magnetic fields look different? Won't they be circles along arrows that are drawn in the figure? I haven't seen anything written about this anywhere."}
{"id":"1673","title":"Coulomb force in SI and cgs","text":"Coulomb force in SI is $ F = \\frac{Q1*Q2}{4\\pi\\varepsilon R^{2}} $ while in CGS $ F = \\frac{Q1*Q2}{R^{2}} $ why is it? I mean doesn't it any make difference in dimension? since $ \\varepsilon $ itself has dimention. And similarly for magnetic force also."}
{"id":"74254","title":"Why is there a factor of $4\\pi$ in certain force equations?","text":"I mean to ask why there is $4\\pi$ present in force equations governing electricity? Though all objects in universe are not spherical and circular, the constant of proportionality in both equations contain $4\\pi$. Why?"}
{"id":"108626","title":"Uncertainty of permittivity of vacuum","text":"**Question** : The value of permittivity of vacuum, $\\epsilon_0$, is given with absolutely no uncertainty in NIST Why is this the case? * * * **More details** : The permeability of vacuum can be given by $$\\mu_0=\\frac{1}{\\epsilon_0 c^2}$$ which comes from the definition of a magnetic field in special relativity, where we solve the problem of a wire with electrons flowing in, and we calculate the force exerted on an external charge moving with some velocity (for details, refer to the book Electricity and Magnetism, E. M. Purcell), and we define a new field called \"magnetic field\" with the form of Lorentz force, where the magnetic field is $$B=\\frac{I}{2\\pi \\epsilon_0 c^2 r}$$ where $I$ is the current due to the flow of electrons in the wire, and $r$ is the distance of the external charge from the wire. And there we get the definition of $\\mu_0$ that makes $B$: $$B=\\frac{\\mu_0 I}{2\\pi r}$$ and there starts the concept \"magnetism\". Why am I giving this detailed example? Because I wouldn't like to get the answer that $\\mu_0$ has no error, and that's why $\\epsilon_0$ has no error, and then we fall into circular logic. So I expect a reason which is independent of $\\mu_0$. Thank you in advance."}
{"id":"128292","title":"Why do we take the value of the constant in Coulomb's law as $\\frac{1}{4\\pi\\varepsilon_0}$?","text":"Why do we take the value of the constant in Coulomb's law as $\\frac{1}{4\\pi\\varepsilon_0}$?"}
{"id":"28673","title":"Coulomb's Law: why is $k = \\dfrac{1}{4\\pi\\epsilon_0}$","text":"This was supposed to be a long question but something went wrong and everything I typed was lost. Here goes.   1. Why is $k = \\dfrac{1}{4\\pi\\epsilon_0}$ in Coulomb's law?    2. Is this an experimental fact?    3. If not, what is the significance of this definition?"}
{"id":"133201","title":"Why do stickers curl?","text":"When peeling a sticker off its base, the immediate reaction is that it curls; why is this? I am having trouble finding an answer to this. Could it be that the glued side expands upon contact with the air? And, on a somewhat related note, is this a similar reason for why a ribbon curls when glided over with a blade?"}
{"id":"30295","title":"Topology needed for Differential Geometry","text":"I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know. I know some basic concepts reading from the Internet on topological spaces, connectedness, compactness, metric, quotient Hausdorff spaces. Do I need to go deeper? Also, could you suggest me some chapters from topology textbooks to brush up this knowledge. Could you please also suggest a good differential geometry books that covers diff. geom. needed in physics in sufficient detail, but not too mathematical? I heard some names such as Nakahara, Fecko, Spivak. How are these?"}
{"id":"11394","title":"Presence Of an Another Universe","text":"> **Possible Duplicate:**   >  Experimental evidence for parallel universes Is there an universe similar to ours somewhere else, I mean I had heard from Einsteins theories that,actually when I am typing this question here, there is an another world somewhere else where I would be typing this question simultaneously. Is this true, if so any evidences."}
{"id":"113092","title":"Why does a system try to minimize potential energy?","text":"In mechanics problems, especially one-dimensional ones, we talk about how a particle goes in a direction to minimize potential energy. This is easy to see when we use cartesian coordinates: For example, $-\\frac{dU}{dx}=F$ (or in the multidimensional case, the gradient), so the force will go in the direction of minimizing the potential energy. However, it becomes less clear in other cases. For example, I read a problem that involved a ball attached to a pivot, so it could only rotate. It was then claimed that the ball would rotate towards minimal potential energy, however $-\\frac{dU}{d\\theta} \\neq F$! I think in this case it might be equal to torque, which would make their reasoning correct, but it seems like regardless of the degrees of freedom of the problem, it is always assumed that the forces act in a way such that the potential energy is minimized. Could someone give a good explanation for why this is? Edit: I should note that I typed this in google and found this page. where it states that minimizing potential energy and increasing heat increases entropy. For one, this isn't really an explanation because it doesn't state why it increases entropy. Also, if possible, I would like an explanation that doesn't involve entropy. But if it is impossible to make a rigorous argument that doesn't involve entropy then using entropy is fine. As a side note, how does this relate to Hamilton's Principle?"}
{"id":"94845","title":"Velocity distribution in Plummer's models and others mass distributions","text":"The Plummer's sphere is an model for the mass density in a globular cluster of stars. For an $N$-body simulation I have initialized the position of $N$ masses with a Monte-Carlo technique but cannot find a way of initializing the velocity initial conditions. Is there a simple function that given a position in a Plummers sphere assigns a velocity to a given mass? Lots of sites list the velocity for a circular orbit but is this a good approximation to a globular cluster and how should it be treated off the $x$-$y$ plane?"}
{"id":"122177","title":"Why is work not related to velocity?","text":"A very simple question; why does it cost me more energy to **very slowly** lift a mass $m$ over my head compared to very fast? The definition of work does not state anything about velocity, only the distance travelled, but I definitely feel more exhausted in the first case."}
{"id":"102599","title":"Numerical problem in solving the Bogoliubov de Gennes equations- methods to solve?","text":"I am trying to solve an assignment on solving the Bogoliubov de Gennes equations self-consistently in Matlab. BdG equations in 1-Dimension are as follows:- $$\\left(\\begin{array}{cc} -\\frac{\\hbar^{2}}{2m}\\frac{\\delta^{2}}{\\delta z^{2}}-\\mu+V\\left(z\\right) & \\triangle(z)\\\\\\ \\triangle(z) & \\frac{\\hbar^{2}}{2m}\\frac{\\delta^{2}}{\\delta z^{2}}+\\mu-V(z) \\end{array}\\right)\\left(\\begin{array}{c} u_{n}(z)\\\\\\ v_{n}(z) \\end{array}\\right)= \\epsilon_{n}\\left(\\begin{array}{c} u_{n}(z)\\\\\\ v_{n}(z) \\end{array}\\right)$$ along with the equations for gap function $\\triangle(z)$ and number density $n(z)$. $$\\triangle(z)=U\\sum_{n}\\left(1-2f_{n,}\\right)u_{n}(z)v_{n}^{\\star}(z)$$ and $$n(z)=2\\sum_{n}|{u_{n}(z)}|^{2}f_{n}+|{v_{n}(z)}|^{2}\\left(1-f_{n}\\right).$$ For the case of solving the BdG equations in Fourier space in Matlab for the case of a periodic potential and periodic gap function (assumed), we can take $$u_{n}(z)=\\sum_{k}\\exp\\left[ikz\\right]U_{n,k}, $$ $$\\triangle(z)=\\sum_{K}\\exp(iKz)T_{K},$$ and $$ V(z)=\\sum_{K}\\exp(iKz)P_{K} $$ where the sum is over the reciprocal lattice vectors $K$ leading to the number equation $$N=2\\sum_{n,k}\\left[f_{n}|{U_{n,k}}|^{2}+\\left(1-f_{n}\\right)|{V_{n,k}}|^{2}\\right]$$ with $ f_{n}$ as the Fermi distribution function. Solving the set of equations self-consistently for a fixed $N$, I am trying to get a value of chemical potential from the number equation each time after solving the eigenvector components $U_{n,k}$ and $V_{n,k}$, but due to the form of the exponentials in the number equation and sum over large number of them, I am unable to get a correct value of chemical potential out of them using Matlab routines as the root of the equation to put it back into the equations for eigenvector components. In most cases, I get random values of chemical potential since the equation is more or less insoluble. How can I avoid this error ? Is there a better way to numerically solve the BdG equations self-consistently ? I also want to do this assignment in real space avoiding finite size effects but started with the Fourier space case to avoid errors associated with discretizing the differential. Please guide and ask for any details you might need. Following is my MATLAB code to solve the equations in real space but the code does not work as fsolve does not find the mu value. http:\/\/postimg.org\/image\/v7amx5vd9\/full\/ ## http:\/\/postimg.org\/image\/do9pyyk7z\/full\/ **If you have solved BdG equations numerically, please tell about the method and steps you used such that the above problems are eliminated.**"}
{"id":"79555","title":"Thermal Conductivity Graph","text":"This is our first time (as an engineer) seeing this type of graph that we can't interpret. This is a rough comparision of thermal conductivity from the Wiki page. http:\/\/en.wikipedia.org\/wiki\/Thermal_conductivity **What we know** The more we go in the log x axis, the more conductive the material is. Say, _silver_ vs _insulation fiber_. **What we don't know** What the length of each bar presents (in terms of y axis). Say, why _silver_ bar is 3 times the _copper_ and what does it mean. ![enter image description here](http:\/\/i.stack.imgur.com\/HsyLk.jpg)"}
{"id":"111669","title":"Isotropy of Space","text":"Weinberg writes in his Cosmology text \"Likewise,isotropy requires the mean value of any three-tensor $t_{ij}$ at $x=0$ to be proportional to $\\delta_{ij}$ and hence to $g_{ij}$, which equals $a^2\\delta_{ij}$ at $x = 0$\" May someone please illuminate the point."}
{"id":"72946","title":"Force acting on a simple rigid body in space","text":"so this is the question that's been bothering me: Say you have a simple rigid body in space that is at rest or travelling with only translational motion at a constant speed. Say that the body is something like a rod and it's not rotating. So, at some point, an external force is acted on the rod, just for an instance and it is not acted on some axes that goes through the center of mass. What will happen?? My guess is that it will start rotating about the center of mass (because of torque) + it will get some translational motion towards the direction of the force. Is that correct? and if so, how much of the force becomes rotation and how much translational movement and why?? please forgive my english and thnx in advance!!"}
{"id":"89570","title":"Why are roofs blown away by wind?","text":"Whenever there are high winds, such as in storms, thin metal roofs on sheds as well as concave roofs on huts are sometimes blown away. One explanation provided to me said that the higher velocity of the air outside causes the air pressure above the roof to decrease and when it has decreased to a certain extent such that the air pressure above the roof is lesser than the air pressure beneath the roof and due to _some kind of osmosis_ , the air particles move from the area of higher pressure (beneath the roof) to the area of low pressure. In this process, the roof is blown away. Another explanation, specifically about the thin metal roofs, said that it was blown away due to the lift caused by the air and this is the same kind of lift you get when you blow on paper. Both these explanations puzzle me. What really bothers me is the basis of the first one, **how can an increase in velocity cause pressure to drop?** I can't seem to correlate that with the Force per unit area definition of pressure. Please, oh great physicists of the internet, help me and every other ordinary person to understand how and why roofs get blown away."}
{"id":"127275","title":"Magnetic declination","text":"i couldnt just figure out when i got to know that declination can be zero also. How can true and magnetic north ever align themselves in a straight line in any place? Also if a compass aligns in the direction of horizontal component of MF at that place does this mean that at any place horizontal component is directed towards the magnetic north ?? Maybe i am misinterpreting this idea of direction. here is what i think : If u look at a bar magnet's field lines not every tangent to the curve will pass through the north pole. So the compass placed at a point will align with the field but it wont point at the north pole always . please explain"}
{"id":"127279","title":"Mathematical formalism to include wave and particle perspectives of light","text":"Does the exist any mathematical formalism (model) describing the behavior of light and incorporating its particle character (divisibility, quantization) and wave character? (i.e. quantized wave model)"}
{"id":"17528","title":"Formation of black holes","text":"If stars start with a finite density and light can escape from them, how can they be compacted to form a mass with infinite density which light cannot escape? The black hole will have the same mass as the original star (correct?) and therefore will act on the photons with the same force of gravity, right?"}
{"id":"17527","title":"Ice cream cone and loop-de-loop","text":"Was siting in class thinking about this problem, did some rough sketches of a solution but never really managed to solve it. ![Ice cream cone and a loop-de-loop](http:\/\/i.imgur.com\/D8XAh.png) > Assume a boy starts at the top of a circle with radius R as described in the > picture. It is a snowy day and the path can be considered without friction. > The boy enters a loop with radius r at the bottom of the hill. At the top of > the loop the boy loses his icecream cone in such a way that it starts > faling. The initial velocity of the icream is 0 m\/s straight down. The problem is to find R expressed by r, such that the boy reaches the icrecream just as he reaches the bottom of the loop. The problem boiled down to finding out how much time the boy uses getting from the top of the loop to the bottom. Any help, solutions or inputs would be great. * * * My attempt, I know that this is most likely 90% wrong By using conservation of mechanical energy. The speed at the bottom of the hill equals $$ v_b^2 = Rmg $$ And the velocity at the top of the loop equals $$ v_t^2 = 2g\\left( R - 2r \\right) $$ Vi know that the aceleration is constant and equals $ g $ (Here is where I think I make my mistake, forgot to acount for the angular velocity) $$ s = \\dfrac{v_1 - v_0}{2} t $$ We use this equation to find out how long it takes the boy to get from the top, to the bottom of the loop. $$ \\large t \\, = \\, \\dfrac{2s}{v_1 - v_0} \\, = \\, \\dfrac{2\\left( \\dfrac{2\\pi r}{2}\\right)}{\\sqrt{2gR} - \\sqrt{2g(R - 2r)}} $$ Now we figure out how long it takes the icream to fall the distance of the diameter or $ 2r $ . $$ s = v_0 + \\dfrac{1}{2}gt^2 $$ $$ t = \\sqrt{\\dfrac{2s}{g}} \\, = \\, \\sqrt{\\dfrac{4r}{g}} \\, = \\, 2 \\sqrt{\\dfrac{r}{g}} $$ By setting these two equations equal each other, and solving for $$ R $ , we obtain that $$ R = \\dfrac{16+8 \\pi^2+\\pi^4) r}{8 \\pi^2} \\cdot r \\approx 2.43 r $$"}
{"id":"92877","title":"\"Single-shot\" Heat engine efficiency limits","text":"The sun is 5778K and Earth is ~290K. Using the sun as the hot reservoir and earth as a cold reservoir we get 95% Carnot efficiency. However, the solar power efficiency limit is only 86%, see: http:\/\/www.energy.udel.edu\/pdf\/Honsberg_UDEI_Symposium.pdf This is not just because of our atmosphere or a non-ideal sun. There is a more fundamental constraint: Carnot heat engines don't care how quickly energy is transferred from the hot to the cold. In our case, however, we are force-fed energy and must accept it as fast as possible. If energy is lost back to the sun we lose efficiency. Thus the term \"single-shot\": we only have one chance to generate power. What is the single-shot efficiency limit as a function of temperature ratio? Assume you are surrounded with black body radiation from all directions at the cold temperature (otherwise you wouldn't even need a sun to get energy!) and an unlimited supply of coolant. The \"sun\" is a hot black body radiator that takes up a small angle of the sky. Efficiency is (useful power)\/(intercepted radiation power). This is a theoretical calculation, there are no restrictions on the design (solar thermal, PV, hybrid, etc)."}
{"id":"92874","title":"How does one calculate the polarization state of random light after total internal reflection","text":"How does one calculate the polarization state of random light after having been totally reflected by a single dielectric interface? Please consider pure specular reflexions from a plane interface between two dielectric mediums of indexes $n_1,\\,n_2$ when the angle of incidence $\\theta_1$ is greater than the critical angle $\\arcsin(n_2\/n_1)$."}
{"id":"96558","title":"The Gunn diode and the two valley theory","text":"In today lecture of microwave and radar my teacher explained about the Gunn diode.   He said it is made up of only one type of matrial e.g. $GaAS$ as shown in (a) part of the image.   ![image 1](http:\/\/britneyspears.ac\/physics\/highfields\/images\/Image93.gif)   He said that there is no depletion layer and no gate so this diode conducts in reverse bias. He explained its working by using two valley theory. I have mainly two questions which emerged from this lecture: >   1. Why a Gunn diode is called a diode as it conducts in both the > directions. > My teacher said that there are two valleys in the conduction band as shown. And the electron in the lower valley has lower effective mass as compared to that of in upper valley. ![image 2](http:\/\/britneyspears.ac\/physics\/highfields\/images\/Image79.gif) So my question is : >   1. Since electron in the lower valley is more near to the nucleus so its > velocity should be more so its effective mass should be more because > $m^*=\\dfrac{m_0}{\\sqrt{1-\\dfrac{v^2}{c^2}}}$in the lower valley >"}
{"id":"123127","title":"Moving towards a clock at .866C","text":"If you set up a clock that sends out a light pulse every second, and move towards it at a speed of .866c, will the clock appear to run faster?"}
{"id":"88625","title":"Force experienced on two particles in a rotating system?","text":"I've a system of two particles of the same mass who rotate in a circle about the centre of mass of the two particles. Is the force experienced by the particles $F=MV^{2}\/r$ or should I use $Torque=$Moment of Inertia*angular acceleration?"}
{"id":"127499","title":"Momentum acceleration in space?","text":"I'm an engineering student and have a few ideas to bat at the scientific community. First premise is based upon common sense. I understand that if I am standing on a skateboard on a relatively smooth surface and a stone of significant mass is thrown to me and I catch it, the momentum of the projectile is transferred to me causing a forward acceleration. My question: Suppose that a system is designed to provided constant acceleration in space utilizing the aforementioned intuitive concept. In the simplest form I can idealize such a system as a basketball constantly striking a flat surface. Opposed to a ball consider an arbitrary mass striking a greater mass. I suppose that if creative means are applied to control the striking action (e.g. magnetism, mechanical means) then this action can be repeated to provide constant acceleration without the use of carrying fuel loads. Would it not be possible to reset the system to an initial state without canceling the resulting acceleration? Due to the length of this question I will post the next question later. Thx"}
{"id":"64872","title":"Why doesn't light kill me?","text":"I was attending my philosophy class and in the middle of student presentations, I found myself mentally wondering off and thinking about light. After a few minutes of trying to piece together how the sun works I came to ask a question I could not answer myself. Why does each individual photon have such a low amount of energy? I am hit by photons all day and I find it amazing that I am not vaporized. Am I simply too physically big for the photons to harm me much, or perhaps the Earth's magnetic field filters out enough harmful causes such as gammy rays?"}
{"id":"90249","title":"Actual meaning of \"Gravitational Potential\"?","text":"In a gravitational field, the gravitational force acting on a body (of mass m) at a point x metres away from the attracting body (of mass M) is $\\frac{GMm}{x^2}$. Integrating this force from a point at infinity to x gives $-\\frac{GMm}{x}$, or the work done by the gravitational field on the mass in moving the mass from a point at infinity to x. Then, the gravitational potential at point x is $-\\frac{GM}{x}$. However, the definition that my school gives is : \"Gravitational potential at a point in a gravitational field is the amount of work done by an **external agent** in moving one unit of mass from a point at infinity to the point in the gravitational field, without any acceleration\". The lecturers explained that as a body moves closer to the attracting body, it accelerates and gains kinetic energy. In order to maintain a constant velocity, negative work must be done by an external agent to remove this extra kinetic energy. Now I am very confused, because both seem valid to me. What does \"gravitational potential\" actually mean?"}
{"id":"121899","title":"Landau & Lifshitz - Euler's equation for one-dimensional flow","text":"One page 5 in Landau & Lifshitz _Fluid Mechanics_ (2nd edition), the authors pose the following problem: > Write down the equations for one-dimensional motion of an ideal fluid in > terms > of the variables $a$, $t$, where $a$ (called a _Lagrangian variable_ > ) is the $x$ coordinate of a fluid particle at some instant $t=t_0$. The authors then go on to give their solutions and assumptions. Here are the important parts: > The coordinate $x$ of a fluid particle at an instant $t$ is regarded as a > function of $t$ and its coordinate $a$ at the initial instant: $x=x(a,t)$. For the condition of mass conversation the authors arrive at (where $\\rho_0 = \\rho(a)$ the given initial density distribution): > $$ \\rho\\,\\mathrm{d}x = \\rho_0 \\mathrm{d}a $$ or alternatively: > $$ \\rho\\left(\\frac{\\partial x}{\\partial a}\\right)_t = \\rho_0 $$ Now the authors go on to write out Euler's equation, where I start to miss something. With the velocity of the fluid particle $v=\\left(\\frac{\\partial x}{\\partial t}\\right)_a$ and $\\left(\\frac{\\partial v}{\\partial t}\\right)_a$ the rate of change of the velocity of the particle during its motion, they write: > $$ \\left(\\frac{\\partial v}{\\partial t}\\right)_a = -\\frac{1}{\\rho_0} > \\left(\\frac{\\partial p}{\\partial a}\\right)_t $$ **How are the authors arriving at that equation?** In particular, when looking at Euler's equation: $$ \\frac{\\partial\\mathbb{v}}{\\partial t} + \\left( \\mathbf{v} \\cdot \\textbf{grad} \\right) \\mathbf{v} = - \\frac{1}{\\rho} \\textbf{grad}\\, p $$ what happens with the second term on the LHS $\\left( \\mathbf{v} \\cdot \\textbf{grad} \\right) \\mathbf{v}$? Why does it not appear in the authors' solution?"}
{"id":"16490","title":"How to turn water opaque by pouring the smallest quantity of matter into it?","text":"Consider a glass of water and a glass of coffee. Their contents differ by no more than a few grams of particles coming from the roasted and ground coffee, yet the former lets almost all visible light pass through, and the latter blocks most of it. I am wondering **what are the most efficient ways to turn water opaque by pouring matter in it** , under normal temperature and pressure. By efficient I mean minimising the mass of added matter. By opaque, let's say 99% of daylight is blocked by two inches of liquid. There is no other reason for my asking than sheer curiosity from the many hours I spend gazing fixedly through the coffee pot."}
{"id":"114178","title":"How does the Gordon Decomposition of Dirac Current give rise to spin angular momentum?","text":"How does the Gordon Decomposition of Dirac Current give rise to spin angular momentum? I used the Gordon Decomposition to split the Probability Current of the Dirac Field into its orbital current and its spin current. I multiplied the currents by $mc$ to covert them into momentum and then crossed the momentum with position to obtain the orbital angular moment and the spin angular momentum. However the spin angular momentum was twice as large as the accepted value of spin angular momentum. I am stuck on how to get rid of the extra factor of two. The Gordon Decomposition splits the probability current into two terms \\begin{align} mc\\langle \\gamma^{0n} \\rangle = \\langle \\gamma^{0} i\\hbar \\partial^{n} \\rangle \\+ \\tfrac{\\hbar}{2}\\partial_{j}\\langle i\\gamma^{0nj} \\rangle_{n\\neq j} , \\end{align} where $\\langle \\gamma^{0} i\\hbar \\partial^{n} \\rangle$ looks like a orbital momentum current and $\\tfrac{\\hbar}{2}\\partial_{j}\\langle i\\gamma^{0nj} \\rangle_{n\\neq j}$ looks like a spin momentum current? The angular moments of $mc\\langle \\gamma^{0n} \\rangle$ can be computed by crossing then with position to obtain the following: \\begin{align} mc \\left( x^m\\langle \\gamma^{0n} \\rangle \\- x^n\\langle \\gamma^{0m} \\rangle \\right) = \\langle \\gamma^{0} i\\hbar (x^m\\partial^{n}-x^n\\partial^{m}) \\rangle \\+ \\tfrac{\\hbar}{2}\\partial_{j} \\left( x^m\\langle i\\gamma^{0nj} \\rangle_{n\\neq j} \\- x^m\\langle i\\gamma^{0nj} \\rangle_{n\\neq j} \\right) , \\end{align}"}
{"id":"65473","title":"Quantum Field Theory and Hilbert space dimensionality","text":"Much (All?) of quantum theory can be done in separable Hilbert spaces with a countable basis.   How about quantum field theory? Is it “quite happy” (mathematically consistent) if everything is countable, or does it “need” to use an uncountable, continuous space (e.g. rigged Hilbert space) for mathematical consistency, or some other reason?"}
{"id":"119513","title":"Extrinsic Photoconductor Charge Carriers","text":"I'm studying some materials on semiconductors, and a section on photoconductivity has left me somewhat confused. In an extrinsic material, is photoconductivity the result of electrons moving from the valence band to the conduction band, or is it the result of dopant atom ionization? If it is the latter, is the complete ionization assumption invalid for photoconducting devices?"}
{"id":"98323","title":"Accelerating cavity in synchotron- how does it actually work?","text":"I wanted to know how a radio frequency accelerating cavity actually works. I know that an electric field is used to accelerate the charged particles. However, is this done by using metal plates with opposite polarities or are there tubes (like in the linac) within the accelerating cavity which have fields in between the gaps?"}
{"id":"46261","title":"Why can we not reduce the size of a system below the correlation length without qualitatively changing its properties?","text":"This question is posed in the context of thermodynamics\/statistical mechanics. Suppose we define the correlation length as the $\\xi$ in the exponential factor $e^{-r\/\\xi}$ that appears in the correlation function $$C(r) = \\langle m(0)m(r) \\rangle - \\langle m \\rangle^2$$ (where $m$ is some relevant order parameter). Then my question is: how can we, from this mathematical definition, argue the following two physical properties. **Suppose we have our thermodynamic system and we take a chunk out of it with side-length L.**   1. If $L > \\xi$, the sample has the **same** qualitative behavior as the original sample;   2. If $L < \\xi$, the qualitative behavior is **different**. If one can add extra conditions for these statements to be true, that would be nice\/welcome."}
{"id":"46260","title":"Conservation of Energy in a Capacitor","text":"Consider a parallel-plate capacitor in free space. A negatively charged point particle with initial velocity $v$ passes through the space between the pair of parallel plates (with an initial path perpendicular to the normal vector of the plates). The point particle accelerates towards the positively charged plate but passes beyond the edge of the plate. How is energy conserved, given that the capacitor does work on the particle by accelerating it in the direction towards the negatively charged plate? EDIT: Was reminded by Art Brown that a negatively charged particle accelerates towards the positive plate."}
{"id":"46267","title":"The Four-Clock Special Relativity Conundrum","text":"Two open-car trains approach each other at fixed velocities. Each has a radar to see how quickly the other train is approaching, but apart from that the trains have no _a priori_ knowledge of each other. Each train has engineers on its first and last cars. Each engineer has an atomic clock and a laser for communicating with their partner on the same train. Long before the trains meet, the engineers on each train use their lasers to adjust their relative separations to exactly one kilometer. Using the synchronization procedure first defined by Einstein, the engineers also exchange time data and use it to adjust their atomic clocks until they are precisely synchronized within their shared frame. The trains meet. The clocks are positioned so that they almost touch as they pass. At the moment of nearest contact, they exchange and record each other's values (timestamps). Some time later, the clocks on the trailing cars meet and perform the same procedure. The delay between the leading and trailing data events can now be measured in two ways. If the two timestamps taken from the first train are compared, the result is the time between the events as measured from the first train. This value is meaningful because the engineers on that train previously synchronized their clocks and stayed within the same frame of reference at all times. If the timestamps from the second train are used, a similar but distinct measurement of the time between the two events can be obtained. Now, three questions:   1. Is there anything wrong or impossible with this experimental setup? If so, what is it?   2. If you accept the experiment as realistic and meaningful, will the delays calculated from the perspective of each of the two trains be the same, or different?   3. If you answered \"different,\" what is the correct procedure for predicting in advance what the ratio of the two delays will be? * * * **2013-01-10 - The Answer(s!)** I awarded the bounty to @FrankH for pages of excellent and educational work worth of reading by anyone who wants to understand special relativity better. However, I've also taken the unusual step of re-allocating the _answer_ designation to @MatthewMcIrvin. There are two reasons: (1) Matthew McIrvin was the first one to spot the importance of the symmetric-velocities frame, which others ended up gravitating (heh!) to; and (2) Matthew's answer is _short_ , which is great for readers in a hurry. FrankH, sorry about the switch, but I didn't realize I could split them before. So: If you are not overly familiar with special relativity and want to understand the full range of issues, I definitely recommend FrankH's answer. But if you already know SR pretty well and want the key insights quickly, please look at Matthew McIrvin's answer. I will have more \"proximate data exchange\" SR questions sometime in the near future. Two frames turns out to be a very special case, but I had to start somewhere. * * * For folks with strong minds, stout hearts, and plenty of time to kill, a much more precise version of the above question is provided below. It requires new notations, alas. To reduce the growing size of this question, I have deleted all earlier versions of it. You can still find them in the change history. * * * **Precise version of the 4-clock conundrum** Please see the first three figures below. They define two new operators, the frame view and clock synchronization operators, that make the problem easier to state precisely: * * * ![Setup of the four-clock conundrum using the frame operators](http:\/\/i.stack.imgur.com\/JTWkQ.png) * * * ![Event T1: Leading edge clocks A1 and B1 pass very closely](http:\/\/i.stack.imgur.com\/347QJ.png) * * * ![Event T2: Trailing edge clocks A2 and B2 pass very closely](http:\/\/i.stack.imgur.com\/EVZtI.png) * * * Note that the four timestamps $\\\\{T_{A1}, T_{B1}, T_{A2}, T_{B2}\\\\}:>\\\\{A,B\\\\}$. That is, the four timestamps are shared and agreed to (with an accuracy of about 1 nanosecond in this case) by observers from _both_ of the interacting frames A or B. Since T1 and T2 are historically recorded local events, this assertion can be further generalized to $\\\\{T1,T2\\\\}:>*$, where $*$ represents all possible frames. (And yes, $:>$ is two eyes looking left, while $<:$ is two eyes looking to the right.) Using the four shared time stamps, define: > $T_{\\Delta{A}} = T_{A2} - T_{A1}$ > > $T_{\\Delta{B}} = T_{B2} - T_{B1}$ My main question is this: * * * > Assuming that $T_{\\Delta{A}} = f(T_{\\Delta{B}})$ exists, what is $f(x)$? * * * **Analysis** (why this problem is difficult) Based purely on symmetry in the setup, the most obvious answer is $f(x)=x$, that is: > $T_{\\Delta{A}} = T_{\\Delta{B}}$ There are some interesting reasons to be troubled by that seemingly straightforward and even obvious conclusion, not the least of which is that it violates the whole concept of special relativity (the Dingle heresy). That is, if you assume $f(x)=x$ and follow that line of logic through to its logical conclusion, you quickly end up with time that flows the _same_ for all frames -- that is, no relativity. Such a conclusion is in flat violation of over a century of very detailed experimental evidence, and so is just not supportable. The following four figures show why it's so hard to assert that $T_{\\Delta{A}} = T_{\\Delta{B}}$ without violating special relativity. * * * ![View of Event T1 from Frame A](http:\/\/i.stack.imgur.com\/G59Bl.png) * * * ![View of Event T2 from Frame A](http:\/\/i.stack.imgur.com\/j2K87.png) * * * ![View of Event T1 from Frame B](http:\/\/i.stack.imgur.com\/IFj5Q.png) * * * ![View of Event T2 from Frame B](http:\/\/i.stack.imgur.com\/jmoYk.png) * * * While the above figures accurately capture the reality of relativistic contraction of both distance and time, the problem in this case is simple: How do you decide which frame to select? The experiment as described can only give one outcome. Which one will it be?"}
{"id":"89278","title":"What were the immediate consequences Yang-Lee work on Weak Interaction?","text":"I am studying the history of Modern Physics and Yang-Lee earned their Nobel the next year after the Cobalt experiments. I am familiar with the chronology, but am not clear what those findings meant to the physics community to warrant such recognition for two relatively' unknown physicists. Other household names had to wait decades for more seemingly significant work. So can someone clarify what their work meant to the eventual development of the Standard Model?"}
{"id":"46269","title":"How is wavefunction probability redistributed after partial wavefunction collapse?","text":"Suppose I set up the double-slit experiment using photons as my particle. Behind the left slit I place a beam splitter that points some of the light off in the direction of a camera (represented as `C` in the diagram below). We'll say that the beam splitter splits a light beam passing through it into two equal beams. The light that passes through the beam splitter unaffected and the light that passes through the right slit continue on to another camera (represented as `CCC` in the diagram below). We'll say that camera `C` is closer to the beam splitter than camera `CCC` is.                    CCC          | |          | |        C-\\ |          | |     -----|-|-----           |           |         Laser      If I fire a single photon through this apparatus, there's about a $1\/4$ chance that it will be detected by camera `C` and about a $3\/4$ chance that it will be detected by camera `CCC`. When the wavefunction reaches camera `C`, the photon is forced to either be detected by camera `C` or have the part of the wavefunction leading to camera `C` disappear. **Question:** In the case that the photon is not detected by camera `C`, how is the $1\/4$ probability from that branch of the wavefunction redistributed to the remaining two branches of the wavefunction, the branch went through the left slit and straight through the beam splitter and the branch that went through the right slit? The first starts with about $1\/4$ probability and the second starts with about $1\/2$ probability. Does the first get a new probability of $1\/3$ and the second a new probability of $2\/3$? Does the first get a new probability of $1\/2$ while the second keeps its old probability of $1\/2$?"}
{"id":"56901","title":"Potential due to a spherical surface charge","text":"The potential at the surface of an insulating sphere (radius R) is given by   $$V(R,\\theta) = k \\cos(3\\theta)$$ where $k$ is a constant. Use separation of variables to find the potential inside the sphere (r $\\leq$ R) and outside the sphere (r $\\geq$ R), and then use your answer to determine the surface charge density $\\sigma(\\theta)$ on the sphere. Assume there is no charge inside or outside the sphere. My approach: I will set my reference point at infinity, therefore the potential at infinity is 0, to set that as a boundary condition. Essentially, the approach in finding the surface charge density would be to find the potential function outside the sphere, then integrate that with the surface area. I am following Griffiths for E&M, but he doesn't include a specific example I can refer to for the separation of variables, and how to apply it. I can follow his examples for following potentials pretty easily, but cannot understand the first steps to set up this problem with SOV."}
{"id":"56900","title":"What is the initial velocity of a projectile so that it passes through a target point in its trajectory?","text":"Let's say I have a projectile being thrown by a player in my 2-D game. I want to work backwards and find the initial velocity to apply to the projectile such that it passes through a target point in space given that the launch point might not be at $y=0$. Some applications of this would be a basketball shot simulator or a pub darts simulation. Given:   * initial or launch angle: $\\theta_o$   * initial or launch height: $y_o$   * target point: ($x_f$, $y_f$)   * constant acceleration due to gravity ($g$)   * air resistance ignored I need to find:   * initial launch velocity: $v_o$ Here's an image of what I'm talking about. ![enter image description here](http:\/\/i.stack.imgur.com\/uRRkX.png) I am interested in learning the derivation of the answer not just a formula that I plug-and-chug. Thanks!"}
{"id":"18588","title":"Why are differential equations for fields in physics of order two?","text":"What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations? * * * If someone on the street would flat out ask me that question, then I'd probably mumble something about physicists wanting to be able to use the Lagrangian approach. And to allow a positive rotation and translation invariant energy term, which allows for local propagation, you need something like $-\\phi\\Delta\\phi$. I assume the answer goes in this direction, but I can't really justify why more complex terms in the Lagrangian are not allowed or why higher orders are a physical problem. Even if these require more initial data, I don't see the a priori problem. Furthermore you could come up with quantities in the spirit of $F\\wedge F$ and $F \\wedge *F$ and okay yes... maybe any made up scalar just doesn't describe physics or misses valuable symmetries. On there other hand in the whole renormalization business, they seem to be allowed to use lots and lots of terms in their Lagrangians. And if I understand correctly, supersymmetry theory is basically a method of introducing new Lagrangian densities too. Do we know the limit for making up these objects? What is the fundamental justification for order two?"}
{"id":"78672","title":"Why there is the requirement for derivatives no higher than second order in free quantum field equations?","text":"Why there is the requirement for derivatives no higher than second order in free quantum fields equations? We can get the equations for the free fields of an arbitrary spin by using the requirements of linearity of the equations, derivatives no higher than second order. But why there is the second requirement? Is it connect with initial conditions $A_{\\mu}(0), \\partial^{\\mu}A_{\\nu}(0)$ for the field? For example, Schrodinger equation is first-order time-derivative equation, because we have the postulate that if we know the $\\psi (\\mathbf r , t)$ in the present, we know it in the future."}
{"id":"90190","title":"Why do fundamental physical laws involve the second derivative?","text":"The title says it all. This is a question that has been nagging at me for some time. Mathematically, the first derivative is not really any different from the second derivative, or the $k$-th. So I ask, what is so special about the second derivative?"}
{"id":"57051","title":"Fermi Walker vs. Fermi transport","text":"A vector field $f^\\mu$ is said to be Fermi-Walker transported along a curve $\\gamma$ parametrized with $\\tau$ if the following holds $$\\frac{\\mathrm{D}}{\\mathrm{d}\\tau}f^\\mu = -(a^\\mu v^\\nu - a^\\nu v^\\mu) f_\\nu,$$ where $v^\\mu$ is the tangent vector $\\gamma$ and $a^\\mu$ is its derivative. The uppercase D denotes the covariant derivative. This is the usual transport law for nonrotating tetrads. However, in \"Introduction to General Relativity\" Lewis Ryder mentions the Fermi transport for which $$\\frac{\\mathrm{D}}{\\mathrm{d}\\tau}f^\\mu = v^\\mu a^\\nu f_\\nu.$$ What is the significance of this transport? Are there any applications of it?"}
{"id":"11619","title":"Mass of a galaxy via Luminosity","text":"Is there a way of calculating the mass of a galaxy, or even a nebula from the luminosity? EDIT I'm deleting this, and moving the question to Astronomy Stack Exchange - thanks david"}
{"id":"27996","title":"Conservation of angular momentum in helicopter","text":"I have a small RC-controlled toy helicopter with removable tail rotor. Suppose I remove the tail rotor, hold the tail with my hand, start the rotor until it moves with constant angular velocity and then let it go. Suppose that the heli remains on the floor all the time. Then the body of the helicopter begins to turn into the opposite direction of the rotor. My problem is to explain this in terms of angular momentum. At first this seemed to contradict angular momentum conservation because I assumed that the helicopter is a closed system and there is no external torque, but the rotor keeps the same angular velocity while the body starts to turn into the other direction. My second thought was that I have to take the air into account. If I do that it seems plausible to me to explain it as follows: The rotor transfers angular momentum from to the surrounding air. However to keep it running at constant angular velocity, the motor has to transfer angular momentum to the rotor, which is only possible if the body of the helicopter gets the same amount of angular momentum as goes to the rotor, but into the opposite direction. But then I thought, the body should turn faster and faster to infinity which obviously doesn't happen. So I guessed I should also take the friction with the floor into account where the angular momentum of the body is transferred to. This leads me to the following three questions:   1. Is my reasoning described above correct or I am completely wrong?   2. If I would do the experiment on an almost frictionless turntable (which I don't have available at the moment), would the body of the helicopter indeed become faster and faster almost to infinity?   3. If I would do the same thing in a vacuum, what would happen? I guess (see my first thought above), that the body wouldn't turn at all. Is that correct? So it would be great if someone could clarify this in terms of angular momentum. Since the same problem should occur in real helicopters, I guess that there should be some detailed literature about it's physics, but I didn't find any resources about this. So if you have any references for more details or further reading on this problem, it would be also great if you could post it."}
{"id":"103261","title":"Why does $c_{-k,-\\sigma}$ create a particle with momentum $k$?","text":"In Mudelung's book, Introduction to Solid-State Theory, I am confused by the following statement. > For many applications a further simplification is helpful. The concept of > the hole presents us with the confusing situationthat a hole in the state > $\\mathbf{k},\\sigma$ has a momentum $-ħ\\mathbf{k}$, while an electron has a > momentum $ħ\\mathbf{k}$. This asymmetry can be avoided by defining new > _quasi-particles_ which have momentum $ħ\\mathbf{k}$ inside and outside the > Fermi sphere. When we note that the creation of a particle with > $+ħ\\mathbf{k}$ is achieved by the operator $c_{\\mathbf{k}\\sigma}^+$ when > outside the Fermi sphere and by $c_{-\\mathbf{k},-\\sigma}$ when inside the > Fermi sphere, it is only a small step to define the following operators: $$ > \\begin{align} > \\alpha_{\\mathbf{k}\\sigma}^+=u_\\mathbf{k}c_{\\mathbf{k}\\sigma}^++v_{-\\mathbf{k}}c_{-\\mathbf{k},-\\sigma} > \\\\\\ > \\alpha_{\\mathbf{k}\\sigma}=u_\\mathbf{k}c_{\\mathbf{k}\\sigma}+v_{-\\mathbf{k}}c_{-\\mathbf{k},-\\sigma}^+ > \\end{align} $$ with $$ \\begin{align} u_\\mathbf{k}=1,\\; > v_\\mathbf{k}=0\\quad\\text{ for }\\quad k>k_\\text{F}, \\\\\\ u_\\mathbf{k}=0,\\; > v_\\mathbf{k}=1\\quad\\text{ for }\\quad k<k_\\text{F}. \\end{align} $$ He showed the motivation of introducing Bogoliubov tansformation. But I don't understand his statement. \"Creation of a particle with $+\\mathbf k$ is acvieved by the operator $c^{+}_{\\mathbf k,\\sigma}$ when outside the Fermi shere and by $c_{-\\mathbf k,-\\sigma}$ when inside the Fermi sphere\". $c^{+}_{\\mathbf k,\\sigma}$ means create a particle with $\\mathbf k$. So, why must we distinguish ouside and inside the Fermi sphere? And why does annihilation operator $c_{-\\mathbf k,-\\sigma}$ inside the Fermi sphere have the same effect as creation operator outside the Fermi sphere?"}
{"id":"60643","title":"Calculate the center of mass of a semicircle","text":"How I determine the center of mass of a semicircle using the definition of center of mass? I only know solve this using the Pappus theorem. Consider that the semicircle is centered on the origin and a homogeneous mass distribution."}
{"id":"53280","title":"Finite amount of matter and space","text":"(Please bear with me.) If the universe included multi-universes, and black holes transferred matter between them, AND yet there was a large, but FINITE amount of matter and space, How would that max-limit change our understanding\/perception or theories of the universe? Would it make any difference or change current concepts?"}
{"id":"54779","title":"A 0.1kg ball of dough is thrown up with a velocity of 15m\/s. What is the momentum halfway up?","text":"I know that $p=mv$ and (0.1kg)(15m\/s)=1.5 kg m\/s and the momentum at the vertex is 0, but what is the momentum halfway up?"}
{"id":"122810","title":"Pressure vs Volume Calculation for air","text":"I have 1 litre of air in a sealed container at atmosphere pressure (approx 15 psi absolute). I want to reduce pressure in the container by 1 psi ( 14 psi absolute or -1 psig).   * If I understand correctly that 1 psi reduction is 6.66 % reduction in pressure compared to standard atmospheric pressure,    * Applying the same to volume would it be correct to say that if I increase the volume of the container by 6.66 % the pressure would drop by 1psi for air OR   * If i remove 6.66% volume of air in the container I would still be able to achieve the 1psi drop in pressure? Do some other factors come in to play for pressure \/ volume calculation...?"}
{"id":"20411","title":"What is known about quantum electrodynamics at finite times?","text":"I'm aware that we can describe the time evolution of states\/operators (choose your favourite picture) of non interacting quantum fields and that perturbation theory is very effective in computing S matrix elements between free states in the remote past and free states in the remote future. Clearly the non-perturbative description of what's going on at finite times, where the interaction is active is intractible, but my question is - are there simplified toy models (scalar electrodynamics ? reduced numbers of dimensions ?) where we _can_ describe what's happening non perturbatively. Even if nothing like electrodynamics has been treatable like this, any results on the other \"textbook\" QFTs (like $\\phi^4$) would be interesting."}
{"id":"68919","title":"Elastic vs Inelastic vs isospin violating scattering particle physics models","text":"I'm looking for a nice paper that explains the difference between three particle physics models for spin-independent dark matter interaction with nuclei: elastic, inelastic and isospin violating scattering. I've found a nice paper that is giving a nice summation of the current results in direct dark matter searches ( http:\/\/arxiv.org\/abs\/1210.4011 ) and want to read up some more. Cheers, Adnan"}
{"id":"17109","title":"Why is the charge naming convention wrong?","text":"I recently came to know about the Conventional Current vs. Electron Flow issue.   Doing some search I found that the reason for this is that Benjamin Franklin made a mistake when naming positive and negative charges.   There is even this little comic about that http:\/\/xkcd.com\/567\/ My question is, how can a naming convention be wrong??   Perhaps I don't understand what is the actual mistake here, I don't know. But I don't see how a naming convention could be wrong or right. There is no right and wrong about that. It could've been any other name, like charge A and charge B. I'll appreciate anyone can help me understand what is wrong in all this."}
{"id":"75467","title":"Why are electrons defined to have negative charge?","text":"We normally think of the \"default\" or \"root\" state of things as being on the positive side of the spectrum. For example, we don't normally use a `+` symbol to indicate the sign of positive numbers, whereas negative numbers are invariably preceded with `-` to indicate their negative sign. Similarly, positive temperatures (in °C or in F) are those that we tend to encounter more typically. Why, then, do we define the basic unit of _electricity_ as something with a negative charge?"}
{"id":"68471","title":"Who (and Why) started the \"electrons are negative, protons are positive\" convention?","text":"For some reason everyone labels electrons using a minus sign and protons using a positive sign, even though the opposite seems more intuitive: ![enter image description here](http:\/\/i.stack.imgur.com\/Palaa.png) ![enter image description here](http:\/\/i.stack.imgur.com\/RTgX7.png) Who started the convention that electrons should be \"negative\" and protons should be \"positive\"? What's the reason for choosing so?"}
{"id":"80503","title":"positive charge , current and electron flow in a simple circuit …confusion","text":"my question is that in a simple circuit one wire attached with battery cell ,and then electrons start flowing from lower potential to higher and as we know in metal wire only electron is the thing which is carrying charge ,,, then why we say there is current and it flow opposite to electron ??? even there is no other thing , please dont go in other stuff yeah i know positive charge also moves some where like alpha particals in some cases but here in simple circuit why they dont just say electron flow ? why current and why its opposite ???"}
{"id":"118953","title":"Semileptonic decays of the $B_c$ meson","text":"I am struggling with calculating the exclusive semileptonic $B_c^+\\rightarrow J\/\\psi l^+\\nu_l$ decay. I learnt that the amplitude is given by a product of the leptonic current $L^{\\mu}$ and the hadronic current $H^{\\mu}$ $$ \\mathcal{M}(B_c\\rightarrow J\/\\psi l^+\\nu_l)=\\frac{G_F}{\\sqrt{2}}V_{cb}L^{\\mu}H_{\\mu} $$ where $V_{cb}$ is the CKM parameter, $L^{\\mu}$ and $H^{\\mu}$ are expressed as $$ L^{\\mu}=\\bar{u}_l\\gamma^{\\mu}(1-\\gamma^5)v_{\\nu},\\quad H^{\\mu}=\\langle J\/\\psi|J^{\\mu}(0)|B_c\\rangle $$ where $J^{\\mu}$ is the $V$-$A$ weak current. However, I did not know how this result can be derived. Could anyone provide some help? There is a second problem. On the tree-level, we have the following Feynman diagram ![enter image description here](http:\/\/i.stack.imgur.com\/l84vp.png) If we calculate $\\bar{b}\\rightarrow\\bar{c}l^+\\nu_l$ as a three-body decay in the electroweak theory (not the four-fermion approximation adopted above), how does it relate to $B_c^+\\rightarrow J\/\\psi l^+\\nu_l$?"}
{"id":"79620","title":"How does one determine black hole mass based on the curvature of a photon?","text":"1. How much mass would a black hole need to create a Schwarzschild radius that would trap a photon, whereby the photon would (to an outside observer) be continually curved 0.004km\/s at the horizon? (assume a non-spinning black hole, then contrast that with a photon traveling in either direction along the equator of a spinning black hole) Eg. [![Example](https:\/\/drive.google.com\/uc?export=download&id=0B7P471Xpxoc1N2FoN1FtRmttV2c)]"}
{"id":"79626","title":"Finding the tangential force experienced by a bob of mass m on a simple pendulum via the gradient\/nabla operator)","text":"The problem was posed as follows. Given a pendulum of length $L$ with a mass $m$ attached to it, which forms an angle $\\theta$ from the y-axis to the direction of swinging. First we had to find the potential energy as a function of the angle, which was trivial enough. $$U = mgL(1 - cos(\\theta))$$ But the next question was to develop the gradient for this potential energy function via the arc length $s = L\\cdot\\theta$, where $s$ is the arc length. I calculated the gradient by using the trigonometric definition of $\\cos(\\theta)$, but I developed the gradient by changing the variable of my function $U(\\theta)$ to $U(x,y)$ after which I calculated it further, and found that $F_{tan} = -mg\\sin(\\theta)$, as desired. But is there an alternative way to calculate it, by using the gradient, and by utilizing the arc length formula above? The question explicitly asks to develop the gradient via the arc length rather than $x$ and $y$ components."}
{"id":"67691","title":"What is the largest diameter an optical fiber can have? - for nonimaging visible light transmission","text":"Is there any theoretical limitation on the size of the diameter of an Plastic optical fiber? I would like to transmit visible sunlight through it. I see bundles of small cores available but wouldn't it be more efficient to make a very large single core?"}
{"id":"92462","title":"Can't understand the principle of least action","text":"I tried many hours to understand the principle of least action, and those hours become days... and I still didn't understand that principle\/ and how it relates to Newtonian mechanics? Could someone please explain it to me easily? Comment: for example if I have in _énoncé_ , light travel between point $a$ and point $b$ in time $s$, we should formulate the equation of motion where the path is the shortest such that it takes $s$ time to travel from point $a$ and point $b$ by that time? Also how does this relate to Newtonian mechanics and how is it better?"}
{"id":"67699","title":"Navier-Stokes equations: conservation of momentum","text":"The first Navier-Stokes equation (conservation of mass) says: $\\vec \\nabla \\cdot \\vec v=0$ For a stationary flow, the l.h.s of the second equation is (conservation of momentum): $\\rho \\frac{D\\vec v}{Dt}=\\rho (\\underbrace{\\frac{\\partial \\vec v}{\\partial t}}_{=0} + (\\vec v\\cdot \\vec \\nabla) \\vec v)=\\rho ( \\underbrace{(\\vec \\nabla \\cdot \\vec v)}_{=0??} \\vec v)\\stackrel{??}{=}0$ I find that the l.h.s of the conservation of momentum equation is always equal to zero for a stationary field. I know this isn't true but where am I wrong in this reasoning ?"}
{"id":"66342","title":"Proton mass and Weizsäcker's formula","text":"I'm _so_ confused in the use of nuclear masses and atomic masses. I have two questions. From the book \"Outline of Modern Physics\" by Ronald, I understand that the _semiempirical mass formula_ (Weizsäcker's formula) is $$M=Zm_{p}+(A-Z)m_{n}-b_{1}A+b_{2}A^{2\/3}+b_{3}Z^{2}A^{-1\/3}+b_{4}(A-2Z)^{2}A^{-1}+b_{5}A^{-3\/4} \\qquad (1)$$ and this formula is for the dependence of the mass of a nucleus on $A$ (mass number) and $Z$ (atomic number), i.e. is the formula that gives you (approximately) the mass of a _nucleus_. **1st question** $M$ is the nuclear mass or the atomic mass? (I understand that the difference between these two masses are the masses of the electrons. Obviously it's small, but is nonzero.) Then, he defines the _average binding energy per nucleon_ like $$BE=[Zm_{p}+(A-Z)m_{n}-M]c^{2}\/A \\qquad (2)$$ So far so good. Then, the book tries to do an example, and calculate the binding energy per nucleon for $_{42}^{98}Mo$ and writes $$BE=[Zm_{p}+(A-Z)m_{n}-M_{nuc}]c^{2}\/A \\qquad (3)$$ and says: \"where the atomic masses are used for $m_p$ and $M_{nuc}$ (so that the electron masses cancel).\" And insert the numbers. $$BE=[(42(1.007825u)+56(1.008665u)-97.905409u)\/98] \\times 931.5\\frac{MeV}{u} \\qquad (4)$$ But, according to Wikipedia the isotopic mass of $_{42}^{98}Mo$ ( _atomic_ mass) is 97.9054082 u, and the mass of the proton is 1.007276466812 u. So my **2nd question** is: why does the book uses the value 1.007825u for the mass of a proton instead of the value 1.007276466812 u, and why does it uses the atomic mass instead of the nuclear mass in $M_{nuc}$? **Note** The book uses the value 1.007825u for the mass of a proton and the atomic mass instead of the nuclear mass in $M$, in other examples. Why?"}
{"id":"71907","title":"Alpha decay, why does it occur?","text":"I was reading about alpha decay and why it happens. The strong force holds protons and neutrons together, but I don't get why does an atom emit helium nucleus when it has too many protons&neutrons? I mean why isn't the strong force holding on to it?"}
{"id":"111302","title":"Changing Coordinate Systems in Two-Loop Integrals","text":"Suppose we have the two-loop integral $\\int \\mathrm{d} ^ 4 k _ {2} \\int \\mathrm{d} ^ 4 k _ {1} \\, f(k _ {1}, k _ {2})$, where $k _ {1}$ and $k _ {2}$ are four-dimensional vectors in Euclidean space. In the first integration with respect to $k _ {1}$, I take $k _ {2}$ to be the z-axis. Then $k _ {2} = k _ {1} \\cos \\omega$ and the four-dimensional spherical volume element is $\\mathrm{d} V = |k _ {1}| ^ {3} \\sin ^ 2 \\omega \\sin \\theta \\, \\mathrm{d} |k _ {1}| \\, \\mathrm{d} \\omega \\, \\mathrm{d} \\theta \\, \\mathrm{d} \\phi$, where $|k _ {1}|$ is the Euclidean norm of $k _ {1}$. When we perform the second integration $\\int \\mathrm{d} ^ 4 k _ {2} \\, g (k _ {2})$, where $g (k _ {2})$ is the result of the first integration, is it allowed to change coordinate systems and take $k _ {2}$ to be the radius in $S ^ 3$? That is, can we write the spherical volume element for the second integration as $2 \\pi ^ 2 |k _ {2}| ^ {3}$, where $|k _ {2}|$ is the Euclidean norm of $k _ {2}$?"}
{"id":"118959","title":"At an instant, does a system of gravitational charges exhibit equivalent behavior to a time-reversed system of like electric charges?","text":"**Question:** In principle, does a system of gravitational charges exhibit equivalent behavior to a time-reversed system of like electric charges? _(At a single instance in time?)_ **Additional Notes:** I am aware that the evolution of this system would not behave the same because orbits cannot manifest in a simple system of like charges due to reasons regarding entropy; Just because it is entropically desirable to evolve from the big-bang state to the current universal state, it is not entropically desirable to evolve from the current universal state to the big-bang state just because we reversed the flow of time. (I intend to reverse the flow of time so that the fields reverse but not the global increase of entropy, this is why I specify _At a single instance in time?_ ). I would ask that SE users to consider that the argument of playing a tape backwards is an ill conceived method for generating an answer since it always admits an unnatural evolution of state of the system (One that would never play out if entropy was increasing, I only reverse time for the sake of reversing the fields). For example if I tape an ink drop falling in water and watch it in reverse, it becomes immediately become apparent to me that it is being played in reverse because the system evolves in a way which violates the second law of thermodynamics. Even though this is true, what I can say with confidence is that every electric field of every particle will reverse in sign and the same can be said about the gravitational field. That is the true purpose of the question. With that being said, I specify \" _(At a single instance in time?)_ \" because I am more concerned with this idea on a fundamental level _(i.e. The physical properties of the fields)_."}
{"id":"34792","title":"\"Dynamic Albedo of Neutrons\" and all these Farnsworth fusors","text":"I've checked specs of Russian \"Dynamic Albedo of Neutrons\" module on Mars MSL rover - it can do 10^8 neutron pulses at 10 pulses per second. It works on D-T reaction in tiny linear accelerator, and power consumption is quite low (14W). If you check specs for all these fusors (Farnsworth, Polywell, e.t.c) - they are happy to have 10^6 neutrons (best ones barely scratch 2*10^8) per second while having huge power consumption and short construction lifetime (<1hour). Does that mean that fusors are just a waste of paper from the science prospective, and real scientists routinely make neutron sources based on nuclear fusion with much better parameters\/efficiency? **Update:** Fusor: http:\/\/en.wikipedia.org\/wiki\/Fusor ![enter image description here](http:\/\/i.stack.imgur.com\/LyXIY.jpg) DAN: http:\/\/mars.jpl.nasa.gov\/msl\/mission\/instruments\/radiationdetectors\/dan\/ More info on Russian: http:\/\/www.federalspace.ru\/main.php?id=58"}
{"id":"61694","title":"Does tapping at the side of a bottle prevent shaken soda from bubbling over?","text":"Anecdotal evidence has it that a bottle of soda that was heavily shaken will not bubble over if tapped at the side multiple times. Yet I wonder: Has the tapping really any effect? Or could it be that the mere time that passes while tapping at the side of the bottle has the carbon dioxide not mixed with the soda anymore?"}
{"id":"108088","title":"Why is energy conserved?","text":"From basic principals, how does one prove that energy is conserved? Or a little more specifically - Why does this hold: $$\\Delta \\mbox{ PotentialEnergy} + \\Delta \\mbox{ KineticEnergy} = 0 $$ Or, for extra credit, why does this hold: $$\\Delta \\mbox{ PotentialEnergy } + \\Delta \\mbox{ KineticEnergy} + \\Delta \\mbox{ ThermalEnergy } = 0 $$"}
{"id":"296","title":"Is energy really conserved?","text":"In high school I was taught energy was conserved. Then I learned that nuclear reactions allow energy to be converted into mass. Then I also heard that apparently energy can spontaneously appear in quantum mechanics. So, are there any other caveats with the conservation of energy?"}
{"id":"19216","title":"Why can't energy be created or destroyed?","text":"My physics instructor told the class, when lecturing about energy, that it can't be created or destroyed. _Why_ is that? Is there a theory or scientific evidence that proves his statement true or false? I apologize for the elementary question, but I sometimes tend to over-think things, and this is one of those times. :)"}
{"id":"47442","title":"Conservation of Energy in the Universe","text":"> **Possible Duplicate:**   >  Is energy really conserved?   >  Why can’t energy be created or destroyed? One of the laws of the universe that dazzles me the most is the law of conservation of energy. I however have a couple of questions regarding this law. Since Einstein's equivalence tells us that matter and energy are different manifestations of the same thing:   * Does this mean that the amount of energy + matter has been the same since the beginning of the universe? If this is true, then another question pops up. Since the universe is expanding at an accelerating rate, and the law of conservation of energy tells us that the amount of energy can't increase:   * does this mean that the universe is getting emptier?  Also, an intertwined question:   * Is there some sort of average 'energy density' in the universe? If so, can we notice the effects of the accelerating expansion of the universe by looking at this 'energy density'? (this might sound like\/ be a stupid question)."}
{"id":"67966","title":"Fluids in thermodynamic equlibrium","text":"I am reading about the Euler Equations of Fluid dynamics from Leveque's numerical methods for conservation laws. After introducing the mass, momentum and energy equations, some thermodynamic concepts are discussed, to introduce an equation of state. He says               In the euler equations we assume that the gas is in chemical and thermodynamic     equilibrium and that the internal energy is a known function of pressure and density.      After this , the usual thermodynamics-related EOS discussions are carried out. Now chemical equilibrium I understand (number of moles of the chemical constituents do not change), however I don't understand how the assumption of thermodynamic equilibrium can be imposed. From what baby thermodynamics I know, any thermodynamic analysis is always calculated for quasi-static processes, like 'slowly' pushing a piston in a cylinder of gas. But in fluid _dynamics_ fluids are **flowing** and that too rapidly and from intuition there will not be any thermodynamic equlibrium during fluid flow. Where is my understanding going wrong?"}
{"id":"24410","title":"How does one get the value of acceleration of gravitation on earth accurately by experiment without electronic device?","text":"How does one get the value of acceleration of gravitation on earth accurately to 5 significant digits by experiment without electronic device?"}
{"id":"118477","title":"Is every material able to exist in every state of matter?","text":"I was shocked while reading Kittel's \"Introduction to Solid State Physics\", that the solid state of noble gases is a well described and makes one of the fundamental achievements of solid state physics. So I wonder: is every possible element and every possible compound able to exist in any of the 3 states of matter?"}
{"id":"99397","title":"How do draw real path which represents the solution of the Landau-Lifshitz equation","text":"![enter image description here](http:\/\/i.stack.imgur.com\/Zux0C.png) how to draw two images look like images below with gnuplot."}
{"id":"23298","title":"Good algorithm for in-experiment 1-D optimization?","text":"I'm running an experiment -- for the question, it doesn't matter which one, but I'm measuring an optical intensity $I$ as a function of two parameters: reflection angle $\\theta$ and wavelength $\\lambda$. I have motion control in place to move the setup to an angle $\\theta_0$, and then I measure $I(\\theta_0, \\lambda)$ all at once using a spectrometer. I then move to the next angle $\\theta_1$ and repeat. Due to the beam travelling through different media at different angles, I have to move the position of the detector $p$ slightly for each angle, which is also automated. I should move the detector so that $M(p) = \\sum_\\lambda I(\\theta_n, \\lambda, p)$ is maximized. (M stands for \"figure of Merit\".) $M(p)$ approximately has the form of a Gaussian plus noise, but I should be able to maximize it without caring what form it has. I have an amateurish algorithm in place to search for the proper position $p$ in order to maximize $M(p)$. The quick-n-dirty algorithm steps $p$ in one direction by a step size $\\Delta p$, until the value of $M(p)$ becomes smaller than a previous value. Then it goes back one step and tries a smaller step size $\\Delta p$ in the other direction. As you can see, it doesn't account for measurement noise. My thoughts on how to improve it were in the direction of measuring a small number of points spaced $\\Delta p$ apart and then fitting a parabola through them. My question is, before I sit down and design a better algorithm, can anyone suggest an already-existing algorithm? I don't think a feedback algorithm (such as PID control) is appropriate, since I'm not trying to maintain a certain setpoint under perturbation of the system -- I just need to optimize to one value for each measurement. For bonus points, can someone point me to some papers on this subject?"}
{"id":"28746","title":"What does it mean that particles are the quanta of fields?","text":"I saw the question What are field quanta? but it's a bit advanced for me and probably for some people who will search for this question. I learned QM but not QFT, but I still hear all the time that \"particles are the quanta of fields\" and I don't really understand what it means. Is there a simple explanation for people who know QM but not QFT?"}
{"id":"28742","title":"Derivation of the supergravity action in 11D","text":"The Einstein-Hilbert action of general relativity is uniquely determined by general covariance and the requirement that only second derivatives in the metric appear. Yang-Mills theory can be motivated in a similiar way. In the original paper of Scherk, Julia, Cremer there are some arguments given from which they deduced the form of the action. They are only sketched however. Is there a more complete exposition of the derivation in the literature, or possibly even a uniqueness result as in the case of general relativity or Yang-Mills theory?"}
{"id":"28743","title":"From knowing just the change in kinetic energy, can we find the friction force and engine power?","text":"I understand this topic well enough to get all the task done because they aren't very creative. But for my exam I think I should have this clear. During the acceleration the force from the engine is of course bigger than air resistance and friction. This force, can we find it? And then the entire force the engine applies for the acceleration. Not just the stub you after subtracting for friction and air resistance. $W = F \\times s$   $F = m \\times a$ We have all the work done by forces at work, and the stretch of road is easy to calculate. If I now do this $\\frac{F}{m} =a $ will that output be the correct acceleration? And this force that we found, is that a sum force? Because if that's the force sum on the car I can't find the engines power output which is what I want. And what about the friction force at work, I think we can't find it when we just have the change in kinetic energy. Is that right? Primarily I would like to know if the change in kinetic energy can be tied somehow to the engines output during the acceleration. The book has this nice equation too: $P = F \\times v $   But that's just constant speed. Since we know the time maybe this can be used: $P = \\frac{W}{T}$   That just seems a little too easy. Edit: I got a B on the exam, which means I'll be at again this fall. Not due to this question. Haha."}
{"id":"128408","title":"Is this really a golden ratio spiral?","text":"In this blog post, I found this picture: ![enter image description here](http:\/\/i.stack.imgur.com\/Qi3C2.jpg) Does the water really form golden ratio spiral in such cases? Or is the photo just a provocative example, without physics grounds for claims about \"goldness\" of the spiral?"}
{"id":"91846","title":"Normalization for QFT single particle destruction operator","text":"I don't understand a particular statement in the QFT book by Klauber. The particular page I'm having difficulty on is page 67 of chapter 3 (PDF link). The big picture is that the author wishes to investigate what the (operator) solutions to the Klein-Gordon equation, $\\phi(x)$ and $\\phi^\\dagger(x)$, do when acting on the vacuum state $|0\\rangle$. As prep for this, he creates a \"general single particle state\" (\"general\" meaning non $\\mathbf{k}$-eigenstate) by operating on the vacuum with the operator $$C\\equiv\\sum_\\mathbf{k}A_\\mathbf{k}a_\\mathbf{k}^\\dagger,\\tag{3-108}$$ $$C|0\\rangle=\\sum_\\mathbf{k}A_\\mathbf{k}a_\\mathbf{k}^\\dagger|0\\rangle=A_1|\\phi_1\\rangle+A_2|\\phi_2\\rangle+\\cdots\\equiv|\\phi\\rangle\\tag{3-109}$$ Each $A_\\mathbf{k}$ is just a number, the absolute value square of which represents the probability of finding the $\\mathbf{k}$ eigenstate for the single particle. The new state $C|0\\rangle=|\\phi\\rangle$ is interpreted as a single particle state in a superposition of $\\mathbf{k}$-eigenstates $|\\phi_k\\rangle$. The subscript $\\mathbf{k}$ represents different momenta. For probability\/normalization arguments, the numbers $A_\\mathbf{k}$ should obey $$\\sum_\\mathbf{k}\\left|A_\\mathbf{k}\\right|^2=1.\\tag{3-110}$$ I feel like I understand the above statements. The author then introduces the \"general single particle destruction operator\" $$D\\equiv\\sum_\\mathbf{k}a_k,\\tag{3-111}$$ and shows that when applied to our general single particle state $|\\phi\\rangle$ above, the vacuum is (re)produced: $$\\begin{eqnarray} D|\\phi\\rangle&=&\\left(\\sum_\\mathbf{k}a_k\\right)A_1|\\phi_1\\rangle+\\left(\\sum_\\mathbf{k}a_k\\right)A_2|\\phi_2\\rangle+\\cdots\\\\\\ &=&A_1\\underbrace{a_1|\\phi_1\\rangle}_{=|0\\rangle}+A_1\\underbrace{a_2|\\phi_1\\rangle}_{=0}+A_1\\underbrace{a_3|\\phi_1\\rangle}_{=0}+\\cdots+\\\\\\ &\\ &+A_2\\underbrace{a_1|\\phi_2\\rangle}_{=0}+A_2\\underbrace{a_2|\\phi_2\\rangle}_{=|0\\rangle}+ A_2\\underbrace{a_3|\\phi_2\\rangle}_{=0}+\\cdots+\\\\\\ &\\ &+\\cdots\\\\\\ &=&\\underbrace{\\left(A_1 + A_2 + \\cdots\\right)}_\\text{can normalize = 1}|0\\rangle. \\end{eqnarray}\\tag{3-112}$$ (Note the subtle but important differences in the underbraces; some are $0$ while others are $|0\\rangle$.) **The part I am struggling with** is understanding how the underbrace \"can normalize = 1\" at the end of $\\text{(3-112)}$ can be true given $\\text{(3-110)}$. It seems to me that the $A$ terms appearing at the end of $\\text{(3-112)}$ are the same ones defined in the construction operator $C$ and normalized so that their absolute values squared sum to $1$. How can their just-plain sum also be of magnitude $1$? I know that one would *like * the underbraced term to sum to zero, but I don't see how that can be. * * * It was suggested I consider the quantity $\\langle\\phi|D^\\dagger D|\\phi\\rangle$. Here is my attempt to calculate it. $$ \\begin{eqnarray} \\langle\\phi|D^\\dagger D|\\phi\\rangle&=&\\langle0|(A_1^\\dagger+A_2^\\dagger+\\cdots)(A_1+A_2+\\cdots)|0\\rangle=\\langle0|\\sum_\\mathbf{j}\\sum_\\mathbf{k}A_\\mathbf{j}^\\dagger A_\\mathbf{k}|0\\rangle\\\\\\ &=&\\sum_\\mathbf{j}\\sum_\\mathbf{k}A_\\mathbf{j}^\\dagger A_\\mathbf{k}\\underbrace{\\langle0|0\\rangle}_{=1}=\\underbrace{\\sum_\\mathbf{j}\\sum_\\mathbf{k}A_\\mathbf{j}^\\dagger A_\\mathbf{k}}_\\text{Can't simplify}\\ne1 \\end{eqnarray} $$"}
{"id":"55021","title":"Solving systems of equations in dynamics","text":"I have an exam in two days for first year university physics. Often for dynamics problems, I am required to solve algebraic systems of equations by hand, and this can be very daunting. When I see the solutions, however, the steps that the solver took to seem very clean and almost obvious. Are there some rules of thumb that physicists use to solve small systems of equations, either by elimination or substitution? Here is an example. Find $\\frac{m_1}{m_2}$ in terms of only $\\theta$ where the known quantities are $\\theta$, $m_1$, $m_2$. $$F_T \\sin \\theta = m_1 g$$ $$F_T \\cos \\theta = m_1 a$$ $$F_T \\sin \\theta + F_N \\cos \\theta = m_2 g$$ $$F_N \\sin \\theta - F_T \\cos \\theta = m_2 a$$"}
{"id":"56598","title":"Gauge invariance and diffeomorphism invariance in Chern-Simons theory","text":"I have studied Chern-Simons (CS) theory somewhat and I am puzzled by the question of how diff. and gauge invariance in CS theory are related, e.g. in $SU(2)$ CS theory. In particular, I would like to know about the relation between large gauge transformations and large diffeos. If you know any good sources, I would be really grateful. Thank you!"}
{"id":"73278","title":"Simulation of an one dimensional driven diffusive system","text":"I'm currently writing a simulation in python with scipy and matplotlib to reproduce an one dimensional driven diffusive system described in this paper from M.R. Evans et al. The system consists of positive, negative and hole particles. On the left side of the system the positive (negative) particles are produced on the left (right) side with a possibility a and destroyed at the right (left) side with the possibility b. In some cases the system should show a flip between positive and negative high density states and I'm trying to reproduce this behaviour with my simulation. But all I can see is an increasing current within my simulation and I can't find any problems in my code explaining such behaviour. Does anyone here have some experience simulation such or similar systems? Cheers, Florian"}
{"id":"8477","title":"Why can't light escape from a classical black hole?","text":"Photons do not have (rest) mass (that's why they can move at speed of \"light\"). So, my question is how the gravity of classical$^1$ black hole can stop light from escaping? \\-- $^1$ We ignore quantum mechanical effects, such as, Hawking radiation."}
{"id":"35445","title":"Kepler problem: flows generated by constants of motion","text":"This is part of an admission exam problem, found at http:\/\/www.sissa.it\/mp\/admission\/tests\/\/2008_common.pdf Consider the Hamiltonian of Kepler problem $$H(\\boldsymbol{r},\\boldsymbol{p})= \\frac{|\\boldsymbol{p}^2|}{2\\mu} +\\frac{\\alpha}{|\\boldsymbol{r}|}, \\qquad \\mu>0>\\alpha,$$ where $\\boldsymbol{r}\\in M=\\mathbb{R}^3\\setminus\\\\{ 0 \\\\}, \\ (\\boldsymbol{r},\\boldsymbol{p})\\in T^*M$ and $|\\boldsymbol{r}|=\\sqrt{r_1^2+r_2^2+r_3^2}$. The quantities $$\\boldsymbol{m}=\\boldsymbol{r}\\times\\boldsymbol{p}, \\qquad \\boldsymbol{W}=\\boldsymbol{p}\\times\\boldsymbol{m}+ \\mu\\alpha\\frac{\\boldsymbol{r}}{|\\boldsymbol{r}|}$$ are constants of motion, as is well known. It is stated then that the flows generated by the functions $m_i$ and $W_i,\\ i=1,2,3$ are canonical transformations. **I don't understand** is what is meant by this statement: I mean, I know what a canonical transformation is, but I would appreciate some explanation or reference about this precise statement. Thanks in advance for the help, and I hope this is formulated in compliance with the rules of this community."}
{"id":"32795","title":"Is Connes model a composite Higgs in disguise?","text":"Most of the 5-dimensional Higgs models can be seen, if I understand correctly, as models where the Higgs is a composite. Now, is this true for Connes models? It is a model of extra dimensions too, in some sense. And when you look at the papers, at some moment a product of two symbols is substituted by a single $\\phi$, so it seems that some composition is at work."}
{"id":"32790","title":"Wilson loops and gauge invariant operators (Part 1) ","text":"I guess the Hilbert space of the theory is precisely the space of all gauge invariant operators (mod equations of motion..as pointed out in the answers)   * Is it possible that in a gauge theory the Wilson loops are the only observables?  (...I would vaguely think that if a set of Wilson loops one for every cohomology class of the space-time is the complete set of observables then this is what would be \"a\" way of defining a Topological Field Theory but may be this is also possible for pure gauge theories in some peculiar limit or on some special space-time geometries..)   * When the above is not true then what are all the pure gauge theory observables?..I guess its only the local observables that is missed by the Wilson loops..   * In general is it always true that all gauge invariant observables are precisely all the polynomials in the fields which are invariant under the action of the gauge group? (..and this is a well studied question in algebraic geometry under the name of Geometric Invariant Theory?..)    * If one has matter in the theory then I guess the baryons and the mesons are the only matter observables? I guess there is no gauge group dependence on their existence?  (..though baryons can always be defined for any anti-symmetric combination of the flavour indices I guess mesons can be defined only if equal amount of matter exists in the conjugate representation of the gauge group also..right?..)   * Why are gauge traces of arbitrary products of matter fields neither baryons nor mesons? (...in arbitrary gauge theories is it legitimate to identify these states as ``chiral primaries\" in any sense?..)"}
{"id":"23876","title":"Coulomb potential energy functional derivative","text":"I'm having problem understanding how to compute a functional derivative when it's involved more than one integral, such as the coulomb potential energy functional: $$ J[\\rho] = \\frac 12\\int \\frac{\\rho(r)\\rho(r')}{|r - r'|} drdr' $$ According to the functional derivative formula I should do something on these lines: $$ \\frac {\\delta J}{\\delta \\rho (r)} = \\frac{\\partial}{\\partial \\rho(r)} [ \\frac 12 \\int \\frac{\\rho(r) \\rho(r')}{|r - r'|}dr' ]$$ In my wrong reasoning I would simply take $\\rho(r) $ out of the integral and apply the derivative: $$\\frac {\\partial \\rho(r)}{\\partial \\rho(r)} \\frac 12 \\int \\frac{\\rho(r')}{|r - r'|} dr' = \\frac 12 \\int \\frac{\\rho(r')}{|r - r'|} dr'$$ Which is wrong because the correct result should be: $$\\int \\frac{\\rho(r')}{|r - r'|} dr'$$ 1) I'm quite confused by the notation and how to treat a partial derivative by $\\rho(x)$ 2) What's the correct way to handle and compute functional derivatives in these cases? I'm actually in a similar situation with a much complex derivative, such as the same thing in the density matrix formalism: $$ J[\\gamma_1] = \\frac 12 \\int \\frac{\\gamma_1 (x_1', x_1) \\gamma_1 (x_2', x_2) \\delta(x_1' \\- x_1) \\delta(x_2' \\- x_2) dx_1 dx_1' dx_2 dx_2'}{|x_1 - x_2|}$$ $$\\frac {\\delta J[\\gamma_1]}{ \\delta \\gamma_1 (x_1'. x_1)}$$"}
{"id":"23874","title":"Finding force exerted in an Inelastic Collision","text":"I did a lab today in Physics in which we launched ball from a spring loaded cannon directly into a pendulum that captured the ball, held it, and swung upwards with it (representing a totally inelastic collision). One question in particular has confused me: > If the collision between the projectile and pendulum had lasted 1 > millisecond, what would the average force have been which the projectile > exerted on the pendulum for the long-range case? My attempt at a solution is as follows: From all the searching I've been doing online, I've found the equation $F = {{p_f}-{p_i}\\over {t}}$. I know $p_f$, $p_i$, and I'm given t. Is my understanding right? Can I go right ahead and crunch these numbers, or do I have an incorrect equation?"}
{"id":"61522","title":"Extended Born relativity, Nambu 3-form and ternary (n-ary) symmetry","text":"**Background:** Classical Mechanics is based on the Poincare-Cartan two-form $$\\omega_2=dx\\wedge dp$$ where $p=\\dot{x}$. Quantum mechanics is secretly a subtle modification of this. By the other hand, the so-called Born-reciprocal relativity is based on the \"phase-space\"-like metric $$ds^2=dx^2-c^2dt^2+Adp^2-BdE^2$$ and its full space-time+phase-space extension: $$ds^2=dX^2+dP^2=dx^\\mu dx_\\mu+\\dfrac{1}{\\lambda^2}dp^\\nu dp_\\nu$$ where $$P=\\dot{X}$$ Note: particle-wave duality is something like $ x^\\mu=\\dfrac{h}{p_\\mu}$. In Born's reciprocal relativity you have the invariance group which is the _intersection_ of SO (4 +4) and the ordinary symplectic group Sp (4), related to the invariance under the symplectic transformations leaving the Poincaré- Cartan two-form invariant. The intersection of SO(8) and Sp(4) gives you, essentially, the unitary group U (4), or some \"cousin\" closely related to the metaplectic group. We can try to guess an extension of Born's reciprocal relativity based on higher accelerations as an interesting academical exercise (at least it is for me). In order to do it, you have to find a symmetry which leaves spacetime+phasespace invariant, the force-momentum-space-time extended Born space-time+phase-space interval $ds^2=dx^2+dp^2+df^2$ with $p=\\dot{x}$, $ f=\\dot{p}$ in this set up. Note that is is the most simple extension, but I am also interested in the problem to enlarge it to extra derivatives, like Tug, Yank,...and n-order derivatives of position. Let me continue. This last metric looks invariant under an orthogonal group SO (4+4+4) = SO (12) group (you can forget about signatures at this moment). One also needs to have an invariant triple wedge product three-form $$\\omega_3=d X\\wedge dP \\wedge d F$$ something tha seems to be connected with a Nambu structure and where $P=\\dot{X}$ and $F=\\dot{P}$ and with invariance under the (ternary) 3-ary \"symplectic\" transformations leaving the above 3-form invariant. **My Question(s):** I am trying to discover some (likely nontrivial) Born- reciprocal like generalized transformations for the case of \"higher-order\" Born-reciprocal like relativities (I am interested in that topic for more than one reason I can not tell you here). I do know what the phase-space Born- reciprocal invariance group transformations ARE (you can see them,e.g., in this nice thesis BornRelthesis) in the case of reciprocal relativity (as I told you above). So, my question, which comes from the original author of the extended Born-phase space relativity, **Carlos Castro Perelman in** this paper, and references therein, is a natural question in the context of higher- order Finsler-like extensions of Special Relativity, and it eventually would include the important issue of curved (generalized) relativistic phase-space- time. After the above preliminary stuff, the issue is: > What is the intersection of the group SO (12) with the _ternary_ group which > leaves invariant the triple-wedge product > > $$\\omega_3=d X\\wedge dP \\wedge d F$$ More generally, I am in fact interested in the next problem. So the extra or bonus question is: what is the (n-ary?) group structure leaving invariant the (n+1)-form $$ \\omega_{n+1}=dx\\wedge dp\\wedge d\\dot{p}\\wedge\\cdots \\wedge dp^{(n-1)}$$ where there we include up to (n-1) derivatives of momentum in the exterior product or equivalently $$ \\omega_{n+1}=dx\\wedge d\\dot{x}\\wedge d\\ddot{x}\\wedge\\cdots \\wedge dx^{(n)}$$ contains up to the n-th derivative of the position. In this case the higher- order metric would be: $$ds^2=dX^2+dP^2+dF^2+\\ldots+dP^{(n-1)}=dX^2+d\\dot{X}^2+d\\ddot{X}^2+\\ldots+dX^{(n)2}$$ This metric is invariant under SO(4(n+1)) symmetry (if we work in 4D spacetime), but what is the symmetry group or invariance of the above (n+1)-form and whose intersection with the SO(4(n+1)) group gives us the higher-order generalization of the U(4)\/metaplectic invariance group of Born's reciprocal relativity in phase-space? This knowledge should allow me (us) to find the analogue of the (nontrivial) Lorentz transformations which mix the $X,\\dot{X}=P,\\ddot{X}=\\dot{P}=F,\\ldots$ coordinates in this enlarged Born relativity theory. **Remark:** In the case we include no derivatives in the \"generalized phase space\" of position (or we don't include any momentum coordinate in the metric) we get the usual SR\/GR metric. When n=1, we get phase space relativity. When n=2, we would obtain the first of a higher-order space-time-momentum-force generalized Born relativity. I am interested in that because one of my main research topics are generalized\/enlarged\/enhacend\/extended theories of relativity. I firmly believe we have not exhausted the power of the relativity principle in every possible direction. I do know what the transformation are in the case where one only has X and P. I need help to find and work out myself the nontrivial transformations mixing X,P and higher order derivatives...The higher-order extension of Lorentz-Born symmetry\/transformation group of special\/reciprocal relativity."}
{"id":"51612","title":"In which direction is the acceleration directed in a non uniform circular motion?","text":"Acceleration is directed towards the center of the circle in a uniform circular motion. Is it same for the non-uniform circular motion?"}
{"id":"61298","title":"Ex 0.2.1 in Sachs and Wu's textbook","text":"In the next attachements are: 1\\. Exercise 0.2.5 which I want help with.   1. Proposition 0.2.1 and its proof. Now, basically a few things are changed in the theorem, I don't think I can use here the definition of s(t) in the proof of prop0.2.1 cause its s(t)=0, I don't think I can use this trick here. Other thoughts that I had, obviously if I plug m=0 into prop0.2.1 I get that I should have: $$\\frac{d\\gamma^1}{du}=\\pm \\frac{d\\gamma^2}{du}$$, and $$\\frac{d\\gamma^2}{du}=a$$. My question is how do I satisfy condition b in the theorem, I guess this x should be $$\\pm Id +constant$$ ![1](http:\/\/i.stack.imgur.com\/uNJZS.png) ![2](http:\/\/i.stack.imgur.com\/Gftvm.png) ![3](http:\/\/i.stack.imgur.com\/Eo0PX.png)"}
{"id":"130099","title":"Angular momentum of the electric field of a point-like electric charge and the magnetic field of a monopole","text":"I am currently reading \"Magnetic Monopoles\" of Ya. Shnir. My problem is I can not retrieve a result the author provides in the first chapter of the first part. In this chapter, he studies the non-relativistic scattering of an electric charge on a magnetic one. The author writes [p.5, near eq. (1.13)]: > ... the appearance of an additional term in the definition of the angular > momentum $(1.11)$ originates from a non-trivial field contribution. Indeed, > since a static monopole is placed at the origin, its magnetic field is given > by $(1.1)$. Then the classical angular momentum of the electric field of a > point-like electric charge, whose position is defined by its radius vector > $\\mathbf{r}$, and the magnetic field of a monopole is a volume integral > involving the Poynting vector > > \\begin{align} \\tilde{\\mathbf{L}}_{eg} &= \\dfrac{1}{4\\pi}\\int \\mathbf{r'} > \\times \\left [ \\mathbf{E} \\times \\mathbf{B}\\right] d^3r'\\tag{L.1}\\\\\\& = - > \\dfrac{g}{4\\pi} \\int d^3r' \\left( \\mathbf{\\nabla}'\\cdot \\mathbf{E}\\right) > \\hat{\\bf r}' \\tag{L.2}\\\\\\ &= -eg\\hat{\\bf r} \\tag{L.3} \\end{align} > > where we perform the integration by parts, take into account that the fields > vanish asymptotically and invoke the Maxwell equation > > \\begin{equation}\\left(\\mathbf{\\nabla}' . \\mathbf{E} \\right) = 4 \\pi e > \\delta^{(3)}\\left( \\mathbf{r} - \\mathbf{r}'\\right)\\end{equation} ... The magnetic field is $\\mathbf{B} = \\dfrac{g}{r^3} \\mathbf{r} \\tag{1.1}$ The generalised angular momentum is $\\mathbf{L} = \\mathbf{r} \\times m\\mathbf{v} - eg \\hat{\\bf r} \\tag{1.11}$ The author gives how he got $(L.2)$ from $(L.1)$ but I do not know how to do? Have you any idea?"}
{"id":"36233","title":"Why does your reflection stay the same size when you move further away from the mirror?","text":"This was an experiment I saw in my son's workbook. It said to mark out the top of your forehead and the bottom of your chin on a mirror using a whiteboard marker. Then slowly move backwards, and investigate what happens to the size of the reflection subjective to the two marks made. It actually got me quite flabbergasted. I always thought the reflection would get smaller as you moved away from the mirror. Why is this?"}
{"id":"130090","title":"Max speed as a function of engine power","text":"My question started out as finding the maximum speed of a go kart, taking into account only the drag forces as force opposing the motor. I've done some investigation to find: $$ F_{drag}=\\; \\frac{\\rho v^{2}\\ C_{x} A_{f}}{2} $$ Where $C$ = drag coefficient, and $\\rho$ is the density of air. Further, with $ F_{motor}=\\dfrac{P_{motor}}{v} $ and the condition that at maximum speed, the acceleration will be 0, net force will be 0 as well, so $F_{motor}=F_{drag}$. Combining equations, $F_{drag}=\\dfrac{P_{motor}}{v}$, so $\\dfrac{P_{motor}}{v}=\\; \\dfrac{\\rho v^{2} C A_{f}}{2}$ Simplifying gives $P_{motor}=\\; \\dfrac{\\rho\\cdot v^{3}C\\cdot A_{f}}{2}$ Solving for $v$ yields: $$v=\\; \\sqrt[3]{\\dfrac{2P_{motor}}{\\rho C A_{f}}}$$ Converting to units of miles per hour for $v$, and horsepower for $P_{motor}$: $$v=\\; 2.2 \\sqrt[3]{\\dfrac{HP_{motor} \\cdot 745\\cdot 2}{\\rho\\mbox{C}A_{f}}}$$ From this pdf, I've found values of $\\rho = 1.2 \\frac{kg}{m^{3}}$, $C=.8$, and $A_{f}=.57$. So! Questions:   1. Are my equations correct? I'm particularly concerned about the third root portion, given that drag is a quadratic. Doesn't connect in my mind, so perhaps my substitutions are faulty?   2. Are the values I have for those constants for ideal values, (i.e. perfectly enginneered kart, very unlike what I'd be able to construct myself). If so, by what general percent will my values be different?   3. Finally, is it unrealistic to call all forces other than drag insignificant? If so, by what general percent will my values for speed be too high? For reference, this calculation has a 5 hp engine at a max speed of 52 mph, 15 hp at 75 mph, and 35 hp at 100 mph."}
{"id":"121986","title":"Why lunar day lasts exactly one month?","text":"Therefore we don't see dark side of the moon. Is the core of the Moon closer to the Earth than the center of the moon? Or what is the reason? ![Lunar motion](http:\/\/i.stack.imgur.com\/xZ3f6.jpg)"}
{"id":"128263","title":"How to get funding for research on something that can revolutionize the quantum world!","text":"I think this can revolutionize the quantum world! Any ideas on how to impress physicists to get a full fledged funding for research?"}
{"id":"74135","title":"Moduli potential in Type IIB String Theory","text":"In the book _String Theory and M-Theory_ by K. Becker, M. Becker and J.H. Schwarz:   1. Why is the potential for moduli given by eq (10.168): $$\\tag{10.168 }V(T,K) ~=~ \\frac1{4\\mathcal{V}^3} \\Big( \\int_{CY_4} F \\wedge \\star F - \\frac16 \\chi T_{M2} \\Big)?$$   Maybe the answer of this question is trivial, but I cannot see how $T$ and $K$ enter this picture and get $V(T,K)$ as their potential.   2. How can one arrives at eq (10.181): $$\\tag{10.181} V~=~e^{\\mathcal{K}} \\Big( G^{a \\bar{b}} \\mathcal{D}_a W \\mathcal{D}_\\bar{b} \\bar{W} - 3|W|^2 \\Big)?$$"}
{"id":"75845","title":"Vacuum Expectation Value and the Minima of the Potential","text":"Often times in quantum field theory, you will hear people using the term \"vacuum expectation value\" when referring to the minimum of the potential $V(\\phi )$ in the Lagrangian (I'm pretty sure every source I've seen that explains the Higgs mechanism uses this terminology). However, a priori, it would seem that the term \"vacuum expectation value\" (of a field $\\phi$) should refer to $\\langle 0|\\phi |0\\rangle$, where $|0\\rangle$ is the physical vacuum of the theory (whatever that means; see my other question). What is the proof that these two coincide?"}
{"id":"75844","title":"If you flew into a black hole","text":"Would you outlive everyone? I'm coming from the point of view that time would be experienced more slowly (although not from your point of view) the denser the gravity gets."}
{"id":"75846","title":"A dielectric table is being inserted between a plate capacitor and $\\triangle U<0$ how to deduce the table is attracted to the plates?","text":"I am practicing for an exam in my Physics $2$ course. One of a previews exam questions described a plate capacitor and asked to calculate the initial energy $U_{0}$, then a dielectric table was inserted between the plates and I was asked to calculate the energy, $U_{1}$, at this state. I got that there was less energy in the final state than there was at the initial state, i.e $$ \\triangle U=U_{1}-U_{0}<0 $$ The question asked if the table is attracted to the plates or repealed from it, and the answer claimed that since there was less energy at the final state then it means that the table is attracted to the plates. I lack intuition on this and I don't understand how the conclusion was made. I tried to think it with terms of work done by the electrical field: If I assume that the table is attracted to the plates then I think that the work is negative (since I don't have to do actual work because there is an attraction), and since there was negative work done $W<0$ I have $$U_{1}=U_{0}+W<U_{0}$$ The other case leads to $U_{1}>U_{0}$ and so I get that the first is the one that occurs. But I am not to sure about this argument since $$W=\\int F\\cdot dl$$ and since there is an attraction the direction of the path is the same as the direction of the force and so $W>0$. Can someone please help me understand why there is an attraction ?"}
{"id":"75848","title":"A question about an identity in deriving Born-Infeld action","text":"I have a question in David Tong's Example Sheet 4 Problem 5b, how to verify the last equation (*) on p.2? (There is a solution for example sheet 3, but seems to be no solution for example sheet 4.) > Problem 5b: > > Show that the equations of motion arising from the Born-Infeld action are > equivalent to the beta function condition for the open string,   >  $$\\beta_\\sigma\\left(F\\right)=\\left( \\frac{1} {1 -F^2} \\right)^{\\mu > \\rho}\\partial_\\mu F_{\\rho\\sigma }=0 $$ **Note:** To do this, it will prove > very useful if you can first show the following results:   >  $$∂_μ\\left[\\operatorname{tr} \\ln(1 − F^2)\\right] = −4 ∂_\\rho > F_{μ\\sigma}\\left(\\frac{F}{1-F^2}\\right)^{\\sigma \\rho } $$ > > which requires use of the Bianchi identity for $F_{\\mu \\nu }$ and > > $$ \\tag{*} \\begin{align} \\partial _\\mu \\left( \\frac{ F}{1-F^2} > \\right)^{\\mu\\nu} &= \\left( \\frac{ F}{1-F^2} \\right)^{\\mu\\rho} \\partial_\\mu > F_{\\rho\\sigma} \\left( \\frac{ F}{1-F^2} \\right)^{\\sigma \\nu} \\\\\\ &\\qquad+ > \\left( \\frac{1} {1 -F^2} \\right)^{\\mu \\rho} \\partial_\\mu > F_{\\rho\\sigma}\\left( \\frac{ 1 }{1-F^2} \\right)^{\\sigma \\nu} \\end{align}$$ In addition, as given in question 5a > $$F_{\\mu\\nu} = \\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu} $$ My attempt to prove the problem: LHS $$\\partial_{\\mu} \\left( \\frac{ F}{1-F^2} \\right)^{\\mu\\nu} = \\partial_{\\mu} \\left[ F^{\\mu}_{\\alpha} \\left( \\frac{1}{1-F^2} \\right)^{\\alpha \\nu} \\right] = \\left( \\partial_{\\mu} F^{\\mu}_{\\alpha} \\right) \\left( \\frac{1}{1 -F^2} \\right)^{\\alpha \\nu} + F^{\\mu}_{\\alpha} \\partial_{\\mu} \\left( \\frac{1}{1 -F^2} \\right)^{\\alpha \\nu} \\tag{1} $$ Using the formula in matrix cookbook for the derivative of inverse matrix, Eq. (53) Eq. (1) becomes $$\\left( \\partial_{\\mu} F^{\\mu}_{\\alpha} \\right) \\left( \\frac{1}{1 -F^2} \\right)^{\\alpha \\nu} + 2 F^{\\mu}_{\\alpha} \\left[ \\frac{1}{1 -F^2} \\left( \\partial_{\\mu} F \\right) F \\frac{1}{1-F^2} \\right]^{\\alpha \\nu} $$   $$= \\left( \\partial_{\\mu} F^{\\mu}_{\\alpha} \\right) \\left( \\frac{1}{1 -F^2} \\right)^{\\alpha \\nu} + 2 \\left( \\frac{F}{1 -F^2} \\right)^{\\mu \\rho} \\left( \\partial_{\\mu} F \\right)_{\\rho\\sigma} \\left(\\frac{F}{1-F^2}\\right)^{\\sigma \\nu} \\tag{2} $$ The second term in Eq.(2) cancels the second term in the RHS in the problem sheet equation. We then need to show $$ \\left( \\partial_{\\mu} F^{\\mu}_{\\alpha} \\right) \\left( \\frac{1}{1 -F^2} \\right)^{\\alpha \\nu} + \\left( \\frac{F}{1 -F^2} \\right)^{\\mu \\rho} \\left( \\partial_{\\mu} F \\right)_{\\rho\\sigma} \\left(\\frac{F}{1-F^2}\\right)^{\\sigma \\nu} - \\left( \\frac{1}{1 -F^2} \\right)^{\\mu \\rho} \\left( \\partial_{\\mu} F \\right)_{\\rho\\sigma} \\left(\\frac{1}{1-F^2}\\right)^{\\sigma \\nu} =0 \\tag{3} $$ then I didn't find a way to show Eq. (3) hold. I tried to combine the second and third term, and rearrange them, but didn't got a simple expression."}
{"id":"98295","title":"A question about the constraints in BRST-Fock theories","text":"In _BRST Symmetry in the Classical and Quantum Theories of Gauge Systems_ , Henneaux says the Fock representation is not applicable to an odd number of constraints. Then he goes on to say that the Kugo-Ojima quartet requires the constraints to be in pairs. For BRST theories, when are they not in pairs?"}
{"id":"98297","title":"Looking for solutions to problems in Feynman and Hibbs path integral and QM text","text":"I've just started reading Feynman and Hibbs path integrals and Quantum mechanics after a decade hiatus from my undergraduate math degree (including a few semesters of physics for engineers). It would be tremendously helpful to see the step by step solutions to some of the first set of problems (p27-28, #2.1 and on) As many as folks are willing to post! Or if anyone knows of the solutions being available on-line that would also be helpful. The specific concept question is finding the extremum of functional that is the integral of the lagrangian of a system (classical action). The easiest example is 2.1, show that for L=m\/2 (dx\/dt)^2, the extremum is m\/2*(Xb- xa)^2\/(tb-ta). I have tried manipulating the integral by replacing 'partial of L with respect to x' with 'd\/dt (partial L with respect to dx\/dt) but haven't gotten there. I know it's the easiest problem but I think if I see an example, I will be able to apply what I learn to the harder subsequent problems I hope that's enough to come off \"hold\" Thanks!"}
{"id":"98293","title":"Capacitors in series: Why is the equivalent charge the same as the individual charges","text":"Consider two parallel-plate capacitors $C_1, C_2$ in series. For the \"equivalent\" circuit, clearly $$ Q_{equiv} = C_{equiv}V$$ should hold, where V is the total voltage drop between input and output. It is also obvious that $$ Q_1 = Q_2 $$ where $Q_1$ and $Q_2$ are the charges (plus and minus on opposite sides) accumulated on each of the two sets of plates. Obviously in general, $$ V_1 \\neq V_2 \\qquad \\text{}$$ However, just because we have two capacitors with equal charge inside the circuit why is it true that for the equivalent circuit $$Q_{equiv} = Q_1 = Q_2$$ I see no obvious reason why this should be true, and this is _assumed_ , though not explained in the discusssion below: http:\/\/farside.ph.utexas.edu\/teaching\/302l\/lectures\/node46.html"}
{"id":"70728","title":"Reconciling topological insulators and topological order","text":"We make an important distinction between the topological insulators (which are essentially uncorrelated band insulators, \"with a twist\") and topological order (which covers a variety of exotic properties in certain quantum many- body ground states). The topological insulators are clearly \"topological\" in the sense of the connectedness of the single particle Hilbert space for one electron; however they are not \"robust\" in the same way as topologically ordered matter. My question is this: Topological order is certainly the more general and intriguing situation, but the notion of \"topology\" seems actually less explicit than in the topological insulators. Is there an easy way to reconcile this? Perhaps a starting point might be, can we imagine a \"topological insulator in Fock space\"? Would such a beast have \"long range entanglement\" and \"topological order\"? **Edit:** While this has received very nice answers, I should maybe clarify what I'm looking for a bit; I'm aware of the \"standard definitions\" of (symmetry protected) topological insulators and topological order and why they are very different phenomena. However, if I'm talking to nonexperts, I can describe topological insulators as, more or less, \"Berry phases can give rise to a nontrivial 'band geometry,' and analogous to Gauss-Bonnet there is a nice quantity calculable from this that characterizes instead the 'band topology' and this quantity is also physically measurable\" and they seem quite happy with this. On the other hand, while the connection to something like Gauss-Bonnet might be clear for topological order in \"TQFTs\" or in the ground state degeneracy, these seem a bit formal. I think my favorite answer is the adiabatic continuity (or lack thereof) that Everett pointed out, but now that I'm thinking about it perhaps what I should have asked for is -- What are the _geometric_ properties of states with topological order from which we could deduce the topological order with some kind of Chern number (but without starting from a Chern-Simons field theory and putting in the right one by hand ;) ). Is there anything like this?"}
{"id":"114829","title":"Redshift Mechanism","text":"How does the stretching\/expansion of space cause redshift in light from distant stars? What is the mechanism that causes the electromagnetic crests to be further apart?"}
{"id":"37546","title":"What is the relationship between the Higgs field and quarks?","text":"I have some difficulty considering the relative size of each and the meaning behind the shape of Higgs boson. I ask relating to the _structures_ of both the Higgs field and quarks. How is it that the structure of a Higgs boson flows into that of, for instance, a bottom-antibottom quark pair? Essentially I am asking (or at least think I am asking): If the interactions for the field to exist occurred at some point in the universe's past, the particle is expressing it's shape in relation to the field, etc, etc.. Does this mean {when viewing some of the type of symmetries seen in readouts of the possible Higgs boson decay} quarks themselves are further expressions of the same field's shape or instead some manner of deformation? ![Now well known shape of a Higgs Boson, computer generated from Wikipedia](http:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/1c\/CMS_Higgs- event.jpg\/650px-CMS_Higgs-event.jpg) This now fairly well known image from Wikipedia is a computer generated Higgs boson demonstrating simulated decay trajectories. This has often given me some considerations and can hopefully serve to slightly illustrate the structures I'm inquiring. (Knowing this is neither the boson or the quarks themselves)"}
{"id":"75269","title":"Distinguishing between an entangled and non-entangled state (mainly $S(H_A) \\otimes S(H_B)$ vs. $S(H_A \\otimes H_B)$)","text":"Say I have two quantum systems $A$ and $B$ I can look at the joint (composite) system $AB$ which is given by $H_{AB} \\in H_A \\otimes H_B$ Measuring a subsystem with respect to a collection of measurement matrices $\\textbf{M} = \\\\{M_i\\\\}_{i \\in I} \\in Meas_{I}(H_A)$ acts as measuring $AB$ with respect to $\\textbf{M} \\otimes \\mathbb{I}_B = \\\\{M_i \\otimes \\mathbb{I}_B\\\\}_{i \\in I}$ Q: Can I do this with entangled states? I know that making a measurement causes the entangled state of say 2 qubits to decompose in to two states. If I understand correctly, they are not entangled anymore. Now because of this we can seperate the states in to a sum of the product states $|AB> = \\sum \\alpha_j|j> \\otimes |\\psi_j>$ over all basis states $|j> \\in S(H_A)$. What this says to me is that we can some how distinguish between these qubit states (hence they are not entangled anymore). Do I have this right? This leads me in to the difference between $S(H_A) \\otimes S(H_B)$ vs. $S(H_A \\otimes H_B)$. Now the first case is not entangled and we have the \"product states\" of two wavefunctions\/state vectors , but in the second case we have some sort of combination (composite?) of states? I would guess this is when the states are entangled and after measurement they decompose in to say $|\\psi> \\otimes |\\phi> \\in S(H_A) \\otimes S(H_B)$. Now if I look at an entangled 2-qubit state that I know is entangled $|\\Phi> = |\\Phi^+> = \\frac{1}{\\sqrt{2}}(|0>|0> \\+ |1>|1>) \\in H_A \\otimes H_B$ (as written out in some notes) I see that we can have an entangled state embedded somehow in the tensor of two state vectors in Hilbert space. I don't know if this is supposed to be $S(H_A \\otimes H_B)$ or $S(H_A) \\otimes S(H_B)$ or something else completely. Edit: I removed some tensor math that was incorrect. I had initially thought that if some mixture of states could be decomposed and have just a $\\otimes$ and nothing such as an addition or subtraction operator it. I know that is very rudimentary but from my basic understanding of product states: http:\/\/en.wikipedia.org\/wiki\/Product_state I see that if a probability density can be written as the tensor product of two different probability densities, then we have a non-quantum correlation, although in the wiki article above, this is neither quantum or classical in nature. I however also see that a mixed state such as $\\rho_{AB}=\\frac{1}{2}(|0_A0_B⟩⟨0_A0_B|+|1_A1_B⟩⟨1_A1_B|)$ only has classical correlations. Furthermore I have found that Two states $\\rho, \\sigma$ are called $\\textbf{perfectly distinguishable}$ if there exists a measurement $M \\in Meas(H) \\text{ with } I = {0,1,...}$ such that $p_0(M,\\rho) = 1 = p_1(M,\\sigma)$. Now from my reading, in the case of a von Neumann measurement $M = \\\\{M_i\\\\}, p_i$ simplifies to $p_i = tr(M_i \\rho)$ and in the case of a complete von Neumann measurement $M = \\\\{ |i \\rangle \\langle i | \\\\}$, $p_i$ simplifies to $p_i = tr(|i \\rangle \\langle i | \\rho) = \\langle i | \\rho |i \\rangle$ So basically, I just have to take the traces of the respective measurements on probability densities, if they are equal (hence perfectly distinguishable) then the system of, say qubits, are not entangled, otherwise they are? Is this correct? Thank you, Brian"}
{"id":"41221","title":"Does the nonlocality of the preferred basis mean QM is nonlocal?","text":"Take the Mach-Zehnder interferometer as an example. A photon passes through a beam splitter, is reflected off mirrors, and interferes with itself at another half-silvered mirror. No measurements or disturbances are made in between. The preferred basis of the photon is given by which path it takes after passing the second beam splitter. The problem is, this preferred basis is nonlocal in between. Is there a problem with locality here, or is silly ol' me just plain confused?"}
{"id":"41226","title":"Role of unit vectors in cylindrical coordinates","text":"I know how the unit vectors are defined in cylindrical coords. If I have a point P, how do I express it as a combination of the unit vectors uρ, uφ and uz. In the case of Cartesian coordinates this combination is linear. But what about cylindrical coords? Does such a combination exist for them? And, BTW, could you suggest a simple Physics problem where the use of cylindrical coordinates is convenient or, in general, the reason to choose them?"}
{"id":"134577","title":"In QFT, why do fermions have to anticommute in order to insure causality?","text":"I have seen this question and I believe I understand the answer to it. However, AFAIK, only for bosons the causality condition is a vanishing commutator. For fermions we expect the _anticommutator_ $[\\phi,\\phi^\\dagger]_+$ to turn zero. The answer given to the question above does not seem to address this."}
{"id":"67481","title":"Proving Simple Harmonic Motion (direction of acceleration)","text":"A particle of mass $m=0.5kg$, is attached to a spring of natural length $l=0.6m$ and modulus of elasticity $\\lambda=60N$, and the setup is on a horizontal smooth table. The other end of the spring is attached to a fixed point $A$ on the table. The particle is then pulled so that the distance $AP=0.9m$, and is then released from rest. ($P$ is to the right of $A$, and the tension in the spring is $T$). If you take $x$ (displacement from the centre of oscillation) as increasing to the right, then you can prove SHM if you can get equation that is like this: $$a = -w^2$$ So $$-T =ma$$ (The $T$ is negative because $x$ increases to the right making right positive, and $T$ is to the left, and $a$ is positive because in SHM $a$ is always in the direction of $x$ increasing.) $$-\\frac{\\lambda x}{l} = ma$$ $$-\\frac{60x}{0.6} = 0.5a$$ $$-200x = a$$ (which fits the equation at the top so its SHM.) However if you take $x$ to increase to the left: $$T = ma$$ ($T$ is positive because $x$ increases to the left making left positive and $T$ is to the left, $a$ is positive because, again, in SHM $a$ is always in the direction of $x$ increasing.) $$\\frac{\\lambda x}{l} = ma$$ $$\\frac{60x}{0.6} = 0.5a$$ $$200x = a$$ (which doesn't fit the equation because there is no negative.) So I don't understand what I'm not getting right in the second part? how can changing the defining of the direction of $x$ increasing have such an effect? (Instead of changing the direction of $x$ increasing I could have said the particle $P$ is pushed so that the spring is compressed and $AP=0.3m$ making the $T$ act towards the right which is the same direction as $x$ increasing)."}
{"id":"86510","title":"Maxwell's Equations using Differential Forms","text":"Maxwell's Equations written with usual vector calculus are $$\\nabla \\cdot E=\\rho\/\\epsilon_0 \\qquad \\nabla \\cdot B=0$$ $$\\nabla\\times E=-\\dfrac{\\partial B}{\\partial t} \\qquad\\nabla\\times B=\\mu_0j+\\dfrac{1}{c^2}\\dfrac{\\partial E}{\\partial t}$$ now, if we are to translate into differential forms we notice something: from the first two equations, it seems that $E$ and $B$ should be $2$-forms. The reason is simple: we are taking divergence, and divergence of a vector field is equivalent to the exterior derivative of a $2$-form, so this is the first point. The second two equations, though, suggests $E$ and $B$ should be $1$-forms, because we are taking curl. Thinking of integrals, the first two we integrate over surfaces, so the integrands should be $2$-forms and the second two we integrate over paths and so the integrands should be $1$-forms. In that case, how do we represent $E$ and $B$ with differential forms, if in each equation they should be a different kind of form?"}
{"id":"83593","title":"The speed of light and fields","text":"Does the speed of light apply to the speed of (waves?) in every known field, or does it only apply to the electromagnetic field?"}
{"id":"83590","title":"Hilbert Space of (quantum) Gauge theory","text":"Since quantum Gauge theory is a quantum mechanical theory, whether someone could explain how to construct and write down **the Hilbert Space of quantum Gauge theory with spin-S**. (Are there something more rich\/subtle than just saying the Hilbert Space are composed by the state space of infinite many sets and infinite many modes of harmonic oscillators? - i.e. more rich\/subtle than usual spin-0 scalar fields' Hilbert space?) Is the quantum Gauge theory's Hilbert Space written **in a tensor product form or not** (eg. thinking about put this gauge theory **on the lattice** )? Whether there are differences for this procedure construction of Hilbert space for these three cases: (1) spin-1 quantum Gauge theory with Abelian $U(1)$ symmetry (2) spin-1 quantum Gauge theory with non-Abelian (such as $SU(N)$) symmetry (3) spin-2 quantum Gauge theory (Gravity? or anything else) Also, whether gauge-redundancy plays any roles? Is there similar thing like Faddeev-Popov ghosts happened in the path integral formalism, when one dealing with gauge-redundancy?"}
{"id":"11966","title":"How large is the information collected from an inverse femtobarn of  collisions?","text":"I ran into this while looking at measures of humongous amounts of data. How does the information (data) collected in an inverse femtobarn exposure compare to a gigabyte of data ?"}
{"id":"38743","title":"Do resistor-based fan regulators save no power at all?","text":"I have heard that the traditional resistor-based fan speed regulators are inefficient. In fact, I have noticed that such regulators tend to get hot when the fan is set to low speed. However, does that mean that they save **_no power at all_** at low speeds, or is it just that power is saved but the amount of power saved is far less than ideal? I was expecting that since power P = V \/ R^2, the overall power consumption of the fan plus regulator system would be comparatively lower at lower speeds."}
{"id":"38742","title":"Relationship between Alcubierre drive space-time evolution and speed of gravity","text":"The top rated answer to this question about the Alcubierre drive asserts, \"spacetime can dynamically evolve in a way which apparently violates special relativity,\" but according to the Wikipedia article on the speed of gravity, changes in the gravitational field propagate at the speed of light. The second answer under this question puts the limitation in broader terms: \"distortions of spacetime are as limited to travel to the speed of light as any other physical influence.\" These statements appear to contradict one another. My question is: Does the Alcubierre drive depend on space-time distortions propagating at superluminal speeds, and is that possible under general relativity? If Alcubierre drive distortions of space-time can propagate at superluminal speeds under GR then why can't gravitational waves (another traveling distortion of space-time) also propagate at superluminal speeds?"}
{"id":"30537","title":"Is the Schrödinger equation derived or postulated?","text":"I'm an undergraduate mathematics student trying to understand some quantum mechanics, but I'm having a hard time understanding what is the status of the Schrödinger equation. In some places I've read that it's just a postulate. At least, that's how I interpret e.g. the following quote: > Where did we get that (equation) from? Nowhere. It is not possible to derive > it from anything you know. It came out of the mind of Schrödinger. -- > Richard Feynman (from the Wikipedia entry on the Schrödinger equation) However, some places seem to _derive_ the Schrödinger equation: just search for \"derivation of Schrödinger equation\" in google. This motivates the question in the title: Is the Schrödinger equation derived or postulated? If it is derived, then just how is it derived, and from what principles? If it is postulated, then it surely came out of somewhere. Something like \"in these special cases it can be derived, and then we postulate it works in general\". Or maybe not? Thanks in advance, and please bear with my physical ignorance."}
{"id":"103146","title":"Phase Plot for Harmonic Oscillator","text":"This is probably gonna be a dumb question but I don't know exactly where I am making the mistake. I have been taught in highschool that simple harmonic oscillator phase plot is the $sin(\\omega t)$: ![enter image description here](http:\/\/i.stack.imgur.com\/v8EOE.png) However now that I am thinking about it, the phase is actually only the $\\phi=2\\pi f t$ part and therefore the real plot should look like this: ![enter image description here](http:\/\/i.stack.imgur.com\/syt8f.png) What is the problem with the second plot? Shouldn't the second one be the phase plot? Note: Both of the plots are for the oscillator with $f=0.25$ and $t\\in[0,10]$."}
{"id":"103142","title":"Physical significance of Taylor and Maclaurin series - What is the significance of defining a Maclaurin series in Mathematical Physics?","text":"In physics, usually Taylor series is used to express a quantity which keep changes with coordinate. For example the potential energy of a molecule changes with coordinate, so we express the potential energy as a Taylor expansion with respect to the coordinate about the equilibrium configuration. I am confusing this with the Maclaurin series in mathematics. If the equilibrium position is zero, it is Maclaurin series. We have the freedom to fix the equilibrium position at zero. In short, what is the physical significance of Maclaurin series? or else, what is the need of defining a Maclaurin series? Any physical interests on Maclaurin series? One more question I would like to add. If a quantity is changing with respect to its coordinate and doesn't have an equilibrium at any of its configuration, then how we can express it? Taylor series can be applied only if the quantity is changing about an equilibrium configuration?"}
{"id":"63764","title":"If particles can find themselves spontaneously arranged, isn't entropy actually decreasing?","text":"Take a box of gas particles. At $t = 0$, the distribution of particles is homogeneous. There is a small probability that at $t = 1$, all particles go to the left side of the box. In this case, entropy is _decreasing_. However, it is a general principle is that entropy always _increases_. So, where is the problem please?"}
{"id":"60411","title":"Does entropy really always increase (or stay the same)?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/FVjLL.jpg) Consider this image. If the big (grey) molecules were all to spontaneously move to the left, and the small ones were to move to the right, there would be an increase in order. While unlikely, wouldn't this mean that entropy decreases?"}
{"id":"21028","title":"Second law of Thermodynamics: Why is it only \"almost\" always true that entropy is non-decreasing?","text":"Wikipedia - Second law of thermodynamics: > ...the entropy of any closed system not in thermal equilibrium **almost** > always increases. I understand that the second law of thermodynamics is based on the statistical unlikelihood of fast-moving molecules to aggregate. However, if it is only \"almost always\" then why is the phenomenon stated as a law? Is it because we have not yet observed the unlikely aggregation?"}
{"id":"83041","title":"How light causes increase in temperature","text":"Temperature is the measure of movements of atoms. So if something is said to have high temperature it means that its atoms are moving fast or have high KE energy. There are basically two ways heat can be travels from on object to another. 1) conduction. 2) Radiation. My question is how does radiation causes increase in KE energy of atoms, when photons only interact with electrons? Or how does electrons going up in energy state translates to atom having more KE energy."}
{"id":"19825","title":"Torque Required For a Motor to Move an Object on Wheels?","text":"I've been attempting to calculate how much torque a motor needs to produce in order to start a stationary object on wheels moving. (The torque is being applied to the rear 2 wheels, the front 2 are on bearings.) I keep seeing Torque = Force * Radius (of the torqued wheel). I can't figure out how to calculate the Force in this equation though, so that I may find the Torque. The radius of the wheel is 3cm. The weight of the entire object (including wheels and everything else) is 5kg. I don't need an incredibly accurate result, so I haven't even been trying to factor in friction from the non-powered front wheels' bearings. I tried something with a friction coefficient and gravity, but the more I read the less I believe that my calculation was correct. Can anyone point me in the right direction?"}
{"id":"17319","title":"Angular Momentum Operator","text":"I'm looking at a review question I was given and it quite frankly has me stumped. \"Using matrix representations find $L^{3}_{x},L^{3}_{y},L^{3}_{z}$ and from these show that $L_{x}, L_{y},L_{z}$ satisfy the same algebraic equations.\" What has me stumped is the $L^{3}_{x},L^{3}_{y},L^{3}_{z}$, I'm not even sure how it makes sense to talk of the cube of L? Any pointers in the right direction here would be incredibly helpful!"}
{"id":"23797","title":"What does it mean for two objects to \"touch\"?","text":"If you've ever been annoyingly poked by a geek, you might be familiar with the semi-nerdy obnoxious response of > \"I'm not _actually_ touching you! The electrons in the atoms of my skin are > just getting really _close_ to yours!\" Expanding on this a little bit, it seems the obnoxious geek is right. After all, consider Zeno's paradox. Every time you try to touch two objects together, you have to get them halfway there, then quarter-way, etc. In other words, there's always a infinitesimal distance in between the two objects. Atoms _don't_ \"touch\" each other; even the protons and neutrons in the nucleus of an atom aren't \"touching\" each other. So what does it mean for two objects to touch each other?   1. Are atoms that join to form a molecule \"touching\"? I suppose the _atoms_ are touching, because their is some overlap, but the subatomic particles are just whizzing around avoiding each other. If this is the case, should \"touching\" just be defined relative to some context? I.e, if I touch your hand, our hands are touching, but unless you pick up some of my DNA, the molecules in our hands aren't touching? And since the molecules aren't changing, the atoms aren't touching either?    2. Is there really no such thing as \"touching\"?"}
{"id":"71846","title":"Is the electromagnetic force responsible for contact forces?","text":"It is commonly stated that there are four fundamental forces, or interactions, in nature. It is natural to consider which of those is responsible for the normal force we meet in elementary physics. What is the nature of the force exerted on a body by the floor on which it rests? Of the four forces, the only one that seems relevant is the electromagnetic force, and I always assumed the repelling force in question is ultimately the EM one. The Wikipedia article on contact forces seems to agree: > Molecular and quantum physics show that the electromagnetic force is the > fundamental interaction responsible for contact forces. An alternative explanation is that the force results from Pauli's Exclusion Principle; however, reasonable as it may seem, it doesn't seem to answer the question of what _force_ is actually at play here. The exclusion principle isn't a force, or at least it's not normally described as such. I've recently come across a rather forcefully presented argument that states that the normal force is a macroscopic force that is not directly reducible to the EM force, or to any of the other forces; rather, it does indeed result from Pauli's Exclusion Principle by way of interaction of electron wave functions. The argument further claims that attempts to correct the Wikipedia article have met with the usual \"source needed\" objections, and that this topic is either misunderstood by or poorly covered in the standard texts. (The argument can be found here). Which way is it?"}
{"id":"15052","title":"Graphene space elevator possible?","text":"I just read this story on MIT working on industrial scale, km^2 sheet production of graphene. A quick check of Wikipedia on graphene and Wikipedia on space elevator tells me > Measurements have shown that graphene has a breaking strength 200 times > greater than steel, with a tensile strength of **130 GPa** (19,000,000 psi) and > The largest holdup to Edwards' proposed design is the technological limit of > the tether material. His calculations call for a fiber composed of epoxy- > bonded carbon nanotubes with a minimal tensile strength of **130 GPa** (19 > million psi) (including a safety factor of 2) Does this mean we may soon actually have the material for a space elevator?"}
{"id":"6562","title":"Singing: Resonance body open-closed or closed-closed?","text":"Googling yields contradictory results, so here my question: When I sing, my vocal chords vibrate, but my whole body is the resonance body, right? So I would say that when I think about standing waves etc. I must think of a resonantor that's closed (i.e. hard boundary conditions) at both ends, because my bottom end is certainly closed, and the top end has the vocal chords which are vibrating similar to woodwind instruments, so it's closed on that end as well. If I am mistaken, could someone point out the physics behind this?"}
{"id":"99568","title":"Energy to move The Earth to mars' orbit?","text":"How much energy would it take to add to the earth's orbital velocity to push the earth into mars' orbit? My thoughts: So, in 1-2 billion years, the sun will gradually get brighter. Suppose a future civilization wanted to gradually move the earth farther from the sun. And for, now, please ignore that Mars is already in Mars' orbit. let's pretend there is room there. I'm wondering what the answer looks like, in power terms, over the course of a billion years? are we talking more power than modern civilization, or if we have a billion years can we do it with just a small generator's worth of power output?"}
{"id":"118901","title":"Photoelectric effect at low frequencies","text":"Through the photoelectric effect among many others we learn that light is actually comprised of discreet quanta of energy. That's because of the energy of the emitted electrons as well as the minimum frequency of incident light that is able to start the process. A photon of frequency (energy) less than the threshold frequency is not able to free the electron from the surface. However if the required energy to free the electron was greater than this frequency by an integral value why couldn't the electron simply absorb two photons or three photons (or any required number of photons) so as to be released? In this case even with frequencies lesser than the threshold frequency there would be photoelectric effect."}
{"id":"39168","title":"Schrodinger and thermodynamics","text":"I heard that Schrodinger pointed out that (classical\/statistical) thermodynamics is impaired by logical inconsistencies and conceptual ambiguities. I am not sure why he said this and what he is talking about. Can anyone point some direction to study what he said?"}
{"id":"39162","title":"Metric tensor and its inverse","text":"1. Is it **always** allowed to represent the metric tensor $g_{\\mu \\nu}$ in General Relativity as a $4\\times 4$ matrix?   2. If the last one is represented for example with a $4\\times 4$ matrix $\\mathbf{A}\\in{\\mathbb{R}^{4\\times 4}}$ is it always true that its metric inverse $g^{\\mu \\nu}$ is the **ordinary** matrix inverse $\\mathbf{A}^{-1}$ which one can calculate with ordinary linear algebra methods? Maybe this question may sound silly, but I've understood that in differential geometry, also simple things become complicated..."}
{"id":"46688","title":"Why don't black holes have magnetic hair?","text":"> **Possible Duplicate:**   >  What happens to an embedded magnetic field when a black hole is formed from > rotating charged dust? It is well stablished that the only hair a black hole can have is:   1. mass   2. angular momentum   3. electric charge But I can't help but wonder about the last two: a rotating charge will generate a current, that will generate a magnetic field. So, how is it even possible that the electric charge and angular momentum together will not generate a magnetic field? Since we are forbidden by the event horizon to measure any distribution of the electric charge, it means that the charge must be evenly distributed over the event horizon, and for sure we know that a rotating charged shell will generate a magnetic field What is amiss in this picture?"}
{"id":"19600","title":"Why is Mendel Sachs's work not taken seriously? Or is it?","text":"Back in college I remember coming across a few books in the physics library by Mendel Sachs. Examples are: _General Relativity and Matter_ _Quantum Mechanics and Gravity_ _Quantum Mechanics from General Relativity_ Here is something on the arXiv involving some of his work. In these books (which I note are also strangely available in most physics department libraries) he describes a program involving re-casting GR using quaternions. He does things that seem remarkable like deriving QM as a low- energy limit of GR. I don't have the GR background to unequivocally verify or reject his work, but this guy has been around for decades, and I have never found any paper or article that seriously \"debunks\" any of his work. It just seems like he is ignored. Are there glaring holes in his work? Is he just a complete crackpot? What is the deal?"}
{"id":"19602","title":"Gauge invariant Chern-Simons Lagrangian","text":"I have to prove the (non abelian) gauge invariance of the following lagrangian (for a certain value of $\\lambda$): $$\\mathcal L= -\\frac14 F^{\\mu\\nu}_aF_{\\mu\\nu}^a + \\frac{k}{4\\pi}\\epsilon^{\\mu\\nu\\rho}\\operatorname{tr}[A_{\\rho}\\partial_{\\mu}A_{\\nu} + \\lambda A_{\\mu}A_{\\nu}A_{\\rho}]$$ Is there an easier way than expliciting every component of A, going to first order gauge transformation, ... ? Cause it seems really ugly to me."}
{"id":"46467","title":"How neutrinos can be harmful?","text":"What are the circumstances in which neutrinos can harm humans or even kill them.?"}
{"id":"81356","title":"Uncertainty in momentum of an excited electron trapped in a box (1D)?","text":"An electron is trapped in a one-dimensional well of width $0.132\\,$nm. The electron is in the ninth excited state ($n=10$) state. What is the uncertainty in its momentum? The problem gives a hint to use the equation $\\Delta{p_x}=\\sqrt{(p_x^2)_{avg}}$ (the root mean square of the momentum of the particle). I know the wavefunction of the electron is this: $$\\psi(x)=\\sqrt{\\frac{2}{L}}\\sin\\left(\\frac{10\\pi}{L}x\\right)$$ But since the probability density function is a function of position, and not momentum, how do I calculate the RMS of the momentum from this?"}
{"id":"112113","title":"What is the difference between Biot-Savart law and Ampere's law?","text":"What is the difference between these laws? Which law is more useful? When to use Ampere's law and when to use Biot-Savart law?"}
{"id":"41964","title":"Thermodynamically reversed black holes, firewalls, Casimir effect, null energy condition violations","text":"Scott Aaronson asked a very deep question at Hawking radiation and reversibility about what happens if black hole evolution is reversed thermodynamically. Most of the commenters missed his point entirely. Of course it's overwhelmingly improbable statistically to set up the initial conditions needed to get a thermodynamically reversed black hole evolution. But that's not the point of the question at all. It is theoretically possible to have a thermodynamically reversed black hole evolution as a valid evolving state by CPT symmetry, even if it's improbable for all practical purposes. As it exists as a possible solution, we can ask questions about the properties of such a solution. In a thermodynamically probable evolution, we can have a massive object with a significant fraction of the black hole's mass fall into a black hole, and the blak hole automatically increases in mass by a huge fraction over a timescale of order of the Schwarzschild radius R. Then, it slowly evaporates away Hawking radiation with a remaining lifetime of order $R^3$. In the thermodynamically reversed version of this event, over a time period of order $R^3$, thermal radiation is fed into a black hole which increases in size very gradually as a result. There is an incredible fine-tuning of the ingoing radiation fed in with fine-tuned higher order multipartite entanglement (this is a theoretical state, not a practical engineering one) so that as a result, over the time period of order $R^3$, no Hawking radiation is _emitted_ from the black hole because of fine-tuned quantum cancellations between the contributions from the black hole, and the contribution from frequency mixing of the radiation fed into the hole. Then, after a time period of order $R^3$, suddenly over a time period of order $R$, the black hole emits as \"Hawking radiation\" a massive body a huge fraction of its mass, and the black hole's mass decreases accordingly. The area of its event horizon goes down by a huge fraction over a time period of order R. According to Raychaudhuri's optical equation, this requires a significant negative null energy. This is too large to come from Casimir effects, as that would require a time period of order $R^3$ to lose that much black hole mass. So, what is the origin of this huge negative null energy? Let's look at what happens just outside the black hole before the massive body is ejected. In the semiclassical approximation, the massive body must have come from a firewall ( What are cosmological \"firewalls\"?) at an exponentially small distance above the event horizon. This exponential factor comes from the very high boost factor and Lorentz contraction. Semiclassically, this firewall couldn't have originated from entangled Hawking pair production just outside the horizon. So, it must have originated from the fine-tuned radiation fed into the hole. What if a probe is sent during an earlier time to measure this firewall? Unfortunately, this probably can't be done. See, we are conditioning a significant part of the future state by stipulating that a huge massive body is ejected as Hawking radiation. To further specify a probe hovering just above the horizon requires in addition a significant condition on the past. Ordinarily, with no thermodynamic conspiracies, we only have past conditions and no future conditions, and so, there's no problem. However, such fine-tuned conditioning of both the past and future might actually be impossible?"}
{"id":"17521","title":"Electron Positron annihilation Feynman Diagram","text":"![](http:\/\/upload.wikimedia.org\/wikipedia\/commons\/b\/ba\/Feynman_EP_Annihilation.svg) I am having some trouble understanding this fenyman diagram, it seems to indicate that the electron produces the positron, as the arrow of the positron is pointing from the electron. Additionally the arrow is in directed downwards, implying that the particle is going backwards in time? Is this diagram wrong, or does the arrow mean something else, or does the positron go back in time?!?"}
{"id":"13451","title":"Why is time special?","text":"In Special Relativity, the spacetime interval between two events is $s^2 = -(c{\\Delta}t)^2+({\\Delta}x)^2+({\\Delta}y)^2+({\\Delta}z)^2$ giving the Minkowski metric $\\eta_{\\mu\\nu}=\\text{diag}(-1, 1, 1, 1)$. What is the justification for making time have a negative coefficient, and how closely is that related to the 2nd law of thermodynamics? Sure, by letting $\\eta = \\text{diag}(1, 1, 1, 1)$, we get a pretty boring spacetime, and the boosts in the Poincaré group become trig instead of hyperbolic functions, but what's the physical reasoning behind this?"}
{"id":"12249","title":"What is the volume of a parcel of air of a giving weight at average sea level pressure","text":"I know this is probably some basic math once you have the figures, but my Google-fu is failing me. Say I have **5 pounds of air**. Air meaning, the typical composition of the atmosphere of Earth. And by 5 pounds I mean the weight of the mass of the air, not it's pressure or anything like that. I want to know approximately how much **volume** that amount of air would take up at a given pressure -- specifically, sea level pressure. I know that can vary -- lets just say its a typical warm summer day, or just an average. I'm not looking for exact numbers. Actually, I'd rather a subjective interpretation of the exact figure if you can offer one. Something like 'about the volume of the air in a 12x12 room', or what have you. I have no idea how big 5 pounds of air is."}
{"id":"101459","title":"Why are radio waves in the 1.43 - 2.5 Mhz range invisible?","text":"Visible light diapason is 400 - 700 nm which is 1.43 - 2.5 Mhz. If using an antenna I would broadcast steady sinusoidal wave in this range, why the EM emitted by the antenna are not visible? Suppose the power source for antenna in a range of Kilo or Mega watts"}
{"id":"62402","title":"Can a magnet spin a small machine?","text":"If you were to build an upright machine that had a rod fitted into a ball bearing at the base and was secured to another rod at the top of the initial rod extending perpendicularly. To which you attached a magnet, and around this machine you also placed several upright rods containing magnets that would push\/pull the magnet attached to the rod, would this cause the initial magnet to spin the machine for some period of time? (Assuming I'm giving the machine a push to start the motion initially) Inevitably, provided the above would work, friction would lead to an eventual slowing. With that said is there anyway to postpone that slowing for a reasonable period of time? Can magnetic shielding be used in any fashion to maintain the motion for longer?"}
{"id":"81973","title":"Vortices and chemical potential in topological superconductors","text":"I am trying to read up some review articles about Majorana physics in topological material, but I am not really familiar with the condensed matter terminology (with condensed matter in general I should say) since I come from a high-energy background, so I come across quite a lot of vocabulary and visualization issues. From the little that I know from semi-conductor physics, a gap is an energy difference between two bands such that a particle cannot go from one band to the other without being given at least that amount of energy. With that picture of band structure in mind, I don't really know what a gap potential is, nor how it can have vortices in it. For instance, in a $p_x + i p_y$ superconductor in 2D, it is said that Majorana fermions appear in vortices in the superconducting pairing potential, or when the gap is closed by variations in the chemical potential. I am wondering if there is an intuitive picture of what a \"pairing potential vortex\" is, without getting into solving the BdG equations for the superconductor, and on how Majorana fermions actually appear in them? Moreover, another question that pops into mind is related to the use of chemical potential in the Hamiltonians describing superconductors. Statistical mechanics tells us that the chemical potential is the energy necessary to add one particle to a system from a reservoir, and also conveniently describes the energy costs related to diffusion processes in solutions. How does one interpret the chemical potential in a superconductor then? I have read somewhere that a non-homogeneous chemical potential $\\mu(x)$ is a sign of an electric field $E(x) \\sim \\mu'(x)$, so it seems there would be a relation between $\\mu$ and the electrostatic potential, but I don't find anything information that explains the relations between all these quantities in superconductors."}
{"id":"20656","title":"Does a straight water hose issue water at a greater pressure than a Coiled water hose of same diameter and length?","text":"I have a one BHP water pump, the water pressure of a coiled hose connected to the water pump output side was not that great. Would an unwound water hose produce greater water pressure? [Friction Losses?] Thanks, Alan"}
{"id":"20651","title":"Is the long range neutron-antineutron interaction repulsive?","text":"I can model this interaction as Zee does in \"Quantum field theory in a nutshell\". In chapter I.4 section \"from particle to force\" he uses two delta functions for the source. The integral gives $E=-\\frac{1}{4\\pi r}e^{-mr}$ I consider $J=\\delta^{(3)}(x-x_1)-\\delta^{(3)}(x-x_2)$. Im assuming that this represents a particle and an antiparticle. I have done the calculations quantum and classically and I obtain $E=\\frac{1}{4\\pi r}e^{-mr}$ So as this simple model is used to describe the long range nucleon interactions I think I could conclude that this is the potential interaction between a neutron and an antineutron, a repulsive Yukawa potential. I want to know If Im right. Textbooks dont talk about this and Zee seems to say that the force is always attractive."}
{"id":"14695","title":"Why does the fundamental mode of a recorder disappear when you blow harder?","text":"I have a simple recorder, like this: ![enter image description here](http:\/\/i.stack.imgur.com\/eYO9G.jpg) When I cover all the holes and blow gently, it blows at about 550 Hz, but when I blow more forcefully, it jumps an octave and blows 1100 Hz. What's the physical difference between blowing gently and blowing forcefully that the recorder suddenly jumps an octave and the fundamental mode is no longer audible?"}
{"id":"80000","title":"How Galaxy is formed?","text":"Given the distance among stars (the most massive objective in the space) is so huge, the difference of order of magnitude is about 7. And also, since gravity is such a weak force, how is it likely for gravity to shape galaxy the way it is? Would it make more sense to say galaxy is collapsing? I would like to know if my reasoning is improper, where goes wrong, and how to test my reasoning or your reasoning with experiment?"}
{"id":"20752","title":"Electrodynamics textbook that emphasizes applications","text":"Which textbook in Electrodynamics which emphasizes practical applications and real life examples would you recommend for undergraduates?"}
{"id":"100345","title":"Determining the Hodge numbers of some orbifold examples","text":"I'm currently reading about complex geometry in order to get a feeling of how to determine the Hodge numbers, e.g. of certain orbifold constructions. Since I'm a physicist with no deeper mathematical knowledge in algebraic geometry and complex geometry, I apologize for the following possibly trivial questions. In the following I'm referring to two examples of the string theory book by Becker², Schwarz. Example 1 (p. 368): $T^4\/\\mathbb{Z}_2$, i.e. $z_1 \\sim z_1+1$ and $z_2 \\sim z_2 + i$, where the $\\mathbb{Z}_2$ isometry is generated by $\\mathcal{I}:(z_1,z_2) \\to (-z_1,-z_2)$. First of all, it is easy to see that there is no invariant $(1,2)-$form, hence $h^{1,2}=0$ and that $dz_1 \\wedge dz_2$ is the only $(2,0)-$form, i.e. $h^{2,0}=1$. Moreover, there are four invariant $(1,1)-$forms (they're obvious, so I'm not gonna write them down) which contribute to $h^{1,1}$. But there are in total also $16$ fixed points of the orbifold which can be blown up à la Eguchi-Hanson. It is claimed that $h^{1,1}=4+1 \\times 16 =20$ because there is one $(1,1)-$cycle for every blown up singularity. Sorry for this questions: How do I see that there is just one $(1,1)-$form for every singularity? And why aren't there cycles from the EH-Space that contribute to $h^{2,2}$ and the other Hodge-numbers? Example 2 (pp. 372): Now we consider $T^2 \\times T^2\/ \\mathbb{Z}_3$ with the action $(z_1,z_2) \\to (\\omega z_1, \\omega^{-1}z_2)$ with $\\omega =\\exp(2\\pi i\/3)$. One has then $9$ singularities.   In this case there are $2$ invariant $(1,1)-$forms. It is claimed that there are now $2$ $(1,1)-$cycles for every blow-up, i.e. $h^{1,1}=2+ 2 \\times 9 =20$. Why are there now two $(1,1)-$cycles for each singularity? And why aren't there other cycles contributing to the remaining Hodge-numbers? I would be happy if someone could explain me the intuition and the maths behind this counting. psm"}
{"id":"133012","title":"The relation between energy quanta of an Einstein solid and the equipartition value of heat capacity","text":"Consider an Einstein solid with quantized energy values $U=q\\epsilon$ and $N$ oscillators. I calculated some values of an Einstein solid numerically through a function in R (at the bottom of the post). In an exercise I'm supposed to estimate $\\epsilon$. Apparently the answer is that I can estimate it by showing that $kT\/\\epsilon$ is approximately $1\/3$ when the heat capacity reaches half of its maximum value (this value is called the equipartition value). I find this conclusion really confusing, since the energy quanta don't have anything to do with the heat capacity (or do they?). The **exact** wording in the solution of the exercise is > Note that in the high-temperature limit the heat capacity approaches the > result predicted by the equipartition theorem, $C=Nk$ (since there are two > degrees of freedom per oscillator). Below $kT=\\epsilon$. however, the heat > capacity falls off dramatically. (...) The value of $\\epsilon$ can be > estimated by noting that the heat capacity reaches half of its equipartition > value at $kT\\approx\\epsilon\/3$. Could anyone please explain how does conclusion works? I can't get a grasp on understanding the relation between the energy quanta and the equipartition value of heat capacity. The function I wrote to produce a table uses column names such as `C_Nk` where I in fact mean $C\/Nk$, R just doesn't handle slashes in function names so well:               install.packages('gmp');install.packages('Rmpfr');library('gmp');library('Rmpfr')     EinsteinProperties <- function(N){          q_total=100          q=0:q_total          Omega=formatMpfr(chooseMpfr(N+q-1,q),digits=2)          S_k=formatMpfr(log(chooseMpfr(N+q-1,q)),digits=4)          kT_eps=rep(0,length(q))          C_Nk=rep(0,length(q))          for(i in 0:q_total){              if(i>0&&i<q_total){                  kT_eps[i+1]=formatMpfr(2\/(log(chooseMpfr(N+(i+1)-1,i+1))-log(chooseMpfr(N+(i-1)-1,i-1))),digits=2)              }else if(i==0||i==q_total){                  kT_eps[i+1]=0              }          }          for(i in (0:q_total)){              if(i==1){                  C_Nk[i+1]=formatMpfr(2\/(2\/(log(chooseMpfr(N+(i+2)-1,i+2))-log(chooseMpfr(N+(i)-1,i))))\/N,digits=2)              } else if(i>1&&i<q_total){                  C_Nk[i+1]=formatMpfr(2\/(2\/(log(chooseMpfr(N+(i+2)-1,i+2))-log(chooseMpfr(N+(i)-1,i)))-2\/(log(chooseMpfr(N+(i)-1,i))-log(chooseMpfr(N+(i-2)-1,i-2))))\/N,digits=2)              } else {                 C_Nk[i+1]=0              }          }          table <- cbind(q,Omega,S_k,kT_eps,C_Nk)          layout(matrix(1:2,1,2))          graph_S_k <- plot(q,S_k,type=\"l\",xlab=\"U\/eps\",ylab=\"S\/k\",main=\"Entropy\",xlim=c(0,100))          graph_C_Nk <- plot(kT_eps[2:q_total],C_Nk[2:q_total],type=\"l\",xlab=\"kT\/eps\",ylab=\"C\/Nk\",main=\"Heat capacity\",xlim=c(0,2),ylim=c(0,1))          return(table)     }     EinsteinProperties(50)"}
{"id":"121617","title":"Energy of damped harmonic oscillator begins to increase with very large Q in numerical integration","text":"I have numerically integrated the (reduced) homogeneous equation of a damped harmonic oscillator in order to see how the error propagates. $$\\frac{d^2 X}{d\\phi^2} + \\frac{1}{Q}\\frac{dX}{d\\phi}+X(\\phi) = 0$$ where $\\phi = 2 \\pi \\tau$ and $\\tau$ is the reduced time In the process I have also extracted the logarithm of the energy and plotted it against time, for different values of the quality factor, Q. One can expect the energy to decay slower for larger Q. $$ E = P^2 + X^2$$ where $P = \\dot{X}$ What I found is that for very large values of Q, the energy of the oscillator begins to **increase**. I checked my code several times and I couldn't see anything wrong with it so I suspected that it purely something to do with numbers so I began playing around with them, but first here is the graph. ![enter image description here](http:\/\/i.stack.imgur.com\/9LTVF.png) What I found is odd; say I'm using a step size of $1 \\times 10^{-n}$ . If Q's magnitude matches or exceeds $10^{n}$ the energy begins to increase! What is causing this? **What is the connection between Q and the step size?** **Also, should the energy be oscillating?** (If you zoom in on the red and green lines on the graph you will see that it is oscillating much slower there.) * * * Here is how I am integrating the equation for those interested:   1. Let $\\dot{X} = P$    2. Thus original equation becomes $\\dot{P} + \\frac{1}{Q} P + X = 0$    3. Now we have two 1st order DEs.    4. Using discrete steps we can roughly find the next point of $X$ and $P$ by considering that $\\dot{X} = \\frac{dX}{d\\phi} = \\frac{X_{n+1} - X_n}{\\Delta \\phi}$ and approaching the other eqution in the same manner.    5. We can then rearrange for $X_{n+1}$ and $P_{n+1}$ and create a code that calculates those points, given some basic initial conditions."}
{"id":"24731","title":"Does the current acceleration of universe imply that our universe is open?","text":"Does the current acceleration of universe imply that our universe is open? If the universe is closed, from the Friedmann's equation, the acceleration of universe wouldn't be possible, would it be? (Of course, except for the very earlier time of inflation era.)"}
{"id":"8636","title":"Why does the water in the toilet move around so much on stormy days?","text":"On calm days, the water in the toilet looks completely still. But when it's rainy and windy out, the water looks like it moves and pulsates. Why is this?"}
{"id":"95368","title":"Calculating dragging frame of satellite orbiting Earth","text":"Say there is a satellite polar-orbiting the Earth at 600km. How much would the satellite be dragged additionally due to dragging force? Such that $$ \\Omega = \\dfrac{r_s \\alpha r c}{\\rho^2(r^2 + \\alpha^2) + r_s \\alpha^2 r sin^2 \\theta}$$ such that: $r_s = \\dfrac{2GM}{c^2}$ is the Schwarzchild radius. $\\alpha = \\dfrac{J}{Mc}$ $\\rho^{2} = r^{2} + \\alpha^{2} \\cos^{2} \\theta$ $\\Delta = r^{2} - r_{s} r + \\alpha^{2}$ I'm a bit confused how I'm supposed to use this formula as $\\Omega$ denotes angular speed not distance doesn't it? Thank you so much for any help"}
{"id":"119330","title":"Reconciling \"The Big Crunch\" with the 2nd Law of Thermodynamics","text":"Assume \"The Big Crunch\" scenario (the universe will collapse to a singularity). In this case, I think of the entire universe as an isolated system; in the \"Big Crunch\" scenario, it seems to me gravity itself would act as the \"walls\" of the system, keeping all the matter and energy inside. Thus, by the second law of thermodynamics, entropy will increase. However, it seems to decrease (everything is being smashed together). I know black holes get around this through the emission of Hawking radiation, but this is the emission of radiation from black holes to the universe. In the big crunch scenario, by definition, there is no universe for the radiation to escape to because the universe itself is collapsing, taking all radiation with it."}
{"id":"129946","title":"Interferometer problem about periodic fringe patterns","text":"Please,kindly help me; my teacher assigned me problem set, and she never went over anything about interferometers in class. Also, there is almost no information about about how to solve interferometer problems in my book. I have no idea how to set this problem up, and I have no intuition concerning how I should solve it... Can someone help me set this problem up and explain it? I know if this example is solved, I will be able to solve the rest of the problems: Sodium D lines, of wavelength(a)=589.0 nanometers and wavelength(b)=589.6 nanometers are used on a Michelson interferometer. When the first mirror moves, these lines periodically appear and disappear. Explain, in detail, this phenomenon and write an equation that would allow you to know how much the mirror must move to make the lines reappear and disappear one time."}
{"id":"111492","title":"Euler-Lagrange equation for continuous systems","text":"I'm having a little trouble with wrapping my head around a part of a method which is fairly 'new' in some fashions to me. I imagine it should be fairly obvious, but I am not seeing something at the moment. Let's take the Lagrangian density to be $$\\mathcal{ L}={1\\over{2}}\\left(\\mu\\dot{\\phi}^2-Y(\\partial_x\\phi)^2\\right),$$ where $\\mu$ is mass density of rod and $Y$ is Young's modulus. One \"should\" be able to put this into the equation $${\\partial\\over \\partial t}\\left({\\partial \\mathcal{L}\\over{\\partial(\\partial_t \\phi)}}\\right)+{\\partial\\over \\partial x}\\left({\\partial \\mathcal{L}\\over{\\partial(\\partial_x \\phi)}}\\right)={\\partial \\mathcal{L}\\over{\\partial \\phi}}$$ to get the equation of motion for the rod $$\\mu\\ddot\\phi=Y\\partial_{xx}\\phi,$$ right? The trouble is I'm not exactly seeing where to go with the LHS of the E-L equation applied to $\\mathcal L$ at the moment. I'm hoping someone would be so kind as to demonstrate that in a few lines."}
{"id":"9992","title":"Any resources to learn how to interpret formulae\/diagrams?","text":"Are there any materials exploring intermediate levels (of any sub-subject) which give plain English descriptions along side formulas\/equations\/\/diagrams\/notations?"}
{"id":"126397","title":"Space Warps & Frame Invariance","text":"The Alcubierre Warp Drive in Higher Dimensional Spacetime (H. G. White & E. W. Davis), among other sources, proposes that the x-axis symmetry paradox for Alcubierre-style warp drives can be resolved by interpreting the warp field as producing a scalar boost on a pre-existing velocity. If that is the case, how can it obey frame invariance? In other words, how does the boost \"know\" which velocity, in which inertial frame, it's supposed to be multiplying?"}
{"id":"95982","title":"What force keeps electrostatic charge from exiting a metals surface","text":"I understand that in electrostatics all charges are 'static' due to equal forces acting upon them. This then leads to the fact that the arising electric field must be perpendicular to the metals surface at the surface. What i don't understand is what force is counteracting the electric field. I have read that there is a potential well but am not sure how this works. Could someone point me in the right direction? Thanks."}
{"id":"95981","title":"How does QFT help with defining measurement in Quantum Mechanics?","text":"I learned that a quantum system has an \"overall state\", a vector in a Hilbert space. That Hilbert space can be decomposed in a basis of \"basic states\". For example if in the Universe there is only a free electron with its own spin, and I pick an axis $z$, there is an infinite basis of \"position+spin\", let's say $|(0,0,0);1\/2>, |(0,0,0);-1\/2>, |(0.1,0,0);1\/2>\\ldots$ I cannot obviously write all possible positions, but that was to give the idea. Some $\\psi (\\vec{r},s)$ contains the coefficients for all the elements of this basis and is given. This is the \"overall state\" of my \"one-electron Universe\" I was said that there is \"measurement\" that makes the $\\psi$ collapse to one of the \"basic states\" and can give the observer the result (position + spin in the $z$ axis) Since a measurement involves an interaction with some sort of detector I figured out that I would need QFT (the possibility to have multiple particles and interaction) to understand measurement and add a detector into my Universe. I thought that the dynamic of the interaction between the particles of the detector and the electron could act in a way that the $\\psi$ collapsed (I hoped that even in QFT there was something analogous to the original $\\psi$ of the electron). Now I couldn't study QFT but from what I grasped it is much more complicated and the reason of the collapsing of the wave function is not explained by an \"interaction\". The wave function itself is subjective and depends on the knowledge of the observer. So I am lead to believe that QFT does not contain anything that explains the collapse of the wave function. Is it true? I know that I presented a rather subjective view. I hope that they are shared by many and that an answer that points out my misunderstandings could help others. I think that many people in the quest for a more \"classical\" explanation of QM resort to thinking that when they would see how the interactions work \"really\" so as to describe a fully fledged measurement equipment (photons, QED, QCD...) they would figure that out."}
{"id":"114517","title":"LIGO sensitivity in terms of minimum received power per area","text":"I've been looking at LIGO figures for gravitational wave sensitivity here and it seem to be displayed in an adimensional strain ratio (which I assume that is more or less equivalent to metric perturbation $h_{\\mu \\nu}\/g_{\\mu \\nu}$, correct me if I'm wrong) What I'm wondering is how can I convert this sensitivity measure into something like watts\/area, as a measure of energy density flux of gravitational waves."}
{"id":"61322","title":"A fundamental equation for solitary wave and dimension analysis","text":"According to the scalar Field theory we write Lagrangian as $$\\mathcal{L}=\\frac{1}{2}\\partial^\\mu \\phi \\partial_\\mu \\phi -\\frac{m^2}{2}\\phi^2 -\\frac{\\lambda}{4!}\\phi^4 \\tag {1}$$ What I want to do is to get a equation of solitary wave (solition) from the above Lagrangian which will lead for solving many solitary problems like life-time, dimension analysis etc. My confusion is choosing the potential because it may vary and depend on the author's selection. What are the physical constraints on choosing an appropriate potential? Another rather technical expectation is solitary wave is nonlinear equation so how will I write down the equation as 2nd order differential equation with respect to time?"}
{"id":"116912","title":"What happens between two harmonics?","text":"I know that standing waves in a simple harmonic system occur when the \"echo\" of a wave overlaps completely with the original wave at a certain point in time, doubling the amplitude. And then, because both waves have the same speed and frequency but opposite direction, they move away from eachother until they cancel eachother out, and then keep moving until they overlap again to amplify the signal. But what happens halfway between two harmonics? Would it be like a half standing wave, or would the signal cancel out?"}
{"id":"126930","title":"Noether's Theorem: Foundations","text":"I'm wondering on what principles Noether's theorem foots. More precisely: _The action is a functional on the fields only. Why do we consider then variations of the space time too? In principle careful considerations, however, seem to untangle them as special field variations. So what's going on here truly?_"}
{"id":"82095","title":"Does Archimedes law depend of lack of matter?","text":"I would like to know if Archimedes law take in equation the lack of matter. Fluid density at 1. An object with density of 0 (or near 0) is put in liquid, it's fixed. The external sphere where liquid is inside is fixed too. The lack of matter change the attraction in all the sphere itself, no ? Density change with depth, no ? How forces can be calculate ? ![e](http:\/\/i.stack.imgur.com\/Jn1KB.jpg)"}
{"id":"82097","title":"Speed of sound in non-newtonian fluids","text":"Recently I saw some videos of non-newtonian (shear-thickening) fluids like corn starch mixed with water(sometimes known as oobleck), where the fluid is placed on top of a speaker cone and starts to create odd shapes as it becomes thicker in some places. Now I wondered what would happen in the following situation: Suppose a room with reflective walls is fully filled with such a fluid. On one side of the room a sound is produced by a speaker. On the other side of the room is a microphone. How will a sound wave propagate through such a substance? Does the fluid increase its viscosity before, during or after a wave moves through it? What will the speed-of-sound be in such a case? I've read that the speed of sound is much faster in solids than in liquids, but how is that for this kind of substance?"}
{"id":"69622","title":"Is there a way to fill Tank 2 from Tank 1 through Gravity alone?","text":"I am a newbie in water system design but I am currently faced with the exact situation below on my land, and I need to know whether gravity alone is sufficient in order to fill Tank 2 from Tank 1, as I already experienced backflow. Please have a look at this image: ![enter image description here](http:\/\/img819.imageshack.us\/img819\/184\/qhzf.png) _Note: let's not worry about how Tank 1 is being filled as it probably does not matter - I just made sure I never used a pipe diameter smaller than 0.5\" between Tank 1 and Tank 2._ As you can see:   * My water source is the overflow of Tank 1   * I have put a 1\" diameter pipe in the first 'downhill' section of my path   * The second, longest section of the path is a pipe of 0.5\" diameter   * The system is 'powered' through gravity alone **Questions**   * Is there any missing variable (eg path length?) on my drawing in order to resolve this system? (What are the important variables?)   * The flow at the source (the overflow of Tank 1) can be nearly 0. How does this variable influence the system? (Can a larger flow at the source help going up the 'uphill' sections between the tanks?)   * Is the 'head' variable important at all?   * I used a bigger pipe for the first section in an attempt to 'make more weight' and create sufficient pressure to go up the small uphill that follows. Should I limit the use of that bigger pipe strictly to the downhill section, and does it actually make sense to have used a bigger pipe at all?"}
{"id":"2439","title":"Solving straight-line motion question for time","text":"I apologise in advance if this question doesn't appeal to the advanced questions being asked in this Physics forum, but I'm a great fan of the Stack Exchange software and would trust the answers provided here to be more correct than that of Yahoo! Answers etc. A car is travelling with a constant speed of 80km\/h and passes a stationary motorcycle policeman. The policeman sets off in pursuit, accelerating to 80km\/h in 10 seconds reaching a constant speed of 100 km\/h after a further 5 seconds. At what time will the policeman catch up with the car? The answer in the back of the book is 32.5 seconds. The steps\/logic I completed\/used to solve the equation were:   \\- If you let x equal each other, the displacement will be the same, and the time can be solved algebraically. Therefore:   $$x=vt$$ As the car is moving at 80km\/h, we want to convert to m\/s. 80\/3.6 = 22.22m\/s   $$x=22.22t$$ As for the policeman, he reaches 22.22m\/s in 10 seconds.   $$\\begin{aligned} x &= \\frac12 (u+v) t \\\\\\ &= \\frac12 \\times 22.22 \\times 10 \\\\\\ &= 111.11 \\mathrm m \\end{aligned}$$ The policeman progresses to travel a further 5 seconds and increases his speed to 100km\/h.   100km\/h -> m\/s = 100 \/ 3.6 = 27.78m\/s. $$\\begin{aligned} x &= \\frac12 (u+v) t \\\\\\ x &= \\frac12 \\times (22.22 + 27.78) \\times 5 \\\\\\ x &= \\frac12 \\times 50 \\times 5 \\\\\\ x &= 250 \/ 2 \\\\\\ x &= 125 \\mathrm m \\end{aligned}$$ By adding these two distances together we get 236.1m. So the equation I have is:   $$22.22t = 27.78t - 236.1 $$ Which solves to let t = 42.47s which is really wrong."}
{"id":"16351","title":"Mass converted to energy in a common fire?","text":"In a common wood fire such as a campfire, is matter converted to energy or is it simply an exothermic chemical reaction and all the mass can be accounted for in the ash and soot?"}
{"id":"93976","title":"Possible spin states?","text":"Given a system of two particles with spin up and down, I have troubles to understand the possible states of this system. I would have normally thought, that the possible states are the tensor products of the states of each particle with one of the other. This would have given us the states that are listed in Wikipedia under: substituting in the four basis states. So why do we define these new states(three with total spin $1$ and one with total spin $0$) that are given by triplet and singlet representations? So maybe you don't know what I am asking so I repeat my question: What is the meaning of these triplet and singlet states, because I would have thought that the four states listed above are the possible states of a system? The reference to what I am talking about: wikipedia reference"}
{"id":"117356","title":"4-acceleration of rotating frame","text":"Consider the 3-dimensional Minkowski space $$ ds^2=dt'^2-dr'^2-r'^2d\\phi'^2 $$ Now we transform it into a rotating frame: $$ t'=t,r'=r,\\phi'=\\phi+\\omega t $$ Then the metric becomes $$ ds^2=(1-r^2\\omega^2)dt^2-dr^2-2\\omega r^2d\\phi dt-r^2 d\\phi^2 $$ Consider a static observer in the rotating frame with coordinate $(r,\\phi)=(R,0)$. Could anyone show me how to compute the 4-acceleration of this observer? I am expecting the answer to be $(0,-\\omega^2 R,0)$ based on my knowledge of Newtonian mechanics. But my calculation does not give this. I have calculated the 4-velocity to be $$ u=\\left(\\frac{1}{\\sqrt{1-R^2\\omega^2}},0,0\\right) $$ Then I use the formula $$ a^\\mu=u^\\nu\\nabla_\\nu u^\\mu=u^t\\nabla_tu^\\mu=u^t(\\partial_tu^\\mu+\\Gamma^\\mu_{t\\alpha}u^\\alpha)=u^tu^\\alpha\\Gamma^\\mu_{t\\alpha} $$ Then it turns out only $\\alpha=t,\\mu=r$ is meaningful, and $$ a^r=-\\frac{2R\\omega^2}{1-R^2\\omega^2}R^2(1-2\\omega^2R^2) $$ which I think is definitely wrong..."}
{"id":"107474","title":"Why does milli- mean 1\/1000","text":"I suppose this is also an English question, but I'll ask it here first. Why does the milli- prefix mean 1\/1000 when it sounds so much like million? According to the internet, this dates back to the 18-19th centuries. Did it make more sense back then? Are we stubbornly holding on to meanings from borrowed prefix's languages (such as milli meaning 1000 in Latin)? Not that I'm qualified in any way to make this kind of statement, but if I were able to rename our unit prefixes, I would shift everything from milli- to femto- down 3 orders of magnitude and eliminate atto-. The resulting pairs would be:               thousand-    kilo    (placeholder)     million-     mega    milli     billion-     giga    micro     trillion-    terra   nano     quadrillion- peta    pico     quintillion- exa     femto     sextillion-  zetta   zepto     ...      Which to me, sounds better. But I suppose I answered my own question by saying that since they are the units everyone uses, social inertia prevents us from changing. Would this be correct?"}
{"id":"9606","title":"Glycerol: refractive index & absorption spectra in 0.2-0.4um range","text":"Could anyone suggest where can I find absorption spectra & refractive index of Glycerol? I am specifically interested in UV range, 200-400nm, everything I was able to find out was for standard conditions only... Is there any software which can get these parameters through simulation?"}
{"id":"133113","title":"How would normal matter behave under conditions found in the core of the sun?","text":"Considering how low the power density is at the sun's core, I seem not to be able to expect what would happen to matter in case it was thrown inside the sun's core. For example, let's assume an Earth-like planet is placed at the center of the Sun's core, and with the power density of 276.5 $watts$\/$m^3$ which is very low to even raise the temperature of the Earth by any noticeable degree. That was one thought. The other thought was that, the temperature in the core is already 15 million degrees K, so any matter there should get close to this temperature quickly enough. So now I'm very confused, like will the planet stay there intact for thousands or even millions of years until it has accumulated enough energy to melt or vaporize ? Or there are other types of energy absorption the planet would experience and thus have a shorter time staying as one piece in the core ? Just for trying to get a practical and numerical answer to this question, let it be : how long can the Earth as a whole survive in the core of the sun ?"}
{"id":"63416","title":"Mathematical proof of non-negative change of entropy $\\Delta S\\geq0$","text":"I understand that we can prove that for any process that occurs in an isolated and closed system it must hold that $$\\Delta S\\geq0$$ via Clausius' theorem. My question is, how can I prove this in a mathematical way?"}
{"id":"87207","title":"Why force $F$ is $ma$ but not $md$ or $mv$? How can I observe and understand \"force\" in real life?","text":"As a layman, i can calculate approx \"displacement\" just by observing the moving object. And accurately by using a simple \"scale\". Similarly, again, I can calculate angle from origin by using displacement in $x$ and $y$ dimensions. Similarly I can use a stopwatch and scale to understand velocity. But when I read about force, first of all it confuses with the english word \"force\", we use in real life. To some extent I am sure, it has nothing to do with that. So exactly what is it in the real sense. How can a layman, see force as, just like he can see displacement or angle? Or force is just a quantity defined by physicists to simplify the combinations of $ma$, they might be facing every time. And thus came up with term \"force\" ( which is similar in spelling to english word in oxford dictionary). And lastly, why not force has just been called as something proportional to mass and it's displacement or velocity. Why something at the level of change of velocity has been used to define it."}
{"id":"113986","title":"Regarding the theory of the origin of water on earth through meteorites, why wouldn't the water evaporate on impact?","text":"Water on earth has been theorized to have come through comets trapped inside crystals. But why wouldn't that water evaporate on impact, and wouldn't the atmosphere at that time allow the vapours to escape Earth? Also, what is the current scientific opinion of the validity of this theory ?"}
{"id":"98941","title":"$\\tau$ pair production question","text":"There's a question on my homework about the process $e^{-} e^{+} \\rightarrow \\tau^{+} \\tau^{-}$. Specifically, it is claimed that the minimum energy required of the colliding positron and electron beams is slightly less than twice the $\\tau$ mass, and I am asked to explain this deviation and compute it. In the context of this course, we have only discussed the relativistic kinematics of such processes (which would predict a minimum energy of twice the $\\tau$ mass), so I am not sure what might be responsible. I'm grasping at straws here, but might it have something to do with the Coulomb interaction between the opposite charges? Thanks in advance."}
{"id":"127472","title":"Is the potential in Schrödinger equation an operator?","text":"In the Schrödinger equation in the position representation $$ i\\hbar\\frac{\\partial}{\\partial t}\\Psi(x,t) ~=~[\\frac{-\\hbar^2}{2m}\\nabla^2+V(x,t)]\\Psi(x,t), $$ is the potential $V(x,t)$ an operator acting on $\\Psi$ or merely a scalar?"}
{"id":"108056","title":"Self-inductance of a toroidal inductor","text":"I am trying to determine the self-inductance of a toroidal coil of mean radius $R$ with $N$ loops of radius $a$ with a current $I$ flowing within them. I have calculated the magnetic field by noting that it must be in the circumferential direction (i.e. in the $\\hat{\\boldsymbol{\\theta}}$ direction) and then using Ampéres Law to get: $$\\oint_{\\partial \\Sigma} \\mathbf{B}\\cdot \\mathrm{d}\\boldsymbol{\\ell}=\\mu_{0}I_{\\text{enc}} \\implies \\mathbf{B}=\\frac{\\mu_{0}NI}{2\\pi r}\\hat{\\boldsymbol{\\theta}}$$ I know that we can get the self-inductance $L$ by using the relation: $$\\Phi=LI$$ And we know that: $$\\Phi=N\\iint_{\\Sigma}\\mathbf{B}\\cdot\\mathrm{d}\\mathbf{A}$$ Where $\\Sigma$ is the area of a single loop. However, I am not sure how evaluate the surface integral, I'm not sure if perhaps I should be using the mean radius $R$ to give us: $$\\Phi=\\frac{\\mu_{0}N^{2}}{2\\pi R}\\pi a^{2}I=\\frac{\\mu_{0}a^{2}N^{2}}{2R}I$$ Or if I need to evalute the integral in a more complete way?"}
{"id":"108058","title":"Gravity through a portal","text":"Ok, this is definitely a silly question and is partly inspired by the last 20 minutes or so of the film _The Avengers_ , so look to that if you can't picture the situation. In the film a portal is opened up above new york, to somewhere in deep space. I.e. from an area where the gravitational field is effectively uniform and field lines would be parallel, to an area with negligible net gravitational acceleration. My question is, assuming this portal has literally linked the two bits of space, it makes sense that the gravity of the earth would be transmitted through the portal in some way, but in what form would the field take on the other side of the portal? Would it behave like a point source, or similarly to a diffraction slits? Would a mass slightly off to one side from the portal experience a force? Let's ignore all the practical factors like the air rushing out into the vacuum and the question of what happens on the \"other side\" of the portal (the bit you'd be looking at if you flew up from new york to above the portal) although there's plenty of scope for interesting things happening there as well."}
{"id":"23469","title":"Is fire plasma?","text":"Is Fire a Plasma? If not, what is it then? If yes why, don't we teach kids this basic example? ![enter image description here](http:\/\/i.stack.imgur.com\/MLPSn.jpg) **UPDATE:** I probably meant a regular commonplace fire of the usual temperature. That should simplify the answer."}
{"id":"114986","title":"A photon travels in space for 10 billion years. What are the odds it will arrive here without interacting with a atom on the way?","text":"Space isn't a perfect vacuum and I wonder how an image of a galaxy can travel billions of years without becoming diffused by photon collisions with space matter?"}
{"id":"79296","title":"Simple Quantum Mechanics Question about The Commutator of Translation Operators","text":"Say there is $\\hat{J} = \\exp[-i \\hat{p} l\/ \\hbar]$ and $\\hat{U}= \\exp[-i\\hat{H}t\/ \\hbar]$, where $\\hat{H}$ is time-independent. Can we say anything about $[\\hat{J},\\hat{U}]$? Is it zero? How do we show this? For example if $\\hat{H} = \\hat{p}^2 \/2m + m\\omega^2 \\hat{x}^2\/2$."}
{"id":"70891","title":"String theory and the SM spectrum","text":"Long ago, I realized this: (super)string theory can NOT give a well- defined\/unique prediction of why the electron (muon, tau) or the neutrino (any flavor) masses have the masses we measure. String theory can NOT give a concrete prediction of any SM particle mass (even resonances like the Higgs and other particles). The question is: Is this lack of predictive ability of the mass of the SM \"elementary particles\" a hint that string theory in its current status is not complete or wrong? Or perhaps even better: **Even when string theory provides a framework in which you can accomodate every Standard Model field, and hence, a priori you can have the SM spectrum, we have not a string theory explanation of the values of the masses and charges. Should a explicit derivation of particle properties like (rest) masses and (bare) charges (like the electron charge value) be \"predicted\" and \"derived\" in a more fundamental version of the theory?**"}
{"id":"81087","title":"Incorrect IR temperature reading on stainless steel?","text":"Using a Fluke 62 Mini Infrared Thermometer today on a stainless steel pipe with coolant I noticed that it gives incorrect temperature readings. There is a temperature sensor inside the coolant saying about 90° Celsius. I spray painted a spot on the pipe and the IR thermometer instantly gave more correct readings on the painted spot. So why? Why is it the coolant fluid is about 90° Celsius (both the painted spot and the temperature sensor suggest that), while the unpainted clear stainless steel says about 50° Celcius? I.e. apx 55%"}
{"id":"118768","title":"Euler Lagrange equation in different frames","text":"Suppose I have an inertial frame with coordinate $\\\\{q\\\\}$. Now I define another reference frame with coordinate $\\\\{q'(q,\\dot q,t)\\\\}$. I obtain the equation of motion in $\\\\{q'\\\\}$ in two different ways:   1. First obtain the equation of motion in $\\\\{q\\\\}$ by the Euler Lagrange equation $$\\frac{d}{dt}\\left(\\frac{\\partial L}{\\partial \\dot q}\\right)-\\frac{\\partial L}{\\partial q}=0$$ and then rewrite the equation in terms of $\\\\{q'\\\\}$.   2. First transform $L(q,t)$ to $L'(q',t)=L(q(q',t),t)$ and then obtain the equation of motion $$\\frac{d}{dt}\\left(\\frac{\\partial L'}{\\partial \\dot q'}\\right)-\\frac{\\partial L'}{\\partial q'}=0$$ Are the two answers just the same?"}
{"id":"34618","title":"Why does quantum mechanics invalidate one of locality and realism?","text":"I am approaching this from an intuitive perspective and I don't speak the language. However, I have been doing a lot of reading about Bell's Theorum and how invalidates either locality or counterfactual definiteness. From my point of view, locality is the concept that there is no way that two particles can interact with each other \"instantaneously\" at a distance (i.e. faster than light.). Realism\/counterfactual definitiveness is the concept that all particles have some true value for all variables at some instant regardless of whether or not that value was observed. Let's assume for the sake of this question that locality is true, which I believe is the current popular model in QM. Assuming locality, why do Bell's Inequalities show that conterfactual definiveness must be false? **Edit to respond to below post:** This is crazy. Let me see if I am following you. I'll use your framework. I'm not really sure what spin is but I'm seeing it as an observable phenomenon that exists as a function of a specific direction in a given particle at an instant of time, and the spin itself at any angle changes (randomly?) over time. A and B are, we are assuming, perfectly correlated so so the spin at angle θ in A is definitely the same as the spin at angle θ in B. In any given electron (let's just take A) we can define some angle θ as the angle between two 'directions' on that electron that have an exactly 99% correlated spin. There's no way we can measure those two directions on the same particle at the same time, but since A and B are perfectly correlated, we can measure A at 0 degrees and B at θ degrees and experimentally we would see that they are the same spin This is crazy. Let me see if I am following you. A and B are, we are assuming, perfectly correlated so so the spin at angle θ in A is definitely the same as the spin at angle θ in B. In any given electron (let's just take A) we can define some angle θ as the angle between two 'directions' on that electron that have an exactly 99% correlated spin. There's no way we can measure those two directions on the same particle at the same time, but since A and B are perfectly correlated, we can measure A at 0 degrees and B at θ degrees and they will be the same spin 99% of the time. Similarly, if we measure the spin at θ in A and at 2θ in B, we will see that these two values are the same spin 99% of the time. Assumingly, this will work for any two angles that are exactly θ apart -- with enough measurements they will be correlated exactly 99% of the time. However, if we measure the spin at 0 in A and at 2θ in B, they are only correlated 96% of the time. If V0 is at most 1% off from V1, and V1 is at most 1% off from V2, V0 should be anywhere from 0-2% off from V2. But, we find, according to QM and experimentation, it is not! It is less heavily correlated than that. This is where I lose you. Why does saying no to realism resolve this issue?"}
{"id":"50045","title":"What's so special about wave solutions of EM?","text":"Maxwell's equations allow for wave solutions via oscillations between electric and magnetic field content. Couldn't we generate electric waves also if that solution didn't exists? Imagine there was no magnetic field, only the electric field. If at one point in space we generated an electric field nicely varying in a sinus pattern of constant frequency, wouldn't this propagate with light speed, just like a plane EM-wave of that frequency, even though there was no magnetic field?"}
{"id":"32618","title":"Is temperature quantized?","text":"I'm learning quantum mechanics on my own. I've known that energy is quantized and I've started wondering about temperature. From thermodynamics we have: $$U=\\frac{3}{2}NkT $$ (for ideal gas, of course)   Both U and N aren't continous, so i think T shouldn't be, too. Is that formula correct also for quantum mechanics?   I'm really sorry because of my language, I'm still working on it."}
{"id":"128492","title":"Quantum Boltzmann Equation","text":"What is the Quantum Boltzmann equation and what does it describe? I think it describes the propagation of electrons and photons but I am not sure."}
{"id":"22281","title":"Relationship between nuclear spin and nuclear magnetic moment?","text":"We know that nuclear magnetic moment can be expressed in terms of the expected value for nuclear spin as: $$\\langle\\mu\\rangle =[g_lj+(g_s-g_l)\\langle s_z\\rangle]\\frac{\\mu_N}{\\hbar}$$ (Cf. Krane), where $\\vec{j}$ is the _total_ angular momentum, $\\vec{l}+\\vec{s}$. How does the expected $\\langle s_z\\rangle$ value relate to the $\\vec{j}$-component of spin, $\\langle s_j\\rangle$? Krane mentions that only that value is needed, given that it remains constant."}
{"id":"57145","title":"Electronic filter","text":"Can you explain, please, **step-by-step** how an electronic filter does work? For example, high pass filter. I know It's a trivial things, but I can't get it completely. Don't bring me formula and etc. Just explanation in three words."}
{"id":"94512","title":"Non-Conservative Behaviour of Static Electric Field","text":"Static electric fields are supposed to be conservative in nature and therefore give $0J$ work if traversed over a loop. However in the following problem I got non zero work by a static electric field. How can I explain this work ? ![enter image description here](https:\/\/lh4.googleusercontent.com\/-S3oWGGy_30Y\/UuLvVIXZnFI\/AAAAAAAAAhE\/kRutOGK9Q88\/s1668-Ut\/2014+-+2)   **Note :** This is done in field of line charge, diagram shows a cuboid type structure but that is just for differentiating between traversed path and line charge. I am writing some of the important parts\/equations here, but I have written the whole derivation here. $$W_{AB} = \\frac{q\\lambda}{4\\pi\\epsilon_0} [ln|\\frac{(\\sqrt{(L)^2+(r_1)^2}+L)(r_1)}{(\\sqrt{(\\frac{L}{2})^2+(r_1)^2}+\\frac{L}{2})^2}|]$$ $$W_{BC} = \\frac{q\\lambda}{4\\pi\\epsilon_0}[ln|\\frac{(\\sqrt{(\\frac{L}{2})^2+(r_1)^2}+(\\frac{L}{2}))(r_2)}{(\\sqrt{(\\frac{L}{2})^2+(r_2)^2}+(\\frac{L}{2}))(r_1)}|]$$ $$W_{CD} = \\frac{q\\lambda}{4\\pi\\epsilon_0} [ln|\\frac{(\\sqrt{(\\frac{L}{2})^2+(r_2)^2}+\\frac{L}{2})^2}{(\\sqrt{(L)^2+(r_2)^2}+L)(r_2)}|]$$ $$W_{DA} = \\frac{q\\lambda}{4\\pi\\epsilon_0}[ln|\\frac{(\\sqrt{(L)^2+(r_2)^2}+(L))(r_1)}{(\\sqrt{(L)^2+(r_1)^2}+(L))(r_2)}|]$$ $$W=W_{AB}+W_{BC}+W_{CD}+W_{DA}$$ $$W = \\frac{q\\lambda}{4\\pi\\epsilon_0} [ln|\\frac{(\\sqrt{(\\frac{L}{2r_2})^2+1}+\\frac{L}{2r_2})}{(\\sqrt{(\\frac{L}{2r_1})^2+1}+\\frac{L}{2r_1})}|]$$ If field were to behave conservatively $W=0$ must have been satisfied, but unless $r_1 = r_2$, $W \\neq 0$. Thus field is not behaving conservatively. PS : This is not a homework question ! I am trying to explain the behaviour."}
{"id":"73334","title":"Projectile Motion Question involving a ball and a ramp inclined at an angle","text":"The question is to finde the initial horizontal velocity of the ball at end of the ramp, where it is released. I know how to do this using gravitational potential energy and kinetic energy ($v=\\sqrt{2gh}$), assuming all potential energy is converted into kinetic energy but the question is asking me to find the error in the experiment. It gives me the values on $y$ as the ramp is lifted up (it is lifted up about 10 cm each time) and the corresponding values of $x^2$. I drew a graph of $x^2$ vs $y$ and found the gradient. How can I use this to find the experimental initial horizontal velocity? This sketch shows what the experiment looks like: ![Experiment](http:\/\/i.stack.imgur.com\/RfGvy.png)"}
{"id":"73333","title":"Colors of light","text":"White light is always said to contain all the different wavelengths of light. Why, then, can we 'make' new colors simply by adding wavelengths? Is it just a matter of our perception, that, when two colors are added together, they appear to be the wavelength of another color. Or are the waves actually forming into another, differently 'waving' wave, with a different wavelength?"}
{"id":"48292","title":"Temperature below absolute zero?","text":"I saw this Nature article today, which cites e.g. arXiv:1211.0545. And it makes no sense to me. The temperature of a collection of particles is the average kinetic energy of those particles. Kinetic energy cannot be less than zero (as far as I'm aware), so I don't understand what this article is trying to say, unless they're playing around with the conventional definition of \"temperature\". The only thing I can thing of is if you have something like: $$\\frac{1}{kT} ~=~ \\left(\\frac{\\partial S}{\\partial E}\\right)_{N,V}$$ And they've created a situation where entropy decreases with increasing energy."}
{"id":"48615","title":"Less than absolute zero possible?","text":"> **Possible Duplicate:**   >  Temperature below absolute zero? According to this article http:\/\/www.sciencemag.org\/content\/339\/6115\/52 (preprint: http:\/\/arxiv.org\/abs\/1211.0545) it is. What do you think. Is this team of scientists a faker team or you could actually go under absolute zero? Could this be considered a way to dark energy something which is less than absolute zero?"}
{"id":"41780","title":"Quantum phyics project for a high schooler","text":"> **Possible Duplicate:**   >  Study Quantum Physics I am a high schooler who is interested in physics and mathematics, and I have a kind of 'high-school thesis' coming up in a year and a half or so. I want mine to be about Quantum Physics, and I have already prepared by self-studying Linear Algebra (and I'm planning on starting with Differential Equations too). I just have a couple of obstacles:   * If I know Linear Algebra and (Ordinary & Partial) Differential Equations, what parts of Quantum Physics will be in my reach?   * What other fields of mathematics would you recommend me studying to gain a grasp of rudimentary quantum physics and to make me able to make a thesis on quantum physics?   * What would you guys consider interesting for a thesis on quantum physics? What are fascinating experiments, concepts, results, etc. of quantum physics that would be worth making a thesis about?"}
{"id":"31822","title":"Why is pressure in a liquid the same in all directions?","text":"I'd like answers both in the more intuitive side an on the more precise side. Thinking of water as \"cubes\" of water, for example, would allow pressure in the z axis to be independent of the y or x axis. Choosing other \"shapes\", like a triangular prism, yield different results. Will this fact, then, be dependent on the format of the molecules of the liquid ? I've heard of the rotational symmetry of liquids. What is its precise statement ? Why is it true ?"}
{"id":"31820","title":"use of Mobile phone on Petrol pump","text":"I was trying to find out the cause behind the Explosion at petrol pump due to the Mobile phones but eventually it turns out it is not Radio Frequency which may cause the explosion rather it is totally the static electricity which may cause serious explosion and ignition of vapors, The other reason I found is, due to the spark by the mobile phone battery in case the mobile has fallen out of your pocket and it seems more legit. The second scenario, Its been said that Electromagnetic Waves absorb in liquid across a wide wavelength , is there any possibility that the absorption of radio Frequency in Petrol may cause any explosion ? My question is, anything can produce Static electricity then why mobile phones are prohibited ? Do they produce more Static Electricity than other equipments ?"}
{"id":"126773","title":"Physical reason for annihilation?","text":"What is the fundamental reason as to why matter and antimatter annihilate? Is it because both particles and antiparticles are excitations of quantum fields, and the annihilation process corresponds to a transition to the _ground state_?"}
{"id":"86894","title":"Casimir force between plate and sphere","text":"In our quantum optics lecture we derived the formula for the Casimir force between two plates (or at least the force per unit surface, since for two infinite plates we would have an infinite force). It was then said, that in experiments one often uses a sphere and a plate to measure the force, where the small distance between the sphere and the plate is varied by a piezo. I'm not particularly interested in how exactly the force is measured but rather how to theoretically calculate it in this case. Our professor said, the measured forces agree with the theory up to 1% so I guess, there is a way to calculate it. Let's say I have a sphere of radius $R$ in a distance $d$ from a plate where $R\\gg d$. How can I theoretically calculate the Casimir force between a plate and a sphere?"}
{"id":"77655","title":"How do we know we've unified two interactions?","text":"1. What is the precise definition of unification of fields (in classical and quantum mechanics)?   2. In general, does unification of a field mean that we can write both of them at both sides of an equation (like Maxwell's laws)? Or does it mean that one of them can produce the other (like $E$ and $B$)?   3. Is there any intuitive explanation of how electroweak unification works? Like an electric charge will feel a weak field or that a flavoured particle will produce a weak field?"}
{"id":"101289","title":"Density Matrix Characterization","text":"I am working a two dimensional Hilbert space with basis {$|0\\rangle, |1\\rangle$} and I am trying to show that the density matrix is characterized by 3 real numbers and show that these three numbers are the expected values of the Pauli matrices. I understand that a density matrix is given by $M_{ij} = \\langle i|\\rho|j\\rangle$ where $|i\\rangle$ is an orthonormal basis of $H$ and $\\rho$ is the density operator. How do I go about showing this?"}
{"id":"98553","title":"What is the difference between QFT and elementary particle physics?","text":"I'm a little unclear as to how QFT differs from Elementary particle physics. They both use pictorials of Feynman graphs, is it that Elementary particle physics assumes the point particle perspective, while QFT treats them as fields?"}
{"id":"112359","title":"Recommendation: Advanced topics in quantum field theory","text":"I have read Srednicki's Quantum Field Theory book. I want to learn more about advanced topics in field theory, such as geometry and topology in field theory, topology defect, anomaly, soliton, instanton, renormalization，advanced application in condensed matter physics and so on. Does anyone have anything recommended? Thanks"}
{"id":"6377","title":"Top spun up with string under tension problem","text":"> **Possible Duplicate:**   >  Homework about spinning top I have a top with an unknown mass. It has a moment of inertia of 4.00 * 10^-7 kgm^2 a string is wrapped around the top and pulls it so that its tension is kept at 5.57 N for a distance of .8 m. Could somebody help me derive some equations to help with this? Or to get me in the right direction? I have been trying to derive some sort of equations from KEr = 1\/2 * I * w^2 but I cant get anywhere without ending up at radius = radius or mass = mass."}
{"id":"130119","title":"Diffusion in the standard map","text":"Consider the standard map (also known as Chirikov map): $$ p_{n+1} = p_n + K \\sin(\\theta_n) \\\\\\ \\theta_{n+1} = \\theta_n + p_{n+1} $$ I know that the diffusion coefficient according to the Einstein relation is defined through the relation: $$ \\sigma^2 = 2Dt $$ being $\\sigma$ the variance of the position in the case of the $1$-D random walker. Can you figure out why the diffusion coefficient for the standard map is given by: $$ D = \\lim_{n \\rightarrow \\infty} \\frac{<(p_n - p_0)^2>}{n} \\quad ? $$ I mean, the $n$ makes sense, but what about the variance, why this is given just in terms of $p$, what happens with the $x$."}
{"id":"76497","title":"Question about gravity felt at various radii within a massive sphere","text":"Imagine there's a sphere of radius $R$ which has constant density $\\rho$ and you can stand anywhere inside of the sphere. Wherever you stand within the sphere, you'll only feel gravity coming from the mass at smaller radii than the point at which you're standing. That is, you could be standing at some point with radius $r$ and if somebody come along and added more mass to the sphere so that the radius was extended to $R'>R$ you would feel no effect because you're insensitive to it. Puzzle: What happens if somebody comes by and adds mass to the sphere until it extends out to infinity? Now the mass distribution is uniform everywhere in space so you'd expect no gravitational pull in any direction, by symmetry. What happens? Do you suddenly stop feeling a force? If you have an issue with extending the mass to infinity then imagine that we've added a point to $\\mathbb R^4$ so that we're on $S^4$. What's the intuitive way to understand what's happening in this limit? What's the technical explanation? Is it a boundary value issue?"}
{"id":"104117","title":"Understanding the Selection Rules of a Spin-Forbidden, Magnetic Dipole Transition in Molecular Oxygen","text":"I am studying the transition from the second excited electronic state of molecular oxygen, $b^1\\Sigma_g^+$ , to the ground state, $X^3\\Sigma_g^-$. I know that the ground state has total angular momentum $J=1$, total spin $S=1$, and three spin sublevels $(m_s=-1,0,+1)$. The upper state has $J=0$, $S=0$, and one sublevel with $m_s=0$. I am specifically interested in the transition from the upper state to the ground state $m=\\pm1$ level and I will refer to it as the $b-X,1$ transition. I would like to understand the the selection rules that govern this transition and the language that is used to describe this transition. Here is what I know so far about this transition.   * Brecha, Pedrotti, Krause, \"Magnetic rotation spectroscopy of molecular oxygen with a diode laser\" JOSA 1998: They describe this transition as \"doubly weak\" and I quote, \"First, it is a magnetic dipole transition and thus is roughly 5 orders of magnitude weaker than a normal electric dipole transition. Second, the transition is a singlet-triplet intercombination band, making it 3 orders of magnitude weaker still.\"    * Minaev, Agren, 1997: They describe this transition as \"magnetic dipolar\" and \"spin-forbidden.\" They also say that it is spin-orbit coupling that accounts for this \"doubly weak\" transition being as large as it is.    * Wikipedia-Selection Rules: Provides the rules for Magnetic Dipole (M1) transitions and discusses Spin-Obrit (LS) Coupling but I do not understand where this came from.    * Sannigrahi, \"Derivation of Selection Rules for Magnetic Dipole Transitions,\" 1982: This is pretty clear but just seems to say that $\\Delta m_s=\\pm1$. This is true for the $b-X,1$ transition but how do I tie in $L$ and $J$ into the selection rules? Most of what I have found on the internet and in textbooks relating to selection rules is directed to electric dipole transitions. Where I have found something discussing magnetic dipole transitions, it is either a summary of the rules or I do not have the required background to understand it.   * What does it mean for a transition to be spin-forbidden?   * What is a singlet-triplet intercombination band? Does this change the selection rules? How is this related to being spin-forbidden?   * What does spin-orbit coupling have with this? I look forward to any feedback on this topic. # Follow up Questions ## Meaning of Spin-Forbidden Does 'spin-forbidden' just mean that the transition from a $J=1$ to $J=0$ state is not allowed because the selection rules for magnetic dipole transitions say that $J$ cannot change? I expected 'spin-forbidden' to imply something about the change of the spin between the initial and final states. For example, suppose I have a time-dependent perturbation like $V_{md}(t) = \\frac{e}{m} \\vec{S}\\cdot \\vec{B}(t)$ and I am interested in the transition rate between the initial state $\\left| s m_s \\right\\rangle$ and the final state $\\left| s' m_s' \\right\\rangle$ with the quantization axis in the z direction. As you pointed out, the transition rate will be proportional to $\\left\\langle s' m_s'\\right| \\vec{S} \\cdot \\vec{B}(t) \\left| s m_s \\right \\rangle$. Now if $\\vec{B}(t)$ is circularly polarized, the rules for an allowed transition will be $s'-s=0$ and $m_s'-m_s = \\pm1$. For $\\vec{B}(t)$ polarized in the z direction, $s'-s=0$ and $m_s'-m_s = 0$ so there is not a transition to other states. I would think that if $s'-s=0$ and $m_s'-m_s = \\pm1$ are **not** true (like if $s'=1$ and $s=0$), then the magnetic dipole transition between $\\left| s m_s \\right\\rangle$ and $\\left| s' m_s' \\right\\rangle$ would be called 'spin-forbidden.' The same arguments could be made for $\\vec{L}$ or $\\vec{J}$ as I did with $\\vec{S}$. Would you also call a magnetic dipole transition between $\\left| L=0 \\right\\rangle$ and $\\left| L=1 \\right\\rangle$ 'spin-forbidden?' ## Spin-Orbit Coupling Now the difference in the energies of the singlet $(b)$ and triplet, ground state $(X)$ is $1.63\\text{ eV}$. That seems too large to be due to spin-orbit coupling breaking a degeneracy. If I was pretending this was a hydrogen atom, I would say this was like a transition where the principle quantum number $n$ changed. I am not sure how to talk about this in a molecule. This isn't key to the rest of your explanation but I did want to clarify to make sure we were on the page. ## Singlet-Triplet Intercombination Band Do you know what the expression 'singlet-triplet intercombination band' is referring to? After your explanation, it seems to refer to the mixing of the singlet and triplet states due to spin-orbit coupling (SOC). Is this true? ## Mixing of unperturbed states How did you know that the perturbed upper state could be written as a combination of the unperturbed upper state and the $M_s=0$ ground state? It makes sense that the states would get mixed up by SOC but I don't know how. If this is a messy explanation, don't worry about it. **I would like to reiterate that your (George G's) explanation has been incredibly helpful. Thank you.**"}
{"id":"122720","title":"Is there some quantum potential producing exponential eigenvalues?","text":"Usual central potentials produce quantum spectra with energy levels going as $n$, $n^2$, $n^3$ and so on, being $n$ the quantum number of the orbit. In the other extreme we have \"dirac-delta\" potentials which have only a single discrete eigenvalue. I was wondering, what kind of potential do we need for producing an exponential $e^n$ set of discrete eigenvalues?"}
{"id":"74302","title":"OPE and 4-point correlation function in CFT_d","text":"I'm reading this paper where to determine the coefficient $C^{\\phi\\phi O}(x_{12},\\partial_2)$ of the OPE (p.10) $$\\phi^\\alpha (x_1)\\phi^\\beta (x_2)=C_\\phi \\frac{\\delta^{\\alpha\\beta}}{x_{12}^{2\\eta}}+C^{\\phi\\phi O}(x_{12},\\partial_2)O(x_2)\\delta^{\\alpha\\beta}$$ they require consistency of the OPE with the 3-point correlation function, i.e. comparing the fixed form of the 3-point correlation function to what happens when we apply the OPE to two channels in the 3-point function and then use the form of the 2-point function. Anyway, to solve the form of the coefficient that satisfies eq. 2.29 they use the integral representation $$\\frac{1}{(ab)^\\rho}=\\frac{1}{B(\\rho,\\rho)}\\int_0^1 dt \\frac{[t(1-t)]^{\\rho-1}}{[at+(1-t)b]^{2\\rho}}$$ which comes from the book Table of Integrals, Series and Products by I.S. Gradshteyn and I.M. Ryzhik. However, looking at that book (I only have access to the 7th ed.) I only found the relation $$B(\\mu,\\nu)\\frac{1}{(a+c)^\\mu(b+c)^\\nu}=\\int_0^1 dx x^{\\mu-1}(1-x)^{\\nu-1}\\frac{1}{[ax+(1-x)b+c]^{\\mu+\\nu}}$$ which reduces to the desired relation when $c=0$ however, in the book it says that the relation is valid for $c>0$, so how can they use that relation?"}
{"id":"128099","title":"Why infrared absorption is a nonlinear technique?","text":"I am looking for a good explanation and\/or reference quotation explaining why infrared absorption technique is essentially nonlinear (eg. for carbon monoxide quantification). When using UV\/visible\/near-IR absorption technique, Beer-Lambert Law often is valid and then you have quite straight way to convert transmittance into absorbance which linearly respond to concentration (provided BLL limitations are not met). Above technique relies on electronic transitions, when IR absorption technique relies on vibrational and rotational transitions for molecules which has permanent or transient dipole moments. I would like to figure out why such technique is essentially nonlinear."}
{"id":"128090","title":"How fast do molecules move in objects?","text":"I guess it depends on the heat or the type of the material but can you give some examples or formulas to calculate it ? The best example would be the average speed of the air molecules (all types in the air) at room temperature or water molecules at human body temperature."}
{"id":"60758","title":"Are the $10^{500}$ different string theories being whittled down?","text":"An example of a test: Ask each variant whether its estimate of the electron mass lies within $\\pm\\,x\\%$ of the known value. This surely can't take long per theory. Although $10^{500}$ is huge, whittling them down could be essentially a background task for a few thousand computers. But is it being done, and if not, why not?"}
{"id":"39706","title":"What equation of state is needed for liquid states?","text":"I'm familiar with the ideal gas law $$PV=nRT$$ but I don't think it applies to liquids like water. If I'm wrong, please correct me! If I'm right, then what equation of state applies to liquids such as water?"}
{"id":"39707","title":"Can a wavefunction be solved to any arbitrary precision, given enough computer time?","text":"I learned that the wavefunction for the hydrogen atom can be solved analytically (we did the derivation in class), but that for more complicated atoms it is \"impossible\" to solve and that only approximations can be obtained. Well, I'm not sure I like the vague wording with \"impossible\". Basically, a hydrogen atom can be solved in terms of elementary functions. Elementary functions, as far as I know, are just arbitrarily defined (Is $\\sqrt{}$ one? What about $\\text{erfc}$?). For instance, I could say that $\\sin{x}$ cannot be solved exactly except in a few special cases (x = some rational multiples of $\\pi$). To analogize, a general quintic equation cannot be \"solved\" exactly using elementary functions -- that is, unless you define a Bring radical as a new elementary function. To me, it's really arbitrary and all of these solutions come down to numerical, iterative refinement. Solved exactly vs solved approximately seems to be an ill-defined concept. So what I am asking -- given enough computer time -- can I solve the Schrodinger equation of, say, Lithium, to whatever precision I like? Much as how I can solve $\\sin{x}$ to however many decimal places I want? Or is there some _other_ limit to such a calculation?"}
{"id":"71376","title":"Can I levitate soon?","text":"Whereas I have seen frogs fly in high magnetic fields and Sumo ringers stand on high-temperature superconductors it seems that these are not the only forms of levitation. Recently a group of physicicsts managed to levitate coffee without a magnetic field (PNAS link, only abstract is free though). So while I am still trying to grasp how this is even possible my question is a bit more far fetched: Can I levitate using this approach?"}
{"id":"70975","title":"Is the conjectured noncommutative heavy scalar \"brother\" of the already detected Higgs boson is a pseudo scalar?","text":"This is a technical (may be trivial?) question about this sigma scalar field advertised by Chamseddine and Connes to improve the electroweak vacuum stability involved by the weak mass of the already found Higgs-like particle. **Is it a scalar or a pseudo scalar field?** (motivation: I was looking for a possible connection with Majoron models)"}
{"id":"71080","title":"About the stability of the ground state of the bosonic string","text":"In Polchinski's string theory vol 1, p. 23, it is said \"It is a complicated question whether the bosonic string has any stable vacuum, and the answer is not known.\" The book was published on 1998. What is the current answer of this question? Is bosonic string a realistic theory nowadays? In addition, the ground state here is $$|0;k \\rangle $$, not $$| \\mathrm{vacuum} \\rangle$$. Is that because the latter one is absolute nothing so we don't study?"}
{"id":"37739","title":"Gauss law in classical U(1) gauge theory","text":"I can see that $a_{0}$ is not an independent field and Gauss law is a constraint on the theory arising from field equations. But, I don't get the geometrical picture. Let $A$ be the space of all field configurations and let $G$ be the group of gauge transformations. Then, the actual configuration space $C=A\/G$ where we consider $A$ modulo the action of gauge transformations. So, first of all, why don't we do our variational calculus to obtain field equations on configuration space? It has always been taught to me that I should consider variation of \"paths\" in configuration space only while applying Hamilton's principle. Instead, we first obtain field equations regarding $A$ as our configuration space and then \"project\" them perpendicular to gauge orbits using Gauss law. Are these two methods equivalent? Moreover, I don't understand how Gauss law really projects my fields orthogonal to gauge orbits. According to Manton's book (see the relevant pages scanned below) on topological solitons, he starts with a variation of the fields $\\phi$ and $a$ in a small interval of time. Then, he claims that a \"naive contribution to the kinetic energy\" would be some expression involving these variations (see below) which is non-zero for infinitesimal gauge transformations also. But, when I look at the expression of kinetic energy (which is $T=\\int(\\frac{1}{2}e_{i}e_{i}+\\frac{1}{2}\\bar{D_{0}\\phi}D_{0}\\phi)d^{d}x$) in terms of fields, then I find that the expression is gauge invariant. I don't see why the particular expression $\\frac{1}{2}\\int\\frac{1}{(\\delta t)^{2}}(\\delta\\vec{a}.\\delta\\vec{a}+\\bar{\\delta\\phi}\\delta\\phi)d^{d}x$ be the change in kinetic energy. In short, I want to understand this geometric insight into Gauss law whose details are attached below. Also, I wonder if this Gauss law has a relation to the Gauss law I learned while doing high school electrodynamics. Any information or resource on this would be very helpful. ![enter image description here](http:\/\/i.stack.imgur.com\/3rP4Y.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/nRlWL.jpg)"}
{"id":"67107","title":"From where do the permanent magnets get energy from?","text":"I have a doubt about permanent magnets. If a magnet is permanent it can attract some materials permanently. Attracting something involves energy. If a permanent magnet can do this forever, from where does this energy come from? How can it not run out of energy? Doesn't it contradict with the laws of energy?"}
{"id":"30709","title":"What is the typical orbital life of an artificial satellite?","text":"The orbit of satellites around Earth eventually decays, or so I read. This is typically caused either by atmospheric drag, or by tides. I would assume most satellites have a limited service life in orbit. Hence the question - What is the typical orbital life of a satellite? How long before it's successor must be launched into orbit ? EDIT: For instance, a weather satellite"}
{"id":"15482","title":"How do I calculate the position on the Bloch sphere of a quantum gate with a given diagonal matrix?","text":"In quantum computation there are several principal quantum gates that have corresponding matrix representations. One of these is the Z gate, whose matrix is $\\left[\\begin{smallmatrix} 1 & 0 \\\\\\ 0 & -1 \\end{smallmatrix}\\right]$. ... anyway, I've found the eigenvalues (equal to +1, -1) using the characteristic equation, and used them to derive the corresponding eigenvectors, which come together quite nicely in a 2×2 matrix $\\left[\\begin{smallmatrix} 1 & 0 \\\\\\ 0 & 1\\end{smallmatrix}\\right]$, equal to the identity. So, in diagonalizing this matrix, I find that the diagonal matrix $D$ is the same matrix as the one for gate $Z$. ... the next step and where I'm stuck is to find the corresponding point on the Bloch sphere for this gate. In order to do that, I need to compute how to take the diagonalized matrix call it $D_z$ and derive two things: **(a)**  its diagonal representation $| 0 \\rangle \\langle 0 | - | 1 \\rangle \\langle 1 |$, and **(b)**  the normalized eigenvalues $a, b$ for $Z$, where $Z = a|0\\rangle + b|1\\rangle$ and must be orthonormal i.e. $a^2 + b^2 = 1$. The $a$ and $b$ terms correspond to the probabilities of measuring 0 or 1 for the state, respectively (I think). After I have the values for $a$ and $b$, I'll be able to locate the gate on the Bloch sphere because the calculation of its coordinates on the sphere is straightforward: $a = \\cos(\\theta \/ 2)$, and $b = e^{i\\phi}\\sin(\\theta\/2)$."}
{"id":"15480","title":"Does the anthropic principle entail the existence of a huge multiverse?","text":"As many cosmologists have pointed out, the universe is extremely fine-tuned for life in so many respects, and life, especially intelligent life, is extremely unlikely to emerge. The strong anthropic principle tries to explain it statistically by an extremely huge multiverse ensemble. However, are there alternative explanations? One is an intelligent designer, but given how hard it is to create intelligent life, It would probably have to engage in significant trial-and-error by simulating many prototype universes before stumbling upon one that works. This is probably true no matter how intelligent It is. So, the multiverse reemerges through the back door. This still applies even if the Designer is a blind naturalistic algorithm. Can you think up of some mechanism for intelligent life to emerge which doesn't require a huge multiverse ensemble?"}
{"id":"103207","title":"Modal analysis with aerodynamic damping","text":"I'm using modal decomposition to predict the steady state response of a beam structure to harmonic loading. The structure itself is very lightly damped, but we know from experiments that the aerodynamic drag force as it moves through the air is very significant. I have the mass and stiffness matrices for the beam ($\\boldsymbol{[M]}$ and $\\boldsymbol{[K]}$) and so I am able to obtain the eigenvectors and eigenvalues ($\\boldsymbol{[V]}$ and $\\boldsymbol{[D]}$). The eigenvalues yields a vector containing the natural frequencies of the system (the square root of the diagonal entries of $\\boldsymbol{[D]}$, I'll call it $\\boldsymbol{\\omega}$. The system is excited by a force vector in the global reference frame $\\boldsymbol{F_0}$ at an excitation frequency $\\omega_F$. There is a rotation matrix which rotates the beams from the local to the global reference frame $\\boldsymbol{[R]}$. I have successfully used the procedure to calculate the steady state response without aerodynamic damping:   * Get eigenvalues and eigenvectors $\\boldsymbol{[V]}$ and $\\boldsymbol{[D]}$   * Get generalised force vector $\\boldsymbol{Q}=\\boldsymbol{[V]}^T \\boldsymbol{F_0}$   * Calculate modal participation factor vector $\\boldsymbol{q}$. The individual elements are calculated as follows: $$q_i=\\frac{Q_i}{{\\omega_i}^2}\\frac{1}{\\sqrt{[(1-(\\omega_F\/\\omega_i)^2)^2+(2\\zeta_i\\omega_F\/\\omega_i)^2]}}$$   * Calculate response from $\\boldsymbol{x}=\\boldsymbol{[V]q}$ Of course, this drastically overpredicts the amplitude of the oscillations because there is no aerodynamic damping. I've found a few examples for single d.o.f. systems in which the energy lost from the system is calculated (by integrating the drag force x the velocity). The integration results in the following equation for energy loss: $$\\Delta E=\\frac{8}{3}\\frac{1}{2}\\rho A C_D \\omega^2 X^3$$ This can then be converted to an equivalent damping coefficient as follows: $$C_{eq}=\\frac{\\Delta E}{\\pi \\omega X^2}$$ Which can be converted to a damping coefficient by doing: $$\\zeta=\\frac{C_{eq}}{2 \\omega_n}$$ I've modelled a simple spring-mass system with air resistance and it works fine (I have to iterate to find the final position because the amplitude calculation requires knowledge of the damping which requires knowledge of the amplitude so there is a circular reference, but that's no biggy!) Could someone please explain the equivalent for a multi-d.o.f. system? My initial forays have been unsuccessful, with much higher drag coefficients than would really occur."}
{"id":"86705","title":"Does sound cancel itself out?","text":"If there are two 10 x 2 x 1 foot rectangles in space and they are lined up so if they hit each other there will be no spots that are not hit in the front of the rectangle. Then they are pushed forward with the same exact same force, (this is space and there is no resistance) and they hit, will the tiny shockwaves of sound moving through them hit each other (they are exactly the same kind of sound wave), will the sound hit it's opposite would they cancel themselves into heat on contact? I know that the sound does not cancel itself out. You cannot say the wave of atoms hitting each other collides with a wave that is exactly the same and stops both of them in a word. Can you?"}
{"id":"86706","title":"Magnetic core, and a DC solenoid","text":"A DC solenoid creates a magnetic field B. A highly permeability magnetic core is added to solenoid to increase B. Now, there is change in flux, due to the increase in magnetic field. What would be the result for the circuit? Would more power drawn support the new stronger field? Or P would be the same before\/after?"}
{"id":"47813","title":"What does a subatomic charge actually mean?","text":"I was recently reading a popular science book called The Canon - The Beautiful Basics of Science by Natalie Angier, and it talks about subatomic particles like protons, neutrons and electrons in chapter 3. I came across this section on subatomic charges that made me wonder about the nature of the positive and negative charges that we associate with protons and electrons respectively. > When you talk about a fully charged battery, you probably have in mind a > battery loaded with a stored source of energy that you can slip into the > compartment of your digital camera to take many exciting closeups of > flowers. In saying that the proton and electron are charged particles while > the neutron is not, however, doesn't mean that the proton and electron are > little batteries of energy compared to the neutron. A particles's charge is > not a measure of the particles's energy content. Instead, the definition is > almost circular. **A particle is deemed charged by its capacity to attract > or repel other charged particles.** I found this definition\/description a bit lacking, and I still don't grasp the nature of a \"subatomic charge\", or what do physicists mean when they say that a proton is positively charged and electron is negatively charged?"}
{"id":"72852","title":"Lagrangian and conservation of energy","text":"If Lagrangian of the motion is $$\\mathcal{L}=\\frac{1}{2}m\\left(a^2\\dot\\phi^2+a^2\\dot\\theta^2\\sin^2\\phi\\right)+mga\\cos\\phi,$$ how can I show that total mechanical energy is conserved? I've read this: > _If the time $t$, does not appear [explicitly] in Lagrangian $\\mathcal{L}$, > then the Hamiltonian $\\mathcal{H}$ is conserved. This is the energy > conservation unless the potential energy depends on velocity._ Potential energy of this motion doesn't depend on velocity. Also, $t$ does not appear explicitly in Lagrangian. Is this enough to say that total mechanical energy is conserved?"}
{"id":"11334","title":"Why and how exactly is electric motor torque limited?","text":"Inspired by this question and specifically this answer to it. From my experience there's always some very specific limit to how much torque an electric motor can output. For example, an electric drill will often have a manually switched mechanical transmission - if one needs to drill some relatively weak materials (like wood) he will use the setting that outputs lets torque at higher RPM and if one want to drill steel or mix cement mortar he will use the settings for more torque at lower RPM. The relation between RPM and torque is more or less clear if one imagines a set of two gears of different diameters and thinks that their radiuses are lever arms - three times more RPM automatically induces three times lower torque and vice versa. But where does the limit to any given electric motor torque come from? Say I have some specific motor right now in front of me and it can output 40 Newton- meters at 500 RPM. Why exactly 40 and not more?"}
{"id":"43221","title":"How to explain the different forms of the Hamilton-Jacobi equation?","text":"In Arnold's _Mathematical Methods of Classical Mechanics,_ he derives the Hamilton-Jacobi equation (HJE) using a generating function $S_1(Q, q)$ to get $$ H\\left(\\frac{\\partial S_1(Q, q)}{\\partial q}, q, t \\right) ~=~ K(Q, t). $$ However, this is different from what I've seen in other physics texts. For example, Goldstein uses the generating function $S_2(q, P, t)$ to get the equation $$ H\\left(\\frac{\\partial S_2(q, P, t)}{\\partial q}, q, t\\right) ~=~ - \\frac{\\partial S_2(q, P, t)}{\\partial t}. $$ Why is there this difference? Are the two equations saying the same thing?"}
{"id":"89108","title":"What's the size of a 'locally' flat space-time patch in a black hole background?","text":"A black hole collapses and emits Hawking radiation with average wavelength 1\/M. The observer at I^+ sees it, so do the local observers (infalling or hovering). None of these cases should violate the equivalence principle. What is the size of the 'local' patch associated with the observer, the size beyond which he is allowed to see deviations from flat space because curvature effects become measureable? I'm confused because on the one hand I'd have thought this size goes to infinity if the observer moves towards I^+, but on the other it seems to me it should stay at 1\/M so that the (average) Hawking radiation has wavelengths the size of the locally flat patch. What am I getting wrong? (Forget about firewalls, just thermal radiation.)"}
{"id":"19106","title":"Did Newton discover gravity waves without realizing it?","text":"Newton's theory explained the changes in sea level by the effect of gravity (moon\/sun). Now we are trying to detect gravitational waves from distant cosmic surces. But, apart from the differences in \"signal frequency\" (extremely small in case of sea effects (10-4 \/ 10-5 Hz or so), maybe higher in case of \"signals\" from cosmic sources (in the order of 1 Hz?) what is the difference in principle? At least in principle, can we say that the change in sea level is an effect of gravity waves?"}
{"id":"88852","title":"Is there a physical reason why the detection pattern of microphones and antennas is a cardioid?","text":"I was wondering if there's an underlying physical reason why detection in microphones and antennas is a cardioid, or if it's just that a cardioid happens to be the mathematical object that best approximates the shape of the detection pattern. In other words, is there an \"ab initio\" reason, or it's an empirical fact?"}
{"id":"103336","title":"Was Einsteins work with relativity necessary for successful space travel?","text":"So I know that Einstein and general relativity had huge impacts on the way we view the world, but how crucial were these scientific advancements to the success of our space programs? Would Newtonian physics sufficed, or would using those formulas and methodologies when travelling off-earth and to the moon resulted in catastrophic failures?"}
{"id":"47103","title":"Can one predict mass distribution in a solar system given the mass of the star?","text":"I'm trying to write a program to procedurally generate solar systems at a reasonably high level (I'm not interested in doing an n-body dust particle formation simulation). Is there some sort of distribution that defines where I am most likely to find mass inside a solar system? i.e. should I expect heavier planets further out... or is it completely random?"}
{"id":"47108","title":"Wigner's friend and quantum Zeno effect","text":"Suppose Winger's friend is placed into a black box, thoroughly isolated from the outside world. He constantly observes an atom with a delay of some microseconds. According to Zeno effect, atom's lifetime will increase. He writes his observations to a log. After a day passed, the box is opened and Wigner's friend asked when the atom decayed. Will the increased atom lifetime be seen by the observers who were outside of the box?"}
{"id":"11481","title":"Where does the minus sign appear from in the metric tensor?","text":"Trying to understand Schutz's AFCIGR, where does the minus sign appear from in the metric tensor? ![metric](http:\/\/i.stack.imgur.com\/jXilu.gif) I understand that this expresses the invariance of the spacetime interval. Schutz says (I think) that the metric is a (0,2) tensor. I assume that means it is the product of two one-forms, so presumably one of these one-forms has a -ve time component. What does that mean? Why don't both one-forms have a -ve time component? Looking at a Minkowski diagram what is a straightforward way to understand\/visualise those one-forms? At my level, two things have been multiplied together to give a 4x4 matrix which has -1 in the top left corner. What are those two things and why do they give a -1 time component?"}
{"id":"43487","title":"Reference for the predictability of rigid body dynamics","text":"I'm looking for a reference, journal article, paper, etc. that supports the idea that classical mechanics, in particular rigid body dynamics, is largely predictable. A view coming from the background of computer physics simulations would be an added bonus."}
{"id":"43481","title":"A confusion from Weinberg's QFT text (a vanishing term in Lippmann-Schwinger equation)","text":"I was reviewing the first few chapters of Weinberg Vol I and found a hole in my understanding in page 112, where he tried to show in the asymptotic past $t=−∞$, the in states coincide with a free state. In particular, he argued the integral $$\\tag{1} \\int d\\alpha\\frac{e^{-iE_{\\alpha}t}g(\\alpha)T_{\\beta\\alpha}^+\\Phi_\\beta} {E_\\alpha- E_\\beta+i\\epsilon}$$ would vanish, where $d\\alpha=d^3\\mathbf{p}$ (also involves discrete indices like spin, but of no relevance here). In his argument, he used a contour integration in the complex $E_{\\alpha}$ plane, in which the integral of central interest is the integration along real line $$\\tag{2} \\int_{-\\infty}^\\infty dE_\\alpha\\frac{e^{-iE_{\\alpha}t}g(\\alpha)T_{\\beta\\alpha}^+\\Phi_\\beta} {E_\\alpha-E_\\beta+i\\epsilon}$$ I don't see how to obtain (2) from (1), since the lower bound of energy is the rest mass, in the best case I could get something like $\\int_{m}^\\infty dE_\\alpha\\cdots$, but how could one extend this onto the whole real line."}
{"id":"43483","title":"Rotating sphere and circular trajectory: minimum speed","text":"I have a sphere (mass = 3 kg), constrained to a fixed length rope, rotating (radius = 5 m) on a vertical plane. My textbook ask me about the minimum speed in the highest point in order to keep the circular trajectory. Now, I know that in the highest point (v=speed): $$F_c=mg+T$$ $$\\frac{mv^2}{r}=mg+T $$ $$ \\frac{3}{5}v^2=29,43+T$$ I know that with a low speed I have a low Tension, so let's put T=0 and go on: $$v^2=49,05 \\rightarrow v=7$$ The result is correct, but I have a little doubt: if the Tension is 0, why the sphere doesn't go along the tangent or start falling down? Exactly what is forcing the sphere to preserve the circular trajectory? In my ignorant opinion, the minimum speed is a little more than 7.00 m\/s, because the tension mustn't be 0, even a very small number, but not null.."}
{"id":"80703","title":"How to obtain stabilizer's generators of a QEC code","text":"The theory of QEC with stabilizer codes defines an alternative way to represent a quantum state in terms of operators. To understand better what I am concerning about, let's consider the 7-qubit Steane code: $$ \\left|0\\right\\rangle_L \\equiv \\frac{1}{\\sqrt{8}}(\\left|0000000\\right\\rangle + \\left|1010101\\right\\rangle + \\left|0110011\\right\\rangle + \\left|1100110\\right\\rangle + \\left|0001111\\right\\rangle + \\left|1011010\\right\\rangle + \\left|0111100\\right\\rangle + \\left|1101001\\right\\rangle) $$ $$ \\left|1\\right\\rangle_L \\equiv \\frac{1}{\\sqrt{8}}(\\left|1111111\\right\\rangle + \\left|0101010\\right\\rangle + \\left|1001100\\right\\rangle + \\left|0011001\\right\\rangle + \\left|1110000\\right\\rangle + \\left|0100101\\right\\rangle + \\left|1000011\\right\\rangle + \\left|0010110\\right\\rangle) $$ Papers and books directly define the relative generators as: $$ K^1 = IIIXXXX $$ $$ K^2 = XIXIXIX $$ $$ K^3 = IXXIIXX $$ $$ K^4 = IIIZZZZ $$ $$ K^5 = ZIZIZIZ $$ $$ K^6 = IZZIIZZ $$ where I is identity matrix and X,Z Pauli's matrices. Therefore, my questions are: how these generators have been calculated? How can I calculate the generators for a generic superposition of n qubits? Thank you"}
{"id":"20155","title":"Why has the ether been disregarded as a valid medium through which light can propagate?","text":"> **Possible Duplicate:**   >  Ether theory acceptance Although the Einstein's theory of relativity seemed to make the concept of an ether obsolete, did it necessarily invalidate it? Are there any experiments which have truly invalidated the ether, with undeniable proof? I'm very curious as I haven't found any, besides of course the Michelson–Morley experiment, which I don't think offers irrefutable evidence that contradicts the ether's existence. May someone explain why this has been the definitive conclusion (or offer some new insight)?"}
{"id":"12680","title":"Can we make a change of variables (for example to polar coordinates) into a divergent integral?","text":"I know that if the integral is convergent we can always make a change of variable to make it better, however what happens with DIVERGENT integrals? can we make a change of variable into a divergent integral after having it regularized? I mean we insert the regulator $(q+a)^{-s}$ or similar on each variable and then make the change of variable to polar, cilindrical or other"}
{"id":"12683","title":"Electric field of not-grounded conducting plate with a given potential?","text":"I have been trying to find an equation (or some solution) of how to calculate the electric field strength (in N\/C) of a conducting rectangular (nearly flat) plate which has non-zero potential to it, say the plate has a potential of 5V. I have searched over the internet (and read Griffiths book on the method of images section), and it seems one would need to apply the method of images to solve the problem. However, since almost all examples regarding method of images is about \"grounded\" plates at 0V, it is very difficult for me to understand how to use method of images to solve for the electric field of a plate at 5V? Also, I would like to mention that within the electric field of the 5V conducting plate, there is a small freely-moving positively charged particle in the vicinity (I assume because the charge on this small particle is the same as the charge on an electron\/proton-- it can be regraded as a test charge and therefore, modification to the combined electric field between the conducting plate and the charge is not necessary?-please correct me here if I am wrong.)"}
{"id":"83193","title":"How to interpret the factor $\\frac{(\\vec v\\cdot \\hat n)}{|\\vec v| 4\\pi r}$?","text":"There is a small area element of size $da$ and normal vector $\\hat n$. I understand that the particles with speed $|\\vec v|$ that hit this area element in the time interval $(t,t+\\delta t)$ lie in a spherical shell with radius $r=|\\vec v|t$ at $t=0$. Let the number of particles with energy $E(|\\vec v|)$ within a solid angle $d\\Omega$ be $dN$. Why then is the total number of particles hitting $\\hat n da$ with velocity $|\\vec v|$ (assuming isotropy of the velocity distribution) equal to $$\\frac{(\\vec v\\cdot \\hat n)}{|\\vec v| 4\\pi r}da \\,\\,dN$$? I don't quite understand how to get the factor $\\frac{(\\vec v\\cdot \\hat n)}{|\\vec v| 4\\pi r}$. Thanks."}
{"id":"93611","title":"Can Massless Particles Have Dimensions?","text":"We have been taught that anything occupying space\/volume has mass. Light has both a particle nature-photon and wave nature. Since It has an momentum of h\/λ, energy of hc\/λ. Since a photon carries energy and experiments have put an upper limit to its mass of 1e-18 eV\/c^2, does it possess dimensions, a shape ?"}
{"id":"69300","title":"Mass loss in Red Giants via dusty-winds and chromosphere activity","text":"I'm reading some literature on mass loss in the RGB\/AGB branches and so far I'm getting a lot of information regarding mass loss via dusty- winds\/pulsations but almost no explanation of mass loss by 'chromosphere activity'. I asked my mentor about it and he couldn't come up with anything so I'm turning to whomever might be able to throw a few references or explanations my way."}
{"id":"69309","title":"vertical wind gradients in the atmospheric boundary layer","text":"I'm reading the following paper: > Intercomparison of Bulk Aerodynamic Algorithms for the Computation of Sea > Surface Fluxes Using TOGA COARE and TAO Data and am having trouble working through some of the equations, I have copied these equations below: ![enter image description here](http:\/\/i.stack.imgur.com\/ymp8m.png) Would someone be able to show me the process of how integrating (1) to (6) gives (7)? Any help would be appreciated."}
{"id":"69308","title":"Is a 3 LED flashlight brighter than a single LED one? (same LEDs, same power through each LED)","text":"Say I have one LED in a flashlight, it produces some light. Now, if I have 3 of those LEDs in the same flashlight, _each_ LED receiving the same amount of energy than the single LED was receiving in the model described above (thus producing, individually, the same amount of light), will I get (perceive) more light? Will it be brighter? (considering the 3 LEDs very close to each other, same direction, etc.) **Quickly said** : does the perceived light \"sum up\"?"}
{"id":"9375","title":"Anti-matter repelled by gravity - is it a serious hypothesis?","text":"> **Possible Duplicate:**   >  Why would Antimatter behave differently via Gravity? Regarding the following statement in this article: > Most important of these is whether ordinary gravity attracts or repels > antimatter. In other words, does antihydrogen fall up or down? Is this a seriously considered hypothesis? What would be the consequences on general relativity? If this is seriously studied, can you point to some not-too-cryptic studies on the (anti ;-)matter?"}
{"id":"39293","title":"Complete annihilation of matter-antimatter","text":"> **Possible Duplicate:**   >  More on matter and anti-matter Everyone knows that when matter and anti-matter come in contact, they result in pure energy.   I want to know that if atoms and anti-atoms of two different elements are brought in contact, and one having much more charge and mass than the other..will it still result in pure energy because in this case equal amount of opposite charge is not present to counter each other.   Will it result in leftovers ?"}
{"id":"109742","title":"What would be the speed of a hypothethetical object created during the big bang","text":"What would be the speed of a hypothethetical object created during the big bang, and that has been moving around in the universe and that is now close to the earth, so the time elapsed (from the hypothetical object's perspective) is around 6,500 years, that is the age of the universe according to some people?."}
{"id":"49999","title":"What is the furthest object from which fermion rays were detected?","text":"What is the furthest object from which non-electromagnetic cosmic rays were detected?"}
{"id":"10852","title":"How are these balls reflected after they hit each other?","text":"> **Possible Duplicate:**   >  Physics of simple collisions I have 2 photos of the balls, one before the collision and one after the collision. They do a elastic collision. I want to know how is the reflected value calculated. I know it uses dot product and such stuff. But it all is very vague to me. Can anybody show me a working calculation and explain me based on this figures? I have been trying to understand this for past 2 days but I can't get how it all works. This is the image before collision: ![enter image description here](http:\/\/i.stack.imgur.com\/nNpCv.jpg) This is the image after collision: ![enter image description here](http:\/\/i.stack.imgur.com\/vlq45.jpg) Thanks in advance :)"}
{"id":"6646","title":"Physics of simple collisions","text":"I'm building a physics simulator for a graphics course, and so far I have it implementing gravitational and Coulomb forces. I want to add collisions next, but I'm not exactly sure how to go about doing it. A quick summary of how this is working so far is: All objects are spheres of a set radius, mass and charge. The mass's and charges of the spheres are treated as point charge\/mass for the calculation. Every step in time (about 1\/50th of a second) the forces acting on each object are calculated in a nice big nested for loop that figures out the coloumb and gravitational force between 2 objects, for every set of 2 objects, and then they are summed together. I use this net force to determine the acceleration, and the rest is fairly obvious from there. What I want to add in is collisions. I can deteremine pretty easily if a collision is happening (if the distance between them <= radius of one + radius of other), what I am not so certain of is how I should add in the collision force (and would need to do it component wise). I want elastic collisions, though for now I'd just be happy with getting conservation of momentum The information I have easily available are: the velocity of each object (and I mean velocity, speed and direction), the position of each object, the mass\/charge (charge obviously not needed here) and the Net force calculated so far for each object for the next step in time. I dont need exact formulas (would prefer if they arent exact), just need a nudge in the right direction."}
{"id":"107193","title":"Collision of two balls of finite radius","text":"This is your typical elastic collision problem except the balls have finite radius. To be clear:   * two billiard balls in the plane each with radius $\\sigma$   * move at constant velocities $v_1, v_2$   * starting at positions $x_1, x_2$ What is the criterion for there to be any collision at all? When does it happen? And at what angle to they collide? I need it to write a computer simulation of the hard sphere model. ![enter image description here](http:\/\/i.stack.imgur.com\/Lkndc.png)"}
{"id":"8107","title":"The physics of two circles colliding","text":"> **Possible Duplicate:**   >  Physics of simple collisions I have looked around the internet and gradually built up a better understanding. However I'm still not 100% clear on the physics. If I have 2 identical circles (in a 2D world, so they are on the same plane) and they collide, from any 2 directions, how do I calculate which direction they will travel in after the collision. My initial though was to get the tangent between the 2 circles, and each circle would bounce off the tangent in the same a ball bounces off a flat object (angle into the collision equals the angle out, but reversed around the normal): (where the red lines are parallel, and the line between the center of the circles is perpendicular to the red lines. And the angles x1 and x2 are equal in value, as are y1 and y2, but x1 and y1 are not [necessarily] equal) ![](https:\/\/docs.google.com\/drawings\/pub?id=13Kw1gxNjd_7Pd181QdVFzoOAtWXw4LO1mJerdnFrVj4&w=521&h=497) But after reading articles on the internet, it doesn't appear this is right? (NOTE: This is not homework, but rather learning by example)"}
{"id":"14270","title":"Bouncing back of a ball","text":"> **Possible Duplicate:**   >  Physics of simple collisions Let the unit vector along the positive x- axis be i and that along the Y-axis be j. Let us consider a rigid wall with the normal to the wall being along Y-axis. Let a ball with velocity V1 i - v2j move towards the wall. After hitting the wall will it bounce back with a velocity v1i +v2j or v2j-v1i?"}
{"id":"13222","title":"Is accelerating particles through a chain of accelerators a continuous or batch process?","text":"At the Advanced Photon Source, they use two accelerators before injecting the electrons into the large storage ring. Is the addition of particles to the storage ring done in \"batches\" (however small\/short they may be), or continuously? By a \"batch\", I'm thinking where the LINAC would throw however many electrons into the booster ring, which would then accelerate them all together, then dump them all into the storage ring when they're at the appropriate energy."}
{"id":"85991","title":"If I lift a box vertically, why is the work I do equal to the distance I lift it times the force of gravity on the box?","text":"I have problems fully understanding the concept of work, so please forgive me if this is simple. If I take a box of mass $ m $, and lift it a distance $ d $ vertically, **why is the work I have done equal to $ gmd $, where $ g $ is the force gravity exerts on the box?** I understand that work is equal to force times distance--so I'm not asking about the definition of work--but if I exert an upward force equal in magnitude to gravity's, won't the box remain motionless, i.e., net zero force, in which case the velocity is constant, and displacement and work done will be equal to zero? Edit: To be clear, **what I'm asking _is not a duplicate_ of \"Why does holding something up cost energy while no work is being done?\", because I'm not asking about work done on an object with zero displacement, nor is it a duplicate of \"What exactly is F in W=∫baFdx?\", because I'm not asking about the distinction between the work done by an individual force and net force.**"}
{"id":"108602","title":"Keep an object at a certain hight, does it require energy?","text":"If i have an object with a defined mass, like a stone, and put it on the shelf it will sit there forever (lets forget all external influences). But if I have to hold up the stone I will eventually get tired, feel pain, and give up. Obviously I am using up energy to keep the stone where it is. But, if it takes energy to keep an object at a certain hight it could not sit on the shelf forever because it would need infinite energy. What am I missing?"}
{"id":"59685","title":"So there are 6 quarks, what are anti-quarks considered then?","text":"I just recently got into particle physics and the quantum world and I love it. So my first big question is. I watch all these videos and people explain the quarks (up, down, top, bottom, strange, charm). And they all say there are 6 quarks. But every so often someone speaks of an anti-quark. What is this anti- quark if there are only 6 quarks? Is it anti-matter? Is it still a quark? (if so that means there are 12 quarks?) Second question, more for clarification. So there are force carriers and particles. Force carriers are bosons, they carry the strong weak and electromagnetic force and the gravity force carrier is still a mystery as to what carries it (mystery as in we just have not observed the gravity force carrying particle; i.e. higgs\/gravitron)? Non-force carriers are leptons and are comprised of only quarks? And quarks can then make more massive particles like protons (uud)?"}
{"id":"59686","title":"S-Wave for minimally coupled scalar field","text":"This question is in reference to the paper here (Equation 3).The extremal 3-brane metic in $D=10$ can be written as: \\begin{equation*} ds^2 = A^{-1\/2}(-dt^2 +dx_1^2 +dx^2+ dx^3) + A^{1\/2}(dr^2 +r^2 d\\Omega_5^2) \\end{equation*} where \\begin{equation*} A = 1+ \\frac{R^4}{r^4} \\end{equation*} In this background the $s$-wave of a minimally coupled massless scalar satisfies: \\begin{equation*} \\left[\\rho^{-5}\\frac{d}{d\\rho}\\rho^{5}\\frac{d}{d\\rho}+ \\frac{(\\omega R)^{4}}{\\rho^{4}}+1\\right]\\phi(\\rho) =0 \\end{equation*} How do I derive this result?"}
{"id":"44360","title":"Is a soundproofed wall really only as strong as its weakest area?","text":"I've seen a few questions about sound waves and sound travel here on Physics SE, so I'm hoping this question is a good fit for this site. During my internet research on soundproofing, I've come across many acoustics gurus who say that, when soundproofing a wall, you have to account for every single little opening such as a wall or ceiling even if the opening is as tiny as the shaft of a screw. If you don't, you may as well not soundproof the wall at all. Since I have approximately zero understanding of the physics of sound waves, my incredulity brings me to doubt those words. I imagine the contact of sound waves upon a wall being much like taking a handful of pebbles and slinging them against the same wall. When the pebbles leave your hand, they spread out. If I were to sling a handful of pebbles at a group of people from a distance, the chances are high that everyone's going to get hit. If you put up a solid wall in front of them, however, no one will feel a thing as the wall will stop 100% of the pebbles. If you cut a hole 2' in diameter and sling the pebbles, perhaps only a few people will be hit since only the pebbles that make it through the hole will have a chance of striking anyone. If I were to apply the \"weakest link\" argument here, it would suggest that slinging a handful of pebbles at a wall with a small hole in it would be the same thing as slinging the same pebbles as if the wall weren't there. The only way this could be true is if all the pebbles spread out after leaving your hand, magically coalesced enough to fit through the hole, then spread out again to hit the group of people. Now back to sound travel. How could a small, screw-sized hole totally negate the _entire_ soundproofing effort? While I can understand that a little sound would still make it through that hole, I am not seeing how that little leak would make it as if the wall isn't there at all. Can someone explain how a small opening in a well-soundproofed wall affects sound travel?"}
{"id":"110887","title":"Snowmaking in the tropics - an estimate of water evaporation","text":"If I set up a snowmaker in the tropics and sprayed water with it how much water would I evaporate? How would I calculate?"}
{"id":"89849","title":"Electric potential difference from electric field of isolated spherical conductor","text":"I'm wondering if my train of thought is correct. Say you have Point $A$ which is $A$ distance away from the center of an isolated spherical conductor, and Point $B$ which is $B$ distance away from the center. If I'm to find the electric potential difference, $V_A-V_B$, I know I can use the relationship $V_A-V_B= -\\int_B^A \\vec{E} \\cdot d\\vec{s} = -\\dfrac{kQ}{r^2} \\int_B^A d\\vec{s}$ What I'm wondering is if $d\\vec{s}$ is equal to $dr$ in this case. I don't really see any other thing it could be. Using that idea, things simplify down to, $\\rightarrow = -\\dfrac{kQ}{r^2} \\cdot r \\bigg|_B^A \\rightarrow -\\dfrac{kQ}{r} \\bigg|_B^A$ Is this correct? I would also appreciate an explanation as to why it is or isn't, since I'm a obviously a little unsure of myself."}
{"id":"134738","title":"Inclusive and exclusive searches","text":"Please, I would like to know what means inclusive and exclusive searches in High Energy Physics. Thanks in advance. Fábio."}
{"id":"78425","title":"Translation operator to higher order","text":"IN QM, the space translation operator, or generator of Translations is set to be Ie-ie\/h_bar*P up to order e. Now my question is what is the physical justification of only going up to order e and do the higher order terms have any meaning? Is it correct to say(if we are only including up to order e) that we are doing a constant linear translation(equal distances in equal times) and hence assuming there is no change in how fast we do it?"}
{"id":"23640","title":"what interactions would take place between a free proton and a dipolariton?","text":"What interactions can be expected to take place between a free proton and a dipolariton, (a) at high energies and (b) at lower energies? A dipolariton is a bosonic quasi-particle mentioned in a recent article in Science. It is a static dipole that consists of an electron bound together with a photon. I suspect the proton and dipolariton would be attracted to one another, but I'm unsure of the specifics."}
{"id":"89843","title":"How close would Earth have to be for us to detect it was habitable, and then inhabited?","text":"Given our current technology (or technology that is near implementation), how close would a clone of our Solar System (and so also Earth) have to be to us in order to detect that the cloned Earth was in habitable, and also how close would we have to be to detect that there is life on the planet (excluding radio signals the cloned humans have broadcast into space)? Basically I'm asking if we assume the worst case scenario where life only exists on Earth-like planets, and that the life is the same as ours (ie is inteligent, builds cities, etc), at what range with our current technology and methods\/techniques would we be unable to detect a planet and civilisation the same as our own (seeing as it's the only civilisation we know about)? EDIT: Another way to put this, assume every star system is identical to the Solar System, using our current technology\/techniques what is the furthest planet we could \"see\" that is habitable, and what is the furthest planet we could \"see\" that is inhabited by a species identical to our own (so the identical Earth 500 lightyears away would actually be in the year 2513, so we'd \"see\" it in 2013)."}
{"id":"23647","title":"Is it possible to witness a circular rainbow?","text":"What conditions would make it possible to see a naturally occurring fully 360° circular rainbow? Would it even be possible?"}
{"id":"20071","title":"Do Maxwell's Equations overdetermine the electric and magnetic fields?","text":"Maxwell's equations specify two vector and two scalar (differential) equations. That implies 8 components in the equations. But between vector fields $\\vec{E}=(E_x,E_y,E_z)$ and $\\vec{B}=(B_x,B_y,B_z)$, there are only 6 unknowns. So we have 8 equations for 6 unknowns. Why isn't this a problem? As far as I know, the answer is basically because the equations aren't actually independent but I've never found a clear explanation. Perhaps the right direction is in this article on _arXiv_. Apologies if this is a repost. I found some discussions on PhysicsForums but no similar question here."}
{"id":"75782","title":"Why does electric field intensity $E$ can be uniquely determined by its divergence and curl?","text":"My question is, the number of following equations $$\\nabla\\cdot E=\\frac{\\rho}{\\varepsilon}$$ $$\\nabla\\times E=-\\frac{\\partial B}{\\partial t}$$ is 4 while the number of unknown variables $E=(E_1,E_2,E_3)$ is 3. Intuitively, the equation is overdetermined and the solution may not exist unless four equations are correlated. Is my intuition right?"}
{"id":"96007","title":"Lowest temperature possible in the universe?","text":"The third law of thermodynamics states that nothing can reach to absolute zero temperature. What is the lowest possible temperature that can be in the universe? Has any experiment reached to a billionth of a Kelvin? Is there any restriction on how low it can be? Is $10^{-1000}\\;\\rm K$ possible? Or is there a lowest quantum of temperature?"}
{"id":"32458","title":"Why Gravity attracts all objects with the same speed?","text":"Why Gravity attracts all objects with the same speed? Is this question was solved? What is the exact answer?"}
{"id":"29306","title":"Fitting of data to a model","text":"Imagine that I have some observable value predicted with a theory for some process to be: $1+a x + b x^2$ and observed value is 1.3 with an error 0.2; a and b are some numerical constants. I also have another observable influenced by the same x to be $1 + cx+ dx^2$ and the measured value 1.2 with an error 0.6, c and d are some numerical constants. My question is: How to calculate value of x that explains both measurements the best? I guess this is called benchmark point, but how to get it and how to quantify it?"}
{"id":"8402","title":"What affects the damping of a spring?","text":"What variables affect the damping of a spring executing simple harmonic motion? What are the independent variables, and what variables would need to be controlled in an experiment? I'm attempting to complete an investigation where I measure the decrease in amplitude of a damped spring, and to prove the relationship between variables in the motion. Thanks!"}
{"id":"111753","title":"How did Einstein come up with his postulates for Special Relativity using thought experiments on simultaneity?","text":"In his popular book on relativity, in chapter IX, \"The relativity of simultaneity\", Einstein describes an experiment in which a flash happens simultaneously on A and B, as defined by the fact that an observer at the middle point M can see the light coming from A and B at the same moment: > >               --train--> >   > > \\---embankment---A-------M-------B---- Then, he proceeds to say that an observer on the train \"is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A\". So Einstein concludes that simultaneity is not absolute. And an explicit calculation using Special Relativity principles confirms this. The conclusion is that: > So the answer is that the observer on the train sees the lightning strike > the front of the train at $t' = -\\gamma\\tfrac{vd}{2c^2}$ and the rear of the > train at $t' = \\gamma\\tfrac{vd}{2c^2}$. The time between the lightning > strikes is $\\gamma\\tfrac{vd}{c^2}$. By setting $\\gamma=1$, we obtain the result in Galilean Relativity (ie: \"The time between the lightning strikes is $\\tfrac{vd}{c^2}$\"), which is the theory of space time before Einstein came out with Special Relativity. The point is that regardless of whether we are calculating using Galilean Relativity or Special Relativity, the conclusion is that different people on different speed will always have different notion of simultaneity. Since this is a conclusion known to everyone before Einstein, how did Einstein use this insight to derive his Special Relativity? Truly, I fail to see how can this thought experiment on simultaneity can give Einstein insights into Special Relativity. Anything I miss?"}
{"id":"8407","title":"Why electrons behave as a particle and also as a wave?","text":"Why do electrons (and other very small particles) sometimes behave as particles (i.e. when we are not looking at them) where as other times they behave as waves?"}
{"id":"9818","title":"Logical positivism and black hole interiors","text":"A black hole exists. Eventually, it will completely evaporate away. Alice falls into the hole. Imagine you are Alice. According to logical positivism, the interior of the black hole exists. But you will die at the singularity. You can't measure anything after you're dead, so any event outside after the black hole has vanished doesn't exist. Bob stays outside the hole. Imagine you are Bob. The interior doesn't exist. Enter Quentin. Right now, he holds a qubit in the state ${1\\over\\sqrt 2} (|0\\rangle + |1\\rangle)$. He will measure the value of the qubit shortly in the future. If 0, he will jump into the hole. If 1, he will stay out forever. Imagine you are Quentin. Right now, does the interior exist? Curiouser and curiouser."}
{"id":"9814","title":"Why can't my eye see itself in the mirror through polarizing 3D-glasses?","text":"I found a pair of polarizing \"3D glasses\" lying around, and tried to look at myself in the mirror while wearing them. To my utter confusion, when closing the left eye and only looking through the right eye, I could not see the right eye in the mirror. The light could not pass through the same polarized lens twice. (I could, however, see the closed left eye clearly.) I would expect the opposite to be true, as light going out the right lens with polarization X and coming back in with the same polarization X should pass through unaffected."}
{"id":"9815","title":"Boundary layer theory in fluids learning resources","text":"I'm trying to understand boundary layer theory in fluids. All I've found are dimensional arguments, order of magnitude arguments, etc... What I'm looking for is more mathematically sound arguments. Not rigorous as in keeping track of all epsilons and deltas, but more rigorous than an heuristic argument. Hope you understand what I mean. Some free resources available on the web would be preferred, but if you can suggest book titles that's also helpful. Thanks. Edit: The applications I have in mind are the calculation of damping in surface waves in basins of various shapes (circular, rectangular, etc)."}
{"id":"92365","title":"Questions about Statistical Mechanics","text":"1. For grand partition ensemble, is it true that the introduction of chemical potential allows us to have the sum of number of the particles in each state to be the total number of particles (\"On average\")? and this fluctuation in the constraint of $N$ will shrink as the number of particles you got is large.   2. Next question is about Boson gas, why would we have to consider the chemical potential to be negative or zero? Is there anything to do with the reference to the ground state energy level?   3. What would it mean if we were to integrate the energy of those states below ground states? NB For last question, I mean when I were to find the total energy of this boson gas system, which I know below a critical temperature Tb, the Bose Einstein condensation becomes important, and we would do the separate sum thing as usual ![enter image description here](http:\/\/i.stack.imgur.com\/ufZ9N.png)(see first Pic) , Now I wonder why when we do the sum of the energy we integrate the states above ground state , does it mean all the particles at ground state have zero energy?? and we only have to concern those above the ground state? ![enter image description here](http:\/\/i.stack.imgur.com\/d3ALk.png)"}
{"id":"66445","title":"Lorentz force for electrically conducting fluid flow in homogeneous magnetic field","text":"I am mathematician and have paper which models situation when homogeneous magnetic field is applied to moving electrically conducting fluid. There is such Lorentz force formula on which all the work is made: $$F= \\sigma (E + V \\times B) \\times B $$ But I think that the last $\\times B$ is not needed. Is this correct formula for this case or the work is not correct?"}
{"id":"66449","title":"Is it possible to calculate distance if non-constant acceleration is unknown?","text":"I know that the distance travelled in non constant acceleration is $d=\\int_a^b f(x)\\mathrm{d}x$, but is it possible to calculate the distance without knowing the value of $f(x)$?"}
{"id":"127168","title":"Twins Paradox Paradox","text":"I've recently has special-relativity explained to be in a rather elegant way. All objects travel at the speed of light in space time. Thus, when you travel faster through the three dimensions of space, the speed at which you travel through time decreases. Photons do not experience time because all their velocity is in the spatial directions and none of it is in time. Considering the 4-dimensional universe this invokes, how can I ever interact with anything that has traveled at a different velocity than me unless it changes direction to meet me? In the terms of the twin paradox, how is it possible for the twins to meet? Twin A, stays on earth traveling at a fixed speed in spacetime. Twin B leaves earth traveling faster in space and slower in time but still traveling at c in spacetime. Some basic geometry tells me that if two objects forced to travel at a fixed speed from the same origin in changeable directions (in any number of dimensions) that there is no way their future locations can be the same unless they **both** change direction to provide the possibility of intersection. If Earth is traveling at a fixed speed and Twin B is traveling at a fixed speed in a space (no matter the dimension) then there is no way they could ever be in the same location in spacetime again unless the earth changes direction and meets Twin B in the middle. Wouldn't they be forever doomed to be in different places along the axis of time unless Twin A goes on a relativistic voyage to allow Twin B to catch up? Should we not expect time to behave the same way as the spacial dimensions?"}
{"id":"52972","title":"Why the peak of spectrum gets vague when the dimension is lower?","text":"In a many-body system, we can know the spectrum function at a particular temperature from Green function. It means density of states. A peak of spectrum represents one mode. My question is that in the same model why the peak of the spectrum gets broader with lower dimension?"}
{"id":"52976","title":"How can the speed of light be a dimensionless constant?","text":"This is a quote from the book _A first course in general relativity_ by Schutz: > What we shall now do is adopt a new unit for time, the meter. One meter of > time is the time it takes light to travel one meter. The speed of light in > these units is $$\\begin{align*}\\end{align*}$$ > > $$\\begin{align*} c &= \\frac{ \\text{distance light travels in any given time > interval}}{\\text{the given time interval}}\\\\\\ &= \\frac{ \\text{1m}}{\\text{the > time it takes light to travel one meter}}\\\\\\ &= \\frac{1m}{1m} = 1\\\\\\ > \\end{align*}$$ > > So if we consistently measure time in meters, then c is not merely 1, it is > also dimensionless! Either Schutz was on crack when he wrote this, or I'm a dope (highly likely) 'cos I can't get my head around this: The space-time interval between different events at the same location measures time, and between different events at the same time measures space. So they're two completely different physial measurents: One is a time measurement using a clock, the other a space measurement using a ruler. In which case the units of $c$ should be $ms^{-1}$ Does Schutz correctly show how $c$ can be a dimensioness constant?"}
{"id":"75416","title":"Hemisphere irradiance","text":"How do I calculate sky irradiance from radiance (L) from a hemisphere above a surface which is tilted relative to the normal (x=0,y=0,z=1). I have L as a function of zenith (0 to 180deg) and azimuth (0 to 360 deg)? Any known MAtlab functions doing this kind of thing?"}
{"id":"114705","title":"Mistake in the Feynman Lectures Volume 1 Ch. 18-2 - Rotation of a rigrid body","text":"I just read http:\/\/www.feynmanlectures.caltech.edu\/I_18.html#Ch18-S2 In my opinion, in this chapter the equations 18.6 and 18.7 are wrong. Have a look at the Picture http:\/\/www.feynmanlectures.caltech.edu\/I_18.html#Ch18-F1 . > If $OP$ is called $r$, then the length $PQ$ is $rΔθ$, because of the way > angles are defined. The change in $x$, then, is simply the projection of > $rΔθ$ in the $x$-direction: $$\\Delta > x=-PQ\\sin\\theta=-r\\,\\Delta\\theta\\cdot(y\/r)=-y\\,\\Delta\\theta$$ That is equation 18.6. _It is wrong because $\\Delta\\theta$ doesn't mean that $\\theta_{Q}-\\theta_{P}$ is infinitesimally small and equation 18.6 is only valid for infinitesimally small $\\Delta\\theta$._ An example is: I take $P$ to be $(1|0)$ and $Q$ to be $(0|1)$ than $\\Delta x$ is $-1$ but calculated using equation 18.6 it should have been $-y\\Delta\\theta=(-1)(\\pi \/2)\\approx -1,57$, which is wrong. In the equations $d\\theta$ should have be used instead of $\\Delta\\theta$. **Is my reasoning correct?**"}
{"id":"96066","title":"Proof for torque=force*perpendicular distance to line of action of force","text":"I want to derive from first principles the proof for magnitude of torque about a point O = perpendicular distance from O to the line of action of the force * magnitude of the force. I want to derive this result for a general 3-D case for any arbitrary point O and force **_F_** . I need to prove this mathematically without knowing the cross-product magnitude in terms of the angle between the vectors..... that is given any force F = Fx i + Fy j + Fz k and any point O and using torque= r cross F & using the determinant expansion for the same & also the expressions for magnitude of a vector, I need to prove this.... In other words, I need to prove that the magnitude of the cross- product is indeed F*r*sin(theta"}
{"id":"106264","title":"What is the maximum range of a bullet flying through the sky?","text":"**How far would my bullet fly with the following characteristics?**   * 1.2km\/second initial velocity   * 45 degree angle into the sky   * 'Normal' atmospheric conditions   * No wind   * Typical high-powered rifles have 5-10 gram bullets I've tried various online calculators but they only do vacuum, not normal atmospheric conditions, so the range is 102km."}
{"id":"106260","title":"Poincaré symmetry and linearized gravity","text":"When working with linearized gravity, is Poincaré symetry assumed to be the symmetry of space-time?"}
{"id":"119632","title":"Why do ferromagnetic thermocouples change their behaviour near the Curie point?","text":"Having read Wikipedia's current explanation that certain types of thermocouple (specifically Type K thermocouples) experience a step-change in their potential when they reach the Curie point of the constituent iron, it does not explain the reason, nor why the Curie point is important for the thermoelectric properties of ferromagnets. There seems to be no more of a hint on Wikipedia's page about the thermoelectric effect, nor for the page on the Curie temperature. Searching the internet more generally for information only seems to find similar assertions without explanations, or abstracts from academic papers hidden behind paywalls. Can anyone explain this connection in terms a layman reading Wikipedia would understand, or failing that an undergraduate physics student?"}
{"id":"116709","title":"Optics, What is a Foil?","text":"I was using a program for simulating optical elements; mirrors, gratings, foils, crystals, slits, and zoneplates are what it mentioned. Does anyone know what a foil is in terms of optics? I know what aluminum foil is, obviously, but in terms of an optical system I don't know what a foil, or foil, would be, and what it's unique optical properties are."}
{"id":"24919","title":"What age to start kids with astronomy","text":"My son is getting interested in astronomy. I was thinking about getting him a scope when he gets a bit older, he is 4 now. At what age have other people gotten their kids telescopes? (I have a 10\" dob that we use together) Also at what age is taking a kid on a dark sky camping trip start to make sense."}
{"id":"75471","title":"What prevents this magnetic perpetuum mobile from working?","text":"As a child, I imagined this device, which may seem to rotate indefinitely. I have two questions.   1. **Is this perpetual motion machine already known?** If it is, could you please give some references?   2. **What is the exact mechanism that makes it stop?** By this, I mean an explanation, not simply \"because it would break energy conservation\". Of course energy conservation is true, and of course cannot rotate indefinitely. But for any known (presumed) perpetuum mobile, there was an explanation, usually based on showing that the force generating the motion is balanced by another force. For example, Stevin obtained the laws of the inclined plane from the perpetual motion device ![Magnetic perpetuum mobile](http:\/\/i.stack.imgur.com\/rhhTv.gif) I would like to make some observations.   * I don't try to convince anyone that it will move forever, because **I don't believe in breaking the energy conservation**. It is true that great physicists like Bohr, Kramers and Slater admitted the possibility, and nowadays some who think that there may be energy exchanges between parallel worlds in MWI believe, but I don't.   * But I don't consider an enough explanation simply to refer to the energy conservation. I am interested in an explanation showing exactly how the magnetic forces making it rotate, are balanced.    * If the forces are balanced, **only then** friction will make it slow down and stop. I don't think that we can explain only by referring to friction, which in principle can be made as small as needed. There has to be a balance of forces.   * Why spending time trying to understand or explain something that admittedly can't work? Well, even though perpetual motion machines cannot actually work, I think they may be interesting as puzzles."}
{"id":"99113","title":"Why is there water coming out of a car’s tail pipe?","text":"I notice yesterday that my neighbor’s car had water coming out of the exhaust pipe in the morning. My first response was since the hot exhaust is hotter than the cold tail pipe, heat is transferred from the hot exhaust through the pipe, and with enough moisture in the exhaust, enough heat leaves such that the humid air is condensed forming water drops.   1. Is my thinking correct here? If not, please correct me.   2. Does the water come from (i) a chemical byproduct of gasoline combustion or (ii) the humidity of the air (i.e. I've only noticed this in the winter, not the summer so I assume that during the summer this would not happen since the tail pipe is “already warm enough.”)"}
{"id":"91954","title":"How is electric flux related to permittivity?","text":"How is Gauss' law related to permittivity? I know that it equals $1\/\\epsilon_0$ times the magnitude of the charge enclosed. But, I'm unable to understand what it actually means. Can someone intuitively explain it to me?"}
{"id":"24006","title":"Reference paper to support information -- energy relation ($kT \\ln2 \\rm\\frac{J}{bit}$)","text":"In answer to Maxwell's Demon Constant (Information-Energy equivalence) there is stated that one bit of information allows to perform $kT \\cdot \\ln2$ Joules of work. Which paper supports the thesis? (there are many publications on Maxwell daemon, Szilard engine, Landauer's principle)."}
{"id":"21249","title":"static friction problem without mass","text":"I am having trouble with a physics problem. The problem is as follows: A car is traveling at 54.0 mi\/h on a horizontal highway. A) If the coefficient of static friction between road and tires on a rainy day is 0.103, what is the minimum distance in which the car will stop? B) What is the stopping distance when the surface is dry and µs = 0.595? This is what I have so far for A)               fs = µs * Fn          Fn = Mg     fs = Ma          Ma = µs * Mg     a = µs * g          V = 54 - a * t     0 = 54 - a * t     t = 53.497 seconds          dist = 54*t - (a\/2)*t^2     = 1444.42243 miles     = 2324572.57 meters      This is not the correct answer. Any help would me much appreciated."}
{"id":"67181","title":"Where can I find the full derivation of Helfrich's shape equation for closed membranes?","text":"I have approximately 10 papers that claim that, from the equation for shape energy: $$ F = \\frac{1}{2}k_c \\int (c_1+c_2-c_0)^2 dA + \\Delta p \\int dV + \\lambda \\int dA$$ one can use \"methods of variational calculus\" to derive the following: $$\\Delta p - 2\\lambda H + k(2H+c_0)(2H^2-2K-c_0H)+2k\\nabla^2H=0$$ But I'm having a lot of trouble tracking down the original derivation. The guy who did it first was Helfrich, and here's his and Ou-yang's paper deriving it: http:\/\/prl.aps.org\/abstract\/PRL\/v59\/i21\/p2486_1 . However, they don't show an actual derivation, instead saying \"the derivation will appear in a full paper by the authors\" or something like that. Yet everybody cites the paper I just linked for a derivation. Does anybody know a source that can derive this, or can give me some hints to figure it out myself? To be honest I can't even figure out how to find the first variation. Edit: So, after some careful thought and hours and hours of work and learning, I realized that the answer that got the bounty was wrong. The author stopped replying to my messages after I gave him bounty.... thanks guys. That said, I've almost got it all figured out (in intense detail) and will post a pdf of my own notes once I'm done!"}
{"id":"62175","title":"General definition of an event horizon?","text":"Horizons are in general observer-dependent. For example, in Minkowski space, an observer who experiences constant proper acceleration has a horizon. Black hole horizons are usually defined as boundaries of regions from which no lightlike curve can reach null infinity $\\mathscr{I}^+$. But how can this be interpreted in terms of an event horizon for an observer? Immortal material observers end up at timelike infinity $i^+$, not $\\mathscr{I}^+$. Is there some nice way of unifying both cases? In other words, is there a general definition of an event horizon that has both these types of horizons as special cases? [Edited to clarify the question and remove a mistake about the dimensionality of $i^+$ versus $\\mathscr{I}^+$.]"}
{"id":"68208","title":"In a turning plane, will the vector of combined centrifugal plus gravity force experienced by passengers be perpendicular to the floor?","text":"In a turning plane, will the vector of combined centrifugal plus gravity force experienced by passengers be perpendicular to the floor? In other words, will the passengers experience the feeling of inclination without looking into windows? I think that the reaction of air is perpendicular to the wings and as such, there will be no feeling of inclination. Am I right?"}
{"id":"4364","title":"Does the positive mass conjecture indicate a necessity of interactions in our universe?","text":"The positive mass conjecture was proved by Schoen and Yau and later reproved by Witten. Total mass in a gravitating system must be positive except in the case of flat Minkowski space, where energy is zero. Since QG is intended to be a theory of interaction with force particles called gravitons, one may begin to wonder if the interactions are in fact the important defining features of the space in question. So does a theory with interactions also require that space be curved?"}
{"id":"37861","title":"Spooky action appears to contradict Relativity of time order of multiple events","text":"It is well known that in special relativity observers can disagree on the time ordering of two events. It is also well known that entangled particles exhibit so called spooky action at a distance. Today I read in the New Scientist and on the arXiv that although the order of two events can be arbitrary, this is not so for multiple events. As the number of events increase the number of combinations of possible orders increase but not every one of these combinations will actually be possible to always observe. According to the NS article it is possible to entangle three particles a,b and c and then have them in such a way that the collaspe of c cannot be observed to precede the collapse of both a and b by any observer even though c spookily causes the collapse of a and b. Is this an outright contradiction between relativity and quantum mechanics or just a paradox ? What is the explantion ?"}
{"id":"37869","title":"A good example of a nonlinear symplectomorphism?","text":"What is a good example of a simple, physically useful _nonlinear_ symplectomorphism $\\kappa: \\mathbb{R}^{2n} \\rightarrow \\mathbb{R}^{2n}$? I'm not much of a physicist, and all the examples I've worked have been purely mathematical and not connected to any physical problems. Obviously linear examples are well known and useful. Actually, it would suffice if I could see some nice example of a nonlinear diffeomorphism (i.e. change of coordinates) $\\varphi: \\mathbb{R}^3 \\rightarrow \\mathbb{R}^3$ given by $x \\mapsto y$, as one can then lift this diffeomorphism to a symplectomorphism by transforming the momenta via $p_y := (d\\varphi^{-1})^{*}p_x$. Thanks in advance."}
{"id":"3107","title":"When one thinks of a field of operators in QFT, is it reasonable to think of a matrix being associated with each point in space time?","text":"Is it correct to visualize operators existing as matrices parameterized by spacetime coordinates in the context of QFT?"}
{"id":"3101","title":"Is it a goal of modern physical theory to avoid big bang singularities and how do they approach the problem?","text":"During discussion on another question, the question of singularties in modern physical theories arose. The big-bang is an obvious singularity in modern conceptions of the cosmos. Is it a goal of modern theories to find a way to remove the big bang singularity from theory, and how is that accomplished in theories being proposed today?"}
{"id":"79311","title":"Depletion region","text":"In a semiconductor diode. There is a depletion region formed, it is formed when electrons from n type side migrate to p type side, now which electrons transfer? Valence band ones present at the boundary or the conduction band ones? If the conduction band ones only migrate, then are there enough present on the boundary such that they can create a field at the boundary that they can stop this diffusion process? By boundary i mean the border of n type and p type material."}
{"id":"131726","title":"Energy from the Feynman-Kikuchi Partition Function","text":"The Feynman-Kikuchi Partition function is given as $$Z_{FK}=K_\\beta \\int dx \\eta(x) \\exp \\left(-\\frac{x}{\\beta}\\right) $$ where $K_\\beta$ is a normalization constant and $$\\eta(x)=\\frac{1}{N!}\\sum_P \\delta\\left(x-(R_0-PR_0)^2\/4\\lambda^* \\right)$$ where $R_0$ is a set of particle coordinates for a perfect cubic lattice, the sum over P means over all perturbations, and $\\lambda^*$ is defined as $$\\lambda^*=\\frac{\\hbar}{2m^*}$$ where $m^*$ is the reduced mass. My questions is this: in Ceperley's notes \"Path Integrals in the theory of condensed helium\", he states that, if we differentiate the partition function above with respect to $\\beta$, we obtain the energy, in the form $$\\epsilon=\\epsilon_0+\\frac{3T}{2}-\\frac{T^2 \\langle x\\rangle}{N}$$ I can't see how he would get that--I know that I have to calculate $$\\epsilon=-\\frac{1}{Z}\\frac{\\partial Z}{\\partial \\beta}$$ But wouldn't that just give me $$\\epsilon=-\\frac{1}{\\beta^2}\\frac{\\sum_P (R-PR)^2\\exp\\left(-\\frac{1}{\\beta}(R-PR)^2 \\right)}{\\sum_P \\exp\\left(-\\frac{1}{\\beta}(R-PR)^2 \\right)}$$ I just don't see how that would give me Ceperley's results. Thank you in advance."}
{"id":"69241","title":"Pauli principle for particles very far apart from each other","text":"Can two electrons be in the _same state,_ when they belong to two different atoms, which are \"far enough\" (whatever that means) apart from each other? With \"same state\" I mean that (as far as specifiable) the states are really identical, except for the position of the electrons. More specifically:   I am still not clear of how the separation of particles is taken into account for the Pauli exclusion principle. E.g. in a crystal the electrons seem close enough for the exclusion principle to become meaningful, for particles \"a universe apart\" from each other, it seems that it is kind of redundant (is there specific maths to that?), but what about everything in between? (This is closely related to the talk Brian Cox gave, see this question: physics.stackexchange.com\/q\/18527\/16689 )"}
{"id":"134224","title":"Why black body radiation is all over the frequency range","text":"I was studying black body radiation and how quantization of energy solves the problem of ultraviolet catastrophe. But I have a very fundamental doubt. A black body can be assumed as a cavity with a small hole with radiation leaking out of it. As the temperature of the black body is increased we can assume the the charge particles, electrons, on the metal surface will behave as harmonic oscillators and the energy of the harmonic oscillation will be equal to the energy density of the radiation inside the cavity at thermal equilibrium. My doubt is that at thermal equilibrium, the cavity(assuming it to be uniform) will have a uniform temperature. Since the oscillations of the charged particles is due to the thermal agitation, how can the charge particles radiate all over the frequency range? The temperature is uniform over the cavity, wont they all be experiencing same thermal agitation and oscillate at same frequency"}
{"id":"91573","title":"How to explain polarization in Zeeman effect","text":"In the Zeeman effect, when we observe along the $B$ field, the polarization of light should be circular polarized. It can be understood by the conservation of angular momentum. $\\Delta m=0$ can not be observed and $\\Delta m=\\pm1$ can be observed and thus circular polarization. However, I can't understand what we observe perpendicular to $B$ field. It should be linear polarized light. So the state of the photon should be something like $|\\mathrm{linear~polarized}\\rangle=\\frac{1}{\\sqrt{2}}|x_+\\rangle+\\frac{1}{\\sqrt{2}}|x_-\\rangle$($|x_\\pm\\rangle$ are circular polarized states in $x$ direction. We assume $B$ is parallel to $z$ axis) ,depends on $\\Delta m$. How could it be like this? Why the light is linear polarized and can't be circular polarized? I am really confused."}
{"id":"91572","title":"Spreading UV light? Fresnel lenses?","text":"I'm making a little UV exposure box and I'm looking for a way to evenly spread out the UV light. I've used Fresnel lenses to do just this in the past, but that was for visible light. But from what I've read, not all things work at all wavelengths and some materials will block UV. So my first question, would a Fresnel lens still work and evenly spread out UV light? What material should I look for? The lens I've seen are PVC and from what I've seen elsewhere, PVC has good UV resistance, but I don't know if that means it will just hold up to UV or if it will block UV all together. Any tips or suggestions would be great thank you"}
{"id":"25834","title":"What is the simplest way to prove that the Earth orbits the Sun?","text":"Assume you're talking to someone ignorant of the basic facts of astronomy. How would you prove to them that the Earth orbits the Sun? Similarly, how would you prove to them that the Moon orbits the Earth?"}
{"id":"80028","title":"What does the complex electric field show?","text":"We have a complex electric field. Is there any definition for absolute and imaginary part of a complex electric field? What do they stand for?"}
{"id":"101647","title":"Coherent State in 2 dimensions","text":"I am looking at a 2D harmonic oscillator $$H=\\frac{1}{2m}(p_x^2+p_y^2)+\\frac 12m(\\omega_x^2x^2+\\omega_y^2y^2)$$ Where $\\omega_x=5\\omega_y$. I am told that the oscillator is prepared in a coherent state with the following qualities: $$\\langle x(0)\\rangle=x_0$$ $$\\langle p_x(0)\\rangle=0$$ $$\\langle y(0)\\rangle=0$$ $$\\langle p_y(0)\\rangle=p_0$$ I am looking for the time dependent state of the system. My approach: Seeing as this is a coherent state, the expectation values act as classical variables, this means that the position and momentum will be given as $$q(t)=q(0)\\cos(\\omega t)+\\frac{1}{m\\omega}p(0)sin(\\omega t)$$ $$p(t)=p(0)\\cos(\\omega t)-m\\omega x(0)\\sin(\\omega t)$$ Additionally, the coherent state is given as: $$\\alpha (t)=\\frac{1}{\\sqrt{2m\\omega\\hbar}}(ip(t)+m\\omega x(t))$$ So if I want to find the state as a function of time, do I find the alpha for both x and y and add them together?"}
{"id":"101642","title":"How would an electron bunch\/beam look different in the rest and lab frames?","text":"With respect to special relativity, I was wondering how the spatial dimensions would differ between the rest and LAB frame of an electron beam. System: Electron bunch\/beam traveling in linear motion. How large\/long would the bunch appear in the point of view of an electron (rest frame). How large\/long would the bunch look in the lab frame? How would the answers change if the electron bunch\/beam was accelerating?"}
{"id":"109112","title":"Proving $RTln(a_i)$ equal to non PV work done by system,where $a_i$ is activity","text":"First of all many here will might say that it is Chemistry question.If you think so then read this article and then answer question. Question is --- We know chemical potential is defined as $\\mu_i =\\mu_i^{standard}+RTln(a_i)$ .Here $a_i$ is activity of $i^{th}$ component of solution.In case of gases instead of $a_i$ it is $f_i$ that is fugacity of $i^{th}$ gas out of mixture of gases.Activity is here nothing but activity coefficient multiplied by concentration or in case of fugacity it is product of fugacity coefficient multiplied by partial pressure of gas. But now this is not the only definition of Chemical potential.Chemical potential also means **partial molar gibbs free energy** when pressure and temperature of system are constant.We know that Gibbs free energy of system at constant temperature and pressure also represents Non-PV work done by system.Now how can the term $RTln(a_i)$ can be correlated to Non-PV work done by system in order both two definitions will be equivalent."}
{"id":"109116","title":"How to find solutions to the gravitational potential metric h","text":"I'm working on a problem in which a star of mass M1, radius R1 is surrounded by a thin shell of mass M2, , radius R2. I want to find the solutions to the gravitational potential h in the region in between the star and the shell and region outside both. Can we assume the mass is stationary, and not too large, and thus use the analogous Poisson's equation (Laplacian of potential = 4*pi*G*rho)? Or is there some physical reason that we should use the potential equation in the Hilbert gauge (d'Alembertian of tensor potential = 4*pi*G*(energy momentum tensor)\/(c^2)? My thinking is that Poisson's is totally appropriate for this simple situation of a stationary star with a shell mass around it. Now, I don't see how to use Poisson's equation. I'm getting tripped up with how to consider the two masses. What if, for example, it's a very dense shell which is in fact more massive than the star. That would give a potential radially outward. How do I find a solution to the Laplacian?"}
{"id":"76804","title":"Mixing of quarks, neutrinos... and leptons?","text":"This is a quite simple question: quarks do mix (through the CKM matrix), neutrinos do mix (through the PMNS matrix). Then **why do charged leptons not mix?**"}
{"id":"128972","title":"Does supercavitation create vacuum?","text":"While reading the amazing things that the mantis shrimp can do, such as moving two of their limbs so quickly that the water around them 'supercavitates', a friend of mine told me that those limbs were creating vacuum, in the water. Does this creation of vacuum happen? Or is it something else? If so, am I creating vacuum when boiling water?"}
{"id":"74404","title":"Energy of a string","text":"What is the correct definition of the energy of a string ? I suddenly get confused with the definition of the energy of a string. Considering, for instance, a bosonic open string in the light-cone gauge, We have $H = p^-=p_+$, the hamiltonian, and we have $p^0$, which appears in the expansion of the string in light-cone gauge $X^o(\\sigma, \\tau) = x^0 + \\frac{p^0}{p^+}\\tau + ...$. a) In Zwiebach's First course in string theory (1), there is a calculus of the entropy of a string, which begins (Chapter $16.2$ p $354$) by a function partition for a non relativistic string with fixed points (the quantum violin string). For a given energy $E = N \\hbar w_0$, where $N$ is the level, the entropy is computed (this is direcly linked to the number of partitions p(n)). More precisely, an expression for the entropy, $\\sim T$, and the energy ,$\\sim T^2$, are obtained, se we get a $S \\sim \\sqrt{E}$, and plugin the $E \\sim N$ expression, we get a $S \\sim \\sqrt{N}$. \"Suddenly\", in chapter $16.3$ page $361$, it is said that \"we now return to relativist strings that carry no spatial momentum, and this happens if the open string endpoints end on a D-0 brane, so the energy levels are given by the rest mass of its quantum states\". Of course, we have $m^2 \\sim N$ (for large $N$), so $m\\sim \\sqrt{N}$, so if the energy ($p^0 ?$) is identified with $m$, we have $S \\sim E$ Now, if we take the hamiltonian $H$, for large $N$, we have $H \\sim N$, so $S \\sim \\sqrt{H}$ So, this confused me, because I have $2$ law expressions for the entropy, depending on the definition of the energy. b) An other problem, is saying that the energy of the string is proportionnal to its length. If we look at $H$, we have $H=p^- \\sim \\frac{1}{l}$ (with zero trans verse momenta $p^i$), while $p^+ \\sim \\frac{l}{\\alpha'}$. So, may I write that the correct energy is $p^o \\sim (p^+ + p^-)?$"}
{"id":"74405","title":"Scaling of non-gravitational energy in a black hole","text":"When looking at a Schwarzschild black hole, for instance, we know that we may apply black hole thermodynamics. We may define a entropy of the black hole which scales like the area of the horizon : $$S \\sim R_s^2$$. It is understood in the more general context of the holographic principle which states that \" the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon\" Now, the non-gravitationnal energy $E_{ng}$, so the mass $M$ for the Schwarzschild black hole, has a different scaling : $$E_{ng} \\sim R_s$$ So, does that mean that the energy is encoded in a one-dimensional object (perimeter, loop, string, radius), and is it a different \"holographic\" principle ?"}
{"id":"131167","title":"The Spinning Log \"Perpetual Motion\" problem, and my attempt at a solution","text":"So I was introduced to this \"perpetual motion\" riddle a few weeks ago. The problem goes like this: we all know perpetual motion machines are not possible, but this riddle _seems_ like it should work as a perpetual motion machine - the problem is to explain _why it doesn't work_. **Here's the situation:** (1) You take a room, and put a wall down the middle of the room - splitting it in to two equal half-rooms. (2) Take a perfectly cylindrical log whose length is the same as the length of the wall we just put in to make two half-sized rooms, and cut out a place halfway up the wall so that the log slips right in, length wise (if you were looking at the middle wall from the inside of one of the rooms, you would not see the ends of the log at all, you would a half-cylinder running the length of the wall - if you looked from the other room, you would see the other side of the cylinders length). (3) Make sure the space between the log and the wall is air-tight (the log is treated to absorb no water, there is no friction between it and the wall, etc), and fill one of the half-sized rooms all the way to the top with water. Now you have one side of the log exposed to air, and one side totally surrounded by water. The idea is that, since everyone has experience that logs rise in water, the side exposed to the water should \"rise\" and the log should spin, creating a perpetual motion machine. Now, I have two explanations of why it _doesn't_ work. One I came up with myself, and one that is proposed in the solution on the page in which I found the problem. **My Solution:** Water pressure increases as you increase in depth, so the water molecules hitting the log deeper in the water would have more force that the water molecules hitting the log less deep, but at each infinitesimal depth increase, the water is hitting the log at all angles, and the force from the molecules coming on from all directions would cancel out to leave a force pointing toward the center, at every place the log is exposed to water. Since all of the net forces from the water at each point exposed to water is pointing toward the center, there can be no net torque and the log cannot spin. The **net** force would be a slight force directly upwards (due to the difference in size of the force vectors pointing inwards between the less dense water up top and the more dense water on the bottom), but since the log is held in place, it cant move upwards. **The Proposed Solution** \"To understand why this would not work, we need to look at how buoyancy itself works. When a lighter than water object (lets say a hollow ball) is submerged it is pushed up by the water apparently in spite of gravity. In fact the opposite is true. It is gravity pulling down on the water that pushes the ball up. The kinetic energy to lift the ball comes from an equal volume of water falling to occupy the space where the ball just was. As the ball moves up, the evacuated area is filled with water from above it; therefore, this water is falling. When the water was above the ball it had potential energy that is exchanged for kinetic as it falls to fill the void left by the ball thus providing energy to lift the ball. The opposite is true for a heavier than water object (lets say a brick), but it's still the same principle. As the brick falls through the water, it is filling space that was once filled by water. As the brick falls, it is providing energy to lift the water to fill the space it just evacuated. Now back to our spinning log example. If the log were to spin, it would not be evacuating any space for the water to fill. Even though a different part of the log would be filling that space, it would still be the same space it was filling before. There would be not downward movement (falling) of water to convert potential energy into kinetic, so there is no energy to cause the log to spin.\" **Here's my problem** I've never understood this explanation of buoyancy. Why, just because an object is less dense than water, is the water \"rushing below it\" to fill the space evacuated by the object as it rises? Why doesn't the object just stay where it is? It seems like you're using buoyancy to explain buoyancy - the less dense object floats upward due to buoyancy and the water rushes in below to fill the evacuated space, which causes the buoyant force upwards... what? Why is the water rushing below less dense objects, and not rushing below more dense objects? Am I missing something in the explanation? I understand that when you submerge an object in water, it displaces an equal volume of water, and the weight of the water displaced equals the bouyant force upwards, but I just don't understand why the water is \"rushing underneath and pushing the object up\"."}
{"id":"6658","title":"What is the closest general-relativistic equivalent of a \"time slice\"?","text":"In a newtonian universe, one can talk of a \"time slice\", that is, the state of the universe at a given point in (global) time. In a \"typical\" classical universe, a time slice would contain enough information to fully compute the state of the universe at any other point in time, backwards or forwards. So, disregarding what we know of quantum effects and talking purely of a general-relativistic universe, which concept is closest to a time slice?"}
{"id":"6654","title":"Is it possible for one side of the universe to \"meet\" the other?","text":"I've variously heard the shape of the universe being described as multi- dimensional, like a helix or mobius strip, and super string theorem seems to say there are lots of universes all piled up next to each other in vibrating planes. My question is, can the edges of our universe ever join up? Are they already all joined up? as in is our universe really in the shape of a giant donut (yum). or is this shear lunacy? I guess I am assuming a model of the universe where the \"edge\" is not necessarily expanding constantly \"out\" into infinite amounts of empty vacuum, but rather one where the edges are the boundaries to that expansion (so that it seems from an observable position that the universe is expanding, when really its flexing inside these boundaries). As indicated in the comments no doubt an inaccurate description, but I was imagining it like a balloon."}
{"id":"92008","title":"Neutrino mass and energy question","text":"If a neutrino has mass then it travels less than the speed of light. Suppose I boost myself to the rest frame; i.e. bring it to rest in the laboratory. Now if it oscillates between different states and masses sitting there, where does the oscillating excess\/loss energy and mass reside? In some internal state? I have a very limited knowledge of QM but in SR it would seem strange that \"mass\" would move into another place. A pointer to where I should start studying would (I hope) be sufficient."}
{"id":"24776","title":"Scale invariance Vs Conformal invariance","text":"> **Possible Duplicate:**   >  Why does dilation invariance often imply proper conformal invariance? What exactly is the difference between the two? Can someone give an example of a theory which is scale invariant but not conformally invariant?"}
{"id":"6389","title":"How could this person have discovered the resonant frequency from this string of magnets?","text":"I stumbled onto this page http:\/\/mylifeisaverage.com\/story\/1364811\/ and the post states that they were > all making strings and shapes with these sets of 216 really small spherical > earth magnets. What did I do? I strung them together and found the string's > resonant frequency. It was 15 Hz so I'm wondering how the poster could have figured this out?"}
{"id":"43977","title":"How would an observer perceive movement on a train that's travelling near the speed of light?","text":"Person A is on Earth and a train (or whatever you want to imagine) travels past him at near the speed of light. How would person A perceive movement on the ship? If time is slowed on the ship from the perspective of Person A, then if Person B were washing the windows or something (moving his hands up and down), would they look like they were going really fast, slow, or normal? If time is slowed from the perspective of Person A, then Person B's hands would be moving a certain distance (let's say the window is a meter) in a really short amount of time, no? And that would mean it looks like it is going fast. But then since Person A is perceiving the time as moving slower, it would look slower, no? Would they cancel and movement on the ship looks normal? I tried to see if the Wikipedia page could help, but it didn't seem to say anything about a stationary observer and a fast-moving vessel."}
{"id":"43971","title":"Why doesn't a neon sign seem that hot?","text":"I heard that neon signs contain plasma, why aren't they hot? is it because the electrons and ions do not hit the lamp's wall? Is it because it is non thermal plasma and electrons and ions are not in thermal equilibrium? If that is the case do the electrons and ions and neutral atoms (all of them) hit the lamps wall?"}
{"id":"43973","title":"Applying angular velocity to a rotation matrix","text":"I have a very simple question. In our project we store an object's orientation as a 3x3 matrix which holds the orthonormal base of that object's local space. For instance if the object is aligned with the axis of the world, then its orientation matrix is :               1 0 0     0 1 0     0 0 1      Now we store the angular velocity as a vector that contains values in rad\/s around each of the world's axis. For example if the object should rotate around the world's y axis at 20 rad\/s, then the angular velocity is               0 20 0      My question is then, say we want to apply this velocity over 1 second to the orientation matrix, how would we do this?"}
{"id":"86911","title":"What is the maximum mass that the airplane can have and still maintain enough lift to fly?","text":"A commercial airplane travels at a speed which is 85% of the speed of sound. The wings of the airplane are designed such that the bottoms of the wings are flat and the tops of the wings are curved so that air travelling above the wings follows a path which is 15% longer than the straight path of the air below the wings. The wings are approximately rectangular in shape with a length of 35 m and a width of 8 m. The thickness of the wings is negligible. What is the maximum mass that the airplane can have and still maintain enough lift to fly?"}
{"id":"128356","title":"Gravitational slingshot maximum","text":"I have recently read an article about gravitation slingshot assist used by Voyagers 1-2, and was thinking on why this hasn't been used for travel between solar and other systems.   I mean sligshot can be done as many times as it is necessary to get speed of lets say half the speed of light that would allow to travel to Alpha Centauri in ~10-20 years can it not? There must be a flaw in my thinking that 3 or 4 planets can be re-used to get to necessary speed otherwise it would already have been done (drawing below). Even if planets would align differently I should always be able 'find' the planet that would allow me to jump to one that is closer to the sun, and repeat the acceleration again and again. ![enter image description here](http:\/\/i.stack.imgur.com\/t52Vs.png) **What maximum (theoretical) speed could be achieved using planets of solar system as sligshot and how much would this speed wary from planetary alignment and what realistic speed could be achieved?** UPDATE: To be more specific on the second part of the question Lets say craft weight's 500kg at starting speed of 30,000 km\/h initially it slings around Mercury (`radius 2440km`), Venus (`radius 6052 - 300 (atmosphere) = 5750 km`), and Earth (`radius 6378 - 300(atmosphere) = 6050km`) until diameter of planets is to wide to not to crash craft on surface. Then it flies to the moons of Saturn - Titan (`radius 5150km`), Rhea (`1527km`), Lapetus (`1470km`), Dione (`1123km`), Tethys (`1062km`), Enceladus(`504km`), Mimas (`396km`) and starts slinging there until diameter is to wide too. What approximate maximum speed could it get to leave the solar system?"}
{"id":"128355","title":"Has the possibility of Higgs boson being a composite particle excluded already?","text":"I heard some theory, such as technicolor, predicts the Higgs-like particle discovered at LHC should be a composite particle (correct me if I am wrong). Has this possible been completely excluded already?"}
{"id":"121483","title":"How much electricity could be generated by cloths made of thermocouples?","text":"Lets say that we made a glove, shirt, pants and a hat out of the most effective thermocouple material available today. How much electricity would be generated by each, by a healthy person, on a cool day (98 degrees F vrs 79 degrees F?)"}
{"id":"45695","title":"How does a magnet work?","text":"I'm having trouble understanding how a magnet (not the field that is generated as a result but the material itself) work. The particles are aligned in a specific direction to give rise to force but I don't see how this alignment gives rise to \"attraction\" or force."}
{"id":"134266","title":"Spin in relativity","text":"Mass and spin of the particle are used in classification of elementary particles. The mass is defined to be a Lorentz invariant quantity. On the other hand, the spin is a spacelike 4-vector and cannot be defined as an invariant quantity. My question is, what would be a convenient definition of a relativistic spin 3-vector? As is known, 3-vectors get contracted\/dilated under Lorentz transformations. So, is the 3-spin defined to be equal to the rest frame 3-spin, or do we allow it to contract\/dilate? It seems to me that the first option is generally used, especially when considering Thomas precession and similar effects. Although, I haven't seen any relevant discussion in textbooks on this topic."}
{"id":"134267","title":"Finding the current in a parallel circuit","text":"Two resistors are joined as shown. The top resistor receives a current of 3 A. What is the current in the other resistor? What is the current that enters at junction A? I'm confused on how the two currents are 3A and 1A because I thought in a series circuit all the currents are the same. Why is the answer to the current entering junction A 4 A and not 1 A? ![](http:\/\/i.stack.imgur.com\/XviUa.jpg)"}
{"id":"98945","title":"Infinite heat capacity or susceptibility means fluctuation on all scales","text":"I remember reading in an introductory text to phase transition (sorry I don't remember the name) that at a second order phase transition the specific heat and the magnetic susceptibility become infinite and that this implies fluctuations on all length scales. I understand how this implies strong fluctuations as they are respectively linked to the variance of the energy and of the magnetization, or that being infinite it's not possible to construct a preferred length scale with them; but I'm not sure to understand why it forces the fluctuation to happen at ALL the scales. Do you have an intuitive argument or example for that? Thank you in advance."}
{"id":"122687","title":"Does Heisenberg equation of motion imply the Schrodinger equation for evolution operator?","text":"Let us choose to postulate (e.g. considering the analogy of the Hamiltonian being a generator of time evolution in classical mechanics) $$ i\\hbar \\frac{d\\hat{U}}{dt}=\\hat{H}\\hat{U}\\tag{1} $$ where $\\hat{U}$ is the (unitary, linear) evolution operator and $\\hat{H}$ the Hamiltonian (the most general version of which; i.e. time-dependent with instances at different times non- commuting). In S-picture, one can easily show that (1) is **equivalent** to $$ i\\hbar \\frac{d}{dt}|\\psi_S(t)\\rangle=\\hat{H}_S|\\psi_S(t)\\rangle\\tag{2} $$ where $\\psi$ is a state, $|\\psi_S(t)\\rangle = \\hat{U}(t)|\\psi_S(0)\\rangle \\equiv \\hat{U}(t)|\\psi_H\\rangle$ and $\\hat{H}_S:=\\hat{H}$. In H-picture, it is straightforward to show that (1) **implies** $$ \\frac{d\\hat{A}_H}{dt} = \\frac{\\partial\\hat{A}_H}{\\partial t}+\\frac{1}{i\\hbar}[\\hat{A}_H,\\hat{H}_H]\\tag{3} $$ where $A$ is an observable and $\\hat{A}_H(t)=\\hat{U}(t)^\\dagger \\hat{A}_H(0)\\hat{U}(t)\\equiv\\hat{U}(t)^\\dagger \\hat{A}_S\\hat{U}(t)$ (and also $[\\hat{H}_{H},\\hat{H}_H]=0$ implying that the time dependence of $\\hat{H}_H$ is purely explicit, i.e. $\\hat{H}_H=\\hat{H}_S\\equiv \\hat{H}$ with $[\\hat{H},\\hat{U}]=0$). **My question:** is it possible to obtain (1) from (3), i.e. to show that (1) is **equivalent** to (3)? Some thoughts on this: it is extensively mentioned in literature that both pictures yield same answers. Therefore, it should be possible to obtain (1) from (3) since (1) and (2) are equivalent. Assuming (3), the best I can get is that given an observable $A$, the operator $$ \\hat{C}:= \\hat{A}_S\\left(\\frac{d\\hat{U}}{dt} \\hat{U}^\\dagger - \\frac{\\hat{H}}{i\\hbar}\\right) $$ must be skew-Hermitian."}
{"id":"5132","title":"What is a good reference for the quantum mechanical description of lasers?","text":"I am currently taking a graduate level class on lasers. The primary focus of the class is on the design and engineering aspects of lasers, e.g. resonator design. However the first portion of the class is an overview of the quantum mechanics of laser processes in semi-classical terms: transition probabilities for a 2-level dipole atom in an external (classical) field, perturbation theory, and matrix QM. I have an undergraduate degree in physics, so the basics of QM are familiar to me, but it has been years since I seriously applied any of that knowledge. In addition, the professor is moving through this material very rapidly, as it is not the primary focus of the class. I can follow well enough that I'm not worried about my grade, but I'm feeling a bit cheated out of a more thorough understanding of this topic, which I would really like to have. Not only is this subject matter an appropriate level of advancement over my existing knowledge, it is also the basis for something (lasers) which I will likely spend a career dealing with. Can anybody recommend a text which covers this topic, and would be appropriate for somebody with my background? I would value a clearly written description over absolute rigor, but I am by no means looking for a laypersons description of QM. For example, I am quite comfortable already with the notation and concepts in the textbook \"Quantum Mechanics of atoms, molecules, solids, nuclei, and particles\" By R. Eisberg and R. Resnick. The ideal reference would be at a similar level of sophistication, but provide a more in-depth look at laser processes in particular."}
{"id":"71120","title":"Reason for different type of energy transfer for two kinds of collisions","text":"According to my physics book, if an electron were accelerated with 15 MeV of (kinetic?) energy and collided into a 100g thermally insulated copper block (not sure if the fact it is thermally insulated is relevant) the energy would transfer into thermal energy on the copper block and the copper block would heat up by a few degrees. However, if I were to collide into that same copper block with a bowling ball travelling with 15 MeV of kinetic energy then that energy would be transferred into kinetic energy on the copper block and the copper block would move. It seems to me, that in both cases, there is a thing smashing into another thing with 15 MeV of energy. When the first thing is small (e.g. electron) the energy is transferred into thermal energy. When the first thing is large the energy is transferred into kinetic energy. Why doesn't the electron's energy get transferred into kinetic energy as well?"}
{"id":"105573","title":"Why does the light travel slower in denser medium?","text":"Wikipedia says that \"in general, the refractive index of a glass increases with its density.\" And the refraction index of water vapor is less than ice, and even less than liquid water. Is there any simple explanation to that?"}
{"id":"466","title":"What is the mechanism behind the slowdown of light\/photons in a transparent medium?","text":"So light travels slower in glass (for example) than in a vacuum. What causes light to slow down? Or: How does it slow down? If light passes through the medium, is it not essentially traveling in the \"vacuum\" between the atoms?"}
{"id":"44115","title":"Difference in velocity of light in change in medium","text":"It is often seen that according to physics the light changes it's velocity according to the medium through which it is traveling. So can it be explained that why so happen?"}
{"id":"134481","title":"Dna Fingerprint","text":"I recently came across an article indicating that the half life of DNA in the most ideal situations is 521 years (http:\/\/www.nature.com\/news\/dna- has-a-521-year-half-life-1.11555). However, since human's DNA is 99.9% identical (and 98% identical to chimps), I can't quite understand how DNA evidence is often used in old samples. There has been DNA extracted from 100,000 year old Neanderthals, and DNA evidence has been used in cases over 50 years old. However, with a 521 year half life, after only 0.75 years, there would only be 99.9% of the DNA left. Surely if 99.9% of a human's DNA is identical after only this short amount of time wouldn't it be impossible to distinguish any sample from any other? After 50 years only 93% would remain the same, meaning it would be much less similar to a human than just any chimp I would think. And the 100,000 year old Neanderthal would only have 1.6*10^-57 % of the original sample left. How is there possibly any sort of useful information left? All in all, how can DNA be distinguished from other samples when they are all so similar and the half life is so low?"}
{"id":"81384","title":"Newton's Third Law Clarification","text":"Assuming you place an object so heavy on a table that it breaks it, then according to newtons third law the forces must cancel out (equal magnitude and opposite direction), but if this is true, then how can the object break the table in the first place?"}
{"id":"99917","title":"Two suns, one moon, and one planet?","text":"I have a question about how would seasons and the moon cycle be affected in a system where one planet orbits Sun #1, and Sun #1 orbits a second sun. Online I found this description: > \"Type II: \"Close Stars\" - The suns may occupy different parts of the sky, > indicating that one sun orbits the other farther out than the planet.\" (I > had a sketch, but didn't know how to upload it) To make it easier to explain I'm going to assume this system's Sun #1, planet, and moon are equal mass and size of Earth, it's moon and it's sun. The planet is in a habitable zone of Sun #1 but outside the gravitational pull of the second sun. So I guess my questions are:   * How would the moon cycle be affected when the planet orbits between both suns? Would there be times when it seemed like the sun never set or would days be twice as short, because there is a sun on both sides of the planet for 3-4 months out of the year?   * Would the same situation described above cause a significant shift in seasons(summer, winter,spring) when the planet is between two suns?   * Does the size of a sun, or it's mass more dictate it's gravitational pull? Is it even possible for a system such as this to survive or would it eventually get stuck in between both suns gravitational pull, or switch to a figure 8 orbit, and eventually be thrown out of both suns orbit."}
{"id":"44168","title":"Minimal coupling of an atom to the EM field","text":"The Hamiltonian of an atom coupled to an EM field, both described quantum mechanically is: $$H = \\frac{1}{2m}(\\hat{p}-q\\hat{A})^2 = \\frac{\\hat{p}^2}{2m} -\\frac{q}{m}\\hat{p}\\hat{A}+\\frac{q^2}{2m}\\hat{A}^2$$ Under the condition of transversality ($\\hat{p}$ and $\\hat{A}$ are vectors). I have only seen it treated in the dipole approximation where the last term becomes a constant and so unphysical; but what is its general interpretation? Is it shifting the energy of the free photons, it doesn't look diagonal in number of occupation basis."}
{"id":"44169","title":"(Earth's) magnetic field","text":"Always when we want to represent the magnetic field of the earth we see a similar image: ![enter image description here](http:\/\/i.stack.imgur.com\/BRFM7.gif) My question is, what exactly does this show? What are the blue and orange lines, what do they represent? Why are they curved like that? I have a vague idea about field lines, but I do not fully comprehend this. An explanation would be appreciated"}
{"id":"133242","title":"Quantum Scales and the Flatness of Space-time","text":"I know that on the smallest scales, general relativity predicts that space- time is flat. But I've also read that space-time can be described as a sort of \"quantum foam\" for distances smaller than the Planck length. Isn't this something of a contradiction (one more way in which quantum mechanics and general relativity disagree)? To make matters worse, theories about quantized space-time means that space-time could be \"discrete\", and not smooth at all. So which principle does the physics community think is right? The idea of smooth, locally flat space-time, or the idea of \"block-like\", discrete space- time?"}
{"id":"15564","title":"Very basic question: When to use $s=vt$, $s=1\/2vt$, $s=at$ and $s=a\/t^2$?","text":"Very basic question: When to use $s=vt$, $s=\\frac{1}{2}vt$, $s=at$ and $s=\\frac{a}{t^2}$? What was the difference between those?"}
{"id":"8240","title":"CFT and the Coleman-Mandula Theorem","text":"The Coleman-Mandula theorem states that under certain seemingly-mild assumptions on the properties of the S matrix (roughly: one particle states are left invariant and the amplitudes are analytic in external momenta) the largest possible Lie algebra of symmetries of a (non-trivial) S matrix is given by Poincaré times an internal symmetry.   On the other hand, there are (interacting) field theories whose Lagrangians are symmetric under the conformal extension of the Poincaré group, and in some rare case this property is retained even at the quantum level.   Why (interacting) conformal invariant QFTs do not contradict the theorem? Is it possible to define an S matrix in these theories? I have read somewhere that they do not admit a particle interpretation, what does it mean exactly?"}
{"id":"15562","title":"Can the Metropolis-Hastings algorithm be generalized to quantum systems?","text":"The Metropolis-Hastings algorithm is an efficient way of simulating classical ensembles using the Monte Carlo method. Is there a generalization of this algorithm to quantum systems? What I DON'T have in mind is Wick rotation to a classical Euclidean system."}
{"id":"15563","title":"Conversion formula from spectrophotometer readings to any standard color space profile?","text":"I have readings of colors from a spectrophotometer that records across the human visible color spectrum 390nm to 790nm intervals. I'd like to convert this into any color space, could be CIE XYZ, HSL, CIE-Lab, etc. Any color temperature is okay."}
{"id":"123226","title":"A sonar continuously emits x(t), a general but known waveform and is reflected by a target. hypothetical question","text":"A sonar continuously emits x(t), a general but known waveform that is reflected by a target and received by the sonar. Both the sonar and the target move in the 3-dimensional space in a general but known manner. That is the positions of the sonar and the target are known. What would be the relation between and the transmitted and received signasl? What would happen if the sound speed, sonar speed and the target speed are all comparable? Is it possible that the received signal contains something like multipath? that is some past signal points of different times are received at the same time? Like receiving multiple balls thrown at different times being received at the same time."}
{"id":"44830","title":"Schrödinger operator with a potential defined implicitly","text":"let be the problem $$ -\\frac{d^{2}}{dx^{2}}y(x)+f(x)y(x)=E_{n}y(x)$$ however we have a problem, we do not know the potential but its inverse $$ f^{-1}(x)=g(x) $$ we know $ g(x) $ but not $ f(x) $ what happense then since a) the function $ f(x) $ may be multi-valued b) the function $ f(x) $ may not exists even if we know $ g(x)$ howe can we analytically solve $ f^{-1}(x)=g(x)$ so we get $ f(x) $ ?? in the case $ f(x)=f(-x) $ is even."}
{"id":"21288","title":"Where is a good place to learn (classical) optics?","text":"I need to learn basic optics for a high school academic competition. Does anyone know any good places, preferably free and online, to learn the basics of optics, like lenses, angles of incidence, etc.? I can't seem to find an MIT OpenCourseWare course addressing the issue."}
{"id":"43133","title":"Good books about waves and optics","text":"> **Possible Duplicate:**   >  What’s a good textbook to learn about waves and oscillations?   >  Where is a good place to learn (classical) optics? I'm looking for a good book about waves and optics, only basics things and a lot of tricks to solve exercises."}
{"id":"15970","title":"What's a good textbook to learn about waves and oscillations?","text":"I'm taking a course on waves and oscillations using Crawford from the Berkeley series (out of print excluding international copies), and would like to know if anyone has any suggestions for a better book.   We cover:   * coupled oscillators,   * forced oscillators,    * Fourier analysis,    * traveling waves,  and some other topics. Any help with the topic would be appreciated."}
{"id":"126764","title":"Why does the probability of obtaining a value of a measurement follow from Dirac's general assumption?","text":"In Dirac's The Principle of Quantum Mechanics he makes the general assumption that \"if the measurement of the observable $\\xi$ for the system in the state corresponding to $|x\\rangle$ is made a large number of times, the average of all the results obtained will be $\\langle x|\\xi|x\\rangle$, provided $|x\\rangle$ is normalized.\" Then he states that the average value of any function of $\\xi$, $f(\\xi)$ is $\\langle x|f(\\xi)|x\\rangle$. Now, taking $f(\\xi)$ to be the function that is equal to unity when $\\xi=a$ and to be zero otherwise, noted by $\\delta_{\\xi a}$, the average value of this function is the probability $P_a$ of obtaining the result $a$ when we perform a single measurement of the observable $\\xi$. Now I know that this really gives the probability, because $\\langle x|\\delta_{\\xi a}|x\\rangle=|c_a|^2$, where $c_a$ is the coefficient of the eigenstate of $\\xi$ corresponding to the eigenvalue $a$ in the expansion of $|x\\rangle$. But Dirac seems to \"know\" that this is the probability without \"knowing\" that $|c_a|^2$ is the probability of obtaining a certain result. So my question is: how do we know that this expression $\\langle x|\\delta_{\\xi a}|x\\rangle$ is really the probability $P_a$ without using the expansion coefficients?"}
{"id":"134964","title":"Question about torque and center of mass","text":"If a yo-yo's string is not attached to anything and the yo-yo is dropped, it is obvious it will fall at $g$. In this scenario, Tension of string = 0. If a yo-yo's string is attached to a cieling and it is dropped, what is $R$ (path of center of mass)? Well, we know from experience that the yo-yo is going to fall at least, but is it going to fall at $g$? We know it is rotating as well. What would $T$ (tension of string) be? Would $T$ = yo-yo's mass * gravity? Assume that there is no friction. A related question is: if there is a stick in space and a force is applied to the center of mass, and for another stick in space an equivalent force is applied to the edge of the stick, will both sticks' center of mass move the same? Would the 2nd stick be rotating?"}
{"id":"103085","title":"Pathria's \"Statistical Mechanics\" first edition","text":"Does anyone know where I could find and purchase the book \"Statistical Mechanics\" by R. Pathria, in 1st edition (the 2nd and the 3rd are readily available, but I really need the first). I believe the first edition was published in 1972. I have searched high and low for it, but can't seem to find it even on Amazon."}
{"id":"65169","title":"Symmetry of the stress tensor","text":"When presenting the stress tensor (say in a non-relativistic context), it is shown to be a tensor in the sense that it is a linear vector transformation: it operates on a vector $n$ (the normal to a surface), and returns a vector $t_n$ which is the traction vector. It is then shown that conservation of angular momentum leads to symmetry of the matrix. However, tensors are more more naturally presented a multilinear functions. I wonder:   * What type of tensor is the stress tensor? Is $n$ a vector or a co-vector? What about $t_n$?   * Is there a way to understand the symmetry when thinking of the stress tensor as a function of two vectors (or two co-vectors), under which it will seems intuitive why $\\sigma(A,B) = \\sigma(B,A)$? _Edit_ : To clarify, let's look, for example, at the 1st coordinate of the traction vector $t_n$ of an arbitrary normal $n$: This is $\\left<e_1, \\sigma(n)\\right>$. From symmetry, this is equivalent to $\\left<n, \\sigma(e_1)\\right>$ - the inner product of $n$ with the traction vector of a surface orthogonal to $e_1$. Mathematically, I understand why this is correct. But is there any intuitive meaning as to why these two quantities are the same?"}
{"id":"103089","title":"Can resistance of wire be ignored","text":"I was doing some physics homework involving direct current circuits and resistors in series, and I started to question the accuracy of the following property of resistors in series, namely that the voltage across the battery is equal to voltage drop across the resistors. But what about the resistance in the wire itself. I wouldn't think that you could just ignore that, especially for very long wires or wires with a significant amount of resistivity. So my question is, why don't text books write $\\Delta V=IR_{1} + IR_{2}+...+IR_{n} + IR_{w}$ where $R_{w}=$ The resistance in the wire"}
{"id":"5692","title":"Diff(M) and requirements on GR observables","text":"This question is kind of inspired in this one: Diff(M) as a gauge group and local observables in theories with gravity The conundrum i'm trying to understand is how is derived the (quite) extraordinary statement that in GR there are no local observables. I just want to stress that this is indeed an extraordinarily counter-intuitive assertion (with extraordinarily dramatic consequences for any compatible theory of quantum reality), and it deserves an extraordinarily robust explanation. Among the statements i see are mostly involved in this argument are:   * _**It is usually argued that Diff(M) is a gauge transformation._** This point i don't have any sort of issue; the atlas of reference frames we use to describe space-time are indeed a human convention and physical laws should not depend on such conventions   * _**A physical observable should be invariant under any gauge transformation._** whaaa? i mean, this is clearly preposterous non-sense; actually i think this is more dangerous than simple non-sense, is just circular self-justification. This is actually saying that observables should be scalars? I can provide you a proof that this is non-sense by saying that the momentum of a particle is an observable in my reference-frame, and in other Diff(M) gauge (that is, another reference frame) i will see a different momentum of the same observable. All eigenvalues and eigenvectors transform according to the vector representation of the Poincare group. You are now of course free of dismiss such 'proof' as non-sense because my assumption that the momentum of a local particle is an observable is non-sense, but then, how are you sustaining that part of the argument, without actually saying again that Diff(M) is a gauge transformation? how do you avoid the circularity in this argument? U(1) gauge is a bad example because this is an internal symmetry; Diff(M) are space-time symmetries and i don't think that what is true in there (A eletromagnetic vector potential being unobservable, but B and E being observable, hence observables are invariant under U(1) )   * _**It is usually argued that observables in GR formally exists in the asymptotic boundary of the space-time._** Is there an argument for this that is unrelated\/independent to the previous point?"}
{"id":"96269","title":"can we PHYSCALLY (not by mathematics) justify that $ \\zeta (-s)= 1+2^{s}+3^{s}+4^{s}+... $","text":"the values $ \\zeta (-1)= -1\/12 $ and $ \\zeta (-3)= 1\/120 $ give accurate results for casimir and to evaluate the dimension in bosonic string theory so is there a physcial JUSTIFICATION to justify that in phsyics (not mahthematics) every time we see a divergent series like $ 1+2^{s}+3^{s}+......... $ otherwise how it would be possible that zeta regularization gave only correct results for the series $1+2+3+4+5+..... $ and $1+8+27+64+.. $ but not for example for $ 1+4+9+16+25+..=0 $ why a matheamtical fucntion would be useful only for certain values but not for others :("}
{"id":"89317","title":"Understanding the forces involved in a bounce from a collision?","text":"> A rock is thrown towards a window, and hits it. The window does not break > and the rock reverses its direction and falls under the window. My question is, under this circumstance since the rock reversed it's direction of movement I want to say that the force exerted by the window was larger than that of the rock because if they were equal, the rock would simply stick to the window and slide down until it reaches the bottom. However, the answer to this problem claims that the window and the rock had the same force. Can someone explain me what is going on?"}
{"id":"16142","title":"Is QFT mathematically self-consistent?","text":"After recently going through a short program of self-study in quantum mechanics, I was surprised to find a quote attributed to Feynman essentially saying he was extremely bothered by the computational process of renormalization. It's \"dippy\" that anyone should have to subtract one infinity by another in order to arrive at a finite answer. What's he referring to there, in rough terms? And what's the latest in attempts to replace this computational procedure with something more physically plausible, so that the theory can have more meaning and less dippiness?"}
{"id":"113067","title":"Plotting a bandstructure along High-symmetry points when kx,ky,kz is known","text":"Suppose you know kx,ky,kz points along with the corresponding energies. Basically, you know about the 4-D E(k) dispersion. How you do then convert that data into the bandstructure plots you commonly see which show E(k) along a closed path through high-symmetry points through the Brillouin zone?"}
{"id":"52408","title":"Which is the axis of rotation?","text":"This should be simple, but it keeps bothering me. If a rigid body has no fixed axis, and a torque (defined relative to a point $A$) is applied, it will rotate around $A$. But often I can also calculate the torque relative to another point $B$ (which often seems to be non-zero too). So does this mean that the rigid body will have an angular acceleration about both axises? This seems a bit strange to me. (For a fixed axis I assume that a rotation around any axis (other than the fixed axis) is impossible, because there will always be zero torque around those axes.)"}
{"id":"17237","title":"Differential Equation in Spherical Harmonics Derivation","text":"I've been reviewing derivations of the spherical harmonics in quantum mechanics; mostly as review but also to make sure I understand where the concepts arise from. However, every derivation I've seen makes use of the following differential equation\/ identity that seemingly comes from nowhere. I've not yet been able to figure out where it comes from in looking through various texts. So, here I am to ask: where does this come from? $$\\frac{d}{d\\theta}+l cot(\\theta) \\equiv \\frac{1}{(sin(\\theta))^l} \\frac{d}{d\\theta}(sin(\\theta)^l)$$"}
{"id":"56962","title":"Electric field due to nonconducting sphere","text":"For calculating electric field outside a nonconducting sphere with a hollow spherical cavity. When I use the rule (Charge density= $dQ\/dV$), I don't know exactly what is $dV$, is the volume here refers to the volume of the Gaussian surface ($V= 4\/3 \\pi r^3$) so that $dV$ will be = $ \\pi r^2 dr$, or the $V$ is the volume containing the charges only, so it will be =$ V_0 – V_1 = 4\/3 \\pi r_0^3 - 4\/3 \\pi r_1^3 $. Thus, since $ r_0$ and $r_1$ are constants, therefore $dV$ will be = 0? Note: $r_0$ is the radius for the whole sphere, $r_1$ is the radius for the cavity, and $r$ is for the Gaussian surface."}
{"id":"11693","title":"How to calculate heat exchange\/deltaT in a moving volume of fluid?","text":"I am building a preheater for a maple syrup evaporator and am going to use the steam generated by the heating process to pre-heat the incoming sap from, say 5 degrees C to (hopefully) something on the order of 80-90 C. (the goal is to heat the sap\/fluid before it gets into the evaporating pan so that we are more efficient - the steam used is \"free\" - it is a waste product of the process and capturing the heat and subsequent condensation for hot water is desirable) My flow rate through the copper tubes will be on the order of 35 to 56 liters per hour (10 to 15 gallons per hour) (this is for the end points of the assembly) I only know the outside diameter of the copper pipes - they will be 1\/2\" OD.   There will be 5 parallel lines inside a hood and at each end a manifold - the incoming cold sap comes into the low side manifold and makes its way up to the high side manifold in one of the 5 pipes. The manifolds are 20 inches long and are perpendicular to the other pipes which are 40 inches long. The heat exchanger will sit in a hood to provide a little back pressure for the steam and to collect the steam and force it to contact the pipes. The boiling surface area of the sap underneath the heat exchanger is larger than the heat exchanger and the hood but most of the steam rises through the hood and will transfer some heat to the copper which in turn heats the fluid. So how can I calculate the some likely heat exchange\/output temperature? It seems to me that the issues are:   * efficiency of heat transfer from steam to copper   * efficiency of heat transfer from copper to fluid   * how much time the fluid is in the lines I guess I am looking for an upper bounds and lower bounds (I know I won't get the fluid over 100 C - that's about all I know for sure - well - I guess I can also be sure that the temperature will rise some - but not sure how much.) Many people report output temps on the order of 95C but I am skeptical. What can I do to improve the heat transfer to maximize the scavenged heat? I can try to find fins from a baseboard radiator, however this is for food product - so I am unsure if I will do that. Some pictures of something I would like to do: http:\/\/s662.photobucket.com\/albums\/uu349\/bkm_photos\/Oil%20Tank%20Evaporator\/?action=view&current=evap-096.jpg&newest=1#!oZZ80QQcurrentZZhttp%3A%2F%2Fs662.photobucket.com%2Falbums%2Fuu349%2Fbkm_photos%2FOil%2520Tank%2520Evaporator%2F%3Faction%3Dview%26current%3Devap-087.jpg%26newest%3D1 and http:\/\/www.leaderevaporator.com\/p-286-parallel-flow-sap-pre-heater.aspx Wood is used as the heat source and one can assume a constant fire\/heat source that will be providing 10 to 15 gallons of evaporation per hour from a 24\" by 60\" pan. **EDIT (answering a comment)** The wall thickness of the copper pipe is 0.569 inches. There is only one fluid being heated - that is the sap from a maple tree. It is water with sugar in it - and by boiling it we can remove most of the water until it turns to maple syrup. The level of the fluid (sap) in the evaporator pans is kept constant by a float valve. The sap in the pan boils - that is where the steam comes from. So the rate of evaporation will be the same (approximately) as the flow through the heating tubes. The heat is generated by a wood fueled fire in a firebox just below the evaporator pans that hold the sap. Maple sap is held in a storage tank above the whole contraption. Gravity feeds it into the evaporator (a 5'x2' pan) through the preheater\/heat exchanger. The fluid flow\/level is regulated by a float valve. Steam is a by-product of the process - and using the wasted heat from the steam (allowing it to condense on the copper pips and transferring heat then drawing that condensation away so as not to have to re-heat it) makes the process a little more efficient. One study that was done shows that there is approximately 15% improvement in boiling rate (best case) for preheating the sap before it goes directly into the pan that gets heated."}
{"id":"69114","title":"potential difference problem?","text":"As far as i know potential difference between 2 points is defined as the negative line integral of electric field between those 2 points. I also know that when magnetic field changes curl of electric field is not 0 and potential difference makes no sense. But when we have an inductor in a rlc circuit then people always say that there is a potential drop across the ends of inductor,but since magnetic field is changing potential difference should make no sense. please help."}
{"id":"64347","title":"What is known about the Magnetic North Pole's location before 1800?","text":"I was recently startled to find that the Earth's North Magnetic Pole is moving at upwards of 40 km per year, with an additional ~80 km daily elliptical drift about its mean position due to variations in solar wind. The Wikipedia article on the NMP has a wealth of information, including this image: ![enter image description here](http:\/\/i.stack.imgur.com\/OQtk0.png) I'm particularly surprised at the existence of \"modelled\" data, showing a lot of detail, about the NMP's location for centuries before anyone went there. Neither that Wikipedia article nor this one on the Earth's magnetic field, which include the image, mention where the modelled data come from. I get the impression from sources like this that the model is derived from historical measurements of magnetic declination. If that is the case I would be disinclined to trust the model much: the Earth's magnetic field has a complex enough structure that the magnetic north and south are not antipodal, with nontrivial spatial variations in magnetic declination at any given time. Has anyone seen references that explain this model and what measurements they are based on?"}
{"id":"54805","title":"I need help with the following Physics' questions","text":"I need to know which formulas and the step-by-step answers for the following physics questions: > _1\\. You throw a ball upward with an initial speed of 7.0 m\/s, and it > returns to your hand o.92 seconds later, ( assuming air resistance can be > ignored)._ > >   * _(a) What is the average acceleration vector of the ball?_ >   * _(b) Find the highest up ward distance that ball can get._ > > > _2\\. A ball was thrown at an angle of 45.0 degrees above the horizontal, > that it traveled a horizontal distance of 496ft, and that it was caught at > the same level from which it was thrown._ > >   * _(a) What was the ball's initial speed?_ >   * _(b) How long was the ball in the air?_ >"}
{"id":"54803","title":"Calculating the dimensional wall-normal coordinate for a self-similar compressible boundary layer using Levy-Lees transformation","text":"How can I convert my self-similar boundary layer solution that is a function of the nondimensional wall-normal coordinate $\\eta$ to be a function of dimensional $y$? For instance, if I determine from my boundary layer solution that $\\delta_{99}$ occurs at $\\eta = 5$, for a given set of dimensional parameters how do I determine the corresponding physical coordinate? This is very straight forward for Blasius flow but is not as obvious to me for compressible flow. The definition of $\\eta$ is shown below. The integral dependence on $y$ is what is throwing me off. $$\\eta = \\frac{U_e}{\\sqrt{\\int_0^x\\rho_eU_e\\mu_edx}}\\int_0^y\\rho dy$$"}
{"id":"62971","title":"Determining the length of a Torsional Pendulum","text":"Currently working on this question, however I'm not sure how to solve it. > As a pendulum swings in simple harmonic motion at the surface of the Earth, > the angle the pendulum makes relative to its equilibrium position is given > by $\\theta(t) = (0.140 \\:\\mathrm{rad})\\cos(5.72t)$ where t is in seconds. > What is the length of this pendulum? > > a) 0.140 m > > b) 0.250 m > > c) 0.300 m > > d) 0.439 m > > e) 0.801 m The equation provided indicates that this is a torsional pendulum, correct? Therefore the length of the string with torsion supporting the disc at the bottom does not matter at all? If so, why?"}
{"id":"12341","title":"Poincare group vs Galilean group","text":"* One can define the Poincare group as the group of isometries of the Minkowski space. Is its Lie algebra given either by the equations 2.4.12 to 2.4.14 (..as also given in this page - http:\/\/en.wikipedia.org\/wiki\/Poincaré_group..) or equations 2.4.18 to 2.4.24 of Weinberg's volume 1 of his QFT books? What confuses me is that in deriving the commutation relations between $J^{\\mu \\nu}$ and $P^\\mu$ he did use quantum theoretic arguments about the Hilbert space operator $U$ but I guess there is nothing quantum about the Lie algebra he derives in the aforementioned equations. Is that right?   * This quantum confusion steepens when one looks at the $K_i$ (..relatvistic boost along the $i^{th}$ spatial direction..), $P_j$ (..linear momentum along the $j^{th}$ spatial direction..) commutator being non-zero. This is justified by saying that the exponential action of the boosts and the translations on the Hilbert space states do not commute and that is being reflected here. (..they pick up an extra phase proportional to the mass and the dot product of the boost velocity and the displacement vector..)  But if the afore mentioned equations are really the Lie algebra of the isometry group of the Minkowski spacetime then in the Galilean limit shouldn't they be instead reflecting the fact that Galilean boosts and translations when acting on the spacetime coordinates do infact commute. But the $K_i$ and $P_j$ commutation continues to be non-zero even when the Galilean limit is taken on page 62. This makes me strongly suspicious that the equations 2.4.12 to 2.4.14 are not the Lie algebra of the isometry group of the Minkowski spacetime but are the Lie algebra of the group whose elements are $U(\\Lambda, a)$ (..using Weinberg's notation..)…right?   * So is the \"low velocity\" limit taken on page 62 recovering non-relativistic quantum theory ? (and not Newtonian physics)   * On page 89 of the same book he derives the topology of the inhomogeneous Lorentz group as being $R^4 \\times R^3 \\times S^3\/Z_2$. Since this is a connected manifold, I guess that by the term ``inhomogeneous Lorentz group\" he is meaning only the proper orthochronous component of the full relativistic symmetry group. right?    * I can't see how the above topology matches with the semidirect product structure for posibly the same thing as given on this Wikipedia page -http:\/\/en.wikipedia.org\/wiki\/Poincaré_group ?  When people talk of the Poincare group is the full symmetry group of relativity is what is being referred to or is it just its proper orthochronous component (and not the other 3 components) ?   * I am familiar with the notion of \"central charge\" as in the \"first\" term on the RHS of the TT OPE of CFT…what also has the interpretation as the zero-point energy when doing the plane-minus-point<->cylinder conformal transformation. In this light it is not clear to me as to what is meant when one says that one can add in the \"mass\" as a central charge to the Galilean group..an extra generator which commutes with all the rest so that with this \"central extension\" the free particles will lie in the unitary representations of the Galilean group rather than the projective representations before the extension. I would be grateful if someone can shed light on this issue and help reconcile the two \"different\" notions of central charge.   * When one takes a \"low velocity\" limit of the Poincare algebra to get the Galilean algebra then is one taking just a non-reltivistic limit or is one also taking a non-quantum limit ?  (..I guess this will depend on my first query about whether what Weinberg calls as the Poincare algebra in the quoted equations has any quantum effect encoded in it {as it seems to be!} or is it just the Lie algebra of the isometry group of the Minkowski spacetime..)"}
{"id":"12346","title":"Ion Drive Propulsion Top Speed","text":"I would like to know if there is some formula \/ graph which would provide \/ show the efficiency of a certain type of propeller in space. Specifically, I'm interested in the acceleration attainable at certain speeds. I'm writing a science fiction book and I'm trying to make it as correct as possible, fact wise. The propeller I'm talking about is the ion drive Now, my book takes place in a world where fusion power is finally ours. So, please, let us assume that we have unlimited energy so you could power a dozen huge ion drives non stop. OK, there's the question of argon\/xenon fuel, let's assume we have 1 year of that. **So... the question is... what speed could you reach?** If a continuous acceleration of $10\\frac{\\mathrm{m}}{\\mathrm{s}}$ is applied (I put that number because it would also constitute an advantage for my crew - living in Earth's gravity), that would mean that a ship would reach the speed of light in just 347 **DAYS** But I know that's impossible because the EFFICIENCY of the ion drive would DECREASE as the ship's speed would approach the exhaust speed of the drive's \"nozzle\" (well, it doesn't have a nozzle per-se, as you can see in Wiki, but anyway...) * * * Please do not fear to elaborate on top of my question. Let's suppose for example that maybe the ion drives of the future have a much higher thrust\/efficiency\/nozzle exhaust speed. This isn't only about currently POSSIBLE facts but also about THEORETICAL limitations which might be overcome in the future (such as fusion energy)."}
{"id":"118867","title":"Where did the energy of the charge go?","text":"Suppose there is a positron and an electron, and they both collide, and we get $E=2mc^2$ of energy from the collision. Now, the charge also got vanished. Now suppose, I create neutrinos from the collision's energy, which are neutral and that doesn't violate conservation of charge, now the mass-energy of the neutrinos is same as the mass energy of the positron and electron destroyed. But instead if I make electron and positron pair again from that energy, it doesn't violate conservation of charge either, but this implies that creating one kind of charge requires positive energy and creating the other requires negative energy in some form. Isn't it ?"}
{"id":"53462","title":"Are group representations possible when the solution space is not a vector space?","text":"As far as I understand, the motivation for using representation theory in high energy physics is as follows. Assume that a theory has some (internal or external) symmetry group which acts on a vector space. Then fields satisfying the theory will have to transform under some representation of that symmetry group, by construction. What happens if we have some internal or external symmetry structure that is no longer acting on a vector space? The gauge group diffeomorphisms of general relativity spring to mind. Is there some more general 'representation' type theory which comes to our aid? And are there any examples of internal symmetries where this viewpoint is needed? Apologies if this question is imprecise or flawed - I'm just starting to get my head around the foundations of the subject! Many thanks in advance!"}
{"id":"49818","title":"How does sound travel in space?","text":"In relation to this question: How can a black hole produce sound? Which notes that the hole \"produces\" sound. The top answer states that: > What you think of as the hard vacuum of outer space could just as well be > seen as a very, very, very diffuse, somewhat ionized gas. That gas can > support sound waves as long as the wavelength is considerably longer than > the mean free path of the atoms on the gas. I get that there is \"stuff\" in space, what I don't get is how sound travels it. I learnt in high school that sound was a wave - as an example, you could fix a shoelace at one end and vibrate the other - voila, a wave forms. And then sound kind of moves in the same way through, say, air, because of the slight molecular attraction between individual molecules is enough to create a similar waveform. And then you get to space, and there aren't any molecules nearby to irritate each other, so there's no sound. If that's the case, why does it matter _how_ long the wavelength is, if there isn't a molecule nearby for the initial molecule to affect, how can sound travel? I'd understand a _little_ better if the particle a just _hit_ particle b (like a pool ball) and b carried the sound - but that has little to do with waves and wavelengths. Or is it that the \"jet\" that is thrown out travels along as a contiguous blob, with a sound wave embedded within it?"}
{"id":"49811","title":"What is the first paper to report observations of polaritons?","text":"I am seeking references to the first articles regarding the observation of polaritons."}
{"id":"68448","title":"Is there a non-technical way to understand why proving confinement is difficult?","text":"As far as I understand, the basic reason why we are not able to solve QCD at low energies is because it is a strongly coupled quantum field theory, and a part from exceptional situations in simplified models (involving for instance supersymmetry), we have no tools to fully analyze such regimes. Correct me if I am wrong, but this is a rather technical obstruction. Is there thus a simple, non-technical reason why proving confinement is such a difficult problem?"}
{"id":"68445","title":"Imposing anti-commutation relations on fermionic quasi-particles","text":"In many theories of CMT, we assume the nature of quasi-particles (without giving proper justifications). For example, we assume nature of quasi- particles to be fermionic in case of a interacting fermion system we began with and impose anti-commutation relations accordingly. Like in BCS theory, while using the Bogoliubov-Valatin transformation to diagonalize the Hamiltonian, we assume that the new operators are also fermionic in nature. Please explain more on this step and how is it justified."}
{"id":"67619","title":"Thermal Penetration Depth discrepancy","text":"I've been working on a project that involves thermoacoustics, and one of the commonly-used values in this field is know as the thermal penetration depth. It is calculated as follows: $$\\delta_k = \\sqrt{2K \/ (C_p 2 \\pi f \\rho)}$$ where $K$ is the thermal conductivity of the fluid in question (helium) $C_p$ is the isobaric specific heat of the fluid $f$ is the frequency of vibration $\\rho$ is the density of the fluid For my project, I've been using this document as a reference guide In it, the given $\\delta_k$ is 0.1 millimeters. The given frequency is 400 $Hz$. The fluid is helium at 10 bar. Based on this information, I found the density of helium (~1.6 $g\/mL$), the thermal conductivity of helium (~0.15 $W\/mK$), and its isobaric specific heat (5.193 $J\/gK$). However, when I plug these values into the equation, I get delta_k = 0.00379 mm, which is considerably off. Am I doing something wrong here? Is this widely-cited paper incorrect on such a basic fact?"}
{"id":"44446","title":"Momentum And A Car Collision","text":"I am studying an example problem, concerning the very topic mentioned in the title. In this example problem, a car has a head-on collision with the wall; the initial and final velocity are known, as well as the mass. My question is, why does the car rebound off of the wall? Why doesn't the normal force of the wall reduce the car's velocity to zero, with the car remaining there, instead of reducing the car's velocity to zero, and then giving it a velocity in the opposite direction?"}
{"id":"26798","title":"Calculations of apparent magnitude","text":"I was attempting to do some calculations of apparent magnitude to help solidify my understanding of the topic, but have been running into some confusion. According to Wikipedia, the apparent magnitude can be given as: $m_x = -2.5\\log_{10}(F_x\/F^0_x)$ where $F_x$ is the observed flux and $F^0_x$ is a reference flux (in other words, this equation provides the _difference_ of apparent magnitude between two observed values). Also, this is assuming that the same wavelength band is used in both flux measurements. Flux, in turn, can be calculated as: $F = \\frac{L}{A}$ where $L$ is the star's luminosity and $A$ is the flux density. Since stars act as point sources, this can be simplified to: $F = \\frac{L}{4\\pi r^2}$ where $r$ is the distance to the star. Since, historically, Vega has been used as the reference zero-point (having an apparent magnitude around 0.03), I tried doing a simple calculation to find out the apparent magnitude of Fomalhaut using the values for luminosity and distance given in Wikipedia for both of them. First, the flux of Vega: $F_{Vega} = \\frac{37\\,L_\\odot}{4\\pi (25.3\\,ly)^2}$ $F_{Vega} = 4.5999\\times 10^{-3}\\,L_\\odot\/ly^2$ Next, the flux of Fomalhaut: $F_{Fomalhaut} = \\frac{17.66\\,L_\\odot}{4\\pi (25\\,ly)^2}$ $F_{Fomalhaut} = 2.2485\\times 10^{-3}\\,L_\\odot\/ly^2$ Now, to calculate the apparent magnitude: $m_{Fomalhaut} = -2.5\\log_{10}(\\frac{2.2485\\times 10^{-3}\\,L_\\odot\/ly^2}{4.5999\\times 10^{-3}\\,L_\\odot\/ly^2})$ $m_{Fomalhaut} = 0.7777$ Huh?? Fomalhaut's apparent magnitude is supposed to be **1.16**. Even correcting for Vega's offset of 0.03, we still come up with **0.8077**. Why are the calculations failing? I don't think I've made a mistake in the mathematics. Am I using the wrong values?"}
{"id":"67611","title":"Formation on image on screen?","text":"When we first learn elementary geometrical optics, the first thing taught to us is the two broad divisions of the type of reflection.-Regular and Irregular or Diffused.   The difference in cause for both of these is that in case of regular reflection an incoming ray is perfectly reflected in a specific direction but in diffused reflection, the reflected ray are diffused in several direction probably owing to minor irregularities of the surface. Hence the diffuse reflection does not form our image as a mirror does but instead produces general luminance. But then in case of multi-media-projectors or when we use a paper or a wall as a screen for real images, an otherwise diffuse surface (wall, paper, etc) produces an image which can be seen by us. Is this not only possible when the incoming rays are REGULARLY reflected on to our eyes?"}
{"id":"131129","title":"How to explain spin of electron?","text":"How can we explain spin of electron, or the spin of other fundamental particles? If we think the spin of electron is similar to the spin of a ball or planet we make a mistake. We say it is an intrinsic property. However, in calculating magnetic momenta and other cases we consider it as spinning entity. It's too difficult to abandon the model which we see. Without which to explain abstract one becomes challenging."}
{"id":"24326","title":"Knots and strengh of a rope","text":"I read a few times that a knot can reduce the strenght of a rope, but I can't understand why this happens. Can someone explain me what happens to a rope tied with a generic knot and stretched? Is there a way to calculate the reduction of resistance from the form of the knot?"}
{"id":"31162","title":"What would happen to Earth if the Sun suddenly disappeared?","text":"> **Possible Duplicate:**   >  If the earth left the solar system for interstellar space. How long would > it take for atmosphere to freeze Suppose the Sun simply vanished, what would be the consequences to Earth?"}
{"id":"68995","title":"How can we describe the electrons of multi-electron atoms (i.e. not Hydrogen) when equations\/analytic solutions only exist for Hydrogen?","text":"I've been digging into emission spectra of different elements and found that such things as the Rydberg equation, Bohr's model, and quantum mechanics can only fully describe the single electron in the Hydrogen atom. How did we then make the leap to s,p,d,f shells of multi-electron atoms? How accurate is our analysis of these more complicated elements? Rydberg Equation (side-note: Is this an empirical 'data-fitting' equation? What's the significance of that?) $$\\frac{1}{\\lambda}=R_H\\left( \\frac{1}{n_1^2}-\\frac{1}{n_2^2}\\right)$$ Hydrogen: ![enter image description here](http:\/\/i.stack.imgur.com\/9yWL5.jpg) Helium: ![enter image description here](http:\/\/i.stack.imgur.com\/EyQi5.jpg) Iron: ![enter image description here](http:\/\/i.stack.imgur.com\/Tx0Vp.jpg) Potassium: ![enter image description here](http:\/\/i.stack.imgur.com\/g23gw.jpg)"}
{"id":"24495","title":"metric signature explanation","text":"Can anyone explain what metric signature is? I have a basic knowledge regarding tensors, btw. Also, how is it related to fundamental understanding of general relativity? Thanks."}
{"id":"91776","title":"Real and imaginary parts of dielectric constant vs refractive index?","text":"So for a complex dielectric constant $\\epsilon = \\epsilon_a + i\\epsilon_b$, the wave vector and index of refraction are related to it through $k = \\frac{\\omega}{c}n$ and $n = \\sqrt{\\frac{\\mu \\epsilon}{\\mu_0 \\epsilon_0}}$. According to Jackson, the real part of the dielectric is related to polarization and anomalous dispersion, while the imaginary part is associated with dissipation of energy into the medium. If you write the wavevector as $k = \\beta + i \\alpha\/2$ and plug it in the general wave formula (just in 1D right now) of $e^{ikr} = e^{-\\alpha r\/2}e^{i\\beta r}$, the intensity drops as $e^{-\\alpha r}$, so $\\alpha$ is the attenuation constant, which tells you how quickly the wave dies out in the medium. But, if you plug that form of $k$ into the above equations to solve for $\\alpha$ and $\\beta$ as a function of $\\epsilon_a$ and $\\epsilon_b$, you find that $\\alpha$ and $\\beta$ are both a function of both $\\epsilon_a$ and $\\epsilon_b$. This is counterintuitive to me, because intuitively I'd think that the attenuation constant $\\alpha$ would only be based on $\\epsilon_b$, due to dissipation, and the same with $\\beta$ and $\\epsilon_a$. Can anyone give a good physical explanation for this \"mixing\"?"}
{"id":"115152","title":"Why is the phase velocity used in the definition of the refractive index?","text":"I'm aware of the so-called group index but why is the phase velocity used in the standard definition of the index of refraction? What advantage does this offer?"}
{"id":"115156","title":"What would be measured, under current technology, of a particle moving faster than light? (Somewhat duplicate)","text":"Sorry, since the question has been posed, but I insist that others posed not from this position. Can any listener deduce whether we could measure, by analyzing electromagnetic disturbance, a particle$^1$ traveling faster than light? (or current measuring device, for the sake of assuming for questioning) Studying to understand cosmic rays, inductance, and capture technologies by the artifacts of Tesla.. The name most studied by us uneducated electrical folk. \\-- $^1$ For the 2011 OPERA neutrino experiment, see this Phys.SE post."}
{"id":"60566","title":"Dimensional regularization and IR divergences and scale invariance","text":"I want to know if dimensional regularization has any issues if the theory has IR divergences or is scale invariant.   * Does dimensional regularization see \"all\" kinds of divergences?  I mean - what does it _exactly_ mean when one says that power law divergences and IR divergences disappear in the dimensional regularization. So is more regularization needed in general over and above dimensional regularization?   * Does anything about the divergences get specially constrained if the theory is scale invariant?  I have often heard it being said that dimensional regularization \"preserves\" scale invariance."}
{"id":"93739","title":"Kinetic energy conservation in a collision","text":"Kinetic energy conservation in collision. In any type of collision type of collision if we take the system to comprised of both bodies then the net external force is zero. So the work done by external force is also zero. By the relation $W=K_f-K_i$ as $W$ is zero hence kinetic energy is conserved. But this not the case. Can you tell me what is the flaw in my argument? Some people say that the argument has neglected the fact that some kinetic energy is transformed into heat. But heat is created due to the interactions between bodies during collision which is thus an internal force and thus shouldn't affect kinetic energy."}
{"id":"51569","title":"Does chaos theory occur in quantum mechanics? Or in any non-newtonian physics?","text":"Does chaos theory occur in quantum mechanics? Or in any non-newtonian physics? Apart from perhaps thermodynamics?"}
{"id":"28165","title":"Is the collapse of the wave function inherently time asymmetric?","text":"Schroedinger's equation, as we all know, is time symmetric. In quantum field theory, we have to come up with a more sophisticated CPT reversal, but the essential point remains unchanged. However, the collapse of the wave function in the Copenhagen interpretation is manifestly time asymmetric. Correct me if I'm wrong, but can you uncollapse a wave function, converting it from an eigenfunction to a superposition of eigenfunctions? Is this asymmetry connected with the thermodynamic arrow of time and the second law in statistical mechanics, or are they independent? How would an uncollapse look like, and can we experimentally arrange for an uncollapse? Why are there more collapses than uncollapses? If an observer unobserves a quantum thingie, does that thingey uncollapse?"}
{"id":"129254","title":"What if an asteroid the size of the moon hitting earth","text":"Would we all parish due to excessive heat? Or would that be only limited to the area near the impact while as the people on rest of the earth would die from other phenomenons such as mega earthquakes, volcanic activities, tsunamis etc. Does it matter where the impact is - if it landed in antarctica we would have massive floods but if it landed in the middle of eurasian continent, the effect would be different?"}
{"id":"130361","title":"Effect of variable permittivity","text":"If I immerse a rod vertically in a liquid with a relative permittivity gradient (the permittivity decreases with depth), will the rod stretch (will the spacing of the atoms in the rod be affected by the varying permittivity)?"}
{"id":"74648","title":"What is it called when two particles are associated so that what happens to one happens to the other?","text":"There was some experiment that I read about some time back in which two particles (or the same particle, but split into two) were sent in opposite directions, but when something happened to one, it happened to the other at the same moment, as if they were the same particle. I thought the word for this was \"entrainment\", but looking that up does not lead back to this experiment. What is the term for this, assuming that I am not remembering something that never happened."}
{"id":"110474","title":"Yet another Young Double Slit Experiment","text":"If I have a laser light incident on the double slit at an angle O < 90 what kind of interference pattern will I see? Will i see the same with less intensity or different pattern?"}
{"id":"73146","title":"Is current related to the length of the conductor?","text":"Ohm's Law tells us that $V = IR$. This implies that $I \\propto \\frac{1}{R}$. But, $R \\propto l$, where l is the length of the conductor. This would mean that $I \\propto \\frac{1}{l}$. But this does not fit with the definition of current that says that current is the amount of charge $Q$ passing through a given point per unit time. Clearly, the amount of charge passing through a given point per second does not vary with the length of the conductor. How can this contradiction exist?"}
{"id":"93192","title":"Divergent issue of Madelung's constant","text":"This is a question triggered by this post Madelung's constant is defined to the coefficient of electrostatic potential energy in a ionic crystal. In the example of $NaCl$, \\begin{equation} M = \\sum_{ijk}{}^{'}\\frac{(-1)^{i+j+k}}{\\sqrt{i^2+j^2+k^2}} \\end{equation} is conditionally convergent. > _Since this sum is conditionally convergent it is not suitable as definition > of Madelung's constant unless the order of summation is also specified. > There are two \"obvious\" methods of summing this series, by expanding cubes > or expanding spheres. The latter, though devoid of a meaningful physical > interpretation (there are no spherical crystals) is rather popular because > of its simplicity. Thus, the following expansion is often found in the > literature:[2]_ $$ M = -6 +12\/ \\sqrt{2} -8\/ \\sqrt{3} +6\/2 - 24\/ \\sqrt{5} + > \\dotsb = -1.74756\\dots. $$ _However, this is wrong as this series diverges > as was shown by Emersleben in 1951.[3][4] The summation over expanding cubes > converges to the correct value. An unambiguous mathematical definition is > given by Borwein, Borwein and Taylor by means of analytic continuation of an > absolutely convergent series._ I have the following questions. 1) Expanding sphere leads to a divergent series. OK, what if I make a perfect spherical sample of $NaCl$, will the experimentally measured Madelung's constant to be infinity? ( **EDIT** : I maybe have some misunderstanding of divergence: it could be the case that the series is bounded but don't have a definite limit. So is the series for expanding sphere bounded? and if there is no definite limit, what's the experimentally measured value for a perfect spherical crystal?) 2) What physical principle dictates the order of summation? or why does finite number obtained by analytic continuation should be consistent with the observed value? 3) What is the role of charge neutrality here? I ever programed to compute Madelung's constant using the fractional charge idea((assign $\\frac{1}{8}$ charge to the corner, $\\frac{1}{4}$ to the edge, $\\frac{1}{2}$ to the face ), which makes the expanding cube charge neutral. Does that mean charge neutrality is one of conditions that _must_ be enforced? or it is just for the sake of computational efficiency?"}
{"id":"93199","title":"Does the Relativity Principle of Special Relativity imply homogeneity and isotropy of all the reference frames?","text":"In Rindler's book: Relativity, Special, General and Cosmological, is stated on page 40 that the Relativity Principle (RP), when applied to just one Inertial Frame (IF), guarantees the homogeneity and isotropy of tha IF. By inertial frame Rindler means an ideal infinity extended rigid body moving freely in a world without gravity. This is distinct from an inertial coordinate system, that should be understood as an IF plus, in it, a choice of standard coordinates $x$, $y$, $z$ and $t$. As he says, the RP concerns inertial coordinate systems: the laws of physics are invariant under a change of inertial coordinate systems. I can't understand why this imply homogeneity and isotropy of an IF. If I suppose the existence of an special direction in some inertial reference frame, I could imagine some physical law governig the propagation of some signal (it can be light if you want, but it's not necessary), and if by measuring the velocity of this signal in two different directions and I get two different results, this would violate the isotropy of the IF and at the same time I could write the physical law in an invariant way under coordinate changes inside de IF (sure, it would depend on the special direction) and this would be in accordance with the RP as stated above. What is wrong with my reasoning?"}
{"id":"56515","title":"How is Doppler redshift of distant galaxies established?","text":"Doppler redshift of distant galaxies gave first hint that the universe is expanding. I am curious to know how this redshift is actually measured and interpreted from observation. Suppose I observe some visible line say red with wavelength \\lambda_1. This is not enough to tell that this line is redshifted, i.e., the original line emitted from that galaxy had a smaller wavelength. How do I know the value of that particular line when it was emitted from that galaxy? It appears that mere observation of a line is not enough. So how do we proceed?"}
{"id":"98745","title":"How is matter stored in black hole?","text":"A black hole engulfs the matter nearby, how does it store the mass inside? We know that matter is composed of particles, does a black hole store the mass in massive particles? Or can we assume that it's composed of extremal black holes? what are theories about it? EDIT: I checked out Black-holes are in which state of matter? it doesn't provide a definite answer."}
{"id":"32173","title":"maximum distance between accelerating objects started at different times","text":"Let there be two objects that have zero relative velocity with respect to each other in an inertial frame. If they both undergo identical accelerations, but one starts the acceleration at t = T1 and the other starts the acceleration at t = T2, then per special relativity the maximum distance that rest frame can measure between the two objects is (T1-T2) * c What acceleration equations show that the separation between these two objects as t approaches infinity approaches c * (T1-T2)"}
{"id":"104092","title":"Standing sound wave tube","text":"If there was a standing sound wave tube and a flammable gas was introduced then ignited, would the combustion be more forceful and more efficient since its following a standing wave, than just a gas ignited within a tube?"}
{"id":"104527","title":"What is weak coupling of photon polarization to a pointer?","text":"This question is refered to those who are familiar with the concept of weak measurement. **In short:** How can the polarization of a photon be coupled to the position of a pointer state? What is the pointer state? The position of some particle? How to realise that in a lab? **In detail:** Let's say we want to weakly measure the polarization of a photon. We write the observable $A=|H\\rangle\\langle H| - |V\\rangle\\langle V|$ and the initial state shall be $|\\psi\\rangle = \\alpha |H\\rangle + \\beta |V\\rangle$. Now we weakly couple it to a pointer by sending it through a birefringent element, i.e. depending on the polarization the photon needs more time to pass the birefringent element. Formally: The Hamiltonian whichs represents this coupling is given by $H=\\lambda A \\otimes P$ where $\\lambda$ is the coupling strenght and $P$ the momentum operator of the pointer. Now $\\exp{(-\\text i H t)}$ will act as follows: $|\\psi\\rangle \\otimes |g\\rangle = (\\alpha |H\\rangle + \\beta |V\\rangle) \\otimes |g\\rangle \\;\\longrightarrow\\; ( \\alpha |H\\rangle \\otimes|g_+\\rangle + \\beta |V\\rangle\\otimes|g_-\\rangle)$ where the first degree of freedome is clearly the polarization of the photon and the second is the pointer state (a gaussian say). Both $g_+$ and $g_-$ refere to small shifts due to the birefringent element. Then there is this well known fact that a post-selection whichs leads to an imaginary weak value $A_w$ causes a shift in the momentum-space of the pointer beeing proportional to $\\text{Im} A_w$. The weak value is defined as $A_w = \\frac{\\langle \\phi|A|\\psi\\rangle }{\\langle \\phi|\\psi\\rangle}$ where $|\\phi\\rangle$ is the post-selected polarization state of the photon. (reference: http:\/\/arxiv.org\/abs\/0706.4207 or http:\/\/arxiv.org\/abs\/0911.5139) Mathematically this is easy to understand, but I hace a very conceptual problem with all that: How do I have to understand the pointer state? Is it a position of a particle? If yes, how can its position be coupled (in a physically understandable way) to a photon? And how can a post-selection affect its momentum?"}
{"id":"88145","title":"Why are rockets so big?","text":"I'm curious why rockets are so big in their size. Since both the gravitational potential one need to overcome in order to put thing into orbit, and the chemical energy burned from the fuel, are proportional to the mass, so if we shrink the rocket size, it would seem to be fine to launch satellites. So why not build small rocket say the size of human? I can imagine small rocket would be easier to manufacture in large quantities and easier to transport. And maybe someone can make a business out of small rocket, carrying one's own satellite."}
{"id":"134709","title":"Moduli spaces in string theory vs. soliton theory","text":"In both string theory and soliton theory, moduli spaces are frequently used. As far as I known, for soliton theory, moduli spaces are something like collective coordinates for solitons, and for string theory, moduli spaces is the spaces of all metrices divided by all conformal rescalings and diffeomorphisms. It seems like these two definitions(?) of moduli spaces are quite different, but the same terminology is used in both cases. I also learned that the name 'moduli spaces' comes from abstract geometry, but I don't know if that's any help here. My question is the following: Could anyone provide an intuitive connection between the two uses of moduli spaces, or highlight the differences?"}
{"id":"98657","title":"What is a phrase for testing for a certain result?","text":"Is there a word or phrase for when someone is testing for a certain result thereby skewing his findings?"}
{"id":"4935","title":"Is there a relativistic (quantum) thermodynamics?","text":"Does a relativistic version of quantum thermodynamics exist? I.e. in a non- inertial frame of reference, can I, an external observer, calculate quantities like magnetisation within the non-inertial frame? I'd be interested to know if there's a difference between how to treat thermodynamics in a uniformly accelerated reference frame and in a non- uniformly accelerated reference frame. Thanks!"}
{"id":"79937","title":"Two hanging masses connected by springs","text":"I had this problem for a candidacy exam, but wasn't able to get the complete answer. Their spring constants and masses are not the same, find the equilibrium position and frequencies of the system. Apparently once you find their equilibrium position, i.e. when gravity and the spring forces cancel out, you can transform to about that position and ignore gravity. Why do you get to ignore gravity? This is a normal modes problem, but I couldn't justify why gravity just disappears, but it does. Anyone have a simple reason why?"}
{"id":"79936","title":"Why is the binding energy per nucleon of helium-3 less than that of helium-4?","text":"I'd guess it has to do with the structure of helium-3 allowing for greater Coulomb repulsion between the protons, but I'm unsure."}
{"id":"2100","title":"What cools a drink?","text":"When you stick ice in a drink, AFAICT (the last physics I took was in high school) two things cool the drink:   * The ice, being cooler than the drink, gets heat transferred to it from the drink (Newton's law of cooling). (This continues after it melts, too.)   * The ice melts, and the cool liquid mixes with the drink, which makes the mixture feel cooler. My first question is, to what extent does each of these cool the drink: which has a greater effect, and how much greater?      Secondly, how much of the cooling by the first method (heat transfer) is without melting the ice? That is, is there any significant amount of heat transfer to each speck of ice before it's melted, and how much does that cumulatively affect the drink's temperature?      I suppose all this will depend on a bunch of variables, like the shape and number of ice cubes and various temperatures and volumes. But any light that can be shed, let's say for \"typical\" situations, would be appreciated."}
{"id":"99830","title":"Why is classical mechanics determinism based on position and momentum only and not forces and scattering rules?","text":"Consider a closed system (say a box) of $n$ particles. There is a well-known idiom\/meme\/law in classical mechanics that says that the position and momentum of those $n$ particles is all that is needed to determine the future and past configuration of the system. (A minor question, does this idiom have a name?) Why do we not consider two other pieces of information to be as important as the position and momentum of the $n$ particles?   * The force field (force as a function of space, and preferably of time too).   * The particle interaction rules (When two particles touch each other, do they whiz by or do they scatter? if they scatter what are the rules for determining angles?)"}
{"id":"106016","title":"Quantum Expectation Values","text":"I'm having trouble understanding the motivation for the definition of the expectation of a self adjoint operator $A$: $$\\langle A \\rangle _\\psi=\\int_{\\mathbb{R}}\\psi^*A\\hspace{0.2cm} \\psi \\hspace{0.2cm} dx$$ where $\\psi(x,t)$ is a normalised state. I can understand the expectation of the position operator in terms of basic probability: one of the assumptions of QM being that $|\\psi|^2$ is the probability to find the particle at $x$. The expected value is just the sum of the positions multiplied by their probabilities: $$\\langle x \\rangle _\\psi=\\int_{\\mathbb{R}}x \\hspace{0.1cm} |\\psi(x,t)|^2 dx=\\int_{\\mathbb{R}}\\psi^* \\hspace{0.2cm} x \\hspace{0.1cm} \\psi \\hspace{0.1cm}dx$$ by commutativity. I don't understand how this might work with the momentum operator, for example. The lecture notes for my course say that the expectation of self adjoint operators are defined in analogy to this, so that $$\\langle p \\rangle _\\psi=\\int_{\\mathbb{R}}\\psi^*\\hspace{0.1cm}p \\hspace{0.1cm}\\psi\\hspace{0.1cm} dx$$ I can't satisfy myself with this explanation; $p$ is a differential operator and so can't be moved about within the integral like $x$ can. For example $$\\int_{\\mathbb{R}}\\psi^*\\hspace{0.1cm}\\psi\\hspace{0.1cm} p\\hspace{0.1cm} dx$$ makes no sense to me, as the operator hasn't been applied to anything. In any case, $\\int_{\\mathbb{R}}|\\psi(x,t)|^2 p dx$ doesn't mean anything to me probability-wise. So basically I'm looking for an explanation as to why the expectation of self- adjoint operators are so defined. Thanks for any replies!"}
{"id":"106017","title":"Restrained expansion of a piston\/cylinder","text":"Suppose an ideal gas in a piston cylinder has some initial pressure $p_1$, volume $V_1$, and temperature $T_1$. A pin, which holds the piston in place, is suddenly removed, and the gas quickly expands. Once equilibrium is reached, the final pressure of the gas is $p_2$. I am looking at a textbook solution which claims that the work done during this process is $$W = \\int p\\,dV = p_2\\left(V_2-V_1\\right),$$ which implies that the pressure is constant ($p=p_2$) throughout the whole process, as soon as the pin is released. Is this really the case, and are there not irreversibilities in this process that must be considered?"}
{"id":"53878","title":"Truck driven from a small motor that can carry a heavy load, yet can travel fast.","text":"I am building a truck from trash as materials. I have one small motor and a a few small gears, but no other engineered materials are allowed. The truck must carry a load for a distance of 3m. The winning truck is the truck with the greatest ratio of load carried:time. What do you think would be the best approach to this project in terms of the gears and the radius of the driven wheels? I don't know what would be the best approach in terms of torque and angular velocity. Any other tips or inputs would be greatly appreciated! Thanks!"}
{"id":"53875","title":"Why must the gravitational wave components be much less than unity?","text":"We start with the metric tensor \\begin{equation} g_{\\mu\\nu}(x) = \\eta_{\\mu\\nu} + h_{\\mu\\nu}(x) \\end{equation} in the linearised theory, or \\begin{equation} g_{\\mu\\nu}(x) = \\bar{g}_{\\mu\\nu}(x) + h_{\\mu\\nu}(x) \\end{equation} in the more general case with a curved background. However, in both cases, we apparently need \\begin{equation} \\vert h_{\\mu\\nu}\\vert \\ll 1. \\end{equation} This condition demands that the components of the perturbation are much less than 1, in the coordinate system where the components of the background metric tensor are of order 1. Why do we need this condition?"}
{"id":"36415","title":"What will be the shape states of water in a centrifuge in 0G","text":"Consider the concept of these videos http:\/\/www.youtube.com\/watch?v=Zip9ft1PgV0&sns=em and http:\/\/www.youtube.com\/watch?v=cUhgKFV5Ri4&sns=em but set in a zero gravity environment and in a fully encapsulated and high pressure container. My questions are as follows: What shapes will the water conform to at increasing speeds? Will the water conform to a parabola and then ultimately a hollow cylinder as it does on earth? Is it possible that without an opening to spit and by continuously adding more water to the centrifuge, that the water could ultimately fold in on itself and conform to various states of torus, by creating vortices within itself? [Edit] Also, taking into account 0G, will the water even conform to its bounding container in a manner in which it receives traction and can be spun?"}
{"id":"104664","title":"electron pathways","text":"if there are two identically equal resistance paths, and you have a large group of electrons, will half of the electrons travel lets say left and half right? because I've been thinking that maybe the electrons travel down lets say the left one till there is a large enough amount of electrons that causes the resistance to change by just a fraction, thus forcing the remainder to switch and go right, and so on till all the electrons have traveled.. is this correct?"}
{"id":"112291","title":"Empty batteries magically resurrect after reinserting them","text":"It happened to me quite often (most recently with my wireless keyboard) that a battery stopped to work and then, if I unplug it and then plug it back in it works again, not just for a couple of minutes but even for a day or two, then you can repeat the process but this time it will last less and so on until it finally dies. The same trick works for my tv control. I can't really understand why, can somebody help?"}
{"id":"34816","title":"Is it possible for a black hole to form for an observer at spatial infinity?","text":"To my knowledge if you calculate the coordinate time (time experienced by an observer at spatial infinity) it takes an infinite amount of time for an object to fall past the horizon of a Schwarzschild black hole. Doesn't this imply that it takes an infinite amount of coordinate time for a Schwarzschild Black Hole to form since the last bit of in-falling matter won't ever ross the horizon as observed by someone at spatial infinity? If so, is it possible for other types of black holes (Kerr etc.) to form in finite coordinate time?"}
{"id":"133043","title":"Ice and liquid water interacting across a boundary","text":"Imagine we have two thermodynamic systems, one a mass of ice and the other an equal mass of liquid water, with both at 273.16K. Each system is isolated, except that they can interact with each other across a boundary that permits the exchange of heat but not matter or work. What will the two systems look like at equilibrium? Somehow I want to automatically imagine that each system will be identical, a combination of liquid water, ice, and water vapor at 273.16K. But if this is true then the two systems were initially at the same temperature but not in thermodynamic equilibrium, an apparent violation of the zeroth law of thermodynamics."}
{"id":"112745","title":"Dingle vs. Bondi: Twin Paradox Debate on BBC radio?","text":"Herbert Dingle and H. Bondi debated the twin paradox on BBC radio before 1971. Does anyone have a link to the audio of this debate? thanks"}
{"id":"61497","title":"Infinite degeneracy","text":"Is something special for a quantum system with infinite degeneracy like free particle levels? $E=\\frac{\\hbar^2 \\vec{k}.\\vec{k}}{2m}$ Edit: I mean what is physical (or mathematical) significance of infinite degeneracy?"}
{"id":"54240","title":"Quantum Entanglement Versus Inflation in the Early Universe?","text":"**Quantum entanglement** is one of the most fascinating and mysterious phenomena in nature. It needs no interactions, or any sort of exchange for it to take place. It is possible, not against any rules of physics as far as we know, that all matter that was created in the early universe was in an entangled state. **The question is:** Is it possible to explain the uniformity and isotropy of matter in the universe, by means of quantum entanglement in the early 'days' of universe? If that could be possible, would it mean that there would be no need for the inflationary model any more? If this problem has been researched in detail, any references posted will be appreciated."}
{"id":"18227","title":"Physics history book with some math","text":"I am looking for a book explaining physics from, say, Galileo and Newton till now; the book should be written using some math, similarly to my physics books when I was at the university (Halliday Resnik). I am searching a book that take care of explaining the true \"evolution\" of reasoning, discoveries, experiment to prove or disprove something. I do not like a book that gives definitions without explaining what had happened. I also like to find, on that book, explanations on how actual measurements are taken (how can we measure the speed of light, the mass of celestial bodies, the attributes of electron, and how were the measurements taken in the past - how was possible to measure the speed of light in 1700-1800?). I am not very interested in recent theories like strings, branes.... so it is OK if these are not discussed. I like to find a lot of discussions on concept that are related, like for instance the implications on space dimensionalities, ether non-existence, strange facts, missing explanations."}
{"id":"71455","title":"What does \"clockwise\" mean, exactly?","text":"I am in the middle of a discussion with a friend about the meaning of the term \"clockwise\". Wikipedia indicates that a clockwise rotation goes as **top-right-down-left**. However, my friend argues that \"the clock is facing opposite to you. So if you rotate top-right-down-left facing the side the clock is facing then it is anti-clockwise.\" Our question, therefore, is > does \"clockwise\" mean **top-right-down-left** or **left-down-right-top**?"}
{"id":"8659","title":"Invariant spacetime - distance - Circular Motion","text":"I understand that the closer something travels to the speed of light, that time will stretch by a factor, and distance will compress by the same factor. My question is, if something travels in a circle, close to the speed of light, what does the distance of the journey look like to them? They measure that the trip took them 10 minutes. And an outside observer says that the journey took 20 minutes, and the outside observer measured that they did, for example, 1000 laps of a circle circumference 1000 km. So if the plane had a distance trip counter, what would it read? And if they were looking out of the window, would the circle still look like it had a circumference of 1000km?"}
{"id":"8671","title":"A question about the relativity of time","text":"> **Possible Duplicate:**   >  Invariant spacetime - distance - Circular Motion I understand that the closer something travels to the speed of light, that time will stretch by a factor, and distance will compress by the same factor. My question is, if something travels in a circle, close to the speed of light, what does the distance of the journey look like to them? They measure that the trip took them 10 minutes. And an outside observer says that the journey took 20 minutes, and the outside observer measured that they did, for example, 1000 laps of a circle circumference 1000 km. So if the plane had a distance trip counter, what would it read? And if they were looking out of the window, would the circle still look like it had a circumference of 1000km?"}
{"id":"9689","title":"Paradox of the Relativistic Record Player","text":"> **Possible Duplicate:**   >  Invariant spacetime - distance - Circular Motion This is a question that I thought up a few years ago when I was taking mechanics. I asked the professor but didn't really get a straight answer. Imagine a record spinning at relativistic speeds. In the lab frame, the circumference of the record should decrease according to the Lorentz contraction. However the radius of the record should remain fixed since it is orthogonal to the direction of motion. So does the shape of the record appear distorted since the circumference is smaller but the radius is the same? What would it look like? I have a feeling the answer is obvious and\/or well known since it would seem to be similar to the situation of relativistic particles traveling around a circular accelerator but I can't think of how it can be solved."}
{"id":"76103","title":"What are the relativistic effects of expanding spacetime?","text":"This is a question I've been mulling over for a while and I'm hoping someone here can point me in the right direction. Sorry if it's a bit of a novice question. For the record, I don't fully know GR, but don't let that stop you from using it in the answer. Since the universe is expanding - that is, the spacetime metric is expanding by way of a near-exponentially increasing scale factor - we can say that the distance between any two non-bound objects is increasing over time. Herein lays my dilemma; if there were two objects separated by a large distance that had no relative velocities initially, after a long time, the effects of expansion would cause them to have large apparent velocities away from each other. Given that there hasn't been any acceleration to cause these velocities, are there still relativistic effects in play? That is, is there time dilation between the two frames? Furthermore, given long enough time, the rate of increasing distance between the two objects could place them outside of their visible horizon (ie they are travelling away from each other at superluminal velocities). Since there was still no acceleration to achieve this feat, what can one say about the relativistic effects in this case? At first I thought this was an easy question. I thought of course there would be relativistic effects and when the objects go superluminal, the visible horizon is there to ensure there can never be causal contact and thus preserve physics. But then I thought what if spacetime stopped expanding abruptly (seems crazy but as far as I know, nothing makes this completely impossible)? Since there was no initial relative velocities, wouldn't the two objects return to being in the same inertial frame? And seeing as none of them experienced any sort of acceleration, how then could we describe their two final states? By which I mean, if we were to assume there were relativistic effects during transit, how would we overcome such simple paradoxes like the twin paradox, or other relevant ones? At this point, I'm stumped. I even attended a lecture by Miguel Alcubierre since he would have had to consider these types of effects in his design... No help. Equations are great to illustrate a point, but I'm really going to need a conceptual answer as well to fully understand this."}
{"id":"39475","title":"Einstein's mass-energy relation","text":"Suppose we have 1 kg of wood and 1 kg of uranium and if we need to find out how much energy would each of the substance give, we'd have to use Einstein's mass-energy relation as follows: In the case of wood, $E_{wood} = 1 × (3×10^8)^2 = 9 × 10^{16} J$ In case of Uranium, $E_{uranium} = 1 × (3×10^8)^2 = 9 × 10^{16} J$ _Question:_ According to the equation, both give the same amount of energy. But in reality, 1 kg of uranium will give a lot of energy when compared to 1 kg of wood. What have I missed? Any explanation regarding the $E=mc^2$ _problem_ would be helpful..."}
{"id":"39473","title":"positive energies","text":"if the potential is bounded below $ V(x)=V(-x) \\ge a $ for some real number a and we can be sure that as $ x\\rightarrow \\infty $ we know that $ V(x) \\ge 0 $ then does it mean that the energies for our one dimensional system will be positive ?? my idea WKB quntization $ N(E)= 2\\int_{0}^{a} \\sqrt {E_{n}-V(x)} $ here $ V(a)=E $ is a turnign point so if the potential is positive for big 'x' then the energies should be also positive otherwise the epxresion inside the integral would be complex am i right ?"}
{"id":"47443","title":"Is the superposition principle universal?","text":"In David J. Griffiths' _Introduction to Electrodynamics_ , he claims that the superposition principle is not obvious but has always been found to be consistent with the experiments. So I was wondering have we found some physics quantities which do not follow superposition principle? If we have not till now why can't we generalize and make it into a law? More specifically: Griffiths was talking about electromagnetic force. My question is about the existence of something like mass or charge and which doesn't follow this superposition principle."}
{"id":"72379","title":"Dominos vs. Conservation of Energy","text":"In this video a single flick of a finger tips 116000 dominos. Domino video I understand the work that needs to be done to move 116000 pieces (at least 100 kilos) of plastic is greater then that expended by flick of a finger. How does the energy conservation law apply here?"}
{"id":"72377","title":"how must i understand this 2-loop integral?","text":"let be the 2-loop integral... $$ \\int d^{d}l\\int d^{d}k \\frac{1}{k^{4}(k+p)^{2}(k+l)^{2}}=I(p)$$ dimensional regularization over the variable 'l0 to evaluate $$ \\int d^{d}l\\frac{1}{(k+l)^{2}}=G(k,d)$$ this is the regularizñation of the subdivergence assumed to exist by dimensional rgularizaton then the final step would be $$ \\int d^{d}k \\frac{G(k,d)}{k^{4}(k+p)^{2}}$$ where $ G(k,d) $ must be taken in finte part to eliminate the pole when $ d=4-\\epsilon $"}
{"id":"115218","title":"Integration over $S^2$ in electrostatics","text":"I'm studying for a test in electrostatics and I'm always failing on putting up the correct integrals. In one problem I have the surface of a sphere with radius $a$ and an opening angle of $2\\theta$. There is a total charge $Q$ evenly distributed on the surface. I want to calculate the potential. In order to calculate the potential I first have to calculate the surface charge distribution, $\\sigma(r)$, and this is where I always fail. I've done like this: $Q=\\sigma A=\\sigma 4\\pi r^2$ Then I tried to convert this into spherical coordinates by changing $r$ into $r=a \\sin\\theta \\cos\\phi +a \\sin\\theta \\sin\\phi +a \\cos\\theta$ The correct integral should be $Q=\\sigma a^2\\int_0^\\theta \\sin \\theta d\\theta \\int_0^{2\\pi}\\\\!d\\phi$ Can someone explain how to put up integrals like these?"}
{"id":"100416","title":"Parity of proton is 1?","text":"I have found from Wikipedia that \"a parity transformation is the flip in the sign of spatial coordinates\". Now when we operate parity operator, does that mean we are taking any physical entity at ${\\bf x}$ to $-{\\bf x}$. Or we are just reverting axes of the co- ordinate system? However if we take parity transformation as active transformation then what does it mean that parity of proton is 1? Doesn't it anyway depend on the **origin** of the co-ordinate system? Please elucidate the meaning of the parity. I get too much confused whenever I hear \"parity\"!"}
{"id":"100417","title":"PDE from dispersion relation?","text":"Suppose I have knowledge of a system's dispersion relation $f(\\omega,k)$. Is it possible to recover the underlying PDE describing the system? Can I simply use the replacement $k=-i\\nabla$, $\\omega=i\\frac{d}{dt}$ to go back? I came across one source which claimed that the original PDE could only be recovered to a certain extent. An example given was the Dirac and Klein Gordon equations which both satisfy the same dispersion relation. But I didn't quite follow. Example: I have a polynomial dispersion relation of the form $\\omega^2+c^2k^2=1$. Can I automatically say that the underlying PDE is $\\frac{d^2u(x,t)}{dt^2}+c^2\\nabla^2u(x,t)+u(x,t)=0$? Or is there a subtlety I'm missing? Thanks"}
{"id":"133954","title":"How do objects heat up?","text":"If every body emits radiation at a given frequency and temperature exactly as well as it absorbs the same radiation, how do objects heat up?"}
{"id":"119878","title":"What is the density and energy of a photon?","text":"As I understand, photons are considered mass-less, which is a necessary condition for moving at the speed of light. However, does that mean their density is 0, as they will occupy some volume. If their density is zero, that means there is no matter inside a photon. Thus, shouldn't a photon be able to pass through matter instead of colliding with it? As $E = mc^2$, shouldn't a photon have zero energy, as it has zero mass?"}
{"id":"86034","title":"Power and magnetism","text":"If we have a solenoid and a magnetic field passes through it, a DC voltage will be produced in the wire. If we want to calculate the power, we find out the current using Ohm's law. I know there is power loss due to the resistance (joule effect). But what about the magnetic field due to current (lenz's law), does it contribute in the power loss?"}
{"id":"73424","title":"Deriving entanglement entropy from Renyi entropy","text":"My questions are based on this paper - http:\/\/arxiv.org\/abs\/0905.4013   * Firstly I want to know as to whether some assumptions are needed about the relationship between the systems $A$ and $B$ for the Hilbert space to factor as tensor products as (\"assumed\"?) on page 3?  I mean assume the more common reverse scenario - if you are given a system C and you decide to call some part of it as $A$ and the rest as $B$ then does it automatically mean that the Hilbert space of C factors between A and B? (...that doesn't intuitively feel to be true...then what exactly is the assumption being made here?..)   * Secondly given the definition of $S_A$ and $S^{(n)}_A$ as in equations 2 and 3 how does this claimed equality follow that, $S_A = \\lim _{n \\rightarrow 1} S^{(n)}_A = - \\lim_{n \\rightarrow 1} \\frac{\\partial \\rho^n_A }{\\partial n }$ I am unable to see the proof of the above 2 equalities. It would be great if someone could help."}
{"id":"107172","title":"Alternate young double slit experiment","text":"What will happen in young's double slit experiment, when instead of screen I put a black screen with a hole and a second screen behind the black one. Will it still form interference pattern albeit a faint one or no interference pattern will be observed?"}
{"id":"132606","title":"Hydrogen extraction efficiency of newish nano particle AA battery","text":"The internet is ablaze with the new nano particle based extraction method of converting water to hydrogen. In 2002\/2003 when I was 16 there was a similar craze about a lawn mower motor which supposedly ran on water. Indeed there were even conversion kits on the internet for vehicles. They didn't work because of a fundamental inefficiency of the conversion. Will this new nanotechnology improve the process so that cars can run on converted water or does there still exist a limit?"}
{"id":"101953","title":"Definition of Ampere","text":"On Wikipedia it says: > This force is used in the formal definition of the ampere, which states that > it is \"the constant current that will produce an attractive force of $2 × > 10–7$ newton per metre of length between two straight, parallel conductors > of infinite length and negligible circular cross section placed one metre > apart in a vacuum.\" In reference to the definition of an Ampere, why was $2 × 10–7$ chosen?"}
{"id":"62353","title":"What is the reasoning behind hole carriers being able to carry heat?","text":"In the Peltier effect, we consider charge carriers being able to carry heat. As for electrons or ions, this attitude makes sense, since external electric potential drives particles with mass in a direction and effectively transfers heat from one point of material to another. But for holes, this situation is only virtual, holes moving in a direction is only reformulation of the fact that the electron making up the hole environment are traveling opposite direction. I just cannot grasp the concept that the holes can indeed carry heat. I know one can argue that the holes have effective mass, but effective mass is only a measure of how much can you accelerate \/ decelerate the electron (or hole) in a crystal field. Can you help me out please?"}
{"id":"103859","title":"Harmonic Oscillator potential, proof that Gaussians remain Gaussians?","text":"I read in several papers that for a Harmonic Oscillator Hamiltonian in the time dependent Schrödinger equation a Gaussian wave packet remains Gaussian. Unfortunately I could not find any proof for this statement and trying to verify it myself I did not succeed. If I make a general ansatz with a spherically symmetric Gaussian wave packet with time dependent width and time and space dependent phase $$ \\psi(t,\\vec x) = (\\pi a(t)^2)^{-3\/4} \\exp\\left(-\\frac{x^2}{2 a(t)^2} + i \\phi(t,\\vec x)\\right)$$ and insert it into the Schrödinger equation $$ i \\dot{\\psi}(t,\\vec x) = -\\frac{1}{2m} \\Delta\\psi(t,\\vec x) + \\frac{k}{2} x^2 \\psi(t,\\vec x) $$ I get relatively complicated differential equations involving first time derivatives of a and $\\phi$ as well as first and second order spatial derivatives of $\\phi$. I failed to solve those equations or even show that a solution exists. Is there an easy way to show this? Is there any reference where this is shown? Which are the solutions for $a(t)$ and $\\phi(t,\\vec x)$ for a Gaussian (given initial conditions $a(0) = \\sigma$ and $\\phi(0,\\vec x)=0$)?"}
{"id":"62309","title":"Faraday tensor, antisymmetric electromagnetic tensor","text":"I want to write $F^{\\mu \\nu}F_{\\mu \\nu}$ in terms of $F_{\\mu \\nu}F^{\\mu \\nu}$. How to do it?"}
{"id":"78158","title":"Why linear wave equation does not have solitonic solutions?","text":"As many people define solitary waves they are localized pulses that propagate without changing the shape. As far as I know the same pulses exist in ordinary wave equation ! why should we look for solitons in nonlinear wave equation?"}
{"id":"110172","title":"Relativity addition and signs","text":"I have just covered a very brief module on special relativity as a part of my physics course. I have also done some extra reading mostly; Morrin's Classical Mechanics. While I found the book really illuminating in some aspects, I still feel that regardless of how hard I try there is something with relativity that prevents me form doing anything but the simplest questions. I was trying to pinpoint my problem and I think that a big part of it is velocity addition. I understand that the Galilean transformation would predict the $$V_{A}=V_{B}-V_{rel}$$ provided that A and B are two frames of reference. I also understand that we need to use the Lorentz transformation to get the velocity transformation in relativity; $$ \\begin{pmatrix} c \\Delta T_A \\\\\\ \\Delta x_A\\\\\\ \\end{pmatrix} \\begin{pmatrix} \\gamma & \\gamma \\beta \\\\\\ \\gamma \\beta & \\gamma\\\\\\ \\end{pmatrix} = \\begin{pmatrix} c \\Delta T_B \\\\\\ \\Delta x_B\\\\\\ \\end{pmatrix} $$ Transforming the velocity u measured in frame to frame B; $$u = \\dfrac{\\Delta x_A}{\\Delta t_A} = \\dfrac{v_B + u_{rel}}{1+\\dfrac{v_B u_{rel}}{c^2}}$$ But as far as I understand we could equally reverse the frames A and B and simply transform the other way around which means we need the inverse of the transformation matrix; \\begin{pmatrix} \\gamma & \\- \\gamma \\beta \\\\\\ \\- \\gamma \\beta & \\gamma\\\\\\ \\end{pmatrix} This will yield the formula; $$u = \\dfrac{\\Delta x_B}{\\Delta t_B} = \\dfrac{v_A - u_{rel}}{1-\\dfrac{v_A u_{rel}}{c^2}}$$. However since the naming of frames is arbitrary, how do I know which of the two formula to use, the one with the all plus and the all minus signs. I have tried to look on the internet for the explanation of this, but I could not find anything. Also provided that I know which equation to use how, do I use it what is the sign convention for the velocities? Thank you very much for all the help and sorry for the long post P.S. I would be also very grateful if someone could point me to some good and simple resources on relativistic dynamics especially collisions. Thanks again."}
{"id":"135076","title":"Formation of atoms question","text":"Could you please, explain to me the logic of the folllowing process as you would do to your 8 y\/o sister: > Ubiquitousness and stability of atoms relies on their binding energy, which > means that an atom has a lower energy than an unbound system of the nucleus > and electrons. Where the temperature is much higher than ionization > potential, the matter exists in the form of plasma – a gas of positively- > charged ions and electrons. When the temperature drops below the ionization > potential, atoms become statistically favorable. Source: wiki How that that binding energy can be explained in plain English? Is it somewhat intrinsic to atoms, with each kind having its own level of binding energy? I understand that this can be translated in the language of closed systems using terms as Gnatt energy (G), etc. - but how?"}
{"id":"128788","title":"Chemical potentials for multicomponent solids","text":"I have a problem understanding how to deal with the chemical potential when two thermodynamic systems are in contact. For simplicity, let us consider $T=0\\ K$. We have a system that is composed of two parts - I and II - where I occupies a space of fixed volume and II occupies the remaining space (essentially going to infinity in all directions). Both parts are initially filled with a crystal that is made up of a finite number of atomic types $i$. This is an equilibrium configuration (1). Now, I apply a long-range displacement field that originates in part I. If the boundary between parts I and II is permeable for the matter, some atoms move across this boundary. Here, part II can be thought of as a reservoir for the matter. The system eventually settles into another equilibrium (2), where the number of atoms $n_i$ in the part I is different from the above, i.e. there is nonzero $$\\Delta n_i=n_i^{(2)}-n_i^{(1)}\\ .$$ The free energy difference between the two equilibrium states of part I is then $$\\Delta F \\equiv F^{(2)}-F^{(1)} = V\\sum_{ij}\\sigma_{ij}\\Delta\\epsilon_{ij}+\\sum_i\\mu_i\\Delta n_i\\ .$$ In each of the two configurations (1) and (2), the parts I and II are in contact equilibrium, which means that $\\mu_i^{I(1)}=\\mu_i^{II(1)}$ and $\\mu_i^{I(2)}=\\mu_i^{II(2)}$, where the number in the bracket refers to the configuration. So, we have two chemical potentials for each $i$, one corresponding to configuration (1) and the other (generally different) for the configuration (2). Moreover, the particles move into I from the immediate neighborhood of the boundary I-II, which implies that $\\mu_i$ should correspond to the atoms in the part II that are close to this boundary. However, how am I to understand a single $\\mu_i$ in the equation above?"}
{"id":"135384","title":"what kind of transformation can be applied to qubits?","text":"I had a doubt on what kind of transformations can be applied to qubits. I understand that the transformations need to be reversible , but they also have to preserve the norm thats why the transformations need to be unitary as $U^{\\dagger}U=I$.      But then I was reading the paper 'Quantum Mechanics Near closed Timelike Lines , by D.Deutsch'.In it he has applied a transformation give by   $|x\\rangle |y\\rangle \\Rightarrow |x\\rangle |y \\vee f(x)\\rangle$ , where $\\vee$ stands for XOR and $x,y \\in \\\\{0,1\\\\}^{n}$ and $f:\\\\{0,1\\\\}^{n} \\rightarrow \\\\{0,1\\\\}^{n}$      Now I can see that that the transformation is reversible ( apply twice and get the same thing ) but it is not unitary , am I missing something ?"}
{"id":"103718","title":"Aufbau principle in modern quantum theory","text":"What is the rigorous definition of the Aufbau principle and the mathematical model used for its description? From Wikipedia, we have that the principle postulates a hypothetical process in which an atom is \"built up\" by progressively adding electrons. For modeling the postulate, we use the Madelung rule, Pauli exclusion and \"if two same energy orbitals are available, choose the less occupied one)\". So, $n+\\ell$ ordering with constraints. And I also know that the Aufbau principle has its exceptions."}
{"id":"121692","title":"Physical, intuitive reason for divergence of dielectric constant at electronic percolation transition?","text":"Several papers such as this (warning, PDF) and this (PDF again) talk about how, near the electronic percolation transition for a metallic 2D film, the real part of the dielectric constant diverges (which, if you read those papers, gives rise to some neat effects). I'm having trouble understanding how, though, aside from just the math. The 2nd paper mentions a more qualitative interpretation: > Near the percolation threshold the metallic clusters are separated by thin > dielectric regions. Each pair of nearest clusters forms a condenser [i.e., > capacitor] whose effective surface tends to infinity near the percolation > threshold. Then the effective capacity of the system diverges, too. Ok, let's assume for simplicity the \"dielectric\" here is vacuum, and use the form of a parallel plate capacitor, $C=A\\epsilon_0\/d$. By \"effective surface tends to infinity\", I'm guessing they mean that $d$ is going to zero (because the distance between clusters is going to 0), and thus $C$ is increasing to infinity. From there, I'm a little confused. My best guess is that the impedance goes as $Z\\propto 1\/(i\\omega C)$, and conductivity $\\sigma \\propto 1\/Z$, and dielectric constant $\\epsilon \\propto i\\omega \\sigma$, so when the C increases, Z decreases, the imaginary part of $\\sigma$ increases, and thus the real part of $\\epsilon$ increases. But I'm not really sure about that, and I'm having trouble reconciling it with what must happen after the percolation transition. Does anyone have a better physical understanding of this? thank you!"}
{"id":"134034","title":"For Faraday's law, why does the emf decrease as you increase the area of the loop?","text":"I've only recently started learning basic electrodynamics, but I don't understand why a loop of coil with a small area and a magnet falling through will produce a larger emf than a loop of coil with a larger area with the same magnetic field falling through. To clarify, lets say you have a loop of coil and you drop a magnet through the coil. This is will produce an emf according to Faraday's law. Now if you have a loop of coil with a larger radius, and you drop the same magnet through at the same speed, it will produce a _smaller_ emf. Why is this?"}
{"id":"43592","title":"Wave\/particle duality","text":"Apologies if this has been asked before (I did check and I believe it wasn't). I have a question about the particle\/wave duality of photons (or other particles). Depending on what and how we measure the photon turns out to be either a wave or a particle. Recently I saw some web page (and I can't remember where) that maybe neither is true. Both the wave perception and the particle perception are just that: perceptions, strengthened by our perhaps limited ability to observe. What if the reality of the photon is something else, something \"on top\" of our two perceptions?? Could someone direct me to a site that could tell me more about it (or maybe debunk the whole idea?)?"}
{"id":"122408","title":"Why aren't all photons virtual particles even in the \"vacuum\" of empty space?","text":"I'm thoroughly confused about the nature of electromagnetic radiation. Light is supposed to exhibit both wave and particle characteristics. But does that mean that it is _both_ a wave and a particle or _neither_ a wave nor a particle? If the latter, then it is not a \"real\" particle at all, but only occasionally behaves as one (hence, a \"virtual\" particle). And if the former, then it is, depending on the experimental setup being used to detect it, a \"real\" particle, but one that only exists (as a particle) for a limited time, which again, corresponds to the definition of a _virtual_ particle (\"a transient fluctuation that exhibits many of the characteristics of an ordinary particle, but that exists for a limited time\" - Wikipedia). So, if photons invariably appear to exhibit the characteristics of virtual particles, why are they not actually virtual particles (even in a vacuum)?"}
{"id":"133834","title":"Particles acting like waves","text":"Wave–particle duality is kinda bothering me... I read that electrons can act like waves, but I know that electrons are actually particles. The theory says that if you have not observed the particle it acts like a wave and can explore all classically available particle trajectories simultaneously and when you observe it it ends up in just one place. But doesn't that just mean that this particle is a particle the whole time, but you can't know it's location without observing it and the wave just presents a range of possible locations where it might be if you do observe it? And if so why would you call it \"Wave–particle duality\" if this particle is just a particle doing it's thing and a wave is just a best guess of where it is?"}
{"id":"90760","title":"Does radio use virtual photons?","text":"In radio communication each accelerated electron in the transmitter antenna interacts with an electron in the receiver antenna by exchanging a photon. Is that photon always a virtual photon as described in the diagram below rather than a \"real\" photon? ![enter image description here](http:\/\/i.stack.imgur.com\/i4Zqd.gif) As I understand it, Feynman's definition of a virtual photon in his book QED is any exchanged photon that does not appear in the initial or final conditions. Therefore by his definition radio works by using virtual photons."}
{"id":"29852","title":"The role of dark matter in black holes and star formation","text":"In my understanding, there exists a critical mass for which a star needs to be in order for it to collapse into a black hole. This also applies to a certain critical density of gas in order for stars to form in the first place. However, given a massive star or a massive amount of gas, it must have a relatively large gravitational field and therefore will interact with dark matter, attracting dark matter towards itself. This makes me question, doesn't the critical mass or critical density that we've figured out before our knowledge of dark matter now need to take that into account? OR Shouldn't the critical mass and critical density resemble the following pseudo equations? Critical mass (normal and dark) for a star to turn to black hole = A*normal matter mass + B*dark matter mass, where A and B are fractional amounts Critical density for a cloud of gas (normal and dark) for star formation = C*normal matter mass + D*dark matter mass, where C and D are fractional amounts So if the answer is yes, then what are the fractions, A, B, C, and D? Moreover, What kind of experiment or calculation would allow us to figure out these fractions?"}
{"id":"13883","title":"Matter-wave interference from free falling cold atoms","text":"and another exam question, this is about current research: > Interference of matter waves has been studied using ultra-cold atoms. The > phase of a matter wave for free-falling cold-atoms at time $t$ and height > $y$ is given as $$\\phi(y,t) = \\phi(y_0,t) + Et - \\int_{y_0}^y dy' k(y')$$ > where $E$ is the total energy of individual coherent atoms and $\\hbar k(y)$ > is the semi-classical momentum of falling atoms at height $y$. A recent > experiment observed coherent matter waves continuously emitted from two > vertical micro-traps (you can neglect the size of the traps) with height > $-\\lambda\/2$ and $\\lambda\/2$. The atoms are emitted at rest from the traps. > > A) For a given height $y$ below the two micro-traps, derive the condition of > constructive and destructive interference. Well, from basic wave mechanics I know that the phase difference of the two waves has to be an even (constructive) or odd (destructive) multiple of $\\pi$, but apart from that I find the question poorly worded: What does $t$ refer to? Total time in laboratory frame, or time since leaving the trap? And what about the function $\\phi(y_0,t)$? If I don't know its form, can I say anything meaningful about the interference pattern at all?"}
{"id":"57881","title":"Non conducting charged planes","text":"I have two parallel non conducting charged planes with opposite charges $6\\mu C\/m^2$, area $A = 3m^2$ and distance between the planes $d = 0.004 m$. I know the potential between these two planes is $$V = \\frac{\\sigma\\times d}{\\epsilon_0}$$ But if I put a conducting sheet of thickness $h = 0.001 m$ with the same area $A = 3m^2$ between these two planes, the new potential between the the planes would be $$V = (2\\times\\frac{\\sigma}{2\\epsilon_0} + 2\\times\\frac{2\\sigma}{2\\epsilon_0})\\times d$$ Is this right? or Am I wrong somewhere? I mounted this schema. ![](http:\/\/i.stack.imgur.com\/j1peM.jpg) Is that right?"}
{"id":"87144","title":"Work done on gas by piston","text":"In this problem there is a cylinder that is closed at both ends and has thermally insulated walls. It is divided into 2 parts by a movable, frictionless, thermally insulated piston. There is a heating coil in the left side of the cylinder and there are $N$ molecules of an ideal gas in each compartment. The original temperature and volume on each side are To and $V_o$. The left side is slowly warmed with the coil until its pressure doubles. How much work was done on the gas on the right? I calculated the final temperature and volume of each side after the left side is heated but I'm unsure about the work calculation. Would the work simply be $W = -P(V_f - V_o)$ where $P$ is 2 times the original pressure (so its $2NkT_o\/V_o$) or do I have to do something with the pressure since it changes from $P$ to $2P$?"}
{"id":"87141","title":"Confusing Classical Mechanics Question","text":"I don't want this to be a \"do my home work question\" so please tell me how I can make this question helpful for other people. In my physics assignment I found the question below. I'd think that both FBD's and $F_{net}$s are the same for each scenario; since the track is friction- less the rider doesn't need to input anything to maintain a constant velocity. If it helps, the answers for 'c' were found to be: $ac = g\\tan(\\theta)$, $a = g\\sin(\\theta)$ for each respective scenario. Any help is much appreciated. ![Problem text](http:\/\/i.stack.imgur.com\/7A93J.png)"}
{"id":"68930","title":"How much faster would a Clock without gravity run?","text":"Pardon the misleading title. It is to my understanding that moving\/heavy clocks run slow. The Earth itself is under gravitational influence from many sources, and is moving. Is there a way to know how much 'faster' a clock would run if those influences were removed? I don't need a precise number, an order of magnitude is fine, just to get the idea. Any insight is appreciated."}
{"id":"95274","title":"Shot noise at high frequencies (can it really be ideal white noise?)","text":"Quantum shot noise (either optical intensity noise or electrical current noise) described by the noise spectral density of $2 e I$ (electrically) or $2 h \\nu P$ (optically). So it is white noise. I know this basically comes from the derivation, where we model electron or photon events as infinitely short (with delta impulses). See e.g. Eq 1 in http:\/\/123.physics.ucdavis.edu\/shot_files\/ShotNoise.pdf. $$I(t)=\\sum_j q \\;\\delta(t-t_j)$$ So far so good, but can this really be true in an exact sense? A consequence would be that with growing bandwidth, the fluctuation (variance) grows indefinitely. That cant be physically correct. So things I would like to know specifically:   * Are there any experimental results where the spectral noise density of shot noise is measured? If e.g. there is a (nonzero) transit time of electrons, we should see some interesting things happen in the noise density at the inverse transit time. has something like this been observed?    * Is there a \"more correct\" (quantum physical) theory to derive shot noise? I.e. gives the correct spectral density? If yes, what is it and what is the idea behind the theory?   * Might there be a fundamental reason why it is impossible to measure shot noise at high frequencies? e.g. there is a fundamental limit for the gain bandwidth product of amplifiers (with which we could observe the shot noise at arbitrary frequencies in, for example, a diode)?"}
{"id":"122345","title":"Why can't the work done by a non-conservative force be zero?","text":"Why can't the work done by a non-conservative force be zero? The displacement along a closed path is always zero. So, whatever be the type of force, variable or constant, the work has to be zero. Why do we need to calculate the work done for individual paths?![enter image description here](http:\/\/i.stack.imgur.com\/jtrI2.jpg) This is a non-conservative force that starts from $A$ moves via Path 1 to $B$ and then back to $A$ via Path 2. Since the displacement is anyways going to be zero, why can't work done be zero?"}
{"id":"118831","title":"the energy relations in oceanography description","text":"I'm reading the following paper and do not entirely understand a point that the author is trying to make. In page 99 (2 of article) the author refers to the following equation: $$ Q^{*} = \\beta S^{*}e^{-\\beta z} + 2B^{*} \\delta \\left(z\\right) $$ and in the following paragraph he states that \"The first term on the right represents the convergence of the penetrating component S^{*} of solar radiation\" My immediate question is does \"first term\" refer to the $2B^{*}\\delta\\left(z\\right)$ or does it refer to \"2B\"? i.e. does rhs mean rhs of the equal sign or what? In addition, can anyone explain, in layman's terms, what this equation is referring to?"}
{"id":"123691","title":"How can I calculate the relationship between power and acceleration for a beam driven photon sail?","text":"I am writing a novel, and although I have a background in physics, I am unsure of the exact equations to use. Specifically: **For a photon-sail ship, how powerful would a driving laser need to be in order for the ship to reach an acceleration of ~0.5g?** Assuming we are only talking within the solar system here, and assuming that the ship is heavy enough to carry passengers. Would such a thing ever be feasible? I am trying to strike the right balance between fun and feasibility for some method of regular interplanetary transport (think the space equivalent of commercial jets)."}
{"id":"83652","title":"Separation of perturbative and non-perturbative contributions in partition function computation","text":"The following is defined, where $\\epsilon \\to 0^+$ is a cutoff: $$ \\mathcal{F}(Z)=\\int_{\\epsilon}^\\infty \\frac{\\mathrm{d}s}{s} \\frac{1}{\\sinh^2 s\/2} e^{-sx}. $$ Question: how do we see that $\\mathcal{F}$ has a perturbative + non- perturbative expansion (in $x \\gg 0$) given by $$ \\mathcal{F}(Z) = \\sum_g \\mathcal{F}_g x^{2-2g} + O(e^{-x}),$$ for some coefficients $\\mathcal{F}_g$ ? Reference: the issue is discussed in arXiv:hep-th\/9809187 eq. (3.2) and below. The book _Quantum Field Theory_ by Itzykson and Zuber eq. (4-117) and below should help clarify. It should be possible to see this kind of expansion by rescaling the variable $s$ in the integral and making a Taylor expansion of the $\\sinh^2$ in the denominator. The non-perturbative terms should arise from picking up poles at imaginary values of $s= 2\\pi i n.$ Note that the $\\epsilon$ parameter is supposed to ensure convergence of the integral. * * * I think I found a partial answer in the paper http:\/\/arxiv.org\/abs\/0907.4082 paragraphs 2.2 and 2.4, although I'm not 100% sure how one relates formula (2.20) here with (2.1) in http:\/\/arxiv.org\/abs\/hep-th\/9812127"}
{"id":"66813","title":"Spectral Decomposition Grover operator","text":"I am studing the Grover operator. Let be $x_0$ the marked element. Let be $t_f = \\pi\/2\\omega$. The measurement of the register in the computational basis return $x_0$ with probability $$p_{x_0}(t_f)=\\langle x_0|U^{t_f}|D\\rangle|=1-1\/N$$ Where $|D\\rangle$ is a diagonal state state, $U$ the grover operator, $\\cos \\omega = 1-2\/N$ and $N=2^n$. My lecture say $p_{x_0}(t_f)$ is a lower bound for $p_{x_0}(\\lfloor t_f\\rfloor).$ $\\lfloor t_f\\rfloor=\\lfloor (\\pi\/4)\\sqrt{N}\\rfloor + O(1\/\\sqrt{N})$ was obtained taking the asymptotic expansion in $N$. My question is Why $p_{x_0}(t_f)$ is a lower bound for $p_{x_0}(\\lfloor t_f\\rfloor).$?"}
{"id":"118460","title":"Moon's pull causes tides on far side of Earth: why?","text":"I have always wondered and once I even got it, but then completely forgot. I understand that gravity causes high and low tides in oceans, but why does it occur on the other side of Earth?"}
{"id":"66815","title":"Infinite Potential Well Energy for Piece-wise Constant Wave Function","text":"I'm trying to compute the expectation value of energy for a certain state in an infinite potential well but I'm getting contradictory answers. The well has potential \\begin{align} V(x) = \\left\\\\{ \\begin{array}{lr} 0 & : 0 < x < L\\\\\\ \\infty & : \\text{ elsewhere} \\end{array} \\right. \\end{align} which has eigenstates $\\phi_n(x) = \\sqrt{\\frac{2}{L}} \\sin(\\frac{n \\pi x}{L})$ with corresponding energies $E_n = \\frac{n^2 \\pi^2 \\hbar^2}{2mL^2}$. Now, consider the state \\begin{align} \\psi(x) = \\left\\\\{ \\begin{array}{lr} 0 & : 0 < x < L\/2\\\\\\ \\sqrt{\\frac{2}{L}} & : L\/2 \\leq x < L\\\\\\ 0 &: \\text{ elsewhere} \\end{array} \\right. \\end{align} We want to compute $\\langle H \\rangle$ for this state. One way to do this is simply using the definition: $\\langle H \\rangle = \\int_{0}^L \\psi^*(x) H \\psi(x) dx$. The problem with this though is that $\\psi(x)$ is piecewise constant, and therefore this will give you $0$. The other option is to expand $\\psi$ in terms of the energy eigenbasis by computing the coefficients $c_n = \\langle \\phi_n | \\psi \\rangle$, and getting $\\langle H \\rangle = \\sum_{n=1}^\\infty | c_n |^2 E_n$. As $E_n > 0$ for every $n$, this quantity will be strictly greater than $0$ and therefore will differ from the previous answer. What is the discrepancy? It surely has to do with the discontinuity at $x = L\/2$, but I can't figure out how to deal with it."}
{"id":"126173","title":"Velocity of a Charged particle in a magnetic field according to Biot-Savart Law","text":"According to Biot-Savart Law, if there is a charged particle in motion, there will be a magnetic field. My question is whether the counterpart of this law also holds true, i.e. if there is a magnetic field, whether there will be a charged particle be in motion. Let me explain my question a bit more clearly. First of all I don't know whether we can have a single charge in motion and another at rest in reality. If we can have then there may be two cases: Case 1: At time t=0 a charged particle is placed at rest at point P where B=0 and at time t>0 an external B field is created in a region R covering the point P. Case 2: At time t=0 a magnetic field B is created in a region R. At time t>0 a charged particle is placed at rest at point P in R. Note of Caution: The point P is chosen in such a way that there won't be any external electric field at P. Now come to the question: Will the charged particle move? If yes, under which case? If 'NO', whether the charged particle will absorb any energy or not?"}
{"id":"127330","title":"Interpretation of a density matrix as an observable","text":"In quantum mechanics, any density matrix (or density operator) is Hermitian. Observables are also represented by Hermitian operators. So it follows that a density matrix can also be interpreted as an observable. What is the physical meaning of this observable?"}
{"id":"28811","title":"Please explain this circular movement problem","text":"I was working over some problems from my physics textbook, and I came across this one, it involves circular movement: A car with the mass of 1t is moving over a hill with the velocity of 20 m\/s. The radius of the hill (which can be viewed as a half-circle) is 100m. When it comes to the top of the hill, with what force does it act upon the hill? The correct answer is, apparently, 6kN. How do I solve this? The situation can be illustrated like this I guess: ![enter image description here](http:\/\/i.stack.imgur.com\/zkukZ.png)"}
{"id":"28815","title":"Optical Tunneling","text":"Is there light tunneling taking place in optically transparent mediums like glass? wherein light travels a larger path in these mediums without interacting with atoms and without any change in velocity, frequency, wavelength or momentum? but just getting affected by the field of atom and following the least resistant path through the medium?"}
{"id":"38594","title":"Software for simulating the motion of a test particle in a force field","text":"> **Possible Duplicate:**   >  Software for simulating 3D Newtonian dynamics of simple geometric objects > (with force fields) Is there any software for linux, which can simulate the motion of a test particle in a force field. I.e. I want to define a force field and the initial definitions of the test particle and the software simulates the moving particle (based on the (numerical) solutions of the corresponding differential equations of motion). Would be great if the software could display the vector field and the field lines as well."}
{"id":"46811","title":"Non reflecting boundaries in waveguides","text":"Can someone please explain the Sommerfeld radiation condition and what is the alternative non-reflecting boundary conditions for waveguides of general geometries?"}
{"id":"127598","title":"In terms of physics, does the phrase \"time slows down\" mean the same thing as \"things happen more slowly?\"","text":"The common definition of \"time\" is a type of measurement, like size. But the sentence \"size gets bigger\" doesn't make any sense. Is \"time slows down\" an odd phrasing of \"events occur more slowly\" or is there a deeper meaning to the phrase?"}
{"id":"88748","title":"Einstein equation and scalar field stress-energy tensor","text":"Let's have interaction between gravitational and scalar real fields. For an action of gravitational field in vacuum I add term $S_{m} = \\int d^{4}x\\sqrt{-g}L_{m}$, where $$ L_{m} = \\frac{1}{2}g^{\\mu \\nu}\\partial_{\\mu}\\varphi \\partial_{\\nu} \\varphi - V(\\varphi ) $$ So the variation $\\delta S_{m}$ must give $\\frac{1}{2}\\int d^{4}x \\sqrt{-g}\\delta (g^{\\mu \\nu})T_{\\mu \\nu}$, where $T_{\\mu \\nu}$ refers to the stress-energy tensor of scalar real field. I tried to do it, but this leads me to the answer $$ \\delta S_{m} = \\int d^{4}x\\delta (\\sqrt{-g})L_{m} + $$ $$ +\\int d^{4}x \\sqrt{-g}\\left(\\frac{1}{2}\\delta (g^{\\mu \\nu})\\partial_{\\mu} \\varphi \\partial_{\\nu} \\varphi - g^{\\mu \\nu} \\partial_{\\mu}\\varphi \\partial_{\\nu} \\delta \\varphi - \\frac{\\partial V( \\varphi )}{\\partial \\varphi }\\delta \\varphi\\right)= $$ $$ =\\frac{1}{2}\\int d^{4}x\\sqrt{-g}\\delta (g^{\\mu \\nu})T_{\\mu \\nu} + \\int d^{4}x\\sqrt{-g}\\left( g^{\\mu \\nu} \\partial_{\\mu}\\varphi \\partial_{\\nu} \\delta \\varphi - \\frac{\\partial V( \\varphi )}{\\partial \\varphi }\\delta \\varphi\\right). $$ What to do with the second integral? Does it equal to zero according to Euler-Lagrange equation, or I can't say so?"}
{"id":"88749","title":"Who is usually credited for the creation of QFT?","text":"I read in a book just now that says: > [...] but it was not explained until the invention of quantum field theory > by Richard Feynmann [ _sic_] and others in the 1940's. I have been under the impression that it was Paul Dirac and Hermann Weyl in the early to mid thirties who developed the foundations of QFT."}
{"id":"121994","title":"I am trying to calculate how $<r>$ in the hydrogen atom evolves with time","text":"I am working on the Hydrogen atom and I was trying to calculate $\\frac{d<r>}{dt}$ using $$\\frac{d<r>}{dt} = \\frac{i}{\\hbar} <[\\hat{H} , \\hat{r}]>.$$ Here $r = \\sqrt(x^2 + y^2 + z^2)$ and $H = \\frac{p^2}{2m} + V$ where $p^2 = -\\hbar^2 \\nabla^2 $. Now according to Ehrenfest's theorem should behave classically and give me some equivalent of velocity, and indeed I do get something but it does't resemble velocity: $\\frac{-\\hbar^2}{2m} (2\\nabla r \\nabla f + f \\nabla^2 r)$ where $f$ is a test function. Steps: $[H ,r]f = [\\frac{p^2}{2m} + V , r]f = \\frac{p^2(rf)}{2m} + Vrf - \\frac{rp^2(f)}{2m} - rVf = \\frac{1}{2m}[p^2 ,r]f = \\frac{1}{2m}[-\\hbar^2\\nabla^2 , r]f = \\frac{-\\hbar^2}{2m}[\\nabla^2 , r]f $$= \\frac{-\\hbar^2}{2m} (\\nabla^2(rf) - r\\nabla^2(f)) = \\frac{-\\hbar^2}{2m} (\\nabla r\\nabla f + r\\nabla^2f + \\nabla f \\nabla r + f\\nabla^2 r - r\\nabla^2f) = \\frac{-\\hbar^2}{2m} (2\\nabla r \\nabla f + f \\nabla^2 r) $ Am I doing something wrong?"}
{"id":"61899","title":"Why do we still need to think of gravity as a force?","text":"Firstly I think shades of this question have appeared elsewhere (like here, or here). Hopefully mine is a slightly different take on it. If I'm just being thick please correct me. We always hear about the force of gravity being the odd-one-out of the four forces. And this argument, whenever it's presented in popular science at least, always hinges on the relative strength of the forces. Or for a more in depth picture this excellent thread. But, having had a single, brief semester studying general relativity, I'm struggling to see how it is viewed as a force at all. A force, as I understand it, involves the interaction of matter particles with each other via a field. An energy quantisation of the field is the force carrying particle of the field. In the case of gravity though, particles don't interact with one another in this way. General relativity describes how space-time is distorted by energy. So what looked to everyone before Einstein like two orbiting celestial bodies, bound by some long distance force was actually two lumps of energy distorting space-time enough to make their paths through 3D space elliptical. Yet theorists are still very concerned with \"uniting the 4 forces\". Even though that pesky 4th force has been well described by distortions in space time. Is there a reason for this that is understandable to a recent physics graduate like myself? My main points of confusion:   * Why is gravity still viewed as a force?   * Is the interaction of particles with space time the force-like interaction?   * Is space-time _the_ force field?   * If particles not experiencing EM\/weak\/strong forces merely follow straight lines in higher-dimensional space (what I understand geodesics to be) then how can there be a 4th force acting on them?  Thanks to anyone who can help shed some light on this for me!"}
{"id":"65732","title":"Eigenvectors of a 4D rotation, and their interpretation","text":"Let us define a 4D rotation by using two unit quaternions: $$\\mathring{q}_l=\\frac{a+ib+jc+kd}{\\left|a+ib+jc+kd\\right|}$$ and $$\\mathring{q}_r=\\frac{e+ib+jc+kd}{\\left|e+ib+jc+kd\\right|}.$$ They differ only by the real term (the denominators are only for normalization). And let us consider two cases: the $a,b,c,d,e \\in \\mathbb{R}$ in the first case and $a,b,c,d,e \\in \\mathbb{C}$ in the second case - that would be biquaternions. Now let us consider a rotation of an arbitrary (non-unit) quaternion $\\mathring{q}$ so that: $$\\mathring{q}_{rotated}=\\mathring{q}_l\\,\\mathring{q}\\,\\mathring{q}_r^* $$ (note the conjugate of $\\mathring{q}_r$). I can represent this 4D rotation using a $4 \\times 4$ rotation matrix, and thus I can find eigenvalues and eigenvectors of this matrix. The question is - what is the interpretation of eigenvectors of such rotation? In 3D rotations the invariant vector (a vector that is not being rotated) is the rotation axis, and the eigenvalue has to be 1 (because it is a rotation, so no particular interpretation here). In 4D rotations I know that there are two planes around which the rotation occurs, but are those planes the invariant of the rotation? If yes, then how can I describe this plane as an eigenvector quaternion - would such quaternion be a \"normal vector\" of this plane, analogous to 3D space where normal vector of a plane is just a 3D vector? How is that different between quaternions and biquaternions? Please note that since $e \\neq a$ the normalized quaternion $\\mathring{q}_r$ will be very different from $\\mathring{q}_l$ - all the imaginary components of unit quaternion (those created by $b,c,d$) will be in fact different, thanks to normalization."}
{"id":"65734","title":"Lightning and Charge Displacement","text":"There is something I don't really understand about flashes of lightning. When a flash occurs, how come electricity be transferred at the speed of light since electricity's displacement is very slow ? It is not that I don't understand anything about that. I've read wikipedia and the others. But on that particular point there's something I really don't get ... But the question could be for lighting as for any charge strike between two points of course."}
{"id":"65735","title":"Layered CMOS structure question","text":"I am trying to understand the workings of a CMOS image sensor. I understand that increasing wavelength results in an increased penetration depth in the silicon often used in CMOS image sensors. What I am in need of assistance in understanding is how the wavelengths (corresponding to the primary colours, red green and blue) relate to the vertical structure of the CMOS sensor. After reading several articles and performing some experiments, I am aware that CMOS sensors are indeed sensitive to UV wavelengths, but I am trying to understand how I was able to get a reddish-pink response using an incident UV source."}
{"id":"69630","title":"Why does a glass window reflect white objects white from an atomic scatterers’ viewpoint?","text":"Related: Is a white object always white? If you are standing in front of a glass window during the day, you can see your dim white t-shirt’s reflection in the window. The reflection is dim because only 4% of the light is reflected (assuming you’re perpendicular to the window) while the rest is transmitted through the glass. If a white object is placed in front of this glass pane, it’s obvious that white light is reflected as white and not colored. Why is this? I vaguely remember that visible light that is reflected penetrates glass about λ\/2, so that means that blue light penetrates the least while red light penetrates more. If I shine white light onto a glass window (assume RGB color model), I have to account for all three colored scatterers in roughly equal amounts of intensity to produce white. Here is what I think. • Blue light penetrates shorter distances into the glass window’s atomic layers and recruits less atomic scatterers that back scatter more intensity blue light. I assume Rayleigh scattering is valid since wavelengths of light are much greater than the atomic spacing’s in glass, therefore $I ∝ 1\/λ^4$. • Green light penetrates deeper into the glass window’s atomic layers and recruits more atomic scatterers that back scatter more “less intensity” green light according to $I ∝ 1\/λ^4$?. So the larger number of green atomic scatterers scattered less intense green light but there are more green atomic scatterers. • In a similar fashion, red light would recruit an even larger number of red atomic scatterers (but less intense than green or blue), however, the increased number of red atomic scatterers compensates for a more overall intense red intensity. In summary, there are roughly equal amounts of blue, green and red light reflected such that a glass window reflects white objects white. Here is what I am asking: is this correct way to view this? I was unable to find this anywhere online or on . I would greatly appreciate feedback, especially if this is not correct."}
{"id":"14378","title":"Why does density decide whether something floats or sinks?","text":"> **Possible Duplicate:**   >  Balloons and lifting gases I understand that density of an object dictates whether or not it will float, but what I don't get and can't get any satisfactory explanation... What causes an object of lower density to float? Can someone clarify? Differences in pressure ( a little more on that) ? Thus far, I've rationalized it in this way ( I sense it's wrong, but still): Object with a lower density has a spread out mass distribution along the volume and therefore, the contact surface. Since water's density is 1 g\/cm$^3$ ( our definition of mass says that 1 g is equal exactly to a 1 cm$^3$ volume of water)... When you have an object with a lower than water density, it exerts a force on the water due to gravity, but because at every specific point the weight of the object is always less than the one of water, it is unable to clear its volume of water out. I know, doesn't make any sense, please help! Every explanation on the Web just says it's dependent on the density of the object and that's that."}
{"id":"81095","title":"How does the entropy of an isolated system increase?","text":"The change of entropy is defined $$\\Delta S = \\int \\frac{dQ_\\mathrm{rev}}{T}.$$ If a system is isolated the heat transfer between the system and the surroundings is zero ($dQ = 0$), thus $\\Delta S = 0$. However, it is commonly stated that the entropy of an isolated system can increase. How is this possible, given the above definition of entropy?"}
{"id":"11715","title":"List of immiscible liquids","text":"I'm looking for a list of immiscible (and miscible) liquids. I currently developing a game that will use the property of differents liquid, some of them will mix, other will not (positioning in the recipient is important so density is). I'm searching but not found anything very relevant. Of course I know water, oil (and mercury). But I'm looking for more than two (at least 8 differents elements in three or four groups). I can always invent them but I'd like to keep things real as far as I can. The elements must be at the liquid state in normal conditions."}
{"id":"17008","title":"centrifugal force in static frame of reference","text":"The other day we derived Kepler's third law. $$ \\left( \\frac{T_1}{T_2} \\right)^2 = \\left( \\frac{r_1}{r_2} \\right) ^3 $$ In order to derive this, you can look at a given planet that revolves around the sun. If you assume that the sun is way heavier than the planet and that the planet moves on a circle, you can simply state the force on the Planet by the sun: $$ F_G = G \\frac{mM}{r^2} $$ Where $F_G$ is the force on the planet, $G$ is the gravitational constant, $m$ is the mass of the planet, $M$ the mass of the sun and $r$ the distance of the planet to the sun. Our tutor said that you now have to have a centrifugal force $F_c$ that is equal in absolute value to $F_G$, but opposite in direction so that the sum of the forces would be zero. The absolute value of this $F_c$ would then be equal to $F_G$: ![](http:\/\/wstaw.org\/m\/2011\/11\/15\/m9.png) Then you plug in the formulas for the respective forces: $$ |\\vec{F_c}| = |\\vec{F_G}| $$ $$ m \\frac{v^2}{r} = G \\frac{mM}{r^2} $$ $$ \\frac{v^2}{r} = G \\frac{M}{r^2} $$ $$ v^2 = G \\frac{M}{r} $$ $$ v = \\sqrt{G \\frac{M}{r}} $$ Where $v$ is the tangential velocity of the planet. Together with $v=\\frac{2\\pi r}{T}$, where $T$ is the revolution period, you get: $$ \\frac{2\\pi r}{T} = \\sqrt{G \\frac{M}{r}} $$ $$ \\frac{4\\pi^2 r^2}{T^2} = G \\frac{M}{r} $$ $$ \\frac{r^3}{T^2} = G \\frac{M}{4\\pi^2} $$ Since the right part is constant for each solar system, you can form a ratio and optain Kepler's law. So this approach does work. * * * I think that this view is only valid if you construct it from a rotating observation point, i. e. standing on the planet and facing the sun at all times. Our tutor said that this is from a static (with respect to the sun) observation point. In the rotating system, you do have the centrifugal force, but it is not a real force in this sense. So by saying that you have a centrifugal force, I think that you are in this rotating reference frame. And in the rotating frame, the planet is not moving at all. And by not moving in this rotating frame, it does revolve around the sun in the static frame. Since the planet is not moving, there must not be any force on it. Constructing a force equilibrium is the way to achieve this. But our tutor told me that there is a force equilibrium in the static reference frame as well. I say that if you have an equilibrium there, there is no net force, therefore the planet does only move along a straight line in a static frame. I was then told that although there is no net force, the forces are still there. At that point, I think that this is converting dynamics into statics by adding opposing forces so that everything cancels out. Say we _do_ have a net force (gravity) on the planet. ![](http:\/\/wstaw.org\/m\/2011\/11\/15\/m10.png) This force is something like this, if the planet has rotated $\\theta$ around the sun: $$ \\vec{F_G} \\propto \\left( \\begin{matrix} -\\cos(\\theta) \\\\\\ -\\sin(\\theta) \\end{matrix} \\right) $$ If you integrate this, you will get the velocity and the position of the planet. $$ \\int \\vec{F_G} \\, \\mathrm d\\theta \\propto \\left( \\begin{matrix} -\\sin(\\theta) \\\\\\ \\cos(\\theta) \\end{matrix} \\right) $$ This seems very good, since it goes along the circle, so this is the tangential velocity. $$ \\iint \\vec{F_G} \\, \\mathrm d\\theta \\propto \\left( \\begin{matrix} \\cos(\\theta) \\\\\\ \\sin(\\theta) \\end{matrix} \\right) $$ And that is simply the position around the unit circle. You can substitute $\\theta$ with $\\omega t$ if you wish to express the angle with time. So I claim that you need a net force in order for the circular trajectory to appear at all. My tutor told me that if the force is too high (too weak), the planet will spiral into (away from) the sun, and therefore the equilibrium has to be there. That makes no sense to me. What makes sense to me is that if the trajectory is deflected too much (not enough) to form a circle, there is no circle. * * * If you have no force, integration turns out quite different: $$ F = 0 $$ $$ \\int F \\, \\mathrm dt = \\underbrace{0t}_{a=0} + \\underbrace{c_1}_{v_0} $$ $$ \\iint F \\, \\mathrm dt = \\underbrace{0t^2}_{a=0} + \\underbrace{c_1t}_{v_0} + \\underbrace{c_2}_{d_0} $$ The latter is your $d = v_0 t + d_0$, which models a linear, unaccelerated movement. In the rotating system, $v_0$ is just zero, and it works out. In the static system, that would mean that the planet either remains still or moves on a straight line -- which is not what is really happening. * * * So I claim that you must _not_ have a force equilibrium for the earth to rotate -- that is the basic disagreement we have. Who is right? Or are we both right, just looking differently at the same problem?"}
{"id":"64062","title":"Field operator eigenvalues","text":"For an harmonic oscillator we can write the Hamiltonian eigenvalues in the basis of the amplitude eigenvalues : for example the ground state is a gaussian : $⟨x|0⟩=a.e^{-b.x^{2}}$. I was wondering if we can do that for the energy states of a field : can we calculate $⟨a|n⟩$ with |a⟩ an eigenvalue of the field operator $\\phi$ (let's say a Klein-Gordon field) ? I failed to carry out the ladder operator method, but I think we should be able to calculate (or at least have implicit equation of) any operator eigenvalue in any other operator eigenvalue basis, or is there something obvious that I missed ? Because we always play to write state in all the basis we can : spin on position, energy on spin etc ..., but I have never seen in classes or in books energy states of a field written in the field operator basis. EDIT : I'll look in the reference given by Peter : \"Weinberg I\" and try to explain how to do. EDIT : I didn't find the answer in the given reference :("}
{"id":"52670","title":"How photons move along with EM wave?","text":"So the wave moves like. a wave, it moves up and down, up and down. But how do photons move? Do they follow the same path or do they just go straight forward without oscillating?"}
{"id":"96510","title":"Software for calculating Feynman Diagrams","text":"Is there a software (open source preferred) where I would input something like \"Ingoing: a fermion $(p1, s1)$ and a photon $(p2, s2)$. Output: A fermion $(k1, r1)$ and a photon $(k2, r2)$\" and I would could then get each diagram (that is, each term in the Dyson series) up to whichever order I choose, also showing terms that evaluate to zero (just to see that they are there)? I have tried to compute the above diagram to 2nd order and I'm just baffled with how many details there are to collect, this seems like the perfect job for a computer, but I can't find anything that would do such a symbolic calculation (hopefully showing the steps along the way) and not just the final result."}
{"id":"52049","title":"Understanding that nothing can go beyond the speed of light","text":"> **Possible Duplicate:**   >  Is the “How to break the speed of light” minute physics video wrong? Radius of the moon - 1737 km. Thus circumference of the side that can been seen from earth - pi * 1737 = 5457 km. Speed of light - 300 kmps So if I point a lazer at one edge of the moon from earth and then move it to the other side in one second I will ultimately make the lazer spot travel 5457km in 1 sec. Thus the speed of the light spot is 5457 kmps! What is it that I am missing interms of concepts here? When we say nothing can go beyond the speed of light then what does \"nothing\" mean?"}
{"id":"62723","title":"Is it possible for a charged, fast-moving object to slow down and enter geo-stationary orbit?","text":"I've had a wild idea which I can not discuss at length in this forum, but it comes down to the following problem: A sphere of radius R=~10μm and mass m=~10-16Kgr is travelling towards the earth at v = ~10^8m\/sec. The sphere carries a charge Q and intersects the earth's magnetic field perpendicularly, at a distance r, such that it enters an elliptic orbit. Given an electron wave function of ~5eV, is there any possible configuration of Q and r that would allow the sphere to end up in a stable geo-stationary orbit? Would releasing charges at particular points of each rotation make any difference? Even though I did get a physics BS 13 years ago, I took a different direction and the calculations required are way beyond my current capabilities. If anyone finds the problem interesting, I'll be glad to hear from you."}
{"id":"80324","title":"Energy transfers in boiling a liquid?","text":"When you boil a liquid the KE stays the same (as temp is the same), so where does the extra energy go? I know the extra energy put in breaks bonds between the molecules\/ atoms, but once it has broken the bonds where does the energy go? And where does the potential energy in a liquid go- as (ideal) gases don't have PE? I've heard it's turned to latent heat, but what is this, how is it stored?"}
{"id":"14633","title":"Transforming a sum into an integral","text":"I posted this in the mathematical forums. Maybe you will help me. I found an hard article http:\/\/prola.aps.org\/abstract\/PR\/v105\/i3\/p776_1 of yang huang and luttinger. The authors begins with the sum: $$\\frac{{E_2 }}{{E_0 }} = \\frac{{16\\pi ^2 a^2 \\lambda ^2 }}{{V^2 }}\\sum\\limits_{} {'\\frac{{\\langle n_\\alpha \\rangle \\langle n_\\gamma \\rangle \\langle n_\\lambda \\rangle }}{{\\frac{1}{2}\\left( {k_\\alpha ^2 + k_\\beta ^2 - k_\\gamma ^2 - k_\\lambda ^2 } \\right)}}\\delta \\left( {\\vec k{}_\\alpha + \\vec k{}_\\beta - \\vec k{}_\\gamma - \\vec k{}_\\lambda } \\right)} $$ which represents the second order perturbation term of the energy of a bose gas. ${\\langle n_\\alpha \\rangle }$ is: $$\\langle n_\\alpha \\rangle = \\sum\\limits_{n = 0}^\\infty {n\\left( {ze^{ - \\beta \\varepsilon _\\alpha } } \\right)} ^n \/\\sum\\limits_{n = 0}^\\infty {\\left( {ze^{ - \\beta \\varepsilon _\\alpha } } \\right)} ^n = \\frac{{ze^{ - \\beta \\varepsilon _\\alpha } }}{{1 - ze^{ - \\beta \\varepsilon _\\alpha } }} $$ In the sum the terms with a vanishing denominator are omitted. There is also the restriction ${\\vec k{}_\\alpha \\ne \\vec k{}_\\beta }\\ \\ $ and ${\\vec k{}_\\gamma \\ne \\vec k{}_\\lambda }\\ \\ $. Now the passage that is unclear to me is the passage to the integral: $$\\frac{{E_2 }}{{E_0 }} = \\frac{{16\\pi a^2 \\lambda ^2 }}{{V^2 }}\\left( {\\frac{{4V^3 }}{{\\pi ^3 \\lambda ^5 }}} \\right)\\sum\\limits_{i,j,k}^\\infty {\\frac{{z^{j + k + l} }}{{\\left( {j + k + l} \\right)^{1\/2} \\left( {j - k} \\right)l}}} \\frac{{\\partial J}}{{\\partial u}} $$ where: $$J = \\int\\limits_0^\\infty {\\int\\limits_0^\\infty {dqdq\\frac{{\\cosh \\left( {upq} \\right)}}{{q^2 - p^2 }}} } e^{ - vq^2 - wp^2 } $$ and: $$u = \\frac{{\\hbar ^2 \\beta }}{{2m}}\\left( {\\frac{{2\\left( {j - k} \\right)l}}{{j + k + l}}} \\right) $$ $$v = \\frac{{\\hbar ^2 \\beta }}{{2m}}\\left( {\\frac{{\\left( {j + k} \\right)l}}{{j + k + l}}} \\right) $$ $$w = \\frac{{\\hbar ^2 \\beta }}{{2m}}\\left( {\\frac{{\\left( {j + k} \\right)l + 4jk}}{{j + k + l}}} \\right) $$ $\\frac{{\\partial J}}{{\\partial u}}$ is calculated here: http:\/\/math.stackexchange.com\/questions\/63534\/difficult-integral Maybe we don't have to do thermal average of n and we need : $$n = \\frac{1}{{z^{ - 1} e^{\\varepsilon \/kT} - 1}} = \\sum\\limits_{n = 1}^\\infty {\\left( {ze^{ - \\varepsilon \/kT} } \\right)^l } $$ so: $$\\sum\\limits_{} {'... = \\sum\\limits_{i,j,k = 1}^\\infty {\\sum\\limits_{} ' } } z^{ijk} \\frac{{e^{ - \\frac{{\\hbar ^2 k_a^2j }}{{2m}}} e^{ - \\frac{{\\hbar ^2 k_\\gamma ^2k }}{{2m}}} e^{ - \\frac{{\\hbar ^2 k_\\lambda ^2l }}{{2m}}} }}{{\\frac{1}{2}\\left( {k_a^2 + k_b^2 - k_\\gamma ^2 - k_\\lambda ^2 } \\right)}} $$ This explain the factor $z^{ijk}$ and the sum. How can we continue?Now we have to transform the sum' in a integral. The density of state depends of $kdk$ so i think we have to evaluate: $$\\int\\limits_0^\\infty {\\frac{{e^{ - \\frac{{\\hbar ^2 k_a^2j }}{{2m}}} e^{ - \\frac{{\\hbar ^2 k_\\gamma ^2k }}{{2m}}} e^{ - \\frac{{\\hbar ^2 k_\\lambda ^2l }}{{2m}}} }}{{\\frac{1}{2}\\left( {k_a^2 + k_b^2 - k_\\gamma ^2 - k_\\lambda ^2 } \\right)}}k_a k_b k_\\gamma k_\\lambda } dk_a dk_b dk_\\gamma dk_\\lambda $$ We have to evaluate two integrals, i think. Do you have any suggestion?"}
{"id":"57843","title":"Why isn't perpetual motion possible, even though we are so technologically advanced?","text":"Why perpetual motion wouldn't be possible if we are so technological advanced? It is just a thing that I was wondering for too long. I mean, we are able to create so powerful permanent magnets, like neodymium magnets which can store an massive quantity of energy for a long period of time, but we are not able to use that energy at all. Why?"}
{"id":"112505","title":"Black Body and Electron","text":"My questions are:   1. How does a black body absorb photons?    2. Why does a black body absorb the most photons of all objects (e.g. those with another color)?    3. Are there any relationship between the electron and a black body?"}
{"id":"8800","title":"Fourier transform and commutation of functions","text":"The book I am reading takes the unjustified step $$e^{-\\frac{i}{\\hbar}\\vec{p}\\cdot\\vec{r}}f(\\vec{p}) = f(\\frac{\\hbar}{i}\\vec{\\nabla})e^{-\\frac{i}{\\hbar}\\vec{p}\\cdot\\vec{r}}$$ and similarly, he uses elsewhere $$e^{-\\frac{i}{\\hbar}\\vec{p}\\cdot\\vec{r}}g(\\vec{r}) = g(i\\hbar\\vec{\\nabla}_p)e^{-\\frac{i}{\\hbar}\\vec{p}\\cdot\\vec{r}}$$ I have a gut feeling this follows somehow from the commutation relation $[r^i,p_j] = i\\hbar\\delta^i_j$ but I cannot justify this. Anyhow, the book has not introduced the standard commutation relations, and is working only with the fact that $\\phi(\\vec{p},t)$ and $\\psi(\\vec{r},t)$ are Fourier transforms of each other, and $f(\\vec{p})$ and $g(\\vec{r})$ are analytic. Could someone give me a hint?"}
{"id":"131714","title":"Experiment demonstrating interference patterns of neutrons","text":"The question is about **experiments** (and references) that demonstrate interference patterns of neutral (chargless) particles (fermions), especially neutrons. Like **double-slit experiments for neutrons** (or equivalent). Searching around did not yield sth (or maybe i did not notice it). My understading is that similar experiments have only been done for charged particles (i.e electrons, protons) or bosons (i.e photons). Is this correct?"}
{"id":"111260","title":"How do traveling waves pass through a standing wave node, if the node doesn't move?","text":"I'm having trouble with the explanation that a standing wave in a string is the superposition of traveling waves. ![standing wave](http:\/\/www.physicsclassroom.com\/Class\/waves\/u10l4c5.gif) The nodes in the diagram above are points where the particles of the string's medium undergo zero displacement, i.e. they do not move at all. But if they do not move, _how is the disturbance of (any internal traveling wave) propagated past the node_? The usual explanation for how a wave is propagated is that when one particle is disturbed (say, moved up), it exerts a pull on another, which in turn exerts a pull on the next one, and so on. In other words, to exert a pull or push on the next particle there must be some movement\/disturbance of the previous one. But the particle(s) at the node point do not move at all, so how does the disturbance of a traveling wave propagation pass through them? (I'm trying to understand the picture in terms of the mechanical forces between particles)."}
{"id":"92698","title":"Phonon-phonon-interaction as higher order terms in the potential","text":"Is there a simple way to understand why phonon-phonon-interaction is described by higher order terms in the potential? I mean: Having a quadratic potential is essential for the definition of the phonons as excitations of an harmonic oscillator. How can you suddenly talk about (interacting) phonons when you don't have a simple quadratic potential anymore..."}
{"id":"107351","title":"Solving the Schrodinger equation with appropriate symmetry","text":"In the paper _Markov Fields_ by Edward Nelson the introduction section claims that analytically continuing a Markov process with appropriate symmetry properties yields the solution of the Schrodinger equation. What is the appropriate symmetry required by a Markov process to solve the Schrodinger equation? Is the symmetry that of the Euclidean group? Furthermore, how would you show a Markov process with appropriate symmetry is a solution to the Schrodinger equation? Link to the paper: http:\/\/www.mathunion.org\/ICM\/ICM1974.2\/Main\/icm1974.2.0395.0398.ocr.pdf"}
{"id":"102700","title":"Experimental results regarding non-extensivity in small systems","text":"From a statistical mechanics point of view, the _extensivity_ of thermodynamics arises from the fact that we can neglect interaction terms in the Hamiltonian. This allows us to infer that when we put two systems in contact, the total energy is approximately equal to the sum of the individual systems' energy. If the systems are in thermal equilibrium it follows that the entropy is extensive as well. However, it is well known that if the interaction terms cannot be neglected then these extensivity results do not follow. In theory, this should typically be the case for very small systems. (It should also be the case for systems with very long range interactions, such as gravitational systems, but for this question I'm more interested in systems that are made of atoms and interact electromagnetically.) I'm interested in whether there has been any experimental work designed to measure this effect experimentally. In principle it could be done with calorimetry, but I'm not sure how sensitive the measurements would need to be. There might be ways to multiply the effect into a larger one, such as letting a very large number of tiny particles come into contact, rather than just two. Indeed, it might be a well-known effect that just isn't usually explained in these terms. I'd be particularly interested in work that measures the non-extensivity of _both_ internal energy _and_ entropy, in the same system, using independent measurements. However, any empirical work that addresses this issue is of interest. From searching Google I did come across a few experimental papers that mention non-extensivity. However, they seem to be concerned with testing the predictions of Tsallis entropy maximisation, which isn't what I'm interested in here. I'm really looking for calorimetric measurements of the non- additivity of the Hamiltonian in small systems, and the resulting changes in thermodynamic properties."}
{"id":"72174","title":"Why do moving particles emit thermal radiation?","text":"While answering another question about heat in an atom, the discussion in the comments led to the question of how heat is related to thermal radiation picked up by infrared cameras. The answer is that molecules undergoing translational motion (which therefore have a temperature) emit energy in the IR range and the intensity of the radiation in that range is related to the temperature. What is the actual mechanism for the emission of this energy? What is it about translation that causes the emission? Does it require collisions with other atoms\/molecules to cause a change in momentum and the emission of thermal energy (thereby slowing down\/cooling the bodies involved in the collision)? Does that mean in rarefied conditions where the mean-free-path is relatively large, the rate of IR emissions decreases (while the intensity is still only dependent on the temperature)?"}
{"id":"1503","title":"Does the energy of a magnetic field decrease when it moves a conductor carrying a current?","text":"When a charged particle moves in an electric field, the field performs work on the particle. Thus, the energy of the field decreases, turning into kinetic energy of the particle. Does the magnetic field of a permanent magnetic similarly lose energy and perform work when moving a conductor with a current?"}
{"id":"80040","title":"Is matter a continuous part of the field of space-time?","text":"I recently found this quote by Einstein (in _On the Generalized Theory of Gravitation_ , 1950), and it seems to me like he is saying that matter is a part of the field of space-time, and is nothing more than an area of the field with very high energy. Is this correct? > According to general relativity, the concept of space detached from any > physical content does not exist. The physical reality of space is > represented by a field whose components are continuous functions of four > independent variables-the coordinates of space and time. It is just this > particular kind of dependence that expresses the spatial character of > physical reality. Since the theory of general relativity implies the > representation of physical reality by a continuous field, the concept of > particles or material points cannot play a fundamental part, nor can the > concept of motion. The particle can only appear as a limited region in space > in which the field strength or the energy density are particularly high."}
{"id":"77522","title":"The electron and classical dynamics","text":"Can I use Newtonian laws of dynamics and kinematics applied to electrons (like $F = ma$ and $s = s_0 + v_0 t + at^2\/2$)? If not, why not? I know that everything in principle behaves in a quantum way, even macroscopic bodies, but is the level of precision of using classical mechanical on electrons acceptable?"}
{"id":"22201","title":"Can a static magnetic field turned into a static electric field? or vice versa?","text":"Consider some positive charge that is distributed uniformly over a very long line along the z-axis. If I am stationary with respect to the line then there is only static electric field which has cylindrical symmetry. Assume now that I am moving with some constant velocity which has only a (positive) component along the z axis. With respect to me there is now current running down the (negative) z-axis, hence I expect to find a static magnetic field in my moving inertial frame. My question is, do I still find (a static) electric field as well in this case? [Because on one hand I find the answer to be NO, since static electric fields should come only from static charges, and the charges here will not be static. On the other hand, although the line carries current, the line is still charged (which is different from currents running in conducting wires where the wires are neutral at all times). That means the answer is YES and one expects to see static electric field because the line is continuously charged.] So is the answer yes or no (and why)?"}
{"id":"22207","title":"Ropes and Pulleys - Really unintuitive answer","text":"I usually don't want to do this, but please go to this link, the solution is too big to post it here http:\/\/engineering.union.edu\/~curreyj\/MER-201_files\/Exam2_2_26_09_Solution.pdf And go to page 5 of the pdf. Briefly, the problem say > Determine the velocity of the 60-lb block A if the two blocks are released > from rest and the 40-lb block B moves 2 ft up the incline. The coefficient > of kinetic friction between both blocks and the inclined planes is $\\mu_k$ = > 0.10. **Things I am confused with the solution** 1.First of all, I seriously thought lb was mass not force. after some googling, it turns out they are used interchangeably... 2.Where did they even get $2s_a + s_b = 0$ from? Why did they determine the change in distance this way? My first assumption was that if block B moved up 2ft, then block A should move down 2ft (the rope must \"move\" 2ft too right?). Then I wasn't sure, so I did a few triangles and found that the angle made a difference 3.Where did $2v_A = -v_B$ come from? 4 The FBD for block A is confusing, why is the friction force $F_A$ in the direction of the ropes? I thought it was block B that is going down? Am I the only one who had trouble deducting that the pulley and block A are the same object? 5.Look at the final answer, how could $v_b$ be negative? The problem says block B goes UP. If you are wondering, this is not homework. I am just interested in this problem and it is out of curiosity and very confused with the concepts. I've had at most introductory physics experience, but I think I should have been able to solve this still. Thank you for reading"}
{"id":"102708","title":"How to derive Schrödinger equation?","text":"How is the Schrödinger equation $$\\frac {\\partial }{\\partial t}\\psi=-\\frac {i }{\\hbar }H{\\psi }$$ being derived?"}
{"id":"3881","title":"Why the Hamiltonian near degeneracies should be proportional to Pauli matrices?","text":"When a quantum system have a double degenerescence at one point, the Hamiltonian should be proportional to Pauli matrices near this point (also known as _diabolic point_ ) [Ref.]. But, why the Hamiltonian have this form?"}
{"id":"79470","title":"$\\gamma$ in Newton's Second Law of Motion in Differential Form","text":"I am teaching myself Differential Equations from a website. In the website I am up to Direction Fields and an example of a differential equation is Newton's Second Law of Motion. It is written on the website like this: $$m\\frac{dv}{dt}=mg-\\gamma v$$ I know that $m$ is the mass, $g$ is the gravitational acceleration, $v$ is the velocity, and $t$ is the time, but what does $\\gamma$ stand for?"}
{"id":"116869","title":"I would like to know, what would happen if a primordial black hole mass of Earth, would impact with the Earth","text":"Sorry for any mistakes, I'm Italian Hello everyone, I'm writing a novel, in which the Earth is devastated by enormous geological cataclysms (something like the end of the world), and to implement my cataclysm, I thought of a primordial black hole with a mass similar to that of the Earth. At first I thought to stabilize the black hole inside the Earth, but it would have destroyed the planet, and I do not want to destroy it; or so thought to pass away from one side to the Earth from the black hole, causing very powerful shock waves, which first pierce the mantle, causing the eruption of every volcano on earth, and then the crust collide, greatly destabilizing the tectonic plates , causing massive earthquakes on a global scale, and triggering the formation of giant tsunami impacting against every continent. such a scenario is likely, or not. ps: I hope that the question is connection to physics."}
{"id":"56493","title":"Is a quantum system mandatory for generating true random sequence?","text":"Is a quantum system necessary if we want to generate true random sequence? The mathematical framework used for classical mechanics doesn't involve any random value. But the mathematical framework of quantum mechanics involves randomness by definition. Can we argue based on these information that a true random number generator must use quantum mechanics? If anyone claims that s\/he has a true random number generator and fails to prove that s\/he is exploiting quantum mechanics can I discard the claim on the basis that s\/he is not using quantum mechanics?"}
{"id":"114013","title":"Cosmic Microwave Background and heat flow","text":"Can the fact that energy is distributed homogeneously in the universe be explained through heat flow and not the Cosmic Microwave Background(or in other words, can we say CMB is a result of heat flow and not the universe expanding from a singularity)?"}
{"id":"57423","title":"Couldn't we measure electrons with good enough technology?","text":"I am a bit confused about the Heisenberg's Uncertainty Principle - just read about it in _How to Teach Physics to Your Dog_ , by Chad Orzel. He states that the reason electrons can't be measured is because the photons used to measure electrons, collide with them causing changes in momentum and velocity. Couldn't this simply be fixed by creating a vacuum and using a different method of observation? Maybe something that doesn't exist today? **Edit:** A full disclaimer: I never took a physics course higher than a basic intro to physics. All the answers below are still referring to light (photons) as the method for measuring the momentum\/location of the particle. My question is: is it not possible to create a vacuum without any possible light source from entering; using this vacuum we can device a more advanced method which does not affect the momentum or position of the electrons."}
{"id":"51710","title":"Magnetic force on capacitor","text":"I have this problem: I have a capacitor formed by two parallel square-shaped plates, of side $a$, and with distance $d$ between them. They're asking me out of other things, the magnetic force as a function of some things. I calculate de magnetic force by calculating the gradient of the magnetic energy, given by: $$E=\\frac{1}{2}\\iiint_VB\\cdot H\\;dV$$ I have calculated $B$ and $H$, that form, during the charge of the capacitor, loops around the center proportional to $r$, being $r$ the distance from the central axis. The problem is that I have to integrate that to a cubic volume: $a^2d$, one of those variables is independent, but the other two are not, and this problems never involve complicated integrals, such that one: $$\\int_0^a\\int_0^a rdxdy$$ If the plates were circles it would be trivial, but being squares, I'm sure I'm missing something."}
{"id":"128548","title":"Superpositions and expectation values in quantum mechanics","text":"When the wavefunction of a particle is not an eigenfunction of an operator, the property to which the operator corresponds does not have a definite value, Why? Also consider a linear combination of basis functions: $$\\psi=c_1\\psi_1+c_2\\psi_2+...=\\sum_kc_k\\psi_k$$ When the momentum is measured, in a single observation one of the eigenvalues corresponding to the $\\psi_k$ that contribute to the superposition will be found, why?"}
{"id":"28069","title":"Why is boiling water loud, then quiet?","text":"Water in my electric kettle makes the most noise sixty to ninety seconds before the water comes to a full boil. I have been fooled many times by the noisy kettle, only to discover that the water was not yet hot enough for tea. The kettle is only at a full boil after the noise has _subsided_. I have noticed the same phenomenon with many other kettles, including conventional kettles on kitchen ranges; it is not a peculiarity of this electric kettle. Why does the boiling become quieter as the water reaches full boil?"}
{"id":"9333","title":"When water is about to boil","text":"Have ever noticed? When water is about to boil, no matters the kettle, there is some **sound** I have no idea where it comes from, sometimes long before it boils. Is there any explanation for this phenomena?"}
{"id":"78987","title":"Does the low-energy gauge structure depend on the choice of $SU(2)$ gauge freedom?","text":"The starting point and notations used here are presented in Two puzzles on the Projective Symmetry Group(PSG)?. As we know, Invariant Gauge Group(IGG) is a _normal subgroup_ of Projective Symmetry Group(PSG), but it may not be a _normal subgroup_ of $SU(2)$, like $IGG=U(1)$. But this may results in a trouble: By definition, we can calculate the $IGG$ and $IGG'$ of the $SU(2)$ gauge equivalent mean-field Hamiltonians $H(\\psi_i)$ and $H(\\widetilde{\\psi_i})$, respectively. And it's easy to see that for each site $i$, we have $U_i'=G_iU_iG_i^\\dagger$, where $U_i'\\in IGG'$ and $U_i\\in IGG$, which means that $IGG'=G_i\\text{ }IGG \\text{ }G_i^\\dagger$. Now the trouble is explicit, if $IGG$(like $U(1)$) is not a _normal subgroup_ of $SU(2)$, then $IGG'$ may **not** equal to $IGG$, so does this mean that _two $SU(2)$ gauge equivalent mean-field Hamiltonians $H(\\psi_i)$ and $H(\\widetilde{\\psi_i})$ may have different IGGs_ ? Or in other words, does the low-energy gauge structure depend on the choice of $SU(2)$ gauge freedom? Thank you very much."}
{"id":"78989","title":"Landau Lifshitz energy for uniform rotation","text":"Landau Lifshitz claim in their Mechanics book (39.11) that for a uniform rotation we have $ E = \\frac{mv^2}{2} - \\frac{m}{2} (\\omega \\times r)^2 + U,$ where the rotation is given by $v' = v + \\omega \\times r.$ This does not make sense to me, since this centrifugal term is negative( I mean, compared to the energy of the rotating system the energy in the inertial system should be HIGHER, since there is an additional kinetic energy of the rotation) as I had exected something like $ E = \\frac{m(v + \\omega \\times r)^2}{2} + U$ in the inertial frame. Then they claim that in the rotating system(in that case they use the notation $v_0$ instead of $v'$ we have $ E = \\frac{mv'^2}{2} - m v' \\omega \\times r+ U,$ ( This looks reasonable to me, as kinetic energy is less than the inertial one), but still,in that case I had expected something like $ E = \\frac{m(v' - \\omega \\times r)^2}{2} + U,$ so where am I wrong(especially about my reasoning in the first place) and why is it wrong, just to plug in the velocity of the transformation."}
{"id":"27350","title":"realization of: CFT generating fuction = AdS partition function","text":"An important aspect of the AdS\/CFT correspondence is the recipe to compute correlation functions of a boundary operator $\\mathcal{O} $ in terms of the supergravity fields in the interior of the $AdS_{n+1}$ (as we approach the boundary). Namely, $\\big< \\exp \\int_{\\mathbb{S}^n} \\mathcal{O} \\phi_{0}\\big > = \\mathbb{Z}_{s} (\\phi \\big|_{\\partial(AdS)} = \\phi_0)$, where $\\mathbb{Z}_s$ is the supergravity partition function. The review papers I have found (and Witten's original paper as well) explain how to use the above formula but fail to provide a satisfactory explanation why the formula ought to work, or even how it came about. Can anyone explain if there is a logical (and\/or insightful) path that would lead to the above correspondence between the generating function of the $n$-points correlators and supergravity\/string theory?"}
{"id":"25264","title":"Why don't more rocky planets\/moons have appreciable atmospheres?","text":"It seems obvious why mercury has no atmosphere, given its proximity to the sun --but yet Venus is also fairly close, and has an extremely dense atmosphere. Titan is a large moon with an atmosphere thicker than Earth's, and then Triton, a more distant large moon, has only a tenuous Nitrogen atmosphere. Why aren't more rocky bodies able to maintain an atmosphere? What factors affect whether a planet or moon is able to hold on to its atmosphere? What factors determine whether a planet or moon develops an atmosphere to begin with?"}
{"id":"32034","title":"Is our universe the only universe out there or are there also other possible universes?","text":"There is the specific universe we live in, the \"actual\" universe with planet Earth, humans evolving, and us discussing metaphysical debates on physics.se in 2012, etc. There might or might not also be other possible\/potential worlds out there, some differing majorly, like in the laws of physics, or minorly, almost like our current world, but maybe I didn't pose this question on physics.se. According to many worlds advocates, the other possible worlds also exist out there in Platonic space. Is this world we find ourselves the only universe out there, or do other possible worlds also exist?"}
{"id":"56729","title":"Quantum Chemistry Kronecker Delta formality","text":"In semi-empirical quantum chemistry, one frequently encounters the so called _zero differential overlap_ approximation $$\\langle \\mu \\nu | \\lambda \\sigma \\rangle = \\delta_{\\mu\\nu}\\delta_{\\lambda\\sigma} \\langle \\mu \\mu | \\lambda \\lambda \\rangle .$$ Why is it rather not written as $$\\langle \\mu \\nu | \\lambda \\sigma \\rangle = \\delta_{\\mu\\nu}\\delta_{\\lambda\\sigma} \\langle \\mu \\nu | \\lambda \\sigma \\rangle = \\langle \\mu \\mu | \\lambda \\lambda \\rangle $$ since on the right hand side of the first equation there are no $\\nu$ nor $\\sigma$ contained anymore. So either all four variables **plus** the Kronecker Deltas (middle expression of second equation), or only the \"remaining\" variables after evaluation of the Kronecker deltas (last expression of second equation)."}
{"id":"56728","title":"Why does an airplane need to climb during a takeoff even if it is in emergency situation?","text":"Right after take-off (which means an airplane already exceeded V1) it is recommended that an airplane keeps climbing even when emergency occurs. Beside worries of crashing into houses and buildings by flying low altitude, why should an airplane not stop climbing up? Why should it climb up and then, lowers to some altitude and then land?"}
{"id":"101960","title":"Why doesn't diamond glow when hot?","text":"In an answer to this SE question, the respondent explains that heating a perfect diamond will not cause it to glow with thermal blackbody radiation. I don't quite follow his explanation. I think it comes down to: there is no mechanism for diamond to generate light in the visible region of the spectrum. He mentions that interband transitions are well out of the visual range, so there will be no contribution from that. He mentions that the Debye temperature for diamond is > 2000 K. I presume that the argument here is that optical phonons will be frozen out, too. (But diamond doesn't have infrared-active phonons, does it?) So is that why hot diamond doesn't glow? I suppose that if one considers real (not ideal) crystals, imperfections, impurities, and the existence of surfaces lead to the possibility of emission mechanisms, and thus glow. In fact it might be the case that a finite but otherwise perfect crystal might have an extremely faint glow. Is this basically the reason that hot diamond does not glow? Further elucidation welcome."}
{"id":"105245","title":"What materials have an electroluminescent threshold in the millivolt or microvolt range?","text":"Does anyone know any materials that have an electroluminescent threshold in the millivolt or microvolt range? Specifically, a stable solid state material. I hope to deposited the material as a thin film. Thank you for the help!"}
{"id":"75383","title":"Deriving the curve of a cantilever","text":"Essentially, there is a beam of length L and negligible mass sticking out of a wall with a mass Mg hanging at the end of it. We are given an equation for elastic energy (which I don't think needs to be modified), and we have to come up with the potential energy contribution from the mass, and then hopefully be able to minimize the energy using variational techniques. The elastic energy per unit length can be approximated to be $U[y]=\\int_0^L \\frac{1}{2} EI (y'')^2 dz$ If I can properly define the potential energy of the mass, I think I can do the rest, but I can't figure out the right term. Thanks for reading, and please comment if there is more I should add, including my work involved getting the (incorrect) gravity terms."}
{"id":"105243","title":"What are the implications of the possibility that the BICEP2 results are caused by a self ordering scalar field transition?","text":"I've found this interesting paper that mentions another possible way to interpret the recent BICEP2 results, and that hadn't been ruled out yet 1. As interesting as the possibility that the BICEP2 team has observed CMB polarization due to gravitational waves resulting from a self ordering scalar field transition sounds, I would like to know what its implications\/significance would be!"}
{"id":"134637","title":"What is the composition of the universe's population of neutrinos?","text":"I believe earth-based detectors measure mainly solar neutrinos, which have energies on the MeV scale of nuclear physics, are directed from the sun, and have flavors determined by the sun's nuclear reactions plus neutrino oscillations. In addition to this, I recently learned from a physics.SE answer that the universe contains neutrinos that are thermalized with respect to the 3 K background temperature of the universe. I assume these latter can't be detected with present technology...? What is our best present knowledge of the universe's population of neutrinos? Energy spectrum? Flavors? Directions of motion? (Are they isotropic?) Spatial distribution? Are there antineutrinos in the mix? Is the neutrino contribution to the cosmological stress-energy tensor primarily from one component of this population?"}
{"id":"26002","title":"\"Blue Bumper\" Stars","text":"I was recently overviewing various massive compact halo object studies (the Anglo-Australian MACHO collaboration and the French I\/II EROS collaboration), and they frequently reference \"blue bumper stars,\" irregular variable stars which produce light curves very similar to gravitational microlensing event. Further searches for more information about them was mostly fruitless, producing a few conference proceedings:   * The MACHO Project LMC variable star inventory: Aperiodic blue variables (code _1995llnl.reptR....P_ )   * The MACHO Project LMC Variable Star Inventory: Aperiodic Blue Variables (code _1995AAS...18710202P_ ),  but little else. Is there any more information on this type of variable star, especially a more rigorous and targeted study of them? Also, how is it possible to get the full text for either of those articles?"}
{"id":"134638","title":"Question about electromagnetic Spectrum","text":"I have question related to electromagnetic Spectrum. If energy of photon is $E=m_e c^2$, to which part of the electromagnetic spectrum does it belong ?"}
{"id":"113376","title":"How do I place point charges when using the method of images with a sphere?","text":"> We have a point charge $q>0$ in vacuum at a distance $a$ from the center $M$ > of an isolated metal sphere that has a total charge $-q$ and radius $R$. > ($a>R$) > > A) Use the method of images to find a charge configuration that has the same > boundary conditions. (hint: you need 3 charges) For question A I know that my point charge $q$ should stay in the same place and that I need to place 2 other point charges inside the sphere. But I can't figure out what the charge and place of these two charges should be. My book (introduction to Electrodynamics, Griffiths) has an example where the sphere isn't charged but they seem to magically find the answer. They place a new charge $q'=-\\frac{R}{a}q$ at a distance $b=\\frac{R^2}{a}$ but they don't explain how this is a logical configuration. Can someone please explain this to me, as I would like to know how to place my point charges for different situations and not just one."}
{"id":"11268","title":"Derivation of the oil drop experiment","text":"The oil drop experiment performed by Robert Millikan in 1909 enables us to calculate the elementary charge. The following forces have effect on the droplet:   1. $F_\\mathrm{G} = m g = \\frac{4}{3} \\, \\pi r^3 \\rho_\\mathrm{Oil} g$   2. $F_\\mathrm{A} = \\frac{4}{3} \\, \\pi r^3 \\rho_\\mathrm{Air} g$   3. $F_\\mathrm{E} = q E = \\frac{qU}{d}$   4. $F_\\mathrm{R} = 6\\,\\pi\\eta r v$ However, which steps of calculation are necessary to finally calculate the elementary electric charge (charge of the electron) or at least a multiple of it? $$q = \\frac{9 \\pi d}{2 U} \\sqrt{\\frac{\\eta^3 (v_1 - v_2)}{\\rho^* g}} (v_1 + v_2)$$"}
{"id":"3576","title":"How to mathematically formulate the Two Slit Experiment in Special Relativity?","text":"How to mathematically formulate the Two Slit Experiment in a Lorentz invariant framework. Is there a paper about this?"}
{"id":"70605","title":"Number operator and Dirac field (with anticommutation relations)","text":"Before using anticommutation relatives the energy, momentum, charge and number operators of the Dirac field have following expressions: $$ \\hat {H} = \\int \\epsilon_{\\mathbf p}\\left( \\hat {a}^{+}_{s}(\\mathbf p )\\hat {a}_{s}(\\mathbf p ) - \\hat {b}_{s}(\\mathbf p )\\hat {b}_{s}^{+}(\\mathbf p ) \\right)d^{3}\\mathbf p, $$ $$ \\hat {\\mathbf P} = \\int \\mathbf p \\left( \\hat {a}^{+}_{s}(\\mathbf p )\\hat {a}_{s}(\\mathbf p ) - \\hat {b}_{s}(\\mathbf p )\\hat {b}_{s}^{+}(\\mathbf p ) \\right)d^{3}\\mathbf p, $$ $$ \\hat {Q} = \\int \\left( \\hat {a}^{+}_{s}(\\mathbf p )\\hat{a}_{s}(\\mathbf p ) + \\hat{b}_{s}(\\mathbf p )\\hat{b}^{+}_{s}( \\mathbf p )\\right)d^{3}\\mathbf p, $$ $$ \\hat {N} = \\int \\left( \\hat {a}^{+}_{s}(\\mathbf p )\\hat{a}_{s}(\\mathbf p ) + \\hat{b}_{s}(\\mathbf p )\\hat{b}^{+}_{s}( \\mathbf p )\\right)d^{3}\\mathbf p. $$ After using anticommutation relations and \"neglecting\" of infinite constant $\\delta (0)$ (as we can do in a case of free field) in an expressions of energy and momentum operators, they transformed into $$ \\hat {H} = \\int \\epsilon_{\\mathbf p}\\left( \\hat {a}^{+}_{s}(\\mathbf p )\\hat {a}_{s}(\\mathbf p ) + \\hat {b}^{+}_{s}(\\mathbf p )\\hat {b}_{s}(\\mathbf p ) \\right)d^{3}\\mathbf p, $$ $$ \\hat {\\mathbf P} = \\int \\mathbf p \\left( \\hat {a}^{+}_{s}(\\mathbf p )\\hat {a}_{s}(\\mathbf p ) + \\hat {b}^{+}_{s}(\\mathbf p )\\hat {b}_{s}(\\mathbf p ) \\right)d^{3}\\mathbf p . $$ But I have a couple of questions in connection with number and charge operators.   1. What can I do with infinite constant into espressions of charge and number operators? For example, $$ \\hat {N} = \\int (\\hat {a}^{+}_{s}(\\mathbf p )\\hat {a}_{s}(\\mathbf p ) + \\delta (0) - \\hat {b}^{+}_{s}(\\mathbf p )\\hat {b}_{s}(\\mathbf p ))d^{3} \\mathbf p . $$ Can I neglect summand with infinite constant in a reason that it is connected with infinite \"vacuum\" energy of field?   2. (Under the condition of neglecting infinite constant). When I acted on one-antiparticle state $|E_{\\mathbf k }\\rangle = \\hat {b}^{+}_{s'}(\\mathbf k )| \\rangle$ by number operator, I got minus: $$ \\hat {N}{b}^{+}_{s'}(\\mathbf k )| \\rangle = -\\int \\hat {b}^{+}_{s}(\\mathbf p )\\hat {b}_{s}(\\mathbf p )\\hat {b}^{+}_{s'}(\\mathbf k)d^{3}\\mathbf p | \\rangle = -| E_{\\mathbf k } \\rangle . $$ But $\\hat {b}^{+}_{s}(\\mathbf p )$ is creation operator (i.e. it was the creation operator before using anticommutation relations; the relations changed this (?)). Did I make the mistake?"}
{"id":"70600","title":"How do they measure the rotational speed of cricket balls?","text":"The 2013 Ashes series (a cricket thing between England and Australia) are underway as of today and it seems they have a new (to me) gimmick. It appears that they are now able to measure almost instantaneously (within a few seconds) the rotational speed of balls bowled by spin bowlers. ![enter image description here](http:\/\/i.stack.imgur.com\/lSivF.jpg) On the (golf) driving range, I was able to get similar readings, but they were from a machine only a few feet away from the tee. I don't quite understand that either, but it seems at least easier. Although there, they were even able to determine the rotational axis as well. ![enter image description here](http:\/\/i.stack.imgur.com\/tpu8i.jpg) But in cricket they also have a rotational axis along the direction of the ball, which is even more difficult to measure (unless they have a third radar somewhere perpendicular- _ish_ to the pitch). _I think_ I now see that they placed very small black boxes between two of the stumps on either end of the pitch. I guess that must be the first two radars. (One of which will be useless because the batsman is standing in front of it.) ![enter image description here](http:\/\/i.stack.imgur.com\/KaMpw.jpg) So, essentially: How do they do it? And, how accurate can it be, given that the reflection of the \"sides\" of the ball may be poor? Or are they using (visible-light) camera images, as suggested by the comments of udiboy and Deer Hunter below? (I'm fairly confident the golf system doesn't do that, but I don't _know_. Anyways, to make sure, the question is primarily about the cricket.) * * * _Added later._ I found that they measure spin rates in baseball as well: http:\/\/sportsillustrated.cnn.com\/2011\/writers\/tom_verducci\/04\/12\/fastballs.trackman\/index.html. The http:\/\/trackman.dk site refers to these technologies and (patent\/patent application) numbers: >   * Measuring spin rate of sports balls by radar using multiple harmonic > spectrum traces. (US2009\/0075744, EP1698380, DE602006009719.0, > GB\/EP\/1698380, ZL200680006869.0, JP2008\/538085A and KR10\/0947898) > >   * Measuring spin axis orientation of sport balls from trajectory > measurements by radar. (US2009\/0075744, DE602006015036.9, GB\/EP\/1853362, > ZL200680006869.0, JP2008\/538085A and KR10\/0947898) > >"}
{"id":"35339","title":"state for a classical particle","text":"> **Possible Duplicate:**   >  Why are coordinates and velocities sufficient to completely determine the > state and determine the subsequent motion of a mechanical system? In classical mechanics, if all the generalized coordinates and the generalized velocities are simultaneously specified, the state of the system is completely determined and one can calculate its subsequent motion. That also means if we know generalized coordinate $q$ and generalized velocity $\\dot{q}$, then we can calculate acceleration $\\ddot{q}$. How to prove this?"}
{"id":"33785","title":"The electron jumps and lets loose photons","text":"Where is the source of the photon. If the photon propagates from within the electrons transit does this point to some sort of field? Does the energy come from a boundary being broken in laymens terms a sound barrier type effect the electron performs on its shell as it changes orbit?"}
{"id":"33786","title":"What is the Zero-dispersion wavelength?","text":"There are a decent number of complex mathematical descriptions of the zero- dispersion wavelength online, but I can't seem to find any basic explanations. So to be redundant, what is it?"}
{"id":"33780","title":"Is quantum perturbation theory taught in college?","text":"Is perturbation theory usually taught in undergraduate physics, and how much of it is taught in quantum mechanics courses? Also, how much of quantum field theory would be taught in undergraduate physics?"}
{"id":"18584","title":"Gravity waves detectors; are they all similar?","text":"Are the gravity waves detectors all working on the same principle\/effect ?"}
{"id":"14600","title":"Elastic collisions with neutrinos","text":"1: Do neutrinos undergo **elastic** collisions with fermions? 2: Would this imply a **variable** speed for neutrinos? 3: Can neutrinos transfer **momenta** in interactions?"}
{"id":"38666","title":"Scale invariance symmetry as a simple argument in an electrostatics problem","text":"In the comments to this post, it was hinted that proving that the force acting on a charge at a vertical distance from a uniformly charged plane is independent of that distance can be done by recalling the scale invariance symmetry. Can someone explain that to me?"}
{"id":"83496","title":"How can a rapid change in the volume of a gas cause changes in its temperature?","text":"We were learning about Boyle's law (pressure is inversely proportional to volume of a gas) and in the experiment to prove the law, we were told that we cannot change the volume of a gas too rapidly without affecting its temperature. I have two questions about this:   1. Why does temperature of a gas change when its volume changes very rapidly?   2. This process (rapid changes in volume) is used in the liquefaction of gases. How is this done?"}
{"id":"89630","title":"How are physics and computer science getting united?","text":"How is theoretical computer science getting united with physics? Phenomena like Quantum Computing uses Quantum Mechanics to be able to compute things, how are computers helping not just to model our equations but actually predict new equations, helping us to see the computational aspect of nature and how various things are being looked at from new perspectives using Computer Science?"}
{"id":"54205","title":"Stability of nucleii and $A=5$","text":"_Why there is no stable nuclei with $$A=5$$ in nuclide the chart and so in nature like we know it?_"}
{"id":"30788","title":"Would oxygen in a container condense automatically at night in the exosphere?","text":"If a container filled with pure oxygen and fitted with leak-proof valves were to be dumped into the exosphere at night, would the contained oxygen condense?"}
{"id":"109007","title":"Thermal resistance of thermal interface materials?","text":"Thermal conductivity are often used for surfaces between the computer chip and the heat sink to increase heat transfer and they want high thermal conductivity to decrease the thermal resistance. By $$\\Delta T=RQ$$ and $Q$ is constant by the chip. When we decrease R, and keep Q constant from the chip, we decrease $\\Delta T$ between the chip and the heat sink. This is where I get confused because I assumed we would want the temperature difference between the chip and heat sink to be as high as possible. If the two temperatures are close to each other then wouldn't that make the chip heat up and exceed the operating temperature? I thought we would want the heat sink to be much colder than the chip so the chip will cool. Can someone clarify?"}
{"id":"89181","title":"How is the Earth heated by a Full Moon?","text":"While the moon is certainly not a good reflector of solar radiation, surely the radiation it reflects back heats the Earth (even if it is a terribly small amount). How would one go about calculating (or estimating) this heating contribution on a night with a Full Moon?"}
{"id":"88534","title":"Death by neutrinos - polonium go home","text":"In Randall Munroe's _What If?_ He is calculating the Lethal Neutrinos dose. > If you observed a supernova from 1 AU away—and you somehow avoided being > being incinerated, vaporized, and converted to some type of exotic > plasma—even the flood of ghostly neutrinos would be dense enough to kill > you. **How do I stay alive to be killed by neutrinos?** Can I pick a large supernova or some other cosmic event, and hide behind a nearby neutron star?"}
{"id":"87551","title":"Why are Hermitian operators linked to observables?","text":"In Quantum Mechanics, why is it that a self-adjoint operator is linked to an observable? What makes it measureable? And why isn't a non-Hermitian operator linked to an observable? Also, what type of observables are we talking about here? Particles?"}
{"id":"17175","title":"Neutrino path bent by gravity?","text":"> **Possible Duplicate:**   >  Neutrino unaffected by gravity Other probably closely related questions here: Link Neutrinos unaffected by gravity   Link Superluminal Neutrinos My question is subtly different (I hope). Gravitational lensing demonstrates that light is bent by massive bodies. If neutrinos and light are bent by the earth's gravity would we be able to measure that? I suppose we would need to take measurements of the difference between traveling into\/out of our gravity well and across the gravity well. A good deal of discussion is going on about the Gran Sasso experiment and one of the points that keeps being mentioned is the time it took for neutrinos to arrive from a supernova explosion. The light\/neutrinos from the supernova are traveling **into** our gravity well and will therefore not be bent by the earth's gravity while the Gran Sasso neutrinos are traveling **across** and will be. What _should_ the difference be?"}
{"id":"88588","title":"Rotation of a slipping ladder","text":"Imagine a ladder leaning against a wall. All surfaces are smooth. Hence the ladder will slip and fall. While falling it rotates because there are external torques acting on it. My question is about which axis does the ladder rotate?"}
{"id":"87559","title":"Do nanoscopes exist?","text":"We are mostly all familiar with a microscope, and know that it helps to see MICRO components, like stuff that is photolithographically etched on silicon semiconductor die. (The latter can also be nano, but let's bide that.) Anyways, we rarely hear of nanoscopes, i.e. a lens that can digitally reflect traces of nanometer objects, like quarks, hadrons, and valence electrons. I want to purchase a good microscope, but that will only let me see cells, substrate micro-components, etc. I want to see the actual atoms that make up the silicon, the atoms that make up the doped logic gates in my tablet, the atoms that make up my arm, and the atoms in the air that surrounds our very demanding atmosphere of carbon dioxide! So I ask again, with endless respect for science, do nanoscopes exist, and if they don't, how close can I get to the atoms, quarks, and hadrons, and how do I tell them all apart!"}
{"id":"39206","title":"Work done by introducing a spin in supersposition into a Magnetic Field","text":"A spin is created in a superposition of up and down states. A magnet is moved very slowly, towards the spin. What is the work done by the magnet. It may be helpful to imagine that the magnet is connected to a spring that expands or contracts depending on work done by the magnet."}
{"id":"31433","title":"Is the Higgs mechanism a fundamental interaction?","text":"Is the Higgs mechanism a fundamental interaction of the same standing as the strong, weak and electromagnetic interactions? If not, is it mediated by the weak interaction? It seems that all the massive particles participate in the weak interaction, and the massless ones do not, but this might just be a coincidence."}
{"id":"27489","title":"Do any entanglement measures for mixed states exist that use only single site correlation functions?","text":"For a pure state $\\rho_{AB}$, the entropy of entanglement of subsystem $A$ is \\begin{equation} S( \\rho_A) = -tr (\\rho_A \\log \\rho_A) \\end{equation} where $\\rho_A$ is the reduced density matrix of A. For a single site of a spin chain, $\\rho_A$ can be written in terms of single site correlation functions $\\langle \\sigma_l^\\alpha \\rangle$ where $\\alpha = x,y,z$. Are there any entanglement measures for mixed states that use only the same correlation functions, $\\langle \\sigma_l^\\alpha \\rangle$ where $\\alpha = x,y,z$?"}
{"id":"103953","title":"What do the BICEP2 results mean for string gas cosmology and the ekpyrotic universe?","text":"The imprint of gravitational waves created shortly after the big bang may offer direct evidence for inflation theory, according to a discovery by the BICEP2 experiment at the South Pole and released today. What this means for those alternatives theories to inflation such as string gas cosmology and the ekpyrotic universe? What would happen with these two cases particularly?"}
{"id":"131453","title":"Circuit breaker Trips during thunderstorm","text":"The circuit breaker at the electrical mains trips at home when there is a thunderstorm outside. Why does this occur?"}
{"id":"127866","title":"Why do these papers show the wrong concavity to the conductivity near the percolation threshold?","text":"I'm looking at two papers in particular: A. L. Efros and B. I. Shklovskii, Critical Behaviour of Conductivity and Dielectric Constant near the Metal-Non- Metal Transition Threshold, Phys. Status Solidi B 76, 475 (1976) and Measurement of the conductivity exponent in random percolating networks of nanoscale bismuth clusters . In percolation theory, for concentrations $p$ higher than the critical concentration $p_c$, the conductivity $\\sigma \\propto (p-p_c)^t$ for $p>p_c$, where $t$ is 1.3 for 2 dimensions. It's often simplified so that the substrate is a perfect insulator, so you have $\\sigma=0$ for $p<p_c$. So it should look pretty simple, something like: ![enter image description here](http:\/\/i.stack.imgur.com\/VeCeo.png) My point is, concave up. However, looking at the two papers I mentioned above, they have these graphs: ![enter image description here](http:\/\/i.stack.imgur.com\/FdxOx.png) with:![enter image description here](http:\/\/i.stack.imgur.com\/azYOy.png) and: ![enter image description here](http:\/\/i.stack.imgur.com\/FiDoY.png) Both of which are concave _down_ in the region $p>p_c$. Is there something going on, or is this a series of errors? I suspect the Efros paper may have just had an error, as the caption corresponding to the dotted line (1) points to the region of $x<x_c$, where they explicitly said that equation doesn't apply. It does have a very strange scale though. I have no idea what's going on in the graph of the second paper. Is there any explanation besides an error? Thank you!"}
{"id":"127867","title":"Definition of causality relation","text":"It seems to me that special relativity has a weird definition of the causality relation. In that theory, the only thing that matters is the space-time distance between events. But I don't think this is the case. It seems to me that if I am pressing a few keys on my keyboard and then causing this question to appear on your screen, my action has a greater locus of influence than just me sleeping in bed. So, what exactly is the definition of the causality relation?"}
{"id":"69434","title":"Temperature loss of a moving object","text":"I would like to know how to calculate the temperature change of a moving object. For example if I throw a ball that is warmer than the surrounding air, how much would it lose form it's temperature in every second? I searched for two days now, but I have only partial solutions :(. I would need a formula that takes in account the followings:   * The material of the ball (Iron, heat conductivity?)   * Area of the ball   * Speed of the ball (true airspeed)   * Material of the surrounding material (air, heat conductivity?)   * And ofc temperature of the ball and the surrounding material.   * anything else I forgot and neccessary :):) It dosen't need to take in account friction generated heat and other things, I only need the temperature loss :) BIG TANKS!"}
{"id":"89475","title":"What exactly constitutes a measurement?","text":"As far as I understand it, quantum mechanics only predicts the probabilities of finding a system in some state of some observable when we make a measurement. So, what exactly constitutes a measurement that will make the system to collapse?"}
{"id":"86208","title":"Proper name for a thermodynamic process with constant internal energy $U$","text":"Back in the day I learned that a few special thermodynamical processes have special names. For example, if one keeps $P$ constant, the process is called isobaric, if one keeps $T, V$ or $S$ constant, one gets, correspondingly, isothermic, isochoric or isentropic processes. Similarly, if one keeps $\\dfrac{\\mathrm{d} \\ln P}{\\mathrm{d} \\ln \\rho}$ constant during the process, it is called polytropic, and if $\\delta Q = 0$ at any time, the process is called adiabatic. Now, the question: what is the process called, if one keeps internal energy $U$ constant?"}
{"id":"81255","title":"Why aren't atoms affected by gravity, but molecules are?","text":"I read this book here: http:\/\/tiny.cc\/Gravity-Atom-Myth It claims that while gravity does affect individual atoms, it's not quantified like molecular mass due to lack of binding proteins which render air-oxygen trajectories improbable."}
{"id":"4709","title":"Will watering tea down make it clearer?","text":"If I poured water into my tea, would I see more or less of the bottom of the tea-cup? Intuitively, there would be as many particles blocking as many photons, and so I'd see the bottom just as clearly as before."}
{"id":"69380","title":"What are common methods for calculating the time dependency of elliptical orbit?","text":"After playing a game called \"Kerbal Space Program\" I got interested in orbital mechanics and started messing with simplified calculations to determine $\\Delta v$ requirements. In which I compared two trajectories to get into orbit\/land from orbit to see how much it would matter if you would burn vertical or horizontal (Hohmann) at the surface. Graphs of some results can be seen here. However after this I also wanted to know how much time it would take to perform these maneuvers. I assumed that changes in velocity are instantaneous, so this would be equal to the time between the two changes in velocity. For the Hohmann-like-transfer it is simple, since you travel from periapsis tot apoapsis, which takes half of the orbital period (due to symmetry). But for the other transfer, which applies a vertical change in velocity at the surface, it is very difficult (if there is a sidereal rotational velocity). I tried solving it using Kepler's second law, given this formula: $$ r=\\frac{a(1-e^2)}{1+e\\cos{\\theta}} $$ Since the area underneath the path from $\\theta_0$ to $\\theta_1$ can be calculated like this: $$ A_{\\theta_0,\\theta_1}=\\int_{\\theta_0}^{\\theta_1}{\\frac{1}{2}r^2d\\theta} $$ Which can be used to determine the amount of time it would take, since the total area of the ellipse would take one orbital period $T=2\\pi\\sqrt{\\frac{a^3}{\\mu}}$: $$ t_{\\theta_0,\\theta_1}=T\\frac{\\int_{\\theta_0}^{\\theta_1}{\\frac{1}{2}r^2d\\theta}}{\\int_{0}^{2\\pi}{\\frac{1}{2}r^2d\\theta}}=T\\frac{\\int_{\\theta_0}^{\\theta_1}{\\frac{1}{2}r^2d\\theta}}{\\pi a^2\\sqrt{1-e^2}} $$ I am not that good at solving integrals, so I both tried to solve it with MATLAB and WolframAlpha, which gave me this as the result: $$ \\int{\\frac{1}{2}r^2d\\theta}=a^2\\sqrt{1-e^2}\\left(\\tan^{-1}\\left(\\frac{\\sqrt{1-e^2}\\tan{\\frac{\\theta}{2}}}{1+e}\\right)-\\frac{e\\sqrt{1-e^2}\\sin{\\theta}}{2(1+e\\cos{\\theta})}\\right) $$ So: $$ t_{\\theta_0,\\theta_1}=\\sqrt{\\frac{a^3}{\\mu}}\\left[2\\tan^{-1}\\left(\\frac{\\sqrt{1-e^2}\\tan{\\frac{\\theta}{2}}}{1+e}\\right)-\\frac{e\\sqrt{1-e^2}\\sin{\\theta}}{1+e\\cos{\\theta}}\\right]^{\\theta_1}_{\\theta_0} $$ However when I tried to determine the integral with symbolic parameters and bounderies, MATLAB returned \"Warning: Explicit integral could not be found.\" probably because there is not a general solution for both closed and open orbits. But if I use this formula and filled in the boundaries myself, it seemed that I got wrong results. However when I let MATLAB calculate the integral for given values of $a$, $e$, $\\theta_0$ and $\\theta_1$ I do get results which seem correct. So is this a good approach to find a time dependency of an (elliptical) orbit. If so is there an (continuous) equation derivable from this. And what might be a better approach? **EDIT:** The unexpected results form this formula might be caused by the fact that $\\tan^{-1}$ returns results from $-\\frac{\\pi}{2}$ to $\\frac{\\pi}{2}$. Since when I calculated the time from $\\theta=0$ to $\\theta=\\pi$ I got results which agree with the conservation of energy and angular momentum. For this is used the following equations to determine the time derivatives: $$ \\omega=\\frac{\\delta\\theta}{\\delta t}=\\frac{\\delta \\theta \/ \\delta \\theta}{\\delta t \/ \\delta \\theta}=\\sqrt{\\frac{\\mu}{a^3\\left(1-e^2\\right)^3}}\\left(1+e\\cos{\\theta}\\right)^2 $$ $$ \\dot{r}=\\frac{\\delta r}{\\delta t}=\\frac{\\delta r \/ \\delta\\theta}{\\delta t \/ \\delta\\theta}=\\omega\\frac{\\delta r}{\\delta\\theta}=\\omega\\frac{ae(1-e^2)\\sin{\\theta}}{\\left(1+e\\cos{\\theta}\\right)^2} $$ Which yields that the specific orbital energy $\\epsilon$ and specific angular momentum $h$ are constant when substituting theses therms into there expressions: $$ \\epsilon=\\frac{v^2}{2}-\\frac{\\mu}{r}=\\frac{\\dot{r}^2+\\omega^2r^2}{2}-\\frac{\\mu}{r}=-\\frac{\\mu}{2a} $$ $$ h=\\omega r^2=\\sqrt{a\\mu\\left(1-e^2\\right)} $$"}
{"id":"135223","title":"Question about derivation of tensor in Di Francesco's CFT","text":"This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the 2-point Schwinger function in two dimensions. He writes that the most general form of the tensor is $$S_{\\mu \\nu \\rho \\sigma} = (x^2)^{-4} \\left\\\\{ A_1 g_{\\mu \\nu} g_{\\rho \\sigma} (x^2)^2 + A_2 (g_{\\mu \\rho}g_{\\nu \\sigma} + g_{\\mu \\sigma}g_{\\nu \\rho})(x^2)^2 + A_3(g_{\\mu \\nu}x_{\\rho}x_{\\sigma} + g_{\\rho \\sigma}x_{\\mu}x_{\\nu})x^2 + A_4 x_{\\mu}x_{\\nu}x_{\\rho}x_{\\sigma}\\right\\\\}$$ This I understand and have obtained this result myself. What I don't understand however, is why he has neglected the following term since it seems to satisfy all the constraints presented on P.108: $$S_{\\mu \\nu \\rho\\sigma} = A_5 (x^2)^{-3} (g_{\\mu \\sigma} x_{\\rho}x_{\\nu} + g_{\\mu \\rho}x_{\\sigma}x_{\\nu} + g_{\\nu \\sigma}x_{\\rho}x_{\\mu} + g_{\\nu \\rho}x_{\\sigma}x_{\\mu})$$ In another thread I posted here, I wondered whether this could be reduced to terms already present in the form Di Francesco gave, but I was quickly reassured this to not be the case. So, if anyone is familiar with his book and would be willing to clarify this it would be great. I asked a professor at my university and he was not sure either why it has been neglected, so I thought I would pose the question here."}
{"id":"135222","title":"Why can't electrons fall into the nucleus?","text":"I read a book on pop sci book on quantum mechanics and the author said that electrons do not fall into the nucleus due to quantum mechanics- which principles suggest this (I think it was Heisenberg's Uncertainty and Pauli's Exclusion Principle) and why? Also, I've heard that if Bohr's planetary model were correct, then electrons would lose energy\/momentum and fall in- is this true and again, which physics principles say this?"}
{"id":"135225","title":"Is there a classical analog to quantum mechanical tunneling?","text":"In comments to a Phys.SE question, it has been written: > _'Tunneling' is perfectly real, even in classical physics. [...] For > sufficiently large temperatures this can put the system above a hump in its > potential energy._ and > _the only difference between the classical case and the quantum mechanical > one is that classical physics is a random walk in real time, while QM is a > random walk in imaginary time._ I understand that in a system of particles with finite temperature some particles can overcome a potential barrier. That's how I interpret the first statement. I don't understand the business of \"random walk in imaginary time\". Can someone explain? **Update** What I was originally looking for was 1.) classical system that can transport mass through a forbidden region and 2.) explanation of \"random walk in imaginary time\". So far, I don't see anything for question 1.), but I think I'll grok 2.) if I invest some time and energy."}
{"id":"78317","title":"How does Newton's first law asserts the existence of inertial frames?","text":"Recently I've seem here one answer telling that Newton's first law really assures the existence of inertial reference frames. But how is that? I really can't see it. As I know, Newton's first law says: > _Every body continues in its state of rest, or of uniform motion in a right > line, unless it is compelled to change that state by forces impressed upon > it._ Now, what I've seem done is to use this to define that whenever this law holds in some reference frame we call it inertial. But how this law states that such a frame does exist?"}
{"id":"74683","title":"Is Newton's first law something real or a mathematical formalism?","text":"Why do objects always 'tend' to move in straight lines? How come, everytime I see a curved path that an object takes, I can always say that the object tends to move in a straight line over 'small' distances, but as you take into account the curvature of the path, a force acting on the particle appears. I mean, I can always take a small enough portion of the curve, zoom in enough, and conclude that the object is moving in a straight line, but then as I zoom out I find out that a force is acting on the particle. The force of gravity is everywhere and, no matter how weak it is, it will make the particle take a path which is different from a straight line. This is my question: since particles are, in reality, never moving in straight lines, is Newton's first law a mathematical formalism or some true property of material objects?"}
{"id":"13556","title":"Why does only one side of a neon lamp glow?","text":"When applying DC to a neon lamp, only one electrode glows: ![http:\/\/commons.wikimedia.org\/wiki\/File:Neonlamp3.JPG](http:\/\/i.stack.imgur.com\/3poGs.jpg) The voltages across the lamps are left: DC (left lead positive), middle: DC (right lead positive), and right: AC. But... why? The electrodes are the same shape, so the electric field around them should be same shape, and the gas should break down at the same electric field strength. It seems like the fields would be symmetrical. Is there a difference in threshold between positive and negative coronas? If so, do both sides light up at high enough voltage? Or maybe only one type of corona is possible in neon since it's a noble gas? If it contained air would it glow at both electrodes?"}
{"id":"19897","title":"Can a solar thermal collector designed for use in homes ever power a stirling engine designed for use in homes?","text":"Can a solar thermal collector common on private roofs today designed for collecting heat for the household ever collect enough heat to power a stirling engine of say this model: http:\/\/www.whispergen.com\/main\/achomesspecs_info\/ Normally a stirling engine of this model is powered by combustion of natural gas."}
{"id":"19890","title":"Temperature change effected by electric heater","text":"A 40-gallon electric water heater has a 10kW heating element. What will the water temperature be after 15 min of heating if the start temp is 50F degrees. There must be an equation. I can't find it in notes or text."}
{"id":"7899","title":"How do we know the size of the universe?","text":"Ok, from astronomical observations we can tell that the observable matter is separating - so rewind the clock about 13.7 billion years and it was all at a single point. However, how do we distinguish between the following two options:   1. Universe is expanding    2. Matter distribution is increasing into infinite void **Clarification 1:**   (My notion of) The traditional notion is that all time\/space\/matter was created at the instant of the big bang. I.e. BB was inital conditions of: $t=0$, $V=0$, $E=very big$ People say \"the universe is expanding\", rather than \"observable matter is separating\". Why is this? How do we know that the big bang event wasn't started by all matter condensed at a single point within a larger (otherwise empty) universe? How do we know that BB wasn't: $t=0$, $V(universe)>0$ but $V(matter)=0$, $E=very big$?"}
{"id":"101559","title":"Plasma Treatment of LaAlO3: Surface Roughening","text":"I'm trying to understand something I've observed in exposing LaAlO3 substrates to an oxygen plasma (yielding atomic oxygen). In literature, these substrates are frequently \"cleaned\" in a oxygen rich environment by exposing the surface to atomic oxygen (via plasma source) at substrate temperatures greater than 700 C. This typically results in a clear diffraction pattern indicative of a flat surface. In my experiments, I'm noticing that initially, the surface improves upon plasma exposure (enhanced diffraction intensity), but after a certain point, the intensity of the diffraction pattern begins to fade, and (presumably) the surface becomes disordered. I'm looking for insight into why this might be happening, because typically the surface will be exposed to the atomic oxygen plasma during growth, but this seems to be damaging my film. My first guess would be that initially, atomic oxygen reacts with surface species to produce a clean surface, and, after a point, begins to somehow erode the surface, reducing crystallinity."}
{"id":"62527","title":"How does the energy in a standing wave travel beyond a node?","text":"In a standing wave, how does energy travel past a node? It should just get reflected. Assume the case of first overtone and you strike the string at a place. How will energy distribute itself? If it distributes over the whole string, then how is it the wave traveled past a node? Basically the problem is a standing wave has been set up in the string . Now you strike the string at a place, and this creates a new pulse. Now this pulse will travel past the next node in the string (not the boundary) or will it reflect off even that node and remain confined to a particular region of the string?"}
{"id":"25252","title":"When will the Moon reach escape velocity?","text":"From what I know, the Moon is accelerating away from the Earth. Do we know when it will reach escape velocity? How do we calculate this?"}
{"id":"95734","title":"Rigid body problem","text":"I have some doubts about the next excercise: > _A bar of length $2a$ and mass $m$ moves freely with both of its extremes on > a ring of radius $\\sqrt2a$. The ring can rotate freely in a certain > diameter, remaining the center fix. Find the equations of motion._ I did it using the balance of angular momentum, but I want to try to do it with the Lagrangian. But I have a problem. The energy of the ring is easy to calculate (just a rotation), but the energy of the bar I am not sure how to do it. ¿Is a translation of the center of mass plus a rotation? ¿Or just translation? ¿Or just a complicated rotation? I did this picture to illustrate: ![enter image description here](http:\/\/i.stack.imgur.com\/DhYke.png) Thanks!"}
{"id":"46632","title":"Tensor Introduction","text":"I have recently started learning about tensors during my course on Special Relativity. I am struggling to gain an intuitive idea for invariant, contravariant and covariant quantities. In my book, invariant quantities are described as being those physical quantities associated with a point in space e.g. temperature at point P. Thus, when changing coordinate system, the value associated with the point is unchanged: you may represent the point by a different set of coordinates, but on locating the point in the new coordinates, the temperature value will still be the same. Here comes my struggle: the archetypical example of a covariant quantity is the gradient vector. Is it not the case that we require it to transform between coordinate systems in a way that preserves the vector associated with each point? Thus, in a way, the gradient vector is an invariant quantity? Why then are many derivations in my course driven by the fact that physical quantities like Action, Lagrangians etc should be \"Lorentz invariant\"?"}
{"id":"23028","title":"Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?","text":"The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\\\\{\\vert n \\rangle\\\\}_{n = 0}^{\\infty}$ each with corresponding eigenvalue $E_n = \\omega \\left(n + \\frac{1}{2}\\right)$. Refer to this construction as the **abstract solution**. How does the abstract solution also prove uniqueness? Why is there only one unique sequence of countable eigenstates? In particular, **can one prove the state $\\vert 0\\rangle$ is the unique ground state without resorting to coordinate representation?** (It would then follow that the set $\\\\{\\vert n \\rangle\\\\}_{n = 0}^{\\infty}$ is also unique.) The uniqueness condition is obvious if one solves the problem in coordinate representation since then one works in the realm of differential equations where uniqueness theorems abound. Most textbooks ignore this detail (especially since they often solve the problem both in coordinate representation and abstractly), however I have found two exceptions:   * Shankar appeals to a theorem which proves one-dimensional systems are non-degenerate, however this is unsatisfactory for two reasons:     1. Not every one-dimensional system is non-degenerate, however a general result can be proven for a large class of potentials (the SHO potential is in such a class).     2. The proof requires a departure from the abstract solution since it classifies the potentials according to their functional properties.   * Griffiths addresses this concern in a footnote stating that the equation $a \\vert 0\\rangle = 0$ uniquely determines the state $\\vert 0\\rangle$. Perhaps this follows from the abstract solution, however I do not see how."}
{"id":"134170","title":"Construction of Bound states in Lattice QCD","text":"How the proton state is found and renormalized in lattice QCD? Is there any literature shows the method step by step? What about other bound states in QCD if we somehow know their quantum numbers?"}
{"id":"62294","title":"Sign issue on electrostatic potential energy","text":"I have been working on a problem about finding the electrostatic potential energy stored on a capacitor of concentric spheres with inner radius $a$ and outer radius $b$ and with charge $Q$. I've got to first calculate the energy using the capacitance and then integrating the energy density. I did the following: first of all I've used Gauss' Law to find the electric field a distance $r$ from the center of both spheres with $a < r < b$. As expected I've got the field of a point charge $Q$ at the center. Then I've integrated the field along a segment joining the two spheres and I've got the following difference of potential $$V=\\frac{1}{4\\pi\\epsilon_0}\\frac{Q(a-b)}{ab}$$ Then I've found the capacitcante using $Q=CV$ and finally I've found the energy using $U=Q^2\/2C$. That's fine, I've got the value: $$U_1=\\frac{1}{8\\pi\\epsilon_0}\\frac{Q^2(a-b)}{ab}$$ Now the second part, I should find the same value integrating the energy density. So, the energy density is $\\mathcal{u}=\\epsilon_0E^2\/2$ and hence it is: $$\\mathcal{u}=\\frac{1}{2}\\epsilon_0 \\frac{1}{16\\pi^2\\epsilon_0^2}\\frac{Q^2}{r^4}$$ Since I must integrate on the region between the two spheres I've used spherical coordinates and computed the following integral getting the energy: $$U_2=\\int_0^\\pi\\int_0^{2\\pi}\\int_a^b \\frac{1}{2}\\epsilon_0 \\frac{1}{16\\pi^2\\epsilon_0^2}\\frac{Q^2}{r^4}r^2\\sin\\phi dr d\\theta d\\phi$$ And this integral gave simply: $$U_2 =\\frac{1}{8\\pi\\epsilon_0}\\frac{Q^2(b-a)}{ab}$$ But wait a moment, I've got $U_2 = -U_1$ instead of $U_2 = U_1$ as expected. I've calculated it once again and once again and I've got the same problem. Can someone point out what's happening? Where's my mistake? Thanks in advance for your help!"}
{"id":"62297","title":"Given a potential energy function, find expression of the force of a particle?","text":"This comes from an AP review packet. I'm given a potential energy functon, $$U(r)=br^{-3\/2} + c,$$ where $b$ and $c$ are constants, and need to find the expression for the force on the particle. There's a graph of $U(r)$ given with the problem, but I'm not sure if it's needed or not. I'm just looking for how to go about solving this problem."}
{"id":"105144","title":"Intro to Solid State Physics","text":"I didn't see this listed on the books page so here it is. I'm currently in an introductory Solid State course, and we are using Kittel's book. I have been having a rough time with this book although I am starting to get used to it as we get farther in. What are good introductory solid state books?"}
{"id":"104648","title":"What does Bell's Theorem really violate?","text":"Well this is a fairly straightforward question. I know it states that either hidden variables is wrong or Quantum Mechanics. But indirectly hidden variables is a part of QM due to uncertainty. So what exactly does it violate. I am just a bit confused."}
{"id":"29007","title":"Introduction to string theory","text":"I am in the last year of MSc. and would like to read string theory. I have the Zwiebach Book, but along with it what other advanced book can be followed, which can be a complimentary to Zwiebach. I would like a more mathematically rigorous book or lecture notes along with Zwiebach. Specifically, mention whether the book discusses string theory   * Rigorously?   * Intuitively? What's the scope of the book? Does it cover the advanced materials, e.g. Matrix string theory, F-theory, string field theory, etc. Maybe even String Phenomenology?"}
{"id":"119384","title":"Prerequisites and introduction to string theory","text":"Can someone please give me the prerequisites and mathematics required for string theory? Are there some good references to study it, both online and in a book? Please consider I am a newbie in string theory, and only about up to the level explained in Hawking's Brief History of Time and Penrose's Road to Reality. I would prefer something which does not omit rigor, but builds it up.Also, are there good online lectures on this? (I have covered most of GR, but I am not really comfortable with advanced Quantum mechanics. I have studied QM only from Griffith's book, and a few McGraw hill lecture notes.)"}
{"id":"59601","title":"Gravitational potential energy","text":"Consider two places next to each other: Place 1, where there is a gravitational field whereas Place 2 - there's no field. Now if we lifted a box in place 1, it gains potential energy. Then, we move this box horizontally to place 2. What happens to this energy?"}
{"id":"59604","title":"Does quantum reversibility require many worlds?","text":"The source S sends a photon into the beam splitter below. There is a 50% chance that it will be detected at A and a 50% chance it will be detected at B.               S -----------\\---------> A                  |                  |                  |                  v                  B      Now if we assume that physics is (generally) reversible we should be able to time-reverse this process. This would imply that if we send in a photon at A or B it should _always_ appear back at S. How can this happen? We know that if we send in a photon at A (or B) we would expect it to appear 50% of the time at S and 50% at C as below.                            C                                         C                  ^                                         ^                  |                                         |                  |                                         |                  |                                         |     S <----------\\--------- A               S <------------\\                                                            |                                                            |                                                            |                                                            B      In order that the photon always appears back at S and never at C we need a superposition of the two situations above such that the paths to S constructively interfere and the paths to C destructively interfere. Thus it seems that in order to retain time-reversibility we need to assume a many-worlds view. As a photon is sent in at A (or B) it must be assumed that another photon is simultaneously sent in at B (or A) and both states weighted with the correct amplitude such that a photon appears with certainty back at S. What do people think?"}
{"id":"59607","title":"How light know which path is smallest?","text":"We know from fermat's principle that light follows the smallest path. But how light know that which path is smallest?"}
{"id":"41111","title":"Pressure in waves on a string","text":"We know that when we speak sound waves are created. The air particles compress and rarefy and pressure is more at the nodes and less at anti-nodes. But can we say the same thing about waves on a string,- that pressure is more at the nodes than the anti-nodes?"}
{"id":"129044","title":"Energon: is it possible?","text":"I'm always quite curious about the \"Energy cube\" in Transformers, or namely Energon. Is it really possible to store energy, such as electricity, into such a compact form? safe to distribution, and seems nothing left after being consumed? ps. Wikipedia has a page for Spark, which is more for transformer's soul. I'm not asking for that yet."}
{"id":"97893","title":"Does humidity make winter air feel colder?","text":"I sometimes hear stories where people compare their feelings in winter in different places in the world. It goes like > in city X the temperature was the same as in city Y, but the humidity made > me feel much colder... or > oh well, -20°C would be cold, but the humidity was low, so it felt OK so it implies that humidity somehow makes it feel colder. I am talking about temperatures below freezing (-30...0°C). Does this have any physical explanation, or is it some sort of psychological phenomenon?"}
{"id":"97894","title":"Looking for the actual reason of refraction explained precisely without analogies","text":"I'm a high school teacher trying to teach my students (15year olds) about refraction. I've seen a lot of good analogies to explain why the light changes direction, like the marching band analogy, that the light \"choose\" the fastest way etcetc, and for most of my students these are satisfying ways to explain the phenomenon. Some students, however, are able to understand a more precise and physically correct answer, but I can't seem to find a good explanation of why the lightwaves actually changes direction. So what I'm looking for is an actual explanation, without analogies, of how an increase\/decrease in the speed of a lightwave cause it to change direction. Thanks a lot"}
{"id":"8770","title":"If you put a latex balloon in a vacuum, how much would it expand?","text":"If you put a latex balloon in a vacuum, how much would it expand? And would it pop? Assume it doesn't leak. EDIT: Some numbers: Ambient pressure is 100 KPa, balloon is perfectly spherical with a diameter of 300 mm, deflated it has a diameter of 25mm, temperature is always at equilibrium. Question components stated more formally:   * What pressure does the balloon need to have been inflated to to reach the 300 mm radius?   * What is the relationship between the (gauge) pressure and volume? _I think this simply comes down to PV=nRT_   * If there is some sort of spring constant involved, what is this value for a typical latex balloon?   * What is the relationship between the pressure (or volume) and the tension on the material the balloon is made from?      * What is the limit the tension can reach before popping? Semi-related: how much would a balloon expand if you sealed it deflated and put it in a vacuum?"}
{"id":"134761","title":"Are there entire anti-matter galaxies?","text":"Would we be able to tell if an entire neighboring galaxy was made up of anti- matter? What would we see in its signals? It seems like the anti-matter could be stable if it was completely surrounded by other anti-matter."}
{"id":"44095","title":"Definition of scattered particle?","text":"Compare the number of scattered particles:   $N_s=Fa\\int\\sigma(\\theta)d\\Omega$   With the total number of incident particles:   $N_{in}=Fa$ Here, F is the flux of incoming beam, a the area. sigma the crossection and omega the solid angle. Why isnt $N_s=N_{in}$? How does one define which particles are scattered and which are not, arent they all interacting with the target to some degree? Isnt particles conserved normally? Do almost all the particles either pass right through almost undetected or are scattered signficiantly, so what we are really integrating over is a sphere surrounding the target except a spot of area $a$ where the beam exits?"}
{"id":"95004","title":"A theory about a travelling wave due to a vibrating rope seems to be contradictory, please spot what is wrong?","text":"Consider a travelling wave produced by vibrating one end of a rope while the other end is made to freely move along a vertical line. Mathematically, the equation of the traveling wave that also represents the equation of motion of each point on the rope is given as follows. $$y(x,t)=A\\sin(kx-\\omega t)$$ In the book I read, a certain particle on the rope moves up and down perpendicular to the propagation vector as wave does not transfer material but energy. I don't agree with the statement that \"a certain particle on the rope moves up and down perpendicular to the propagation vector\". It is because I think to maintain the same point on the rope to move up and down, the arc length of $y(x,t)$ must be equal to that of $y(x,t+\\Delta t)$ for $0\\leq x \\leq a$ and any $\\Delta t$. However, according to the Notes below, their arc length are not generally equal. ![enter image description here](http:\/\/i.stack.imgur.com\/8B30K.png) From the figure above, how can $P$ and $Q$ be the same point of the rope while the red and blue curves have different length? So my question are: **Does each point of the rope move up and down?** ## Notes From $x=0$ to $x=a$, generally the arc length of $y=\\sin x$ is not equal to that of $y=\\sin(x-x_0)$. Let $S(x_0)$ be the arc length of $y=\\sin(x-x_0)$ for the given interval. \\begin{align} S(x_0) &= \\int_0^a\\sqrt{1+\\left(\\frac{dy}{dx}\\right)^2}\\, \\textrm{d}x\\\\\\ &= \\int_0^a\\sqrt{1+\\cos^2(x-x_0)}\\, \\textrm{d}x\\\\\\ \\end{align} If $S(0)=S(x_0)=\\text{const}$ for any $x_0$ then $\\frac{\\textrm{d}S(x_0)}{\\textrm{d}x_0}=0$. Thus I have to check whether or not $\\frac{\\textrm{d}S(x_0)}{\\textrm{d}x_0}=0$. \\begin{align} \\frac{\\textrm{d}S(x_0)}{\\textrm{d}x_0} &= \\frac{\\textrm{d}}{\\textrm{d}x_0}\\int_0^a\\sqrt{1+\\cos^2(x-x_0)}\\, \\textrm{d}x\\\\\\ &= \\int_0^a\\frac{\\textrm{d}\\left(\\sqrt{1+\\cos^2(x-x_0)}\\right)}{\\textrm{d}x_0}\\, \\textrm{d}x\\\\\\ &= \\int_0^a\\frac{\\cos(x-x_0)\\sin(x-x_0)}{\\sqrt{1+\\cos^2(x-x_0)}}\\, \\textrm{d}x\\\\\\ &= \\int_{-x_0}^{a-x_0}\\frac{\\cos y\\sin y}{\\sqrt{1+\\cos^2y}}\\, \\textrm{d}y\\\\\\ &= \\sqrt{1+\\cos^2 x_0}-\\sqrt{1+\\cos^2(a-x_0)}\\\\\\ &\\not = 0 \\end{align} It implies that generally they are not equal."}
{"id":"119185","title":"Which symmetry for which distance function","text":"For evaluating the electric field of some charge distribution one can use $$\\phi(r):= \\frac{1}{4 \\pi \\varepsilon_0}\\int_{\\mathbb{R}^3} \\frac{\\rho(r')}{||r-r'||_2} dr'.$$ My question is: What symmetry do we need to have that we can write in spherical coordinates $$||r-r'||_2 = \\sqrt{||r||_2^2+||r'||_2^2-2||r||_2||r'||_2\\cos(\\theta')}~?$$ This is of course not the most general way to express this distance, as the $\\phi$ dependence is missing. So, under what conditions can the distance be expressed like this? Notice that $\\theta'$ is the respective angle in spherical coordinates, so it's NOT the angle between $r$ and $r'$. So in particular, your answer should clarify, why we can evaluate for example the electric potential of a sphere by integrating: $$\\frac{1}{4 \\pi \\varepsilon_0} \\int_0^{2\\pi}\\int_0^\\pi \\int_0^{\\infty} \\frac{\\rho(r')||r'^2|| sin(\\theta')}{\\sqrt{||r||_2^2+||r'||_2^2-2||r||_2||r'||_2\\cos(\\theta')}}d||r'||_2 d\\theta'd\\phi',$$ but need to refer to a more general equation in this example, where the $\\phi$ angle is used too: excercise 14b)"}
{"id":"28371","title":"Turbulence parameterization from gravity - fluid dynamics correspondence","text":"I`m looking for a nice introductary reference that explains how the turbulence coefficient or any kind of turbulence parameterization (in view of applications to atmospheric turbulence for example) can be derived from the gravity - fluid dynamics correspondance, such that even I can get it. I mean, if something like this exists ... I`m basically quite familiar with the hydrodynamic part (NS equation, etc) of this correspondance whereas about the other side I feel a bit more shaky ... I`m finally looking for a citable reference, but any \"reasonable\" source (slides of a talk, video, ect) that explains how a turbulenc coefficient \/ parameterization can be obtained would be welcome and appreciated. Edit To clarify what I mean, relevant papers for the topic are for example here, here, and jep this one linked to by Mitchell."}
{"id":"11170","title":"A tunnel inside the Earth (but not an ordinary tunnel)","text":"I decided to dig a tunnel inside the Earth. In equatorial plane. It should be designed in such a way that it follows the Coriolis effect. That means if, say a stone is dropped from rest into the tunnel, then it travels in the tunnel without touching the walls of the tunnel until it reaches the tunnel's lip on the other side somewhere near the surface of the Earth. So the question itself is an equation of the curve of the tunnel. I suspect that there is no nice closed form for the curve. But who knows? Nevertheless, it would be good to know how deep the tunnel goes and where it rises again to the surface and the travel time. You could say that it would be impossible to do this on Earth, but you actually could do this on the Moon."}
{"id":"11177","title":"Can space-time be defined by the requirement that the physical laws are simple?","text":"When I was student I was told that time is defined by the requirement that the physical laws are simple. For example, in classical mechanics time can be defined by the requirment that the velocity of an isolated body is constant. One could generalize this approach by assuming that _space-time_ is defined by the requirement that the physical laws are simple. This can be easily expressed in mathematical language as follows. In general relativity the universe is represented by the triple $(M, g, T)$ where $M$ is a four-dimensional differentiable manifold, and $g$ and $T$ are a Lorentzian metric and a tensor field on $M$ satisfying Einstein's equation. According to the above approach, we could say that the \"true\" physical reality of the universe is represented by the pair $(M, T)$, while the metric $g$ emerges as an appeareance from the requirement that evolution appears simple, i.e., that $g$ and $T$ satisfy Einstein's equation. From the mathematical point of view, this approach poses for example the following problems: does any tensor field $T$ admit a metric $g$ such that Einstein's equation is satisfied? To what extent a tensor field $T$ univocally determines the metric $g$? Does this approach make sense, at least from the mathematical point of view?"}
{"id":"126193","title":"About the Hayden-Preskill circuit","text":"Can someone summarize as to what are the problems and\/or the open questions with the Hayden-Preskill circuit? (in the context of understanding black-holes or as a computer science question)It gives a framework to see how the black- hole can thermalize in a $log (entropy)$ time scale. What next? To start off, The sort of problem with the Hayden-Preskill circuit is that it works via random unitary transformations on disjoint pairs of qubits and hence doesn't have an Hamiltonian interpretation. I believe one has always believed that matrix models (of matrix size $\\sqrt{entropy}$) can saturate this logarithmic thermalization bound via Hamiltonian evolution.   * Has this above belief about the matrix models been proven yet?    * Is there an intuitive explanation as to why a theory with finite dimensional matrices should behave like an infinite dimensional system?"}
{"id":"119668","title":"Did cosmological natural selection get a reprieve?","text":"The March 2014 issue of Physics Today has an article by Lee Smolin in which he argues that natural laws must change over time. As examples of such theories, he gives Penrose's CCC and his own cosmological natural selection (CNS): http:\/\/arxiv.org\/abs\/hep-th\/0612185 My understanding was that CNS had already been falsified in 2010 by the discovery of a 2-solar-mass pulsar: http:\/\/arxiv.org\/abs\/1010.5788 But in the Physics Today article, Smolin says that CNS makes predictions and that, \"The two main predictions,[4] first published in 1992, have survived despite several chances to falsify them since.[7] One of those is actually easy to state: The upper mass limit of neutron stars is at most two solar masses.\" The footnotes are: [4] L. Smolin, Class Quantum Grav 9, 173 (1992); The Life of the Cosmos, Oxford U Press, New York (1997) [7] L. Smolin, http:\/\/arxiv.org\/abs\/1201.2632 ;http:\/\/arxiv.org\/abs\/0803.2926 . The two arxiv papers are from 2012 and 2008, respectively. Their abstracts don't mention anything about empirical tests of the theory. I searched the text of the 2012 paper for the word \"neutron,\" and found only one mention of neutron stars, which is in a reference to this 1997 paper:http:\/\/arxiv.org\/abs\/astro-ph\/9712189 ; since it's from 1997, it's from before the discovery of the 2-solar-mass pulsar. What's up here? Did Smolin move the goalposts? Did he decide that the 2-solar- mass limit was fuzzy rather than sharp? Was the mass of this neutron star found to be a mistake? Are the error bars on its mass too big to allow it to falsify CNS? The article also describes Penrose's CCC as alive and kicking, whereas in fact I think it's clearly been dead for several years. Can anyone edit my tag"}
{"id":"28994","title":"What is the meaning of pressure of a solid?","text":"Currently I'm taking an introductory course on thermodynamics. I've got a problem with understanding what is the meaning of pressure of a solid body. The question arose when I looked at phase diagram in P, T coordinates. When we're talking about pressure of gas or liquid, it can be defined thus: it is the force per unit area with which the molecules of the substance hit the surface. This force is due to the molecular motion. But in the case of solids, molecules don't \"move\", they just oscillate about their fixed positions. Therefore, they don't hit (in usual meaning) a testing surface and we have problems with pressure's definition. How then the pressure can be defined?"}
{"id":"118638","title":"Problem about thermodynamic relations","text":"We know that for constant pressure thermodynamic processes, $dH=dQ_p$. My question is, does it implies that only reversible work is possible in this processes so that $dw=0$ because $dv$ is zero? In addition, does $Q_v$ necessarily be reversible heat transfer in this case? What if the heat transfer is irreversible? Similar question for $dU=dQ_v$, does the process need to be reversible? Another question is, do the above relations have anything to do with whether or not the system is an ideal gas? I have heard from a lecture that $du≠dQ_v$ in Joule's free expansion for non-ideal gas. Consider reversibility and whether it's ideal gas there are four combinations of situations (ideal gas reversible,ideal gas irreversible, non-ideal gas reversible, non-ideal gas irreversible). I get confused with how these factors affect the thermodynamic relations."}
{"id":"51269","title":"Showing that the Ricci scalar equals a product of commutators","text":"I have to compute the square of the Dirac operator, $D=\\gamma^a e^\\mu_a D_\\mu$ , in curved space time ($D_\\mu\\Psi=\\partial_\\mu \\Psi + A_\\mu ^{ab}\\Sigma_{ab}$ is the covariant derivative of the spinor field and $\\Sigma_{ab}$ the Lorentz generators involving gamma matrices). Dirac equation for the massless fermion is $\\gamma^a e^\\mu_a D_\\mu \\Psi=0$. In particular I have to show that Dirac spinors obey the following equation: $$(-D_\\mu D^\\mu + \\frac{1}{4}R)\\Psi=0 \\qquad (1)$$ where R is (I guess) the Ricci scalar. Appling to the Dirac eq, the operator $\\gamma^\\nu D_\\nu$ and decomposing the product $\\gamma^\\mu \\gamma^\\nu$ in symmetric and antisymmetric part I found: $$D_\\mu D^\\mu\\Psi + \\frac{1}{4}[\\gamma^\\mu,\\gamma^\\nu][D_\\mu, D_\\nu]\\Psi=0$$Now I have troubles to show that this last object is related with the Ricci scalar. Can somebody help me or suggest me the right way to solve Eq. $(1)$?"}
{"id":"51262","title":"Are the speeds of the different wavelengths of visible light different or varying in a medium such as air?","text":"Are the speeds of the different wavelengths of visible light different or varying in a medium such as air? If so, please inform by how much? Also, even if the wavelength speeds vary minimally, please inform."}
{"id":"25061","title":"In what ways can a lunar eclipse occur?","text":"In what ways can a lunar eclipse occur? Also, on what percentage of the Earth are they usually viewable? I am aware that there are multiple configurations that constitute a lunar eclipse (umbral, penumbral, partial) and would like more information about each and how they occur."}
{"id":"103948","title":"Power dissipated through resistor in RC circuit with charged Capacitor","text":"so here's the problem: A simple RC circuit where the capacitor has been charged, and two resistors are in parallel configuration. How does one find the power (as function of time) through any of the two resistors? ![enter image description here](http:\/\/i.stack.imgur.com\/mOGjJ.jpg)"}
{"id":"98312","title":"Ensuring globally hyperbolic geodesically-complete spacetimes","text":"Let's say we have an incomplete spacetime A that is globally hyperbolic, does there necessary exist a globally hyperbolic completion? My guess is no, in which case what further restrictions can be placed on A to ensure that it can always be extended to geodesically-complete globally hyperbolic spacetime?"}
{"id":"98316","title":"Frequency of rotating coil","text":"Given a coil initially in the x-y plane, rotating at angular frequency $ \\omega $ about the x-axis in a magnetic field in the z-direction. This uniform time varying magnetic field is given by $B_z (t)=B(0)cos(\\omega t) $ I am required to show that there is a voltage of frequency $2\\omega $ across the loop. Clearly when t=0 the flux is at a maximum, but I dont understand how to relate to the frequency? If the frequency is just the inverse of the period then $f=\\omega \/ 2\\pi $ ? Clearly I am not understanding something. How does the voltage affect the frequency?"}
{"id":"51847","title":"Why don't tuning forks have three prongs?","text":"I was reading Why tuning forks have two prongs?. The top answer said the reason was to reduce oscillation through the hand holding the other prong. So if having 2 prongs will reduce oscillation loss, surely a 3-pronged tuning fork would be even more efficient. Why don't you see more 3-pronged tuning forks?"}
{"id":"16259","title":"Does thermodynamic processess affect phase transition in solid state metals","text":"I read an article about phase transitions and I read about thermodynamic processes such as Adiabatic process, Isochoric process, Isobaric process, Isothermal process Do these processes affect phase transition in solid state metals? The reason I ask is because when these thermodynamic processes are being defined or explained, a fluid is used to explain the concept behind them. Take for example when a solid state metal is in use (say semiconductor chip) a lot of heat is evolved or given out, if the heat is too much, and the metal is considerable small, it might melt, thus going from solid of a liquid form. How are these thermodynamic processes modeled? Is there any reference that you can provide?"}
{"id":"116785","title":"Deriving Feynman rules from Renormalized Lagrangian","text":"In the context of Renormalized Pertubation Theory Peskin Schröder says:   The Lagrangian $$ \\mathcal{L}=\\frac{1}{2} (\\partial_\\mu\\phi_r)^2-\\frac{1}{2}m^2\\phi_r^2-\\frac{\\lambda}{4!}\\phi_r^4 + \\frac{1}{2} \\delta_Z(\\partial_\\mu\\phi_r)^2 -\\frac{1}{2}\\delta_m^2\\phi_r^2-\\frac{\\delta_\\lambda}{4!}\\phi_r^4 $$ gives the following set of Feynman rules:   \\------------>------------ = $\\frac{i}{p^2-m^2+i\\epsilon}$   \\------------X------------ = $i(p^2\\delta_Z-\\delta_m)$   and the two 4-vertices.      The question is: Why look the Feynman rules for the first and the fourth term of the Lagrangian look so different? I believe the answer is connected to the fact that one has to bring the kinetic term of the Lagrangian to its canonical form $\\frac{1}{2} (\\partial_\\mu\\phi_r)^2$ and has to interpret everything else as (possibly momentum dependent) vertices. How does this look in formulae?"}
{"id":"65504","title":"Collision between a photon and an electron","text":"Looking through this AP Physics question, I was struck by how the 'collision' between a photon and electron looks so much like a macroscopic collision. Is this even physically possible? Look at the last page of this pdf: http:\/\/lodischool.tripod.com\/dovesol\/DOVE02SOL.pdf EDIT: Some more questions: How could a photon _collide_ with an electron, when their positions cannot be determined exactly? Also considering how very small the electron is, I doubt that it is even possible to make the two collide; and if it is, how could you possible detect that? It also seems as though the photon and electron are acting as particles, which seems to me not to be the whole story. What if I put the electrons behind a double slit apparatus, and treat individual photons as particles? Based on this \"compton scattering,\" it's possible for the photon to be deflected any which way. I could claim that the diffraction pattern observed in the double-slit experiment is due to compton scattering, among other factors. Prove me wrong!"}
{"id":"64457","title":"Distribution of pressure inside a capsule","text":"How would pressure of an ideal gas be distributed over the inside of a capsule (a cylinder with semi-spheres on the ends)? What about the strain on the material? Is there a general formula for how much?"}
{"id":"64459","title":"Fermi level for the bulk of topological insulator","text":"![enter image description here](http:\/\/i.stack.imgur.com\/qJfUU.png) \"Fermi level\" is the term used to describe the top of the collection of electron energy levels at absolute zero temperature. Why does the Fermi level for the bulk of topological insulator fall within the bulk band gap? There is no energy level in the band gap."}
{"id":"81758","title":"Confusion about the probability cloud","text":"What is the meaning of the electron probability cloud? I understood it to mean that the electron has a probability to be found in a certain postion before measurement, but now after reading experiments involving Schrödinger's cat type states (with bonding and antibonding gaps in $\\mathrm{H_2}$ and molecule and squid measurements where the wavefunction interferes and separates out the bonding band from the antibonding band) and that this physical effect of the supercurrent moving in both directions across the squid junction proves the electron is really everywhere in the cloud, smeared out and not just a probability cloud, which interpretation is right? Does the above experiement prove the electron is everywhere and smeared out in the wavefunction?"}
{"id":"107789","title":"Finding Time & Required Angle for Projectile Motion with Quadratic Air Resistance","text":"This is a physics\/calculus\/computer science problem, but I think I'll get better results in the Physics SE. I have a fun little project I've been working on (hobby, not homework\/production), that models the flight of a projectile in various environments (2 Dimensional). Everything is nice and shiny with gravity and no air resistance. But of course, I've run into the classical Quadratic Air Resistance problem. Now I have no problem with just accurately calculating various position: I have a 4th Order Runge-Kutta Method which works just fine for that. What I would like is, given an (x,y) coordinate pair, find the Required Angle needed to hit that coordinate, as well as the time needed to get there. Currently I'm working on an algorithm that starts by finding two angles, one which overshoots, and another that undershoots, then using a binary search to narrow the range between them by plotting using the Runge-Kutta method, and increasing the stepsize until I've reached an acceptable accuracy. Unfortunately I fear this will be much slower than I like, not to mention that one angle might overshoot one time, then when I increase the stepsize, could undershoot it, which would just make things crazier than I think they should be. I've googled this quite a bit, but sadly couldn't find anything that, you know, _helps_. If you guys know of any algorithms that might be helpful, I would greatly appreciate it. Even better would be a way to rework the Differential Equations to simplify the search, but sadly I am no where near good enough at Calculus for that to be a viable option. Thanks, Jacob"}
{"id":"22871","title":"All matter has a mass but does all matter have a gravitational pull?","text":"_I know that all planets and stars have a gravitational pull but does a simple much smaller object have a gravitational pull for example a pebble?_"}
{"id":"130904","title":"Does the mass distribution matter in (Schwarzschild) black holes?","text":"Is it possible that from the same initial mass different black hole radius will be created due to different mass distribution during black hole creation? If mass is concentrated more on the outside bigger event horizon will be created? If mass is concentrated more in the center smaller black hole will be created?"}
{"id":"94152","title":"Rubbing a rod with silk?","text":"> Rubbing a glass rod with silk causes charges to be exchanged and > consequently both objects get charged. Why do the objects have to be \"rubbed\"? I get that one has a stronger pull on the electrons than the other, but shouldn't just allowing the objects to make contact be enough.? I would appreciate a \"visualization\" of whats happening. Similar questions: Why two objects get charged by rubbing? How does rubbing cause the transfer of electrons from one object to the other? **Neither question addresses why the objects need to be rubbed instead of just making contact.**"}
{"id":"98863","title":"What's the dimensionality of a solid angle?","text":"I haven't seen this explained clearly anywhere. Solid angles are described usually as a fraction of the surface area of a unit sphere, similar to how angles are the fraction of the circumference of a unit circle. However, I don't know how solid angles are actually quantified. Are solid angles just a single number, the describes this fraction of the area? It's confusing to me since often times, I've seen integrals that integrate over a sphere using solid angles, which seems to imply that solid angles are multi-dimensional quantities (e.g. when integrating using spherical coordinates, the solid angle would have to consist of the azimuthal and polar angles covered by the differential solid angle). Following from this, how would you write down a solid angle that covers the entire surface of a unit sphere?"}
{"id":"110790","title":"How do the single photon energy and em-signal energy correlate?","text":"If the photon (as a quantum of the electromagnetic field) has no defined(?) amplitude, how does (or where from?) the electromagnetic wave's amplitude appear? The formulation of the question is not quite precise, but the main idea is if we cannot define the single photon's amplitude, since $E=hf$ only (are there any other equations?), how can we talk about an amplitude of an EM wave, which is carried via photons (EM quanta)?"}
{"id":"90644","title":"Derivation of Noether's theorem - A problem with physical significance","text":"My question is about the field theoretic version of Noether's theorem. I am deeply troubled by one of the hypotheses of the theorem. As it is the standard textbook for Lagrange mechanics, I'll follow Goldstein's account (starting p. 588 in the second edition of \"Classical Mechanics\"). I have no problem with condition 1 since I work in Minkowski space. I am completely okay with condition 2, which amounts to asking that the equations of motion be the same for two observers who use different systems of coordinates to describe the same spacetime and different functions to describe the same fields. However, I can't make any sense of condition 3. I don't see what its physical meaning can be. I haven't seen it explained convincingly anywhere, and can't seem to figure it out for myself. For those who don't have any access to Goldstein's book but feel they might be able to help, condition 3 is the requirement that the action integrals be equal for the two aforementioned observers. I hope someone has some fantastic insight on this! :-)"}
{"id":"58094","title":"Motion of mercury","text":"I studied that mercury motion around the sun slightly displace by a certain value in each year. But, this is not predicted by kepler until general theory of relativity. What does general theory does with mercury."}
{"id":"26408","title":"What did general relativity clarify about Mercury?","text":"I frequently hear that Kepler, using his equations of orbital motion, could predict the orbits of all the planets to a high degree of accuracy -- _except Mercury_. I've heard that mercury's motion couldn't be properly predicted until general relativity came around. But what does general relativity have to do with Mercury's orbit?"}
{"id":"57868","title":"General Relativity & Kepler's law","text":"According to Kepler's law of planetary motion, the earth revolves around the sun in an elliptical path with sun at one of its focus. However, according to general theory of relativity, the earth revolves around the sun in the curved space and it revolves in a circular path. Which is correct? Does the planet moves in a circular path or an elliptical path?"}
{"id":"2685","title":"What is information?","text":"We're all familiar with basic tenets such as \"information cannot be transmitted faster than light\" and ideas such as information conservation in scenarios like Hawking radiation (and in general, obviously). The Holographic Principle says, loosely, that information about a volume of space is encoded on its two-dimensional surface in Planck-sized bits. In all these contexts, I can take \"information\" to mean predictive or postdictive capability, i.e. information is what enables us to state what the outcome of a measurement was or will be (locally). But what _is_ information, exactly? Do we have any kind of microscopic description of it? Is it just a concept and, if so, how can we talk about transmitting it? I suspect this is probably as unanswerable as what constitutes an observer\/measurement for wave function collapse, but I'd love to know if we have any formulation of what information is made of, so to speak. If I'm talking nonsense, as I suspect I may be, feel free to point this out."}
{"id":"14785","title":"2d soft body physics mathematics","text":"> **Possible Duplicates:**   >  Modern references for continuum mechanics   >  Good books on elasticity The definition of **rigid body** in Box2d is > A chunk of matter that is so strong that the distance between any two bits > of matter on the chunk is completely constant. And this is exactly what i don't want as i would like to make **_2D_** (maybe 3D eventually), elastic, deformable, breakable, and even sticky bodies. What I'm hoping to get out of this community are resources that teach me the math behind how objects bend, break and interact. I don't care about the molecular or chemical properties of these objects, and often this is all I find when I try to search for **how to calculate what a piece of wood, metal, rubber, goo, liquid, organic material, etc. might look like after a force is applied to it.** Also, I'm a very visual person, so diagrams and such are EXTREMELY HELPFUL for me."}
{"id":"120071","title":"Does time really exist?","text":"Does time really exist? or is it a human invention and convention? What is the truth? are there time particles? please remember, I'm looking for constructive answers, not condescending and void ones."}
{"id":"131810","title":"What makes us move in time?","text":"Time is considered to be a dimension, and we are moving at certain rate in one direction in time. **What force makes us move in time?** I mean it must be ether time moving or us moving in time so there has to be some force that 'pushes'\/'pulls'? Was this 'time inertia' acquired during big bang, or nothing is moving and I am just being silly?"}
{"id":"16048","title":"What are quarks made of?","text":"So atoms are formed from protons and neutrons, which are formed from quarks. But where do these quarks come from? What makes them?"}
{"id":"64126","title":"What is the smallest existing thing in theory and law?","text":"What is the smallest existing thing in theory and law?"}
{"id":"17421","title":"Are quarks and leptons actually fundamental particles?","text":"> **Possible Duplicate:**   >  What are quarks made of? Are quarks and leptons actually fundamental, or are they made up of even more fundamental particles? And is it true that many consider quarks and leptons are so small that they may be thought of as geometrical points in space with no spatial extension at all?"}
{"id":"68288","title":"Non-dimensionalization for spatially varying material parameters","text":"For a homogeneous material of length $L$, we can write the heat equation as $$\\rho c\\frac{dT}{dt}=k\\frac{du^2}{dx^2}, \\text{ } x\\in (O,L)$$ where $T$ is the temperature, $\\rho$ is the thermal conductivity, $c$ is specific heat, and $k$ is the thermal conductivity. To non-dimensionalize it, I know that I can define a new variable for space $\\hat{x}=\\frac{x}{L}$, time $\\hat{t}=\\frac{kt}{L^2\\rho c}$ and temperature $\\hat{T}=\\frac{T}{T_0}$ where $T_0$ is the initial temperature, using the chain rule, obtain the simplified form $$\\frac{d\\hat{T}}{dt}=\\frac{du^2}{dx^2}.$$ When the material is heterogeneous, we often assume that the thermal conductivity varies in space and is modeled by the function $k=k(x)$. When this is the case, I'm not quite sure what is the standard approach non- dimensionalization. I conjecture that the process should still be similar even though $k(x)$ is not a constant. Should I choose a reference thermal conductivity $k$ from the range of values $k(x), x\\in (0,L)$ follow the same proceedure? Is there an alternative methodology to non-dimensionalize in the case of spatially varying material parameters?"}
{"id":"24369","title":"Angular Momentum Addition Theorem - Sanity Check","text":"Looking back at my quantum mechanics notes, the angular momentum addition theorem is listed as: $j=j_1+j_2,j_1+j_2-1, ..., |j_1-j_2| $ (Using conventional notation) , but I'm a little unsure how to interpret the introduction of the modulus operation ($|...|$) and couldn't easily find any examples. I'm assuming you apply the modulus to any expression which would otherwise yield a negative value for $j$? I'd appreciate a nod from someone in the know :-)."}
{"id":"132343","title":"Prove that a derivative with respect to a covariant 4-vector is a contravariant vector operator","text":"In special relativity, I know you can prove that the derivative with respect to a contravariant 4-vector component transforms like a covariant vector operator by using the chain rule, but I can't work out how to prove the inverse, that the derivative with respect to a covariant 4-vector component transforms like a contravariant vector operator."}
{"id":"79395","title":"Work Done On a Body When It is Not in Contact with the agent of the Force","text":"_Is it necessary for work to be done on a body that the agent of the force remains in contact with the body?_ For example, if I hit a football with my foot with a small amount of force and it moves a certain distance on the ground, then what could we say about the 'work' in this case if there would be no friction and the air resistance to stop the ball?"}
{"id":"91064","title":"Improvement of microscope resolution with Oil","text":"Why and when do we need to place oil over the sample to achieve higher optical resolution ? Is this idea is valid for the enhancement of all optical microscopy techniques and magnification scales ?"}
{"id":"27577","title":"Analytic continuation of imaginary time Greens function in the time domain","text":"Consider the imaginary time Greens function of a fermion field $\\Psi(x,τ)$ at zero temperature $$ G^τ = -\\langle \\theta(τ)\\Psi(x,τ)\\Psi^\\dagger(0,0) - \\theta(-τ)\\Psi^\\dagger(0,0)\\Psi(x,τ) \\rangle $$ It is well known that we can obtain the retarded Greens function by performing Fourier transformation into frequency space and performing the analytic continuation $iω \\to ω + i\\eta$. What I would like to do is to perform the analytic continuation directly in the form $iτ \\to t$, but I don't know how to deal with the $\\theta(τ)$ terms. > How to perform the analytic continuation $iτ \\to t$ of the step function > $θ(τ)$? In my case, I am dealing with a chiral Luttinger liquid, giving something like $$ G^τ(x,τ) = -\\left[\\theta(τ)\\frac i{iλ + ivτ - x} - \\theta(-τ)\\frac i{iλ - ivτ - x}\\right] $$ where $λ \\approx 0$ is an infinitesimal but important regularization. Of course, the analytic continuation into the time domain is going to look something like $$ \\frac1{iλ + vt - x} $$ but I'm interested in the precise form. Also, I'm ultimately interested in the spectral function, so I don't mind if analytic continuation gives me yet another variant of a Greens function, but I would like to obtain it precisely from the imaginary time Greens function without going through a tedious Fourier transform. For instance, Giuliani and Vignale's book \"Quantum Theory of the Electron Liquid\" uses the Greens function $G_{>}(x,t)$ to great effect (equation (9.133))."}
{"id":"108570","title":"Photon Angular Momentum","text":"Essentially I am wanting to evaluate $$\\langle j\\, m \\mid a^\\dagger(\\mathbf{k}, \\lambda) \\mid 0 \\rangle \\,,$$ where $\\lambda$ indicates the circular polarization (about $\\mathbf{k}$). We have that $\\mathbf{J}= \\mathbf{L} + \\mathbf{S}$. It's straightforward to show that circular polarization corresponds to definite spin projections along the $\\hat{\\mathbf{k}}$ axis (the direction of propagation). However, I don't yet know how to find the $\\mid \\ell\\, m_\\ell \\rangle$ projections. I don't know how how to express $\\mid \\ell\\, m_\\ell \\rangle$ states in terms of Fock states, for example. With $\\hbar = c = 1$, $$\\mathbf{L} = \\frac{1}{4\\pi}\\int \\mathop{d^3r} \\sum_i E_i\\left(\\mathbf{r} \\times \\boldsymbol{\\nabla} \\right)A_i $$ $$\\mathbf{S} = \\frac{1}{4\\pi}\\int \\mathop{d^3r} \\mathbf{E} \\times \\mathbf{A} = -i \\int \\mathop{d^3k} \\mathbf{a}^\\dagger\\left(\\mathbf{k}\\right) \\times \\mathbf{a}\\left(\\mathbf{k}\\right)\\,. $$ The above uses the following definition $$\\mathbf{a}\\left( \\mathbf{k}\\right) = \\sum_{\\lambda=\\pm1} \\boldsymbol{\\epsilon}_\\lambda\\left( \\mathbf{k}\\right) a_\\lambda\\left( \\mathbf{k}\\right)$$ So, any help determining the angular momentum of a photon would be appreciated (incuding mention of references dealing with this subject). I have been using the Coulomb gauge."}
{"id":"109448","title":"Thermodynamic entropy vs. quantum mechanical entropy","text":"Is there a fundamental difference in the definition of entropy when considering the classical thermodynamic picture vs. the quantum mechanical picture, or are they both fundamentally equivalent?"}
{"id":"109449","title":"Dipole matrix elements through parity argument","text":"I am trying to find the following dipole moment matrix element $(|n,\\ell,m\\rangle)$. $$e\\langle1,0,0|\\vec r|2,0,0\\rangle$$ I believe that I can say this matrix element is zero because of parity. The wavefunctions have parity $(-1)^\\ell$ and seeing as each has $\\ell =0$, they are even parity. Then r is odd, as it sends $\\vec r \\rightarrow-\\vec r$. This means the entire expression is odd, therefore the matrix element is 0. Is my reasoning sound?"}
{"id":"116260","title":"Constant Acceleration Question","text":"A Cessna 150 aircraft has a lift-off speed of approximately 125 $kmh^{-1}$. What minimum constant acceleration does this require if the aircraft is to be airborne after a take-off run of 129 m? So I wanted to use the formula that $x=\\cfrac{(v_{final}^2 - v_{initial}^2)}{2a}$ To solve for the constant acceleration but I don't have a final velocity from this problem? My initial velocity is 125 $kmh^{-1}$ and my $x$ (displacement) would be 129 meters but how do I find the constant acceleration?"}
{"id":"29100","title":"Equivalent Rotation using Baker-Campbell-Hausdorff relation","text":"Is there a way in which one can use the BCH relation to find the equivalent angle and the axis for two rotations? I am aware that one can do it in a precise way using Euler Angles but I was wondering whether we can use just the algebra of the rotation group to perform the same computation?"}
{"id":"29104","title":"Why $\\frac{d}{dt}r_{a}\\nabla_{a}U_{ab}+\\frac{d}{dt}r_{b}\\nabla_{b}U_{ba}=\\frac{d}{dt}U_{ab}?$","text":"In classical mechanics for two mass particles $a$,$b$ we assume the symmetric potential arising from $F_{ab}$ and $F_{ab}$ given by $$U_{ab}(r)=-\\int^{r}_{r_{0}}F_{ab}(r')dr'$$ and $$U_{ba}(r)=-\\int^{r}_{r_{0}}F_{ba}(r')dr'$$ The book **mechanics** by _Florian Scheck_ gives $$\\frac{d}{dt}r_{a}\\nabla_{a}U_{ab}+\\frac{d}{dt}r_{b}\\nabla_{b}U_{ba}=\\frac{d}{dt}U_{ab}$$ because $$[\\frac{d}{dt}r_{a}\\nabla_{a}+\\frac{d}{dt}r_{b}\\nabla_{b}]U_{ab}=\\frac{d}{dt}U_{ab}$$ I am confused how we get the summation form. My derivation goes as follows: Notice $F_{ab}=-\\nabla_{b} U_{ab}$. Thus we should have $$\\frac{d}{dt}U_{ab}=\\frac{d}{dr}*\\frac{dr}{dt}U_{ab}=\\frac{dr}{dt}\\frac{d}{dr}U_{ab}=\\frac{dr}{dt}[-F_{ab}]=\\frac{d}{dt}[r_{a}-r_{b}][-F_{ab}]=[\\frac{d}{dt}r_{a}\\nabla_{a}+\\frac{d}{dt}r_{b}\\nabla_{b}]U_{ab}$$ My simple question is just whether my derviation is correct, for I assume $r=r_{a}-r_{b}$ at here."}
{"id":"79539","title":"Is imperative magnetic flux the capacity of individual atoms, or the constituent valency of chemically-bonded molecules within a vacuum?","text":"If magnetic energy depends on the electron poles within two-fields within a permanent magnets void, how do invidiual atoms react within the attraction or repulsion of poles, and what incurrence does this live by in regards to molecular struction?"}
{"id":"109199","title":"Why is $\\vert I=1,I_3=1\\rangle = -p\\bar n$","text":"My book doesn't explain well how to build a doublet of antiparticles that transforms the same way the particle doublet $(p,n)^T$ (proton neutron) does. They claim $$\\tag 1 \\vert I=1,I_3=1\\rangle = -p\\bar n$$ for a composite nucleon-antinucleon system. **Why is $(1)$ true?** Perhaps it's just bad notation in the book? I got confused because the Clebsch-Gordan for $(1)$ comes with a $+1$ and not a $-1$, but perhaps one should insclude this negative sign into the CG coeff? That is $(1)$ should be $$\\tag 2 \\vert I =1,I_3=1\\rangle =\\underbrace{\\vert 1\/2,1\/2\\rangle}_{p}~\\Big(\\underbrace{-\\vert 1\/2,1\/2\\rangle}_{-\\bar n}\\Big)?$$"}
{"id":"34557","title":"Efficiently distinguishing mixed quantum states?","text":"Assume we know two different mixed states, p and q, and an efficient (quantum) algorithm for creating such two. Does it follow that there exists a computationally efficient method\/measurement for optimally(that is, according to the trace norm distance) distinguishing between the two?"}
{"id":"109196","title":"How could there be a truly \"pure\" state?","text":"If the Universe did start from a single point, then wouldn't all particles be fundamentally entangled? How then could there be a truly \"pure\" state?"}
{"id":"14722","title":"Behaviour of liquid in vaccume","text":"Is it possible for a liquid to exist in a high quality vacuume? For example, a few Torr. If so what are the methods for doing this?"}
{"id":"100373","title":"What is \"thermal undulation\" in the context of lipid bilayers?","text":"What is thermal undulation in the context of lipid bilayers? Is it another word for \"thermal fluctuation\"?"}
{"id":"92087","title":"How does this formula for calculating the \"mass sum\" in a collision translate to 3D?","text":"According this tutorial, formula number 5: $$j = \\frac{-(1 + e)((V^{A} - V^{B}) * t)}{\\frac{1}{mass^{A}} + \\frac{1}{mass^{B}}}$$ translates into formula number 6: $$j = \\frac{-(1 + e)((V^{A} - V^{B}) * t)}{\\frac{1}{mass^{A}} + \\frac{1}{mass^{B}} + \\frac{(r^{A} \\times t)^{2}}{I^{A}} + \\frac{(r^{B} \\times t)^{2}}{I^{B}}}$$ when dealing with oriented bodies. $j$ is a scalar by which you divide normal and friction impulses to simulate them. This is only valid for the 2D case, though, where the inertia is a scalar. What if I have the inertia as a tensor (more specifically a 3x3 matrix)?"}
{"id":"77285","title":"Gauss's \/ Divergence theorem in Classical electrodynamics for the Electric field","text":"Can somebody explain the proof of Gauss's theorem \/ divergence theorem taking the vector as electric field $$\\iiint(\\nabla\\cdot\\vec E)\\mbox{ d} V=\\iint \\vec E \\cdot\\hat{n} \\mbox{ d} A?$$"}
{"id":"66764","title":"Bass and Treble-Car Steroes","text":"In a car which phenomenon, diffraction or the resonant frequency of the car, lends itself more to the ability of bass to go farther? Related Answer: Why do bass tones travel through walls?"}
{"id":"24709","title":"The discontinuity of Electric Field","text":"''electric field always undergoes a discontinuity when you cross a surface charge $\\sigma$'' GRIFFITHS In the derivation; Suppose we draw a wafer-thin Gaussian Pillbox, extendind just barely over the edge in each direction. Gauss law states that: $\\int_{S} E \\cdot A = Q_{enc}\/ \\epsilon $ and so $E_{above}^{perp} - E_{below}^{perp} = \\sigma\/ \\epsilon $ My question is why not $2A$ ? $\\int_{S} E \\cdot A = 2EA$ because the top area of pillbox and the bottom area of pillbox, just as because the 2 parts of the flux... SO.. WHY NOT : $E_{above}^{perp} - E_{below}^{perp} = \\sigma\/ 2\\epsilon $ And why there is tangencial component of electric field; not just perpendicular to the surface, which can be seen as flat just looking very close to the surface."}
{"id":"86214","title":"Quantum Theory as a framework for other theories of nature","text":"We know that Quantum Theory should be considered as a framework in which all other theories\/forces (Strong, Weak, EM and Gravity) exist. For example, we have the Quantum Chromodynamics, Quantum Flavordynamics (Electroweak), Quantum Electrodynamics (but still no Quantum GR). When I think about this, it then strikes me why gravity, and specifically special relativity, is part of the framework itself (because QFT is based on QM and SR). Why would a theory like SR (and maybe in future GR) be part of the framework? This looks like a circular logic. I would appreciate it if someone can explain? My understanding is that gravity is different because it exist everywhere, and are not a result of a charge (electric, color or flavor). That's why we cannot speak about a quantum theory of EM without considering gravity, but we can do it without considering strong or weak force. Is this correct? EDIT: it would make more sense to me to see Gravity\/GR considered only as a framework (a geometric one) rather than a force, or it can be considered a fictitious force, as with the centrifugal force. see first paragraph of first answer here."}
{"id":"110947","title":"Upward bouyant forces?","text":"When an object is submerged in a fluid (e.g., water), there is a pressure on the object due to weight of water by p=mgh, but why is there an upward buoyant force?"}
{"id":"22018","title":"Does existence of magnetic monopole break covariant form of Maxwell’s equations for potentials?","text":"Absence of magnetic charges is reflected in one of Maxwell's fundamental equations: $$\\operatorname{div} \\vec B = 0 \\text{ (1).}$$ This equation allows us to introducte concept of vector potential: $$\\vec B = \\operatorname{rot} \\vec A.$$ Using this concept, it is possible to express Maxwell's equations in a graceful symmetric form: $$\\nabla^2 \\vec A - \\frac{1}{c^2}\\frac{\\partial^2 \\vec A}{\\partial t^2} = = - \\frac{\\vec j}{\\epsilon_0 c^2} \\text{ (2)}$$ $$ \\nabla^2 \\phi -\\frac{1}{c^2}\\frac{\\partial^2 \\phi}{\\partial t^2} = - \\frac{\\rho}{\\epsilon_0} \\text{ (3)} $$ Noticing, that vector $\\vec A$ and scalar $\\phi$ potentials, as well as electric current density $\\vec j$ and charge density $\\rho$, form a 4-vector in Minkovsky space-time. Therefore, Maxwell's equations can be expressed in a covariant form, using dalambertian: $$\\nabla_{\\mu}\\nabla^{\\mu} A_{\\nu} = \\frac{j_{\\nu}}{\\epsilon_0} \\text{ (4)}$$. If magnetic monopols exist, Maxwell's equation (1) will look as: $$\\operatorname{div} \\vec B = \\mu_0 c \\rho_{magnet}$$ As the divergence of $\\vec{B}$ isn't equal to zero, it impossible to introduct concept of vector potential. Thus, the equation in the form of (4) will not be possible to express."}
{"id":"73073","title":"Atoms' excitation energies as derived from Frank-Hertz experiment data","text":"I'm analyzing the experimental data obtained during Frank-Hertz experiment (conducted with Hg atoms): ![enter image description here](http:\/\/i.stack.imgur.com\/D3K82.jpg) Accelerating voltage values were multiplied by 0.1 during measurement (i.e. the mean value of energies differences if not 0.508 eV, but 5.08 eV). The output voltage was measured on resistive load of the anode, therefore it is proportional to anode's current. The first five minimas in output voltage (top graph) have constant voltage difference - this value is used to calculate the first excitation energy of Hg atom (bottom graph). The sixth minima, however, can be seen to have larger increase in output voltage(top graph - the first five voltage peaks may be fitted by a linear graph, whereas the sixth is no longer fits to this graph) and a higher voltage offset from the previous minima (bottom graph). **Question 1:** Is the sixth minima corresponds to the higher order excitation of Hg, or ionization, or else? **Question 2:** The curve is clamped at $V_{out} = 5V$. Is there a physical effect which causes this, or it is just the limit of measurement equipment, or else?"}
{"id":"73070","title":"How to work out momentum when there are velocity and mass changes","text":"I have a pretty simple homework question, but I can't rap my head around it. In the question a swimmer of $55 \\mbox{ } \\mathrm{kg}$, jumps off a stationary raft of $210\\mbox{ }\\mathrm{kg} $. The swimmer jumps off the raft with a speed of $4.6 \\mbox{ } \\mathrm{ms}^{-1} $. I need to work out the recoil velocity of the raft. So because Momentum before = Momentum after, I went: $p_i = 0$ Therefore $0 = p_f$ and $p_f = mv$, $m = 210\\mbox{ }\\mathrm{ } $, so the $v$ would have to equal $0$. Making the recoil velocity equal $0$. However that doesn't seem right. Could use some clarification or help, thanks."}
{"id":"31909","title":"Why is Titan able to maintain an atmosphere, and not Luna?","text":"I just read stability of hypothetical lunar atmosphere. From the correct answer, i understand, the low escape velocity from Luna is part of the reason it is unable to retain an atmosphere. Titan apparently has a comparable escape velocity   * Titan = 2.65 km\/sec    * Luna = 2.4 km\/sec ; yet Titan maintains an atmosphere. Why? What have I missed? Does Luna's relative proximity to Sol make the difference?"}
{"id":"31907","title":"Is a photovoltaic cell on an artificial satellite the same construction as one used on Earth's surface?","text":"Are photovoltaic cell arrays on a satellite the same that are used within Earth, or is there some difference in their construction given the differing environment in which they are to operate? Does the efficiency\/life of a photovoltaic cell change depending upon whether it is within\/without Earth's atmosphere?"}
{"id":"93686","title":"Explain Turbulence","text":"I'm a high school student. I still don't understand what turbulence is. Please can you explain what it really is? This is what I think it is: rotating motion of water when a particle travels at a velocity of $V$."}
{"id":"74488","title":"Consistency of equation with special relativity?","text":"The following is the equation which, I want to know, if it is valid in relativistic domain. Consider two equal charges moving in same direction with velocity $v$ and charge $q$ at a separation of $d$. The magnetic force acting between them is $(1\/4\\pi\\epsilon_0 c^2)v^2q^2\/d^2$ and the electrostatic force is $(1\/4\\pi\\epsilon_0 )q^2\/d^2$.the ratio of the magnetic (attractive) force and the electrostatic (repelling) force is $v^2\/c^2$ and therefore the conclusion provided was that two individual charges with same (or any) velocity can never attract. Is this equation valid relativistically. Moreover, since magnetic field is just the relativistic counterpart of electrostatic force, can we find a frame of reference wherein in the case stated, all the force is magnetic and must be attractive? I am comfortable only with basic special relativistic mechanics but an intuitive understanding would be better."}
{"id":"98456","title":"Do we have 2 minutes of extra morning?","text":"my physics teacher told me about the refraction and its applications one of them was   **2 minutes of early sunrise** and after she explained this effect she concluded that days are   **2 minutes longer** than one would naively presume. However, I think that **her conclusion is wrong** because   **if sunrise is considered then sunset should also be considered** and according to me **sunsets should be 2 minutes late** therefore **the day time is increased by 4 minutes and not 2 minutes** over the naive calculation. According to me, the situation looks something like this:   ![enter image description here](http:\/\/i.stack.imgur.com\/O1YUG.jpg)   Is this idea of mine correct or not? And if we both are wrong, then what should be the right conclusion and why ?"}
{"id":"100951","title":"What was Feynman's \"much better way of presenting the electrodynamics\" -- which did **not** appear in the Feynman lectures?","text":"Does anyone know what Feynman was referring to in this interview which appears at the beginning of The Feynman Tips on Physics? Note that he is referring to something that did not appear in the Feynman lectures. > I didn't like to do the second year, because I didn't think I had great > ideas about how to present the second year. I felt that I didn't have a good > idea on how to do lectures on electrodynamics. But, you see, in these > challenges that had existed before about lectures, they had challenged me to > explain relativity, challenged me to explain quantum mechanics, challenged > me to explain the relation of mathematics to physics, the conservation of > energy. I answered every challenge. But there was one challenge which nobody > asked, which I had set myself, because I didn't know how to do it. I've > never succeeded yet. Now I think I know how to do it. I haven't done it, but > I'll do it someday. And that is this: How would you explain Maxwell's > equations? How would you explain the laws of electricity and magnetism to a > layman, almost a layman, a very intelligent person, in an hour lecture? How > do you do it? I've never solved it. Okay, so give me two hours of lecture. > But it should be done in an hour of lecture, somehow -- or two hours. > > **Anyhow I've now cooked up a much better way of presenting the > electrodynamics, a much more original and much more powerful way than is in > the book.** But at that time I had no new way, and I complained that I had > nothing extra to contribute for myself. But they said, \"Do it anyway,\" and > they talked me into it, so I did. Did this approach to teaching electrodynamics appear in any of his later writing?"}
{"id":"130256","title":"Rotating a complex number","text":"Let us begin in a two-dimensional Euclidean plane. The vector is e.g. $\\vec{V}(x,y)$ It is often useful – but in this case, it's just a mathematical trick that doesn't make the complex numbers \"fundamental\" – to combine the components into a complex number, $z=x+iy$. The two-dimensional rotations in $SO(2)$ are fully determined by the angle $δ$. And the matrix acting on the vector $\\vec{V}(x,y)$ $M= \\begin{bmatrix}+\\cosδ & −\\sinδ\\\\\\\\+\\sinδ & +\\cosδ\\end{bmatrix}$ may be fully replaced by the complex coefficient $e^{iδ}$ that multiplies our complex coordinate $z$. Instead of $z$, however, we could have dealt with its power $z^p$. The rotation could be described by the complex transformation $z^p→z^{′p},z^{′p}=e^{ipδ}z^p$. In this context, the complex number $e^{ipδ}$ plays the role of the \"transformation matrix\" that rescales the one and only component of our tensor-spinor-whatever, $z^p$. In our overly trivial two-dimensional context, the coefficient or exponent $p$ may be anything you want. But the value $p=1\/2$ may be identified with the spinors in two dimensions. The spinor in two dimensions may be represented as $\\sqrt{z}$ where $z=x+iy$ encodes a vector; so the spinor $z^{1\/2}$ is literally the square root of a vector (translated to a complex number) in this case. So I got all this from a blog that tries to explain spinors which unfortunately I don't remember. I don't understand why the $p$ in $e^{ipδ}$ has to be there since any complex number can be rotated without the need for that coefficient. Is there some rule for rotating complex numbers that are raised to higher powers $p=2,...,n$ or am i missing something?"}
{"id":"95609","title":"Textbook on the Geometry of Special Relativity","text":"I am looking for a textbook that treats the subject of Special Relativity from a geometric point of view, i.e. a textbook that introduces the theory right from the start in terms of 4-vectors and Minkowski tensors, instead of the more traditional \"beginners\" approach. Would anyone have a recommendation for such a textbook ? I already have decent knowledge of the physics and maths of both SR and GR ( including vector and tensor calculus ), but would like to take a step back and expand and broaden my intuition of the geometry underlying SR, as described by 4-vectors and tensors. What I do **_not_** need is another \"and here is the formula for time dilation...\" type of text, of which there are thousands out there, but something much more geometric and in-depth. Thanks in advance."}
{"id":"104014","title":"Direction of expansion of the universe","text":"From what I understand the expansion of the universe has no \"center\". If we're flying through space away from the \"center of the big bang\", there's basically no way to tell. Every two given points in space gets farther away from each other, and we can pick any point as center if we like. I also understand that the speed of light is not relative to the speed of the source emitting the light. If I go on a train in $c\/2$, turn on a flash light pointing forward, the light emitted from the flashlight will still travel at the speed of $c$. Now here's my question: Why can't we set up a sphere with photodetectors with synchronized clocks on the inner walls, turn on a light in the center, record the exact time at which each photodetector detects the light, and compare the times to figure out if the sphere was traveling in some certain direction? I mean if we turn on a lightbulb and light travels with the speed of $c$ in all directions at the same time, my intuition tells me that we should be able to figure out some form of \"reference stand still\". (Tagging this with general relativity because I _suspect_ that it's impossible to set up the experiment the way I like due to relativity.)"}
{"id":"122603","title":"Why is the inner product between divergence-free current$\\vec{J}$, (satisfied $\\nabla\\cdot\\vec{J}=0$) and a gradient field$\\nabla \\varphi$ zero?","text":"I read a statement saying that the inner product between divergence-free current and a gradient field is zero. Divergence-free surface current is $\\nabla\\cdot\\vec{J}=0$, and $\\vec{J}$ could be represented as $\\vec{J}=\\nabla\\times(\\psi\\hat{n})$, where $\\hat{n}$ is the normal vector of the surface. So the statement becomes: $\\nabla\\times(\\psi\\hat{n})\\cdot \\nabla \\varphi=0$. I think according to the identity: $$\\nabla\\cdot(\\vec{A}\\times\\vec{B})=\\vec{B}\\cdot(\\nabla\\times\\vec{A})-\\vec{A}\\cdot(\\nabla\\times\\vec{B})$$ we have $$\\nabla\\times(\\psi\\hat{n})\\cdot \\nabla \\varphi=\\nabla\\cdot(\\psi\\hat{n}\\times\\nabla\\varphi)+\\psi\\hat{n}\\cdot\\nabla\\times\\nabla\\varphi=\\nabla\\cdot(\\psi\\hat{n}\\times\\nabla\\varphi),$$ but what next? **Update** Thank you Luboš Motl. I suppose I now understand why, but I have no enough points to reply below, so just update here my answer. Target is to prove $\\int_s \\vec{J}\\cdot\\nabla\\varphi ds=0$ The whole process is as follows: First, $\\vec{J}$cannot go across the surface edge, so $\\vec{J}\\cdot\\hat{t}=0$, where $\\hat{l}$ is the surface edge direction and $\\hat{t}=\\hat{l}\\times\\hat{n}$ is the edge out direction Second, according to the identity $\\nabla\\cdot(\\vec{A}\\times\\vec{B})=\\vec{B}\\cdot(\\nabla\\times\\vec{A})-\\vec{A}\\cdot(\\nabla\\times\\vec{B})$, we have $\\vec{J}\\cdot\\nabla\\varphi=\\nabla\\times(\\psi\\hat{n})\\cdot \\nabla \\varphi=\\nabla\\cdot(\\psi\\hat{n}\\times\\nabla\\varphi)+\\psi\\hat{n}\\cdot\\nabla\\times\\nabla\\varphi=\\nabla\\cdot(\\psi\\hat{n}\\times\\nabla\\varphi)$ since $\\nabla\\times(f\\vec{A})=\\nabla{f}\\times\\vec{A}+f(\\nabla\\times A)$ $\\psi\\hat{n}\\times\\nabla\\varphi=-\\nabla\\times(\\varphi\\psi\\hat{n})+\\varphi\\nabla\\times(\\psi\\hat{n})$ Then $\\nabla\\cdot(\\psi\\hat{n}\\times\\nabla\\varphi)=\\nabla\\cdot(-\\nabla\\times(\\varphi\\psi\\hat{n})+\\varphi\\nabla\\times(\\psi\\hat{n}))=\\nabla\\cdot(\\varphi\\nabla\\times(\\psi\\hat{n}))$ finally, $\\int_s \\vec{J}\\cdot\\nabla\\varphi ds=\\int_s\\nabla\\times(\\psi\\hat{n})\\cdot \\nabla \\varphi ds =\\int_s \\nabla\\cdot(\\varphi\\nabla\\times(\\psi\\hat{n}))ds=\\oint_l \\varphi\\nabla\\times(\\psi\\hat{n})\\cdot\\hat{t}dl=\\oint_l \\varphi\\vec{J}\\cdot\\hat{t}dl=0$ I think here the important things are:   1. Generally speaking, divergence-free current usually can be expressed as $\\vec{J}=\\nabla\\times\\vec{T}$, and $\\vec{J}=\\nabla\\times(\\psi\\hat{n})$ is specially for surface current.   2. the $\\hat{n}$ is only valid on the surface(there is no meaning of $\\hat{n}$ for point in side of a body). the integral is on the surface rather than on the body. According to the original article, it is just talking about PEC and surface current."}
{"id":"74226","title":"Could spontaneous symmetry breaking happen again in our universe?","text":"It is generally believed that $10^{-35}$ seconds after the Big Bang, the symmetry of a GUT was broken and after $10^{-12}$ seconds the electroweak force was broken: \\begin{equation} \\mathrm{SU(2)} \\times \\mathrm{U(1)} \\rightarrow \\mathrm{U(1)} \\end{equation} This symmetry breaking is a result of the universe cooling down and undergoing a phase transition. I'm aware that the temperature of the universe it about $2.7$ Kelvin, so the temperature of the universe cannot decrease much more, but I was wondering if there is a chance that another phase transition might happen again in the future?"}
{"id":"38945","title":"Good introductory books on AdS\/CFT correspondence","text":"> **Possible Duplicate:**   >  Introduction to AdS\/CFT Since my question in a similar topic was deleted, I'll ask away and hope ppl won't come here telling me: this was already asked! :\\ I have a course in advanced gravitation, and I was thinking of doing this for a seminar, and maybe something about it for a masters thesis (but there is still time for that), so I'd need something that's readable and something that can introduce someone into this field of physics. I have some knowledge of particle physics (mostly what I've learned through Griffith's book), and I've started to read Introduction to AdS-CFT, lectures by Horatiu Nastase that I found on ArXiv, and so far I can follow what he's talking about (I am familiar with path integral notation, Feynman diagrams...). But I'd like some 'simple' books, so that when I'd be giving presentation, my colleagues won't be blindly looking at me and wonder what the hell I'm talking about :D Any suggestion is appreciated :)"}
{"id":"87370","title":"electric potential energy symbol in Schrodinger equation","text":"In my introductory physics class, $V$ is the symbol for electric potential (joules per coulomb) and $U$ is the symbol for electric potential energy (joules). Since the Schrodinger equation is the sum of Kinetic and Potential energies in the system, $V(r)$ must represent $U$... if so, is there any particular reason why $V(r)$ is used as opposed to $U$?"}
{"id":"87371","title":"Why a system should be at its lowest energy state for its stability?","text":"Every possible reaction in chemistry is to attain stability. In physics, the alignment of an electric dipole in an external electric field and in all other physical systems (at least those I study in high school) attains stability at the lowest energy state. But, why is it so?"}
{"id":"3445","title":"References for ADM formalism and cosmological perturbation theory","text":"What would you consider the best online resources for learning the 3+1 ADM formalism and gauge invariant perturbation theory in cosmology? (Assuming intermediate level GR and QFT familiarity)"}
{"id":"106878","title":"What is the relationship between the verbal definition and the mathematical definition of some quantities?","text":"I know this is probably an easy question, but it's been a while since I've studied physics and I've started reading some circuit analysis textbooks. I'm finding hard to understand the relationship between between the verbal definition of quantites and the mathematical definitions. For instance, in the Sadiku's book \"Fundamentals of electic circuits\", I've got the following verbal definiton for voltaje (literal) \"Voltage (or potential difference) is the energy required to move a unit charge through an element, measured in volts (V).\" And then, it says that this mathematically \"means\" $$ v_{ab} \\triangleq \\frac{dw}{dq} $$ I can't understand well the relationship between this two \"definitions\" could someone explain further the relationship?"}
{"id":"41359","title":"Kepler Orbital Elements to Cartesian (xyz)","text":"I'm not really sure if this is the place to ask this, but anyway here is my question: Let's say I have the Kepler orbital elements of the ISS, for example, (NASA stuff). Now I want to compute the coordinates relative to the earth at a specific time so that it can be displayed, like in a sky map (I am actually experimenting with Google Sky Map). I found many websites discussing Kepler orbital elements, but I have only found 2 pdf's that talk about this conversion: http:\/\/www.phas.ubc.ca\/~newhouse\/p210\/orbits\/cometreport.pdf and I can't post a 3rd link yet. I do not fully understand which values I need to calculate, and which are given by NASA. I am also assuming that the data given by NASA is valid for the vector time provided, meaning that time obviously has to come into the calculations somewhere. Thanks, I would appreciate some clarification."}
{"id":"122533","title":"Gauge Invariance of Yang Mills Lagrangian","text":"I am trying to show the invariance of the following Yang Mills Lagrangian: $$L= -\\frac{1}{4} F^a_{\\mu \\nu} F_a^{\\mu\\nu} + J_a^\\mu A_\\mu^a$$ under the following gauge transformation ($\\theta$ being a rotation in color space and $g$ related to the structure constant): $$L \\rightarrow -\\frac{1}{4} \\left( F^a_{\\mu\\nu} +\\epsilon ^a_{jk} \\theta^j F^k_{\\mu \\nu}\\right)\\left(F_a^{\\mu\\nu} + e_a^{jk}\\theta_j F_k^{\\mu\\nu}\\right) + \\left( J_a^\\mu + \\epsilon_a^{jk} \\theta_j J_k^\\mu \\right) \\left( A_\\mu^a + \\epsilon^a_{jk}\\theta^jA_\\mu^k -\\frac{1}{g} \\partial^\\mu \\theta^a \\right),$$ where each term is now transformed accordingly. I was able to simplify the above to and obtain: $$L \\rightarrow -\\frac{1}{4} \\left( F^a_{\\mu\\nu}F_a^{\\mu\\nu} + \\epsilon^a_{jk} \\theta^j F^k_{\\mu\\nu} \\epsilon_a^{j' k'} \\theta_{j'} F_{k'}^{\\mu\\nu}\\right) + J_a^\\mu A_\\mu^a - J_a^\\mu \\frac{1}{g} \\partial^\\mu \\theta^a + \\epsilon_a^{jk} \\theta_j J_k^\\mu \\epsilon^a_{j'k'} \\theta^{j'}A_\\mu^{k'} - \\epsilon_a^{jk} \\theta_{j} J_k^{\\mu}\\frac{1}{g} \\partial^\\mu \\theta^a.$$ How could I possibly reduce it to a form similar to the original, untransformed Lagrangian? There are about 4 terms I can't get rid of, though it has been suggested to me that I use the equation of motion of YM, which I have handy but can't seem to use them appropriately. Any help would be greatly appreciated. Also note that I may end up with a boundary term which would vanish when varying the action, thus possibly giving me say 3 terms instead of the original 2 (which is fine, though I can't identify them yet)."}
{"id":"41352","title":"What is the length of 1 second in meters","text":"If time is treated as a fourth dimension of spacetime, what is relation between length and time units? Or in other words, how can I convert time units to length units, for instance seconds to meters?"}
{"id":"41351","title":"Why the energy of a marshmallow is so huge?","text":"In this comment in a blog kudzu computes the energy of a marshmallow with mass $M=25 grams$ by using $E=mc^2$: $E=Mc^2 = 2.247\\times 10^{+15} Joules$ I may be wrong but this seems like a huge energy for a marshmallow. Can anybody help me put this in context."}
{"id":"15878","title":"Do we have control on what an electron transition emits: light or heat?","text":"I don't know quantum mechanical model. So, I'm referring to just bohr's model of atom. Any atom emits energy when it makes transition from higher excited state to lower excited state. Now, some times they say that this energy is light energy and some times heat energy. I'm confused. What decides the emitted energy will be light or heat? Do we have control over what kind of energy it emits? I mean can I make an atom emit light and not heat? or heat and not light? What are the factors influencing this? **EDIT:** To clarify the issue of what exactly do I mean by heat. Well, I myself am not quite sure. I just thought heat as just heat. This question araised from my previous question where I discovered that in a filament bulb 98% of energy is converted into heat energy and rest into light energy. But where as in CFL about 40% to light and rest into heat. I know these conversion is because of electron transistions. But whats controlling light & heat components?"}
{"id":"83286","title":"Exercise about Lagrange-Euler equations","text":"I'm solving an exercise about the Lagrange-Euler equations, that states the following: > _Let $\\gamma (t) = \\\\{ (t,q) : q = q(t), t_0 \\leq t \\leq t_1\\\\}$ be a curve > in $\\mathbb{R} \\times \\mathbb{R}^2$. Further let $F(q,\\dot{q},t)$ be the > function from $\\mathbb{R}^2 \\times \\mathbb{R}^2 \\times \\mathbb{R} > \\rightarrow \\mathbb{R}$ for which the functional $\\Phi = \\int_{t_0}^{t_1} > F(q,\\dot{q},t) dt$ is the length of the curve._ > > _(a) Which is the form of $\\Phi$ in cartesian coordinates? Which is its form > in polar coordinates?_ > > _(b) Give the Euler-Lagrange equations in both coordinate systems._ > > _(c) Solve the differential equations in both coordinate systems and show > that the solutions are the same._ Now, my problem begins with giving the form of $\\Phi$. I found that the element of length in cartesian coordinates is $ds = \\sqrt{dx^2 + dy^2}$, so with $$\\int ds = \\int \\frac{ds}{dt} dt = \\int \\sqrt{\\left(\\frac{dx}{dt}\\right)^2 + \\left(\\frac{dx}{dt}\\right)^2} dt,$$ We find that $\\Phi = \\int_{t_0}^{t_1} ||\\dot{\\gamma}(t)|| dt$. Now, my plan is finding the element of length in polar coordinates, and plugging in the respective expressions in terms of $\\gamma$. The problem is that I don't see how to find the element of length in polar coordinates. I looked it up on Wikipedia, and found $ds^2 = dr^2 + r^2 d\\theta^2$. Now, for $dr^2$ I would plug in $||\\dot{\\gamma} (t)||^2$, for $r^2$ I'd set $||\\gamma (t)||^2$, and for $d\\theta$ I have no idea. Can you help me, especially with the derivation of the polar line element and the form of $\\Phi$ in polar coordinates?"}
{"id":"30629","title":"Why don't most physics programs study the primary sources?","text":"**Why don't most physics programs study the primary sources?** In other words: Why don't they include Newton's _Principia_ , Lagrange's _Analytical Mechanics_ , etc., in the curricula?"}
{"id":"83280","title":"Classical Information carrying capacity of two states","text":"What classical information is carried by $\\alpha|0\\rangle+\\beta|1\\rangle$ and $\\alpha|00\\rangle+\\beta|11\\rangle$? How to quantify it? To be specific, A GHZ state, $\\frac{1}{\\sqrt 2}[|000\\rangle+|111\\rangle]$ can deterministically teleport both the states. Both the states appears to be same from the teleportation point of view. Does they carry same information?"}
{"id":"107684","title":"What does really mean by- power of a number or an exponential function is dimensionless?","text":"Is power of only a number or an exponential function is dimensionless? If power of any other thing can also be dimensionless then please explain with examples."}
{"id":"86422","title":"Problem in Youngs double slit experiment","text":"![double slit ray diagram](http:\/\/i.stack.imgur.com\/Whguh.gif) This is from Young Double slit experiment. But How to prove the the two $\\theta$ are equal, I meant, how $\\angle EAD= \\angle PEC$? I see from the both triangle have $90^0$ but what about others?"}
{"id":"64436","title":"Why do wind power plants have just 3 blades?","text":"Why do wind power plants have just 3 blades? It seems that adding more blades would increase the area that interacts with the wind and gather more energy. ![enter image description here](http:\/\/i.stack.imgur.com\/KOC1w.jpg) (Image from Wikipedia.)"}
{"id":"13843","title":"How does the internuclear repulsion vary in Hydrogenic atom collision?","text":"Hydrogen fusion requires two hydrogen nuclei to get close enough (typically a few fm) to fuse. Much of the problem of creating a fusion reactor is overcoming the Coulomb repulsion between a pair of nuclei - the millions of degrees for Maxwellian distributions, the Bremstrahlung losses for inertial confinement. If we could align the paths of two neutral Hydrogen **atoms** (of whichever isotopes), what would the repulsion look like between them as they approach collision? Obviously at long range there is negligable force as both are neutral. But as they approach each other, what happens to the electron distribution? Intuitively, I expect a bonding cloud to form between the nuclei, and antibonding clouds beyond them. This would presumably attract at first until reaching the usual Hydrogen covalent bond length, after which the internuclear repulsion would increasingly dominate. But how does that compare to bare ionic collision? How much lower is the potential barrier? Obviously if it was significantly lower and we could somehow engineer the collision to achieve fusion, the cross section would be larger than ionic fusion, but how much? Or would the barrier be just as high over the final few femtometers?"}
{"id":"13842","title":"The quantized energy level E depends on which power of n?","text":"A particle in one dimension moves under the influence of a potential $V(x)= ax^6$, where $a$ is a real constant. For large $n$, what is the form of the dependence of the energy $E$ on $n$?"}
{"id":"13848","title":"What interpretive difference is there between defining a function with or without a differential as a postfactor?","text":"I have thought about this and looked for answers for a long time now, but I do not have any name or label for this problem, which is the reason for the long title of this question. I have come across this many times, but I have never found any rationale for this, i.e. a clear interpretation or explanation, so here are a few examples of what I mean. I suspect that they have different answers. I) In e.g. Shapiro and Teukolsy's book _Black Holes, White Dwarfs and Neutron Stars_ , (1983), the Salpeter birthrate function for stars is defined as $\\psi_s\\mathrm{d}\\left(\\frac{M}{M_\\odot}\\right) = 2\\times10^{-12}\\left(\\frac{M}{M_\\odot}\\right)^{-2.35}\\mathrm{d}\\left(\\frac{M}{M_\\odot}\\right)\\mathrm{\\ stars\\ pc}^{-3}\\mathrm{\\ yr}^{-1}$ Questions: why does it matter that those differentials are included? Why make the statement look messier by multiplying both sides of the equation with something that obviously cancels out? If it is just to show what $\\psi_s$ depends on, then it makes mre sense to me to define the function like so: $\\psi_s(M) = 2\\times10^{-12}\\left(\\frac{M}{M_\\odot}\\right)^{-2.35}\\mathrm{\\ stars\\ pc}^{-3}\\mathrm{\\ yr}^{-1}$ What is (or may be) the answer? II) When defining the equation of continuity for brownian motion particles in 1D space, I have a lecture note that reasons in the following way: Given the setup                        |  n(x,t) |              |         |         J(x,t) ---->     ----> J(x + dx, t)              |         |              |         |         ---------------------->              x        x+dx      Where $n(x,t)$ is the number density of particles between positions $x$ and $x + \\mathrm{d}x$ and $J(x, t)$ is the flux of these particles at given position $x$ and time $t$. The particle number is conserved, and they are assumed identical and non-interacting with each other. In the notes that I have, the following equation is readily displayed as though it was crystal clear why it was written inthis way: $n(x, t + \\mathrm{d}t)\\mathrm{d}x - n(x, t)\\mathrm{d}x = - (J(x + \\mathrm{d}x, t)\\mathrm{d}t - J(x, t)\\mathrm{d}t)$ or (multiplying the minus into the parenthesis) $n(x, t + \\mathrm{d}t)\\mathrm{d}x - n(x, t)\\mathrm{d}x = J(x, t)\\mathrm{d}t - J(x + \\mathrm{d}x, t)\\mathrm{d}t$ Question: I can not myself justify why it makes sense to multiply with the diferentials as they are here. What is the significance, or rather the line of thought that lies behind the way this equation turns out? I am unsure of my own answer which is only to look at the dimensions of the two functions themselves and then conclude what unit the factor should have on each side in order for the dimensions to make sense. Just to complete the derivation, rewriting the equation by replacing the functions $n(x, t + \\mathrm{d}t)$ and $J(x + \\mathrm{d}x, t)$ with their Taylor expansion (to first order) gives $n(x, t + \\mathrm{d}t) = n(x, t) + \\frac{\\partial n}{\\partial t}\\mathrm{d}t$, $J(x + \\mathrm{d}x, t) = J(x, t) + \\frac{\\partial J}{\\partial x}\\mathrm{d}x$, which, when inserted into the original equation gives $\\frac{\\partial n}{\\partial t}\\mathrm{d}t\\mathrm{d}x = -\\frac{\\partial J}{\\partial x}\\mathrm{d}x\\mathrm{d}t$ so producing, again, a shared factor ($\\mathrm{d}t\\mathrm{d}x$) on both sides of the equation which can just be divided away (right?), and we end with the coninuity equation $\\frac{\\partial n}{\\partial t} = -\\frac{\\partial J}{\\partial x}$ I hope this question makes sense or at least allows you to suggest a label for this kind of problem that I can use to find more information, Or better, help by answering the wuestion here. Cheers."}
{"id":"125897","title":"ODE envolving current","text":"I have to do this exercise: > 9\\. The current $I(t)$ at time $t$ flowing in an electric circuit obeys the > differential equation > > $$I''(t) + RI'(t) + I(t) = \\sin \\omega t$$ > > where $R$ and $\\omega$ are positive constants. The solution can be expressed > in the form $I(t) = F(t) + A\\sin(\\omega t + a)$, where $F(t)\\to 0$ as $t\\to > +\\infty$, and $A$ and $\\alpha$ are constants depending on $R$ and $\\omega$, > with $A > 0$. If there is a value of $\\omega$ which makes $A$ as large as > possible, then $\\omega\/(2\\pi)$ is called a _resonance frequency_ of the > circuit. > > (a) Find all resonance frequencies when $R = 1$.   >  (b) Find those values of $R$ for which the circuit will have a resonance > frequency. I have a doubt about the first item. To find all resonance when $R=1$, I found the particular solution $I_{p}(t)=A\\sin(\\omega t)+B\\cos(\\omega t)$. I got the first and second derivates $$I_{p}'=A\\omega\\cos(\\omega t)-B\\omega\\sin(\\omega t)$$ $$I_{p}''=-A\\omega^{2}\\sin(\\omega t)-B\\omega^{2}\\cos(\\omega t)$$ then, when I substitute in the equation $$I''(t)+I'(t)+I(t)=\\sin(\\omega t)$$ I got the system on $(A,B)$: $$\\left\\\\{\\begin{array}{l} -A\\omega^{2}-B\\omega+A=1 \\\\\\ -B\\omega^{2}+A\\omega+B=0 \\end{array}\\right.$$ so $$A=\\frac{(1-\\omega^{2})}{(1-\\omega^{2})^{2}+\\omega^{2}}$$ I calculated $A$ because it follows $\\sin(\\omega t)$ and with this I can find all resonance frequencies. Well, to find them, I gotta find the values of $\\omega$ which makes $A$ as large as possible. To $A\\to+\\infty$, so, its denominator must goes to 0: $$(1-\\omega^{2})^{2}+\\omega^{2}\\to0$$ But that's not possible. The answer is $\\displaystyle\\frac{1}{2\\pi\\sqrt{2}}$. Comparing with the exercise, I know that $\\omega=1\/\\sqrt{2}$. If the denominator was $(1-\\omega^{2})^{2}-\\omega^{2}$, I got the result, but I don't know what's wrong here. **EDIT:** I found after $$I_{p}(t)=\\frac{1}{\\sqrt{(1-\\omega^{2})^{2}+\\omega^{2}}}\\sin(\\omega t-\\alpha)$$ where $\\alpha=\\arctan(\\omega\/(1-\\omega^{2})$. So, $$A=\\frac{1}{\\sqrt{(1-\\omega^{2})^{2}+\\omega^{2}}}$$ and it's large just if its denominator is small... but how?!"}
{"id":"85998","title":"Determining the limits of an integral","text":"In Griffiths' _Introduction to Electrodynamics_ , Problem 1.28 (the triangular prism question) is especially challenging for me. I do not know how the limits of x are **0** to ( **1-y** ). My concern is the latter limit, **1-y**. Can you explain systemically how this is arrived at? On the face of it, I'm inclined to say the limits of x are **0** and **1**. I can see it's the diagonal across the x- and y-axes that's presenting the extra challenge, but I do not know how to go about establishing that indeed the limit is **1-y**."}
{"id":"117180","title":"What would happen in a collision of an antimatter singularty and a matter singularity?","text":"Would energy be released if a black hole made out of antimatter and another of matter were to collide?"}
{"id":"52524","title":"Is it part of special relativity that mass possessing energy is more dense?","text":"I was reading http:\/\/www.edge.org\/3rd_culture\/hillis\/hillis_p2.html and it says that a charged battery weighs more than a dead one or a rotating object weighs more than a stationary one (i.e. mass containing energy weighs more than mass that doesn't). Is this true in special relativity? Because the energy would be confined to the area of the original mass, wouldn't it become more dense if it is true? Also, if the mass we are speaking of does become more gravity-sensitive on energy gain, does it actually gain physical mass (protons, electrons and neutrons)??"}
{"id":"88959","title":"Drag force on a cone","text":"I was asked to calculate the drag force on a cone with velocity 10 m\/s , everything was okay until i needed to calculate the cross sectional area , the radius of the base was 0.5 m , radius of the top 0.0005 m , given that the cone falls top first, which one should i use ? should i get some kind of an average?"}
{"id":"47026","title":"Is $k_B \\rightarrow 0$ the classical limit of stat. mech., as $\\hbar \\rightarrow 0$ is in QM?","text":"I hear very often among my peers and seniors that just as how $\\hbar\\rightarrow0$ takes me to classical mechanics from quantum mechanics, $k_B\\rightarrow0$ will take me to classical thermodynamics from statistical mechanics. As nice as this sounds, my gut feeling and intuition tells me this is not the right analogy. I think $N\\rightarrow\\infty$, the ensemble size of statistical mechanical system, is closer to $\\hbar\\rightarrow0$ in that respect. Is this correct? If so, what role does $k_B$ play then?"}
{"id":"11561","title":"Asymptotic Invariants in General Relativity","text":"I was trying to understand Witten's proof of the Positive Energy Theorem in General Relativity by reading the original argument given by Witten. I am comfortable with the overall argument, but I would like to understand the following statement, made on the second paragraph of page 394 in the previous link: > \"The only invariants that can be formed from the $1\/r$ term in the metric > tensor are the total energy and the total momentum.\" These invariants (which I will call \"$1\/r$-built\" for short) refer to an asymptotically flat initial value (spacelike smooth) 3-surface in spacetime. Is there an obvious reason why this should be true? Looking at the definition of the ADM-energy and momentum, it seems plausible because the only data we have are the first and second fundamental form, and it should be possible to write any invariant as a combination of these, and consequently, any $1\/r$-built invariant as a function of ADM energy and momentum. However, this reasoning is too hand-wavy, so I wonder if there is a clear cut explanation. My interest in this fact is that although it is not really a logical step in the proof, if true it is probably one of the best ways to motivate a spinorial proof (I was thinking of something along the lines of \"if we can construct a manifestly non-negative $1\/r$-built invariant by means of the asymptotic behavior of spinors, then it should be possible to prove that energy is non- negative by writing this invariant as a function of ADM energy and momentum\"). From a purely physical perspective, if it were true that ADM energy and momentum suffice to specify the system (to order $1\/r$), then they should be the only independent invariants. However, this suggests that an asymptotic observer who knew the total energy and momentum could reconstruct the metric up to order $1\/r$. I was thinking if it is not possible to construct a counterexample by defining an axisymmetric spacetime in which the Killing field of azimuthal symmetry is build up only with $1\/r$ terms and showing that it is asymptotically distinguishable from its non-rotating counterpart. In this spirit, I think it is worth asking a broader version of my question: > What kinds of \"physically interesting\" boundary terms can appear in a > Hamiltonian formulation for an asymptotically flat spacetime manifold in > General Relativity? Thanks in advance."}
{"id":"11562","title":"Is nature symmetric between particles and antiparticles?","text":"Is nature symmetric with respect to presence of particles? Do we have an antiparticle for every particle thought of? Are there any proven examples where we don't have an antiparticle? And what about antiparticle of a photon (we know it can also behave as a particle)?"}
{"id":"89392","title":"Microscopy types and techniques?","text":"I didn't get the basic difference in between the different types of microscopy. for example there are several different microscopy techniques are available such as Bright field, Dark field, Confocal, STED, electron microscopy etc. Is only the illumination system varies in between different techniques or are there any other parameters?"}
{"id":"11565","title":"Spinning bucket of water in zero gravity","text":"Everyone knows how the surface of a spinning bucket of water would look like on earth - parabolic. But what if we turned off gravity (for instance by doing the experiment in a freely falling lift)? Would the surface be still parabolic? I'll explain my confusion in more detail. The velocity of the spinning bucket is transferred to the water by means of frictional forces arising in the boundary between the bucket and water. But these frictional forces exist no matter whether there is gravity or not. So if I consider the whole bulk of water inside the bucket as a single system, this frictional force would give it a positive torque. Thus the water has to rotate. For the sustained rotation of water, a centripetal force has to exist. In normal gravity, the water surface changes its shape into a paraboloid so that there is a net force on any particle directed inward. But in free fall, there is no pressure on a particle inside the liquid. Thus the only force that can supply the centripetal acceleration is inter-molecular force between the particles which is weak to sustain huge velocities. So what exactly happens?"}
{"id":"82420","title":"Angular Momentum with Upper Index","text":"I am asked to show $[L^2,L_i] = 0 $, but with the definition : $L^2 \\equiv L_i L^i$ I tried this: $[L_i L^i,L_i] = L_i [L^i,L_i] + [L_i,L_i]L^i$ We know that : $[L_i,L_i]$ = 0 , so we have, $[L_i L^i,L_i] = L_i [L^i,L_i]$. So I could not understand this term with upper index. What is $L^i$ ?? Any clues?"}
{"id":"131866","title":"Zeroth Law of Thermodynamics, temperature, and ordering","text":"In my thermodynamics course (and in other places on the internet) it is asserted that the Zeroth Law of Thermodynamics can be used to define the concept of temperature. One statement of the Zeroth Law I have seen states that the relation _thermal equilibrium_ on two closed systems brought into diathermal contact defined is in fact an equivalence relation. The argument continues by saying that if we call the equivalence classes so defined _isotherms_ , then we can assign arbitrary numbers to these isotherms and these numbers are what we call _temperature_. We now have a set of numbers assigned to a set of equivalence relations. But what I **don't** see is how this numerical assignment bears any relation to physical temperature. Where is order defined? For example, if one of the classes gets the number \"1\", another gets the number \"2\", and a third \"3\", what in the above derivation shows that the isotherm \"2\" comes between the isotherm \"1\" and the isotherm \"3\"? Maybe more is needed than just the Zeroth Law. If so, what is necessary in order to complete the argument?"}
{"id":"128397","title":"Work done by a frictional force on a cylinder","text":"> _A cylinder of mass $m$ and radius $R$, is rolled on surface with > coefficient of kinetic friction $\\mu_{k}$ about the axis passing through the > center and parallel to the surface, with initial angular velocity > $\\omega_{0}$; the work done by frictional force from start until it begins > to roll w\/o slipping is to be found out._ It is easy to see that $a=\\mu_{k}g,\\alpha=-2\\mu_{k}g\/R,t=\\frac{R\\omega_{0}}{3\\mu_{k}g},d = \\frac{1}{2}at^2 $ are acceleration, angular acceleration, time from beginning to the time of rolling w\/o slipping and distance covered. I am somehow missing why I am getting $$W=\\int F.ds+\\int\\tau.d\\theta=\\mu_{k}mg*d+\\mu_{k}mgR*\\frac{d}{R}\\neq\\triangle K.E.=-\\frac{1}{6}mR^{2}\\omega_{0}^{2}.$$ Please shed some light."}
{"id":"39019","title":"Field due to current in a wire","text":"Suppose a current flows in a straight cylindrical wire so that an electric field $\\textbf{E}$ is maintained in the wire. Will there be an electric field just outside the wire..?"}
{"id":"16720","title":"Find the acceleration based on V² x d graph","text":"How could I find the acceleration of a body knowing only that its position and velocity satisfy $v^2 = f(x)$, where $f(x)$ is a known function of $x$ (position)?"}
{"id":"257","title":"Particle colliders: why do they need an accelerator chain","text":"Particle colliders like the LHC or the Tevatron use a complex accelerator chain to have particles at a given energy before being accelerated. For example:   * The CERN accelerator complex to inject in the LHC: protons are first accelerated by a LINAC, then by a \"booster\" synchrotron, then by a larger synchrotron, and finally by a very large synchrotron where they reach 450 GeV\/c.   * As was discussed in link text , the chain for the Tevatron They both require \"injector\" accelerator(s). ![alt text](http:\/\/i.stack.imgur.com\/PzikJ.gif) My question is simple: why ? Why can't we have a single accelerator with a source and the possibility to accelerate the particles to the top energy. * * * Note: this is also a seeding question, but I am curious how complete and clear an explanation to that can be provided."}
{"id":"27159","title":"Source term of the Einstein field equation","text":"My copy of Feynman's \"Six Not-So-Easy Pieces\" has an interesting introduction by Roger Penrose. In that introduction (copyright 1997 according to the copyright page), Penrose complains that Feynman's \"simplified account of the Einstein field equation of general relativity did need a qualification that he did not quite give.\" Feynman's intuitive discussion rests on relating the \"radius excess\" of a sphere to a constant times the enclosed gravitational mass $M$: for a sphere of measured radius $r_{\\mathrm{meas}}$ and surface area $A$ enclosing matter with average mass density $\\rho$ smoothly distributed throughout the sphere, $$\\sqrt{\\frac{A}{4π}}−r_{\\mathrm{meas}}=\\frac{G}{3c^2}\\cdot M,$$ wherein $G$ is Newton's gravitational constant, $c$ is the speed of light in vacuum, and $M=4π\\rho r^3\/3$. I don't know what $r$ is supposed to be, but it's presumably $\\sqrt{\\frac{A}{4\\pi}}$. Feynman gratifyingly points out that $\\frac{G}{3c^2}\\approx 2.5\\times 10^{−28}$ meters per kilogram (for Earth, this corresponds to a radius excess of about $1.5$ mm). Feynman is also careful to point out that this is a statement about average curvature. Penrose's criticism is: \"the 'active' mass which is the source of gravity is not simply the same as the energy (according to Einstein's $E=mc^2$); instead, this source is the energy plus the sum of the pressures\". Damned if I know what that means -- whose pressure on what? So, taking into account Penrose's criticism but maintaining Feynman's intuitive style, what is the active mass $M$?"}
{"id":"12034","title":"Cross-section in relativistic limit: Fermi's golden rule still valid?","text":"In order to calculate the cross-section of an interaction process the following formula is often used for first approximations: $$ \\sigma = \\frac {2\\pi} {\\hbar\\,v_i} \\left| M_{fi}\\right|^2\\varrho\\left(E_f\\right)\\,V $$ $$ M_{fi} = \\langle\\psi_f|H_{int}|\\psi_i\\rangle $$ Very often plane waves are assumed for the final state and therefore the density of states is given by $$ \\varrho\\left(E_f\\right) = \\frac{\\mathrm d n\\left(E_f\\right)}{\\mathrm d E_f} = \\frac{4\\pi {p_f}^2}{\\left(2\\pi\\hbar\\right)^3}\\frac V {v_f} $$ I understand the derivation of this equation in the context of the non relativistic Schrödinger equation. But why can I continue to use this formula in the relativistic limit: $v_i, v_f \\to c\\,,\\quad p_f\\approx E_f\/c$. Very often books simply use this equation with matrix element derived from some relativistic theory, e.g. coupling factors and propagators from the Dirac equation or Electroweak interaction. How is this justified? ### Specific concerns:   * Is Fermi's golden rule still valid in the relativistic limit?   * Doesn't the density of final states has to be adapted in the relativistic limit?"}
{"id":"100036","title":"How does pressure factor into mechanics?","text":"We just started learning about pressure in our school, and i am a bit confused. I tried to google extensively, but i didn't really find much. In mechanics, so far in our school we've been taught about Newton's laws of motion and some related stuff, and suddenly we started learning about pressure. I asked this same question to my physics teacher, but he failed to answer. How does pressure actually factor into mechanics? We studied so far that to cause motion, you require a force. Then we study about pressure, and the examples are ones like pressure of gas in a contained beaker and then lowering\/raising it's volume. Suddenly in the applications section however there are things like Paper pins have a low surface area at the end to maximize pressure, and that you can't walk on sand easily because the sand depresses under pressure. I understand the math behind it, but what does the math actually mean? I mean, a force is being applied. It should cause motion, but apparantly if the surface area of the end is small then pressure goes up? The force here remained constant(?) so how does pressure factor into the mechanics? Why did that just happen? I realize i may be way off topic and asking a stupid question, but i am having some trouble comprehending how pressure works. I understand the examples of things like pressure in a container, atm. pressure e.t.c but how does pressure transmit in a case like this? My apoligies if this is a stupid question."}
{"id":"63043","title":"Dark and bright areas around atoms in a scanning tunnelling microscope image","text":"Recently IBM created world’s smallest ever animation on an atomic scale video. Researchers made the animation using a scanning tunnelling microscope to move thousands of carbon monoxide molecules to show a boy dancing, throwing a ball and bouncing on a trampoline. My question is, why in this video we see a pattern of dark and bright circles around each molecule? What do they represent?"}
{"id":"63046","title":"How we approach RLC circult from RLGC model?","text":"In the text, it introduces a practical model to investigate a transmission line (like BNC cable), it considers the transmission line has resistive $R$, inductance $L$, conductance $G$ and capacitance $C$. The model is illustrated as follow ![enter image description here](http:\/\/i.stack.imgur.com\/Iaxe3.jpg) It is easy to derive the (telegraph) equations and figure out the impedance Z to be $$Z = \\sqrt{\\frac{R+iX_L}{G+i\/X_C}}$$ where $i$ is the imaginary unit, $\\omega$ is the angular frequency, $X_L$ is the inductive reactance and $X_C$ is the capacitive reactance. And in other section, it introduce a RC circuit and RLC circuit, in which, the impedance are $$Z_{RC} = \\sqrt{R^2 + X_C^2}, \\qquad Z_{RLC}=\\sqrt{R^2 + (X_L - X_C)^2}$$ It is pretty confusing because from RLGC model, if we make the electrical conductance $G$ to zero and consider no inductance ($L=0$), so the circuit becomes RC circuit, but from the first equation for the impedance given by the RLGC model, the impedance should be $$Z = \\sqrt{-iRX_C}$$ Why are they not the same? How to approach RC and RLC case from RLGC model?"}
{"id":"21639","title":"Thought experiment with entangled electrons","text":"Suppose we start out by having two entangled electrons. We separate them by some distance and we put one electron inside a thin loop of wire connected to an extremely sensitive voltage measuring device at lab 1 and the second electron at lab 2. At this point, no measurement is made. Both electron's spin's are undetermined. Therefore... -The direction of the spin is undetermined. -We do not know if the electron's spin, $s=\\pm\\hbar$. -We do not know its magnetic moment,$ m = (-gu_bS)\/\\hbar$. $S = \\frac{h}{2\\pi}\\sqrt{s(s+1)}$, $g$ is g factor, $u_b$ is Bohr Magneton. -We do not know the Magnetization, $M = (N\/V)m$, where $N$ is number of magnetic moments and $V$ is the volume of the system in question. -We do not know the magnetic field, $B = \\mu_0(H + M)$, where $\\mu_0$ is vacuum permeability, $H = M\/X$, where $X$ is the magnetic susceptibility. The magnetic field of the first electron at lab 1 is undetermined because no measurement has been made, therefore no magnetic field can possibly be present, $B = 0$. (Please correct if wrong) Now, we measure the second electron's spin by sending through a Stern-Gerlach device and having that electron hit a screen to record its spin value, $+$ or $-$, at lab 2. Regardless of whether or not the second electron's spin is up or down, we know that the first electron's spin is now determined. This means that the magnetic field at lab 1 has been determined and therefore a magnetic field must be present. Since there is a change in magnetic field from 0 to some non-zero value, there must be a change in voltage from the law of induction, $V = -\\frac{d}{dt}\\left(BNA\\cos\\theta\\right)$, where $A$ is the area the magnetic flux is going through, and $N$ is the number of coiled wire. This suggest that there is a measurable effect, although extremely small, at lab 1 due to the entanglement breaking in lab 2. My question is, is this theoretically correct? If yes, then I would suggest this as a method of communication by the following. By creating a large ensemble of these entangled electrons, A-A', B-B', C-C', D-D', where A is an electron at lab 1, entangled to a second electron, A', at lab 2, etc. For example, by choosing to measure A' and C' and leaving B' and D' alone at lab 2, we create a measurable effect at lab 1 for the electrons' A and C, voltage is changed. Thus, this would constitute a sent message as (1 0 1 0). Where 1 would be a voltage change and 0 would be no voltage change. This would of course be a one time messaging system, but it still does not negate the fact that it would be able to send a message via entanglement by this specific scheme. This is true only if my scheme is logically and theoretically correct. I am open to scrutiny and correction. Please help me determine if my scheme is wrong. Thank you. :D"}
{"id":"46842","title":"Do all the forces become one?","text":"1. Were the forces of nature combined in one unifying force at the time of the Big Bang?   2. By which symmetry is this unification governed?   3. Are there any evidence for such unification of forces?   4. Has ever been published Theory or experiment in this issue? (Even original researches or unpublished theories. Anything that you can start with.)"}
{"id":"58052","title":"What information about a meteor's trajectory, size, or height can be derived from a single location?","text":"If one sees a meteor, is there any way to get even a rough approximation of its height, entry angle, size, or other characteristic _without triangulation_ from another position? If it appeared as a point source and got uniformly brighter, you'd know to take a step aside. And if it appeared on one horizon, traveled overhead, and disappeared over the other, you'd be able to say \"Well, _that_ was a shallow angle of attack.\" But short of those scenarios (which are, probably for the best, rare), is there anything?"}
{"id":"58056","title":"Landauer's principle vs Rayleigh–Jeans law","text":"Can we argue based on Landauer's principle that if one bit information is changed inside a blackbody, the total radiated energy should be at least or in order of $kTln2$? If it is so, can we also argue that this energy should be distributed over all the modes of the cavity? Furthermore, can it also be argued that this contradicts with the Rayleigh–Jeans law which says the total energy should be infinite?"}
{"id":"45793","title":"Direction of friction on rollers with plank resting on top","text":"In a question, its given that a plank is resting on 2 rollers like: ![enter image description here](http:\/\/i.stack.imgur.com\/afMB1.png) Then the explanation, as part of computing the forces, for the direction of friction on the rollers look like: ![enter image description here](http:\/\/i.stack.imgur.com\/KcJHQ.png) Why are the direction this way? Since the rollers are turning clockwise (plank is moving right), I expected the opposite? Explanation on the directions look like: (see line 5 counting from bottom) ![enter image description here](http:\/\/i.stack.imgur.com\/K5lpx.png) Although, after the computation, the direction of $f_b$ is pointing to the right as its negative."}
{"id":"80119","title":"Question about moments?","text":"Take a look at this diagram: http:\/\/www.chegg.com\/homework-help\/questions-and- answers\/7-pipe-assembly-supports-vertical-loads-shown-determine-components- reaction-ball-socket-jo-q2260091 If you calculate moment around point B, you can ignore the forces of the tension and only the forces at the ball and socket and the weights come into play. Therefore for moment at B to be zero, the Y and X components of the ball and socket have to be zero. However, if the x and y components at the ball and socket are at 0, theres nothing to cancel out the X and Y components of the tension forces, meaning the forces are at unbalanced. What am I missing? **Please answer MY question not the question at the link**"}
{"id":"10764","title":"Nanotechnologies: current problems and general limitations","text":"I have some questions about Nanotechnologies. I'm not physicist and my knowledges about the topic are limited by the \"Engines of Creation\" book by Eric Drexler. So I think that my questions are related mostely to popular science.   1. As far as I know the molecular mechanisms like nanoassemblers and nanofactories exist in nature(I mean alive cells). Why have they not been created artificially by now?   2. In case that they have been artificially created yet, do they have any limitations in developing of macro objects? For instance, macromechanisms which exist in nature like animals are grow very slowly(months and years) and the initial program(i.e. DNA encryption) doesn't allow to make accurate copies. I mean that the twins who are growing from the same genetic material have a significant difference in adulthood(i.e. different phenotypes). Is this phenomena the result of some fundamental low of nature like Heisenberg uncertainty principle or any other that can't be resolved by modern technological methods?   3. What are the top well known laboratories and organisations in the world which are engaged to resolve such kind of problems? Thanks in advance."}
{"id":"53757","title":"Tracing out an observable vs integrating over unitaries","text":"Let $O$ be an observable on a Hilbert space $\\mathcal{H}$, and let $B$ be a subset of the spins composing $\\mathcal{H}$, and let $\\bar{B}$ be its complement. Now define $\\displaystyle O_B = \\frac{1}{\\operatorname{Tr}_{\\bar{B}}\\mathbf{1}_{\\bar{B}}} \\operatorname{Tr}_{\\bar{B}}(O) \\otimes \\mathbf{1}_{\\bar{B}}$. Is this quantity equal to $\\displaystyle \\int d\\mu(U) U O U^\\dagger$? The integral is taken over the set of unitary operators acting on $\\bar{B}$ and $\\mu$ is the Haar measure of $U$. If so, why is this the case? What physics course\/book\/reference introduces these sorts of integrals? Note: this question came up from trying to understand the following paper: http:\/\/arxiv.org\/abs\/quant-ph\/0603121"}
{"id":"86051","title":"Why is the speed of light considered as a fundamental constant if its speed changes with medium resulting in refraction?","text":"I know that the speed of light, the universal constant of gravitation and the Planck's constant are considered to be the three fundamental constants of the universe. But, why is speed of light considered as a fundamental constant? The speed of light changes from medium to medium for satisfying Fermat's principle then, how can it be a constant?"}
{"id":"135213","title":"Gauss's Law :To find the Electric Field for a Non-Conducting Sphere","text":"While determining the electric field in a Non-Conducting Sphere using Gauss's law,why the positive charges are considered inside the surface,but in determining the electric field in a conducting Sphere,why the positive charges are considered outside the surface? And,why if any point charge is inside a sphere,the the net electric field is considered zero?"}
{"id":"13074","title":"Autocorrelation of sound in liquids vs gas","text":"This is just a curiosity and you need to bear with me as my math skills were always sub-par and so is my English academic language. My specialty is Electronics but I have always been a programmer. Years ago I was working on my diploma paper that dealt with locating leaks using auto-correlation.   Basically you have several sound sensors along a pipe which record the sound wave. If a leak appears then you get a peak in the autocorrelation function. Then you perform a 'triangulation' from two or more sound sensors and you can locate the leak pretty accurately. Anyway, my task was mostly to help a PHD student with transposing her algorithms into Matlab.      We have succeeded doing this with air but as soon as we switched to water I could not use any kind of windowing and\/or transformation to get relevant peaks. At some point I just hunted blindly for some Matlab functions that would give me a relevant peak somewhere that would correlate with the distance to the leak, but failed.      I cannot to this day understand why we were not able to succeed, though I do assume that is has to do with water's turbulence. The setup was a running tap routed through metal pipes (about 10cm diameter), with about 5 very sensitive sound sensors (going to about 100kHz) placed about 3 meters apart and some taps along the way that simulate leaks. Everything was placed in a phonic- insulated basement, signals were truncated taken to stabilize against footsteps\/vibrations so measurements were pretty good.      The multiple question is: what could have been done to achieve the goal or at least get closer to it? Is it really achievable (probably not in real life where you have trucks running above)?   Some variables that could influence but for which I lack the knowledge to explain: pipe diameter, water debit rate, tap\/leak rate and diameter, choice of liquid, pipe material, sound spectrum and possible frequency filtering."}
{"id":"109587","title":"Lebesgue integration","text":"I know this question is probably not adequate to this SE either, but let me explain my situation: I'm civil engineering's college, so, there isn't a SE for civil engineering, and my doubts about integration in engineering are pratically pure physics. So, as I said, I'm graduating, but I'm from Brazil, education here is the same thing as nothing (belive me, it really sucks, I see people come out of physics's college without knowing who Maxwell was..), and I read my calculus books and see they all use Riemann integrals (of course, they don't say that..); but, in my searchs, I see a lot of Lebesgue integration, especially concerning problems of calculating the center of mass of an object, in continuum mechanics, etc. So, here's my question:   1. Lebesgue integration isn't the most easy thing in the world, I don't have a lot time, and education here sucks, should I spent my time studying Lebesgue integration instead of Riemann's?    2. Which one do you guys use more in physics with applications in engineering?; of course, the last one it's a little trouble, because engineering just \"borrow\" from physics. But, in general, which compensates more?"}
{"id":"54194","title":"Are there any exceptions to Newton's laws?","text":"So I am studying the laws of Newton and I'm wondering, are there any deficiencies related to the laws? I mean, somewhere where I can't use them or anything?"}
{"id":"94569","title":"Normalization of Momentum Eigenfunctions: the number of particles","text":"After finding the eigenfunctions $u_p(x)=Ce^{ipx\/\\hbar}$ of the momentum operator just like in this UCSD lecture notes, one seeks to normalize them, so one first tries: $$\\int\\limits_{-\\infty}^{\\infty} dx \\, |C|^2 e^{-ipx\/\\hbar} e^{ipx\/\\hbar} = \\int\\limits_{-\\infty}^{\\infty} dx \\, |C|^2 \\rightarrow \\infty $$ which diverges unless $C =0$. Then it is shown that $u_p(x)=\\frac{1}{\\sqrt{2\\pi \\hbar}}e^{ipx\/\\hbar}$ satisfies the normalization condition $\\langle p'|p\\rangle=\\delta(p-p')$ Why does the UCSD page say that the first solutions (the divergent ones) \"are not normalizable to one particle\"? How does the development that follows relate to many particles? Does this have something to do with $\\int\\limits_{-\\infty}^{\\infty}dp \\delta(p-p') =1$? Then all the $p'$ are the other particles? I have no source to show for this, but how would $\\frac{1}{2\\pi \\hbar}$ particles per unit of length and unit of momentum relate? Is it correct to say that $2\\pi \\hbar$ is the expected number of times one repeats the measurement of the momentum? Then where would the 'per unit of length' part come from?"}
{"id":"94563","title":"Why does cold water weigh more than hot water in a fixed volume?","text":"Take a bucket of hot water and the other bucket of cold water. Why does the bucket full of cold water weigh more than bucket full of hot water?"}
{"id":"10182","title":"Convergence of periodic single fermion operators","text":"First, a quick remark: I'm a mathematician, now working on some problems coming from physics (in particular Ising models on quasiperiodic chains). A few things I find rather mysterious. I would appreciate your help. For the purpose of generality, let's consider the following Ising model on a chain of $N$ nodes. $$H_N = - \\sum_{i = 1}^N J_i\\sigma_i^{(x)}\\sigma_{i+1}^{(x)} - \\sum_{i = 1}^N\\sigma_i^{(z)},$$ with $J_i$ depending on the node $i$ (we assume no particular order for generality), and $\\sigma_i^{(x),(z)}$ the Pauli matrices. By Jordan-Wigner, we can consider the corresponding Fermionic operator given by $$\\widehat{H}_N = \\sum_{i,j}\\left[c_i^{\\dagger}A_{ij}c_j + \\frac{1}{2}\\left(c_i^{\\dagger}B_{ij}c_j^{\\dagger}+ H.c.\\right)\\right],$$ where $c_i$, $1\\leq i \\leq N$ are anticommuting Fermionic operators and $\\left\\\\{A_{ij}\\right\\\\},\\left\\\\{B_{ij}\\right\\\\}$, $1\\leq i, j \\leq N$ are the elements of appropriately chosen matrices $A, B$, which depend on $\\left\\\\{J_i\\right\\\\}_{1\\leq i \\leq N}$. We use periodic boundary conditions. Now we can extend $\\widehat{H}_N$ to a lattice of infinite size, by gluing the unit cell of size $N$ infinitely many times. Let us call this new extension $\\tilde{H}_N$. Now the questions: 1) What is $H.c.$? 2) I am interested in the thermodynamic limit $N\\rightarrow\\infty$. Is it obvious whether the sequence of operators $\\left\\\\{\\tilde{H}_N\\right\\\\}$ converges, say in strong operator topology, to some well-defined operator $\\tilde{H}$ as $N\\rightarrow\\infty$? Let me motivate the second question: For a certain sequence $\\left\\\\{J_i\\right\\\\}$, constructed deterministically with certain properties (so-called _quasi-periodic sequence_ ), I believe I can say something about what Physicists call the \"energy-spectrum in the thermodynamic limit\". I'm interested to know whether this energy spectrum is the spectrum (in the usual functional-analytic sense) of some operator $\\tilde{H}$. Thanks for any help!"}
{"id":"29749","title":"Gauge invariant but not gauge covariant regularization","text":"I'm not sure if someone's already asked this before, but I was wondering, in field theory,   1. when we say that a certain field is gauge invariant but not gauge covariant, what does this mean? In particular, in Wikipedia, the regulator of Pauli-Villars is said to be as such.    2. Moreover, as a consequence of not being gauge covariant, the Wikipedia article says that this regulator can't be used in QCD. How to see the link between not being gauge covariant and QCD here? And, why can one use it in QED then?"}
{"id":"119028","title":"Can one construct a new operator in terms of the powers of another operator?","text":"Suppose we have a quantum state, well described by its time-independent wave function Psi. And we have a well-defined Hermitian (self-adjoint) operator $A$. We successfully evaluate the expectation value of the operator $A$. Next we derive the general formula for the higher moments of $A$ (i.e. the expectation value of $A^n$ for $n=2,3,4\\ldots $). In this situation, is it permitted to regard each of the $A^n$ for $n=1,2,3,\\ldots$ as a proper operator by itself? For example, should every $A^n$ have a positive variance and other statistical properties (as long as we restrict ourselves to the state $\\Psi$)? Can one make linear combinations of different powers to construct a new operator, e.g. $B = A + A^2$? Is it allowed to construct new operators acting on $\\Psi$, that are defined in terms of their series expansion in $A^n$? For example, $D = \\exp(CA)$ where $C$ is a constant?"}
{"id":"92241","title":"periodicity of diffraction gratings","text":"Imagine we have a diffraction grating consisting of N slits (being N very large) with a separation of \"d\" from slit to slit. Now, we could regard this grating as a diffraction grating constructed by a repeated double slit, in which case the periodicity would be 2d. I realize this can't be legitimate since the theoretical positions of the maximums would change and this can't be (since the grating is doesn't change no matter how we regard it). Nonetheless, dealing with examples of defectuous grating exercises I have seen examples that regard gratings as repetitions of double slits or triple slits. So, my question is the following, why is it possible in some cases and not in others? I mean, why can't we regard the typical grating mentioned above as a repetition of double slits (or triples or whatever)? (my guess is that maybe we always have to choose the $\\textit{minimum}$ periodicity but I am not sure)"}
{"id":"74892","title":"Radio antenna producing waves in the visible spectrum","text":"If a radio could produce waves in the visible light spectrum, what would the result be? This is a thought experiment that I've pondered for a few years now. I realize there are a few\/many real-world constraints, but if we lifted these constraints for the sake of thought, what could we expect? Personally, I don't see why we wouldn't observe visible light emitting from the antenna, disregarding any light from Blackbody Radiation."}
{"id":"127282","title":"Physics of weird \"boing\" sound in racquetball courts?","text":"While playing racquetball, I frequently hear a very prominent \"boing\" sound (or more formally, a chirp). For example, you can hear it in this video when the ball hits the front wall. Does anyone know what the origin of this sound is, and why the pitch rises? Here is the spectrogram from the above video: ![enter image description here](http:\/\/i.stack.imgur.com\/G2tdWm.png) A careful examination shows that there are at least four linear chirps, which I've highlighted below. If you really listen carefully, all four of these are audible. (However I can only distinguish between the two high frequency chirps when the audio is played at half speed.) ![enter image description here](http:\/\/i.stack.imgur.com\/lzyGMm.png)"}
{"id":"111561","title":"Space-time Topologies?","text":"When it comes to questions of existence of bounds for PDE's and such, one must often make some assumptions regarding the topology of the space-time to use well known theorems. My question is two-pronged: i) I've often read, on Wikipedia (http:\/\/en.wikipedia.org\/wiki\/Spacetime) for example, that space-time is paracompact. I am aware of the mathematical definition and this seems counter-intuitive to me. Since the form of the stress-energy tensor entering Einstein's equations need only satisfy the conservation of energy $\\nabla_{\\mu} T^{\\mu \\nu} = 0$ condition from the Bianchi identities, how does one show this result? Can someone give me a reference for a proof? ii) In (electro)-vacuum, what other topologies are permitted globally? I have seen some papers finding solutions that resemble black holes (in the sense they have singularities, event horizons...) but are topologically not 2-spheres at their cross section (seemingly in violation of Hawking's theorem, see e.g, http:\/\/arxiv.org\/abs\/hep-th\/9808032, but perhaps the fact the space- time is asymptotically anti de-Sitter is why there is no violation). What space-times are compact? The spheres $S^{n}$ are all compact, but presumably even the Schwarzschild space-time is not globally? I think one can only have a compact space-time if its Euler characteristic $\\chi = 0$, can this be translated into a demand on $T^{\\mu \\nu}$? I am particularly interested in compact space-times, for one can then apply the Yamabe problem. Thanks!"}
{"id":"22652","title":"Calculating Ground State Energy in 1D Potential","text":"Given potential $V(x) = Asec(x)$ for $x > 0$. I want to calculate the ground- state energy $E_0$ via the Schrödinger equation. I'm completely stuck on this one. I've set up the time-independent Schrödinger equation, but it can't be solved without using special functions. I don't see how I can calculate the energy without solving the schrodinger equation. Any hints?"}
{"id":"31509","title":"If electromagnetic fields give charge to particles, do photons carry charge?","text":"As I understand these two statements:   1. An electromagnetic field gives particles charge   2. A photon is a quantum of electromagnetic field It must mean that a photon carries charge. But I guess it isn't true. Why not?"}
{"id":"26544","title":"What could this very dark planet be made of?","text":"I was reading about the planet TrES-2b which is less reflective than charcoal. What could possibly be its composition?"}
{"id":"106589","title":"What is the correct way to handle significant figures when calculating compound uncertainties?","text":"When processing experimental data, and calculating an uncertainty value in multiple steps, should intermediary uncertainties be used to a certain number of significant figures or kept to the full precision allowed by the calculator\/computer? I have a number of measured values and their associated uncertainties, and I have processed this data to give me a final value and uncertainty without ever limiting significant figures in either the values or uncertainties at each step. Is this the correct way to go about this?"}
{"id":"80701","title":"Sandstone getting soaked with water","text":"I have seen someone putting a sandstone in water. With only about 10% of the stone sitting in the water. One could see the stone getting soaked with water. So there must be a force, which lets the water climb up through the stone against gravity. What is that force? Or is there some other effect present?"}
{"id":"126487","title":"A question about the Thomson experiment","text":"Recently, I was studying about Thomson's experiment with cathode rays. My textbook shows it like this. ![enter image description here](http:\/\/i.stack.imgur.com\/GrPY3.jpg) It says: > When only electric field is applied, the electrons deviate from their path > and hit the cathode ray tube at point A. Similarly when only magnetic field > is applied, electron strikes the cathode ray tube at point C. But if we apply Fleming's Right Hand Rule, then we get the direction of force in the upward direction, so the rays should deviate towards A but they deviate towards C. I think it is because Fleming's Left Hand Rule is defined for conventional current(flow of positive charges) and what we are dealing with are negative charges. Is that correct?"}
{"id":"65208","title":"How large can planets or moons appear?","text":"In many artistic impressions or movies there are pictures or scenes where the sky is filled with an enormous moon (as seen from a planet) or vice versa. I wonder if there is an upper limit to the apparent size when viewed with the naked eye (no tele lens). Since the Roche Limit forbids celestial bodies coming too close to each other, there certainly is a limit to the apparent size. ![Artistic image of a large planet seen from a moon](http:\/\/i.stack.imgur.com\/B3mEm.jpg)"}
{"id":"114421","title":"Estimation of polarization perturbative term","text":"I'm studying a diagrammatic approach to degenerate electron gas. Now I need to prove that the energy contribution, with an arbitrary potential, of polarization at first perturbative order given by the diagram: ![Diagramma](http:\/\/i.imgur.com\/pECT8IX.png) is null. I expanded it as an integral: $$ \\Pi_{(1)b}^*(q)=\\lim_{p^o\\to 0^+}\\lambda\\int d^4k\\int d^4p\\text{ }(G^{(0)}(k))^2e^{ip^0\\eta}G^{(0)}(p)U_0(k-p)G^{(0)}(k-q) $$ $$ E^{(1)}_b\\propto\\int d^4q\\text{ }U_0(q)\\Pi^*_{(1)b}(q) $$ I've made several tries, but I can't prove it. Have you got any ideas? Thanks in advance."}
{"id":"32241","title":"Does a FTS work on the same principle as a michelson (amplitude division) interferometer?","text":"As far as I can tell within an Fourier Transform Spectrometer the spectral information is gained from changing the path length along one arm, this sounds very similar to a michelson interferometer but using two apertures instead of one. So are there any underlying differences between the two?"}
{"id":"32246","title":"What is the approximate electrical conductivity $\\sigma$ of graphene in S\/m or S\/cm?","text":"I am trying to find an approximate value of the electrical conductivity $\\sigma$ of graphene in units of S\/m or S\/cm. This table on Wikipedia gives $\\sigma$ values for a variety of materials (including references), but I do not see graphene. In the classic 2004 paper on graphene by Novoselov and Geim (Novoselov et al., _Science_ **2004** , _306_ , 666: available here), I see a plot of $\\sigma \\text{ (m} \\Omega^{-1})$ versus $V_g \\text{ (V)}$, where $V_g$ is the gate voltage: ![Electrical conductivity](http:\/\/i.stack.imgur.com\/0H2pT.gif) Based on that plot, what can we say about the electrical conductivity $\\sigma$? One confusing thing about the above plot is that it is in $\\text{m} \\Omega^{-1}$. But $1 \\text{ S} = 1 \\text{ } \\Omega^{-1}$, so in the plot is given in units proportional to S, not S\/m. Why is this?"}
{"id":"104252","title":"Which BICEP2 r value should be compared to Planck's r<0.11?","text":"The BICEP2 paper reports a tensor\/scalar ratio $r = 0.20_{-0.05}^{+0.07}$, but then says: Subtracting the various dust models and re-deriving the r constraint still results in high significance of detection. For the model which is perhaps the most likely to be close to reality (DDM2 cross) the maximum likelihood value shifts to $r = 0.16_{-0.05}^{+0.06}$ Which value is proper to compare with Planck's $r<0.11$?"}
{"id":"25387","title":"Is there evidence of a larger universe?","text":"While mapping out the position, distance and movement of galaxies and quasars at the limits of the observable universe, have astronomers ever observed the motion of these object due to gravitational sources that may reside outside the limits of the observable universe? If so, would this break the principle of information travelling faster than the speed of light? If not, are these objects in fact the furthest objects there are?"}
{"id":"25382","title":"How can we be sure that liquid iron\/titanium droplets can form around particulates on hot Jupiters - just as they do on Earth and Venus?","text":"I know that we have some solar system analogs. But is there a limit to it?"}
{"id":"2188","title":"Does 'electricity' have mass? Is 'electricity' tangible?","text":"**Background:** I'm in a legal academic discussion about the status of electronic 'goods' and whether they qualify as 'goods' in the same way a chair and a pen do. In this context (and specifically at the exact circumstance under discussion), it matters if electricity is 'tangible'. So far, most authors have blindly assumed electricity to be a flow of electrons, making a literal analogy with water, making statements such as:   * Information is stored in capacitors in the form of electrons. When a capacitor is filled more than 50% with electrons, it's considered to be 'on' (a bit with value '1').   * The information represented by a certain current (or rather, a series of on\/off currents) has mass, because it consists of the electrons that flow through the wire.   * A virtual object is tangible because it exists in memory in the form of electrons that are there (or not) in a certain pattern. Now I have a background in informatics but only a basic knowledge of electricity, and as good as none on the fundamental (physics) level. However I still feel this representation is wrong, and that you can't just say that information in a RAM chip has mass because it consists of electrons that are or are not in the capacitors on that chip. I have found hints in that direction on sites such as http:\/\/amasci.com\/miscon\/eleca.html#made, but I can't quite make out what 'electricity' _is_ and how it relates to _current_ and _potential_ and other words that are used interchangeably in these discussions but which are, I think, different things. So my questions are (all just different angles of looking at the same underlying concept):   * What is 'electricity', really, on a fundamental level; but explained in terms a layman can understand? Is there an analogy with other things that is accurate, unlike the 'flowing water' analogy, which is sufficient for high school level but is a simplification? (at least, I think...)   * Do 'electricity', an 'electrical current' and an 'electrical charge' have mass, apart from the object they are embodies in? Does the mass of a copper wire change when you put a current through it, because of the electrons coming in and going out?   * How do electrons fit into this? Is electricity composed of a bunch of electrons that flow through a mass? I think not, reading the link I gave before, but I don't quite understand what their role is.   * Most authors blindly assume that electricity is merely a flow of electrons through mass. In how far and under what assumptions is this correct?"}
{"id":"15436","title":"To what extent is the \"minimal substitution\" or \"minimal coupling\" for the EM vector potential valid?","text":"In all text books (and papers for that matter) about QFT and the classical limit of relativistic equations, one comes across the \"minimal substitution\" to introduce the magnetic potential into the equation (Schrödinger\/Dirac\/Klein-Gordon) through: $$ \\hat{p}^2 \\rightarrow (\\hat{p} - e \\hat{A})^2$$ The wording suggests that this is an approximation for small? electromagnetic fields (or at least not strongly coupled). I understand this was chosen such that the \"classical\" Lorentz force is retrieved from the Hamiltonian, but not why exactly this form and not another that leads to the same result.   1. To what extent is this \"approximation\" valid?   2. How can one improve this \"minimal substitution\"? Is there a more general expression, intuitively something like a series of the vector potential?"}
{"id":"57640","title":"Classical scattering of two particles by a Yukawa potential","text":"A point-like particle $A$, coming from minus spatial infinity, heads at another one, $B$, with an impact parameter of $b$. Initial momenta are $p_A$ and $p_B=0$. They repel each other via a Yukawa potential $$\\ \\ V(r_A,r_B)= +g^2\\frac{e^{-\\frac{|r_A-r_B|}{\\lambda}}}{|r_A-r_B|} \\ge 0.$$ > How does the angle of the incoming particle $A$ change, relative to how it > was before?"}
{"id":"60078","title":"How does current flow from the emitter, through the base and to the collector in a NPN transistor?","text":"So, I understand that for a NPN transistor to work the emitter-base junction needs to be forward biased and the collector-base junction needs to be reverse biased. I understand how current flows from the emitter to the base when forward biased, but I can't seem to wrap my head around how it flows from the emitter, through the base, and into the collector. From what I understand about the depletion region of a PN junction diode, when forward biased, the electrons in the N-type material are repulsed by the negative charge of the supply (say a battery) and the holes in the P-type material are repulsed by the positive charge from the supply, thus narrowing the depletion region and allowing electrons to flow from the N region to the P region. I also understand that the holes and electrons are attracted to the source charges when reverse biased, widening the depletion region and blocking any electrons from flowing. What I don't understand is what happens to the depletion region of the collector-base junction that allows electrons to flow from the emitter to the collector even though the collector-base junction is reverse biased. Take a simple common emitter circuit for example. I apparently am not cool enough to post pictures yet so you'll have to use your imagination. So pretend we have the base connected to the positive terminal on a 1v battery (Vbb). The collector is connected to the positive terminal on a 10v battery (Vcc). The emitter and both negative terminals are connected to ground. Also pretend that there is some kind of current limiting. How does current get from the emitter through to the collector? I can see how they are at different potentials, but I don't understand how this overcomes the reverse biasing of the collector- base junction. Can anyone tell me what actually happens at the junction to allow current flow and how changing Vbb will affect that current?"}
{"id":"102967","title":"Bogoliubov transformation with a slight twist","text":"Given a Hamiltonian of the form $H=\\sum_k \\begin{pmatrix}a_k^\\dagger & b_k^\\dagger \\end{pmatrix} \\begin{pmatrix}\\omega_0 & \\Omega f_k \\\\\\ \\Omega f_k^* & \\omega_0\\end{pmatrix} \\begin{pmatrix}a_k \\\\\\ b_k\\end{pmatrix}$ where $a_k$ and $b_k$ are bosonic annihilation operators, $\\omega_0$ and $\\Omega$ are real constants and $f_k$ is a complex constant. How does one diagonalise this with a Bogoliubov transformation? I've seen an excellent answer to a similar question here, but I'm not quite sure how it translates to this example. Any hints or pointers much appreciated."}
{"id":"106760","title":"What is the symmetry associated with electric charge conservation","text":"Is there a kind of symmetry that yields the conservation of charges? and if so , how it works for both type of charges? Electrical Charges"}
{"id":"113796","title":"Feynman Rules in Momentum space","text":"What's the difference between Feynman rules in momentum space for $\\phi^3$ theory and for $\\phi^4$ theory? I know it's only a slight difference and perhaps found in the vertex factor? But for some reason I can't seem to find this difference anywhere."}
{"id":"63527","title":"Acceleration by spherical particles (micron-scale) by an external force","text":"I am looking for an expression for the velocity of a micron sized (1 - 10 micron diameter) sized particles under accelerating forces. I have aerosols in mind. This is what I have in mind The resisting force acting on the particle is $F_D = 3 \\pi \\mu Vd$ so its (mechanical) mobility is $\\frac{V}{F_D} = \\frac{1}{3 \\pi \\mu d}$ This expresses velocity of a particle per unit force. Does that mean that if the same force acts on a bunch of different sized particles, the smaller particles should move faster due to their smaller size and the dependance is $\\frac{1}{d}$? Is this problem perhaps more complicated than this (full F=ma treatment)"}
{"id":"63529","title":"``integrated vertex operators\" in 1-loop open\/closed bosonic string amplitude","text":"This question is in reference to the first ~15 minutes of this String Theory lecture by Prof.Shiraz Minwalla, http:\/\/theory.tifr.res.in\/Videos\/strings28_24sep08.mp4 Can one give a reference (paper\/review\/book) for this (to my mind very technical) issue being explained here? I have seen Shiraz mention this issue in the previous 2-3 lectures too and its one of those rare concepts for which I could not understand his explanation. I can't relate as to what in Polchinski's book corresponds to this. (though this course claims to follow that book). * * * The final answer that Shiraz wants to get comes as two integrals (one for open and one for the closed case) on the top of the LHS of the board at 15:32 of the video. He claims something about being able to rewrite the moduli space integral in such a way that the b-c 0-mode ghost expectation value decouples from the \"integrated vertex operators\". I can't understand the point that he is making here about why since the ghost 0-mode is a constant one can evaluate it any any point and that somehow leads to some simplification etc. The initial few minutes of motivation part of the lecture is also not clear to me as to what is he trying to do - about the independence or not of the holomorphic and anti-holomorphic part of the b-ghosts and their boundary values on the worldsheet."}
{"id":"24169","title":"Explain these graphs of rotation and velocity of pucks on air hockey board","text":"I've been tasked to do a simple experiment on the elasticity of collisions. For this experiment I used two \"pucks\" (very light circular metal pieces of certain height but hollow) and a table that works like air hockey tables (decreases friction by blowing air from underneath). One puck was placed on this board and the other one was shot into it. Each puck had two reflective markers on them, one in the center and one on the edge. The positions of these markers were logged by two cameras shooting infrared light. I am now trying to understand this position data. This is the data I have (I'm using Mathematica):               m1Vel = Differences \/@ {m11x, m11y};     m2Vel = Differences \/@ {m21x, m21y};     m12Vel = Differences \/@ {m12x, m12y};     m22Vel = Differences \/@ {m22x, m22y};     m1DeltaX = m11x - m12x;     m1DeltaY = m11y - m12y;     m2DeltaX = m21x - m22x;      m2DeltaY = m21y - m22y;     angularVel[dy_, dx_] := Differences@ArcTan[dy\/dx]     vectorNorm2[list_] := Sqrt[list[[1]]^2 + list[[2]]^2];      Using this to plot position data for the pucks M1 and M2:               ListLinePlot[{m11x, m11y, m21x, m21y},       PlotLegend -> {\"M1 X\", \"M1 Y\", \"M2 X\", \"M2 Y\"}, LegendSize -> 0.5,       LegendPosition -> {1.1, 0}]      ![Position data](http:\/\/i.stack.imgur.com\/0a7e3.png) And then approximate the velocity for each puck, for the marker in the center:               ListLinePlot[{vectorNorm2[m1Vel], vectorNorm2[m2Vel],        vectorNorm2[m1Vel] + vectorNorm2[m2Vel]}, PlotRange -> Full,       PlotLegend -> {\"M1 v\", \"M2 v\", \"M1+M2 v\"}, LegendSize -> 0.5,       LegendPosition -> {1.1, 0}]      ![Velocity for center marker](http:\/\/i.stack.imgur.com\/r1jRV.png) And for the marker on the edge: ![Velocity for edge marker](http:\/\/i.stack.imgur.com\/FDS8P.png) And finally the rotation of each puck, using the approximation that the angle from a horizontal line is $\\mathrm{arctan}(\\frac{\\Delta x}{\\Delta y})$ and the angle velocity therefore the difference between the angle at one point and the angle at the next point, as seen in the function angularVel above.               ListLinePlot[{MovingAverage[angularVel[m1DeltaY, m1DeltaX], 10]^2,        MovingAverage[angularVel[m2DeltaY, m2DeltaX], 10]^2},       PlotRange -> Full, PlotLegend -> {\"M1 w\", \"M2 w\"}, LegendSize -> 0.5,       LegendPosition -> {1.1, 0}]      ![Rotation of each puck](http:\/\/i.stack.imgur.com\/bdVLN.png) **Alright, so what's the matter?**   1. I was expecting both of the velocity graphs to look like the first. Since the kinetic energy is proportional to the velocity squared, it is unacceptable to me that it goes up and down in the second velocity graph. It should decrease in a monoton manner. The first collision is with the other puck, but there are collisions after that with the walls.   2. The rotational energy is proportional to the angular velocity. I get that if in the collision with a wall some energy is transferred from translation to rotation and that the rotational energy therefore does not decrease in a monoton manner, but these really sharp peaks (even sharper without the moving average) I cannot understand. Since I'm studying the elasticity I really need only to understand what happens in the first collision. But I feel like a fraud if I write something up about that, neglecting the rest of the graph with all of its peculiarities. **If you had to explain these things in a report, what would you write?**"}
{"id":"71987","title":"Time ordering and time derivative in path integral formalism and operator formalism","text":"In operator formalism, for example a 2-point time-ordered Green's function is defined as $\\langle\\mathcal{T}\\phi(x_1)\\phi(x_2)\\rangle_{op}=\\theta(x_1-x_2)\\phi(x_1)\\phi(x_2)+\\theta(x_2-x_1)\\phi(x_2)\\phi(x_1),$ where the subscript \"op\" refers to operator formalism. Now if one is to take a time derivative of it, the result will be $\\frac{\\partial}{\\partial x_1^0}\\langle\\mathcal{T}\\phi(x_1)\\phi(x_2)\\rangle_{op}=\\langle\\mathcal{T}{\\frac{\\partial \\phi(x_1)}{\\partial x_1^0}}\\phi(x_2)\\rangle_{op}+\\delta(x_1^0-x_2^0)[\\phi(x_1),\\phi(x_2)]$, the delta function comes from differentiating the theta functions. This means time derivative does not commute with time ordering. If we consider path integral formalism, the time-ordered Green's function is defined as $\\langle\\mathcal{T}\\phi(x_1)\\phi(x_2)\\rangle_{pi}=\\int\\mathcal{D}\\phi\\phi(x_1)\\phi(x_2)e^{iS(\\phi)}$. Of course $\\langle\\mathcal{T}\\phi(x_1)\\phi(x_2)\\rangle_{op}=\\langle\\mathcal{T}\\phi(x_1)\\phi(x_2)\\rangle_{pi},$ as is proved in any QFT textbook. However in path integral case time derivative commutes with time ordering, because we don't have anything like a theta function thus $\\frac{\\partial}{\\partial x_1^0}\\int\\mathcal{D}\\phi\\phi(x_1)\\phi(x_2)e^{iS(\\phi)}=\\int\\mathcal{D}\\phi\\frac{\\partial}{\\partial x_1^0}\\phi(x_1)\\phi(x_2)e^{iS(\\phi)}$. I did a bit googling and found out that for the path integral case the time-ordered product is called \"$\\mathcal{T^*}$ product\" and operator case just \"$\\mathcal{T}$ product\". I am not that interested in what is causing the difference(still explanations on this are welcomed), because I can already vaguely see it's due to some sort of ambiguity in defining the product of fields at equal time. The question that interests me is, which is the right one to use when calculating Feynman diagrams? I did find a case where both give the same result, i.e. scalar QED(c.f. Itzykson & Zuber, section 6-1-4), but is it always the case? If these two formulations are not effectively equivalent, then it seems every time we write down something like $\\langle\\partial_0\\phi\\cdots\\rangle$, we have to specify whether it's in the sense of the path integral definition or operator definition. **EDIT:** As much as I enjoy user1504's answer, after thinking and reading a bit more I don't think analytic continuation is all the mystery. In Peskin&Schroeder chap 9.6 they manage to use path integral to get a result equivalent to operator approach, without any reference to analytic continuation. It goes like this : Consider a T-product for free KG field $\\langle T\\\\{\\phi(x)\\phi(x_1)\\\\}\\rangle=\\int\\mathcal{D}\\phi\\phi(x)\\phi(x_1)e^{iS(\\phi)}$. Apply Dyson-Schwinger equation, we get $\\int\\mathcal{D}\\phi(\\partial^2+m^2)\\phi(x)\\phi(x_1)e^{iS}=-i\\delta^4(x-x_1)$, then they just assume the $\\partial^2$ commute with path integration(which is already weird according to our discussion) and they conclude $(\\partial^2+m^2)\\int\\mathcal{D}\\phi\\phi(x)\\phi(x_1)e^{iS}=(\\partial^2+m^2)\\langle T\\\\{\\phi(x)\\phi(x_1)\\\\}\\rangle=-i\\delta^4(x-x_1)$. This is just the right result given by operator approach, in which $\\delta(x^0-x_1^0)$ comes from $\\theta$ function. Given my limited knowledge on the issue, this consistency looks almost a miracle to me. What is so wicked behind these maths? **Response to @drake** :If $a$ is a positive infinitesimal, then $\\int \\dot A(t) B(t) \\,e^{iS}\\equiv\\int D\\phi\\, {A(t+a)-A(t)\\over a}B(t)\\,e^{iS}=\\frac{1}{a}\\langle T\\\\{A(t+a)B(t)\\\\}\\rangle-\\frac{1}{a}\\langle A(t)B(t)\\rangle$, notice the second term has an ordering ambiguity from path integral(say $A=\\dot{\\phi},B=\\phi$), and we can make it in any order we want by choosing an appropriate time discretization, c.f. Ron Maimon's post cited by drake. Keeping this in mind we proceed: $\\frac{1}{a}\\langle T\\\\{A(t+a)B(t)\\\\}\\rangle-\\frac{1}{a}\\langle A(t)B(t)\\rangle\\\\\\=\\frac{1}{a}\\theta(a)\\langle A(t+a)B(t)\\rangle+\\frac{1}{a}\\theta(-a)\\langle B(t)A(t+a)\\rangle-\\frac{1}{a}\\langle A(t)B(t)\\rangle\\\\\\=\\frac{1}{a}\\theta(a)\\langle A(t+a)B(t)\\rangle+\\frac{1}{a}[1-\\theta(a)]\\langle B(t)A(t+a)\\rangle-\\frac{1}{a}\\langle A(t)B(t)\\rangle\\\\\\=\\frac{\\theta(a)}{a}\\langle [A(t+a),B(t)]\\rangle+\\frac{1}{a}[\\langle B(t)A(t+a)\\rangle-\\langle A(t)B(t)\\rangle]$ Now taking advantage of ordering ambiguity of the last term to make it $\\langle B(t)A(t)\\rangle$(this amounts to defining A using backward discretization, say $A=\\dot{\\phi}(t)=\\frac{\\phi(t+\\epsilon^-)-\\phi(t)}{\\epsilon^-}$), then the finally: $\\frac{\\theta(a)}{a}\\langle [A(t+a),B(t)]\\rangle+\\frac{1}{a}\\langle B(t)[A(t+a)-A(t)\\rangle]\\to \\frac{1}{2a}\\langle [A(t),B(t)]\\rangle+\\langle B(t)\\dot{A}(t)\\rangle$.(Here again a very dubious step, to get $\\frac{1}{2a}$ we need to assume $\\theta(a\\to 0^+)=\\theta(0)=\\frac{1}{2}$, but this is really not true because $\\theta$ is discontinuous) However on the other hand, since $a$ was defined to be a postive infinitesimal, at the very beginning we could've written $\\frac{1}{a}\\langle T\\\\{A(t+a)B(t)\\\\}\\rangle-\\frac{1}{a}\\langle A(t)B(t)\\rangle=\\frac{1}{a}\\langle A(t+a)B(t)\\rangle-\\frac{1}{a}\\langle A(t)B(t)\\rangle$, then all the above derivation doesn't work. I'm sure there are more paradoxes if we keep doing these manipulations."}
{"id":"71988","title":"Irreducible Representations of SO(n) tensors","text":"My interest is purely in $\\text{SO}(n)$ tensors and how one works out their irrep decomposition. For instance, for rank 2 tensors we simply split into an antisymmetric part, a traceless symmetric part and the trace. Is there a more general, recursive procedure for higher rank tensors? Even if not, what is the usual method when trying to do it for say rank 3 or 4? Any pointers to the literature would be more than helpful."}
{"id":"68323","title":"In Newtonian pressure, what type of function is force?","text":"This is pressure in Newtonian mechanics: $$P=\\frac {dF}{dA}.$$ What does this mean? (Doesn't it mean that force is a function of area?) What type of function is force?"}
{"id":"67312","title":"How are excess charges distributed over non-spherical conductors?","text":"My textbook gives the following explanation on how excess charges are spread over conductors: > The excess charge on an isolated conductor moves entirely to the conductor's > surface. However, unless the conductor is spherical, the charge does not > distribute itself uniformly. I was studying for a test and there was the following question, regarding the thin-wall conducting cylindrical shell below (which is coaxial to the rod inside it): ![enter image description here](http:\/\/i.stack.imgur.com\/jiLmD.png) $Q_1 > 0, Q_2 < 0$ > What is the charge on the interior and exterior surface of the shell? At first I thought the excess charge on the interior surface would be zero, since all the excess charge would move to the outer surface of the shell. But according to my textbook, it is as follows: > We consider a cylindrical Gaussian surface whose radius places it within the > shell itself. The electric field is zero at all points on the surface since > any field within a conducting material would lead to current flow (and thus > to a situation other than the electrostatic ones being considered here), so > the total electric flux through the Gaussian surface is zero and the net > charge within it is zero (by Gauss's law). Since the central rod has charge > $Q_1$, the inner surface of the shell must have charge $Q_{in} = -Q_1 = > -3.40 \\times 10^{-12}\\: \\mathrm{C}$. > > Since the shell is known to have total charge $Q_2 = -2.00\\: Q_1$ it must > have charge $Q_{out} = Q_2 - Q_{in} = - Q_1 = -3.40 \\times 10^{-12}\\: > \\mathrm{C}$ on its outer surface. So there is $-Q_1$ excess charge on the inner surface because said charges are being attracted by the rod? What if there was no rod, how would the excess charge be distributed over the shell's surface? Can I tell how the excess charge will be distributed over the surface of any non-spherical conductor, or only in special cases such as the above?"}
{"id":"31254","title":"Can the lightning be captured and used as power source?","text":"I would like to update my knowledge in this area, that is really out-of-dated and stopped somewhere like ten years ago. I asked the very same question on my physics lecture at my studies and got the answer, that although some tests and experiments were made (by French scientists?), there is no material, either natural or man-made, that would suffer direct lightning hit and therefore could be used to \"capture\" and store energy from lightning, for future use or processing of any kind. Can someone tell me, how does it looks now? Has anything changed during last ten years in this area?"}
{"id":"129921","title":"What does the speed of light have to do with mass and energy?","text":"I've been thinking about Einstein recently, and I came back to the famous equation e=mc^2. Why does this work? It's saying the difference between mass and energy happens to be the speed of light squared? I'm sure there's a larger explanation behind it, but the simplicity of the equation seems improbable."}
{"id":"96374","title":"Reference request for supersymmetry","text":"I have a good background in linear algebra, Topology and Differential geometry. I would like to understand the concept of supersymmetry in theoretical physics, and hopefully read Witten's landmark paper _Supersymmetry and Morse theory_ sometime later. Please suggest suitable references. Thanks"}
{"id":"109879","title":"Does the LSZ reduction method prove gauge-independence in massless gauge theories?","text":"I've been working my way through L. Baulieu's excellent paper [ _Perturbative gauge theories_ , Physics Reports, Volume 129, Issue 1, December 1985, Pages 1-74]. Towards the end, he goes on to prove that renormalized Yang-Mills theories that are BRST invariant exhibit gauge independence and unitary. At the end of the proof he states: > _One should note that we have implicitly assumed in this demonstration the > existence of the S-matrix, i.e., the applicability of LSZ reduction formula > [...] the proof is only valid when all gauge bosons are massive. Whenever > massless gauge bosons are present in theory, all our arguments become > formal._ In the case of QCD, the gluons are massless, so does the LSZ formula not apply to QCD? Is there a formal argument that proves gauge-independence as he suggests?"}
{"id":"100154","title":"Good Source to Understand Angular Momentum","text":"I am looking for a good source to understand angular momentum. I know the basics but I am looking for a sound in-depth knowledge like directions of angular momentum, when it is not parallel to angular velocity, differences between normal angular momentum and spin angular momentum, how one is intrinsic property and other is not, etc. Can you please help? Any book, any video, any website etc.? Thanks for your help."}
{"id":"91186","title":"Is it expected tha all stellar black holes will be spinning near the maximum allowed $\\omega$-velocity?","text":"Using a bit of classical reasoning I'm imagining black hole formation to be much like an ice skater pulling in her arms: ![skater pulls in her arms to increase angular velocity](http:\/\/i.stack.imgur.com\/9fiOU.png) Now, the size difference between a star and its black hole can't even be effectively captured in an image. The black hole for our sun would be much less than a pixel in this image: ![size comparison of our sun and planets](http:\/\/i.stack.imgur.com\/XoKDr.png) That suggests to me that even a very slowly rotating stars would have much more angular momentum than could be supported by their resulting black holes. I haven't done the calculation because I don't really understand the Kerr metric but even with a bunch of classical hand-waving I'd think that just about every black hole formed in a stellar-collapse would be spinning maximally. So my question is, do we expect nearly all black holes to be spinning maximally? If so, (roughly) how much angular momentum is lost because the star had much more than the black hole could support? And, how is all of this extra angular momentum shed during collapse? Is it just in the form of tons of matter being ejected until the the angular momentum is low enough to allow for the formation of a Kerr black hole?"}
{"id":"109876","title":"Doubt about states of matter","text":"What does exactly means \"Small molecules may appear as solid, liquid, and gaseous phases without losing their molecular integrity\"?, I can't image how just a molecule can be a gas, liquid or solid. Or in fact the previous phrase refers to the state of aggregations of those small molecules? My source is this article: B Wunderlich, A classification of molecules, phases, and transitions as recognized by thermal analysis, Thermochimica Acta, Volumes 340–341, 14 December 1999, Pages 37-52 If you could suggest me a reference that extend the previous one to more modern states of matter like Bose-Einstein condensates, i would really appreciate it."}
{"id":"91231","title":"Microscopic explanation of optical activity","text":"The origin of linear birefringence in crystal can be easily explained by the symmetry of the crystal. However, it seems it is hard to be applied in circular birefringence (i.e. optical activity), since there is no macroscopic symmetry in sugar solutions. So I wonder how we explain the levorotatory and the dextrorotatory in atom level POV?"}
{"id":"91232","title":"Temperature of thermally isolated space region","text":"If we thermally isolate a region in space, say using a hypothetical material of $0$ conductivity, and measure the region's temperature, will it be 2.7K?"}
{"id":"103766","title":"Why does water gulp out of a water bottle with a narrow opening instead of a steady flow?","text":"For example, take a water bottle. Fill it with water and then turn it upside down. Instead of flowing steadily downward, it gulps down in parts. Why?"}
{"id":"66045","title":"$Q$-value in nuclear reaction $^{9}Be\\left(\\gamma,n\\right)^{8}Be$","text":"I have two questions. **First** , I understand that in a nuclear reaction $$Q:=K_{after}-K_{before}\\equiv E_{0,before}-E_{0,after} \\qquad (1)$$ where $K$ is the total kinetic energy, and $E_0$ is the total rest energy. My question is, in the reaction $^{9}Be\\left(\\gamma,n\\right)^{8}Be$, where should I put the energy of the $\\gamma$-photon? I mean, conservation of energy says $$Mc^{2}+h\\nu=mc^{2}+M'c^{2}+K_{n}+K' \\qquad(2)$$ where $M$=mass of $^{9}Be$, $m$=mass of the neutron, $M'$=mass of $^{8}Be$, and $K_{n}$, $K'$ are the kinetic energies of the neutron and of the $^{8}Be$, respectively. Then, using (1) (with the rest energies version) I wrote $$Q=(M-m-M')c^2 \\qquad (3)$$ but this is equivalent to: (using (2)) $$Q=K_{n}+K'-h\\nu \\qquad (4)$$ So, my **1st question** is: is (4) the correct expression for $Q$? **Second** , I want to calculate the kinetic energies $K_n$ and $K'$. In order to do this, I'm thinking to use (4) (if it's correct) with $K_{n}=\\frac{1}{2}mv^{2}$ and $K'=\\frac{1}{2}M'V^{2}$, and the conservation of linear momentum $$\\frac{h}{\\lambda}=mv+M'V \\qquad (5)$$ Assuming that the neutron and the $^{8}Be$ continue moving in the direction of the $\\gamma$-photon. So, my **2nd question** is: can I make this last assumption? Or in other words, is (5) a correct expression for the linear momentum conservation?"}
{"id":"29961","title":"Transforming a sound wave into a literal light wave. Is it possible?","text":"Literally transforming sound into an actual light wave seems almost impossible. But transforming the sound wave into a light wave while containing a single mass? I know that sound can contain a mass of solid, liquid or a gas type. but does light have a mass at all? Is it possible to literally transform sound into light while containing a single mass? I have a experimental project. I want to know if it's possible in the slightest way? let me figure out how, just for the fun of the research."}
{"id":"111076","title":"Is light and sound waves interconvertible?","text":"Light is produced when an air bubble underwater is collapsed with sound wave. Why does it happen so?"}
{"id":"68381","title":"Where did Schrödinger solve the radiating problem of Bohr's model?","text":"One of the problems with Bohr's theory to describe the hydrogen atom, was that the electron orbiting around the nucleus has an acceleration. Therefore it radiates and loses energy, until it would collapse with the nucleus. Now Schrödinger describes the electron as a wave function. His theory is able to describe all atoms (in contrast to Bohr's model), but how is the radiation problem solved? I understand that the wave has not an exact position in the time anymore. But the electron still \"moves\", so it has an acceleration anyway (because of vibrations or so). Why is in this theory the electron not radiating anymore? And if it is, why doesn't the atom collapse?"}
{"id":"132688","title":"Is the harmonic oscillator potential unique in having equally spaced discrete energy levels?","text":"I was wondering if the good old quadratic potential was the only potential with equally spaced eigenvalues. Obviously you can construct others, such as a potential that is infinite in some places and quadratic in others, but that's only trivially different. I am not referring to equally spaced as a limiting behavior either, I mean truly integer spaced. Any ideas? If not, is there a proof for its uniqueness?"}
{"id":"65124","title":"can be exist the negative mass?","text":"I'm not sure about this but I guess there must be negative masses in the universe because of the symmetry. If the gravity is one of the main forces in nature it must has negatives mass to be able to uniform with other forces. I think the m^2 in the relative energy E^2=P^2c^2+(m^2)c^4 refer to it and it must corrected that E=|m|c^2"}
{"id":"86285","title":"If space and time are equivalent, what's Spin in time dimension","text":"This troubles me: We are talking about time and space being equivalent, but still only consider Spin in the $x$, $y$ or $z$-direction. What's Spin in time dimension? Is it distinction between particles and antiparticles?"}
{"id":"69302","title":"Writing wave functions with spin of a system of particles","text":"Suppose I have 2 fermions in a potential $V(x)$. Both particles are moving in one dimension: the $x$ axis. Then, neglecting the interaction between the particles, the spatial wave function of the system would be of the form $$\\psi_{n_{1}}(x_{1})\\psi_{n_{2}}(x_{2}) $$ Now, if I'm considering particles with spin 1\/2, the notation $\\alpha(1)$ indicates that the particle 1 has spin up, and $\\beta(2)$ denotes the particle 2 having spin down. Now, I want to write the complete wave function, a function of the form $$\\psi_{n_{1}n_{2}s_{1}s_{2}}(x_{1},x_{2},s_{1},s_{2})=\\psi_{n_{1}}(x_{1})\\psi_{n_{2}}(x_{2})F(\\alpha,\\beta)$$ where $F(\\alpha,\\beta)$ is a function of the spin of the system. To this end, I have that the only physically possible functions $F(\\alpha,\\beta)$ are: Symmetric: $\\chi_{\\alpha}:=\\alpha(1)\\alpha(2),\\quad\\chi_{\\beta}:=\\beta(1)\\beta(2),\\quad\\chi_{+}:=\\frac{1}{\\sqrt{2}}\\left[\\alpha(1)\\beta(2)+\\alpha(2)\\beta(1)\\right]$ Antisymmetric: $\\chi_{-}:=\\frac{1}{\\sqrt{2}}\\left[\\alpha(1)\\beta(2)-\\alpha(2)\\beta(1)\\right]$ In order to write down the complete wave function with spin, I understand I have to consider the energy levels. For example the ground state: $\\psi_{1}(x_{1})\\psi_{1}(x_{2})$. If $\\psi_{1}(x_{1})\\psi_{1}(x_{2})$ is symmetric ( _as I understand it is_ ), then I must multiply this function times the antisymmetric function $\\chi_{-}$ (in order to get an antisymmetric wave function, for two fermions). If $\\psi_{1}(x_{1})\\psi_{1}(x_{2})$ is antisymmetric (and I understand this is impossible, since the ground state is not degenerate), then I'd have 3 wave functions, obtained by multiplying $\\psi_{1}(x_{1})\\psi_{1}(x_{2})$ times $\\chi_{\\alpha}$, $\\chi_{\\beta}$ and $\\chi_{+}$. Now, for the 1st excited level, say $\\psi_{2}(x_{1})\\psi_{1}(x_{2})$, **my question is** , what happens when this function is not symmetric neither antisymmetric? I mean, I could built a symmetric $$f_S=\\frac{1}{\\sqrt{2}}\\left[\\psi_{2}(x_{1})\\psi_{1}(x_{2})+\\psi_{2}(x_{2})\\psi_{1}(x_{1})\\right]$$ or an antisymmetric $$f_A=\\frac{1}{\\sqrt{2}}\\left[\\psi_{2}(x_{1})\\psi_{1}(x_{2})-\\psi_{2}(x_{2})\\psi_{1}(x_{1})\\right]$$ wave function. But which one of these must I choose? Or, must I calculate the resultant complete wave functions with both? Then, when I count the states with the 1st excited energy, I'd have 4 instead of 1 or 3."}
{"id":"130688","title":"Which way does the scale tip?","text":"I found the problem described in the attached picture on the internet. In the comment sections there were two opposing solutions. So it made me wonder which of those would be the actual solution. So basically the question would be the following. Assume we would have two identical beakers, filled with the same amount of the same liquid, lets say water. In the left beaker a ping pong ball would be attached to the bottom of the beaker with a string and above the right beaker a steel ball of the same size (volume) as the ping pong ball would be hung by a string, submerging the steel ball in the water as shown in the picture. If both beakers would be put on to a scale, what side would tip? According to the internet either of the following answers was believed to be the solution.   1. The left side would tip down, because the ping pong ball and the cord add mass to the left side, since they are actually connected to the system.   2. The right side would tip down, because of buoyancy of the water on the steel ball pushing the steel ball up and the scale down. Now what would the solution be according to physics? ![enter image description here](http:\/\/i.stack.imgur.com\/Mj3Y0.jpg)"}
{"id":"65335","title":"How do moving charges produce magnetic fields?","text":"I'm tutoring high school students. I've always taught them that: > A charged particle moving without acceleration produces an electric _as well > as a magnetic field_. It produces an electric field because it's a charge particle. But when it is at rest, it doesn't produce a magnetic field. All of a sudden when it starts moving, it starts producing a magnetic field. Why? What happens to it when it starts moving? What makes it produce a magnetic field when it starts moving?"}
{"id":"75082","title":"Why are magnetic fields only produced by moving charges?","text":"Why do charged particles only produce magnetic fields while in motion?"}
{"id":"110805","title":"Is a magnetic field just a moving charge?","text":"I am trying to understand electromagnetism. I would like to pare myself down to the minimum number of necessary concepts. From my physics education (high school only), I understand that a moving charge **creates** a magnetic field. The implication is that magnetism is a discrete concept that I also need to understand. However, is this view actually confusing: is it the case that a magnetic field **is** a moving charge? Either way, a link to a primer accessible to someone with a engineering degree level maths would be wonderful. If it is the case magnetism **is** a moving charge, then an introduction following this approach would be wonderful. Thank you!"}
{"id":"87818","title":"Why don't stationary electric charges possess a magnetic field?","text":"Why don't stationary electric charges posses magnetic field, while moving charges do?"}
{"id":"51346","title":"No magnetic field from a static charge - Is there a simple physical argument to show why?","text":"For a charge moving in an electric field $\\vec E$, its equation of motion is given by the electric part of the Lorentz force $$\\frac d {dt}\\gamma m \\vec v = e\\vec E$$This comes from the conservation of relativistic energy in a static electric field. But a magnetic field would still make this conservation law true since the magnetic force is always orthogonal to the velocity of the charge and therefore doesn't change its energy. Is there a simply physical argument that shows why a static charge doesn't create a magnetic field?"}
{"id":"71763","title":"Why doesn't mass of bob affect time period?","text":"Please correct me if I'm going wrong -   By the gravitation formula: $F = \\frac{G m_1 m_2}{r^2} $, So if the mass of a bob is greater then the torque on it should increase because the Force increased now by a very small magnitude. So it should go faster and thus the Time period should be lesser. But my Physics book says that Time period is only affected by effective length and $g$ Why doesn't mass of bob affect it?"}
{"id":"95018","title":"What is meant by a local Lagrangian density?","text":"What is meant by a local Lagrangian density? How will a non-local Lagrangian look like and what is the problem that we do not consider such Lagrangian densities?"}
{"id":"6750","title":"How can earthquakes shift the earth's axis?","text":"One often comes across news articles that claim that an earthquake shifted the earth's axis. > > http:\/\/news.google.com\/?q=earthquake%20shifted%20OR%20shifts%20earth%27s%20axis If you ignore the influence of other celestial bodies, an internal event like an earthquake surely can't change the direction of the angular momentum of the Earth (unless stuff is ejected out of Earth), since angular momentum has to be conserved in the absence of an external torque. So the axis has to remain fixed. Am I missing something? Or are geologists trying to say that the resulting movement of tectonic plates causes a change in the point of intersection of the axis (which remains the same) and the plates that include the poles, so that it _seems_ as if the axis has shifted? **EDIT** Some articles mention the value of the shift in the axis and also the change in the length of the day. If, as Ted Bunn's answer indicates below, the shift in the axis isn't actually real but is because of the movement of tectonic plates with respect to the axis, shouldn't the shift be different at the north and south poles? How are the shifts and the change in day-length calculated?"}
{"id":"92763","title":"Stringy corrections to Friedmann equation","text":"Does anyone know a reference or a paper which discusses string theory correction to Friedmann equations?"}
{"id":"32723","title":"Can one make a ball rotate around a vertical axis using only a combination of horizontal axis rotations?","text":"This is a nice problem that I would like to share. **Problem:** In a public garden, there a statue consisting of a spherical stone and a stone cup. The ball is 1 meter in diameter and weighs at least a ton. The cup is an upside-down hemispherical shell, and the ball sits in this shell and fits it almost exactly. Water is pumped into the bottom of shell so that a thin film exists between shell and the ball. The result is a ball that is free to rotate with negligible friction. You only have access to the ball near the top, so while you can push it to make it turn around any horizontal axis, you can't get enough of a grip to make it turn around the vertical axis. Can you impart a net angular momentum around the vertical axis anyway, so that the balls spins around the vertical axis? **Source:** _Vector Calculus, Linear Algebra, and Differential Forms_ by Hubbard and Hubbard."}
{"id":"101185","title":"Difference between RPA and generalized RPA","text":"The random phase approximation (RPA) is an approximation method in condensed matter physics and in nuclear physics. What is the difference between RPA and generalized RPA?"}
{"id":"123703","title":"String Theory and Fourier Analysis","text":"Me and my friend, both many years from learning string theory, had a recent debate about it anyway. He said he already partially discounts it because after learning waves, he believes any function, and thus any kind of physical process can be created through a superposition of an infinite sum of waves. Since string theory to our understanding is made up of primarily oscillating strings, he argues that it is a mathematical necessity for string theory to create equations for the physical world, regardless of whether the theory actually is \"true\" or not. My understanding is that Fourier analysis can only create arbitrary periodic functions, can it in principal create any real world field or particle, etc.? Besides this, I argued that this might be a pointless question anyway, as if we can create a perfect model of the universe with string theory, then the question of whether the universe is really made of oscillating strings is more of a philosophical question that may not mean anything at all. Through reading Brian Green's books I know that M-theory at least incorporates more than just strings, but I have no clue as to how things such as branes work, can his argument even apply to these other fundamental objects in string theory? So my question is, is Fourier Analysis essentially what String Theory is? And if so, does that make it any less of a true physical theory of the universe?"}
{"id":"131338","title":"Which electromagnetic radiation is faster in water, microwaves or light?","text":"Well I've been asked this question, but I haven't been able to come with an answer yet using books and some web searches. The point is as the title says, to answer the question with the whole phenomenon explanation if possible."}
{"id":"131333","title":"How can helicity be conserved but chirality not?","text":"I read in a book that for $\\beta$-decay the electrons have always been found to have an expectation value for their helicity of $h=-v\/c$. Then ist is said in the book, that it follows from this fact that such electrons are in a left-handed chiral state which is characteristic for the weak interaction. In another article I read that the chiral state of an electron is not conserved in time. The electron will soon evolve a component with a right- handed chiral state and it will be a mixture of right- and lef-handed chiral states. Suppose after the decay one electron moves like a free particle.   When it evolves a right-handed chiral component in addition to the left-handed component it starts off with, how can its helicity be conserved?"}
{"id":"75896","title":"Twist of null Killing fields","text":"I have a (hopefully) quick question: is it possible to have a null Killing field $\\xi ^ \\mu$ such that the twist 1-form $\\omega_{\\mu} = \\epsilon_{\\mu\\nu\\alpha\\beta}\\xi^\\nu \\nabla^\\alpha \\xi^\\beta \\neq 0$ but the exterior derivative $(d \\omega)_{\\mu\\nu} = 2\\nabla_{[\\mu}\\omega_{\\nu]} = 0$? Or does $(d \\omega)_{\\mu\\nu} = 0$ always imply $\\omega_{\\mu} = 0$ for a null Killing field?"}
{"id":"75893","title":"Photoelasticity and existence of stress","text":"Shall we consider photoelasticity a non-mathematical, or purely visual, proof of the existence of stresses in mechanical structures undergoing external loading?"}
{"id":"130955","title":"Can we introduce the mass of a quantum field as an interaction?","text":"At a free massless Lagrangian \\begin{equation} L_0 = \\frac 1 2 ( \\partial \\psi)^2 ,\\end{equation} add an interaction term \\begin{equation} L_I = \\frac 1 2 m^2 \\psi^2\\end{equation} where m is small enough to converge its perturbation calculation. From these setup, is it possible to gain the same results as of the free massive Lagrangian? Please point me a book description or some internet site. Thanks."}
{"id":"103103","title":"Info request on studying QIT\/QIS or QM with a Computer Science background","text":"I've been considering a career change for a long time and recently discovered the Two-Slit Experiment, which, to put it frankly, blew my mind. I then started some hefty reading and investigation into all things Quantum (Bell, Campbell, entanglement, QIT), which has led me here. I have 20 years experience in IT working as a programmer, having graduated with a Bachelor in Computer Science back in the early 90s. As mentioned, I have been considering a career change for some time (having become quite burned out in this industry), and have found something that has piqued my interest more than anything else (tho I did consider Astrophysics a couple of years ago). I am not yet sure which area of \"Quantum\" I will be most drawn to, possibly Quantum Information Theory\/Science. I am aware that there is a significant amount of Maths and Physics pre- requisites involved. It seems likely, given that I have not done any Maths or Physics study probably since I was doing my CS undergrad, that I will need to start over with an undergrad in Science\/Physics in order to get the fundamentals. So my questions are:   1. What is the ideal path of education to get to QIT\/QIS or QM?   2. Is the path to QIT\/QIS also via the Physics route? Can I leverage my CS background?   3. Resources (books, online courses, etc) that would help with the transition from CS-type thinking to Physics\/Maths-type thinking. Apologies if this encompasses elements too broad or off-topic, I'm trying to get a better understanding on what is to come following this path. Thanks in advance."}
{"id":"70115","title":"Is there a phenomenon where physicists are only interested in the standard deviation of the quantity to be measured?","text":"or a phenomenon where we can only measure the standard deviation ($\\sigma_w$) of a variable $w$ and not the mean $\\overline{w}$"}
{"id":"109610","title":"Electric and Water Flow Analogy","text":"I’ve been studying electricity lately and it has been quite hard to understand the terminology, and picturing it in my mind. After struggling to understand what each property of electric currents meant, I found an analogy that related electric current to water flow. I am not sure if this analogy is 100% accurate, but it sure has helped me understand the complex (at least for young, starter scientist like me) topic of electricity. This far I have managed to relate in the analogy that: \\-----Electricity-------------------------Water------------ Current (Amps)(A)-------Flow rate (Volume\/Time) Resistance (Ohms)(Ω)----pipe size (Inches) Voltage (Volts)(V)------Water pressure (psi) But I am still missing properties like charge. What properties am I missing? How could it be related to water flow? If it can’t be related, can you please give me a short definition and explanation to better grasp the concept? Please include units in both sides of the analogy. Also, please keep the answer short and simple for us, the inexpert scientist (including me) that struggle to grasp the concept. **Thanks a bunch for your expert help** It is kind of discussed in these articles: Flow of water and flow of electrons, how this analogy works? Power in hydraulic analogy But I couldn’t find (or understand) the answer provided in them."}
{"id":"33098","title":"Why does it seem that the potential difference dependence of capacitance and total energy stored in a parallel-plate capacitor are contradictory?","text":"Consider a parallel-plate capacitor. Charge is stored physically on electrodes (\"plates\") which are flat and parallel to one another. If one electrode has charge $+Q$ and the other electrode has charge $-Q$, and $V$ is the potential difference between the electrodes, then the capacitance $C$ is $$C = \\frac{Q}{V}$$ (This definition of $C$ is given in, for example, _Introduction to Electrodynamics_ by David J. Griffiths.) But, now, let's think about the energy stored in the electric field between the electrodes of this parallel-plate capacitor. As stated in Griffiths on page 105, \"How much work $W$ does it take to charge the capacitor up to a final amount $Q$?\" It turns out that $W$ is $$W = \\frac{1}{2} CV^2$$ So: (i) the capacitor's capacitance $C$ goes like $\\frac{1}{V}$; and (ii) the energy $W$ stored in the electric field goes like $V^2$. **Are statements (i) and (ii) at odds with one another?** I am sure that they _cannot_ be. But conceptually I am having difficulty. We desire high capacitance -- we want to put as much charge on the electrodes as possible, because if we accomplish this, then I think that will increase the energy density of the system. But is what I just said true? If we manage to increase $Q$, then by $V = \\frac{Q}{C}$, the potential difference $V$ between the plates will also increase. This, I think, is why capacitor electrodes are separated by a material (such as a polarizable dielectric material like a slab of plastic); otherwise $V$ will become too large and the breakdown voltage will be reached, generating a spark. But, now, the equation $W = \\frac{1}{2} CV^2$ (where I think that $W$ can be conceptualized as the energy stored in the electric field between the electrodes) seems to say that as $V$ increases, so does the energy $W$, quadratically. So, my question is, **do we want a capacitor to have a _large_ potential difference $V$ or a _small_ potential difference $V$?** If $V$ is large, then $W$ is large (which we want), but $C$ is small (which we do **_not_** want). Am I somehow thinking of two different potential differences $V$ and confusing them?"}
{"id":"33090","title":"Coupling of vector gauge and a massive tensor field","text":"I was reviewing the paper-Coupling of a vector gauge field to a massive tensor field In the calculation I found the term $ 2\\mu^2 \\varepsilon^{ijk} \\dfrac{\\partial_j}{\\partial^2}B_k\\dot{B}$ which needed to be integrated by parts. How can I do this? Please help me. _Note_ : Here $B_k$ is the space component of an antisymmetric tensor field and $\\dot{B}$ is the time derivative of $B_k$. ( _Edit: corrected the index_ )"}
{"id":"122038","title":"Why is parasitic capacitance in inductor said to be in parallel?","text":"Internal resistance of inductance (or other devices) are said to be in series. But parasitic capacitance is said to be in parallel (in case of an inductor). Why is that so? What determines whether an internal property is in series or parallel?"}
{"id":"122035","title":"Behavior of multiple flat coils in a cylinder","text":"I am trying to wrap my head around how this would work. The setup is quite simple - there are multiple flat coils sealed in insulated disks, and an active coil with alternating current underneath. All is placed in a cylinder. The question is, how will the disks behave - all should induce the same magnetic field and thus repel each other. Will this force be stable enough, when the field collapses on each pulse? Thanks!"}
{"id":"122037","title":"Particle Collision with Static System","text":"I have a system of particles with equal distance with each other and another at random positions which is moving with time. What I want to know is :   1. The method by which I can reduce the number of particles from the first system, as I know the maximum motion of any particle is X ( **Meaning: Maximum displacement from starting to end of simulation is X)**.   2. How to efficiently calculate the collision between them, I have heard about quadtree and octree but (as per I understood till now) they are for collision of particle from each other. In my problem,second system of particles doesn't collide with each other. Note: Sorry, if it is a very basic question, I am (very) new to this field. **Update:** ![enter image description here](http:\/\/i.stack.imgur.com\/ZTixa.png) ## **The combined System** ![enter image description here](http:\/\/i.stack.imgur.com\/uEgwX.png) ## **Movable parts (Same color means same radius and mass)** ![enter image description here](http:\/\/i.stack.imgur.com\/i3whf.png) ## **Unmovable Grid (Colors means nothing and yes, it's a cube of spheres)**"}
{"id":"107820","title":"Relativistic addition of velocities of spaceships","text":"If Spaceship 1 is traveling at speed $.5c$ relative to Earth, and Spaceship 2 is traveling at speed $.3c$ relative to Earth in the same direction, what does Spaceship 2 see Spaceship 1's speed as? I don't think it is quite as simple as $.5c-.3c = .2c$."}
{"id":"36113","title":"Was the Universe's entropy equal to zero at the Big Bang? Is zero-entropy state unique?","text":"It is postulated by many cosmologists that at the Big Bang time the universe was in an unusual low entropy state. Does this claim specifically mean that the entropy of the initial universe was zero? Is zero-entropy state unique for given physical laws? Is it possible that entropy was growing always so that only difference in entropy has physical meaning rather than absolute value? Was there ever negative entropy state?"}
{"id":"41527","title":"Find total energy and momentum of an moving electron in a rest frame","text":"I have an electron moving with speed $u'$ in a frame $S'$ moving with speed $v'$ relative to a rest frame $S$. How do I find the total energy and momentum of the electron in the rest frame $S$? I thought the equations were: $E_{total} = \\gamma \\times mc^2$   $p = \\gamma \\times mv$ But, that doesn't look right... Could someone point out to me what is wrong here?"}
{"id":"33255","title":"Are the only observables in string theory the S-matrix?","text":"Is the S-matrix the only observable in string theory? What about time varying spacetime backgrounds, or thermal states then?"}
{"id":"129417","title":"What is pseudo tensor?","text":"What is the pseudo tensor in relativity? How do we transform tensor and pseudo tensor under parity?"}
{"id":"7149","title":"Why does nuclear fuel not form a critical mass in the course of a meltdown?","text":"A BWR reactor core may contain up to 146 tons of uranium. Why does it not form a critical mass when molten? Are there any estimates of the critical mass of the resulting zirconium alloy, steel, concrete and uranium oxide mixture?"}
{"id":"18745","title":"Why quantum states are classified using only conserved quantities?","text":"While studying quantum mechanics from standard textbooks I always felt some conceptual gap that was never mentioned or explained. In what follow I tried to formulate my question, please be patient with me. For a quantum particle in an infinite potential well the stationary states are labelled by the quantum number $n$ which labels the eigenenergies. An eigenenergy, that corresponds to a stationary state, does not change with time, hence is a conserved quantity. For a spinless electron in Coulomb potential, to model the hydrogen atom, again we have the same story, the stationary states are labeled by the quantum numbers $n$, $l$, $m$ which corresponds to conserved quantities. My question is rather general since I am trying to understand conceptually why only conserved quantities are used to label the quantum states. I mean how would someone think in advance that he has to look for conserved quantities, and then use such conserved quantities to label the states ?"}
{"id":"71798","title":"Scanning electron microscope imaging","text":"In a scanning electron microscopy, secondary electrons are defined as the electrons which obey inelastic scattering whereas backscattering electron follow elastic scattering. Now my question > Can backscatter electrons produce same level of solution as secondary > electrons?"}
{"id":"81445","title":"Negligible Mass String","text":"I've recently been working through a lot of physics problems and a lot of them say to assume that the mass of the string used in a problem involving a pulley, for example, is negligible. Why is this important? What would happen if the mass of the string wasn't negligible?"}
{"id":"7491","title":"Is it possible to work on physics independently outside academia?","text":"The traditional physics career is an academic job at some university, with the eventual goal of becoming a tenured professor. Is it possible for a mostly self-educated outsider working outside academia to come up with significant results in physics? Let's take significant to mean accepted in a high ranking peer review journal with a high citation count. Let's just say due to external life circumstances, the academic path is unfeasible. Are there any examples of notable physics results coming from outsiders? e.g. a third class patent clerk coming up with light quanta, Brownian motion and relativity."}
{"id":"44967","title":"Mass points of a Mass-spring model","text":"Let's say I have a mass spring model like the one in the picture below: ![enter image description here](http:\/\/i.stack.imgur.com\/JXnUn.png) So, there are 3 parts of the spring joined together in an _equilateral_ triangular manner. Each of the joints has a mass of $m$. The resting length of each of the springs is $l$. The top joint point of the spring is fixed to the ceiling. Now, if I were to pull both the lower points of the spring model such that each of the springs extend proportionally by $\\Delta l$ change of length, and release. Now, I want to find an equation for Point A (indicated in the picture above) under gravity and sprint forces when springing back to position. The assumption is all the angles remain at $60$ degrees in every iteration when it springs back. What I did is: Let $k$ be the elasticity of the spring. Then, for the X-axis component of the equation, $k \\cdot \\Delta l \\cdot cos(60) + k \\cdot \\Delta l = m\\cdot a_x$ The acceleration of the spring going back to original x position would then be dividing both sides by the mass $m$. For the Y-axis component of the equation with consideration of gravity as $g$, $k \\cdot \\Delta l \\cdot sin(60) + k \\cdot \\Delta l - mg = m\\cdot a_y$ Similar to the X-axis, I thought I would consider the total of the extended length plus the projection from the x-axis onto the y-axis, and then minus the gravity resistance. However, it turns out that I am wrong for the Y-axis component. The given answer is: X: $k \\cdot \\Delta l \\cdot cos(60) + k \\cdot \\Delta l = m\\cdot a_x$ Y: $mg - k \\cdot \\Delta l \\cdot sin(60) = m\\cdot a_y$ I don't understand why is it so for the Y-axis, especially when the the gravity turns out to minus the projection and the extended $\\Delta l$ is not added as part of the force."}
{"id":"44976","title":"BEC for holography?","text":"I am spending some time reading about Bose-Einstein condensation. I want to know if it is possible to use atom lasers to realize the kind of holography traditionally associated with nano-fabrication. Many papers say that the motivation for BEC is the aforementioned holography, but I can't find if they were able to actually realize it. I am very much a novice at searching through academic literature. The closest I have found is the work of some Japanese scientists in the 90s who were able to make 2D hologram, without a laser style beam. * * * Does anybody know if it was done, or possible can comment on some of the challenges."}
{"id":"127517","title":"Correlation in electron gas","text":"In the textbooks that I read (namely Ashcroft\/Mermin , Marder, etc.) it seems that a distinction is made between the correlations in electron gas and a Couloumb interaction between the electrons. What is exactly meant by the concept of correlations? How is that connected to the interactions in electron gas, and how does the screening enters the picture?"}
{"id":"65521","title":"Color of sky scattering by sky dust","text":"> Why does we see sky like blue in maximum time? We know violet has less wavelength, so we should see sky like violet. My assumption is that, violet is not primary color and not sufficient in the spectrum so we see blue because after violet, blue has the shortest wavelength? **Edit: Because our eyes have sensibility for three primary colors so we may see primary color most but, If a people is blue color blind then what sky color will he see?**"}
{"id":"40875","title":"Why is the sky blue?","text":"> **Possible Duplicate:**   >  Why does the sky change color? Basically what the title says. What mechanisms are significant and how do they contribute to make the sky blue. Also when the sky is not blue, like when the sun sets, how does it happen?"}
{"id":"17","title":"Why does the sky change color?","text":"Why the sky is blue during the day, red during sunrise\/set and black during the night?"}
{"id":"1927","title":"Would something like the uncertainty principle arise even if the universe was built on something like Newtonian mechanics?","text":"I am thinking of a (greatly simplified) computer simulation of a universe that followed something like Newtonian rules. Inside the simulation are A.I.s that are made from those same rules, and can only use those rules interact the world around them. Would there be some fundamental limits on what those A.I.s could work out about their universe, like their own version of an uncertainty principle? Sorry for phrasing this question in such a convoluted way. If anyone recognises what I am asking, and can point me in the right direction that would be appreciated."}
{"id":"43317","title":"Chiral anomaly in odd spacetime dimensions","text":"In odd number of space-time dimensions, the Fermions are not reducible ( _i.e._ do not have left-chiral and right-chiral counterparts). Does this mean that there is no such thing as 'chiral' anomalies in odd number of space-time dimensions, when these fermions are coupled to gauge fields?"}
{"id":"14052","title":"Is the wind's force on a stationary object proportional to $v^2$?","text":"I am on a boat docked at Cape Charles, VA, about 30 or 40 miles from the center of Hurricane Irene. This understandably got me thinking about the force of wind on the boat. Since air friction is proportional to the speed squared (except for some friction types I'm sure someone will be kind enough to remind me of), is the wind's force on the boat also proportional to the wind's speed squared? In other words, will a 70 knot wind produce almost twice the force on the stationary boat as a 50 knot wind, all other factors being equal?"}
{"id":"81546","title":"Does rate of acceleration affect the amount of energy used to accelerate?","text":"Assume you start with a car at rest. Then you accelerate the car to 100 kilometers per hour. In terms of the energy needed, does it matter how fast you accelerated the car? In other words, do you need to provide the same amount of energy to accelerate a car from 0 to 100 kilometers per hour whether you do it in 4 seconds or 20 seconds?"}
{"id":"11247","title":"proof of gauge invariance for quantum 1D ring","text":"This is a question on gauge invariance in quantum mechanics. I do some simple math on a 1D wave-function with periodic boundary conditions, and get that gauge invariance is violated. What am I doing wrong? Consider one coordinate dimension configured as a ring. The gauge dependent momentum operator can be written: $p_{op}=-i \\frac{\\partial}{\\partial x} - k$ Units have been chosen so that $\\hbar = 1$, $k$ is an arbitrary real constant different for each gauge and $x$ represents the coordinate. The gauge dependent eigenfunction can be written $\\psi(x)= Ae^{i(n+k)x}$ where A is a constant determined by normalization. As is well known in quantum mechanics, an operator applied to one of its eigenfunctions should yield a real constant eigenvalue multiplying the same eigenfunction: Thus $[-i \\frac{\\partial}{\\partial x} - k] Ae^{i(n+k)x}= nAe^{i(n+k)x}$ so that the real number n is the eigenvalue, which must be determined by the boundary conditions. The boundary condition for this periodic system must be that the wave function should join onto itself smoothly everywhere. Thus, if the coordinate is chosen such that x extends from –$\\pi$ around the ring to $\\pi$ then the eigenfunction in equation 3 must have (n + k) = m, where m is an integer. Under these conditions, the eigenvalue n in equation 3 will be n = m – k. This eigenvalue depends explicitly on k, and so is not gauge invariant. I'm assuming this simple situation should be gauge invariant, but I don't see where I goofed. I'd appreciate any help."}
{"id":"27909","title":"Why is exhaling more forceful than inhaling?","text":"By blowing at pencil, a piece of paper, or another object up to fifty centimeters away, I can cause it to move away from me significantly. But I can't move an object toward myself by inhaling sharply from that distance, even if it is extremely light. Why is that?"}
{"id":"5800","title":"What is the historical origin of the term action","text":"In Physics ordinary terms often acquire a strange meaning, action is one of them. Most people I talk to about the term action just respond with \"its dimension is energy*time\". But what is its historial origin? http:\/\/en.wikipedia.org\/wiki\/Action_(physics)#History doesn't really give much insight, as it lacks citations and depth. So, how did \"action\" become to mean what it means now? Cheers"}
{"id":"90859","title":"Schroedinger field operators and their commutation relations","text":"I've got several questions regarding the so called second quantization of the Schroedinger equation. My professor introduced the field operators for the Schroedinger field by simply stating them as follows: $$ \\hat\\psi (\\vec{r},\\xi)=\\sum\\limits _i \\psi_i(\\vec{r},\\xi) \\hat a_i\\\\\\ \\hat\\psi^\\dagger (\\vec{r},\\xi)=\\sum\\limits _i \\psi_i^\\star (\\vec{r},\\xi)\\hat a^\\dagger_i $$ Where $\\psi_i(\\vec{r},\\xi)$ are the time independent one particle wave functions and $\\hat a_i,\\, \\hat a^\\dagger_i$ the corresponding creation and annihilation operators. Is there a way to explain, why one does this? If I understood correctly what I've been taught so far, in QFT one must find some way to quantize the fields obeying the field equation in question. I do, however, not quite understand why in this particular case it is done like this. Shouldn't the $\\psi_i(\\vec{r},\\xi)$ be the time dependent one particle wave functions? Because I thought the field operators for a system in a box look like this: $$ \\hat\\psi (\\vec{r},\\xi)\\sim\\int\\text{d}^3k\\ \\exp(i\\omega_k t-i\\vec k\\vec x) \\hat a_k $$ My professor then proceeded to prove the (anti)commutation relations between the field operators, postulating the corresponding relations between the fermionic or bosonic creation and annihilation operators: $$ \\left[\\hat\\psi (\\vec{r},\\xi);\\hat\\psi^\\dagger (\\vec{r}',\\xi')\\right]_\\pm= \\left[\\sum\\limits_i\\psi_i (\\vec{r},\\xi)\\hat a_i;\\sum\\limits_j\\psi^\\star_j (\\vec{r}',\\xi')\\hat a^\\dagger_j\\right]_\\pm =\\sum\\limits_i\\psi_i (\\vec{r},\\xi)\\psi^\\star_i (\\vec{r}',\\xi')\\\\\\ =\\delta(\\vec{r}-\\vec{r}')\\delta _{\\xi,\\xi'} $$ Here I do not understand the last step. Is that conclusion possible? And shouldn't or couldn't one postulate the commutation relations between the field operators and arrive at the relations for the creation and annihilation operators?"}
{"id":"122063","title":"How to calculate the equation of projectile motion for curved ramps?","text":"Consider the following cases (these are sections): ![straight line ramp](http:\/\/i.stack.imgur.com\/cnea1.png) ![curved ramp](http:\/\/i.stack.imgur.com\/HfNpp.png) I can find the equation of motion for Case a however I don't know how to deal with Case d. Normally I would need the angle α otherwise I can't manage to find the function for x and y with respect to time. There is also friction involved, and v0 is known. How do I find equation of motion for Case d?"}
{"id":"83073","title":"What makes charges flowing in a circuit with a higher potential difference perform more work?","text":"I understand that a 1.5 V cell will not deliver as much energy per coulomb as a 150 V power supply will. What I do not understand is that why it is so. I am digressing now. If we place two point charges at a distance r, then to increase the force between them, we need to increase their charges. This makes sense. However, if we need to increase the Energy carried by 1 Coulomb of charge in a circuit, we need to increase the potential difference. Here, the number of electrons remain the same but still, somehow, they are able to do more work. The definition of electric potential is \"electric potential at a point is the amount of electric potential energy that a unitary point charge would have when located at that point\" But why does electric potential energy act upon it at all if there is no charge to attract or repel it? These seem to be different questions, but they are all linked to a fact that I cannot understand \"The same amount of charge performs more work when a higher potential difference is applied\""}
{"id":"16688","title":"Is 'now' smeared over time?","text":"Conventional physics as is usually presented in textbooks deals with the evolution of states in phase space parameterized by sharp instances in time, a real parameter. However, quantum fluctuations seem to suggest we have to smear a little over time to average over vacuum fluctuations and the like. What implications does this have over the meaning of 'now' and the nature of time?"}
{"id":"80635","title":"Strange electrical circuit","text":"This is a question I would like to have an explanation with: It's in this PDF, question 9. ![circuit](http:\/\/i.stack.imgur.com\/euGXK.png) > In that circuit, if switch 1 is closed, bulb A burns normally. If 2 is > closed as well, do other lamps burn normally as well? The answer is D: yes, both B and C, but why? Could somebody explain the steps in order to think? This is not homework (even though in principal it is) but I did this voluntarily on my own interest in how more advanced electrical circuits work."}
{"id":"80630","title":"Will there be any force of attraction or repulsion between an electrified body and a non-electrified body?","text":"Up to my knowledge an electrified (charged) body can attract a non-electrified (neutral) body. I thought this because, when we bring a charged (suppose negatively charged) body near a neutral one. Electrified body can attract a non-electrified body by the opposite charge induced on neutral body due to electrostatic induction, then the answer to the above question would be likely to say that electrified body exerts attractive force on non-electrified body.   But going through the Wikipedia's encyclopedia of electric charge, I found the following line: > _No force, either of attraction or of repulsion, can be observed between an > electrified body and a body not electrified.[3]_ My view on the concept is contradictory to the statement. **EDIT:** I have seen a article supporting my view.You can see at the bottom of this link page about interaction between charged(electrified) and neutral(non-electrified)body."}
{"id":"99549","title":"Gossip in Physics","text":"Given, $M_4 = \\Sigma \\times C$, How do you get an effective theory by studying maps $\\Sigma \\rightarrow M_4$ . Technically, the physics in one manifold is supposed to gossip about the overall affairs of people in the entire manifold at least up to a certain limit. To be direct I am trying to know how the algebra bundles come about as a twist in $\\sigma$ models. I am sure my question is not well posed, as usual feel free to edit it."}
{"id":"7963","title":"Madelung constant list (for surfaces as well)","text":"Searching for this on google proved to be quite tedious, but I reckon that someone working with crystals a lot might know this off the top of his head: Is there a good source that lists the Madelung constants for a variety of geometries? I'd be particularly interested in that for a NaCl-type 110 surface."}
{"id":"95439","title":"Affine connection notation","text":"Can ${g}^{\\mu\\sigma}{\\Gamma}^{\\rho}_{\\sigma\\nu}$ be written as ${\\Gamma}^{\\mu\\rho}_{\\nu}$? If so how come this symbol never appears in any GR book?"}
{"id":"12265","title":"Non-interchangeability of time-like intervals","text":"I am reading Landau's Volume 2 of the course of theoretical physics. I have a doubt after reading the first few pages of it which I explain below. Landau first defines intervals and on pages 5 and 6 shows that two events having time like interval between them can never occur simultaneously in any reference system. Then he goes on to construct a 2D space-time graph (for visualization) with an event O occurring at (0,0,0,0). Then he considers any event which occurs in future in that frame and is time-like w.r.t. origin and says on page no. 7, > But two events which are separated by a time-like interval cannot occur > simultaneously in any reference system. Consequently, it is impossible to > find a reference frame in which any of the events in region aOc occurred > \"before\" the event O, i.e. at time t<0. The argument above just proves that because interval square should be positive, i.e. the events can't be simultaneous. But, if I replace the difference in time in the original frame with its negative in my proposed frame and let the space distance between them to be same in both frames, then I get an in my proposed frame an interval which is time like but in it the order of events is changed. Am I making some gross error or Landau has missed some argument?"}
{"id":"10532","title":"Do we need a quantum deformation of the diffeomorphism group in string theory?","text":"Let me justify my question before I go on. In string theory, gravitons are strings extended over space. Longitudinal gravitons are pure gauge modes of the diffeomorphism group. However, in string theory, longitudinal gravitons are also extended objects. A condensate of longitudinal gravitons is equivalent to a diffeomorphism, but this diffeomorphism has to be smeared out over the string scale. Is the diffeomorphism group in string theory a quantum deformation smeared out over space? In the weak field small string coupling limit, the string theory algebra and the classical diffeomorphism algebra ought to coincide, but in this limit, all such algebras over a Poincare invariant background are isomorphic. Away from this limit, what is the form of this quantum deformation?"}
{"id":"10538","title":"Wheatstone Bridge","text":"Why is using a Wheatstone bridge such an accurate way of calculating an unknown resistance? What are the benefits of using it over Ohm's law? It seems that it has something to do with the wires heating up during ohm's law calculations, and the fact that an ammeter\/voltmeter still draw some current and provide some resistance which makes results unreliable."}
{"id":"59248","title":"Why is copper diamagnetic?","text":"Cu has an unpaired electron in 4s, but it is diamagnetic. I thought that it has to be paramagnetic. What am I missing?"}
{"id":"90147","title":"How does one estimate the electrical power of a power plant?","text":"I have two related questions that I would like help on:   1. When a power plant creates power like the Hoover Dam, it can provide 2.07 GW of electrical power. My question is what does this mean? I assume from Faraday’s law that the induced voltage across the generator coil produces an current, and this combination (P = VI) is the output power. So in the case of the Hoover Dam, P = VI = 2.07 GW. I feel that this is naïve thinking but when I do a search to get more information, I am not able to find any. Can someone roughly sketch out how the electrical power of a power plant is computed?   2. If possible, what kind of voltages and currents are power plants producing before the Step Up transformers?"}
{"id":"21468","title":"Frequency of nomad planets passing within 30 AU of the sun","text":"A recent estimate by the Kavli Institute for Particle Astrophysics and Cosmology (a joint institute of Stanford and SLAC) is that there are circa 100000 times as many 'nomad planets' as stars I found \"The Close Approach of Stars in the Solar Neighborhood, Matthews, R. A. J., Quarterly Journal of the Royal Astronomical Society, Vol. 35, NO. 1, P. 1, 1994\" which estimated that the frequency of other stars passing within a given distance to be $$ F_{r}(r) = \\sqrt{2} \\pi r^{2}\\rho_{s}V_{s} $$ where $$ V_{s} \\approx 19.5 \\text{ km}\/\\text{second} $$ and $$ \\rho_{s} \\approx 0.11 \\text{ stars}\/\\text{parsec}^3 $$ resulting in $$ F_{r}(r) \\approx 10^{-5} r[\\text{pc}]^{2} \\text{year}^{-1} $$ Assuming that those estimates are accurate and substituting $$ \\rho_{s} \\approx 11000 \\text{ planets}\/\\text{parsec}^{3} $$ and $$ r[\\text{pc}] \\approx 0.000145 \\text{ parsecs} $$ we get a frequency of $$ F_{r} \\approx (10^{-5})(0.000145^{2})(10^{5})\/\\text{year} $$ or $$ F_{r} \\approx 2 \\times 10^{-8}\/\\text{year} $$ This gives us a net 'close encounter' of the solar system with a nomad planet roughly every 50 million years. Does this seem a reasonable estimate?"}
{"id":"14943","title":"Spontaneous emission and induced radiation","text":"In Einstein A., Zur Quantentheorie der Strahlung, Phys.ZS., 18, 121-137 (1917) spontaneous emission is considered to occur together with induced radiation so that one can write the following condition for equilibrium (which also includes induced absorption): $$ p_n e^{-\\frac{\\varepsilon_n} {kT}} B_n^m \\rho = p_m e^{-\\frac{\\varepsilon_m} {kT}} \\left( B_m^n \\rho + A_m^n \\right) $$ where $p_n$ and $p_m$ are statistical weights of the states $n$ and $m$, $\\rho$ is radiation density of frequency $\\nu$, $A_m^n$ is a constant characteristic of the spontaneous $m \\rightarrow n$ transition (spontaneous emission), $B_m^n$ and $B_n^m$ are constants expressing the change of state under induced emission and absorption. To arrive at Planck's radiation density law it is considered that at high temperatures the above equation becomes $$p_n B_n^m = p_m B_m^n .$$ What is the justification, however, to substitute $B_n^m$ expressed through the latter equality (valid for extreme temperatures) into the initial equation above (valid for lower temperatures)?"}
{"id":"8882","title":"Why absoluteness of time implies galilean transformations?","text":"In Landau course, vol.1 Mechanics, one finds the statement: > ...the absoluteness of time necessarily implies that the ordinary law of > composition of velocities is applicable to all phenomena. I don't see this implication clearly."}
{"id":"45611","title":"How to get energy of collision if you know force of gravity of an object($m \\rightarrow F=mg$)?","text":"How to get energy of collision if you know force of gravity of an object($m \\rightarrow F=mg$)? You get energy of collision by kinetic energy $E_k= \\frac{1}{2}mv^2$, but if you use just force of gravity($F=mg$), how you get then the energy of collision? I know that work(energy?) $W$ made by object is $W=F \\Delta s$, where $F$ is force exerted to object and $\\Delta s$ traveled distance, but this distance makes it hard to calculate any collision energy of a meteorite for example."}
{"id":"28464","title":"Under what conditions a person can lift a car alone?","text":"I'm thinking about starting my work of physics with this question but do not know how to answer."}
{"id":"54015","title":"Why does the presence of a battery change whether a circuit is in parallel or series?","text":"If I take a closed circuit with two capacitors and a voltage difference, the circuit is apparently in parallel, but if I introduce a battery, the circuit is in series. Why does the presence of the battery make a difference? Here's a diagram of what I mean: ![enter image description here](http:\/\/i.stack.imgur.com\/CRwHQ.png) According to my book, the left circuit is in parallel, but the right circuit is in series. I don't see why—after all, for any voltage to run through the circuit on the left, it has to pass through C1 before C2 (or vice versa)."}
{"id":"91438","title":"Do these steps demonstrate that acceleration of charged particle is proportional to current?","text":"One formulation of Maxwell's Gauss Law for electric field is: $$\\bigtriangledown E = 4 \\pi k \\rho $$ This can be worked into the Divergence Theorem as follows: $$\\int\\int_{A} F_\\perp \\:dA= 4\\pi k \\int\\int\\int_V \\rho\\:dV$$ As far as I can tell, the \"outward pointing area normal\" $F_\\perp$ is the same as the current density or flux $J$ magnitude. I.e., $$F_\\perp=J\\cdot n=\\frac{d}{dA}\\left(\\frac{dq}{dt}\\right) \\cdot n$$ $$\\int\\int_{A} J \\cdot n \\:dA\\sim 4\\pi k \\int\\int\\int_V \\rho\\:dV$$ Now, if we assume the vector field is isotropic (flowing across a spherical surface), then $J \\cdot n$ is constant across the surface, and so the LHS integral evaluates to the surface area of the sphere times the constant $J\\cdot n$: $$\\int\\int_{A} J \\cdot n \\:dA=J \\cdot n \\int\\int_{A} \\:dA=J\\cdot n 4\\pi r^2$$ where $r$ is the radius of the sphere. And then we get: $$ J\\cdot n 4 \\pi r^2\\sim 4\\pi k \\int\\int\\int_V \\rho\\:dV$$ $$ J \\cdot n 4 \\pi r^2\\sim 4\\pi k E$$ $$J \\cdot n \\sim \\frac{kE}{r^2}$$ This already seems suspect since I believe by Coloumb's Law the RHS is proportional to the acceleration of a charged particle in the field. So this equation is tantamount to equating current density to acceleration. I do not have enough background in electromagnetism to know if such a statement makes sense or not. But then also consider that by the conservation of charge we know that (continuity equation): $$ \\bigtriangledown J = -\\dot{\\rho} $$ Substituing in the result obtained from the Divergence Theorem ($J\\cdot n\\sim \\frac{kE}{r^2}$) gives: $$ \\bigtriangledown \\frac{k E}{r^2} \\cdot n = -\\dot{\\rho} $$ Working out $\\bigtriangledown \\frac{k E}{r^2} \\cdot n$ gives a mess that doesn't seem related to $\\dot{\\rho}$. Please help show how it is true or where I have gone wrong in my derivation."}
{"id":"115052","title":"Reference for stochastic processes which helps moving from a basic level to a measure theory one","text":"I'm looking for a reference (books, notes, lectures) which helps a physicist to understand the language of measure theory in the context of stochastic processes (in particular markov chains). I've studied markov chains and measure theory but now I'm looking for something which helps me filling the gap and making this two topics converge. I've already read: _Measure, Integral and Probability_ \\- Marek Capinski, Peter E. Kopp Maybe something with direct comparison (which writes the same probability both at a basic level and in measure theory) would be great! Thanks in advance!"}
{"id":"66681","title":"Torque in a heavy and light Body","text":"Suppose we have two wheels attached to axis (each wheel has its own axis). One wheel is heavy compared to the other. The moment arm of force is same for the both wheels. A force F is applied on both the wheels. The applied force is also same for both wheels. The light wheel will rotate fast compared to the heavy wheel. Will the torque for both wheels be same? Torque is also defined as \"The turning effect of a body.\" But the turning effect of both the bodies is different here so, will their torque be different?"}
{"id":"91205","title":"What is Einstein regarding as the problem of physical relativity in his lecture of 1923?","text":"The following is the extract from _Albert Einstein's lecture_ to the Nordic Assembly of Naturalists (July 11, 1923). > _If we consider that part of the theory of relativity which may nowadays in > a sense be regarded as bona fide scientific knowledge, we note two aspects > which have a major bearing on this theory. The whole development of the > theory turns on the question of whether there are physically preferred > states of motion in Nature (physical relativity problem). Also, concepts and > distinctions are only admissible to the extent that observable facts can be > assigned to them without ambiguity (stipulation that concepts and > distinctions should have meaning). This postulate, pertaining > toepistemology, proves to be of fundamental importance._ I am not able understand Einstein's question in the extract, i.e _are there physically preferred states of motion in Nature?_ As what I can think as layman, I didn't understand the meaning of _physically preferred states of motion_. Does Einstein mean physically preferred states of motion as wave motion, particle motion? Einstein uses the term _physical relativity problem_ , what is the problem of _physical relativity_? Why does Einstein regards epistemology principle mentioned in the extract as of fundamental importance? I am the beginner in this concept, if some where I have gone wrong in explaining pardon me."}
{"id":"127094","title":"Schottky Barrier - Why energy band levels at interface are assumed to remain the same that bulk","text":"I have been chewing up some time ago the Schottky-Mott theory of Schottky Barrier height (which ignores the surface states). All the deduction seems to ground on fundamental thermodynamical principles (as the equality of Fermi levels- i.e. equality of chemical potential in equilibrium) but there is something which I can't clearly see and is one of the key points to calculate the height barrier: Why the energy bands on the semiconductor side just at the interface are assumed to be the same that the ones of the isolated semiconductor? It is clear that the band have to bend (because of the electric field) but I see no reason to why the bands values should \"start\" to bend from the original values (the values of the isolated semiconductor). Thanks."}
{"id":"56070","title":"Why path integral approach may suffer from operator ordering problem?","text":"In Assa Auerbach's book (Ref. 1), he gave an argument saying that in the normal process of path integral, we lose information about ordering of operators by ignoring the discontinuous path. What did he want to say? I don't think there is any problem related to the ordering of operators. References:   1. Assa Auerbach, _Interacting Electrons and Quantum Magnetism,_ p.102, just below eq. (10.6)."}
{"id":"59227","title":"US time zones and Daylight saving time, energy efficient?","text":"As a foreigner, United States has a very complex time system for me. `Central Time Zone`, `North American Eastern Time Zone`, `Mountain Time Zone`, `Alaska Time Zone`, `Pacific Time Zone`, `Samoa Time Zone`, `Atlantic Time Zone`, `Hawaii–Aleutian Time Zone`. And also plus `DST` (`Daylight saving time`). I heard people saying this can save a lot of energy. I want to know if there is any **scientific report** to support this. Or it's just people they like this system. Thanks"}
{"id":"92165","title":"What speed should have an alone planet to have a habitable zone due to relic radiation?","text":"What speed should have an alone planet to have a habitable zone due to relic radiation, without any star involvement? How much time the planet will be able to remain in the habitable zone before the speed drops, given the planet has no engines? Are such speeds achievable using gravity assist of the galactic center or similar objects? What would be effect of non-CMB sources like particles and stellar radiation at such speed?"}
{"id":"73626","title":"Nukes with cup-sized mushroom clouds","text":"A few sequential questions:   * Is it possible for a nuclear explosion to be small enough to produce a 250-ml (one cup) mushroom cloud?   * If so, how much uranium would that take?   * How close to the explosion could one be (in normal clothing) not to suffer from burns or excessive radiation exposure?"}
{"id":"73622","title":"Understanding the basics of electromagnetic induction","text":"Suppose two rings are kept facing each other and that one ring have some current which increases constantly. Will the other ring be attracted or repelled? Does this also depend on _how they are kept_?"}
{"id":"74077","title":"Is the Higgs mechanism needed for Quantum Field Theory?","text":"The NY Times article on Firewalls today has the following paragraph: > _Quantum field theory is how the world works [quoting a physicist]. It had a > major triumph just a year ago, when the Higgs boson, a subatomic particle > responsible for the mass of other subatomic particles, was discovered after > a 40-year search, at the Large Hadron Collider at CERN._ Is it correct that   1. the Higgs (boson, mechanism, field, etc) is needed for QFT _per se_ , or    2. is it \"just\" needed for the particular _application_ of QFT known as the standard model, electroweak theory etc?  Couldn't QFT still be the correct framework for \"how the world works\" if the Higgs model turned out to be wrong?"}
{"id":"78903","title":"Can you take components of tension in a string?","text":"Imagine there is a string, not a massless one but a heavy one, which is attached to the ceiling where the angle between the ceiling and the tangent at the end of the string is $\\theta$. (First visualize it, or draw it somewhere). Let at any point the tension in the string be T. Now obviously there are components of forces acting, as T$\\sin\\theta$ is balancing its weight. But if I nudge the string from that point with a very lightly, I actually do not experience a resistive force (I've tried it several times). But I know T$\\cos\\theta$ is acting towards me. So my question is, why do not I experience a resistive force? Also, when we derive the expression for centripetal force, we do not take components of Tension but of its weight, because it \"sounds\" meaningless. Is it logical to take components of T?"}
{"id":"119614","title":"How does the expansion of the universe not violate causality?","text":"It is often said that faster than light travel would violate causality. However, because the universe is expanding, there are actually distant stars that move away from us at a speed greater than the speed of light. Why is this allowed, even though it could violate causality?"}
{"id":"98627","title":"Temperature Dependence of Fission","text":"I've always heard fusion depends on incredible temperatures and pressures, but I'm curious what the temperature dependence of fission is. For instance, if I had uranium and heated it to a sufficiently hot temperature wouldn't it decay at a faster rate at some point? I'm not talking about 300 K, but instead say we had 5800 K or 11600 K or 17400 K. At some point I would expect the decay rate to increase, so does anybody have more information on this?"}
{"id":"114096","title":"How can a vacuum have a breakdown voltage?","text":"This question (What is the capacitance between the Earth and Moon?) on EESE makes me wonder: How can a vacuum have a breakdown voltage? If electrons can find the shortest path through a vacuum, how is that possible? With nothing to carry the charge, I can't understand how the charge can traverse a vacuum. The value for dielectric strength of a vacuum comes from Wikipedia"}
{"id":"9921","title":"Voyager local time dilation (caused by gravity)","text":"Voyager I, as an example, taking account gravity   and setting aside effects of speed as cause of time dilation. If it is very far away from earth and sun, so then there must be a difference in the spacetime curvature there in the ship compared with here in earth, It means a detectable difference between our local clocks and its onboard clocks. Imagine a signal transmision was designed to be at 1 byte for second at local Voyager clock Should we receive it at higher and higher rates ? ( because of solar system gravity decrease as it(Voyager) moves away and it will keep decreasing while it doesn't reach a middle point between another massive object) Or the middle Voyager-Earth light path would compensate the effect, making the high rates generation from low curvature zones being delayed enough for us to receive it at same rate that it was generated?"}
{"id":"107933","title":"How to calculate distance travelled from velocity vector and angle?","text":"I am trying to get the distance a projectile travels, so I can display it in my program. I'm sort of new to physics, but I've tried looking this up. I'm not getting the right results however, and me being new to physics, I'm not sure if I'm just using the wrong formula or what. This is the formula I have tried using: ![](http:\/\/upload.wikimedia.org\/math\/c\/1\/d\/c1da5860501561519415962ddda5e85e.png) As far as I've read, a projectile launched at 15 degrees would travel the same distance as one launched at 75 degrees, however when I run the calculation I get two different distances for those two instances. Is this not the correct formula, or maybe I'm using it wrong? These are the values I get for 15 degrees and 75: 15: velocity vector = (5,-19) magnitude = 20 vCos = -15.2 vSin = 13 gravity = 9.8 output = 41 75: velocity vector = (19,-5) magnitude = 20 vCos = 18.4 vSin = -7.8 gravity = 9.8 output = 2 The projectile starts on the ground and gets launched at the specified angle over a flat surface. Not sure where I'm going wrong. All help is greatly appreciated."}
{"id":"55857","title":"If you had two \"perfectly\" flat surfaces of the same material?","text":"Let's say you had 2 nano-engineered surfaces of diamond which were as 'flat' as possible (of course considering the radii of each carbon atom in the lattice)... would there be any friction between these 2 flat diamond surfaces when rubbed together? My thinking is that there would be very little to no friction due to the electron repulsion and there being no imperfections in the surfaces."}
{"id":"26089","title":"How to identify the objects in an astrophoto, and what portion of the sky it covers?","text":"Given an astrophoto with a resolution between 0.5 and 5 arcseconds per pixel, which ways exist to identify the direction of view, field of view, and objects in the picture? I believe most amateur astrophotographers are in this resolution range, and I believe most of us would like to confirm the captured images. I have read about the http:\/\/astrometry.net project and I'm wondering if there are any similar services available."}
{"id":"26085","title":"Most accurate ways to find the average distance between stars in Milky way galaxy","text":"I've already posted here on quora. But, I'm not totally sure if it's the most reasonable method. Would anyone care to elaborate on how to find the average distance between stars in a given galaxy (For instance, let's take Milky way), somewhat efficiently and accurately - or at least clarify that my method is sufficiently efficient and accurate?"}
{"id":"104771","title":"Advantages of $e^-e^-$ or $e^+e^+$ collisions over $e^+e^-$ collisions?","text":"Is there any (interesting) HEP process whose study would take advantage from $e^-e^-$ or $e^+e^+$ collisions with respect to $e^+e^-$ collisions?"}
{"id":"53933","title":"Divergence theorem over entire space on non euclidean spaces","text":"I'm a physics major so bear with me here on the math. This is related to a problem from the textbook General Relativity - Wald. In classical electromagnetism if we have a vector field say $V$ defined on all of $\\mathbb{R}^{3}$ such that $V\\rightarrow 0$ , dropping off as $O(1\/r^{3})$, in the limit as $r\\rightarrow \\infty $ then when calculating $\\int_{\\mathbb{R}^{3}}\\partial _{i}V^{i}d^{3}x$ one usually takes a closed ball $\\bar{B_{r}}(x)\\subset \\mathbb{R}^{3}$ and, using the fact that $\\bigcup _{r}\\bar{B_{r}}(x) = \\mathbb{R}^{3}$ gives us, with an appropriate limit theorem, that $\\int_{\\mathbb{R}^{3}}\\partial _{i}V^{i}d^{3}x = lim_{r\\rightarrow \\infty }\\int_{\\bar{B_{r}}(x)}\\partial _{i}V^{i}d^{3}x$ to which we can then apply the divergence theorem to state that $lim_{r\\rightarrow \\infty }\\int_{\\bar{B_{r}}(x)}\\partial _{i}V^{i}d^{3}x = lim_{r\\rightarrow \\infty }\\int_{\\partial \\bar{B_{r}}(x)}V^{i}n_{i}d^{2}x = 0$ where the zero comes from the fact that the integral will drop off as $O(1\/r^{2})$ as $r\\rightarrow \\infty $ so the sequence of surface integrals will eventually converge to zero. This is all fine and dandy but my problem deals with a background flat space - time with metric perturbation $(M,\\eta _{ab} + \\gamma _{ab})$ where $\\eta _{ab}$ is the minkowski metric and $|\\gamma _{ab}| << 1$ as usual. We have a space - like hypersurface $\\Sigma $ of this manifold, the Landau Lifshitz pseudo tensor $t_{ab}$, which is divergence free, and the total energy $E = \\int_{\\Sigma } t_{00}d^{3}x$. We must show that $E$ is time translation invariant. Using the facts that $\\partial ^{a}t_{ab} = 0, \\partial _{0}E = -\\partial ^{0}E$ we proceed: $\\partial _{0}E = -\\partial ^{0}E = -\\partial ^{0}\\int_{\\Sigma }t_{00}d^{3}x = -\\int_{\\Sigma }\\partial ^{0}t_{00}d^{3}x = \\int_{\\Sigma }\\partial ^{i}t_{i0}d^{3}x$. We are also given that $t_{\\mu \\nu }\\rightarrow 0$ identically, dropping off as $O(1\/r^{3})$, in the limit $r\\rightarrow \\infty $. This is, of course, very similar in situation to the electromagnetic case and one would ideally like to use the divergence theorem to get the desired result that $\\partial _{0}E = 0$ however here we do not have a prescribed metric $d$ on $\\Sigma$ to make sense of closed balls as far as I can tell and even if there is some natural choice of metric $d$ for $\\Sigma$ how will we know if the closed balls with respect to $d$ will be orientable? Thanks for any and all help and sorry if this was a bit long winded; it is my first post here so I'm not sure how it is meant to work :[; I just wanted to be thorough in explaining my issue. Thanks again."}
{"id":"113043","title":"How does the windchill factor exist above 98.6 degrees?","text":"As I understand it, the explanation of the windchill factor goes like this: > Your body is warmer than the air around it, therefore it loses a certain > amount of heat to the air. This warms up the air immediately around you > first. The heat will then gradually be transferred to the air further away > through convection, but there's so much more of the air than there is of you > that the air essentially acts as a perfect heat sink, and your body heat > isn't likely to raise the overall temperature of the air in even a > moderately-sized room by any measurable degree. > > However, you're still warming the air immediately around your body first. > When the air is moving, it blows away the air immediately around your body, > which is slightly warmer than the rest of the air, and replaces it with the > slightly cooler air that hasn't been warmed up yet. Since the speed of > convective heat transfer is based on the difference between the temperatures > of the bodies involved, this means that you lose heat faster, and feel > chilled more than you would at the same temperature if there was no wind. However, I have memories of days when the temperature was well over 100 degrees, and breezes were still a relief. By the explanation above, wind should have made the heat _even more oppressive_ because the convective heat transfer is flowing in the opposite direction, into my body rather than out of it. What's the principle here that makes this possible?"}
{"id":"105427","title":"An Electric Potential Glued to a Cube-Shaped Insulator to Replicate a Point Charge: Charge Distribution","text":"I have been going back over this problem with a friend for the better part of a day: A potential is glued to a cube-shaped insulator so that outside of the insulator the field is the same as a point particle. How can we calculate the surface charge distribution and the volume charge distribution?"}
{"id":"112391","title":"relation between photon number and energy","text":"Suppose there are two light beams. One is red while the other is violet. The energy of both is the same. Which one of these beams has a larger number of photons, or is the number of photons relevant?"}
{"id":"106689","title":"Consequences of tensor form due to various symmetries","text":"I'm reading a paper on 2D hydrodynamics, specifically on the drag of a rod in a 2D fluid. It is a low-Reynolds number regime, therefore linear hydrodynamics and the velocity of the rod can be expressed via: $$v_i^{rod} = \\mu_{ij}F_j$$ then it says: By in-plane rotational symmetry combined with the $\\hat{n} → -\\hat{n}$ symmetry of the rod, the mobility tensor must take the form: $$\\mu_{ij} =\\mu_\\parallel\\hat{n}_i\\hat{n}_j + \\mu_\\perp(\\delta_{ij}-\\hat{n_i}\\hat{n_j})$$ Here, $\\mu_\\parallel$ and $\\mu_\\perp$ are the mobilities for motion parallel and perpendicular to the rod's long axis, respectively. I'm having a hard time connecting the dots with this logic, can anyone show me how this follows?"}
{"id":"134852","title":"Does the superposition principle affect the space of quantum states?","text":"I am confused about the set of quantum states. I have seen it written that in classical physics, the set of all states is a simplex. (I think this refers to the probability simplex.) In quantum physics, the set of all states is not a simplex. Does this have anything to do with the superposition principle? If we didn't have the superposition principle in quantum mechanics, would the set of states be a simplex, like in the classical case? Thanks for your help!"}
{"id":"83416","title":"What determines the probability of a pair of photons interacting, and producing a positron and an electron?","text":"The second answer to this question describes how this process might occur, and I'm curious for more details about it:   1. What is the probability distribution of the interaction producing electron-positron pairs (and what's the general process for calculating it)?   2. Is it possible to produce beams of positrons and\/or electrons, through this process?  I would love some references that describe this kind of interaction in depth."}
{"id":"99025","title":"Seesaws and Dark Energy","text":"Lawrence Krauss and James Dent recently proposed a mechanism for producing the observed scale of dark energy. This proposal was inspired by the see-saw mechanism that produces light yet non-zero neutrino masses. I can't help but notice there are many reasons to doubt the see-saw mechanism package since it suffers from the the flavor, CP, and gravitino problems and several alternatives have been proposed. Would it be feasible to come up with a mechanism based on the alternatives to the seesaw and hence extend Krauss' and Dent's proposal?"}
{"id":"99021","title":"Has QFT successfully mediated between QM and Special Relativity?","text":"I understand that QFT is the theoretical framework for combining QM and Special Relativity, but as I understand it, though even without proof or experimental confirmations; has QFT managed to \"behind all the rigor\" design a mathematical construct that fully explains and logically computes there conflicts? Or does it continue to struggle for a formulary solution?"}
{"id":"27841","title":"Temperature of the CMB when the Earth formed and the faint young Sun paradox","text":"The cosmic microwave background (CMB) has a modern temperature of about 2.7 K. At the time of the origin of the CMB, about 13.6 billion years ago, it had a temperature of about 3000 K. There is a well-known problem that the Sun should have been too faint at the time the Earth formed, about 4.6 billion years ago, to allow the Earth to avoid becoming a permanent Snowball Earth. This problem is called the faint young Sun paradox. I was thinking about that, and it occurred to me that those two things may not be completely unrelated. What was the temperature of the CMB at the time the Earth formed, and what effect would that temperature have on the equilibrium temperature of the Earth? It seems to me that there should be two effects in play.   1. The equations describing the equilibrium temperature of the Sun then needs to be adjusted for the fact that rather than being a blackbody radiating to a near zero background, it would be radiating to a background with a significant temperature of its own. I would think that would raise the required temperature for the Sun to achieve radiative equilibrium there-by making the Sun brighter.   2. The equilibrium temperature of the Earth should also be boosted by the increased temperature of the CMB for similar reasons of having a less efficient heat 'sink'. Could those two effects together have shifted the equilibrium temperature of the Earth enough to solve the faint young Sun paradox? Edit: Having done some more googling on it, the CMB would only have been about a degree or so warmer than it is now because 4.6 billion years only corresponds to a Z of about 0.4. This isn't anywhere near enough to solve the faint young Sun paradox. I'm going to leave this question in place just so if someone else has the same idea this will explain why it can't solve the paradox."}
{"id":"35987","title":"Why there is a 1\/2 in kinetic energy formula?","text":"> **Possible Duplicate:**   >  Why is there a $\\frac 1 2$ in $\\frac 1 2 mv^2$? Hèllo, I have a question about kinetic energy formula. As you know, in kinetic energy formula, we have: $\\large\\frac{1}{2}mv^2$ Okay. And we know, Joule (energy unit), is: $\\large J= kg~(\\frac{m}{s})^2$ (Guys, please light me up if I'm wrong.) Here's my question: Why do we have a $\\frac{1}{2}$ in our formula? Why do we divide our $mv^2$? Please answer in simple word."}
{"id":"39539","title":"Other ways of checking whether particular system result in non-locality","text":"In quantum mechanics, when hamiltonian $H$ is constrained ($H = \\sqrt{m^2 - \\hbar^2 \\nabla^2} $) so that it would produce simple \"relativistic\" model of quantum mechanics, we can show that it results in non-locality (Reference: $\\nabla$ and non-locality in simple relativistic model of quantum mechanics ) The question is would Taylor-expanding every constraint equation on some quantity\/operator, such as Hamiltonian, show that it will result in non- locality? Or in some case, should we check other expansions\/methods?"}
{"id":"31704","title":"Is my understanding of the mechanics of skidding correct?","text":"I'm trying to understand the mechanics which determine if a car making a turn will skid. Are the following correct or incorrect:   1. A vehicle making a turn will skid unless the centripetal force is adequate to produce the centripetal acceleration. In a simple turn, where $r$ is constant, the force needed is $mr\\omega^2$.   2. If the road is flat, this force can only come from friction created by turning the wheel. The maximum force is $\\text{weight}\\times\\mu_\\text{static}$. All of this force will be directed centripetally, and will therefore be available to prevent the skid.   3. If the road is banked by $\\phi$, the friction force will be lessened by $\\cos\\phi$, but there will also be a centripetal component of the normal force, equal to $\\sin\\phi\\cos\\phi$. Are these all correct? I believe they are but my results using them don't seem to work. **UPDATE:** I believe my mistake was in the direction friction will be in. Determining the direction of friction in this case is tricky, because it has to both counteract the sliding due to normal force of the banked road, and also accelerate in the centripetal direction. It requires 3 dimensions. How do I determine the direction of friction, with respect to $r$ and $\\theta$ vectors?"}
{"id":"107152","title":"Why must an integrating sphere be a sphere?","text":"Why must an integrating sphere be a sphere? Why can't it be an integrating cube? What is the difference? Could I use a cube to measure total illuminance like an integrating sphere does? ![enter image description here](http:\/\/i.stack.imgur.com\/kf0Rz.gif)"}
{"id":"107155","title":"Is there an analogue to the role of vapor in liquids and gases, but for solids and liquids?","text":"It seems common for an ordered phase to have some amount of disorder present. For example, the average moment of a ferromagnet is less than maximum except at T=0 due to the presence of fluctuations. Also, a liquid is typically only in equilibrium if a portion of it is vapor at the right vapor pressure. Is there an analogue for the solid-liquid transition? In other words, for, say, an elemental solid below the melting point, should we expect a portion of its surface to be liquid at any given time, with this portion increasing steadily until the melting point when the whole thing becomes liquid?"}
{"id":"112023","title":"Subnuclear physics vs wave function","text":"This question is more a philosophical question than a physics one. When we appreciate particle physics we study that in order to explain some experimental results we have to introduce a new particle (for instance thinking about neutrinos). This is obviously true but I just want to ask you: is there a wave interpretation of this results? I mean, consider for example the so-called sea quarks, which coming up as a mixture of different quarks, if I describe this processes in a wave model can I see this as an interference of wave function? I am not sure that this question is clear. generalizing, why do not we study subnuclear physics both in particle and wave model, as quantum mechanics suggest?"}
{"id":"112028","title":"Where's the missing helium in the Universe?","text":"I'm confused: Big Bang nucleosynthesis is adamant about the 1 neutron to 7 proton ratio which yields 75% hydrogen to 25% helium (with a nominal amount of partially-reacted deuterium and heavier lithium). But everything I read about the interstellar medium gives a helium figure closer to 10%, so what happened to the missing 60% of the helium in the Universe? Of the gas in the ISM, 89% of atoms are hydrogen and 9% are helium, with 2% of atoms being elements heavier than hydrogen or helium, which are called \"metals\" in astronomical parlance."}
{"id":"34067","title":"Expression for the (relativistic) mass of the photon","text":"I started learning a bit ahead from an old physics book, and they were discussing the photoelectric effect and after that Planck's hypotheses and energy quantas. The book said that the mass of a microscopic oscillator (what is that?) is not continuous, but discrete and the difference between states is an energy quanta: $ \\varepsilon = h\\nu = E_k - E_i $ And since $ E = mc^2 $ then the (relativistic) mass of the photon is $ m = \\frac{h\\nu}{c^2} $ How did they deduce that?"}
{"id":"39287","title":"Particle sources and particle detectors in quantum field theory","text":"I am looking for a resource that clearly exposes the concepts of a particle source and a particle detector in the context of Quantum Field theory. I want to understand Irreversibility in this context."}
{"id":"68510","title":"Need an intermediate resistivity part\/material","text":"I need a part or material for a planned experiment (the experiment is similar to those described in my articles http:\/\/arxiv.org\/abs\/1208.0066 and http:\/\/arxiv.org\/abs\/1109.1626 ). The problem is that the required resistivity (about 0.3 Ohm-cm or of the same order of magnitude) is much higher than that of metals and much lower than that of dielectrics. Eventually, I need a long cylindrical part, about 1.5 mm diameter and about 1 m length. So far I have considered semiconductors, conducting polymers, and absorbing materials of http:\/\/www.eccosorb.com\/Collateral\/Documents\/English- US\/Electrical%20Parameters\/ls%20parameters.pdf . The latter materials seem good, but they are essentially foams, and the required part cannot be machined from them. As for semiconductors and conductive polymers, I don't have a clear idea how to get (to order) a material with the required resistivity and how to make (to order) the required part. I need the above resistivity at a frequency of about 25 GHz, so, in principle, I could use a nonconductive, but absorbing (at the required frequency) material, but I would prefer a material that is conducting for direct current as well, to be able to measure the absorbed power. I would prefer a material with decent mechanical properties, so that I could, e.g., strain (tighten) the cylindrical part. Any advice? EDIT (02\/02\/2014): I have finally obtained the required parts. They are made of doped polysilicon. I am grateful for the answers."}
{"id":"47279","title":"Counterpropagating beams in a ring cavity lasers","text":"Ring cavity lasers usually has a intracavity element like a optical diode to forbid standing wave pattern and, consequently, spacial hole burning and related instabilities. So, my question is: why to beams exist (before install the optical diode-like element) inside the cavity, since (as far as I know) stimulated emission radiation \"follows\" the direction of the pump beam? The beam propagating in the opposite direction is the amplification of the spontaneously emitted radiation amplified by the resonator? ![Laser](http:\/\/i.stack.imgur.com\/7LVu5.gif)"}
{"id":"132189","title":"Vectors-Can anyone explain me the concept of sense in vectors?","text":"Is it same as the direction?Then , why another term \"sense\"is used ,instead of direction? Can anyone illustrate it?"}
{"id":"100682","title":"what is the best way to collimate light emitted by a led?","text":"I'm new to this forum. This is half a question, half a challenge. And it's more engineering then physics but I thought I might get good insights from a physics forum. I would like to cure a UV activated resin using a collimated 390 to 420nm light source. The area I need to cure is 3\" (7.5cm) by 3\". ( althoug the light beam can be bigger then this area and of any shape) so long as it's colimated and as efficient and low cost as possible. I want to get at least 3 watts of light power over that area and it has to have light everywhere and ideally as evenly distributed as possible. Ouff! I'll have to build some sort of DIY apparatus to collimate light coming from 390 to 420nm LED's. There are different types of LEDs on the market. Single point high power ones (1-3W) with viewing angles 40 to 160 degrees ( the angle at which the light cone expands out) or some even higher power arrays of diodes : 5w + Or low power 5mm diameter ones with witch I could build a large array of evenly spaced LEDs. ( but not necessarily) these have viewing angles of 25 to 160 degrees. In all cases, the actual light emmiting part is generally square and very small, then it is encapsulated in another package. I don't know much about optics and I have been pondering about this problem for a while now. How would you go about doing this? From lenses to mirrors, to what type of led tp use, there's pretty much an infinite different ways to approach this and I needed to get the idea out there, so others could share their insight. Feel free to throw any ideas but try to remain as technically and scientifically correct as possible. Also, although the problem makes sense to me maybe I failed to share it with you appropriately, feel free to ask questions or clarifications! EDIT I intend to pass the light through an LCD screen before hitting the resin. Which brings me to another question. At these wavelength, will diffraction come I to play if the LCD \"holes \" the light is passing through are 100 microns large? Edit wavelength can be between 390 and 420 blow that will damage the LCD. Above will not cure the resin."}
{"id":"100685","title":"Does it take infinite energy to create a perfect vacuum?","text":"Question is inspired by a recent burst of perpetuum mobile-type questions. It would be nice if one could simply discard them all by an argument that shows it's impossible to create a perfect vacuum. Intuitively, I have some hope that there will be a thermodynamics\/statistical mechanics argument that we can never even eliminate air friction _completely_ , thereby eliminating all these elaborate constructions requiring specific arguments from the get-go. My question is therefore twofold:   1. Does it take infinite energy to create a perfect vacuum (in a macroscopic box)?   2. If yes, can you include a derivation? If no, can you give an explicit construction with a finite amount of work being done?"}
{"id":"27738","title":"Geometric picture behind quantum expanders","text":"A $(d,\\lambda)$-quantum expander is a distribution $\\nu$ over the unitary group $\\mathcal{U}(d)$ with the property that: a) $|\\mathrm{supp} \\ \\nu| =d$, b) $\\Vert \\mathbb{E}_{U \\sim \\nu} U \\otimes U^{\\dagger} - \\mathbb{E}_{U \\sim \\mu_H} U \\otimes U^{\\dagger}\\Vert_{\\infty} \\leq \\lambda$, where $\\mu_H$ is the Haar measure. If instead of distributions over unitaries we consider distributions over permutation matrices, it's not difficult to see that we recover the usual definition of a $d$-regular expander graph. For more background, see e.g.: Efficient Quantum Tensor Product Expanders and k-designs by Harrow and Low. My question is - do quantum expanders admit any kind of geometric interpretation similar to classical expanders (where spectral gap $\\sim$ isoperimetry\/expansion of the underlying graph)? I don't define \"geometric realization\" formally, but conceptually, one could hope that purely spectral criterion can be translated to some geometric picture (which, in the classical case, is the source of mathematical richness enjoyed by expanders; mathematical structure of quantum expanders seem to be much more limited)."}
{"id":"29226","title":"Can a charged black hole interact via electromagnetism?","text":"> **Possible Duplicate:**   >  Detection of the Electric Charge of a Black Hole Light cannot escape from a black hole. However light is also interpreted as the carrier of the electromagnetic force. So how can a charged black hole interact via electromagnetism with its environment? Can one see from the outside of a charged black hole, that this black hole is actually charged? Or is this information hidden, because light (and therefore information about the electric charge) cannot escape from the black hole?"}
{"id":"91371","title":"Newton's rings experiment","text":"I have performed experiments in my college laboratory on Newton's rings to find radius of curvature of a convex lens used.i always get a dark center.Is it possible to get a bright center?If yes,then how?"}
{"id":"79220","title":"How do the effects of semiconductor doping affect the Hall effect?","text":"For instance, consider number 4 and 5 in the following sample: ![enter image description here](http:\/\/i.stack.imgur.com\/PP8Eo.png) Using the right hand rule, B points downwards, conventional current points to the right (because of the 5V battery), and therefore, the force on electrons points into the page. Electrons are going into the page from the red wire to the black wire and conventional current is going from the black wire to the red wire. But when conventional current goes from ground (black wire) to the higher voltage (red wire), then the voltage must be negative. Therefore, the voltmeter would read a negative reading. However, I am unsure what kind of effects doping the semiconductor would have on the voltmeter."}
{"id":"72230","title":"Difference between a quantum process and a thermal process?","text":"I was reading an article online pertaining to quantum mechanics and I stumbled across these few sentences. > A look at the corresponding energy regimes shows (Beck and Eccles 1992) that > quantum processes are distinguishable from thermal processes for energies > higher than $10^{-2}$ eV (at room temperature). Assuming a typical length > scale for biological microsites of the order of several nanometers, an > effective mass below 10 electron masses is sufficient to ensure that quantum > processes prevail over thermal processes. I would like to know what they mean when they say \"is sufficient to ensure that quantum processes prevail over thermal processes\". Or possibly just what the difference is between a quantum process and a thermal process. Original Text (Section 4.4 Beck and Eccles: Quantum Mechanics at the Synaptic Cleft) http:\/\/plato.stanford.edu\/entries\/qt-consciousness\/#4"}
{"id":"103488","title":"How is the current flow perpendicular to the wire?","text":"This answer gives a great explanation of how surface charge builds up to force the current to move perpendicular to the wire: http:\/\/physics.stackexchange.com\/a\/102936\/41086 However, it fails to address the _magnetic field_. Moving charges generate a magnetic field, so in addition to the electric field creating a force on the charges, so does the magnetic field. The answer doesn't explain what effect (if any) the magnetic field generated by the moving charges in the transient state has on the charge build up or the effect of the magnetic field in the steady state. So could someone give me an explanation of how the current flows perpendicular to the wire, **by taking the generated magnetic field into account**? Essentially, I'm looking for an extension to the answer I linked to above by including information on the steady state and taking the magnetic field into account."}
{"id":"103295","title":"Are there any serious alternatives to QCD nowadays?","text":"I've read several posts here where people talk about the history of the developement of the theory of strong interactions. And they mention Regge theory, pomerons, S-matrix and so on. I'm confused because I see the S-matrix in my QFT books, while wikipedia says: \" it **was** a proposal for replacing QFT...\"? What? Also in the article on Pomeron wikipedia says that it **is still used and that pomeron carries no charge etc...**? **Are pomerons real?** My question is whether there are any serious competitors to QCD today? Perhaps I should split this into several question, but I don't know. I would like to get an overview of the history of the development of the theory of strong interactions **from someone neutral** , but that's too much to ask for I guess. See for instance @Ron Maimon's reply to this question."}
{"id":"30946","title":"why is dark matter the best theory available to explain missing mass problems?","text":"Why is dark matter the best theory to explain the missing mass problem? Why is dark matter mathematically necessary to explain the missing mass problem? On a side not I believe dark matter is definately intuitive the next logical step in physics is to accept the fact the direct observations can not always be the determining factor in the acceptance of a theory. Positivism is not going to be the only driving force in physics forevermore. One theory that seems almost as good is the supersymetric particle. What determination which are actually detectible and how would we distinguish the differences between the two? References:   http:\/\/blogs.scientificamerican.com\/observations\/2011\/04\/14\/underground- xenon100-experiment-closes-in-on-dark-matters-hiding-place\/   http:\/\/www.science20.com\/hammock_physicist\/dark_matter_plot_thickens. http:\/\/www.holoscience.com\/wp\/synopsis\/synopsis-5-electric-galaxies\/5\/"}
{"id":"87886","title":"Newton's third Law?","text":"**Third law:** > When one body exerts a force on a second body, the second body > simultaneously exerts a force equal in magnitude and opposite in direction > to that of the first body. Newton's third law is unintuitive to me. If the earth exerts a gravitational force on me, how do I exert a force on Earth? Why is this true? It doesn't make any sense. How did Newton come to this conclusion?"}
{"id":"27794","title":"Is a photon \"fixed in spacetime\"?","text":"From what I've read, according to relativity, a photon does not \"experience\" the passage of time. (Can we say there is no past\/present\/future for a photon?) Would it be better to say a photon is \"fixed in spacetime\"? If so, how do we explain the apparent movement of a photon? Is everything else moving relative to it in spacetime?"}
{"id":"54162","title":"How does a photon experience space and time?","text":"To an an external observer it appears that time has stopped for photon. But this relation is reflexive, so for an observer travelling with the photon it appears the universe has stopped everywhere. Is this right? Space also gets distorted parallel to the direction of motion, but not perpendicular to it. Does this mean that for an observer travelling with a photon sees spacetime as a flat plane? NOTE: I'm using language _vividly_ not _literally_ when I say a photon experiences space and time. Not that I'm against idealist or panpyschist interpretations of matter or energy come to that."}
{"id":"55393","title":"How do photons experience time?","text":"I know that as velocity approaches the speed of light the time dilation shoots to infinity as shown below. ![enter image description here](http:\/\/i.stack.imgur.com\/eQWHb.png) 1)So I want to know how time is perceived from the point of view of the photon? 2)Since time stops at the speed of light how do photons move? 3)Is this why photons do not decay or are made up of any smaller particles?"}
{"id":"90350","title":"How is the universe is experienced at light speed?","text":"When moving faster, we experience time dilation and space contraction. We often state that a photon does not travel through time, i.e. if it where capable of observation, it would not experience time. But how about space? Would the photon experience a two-dimensional space?"}
{"id":"66602","title":"How can we observe lights properties if it travels at the speed of light, or can we?","text":"Special relativity says that anything moving (almost) at the speed of light will look like its internal clock has (almost) stopped from the perspective of a stationary observer. How do we see light as alternating electric and magnetic fields? Also does light never age?"}
{"id":"65878","title":"If time stops at the speed of light is a photon 'everywhere' at once?","text":"I am not a physicist so excuse my question if it's paticularly stupid. As a particle gets closer to the speed of light time slows down as for that particle as compared to a reference from the surrounding environment. Therefore for a photon travelling at the speed of light time has stopped for the particle or in a sense does not even exist for the particle....correct?? So is it true that a particle travelling at the speed of light would be everywhere that it does exist at the same time from an outside reference? For example if it travels in a line from point A to B at the speed of light wouldn't the particle be at every point along that line between A and B at the same time? I always read that photons exists as a probability wave but is it equally true to say that a photon exists everywhere that it will possibly exist at the same time? Presumably when a photon interacts with another particle (or is observed) it slows down and no longer travels at the speed of light i.e. no longer existing 'everywhere' at once. Would this explain the retrocausality of the delayed double split experiment? I think this would also explain duality. When unobserved the particle is at the speed of light and therefore exists everywhere that it could exist between A and B. When observed it no longer travels at the sppeed of light and therefore only exists at a point. Where's the mistakes? Thanks for your thoughts."}
{"id":"54516","title":"Life of a photon","text":"I am a student of class 12th and as far as i know when anything reaches about 99.99% of the speed of light it starts traveling in time or time for it slows down so that it don't breaks the speed barrier. So according to this light photons are immortal. Please clear my doubt."}
{"id":"130765","title":"Is the quantity $mv^{2}$ also conserved?","text":"The following is my reasoning: Suppose $\\vec{F}$ is a conservative force field. Then the total mechanical energy, $E$, of the isolated mechanical system is a conserved quantity which implies that $E=T+U=k_{0}$, where $k_{0}$ is a constant, $T$ is the kinetic energy and $U$ is the potential energy. Now, $\\nabla{E} =\\nabla{(T+U)}=\\nabla{T}+\\nabla{U}=\\vec{0}$. Hence, $\\vec{F}=-\\nabla{U}=\\nabla{T}$. Now, $U=\\int_{C} \\nabla{T}\\cdot d\\vec{r}= T+k_{1}$, where $k_{1}$ is a constant of integration. Now, $E=T+U=T+T+k_{1}=k_{0}$ which implies that $E=2T=k_{0}-k_{1}$ is constant and since $T=\\frac{1}{2}mv^{2}$ and hence $E=mv^{2}=k_{2}$, where $k_{2}=k_{0}-k_{1}$ is a constant and therefore $mv^{2}$ is a conserved quantity."}
{"id":"81839","title":"Surface gravity for a rotating charged black hole","text":"I have that the surface gravity (at the outer event horizon) for a Kerr-Newman black hole is $$ K_+ = \\frac{r_+-r_-}{2(r_+^2+(J\/M)^2)} = \\frac{\\sqrt{M^2-Q^2-J^2\/M^2}}{2M^2-Q^2+2M\\sqrt{M^2-Q^2-J^2\/M^2}} $$ where $r_\\pm$ are the outer and inner event horizons, $M$ is the mass, $Q$ is the charge, and $J$ is the angular momentum. I have a simple question, almost too dumb to ask: what (natural) units are used here, such that $M$, $Q$, $r_\\pm$, and $J\/M$ are dimensionally equivalent?"}
{"id":"86823","title":"Tesla to Newtons","text":"Is it possible to convert Tesla to Newtons of force?   Or magnetization force H:(A\/m) to newtons?"}
{"id":"74539","title":"Justification for smeared fields in the Wightman axioms?","text":"I just started reading _PCT, Spin and Statistics, and All That._ Can someone explain why we use operator valued distributions to describe fields? I read somewhere that it would take infinite energy to measure an observable at a single point. Why don't we instead use functions from some collection of subsets (i.e. the open sets) of space to the operators to emulate the fact that measurements usually occur over some area? In other words, what is the physical meaning of the test functions used to define the operator valued distributions? Are some of these functions non-physical, possibly too narrow in width that they'd violate some uncertainty principle?"}
{"id":"10332","title":"How fast do I need to go in order to avoid being seen by the Police?","text":"I was driving down the road at roughly the speed of traffic. I saw a police officer parked on the side of the road, and also noticed that a Semi was traveling in the lane right next to him. This got me thinking, is it possible to avoid being seen from the officer by using the Semi Truck and how fast would I have to go? Let's say the officer is parked on the side of the road and the semi is traveling on the lane next to the officer: ![My little Paint Drawing](http:\/\/i.stack.imgur.com\/7EWmd.png) Let's also assume that the truck is traveling at roughly constant 65 mph. Also assume that I see the officer and line up my car to block the officer's view via the semi at roughly a mile away. (Ya I know a little far). What I want to know:   * How fast do I have to go to avoid being seen by the officer?   * Is this speed constant or variable? _Note: I am in NO way advocating speeding on the highways and breaking the law. This was just was just sparked as it somewhat happened while I was driving down the road._"}
{"id":"70093","title":"What is band filling effect?","text":"Could anyone tell me what is band filling effect? I have search Physics.stackexchange and Google, but didn't find any useful information. I'll appreciate it if anyone can tell me the physical image of band filling effect."}
{"id":"121241","title":"Is it reasonable to visualise an additional dimension of time as a part of 4th dimensional spacetime","text":"_[EDIT] In the spirit of asking a \"good question\", this question is considerably more refined that the one I decided to blurt out earlier thanks to a little additional research and nudges from comments._ I'm not a physicist, but love to let ideas roll around in my head - so please excuse any crossed boundaries. I'm not equipped with the mathematics required to examine or perhaps even pose this question in a way many of you are probably used to. So, my question: Spacetime as we know it consists of 3 spatial dimensions + 1 temporal dimension. Is it reasonable to visualise spacetime in our next-highest dimension as consisting of 4 spatial + 2 temporal dimensions? _From another point of view:_ If we view our 3D space as a submanifold of a 4D manifold, is it reasonable to suppose that time as we experience it may be a similar submanifold of time in a higher dimension? _Or:_ **Adding and subtracting spatial dimensions is the easy bit (an axis perpendicular to all other axes in that dimension) ... do we also get to add and subtract temporal dimensions from spacetime as we journey from one dimension to the next?** (For posterity, my original question was phrased as \" _Does the 4th dimension include imaginary time as part of its fabric of spacetime?_ \" - referring to the concept of imaginary time as popularised by Hawking. In case it helps, this annoying little thought experiment arose from musings of the possible mechanics behind quantum entanglement) ========================================================================= _References I've found helpful so far:_ \"Survey of two time physics\": http:\/\/inspirehep.net\/record\/532282?ln=en \"Dual field theories in (d-1)+1 emergent spacetimes from A unifying field theory in d+2 spacetime\": http:\/\/inspirehep.net\/record\/750980"}
{"id":"77465","title":"How to know if something is a primitive concept, a law, a definition or a theorem","text":"Some basic Physics books are often misguiding in the sense that they don't make clear whether something is a primitive concept, a law, a definition or a theorem. This is often a little confusing. I've asked here some moments ago if there's a definition of force, and I've been redirected to a question about whether Newton's Laws are laws or definitions of mass and force. This is just an example, but there are many others, like for instance, the conservation of energy, which some books present as a theorem, and some places says that this is a law of Physics. Time on the other hand, is something that according to Feynman's lectures on Physics, cannot be defined, so we just say how to measure it and leave it as a primitive concept. In this context: how in general can we identify whether something is a primitive concept, a fundamental law of Physics, a definition or a theorem?"}
{"id":"6279","title":"The Galileo thermometer: why do the bubbles float in the middle of the tube?","text":"If the water were uniform temperature, it would have uniform density, so a bubble should either be all the way at the top (if it's lighter than water) or all the way at the bottom (if heavier). But in reality you don't see this neat separation. Sometimes you see bubbles hovering in the middle. Is this because the water temperature is not uniform? Does this mean that with time, if the room temperature stays constant, all bubbles will neatly separate along the top and bottom?"}
{"id":"122912","title":"What is the difference between surface plasmon and surface plasmon polariton?","text":"I'm trying to understand this reading article linked below and I still don't know how to explain this simply, without need to derive everything mathematically. Can someone just write here how do SP's and SPP's differ? http:\/\/dipc.ehu.es\/etxenike\/admin\/documentos\/archivos\/publicaciones\/307RPP2007.pdf Moreover while reading this article : http:\/\/daedalus.caltech.edu\/publication\/pdf\/0801_opinion.pdf I think that they are writing about SPP's despite of using SP term. So maybe there's no difference at all?"}
{"id":"122915","title":"what if the time is zero in one reference frame","text":"Consider measuring the length of an object in another frame of reference.   Of course this should happen at the same instance in the frame of reference the measurement takes place in.   but using Lorentz transformation the time in the other frame of reference would be different and there would be a period   \\begin{equation}\\Delta t^\\prime = \\gamma v\/c^2(x1 - x2)\\end{equation} so we have $\\Delta t = 0$ but $\\Delta t^\\prime$ has a value   doesn't this contradict with the time dilation basic law $$\\Delta t^\\prime = \\gamma \\Delta t$$"}
{"id":"93918","title":"How to calculate the force in an \"impulse based reaction model\"?","text":"I have a sphere moving forward with `v (pre collision) = v1i` A second sphere colliding with the first one has its own pre collision `velocity = v2i` The collision should be not totally inelastic (like, you know, the case of two human bodies colliding each other), but for my purpose I think I can consider the coefficient of restitution = 0 Using the formula $$ I = mv_f - mv_0$$ and replacing `I` with `F * (delta)t` I woul like to know how can I compute that `F`, since I don't know what will be `v(post collision)` because it is my final unknown to compute. And I think that force should be computed with the relative velocity of the two spheres, but I have no idea at the moment on how to proceed. Any suggestion will be strongly appreciated. UPDATE Thank you for your suggestion ja72. So in that case can I write something like this? $$ F *\\Delta t = \\frac{ (\\epsilon+1) \\left(v_2-v_1\\right)}{\\frac{1}{m_1}+\\frac{1}{m_2}} $$ But in this case, how am I supposed to evaluate final velocity of the two spheres?"}
{"id":"80875","title":"What is wrong in following arguments about connection of local gauge invariance and causality?","text":"There is a question and corresponding downvoting of my answer, so I decided to ask this question. There is my answer on it: \"...The most theories of free fields are invariant under global gauge transformations. It can be interpreted as instantaneous \"rotation\" of all space-time relatively gauge space, which contradicts the causality (of gauge transformations) in relativity case, because the speed of interactions is limited by $c$ according to it. So we introduce local gauge transformations, where \"rotation\" depends on space-time point. The gauge states is indistinguishable in a case of free field. But when our global gauge invariance field interacts with other fields, degeneracy is lifted, and this leads to instant changes in interaction. Thus we satisfy causality principle for gauge transformations. Also I can add some example. Let's have global gauge isotopic invariance of QFT. So we identify proton and neutron as the state with some value of isotopic spin, and then if we choose that what to call a proton at one space- time point, we also must choose what to analogically call a proton at other points, so it little contradict the principle of local field theory. If the nucleon interacts with EM field, we may instantly change it state by using global gauge transformations, which also instantly changes the interaction between nucleon and EM field in all points of the space-time. It leads to non- local theory and contradict the causality. Corresponding arguments were used by Yang and Mills in 1954....\" Where did I make the mistake (I really do not see the error)?"}
{"id":"44018","title":"What is voltage: strict but intuitive definition from accumulator's perspective","text":"I know, that voltage is analogous to pressure for charge, but analogies lie. I don't see charge pressing anything and I don't understand definition of $U=A\/q$ (voltage = work\/charge), cause I can't see what is that work. ![enter image description here](http:\/\/i.stack.imgur.com\/EAxef.png) Voltage in a circuit is determined by an accumulator. I suppose, that accumulator is characterized with voltage output and current output. I don't get how they are related, but can imaging some tweaks to the accumulator: 1) If I double the area of both metal plates, contacting with respective solutions, would it double the voltage output? Intuitively it will cause twice the amount of Ions bump into plates and give their charges to them, so it would double the current and won't change any \"charge pressure\", I can think of. 2) If I double the concentration of each solution, would it double the current or charge. My intuition that it is the same as if I doubled the area of plates - it would just double the amount of ions transferring their charge per moment of time. Nernst equation suggests a change of voltage in this case, but I can't see what it is. What I want to understand, is what is voltage in terms of ball-like ions, forces and such intuitive entities. Thanks."}
{"id":"44017","title":"Have I discovered how to calculate the proton's mass using only integers?","text":"Could it be possible that the mass of the proton can be calculated by a series of integer sequences? Or is this just a curiosity? $$\\sum_{m=1}^{\\infty } \\frac{1}{10^{26}(m^2+1)_{2m}}=$$               NSum[1\/Pochhammer[m^2+1,2m], {m,1,\\[Infinity]}, WorkingPrecision -> 50] \/10^26       First seven digits match the proton's mass in kilograms. $1.6726218229590580987863882056891582636342622102204\\times10^{-27}$   $1.672621\\times10^{-27}$ - from OEIS _revised 11\/15\/12_   $1.672621777\\times10^{-27}$ - from Wikipedia What's to say that sometime in the future, the proton's mass won't be made more accurate by adding $4.5\\times10^{-35}$ to the current number? **Edit to explain motivation** Whenever I get a result I don't recognize, I look it up on OEIS. I found this number. I posted on Mathematica.SE with the intention of asking for advice on how to prove that it converges. That would make this number a constant. If this is a \"fluke\" or the result of \"small numbers,\" it's still worth exploring. **Edit: It does converge.** **Final Thoughts** $f_{p}=0.16726218229590580987863882056891582636342622102204$ is the 0-dimensional value of a fractal know as the Hilbert Curve. To get the minimal 3-dimensional value: $f \\times 10^{((dimension+1)!)}$ where $0\\le dimension \\le 3$. This results in the value for a $1\\times 1\\times 1$ cube (coincidentally, the definition of the gram.) To get kilograms: $f \\times 10^{((dimension+1)!+3)}$. I posit that the fractalness is the stabilizing influence on the proton. **Coda** I agree with everyone that I have been wrong-headed about the importance of this constant. I have posted the constant on OEIS A219733. Thanks for your patience."}
{"id":"5228","title":"On a principal bundle, why is the horizontal vector space not unique?","text":"On a principal bundle, at each point you have a tangent vector space. At a given point, the vectors tangent to the fiber form the vertical vector space. Then the vector space at that point is a direct sum of the vertical vector space and what's called the horizontal vector space. This is all standard stuff in gauge theory. What I don't intuitively understand is why is the horizontal vector space not unique?"}
{"id":"41193","title":"Is there stringy Morse theory?","text":"This question is pretty vague and open. I'm just curious if anyone has considered this. Morse theory has a nice physical formulation: a Morse function can be thought of as a potential, so the gradient flow is the force experienced by a particle. The equilibria are the critical points of the potential. If we take a supersymmetric extension of this as a quantum theory, its ground state structure computes the homology of the space the particle moves in. What can strings moving in a manifold subject to a potential tell us about the topology of the manifold?"}
{"id":"112600","title":"Twin Paradox: Whose time is slow?","text":"What I understand about the twin paradox is that if a person stays at rest with something lets say the earth and a traveller moves with a great velocity with respect to the firsy person, then the time of traveller becomes very much smaller than the time of person who stays back, which led to the conclusion that one would age very much and one would not. But if you see from the frame of the traveller, the person on rest on earth seems to be moving very fast, and according to this the person on earth should be younger than the traveller. So, who is actually going to be younger? One of my friends had asked me this question, and I replied that from the frame of the traveller if you measure the velocity of earth bound person you will have to take the same velocity in negative and you will get the same result. Even though he accepted my answer; I am myself not satisfied with it."}
{"id":"70326","title":"Where does additional heat energy come from in isothermal step of Carnot cycle","text":"According to widipedia: > During step 1 (isothermal heat addition or absorption) the gas is allowed to > expand and it does work on the surroundings. The temperature of the gas does > not change during the process, and thus the expansion is isothermal. The gas > expansion is propelled by absorption of heat energy Q1. The gas temperature is the same as the temperature of the reservoir. But where may heat energy Q1 come from as there is no temperature gradient between gas and reservoir?"}
{"id":"126497","title":"Does the Higgs boson give mass to ALL other particles?","text":"The Higgs Field gave mass to other particles via spontaneous symmetry breaking; does this mean it gives mass to all particles that have mass - such as neutrinos, quarks or weak bosons and strong bosons (gluons)? In the articles I've seen its not clear what the criteria is for the Higgs Field to couple to particles to give mass; Anna v's answer suggest that electo-weak vertexes are required; but the linked article suggests it also couples to quarks."}
{"id":"19900","title":"frames of reference","text":"> **Possible Duplicate:**   >  Help Me Gain an Intuitive Understanding of Lorentz Contraction Frames k and k' are inertial frames. Frame k' is moving at a velocity of magnitude v relative to frame k in the direction of the x-axis. there are rigid rods in each frame oriented along the direction of the relative velocity. Someone in frame k measures the rod that is stationary in k as having length L1 and the rod stationary in frame k' having a length L'2. What are the lengths of the rods measured by someone in frame k'?"}
{"id":"87065","title":"Why does adjoint representation matter in some field theories?","text":"Recently I am reading a paper about monopoles. In several cases, it seems that writing fields in adjoint representation of the gauge group makes a difference. Once it leads to different group after symmetry breaking when using other representation. And I also noticed statement like this, \"An important open question is whether an analogous Bogomolny monopole's mass bound can be obtained if the Higgs field is not in the adjoint representation.\" Can anyone kindly shed light on this. Thanks! **Update:** I reckon any field (either EM field in real space or Higgs field in internal isotopic space) be in a certain type of representation space of the symmetry group associated with the Lagrangian or action. This space also dictates some constraints on the fields, e.g., specific tensor or spinor structures ( _anything more???_ ). And what representation space you use contains physics as well, that is to say, we have to check it by experiments. Perhaps this question addresses on a particular case. Either does the explicit and concrete 2nd answer. Is this understanding correct?"}
{"id":"87063","title":"Unit for expressing energy eigenvalue in infinite potential well problem","text":"How are the energy eigenvalues expressed in a infinite potential well problem( Joules\/eV)?"}
{"id":"123791","title":"Rotating hoop with fixed body inside of same mass","text":"Please help me to solve this problem. I am unable to understand which force will cause the hoop to bounce. > A small body $A$ is fixed to the inside of a thin rigid hoop of radius $R$ > and mass equal to that of the body $A$. The hoop rolls without slipping over > a horizontal plane; at the moments when the body $A$ gets into the lower > position, the center of the hoop moves with velocity $v_0$. At what values > of $v_0$ will the hoop move without bouncing? ![enter image description here](http:\/\/i.stack.imgur.com\/o5S2m.gif)"}
{"id":"118503","title":"Can internet speed be faster than speed of light?","text":"We send email in order to transfer information and, over time, the technology used to send email improves to send email at faster rate. Since we use the Internet to send information from one place to another place all over the world, can information speed be faster than light speed via the Internet?"}
{"id":"118507","title":"Brownian motion and physical meaning","text":"I have read Stochastic Differential Equations by Bernt Oksendal   It constructs Brownian motion by Kolmogorov extension theorem by consider $p(t,x,y)=(2\\pi t)^{-n\/2} e^{- \\frac{|x-y|^{2}}{2t}}$   But I can't understand what is the relation to the Brownian motion in physics."}
{"id":"47737","title":"Resonance in a 1 ft granite box","text":"* I have a granite cube made using 6 slabs of granite 1 foot square and 1 inch thick. The top and bottom slabs have a 1 inch margin around the edge. The slabs are just set together, not notched or mortared or anything.   * I also have a concrete block with pipes running through it that generates a sonic pulse\/compression wave from a valve closing. If this were a regular water system we'd all call it water hammer. This water hammer comes _out the top of the block_. It is assumed that this block is not going to self destruct from the water hammer. It pulses about 60 times per minute. There is a working example of this block that I have seen online, so I know how to build it. Now what I'm trying to figure out is what would happen if I would set my granite cube on top of the concrete block. Would the sound reverberate around in the granite box until it blew it apart? Or would it somehow go through the block somewhere else? What would happen? Would it resonate at a certain frequency and produce a hum? Would it make a difference if I took away the bottom slab?"}
{"id":"28910","title":"Crushing a magnetic field","text":"What would happen if you crushed a magnetic field to an ever decreasing size? Thanks. EDIT: How small could the field possibly go? Is there a limit on how small it could get? Is there a maximum field density based on the size?"}
{"id":"82116","title":"Movement of a gyroscope with non-fixed axis","text":"Assume one has a gyroscope rotating around an axis with both ends leaning on a dedicated semiplane as shown on the picture below. There is no friction either between the rotor and the axis or between the axis and the semiplanes. The only force is the gravity acting on the rotor. How will the gyroscope behave if one instantly removes one of the semiplanes? ![enter image description here](http:\/\/i.stack.imgur.com\/pn3ZO.png)"}
{"id":"82118","title":"Understanding the Wave Function and Excited States","text":"A wave function is an infinite dimensional vector space, how can it \"live\" in $\\mathbb{R}^3$? Given the equation that is built like: $$\\Psi (x,t) = \\sum ^{\\infty} _{n=1} c_n \\psi _n (x) e^{-i E_n t \/ \\hbar}$$ How does one \"excite\" a quantum particle in the lab? The excited states simply give a different probability distribution for the particle? For the infinite square well, the wave function in excited state $n$ has exactly $n$ bases, all of which are non zero? If a particle is in its ground state, does that mean that all other bases must be zero? making the wave function dependent only on one basis?"}
{"id":"110733","title":"Is dimensional analysis always sufficient to establish equivalence of quantities?","text":"In dealing with the Biot-Savart law, it was argued that $$ q\\frac{d\\vec{s}}{dt}\\equiv Id\\vec{s} $$ using the fact that the units are equal. Does this kind of argument always work? It seems too simple to be true."}
{"id":"118669","title":"Effects of gravity on light","text":"If gravity can bend light, why can't gravity slow light. At least momentarily? Wouldn't that give the illusion of the universe expansion speeding up?"}
{"id":"19201","title":"Two axes for rotational motion","text":"I understand that angular momentum is a vector, etc.. But, what really happens when some object, say a ball for example, is set to rotate along two axes? What would the resulting motion look like?"}
{"id":"119752","title":"Why is mud muddy?","text":"How to explain properties of mixture of sand with water? Why is it so coherent and slippery at the same time? Is it due to hydrogen bonds? Why is mud of smaller grains more slippery and incoherent and mud of bigger grains is more coherent and less slippery?"}
{"id":"119751","title":"Are standard and isotropic forms of Schwarzschild metric truly equivalent?","text":"My admittedly rudimentary understanding of physical meaning of conformal flatness - as pertaining to a stationary observer exterior to a spherically symmetric static gravitating mass $M$: Locally Euclidean, in that the proper differential volume $dV$ between two concentric spherical surfaces centered about M, having differential proper radial spacing $dR$, is given by the Euclidean formula for enclosed volume $dV$ = $AdR$, where $A$ can be taken as the mean value of the two proper surface areas, and we take the limit as $dR\\rightarrow0$. Correct? And that in GR, SM (standard Schwarzschild metric) is **not** conformally flat since there $dR$ is greater by factor $\\sqrt{-g_{rr}}$, as can be determined by inspection of the line element - e.g.:Wikipedia - archived revision That is, $dV = \\sqrt{-g_{rr}}AdR$ for SM, $ > dV$ (Euclidean) - for a _given_ proper areal difference between shells. It's evident by inspection of the line element for ISM: here, for the equivalent differentially separated concentric shells arrangement as above, $dV = AdR$ asymptotically applies as per Euclidean formula? In other words, _by virtue of it's construction as spatially isotropic_ , ISM necessarily claims a conformally flat metric, in-principle measurably physically distinct from that of SM? How then is it that the two are claimed to be physically equivalent? Also is there a precise technical term and definition specifying departure from conformal flatness here? While the above concentric spherical shells situation is the one I was introduced to, there is surely no reason preventing it being dimensionally reduced to one of proper radial spacing between concentric great circles, and in fact then further reduced to an arbitrarily small local sector cut from such concentric circles. Meaning it must be an in- principle locally observable quantity rather than only determinable globally?"}
{"id":"13757","title":"How was Avogadro's number first determined?","text":"I read on Wikipedia how the numerical value of Avogadro's number can be found by doing an experiment, provided you have the numerical value of Faraday's constant; but it seems to me that Faraday's constant could not be known before Avogadro's number was as it's the electric charge per mole. (How could we know the charge of a single electron just by knowing the charge of a mole of electrons, without knowing the ratio of the number of particles in both?) I just want to know the method physically used, and the reasoning and calculations done by the first person who found the number $6.0221417930\\times10^{23}$ (or however accurate it was first discovered to be). Note: I see on the Wikipedia page for Avogadro constant that the numerical value was first obtained by \"Johann Josef Loschmidt who, in 1865, estimated the average diameter of the molecules in air by a method that is equivalent to calculating the number of particles in a given volume of gas;\" but I can't access any of the original sources that are cited. Can somebody explain it to me, or else give an accessible link so I can read about what exactly Loschmidt did?"}
{"id":"43291","title":"Can additional airfoil enable a commercial liner to reach 100km altitude?","text":"Assuming that it's engines are incapable of dying out at 100km altitude, would mere addition of airfoil area enable a commercial liner e.g. B787 to reach that altitude?"}
{"id":"43293","title":"Could we use particle colliders as fusion generators?","text":"So I know the basic gist is that fusion power's main issue is sustaining the fusion. I also know that there are two methods. The Torus method and the laser method. The torus magnetically contains plasma and heats it with radiation and accelerates the plasma around to make strong enough collisions that protons fuse. The laser method uses 192 lasers and focuses it on tiny frozen hydrogen pellets and aims to initiate fusion each time pellets are dropped. The though struck me when we could sorta combine the two designs together. The torus doesn't have to worry about making fusion happen at a specific location but it has issues in that the plasma is unevenly heated and leaks. On the other hand, the laser design is extremely complicated in the level of precision needed and would have to repeat this for every pellet. This lead me to think to make something precise and contained at the same time. I see that particle colliders are able to direct two beams of protons and have them collide at a specific spot with a very precise energy. Couldn't we tune the energy of the two beams of protons to the energy required for them to fuse? We have the ability to smash them into bits, surely we have the ability to have them fuse. (I'm thinking about the type of collider that circles two beams in opposite directions) It would be at much lower energies than normal colliders and would be very precise and it would be possible to fuse at a specific location that has greater leeway because for protons that missed collision, they'd just circle around again! Thus protons would efficiently be used and very little would be wasted. There wouldn't be problems of plasma leakage because we are focusing them in a thin tight beam. It seems that this idea has girth, or I feel this way at least, can someone back me up by offering some calculations on how to calculate the efficiency? How would I go about calculating the two circling beams of protons and at what specific velocity would be needed? etc."}
{"id":"64509","title":"Different batteries connected in parallel","text":"If we have 2 batteries one of emf x and the other is of emf y and we connect them in series we get an effective emf of x+y. But what if we connect them in parallel, how to calculate the emf now?"}
{"id":"65583","title":"Work on Ferromagnetic Object Due to Solenoid","text":"I've been going through some equations and such trying to determine the work done by a solenoid on a ferromagnetic object. I have the following: Magnetic field due to solenoid: $\\vec{B} = \\langle0,0,\\mu_0nI\\rangle$ (Assuming coils are on xy-plane and current is counter-clockwise) Force of magnetic field: $ F = q\\vec{v} \\times \\vec{B} $ Work: $ W = \\int F \\cdot dl $ Work of Magnetic Field: $ W = \\int_c(q\\vec{v} \\times \\langle0,0,\\mu_onI\\rangle) \\cdot d\\vec{r} $ For one, this seems to indicate a work of 0 if the object is not charged, which I have seen in some places but just doesn't seem right. Also, this does not take into account the properties of the object, such as relative permeability, which I guess could have some effect with the charge value. I'm trying to calculate the acceleration of a ferromagnetic object from a magnetic field, is there a better way to do this? I've thought about the following: $ \\vec{a} = \\frac{q\\vec{v} \\times \\vec{B}}{m} $ However, this is where I started running into the charge issue and thought to calculate it from the work done."}
{"id":"2573","title":"Intelligent(?) Particles","text":"Quite a while ago I read about a series of experiments that basically suggested that a certain kind of particle\/atom\/(something) were \"intelligent\" and could appear in two places at once, or essentially could \"tell the future\" when it came to navigating a \"maze\" ...I think it might have involved lasers or mirrors? Does any one a) know what I'm talking about, and b) have links\/further information on it? Really don't have much more recollection than that I'm afraid. This will probably come across as a rather vague question so my apologies but hopefully someone will know what I'm talking about!"}
{"id":"57218","title":"Deriving the \"total\" Bose Einstein density of states, including the condensate","text":"Is is possible to derive the Bose-Einstein density of states containing the delta function representing the BE condensate?"}
{"id":"60409","title":"How to tackle 'dot' product for spin matrices","text":"I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as $$ H = \\alpha[\\sigma_z^1 + \\sigma_z^2] + \\gamma\\vec{\\sigma}^1\\cdot\\vec{\\sigma}^2 $$ where $\\vec{\\sigma}^1$ and $\\vec{\\sigma}^2$ are the Pauli spin matrices for two particles separately. I think $\\sigma_z$ is the z component, I found that $$ \\sigma_z = \\left( \\begin{matrix} 1 & 0 \\\\\\ 0 & -1 \\end{matrix} \\right) $$ which is 2x2 matrix. I am wondering if the $\\sigma_z$ is the same for particle 1 and 2? if so, $$ \\sigma_z^1 + \\sigma_z^2 = 2\\left( \\begin{matrix} 1 & 0 \\\\\\ 0 & -1 \\end{matrix} \\right) $$ Is that right? The most confusing part is $\\vec{\\sigma}^1\\cdot\\vec{\\sigma}^2$, there are two matrices involved, so how does the dot product work? I am trying solve for the eigenvalues of H, it looks like to me that each $\\sigma_z^1$ and $\\sigma_z^2$ is 2x2, so there are two eigenvalues, is that correct?"}
{"id":"976","title":"What's the highest speed one can obtain safely at home?","text":"Among common household appliances, things one can make from stuff in the garage or hardware store, and reasonably safe, e.g. within reach of hobbyists and high school kids entering a science fair, what is the highest speed of motion of any bit of matter one can obtain? The bit of matter, or surface points of some object, should be macroscopic - big enough to see, time the motion of somehow, make collisions with other bits of matter. Electrons aren't big enough! Fluids are fine too. Guns, dynamite would be too dangerous. The tips of a fan blade or shutter edges in a camera are fine. Using a drill to make something spin real fast might be okay if all mad scientists involved are careful."}
{"id":"62808","title":"Tension of rope in the gravitational field of earth","text":"> Two balls of mass $m$ each one are connected with mass-less rope with the > same length as the radius of earth. The system is in free fall. Prove that > the tension of the rope when the nearest (to the earth) ball's distance from > the earth surface is $R_E\/2$ is: $T = \\frac{32}{225} mg$ ![Illustration](http:\/\/i.stack.imgur.com\/TgZjm.png) What I did is the following: $F_1$ is a gravitation force exerted on the nearest ball by the earth: $F_1=G \\frac{M_Em}{(1.5R_E)^2}$ $F_2$ is a gravitation force exerted on the farthest ball by the earth: $F_2=G \\frac{M_E m}{(2.5R_E)^2}$ $T=F_1-F_2=G \\frac{M_E m}{(1.5R_E)^2}-G \\frac{M_E m}{(2.5R_E)^2}=\\frac{G M_E m}{R_E^2} \\left (\\frac{4}{9} - \\frac{4}{25} \\right)=\\frac{64}{225} mg$ However, my answer is somehow twice bigger than what is expected. Where am I wrong? What am I missing?"}
{"id":"62801","title":"Has martian sunset same spectra than this earthly bluish-violet sunset?","text":"1. Has martian sunset same spectra than this earthly bluish-violet sunset?    2. What about sunset on Mercury?"}
{"id":"45004","title":"Help an aspiring physicists what to self-study","text":"This is probably not the kind of question you'll often encounter on this forum, but I think a bit of background is needed for this question to make sense and not seem like a duplicate: 2012 has been an annus horribilis in my life. I have lost a lot of close relatives in a sudden surge of cardiovascular diseases in my family. I also discovered I inherited genetic diseases and that I'll probably undergo the same fate sooner or later. One of the people I've lost is my father. We used to talk about physics all the time, and ever since I was 5 I kept telling him my dream was contributing to the field. When I lost him a couple of months ago, I used studying as an emotional outlet. He always emphasized the importance of academic excellence, and for this reason I got obsessed with studying and getting high grades even more than I ever did. I now am 1\/80 in a top high school, but I am frustrated enormously. I find that I waste my time at high school, especially since I probably won't have as much time here as many other people do. I find the mathematics and physics boring and easy, and I feel like I'm wasting my time with certain classes which don't interest me at all (for example Latin). So I decided to study physics and mathematics outside of school. My school has been somewhat supportive, granting me a day per week off to do whatever I want, basically. I of course have a considerable amount of free time in addition to that day, since I ace almost every test without too much studying and without making my homework (not because I don't want to, but because I don't need to). I decided to self-study because I decided that life is too short (and mine will be even shorter, if I reach 50 I'd be lucky) to waste time. So my plan is to do at least the first 2 years of undergraduate physics in the 2 years I've got left at my high school. My main objective is to gain a mathematical and physical understanding of quantum mechanics, as advanced as I possibly can. I am currently studying Linear Algebra and Statistics, but I have a problem. I don't know what to study and, especially, in what order to study it. I have read literally read dozens of questions and answers as to what should be the mathematical\/physical background for Quantum Mechanics (my future field of interest). But I find these to be too general, and I often am overwhelmed by it. In the same way you can get overwhelmed when you need to clean your house, but it’s so dirty that you don’t know where to start. So I would like your help. My current mathematical background:   * Basic differential calculus and no integral calculus, we will get that later on this year, however, I think it’s best for me to study it myself before we get it at school since it is crucial in physics. To show my level of differential calculus, this is about the toughest homework question we had to solve algebraically: _Given are the functions $f_p(x) = \\dfrac{9\\sqrt{x^2+p}}{x^2+2}$. The line $k$ with slope $2.5$ touches the function of $f_p$ at point $A$ with $x_A=-1$. Get the function of $k$ algebraically._   * Trigonometry and trigonometric functions. Again, as above, one of the toughest question we had to solve: _Given are the functions $f(x)=-3+2cos(x)$ and $g(x)=cos(x-0.25\\pi)-2$. Get the functions $s(x)=f(x)+g(x)$ and $v(x) = f(x)-g(x)$ in the form $y(x)=a+bcos(c(x-d))$._   * Analytic Geometry (conic sections, tangency, bisections, you know the drill).   * And of course everything below this level. I probably forgot some things, but you can ask my in the comments if I know certain fields. We will get a lot more mathematics in the coming years, but I want you to disregard that fact when answering that questions. I want to self-study as much as I can, and my mathematics teacher is very fond of me, so if I know a topic before we get it in class, he will let me do other mathematics that I want (he even said this). So I won’t lose time by self-studying subjects we’ll get eventually, so don’t worry about that.  My current physics background (names of the chapters we discussed):   * Newton’s laws, Mechanical energy\/forces    * Pressure and Heath    * Signal processing    * Electric currents (Ohm’s law, Series and parallel circuits, etc.)   * Again, everything below this level too (again, I’m probably forgetting stuff). Here exactly the same thing counts as with mathematics, we will get a lot more physics in the coming year, but again, disregard that. My physics teacher adores me, even more so than my mathematics teacher, so again, he won’t mind if I do something else if I know the material he’s discussing already. This is of higher level than American AP classes and British A-levels, keep that in mind. Now my question is, what mathematics and physics do I need to study, and my importantly, in which order do I need to study it, in order to have a basic understanding of quantum mechanics in 2 years? I know basic is a very general term, but I think you people, as people who studied it themselves, know what is realistic and achievable. I know this might seem like a duplicate of hundreds of previous questions, but it isn’t. All the other people asking this question have gotten answers that I don’t find suitable for me. Mostly the answers are from people who assume that you have to ‘have a basic understanding of this, a basic understanding of that’, etc. But how do I know what ‘basic means’? Also, now that you guys know exactly what I know and what I don’t, you can more finely tune the answers into my personal situation. As I said, currently I am doing Linear Algebra and Statistics, so you can omit those 2 from your answers, and start from the point I finished those 2 (which will be around January).   * p.s. If you want to recommend certain books, be my guest. If it's a good book, than money is no issue, I've saved up enough money throughout the years"}
{"id":"48034","title":"Residues in QFT propagator","text":"It is a well known fact that the location of the pole of a propagator (in QFT) can be interpreted as the physical mass. Is there an interpretation for the residue of the propagator? **Note:** I´m thinking of generalised propagator, not necessarily a propagator of a fundamental field."}
{"id":"10479","title":"Is the quantum analog of a probability distribution the wave function or the density matrix?","text":"Classically, probability distributions are nonnegative real measures over the space of all possible outcomes which add up to 1. What they mean is open to debate between Bayesians, frequentists and ensemble interpretations. A degenerate distribution is the least random distribution with a probability of 1 for a given fixed event, and 0 for everything else. What is the analog of a classical probability distribution in quantum mechanics? Is it a wave function augmented with the Born interpretation for probabilities, or is it the density matrix? Does a pure density matrix correspond to a degenerate distribution?"}
{"id":"63171","title":"is the nature of particle beam weapons in science fiction true to the reality of particle physics?","text":"I am referring to the use of specific particle types such as an antiproton beam, positron beam, meson beam or muon beam for example in the likes of shows like star trek. I was curious if a beam of a certain particle would produce specific effects or is this purely wishful thinking. I am by no means a physicist so straightforward answers would be great if possible. If the answer could describe what effect each particle would have on materials, which particles are of particular use and which ones are impractical. Would it matter if the particle is normal or antimatter? I understand there is a lot of variables and factors like the way a particle decays, the particle half life and velocity affects its range distance, the energy of each particle, the materials the particles interact with, etc etc so any input is appreciated. To summarize, i am asking in terms of the realistic attribution of weapon effects to certain particle types and also non-weapon effects."}
{"id":"29273","title":"How does Earth's interior dynamo work?","text":"I'm interested in getting a basic physical understanding of how Earth's magnetic field is generated. I understand that it's a \"dynamo\" type of effect, driven by convection currents in the molten outer core. These currents cause charges to move, and this generates the field. However, what I can't find a good explanation of is why there is a separation of charges in the first place. Presumably, moving neutrally charged molten iron would have the same effect as moving any other neutrally charged thing, i.e. it wouldn't create a field. And presumably, if the fluid wasn't moving then it would become neutral pretty quickly, since molten iron is a good electrical conductor. So am I right in thinking that the charge separation is the result of positive feedback, in that an intial deviation from neutrality would generate a field, and this would (somehow) cause a greater separation of charges, resulting in a kind of self-maintaining charge separation? Or is there another explanation? In either case, does anyone know of a good resource that explains the basic principles in physical terms? I know that the interior dynamo is a very complex phenomenon, but I'd like something that gives a good physical picture of how the electromagnetic and fluid dynamical phenomena interact, rather than diving straight into partial differential equations."}
{"id":"111761","title":"Why is spacetime curved by mass but not charge?","text":"It is written everywhere that gravity is curvature of spacetime caused by the mass of the objects or something to the same effect. This raises a question with me: why isn't spacetime curved due to other forces or aspects of bodies? Why isn't it that there are curvatures related to the charge of a body or the spin of particles or any other characteristics?"}
{"id":"34263","title":"What does the notation $|x_1,x_2\\rangle$ mean?","text":"I would like clarification on an equation in the paper \"Free matter wave packet teleportation via cold-molecule dynamics\", L. Fisch and G. Kurizki, _Europhysics Letters_ **75** (2006), pp. 847-853, DOI: 10.1209\/epl\/i2006-10205-7. The paper talks about entangling two particles translationally, meaning that two particles' position and momenta are correlated such that a precise measurement of particle 1 will cause particle 2's spread in momenta to be uncertain, vice versa. So the equation is equation (2) in the paper, $$\\langle x_1, x_2 | \\Psi \\rangle= N e^{-\\left({x_+}\/{2\\Delta x_+}\\right)^2}N e^{-\\left({x_-}\/{2\\Delta x_-}\\right)^2}$$ where $x_+ = (x_1 + x_2)\/2$, $x_- = x_1 - x_2$, and $N$ is a normalization constant. I'm assuming that the $\\Delta x_\\pm$ are the standard deviations of $x_\\pm$. I've never seen bra-ket notation with \"$\\langle x_1,x_2|$\" in it. This confuses me a lot! It doesn't make sense to have $x_1$ (comma) $x_2$. What the heck does this mean? I am interpreting this as the expectation value of the positions of the two entangled particles where $|\\Psi\\rangle$ is the wave function of two translationally entangled particles. Can someone please help me?"}
{"id":"29182","title":"What would happen if the Earth was tidally locked with the Sun?","text":"I'm thinking of writing a short story set on a version of Earth that is tidally locked to the Sun. I'm not exactly sure how to research the topic. Here's a number of questions about what would happen:   * How hot would the light side get? Are we talking ocean-boiling levels? I imagine that life would eventually flourish, given the massive constant energy source. Is this accurate?     * On that note, I imagine massive thunderstorms along all the coasts due to increased evaporation. How bad would they get? Would the ground ever see the Sun, or only rainfall?   * How cold would the dark side get? Is it conceivable that any life could still exist there? (Life has proven itself quite versitile in the past, i.e. life at the bottom of the ocean.)   * What wind speed would the twilight zone experience? I imagine the atmosphere would transfer heat from one side to the other, but would the wind speeds be bearable? In what direction would air flow?   * I hear that the oceans would recede into disjoint northern and southern oceans if the world stopped spinning. Would this also happen if the Earth became tidally locked?   * Would the Sun create a 'tidal' bulge in the ocean at the apex of the light side? Would this or the above dominate ocean behavior?   * Would we completely lose the magnetic field? Would life be able to survive without such shielding from magnetic radiation?   * Would the Moon eventually unlock the Earth? What state would the Moon have to be in for there to be both a locking between the Sun and the Earth as well as the Earth and the Moon?   * What other radical differences would exist between our Earth and a tidally locked alternative?"}
{"id":"49926","title":"What amount of force is needed to topple a person?","text":"Assuming no sliding and that the shoulder is 1.2m from the feet, what force is required to topple a person weighing 70 Kg standing with his feet spread 0.9 m? If possible, please include an explanation about your answer."}
{"id":"114188","title":"Why is speed\/position relative but acceleration not?","text":"I think i understand it now, if found this: link I know that position and speed are relative. There is no such thing as universal coordinates. Then why is acceleration absolute? Is the 3th and 4th derivative of position to time also absolute? Where\/how can i see this in the math\/laws of motion? For example: you are in a rocket and you have an accelerometer with you, and someone else is standing on an astroid that is not attracted by gravity towards something else, so it has no net force on it. If the engines of the rocket are turned on, then you will see that on the accelerometer, and you will know that it is you that is accelerating, not the astroid. Also if the universe were completely empty, except for a bucket of water. If the water would be pushed towards the edges of the bucket, then we would know that the bucket is rotating around its axis. But what would happen if you could rotate space itself around the bucket? It probably is because $F=ma$ and not something else like $F=m^2a^3+2a+...$, is that correct? Can $F=ma$ be deduces from deeper principles, or is it an experimental law?"}
{"id":"131614","title":"How do photon emitters and photon detectors work?","text":"The interpretation of the double slit experiment is very strange and i want to understand how they did it before I give up my concept of reality. With the double slit experiment a interference pattern is created even when you emit just one photon. However, when you use photon detectors to find out which slit the photon went though, the photon goes in straight line. So firstly I want to know how do the photon emitters work and secondly I want to know how these photon detectors work. How are you even meant to detect a photon without even touching it? also how big the the slits?"}
{"id":"66270","title":"Topological Phases and Confinement","text":"I recently attended a talk in which the speaker defined a topological phase as \"A phase which has a gap above the ground state for bulk excitations in the thermodynamic limit.\" I am interested in what sense then can we think of confinement in non-Abelian Yang-Mills theories as topological phases. What I'm looking for are the analogies; what would the **thermodynamic limit** and the **bulk excitations** mean if we were talking about a YM theory (QCD, for instance). The thermodynamic limit is the number of particles $N\\to\\infty$, which I suppose we can think of as the number of Feynman diagrams (order of the loops) growing as large as possible. \"The bulk\" seems a bit more vague (which is perhaps because of my definition), but it seems like bound states of quarks is the appropriate notion for that. So, is it possible to (correctly) say something like \"QCD is a topological phase for the standard model\"? If not, is there a clear reason why this is not the case?"}
{"id":"66276","title":"Can open, unsafe nuclear fusion reaction burn the atmosphere?","text":"I happened to hear people saying that the nuclear fusion bomb tests could set the atmosphere on fire. I have some serious doubts about that - but I have no facts.   Nuclear fusion reaction requires $15*10^{6}$ kelvins to start. If we produce such temperature in \" _open air_ \" would the atmosphere become a fuel for further fusion? Shouldn't the whole thing just be torn apart by its terrible pressure?"}
{"id":"66274","title":"Quantum Mechanical Effects of an object accelerating near speed of light $c$?","text":"Consider a space ship, undergoing constant acceleration (which for our purposes means that the same amount of energy is being used per second to increase its speed). According to special relativity the ship will accelerate but in such a way that it will approach the speed of light $c$, but never reach the speed itself or cross it. This means however that the more time we spend accelerating the space ship, the closer and closer it will get to the speed of light. Which basically amounts to: our knowledge, or accuracy of our velocity is going to steadily go up. Heisenberg's Uncertainty principle states that there is limit to how accurately we can know our velocity before we need to give up some information about our position? What does this mean in our context? Say we reach the critical threshold where we are accurate to some 35+ significant digits aka 0.999999999999999999999999999999999999999999999999999999999999999999c is our speed. We need to lose accuracy of our position between our start and endpoint (assuming this trip has a definite endpoint)? Whats happening? Is the ship suddenly making jumps to random locations? Is it spreading out like a wave, where we the faster we go, the less likely we are to know when we reached our endpoint? I'm very very curious. * * * ## EXTENSION: Additional Thought Consider an object in circular orbit around a black hole, outside of the event horizon (the black hole itself is stationary) and these are the only 2 systems present, with the object in question having very negligible mass compared to the black hole. As the object is brought closer and closer to the black hole, the centripetal force it experiences, obviously goes up, causing to orbit at a faster speed. The closer you bring it towards the center of the black hole, the faster the object will orbit, allowing you to bring it arbitrary close to value of C. According to heisenberg as the velocity increases to an accuracy beyond the reduced planck's constant (0.9999...)c the position of the object becomes increasingly unknown. It would start to \"smear\" out in a circle around the orbit, transforming into a haze which can be observed. Or So I think... Is this correct intuition?"}
{"id":"98398","title":"Accuracy and Error of Atomic Clocks","text":"I'm quoting a passage from my notes: > The development of clocks based on atomic oscillations allowed measures of > timing with accuracy on the order of $1$ part in $10^{14}$, corresponding to > errors of less than one microsecond (one millionth of a second) per year. I do not understand what the accuracy of $1$ part in $10^{14}$ means. Does it mean that the atomic clocks can tell us the time accurate and ceratain to $10^{-14}s$? How should I understand this? Moreover, what is meant by the error of one microsecond per year? Is it a kind of uncertainty in measurement? How should I understand it? I googled this topic and found information about the atomic clocks and also reviewed the definitions of accuracy and error; however, I'm not able to make any sensisble connection between the concepts. Please help me, thank you."}
{"id":"28583","title":"What is negative about negative energy states in the Dirac equation?","text":"This question is a follow up to What was missing in Dirac's argument to come up with the modern interpretation of the positron? There still is some confusion in my mind about the so-called \"negative energy\" solutions to the Dirac equation. Solving the Dirac equation one finds the spectrum of allowed energies includes both positive and negative solutions. What does this negativity refer to? Given that the Dirac equation is symmetric under charge conjugation, the convention to call one positive and the other negative appears perfectly arbitrary. Would it be therefore correct to refer to the electrons as \"negative energy positrons\" ? In a similar spirit, physicists used to be worried about the \"negative energy\" solutions decaying into infinity through emission of photons. By the same symmetry argument, this should also be a problem for photons. It is not entirely clear to me how the quantization of the field supresses this issue : is the \"photon emission\" for the negative state re-interpreted as photon absorption by a positron ? My understanding is that the whole discussion about \"positive\" or \"negative\" energy solutions is misleading : what matters is the physical content through the QED interaction hamiltonian, which does not predict this infinite descent. Is this correct? Edit: I think I understand the source of my confusion after the comments. If I get it right, the Dirac sea picture is equivalent to the freedom of choice in the formally infinite vacuum energy one observes after quantization of the QED Hamiltonian. Holes in the sea are positive-energy positrons, equivalent to the action of the positron creation operator on the vacuum. Is this correct?"}
{"id":"73781","title":"Why does the thickness of a wire affect resistance?","text":"For small thicknesses of wire, it's pretty obvious why resistance affects thickness. (The electronics squeeze to get through). But after a certain thickness shouldn't the thickness become irrelevant? For example if your trying to pour a bucket of water through a straw, the thickness of the straw is obviously gonna be a bottle neck- the bigger the straw, the easier it is for water to get through. But if you try to pour a bucket of water through a tunnel - the size of the tunnel doesn't really matter, because the tunnel is already big. So after a certain thickness shouldn't the thickness stop mattering?"}
{"id":"77206","title":"Relativity question: frequency shift under constant acceleration","text":"Okay, my buddy tells me this: Let there be a starship, ovoid, and me and my buddy stand each at the extremities of the ship, him below the roof, me on the floor. We start a journey and the ship accelerates at a constant rate, let's say $1g$, in the direction of the axis of the ship. _My buddy says_ : since the ship is accelerating, when he strobe (on off on off) a light from the room, back to me, at a frequency $F$, I'll see a shifted frequency $F' = F + \\delta f$. If he switches the light at $1Hz$, I'll see $1.01Hz$ for instance. _My take on the thing_ tells me that in this case, there's no shift: we're both immobile in our referential, and the light will travel at $c$ in this referential. Who's wrong?"}
{"id":"24781","title":"Why does a coulomb explosion induce fusion?","text":"If you strip the valence electrons apart with a very short intense electromagnetic field the remaining core explodes in a so called coulomb explosion. But experiments have shown that under certain cases fusion does occur in a manner, that the ions are accelerated and you get a non maxwellian velocity distribution. Can you explain the process a little bit more and show how it is possible to overcome the nuclear repulsion for fusion? The wikipedia page to this topic is a little bit missleading, so I give you a paper Efficient fusion neutron generation from heteronuclear clusters in intense femtosecond laser fields that this kind of fusion is indeed possible and experimentally proven."}
{"id":"24789","title":"We tend to think the action of a constant force...?","text":"This is a school exercise. We tend to think that the action of a constant force produces a constant movement speed as well. How can you explain this situation in accordance with Newton's second law? Because as the force will end up in a certain time that means there will not be a movement of constant speed; I'm correct?"}
{"id":"28633","title":"How do we recognize hardware used in accelerator physics","text":"When I see a new accelerator in real life or on a picture, I always find it interesting to see how many thing I can recognize. In that way, I can also get a small first idea of how the accelerator is working. Here is a picture, I have taken of LEIR at CERN![LEIR](http:\/\/i.stack.imgur.com\/d8t7B.jpg) Help me to be able to recognize even more stuff, than I can now(I will post a few answers myself) Suggested answer form:   * **Title**   * **Images**   * **One line description**   * **Link**"}
{"id":"90359","title":"Light pulses and energy-time uncertainty principle","text":"Suppose we have a monochromatic light beam. We put an obstacle between source and observer and remove it repeatedly by certain frequency such that observer sees an oscillating intensity of light. Will the observer see different frequencies or only the original frequency? Does the energy-time uncertainty principle apply in this case?"}
{"id":"91605","title":"Impurities involved in nucleation centres","text":"When magmas begin to crystallise, a first step is the formation of a crystal nucleus around an impurity (nucleation centre), whereby a few of the right atoms get together to form a speck of a crystal. I know that impurities are not always necessary for crystallisation to occur but I'm curious to know what these impurities are?"}
{"id":"91606","title":"Charges and their configurations","text":"Can we treat any charge configuration as small point charges by using superimposition principle to derive electric fields, forces and other things ? For example suppose we have a symmetrically charged cylinder, from the distance outside the cylinder, we can treat it like a line charge along the axis of the cylinder. We know that in the original charge distribution the charges do not sit together at 0 distance from one another, but when we consider it as a line charge we do not think about the quantisation therein; We treat the line charge as one continous thread of point charges placed at 0 distance from one another. Back to my question, since this line charge is just an assumption, can we extend our assumption and treat it like individual point charges sitting next to each other, is there any limitation to this assumption ?"}
{"id":"31984","title":"How do I show that all Brillouin zones have the same volume?","text":"I have read in a few books that all Brillouin zones have the same volume, and I can vaguely see how it works, but have not been able to think up a formal proof. Help?"}
{"id":"35606","title":"Is it possible that QM is just GR?","text":"The more I learn about General Relativity, the more it seems like it isn't fully understood. It seems that before it's full consequences were exhaustively understood, not 10 years after its discovery, QM came on the scene and stole the limelight. Now it seems like a \"boring\" field without much funding, even though all but the most trivial and artificial types of solutions to the field equations are known. Here, for example, @Ron Maimon describes how classically a type of black hole allows solutions in which a particle can cross the event horizon, and then exit the event horizon at an earlier time, seemingly leading to causal paradoxes. It sounds like this is an issue that was never fully resolved. It seems the sort of very messy thing that, once properly understood, could lead to some very odd physical behavior. Is it possible that all particles are just extremal black holes, and that Quantum Mechanics is just an emergent property of the solution to Einstein's field equations for the interactions between extremal black holes going backwards and forwards and time? Does something like Bell's inequality rule out this sort of idea? **EDIT** : There are some papers purporting to do this. Mitchel Porter pointed out these: McCorkle, Hadley I also found: McCorkle And then there is Mendel Sachs who has written a number of books purporting to derive QM from GR."}
{"id":"66494","title":"Physical interpretation of normalization of wave fuctions","text":"Does normalization of wave function mean that we are getting our state vector to unit length? If that's the case what does it mean physically? Also is the underlying vector space finite dimensional? If yes, then what is the dimension and how do I find the basis vectors?"}
{"id":"114113","title":"Determing what is differnece beetwen eigenvalues for attractive Coulomb Field (Hydrogen) calculated by exact method and WKB aproach","text":"I have task to compare eignevalues gained by exact calculation on electron in Coulomb attractive field (hydrogen) and gained by WKB method.. As far as I get, eigenvalues gained with exact method are E=13.6 \/ (n^2) and for WKB method all i find is exact same answer E=13.6 \/ (n+l+1)^2 exept there is l. I Don't understand what does it means exactlly..."}
{"id":"66499","title":"Street Light Interference Phenomenon","text":"Is there a scientific approach that can explain the street light interference phenomenon? Everytime I walk past a Streetlight it turns off."}
{"id":"32956","title":"How does a holographic object change perspective when the image is rotated?","text":"Fundamentally i want to know: **How do holograms work?** The problem with that question is that normally you will end up with pages and pages talking about:   * a laser   * a beam splitter   * a diffuser   * the object being imaged   * object beam   * reference beam   * mirror   * holographic emulsion Even Wikipedia is heavy on _how to make a hologram_, rather than **how does a hologram work**. * * * Other people have mentioned stereoscopic vision; how having two eyes gives the illusion of a 3d object. That is also irrelavent, since someone with one eye (or, in my case, one eye closed) can still experience a hologram. What i am trying to figure out is _how does a hologram work?_. More to the point, how is it that rotating a **flat** holographic sticker allows the virtual object to change orientation - allowing me to see content that was not there a moment ago? * * * Wikipedia has an image that mentions _reconstructing a virtual 3d object_ : ![enter image description here](http:\/\/i.stack.imgur.com\/tEyZM.png) Some problems that that image, though, is that my credit card:   * has no _reference beam_   * is not being viewed at a 45 degree angle (meaning no interference can happen) Assuming i have a holographic image of a simple cube. If i am looking at the holographic plate straight on, i will only see a square (i.e. the face of the cube closest to me): ![enter image description here](http:\/\/i.stack.imgur.com\/BHgQO.png) If i rotate the holographic plate, so the right side of the plate is further away, the virtual cube will rotate, and i will actually be able to see the **_left_** face of the cube: ![enter image description here](http:\/\/i.stack.imgur.com\/1XT9U.png) What is happening in the **flat, 2-dimensional, holographic sticker** that it can display _continuously_ different information as i rotate it? What is the mechanics of this holographic _\"paper\"_ that it can present my eye different images?"}
{"id":"32954","title":"How are quantum potential wells fabricated?","text":"Potential wells, such as infinite and finite potential well, have been the standard examples in quantum mechanics textbooks for tens of years. They started being only theoretical toy models but as time progresses scientists succeeded in fabricated them which resulted in the nanotechnology revolution. 1-Is the fabrication process too technical to be explained to the undergraduate students who are taking a course in quantum mechanics? 2-How such potential wells are fabricated in practice ? 3-Why engineering potentials of certain shapes became only possible recently, what changed in science that made it possible (since obviously on the theoretical level quantum mechanics did not change) ?"}
{"id":"27789","title":"States diagonal in the tensor product of Bell states.","text":"Bell-diagonal states are 2-qubit states that are diagonal in the Bell basis. Since those states lie in $\\mathbb{C}^{2} \\otimes \\mathbb{C}^{2}$, the Peres- Horodecki criterion is a sufficient condition to show separability and it's also pretty easy to check: $\\rho = \\sum_{i \\in [0,3]}\\lambda_i|\\psi_{i}\\rangle\\langle\\psi_{i}| $ is PPT (or separable) if and only if $Tr(\\rho) \\geq 2\\lambda_i \\geq 0$ for every $i$. (Here {$|\\psi_{i}\\rangle$} are the Bell states) In my research I am dealing with a generalization of those states. In particular, my question is about states in $\\mathbb{C}^{2^d} \\otimes \\mathbb{C}^{2^d}$ that are diagonal in the basis given by the $d$-fold tensor product of Bell states. For example, for $d=2$, the states I am considering are diagonal in the basis: $$ |\\psi_{0}\\rangle\\otimes|\\psi_{0}\\rangle,|\\psi_{0}\\rangle\\otimes|\\psi_{1}\\rangle, \\ldots,|\\psi_{3}\\rangle\\otimes|\\psi_{3}\\rangle. $$ I am wondering the following: > are there some nice criteria already known to check when these states are > PPT or separable? Notice that those states are in general different from the states diagonal in what is called the generalized Bell basis in literature."}
{"id":"22362","title":"General procedure for Clebsch-Gordan expansions","text":"I'm wondering if the Clebsch-Gordan series generalize to any orthonormal set of basis functions? If so, how would one go about deriving an expression for an arbitrary set of basis functions (perhaps an example derivation of the well known expression of spherical harmonics would be helpful)? I know it has something to do with being able to compute the multiplicities of the tensor product of the irreducible representations, but I don't know how one would do that."}
{"id":"27781","title":"partial trace with sparse matrices","text":"Let $\\rho_{ABCD}$ be a sparse matrix of 4 systems each in a $d$-dimensional Hilbert space. For $d<7$ in a reasonable time (few seconds) I able to perform the partial trace $\\rho_{AD}$ using the code proposed in http:\/\/www3.imperial.ac.uk\/people\/m.tame\/research. I would need an efficient algorithm for calculating $\\rho_{AD}$ where $d\\geq7$. The algorithm of the site above does not exploit any properties of the matrix and it requires a lot of permutations and rearrangements. Do you know an efficient algorithm for calculating the partial trace of qudit which uses the fact that the matrix is sparse? It would also be interesting if the algorithm can take advantage of parallel computation. Thank you very much in advance for your answers. Regards, Silvio"}
{"id":"27787","title":"Kramer's-Kronig relations for the electron Self-Energy Σ","text":"I'm currently studying an article by Maslov, in particular the first section about higher corrections to Fermi-liquid behavior of interacting electron systems. Unfortunately, I've hit a snag when trying to understand an argument concerning the (retarded) self-energy $\\Sigma^R(ε,k)$. Maslov states that in a Fermi liquid, the real part and the imaginary part of the self-energy $\\Sigma^R(ε,k)$ are given by $$ \\mathop{\\text{Re}}\\Sigma^R(ε,k) = -Aε + B\\xi_k + \\dots $$ $$ -\\mathop{\\text{Im}}\\Sigma^R(ε,k) = C(ε^2 + \\pi^2T^2) + \\dots $$ (equations 2.4a and 2.4b). These equations seem reasonable: when plugged into the fermion propagator, $$ G^R(ε,k) = \\frac1{ε + i\\delta - \\xi_k - \\Sigma^R(ε,k)} $$ the real part slightly modifies the dispersion relation $ε = \\xi_k$ slightly and the imaginary part slightly broadens the peak. That's what I'd call a Fermi liquid: the bare electron peaks are smeared out a bit, but everything else stays as usual. Now, Maslov goes on to derive higher-order corrections to the imaginary part of the self-energy, for instance of the form $$ \\mathop{\\text{Im}}\\Sigma^R(ε) = Cε^2 + D|ε|^3 + \\dots .$$ First, I do not quite understand how to interpret this expansion. > How am I to understand the expansions in orders of $ε$? I suppose that $ε$ > is small, but in relation to what? The Fermi level seems to be given by > $ε=0$. Second, he states that this expansion is to be understood \"on the mass-shell\". > I take it that \"on the mass shell\" means to set $\\xi_k=ε$? But what does the > expansion mean, then? Maybe I am supposed to expand in orders of > $(ε-\\xi_k)$? Now the question that is the most important to me. Maslov argues that the real part of the self-energy can be obtained via the Kramers-Kronig relation from the imaginary part of self-energy. My problem is that the corresponding integrals diverge. > How can $$ \\mathop{\\text{Re}}\\Sigma^R(ε,k) = > \\mathcal{P}\\frac1{\\pi}\\int_{-\\infty}^{\\infty} d\\omega > \\frac{\\mathop{\\text{Im}}\\Sigma^R(\\omega,k)}{\\omega-ε} $$ be understood for > non-integrable functions like $\\mathop{\\text{Im}}\\Sigma^R(ε,k) = ε^2$? It probably has to do with $ε$ being small, but I don't really understand what is going on. * * * I should probably mention my motivation for these questions: I have calculated the imaginary part of the self-energy for the one-dimensional Luttinger liquid $\\xi_k=|k|$ as $$ \\mathop{\\text{Im}}\\Sigma^R(ε,k) = (|ε|-|k|)θ(|ε|-|k|)\\mathop{\\text{sgn}}(ε) $$ and would like to make the connection to Maslov's interpretation and results. In particular, I want to calculate the imaginary part of the self-energy with the Kramers-Kronig relations."}
{"id":"26605","title":"What is Hawking radiation and how does it cause a black hole to evaporate?","text":"My understanding is that Hawking radiation isn't _really_ radiated from a black hole, but rather occurs when a particle anti-particle pair spontaneously pop into existence, and before they can annihilate each other, the antiparticle gets sucked into the black hole while the particle escapes. In this way it appears that matter has escaped the black hole because it has lost some mass and that amount of mass is now zipping away from it. Is this correct? If so, wouldn't it be equally likely that the particle be trapped in the black hole and the antiparticle go zipping away, appearing as if the black hole is spontaneously growing and emitting antimatter? How is it that this process can become unbalanced and cause a black hole to eventually emerge from its event horizon and evaporate into cosmic soup over eons?"}
{"id":"23947","title":"How does the energy of a sound wave decrease with the distance","text":"More precisely, how small is the potency a listener hears, compared to the potency of the emitter. I'd like to present a simple and yet reasonable approximation, to a high school audience (I am a teacher) Intuitively, it should decrease with a square (of distance; because of the expansion of the area the wave covers) and exponentially (because of accumulation losses in the way) .Here, they mention both decreases: http:\/\/www.sfu.ca\/sonic-studio\/handbook\/Sound_Propagation.html . The air absorption component is linear in dB (which, as far as I know, means exponential on energy) I am just not sure of how to put them together, and if there are other ideas that a first approximation should cover."}
{"id":"27257","title":"Discussions of the axioms of AQFT","text":"The most recent discussion of what axioms one might drop from the Wightman axioms to allow the construction of realistic models that I'm aware of is Streater, Rep. Prog. Phys. 1975 **38** 771-846, \"Outline of axiomatic relativistic quantum field theory\". I'm looking for any more recent review that seriously discusses the axioms. A critique of the Haag-Kastler axioms would also be welcome, but I would prefer to stay close enough to Lagrangian QFT to allow relatively immediate characterization of the difference between models of the reduced axiomatic system and the standard models that are relatively empirically successful. I'm specially interested in any reviews that include a discussion of what models are possible if one relinquishes the existence of a lowest energy vacuum state (we know that, at least, this weakening allows the construction of thermal sectors of the free field, and that such a sector contains a thermal state that is thermodynamically stable even though it is not minimum energy, and that a Poincaré invariant \"extra quantum fluctuations\" sector is also possible—I'd like to know what is the full gamut of such models?). [Added: This question was partly inspired by a Cosmic Variance post on the subject of QFT, particularly the link to John Norton, obviously with my research interests added.]"}
{"id":"27252","title":"How to prove quantum N=4 Super-Yang-Mills is superconformal?","text":"I'm especially interested in elegant illuminating proofs which don't involve a lot of straightforward technical computations Also, does a non-perturbative proof exist?"}
{"id":"65910","title":"What force is responsible for anti-gravity?","text":"I've seen several video which claims that is anti-gravity. I am sure at least one of them, use a kind of electricity to lift an object! (triangle lifter), I would guess electricity lift that object by air. But what is the equation?, I have no idea."}
{"id":"102541","title":"Phase Structure of (Quantum) Gauge Theory","text":"> **Question** : How to **classify\/characterize the phase structure of > (quantum) gauge theory**? Gauge Theory (say with a gauge group $G_g$) is a powerful quantum field theoretic(QFT) tool to describe many-body quantum nature (because QFT naturally serves for understanding many-body quantum problem with (quasi-)particle creation\/annihilation). **Classification of gauge theory** shall be something profound, in a sense that gauge fields (p-form $A_\\mu$, $B_{\\mu\\nu}$, or connections of $G_g$-bundle etc) are just mediators propagating the interactions between matter fields (fermion $\\psi$, boson $\\phi$). Thus, effectively, we may \"integrate out\" or \"smooth over\" the matter fields, to obtain an effective gauge theory described purely by gauge fields ($A_\\mu$, $B_{\\mu\\nu}$, etc). **Characterization of gauge theory** should NOT simply rely on its gauge group $G_g$, due to \"Gauge symmetry is not a symmetry\". We should not classify (its distinct or the same phases) or characterize (its properties) ONLY by the gauge group $G_g$. What I have been taught is that some familiar terms to describe the phase structure of (quantum) gauge theories, are: (1) confined or deconfined (2) gapped or gapless (3) Higgs phase (4) Coulomb phase (5) topological or not. (6) weakly-coupling or strongly-coupling > **sub-Question A.** : Is this list above (1)-(6) somehow enough to address > the phase structure of gauge theory? What are other important properties > people look for to classify\/characterize the phase structure of gauge > theory? Like entanglement? How? (for example, in **2+1D gapped deconfined weak-coupling topological gauge theory** with finite ground state degeneracy on the $\\mathbb{T}^2$ torus describes anyons can be classified\/characterized by braiding statistics $S$ matrix (mutual statistics) and $T$ (topological spin) matrix.) > **sub-Question B.** : Are these properties (1)-(6) somehow related instead > of independent to each other? It seems to me that **confined** of gauge fields implies that the matter fields are **gapped**? Such as 3+1D Non-Abelian Yang-Mills at IR low energy has **confinement** , then we have the Millennium prize's Yang–Mills(YM) existence and mass gap induced **gapped** mass $\\Delta>0$ for the least massive particle, both(?) for the matter field or the gauge fields (glueball?). So **confinement** and **gapped** mass $\\Delta>0$ are related for 3+1D YM theory. Intuitively, I thought **confinement $\\leftrightarrow$ gapped** , **deconfinement $\\leftrightarrow$ gapless**. However, in 2+1D, condensed matter people study $Z_2$, U(1) spin-liquids, certain kind of 2+1D gauge theory, one may need to ask whether it is (1) confined or deconfined, (2) gapped or gapless, separate issues. So in 2+1D case, **the deconfined can be gapped**? **the confined can be gapless?** Why is that? Should one uses Renormalization group(RG) argument by Polyakov? how 2+1D\/3+1D RG flow differently affect this (1) confined or deconfined, (2) gapped or gapless, separate issues? > **sub-Question C.** : are there **known mathematical objects to classify > gauge theory**? perhaps, say **other than\/beyond the recently-more-familiar group cohomology** : either topological group cohomology $H^{d+1}(BG_g,U(1))$ using classifying space $BG_g$ of $G_g$, or Borel group cohomology $\\mathcal{H}^{d+1}(G_g,U(1))$ recently studied in SPT and topological gauge theory and Dijkgraaf-Witten?"}
{"id":"78389","title":"Cosmological redshift in a lab?","text":"I want to consider a thought experiment. Lets ignore technical problems of actually performing such an experiment. Consider two photons having the same wavelength. We send 1 photon to a distant galaxy (millions of light years away). The photon would hit a mirror there and return to Earth. On Earth (I assume) a cosmological redshift would be detected. At the same time we keep the second photon on Earth in small container with perfect mirrors where it bounces back and forth. Would the photon in the container exhibit the same cosmological redshift as the photon that traveled to the distant galaxy and back? If so, (now back to technical problems) could the redshift be ever measured in the lab or is the effect many orders of magnitude too small to be ever measured in a lab on Earth, not to mention the problem of constructing a perfect mirror to keep the photon?"}
{"id":"32374","title":"Computing an average escape distance for a particle","text":"Somewhere in a two dimensional convex bulk of particles (pic related) on a random position a reaction takes place and a particle is sent out in a random direction with a constant velocity $v$. What is the average distance such a particle travels until it leaves the bulk? ![enter image description here](http:\/\/i.imgur.com\/PxTWz.gif) Might be codable if one puts a grid over the planes and weightens with a yes\/no function if the stating position is within the bulk. Then I'd cut that object with n corners (8 in the pic) into pizza slices and do some geometry to compute the distances in all directions and integrate over all of these and all staring point. I really wonder if there is a good way to do this on a piece of paper, the problem being how to parametrize the points which are inside and not counting these outside. Monte-Carlo computations for regular polygons would be an interesting semi- solution too. Edit: Not that it matters much, but the question is motivated by the question Mean free path of UV photon and is part of me wondering about the escape route for particle entering a bunch of mass, and there specifically on the the dependence on the two length characteristica cloud dimensions and mean free path derived by the clouds constitution. Given that the direction of a particle after a collision is random and so it will probably have to make a detour through the cloud, what is the relation between the average cloud radius to the mean free patch such that the particle is able to leave after only one reaction. Because a collision is almost a reset, I have the suspicion that the escape time only falls slowly with the number of collisions a particles had to endure. The simulation of this might be a more standard question, i.e. starting at a random position and choosing a random direction after the mean free path traveled, how manny mean free paths does it take for a particle to escape a random polygon with characteristic measure being some multiple of the mean free path."}
{"id":"52983","title":"Why is the BCS trial function valid across the BEC-BCS crossover?","text":"In one of the two main theoretical approaches used in describing ultracold Fermi gases and the BEC-BCS crossover, the so-called BCS-Leggett approach, the starting point is the BCS trial wavefunction: $$ \\mid BCS \\rangle \\equiv \\prod_{\\mathbf{k}} \\left( u_{\\mathbf{k}} + v_{\\mathbf{k}} P^\\dagger_{\\mathbf{k}} \\right) \\mid 0 \\rangle $$ where the $P^\\dagger_{\\mathbf{k}}$ operator creates a Cooper pair. It is often asserted that this wavefunction, which may seem tailored for a BCS-like problem, has far greater validity and can also be successfully exploited in describing the BEC-BCS crossover (see for instance: http:\/\/arxiv.org\/abs\/cond- mat\/0404274). Even looking at the original articles by Leggett and Eagles (cited in the reference above) I cannot see why $\\mid BCS \\rangle$ should be valid in the BEC regime: I am looking for a review article (or even a textbook) addressing this issue."}
{"id":"56608","title":"Photon in a weighted superposition of states","text":"Consider an experiment that produces photons in an entangled state such as $1\/\\sqrt{2}(|{H,H}\\rangle+|{V,V}\\rangle)$. The photons are in a superposition of horizontal and vertical polarization, and the way we analyze this is to say that the photons are in **both** states at the same time. Though this is odd, I can eventually reconcile it. However, the photons can also be in an entangled state such as $\\sqrt{0.2}|H,H\\rangle+\\sqrt{0.8}|V,V\\rangle$. Again, the photons are in **both** states at the same time, but do we say that they is somehow more in one state than the other? How can we think of this unevenly weighted superposition of states?"}
{"id":"57556","title":"Have negative pressures any physical meaning?","text":"Some cubic thermodynamical equations of state predict negative pressures, have negative pressures any physical meaning? Could they be related to negative mass?"}
{"id":"104631","title":"The value of current and voltage in a circuit","text":"![enter image description here](http:\/\/i.stack.imgur.com\/s2KzU.png) The picture is just for example. I don't know why $i$ and $i_1$ have different directions and $i+i_1=0$ according to KCL. Also, the value of voltage at each element. I mean in why the $V$ of the end of $R_1$ is negative but the start of $R_2$ is positive. Is the sign is just to indicate the direction of current, where the voltage is higher than the other at each element but not tell its value is positive or negative. Moreover, does the $V$ value of each element is define by its difference in $V$ which mean $\\Delta V$."}
{"id":"78385","title":"current induced in magnetic field","text":"\"If a wire loop is completely in magnetic field no current is induced as voltage induced is balanced by an equal and opposite voltage. \" My guess for this statement is that the loop wire must be stationary and that voltage is zero if its equal on both sides of the wire my question is if current will stil be induced if the wire loop is moving but is still completely in the magnetic field. (I know how as it enters and leaves induced current moves either clockwise or anti so what wil happen if its moving but not entering or leaving)"}
{"id":"57559","title":"Can we change frame of reference twice in a single problem?","text":"My question has an inclined plane of mass $M$ and simple block kept on it, of mass $m$ (Both on a table). All surfaces are friction-less. Both of the objects would move, block down the incline and inclined plane parallel to the table, somewhat opposite to the block. Can the two equations I make be from the Free Body Diagram (FBD) of incline, in GROUND frame, and FBD of block in the INCLINED-PLANE frame? Or do I need to solve in a single frame(either GROUND or INCLINED-PLANE)?"}
{"id":"19648","title":"Double-slit experiment simulated with water waves?","text":"We all know the explanation videos and other material using the water waves analogy to illustrate the propagation of electrons or photons and the interference patterns measured in the the single-slit and double-slits experiments. Being just an analogy it misses for sure many attributes by which light\/electron waves differ from water waves. BUT: If limited only to use water-waves analogy would it be possible to simulate the impact of the observer\/detector in the experiments by means of water waves (yes, i'm smiling my self at this a little ;) What would be the properties of such a detector-by-means-of-waves needed to simulate the collapse of the interference pattern, for example? In other words is it possible to rebuild the double-slit experiment with such a water-waves setup and with same known results? How about in theory? What (\"strange\") requirements are there? For starters maybe let's drop the switching the slit-result and let's assume when the observer is turned on the water wave shows to be originating always from the same slit... At what earliest point does this \"model\" break? How? Thanks!"}
{"id":"102529","title":"Calculating Lyapunov exponents from a multi-dimensional experimental time series","text":"Wolf's paper Determining Lyapunov Exponents from a Time Series states that: > Experimental data typically consist of discrete measurements of a single > observable. The well-known technique of phase space reconstruction with > delay coordinates [2, 33, 34] makes it possible to obtain from such a time > series an attractor whose Lyapunov spectrum is identical to that of the > original attractor. One of the cited papers, Geometry from a Time Series, elaborates: > The heuristic idea behind the reconstruction method is that to specify the > state of a three-dimensional system at any given time, the measurement of > _any_ three independent quantities should be sufficient [...]. The three > quantities typically used are the values of each state-space coordinate, > $x(t)$, $y(t)$, and $z(t)$. We have found that [...] one can obtain a > variety of three independent quantities which appear to yield a faithful > phase-space representation of the dynamics of the original $x$, $y$, $z$ > space. One possible set of three such quantities is the value of the > coordinate with its values at two previous times, e.g. $x(t)$, $x(t - > \\tau)$, and $x(t - 2\\tau)$. Finally, Rosenstein's paper A practical method for calculating largest Lyapunov exponents from small data sets states that: > The first step of our approach involves reconstructing the attractor > dynamics from a single time series. We use the method of delays [27, 37] > since one goal of our work is to develop a fast and easily implemented > algorithm. The reconstructed trajectory, X, can be expressed as a matrix > where each row is a phase-space vector. That is, $$ X = [X_1\\;X_2\\; > ...\\,X_M]^T$$ where $X_i$ is the state of the system at discrete time $i$. All three papers seem to implicitly assume that the system under study has a multi-dimensional phase space, but that only one dimension can be measured experimentally, and therefore that the full phase space data must be reconstructed from a one-dimensional time series. However, what if the time series is multi-dimensional, indeed of the same dimension as the phase space, to begin with? For instance, consider the problem of showing experimentally that a simple pendulum is not chaotic. The phase space is 4-dimensional ($r$, $\\dot r$, $\\phi$, $\\dot \\phi$) and it is straight-forward to design an experiment which generates a 4-dimensional time series of the values of these variables at each time step. Is it possible to skip the reconstruction in this case, and use $X = [r\\; \\dot r\\; \\phi\\; \\dot \\phi]^T$ in place of the reconstructed trajectory in Rosenstein's paper, with no additional modifications? Is there a simpler way to calculate Lyapunov exponents when the full phase-space state of the system is known?"}
{"id":"105328","title":"What phenoma have \"intuitive\" explanations which are completely wrong?","text":"Some phenomena in physics defy intuition. Others have very intuitive explanations which you could explain easily to a layman. Sometimes, these intuitive explanations are simply wrong. What phenomena in physics have intuitive, but totally wrong, explanations? An explanation you could use to easily convince your non-physicist spouse: something that would make them _think_ they understand what's happening while being incorrect? I'll present one (somewhat contrived) example: > Bernoulli's principle is often used to explain how airplanes fly. After > explaining Bernoulli's principle, the layman might ask \"okay, so why does > the air on top of the wing move faster than the air below it?\" A seemingly > intuitive explanation is that because the wing is curved, the top surface > has a longer length from leading edge to trailing edge than the bottom > surface. Therefore, if two air particles hit the leading edge > simultaneously, and one travels above the wing while the other travels below > it, the one traveling above must move faster in order to arrive at the > trailing edge at the same time as its under-wing counterpart. This > explanation is totally wrong because there is no rule of physics that states > the two particles have to arrive at the same time. (And a non-curved \"barn > door\" wing is quite capable of producing lift.) So the phenomenon here is > genuine -- a wing does generate lift, Bernoulli's principle is involved, and > the air above the wing is moving faster and has lower pressure than the air > below it. But the \"two particles racing each other, ending in a tie\" > explanation is simply wrong."}
{"id":"26126","title":"Where to get hard historical and trend data related to reentry of satellites like UARS","text":"NASA is providing very brief updates on the reentry of the UARS satellite. They also published an extensive study of the Re-entry and risk assessment for the NASA Upper Atmosphere Research Satellite (UARS) The Planetary Society Blog has some nice coverage and a graph of altitude over time: Keeping track of UARS' reentry The well-known satellite tracking site Heavens Above site provides UARS - Visible Passes data which seems relatively up-to-date, but without information on the uncertainty. What I want is actual tabular historical data, e.g. the changes in the Orbital elements over the last months, weeks or days of its flight, and a characterization of how those elements other than altitude change over time. _**Update:_** The Center for Orbital and Reentry Debris Studies has a prediction and map for UARS orbits. _**Update 2:_** One authoritative source is Space-Track, run by United States Strategic Command (USSTRATCOM), but you need to be an \"approved registered user\"...."}
{"id":"26124","title":"Error analysis of flux from telescope-CCD setup (photometric calibration)","text":"I am doing photometric calibration of a telescope and am trying to work out the errors on the flux that I measure. I have attempted to work out the errors due to the process of removing bias, dark frames, etc. using Poisson statistics, but this gives a fractional error of about 0.7, which is poor. I may try this analysis again using Gaussian stats, but I also want a measure of the error on the raw flux in the image (ignoring processing). I have tried to do this by taking many observations of one star to look at the variation of flux; it roughly looks like a Gaussian, but I can't get a fit to it without altering the data. Is there a good way to find a flux variation from data I have taken?"}
{"id":"26122","title":"Could a neutron star be active, fusioning neutrons into something more dense and releasing constant power?","text":"Could a neutron star be a bit like our Sun or any other star? Just to have a different scale: A neutron star fusioning not hydrogen and other elements but neutrons and quarks in its core at a stable, constant speed, would release energy, and the neutron star would therefore be larger than just packed neutrons would be. It would shine with very high frequency electromagnetic waves like gamma rays and far beyond. Finally it would be burning all its \"material\" into someting weird and collapse. Would it be possible for the neutron star to explode as a supernova or gamma-ray burst?"}
{"id":"26120","title":"From Computational Astrophysics to Finance","text":"If for some reason, an Astrophysics graduate decides to shift to Finance will he\/she be in a better position (given his\/her overspecialization) to make a transition compared to his\/her counterparts who've been doing Analytical\/Observational Astrophysics?"}
{"id":"132702","title":"Is quantum indeterministic?","text":"The question might look clear from a viewpoint of a non-physics guy but let me be more specific. Can we say quantum leaps or waves or maybe the universe itself are completely indeterministic or do the scientists say that because of lack of information? I am asking this because there are people telling me the you cannot know whether it is indeterministic because you might not have all the information needed to calculate the behaviour of the phenomenon. Also, some people argue that chaos is completely indeterministic. Which is the case and why?"}
{"id":"79846","title":"What is off-diagonal long range order in superfluid?","text":"From Wikipedia: > _[...]Off-diagonal long-range order (ODLRO) [...] exists whenever there is a > macroscopically large factored component (eigenvalue) in a reduced density > matrix of any order._ How to understand the ODLRO in superfluidity?"}
{"id":"75176","title":"How does fire create thrust in rocket?","text":"All big rockets are burning either gas or fluid to create thrust. While this is so, I have filled up a plastic bottle with air at high pressure, and it can go long distances by blowing the pressurised air at reverse direction. If my bottle can do this without using any fire, why don't rockets just use air? How is the effect of combustion in thrust?"}
{"id":"103199","title":"Capacitor in series?","text":"Say you have two charged capacitors in series. Zoom in on one capacitor. For this specific capacitor, the charge on the two plates will be the same in magnitude, according to my textbook. My teacher said that the charge on the two plates won't be the same if the gap between the the plates is large. In fact, the charge on each plate in a capacitor is _never_ the same. They're only approximated to be the same if the gap is small. Why is the charge not the same on both plates on a capacitor in series if the gap is large? ![enter image description here](http:\/\/i.stack.imgur.com\/u9x6F.png)"}
{"id":"4847","title":"How is the Joukowsky Transform used to calculate the Flow of an Airfoil?","text":"As I read in The Road to Reality by Roger Penrose, the Joukowsky transform $$w(z) = \\frac12\\left( z + \\frac1z \\right)$$ after Nikolai Zhukovsky (transcribed in several versions from Никола́й Его́рович Жуко́вский) can be used to **calculate the flow** of a non-viscious, incompressible and irrotational flow **around an airfoil**. This can be done since the solution of a potential flow around a cylinder is known in full analyticity and the given transform conformally maps a circle on an airfoil-like geometry. I don't understand this argumentation, so: > ### How is the Joukowsky Transform used to calculate the Flow of an Airfoil? An **example** of such a transformation is given in the mentioned Wikipedia article: ![RotatingDisc](http:\/\/upload.wikimedia.org\/wikipedia\/commons\/d\/d5\/Jouktrans.png) Thank you in advance."}
{"id":"30070","title":"Is it possible to study solid state from kittel after taking only one course in quantum mechanics?","text":"I have taken only QM I, which is the 1st half of Griffiths including the chapter on identical particles, will that be enough to understand Kittel's solid state? Should one have also taken a course in statistical mechanics before studying kittel? I have seen in some universities in the states that the prerequisite of an introductory course in solid state physics sometimes are QM I, II, SM, and other times are QM I only (or with SM and QM II are corequisite)"}
{"id":"76282","title":"Magnetic force at the edge?","text":"When sliding a magnet over a ferromagnetic surface there is no magnetic force at all, however,there is a magnetic force only at the edge, when the magnet is being pulled of I assume the magnetic field is trying to resist. However, a fact it noticeable, the sliding\/lateral magnetic force at the edge is a lot less, by a factor of half - ten times less. What could be the reasons that detains the magnitude of the lateral force? Pelase do explain in detail. Thank you."}
{"id":"102670","title":"Tensor product of Hadamard Operators","text":"The Hadamard Operator on one qubit is: \\begin{align*} H = \\tfrac{1}{\\sqrt{2}}\\left[\\,\\left(\\color{darkgreen}{|0\\rangle + |1\\rangle}\\right)\\color{darkblue}{\\langle 0|}+\\left(\\color{darkgreen}{|0\\rangle - |1\\rangle}\\right)\\color{darkblue}{\\langle 1|}\\,\\right] \\end{align*} Show that: \\begin{align*} H^{\\otimes n} = \\frac{1}{\\sqrt{2^n}}\\sum_{x,y}(-1)^{x \\cdot y}\\,\\left|x\\rangle \\langle y\\right| \\end{align*} I can evaluate things like $H \\otimes H$ in practice, but I don't know how to get a general formula for $H^{\\otimes n}$. Are there any tricks I could use?"}
{"id":"113892","title":"Is it possible that black holes are also neutron stars, but so dark that we cannot see them?","text":"1. Since the concept of the singularity in a black hole leads to infinite densities, I wonder if it is really certain that black holes exist?    2. Is there a possibility that massive objects (which are believed to be black holes) are in reality dense stars not emitting light (dark neutron stars)?   3. Is there a possibility to explain the observed facts without the use of black holes?   4. One more question: According to the mainstream, are neutron stars believed to eventually collapse to a black hole, or are they stable objects?"}
{"id":"34924","title":"Physical meaning of the sign basis in quantum mechanics","text":"If we take a hydrogen atom as qubit, let $\\lvert0\\rangle$ = unexcited state   $\\lvert1\\rangle$ = excited state then what is the meaning of measuring the qubit value in the sign basis? If the atom may only be in excited or unexcited state, but $\\lvert+\\rangle$ and $\\lvert-\\rangle$ are superpositions of those states — then what would the outcome of the measurement be — also a superposition of $\\lvert+\\rangle$ and $\\lvert-\\rangle$? Can anyone please help to understand the idea behind the sign basis?"}
{"id":"34921","title":"Quantum mechanics on Cantor set?","text":"Has quantum mechanics been studied on highly singular and\/or discrete spaces? The particular space that I have in mind is (usual) Cantor set. What is the right way to formulate QM of a particle on a Cantor set? I can only guess that: i) There will be no momentum operator. ii) Hilbert space will be spanned by position vectors corresponding to points of Cantor set. iii) Corresponding to any two points $x$ and $y$ in Cantor set there will be a unitary operator $P_{xy}$ (analog of the exponential of momentum operator) such that $P_{xy}|x>=|y>$; $P_{xy}=P_{yx}^{-1}$; and $P_{yz}P_{xy}=P_{xz}$ for all $x,y,z$. But I am not able to see what would be correct notion of free particle, and what would be corresponding Hamiltonian."}
{"id":"36161","title":"What is the mass of the emergent magnetic monopoles in spin ice and how is the mass of an emergent particle determined?","text":"In solid state physics emergent particles are very common.   1. How one determines if they are gap-less excitations?    2. Do the defects in spin ice called magnetic monopoles have mass?    3. What is the mass of emergent particles in the fractional quantum hall effect (FQHE)?  I am not looking for a deep theory, just the general picture."}
{"id":"69984","title":"The mighty man and the bridge","text":"Let us say we have a mighty man crossing a bridge, carrying 4 bags of concrete, each of which weighs 50 pounds. Let us say, for the sake of the argument, that the mighty man himself weighs 300 pounds. The support capacity of the bridge $x$ is less than the 500 pounds they weigh together - therefore, the mighty man cannot carry the concrete in a stack across the bridge - it will fall. However, the mighty man does have the ability to juggle all the concrete bags so that he is only in contact with one of them at a time. Is there such a value of $x$ possible that he can cross the bridge juggling all four bags of concrete at once, or will the bridge necessarily exceed its support capacity?"}
{"id":"103074","title":"Which types of particles are affected by the wave-particle duality?","text":"If we take the double slit experiment as a way of demonstrating the wave- particle duality, which types of particles would show an interference pattern? For example, I know that electrons show such a pattern. But do protons, too? What about atoms? Where is the boundary between \"wavey particles\" and \"classical particles\"?"}
{"id":"6455","title":"Construction of the $\\cal{N}=3$ supersymmetric Yang-Mills Chern-Simons theory in $2+1$ dimensions","text":"I am stuck with understanding the following construction. I am breaking it up into segments which I think can be separately answered. This is related to an earlier question of mine. Note that this previous question was edited a lot after the initial comments came in.   * Firstly I would like to know how the following field multiplet is determined,  The gauge multiplet of $\\cal{N}=3$ in $2+1$ dimensions is apparently , $A_\\mu$, a massive vector field of spin $1$ $\\lambda_a$, $3$ Majorana fermions of spin $\\frac{1}{2}$ $C_a$, $3$ neutral scalar bosons $\\chi$, a Majorana fermion of spin $\\frac{1}{2}$ (I guess there should be another scalar field $D_a$ not mentioned in the above list)   * Its not clear to me as to what are the superfields one has in mind with the above multiplet structure that helps construct the lagrangians below.    * Like in the linked earlier question above one had to \"know\" what exactly is the meaning of doing the dimensional reduction on a given set of supersymmetric transformations, here its not clear to me as to what are the corresponding starting points. (I wonder if I can start from the $\\cal{N}=2$ SYM in $3+1$ lagrangian that is given in say equation $27.9.3$ in Weinberg's QFT book)  Then dimensional reduction of $\\cal{N}=2$ SYM in $3+1$ apparently gives the following $\\cal{N}=3$ SYM in $2+1$, $L_{\\cal{N}=3,2+1,SYM} = \\frac{1}{g^2}Tr\\left [ -\\frac{1}{2}F_{\\mu \\nu} ^2 + (D_\\mu C_a)^2 + (D_a)^2 + i \\bar{\\lambda_a}\\gamma^\\mu D_\\mu \\lambda_a + i \\bar{\\chi}\\gamma^\\mu D_\\mu \\chi \\+ i\\epsilon_{abc}\\bar{\\lambda_a}[\\lambda_b,C_c] -2i\\bar{\\lambda_a}[\\chi,C_a] - \\frac{1}{2}[C_a,C_b][C_b,C_a] \\right ]$ where the notation is, $D_\\mu = \\partial_\\mu -iA_\\mu$, $ab,c=1,2,3$. The gauge group generators in the fundamental representation satisfy $[T^m,T^n]=if^{lmn}T^l$ with normalization as $Tr{T^mT^n}=\\frac{1}{2}\\delta ^{mn}$ and $f^{kmn}f^{lmn}=c_v\\delta^{kl}$. ($c_v$ being the quadratic Casimir) The fields belong to the adjoint representation and $A_\\mu = A_\\mu^mT^m$. The metric is given by $\\eta_{\\mu \\nu} = diag(1,-1,-1)$. The purely imaginary gamma matrices satisfy, $\\gamma^\\mu \\gamma ^\\nu = \\eta^{\\mu \\nu}-i\\epsilon^{\\mu \\nu \\rho}\\gamma_\\rho$.   * It is not clear to me as to how does one argue that the above has $\\cal{N}=3$ supersymmetry.    * The dimensional reduction of the $\\cal{N}=2$ supersymmetry somehow implies that the theory has the not-so-obvious $\\cal{N}=4$ supersymmetry.  * * * One also notes that supersymmetrizing the Chern-Simon's terms one is supposed to get the Chern-Simon's lagrangian as, $L_{\\cal{N}=3,2+1,SCS} = kTr\\left [ \\epsilon^{\\mu \\nu \\rho}(A_\\mu \\partial_\\nu A_\\rho - \\frac{2}{3}iA_\\mu A_\\nu A_\\rho) - \\bar{\\lambda_a}\\lambda_a + \\bar{\\chi}\\chi \\+ 2C_aD_a + \\frac{i}{3}\\epsilon_{abc}C_a[C_b,C_c] \\right ]$   * In all these lagrangians apparently the corresponding $\\cal{N}=2$ theories can be obtained by setting $C_1=C_2=D_1=D_2=\\lambda_3 = \\chi = 0$ and the $\\cal{N}=1$ theories can be obtained from the $\\cal{N}=2$ theories by further setting $C_3=\\lambda_2=0$.  (I am unable to get the equations to wrap around properly!)"}
{"id":"15349","title":"Apple falls for which of these 2 reasons?","text":"Needles to say I am a visitor here. I do not belong to the science world;) But I have read both of these things before:   1. Apple falls to the ground because curved spacetime pushes it there (same force as keeps moon in orbit)   2. Apple 'falls' to ground because the ground is rushing up to meet the apple (which is actually suspended in space) because of Earth's acceleration through space. I don't think these can both be true. I'd appreciate any clarification - thank you."}
{"id":"109761","title":"Moment of inertia of a hollow sphere wrt the centre?","text":"I've been trying to compute the moment of inertia of a uniform hollow sphere (thin walled) wrt the centre, but I'm not quite sure what was wrong with my initial attempt (I've come to the correct answer now with a different method). Ok, here was my first method: Consider a uniform hollow sphere of radius $R$ and mass $M$. On the hollow sphere, consider a concentric ring of radius $r$ and thickness $\\text{d}x$. The mass of the ring is therefore $\\text{d}m = \\frac{M}{4\\pi R^2}\\cdot 2\\pi r\\cdot\\text{d}x$. Now, use $r^2 = R^2 - x^2:$ $$\\text{d}m = \\frac{M}{4\\pi R^2}\\cdot 2\\pi \\left(R^2 - x^2 \\right)^{1\/2}\\text{d}x$$ and the moment of inertia of a ring wrt the centre is $I = MR^2$, therefore: $$\\text{d}I = \\text{d}m\\cdot r^2 = \\frac{M}{4\\pi R^2}\\cdot 2\\pi\\left(R^2 - x^2\\right)^{3\/2}\\text{d}x $$ Integrating to get the total moment of inertia: $$I = \\int_{-R}^{R} \\frac{M}{4\\pi R^2} \\cdot 2\\pi\\cdot \\left(R^2 - x^2\\right)^{3\/2}\\ \\text{d}x = \\frac{3MR^2 \\pi}{16}$$ which obviously isn't correct as the real moment of inertia wrt the centre is $\\frac{2MR^2}{3}$. What was wrong with this method? Was it how I constructed the element? Any help would be appreciated, thanks very much."}
{"id":"89977","title":"Quantum Mechanisms for Isotope Fractionation","text":"Are there any quantum properties that would enable isotope fractionation? For example, atoms with odd versus even numbers of neutrons are fermions and bosons, respectively. Has any work been done using the distinct properties of fermions and bosons to isolate isotopes?"}
{"id":"89974","title":"Lorentz boost matrix in terms of four-velocity","text":"As I understand it, the value of a 4-vector $x$ in another reference frame ($x'$) with the same orientation can be derived using the Lorentz boost matrix $\\bf{\\lambda}$ by $x'=\\lambda x$. More explicitly, $$\\begin{bmatrix} x'_0\\\\\\ x'_1\\\\\\ x'_2\\\\\\ x'_3\\\\\\ \\end{bmatrix} = \\begin{bmatrix} \\lambda_{00}&\\lambda_{01}&\\lambda_{02}&\\lambda_{03}\\\\\\ \\lambda_{10}&\\lambda_{11}&\\lambda_{12}&\\lambda_{13}\\\\\\ \\lambda_{20}&\\lambda_{21}&\\lambda_{22}&\\lambda_{23}\\\\\\ \\lambda_{30}&\\lambda_{31}&\\lambda_{32}&\\lambda_{33}\\\\\\ \\end{bmatrix} \\begin{bmatrix} x_0\\\\\\ x_1\\\\\\ x_2\\\\\\ x_3\\\\\\ \\end{bmatrix} $$ I have seen examples of these components written in terms of $\\beta$ and $\\gamma$, which are defined as $$\\beta=\\frac{v}{c}$$ $$\\gamma=\\frac{1}{\\sqrt{1-\\beta\\cdot\\beta}}$$ where $v$ is the 3-velocity and $c$ is the speed of light. My question is this: How can the components of $\\lambda$ be written in terms of the 4-velocity $U$ alone? I know that $U_0=\\gamma c$ and $U_i=\\gamma v_i=\\gamma c\\beta_i$ for $i\\in\\\\{1,2,3\\\\}$, but I'm having trouble deriving the components for $\\lambda$ using the matrices based on $\\beta$ and $\\gamma$. An example of one of these matrices can be found at Wikipedia. How can I rewrite this matrix in terms of $U$ alone?"}
{"id":"103594","title":"Calculate impulse based on mass, acceleration and time elapsed","text":"I want to move to Sprite Kit framework offered by apple which has physics integrated right into it. The back bone of the physics engine is famous Box2D. Sprite Kit has made it pretty easy by wrapping C++ inside Objective-C but unfortunately some features lack in Sprite Kit. One them is GetReactionForce function, that returns the impulse applied on the body B of a joint. Just disregard the software related terms if you find it difficult to understand. Basically what I need is to create my own function that will calculate impulse. Here are the ingredients I have: Mass, gravity, delta time. P.S. Sorry for the tags, I couldn't find anything more suitable and I have no right to create new ones"}
{"id":"103690","title":"How to calculate tortuosity of a signal?","text":"Not sure which stack exchange forum to put this in. Moderators feel free to migrate it. One definition of the tortuosity $T$ of a curve is: $$T=\\frac{C}{L}$$ Where $C$ is the chord length (distance between the ends of the curve) and $L$ is the total length of the curve. I am trying to calculate the tortuosity of a white noise signal. But it is not clear to me what the unit length should be. Here's what I mean: The white noise is a time series $X(t)$ of length $N$ of normally distributed random numbers with mean $0$ and standard deviation of $1$. To calculate the length $L$ of this signal I do the following: $$dy(t)=\\left| X(t+1)-X(t) \\right |$$ $$L=\\sum_{t=1}^{N-1} \\sqrt{dy(t)^2+dx^2}$$ Where $dx$ is the unit interval. The question is how to select a suitable value for $dx$. Should it just be $1$? Or perhaps the mean or minimum value of $dy(t)$? The chord length $C$ of this signal is $N\/dx$, so it also depends upon the selection of $dx$."}
{"id":"214","title":"Would it help if you jump inside a free falling elevator?","text":"Imagine you're trapped inside a free falling elevator. Would you decrease your impact impulse by jumping during the fall? When?"}
{"id":"19447","title":"Implications of unbounded operators in quantum mechanics","text":"Quantum mechanical observables of a system are represented by self - adjoint operators in a separable complex Hilbert space $\\mathcal{H}$. Now I understand a lot of operators employed in quantum mechanics are unbounded operators, in nutshell these operators cannot be defined for all vectors in $\\mathcal{H}$. For example according to \"Stone - von Neumann\", the canonical commutation relation $[P, Q] =-i\\hbar I$ has no solution for $P$ and $Q$ bounded ! My basic question is :   * If the state of our system $\\psi$ is for example not in the domain of $P$ (because $P$ is unbounded), i.e., if $P\\cdot \\psi$ does not, mathematically, make sense, what does this mean ? Does it mean we cannot extract any information about $P$ when the system is in state $\\psi$ ?"}
{"id":"73994","title":"Quantization of Electron Spin","text":"Why is electron spin quantized? I've seen the derivation for the Hydrogen atom's energy levels, but my professor jumped to electrons having spin 1\/2 or -1\/2 as experimental. Why do electrons obey the same quantization rules for angular momentum as the Hydrogen atom does? Why must the two states be one apart?"}
{"id":"100521","title":"Double-spring mass system","text":"We just had a lesson about elementary mass-spring systems (SHO), and I thought about a horizontal situation with two springs with the test mass oscillating in between. If we are to manually stretch the mass a distance $\\Delta x<\\Delta_0$ , we obtain opposing restoring forces, which yields the familiar SHO equation. But do thesee conditions change when we stretch it beyond $\\Delta_0$(a little beyond, it's still elastic with Hooke's law), in some intervals the restoring forces point in the same direction and in others don't. Does the system retain its \"SHO-ness\"? ![System diagrams](http:\/\/i.imgur.com\/g0rBIf3.png)"}
{"id":"55395","title":"what is the difference between constant and changing magnetic and electric fields? How do they occur? How do they form an electromagnetic wave?","text":"what is the difference between constant and changing magnetic and electric fields? How do they occur? How do they form an electromagnetic wave?"}
{"id":"32974","title":"Does black hole complementarity apply to white holes?","text":"If physics is time reversal invariant, there ought to be white hole complimentarity as well. Imagine a white hole so enormous that it is possible for life to evolve entirely within it for billions of years before emerging to tell their story about the interior of the white hole. Meanwhile, an external observer can observe and keep note of everything in the vicinity of the white hole and what comes out of it. Both observers can meet and compare notes. Can the external observer really claim the white hole interior doesn't exist?"}
{"id":"134922","title":"Is charge density zero in a dielectric material and why?","text":"I'm trying to solve a problem involving parallel capacitor. I can't decide whether to use poisson's formula or laplace's formula. ![enter image description here](http:\/\/i.stack.imgur.com\/hArz0.png) The question is, is there $\\rho_v$ in a piece of dielectric and why?"}
{"id":"43721","title":"How to tell local and non-local in QFT?","text":"I'm taking QFT course in this term. I'm quite curious that in QFT by which part of the mathematical expression can we tell a quantity or a theory is local or non-local?"}
{"id":"4067","title":"Question on particles","text":"Is there any theory in which every particle can be further subdivided into any number of particles and the total number of particles any where in the space time are infinity in theory and only due to practical constraints that we are capable of observing the known particles in the current World. Also could you please explain the nature of the two widely accepted theories which explain the small scale and large scale universe which are QP and GR repectively in the current context of the question."}
{"id":"86328","title":"Acceleration on an inclined rail","text":"I have to use $\\vec{P}+\\vec{R_N}+\\vec{F_f}=m\\vec{a}$ with $\\|\\vec{F_f}\\|=\\mu\\|\\vec{R_N}\\|$ to express the acceleration as $$a_{x'}=g\\sin\\alpha(1-\\frac{\\mu}{\\tan\\alpha})$$ The configuration is (sorry about the poor image) : ![scheme](http:\/\/www.beziaud.org\/~work\/upload\/docs\/131112083008.schema.jpg) I'm totally lost. How could I do that?"}
{"id":"90400","title":"Why Inox Steel doesn't interact with magnets?","text":"My dad has a HUGE magnet on his workshop. I love magnets, and when I saw it, I asked him what it was for. His reply was: \"I don't know why, but inox steel bolts don't get attracted to it, so I use it to identify them.\" Thus I got curious, why a magnet don't attract inox steel bolts? Steel, even if a inox variation is still mostly iron, no?"}
{"id":"90405","title":"Are composite bosons always bosonic (e.g. the pion-cloud surrounding the nuclei)?","text":"The $\\pi$-meson is a boson, but consists of quark-antiquark (fermions). It seems to me that at some energy level (equivalently distance) the inner structure (fermionic nature of the quarks) of the particle in question should become important and the bosonic nature less so. Question:   Can we have a bunch of pions occupying the same quantum state at all temperatures, or is this model bound to fail due to the fermionic nature of its constituents? I'm thinking for example of the pion-cloud (in some models) surrounding the nuclei. EDIT: I found this question which is very related to the present one. Although I should add that my question (regarding the pion-cloud) is more specific and less general."}
{"id":"123779","title":"How can I calculate dew point from relative humidity and temperature?","text":"Meteorilogical stations in my country report relative humidity. I am not sure why as this is actually misleading, as relative humidity will change with temperature. Unless I miss my guess, what is important for human feeling is a dew point. How can I calculate dew point from relative humidity and temperature? PS: I have found quite a nice tool: http:\/\/www.dpcalc.org\/"}
{"id":"10021","title":"Online QFT video lectures","text":"I'm aware of Sidney Coleman's 1975\/76 sequence of 54 lectures on Quantum Field Theory. Are there any other high-quality QFT lecture series available online?"}
{"id":"101342","title":"How to obtain Maxwell's Lagrangian from complex scalar fields?","text":"I've looked in several books and they all show how to obtain electrical interactions by forcing local gauge invariance of any complex scalar field Lagrangian (like Klein-Gordon or Dirac). I manage to separate the new Lagrangian into the original one (the free Lagrangian) and the interaction part of it. But how I get Maxwell's Lagrangian from this? As the fields that are inserted to keep the Lagrangian gauge invariant should have it's dynamics described by Maxwell's. So, how to does the term $F^{\\mu\\nu}F_{\\mu\\nu}$ enter the new Lagrangian naturally? Thanks"}
{"id":"20142","title":"Find the Unit Vector of a Three Dimensional Vector","text":"How can I find the unit vector of a three dimensional vector? For example, I have a problem that I am working on that tells me that I have a vector $\\hat{r}$ that is a unit vector, and I am told to prove this fact: > $\\hat{r} = \\frac{2}{3}\\hat{i} - \\frac{1}{3}\\hat{j} - \\frac{2}{3}\\hat{k}$ I know that with a two-dimension unit vector that you can split it up into components, treat it as a right-triangle, and find the hypotenuse. Following that idea, I tried something like this, where I found the magnitude of the vectors $\\hat{i}$ and $\\hat{j}$, then using that vector, found the magnitude between ${\\hat{v}}_{ij}$ and $\\hat{k}$: > $\\left|\\hat{r}\\right| = \\sqrt{\\sqrt{{\\left(\\frac{2}{3}\\right)}^{2} + > {\\left(\\frac{-1}{3}\\right)}^{2}} + {\\left(\\frac{-2}{3}\\right)}^{2}}$ However, this does not prove that I was working with a unit vector, as the answer did not evaluate to one. How can I find the unit vector of a three- dimensional vector? Thank you for your time."}
{"id":"128744","title":"Kinetic energy and potential energy variation over distance in SHM","text":"When you compute the average potential energy of a horizontal spring mass system from the mean position to the positive amplitude A, the value comes out to be $\\frac{1}{6}kA^2$. For the average kinetic energy over the same range and direction, it is $\\frac{1}{3}kA^2$, which is double the average potential energy. What the physical explanation of the different average values of PE and KE? P.S. No mathematical explanations please, i.e. area under the graph, etc, only explanations in terms physics of the event are appreciated. The question does not involve time averages of PE and KE. Snapshots of derivations can be uploaded if requested."}
{"id":"48634","title":"How do I account for the direction of friction acting on a spring?","text":"I would like to set up the equations of motion for a simple spring oscillator. Let's have a spring lying horizontally; we attach a small mass $m$ to the (massless) spring. The force of the spring on the mass is $$F_\\text{spring} = - k x$$ where $k$ is the spring constant, and $x$ is the displacement from the position of rest. Since I am considering the system in motion, I only need to account for the kinetic friction. I know that the magnitude for the kinetic friction is: $$F_\\text{fric} = F_g \\mu$$ where $\\mu$ is the coefficient of kinetic friction, $F_g$ equals the Normal Force on the surface and is also equal to the gravitational force. But the direction of the force is always antilinear to the direction of motion. How do I set this up correctly in my approach for the equations of motion? My approach is $$m a = -k x - \\operatorname{sign}(x) F_\\text{fric}$$ where $a$ is the acceleration of the mass point, and $\\operatorname{sign}$ is the sign function. Is this correct?"}
{"id":"44739","title":"Regulating the sum in Casimir Force","text":"I am trying to evaluate the Casimir force using a Gaussian regulator (I know there are other much easier ways to do this, but I want to try this!) We then are reduced to evaluating the sum $$ \\sum\\limits_{n=1}^\\infty n e^{-\\alpha n^2} $$ Moreover, I am interested in the series expansion of the above sum around $\\alpha = 0$. Any ideas how I would go about obtaining this sum? PS - I don't want to use the Euler-Mclaurin formula.That was used to show that a general regulator would always give one the same answer, so this would just be a special case of the that proof and not a very novel way. Any other ideas?"}
{"id":"44733","title":"Translate a two dimensional classical Dirac theory to a (1+1)-dim quantum theory","text":"Suppose I have a two dimensional _classical_ Dirac Hamiltonian with $\\Psi=(\\psi_1,\\psi_2)^T$: $$ H=\\int \\mathrm{d}x \\mathrm{d}y \\Psi^\\dagger(\\sigma^x i\\partial_x+\\sigma^y i\\partial_y+m\\sigma^z)\\Psi. $$ The partition function is given by $$ Z=\\int[\\mathrm d\\psi^\\dagger][\\mathrm d\\psi]e^{-H}. $$ How can I relate it to a (1+1)-dim _quantum_ system? A simple replacement $x\\rightarrow t$ seems to be wrong. Because in the standard (1+1)-dim Dirac action, the time derivative term must be $\\Psi^\\dagger\\partial_\\tau\\Psi$, i.e. with identity matrix in front of the time derivative, to ensure the conjugate momenta of $\\Psi$ is just $i\\Psi^\\dagger$."}
{"id":"6330","title":"What happens to internal energy in high-density materials?","text":"Normal-density materials have internal energy, which is the sum of the average energies associated to each of the degrees of freedom. Degrees of freedom can be described as vibrational, translational, and rotational. If you compress this material in such a way that matter cannot escape, you can detect energy escaping in the form of heat. Now imagine an extremely high-density material such as a neutron star or black hole. I suspect it's not possible to describe the internal energy of (or anything about) an individual atom inside a black hole, since it's impossible to inspect beyond the event horizon, and since I assume that atoms don't exist in an distinguishable form there. But presumably the atoms _had_ energy when they became part of the thing. How is that energy accounted for?"}
{"id":"6339","title":"Strange behavior and motion of bubbles in a glass of beer","text":"_Very unintuitive observation:_ I pour myself a Guinness and the bubbles in my glass seem to move down toward the bottom of the glass instead of rising directly to the top of the glass as foam. How can this be explained? Why is it that I observe this behavior drinking Guinness and not other carbonated drinks? What system properties (ie, temperature, nature of the solute and solvent) would affect this behavior?"}
{"id":"59724","title":"Do WIMPs have to interact non-gravitationally with each other or other particles?","text":"I know many collaborations are attempting to detect the interaction of WIMPs with nucleons or with themselves, with the recent result from Ice Cube (http:\/\/physics.aps.org\/synopsis-for\/10.1103\/PhysRevLett.110.131302) showing no evidence for WIMP self-interaction. I am wondering if WIMPs might actually have zero non-gravitational interaction with SM particles and zero self- interaction. Is such a thing possible or likely? If so, what sorts of theories predict such WIMP candidates, and could they be detected indirectly by other means (e.g., cosmological ratio of WIMP mass-energy to ordinary matter mass- energy, or something like that)?"}
{"id":"134723","title":"Is a plasma necessarily made of monoatomic ions?","text":"Is it possible to have a plasma made of polyatomic ions instead of monoatomic ions? I want to know all the details why such a thing may be attainable or not and, if possible, what methods we can use to create such a substance."}
{"id":"121453","title":"Why diaphragm in diffraction experiment using electrons is quantum object?","text":"In the book _Quantum Mechanics - Volume 1_ written by Albert Messiah, page no. 142-143, author says: > _...But the diaphragm is a quantum object, just like the electron. Its > momentum is not defined to better than dp...._ I did not understand why diaphragm is a quantum object ? Also, it is not clear to me what author says below, > _....One must postulate that the measuring apparatus is a quantized object > which also obeys uncertainty relations.....\"_ On the other hand, N. Bohr says that measuring apparatus is always a classical object (i.e. instrument which follows rules of classical mechanics). Any CLEAR explanation?"}
{"id":"54269","title":"Water-cooled fast neutron reactors","text":"Can anyone explain why fast neutron reactor designs use sodium\/lead\/salt cooling, instead of water (heavy\/light)? Is that because neutron absorption by water would not allow to break even in fuel cycle? Will heavy water help here? Or water slows neutrons so efficienly so that even if we reduce amount of water inside the reactor (by increasing flow speed) - it still will significantly lower neutron energy, while sodium does not slowdown neutrons at all?"}
{"id":"11852","title":"How could pinion in automatic quartz watch be rotated at 100K RPM?","text":"Wikipedia article on automatic quartz watch describes the watch mechanism as follows: a rotating pendulum is attached to a pinion and when the wearer moves his hand the pinion is rotated at **_up to 100 thousand revolutions per minute_** and that pinion rotates an electric generator. AFAIK the fastest hard disk drives currently reach only 15K RPM and they have serious problems with heat dissipation - a fan is required for cooling the drive. How can a miniature pinion inside a watch be rotated at such speed without the whole assembly being worn out by friction in no time?"}
{"id":"54264","title":"What is the momentum of these emitted photons?","text":"`5 x 10^7 kg of radioactive material emits energy in the form of photons of red light (700 nm). (Note: photons have no mass.) What is the momentum of each photon?` We just started a new chapter, and in the process of learning it, I came across this question in the back of our book, which I am really having trouble understanding right now. Can anyone suggest how I can go about solving this (note, I would like to actually work out the problem myself; however, I need a sufficiant place to start.) Obviously, the Newtonian equations like `p =m\/v` will not apply here, so I'm somewhat stuck at the moment... Any help is much appreciated, thank you."}
{"id":"97998","title":"Finding the appropriate source term for 2D Pressure Driven Flow in a Square Channel","text":"I'm trying to solve the equation: $$ \\frac{\\partial^2 v_z(x,y)}{\\partial x^2} + \\frac{\\partial^2 v_z(x,y)}{\\partial y^2} = \\frac{\\Delta P}{\\mu \\Delta X} $$ This flow is supposed to be flowing in a square channel (laminar,steady,fully developed). I know I need to solve it numerically and if there was no source term I could just take the average of all four points around each node and get my answer. The problem is that I need to find the appropriate source term and I honestly don't know how to start. I also know I need to make the equation dimensionless. I know the source term can't be found analytically and I honestly am lost there. Can someone explain what the source term is and how I can start my quest to find it?"}
{"id":"3468","title":"What would happen if you put your hand in front of the 7 TeV beam at LHC?","text":"Some speculation here:   http:\/\/www.youtube.com\/watch?v=_NMqPT6oKJ8 Is there a possibility it would pass 'undetected' through your hand, or is it certain death? Can you conclude it to be vital, or only loose your hand? Would it simply make a small cylindrical hole through your hand, or is there some sort of explosion-effect? Assume your hand has a cross section of 50cm², and a thickness of 2cm, how much of the beam's energy would be transferred to your hand?"}
{"id":"44192","title":"Why does $H_2$ form on such a long time scale?","text":"If we were trying to figure out the time scale for a gas-phase reaction between two hydrogen atoms in a molecular cloud (which has density $~10^4\/$cm$^3$), apparently the reaction would happen on a time scale proportional to the inverse of the density multiplied by $10^{15}$ years. Aside from the cloud not being dense and the probability that a collision will surpass the activation energy is small, is the time elongated because you need to induce a dipole moment between two hydrogen atoms to actually bind them together?"}
{"id":"80933","title":"Resistances in a circular loop","text":"![enter image description here](http:\/\/i.stack.imgur.com\/B7pBq.png) I have no idea what formula I'm supposed to use.I just want to know the concept that would allow me to get the answer. I know basic concepts of electricity like charge, currents, etc., but as far resistances go I know only about series and parallel. I tried looking for resistances in a circular loop but I didn't understand what was being explained. Also I didn't find any concept that helped me understand what was to be done, probably because I didn't understand them.I used some help from a different question and got the resistances but my answer wasnt correct."}
{"id":"82020","title":"Electrons of conductors Free?","text":"Are electrons of any conductor really free ? I mean are they always already moving or do they move only when electrostatic field of some sort is applied across them. I suppose if they were always moving each and every conductor must have some varying magnetic field around it at all times which though negligible musy be detected. Is that really so ? Or are the so called free electrons not really free ? Also is there some other experiment which i do not know about by which we have already proven that they are always moving ? does it take into account the earths and other electric fields which may be causing that movement ? Addendum : the question is not duplicate of Are free electrons in a metal really free as that post talks about ease of electrons to interact with external fields, my question is about the nature of motion of electrons in absense of external field interference as in are electrons in free motion even when they are not under the influence of any electromagnetic field."}
{"id":"109303","title":"Has this boundary condition been used in fluid flow?","text":"I would like to know whether anyone has seen a boundary condition used in a fluid flow problem, of the following type. Suppose viscous incompressible fluid is to the left of a plane $x_1=a$, so the outer normal to the fluid region is $n=(1,0,0)$. The condition specifies that the $x_1$ momentum flux is a nonnegative multiple of the velocity: $$ m\\begin{bmatrix} u_1\\cr u_2\\cr u_3\\end{bmatrix} = \\begin{bmatrix}\\rho u_1^2+p-2\\mu u_{1,1} \\cr \\rho u_2u_1-\\mu(u_{1,2}+u_{2,1}) \\cr \\rho u_3u_1-\\mu(u_{1,3}+u_{3,1}) \\end{bmatrix} $$ where $m$ could be a number $\\ge0$ or even a nonnegative 3 by 3 matrix. (This is connected with my question at http:\/\/www.mathoverflow.net\/q\/163772\/) Thanks for any references."}
{"id":"82659","title":"momentum conservation and gluons","text":"The process is the following: $$e^-e^+ \\rightarrow photon \\rightarrow quark + antiquark$$ Regarding the momentum conservation law, how come we have a photon of spin 1 and at the end some meson with spin 0? Are gluons \"behind this\"? If this is correct, at which point are they radiated? From quarks? Or? Is this photon a virtual photon or not? I'm a bit confused here."}
{"id":"129786","title":"What exactly is a virtual displacement in classical mechanics?","text":"I'm reading Goldstein's Classical Mechanics and he says the following: > _Avirtual (infinitesimal) displacement of a system refers to a change in the > configuration of the system as the result of any arbitrary infinitesimal > change of the coordinates $\\delta \\mathbf{r}_i$, *consistent with the forces > and constraints imposed on the system at the given instant $t$*. The > displacement is called virtual to distinguish it from an actual displacement > of the system ocurring in a time interval $dt$, during which the forces and > constraints may be changing._ Then he discusses virtual work and so on. Now, I can't grasp what this thing of virtual really is. By this text there's a diference between one infinitesimal change and one virtual change and I really don't get what this virtual really is. Also, this is based on infinitesimals. How can this be expressed rigourously without refering to infinitesimals? I tried looking on Spivak's Physics for Mathematicians where he considers these virtual displacements as tangent vectors to a certain manifold, but I'm not sure this is the most \"standard\" way to do it rigorously."}
{"id":"129784","title":"Rotating platform with pulleys and stacked blocks","text":"This was a practice test question. > Consider the following figure. (If the frictional coefficients are unclear, > between A and B we have $k_1 = 0.3$, B and platform is $k_2 = 0.2$. If you > further doubts about the diagram ask me in the comments below and I will > clarify.) ![enter image description here](http:\/\/i.stack.imgur.com\/PRMA5.jpg) > The blocks are resting on the rotating platform (bold) attached to the rod > which is rotating at constant angular velocity $\\omega\\ \\mathrm{rad\/s}$. The > masses of blocks and distances from the pulley are as given in the figure. > The distance of the pulley from the rod (axis) is negligible. Block B is on > verge of slipping. Find $\\omega$. What I did : Let mass of $A = m_1$. Distance of $A$ from axis = $r_1$. Mass of $B = m_2$. Distance of $A$ from axis = $r_2$. If block $B$ is on verge of slipping $A$ is also on the verge of slipping. Questionable observation. But leads to correct answer. Anyway, we consider a rotating frame attached to the rod. So we give a centrifugal force to $A$ = $m_1 \\omega^2 r_1$ and to $B$ equal to $m_2 \\omega^2 r_2$. Tension in string $ = T$. We check two cases. Let $B$ slip inwards. Then $A$ slips outwards. So $A$ slips rightwards relative $B$. $A$ was on verge of slipping. Static friction on $A = k_1 m_1 g$. Static friction on $B = k_2 (m_1 + m_2) g$. Then equating forces and solving the system and putting values we get $ \\omega = \\sqrt{22}\\ \\mathrm{rad\/s}$. The other case ($B$ goes outwards) leads to contradiction. Though the result is correct, I have a few questions. Why does the fact that $B$ is on the verge of slipping imply that $A$ is also on the verge of slipping. I mean, suppose we just eliminate the rotation and add an external force on $B$ instead. Let the magnitude of force reach $k_2 (m_1 + m_2) g$. Then why is it necessary that the force of static friction between $A$ and $B$ is $k_1 m_1 g$ ? Am I wrong, and is it just that the answer matched, without correct physics? A detailed explanation of a solution to this problem would be very helpful as I'm a beginner, and I intend to clarify these essential concepts."}
{"id":"126153","title":"How can a gas support tensile stresses?","text":"In working through a rigorous derivation of the compressible Navier-Stokes equations, I find that the momentum flux in the X-direction should be driven not only by the normal pressure gradient $\\frac{\\partial p}{\\partial x}$ and shear stress terms $\\frac{\\partial(\\tau_{yx})}{\\partial x}$ and $\\frac{\\partial(\\tau_{zx})}{\\partial x}$, but also by the gradient of the normal stress $\\frac{\\partial(\\tau_{xx})}{\\partial x}$. It's intuitively clear to me how adjacent lamina moving at different speeds can transfer momentum across their interface, and so the shear stress terms in the momentum equation are readily intelligible. The normal stress term, on the other hand, is far less intuitive because I cannot see how a freely-deforming fluid can support tensile stresses. Positive normal stresses (i.e. compression) are not that hard to understand, but it's proving exceedingly difficult to fully envisage an element of a fluid \"pulling on\" an adjacent element in a way even remotely analogous to the behavior of a solid under the same conditions. I am also unclear on the difference between \"pressure\" and \"normal stress\" in the fluid. How exactly are these terms different? I am interested primarily with gases not liquids."}
{"id":"94876","title":"Why is electric potential 0 in this case?","text":"On a test, we had a question where there are 4 point charges at the vertices of a square. The 2 charges at the upper vertices have charges of `+q` and the 2 charges at the lower vertices have charges of `-q`. The magnitude of the charges are equal. According to the answer sheet, the electric potential is 0 along a horizontal line halfway between the 2 upper and 2 lower charges. Why is it 0? Shouldn't the test charge be attracted to either to top or the bottom depending on its charge?               +q        +q          ------------    <--- 0 electric potential          -q        -q"}
{"id":"129878","title":"How absorption coefficient determines which material is used to make solar cells?","text":"Does the knowledge of the material absorption coefficients aids engineers in determining which material to use in their solar cell designs? If yes, how?"}
{"id":"129873","title":"How do crocodiles jump?","text":"In a video (Here), I saw crocodiles jump vertically about three meters without using any solid surface. The wonderful thing is that when they start to jump, their vertical velocity is approximately zero, unlike fish who jump using initial velocity. It seems that crocodiles create an upward force that counteracts gravity, because when they are rising, their velocity seems to be constant. How is this possible? Could anyone explain this phenomenon using physics laws?"}
{"id":"114351","title":"Feynman's $i \\epsilon$ prescription in loop expansion","text":"I have some questions about the $i\\epsilon$ factor in Feynman diagrams. First, what is the physical meaning of $i\\epsilon$ in loop amplitudes. Second, how does it ensures unitarity? And third, Dyson series assume that incoming and outgoing particles are free, this can be implemented by assuming that the interaction Hamiltonian switches off adiabatically, $e^{-\\eta\\,t}H_{I}(t)$. Is this $\\eta$ related with the $i\\epsilon$?"}
{"id":"111653","title":"How to design a house to be cooled passively?","text":"I live in Louisiana these days, in an area that is known for its numerous antebellum plantation homes (circa early 1800s). While touring one of these homes it was clear that almost everything about the house was designed around keeping cool in the summer. Some examples:   * 4 meter high ceilings to allow hot air to rise to the ceiling.   * Floor to ceiling windows to allow hot air at the top to escape and cool air to be drawn in at the bottom.    * Porches on the sunny sides of the house to prevent sunlight from entering the windows.   * Large central staircases to allow hot air to rise to the second floor, drawing cool air in on the bottom floor.   * Some have a cupola, a central observation room at the top of the house, again to allow hot air to escape at the top of the house and draw air in from the bottom. * * * My question is: Given our modern understanding of thermodynamics, how could one design a home today to be cooled passively? Could we do any better than the plantation owners of the 1800s? Lets define cooling as making the house more comfortable for humans. This means that it is not only important to reduce the temperature, but also to block sunlight and maintain airflow. Also, if possible, it would be very beneficial to extract moisture from the air."}
{"id":"80926","title":"Why the electric bulb turns on almost instantly when the switch is closed?","text":"**The electron drift speed is estimated to be very low.How could there is current almost the instant a circuit is closed??**   ![enter image description here](http:\/\/i.stack.imgur.com\/JOVuz.jpg) By the discussions it is known that **The information about beginning of the flow of current is transmitted through the propagation of electromagnetic waves(electric impulse)and not with the drift velocity of the electrons.**   But I want any one to explain how this process takes place.CURIE:)"}
{"id":"47617","title":"How can I calculate the wave propagation speed in a copper wire?","text":"First of all: I am a computer science student, so I don't have much knowledge of physics. So please keep your answers simple. I recently learned something about circuit design and its problems (differend kinds of hazards). To model the problems, we introduced a \"dead time model\" (\"Totzeitmodel\" in German, I have if it is \"dead time model\" in English)   We added some dead time to each element of the circuit, but we didn't add dead time to the wires of the circuit. I asked the prof. why we didn't add dead time to the wires. He responded that the signal is moving much faster and you can neglect the time that signals need to pass the wires. Now I would like to calculate the speed of the signal (is this the wave propagation speed?) for some very simple settings:   * assume we have a copper wire   * the wire is a perfect cylinder with diameter of 1mm   * the current is 2A   * the voltage is 12V Can you help me with this? Do you need something else to calculate the speed? * * * Notes: I found the wikipedia article Wave propagation speed and some questions on physics.stackexchange.com, but the questions and answers were either too complicated or didn't directly give numbers (like that one) A little side question: When I think about the electric signal, I imagine some elastic balls. When there is a signal at one end, you push the ball. It gets compressed and expands later, which compresses the next ball a bit and it expands, ... This way, the last ball gets moved and the signal arrives at the end. Do I have to get another thinking-model for simple circuits or will I be able to understand simple circuits with this model in mind?"}
{"id":"89506","title":"Axisymmetric fluid flow","text":"I'm having trouble with a boundary condition. In a fluid mechanics problem, I have flow at $z = \\infty$ flowing into a solid plate at $z = 0$ and then flowing radially, and the problem is given as axisymmetric. I know that the velocity field has $v_r$ and $v_z$, and $v_\\theta = 0$. How in the world would I express the symmetry boundary conditions without $\\theta$?"}
{"id":"127205","title":"Possible intergalactic celestial objects","text":"We know there are rouge stars floating in intergalactic space, thought to be caused by galactic collision. What other other classes of celestial object could be found floating around in intergalactic space? Within galaxies there are the following types of object:   * Nebulae    * Stars   * Brown Dwarfs   * White Dwarfs   * Black Holes   * Neutron Stars   * Comets   * Asteroids   * Interstellar rogue planets   * Planetary Systems   * Star Systems   * Star Clusters   * Debris Disks Which of these can possibly exist in intergalactic space? Asteroids and comets are probably quite common, but is it known for entire planetary systems to exist in intergalactic space? Even star clusters? Out of all the possible types, could I also have a link \/ designation to each object thats been observed, floating in intergalactic space. For example, here's an intergalactic star HE_0437-5439"}
{"id":"127206","title":"For a planet which has a temperature gradient, hot in the center and cooler on the surface, why do we get absorption lines?","text":"For a planet which has a temperature gradient, hot in the center and cooler on the surface, why do we see absorption lines? Similarly, why do we see emission lines if the planet is hot on the surface and gets cooler as you move to the center? Note, for this question I am only thinking of the planet as a black body, not as something, for example, transiting a star and observing the spectra (transit spectra) of the light that shines through the planet's atmosphere. The source of the spectra is the planet itself."}
{"id":"25305","title":"How is the Hanbury-Brown and Twiss effect used to measure the size of stars?","text":"I understand what an Hanbury Brown and Twiss (HBT) interferometer _does_ , but how can this be used to measure the apparent angular diameter of some object? What is the mathematical explaination?"}
{"id":"82303","title":"Problem involving 1st law of thermo and ideal gas law","text":"Problem: > $1.0 \\text{ kg}$ of air at pressure $10^6 \\text{ Pa}$ and temperature $398 > \\text{ K}$ expands to a five times greater volume. The expansion occurs such > that in every instance the added heat is a quarter of the work done by the > gas. Calculate the final pressure. $1 \\text{ kmol}$ has a mass of $29 \\text{ > kg}$ and $C_V = \\frac 52 R$. I solved the problem but I have some questions regarding the approaches I made. I interpreted heat being a quarter of the work done every instance as $\\text{d}Q = \\frac 14 \\text{d}W$. By the 1st law of thermodynamics ($Q = W + \\Delta U$), this yields $-\\frac 34 \\text{d}W = \\text{d}U$. But for molar heat capacity we know that $\\text{d}U = nC_V\\text{d}T$, hence $-\\frac 34 \\text{d}W = nC_V\\text{d}T$. By definition, $\\text{d}W = p\\text{d}V$ and therefore $-\\frac 34 p\\text{d}W = nC_V\\text{d}T$. We treat the gas as ideal and thus we can substitute $\\frac {nRT}V$ for $p$, arriving at $-\\frac 34 \\frac{nRT}V \\text{d}V = nC_V\\text{d}T$. Separation of variables yield $-\\frac 34 \\frac{nR}V \\text{d}V = \\frac {nC_V}T \\text{d}T$. We take the definite integrales $ \\displaystyle \\int_{V_1}^{5V_1} \\\\!\\\\!\\\\!\\\\!\\\\! -\\frac 34 \\frac{nR}V \\text{d}V = \\int_{398}^{T_2} \\frac {nC_V}T \\text{d}T$ and can then finally solve for $T_2$ (it turns out to be approximately $245 \\text{ K}$). At last, we can determine the final pressure from $\\displaystyle \\frac {p_1V_1}{T_1} = \\frac {p_2V_2}{T_2}$. * * * **(1)** My main question is whether you guys know of a different way to solve this problem not involving having to solve differential equations? **(2)** I've seen the 1st law written as $\\text{d}U = \\text{d}Q + \\text{d}W$ but if we use that expression we definitely won't get the right answer; what's up with the inconsistency? **(3)** I saw no other option than having to use the molar heat capacity equation $\\text{d}U = nC_V\\text{d}T$ but doesn't this assume constant volume? In our problem the volume is definitely changing, so is it not contradictory to use that very equation?"}
{"id":"82306","title":"Unitary Confirmation","text":"I am asked to show that an new defined operator: $$U_{\\beta} = \\exp(\\displaystyle\\frac{i\\beta L_z}{\\hbar})$$ is unitary, where $$L_z = -i\\hbar\\,\\,(x\\displaystyle\\frac{\\partial}{\\partial y} - y \\frac{\\partial}{\\partial x }).$$ I tried the following: $$ U_{\\beta}^{\\dagger} U_{\\beta} = \\exp \\left( \\frac{i\\beta(-L_z^{\\dagger}+L_z)}{\\hbar} \\right)$$ So I couldn't make the inside of exponential zero."}
{"id":"81625","title":"Does the Casimir effect give the correct value for Dark Energy?","text":"My understanding is that the Casimir Effect is caused by vacuum energy. Quantum mechanics (QED) predicts vacuum energy, but gets the value grossly wrong, by a factor of $10^{120}$. On the other hand, from what I have read, Dark Energy is understood to be caused by vacuum energy. Has anyone checked whether the measured value for the Casimir Effect fits the value required for Dark Energy to make up 70% of the energy in the universe? If it did, that would be an excellent validation of Dark Energy."}
{"id":"92828","title":"Is it reasonable to assume that de Sitter temperature will stop growing when it reaches equilibrium with the temperature of CMB?","text":"Is it reasonable to assume that de Sitter temperature will stop growing when it reaches equilibrium with the temperature of CMB? Currently the de Sitter temperature (the temperature of the Universe's boundary) is $$T_{dS}=2.67\\times10^{-30}\\,\\mathrm K$$ and rises. At the same time, the temperature of CMB is $$T_{cmb}=2.7\\,\\mathrm K$$ and falls. I wonder, does it mean that at the moment they will equate, the de Sitter temperature will stop growth, and so the expansion of the universe will also stop accelerating, thus reaching the thermodynamic equilibrium? In other words, whether the excessive temperature of CMB compared to the de Sitter temperature is the main source of the accelerated cosmic expansion?"}
{"id":"96490","title":"interaction between mathematical structures","text":"From a physicist's perspective there are several situations in which somehow arbitrary choices of mathematical structures can be made. One can describe a system from different perspectives, etc. without changing anything in the physical properties. (see gauge invariance, picture changing etc.) In the situation when these choices impose a mathematical structure one has limitations in the subsequent choices (are these limitations physical???). However, there are situations when this prescription does not hold. The best example is in this case the Black-Hole complementarity principle (if it is well defined in the first place). So, the question about how mathematical structures \"interact\" in a logical way is not a bad one. Again: what is the interaction between, say, locality and Hausdorff-ity in the case of a black hole? Or what is the true relation between metrizability and hausdorff-ity of a space in the context of a black hole? The question about how mathematical structures assigned (innocently) at some point interact in different situations is in my opinion a quite relevant one..."}
{"id":"109185","title":"Faradays Law: How Does Curl E Change Over Time In a Coil?","text":"I think I am currently making a mistake regarding my interpretation of Faraday's Law. $\\vec{\\nabla}\\times\\vec{E}=-\\frac{\\partial\\vec{B}}{\\partial t}$ Assuming we have an electric field $\\vec{E}$ in the plane between 2 particles given by: $\\vec{E}=(\\frac{x}{x^2+y^2}+\\frac{1-x}{(x-1)^2+y^2}, \\frac{y}{x^2+y^2}-\\frac{y}{(x-1)^2+y^2})$ Then $\\vec{\\nabla}\\times\\vec{E}=(0,0,\\frac{y\\cdot(2\\cdot x-2)}{((x-1)^2+y^2)^2}+\\frac{(2\\cdot(1-x))\\cdot y}{((x-1)^2+y^2)^2}) = (0,0,0)$ As expected, now, if there's an electric field through a coil due to potential difference, then why will the curl be non-zero only when the electric field is initiated and closed off from the coil (e.g. connected\/disconnected from a power source)? Currently; I assume that the derivative of the magnetic field converges to 0 over time - because it can not grow indefinitely if we simply have a potential difference (coil connected to a power source). So my question is: How does the electric field (I am correct in that E indeed stands for the electric field, right?) change over time in the time we connect the coil (get potentials at both ends, and a current), until the coil's E curl has stabilized (becomes 0)? It'd be great if someone had an example of an $\\vec{E}(\\vec{r}, t)$ that changes with time so that $\\vec{\\nabla}\\times\\vec{E}$ converges to 0. Where $t_0$ is the connection time and $t_1$ the stabilization time, and $\\vec{r}$ the position in space. Are my questions based on false premises? I feel really confused, I've tried my best to look for answers but I can not seem to find them."}
{"id":"16199","title":"Is there a Newton's third law for the em field?","text":"There is a momentum associated with the em field that ensures the conservation of total momentum for a system of interacting charges. Can the same be done in an analagous way to ensure Newton's third law is also true?"}
{"id":"91692","title":"Interpretation of Dirac equation states","text":"In Pauli theory the components of two-component wavefunction were interpreted as probability amplitudes of finding the particle in particular spin state. This seems easy to understand. But when talking about Dirac equation, we have four-component wavefunction, two of which correspond to usual spin components of Pauli electron, and another two... How do I interpret positron-related components of Dirac electron? Are they probability amplitudes for the particle to appear to be positron? Or maybe to appear to _not_ be positron (taking Dirac sea picture into account)?"}
{"id":"16197","title":"At what size will self-gravitation contribute more to stability than surface tension?","text":"The governments of Earth have embarked on an experiment to place a massive ball of water in orbit. (umm... special water that doesn't freeze) Imagine this to be a fluid with a given density, $\\rho$ ($kg\/m^3$), surface tension, $\\sigma$ ($J\/m^2$), and formed in a sphere of radius $R$ ($m$). I think that the viscosity $\\mu$ is not needed for this question, but correct me if I'm wrong. At what size will the restorative forces from gravity (after some small perturbation) become more significant than that from surface tension? Would the type of perturbation make a difference? Just for fun, here's a video of a ball of water stabilized by surface tension."}
{"id":"56374","title":"Can an invisible light source cast shadows?","text":"Let's assume that we have a mechanism for producing EM radiation suspended in the air, and that that mechanism itself is invisible to the naked eye (e.g. a microscopic light bulb on a microscopic wire or a suspended molecular reaction giving off energy.) When off, the light source would not be visible. When turned on however, any light cast from the source to the environment around it would also reach the viewer's eye, and identify the source of light. Is it theoretically possible to cause the lit light source to cast shadows on the environment that are visible, but for the source itself to remain invisible? As an example, most 3D graphics programs can create invisible light sources which are only identifiable by the scene they illuminate. I imagine this may be possible with certain environments, such as an environment covered in phosphorescent paint and a black light source, where the light source is not seen but the environment which it illuminates is identifiable. But what about in the general case, when we limit the variables to the source of the light itself, stipulating it must work in a general environment?"}
{"id":"56375","title":"How is a Rydberg Blockade Radius defined?","text":"Rydberg blockade is a phenomena in 3 or more level systems of Rydberg dressed atoms."}
{"id":"57029","title":"Annihilation and creation operator - $\\phi$ and $\\pi$ for Klein-Gordon Field","text":"**Introduction and Notation** Let $\\phi(\\vec{x})$ be the real Klein-Gordon (quantum) field, written as: $$\\phi(\\vec{x})=\\int\\frac{d^3p}{(2\\pi)^3}\\frac{1}{\\sqrt{2\\omega_{p}}}\\left(a_{\\vec{p}}+a^{\\dagger}_{-\\vec{p}} \\right)e^{i\\vec{p}\\cdot\\vec{x}} $$ where $\\omega_{p}=\\sqrt{|\\vec{p}|^2+m^2}$, $a_{\\vec{p}}$, $a^{\\dagger}_{-\\vec{p}}$ the annihilation and creation operators, and let $\\pi(\\vec{x})$ the momentum density conjugate to $\\phi(\\vec{x})$, given by $$\\pi(\\vec{x})=\\int\\frac{d^3p}{(2\\pi)^3}(-i)\\sqrt{\\frac{\\omega_{p}}{2}}\\left(a_{\\vec{p}}-a^{\\dagger}_{-\\vec{p}} \\right)e^{i\\vec{p}\\cdot\\vec{x}} $$ **The question** The only non trivial equal-time commutator is $$[\\phi(\\vec{x}),\\pi(\\vec{y})]=i\\hbar\\delta^{(3)}(\\vec{x}-\\vec{y}) $$ As the relation between $\\phi(\\vec{x})$ and the $a,a^{\\dagger}$ is linear, and so is between $\\pi(\\vec{x})$ and them, I'm going to express the commutator obeyed by $a$ and $a^{\\dagger}$. I'm failing to derive the inverse Fourier transform $$ \\int d^3x'\\phi(\\vec{x})e^{i\\vec{p}'\\cdot\\vec{x}'}=\\int\\int d^3x'\\frac{d^3p}{(2\\pi)^3}\\frac{1}{\\sqrt{2\\omega_{p}}}\\left(a_{\\vec{p}}+a^{\\dagger}_{-\\vec{p}} \\right)e^{i\\vec{p}\\cdot\\vec{x}}e^{i\\vec{p}'\\cdot\\vec{x}'}=\\\\\\=\\int d^3x'e^{i\\vec{p}'\\cdot\\vec{x}'} \\int\\frac{d^3p}{(2\\pi)^3}\\frac{1}{\\sqrt{2\\omega_{p}}}\\left(a_{\\vec{p}}+a^{\\dagger}_{-\\vec{p}} \\right)e^{i\\vec{p}\\cdot\\vec{x}}=\\\\\\=(2\\pi)^3\\delta^{3}(\\vec{p}')\\int\\frac{d^3p}{(2\\pi)^3}\\frac{1}{\\sqrt{2\\omega_{p}}}\\left(a_{\\vec{p}}+a^{\\dagger}_{-\\vec{p}} \\right)e^{i\\vec{p}\\cdot\\vec{x}}$$ **What I want is a Dirac delta $\\delta^{3}(\\vec{p}-\\vec{p}')$** , so where is my procedure failing? I know this is a math question, but given the physical context, it may fit here better. Thanks for your time, any hint will be appreciated **EDIT 1** **EDIT 2** EDIT 1 Now as an answer"}
{"id":"116612","title":"Online course on theoretical electrodynamics","text":"I'm looking for good online course for an introduction into theoretical electrodynamics. However, it seems that the MIT opencourseware only contains partial lectures for this topic. Has anyone got a recommendation for me?"}
{"id":"93884","title":"Change of basis in Dirac Notation","text":"> `Question`: An operator $A$ is in a particular basis $|a_i\\rangle$ (where > $i=1,2$), and is represented by $$A=\\begin{pmatrix} 0 & -i \\\\\\ i &0 > \\end{pmatrix}$$ Now, define two new basis vectors $|b_i\\rangle$ by $$\\langle > a_i | b_1\\rangle =\\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\\\ 1 \\end{pmatrix}$$ > $$\\langle a_i | b_2\\rangle =-\\frac{i}{\\sqrt{2}}\\begin{pmatrix} 1 \\\\\\ -1 > \\end{pmatrix}$$ > > What is $A$ in the new basis? `Attempt`: First, I defined the transformation matrix $U$: $$U=\\begin{pmatrix} \\langle a_1 |b_1 \\rangle & \\langle a_1 |b_2 \\rangle \\\\\\ \\langle a_2 |b_1 \\rangle & \\langle a_2 |b_2 \\rangle \\end{pmatrix}=\\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 & -i \\\\\\ 1 & i \\end{pmatrix}$$ If we let $A'$ be the matrix in the new basis, we obtain $$ A'=UAU^\\dagger$$ And it is a simple calculation (by hand or through mathematica) to show that $$ A'=\\begin{pmatrix} 1& 0 \\\\\\ 0 & -1 \\end{pmatrix}$$ However, I wanted to check my work using Dirac notation, so I used the equation $$ A_{ij}'=\\langle b_i |A|b_j\\rangle=\\sum_{n,m} \\langle b_i |a_n\\rangle A_{nm} \\langle a_m | b_j \\rangle$$ A sample calculation of $A_{12}'$ is shown below: $$A_{12}'=\\sum_{nm}\\langle b_1 | a_n \\rangle A_{nm} \\langle a_m | b_2 \\rangle$$ $$=\\langle b_1|a_1\\rangle A_{12}\\langle a_2 | b_2 \\rangle+ \\langle b_1 |a_2\\rangle A_{21}\\langle a_1|b_2\\rangle$$ $$=\\frac{1}{2}(1)(-i)(i)+\\frac{1}{2}(1)(i)(-i)=1$$ Using the same method for the other components, we find $A'$ to be $$ A'=\\begin{pmatrix} 0 & 1 \\\\\\ 1 & 0\\end{pmatrix}$$ My question is which method is incorrect? Do I have the formula for the matrix U wrong, or am I not using Dirac notation right? Thank you in advance."}
{"id":"93886","title":"Rank $L$ spherical harmonic tensor as a $2L+1$ dimensional Cartesian vector?","text":"Rank two Cartesian tensors can be decomposed into $L=0,1,2$ spin like things 3x3=1+3+5. But the second equation below does not transform like a \"tensor\", it looks more like a vector transform in five dimensional Cartesian coordinate. $$\\mathfrak{D}^{+}(R)V_{i}\\mathfrak{D}(R) = \\sum_{j=1,2,3}R_{ij}V_{j}\\tag{1} $$ (three dimensional vector.) $$\\mathfrak{D}(R)T_{i}^{(2)}\\mathfrak{D}^{+}(R) = \\sum_{j=-2,-1,0,1,2}\\mathfrak{D}_{ij}^{(2)}T_{j}^{(2)}\\tag{2}$$ (rank two spherical harmonic tensor.) $$ \\mathfrak{B}^{+}(F)V_{i}\\mathfrak{B}(F) = \\sum_{j=1,2,3,4,5}F_{ij}V_{j}\\tag{3}$$ (a vector transform in 5 dimensional Cartesian space.) $(2)$ and $(3)$ above looks the same. My question is can 3d spherical tensors be regarded as high dimensional vectors? Are there some examples?"}
{"id":"57026","title":"Can there be energy with no force or energy with no power?","text":"I think that both force (number of newtons) and power (p=ui(?)) implies that there is energy so we can't have force without energy and we can't have power without energy(?) But can there be energy that is energy and no force and no power?"}
{"id":"61965","title":"Gas transformation for different molar number","text":"When a gas at normal conditions ( **1atm, 273K, 22.4L** ) we say like the $$\\frac{PV}{T} = 0.0820...$$ And by the know equation: $$PV = nRT$$ Where R is equals to $0.08205...$ and $n$ is the number of mols of the substance. When we want to compare a previous state of a gas with its next state, we use the formula: $$\\frac{P_1 V_1}{T_1} = \\frac{P_2 V_2}{T_2}$$ But the $R$ is just the first state of the gas, already in a constant term ($\\frac{P_1 V_1}{T_1}$ is $\\frac{1 \\cdot 22.4}{273}$ that is $R=0.082...$) So, instead of doing: $$\\frac{1 \\cdot 22.4}{273} = \\frac{P_2 V_2}{T_2}$$ We do: $$R = \\frac{P_2 V_2}{T_2}\\tag{assuming the same n of mols}$$ But let's say we're doing this equality for a gas that has a given number $n$ mols of the substance. By what I know it is $$nR = \\frac{P_2 V_2}{T_2}$$ I have $2$ questions: $(1)$ - How do I know that for a $1\/2$ number of mols (for example), the quotient $\\frac{PV}{T}$ is gonna be exactly $1\/2$ (in other words, how do I know that they are linear) $(2)$ - Why do I have to multiply the $n$ always for the $\\frac{1 \\cdot 22.4}{273}$ and not for the $ \\frac{P_2 V_2}{T_2}$ in the equation? (in other words, if I'm just equating two states of the gas, I guess there should be no problem with which side I multiply by $n$). I don't get what \"multiplying by $n$\" means. Like, the two states of the gas that I'm equating have the same amount of molecules, doesn't mean if they are $1\/2$ mol or anything, they're the same quantity. Thanks!"}
{"id":"65425","title":"How to directly calculate the infinitesimal generator of SU(2)","text":"We commonly investigate the properties of SU(2) on the basis of SO(3). However, I want to directly calculte the infinitesimal generator of SU(2) according to the definition $$X_{i}=\\frac{\\partial U}{\\partial \\alpha_{i}}$$ from Lie group theory. But, where are the problems of the methods I used below? First, I parameterize the SU(2) with the $(\\theta, \\phi, \\gamma) $ like this: $$U=\\begin{bmatrix} e^{i\\theta}sin\\phi & e^{i\\gamma}cos\\phi \\\\\\ -e^{-i\\gamma}cos\\phi & e^{-i\\theta}sin\\phi\\end{bmatrix}$$ and the E is when $(\\theta, \\phi, \\gamma)$= $(0,\\frac{\\pi}{2},0)$. Second, I use the definition of infinitesimal generator like this: $$ I_{1}=\\frac{\\partial U}{\\partial \\theta}|_{(0,\\frac{\\pi}{2},0)}=i\\begin{bmatrix}1 & 0\\\\\\ 0 & -1 \\end{bmatrix}$$ $$ I_{2}=\\frac{\\partial U}{\\partial \\phi}|_{(0,\\frac{\\pi}{2},0)}=i\\begin{bmatrix}0 & i\\\\\\ -i & 0 \\end{bmatrix}$$ $$ I_{3}=\\frac{\\partial U}{\\partial \\gamma}|_{(0,\\frac{\\pi}{2},0)}=i\\begin{bmatrix}0 & 0\\\\\\ 0 & 0 \\end{bmatrix}$$ Here is the question... Why do I get the 0 matrix? We should expect to have the Pauli Matrix. Isn't it?   Where is the problem from?"}
{"id":"109188","title":"How can primordial helium be formed before atoms?","text":"In my physics textbook it describes the events at the beginning of the Universe. I'm confused about the order at a certain point. It says that at some point primordial helium is created, then it says that later atoms are formed. Isn't primordial helium made of atoms? Thank you :)"}
{"id":"21378","title":"Does sending data down a fiber optic cable take longer if the cable is bent?","text":"Ok, so, my simplified understanding of fiber optics is that light is sent down the cable and it rebounds off the sides to end up at its destination. Which got me thinking, if it has to bounce more times (and having a shorter travel between each bounce), does the light (data) take longer to get to the other end of the cable? Like this: http:\/\/i.imgur.com\/pCHUf.jpg A part of me is saying no, because it's still the same distance to travel and bouncing doesn't take up any time, but another part of me is saying yes because the light will have further to travel the more times it bounces, and thus will take longer to get to its destination. I'm swaying towards it taking more time. Thanks!"}
{"id":"21379","title":"Experimental proof of gravitational redshift of light","text":"Has the gravitational red shift been proven for electromagnetic waves only or also for a single photon?"}
{"id":"21373","title":"Reduction of a sum to the first Brillouin zone in a band structure calculation","text":"this might be a \"standard trick\" for many solid state physicists, however it's one that I'm not familiar with so maybe you can help me. Here's the Problem: Suppose we're given a Hamiltonian of the form $H=\\sum_{k} \\epsilon_{k} c^{\\dagger}_{k}c_{k} + \\sum_{k} U c^{\\dagger}_{k+Q} c_{k}$. Here U is some complex (!) number, k is a 2 dimensional wavevector and Q=$(\\pi,\\pi)$. Furthermore we impose $\\epsilon_k = - \\epsilon_{k+Q}$. The first Brillouin Zone is the set $\\\\{ (k_x,k_y) ; |k_x|+|k_y|<\\pi \\\\} $ in 2D-k-space. Now define the 2D-vector $\\Psi_k = (c_k,c_{k+Q})$. Then the Hamiltonian be written as: $H= \\sum_{k}' \\Psi_k^{\\dagger} A_k \\Psi_k $ with the k-dependent 2x2 Matrix $A_k$ defined by: $$ \\left[ \\begin{array}{ c c } \\epsilon_k & U \\\\\\ U^* & \\epsilon_{k+Q} \\end{array} \\right] $$ The prime (') in the sum denotes that it has to be taken over wavevectors in the frist Brillouin zone only! Now multiplying this quadratic form out \"in reverse\" I obtain something like: $H=\\sum_{k} \\epsilon_{k} c^{\\dagger}_{k}c_{k} + \\sum_{k} U c^{\\dagger}_{k+Q} c_{k} + \\sum_{k} \\epsilon_{k+Q} c^{\\dagger}_{k+Q}c_{k+Q} + \\sum_{k} U c^{\\dagger}_{k} c_{k+Q} $ It's not clear to me why the third and fourth term are supposed to vanish in case I'm restricting my sum to the frist Brillouin zone. I hope someone can help. It should be rather technical, but still important I think. Thanks in advance."}
{"id":"21376","title":"What is the origin of the naming convention for position functions?","text":"In physics, position as a function of time is generally called `d(t)` or `s(t)`. Using \"d\" is pretty intuitive, however I haven't been able to figure out why \"s\" is used as well. Is it possibly based on another language?"}
{"id":"116240","title":"Coulomb splitting in atomic physics","text":"In atomic physics, if a configuration with equivalent electrons in some shell (say Neodymium : $[Xe] 6s^24f^4$) gives same $L = \\sum_i l_i$ and $S = \\sum_i s_i$ ($i$ denoting individual spin and orbital momentum of the electrons, and in this example $L = 1$ appears twice with $S = 0$) considering the Coulomb interaction between electrons ($L$ and $S$ being the good quantum numbers in this case), how can these two $(L,S) = (0,0)$ have different energies ? I don't really see what can split these two levels if they start from the same configuration. Does someone know if they actually split ? And if they do, why ?"}
{"id":"56959","title":"effects of sunlight through a glass window?","text":"Both my father and my grandfather where drivers, and over time ended up with a wrinklier left hand compared to the right hand, due to sunlight exposure over 40+ years while holding the steering wheel with the left hand. I googled for sunlight exposure through a glass window and I get contradictory answers either stating either that most glasses filter out UV-light that affects skin tanning and ageing or that the effect of sunlight on skin is pretty much the same if it is direct or through a glass window. So, are the effects of sunlight through a glass window different from direct sunlight exposure?"}
{"id":"94054","title":"Could quarks be free in higher-dimensional space than 3D?","text":"Reading this answer, I now wonder: if quarks are confined by $r^2$ potential, could their potential allow infinite motion in higher-dimensional space? To understand why I thought this might be possible, see what we have with electrostatic potential: in 3D it is proportional to $r^{-1}$. This is just what Poisson equation tells us for point charge. If we solve Poisson equation in 2D space, we'll see potential is proportional to $\\ln\\frac r {r_0}$, and in 1D it's proportional to $r$. We can see that it only allows infinite motion starting form 3D. Could the same hold for quarks, but with some higher than 3D dimension? Or is their potential of completely different nature with respect to space dimensionality?"}
{"id":"56951","title":"The string Poisson bracket","text":"Where does the factor $\\frac{1}{T}$ ($T$ is the string tension) in this Poisson bracket come from? $$ \\\\{X^{\\mu}(\\tau,\\sigma),\\dot{X}^{\\nu}(\\tau,\\sigma')\\\\} ~=~ \\frac{1}{T}\\delta(\\sigma-\\sigma')\\eta_{\\mu\\nu}. $$ I think I can see from remembering the definition of a Poisson bracket (for example in canonical coordinates) why in terms of momentum we have $$ \\\\{P^{\\mu}(\\tau,\\sigma),X^{\\nu}(\\tau,\\sigma')\\\\} ~=~ \\delta(\\sigma-\\sigma')\\eta_{\\mu\\nu} $$ but I don't see why this factor in the first equation has to be there. In addition to deriving it by calculation, is there an intuitive physical way how one can see why the factor of inverse tension has to be there, similar to explaining the appearance of the tension in front of the integral in the action by the fact that it costs energy to stretch the world-sheet?"}
{"id":"98854","title":"What is the physical meaning of the \"decay rate\" in Fermis golden rule?","text":"As far as I understood, Fermi's golden rule gives a prediction of the transition rate in a perturbed quantum system $H_0+V$ between two eigenstates of the unperturbed system $H_0$, say from $\\left| i\\right>$ to $\\left| f\\right>$ with eigenenergies $E_i,E_f$. The prediction is perturbative in the first order of $V$. The result is, that the rate equals $$\\gamma_{i\\to f} = 2\\pi |\\left<f|V|i\\right>|^2 \\delta(E_i-E_f)$$ This rate is defined as $$\\gamma_{i\\to f}=\\frac{d}{dt} |\\left<f|U(t)|i\\right>|^2.$$ **My question:** How should one interpret this differential \"rate\" physically? **My thoughts so far:** Apparently it is the derivative of the _transition probability_ with respect to time, so it is not a probability itself. In literature it is often called _decay rate_ , but for an exponentially decaying probability $p(t)\\sim e^{-\\lambda t}$ the decay rate $-\\lambda$ would not be computed as $dp\/dt$, but $(dp\/dt)\/p$."}
{"id":"102474","title":"relationship between Fredkin gate and the swap and I gates","text":"I need to prove this relationship: $$G_{Fredkin} = I \\otimes |0\\rangle\\langle 0| + G_{swap} \\otimes |1\\rangle\\langle 1|$$ with $$G_{swap} = |00\\rangle\\langle 00| + |01\\rangle\\langle 10| + |10\\rangle\\langle 01| + |11\\rangle\\langle 11|$$ So I think I need to show that both sides are a linear map on $H \\otimes H \\otimes H$ So what is the basis for $H \\otimes H \\otimes H$, and then how do I show that $H \\otimes H = I$? Is this the right approach?"}
{"id":"102475","title":"Why does surface charge not move?","text":"If you have a wire with current flowing through it, and the current flowing the wrong way (not parallel to the wire) surface charge will buildup, generating a field to force the current to flow the right way. What's stopping the surface charge from redistributing? After the correct distribution has been set up for the surface charge to flow, what's stopping the surface charge from redistributing itself over the conductor to maximize distance between like charges?"}
{"id":"102471","title":"Two identical particles with spin $s$. What is the spin of its corresponding \"center-of-mass\" and \"relative\" particles?","text":"Consider a system of two identical quantum particles with spin $s$ and mass $m$. Using center-of-mass coordinates one obtains an equivalent system given by a particle of mass $2m$ and one of mass $m\/2$. Can we talk about the $\\textit{spin}$ of these two new fictitious particles? What can we say?"}
{"id":"100753","title":"Where does this energy equation come from: $E_k=k\\frac{k(q_1q_2)}{r}$","text":"I have this equation but I have no idea where it came from. $E_k=k\\frac{(q_1q_2)}{r}$ I don't understand especially since I have this other equation: $E = k\\frac{Q_{source}}{r^2}$ And cannot find any relation between the two. Please any helps or suggestions would be useful. Thanks! **EDIT:** I have researched this and either am looking in the wrong place or just can't find it anywhere so please help! :)"}
{"id":"71903","title":"Molecular spectroscopy for atomic spectroscopers","text":"My experience is with atomic spectroscopy of alkali atoms, I've recently been asked by a friend to help with some advice on analyzing molecular spectrophotometry data in the context of molecular biology, To fit spectroscopic data to extract density in atomic spectroscopy you typically need to know the relative transition strengths and natural linewidths for all transitions (for hyperfine resolution spectroscopy), know the interaction path length, know the doppler broadening coefficient (and know the temperature), construct a voigt profile and fit. If I wanted to fit spectroscopic data to extract, say, ATP concentration from a spectrophotometer sweep, would I be able to do this by determining the corresponding molecular parameters? Are there caveats to single transition molecular spectroscopy that I should be aware of? It would seem that doppler broadening would be the largest contribution to the linewidth so it I may be able to fit to a simple gaussian profile instead of the more computational intense voigt. In atomic spectroscopy you usually measure a change in coherent beam intensity so the contribution from fluorescence of the atom can be made negligible in the low power unsaturated regime with polarization analyzers and other small trade-tricks. I'm not completely certain how spectrophotometers are set up with that regards, would fluorescence near the measurement frequency be something I need to take into account?"}
{"id":"71904","title":"Falling chain with friction","text":"I really need some help with a physics problem, and I guess my doubt is more conceptual, than question based. But still, let me pose it to you: > _A uniform chain of mass $M$ and length $L$ lies on a rough table, with > coefficient of friction $k$. When 1\/3rd part of the chain is hanging off the > edge, it starts slipping. Find the work done by the frictional force on the > chain, at the instant the last link falls off._ Now, to find the work, I know we must use integration. The work done would be $$\\int k \\, p \\, T \\, g\\, dp \\tag{1}$$ Where $k$ is the coefficient of friction, $p$ is the varying length, $T$ is the mass per unit length ($= \\frac{M}{l}$), and $g$ is the acceleration due to gravity. After applying appropriate limits, we get the answer. The integrand in $(1)$ simply should mean force $\\times$ displacement. Now does my question come in. In the integrand, the Force quantity is defined by the variable frictional quantity, which is equal to $(k \\, p \\, T \\, g)$, so where did the quantity of displacement of the chain go? Shouldn't there be an extra $p$ in the intgrand, such that it becomes $(k \\, p^2 \\, T \\, g)$? Or is the notation $dp$ itself used to signify the displacement? I know the question is a bit complicated, but please help me out. It will be highly appreciated."}
{"id":"17717","title":"laser spectral width vs. linewidth","text":"I have been racking my brains over the differences between laser spectral width and something called the linewidth. The linewidth was written about in detail by Henry in 1982. The spectral width is the width at -20dB down from peak of the wavelength spectrum of the laser. I am looking at some laser data right now that is saying that laser x has 10 kHz linewidth and 60 pm spectral width. You can convert spectral width from frequency to wavelength as in this article and I have done that calculation. By that calculation, a laser with a linewidth of 10kHz should have a wavelength width of about 1x10^-6 nm, not 0.06 nm. Linewidth is often measured with self-heterodyne technique, not a spectrum analyzer. What am I missing?"}
{"id":"53252","title":"Matrix elements of momentum operator in position representation","text":"I have two related questions on the representation of the momentum operator in the position basis. The action of the momentum operator on a wave function is to derive it: $$\\hat{p} \\psi(x)=-i\\hbar\\frac{\\partial\\psi(x)}{\\partial x}$$ **(1)** Is it ok to conclude from this that: $$\\langle x | \\hat{p} | x' \\rangle = -i \\hbar \\frac{\\partial \\delta(x-x')}{\\partial x}?$$ And what does this expression mean? **(2)** Using the equations: $$ \\frac{\\langle x | \\hat{x}\\hat{p} | x' \\rangle}{x} = \\frac{\\langle x | \\hat{p}\\hat{x} | x' \\rangle}{x'} = \\langle x | \\hat{p} | x' \\rangle $$ and $$\\langle x | [\\hat{x},\\hat{p}]|x'\\rangle=i\\hbar \\delta(x-x')$$ one can deduce that $$\\langle x | \\hat{p} | x' \\rangle = i \\hbar \\frac{\\delta(x-x')}{x-x'}$$ Is this equation ok? Does it follow that $$\\frac{\\partial \\delta(x-x')}{\\partial x} = - \\frac{\\delta(x-x')}{x-x'}?$$"}
{"id":"116693","title":"Observation of light bending spacetime","text":"Has radiation or energy bending spacetime ever been observed? If not, is it likely that it ever will, assuming current technology? * * * **Note:** This is not a question of _space_ bending _light_ , but of _light_ bending _space_."}
{"id":"54706","title":"Limitations on the choice of axis of rotation regarding rolling wheels","text":"Consider a situation where a wheel is rolling without friction on a level surface. Call the center of the wheel $C$, the point where the wheel contacts the ground $G$, and some arbitrary other point on the rim of the wheel $P$. The normal force has no torque when $G$ or $C$ is picked as the axis of rotation, but it does if $P$ is chosen for any point (minus the top of the wheel). If a physics problem then says to pick the force that gives the largest torque, how am I meant to choose if the axis of rotation choice changes the answer? The physics book I have states that $G$ or $C$ are points at which an axis of rotation may be placed because they both have the same angular acceleration around each other. This seems fine to me; I can visualize the concept of the $C$ rotating about $G$. The trouble is that it seems to be the same situation for any $P$. Working out the math, it does seem that every point on the rim chosen as the axis of rotation is such that the center of mass rotates about the chosen point at the same speed as the chosen point rotates about the center of mass. What is going on here? How does one 'get a feel' for these problems where it seems that the axis of rotation choice is so critical?"}
{"id":"37904","title":"Thermodynamics - Sign convention","text":"I use the sign convention:   * Heat absorbed by the system = $q+$ (positive)   * Heat evolved by the system = $q-$ (negative)   * Work done on the system = $w +$ (positive)   * Work done by the system = $w -$ (negative) Could anyone please tell me, that volume increased in system does positive or negative work?"}
{"id":"91100","title":"Triangle inequality Clebsch-Gordan coeffcients","text":"The Clebsch-Gordan coefficients can only be non-zero if the triangle inequality holds: $$\\vert j_1-j_2 \\vert \\le j \\le j_1+j_2$$ In my syllabus they give the following proof: $$-j \\le m \\le j$$ $$-j_1 \\le m_ \\le j_1$$ and $$ -j_2 \\le m_2 \\le j_2$$ For maximal values: $m = j$, $m_1 = j_1$ and $m_2 = j_2$ we get: 1) $-j_1 \\le j-j_2 \\le j_1$ which implies $j_2-j_1 \\le j \\le j_1+j_2$ 2) $-j_2 \\le j-j_1 \\le j_2$ which implies $j_1-j_2 \\le j \\le j_1+j_2$ Which should prove the triangle inequality. This proof looks really simple, but I don't completely understand it though. It seems that I'm missing some essential reasoning, and I can't find where. Why for instance do they take for $m_1$, $m_2$ and $m$ all maximal values? Can't I also take $m$ maximal and $m_1$ minimal? This would give bad results though. So I really don't understand it, and I hope that someone can clarify it."}
{"id":"27439","title":"Vasiliev Higher Spin Theory and Supersymmetry","text":"Recently there is renewed interest in the ideas of Vasiliev, Fradkin and others on generalizing gravity theories on deSitter or Anti-deSitter spaces to include higher spin fields (utilizing known loopholes in the Weinberg-Witten theorem by including infinitely many higher spin fields and by working with asymptotic conditions that do not permit S-matrix to exist). There is also a conjecture for a duality for the theory as formulated in asymptotically AdS space with the O(N) vector model in the large N limit. So in this context I am curious if there are supersymmetric generalizations of the theory, and how much supersymmetry can be shown to be consistent with this set of ideas (given that the usual restriction to 32 supercharges comes from forbidding higher spin fields)."}
{"id":"67069","title":"Total current of two sources in series?","text":"Here is a question that a friend asked me. He had to an experiment in school and do some calculations afterwards. Those calculations require maximal current that the DC source can produce. He has measured the maximal currents for two separate sources but forgot to measure it for both connected in series. If maximal current of one battery is $a$ and maximal current of other battery is $b$, what will be the maximal current if one connects these batteries in series? I could imagine the equation system: $$I=\\frac{\\varepsilon_1 + \\varepsilon_2}{r_1+r_2}$$ $$I_1=\\frac{\\varepsilon_1}{r_1}$$ $$I_2=\\frac{\\varepsilon_1}{r_1}$$ where $I$ is for current, $\\varepsilon$ for electromotive force and $r$ for inner resistance. But it seems to have too many unknowns. And I'm not even sure if it's correct to describe the system in this way."}
{"id":"116952","title":"Virtual Particles and Causation","text":"Sometimes when people debate what type of cause a universe with a beginning may have Virtual Particles has been used as an example of a thing that can arise without a cause. So my question would be what exactly is a virtual particle and do they really pop in and out of existence without a cause?"}
{"id":"106575","title":"Darcy Flow in porous material - consider porosity in cross-section area?","text":"According to Darcy's Law, the volumetric flow rate Q of a fluid occuring due to a pressure difference $\\Delta p$ over a distance L and through a cross- sectional area A of a porous medium (volume V) is given by: $$Q = \\frac{k\\cdot A}{\\mu}\\cdot \\frac{\\Delta p}{L}\\tag{1}$$ When calculating the pressure increase caused by that inflow of Q over a time $\\Delta t$, it is typically considered for porous media that only a fraction of its total volume can actually be filled by the fluid, where the fraction of \"free space\" is characterized by the porosity $n = V_{free}\/V_{total}$: $$\\Delta p = K\\cdot Q\\cdot \\frac{\\Delta t}{n\\cdot V_{real}}\\tag{2}$$ My question now is: Why is the porosity considered in the last equation, but not in the cross-sectional area of the first equation? By the very same logic I could say, that only a fraction of the cross-sectional area is \"open\" for fluid to flow into, couldn't I? Especially since (1) is often used for the determination of material permeabilities, but always the total cross-sectional area is used for the calculation. This is just something that seems odd to me and maybe one of you guys has some insight on that and tell me where I'm thinking wrong."}
{"id":"67398","title":"Why is pressure gradient assumed to be constant with respect to radius in the derivation of Poiseuille's Law?","text":"Poiseuille's Law relies on the fact that velocity is not constant throughout a cross-section of the pipe (it is zero at the boundary due to the no-slip condition and maximum in the center). By Bernoulli's Law, this means that pressure is maximum at the boundary and minimum at the center. But in the book I have it is assumed that the pressure gradient is independent of radius (distance from the center of the pipe), and the pressure gradient is thus extricated from a radius-integral. Can anyone justify this?"}
{"id":"78718","title":"How to derive the equation in my question?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/oar2l.jpg) How to derive the equation in my question?"}
{"id":"29023","title":"Grain of sand attracting the sun?","text":"My friend keeps telling me that according to physics... _\"The sun attracts a grain of sand on the earth with the same force that the grain of sand attracts the sun\"_ or _\"A grain of sand on the earth attracts the sun\"_ Is that true? Does in theory a grain of sand really attract the sun?"}
{"id":"78509","title":"Modeling Syringes e.g. with the ideal gas law","text":"Gentlemen I have a similar yet very practical problem that might provide further insight. I'm trying to model a moving plunger in a syringe (something like a piston in a cylinder). At time zero the plunger is at rest, the pressure (behind the plunger) is atmosphere and the volume zero. Then I open (hydrogen) gas which flows at a certain flow rate (which varies in time) and the pressure to move the plunger also varies in time. So my idea is to model using: $$\\frac{\\mathrm{d}}{\\mathrm{d}t}(PV)=\\frac{\\mathrm{d}}{\\mathrm{d}t}RT≡P\\frac{\\mathrm{d}V}{\\mathrm{d}t}+V\\frac{\\mathrm{d}P}{\\mathrm{d}t}=R\\frac{\\mathrm{d}T}{\\mathrm{d} t}$$ If I assume constant temperature $P\\frac{\\mathrm{d}V}{\\mathrm{d}t}+V\\frac{\\mathrm{d}P}{\\mathrm{d} t}=RT$ Is this right? How do I go about solving this if I know the change in flow rate and pressure versus time? Note also that as the volume of gas increases so does m (the mass)....I'm puzzled...has anyone every modeled a syringe in detail?"}
{"id":"87456","title":"Why not formulate Quantum Mechanics using Lagrangians?","text":"As the title implies, why is it that the most common formalisms we use in quantum mechanics prefer to describe systems in the terms of a Hamiltionian instead of a Lagrangian? Is there some convenience to defining our systems one way over the other? Are there cases I'm not aware of where Lagrangian formalism is preferred?"}
{"id":"105902","title":"Calculating the $J$ value for atomic terms, having a lot of trouble with this. Already attempted","text":"I am trying to understand this, and want to be very very clear. This is a homework question but I **already attempted to answer it** , so please don't put this question on hold. The question > What atomic terms are possible for the electronic configuration ns1nd1? > Which term is likely to lie in the lowest energy? Here is my attempt at answering this question. Some of the things I understand like calculating Multiplicity, you do that by the following relationship: > $$S = s_1 + s_2, s_1 + s_2 - 1, \\ldots , |s_1 - s_2|.$$ Then apply it to the multiplicity rule: > $$ 2S + 1$$ I got the values 1, and 3 for the multiplicity and understand that the maximum multiplicity is the one that lies in the lowest energy. I also understand that the maximum $L$ value corresponds to the lowest energy, and since $L = 2$ therefore the term Symbol will be D, and the supercript for S will be either 1, or 3. What I don't get is how to calculate the $J$ values, the $J$ values the solutions manual has are 3, 2, 1 but it didn't explain how they derived those values. Do you use the Clebsch-Gordan relationship for the $J$ values as well, as such? > $$J = L + S, L + S -1, \\ldots |L - S|$$ Really having trouble. Can someone please explain this to me? **Here is my attempt:** ![enter image description here](http:\/\/i.stack.imgur.com\/WYEsS.png) I don't understand how they obtained the $J$ values"}
{"id":"105906","title":"Why p-wave superconductors are rare in nature?","text":"I have the basic question that why so many superconducting materials are s-wave and d-wave pairing, but the p-wave superconductors are so rare in nature? An equivalent question may be that why topological materials are so rare in nature, but the answer to the this question may be more complicated then my first question."}
{"id":"26745","title":"Andromeda\/Milky Way collision: How, and how accurately, can a galaxy's lateral velocity be measured?","text":"Some sources suggest that the Andromeda Galaxy is likely to collide with our own in approximately 3 to 5 billion years. We can estimate the distance to the Andromeda Galaxy using various techniques, including measuring the apparent brightness of Cepheid variable stars; its distance is currently estimated to be about 2.5 million light-years. We can measure its radial velocity (i.e., the rate at which it's either approaching or receding from us) using Doppler shift. One source, the same Wikipedia article I linked to above, indicates that its radial velocity with respect to the Sun is about 300 km\/s in our direction; another article says the radial velocity relative to our galaxy is about 120 km\/sec, also in our direction. (Presumably the difference is due to the Sun's orbital motion around the core of the Milky Way.) But that's just the radial component of the velocity. Taking the 120 km\/sec figure, it could be moving directly toward the Milky Way (more precisely, its core could be moving directly toward the core of the Milky Way) at 120 km\/sec, or it could be moving at a 45° angle at about 170 km\/sec, or any of a number of other possibilities. Without an estimate of the lateral component of the velocity, there's no way to be sure whether the collision will occur or not. I'm reasonably sure we can't measure the lateral velocity directly; 120 km\/sec over a century would cause Andromeda to move only about 0.04 light-year (if my calculations are correct). And yet this Wikipedia article says: > The best indirect estimates of the transverse velocity indicate that it is > less than 100 km\/s. with a reference to \"Abraham Loeb, Mark J. Reid, Andreas Brunthaler and Heino Falcke The Astrophysical Journal, 633:894–898, 10 November 2005\", but the link is invalid. So how can a galaxy's lateral velocity be measured, or at least estimated? How accurate can such an estimate be with current technology? Can we expect improvements in the near future?"}
{"id":"114839","title":"Baryonic density in collision experiments","text":"Does anyone know any way of estimating the net baryon density in collision experiments, e.g. in LHC, RHIC, or the upcoming ones at GSI-FAIR? I have comes across many hand-waving arguments, sample -   1. Electron-positron colliders start with $\\rho_B = 0$, and since conservation laws have to respected, matter and anti-matter must be produced in equal amounts. Hence, barring inhomogeneities, LHC is essentially in the $\\rho_B \\approx 0$ regime.   2. Antikaon-proton interaction is known to be attractive, through various model-dependent calculations, so if we have a ${\\bar K}-p$ collision experiment, it shall involve the formation of quasi-matter having a larger density than usual, and hence, via these kind of reactions, GSI-FAIR plans to study QCD matter at high densities (especially in the Compressed Baryonic Matter experiment), while it remains inaccessible at LHC. (Please correct me if the hand waving is inappropriate.) I am wondering, if there is any way of making this a little more calculation oriented, though I understand that a fully rigorous calculation would be very difficult. (I don't want that, too.) Also, is it possible to have any model-independent answers, since model- dependent answers can be found with a bit of googling, e.g. Phys. Rev. C 72, 034613 (2005), which even has the evolution of these densities in lead-lead collisions. Though these calculations are the order of the day, model- independent calculations, if possible, teach you much more."}
{"id":"114835","title":"Why does quantum zero point energy contribute negative mass to strings?","text":"A string which doesn't have any kind of vibrations will have mass whose square is negative due to quantum zero point energy. But why does it contribute negative rather than positive mass to strings?"}
{"id":"91810","title":"How do I predict volume loss due to evaporation when boiling water?","text":"Suppose I have a pot with diameter $D$ containing a volume of water $V$, being heated by a flame under it. If the ambient air temperature is $T$ and relative humidity is $R$, how can I calculate the expected rate of loss due to evaporation over a period of time, $t$? Assume the period of time begins once the water begins boiling. **Edit:** For simplicity's sake, also assume the flame is just hot enough to get the water boiling."}
{"id":"131002","title":"Is speed an intensive property?","text":"I remember being taught in elementary physics that while it makes sense to add volumes, masses, or heat, it makes no sense to add temperatures. As I wanted to use that to illustate some other issue, I checked it on wikipedia, and discovered the concept of intensive and extensive properties of materials and systems, which is apparently a century old, though I do not remember it being taught to me (it was not as old then :-). The concept seems to be particularly useful in material and systems thermodynamics. But I was wondering how far it extends, and I was keeping in mind my original problem of determining when adding values in a given unit (not necessarily a physical one) made sense or did not. I noticed the fact that the ratio of two extensive variables is intensive. So I started looking, a bit ramdomly I confess, at ratios, and the first that came to mind was speed. And I wondered whether adding speeds made sense. We do that all the time, so it should. But then, my initial problem was not about adding two quantities, but a long list of them (it was a database question). While adding a few speeds (or velocities) makes sense when I move in the bus, or analyse the motion of the Moon in the solar system, I cannot imagine it would ever make sense to add a list of a hundred or a thousand speeds (though computing the average would make sense, as it would for temperature). But I makes perfect sense to sum the masses of a thousand objects. So there is apparently something special, not quite right, about adding speeds. Of course, I know that slightly older results state that adding speeds is not done with simple addition, but I feel that my problem is elsewhere (though there may possibly be a connection). I am aware that temperature in some materials is related to speed of motions inside it, so I am not too surprised. But speed in general seems to go beyond that (sorry for the vagueness). Also speed is not listed by Wikipedia as an intensive property. So, my question is : why does it seem improper to add many speeds (or velocities)? I guess this must also be true of other physical quantities, and I am wondering what is the right way to look at this, and understand it. Is there a more general notion than intensive and extensive?"}
{"id":"50134","title":"Rutherford's Gold Foil Experiment","text":"Can anybody explain how Rutherford bombarded a 0.0004 cm thick gold foil? How did he put it in a photographic sheet? Wasn't the foil too thin to be held? How did he know that the atoms were deflected at various angles? Did he calculate every alpha particle's angle of deflection?"}
{"id":"24421","title":"Variance of Nested Experimental Uncertainty","text":"I have to find the uncertainty of a quantity $Q$ doing two mean values. For example for a set of parameters I measure ten times $Q$, I obtain a mean value $Q_1$ and variance ${\\rm Var}(Q_1)$. Then for a different set of parameters I measure ten times $Q$ and obtain $Q_2$ and ${\\rm Var}(Q_2)$ etc. At the end I compute the mean value which is the sum of $Q_i$ but these 10 from one set of parameter to the other are correlated so I don't know how to compute the variance. Put differently, I don't know how to compute the variance if I have two averages."}
{"id":"24425","title":"How does $F = \\frac{ \\Delta (mv)}{ \\Delta t}$ equal $( m \\frac { \\Delta v}{ \\Delta t} ) + ( v \\frac { \\Delta m}{ \\Delta t} )$?","text":"That's how it's framed in my Physics school-book. The question (or rather, the explanation) is that of the thrust of rockets and how the impulse is equal (with opposite signs) on the thrust-gases and the rocket itself. $( m \\frac { \\Delta v}{ \\Delta t} ) = -(v \\frac {\\Delta m}{ \\Delta t} ) = F_i$ I suppose it's a problem with how I see the transfer of impulse and exactly which part of the equation relates to which part of the physical world (gases, rocket). So we can start from there. _**Title equation:_** $F = \\frac{ \\Delta (mv)}{ \\Delta t} = \\left ( m \\frac { \\Delta v}{ \\Delta t} \\right) + \\left ( v \\frac { \\Delta m}{ \\Delta t} \\right)$ _Grade:_ The equivalent of G-10 in the US."}
{"id":"119376","title":"Does an electomagnetic field \"modulate\" an electric field?","text":"Lets imagine an electromagnetic wave is vertically polarized. If you ignore the magnetic component, the electric field lines would point upwards from the ground or downwards in a cyclical way and of course vary in field strength. Now lets say you attach a battery to two metal plates and you produce an electric field having an horizontal polarization\/direction. Does the presence of an electromagnetic wave \"modulate\" the horizontally orientated electric field, in terms of it's strength and direction?"}
{"id":"102656","title":"Why do we require the generators of $\\mathrm{SU(N)}$ gauge theories to be $N \\times N$ matrices?","text":"I have often read that the generators for $\\mathrm{SU(N)}$ gauge theories must be $N \\times N$ matrices; see for instance these notes at the top of page 3: http:\/\/www.staff.science.uu.nl\/~wit00103\/ftip\/Ch12.pdf‎. Why is this? I don't think this is necessary from a mathematical point of view. For instance, for $\\mathrm{SU(2)}$ we can consider the $2 \\times 2$ generators: \\begin{equation} \\begin{array}{ccc} \\displaystyle J_{1\/2}^1=\\frac{1}{2}\\begin{pmatrix} 0 & 1 \\\\\\ 1 & 0 \\end{pmatrix} \\; ,& \\displaystyle J_{1\/2}^2=\\frac{1}{2}\\begin{pmatrix} 0 & -i \\\\\\ i & 0 \\end{pmatrix} \\; ,& \\displaystyle J_{1\/2}^3=\\frac{1}{2}\\begin{pmatrix} 1 & 0 \\\\\\ 0 & -1 \\end{pmatrix} \\end{array} \\end{equation} However, also the following $3 \\times 3$ generators satisfy the Lie algebra: \\begin{equation} \\begin{array}{ccc} \\displaystyle J_{1}^1=\\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 & 1 & 0\\\\\\ 1 & 0 & 1 \\\\\\ 0 & 1 & 0 \\end{pmatrix} \\; ,& \\displaystyle J_{1}^2=\\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 & -i & 0 \\\\\\ i & 0 & -i \\\\\\ 0 & i & 0 \\end{pmatrix} \\; ,& \\displaystyle J_{1}^3=\\begin{pmatrix} 1 & 0 & 0 \\\\\\ 0 & 0 & 0 \\\\\\ 0 & 0 & -1 \\end{pmatrix} \\end{array} \\end{equation}"}
{"id":"73243","title":"Topological quantum computation : Anyon model","text":"Could someone tell me about Frobenius-Schur indicator and the associated cups and caps notation in context of anyon model. One possible reference could be Parsa Bonderson thesis which is freely accessible here: http:\/\/thesis.library.caltech.edu\/2447\/2\/thesis.pdf"}
{"id":"101103","title":"Why will choice of coordinates impose functional relations on the metric?","text":"I am reading Steven Weinberg's _Gravitation and Cosmology_. On page 10 he says: > In $D$ dimensions there will be $D(D+1)\/2$ independent metric functions > $g_{ij}$, and our freedom to choose the $D$ coordinates at will allows us to > impose $D$ arbitrary functional relations on the $g_{ij}$... Can anyone tell me why the choice of coordinates will impose functional relations on the metric?"}
{"id":"51660","title":"Do electrons in multi-electron atoms really have definite angular momenta?","text":"Since the mutual repulsion term between electrons orbiting the same nucleus does not commute with either electron's angular momentum operator (but only with their sum), I'd assume that the electrons don't really have a well- defined angular momentum (i.e., they do not occupy a pure $\\left|lm\\right>$ state). I would assume that the actual wavefunction is dominated by one such state compared to others, so it is approximately pure, but is there really a points in enumerating electrons according to their momenta, like the s, p, d, f and so on sub-shells?"}
{"id":"98243","title":"How does battery cell size affect voltage drop for a fix current load?","text":"For a fixed current load, will the voltage drop be larger in a small cell or a big cell battery? Why?"}
{"id":"98240","title":"1-dimensional Ring geometry - Group of Translations","text":"I considered a Ring-like one dimensional geometry. In this, if we fix an origin (at some point on the circumference), we can think of **set of all displacements along the circumference** to form a **vector space**. Now one vector can be denoted by (for some reasons that will become clear), $$ \\left( \\begin{array}{ccc} x \\\\\\ 1 \\end{array} \\right) $$ Further one can obtain any other vector in the space by translating the vector, say $ x_0 \\rightarrow x_0+a $. We can use the linear transformation : $$ T(a) = \\left( \\begin{array}{cc} 0 & a \\\\\\ 0 & 0\\end{array} \\right) $$ such that $$ \\left( \\begin{array}{ccc} x + a \\\\\\ 1 \\end{array} \\right) = \\left( \\begin{array}{ccc} x \\\\\\ 1 \\end{array} \\right)+ T(a)\\left( \\begin{array}{ccc} x \\\\\\ 1 \\end{array} \\right) $$ Now the set of all such linear transformations will form a group. Most important part of this transformation is that, if the circumference of the ring is some $L$, then the transformation $T(nL)$ where $ n \\in \\mathbb Z $ should not change the vector. Mathematically, $$ T(nL) \\left( \\begin{array}{ccc} x_0 \\\\\\ 1 \\end{array} \\right) = \\left( \\begin{array}{ccc} x_0 \\\\\\ 1 \\end{array} \\right) $$ Now my question is, with these definitions is the **group of Translations a Compact one** ? And if it is the generator of the translations will have some properties like angular momenta (although this is a generator of translations) ? PS : I hope I am not talking about rotations. I am just talking translations along the circumference of the circle."}
{"id":"4479","title":"Half wave plate and angular momentum","text":"Given:   1. A half wave plate freely floating in space.    2. Circularly polarized light, falling perpendicularly to it. The plate changes polarisation of the beam to the opposite one. Therefore it receives angular momentum and starts to rotate. Where does energy comes from?   What would happen with a single photon passing through the plate?"}
{"id":"25462","title":"Which is the heaviest present day lifter (rocket)? And is it comparable to the Saturn V rocket?","text":"I know of the Ariane 5 ECA, the Delta IV rocket and a few more, but which of the present day's rockets is the top heavy lifter, say, to low Earth orbit (LEO)? Although it is not a certain fact, I would imagine that a very heavy lifter to LEO is also good for placing objects into geostationary transfer orbit (GTO) and could be a good candidate for out of Earth orbit flight (for instance a trip to the Moon)."}
{"id":"4470","title":"twistor-spacetime correspondence","text":"Could someone explain the correspondence between lines in twistor space and minkowski space-time points? a basic derivation would suffice"}
{"id":"4474","title":"Normalizable and non normalizable modes of gauge fields in AdS\/CFT","text":"In Lorentzian AdS space there are both normalizable and non normalizable solutions and we also know (at least for scalar fields in bulk) what do they correspond to in the boundary. But I saw the calculation only for scalar fields. Can someone please give me a reference where people have calculated these modes for a gauge fields, say for a graviton field? McGreevy's lecture note says the relation $\\Delta(\\Delta-D)=m^2L^2$ gets modified to $(\\Delta+j)(\\Delta+j-D)=m^2L^2$ for form $j$ fields. Does this mean for other fields too the normalizable and non normalizable behavior remains the same: namely $Z_0^{\\Delta_+}$ and $Z_0^{\\Delta_-}$, as $z_0\\rightarrow 0$ ($\\Delta_{\\pm}$ are two solutions of course)? How can that be?"}
{"id":"74140","title":"Information escape from a black hole","text":"_Is the following a possible scenario? If not, why not?_ Assume there is a supermassive black hole $Z$ isolated in inter-galactic space. Nearby and stationary relative to $Z$ is observer $A$. A number of light years away spaceship $X$ is travelling at, say, $0.25c$ directly towards the black hole and (eventually) $X$ falls directly into $Z$. $X$'s mass as observed by $A$ will increase as $X$'s velocity approaches $c$ as $X$ approaches $Z$'s event horizon. It seems that $X$'s velocity should be so close to $c$ as it encounters the event horizon of $Z$ that $X$'s apparent mass should cause it to have its own event horizon. I am assuming that $A$ will observe that $X$ becomes a black hole just prior to its event horizon merging with $Z$'s. Now assume that just prior to the merging of the event horizons a spaceship $Y$ is travelling a few light seconds behind $Z$ on the same path, but at about $0.5c$. Just prior to the merging of event horizons $Z$ radios $Y$ (to whom $Z$ will NOT appear to have an event horizon) and $Y$ relays this message to $A$. (Note: This could also happen in reverse, so a fast conversation may be possible.) _What is wrong with this, as it appears to anyone stationary with respect to $Z$, that $A$ and $Z$ are communicating, with information being exchanged from within a black hole?_"}
{"id":"74144","title":"What is the difference between air pressure and atmospheric pressure?","text":"I remember studying that air pressure and temperature are inversely proportional. Now I saw in book that \"Atmospheric pressure decreases as we go higher and higher.\" But in height the temperature is less, and so the air pressure is high. But it is given atmospheric pressure increases with altitude. I googled and understood that **air pressure** and **atmospheric pressure** are different. But I can't understand how each is different from the other in properties.``"}
{"id":"41258","title":"Is airspeed constant or cyclical when flying model aircraft in windy conditions?","text":"I need help to settle an argument about aeronautics. Particularly model aircraft. It has been observed by some that when a model airplane flying with the wind turns back into the wind , some aircraft tend to pitch up and gain altitude indicating an increase in airspeed. Alternately, when flying into the wind, after turning around tend to lose altitude indicating a lower airspeed. Most model pilots say that this is an “illusion” or “pilot error” and that given no change in throttle or thrust the airspeed remains constant. I disagree! It is my theory that in the first case forward momentum is carried through the turn causing a momentary increase in airspeed and when the model turns to fly with the wind it takes a while for the model, now with a slower ground speed to get up to optimum air speed. Thus airspeed is not constant but cyclical. I realize that there are many factors involved and that some of my detractors base their belief on their training in full scale aircraft. Model aircraft tend to be smaller, have lower glide ratios and are often based on high performance aircraft designs as opposed to your typical civilian aircraft. Flying style is also quite different with faster scale speeds and sharper turns for the models. Am I right? Wrong? How do Newton's laws of motion relate to this?"}
{"id":"28642","title":"Using Gauss's Law to calculate electric fields between plates","text":"I have two earthed metal plates, separated by a distance $d$ with a plane of charge density $\\sigma$ placed a distance $a$ from the lower plate. I want to derive expressions for the strength of the electric field in the regions between the plane and the top plate and the plane and the bottom plate. I'm not sure how to apply Gauss's Law to the given situation. I think the best Gaussian surface to use would be a cylinder(?), and then somehow integrate over the two regions. But this would suggest that the electric field between the charged plane and the bottom surface was independent of the total separation, $d$ of the two earthed plates, which I don't think is correct. I don't yet fully understand Gauss's Law, and need more practice with it. Please can someone point me in the correct direction, and give me a hint as to what I should be looking to integrate? With very many thanks, Froskoy."}
{"id":"36192","title":"What is the general formula for a trebuchet?","text":"What I'm really looking for is a formula for a trebuchet that I can input the desired initial velocity after launch and mass of the object, and from that figure out how long the long arm, short arm, sling and pivot need to be, and how much mass I need to have for a counterweight. Does this exist?"}
{"id":"36198","title":"Did Aristarchus take the radius of the Earth into account in calculating the distance to the Moon?","text":"My text says that Aristarchus (310 BC – ~230 BC) measured the \"angle subtended by the Earth-Moon distance at the Sun\" ($\\theta$ in the figure below) to establish the relative Earth-Moon and Earth-Sun distances. ![enter image description here](http:\/\/i.stack.imgur.com\/YAlXy.png) I understand that he must, in fact have used the Moon-Earth-Sun angle, and then subtracted that from 90° to arrive at $\\theta$; but how did he establish the Moon-Earth-Sun angle? The reference points for all three objects is their centers, yet what Aristarchus must have in fact measured was the angle between the Moon and the Sun at the _surface_ of the Earth. Did Aristarchus take this discrepancy into account in his calculations? If so, how?"}
{"id":"36199","title":"Finding the charge density","text":"I can't figure out how to find the charge density for the following problem: > A conductor is placed in an external electrostatic field. The external field > is uniform before the conductor is placed within it. The conductor is > completely isolated from any source of current or charge. > > Assume that at some point just outside the surface of the conductor, the > electric field has magnitude E and is directed toward the surface of the > conductor. What is the charge density n on the surface of the conductor at > that point? I'm supposed to express my answer in terms of $E$ and $\\epsilon_0$. I know that Electric Flux = Normal Angle x Electric Field x cos(theta), but I don't know how to relate this to charge density? What am I missing to solve this problem? This is very basic, I know, but I'm stumped. The hints on Mastering Physics also aren't really helping."}
{"id":"71241","title":"Maxwell equations and Fourier decomposition","text":"I'm currently working on maxwell equations and in order to lower the fields dimension, we perform a Fourier decomposition (according to $\\theta$) due to the system symmetry. For any vector field $\\mathbf{U}(r, \\theta, z)$, we have, $$ \\mathbf{U}(r,\\theta,z) = \\sum_\\alpha \\tilde{\\mathbf{U}}^\\alpha(t,z)e^{i\\alpha \\theta}.$$ Together with the usual Maxwell equations (in cylindrical coordinate system) we decide to work with metallic boundary conditions. Let $\\Omega$ be our domain and $\\Gamma$ its boundary, then, for the electric field, $$ \\mathbf{E} \\times \\mathbf{n} = 0 \\quad \\text{on $\\Gamma$}$$ Where $\\mathbf{n}$ is the normal vector oriented outside $\\Gamma$. Then, my question is how to get the boundary conditions verified by $\\mathbf{\\tilde{E}}$ instead of $\\mathbf{E}$ ? Does it lead to $\\mathbf{\\tilde{E}} \\times \\mathbf{n} = 0$ ? An additional question would be : what about a non homegenous conditions such as $\\mathbf{E} \\times \\mathbf{n} = f$ ? I know that's pretty dumb but I can't convice myself. Thanks in advance."}
{"id":"123347","title":"Why do we remember the past but not the future?","text":"The question is sometimes referred to as the \"psychological arrow of time\" (Hawking, 1985). Here the past is understood as a moment or time when the entropy of the universe was lower, and contrarily for the future. So it is generally thought that PAOT is a consequence of the thermodynamic arrow of time of our universe. If so (maybe not?), how do the two relate? Some explanations in the literature:   1. Practical memory systems work in a way that the formation of new memories entails an overall increase of total entropy of the system and the environment. For example, to create a memory, i.e. to cause our neurons to orient in a particular fashion, requires energy which results in our body heating up a little bit, increasing the total entropy (Hawking, 1985 and 1994); The initialization of memory to make it reusable is an irreversible process that increases total entropy (Landauer, 1961. Wolpert, 1992).   2. More recently, People have argued that even reversible and non-dissipative memory systems are subject to PAOT (Mlodinow and Brun, 2014). The conclusion is arrived by imposing some constraints on what a memory system should be like. Specifically, they argue that a memory should be somehow robust to small microscopic changes in states of the system it records (what they call \"generality\" requirement). But the smallest changes in the future state destroy the thermodynamic arrow of time between now and the future. So any memory of the future of the system \"could remember only one possible configuration of that system\". This fine-tuning disqualifies it as a bona fide memory. My problem with explanation (1) is that even if it's correct, it doesn't seem to be a complete answer in itself. Yes, increase of (new) memory happens only as total entropy of the universe increases. So what? It doesn't have anything to say on the nature of that memory. Why couldn't it occasionally be a memory of the future for that matter? Explanation (2) leaves no such ambiguity. But the generality requirement seems artificial: surely a memory that records the only future configuration of the system remembers the future in a deterministic world, there being no \"what ifs\" regarding that state? Of course, my understanding of the problem is only preliminary. I'd like to know whether there is not some generally accepted explanation, or any other thoughts you have on it."}
{"id":"86540","title":"Etale bundles and sheaves","text":"> Before answering, please see our policy on resource recommendation > questions. Please try to give substantial answers that detail the style, > content, and prerequisites of the book or paper (or other resource). Explain > what the resource is like as much as you can; that way the reader can decide > which one is most suited for them rather than relying on the suggestions of > others. Answers which just suggest a book or paper may be deleted. > > Also note that all answers to this question are automatically **community- > owned** , so they are often subject to major editing, often to make them > comply with the book policy. I am now going through Isham's book _Modern differential geometry for physicists_ and got stuck with the notions of _etale bundle_, _presheaf_ and _sheaf_. Could someone please suggest some other, more intuitive and more accessible references on etale bundles and sheaves, preferably the ones giving more motivation and sufficiently many explicit (and worked-out) examples and, preferably, accessible to theoretical physicists (i.e., not just mathematicians)? P.S. To make things clear, a few math texts I have managed to find so far like Godement's and Bredon's _Sheaf Theory_ (two books with the same title) seem way too tough for me. The part on sheaves in Arapura's _Algebraic Geometry over Complex Numbers_ is somewhat better but still a bit too fast going and with too few examples and not too much motivation. Pretty much the same applies to the part on sheaves (which is too brief anyway) in the Clay Institute volume _Mirror Symmetry_. If there are no suitable books, are there perhaps some good lecture notes on the subject accessible to physicists rather than just mathematicians, from which one get a reasonable intuition on sheaves and stuff?"}
{"id":"132964","title":"An oscillating skateboard","text":"A half pipe of a skateboard park consists of a concrete trough with a semicircular section of radius 5m, I hold a frictionless skateboard on the side of the trough, pointing down toward the bottom and release it; how long will it take to come to back to the point of release? Note: I am supposed to use Newton's laws and polar coordinates. The solution is already given in the book, I am just confused as to why $F_r$, the force in the radial direction, is equal to $mg\\cos\\theta$ minus the normal force (where $\\theta$ is the angle from the origin of the half circle). Also, I am used to using the vector form of Newton's laws in polar coordinates and not using conservation of energy or free fall or anything else."}
{"id":"54323","title":"Collision of two photons","text":"Could someone explain me how will be look like collision of two photons? Will they behave like:   1. Electromagnetic waves, they will interpher with each other and keep they wave nature   2. Particles and they will bounce like classical balls I assume that energy of that system is too small to make creation of pairs possible."}
{"id":"119448","title":"Redshift 1+z - CMB Temperature lower?","text":"I know that $\\frac{\\lambda_2}{\\lambda_1} = 1 + z$ Suppose a galaxy had redshfit $z=3$. Does this mean that the wavelength becomes $4\\lambda$? Then by wien's law where $\\lambda \\propto \\frac{1}{T}$, does this mean that the temperature now observed is $\\frac{1}{4} \\times 2.73 K$?"}
{"id":"86178","title":"What is the apparent viscosity in shear thinning turbulent flow through a pipe?","text":"The explanation of shear rate in laminar flow is straightforward: We imagine small layers of fluid that glide on each other. Now, in turbulent flow, this does not work as there are no layers. If I want to know the apparent viscosity of a shear thinning (or other non-Newtonian) liquid, I need to know the shear rate. I've asked about this before here, and received an answer. However, I can't solve the Navier Stokes equation, so someone has to walk me through it or present me with an answer. Fluid may be assumed to be a power-law fluid."}
{"id":"53075","title":"Sum of intensity of reflected and transmitted waves","text":"**The given state** :   Let $\\psi$ be a wave that passes from medium $a$ to medium $b$.   Let $A$ be the amplitude of $\\psi$.   Let $R$ be the amplitude ratio of the reflected wave $\\psi_r$ and the original one, and $T$ the amplitude ratio of the transmitted wave $\\psi_t$ and the original one.      **The question** : What is the intensity of the transmitted wave?      **An attempt** :   The condition at the boundary of the two media demands that $1+R=T$.   The intensity of a wave is $I_\\psi=|\\psi|^2$.      On one hand we have $I_t\\propto T^2=1+2R+R^2$.   But on the other hand, assuming that the intensity is preserved and that intensity is additive, we also have $I_t=I_\\psi - I_r \\propto T^2=1-R^2$.      Assuming that $R\\neq 0$ there is a contradiction between the 2 answers.   Which one is the right answer?      Thanks."}
{"id":"76734","title":"Electron-Positron Scattering using the Feynman Rules - Integration Q","text":"![enter image description here](http:\/\/i.stack.imgur.com\/neYV0.jpg)   I'm doing independent studies on electron-positron scattering, specifically the annihilation diagram contribution to the M matrix in Bhabha scattering, and this is the equation I recovered with the following initial conditions: An electron, associated with external momentum p_1, and a positron, associated with p_2, annihilate to produce a virtual photon, associated with internal momentum q, that produces a positron-electron pair (p_4 and p_3, respectively). I'm just going to assume you have a running understanding of the Feynman Rules. These are the associated spinors for the particles involved in the interaction. $$u\\to e^- enters\\\\\\\\\\overline u\\to e^- exits \\\\\\ v \\to e^+ exits\\\\\\ \\overline v \\to e^+ enter$$ This is the term added when a vertex is reached, it is a 4 x 4 matrix for fermions. $$-ig_e\\gamma^\\mu$$ The recovered equation: $$ (2\\pi)^4\\int{[\\overline u ^{(S3)} (p_3)(-ig_e\\gamma^\\mu)v^{(S4)}(p_4)}](\\frac{-ig_{\\mu \\nu}}{q^2})[u ^{(S1)} (p_1)(-ig_e\\gamma^\\nu) \\overline v^{(S2)}(p_2)]\\ \\delta^4(q-p_3 -p_4) \\\\\\ \\times \\delta^4(p_1+p_2 -q)\\ d^4q $$ where $g_e$ is the quantum-electrodynamic coupling constant. How do I reduce this down (I see that I can contract the index on $\\gamma^\\nu$) and integrate the derivatives of the Dirac delta function with arguments $q, p_1,p_2,p_3,$ and $p_4$? I've never come across having to integrate derivatives of the delta function. Is it possible to use Eq. (17) on http:\/\/mathworld.wolfram.com\/DeltaFunction.html ? Any help is appreciated."}
{"id":"5552","title":"What physics does occur at short distances in QED?","text":"Let us take the standard QED ($e^-, e^+, \\gamma$) as a model of QFT and ask what is its \"short-distance\" physics? They say the UV infinities appear because we do not know the real physics of short distances and initially we introduce it wrong. OK, but after renormalizations, what physics does remain? Do we replace the unknown\/wrong physics with certain\/right one? Can anybody describe it without appealing to unphysical bare particles? Have we an idea about the real electron from QED? If so, why we cannot use it as the input to construct a reasonable theory from the very beginning? P.S. Moderators, please do not close my questions before they are answered, let people answer."}
{"id":"72927","title":"Can an object be infinitely small?","text":"I read somewhere that the earth has to be smaller than 1 cm to become a black hole, according to Schwarzschild. Since big bang came from a singularity, I am wondering, is there any minimum volume for anything?"}
{"id":"37571","title":"Formulas for compressibility of solids","text":"I am taking a course in mechanics this semester, as well as a course in reservoir physics. Both courses have sections devoted to pressure\/compressibility of solids, but the formulas look slightly different, so I wondered if they really mean the same or not. In my mechanics class I am told that: $$\\Delta P = B \\left(\\frac{- \\Delta V}{V_0}\\right)$$ Where $B$ is a constant known as the **bulk modulus** of a given material. In my reservoir physics class I am presented with the formula: $$c = -\\frac{1}{V} \\left(\\frac{\\partial V}{\\partial p}\\right)_{T}$$ Where $c$ is referred to as **isothermal compressibility**. So my question is - are these formulas basically the same, where $c = \\frac{1}{B}$? And if they are not the same, can someone please explain the difference to me? I would really appreciate if someone could help me with this!"}
{"id":"38249","title":"Underground explosions due to plate tectonics and natural gas pockets","text":"I am not sure if anyone has ever researched this but I am curious about underground reservoirs of natural gas and plate tectonics. Specifically, as the Earth's crust gets pulled down to the mantle do pockets of natural gas get ignited and explode? Or do these pockets of gas get pushed to the surface or further back along the crust as the pressure increases? Given that we have such large underground reservoirs of natural gas that we have discovered I would think that some exist in or near subduction zones. I guess the same question could be asked of reservoirs of oil. But I would think that the natural gas deposits would be a bit more volatile."}
{"id":"72928","title":"Mechanics of Materials (pressure and temperature)","text":"A solid right cylinder of rock core is surrounded by four rods made of mild steel (all-thread rods). The rods are placed equidistantly around the core in a square formation. The tops and bottoms of the four rods are fixed to the top and bottom rigid plates, but the rock is just fitted snugly between the top and bottom plates. The rock is wrapped in an impermeable jacket. The whole system is submerged into a pressure vessel holding water and the water pressure is increased. When the water in the pressure vessel is increased to a certain amount, the fluid in the pore space of the rock is increased (pore pressure) accordingly to give the rock pressure support resulting in an effective stress on the rock. After reaching final pressures the temperature of the pressure vessel and all its contents is then increased. Determine the following quantities: (a) the resulting load $P_s$ in the rock cylinder and $P_r$ in the steel rods; (b) the corresponding axial stresses $\\sigma_s$, $\\sigma_r$; and (c) the axial deformation $\\delta$ of the assembly and (d) the principle stresses on the core. ![fixture](http:\/\/i.stack.imgur.com\/ELknN.jpg) Below is my initial attempt at this problem. I assumed that lateral strain is insignificant but I don't know if that is true. If it is not true then I suppose this problem would have to be solved with iterations? I am hoping someone can tell me if the methodology is correct or how to correctly solve this problem. I've included some values to solve with, but I would ultimately like to have the equations\/methodology to solve for any values of the system. Thanks. When trying to solve this problem I assumed the deformation of the rods are uniform throughout their volume, the rods are prismatic, the compressive load acts at the center of the rods, rods are homogeneous, isotropic, and behave linearly elastically. I made the same assumptions about the rock aswell. The bottom rigid plate is a circle with a diameter $d_p=15$ in., each rod has a diameter $d_r=0.4$ in., and the core has diameter $d_s=7$ in. The rods have a Young's modulus $E_r=30,000$ ksi and the rock has a modulus of elasticity of $E_s=1,450$ ksi. The starting length of both the rods and the core is $L= 20$ in. Before temperature is introduced to the system the entire fixture is subjected to the overburden water pressure $\\sigma_{ob}=25,000$ psi and internal pore pressure $\\sigma_{Pp}=20,000$ psi. Therefore the resulting load on the core and rods imposed by the plate is $$P = \\sigma_{ob} \\cdot A_{plate} + \\sigma_{Pp} \\cdot A_s=\\sigma_{ob} \\cdot \\left(A_{plate}^{bottom}-A_{plate}^{top}\\right) + \\sigma_{Pp} \\cdot A_s $$ $$=\\sigma_{ob} \\cdot \\left(A_s+4A_r\\right) + \\sigma_{Pp} \\cdot A_s $$ $$ = (-25\\, \\text{ksi}) \\left(\\frac{\\pi}{4} \\cdot (7 \\, \\text{in.})^2 + \\pi \\cdot (0.4 \\, \\text{in.})^2 \\right) + (20\\, \\text{ksi}) \\left(\\frac{\\pi}{4} \\cdot (7 \\, \\text{in.})^2 \\right) \\approx -205 \\, \\text{k}$$ The plate is subjected to the force $P$ and the the unkown compressive forces $P_s$ and $P_r$; thus the equation of equilibrium is: $$\\Sigma F_{vert} = 0$$ and $$4P_r + P_s - P = 0$$ This equation, which is the only nontrivial equilibrium equation available, contains two unkowns. Therefore, we conclude that the structure is statically indeterminate. Because the end plates are rigid, the rock cylinder and steel rods must shorten by the same amount. Denoting the shortenings of the rock and steel parts by $\\delta_s$ and $\\delta_r$, respectively, we obtain the following equation of compatibility: $$\\delta_s = \\delta_r$$ The changes in lengths of the cylinder and rods can be obtained from the general equation $\\delta = PL\/EA$. Therefore, in this problem the force- displacement relations are: $$\\delta_s = \\frac{P_sL}{E_sA_s}$$ and $$\\delta_r= \\frac{P_rL}{E_rA_r}$$ We can now solve simultaneously the three stes of equations to obtain the axial forces in the rock cylinder and steel rods: $$\\frac{P_sL}{E_sA_s}=\\frac{P_rL}{E_rA_r} \\rightarrow P_r=P_s \\frac{E_rA_r}{E_sA_s} \\rightarrow 4P_s \\frac{E_rA_r}{E_sA_s}+P_s-P=0$$ Solving for $P_s$ we have $$P_s=P \\left(\\frac{E_sA_s}{4E_rA_r+E_sA_s}\\right)$$ Likewise, solving for $P_r$ we have $$P_r=P \\left(\\frac{E_rA_r}{4E_rA_r+E_sA_s}\\right)$$ Substituting our values we can solve for the resulting loads: $$P_r=(-205 \\, \\text{k}) \\left(\\frac{(30000 \\, \\text{ksi})(\\frac{\\pi}{4}(0.4 \\, \\text{in.})^2)}{4(30000 \\, \\text{ksi})(\\frac{\\pi}{4}(0.4 \\, \\text{in.})^2)+(1450 \\, \\text{ksi})(\\frac{\\pi}{4}(7 \\, \\text{in.})^2)}\\right) \\approx -11 \\, \\text{k}$$ Similarily, plugging values in for $P_s$ we get: $$P_s \\approx -161 \\, \\text{k}$$ The change in length of the core and rods due to the pressure increase will be: $$\\delta_r=\\delta_s=\\frac{PL}{4E_rA_r+E_sA_s}=\\frac{(-205 \\, \\text{k})(20 \\, \\text{in.})}{4(30000 \\, \\text{ksi})(\\frac{\\pi}{4}(0.4 \\, \\text{in.})^2)+(1450 \\, \\text{ksi})(\\frac{\\pi}{4}(7 \\, \\text{in.})^2)} \\approx -0.057 \\, \\text{in.}$$ At this point the temperature is increased to $400\\,^{\\circ}\\mathrm{F}$. The thermal strain relation $\\epsilon_T$ induced on a body is $$\\epsilon_T = \\alpha (\\Delta T)$$ where $\\alpha$ is the coefficient of thermal expansion and $\\Delta T$ is the change in temperature. We assumed $\\alpha_r = 6.5 \\times 10^{-6} \\,^{\\circ}\\mathrm{F}^{-1}$ and $\\alpha_s = 3 \\times 10^{-6} \\,^{\\circ}\\mathrm{F}^{-1}$. The equivalent stress induced due to thermal expansion is: $$\\sigma_{eq} = E\\alpha(\\Delta T)$$ And the equivalent load can be calculated as: $$P_{T} = \\sigma_{eq} \\cdot A$$ Substituting values we can solve for the equivalent loads due to temperature increase in the rods and rock: $$P_T^s = (1450 \\, \\text{ksi})(3 \\times 10^{-6} \\,^{\\circ}\\mathrm{F}^{-1})(400\\,^{\\circ}\\mathrm{F})(\\frac{\\pi}{4}(7 \\, \\text{in.})^2) \\approx 67 \\, \\text{k}$$ $$P_T^r = (30000 \\, \\text{ksi})(6.5 \\times 10^{-6} \\,^{\\circ}\\mathrm{F}^{-1})(400\\,^{\\circ}\\mathrm{F})(\\frac{\\pi}{4}(0.4 \\, \\text{in.})^2) \\approx 9.8 \\, \\text{k}$$ We then add these loads to the loads due to compression to get the final loads: $$P_r + P_T^r = -11 \\, \\text{k} + 9.8 \\, \\text{k} \\approx -1.1 \\, \\text{k (compression)}$$ $$P_s + P_T^s = -161 \\, \\text{k} + 67 \\, \\text{k} \\approx -94.4 \\, \\text{k (compression)}$$ With these values we can calculate the axial stresses: $$\\sigma_s = \\frac{P_s}{A_s} = \\frac{-94.4 \\, \\text{k}}{\\frac{\\pi}{4}(7 \\, \\text{in.})^2} \\approx -2.45 \\, \\text{ksi}$$ $$\\sigma_r = \\frac{P_r}{A_r} = \\frac{-1.1 \\, \\text{k}}{\\frac{\\pi}{4}(0.4 \\, \\text{in.})^2} \\approx -8.76 \\, \\text{ksi}$$ We now solve for the resulting length change in the rods and core do to the temperature increase: $$\\delta_r=\\delta_{r}^i+\\alpha_r(\\Delta T)L'= -0.057 \\, \\text{in.} + (6.5 \\times 10^{-6} \\,^{\\circ}\\mathrm{F}^{-1})(400\\,^{\\circ}\\mathrm{F})(20 \\, \\text{in.} - 0.057 \\, \\text{in.}) \\approx -0.006 \\, \\text{in.}$$ $$\\delta_s=\\delta_{s}^i+\\alpha_s(\\Delta T)L'= -0.057 \\, \\text{in.} + (3 \\times 10^{-6} \\,^{\\circ}\\mathrm{F}^{-1})(400\\,^{\\circ}\\mathrm{F})(20 \\, \\text{in.} - 0.057 \\, \\text{in.}) \\approx -0.034 \\, \\text{in.}$$ From this result we can conclude that the expansion in length of the rods due to the temperature increase causes a seperation between the bottom core face and the bottom fixture plate. This result may have came about due to incorrect assumptions on the values for the young's modulus and\/or coefficient of thermal expansion. This separation will cause a fluid connection between overburden pressure and pore pressure which is not acceptable. If the resulting axial stress on the rock is to be a 1000 psi less than the resulting rock lateral stress, what would be the required material properties for the rock and rods?"}
{"id":"30095","title":"Is there any way to survive solarwinter like in Sunshine - movie?","text":"Is there any way to survive solarwinter like in Sunshine - movie? Solar winter is where for some reason sun looses its capasity to produce radiation( heat etc.). It doesn't loose everything but some of its radiation energy( say 50 %) That causes earth to cool down causing next \"ice age\""}
{"id":"113583","title":"Does the weight of a computer go up as information is added to it?","text":"This probably sounds really naive. But, a strange discussion came up on Quora about computers possibly weighing more when information is added to them. I tried looking around but couldn't find a definitive answer. There are a few threads where people have tried to ask something similar. 1) http:\/\/www.thenakedscientists.com\/forum\/index.php?topic=38844.0 2) http:\/\/www.lolhappens.com\/27706\/does-a-computers-weight-increase-as- information-is-added-to-the-hard-drive\/ Both threads have people arguing about the possibilities, but I'm sure a more definitive, and painfully detailed answer must exist. I hope you have a good laugh and then help me out! I don't have the rep to post more than two links so I'll just put the original thread in the comments. Thank you!"}
{"id":"5883","title":"How does a honeycomb grid affect the travel of light?","text":"A light modifier that is commonly used in studio photography is a honeycomb grid. It narrows the beam of light to a circle with soft edges, as it can be seen here: My question is: how is this happening? A small reporter flash has a rectangular shape, if you place a rectangular shaped grid on it, it produces a \"soft\" circle of light. How is the light travel modified by the structure of the grid?"}
{"id":"5882","title":"Convert running speed uphill to equivilent speed on flat","text":"Given a certain running pace uphill, I want to be able to determine an equivalent pace running with no elevation change. Assumptions: similar effort in both cases (say for example running at 90% max heart rate), ignore wind, slope is constant for simplicity, ignore physiological and bio-mechanical factors, weight of the runner is 135 lbs if that matters. Example: Elevation change +236 feet, distance traveled 1 mile, elapsed time 6 minutes 55 seconds. What could I theoretically run for 1 mile with no elevation change given the same effort?"}
{"id":"5886","title":"What is the effect of polarization on diffraction by a narrow slit?","text":"Consider the well known demonstration of diffraction by a narrowing slit. (See for example the demonstration at the 30 minute mark of this lecture at MIT by Walter Lewin) It is my (possibly mistaken) understanding that the light emerging after the slit becomes substantially slimmer than one wavelength is polarized.   This would seem to imply that light of perpendicular polarization would not be transmitted, thus implying a fairly substantial and dramatic difference in the results of the experiment with parallel and perpendicularly polarized light. That is, instead of spreading out, the light polarized in the wrong direction would essentially just shut down as the slit narrows below one wavelength. Is this true?"}
{"id":"47345","title":"following up dark matter accretion in supermassive black holes","text":"A while ago, there was some conspicuous evidence that supermassive black holes didn't seem to be eating dark matter at the expected rate of 70%-30%, in fact, only 10% of the black hole mass increase could in principle be attributed to non-baryonic mass. Was there some follow-up evidence for or against this observation? because it is 2012 and every astrophysicist that is out there seems to be happy and dandy about dark matter, despite this huge clue (together with MOND, and the negative results from CoGENT) that dark matter doesn't actually exist"}
{"id":"12420","title":"What does an electromagnetic wave look like at a fixed moment in time?","text":"I am curious what the electric and magnetic field's of light look like when time is stopped. A \"photograph\" or illustration\/description of these fields at a moment in time is what I desire. Also, does the picture change at various points in the wave? I assume that this is like a \"time exposure\" of a light wave."}
{"id":"5888","title":"How can super massive black holes have a lower density than water?","text":"I heard on a podcast recently that the supermassive black holes at the centre of some galaxies could have densities less than water, so in theory, they could float on the substance they were gobbling up... can someone explain how something with such mass could float? Please see the below link for the podcast in question: http:\/\/www.universetoday.com\/83204\/podcast-supermassive-black-holes\/"}
{"id":"87790","title":"Exciting Surface Plasmons using ATR","text":"I'm very new to the topic of surface plasmons and I have been reading about different methods of exciting them. There is one method in which a prism is set up to allow phase matching of an incident light ray to the required wave vector for exciting a surface plasmon. The light goes through total internal reflection in the prism and then either tunnels through the metal (on which we want to excite the surface plasmon), or a lower indexed dielectric to excite the surface plasmon. This might be a lack of understanding of attenuated total internal reflection, but apparently this field harbors no energy and 100% of the incident light is reflected. My question is how is energy conserved (how can it excite surface plasmons)? Is it because the energy of the attenuated field only time averages to 0? But then how can all the light be reflected?"}
{"id":"15063","title":"Is this static mechanical balance possible without trickery?","text":"On damnlol.com, I came across this picture: ![enter image description here](http:\/\/i.stack.imgur.com\/Bwxxl.jpg) http:\/\/www.damnlol.com\/hello-god-i-have-a-fault-to-report-7549.html My question is: Is this possible without glue? If not are there similar situations in which something like this is possible ?"}
{"id":"122307","title":"The concept of center of mass behind dropping water from a faucet?","text":"> Water, dripping at a constant rate from a faucet, falls to the ground. At > any instant there are many drops in the air between the faucet and the > ground. Where does the center of mass of the drops lie relative to the > halfway point between the faucet and the ground? (a) Above it (b) Below it > (c) Exactly at the halfway point When I looked at this, I thought the answer is (b) below it since if the water drop and the ground are in a system, then the center of mass will be nearer to the heavier object, which is the ground. But when I looked up the answer it was (a) above it. Could you please explain why? Thank you!"}
{"id":"11293","title":"Determining wave function for term symbol 1D","text":"I am trying to follow a book (Introduction to Ligand Field Theory by Ballhausen in 1962 on pg 15), but it isn't clear how they make a particular leap. **Background** I want to find the wave function for the term $^1 D$. We know that $\\psi(L,M_L,S,M_S) = \\psi (2,2,0,0)$ is made up of three micro states: $(2^+,0^-)$, $(2^-,0^+)$, $(1^+,1^-)$. Thus, a linear combination must be taken of these micro states. Now in the book they say they must be orthogonal to $\\psi(3,2,1,0)$ and $\\psi(4,2,0,0)$. Presumably these are picked because we just determined those via lowering operators in the previous section. I don't take it that there is another reason those two wave functions are mentioned. $(2^+,0^−)$ means that electron 1 has $m_s=+1\/2$ and $m_l=2$ and that electron 2 has $m_s=−1\/2$ and $m_l=0$. We write that $\\Psi=|(\\psi^+_1)(\\psi^−_2)|$ where we have written the short form of the determinantal antisymmetrized normalized wave function, which comes from the diagonal element in the Slater determinant AND separated the orbital- and spin-dependent parts of the wave function (spin is denoted by super plus or minus sign). **Problem** Anyway, I understand that $\\psi(2,2,0,0) = a (2^+,0^-) + b (2^-,0^+) + c(1^+,1^-)$. Then they write \"and we get $a \\sqrt{3} - b \\sqrt{3} + c \\sqrt{8} = 0$ and $a + b = 0$.\" I'm totally lost how they made this leap and why these must be equal to zero. Does this have to do with the orthogonal wave functions just mentioned? Where did the numbers in the sqrt come from? And the minus sign! Greatly appreciated."}
{"id":"82678","title":"Does someone falling into a black hole see the end of the universe?","text":"This question was prompted by Can matter really fall through an event horizon?. Notoriously, if you calculate the Schwarzschild coordinate time for anything, matter or light, to reach the event horizon the result is infinite. This implies that the universe ages by an infinite time before someone falling into the black hole reaches the event horizon, so could that person see the universe age by an infinite time? To be more precise, suppose the observer starts falling from rest at time $t = 0$ and some initial distance $r > r_s$. If we wait for some time $T$ then shine a light ray at the falling observer. Will the light ray always reach the falling observer before they cross the event horizon? If not, what is the formula for the longest time $T$ that we can wait and still be sure the ray will catch the observer? If $T$ is not bounded it implies that observer could indeed see the end of the universe. I can think of a qualitative argument for an upper limit on $T$, but I'm not sure how sound my argument is. The proper time for the observer to fall to the event horizon is finite - call this $\\tau$. The proper time for the light ray to release the horizon is zero, therefore the light ray will reach the observer before they cross the event horizon only if $T < \\tau$. Hence $T$ is bounded and the observer won't see the end of the universe. I think a more rigorous approach would be to determine the equations of motion (in the Schwarzschild coordinates) for the falling observer and the light ray, and then find the condition for the light to reach the falling observer at some distance $\\epsilon$ from the event horizon. Then take the limit as $\\epsilon \\rightarrow 0$. In principle this seems straightforward, but in practice the algebra rapidly defeated me. Even for a light ray the radial distance:time equation isn't closed form (Wolfram claims it needs the $W$ function) and for the falling observer the calculation is even harder."}
{"id":"16607","title":"Renormalization scheme independence of beta function","text":"I have some questions about renormalization. To my understanding, in order to deal with infinities that appear in loop integrals, one introduces some kind of regulator (eg, high momentum cutoff, taking $d\\to d+\\epsilon$, etc.) so that we get a finite answer, which blows up as we remove the regulator. Then we renormalize various coefficients in the Lagrangian in a regulator-dependent way so that, in scattering amplitudes, a finite piece remains as we remove the cutoff. The regulator seems to always require introducing some arbitrary scale (eg, the momentum cutoff or $\\mu^\\epsilon$ for dim-reg), although I'm not sure why this must be the case. Now, this finite piece is completely arbitrary, and depends on what scheme we want to use (eg, on-shell, minimal subtraction, etc). The beta function is then, roughly speaking, the rate of change of this finite piece under changes in the scale in the regulator. My question is, what exactly is the invariant information in the beta function? Under changing renormalization scheme, it obviously changes, but it seems that this roughly corresponds to a diffeomorphism on the space of couplings (is this always true? for example, in on-shell renormalization, if we set the mass to the physical mass and the coupling to the corresponding exact vertex function at zero momentum, these seem to be independent of scale - do the beta functions vanish here?). Also, it seems to depend on exactly what regulator we are using, and how the dimensionful scale enters into it. There are certain quantities, such as the anomalous dimension of a field at a fixed point, which must be independent of all these choices, but I don't understand why this is the case. Finally, and this is more of a philosophical question, what exactly do the infinities in the loop integrals mean in the first place? Why is it that we can remove them by any of these various regulators and expect to get the same answer? I've read something about Haag's theorem, that interacting theories live in a different Hilbert space than the free-field Fock space, and that this is somehow related, but I'm not sure why. Thank you."}
{"id":"88166","title":"Why does a particle fall in a straight line?","text":"In Lagrangian Mechanics we choose the path of least action. Given a uniform gravitational field, and a particle of finite mass; and fixing two points the start & end-point we consider all paths connecting the two points and minimise the action. This turns out to be a Brachistone, as first shown by Bernouilli. When we fix the end-point vertically below; the Brachistone is in fact astraight-line. But is there a principle in Lagrangian Mechanics that allows me to choose the point vertically below? Of course, we know from Newtons Mechanics that this must be the case. But how do we determine that end-point entirely within Lagrangian Mechanics?"}
{"id":"69547","title":"Is pressure invariant under Lorentz transformations?","text":"In an article from the reference there are following words (page 2): \"...pressure P is a Lorentz invariant... the result follows from standart properties of the relativistic stress-energy tensor...\". What properties are used by the authors of an article? For example, I used the expression for stress-energy tensor of an isotropic body: $$ T_{\\alpha \\beta} = (\\varepsilon + p)\\frac{v_{\\alpha }v_{\\beta}}{c^{2}} - g_{\\alpha \\beta }p. $$ If pressure is determined as a 3-trace of this tensor, it's obviously that it isn't Lorentz invariant."}
{"id":"64711","title":"Topological vs. non-topological noetherian charges","text":"What (if any) is the relationship between the conserved (non-topological) noetherian charges and topological charges? Namely, is there any \"generalization\" of the Noether's first theorem that includes topological charges as subcase and it provides some relationship between these two classes of charge?"}
{"id":"21170","title":"Showing constraint is nonholonomic","text":"One example of a nonholonomic constraint is a disk rolling around in the cartesian plane that is constrained to not be slipping. These leads to the constraint $dx - a \\sin\\theta d\\phi = 0$ and $dy - a\\cos\\theta d\\phi = 0$ Where $\\phi$ is the angle of how far the disk has rotated, $-\\theta$ is angle that velocity makes with respect to $x$. We know that these aren't exact differentials because to put it in form $Mdx + Ndy = 0$, we don't have $\\partial M\/\\partial y = \\partial N \/ \\partial x$. But that doesn't mean we can't find some integrating factor to multiply it by and make it an exact differential? So I don't see how we know that is it nonintegrable? If I take $f(x,\\phi)[ dx - a \\sin\\theta d\\phi] = 0$ and try to manipulate it to get something analogous to $\\partial M\/\\partial y = \\partial N \/ \\partial x$, then I get a tricky PDE, which I don't know how to solve."}
{"id":"123104","title":"Why can we allow the speed of light being infinite in case of Surface Plasmons?","text":"I have a problem with understanding of these sentences: > We have indicated in the opening paragraph of the Introduction that surface > plasmon polaritons are solutions of Maxwell’s equations in which the effects > of retardation—the finiteness of the speed of light—are taken into account. > An important subclass of surface plasmon polaritons are surface plasmons. > These can be viewed as the limiting case of surface plasmon polaritons when > the speed of light is allowed to become infinitely large. > > Nano-optics of surface plasmon polaritons (1.1.2) - Anatoly Zayats et.al. How can we allow c be infinite? Why is it correct in case of surface plasmons? The article is available online, easy to find with Google Scholar"}
{"id":"135357","title":"How many seconds should you accelerate to complete the run on time?","text":"Your goal is to complete your 10,000-m run in less than 30.0 min. Apparently you've started too slowly because at 27.0 min, you still have 1100 m to go! Part A You can accelerate at 0.20 m\/s2. For how many seconds should you accelerate (and after this have constant velocity) in order to complete the run on time?"}
{"id":"45693","title":"How quarks converted into leptons","text":"![Pion Plus Decay](http:\/\/hyperphysics.phy- astr.gsu.edu\/hbase\/particles\/imgpar\/pidec1.gif) Since the charged pions decay into two particles, a muon and a muon neutrino, seems quarks disappeared!, The decay proceeds by the weak interaction $W^{+}$ and can be visualized in terms of Feynman diagrams. Isn't it why Quarks are not directly observed!? I read somewhere: > If you are consistent thinker you can go even further and question existence > of quarks themselfs then you will not have a problem with fractional charge"}
{"id":"10602","title":"Calculating threshold energy of particle reactions","text":"{..everything that follows is in the domain of relativistic kinematics..} Say a particle A collides with a particle B at rest and produces particles C and D. What exactly is the definition of \"threshold energy\" for a reaction ? Is it the energy that A should have so that so that the heavier of C and D is produced at rest? Given the masses of all the particles, how does one calculate the threshold energy that A must have to cause this reaction ? I intuitively feel that a threshold energy scenario would mean that C and D are moving along the same direction in which A was coming in. I think it is an unnecessary \"waste\" of energy for C and D to develop momentum in the two transverse directions of A's motion. But I can't prove this in general. I would like to know how this situation is analyzed."}
{"id":"11959","title":"How thermal imaging cameras work","text":"As far as I see from wiki, 'consumer'-grade(non-cryogenic) thermal imaging cameras use microbolometer sensors to get integrated IR intensity over some 5-12um range. But I had an impression, that having just integrated IR intensity you can't get surface temperature precisely, especially with 0.1C precision we see on consumer cameras. IR thermometers (pyrometers) for example use 2-band IR measurement, but I don't see this principle being used in thermal imaging. Or there is some sort of 'know-how' matrix of IR band filters in front of the sensor?"}
{"id":"62458","title":"Induced voltage by a rotating ring","text":"Two concentric rings dielectrics and uniformly charged are suspended on the same floor. The outer ring has a radius R and mass M, while the other has radius r << R and mass M. The outer ring that has overall charge Q, is made to rotate around the axis passing through the center and perpendicular to the plane of the ring itself, with angular acceleration α. Calculate the angular acceleration α' of thr inner ring. * * * My solution is the following: The outer ring produce a variation of magnetic flux on the inner ring equal to               (μ * Q * π^2 * r^2 ) \/ ( α * R )  (*)      The inner ring produce a self-induced variation of magnetic flux for the law of Faraday-Lenz equal to               (μ * Q * π^2 * r' ) \/ α'   (**)      Comparing the ( * ) and ( ** ), I obtain               α' = α * R \/ r      Is it my solution right?"}
{"id":"62457","title":"Questions about angular momentum and 3-dimensional(3D) space?","text":"**Q1:** As we know, in classical mechanics(CM), according to Noether's theorem, there is always one conserved quantity corresponding to one particular symmetry. Now consider a classical system in a $n$ dimensional general coordinates space described by the Lagrangian $L(q,\\dot{q})$, where $q=(q_1,...,q_n)$ and $\\dot{q}=(\\dot{q_1},...,\\dot{q_n})$ are general coordinates and general velocities, respectively. If the system has $SO(n)$ spatial rotation-symmetry, e.g.$\\forall A\\in SO(n),L(Aq^T,A\\dot{q}^T)=L(q,\\dot{q})$, then we can get $\\frac{n(n-1)}{2}$(number of the generators of group $SO(n)$) conserved angular momentum. **My question is as follows** , now note that our spatial dimension is $n$, when $n=3\\rightarrow $ number of angular momentum$(\\frac{n(n-1)}{2})$$=$spatial dimension$(n)$$=$3, otherwise, number of angular momentum$\\neq $spatial dimension, so why spatial dimension 3 is so special? Is there any deep reason for number 3 or it's just an accidental event? Or even does this phenomena have something to do with the fact that we \"live\" in a 3D world? **Q2:** In quantum mechanics(QM), a Hermitian operator $J=(J_x,J_y,J_z)$ is called an angular momentum iff $[J_x,J_y]=iJ_z,[J_y,J_z]=iJ_x,[J_z,J_x]=iJ_y$. **And my question is as follows** , in QM, we can have 1-component momentum operator $\\hat{p}$ , or 2-component momentum operator$(\\hat{p_x},\\hat{p_y})$, and so on. But Why we have never encountered a angular momentum with only two components $J=(J_x,J_y)$? Can we define a 2-component angular momentum? Like in the **CM** case, again the **number 3** is special in **QM** case, why? Thanks in advance. **By the way:** More questions concerning the **definition of rotation groups** for angular momentum can be found here , who are interested in may have a look, thanks."}
{"id":"118905","title":"Tension in an Atwoods machine conceptual?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/jLunv.gif) Assuming $T_1$ is the force that acts on box $1$ and $T_2$ is the force that acts on box $2$. Exactly what causes the Tension? Why does $T_1 = T_2$? The problem is we are told to memorize that $T_1 = T_2$ for mass less ropes, but I do not understand why, especially in an Atwoods machine. When I think about it, I know it has something to do with the force $m_2g$ which causes the box with mass $m_1$ to rise, and vice versa. But I cannot apply physics terminology to this, or really understand whats going on."}
{"id":"118906","title":"Geometric Interpretation of these equations of motion?","text":"I was reading my Engineering Mechanics book, and it derived some strange looking integrals I'll have to apply. I could memorize them, but I'd rather understand them - then I won't have to memorize. A few words mentioning Leibniz notion, and viola - we have these relationships: $$\\int_{v_0}^vdv=\\int_0^ta_cdt$$ $$\\int_{s_0}^sds=\\int_0^t(v_0+a_ct)dt$$ $$\\int_{v_0}^vvdv=\\int_{s_0}^sa_cds$$ I can manipulate them given appropriate data, but they don't really _mean_ much to me. (I _do_ easily understand things like $a=\\frac{dv}{dt}$,however). I anyone could point me to website that discuss or even just show me how to derive these, I would really appreciate it - I find it easier to understand geometric interpretations, but I would appreciate any knowledge you would be willing to share."}
{"id":"118903","title":"Aside from Noether's theorem, what other concepts would explain energy conservation?","text":"Energy is defined more in the mathematical sense, and tends to show true with observations in the physical world. But why is energy conserved aside from \"Noether's theorem\"? In a closed system that has an energy $E$, we know that system will always have this much energy, but why? In the cosmological scale I heard that energy is not conserved, why? Is the universe creating more energy? OR Is the expansion of the universe due to the energy at the beginning that caused it's creation? Could anyone explain their point in layman terms? Without using difficult Mathematics, I'm not as \"qualified\" to understand such rigorous equations yet."}
{"id":"54090","title":"Contact angle of liquid drop on surface","text":"How do you explain point 44 of the attached pdf document on surface tension? Here's the link. How is the direction of surface tension found out? (I know it tangential but in which direction along the tangent?) Also why is surface tension for solid-air surface considered when balancing the forces on liquid? Any help will be appreciated."}
{"id":"62188","title":"Would HgTe be a topological insulator?","text":"In \"Quantum Spin Hall Insulator State in HgTe Quantum Wells\", researchers observed a 2D topological insulator by sandwiching HgTe between CdTe. Is the CdTe really necessary? Would Vacuum\/HgTe\/Vacuum itself be a topological insulator?"}
{"id":"54096","title":"How to design an experiment that shows that a rectangular pulse can be expressed as a series of infinite sinusoids?","text":"Is it possible to design a physical experiment that shows that a time limited signal, such as a rectangular pulse is composed of infinite continuous sine\/cosine waves?"}
{"id":"122570","title":"Which is more fundamental, Fields or Particles?","text":"I hope that I am using appropriate terminology. My confusion about quantum theory (beyond my obvious unfamiliarity with its terminology) is basically twofold:   1. I lack an adequate understanding of how the _mathematics_ of quantum theory is supposed to correspond to phenomena in the physical world   2. I still have an incomplete picture in my mind of how cause and effect relationships occur at the quantum level of reality.  This is why phenomena such as \"entanglement\" make absolutely no sense to me. So, in an attempt to come to some understanding of all of this, I would like to know that if what we conceptualize as a \"field\" is merely an interaction among particles (bosons and fermions in the case of quantum fields), and particles (themselves) are actually fluctuations in \"fields\", then which comes first in the hierarchy of cause and effect relationships, particles or \"fields\"?"}
{"id":"16775","title":"Literature request for books \/ review papers on gravitation, gauge theories and related mathematics","text":"Similar to this reference, are there more such references \/ works [including textbooks] available in the literature? (A list would be greatly welcomed and appreciated.) With great appreciation."}
{"id":"29559","title":"The multiverse of eternal inflation","text":"I don't have a clear picture of what is \"by definition\" the multiverse appearing in the models of eternal inflation. A long time ago I heard that it is a quantum state, but in the cosmology books I have looked into (all treating the subject superficially, may I suspect it's because the object is still not clearly defined?) they define a non spatially homogeneous scalar field on it, entering in the slow-roll inflation regime in some areas, and creating a nucleation this way... So it must also be a manifold, so we can define a coordinate chart on it. Then how many dimensions is it? Because they is also this picture where it solves the landscape problem of string theory and each bubbles becomes a particular vacua of ST (but I am not sure about that neither as I have not read enough of string theory so far). Well to summarize I have a really blurry picture of this mysterious multiverse, so if someone could share his clear definition of it (assuming it exists) I would be really thankful. ps: is it even a physical object? in the sense obeying itself to some physical laws (which ones?) not like Newton's universe for example? is it dynamical?"}
{"id":"123627","title":"Role of the canonical ensemble and electric charge in AdS\/CFT","text":"If we consider a charged black hole in AdS spacetime, we can either do thermodynamics in the grand canonical or the canonical ensemble. In the former, we fix the electrostatic potential $\\Phi=A_t(r=\\infty)$ at the boundary of the bulk such that $\\left<Q\\right>=-\\frac{1}{\\beta}\\left(\\frac{\\partial S}{\\partial\\beta}\\right)_{\\beta}$, where $S$ is the Euclidean action. In the latter, we fix the charge $Q$ of the black hole and we do not consider $\\Phi$ at all. The phase diagram of the black hole is highly dependent on the choice of ensemble, see for example this paper by Chamblin, Emparan, Johnson and Myers. One could therefore expect that this choice also has an influence on the CFT side. In the AdS\/CFT dictionary, the charged black hole gives a global $U(1)$ symmetry on the CFT side. Here $\\Phi$ in the bulk corresponds to a chemical potential $\\mu$ on the CFT, so that we usually consider the grand canonical ensemble when using the correspondence. My questions are as follows:   * Do we ever consider the _canonical_ ensemble in AdS\/CFT?    * If so, what would $Q$ determine on the CFT side (in the same way $\\Phi$ determines $\\mu$)?    * If we work in the grand canonical ensemble, does $\\left<Q\\right>$ play any role on the CFT side, or do we only need $\\Phi$? EDIT: For my last question: I just looked into holographic superconductors and it seems that from $\\Phi$ one can derive both the chemical potential and the charge density of the CFT. The charge density seems to coincide with $\\left<Q\\right>$ in this case (up to some constant factors), but we don't need $Q$ to calculate it as we can derive it from an asymptotic expansion of $\\Phi$. Specifically see page 8 of this paper by Horowitz. I don't think this answers my first 2 questions though."}
{"id":"95319","title":"Difference between the two equations for acceleration","text":"I came upon this while studying S.H.M. Well,is there a difference between writing $$a=\\frac{dv}{dt}\\;$$ and $$a=v\\frac{dv}{dx}\\;$$ do they differ on the basis of one being a vector and the other being a scalar equation?Please explain."}
{"id":"95311","title":"Hawking radiation and entropy","text":"What is your opinion of the hawking radiation mechanism, does that actually lower the entropy of the black hole?"}
{"id":"95316","title":"Angles in cosmology","text":"We have several definitions of distances in cosmology. Take angular diameter distance. $$d_A = a(t)\\int \\frac{dt}{a(t)}\\;.$$ If an object has proper length $L$ then the angle $\\theta$ it is seen at is $$\\theta = \\frac{L}{d_A}$$ This is similar to the way an angle $\\theta$ and arc length $s$ are related in a circle with radius $r$ $$ \\theta = \\frac{s}{r}$$ On cosmological scales distances are so vast that the arc length $s$ is indistinguishable from the proper length $L$ of the star or galaxy observed. All this is fine. But when it comes to the cosmic microwave background, we look at temperature correlations between two directions $\\vec{n_1}$ and $\\vec{n_2}$ with an angle $\\theta$ between them on the sky today. $$ C(\\theta) = C(\\vec{n_1}\\cdot\\vec{n_2})$$ These angles are not small necessarily. For the quadrupole there is $90$ degrees between the two directions. Subsequent Higher multipoles have smaller angles. With such wide angles we should no longer use $$\\theta = \\frac{L}{d_A}$$ but instead use $$tan(\\theta) = \\frac{L}{d_A}$$ Where am I going wrong with this line of thought?"}
{"id":"66605","title":"What is the gamma five matrix $\\gamma_5$?","text":"This Wikipedia page explains that for each of the four main gamma matrices $\\gamma^{\\mu}$, you can find the covariant matrices $\\gamma_{\\mu}$ with the equation $\\gamma_{\\mu} = \\eta_{\\mu\\nu}\\gamma^{\\mu}$. But that formula doesn't make any sense for $\\gamma^5$ because $\\eta_{\\mu\\nu}$ does not have that many indices. So what is $\\gamma_5$?"}
{"id":"61379","title":"What is the maximum tension in N the string can support","text":"I have a length of string and I want to know the maximum tension the string can support. I tie one end of the string to the ceiling and the other end to a glass of mass 100 g. The glass is cylindrical, with a cross-sectional radius of 4 cm and a height of 15 cm. I fill the glass with water (density of 1 g\/cm3) and discover that the string breaks when the glass is 2\/3 full. What is the maximum tension in N the string can support?"}
{"id":"8396","title":"What are the conditions to be satisfied by a theory in order to be a quantum theory?","text":"This is in continuation to my previous question. It is not a duplicate of the previous one. This question arises because of the answers and discussions in that question. Can we call a theory, quantum theory, if it is consistent with HUP? For example, suppose there is a finite and self consistent theory of gravity which incorporates the uncertainty principle. Can we at once call this theory a quantum theory of gravity or does it have to satisfy other conditions too? This question may be too basic but it is intriguing my mind."}
{"id":"8390","title":"What allows the Wave Disk Generator to be so efficient?","text":"Researchers at Michigan State University recently invented the Wave Disk Generator that is supposed to get 60% fuel efficiency. What allows it to be so much more efficient than a traditional Internal Combustion Engine? I am aware that there is better mixing of fuel and air, but surely this alone does not produce the extreme efficiency."}
{"id":"117200","title":"Strong interacting v.s. Strong Coupling v.s. Strong Correlated","text":"One of the active research areas in present is Strong interacting, Strong Coupling, Strong Correlated regime of the phases of matters. It seems to me that some physicists in the fields often mix the usages of these twos: **Strong Coupling, Strong Correlated**. However, in my viewpoint, they are NOT the exactly same, I regard that > $\\bullet$ **Strong Coupling** : implies the large coupling of interactions > comparing to the free part of theory. Say, suppose there is a Lagrangian > description, then the action $S$ $$ S=S_{free} +g S_{interact} $$ the > **Strong Coupling** means $g >>1$. So this can be the confined phases of > QCD, where coupling $g$ of quarks and gluons runs large. \\-- > $\\bullet$ **Strong Correlated** : in my view, usually implies the > **fractionalization of the elementary particles into fractional quantum > numbers**. For example, this happens at 1+1D Luttinger liquids, where spin > and charge can separated their degree of freedom from the elementary > constituents(electrons), but the system needs NOT to be **Strong Coupling**. > i.e. this example is **Strong Correlated** but NOT **Strong Coupling**. This > is about the fields of **Strong Correlated Electron** on arXiv. \\-- **My question** , so what are other **examples of systems** that are: > **1\\. YES Strong Coupling and YES Strong Correlated** > > **2\\. YES Strong Coupling but NOT Strong Correlated** > > **3\\. NOT Strong Coupling but YES Strong Correlated** See also this relevant post."}
{"id":"26372","title":"Did physical models of galaxies come before they were actually observed?","text":"Black holes were first predicted by astrophysics, then observed. Was the existence of galaxies first predicted by astrophysics, or first observed by astronomers?"}
{"id":"93251","title":"Energy of a simple pendulum like device","text":"If there is a device in which a ball is attached to a pivot by a rod of length R and it is left to oscillate by dragging it to one side so that it is completely horizontal, at any point on its decent the ball has some velocity and also has some angular velocity about the pivot. In this case is the rotational kinetic energy about pivot equal to translational kinetic energy of the ball ? Why\/ Why not ? If they are same then what is better way to deal with the calculations of such problems, the rotational KE about the pivot or translation KE of the ball alone ?"}
{"id":"93252","title":"Transformation from waves to matter","text":"Let $|\\psi \\rangle$ represents a wavefunction through and $\\langle \\psi |$ represent the dual   vector. Now there are things such as matter waves, could $|\\psi \\rangle$ represents a matter wavefunction, (of course representation is a definition), which is analogous to that of the massless wavefunction $|\\psi \\rangle$? If so, what would the $\\langle \\psi |\\psi \\rangle = n \\in \\mathbb{R}$ represent, an object? and the outer product $|\\psi \\rangle \\langle \\psi | = \\rho$ would that be similar to the probability of localization of matter? I guess what I am trying to do is understand mathematically how I would be able to represent matter and how, if any, way there is to represent this in an analogous matter with the bra-kets. Seems there should be a relation as if waves are representable and matter is a wave, then matter is representable. Thanks, Brian"}
{"id":"55767","title":"What role has our Moon played in creating a persistent geomagnetic field?","text":"The question comes from a comment by Mark Rovetta on my earlier question about the Earth's core going cold."}
{"id":"55765","title":"Can we excite a nucleus by means of very intense low energy gamma-photon irradiation?","text":"The phenomenon of multi-photon ionization of atoms has been studied, both theoretically and experimentally, for several decades. Intense laser beam devices are the apparatuses used for the experimental study of this phenomenon. **QUESTION:** Would it be possible to use similar excitation processes with nuclei, using \"low energy\" $\\gamma$-photons in order to manufacture nuclear isomers for industrial and medical applications?"}
{"id":"32523","title":"Can you prepare the superposition of two arbitrary light beams?","text":"More specifically, how do you physically prepare the superposition of two beams of polarized light? Suppose beam A and beam B, the two input beams, each have unknown polarizations.   Beam C, the output beam, must be the superposition of beams A and B.   How do you do it? You can create an overlap of two beams with a half silvered mirror at 45%.   You can also creat a localized overlap by crossing the beams with a small angle.   In both cases you get a mixture, not a superposition. How do you get the superposition? Bonus Points: How do you prepare $ a|A \\rangle + b|B\\rangle $?"}
{"id":"56239","title":"When I ride my bike, does half of the energy go into the earth?","text":"When I ride my bike, does half of the energy go into the earth because of Newton's third law? Does the energy in the earth transfer into heat upon my brakes when I utilize them? If the earth was 2000 times less massive than it is, and I applied the same energy to my pedal, would my bike move (relative to an observer not connected to earth) at the same speed as it would if the earth was normal? I think if you think about it these two questions are the same."}
{"id":"110313","title":"photoelectric cell","text":"In the photoelectric cell my teacher says that the electron emission from the cathode depends on the frequency of the incident photon and it doesn't depend on the light intensity (I = nhU\/ta , Right?) so if the light intensity increased the photoelectric current won't change . but what if this increase in the light intensity is due to an increase in the frequency(energy) of each photon. to make a correct statement shouldn't the light intensity be replaced with rate of photons? so that if the number of photons increased even to millions and if the frequency of each was less than the threshold frequency , the photoelectric current won't be affected."}
{"id":"111685","title":"Tidal force of Sun?","text":"As I understand, satellites and the Moon orbiting Earth are in free fall. Isn't the same true for Earth orbiting the Sun? My question is then: How can the Sun's gravity affect tides? Aren't the molecules in the oceans in free fall relative to the Sun?"}
{"id":"57724","title":"Maximum Shear on a Beam - beam with fixed support on one end and hinge on other end","text":"A beam $\\displaystyle 3m$ long with fixed support on one end and hinge on the other end is subjected to a uniform load of $10\\ kN\/m$. What is the maximum shear of this beam? The solution is this one: $\\displaystyle Max.Shear = \\frac{5\\omega{L}}{8}=\\frac{5\\left(10\\right)\\left(3\\right)}{8}=18.75\\ kN$ where $\\omega$ = distributed load $N\/m$ $L=$ length My question is, how did he get this equation? All I know is that hinges have two forces acting on them. Can anyone derive this equation?"}
{"id":"57726","title":"Future light cones inside black hole","text":"In Caroll's Spacetime and Geometry, page 227, he says that from the Schwarzschild metric, you can see than from inside a black hole future events all lead to the singularity. He says you can see this because for $r<2GM$, t becomes spacelike and r becomes timelike. I don't understand his reasoning though. Why does r being timelike mean you can only travel towards the singularity?"}
{"id":"113056","title":"How to show that in 1D scalar potential well there isn't pairs production (Dirac particle)","text":"Let's have the potential $U = -V_{0}$ for $|x| \\leqslant a$ and $U = 0$ for $|x| > a$. The stationary Dirac equation for bound states gives $$ tg(\\frac{p_{2}a}{\\hbar}) = \\frac{2\\Gamma}{1 - \\Gamma^{2}}, \\qquad (1) $$ where $$ \\Gamma = \\frac{p_{1}}{p_{2}}\\frac{E + mc^{2} + V_{0}}{E + mc^{2}}, \\quad p_{1}c = \\sqrt{m^{2}c^{4} - E^{2}}, \\quad p_{2}c = \\sqrt{(E + V_{0})^{2} - m^{2}c^{4}}. $$ How to show that there isn't pair production independently on well's parameters $V_{0}, a$? My attemption. I assume that if scalar potential acts on particle and antiparticle by the same ways, there exists only one case when the production is possible. It is when $E_{1} = E_{2} = 0$, where these energies refer to the particle and antiparticle energies correspondingly. For case $E = 0 (1)$ gives me $$ \\sqrt{V_{0}^{2} - m^{2}c^{4}} = -mc^{2}tg\\left(\\frac{\\sqrt{V_{0}^{2} - m^{2}c^{4}}a}{c\\hbar}\\right), $$ and theoretically it doesn't have solutions for all possible values of $a, V_{0}$. But it seems that it has. Can you help me?"}
{"id":"127380","title":"Does Light (EM Signal) undergo accelration during reflection?","text":"Since Average acceleration is defined as, Average acceleration over a period of time is the change in Velocity divided by the Duration of the Period. Does Light (EM Wave) undergo acceleration during reflection from mirror or any other surface?"}
{"id":"35177","title":"What happens when a photon hits a mirror?","text":"When a photon of light hits a mirror does the exact same photon of light bounce back or is it absorbed then one with the same properties emitted? If the same one is bounced back does it's velocity take all values on $[-c,c]$ or does it just jump from $c$ to $-c$ when it hits the mirror? Or, is the phenomenon of a mirror better explained using a wave analogy? If so, what is this explanation?"}
{"id":"60130","title":"Bloch sphere representation","text":"Suppose you know that a qubit is either is in state $|+\\rangle$ with probability $p$ or in state $|-\\rangle$ with probability $1-p$. If this is the best you know about the qubit's state, where in the Bloch sphere would you represent this qubit?"}
{"id":"134336","title":"Why are orbits elliptical?","text":"Just wanted to know why planetary orbits get to be elliptical? I'm not debating how stable these orbits are (obviously almost completely stable), but why don't planets revolve in the special ellipse known as a circle? Wouldn't the centrifugal\/-pedal forces between the planet and Sun make the planet revolve around the Sun in a perfect circle? Is there a 'third force' that attracts planets at a specific point (a hypothetical reasoning for elliptical orbits)? I'm not trying to disprove Kepler over here, but I don't exactly get it."}
{"id":"82532","title":"How to calculate the radius of a rain drop with variable mass?","text":"I need help with the following problem, please help me get started as I do not know where to begin with One spherical raindrop is falling in the atmosphere. Mass of the raindrop increases proportional with it's area. There's no counterpoise in it's move. I should show that radius of a rain drop is increasing linearly with time."}
{"id":"82530","title":"What field of physics play a major role in water pipe function","text":"Background: I'm a legal MMJ patient and I've been looking into water pipers\/bongs recently. I understand that this question can be a little controversial but where I live this is all legal as long as you have the proper prescription. Last few years there has been quite a lot of \"scientific\" bongs that has been hitting the market and in my to medicate as healthy as possible I would like to understand how they work. I know about vaporizers and I have a few of them, this is a physics question. So to my question. What are the major fields that come into play when talking drag\/resistance, smoke cooling and smoke filtration? My theory is that it is fluid dynamics. Are there any good books to read to fully understand the mechanics behind the water bong? Thank you"}
{"id":"31293","title":"Practical matter of the Higgs-Mechanism","text":"My maybe very naive question is, of what practical importance will the discovery of the Higgs-Mechanism be for our technological advance in the near future?"}
{"id":"31362","title":"What good is extraordinary matter to the consumer?","text":"> **Possible Duplicate:**   >  Practical matter of the Higgs-Mechanism This video claims that for everyday matter you only need electrons, upquarks, and downquarks. So are all these other fermions useless? Has anyone ever made a shovel out of strange-charm-muon atoms or something?"}
{"id":"31897","title":"What's next after Higgs Boson discovery?","text":"> **Possible Duplicate:**   >  Practical matter of the Higgs-Mechanism As everybody knows that the Higgs Boson was discovered on July 4th,2012, I am so curious about it.   1. What are the possible outcome applications of Higgs Boson?   2. How Higgs Boson is related to Quantum Mechanics?   3. Does Higgs Boson fits in General Theory of Relativity?   4. Does Higgs Boson fits in String Theory? Please share the knowledge. I am very curious about it."}
{"id":"127087","title":"Not-so-hot black shirt","text":"As it is getting warmer here by the minute I was asking myself: **Are there materials, that are black (in th visible range) but reflect (most) invisible light?** Furthermore, I asked myself **what fraction of heat could be kept** away using this material in daylight. For bonus points, it would be great if it could be used as some kind of fabric. I am not looking for a specific material but am curious, as e.g. metals do reflect all light up to a certain frequency and thus would not suffy. Therefore it would be nice to understand the underlying mechanism. In the winter of course it would be nice to reverse this, so to have white shirts, still getting hot in the sun."}
{"id":"117324","title":"Equal and opposite electron flows in a magent?","text":"If the electron flow in a magnet is say from South pole to North pole, then I can understand why you can't put the two north ends of magnets together, as the two flows of electrons repel each other. But then why can't I put the two south pole ends of a magnet together? If the electrons flow out the North pole and into the south pole, then what force is repelling the two south poles?? If the answer is to say there are two electron flow, in opposite directions, then why don't they just cancel each other? There must be \"something\" else flowing in the opposite direction of the electrons that causes the two south poles to be repelled. Edit: Perhaps electron flow was bad choice of words. Every diagram of a magnet I see shows the lines of flux flowing from out of North pole and into South pole. Given this flow of \"whatever\", it is easy to understand why the North poles would repel each other. It is also easy to understand why the North pole of one magnet is attracted to the South pole of another. My question really is why do the South poles also repel each other? If the magnetic field is really flowing \"into\" the south pole, then what force causes the two South poles to Repel each other."}
{"id":"105717","title":"Motional EMF and EMF?","text":"What is the difference between motional EMF = $-vBL$ , and Faraday's law of induction $\\displaystyle\\mathcal{E} = \\left|\\frac{d\\Phi_B}{dt}\\right|$? Aren't they the same? What is the relation of Lorentz force to motional EMF?"}
{"id":"105710","title":"The relation between the movement of electrons and energy","text":"So, I've been enjoying reading a lot of helpful posts, but now, I found myself in the need of asking something. I have a hard time grasping the general concept of electricity \/ how the relation between the movement of electrons and energy corresponds. But so far I've reach this conclusion \/ guess: 1) The movement of electrons, should rather be called mechanical energy, than calling it electric energy. Why? Electrons move by pushing to each other, typical mechanical energy example. That electrons has a electric charge, doesn't make the movement of them electric energy. 2) Every single electron has their own electric field. When there is a potential difference between minus and plus (electroncurrent), we say that there will be created an uniform field, because of the field lines between two parallel conducting plates with oppositely charges. Meanwhile, when electrons move in the same direction, they create a magnetic field. **And now comes my guess:** My statement: It is the effects of the magnetic and electric field in the wire, that do the work on electronic devices. We know that fields of all kinds can exert a force on any given object within its reach. So my guess is that this electromagnetic field the wire creates, can effect: a smaller wire in a computer, the wires that goes to the screen, the way the screen lights up. So you could say that the energy \/ what I would believe people in general thinks is 'electricity' exists not only inside the wire, but also outside. So when a device is 'using' \/ 'transforming' the energy to other energyforms, you could think of it this way. 1) Remembering that a current exists in a closed circuit. 2) When you exert a force from a battery or whatever, the electrons move very slowly themselves, but all in all as a system, they move at the speed of light. When they do this, the electromagnetic field will be a natural follower of this event. So when we transform energy\/ use energy, it is the constant need for making the electrons move around, in order to create the electromagnetic field. And when there is no force to push the electrons around, the electromagnetic field disappears - and no work is done, and no money spent. This would also mean \/ or be the reasoning behind why devices \/ light or what so ever, starts in splitseconds. Because the electromagnetic field comes at the speed of light, because the electrons move the way they do. And using the expressions as potential energy, is just a way of saying that you create a strong difference between minus and plus (electroncurrent.) And the electromagneticfield will be \"tapped\" when there is devices nearby, but the current logically remains the same, buut, the electromagnetic field will be weaker near plus, because it has already done a lot of work. But we might be taking about fractions of seconds, so we probably wouldn't register it.. (is this right or just totally wrong reasoning??) So saying electricity, could in daily life relate to: the movement of electrons or the effects of electromagnetic field exertion on electronic devices. Many thanks in advance!"}
{"id":"41589","title":"Reading list in topological QFT","text":"I'm interested in learning about topological QFT including Chern Simons theory, Jones polynomial, Donaldson theory and Floer homology - basically the kind of things Witten worked on in the 80s. I'm looking for pedagogical reviews rather than original articles. Though these things sit at the interface of mathematics and physics I'm interested in them more as a physics student. I remember someone asking for a suggested reading list for topological QFT in mathoverflow. The suggested papers were almost uniformly rigorous mathematics written by mathematicians. I am not looking for something like that."}
{"id":"93900","title":"References for Chern-Simons action $ \\int d^3 x ~\\epsilon^{\\mu \\nu \\rho} A_{\\mu} \\partial_{\\nu} A_{\\rho}$","text":"Can you recommend to me good references on this action? From basics to more advanced stuff."}
{"id":"9635","title":"Correlation between outstanding hints in experimental particle physics","text":"The 115 GeV ATLAS Higgs with enhanced diphoton decays has gone away but there are several other recent tantalizing hints relevant for particle physics, namely   * CoGeNT's 7-8 GeV dark matter particle that seems to support DAMA's and other signals   * D0's fourth-generation top prime quark that may be there at 325 GeV if a 3-sigma signal is right   * CDF's \"new\" force Z prime boson at 144 GeV which could be there if another 3-sigma bump is real   * Tevatron's top-antitop asymmetry above 400 GeV which seems very large   * A related CDF top-quark anomaly and maybe others I missed. My question is simple: > Is there some sensible theoretical basis (e.g. paper) that would > simultaneously explain at least two of the observations above if they were > real?"}
{"id":"63449","title":"Absorption\/Extinction formula of nanoparticles","text":"I know the absorption\/extinction equations in nanoparticle physics should be: $$Q_{abs}=\\frac{1}{2}\\mathbf{Re}\\int \\mathbf{J}_{tot}\\cdot\\mathbf{E}_{tot}^\\ast dV=\\frac{\\omega}{2}\\mathbf{Im}(\\epsilon)\\int|\\mathbf{E}_{tot}|^2dV$$ also, for the extinction, it reads: $$Q_{ext}=\\frac{1}{2}\\mathbf{Re}\\int \\mathbf{J}_{tot}\\cdot\\mathbf{E}_0^\\ast dV$$ But I see in some papers people use the following equations: $$Q_{abs}=\\frac{\\omega}{2}\\mathbf{Im}(\\mathbf{d}\\cdot\\mathbf{E}_{inside}^\\ast)$$ where $\\mathbf{d}$ is the total dipole moment of nanoparticles. Also, for the extinction, it reads: $$Q_{ext}=\\frac{\\omega}{2}\\mathbf{Im}(\\mathbf{d}\\cdot\\mathbf{E}_0^\\ast)$$ I failed to derive the two equations. Can anyone give some help? Or, some reference papers would be also very helpful. Thanks a lot for the help."}
{"id":"112312","title":"Only gravitation and Newton's $2^{\\mathrm{nd}}$ law needed to derive Kepler's laws?","text":"It is known that Kepler's laws of planetary motion can be derived from Newton's laws of motion and his law of universal gravitation. However, are all of Newton's laws of motion necessary? According to this website, all of Kepler's laws can be derived from gravitation and the Newton's second law ($\\mathbf{F} = m\\mathbf{a}$). Does this mean that Newton's first and third laws are irrelevant for planetary motion? Perhaps Newton's third law (action-reaction) is necessary to take into account the motion of the Sun as induced by gravitational pull from the orbiting planets, but when deriving Kepler's laws the Sun is assumed to be fixed..."}
{"id":"112315","title":"Gauss’s Law inside the hollow of charged spherical shell","text":"Use Gauss’s Law to prove that the electric field anywhere inside the hollow of a charged spherical shell must be zero. My attempt: $$\\int \\mathbf{E}\\cdot \\mathbf{dA} = \\frac{q_{net}}{e}$$ $$\\int E \\ dAcos\\theta = \\frac{q_{net}}{e}$$ $$E \\int dA = \\frac{q_{net}}{e}$$ $E\\ 4\\pi r^2 = \\frac{q_{net}}{e}$ and since it is a hollow of a charged spherical shell the $q_{net}$ or $q_{in}$ is $0$ so: $E = 0$. Is my reasoning on this problem correct? Essentially $E$ is $0$ because there is no charge enclosed."}
{"id":"45646","title":"Lorentz Transformations and Aberration","text":"Why does the azimuthal angle, $\\phi$, remain unchanged between reference frames in special relativity? Edit: Here is a link from \"Radiative Processes in Astrophysics\": http:\/\/i937.photobucket.com\/albums\/ad216\/Kyanise\/Capture-5.jpg. I think this comes from the aberration formula, showing dependence only on the polar angle, $\\theta$. The aberration formula is: $\\tan \\theta = \\frac{u_\\perp}{u_{||}} =\\frac{ u' \\sin \\theta '}{\\gamma (u' \\cos \\theta ' + v)}$ where ' indicates a property of the moving frame. But I'm not even sure why this is true. I know it comes from the transforms of the velocities, but why should $\\tan \\theta = \\frac{u_\\perp}{u_{||}}$ anyways? I mean $v$ can be in any direction, right? How come there is no dependence on $\\phi$?"}
{"id":"102158","title":"Question about the vacuum bundle on A- and B-model","text":"Let us consider the topological string A- and B-model (twisted SUSY non-linear sigma model on CY 3-manifold $X$). They are realization of $N=2$ SCFT and there are ground-states vector bundle $\\mathcal{H}$ and vacuum line bundle $\\mathcal{L}$ over the moduli space of the theory. In the A-model, $\\mathcal{H}=H^{even}(X,\\mathbb{C})$ and $\\mathcal{L}=H^0(X,\\mathbb{C})$. In the B-model, $\\mathcal{H}=H^{3}(X,\\mathbb{C})$ and $\\mathcal{L}=H^{3,0}(X,\\mathbb{C})$. The genus $g$ string amplitude is given as a section of $\\mathcal{L}$ in either theory. Mirror symmetry is an identification of the geometry of A- and B-model ground-state geometry on distinct CY 3manifolds. My questions is following. In the A-model, it seems the splitting of the bundle $$ \\mathcal{H}_A=H^{even}(X,\\mathbb{C})=\\oplus_{i=0}^3 H^{2i}(X,\\mathbb{C}) $$ does not vary over the moduli space of the theory (Kahler moduli space). On the other hand, in the B-model, the splitting $$ \\mathcal{H}=H^{3}(X,\\mathbb{C})=\\oplus_{p+q=3}H^{p,q}(X,\\mathbb{C}) $$ varies over the moduli space (variation of Hodge structure). Moreover, $\\mathcal{L}$ is the trivial line bundle in the A-model, while it is not in the B-model. Isn't this contradiction?"}
{"id":"113954","title":"What is the extent of the Galactic Magnetic Field?","text":"What is the extent of the `Galactic Magnetic Field`? What does the `Galactic Magnetic Field` look like from afar (such as half-way between the `Milky Way Galaxy` and `Andromeda Galaxy`)? Does the `Galactic Magnetic Field` interact with other galaxies's magnetic field? Also, is there an `Inter-Galactic Wind` interacting with all of the above (in similar nature to how the `Solar Magnetic Field` interacts with the `Galactic Wind`)?"}
{"id":"98917","title":"Does time exist in a vacuum?","text":"Imagine a perfect vacuum devoid of all matter and radiation. Irrespective of whether such a space can even exist, my intuition tells me that time would not exist within such a space. What this thought experiment tells me is that the human concept of time is basically that it's a measure of motion. Whether its a car speeding down a highway, skin on your face as the years go by, or photons, it's really all about motion. Now what about something like a gravitational field? Could that also be excluded from acting within a perfect vacuum?"}
{"id":"98919","title":"Caculating thermal stresses in a composite rod","text":"**The Question** > Between two rigid walls separated by a distance $l_1+l_2$, two rods of equal > cross-section area $A$, and length, coefficient of linear expansion and > Young's modulus $l_1, \\alpha_1, Y_1$ and $l_2,\\alpha_2,Y_2$. Suppose the > temperature of the entire system is raised by $T$ kelvins. What is the force > the two rods exert on each other?   >  ![enter image description here](http:\/\/i.stack.imgur.com\/XhGMT.png) **Concepts Involved**   1. Tendency to expand of the two rods   2. Stress generated due to the constraint on total length which limits their expansion   3. Different increases in length of both the rods owing to different young's modulus and original lengths  **My efforts** Let at the increased temperature, the length of Rod 1 be $l_1+x$ and that of rod two be $l_2-x$. The \"natural length\" of both the rods at that temperature is $l_1(1+\\alpha_1T)$ and $l_2(1+\\alpha_2T)$. Therefore the longitudnal stress developed in each is:- $\\delta l_1=l_1\\alpha_1T-x$ and $\\delta l_2=l_2\\alpha_2T+x$. For the rods in equilibrium, the forces they exert on each other need to be equal. Therefore, using $F=\\delta l\/l\\times AY$, and equating their forces we get:- $$\\frac{Y_1l_2(l_1\\alpha_1T-x)}{Y_2l_1(l_2\\alpha_2T+x)}=1$$ This can be solved for $x$ which on substituting in $F=(l_1\\alpha_1T-x)\/l_1\\times AY_1$, we should get the force. But no matter how much I rearrange this equation, it does not match with the given answer. **Is there a fault in my arguments\/equations or calculations?** Sorry if there is trivial mistake."}
{"id":"108009","title":"Why do we set $x^0 = ct$ instead of $x^0 = t$?","text":"When we deal with Special Relativity and we start considering spacetime instead of space and time each at once, we usually see books saying that we consider a space with four coordinate $x^\\alpha$ with $x^0 = ct$. We also consider this manifold to be $\\mathbb{R}^4$ and give to it the metric tensor $g = \\operatorname{diag}(-1,1,1,1)$. Now, why $x^0 = ct$ and not just $x^0 = t$? One possible answer might be \"so that all coordinates are measured with the same units\", but isn't there any deeper meaning?"}
{"id":"15303","title":"Why Is String Theory Called A Theory","text":"> **Possible Duplicate:**   >  Laws of Atomic Theory - how is this possible? Generally in science, a theory is effectively a fact. The **theory** of evolution is not a guess, or a hypothesis. It's currently understood to be a fact, backed up by evidence from numerous scientific fields spanning many decades. String **theory** on the other hand seems much more contentious, much less settled in the scientific community. Why isn't it called the string hypothesis?"}
{"id":"3346","title":"How small of a depletion signal can the best modern mass spectrometers detect?","text":"So I know the answer to this question varies widely across mass spec techniques, masses, and, of course, budgets, but my question is about the _best_ case scenario for all of these variables (although I'm not really looking at proteins or anything of that size). I realize that the concept of depletion spectroscopy relies heavily on very sensitive mass spectrometers, and I'm curious about the lower limits of signal detection for such methods. Thus, I'm interested in the sensitivity, not resolution (\"mass resolving power\"), which seems to be a statistic that is difficult to find for most of the systems that I've looked at. Some related follow-up questions are what mass spectrometry methods can and cannot be used with depletion spectroscopy and what methods\/manufacturers are the gold standard in the field of mass spec? Thanks in advance for your answers."}
{"id":"3344","title":"Electrical eddy current visualization or simulation","text":"Eddy currents are induced in a metal plate when it experiences a changing magnetic flux. Is there a realistic visualization or simulation of eddy currents available? The only picture I found, on wikipedia, is a cartoon schematic."}
{"id":"30251","title":"Using Einstein's Relativity: Who is younger?","text":"Suppose we have a person **A** and a person **B**. Person B travels very close to speed of light and never returns. He's constant in speed. Then, we can say two things:   1. B is younger than A.   2. A is younger than B (since we can consider B's reference as inertial). Who is correct between the two?"}
{"id":"31126","title":"How many sigma did the discovery of the W boson have?","text":"When the W boson was discovered in the 1980s, nobody spoke of sigmas. How many sigmas was it at that time?"}
{"id":"67216","title":"About electric current analogy","text":"my teacher gave me this analogy to the electric current , the wire is like a pearl necklace where the pearls can move, the current or the movement of electrons is like putting your fingers between 2 pearls separating them then you start taking one pearl from the right of your finger and push itto the left of your finger starting making the pearls to move, is this analogy right?"}
{"id":"67211","title":"Why is Standard Model + Loop Quantum Gravity usually not listed as a theory of everything","text":"I have often seeen statements on physics.SE such as, > The only consistent theory of everything which we know of to date (2013) is > string theory. Why exactly is this so? Adding the Loop Quantum Gravity Lagrangian Density (the Einstein-Hilbert-Palatini-Ashtekar lagrangian density) to the Standard Model Lagrnagian Density should be able to describe all the interactions and fermions, in my opinion. Maybe it isn't as elegant as string theory since it doesn't really unify all the forces\/interactions and fermions but it is still a complet description, right? Because once the Lagrangian Densities are added, one obtains the following \"Complete Lagrangian Density\": $${{{\\cal L}}_{\\operatorname{complete}}} = - \\frac{1}{4}{H^{\\mu \\nu \\rho }}{H_{\\mu \\nu \\rho }} + i\\hbar {c_0}\\bar \\psi \\not \\nabla \\psi + {c_0}\\bar \\psi \\phi \\psi + \\operatorname{h.c.} + {\\left\\| {\\not \\nabla \\phi } \\right\\|^2} - U\\left( \\phi \\right){\\rm{ }}+\\Re \\left( {\\frac{1}{{4\\kappa }}\\mbox{}^ \\pm\\Sigma _{IJ}^\\mu {{\\rm{ }}^ \\pm }F_{IJ}^\\mu} \\right) $$"}
{"id":"131905","title":"Am I right in saying that $gmm\/r$ is motion due to potential energy?","text":"I was watching a video about the swarzchild radius and it said that potential energy is $gmm\/r$. This cannot be right though because potential energy goes up with distance not down. I'm assuming he meant motion due to potential energy. Also what is the intuition behind the equation $gmm\/r$?"}
{"id":"66164","title":"Stringy corrections of Einstein's vacuum field equations","text":"From string theory, the vacuum field equations obtain correction of the order $O[\\alpha'R]^n$ such that they can be written as $$ R_{\\alpha\\beta} -\\frac{1}{2}g_{\\alpha\\beta}R + O[\\alpha'R] = 0 $$ where $\\alpha' = \\frac{1}{2\\pi T_s}$, when including just the first order term for example. What is the physical interpretation of these corrections, what do they look like more explicitely, and how can they be obtained?"}
{"id":"90045","title":"maths question on levers","text":"> Someone holds a 2kg weight in their hand at a distance of 35cm from the > elbow (fulcrum) and the downward force (load) due to the weight is 20 > Newtons. Calculate the effort in Newtons that the muscles of the forearm > must apply at a distance of 5cm from the fulcrum to hold the weight up. Show > calculations. Anyone any ideas please, I kinda grasp the concept but am unsure of how to show workings"}
{"id":"87807","title":"Decomposition of two particle wavefunction into product of single-particle wavefunctions","text":"Suppose you prepare a two-particle system such that $\\Psi(\\vec{r}_1,\\vec{r}_2, t_0) = \\Psi_1(\\vec{r}_1, t_0)\\Psi_2(\\vec{r}_2, t_0)$. So then, initially $\\Psi(\\vec{r}_1,\\vec{r}_2,t_0) - \\Psi_1(\\vec{r}_1, t_0)\\Psi_2(\\vec{r}_2, t_0) = 0$ But now you let the system evolve in time and $\\Psi(\\vec{r}_1,\\vec{r}_2, t)$ can now no longer be decomposed into a product of two single-particle wavefunctions. What then does, physically, does $\\Psi(\\vec{r}_1,\\vec{r}_2, t) - \\Psi_1(\\vec{r}_1, t)\\Psi_2(\\vec{r}_2, t)$ represent?"}
{"id":"29615","title":"Entropy increase and end of the universe","text":"While taking thermodynamics our chemistry teacher told us that entropy is increasing in day by day (as per second law of thermodynamics), and when it reaches its maximum the end of the world will occur. I didn't see such an argument in any of the science books, is there any probablity for this to be true?"}
{"id":"29614","title":"Rotationally invariant body and principal axis","text":"Suppose a rigid body is invariant under a rotation around an axis $\\mathsf{A}$ by a given angle $0 \\leq \\alpha_0 < 2\\pi$ (and also every multiple of $\\alpha_0$). Is it true that in this case the axis $\\mathsf{A}$ is a principal axis of the rigid body? If so, how to prove it? Do you have any references for a proof?"}
{"id":"73897","title":"Full refraction in fibre optics","text":"Well in a problem I had to calculate the maximum amount of \"reflections\" in a glass fibre optic pipe (index of refraction = 1.3, width of 20 micrometer and length of 1 meter). I am a bit blocked on how to calculate this. I considered first the critical angle $sin(\\theta_{cr}) = \\frac{1}{n}$ Which came out to be 50 degrees. I thought: well at critical angle the whole light beam is reflected. $\\theta_i = \\theta_o = \\theta_{cr}$. And the length of each \"reflection\" would simply be $\\frac{d}{\\tan(\\theta_{cr})}$. Resulting in $\\frac{l}{d}$ reflections in the pipe. However this came a bit over 60 000 reflections. While the answer should be around 42 000. What am I missing here?"}
{"id":"73899","title":"Does tritium hydride exhibit measurable spontaneous fusion via proton tunneling?","text":"In a fascinating 30 June 2013 article in _Nature Chemistry_ , researchers from the University of Leeds found that when molecules of hydroxyl ( **OH** , a fairly stable radical) and methanol ( **CH$_3$OH** ) are cold enough to adhere to each other in deep space instead of bouncing apart, they combine to form methoxy molecules ( **CH$_3$O** ) and water at rates hundreds of times higher than at room temperature. The reason for this unexpected increase in reactivity is both elegant and simple: By being cold enough to cling together loosely for long periods, the integral of very low quantum tunneling rates becomes large enough to enable appreciable rates of combination of the two molecules, despite the reaction having a high classical energy barrier. Now what I find interesting is that there is a simple equivalent of this \"let's stay close for a while\" situation in nuclear chemistry. Specifically, it is the formation of an ordinary chemical molecule of two nuclei that have a high energy potential for fusion. The chemical bond becomes the analog of the gentle \"let's say close\" binding of cold hydroxyl and methanol molecules, and the nuclear fusion reaction becomes the analog of the exothermic recombination of the two molecules, albeit with an enormously higher energy barrier. This analogy leads to only two or three interesting candidates: **TH** , tritium hydride; **TD** , tritium deuteride; and **T$_2$** , a molecule of two tritons. **TH** would seem by far the most interesting, since the probabilities for protons to tunnel over atomic distances are orders of magnitude higher than those of deuterons or tritons. The details will be complicated by the larger fusion cross sections of **D-T** and **T-T** in comparison to **H-T** , however. So my question: Has anyone ever specifically looked for this effect -- that is, for higher rates of spontaneous fusion in **TH** that can be attributed to increased tunneling probabilities due to the proton being kept at an atomic distance from the triton for an indefinitely long time? I did a cursory online-only look for relevant articles, without much success. Nonetheless, my best guess is that spontaneous **TH** fusion rates were likely calculated out to the $n^{th}$ degree decades ago, and likely experimentally verified to the same degree. After all, everything in nuclear chemistry is about tunneling probabilities. Nonetheless, the close analogy of long-term atomic-distance binding of nucleons to this more recent finding on increased tunneling between cold-bound molecules is intriguing. I was curious whether anyone out there is familiar with the topic and can talk about it in this forum."}
{"id":"25947","title":"Identifying objects, for dummies","text":"If I'm a tyro and have a latitude, longitude, time, date, height from the horizon, and compass direction, what means can I use for identifying what I see there? Realize that, as a tyro, I don't have any astronomy books or programs, and don't want to buy any; I might download a free program for my occasional inquiries about the sky, but would prefer a Web site."}
{"id":"3834","title":"Generated wavelength of free electron laser","text":"Could you please help me understand how one can measure output frequency of free electron laser **(provided that we know size of magnetic domains and electron energy)**? This should be a function of magnetic domains size & electron velocity? But electron velocity should drop due to photon emission - does that means that FEL have a \"wideband\" spectrum? Why military users focus on Röntgen radiation while UV in air transparancy window might have less air absorption? Bold is still unanswered :-)"}
{"id":"29949","title":"How were the heavy elements from iron to uranium made?","text":"> **Possible Duplicate:**   >  Age of the Earth and the star that preceded the Sun How were the heavy elements from iron to uranium made? References: http:\/\/www.phy.anl.gov\/accelerator_rd\/index.html Review of proposed Tech: http:\/\/www.annualreviews.org\/doi\/abs\/10.1146\/annurev.nucl.55.090704.151604"}
{"id":"76973","title":"Do bubble universes have a centre?","text":"This question is motivated by Big Bang Physics\/Cosmology. If eternal inflation is a good description of the universe and we live in a bubble universe then presumably our bubble nucleated at a spacetime point. Does this mean it is (approximately) a finite hypersphere and therefore has a centre?"}
{"id":"86686","title":"Assuming that the Cosmological Principle is correct, does this imply that the universe possess an empirically privileged reference frame?","text":"OK...before everyone blasts this with references to the relativistic invariance of the physical laws, time dilation, etc let me add some context. Also, I am an _amateur_ with an interest in physics, so I don't know the details of the physics. However, I have read enough about the cosmological principle and cosmic time to develop this \"confusion\" . Based on evidence gathered to-date, it appears that the universe is both isotropic and homogenous to a high degree, as evidenced by observations and the usefulness of the Freedman-Lemaitre-Robertson-Walker (FLRW) metric. This principle, when combined with the FLRW metric, allows space-time to be divided into non-intersecting slices and therefore establish a \"cosmic time.\" Now, the existance of such a time in no way suggests that it is the \"right\" time or a true \"now\", hence violating relativity -- although some presentist philosophers (e.g., W.L. Craig) have tried to make this argument, but it is not generally accepted in philosophical circles. However, this leaves me confused about an apparent disconnect between relativity and the cosmological principle: Isotropy only holds if we are at rest relative to the cosmic microwave background radiation (universal rest frame), o\/w anisotropy is present - yet each reference frame is supposed to allow equally valid observations. How can these two be reconciled when most reference frames would lead us to conclude that the universe is NOT isotropic? It seems that only by appealing to the idea of being at rest relative to \"universal rest frame\" can we explain away any discrepancies from isotropy as due to \"peculiar motion\". However, doesn't this give this \"universal rest frame\" and its associated time an _empirically_ privileged status, even though physical laws work just fine in every reference frame? For example (pardon any abuse of astronomy): if look off in some part of space and see only quasars, and then in another part of space and see only brown dwarfs, but I measure both as apparently the same distance from me, then can I conclude that we are in motion, since otherwise we would have an empirical contradiction (kind of like finding dinosaur and human remains in the same strata)? Any help on where I am going wrong would be helpful. Intuitively, I don't think there should be a way to establish absolute ordering between space-like separated events, but the above reasoning suggests otherwise."}
{"id":"35333","title":"can hydrogen stay frozen in vacuum?","text":"I've look into the hydrogen state diagram, and it seems that it can be frozen under pressure. > **Question:** Does this mean that hydrogen cannot be kept frozen in a vacuum > chamber?"}
{"id":"100175","title":"Different directions of frictional force when objects are rolling","text":"My textbook has two instances of rolling bodies (smooth rolling). In the first, the body is rolling on the horizontal floor with some acceleration of its centre of mass. In this case, the book says that the friction will act in the direction the ball is accelerating. In the other instance, the ball is rolling down an incline. In this case, the book says that the force of friction acts opposite to $mg\\sin\\theta$, the component of the gravitational force parallel to the incline. I don't understand the difference. I mean, in both the cases, the ball experiences an acceleration due to a force. Why does the direction of friction change? For me the second case was more intuitive. The friction and gravity both induce torques of the same sign on the body. But I'm lost on the first case. The force that produces the acceleration and the frictional force clearly induce opposite torques which should cancel. Or atleast cancel partially. Also, my intuition says that for the first case, friction should act opposite to the direction of acceleration. I tried to get answers on the web, but I couldn't find anything that was explained in a lucid manner. Some websites mentioned that the direction of friction is affected if the force is applied at the axle. I don't see why that should be."}
{"id":"35338","title":"Darwin term and Zitterbewegung","text":"I've noticed that in the discussion of the fine structure of Hydrogen atom standard QM texts claim that the Darwin term, which corrects energy of $\\ell=0$ (or $s$-) states only, is related to the zitterbewegung of the relativistic electron. I'm quite familiar with the zitterbewegung phenomenon, but it is not clear to me how it leads to the correction to the Hamiltonian in the form: $$H_\\text{Darwin}=\\frac{\\hbar^2}{8m^2c^2}\\nabla^2 V$$ Furthermore, this article: http:\/\/arxiv.org\/abs\/atom-ph\/9607005 claims that the Darwin term has its origin in the spin-orbit coupling rather than zitterbewegung. Who is right? Question:   1. What is the physical origin of the Darwin term?   2. What is the history behind the Darwin term?"}
{"id":"55243","title":"The definition of Density of States","text":"The density of states (DOS) is generally defined as $D(E)=\\frac{d\\Omega(E)}{dE}$, where $\\Omega(E)$ is the number of states. But why DOS can also be defined using delta function, as $$D(E)~=~\\sum\\limits_{n} \\int \\frac{d^3k}{(2\\pi)^3}\\delta(E-\\epsilon_n(\\mathbf{k}))?$$"}
{"id":"22627","title":"In What Frame of Reference does the Special Theory of Relativity Operate?","text":"> **Possible Duplicate:**   >  Time Dilation - How does it know which Frame of Reference to age slower? This has bugged me for years. According to the theory of relativity, the faster an object moves, the slower time goes for that object, relative to a stationary observer. However, it is my understanding that motion is necessarily relative to a frame of reference. If you have two objects in space moving away from each other at a combined velocity of 20mph, the rate at which each is moving is totally conditional on what frame of reference you specify.   1. If object A is your frame of reference, than object A is moving at 0mph, and object B is moving at 20mph.   2. If object B is your frame of reference, the opposite is true.   3. If something else is your frame of reference, then the two objects will have some other two velocities, which, when summed, equal 20mph. If this is true, how does physics know how much to slow down time for each object? It's as if they don't actually have set velocities."}
{"id":"131273","title":"Deriving Feynman rules from a Lagrangian for vertex factors for \"more complicated\" interactions","text":"I am trying to derive Feynman rules from a given Lagrangian and I got stuck on some vertex factors. What for example is the vertex factor that corresponds to the four-scalar interaction that is decribed by the following Lagrangian? \\begin{equation} L = -\\frac{1}{4} g_3^2 \\phi^\\dagger \\lambda^a \\phi \\chi^\\dagger \\lambda^a \\chi + \\frac{2}{9} g_1^2 \\phi^\\dagger \\phi \\chi^\\dagger \\chi \\,, \\end{equation} where $\\phi,\\chi$ are complex scalar (color triplet) fields, $\\lambda^a$ are the Gell-Mann matrices, and $g_1,g_3$ are the coupling constants corresponding to $\\text{U}(1)$ and $\\text{SU}(3)$ respectively. If we would have only had the second term here, say, then the vertex factor would simply be found by \"dropping\" the fields and multiplying by $i$. But now there are two terms contributing, and in the first term the Gell-Mann matrices even mix the color components of the scalar triplets. So how do I proceed in this case? And could anyone give me some general strategies on how to derive vertex factors for \"complicated\" interactions? For example, I also find it tricky to get the sign right if there is a derivative in an interaction. (If you are interested in the context of this Lagrangian, for $\\phi = \\tilde{u}_R$ and $\\chi = \\tilde{d}_R$ this Lagrangian describes the interaction between two up squarks and two down squarks in a supersymmetric theory.)"}
{"id":"105160","title":"Radiofrequency attenuation in cold plasma","text":"The missile plume can be considered a cold plasma. If a radio signal passes across the plume it's attenuated. Obviously the attenuation depends on the frequency of the signal. Where can I find some information about the behaviour of the RF across the cold plasma? Thanks"}
{"id":"75991","title":"Solution Maxwell's equations cylinder","text":"I face some trouble solving Maxwell's equations inside a cylinder with perfect conductor boundaries (in 3D) ? We work with cylindrical coordinates $(r, \\phi, z)$ and we make the assumption that fields have a sinusoidal \"$e^{i\\omega t}$\" time dependence. Note that we have a $\\phi$ symmetry. First, and in any coordinates system, by taking the rotational and injecting one equation in the other we reduce Maxwell's equations to the following, $$ \\nabla\\times\\nabla\\times E = -\\partial_t^2 E = \\omega^2 E $$ In vacuum, from the $curl curl$ identity, it leads, $$ \\nabla\\times\\nabla\\times E = \\nabla(\\nabla . E) - \\nabla^2 E = - \\nabla^2 E $$ Where $- \\nabla^2 E$ is the laplacian operator applied to each coordinate. Now, in cylindrical coordinates, we can only compute the $z-$coordinate since, in this case we get the wave equation, $$ \\nabla^2 E_z = \\omega^2 E_z $$ For the other coordinates, the change of coordinates introduce other terms such that (for the $\\phi-$ coordin. ate)$\\frac{E_r}{r^2} - \\frac{2}{r^2}\\frac{\\partial E_\\phi}{\\partial\\phi}$. Then, a fastidious step consists in performing a separation of variable which leads us quite easily to the solution for every separated variable and also to the Bessel differential equation which brings its solution, the Bessel function. Together with boundary conditions we can get the solution according to $z$ but what about the other coordinates ?"}
{"id":"75990","title":"Doubt in law of mutual interaction","text":"Book: Classical mechanics (textbook) by Douglas Gregory (cambridge publications) Law of mutual interaction states that when two particle (let it be P1 and P2) interacts, the particle (P1) induces an instantaneous acceleration (a21) on particle P2 and the particle P2 induces an instantaneous acceleration (a12) on particle (P1). If the (inertial)masses of the particles are same, then the magnitude of acceleration be the same, and the ratios of acceleration will be constant ( for this case it is 1)(consistency relation) That is what Newton's third law says. > My question is, for different (inertial)masses the ratio will be constant ( > but not unity) ( it does not satisfy consistency relation) Am i right? > > If yes My question is consistency relation is important in classical > mechanics?"}
{"id":"77620","title":"What restrictions on time boundary conditions does it have to use Fourier transform to solve wave equation?","text":"The wave equation can be solved using Fourier transform, by assuming a solution of the form of $$\\mathbf{E}(x,y,z,t)~=~\\mathbf{E}(x,y,z)e^{j\\omega t}$$ and then reducing the equation to the Helmholtz equation.   * What are the presumed _restrictions_ on the solution, when solving the equation this way? (e.g., on _time_ boundary condition) I mean can this method give the most general solution (given some boundary conditions)? What _features_ does the solution obtained this way have?   * Does it have any difference with solutions obtained using Laplace transform? (The very same above questions, for Laplace transform.)"}
{"id":"60192","title":"What would be the effect of an excess of up quarks on stellar formation?","text":"Suppose you had 80% up quarks, and only 20% down quarks. How would this affect stellar formation?"}
{"id":"38420","title":"What is crystal field anisotropy or effect ? It forces the magnetic moment to point in particular local direction.. ","text":"Can you give a basic explanation of what is crystal field anisotropy ? What is the reason to arise ? In spin ice it forces the dipoles to point in the local 111 direction. For partially filled rare earth atoms hund rule requires S and L max. This leaves (2s +1)(2l+1) degeneracy which is partially lifted from the LS coupling. When inserted in anisotropic field inside crystal the expectation value , or mean value of L is 0 = 0 and the L is quenched leaving only S so is forced to point in some local direction, but this is if the field removes the deneracy ? Is it this or it is much more complicated ?"}
{"id":"10501","title":"Lightning and nuclear fusion","text":"I'm going to be brief, I just saw a Discovery Channel show that showed a lot of interesting phenomena around lightning (like elves, how cool is that(!)), and got me wondering. 1) Thinking of lightning as a purely mechanical phenomenon, I would think the elves are the \"other side of the momentum balance\". What I mean is this: somewhere in some cloud formation, an event happens that triggers a lightning flash. This means a ton of electrons (and other, associated particles) that start a _very_ fast journey to the surface of the Earth. Their momentum must be balanced by particles going in the opposite direction, hence elves. Am I right? 2) Taking this further, is it possible that the \"trigger\" for the lightning flash could be atomic\/molecular fusion forced by extremely high electrical fields, that may only exist for a nanosecond, causing a ton of energy to be transferred to the electrons around the atoms\/molecules, and we have lightning. I would then think of the energy necessary to force the fusion to occur as a quantum fluctuation as in $$\\Delta E \\Delta t \\leq \\hbar$$ 3) Is every arc caused by a particle collision, somewhat like in a particle accelerator? It seems logical to me: there are more than enough particles in the air to collide with, and all the light could well be some form of Brehm or Cherenkov radiation. The question I seem to be asking is where to find good scientific theory\/information about lightning. Some say we know a lot about it, but I haven't found any good papers explaining it. I have a pretty good background in physics (1st year Master student) and am not afraid of serious literature. Thanks!"}
{"id":"4532","title":"Strings with negative pressure","text":"This question is inspired by the following comment: > the strings in string theory are relativistic and on a large enough piece of > world sheet, the internal SO(1,1) Lorentz symmetry is preserved. That's why > a string carries not only an energy density ρ but also a negative pressure > p=−ρ in the direction along the string. by Lubos at the end of his answer to the question \"What is tension in String Theory?\". I don't see how having a $SO(1,1)$ symmetry for the worldsheet leads to a negative pressure. I have the following questions:   1. Why does a string carry negative pressure?   2. Can a gas of strings then be treated as a substance which has a negative equation of state: $$ w = \\frac{p}{\\rho} = -1 $$   3. If this is possible then the next natural question is: what are the implications for the cosmological constant problem? **Background material to give some context for questions 2 and 3:** One part of the cosmological constant problem is an explanation of what form of matter could seed cosmological expansion. A cosmological constant term $\\Lambda$ in the action for GR is equivalent to having a medium which satisfies the negative equation of state (relation between density and pressure). One simple example of matter with a negative equation of state is a homogenous, isotropic scalar field [for details see any book on cosmology with a chapter on inflation]. If strings carry negative pressure and if a string gas can be treated as matter with $w\\sim-1$ then one would have a far more natural alternative to a scalar field."}
{"id":"104128","title":"How to get discretization coefficients of matrix $A$ in Finite Volume Method (FVM)?","text":"First we have Discretization of the Transport Equation $$ \\frac{\\partial \\rho \\phi}{\\partial t} + \\nabla(\\rho U \\phi) - \\nabla (\\rho \\Gamma_\\phi \\nabla \\phi) = S_\\phi (\\phi) $$ In Finite Volume Method it looks like: $$ \\int_t^{t+\\Delta t} \\left[ \\frac{\\partial}{\\partial t} \\int_{V_p} \\rho \\phi dV + \\int_{V_p} \\nabla \\cdot (\\rho U\\phi)dV - \\int_{V_p} \\nabla \\cdot (\\rho \\Gamma_\\phi \\nabla \\phi)dV \\right]dt = \\int_t^{t+\\Delta t} \\left(\\int_{V_p}S_\\phi(\\phi)dV \\right)dt $$ and after transformations: $$ \\int_t^{t+\\Delta t} \\left[ \\left( \\frac{\\partial \\rho \\phi}{\\partial t} \\right)_p V_p + \\sum_f F\\phi_f - \\sum_f(\\rho \\Gamma_\\phi)_f S.(\\nabla\\phi)_f \\right]dt=\\int_t^{t+\\Delta t}(SuV_p+S_pV_p\\phi_p)dt $$ and time discretization: $$ \\frac{\\rho_p\\phi_p^n-\\rho_p\\phi_p^o}{\\Delta t}V_p+ \\frac{1}{2}\\sum_f F\\phi_f^n-\\frac{1}{2}\\sum_f(\\rho\\Gamma_\\phi)_f S.(\\nabla\\phi)^n_f+ \\frac{1}{2}\\sum_f F\\phi_f^o-\\frac{1}{2}\\sum_f(\\rho\\Gamma_\\phi)_f S.(\\nabla\\phi)^o_f = SuV_p + \\frac{1}{2}S_pV_p\\phi_p^n+\\frac{1}{2}S_pV_p\\phi_p^o $$ For every cell we can make equation: $$ a_p\\phi^n_p+\\sum_N a_N \\phi_N^n = R_p $$ But how we can get elements of matrix $A$ if we don't know $\\phi$? $$ [A][\\phi]=[R] $$ Here Hrvoje Jasak wrote that every coefficient $ a_p $ includes the contribution from temporal derivative, convection and diffusion terms. But what formula of $a_p = ...$? http:\/\/powerlab.fsb.hr\/ped\/kturbo\/OpenFOAM\/docs\/HrvojeJasakPhD.pdf"}
{"id":"128040","title":"Calculate loudness of sound: why am I getting contradictory answers?","text":"I know of events that are happening about 45 KM away from me which are said to be 210 or 213 dB at 75 meters distance from multiple sources. I think that I can hear them, so I did the obvious: $$ \\frac{I_2}{I_1} = \\left( \\frac{d_1}{d_2} \\right)^2$$ $$ I_2 = I_1 * \\left( \\frac{d_1}{d_2} \\right)^2\\text{ dB}$$ For my values: $$ I_2 = 213 \\left( \\frac{75}{45000} \\right)^2\\text{ dB}$$ $$ I_2 = 213 \\left( \\frac{1}{600} \\right)^2\\text{ dB}$$ $$ I_2 = \\frac{213}{360000}\\text{ dB}$$ $$ I_2 \\approx 0.6 \\cdot 10^{-3}\\text{ dB}$$ Well, **0.6 e-3 dB** I should not be able to hear! However, going by the rule of thumb \"twice the distance, minus three decibels\": $$ 75 * 2^x = 45000 $$ $$ 2^x = 600 $$ $$ x = \\log_2(600) $$ $$ x \\approx 9.2 $$ $$ 213 - 3 \\cdot 9.2 = 213 - 27.6 = 185.4 $$ The value of **184 dB** seems too loud, and additionally does not agree with the previous method. Lastly I tried tool for estimating Sound Levels With the Inverse Square Law from the usually-terrific HyperPhysics website. This tool gave a value of **157 decibels**! Another on-line tool gives the same result. **Which result should I trust?**"}
{"id":"83919","title":"does light experience time?","text":"If we positioned a mirror 1 light year away from earth and shot a particle of light at the mirror so that it would reflect and come back to earth, how long would it take for us to receive that particle of light back to earth and if I jumped on this particle of light for the trip how would it effect how much time has gone by for me and the people waiting for me back on earth"}
{"id":"7069","title":"Double-slit experiment with alternating on-off switch","text":"Suppose we perform a double-slit experiment with a detector placed at a position of minimum intensity (maximum destructive interference), off-center where the path lengths differ by half a wavelength. The light source is alternately turned on and off (or blocked and unblocked near the source) and the intensity over time is recorded. I interpret the uncertainty principle to mean that there will be a peak in intensity at the times when the switch is flipped (whether on-to-off or off-to-on). i.e., it will look something like this (in ASCII art):                __________'-----'__________'-----'__________      Is this correct? I have had trouble convincingly explaining my reasons for thinking so. What will be the measured intensity over time and why?"}
{"id":"7068","title":"How much radiation exposure in the US was caused by the 105 nuke tests in the Pacific?","text":"Between 1947 and 1962 the US conducted 105 tests of nuclear weapons in the \"Pacific Proving Grounds\". I'm wondering how much radiation exposure resulted on the west coast of the US. These were part of the 1056 nuclear bombs that the US has ignited over the years (most underground, but two notably in Japanese cities). So how much radiation exposure in the US was caused by the 105 nuke tests in the Pacific? Should the inhabitants of the West coast have taken iodine pills? I'd also like to know how much fallout these bombs produced, as compared to the reactor steam releases in Japan, 2011."}
{"id":"7066","title":"Rigid body dynamics joints","text":"I can't seem to find any info on connected rigid bodies by a joint. Can someone explain the basics to me? I'm trying to do a little research to find out how feasible it would be to implement 3d ragdoll physics for my first person shooter game."}
{"id":"76509","title":"What is the Units for Thermal conductivity?","text":"What are the units for thermal conductivity and why?"}
{"id":"15452","title":"why does what get pushed away when centripetal acceleration is towards the center","text":"If centripetal acceleration is towards the center, then why - when you spin a bucket of water (a classic demonstration) - does the water not get pushed out but rather stays in the bucket without spilling?"}
{"id":"8151","title":"What is the covariant derivative in mathematician's language?","text":"In mathematics, we talk about tangent vectors and cotangent vectors on a manifold at each point, and vector fields and cotangent vector fields (also known as differential one-forms). When we talk about tensor fields, we mean differentiable sections of some tensor power of the tangent or cotangent bundle (or a combination). There are various natural differentiation operations, such as the exterior derivative of anti-symmetric covariant tensor fields, or the Lie derivative of two vector fields. These have nice coordinate-free definitions. In physics, there is talk of \"covariant derivatives\" of tensor fields, whose resulting objects are different kind of tensor fields. I was wondering, what is the abstract interpretation of the general notion of a covariant derivative in terms of (tensor products of) tangent vectors and vector fields."}
{"id":"86736","title":"Is there a better, faster way to do this projectile motion question?","text":"The question is > In a combat exercise, a mortar at M is required to hit a target at O, which > is taking cover 25 m behind a structure of negligible width 10 m tall. This > mortar can only fire at an angle of 45 degrees to the horizontal, but can > fire shells of any velocity. Find the minimum initial velocity required to > hit the target. ![diagram](http:\/\/i.imgur.com\/yoTXyv8.png) I solved it as follows. Forming a parabola with $ r_v $ against $ r_h $: $ r_v = u_vt + \\frac{1}{2}a_vt^{2} $ (1) $ r_h = u_ht + \\frac{1}{2}a_ht^{2} $ but $ a_h = 0 $ so $ r_h = u_ht $ $ t = \\frac{r_h}{u_h} $ (2) Substituting (2) into (1): $ r_v = u_v\\frac{r_h}{u_h} + \\frac{1}{2}a_v\\frac{r_h^{2}}{u_h^{2}} $ (3) and because the angle of inclination is 45° $ u_v = u_h = \\frac{u}{\\sqrt{2}} $ (4) From (4) and (3): $ r_v = r_h^{2}\\frac{a}{u^{2}} + r_h $ (5) Let the distance between the mortar and the building be $ d $. Then when $ r_h = d + 25 $, $ r_v = 0 $. (6) From (5) and (6): $ 0 = (d + 25)\\frac{a}{u^{2}} + 1 $ so $ u^2 = -a(d + 25) $ (7) Substituting (7) into (5): $ r_v = -r_h^{2}\\frac{1}{(d + 25)} + r_h $ (8) We also know that to clear the building, when $ r_h = d $, $ r_v > 10 $. (9) From (9) and (8): $ 10 < -d^{2} \\frac{1}{(d + 25)} + d $ After simplifying... ($ d + 25 $ is positive) $ d > \\frac{50}{3} $ (10) Rearranging (7): $ d = -\\frac{u^{2}}{a} - 25 $ (11) And then from (10) and (11) and with $ a = -9.8 $: $ \\frac{50}{3} < \\frac{u^{2}}{9.8} - 25 $ Simplifying, and with the knowledge that $ u > 0 $: $ u > 20.2073... $ So the minimum initial velocity required to hit the target is **20 m\/s** (2 s. f.). Huzzah! My question is: **is there a faster way to solve the problem?**"}
{"id":"107357","title":"Help understanding the solution to a problem regarding kinetic energy of a group of point charges","text":"The problem provided by my professor goes as follows: \"Now consider a situation in which all charges are equal to q and they simultaneously become \"unglued\". What speed will each charge have when a hexagonal configuration has doubled in size (each side has a length). The work done was found to be $$U=\\frac{kq^2}a\\left(\\frac{5}2+\\frac{2}{\\sqrt2}\\right)$$ So far I have done: $$\\Delta K=-\\Delta U$$ $$K_f=U_i-U_f$$ $$6\\cdot\\left(\\frac{1}2mv^2\\right)= k\\frac{q^2}a\\left(\\frac{5}2+\\frac{2}{\\sqrt3}\\right)-\\frac{kq^2}{2a}\\left(\\frac{5}2+\\frac{2}{\\sqrt3}\\right)$$ However, the solution is: $$6\\cdot\\left(\\frac12mv^2\\right)=k\\frac{q^2}a\\left(\\frac{15}2+\\frac{6}{\\sqrt2}\\right)-\\frac{kq^2}{2a}\\left(\\frac{15}2+\\frac{6}{\\sqrt3}\\right)$$ Can anyone explain why?"}
{"id":"88821","title":"Is it possible to build an optical system that increases the perceived surface brightness?","text":"So is it possbile to build a system from lenses and mirrors that can make faint gas nebulas brighter or can be used as nightvision? If you increase the size of the aperture of a telescope it will collect more light, but the exit pupil will be also bigger, so not all light will enter the eye. In order to direct all light into the eye you'll need to shrink the exit pupil and you'll need a stronger eyepiece. But this will increase the magnification too and the collected light will spread on a larger image so the surface brightness will remain the same. Is it possible to work this limitation around?"}
{"id":"60185","title":"Quantum mechanics - how can the energy be complex?","text":"In section 134 of Vol. 3 (Quantum Mechanics), Landau and Lifshitz make the energy complex in order to describe a particle that can decay: $E = E_0 - \\frac{1}{2}i \\Gamma$ The propagator $U(t) = \\exp(-i H t)$ then makes the wavefunction die exponentially with time. But also, $H$ is non-Hermitian. My question: Do we have to modify the basic postulates of quantum mechanics (as described by Shankar, say, or the earlier sections of Landau & Lifshitz) to describe unstable particles?"}
{"id":"2679","title":"What temperature can you attain with a solar furnace?","text":"A solar furnace is a device that concentrates the sun's light on a small point to heat it up to high temperature. One can imagine that in the limit of being completely surrounded by mirrors, your entire $4\\pi$ solid angle will look like the surface of the sun, at about 6000K. The target will then heat up to 6000K and start to radiate as a blackbody, reaching thermal equilibrium with the sun. The question is: is there any way to surpass this temperature, perhaps by filtering the light to make it look like a BB spectrum at higher temp, then concentrating it back on the target?"}
{"id":"38168","title":"when an object moves downward, is its height negative?","text":"the question is: A ball is thrown directly downward with an initial speed of 8.00m\/s from a height of 30.0m. After what time interval does it strike the ground. so i went through the problem and got the answer 1.79 sec. but after reviewing some notes from my teacher, I got the idea that in equations where an object is dropped down, the height has to be counted as negative. so I went through the problem again using -30m, but now my answer doesn't check out when plugged back in to x - x0 = V0 (t) + 1\/2 at^2 Is there something I'm missing about this idea that height or distance is negative in drop-down questions?"}
{"id":"103360","title":"How to do this index notation differentiation?","text":"I am studying classical Maxwell fields and I am stuck on this differentiating part. How can I derive the result given below ? $$\\dfrac{\\partial}{\\partial(\\partial A_{\\mu}\/\\partial x_{\\nu})} \\left(2\\dfrac{\\partial A_{\\sigma}}{\\partial x_{\\lambda}}\\dfrac{\\partial A_{\\sigma}}{\\partial x_{\\lambda}}-2\\dfrac{\\partial A_{\\sigma}}{\\partial x_{\\lambda}}\\dfrac{\\partial A_{\\lambda}}{\\partial x_{\\sigma}}\\right)$$ The answer is $$4\\dfrac{\\partial A_{\\mu}}{\\partial x_{\\nu}}-4\\dfrac{\\partial A_{\\nu}}{\\partial x_{\\mu}}$$ where $A$ is vector potential and $x$ is four-vector."}
{"id":"103368","title":"What are \"parity considerations\" in deciding the form of the Hamiltonian?","text":"In \"introductory Quantum Optics\", by Gerry and Knight, the Jeynes model is considered. In this model of electron-EM field interaction the electron is approximated by a two state system ($\\lvert g\\rangle$ and $\\lvert e\\rangle$), and the form of the dipole operator $\\hat{d}$ is said to be constrained by parity consideration not to have on-diagonal terms: > Only the off-diagonal elements of the dipole operator are nonzero, since by > parity consideration $\\langle e\\rvert\\hat{d}\\lvert e\\rangle=0=\\langle > g\\rvert\\hat{d}\\lvert g\\rangle$. Why? What does parity have to do with it?"}
{"id":"19175","title":"Is the axial gauge with a $\\xi$ term useful in Yang-Mills theory?","text":"i) Do people use axial gauge with a $\\xi$ term? When $\\xi\\neq 0$, ghosts do not decouple, but maybe it's still useful? ii) Is it proved that the term $\\frac 1 {2\\xi}(n.A)^2$ in the Lagrangian does not renormalize, for $\\xi=0$ and $\\xi\\neq 0$?"}
{"id":"19174","title":"Understanding the relationship between electricity and magnetism","text":"I keep on hearing that magnetism is just another form of electricity and vice versa. If that's the case why can't we use magnets as batteries, and why aren't my batteries magnetic?"}
{"id":"60339","title":"Frequency of a Tuning Fork","text":"**Question:** Which of the following affect the frequency of a tuning fork?   * Tine stiffness   * Tine length   * The force with which it's struck   * Density of the surrounding air   * Temperature of the surrounding air **Answer Attempt:** Based on the formula for the frequency, I know that tine stiffness (or density) affects it, and so does the tine length. I believe the temperature and density of air can have a slight affect as well. What about the force with which it's struck?"}
{"id":"94222","title":"If a material is built to handle tension, would removing the tension damage it?","text":"If an object is designed to cope with large forces such as tension, would removing these forces risk damaging the object? For example: The neck of a guitar is built to handle the tension of steel strings (~800 Newtons), if you removed\/reduced the tension (removed the strings) for a long period of time would this risk damaging the guitar neck?"}
{"id":"116430","title":"Stopping potential in the photoelectric effect, collector work function","text":"In this question I am talking about the following situation: ![enter image description here](http:\/\/i.stack.imgur.com\/Q4brH.gif) Now, I know that the max kinetic energy of the electrons emitted is $KE_{max} = h\\nu - e\\phi_{em}$ where $\\phi_{em}$ is the work function of the emitter electrode (on the left in the diagram). And my lecturer agrees with that, but he tells us that the stopping potential $V_0$ can be found using $eV_0 = h\\nu - e\\phi_{col}$ where $\\phi_{col}$ is the work function of the collector electrode (on the right in the diagram). The emitter and collector electrodes are made from different metals. What I don't understand is why the stopping potential doesn't depend on the kinetic energy of the emitted electrons. **EDIT** I have attached the slide from the lecture course ![Lecture slide](http:\/\/i.stack.imgur.com\/vJ2aF.png)"}
{"id":"94225","title":"Dependence of intensity of a LED on forward bias voltage","text":"Due to increase in forward bias voltage, the intensity of light increases but after a particular value the intensity decreases. Why?"}
{"id":"94226","title":"Fermi Energy and the Electric Potential","text":"In an extrinsic semiconductor the electric potential is: $$\\phi = \\frac{1}{q}(E_{\\mathrm{F}} - E_{\\mathrm{Fi}})$$ where $E_{\\mathrm{F}}$ is the Fermi energy, $E_{\\mathrm{Fi}}$ is the intrinsic Fermi energy, $q$ is the electron charge and $\\phi$ is the electric potential. I am not sure where this equation comes from. I understand why a potential will be created qualitatively, but where does this equation come from quantitatively? From what I understand , the expected value of the electron energy is the Fermi energy. You take $E_{\\mathrm{Fi}}$ as the reference for the electron potential. This is the equilibrium condition of an intrinsic semiconductor. If you dope the semiconductor, you introduce an imbalance and that causes the generation of an electric field, hence the new expected value $E_{\\mathrm{F}} - E_{\\mathrm{Fi}}$. Is that correct?"}
{"id":"128352","title":"From where does the sound come from when two charged objects meet in real life?","text":"I am sure all of us have played with rubbing things and producing static electricity and when I was charging my comb by rubbing it on my hair and watching it attracting a small piece of paper, I heard a very feeble noise just like an electric buzz sound. I could feel the electron \"cloud\" all around the comb and the paper it I could find no reason as to why a sound should be produced. From where is that sound coming from?"}
{"id":"65676","title":"Complex masses for Dirac and Weyl spinors","text":"I'm trying understand how to rotate Dirac fields to absorb complex phases in masses. I have a few related questions:   1. With Weyl spinors, I understand, $$ \\mathcal{L} = \\text{kinetic} + |M|e^{i\\theta}\\xi\\chi + \\textrm{h.c.} $$ The phase removed by separate left- and right-handed rotations, e.g. $\\xi \\to e^{-i\\theta\/2}\\xi$ and $\\chi \\to e^{-i\\theta\/2}\\chi$. These phases cancel in the kinetic terms. Is it correct that with Dirac spinors, $$ \\mathcal{L} = \\bar\\psi |M|e^{i\\theta\\gamma_5} \\psi = \\text{Re}(M)\\bar\\psi\\psi +i\\text{Im}(M)\\bar\\psi\\gamma_5\\psi $$ and the phase is removed by $\\psi \\to e^{-i\\theta\\gamma_5\/2}\\psi$? The appearance of the $\\gamma_5$ in the phase troubles me a little - I suppose this is telling us that Weyl spinors are a more suitable basis than Dirac spinors?   2. If the field is Majorana, $\\xi = \\chi$, and the field can still absorb a phase? I think I must be making a trivial mistake.  For example, Majorana neutrino fields cannot absorb phases, leading to extra CP violation. And in SUSY, the gaugino Majorana soft-breaking masses are e.g. $M_1e^{i\\theta}$. Can their phases be re-absorbed via a field redefinition? I don't think they can. So I must have a mistake."}
{"id":"65674","title":"Is particle entanglement a binary property?","text":"Is the particle entanglement a boolean property? That is, when we consider two given particles, is the answer to the question \"are they entangled\" always either \"yes\" or \"no\" (or, of course, \"we are not sure if it's yes or no\")? Is there such thing as partial entanglement?"}
{"id":"65672","title":"Is there a general systematic approach how to calculate the individual terms in an operator product expansion?","text":"Is there a general systematic procedure or approach to obtain the analytic functions $c_{ijk}$ as well as the corresponding operators $$A_i(z)$$ that appear in the operator product expansion (OPE) $$ A_i(z)A_j(w) = \\sum\\limits_k c_{ijk}(z-w)A_k(w) $$ for any given two operators $A_i(z)$ and $A_i(z)$? I have followd (a limited number of) examples of OPE calculations that involved things like expanding until some correlators for which the expression is known appear and power series expansions of derivatives of operators, but from this I was not able to discern the general way to go."}
{"id":"60487","title":"Hollow stone columns provide more support?","text":"In history class in elementary school I remember learning that the Greeks would build their stone columns hollow because they thought this provided more support. Is it true that a hollow column is stronger? Thanks!"}
{"id":"60485","title":"Why don't you see multiple images of an object?","text":"Consider the ray model of light. Let's say an object such as a pencil is illuminated, and consider one point on that pencil. Since there could be many rays of light bouncing off the same point on the pencil, one ray could hit the left side of your eye and another ray from the same point could hit the right side of that same eye. Why don't you see that point in 2 different places then? Thanks!"}
{"id":"59921","title":"Explanation that air drag is proportional to speed or square speed?","text":"A falling object with no initial velocity with mass $m$ is influenced by a gravitational force $g$ and the drag (air resistance) which is proportional to the object's speed. By Newton´s laws this can be written as:   1. $mg-kv=ma$ (for low speeds)    2. $mg-kv^2=ma$ (for high speeds).  I assume that $k$ is a positive constant that depends on the geometry of the object and the viscosity. But how can one explain that the air resistance is proportional to the velocity? And to the velocity squared in the second equation?"}
{"id":"71043","title":"Resisting force depends on velocity?","text":"Why does resisting force depend on velocity? I think there is no relation between resisting force and velocity of object. Please speak about it logically."}
{"id":"129214","title":"Confused about the theoretical origin of quadratic air drag","text":"Though the mathematical concepts underlying quadratic air drag are quite straightforward (a single variable differential, just like the linear drag equation), my text book (and online text books) completely ignore the reasoning behind why the \"drag\" at high speeds on a trajectory through a fluid is proportional to the velocity squared. I know that this could not just be an empirical result. Is there an intuitive reason for this (perhaps like momentum transfer or kinetic energy?). If possible, I would also like to ask for an intuitive reason behind the units $m^2\/s^2.$"}
{"id":"60480","title":"Does the mass point move?","text":"There is a question regarding basic physical understanding. Assume you have a mass point (or just a ball if you like) that is constrained on a line. You know that at $t=0$ its position is $0$, i.e., $x(t=0)=0$, same for its velocity, i.e., $\\dot{x}(t=0)=0$, its acceleration, $\\ddot{x}(t=0)=0$, its rate of change of acceleration, $\\dddot{x}(t=0)=0$, and so on. Mathematically, for the trajectory of the mass point one has \\begin{equation} \\left. \\frac{d^{n}x}{dt^n}\\right|_{t=0} = 0 \\textrm{ for } n \\in \\mathbb{N}_0\\mbox{.} \\end{equation} My physical intuition is that the mass point is not going to move because at the initial time it had no velocity, acceleration, rate of change of acceleration, and so on. But the mass point not moving means that $x(t) \\equiv 0$ since its initial position is also zero. However, it could be that the trajectory of the mass point is given by $x(t) = \\exp(-1\/t^2)$. This function, together with all its derivatives, is $0$ at $t=0$ but is not equivalent to zero. I know that this function is just not analytical at $t=0$. My question is about the physical understanding: How could it be that at a certain moment of time the mass point has neither velocity, nor acceleration, nor rate of change of acceleration, nor anything else but still moves?"}
{"id":"46065","title":"Does General Relativity encompass Special Relativity?","text":"Can all of the predictions made in Special Relativity (SR) also be made in General Relativity (GR)?"}
{"id":"80773","title":"The example of relativity of simultaneity given by Einstein","text":"I understand (supposedly) the mathematics concerning the relativity of simultaneity in Special Relativity, but I have a nagging question regarding the original example given by Einstein supporting it (I'm only disagreeing with this specific example, not the concept). It is normally given as a person on an embankment and a person on a train. There is a relative speed between them (usually presented as the train passing the embankment). Now, when both people are at the same x-position (x=0), there is a flash of light at x = +dx and x = -dx. The argument as I keep seeing it is that the person on the embankment will say that both flashes reach him at the same time, whereas the person in the train will say that the flash in front of him reaches him before the other because he was moving toward it, and thus the observers will disagree on the simultaneity of the flashes. But given that the flashes occurred at the same distance from each of them, the speed of light is constant in both frames, and either one can claim to be at rest, then won't they, according to SR, necessarily see the flashes as simultaneous (both flashes have to travel the same distance in both frames since at the time of emission, the sources of both flashes were equidistant from both observers). I agree that the person on the embankment will say that the person on the train shouldn't see them as simultaneous (and vice versa) since either observer will see the other moving relative to the sources, but in each of there own frames, they must see the flashes as being simultaneous shouldn't they? Am I just misunderstanding the example? Thanks."}
{"id":"53195","title":"How to solve fixed-fixed beam with finite difference method?","text":"What equations to use on this system to form a matrix $A$ with dimensions $[n,n]$ and load vector $q$ with dimension $[n]$ ? I am trying to get vertical displacement $w$. $$w = A^{-1}\\times q$$ Boundary conditions are as follows: $$w(o) = 0 $$ $$w(L) = 0 $$ $$\\phi(o) = 0$$ $$\\phi(L) = 0$$ It is becouse in any point of beam I can't make equation: $$d^2y\/dx^2*E*I=M=0$$ so I can't get the exact values of displacement. The problem is that everywhere I look for solution it is done on a beam with continous load over entire beam or with at least one joint and I have only half of the beam covered with continous load and no joints. From $d^4y\/dx^4*E*I=q=0$ again I have too many unknown values. ![Scheme](http:\/\/imgur.com\/YkIdCAI.jpg)"}
{"id":"53191","title":"Car tires wearing in dependence on breaking system","text":"Lets say I have two cars.   They are identical in every way, except that Car A has a normal breaking system, where most of the breaking power is inflicted on the front wheels, and some on the back, and Car B has a breaking system where all of the power is inflicted on the front wheels and none on the back. In the long run, in which car will the tires on the back wheels experience more wearing?   (For the sake of comparison, let's say we measure wear by the weight lost from the tires.)"}
{"id":"99225","title":"Calculating energy U from $\\partial U\/\\partial q$","text":"Imagine $N$ oscillators with only two possible energies, $\\epsilon_0$ and $ \\epsilon_1$, with $\\epsilon_1 > \\epsilon_0$. Taking $\\epsilon_0 = 0$ for now I showed $\\Omega(q\\epsilon_1) = \\frac{N!}{(N-q)!q!}$ and then $$\\frac{\\partial S}{\\partial q} = k \\log(N\/q - 1) $$ How can I use the above equation to show that $$U = N\\epsilon_1\\frac{e^{-\\epsilon_1\/(kT)}}{1+e^{-\\epsilon_1\/(kT)}} $$ I tried moving the $\\partial q$ over to the right, and then have $dS = dU\/T$, but i wasn't getting anything meaningful."}
{"id":"53199","title":"In the diode equation, why the exponential $\\exp$ and the ideality factor $n$ are there? What do they represent & what is their significance?","text":"In the Shockley diode equation, why the exponential $\\exp$ and the ideality factor $n$ are there? What do they represent & what is their significance? I have to work on Solar Photovoltaics, and I need to understand the Shockley diode equation clearly."}
{"id":"16831","title":"Planetary model of atom still valid?","text":"When I was in school, I learned (from Democritus) that an atom was similar to a solar system, with the nucleus being the sun, and the electrons being the planets. Of course, there are some differences:   * The \"sun\" isn't a single entity, but a collection of protons and nuetrons.    * Two planets can share an orbit (which might be possible in a solar system too, but it doesn't happen in our solar system).  Is this model still valid? Here are my problems with it:   * In \"Surely You're Joking, Mr Feynman\", Richard Feynman implies that electrons are more a theoretical concept than real objects.    * I have trouble understanding atomic bonds (ionic and covalent) in this model.    * I also have trouble understanding electron \"orbit jumping\" in this model, as well as several other things.  Is there a better model for someone learning this for the first time?"}
{"id":"54971","title":"Reason for considering the positive root","text":"In eqn. (3.11) of Srednicki's QFT book only the positive root is considered; i.e., $ \\omega = + \\sqrt{(k^2 + m^2 )} $ Why the negative root is not considered? And what is the $\\omega$?"}
{"id":"130577","title":"Why does the power in an inductor equal what it does?","text":"I understand that power is that rate at which work is done and that because of this the power in an inductor is equal to $$P=\\frac{d}{dt} \\left(\\frac12Li^2\\right).$$ I also understand that the power is also equal to $$P=Li\\frac{di}{dt}$$ since $L\\,\\frac{di}{dt}=V$ and $Vi$ is power. I understand that since the power is equal to both of these equations that they are equal to each other. The part that I don't get is mathematically how to get from one to the other."}
{"id":"44686","title":"Glueball mass in non-abelian Yang Mills theory","text":"How can the glueball mass be calculated in Yang Mills theory?"}
{"id":"26957","title":"Is there precision experimental evidence for Furry's theorem -- that only even degree VEVs are non-zero?","text":"Is there precision experimental evidence for or contradicting Furry's theorem -- that only even degree VEVs are non-zero, specifically for the EM field?"}
{"id":"90899","title":"Acceleration of particles in a pipe with variable cross section","text":"I am currently making an exercise from Paterson,1983, IV.3.ii. It goes as follows > Water flows steadily and incompressibly along a pipe whose area cross > section $A(x)$ varies slowly with the coordinate $x$ along the pipe. Use the > conservation of mass and\/or incompressibility to calculate the mean velocity > along the pipe at $x$, and calculate the acceleration of a moving particle > moving with this mean velocity. Take the mean velocity to be along the pipe > and depending only on $x$. I would solve the first part of the question as follows. Let $A(0)$ and $v(0)$ be the cross section and the velocity at the begin of the pipe respectively. Since the medium is incompressible, it holds that $A(0) v(0) = A (x) v(x)$ for every $x \\in [0, L]$ where $L$ is the length of the pipe. The local velocity in the x-direction at any point $x$ therefore is $$v(x)=\\frac{A(0) v(0)}{A(x)}$$ So the average velocity of a fluid particle therefore is $$\\langle v(x)\\rangle=\\frac{1}{L}\\int_0^L \\frac{A(0) v(0)}{A(x)} dx $$ So far so good. It's the second part of the question I don't get: How can I know where the particle attains its average velocity and what the local acceleration is. I only know that it exists (intermediate value theorem for differentiable functions)."}
{"id":"110984","title":"Mass eigenstate of neutrinos","text":"Isn't mass a fixed and an intrinsic property of a particle? How can we talk about eigenstates of the mass in the context of neutrinos?"}
{"id":"91498","title":"What are mass eigenstates?","text":"According to Wikipedia > _Neutrino oscillation arises from a mixture between the flavor and mass > eigenstates of neutrinos. That is, the three neutrino states that interact > with the charged leptons in weak interactions are each a different > superposition of the three neutrino states of definite mass. Neutrinos are > created in weak decays and reactions in their flavor eigenstates. As a > neutrino propagates through space, the quantum mechanical phases of the > three mass states advance at slightly different rates due to the slight > differences in the neutrino masses. This results in a changing mixture of > mass states as the neutrino travels, but a different mixture of mass states > corresponds to a different mixture of flavor states. So a neutrino born as, > say, an electron neutrino will be some mixture of electron, mu, and tau > neutrino after traveling some distance. Since the quantum mechanical phase > advances in a periodic fashion, after some distance the state will nearly > return to the original mixture, and the neutrino will be again mostly > electron neutrino. The electron flavor content of the neutrino will then > continue to oscillate as long as the quantum mechanical state maintains > coherence._ What are these mass eigenstates?"}
{"id":"68755","title":"What are the largest thermal gradients achievable in a lab environment?","text":"I am looking for a system capable of creating a gradient of $100\\, \\mathrm{K}\/\\mathrm{\\mu \\textrm{m}}$ on a $30\\, \\mathrm{\\mu}\\textrm{m}$ spacing of a system mounted on a Si-N membrane. My so-called nanoheater is not up to the task."}
{"id":"1971","title":"Approximating mean daily and hourly temperature beyond Fourier series","text":"Summary: What \"well-known\" and short parametrized mathematical function describes daily and hourly temperature for a given location? If you look at the mean daily temperature graph for a given location, it looks like a sine wave with a period of one year. Similarly, the hourly temperature for a given day for a given location also looks like a sine wave with a period of one day. However, closer inspection (Fourier analysis) shows that they're not. There are fairly strong components of frequency 2\/year and 3\/year for the daily temperature, and the hourly temperature also has strong non-single-period terms. Is there a parametrized function that reasonably describes the daily mean temperature and (a separate function) the hourly mean temperature? The parameters would be location-based. I realize I can keep taking more Fourier terms to increase accuracy, but I was hoping for something more elegant. For example, maybe the graph is a parametrized version of sin^2(x) or some other \"well-known\" function."}
{"id":"59832","title":"Math for Thermodynamics Basics","text":"I am studying Statistical Mechanics and Thermodynamics from a book that i am not sure who has written it, because of its cover is not present. There is a section that i can not understand: ${Fj|j=1,..,N}$ $S= \\sum_{j=1}^{N} F_{j}$ $<S>=< \\sum_{j=1}^{N} F_{j}> = \\sum_{j=1}^{N} <F_{j}>$ $\\sigma^{2}_{S} =<S^{2}>-<S>^{2}$ line a: $=\\sum_{j=1}^{N}\\sum_{k=1}^{N} <F_{j}F_{k}> \\- \\sum_{j=1}^{N} <F_{j}>\\sum_{k=1}^{N}<F_{k}>$ line b: $=\\sum_{j=1}^{N}\\sum_{k=1(k\\neq j))}^{N} <F_{j}><F_{k}> +\\sum_{j=1}^{N} <F_{j}^{2}> \\- \\sum_{j=1}^{N} <F_ {j}>\\sum_{k=1}^{N}<F_{k}>$ line c: $=\\sum_{j=1}^{N} (<F_{j}^{2}>-<F_{j}>^{2})$ $=\\sum_{j=1}^{N} \\sigma_{j}^{2}$ My question is what happened after line a to line b and after that to line c? My other question is, i have a little math, what should i study to understand such thermodynamics root math studies, calculus 1 or 2 or what else, can you specify a math topic? Thanks"}
{"id":"13275","title":"Value of Ramanujan Summation In Quantum Mechanics","text":"In mathematics, sum of all natural number is infinity. but Ramanujan suggests whole new definition of summation. \"The sum of $n$ is $-1\/12$\" what so called Ramanujan Summation. First he find the sum, only Hardy recognized the value of the summation. And also in quantum mechanics(I know), Ramanujan summation is very important. **Question.** What is the value of Ramanujan summation in quantum mechanics?"}
{"id":"20034","title":"What practical issues remain for the adoption of Thorium reactors?","text":"From what I've read on thorium reactors, there's enormous benefit to them. Their fuel is abundant enough to power human civilization for centuries, their fission products are relatively short-lived, they're far less prone to catastrophic failure, and they don't produce anything you could feasibly use as a source of material for nuclear weapons. So what technical issues need to be resolved so that Thorium reactors become practical and put into wide use? Is it purely engineering issues that need to be overcome? Or are there problems of physics as well? If so, what are the technical problems and what research is occurring to overcome them? If none of the problems that face thorium reactors are insurmountable, then why aren't they the focus of research and development that nuclear fusion is? Are there real environmental issues? (If so what are they?)"}
{"id":"44356","title":"Why, in EXAFS spectrum, does the absorption coefficient monotonically decrease with increasing photon energy?","text":"In atomic physics, it is common knowledge that following the absorption edge, where the photon energy equals the binding energy of a core electron, a monotonic decrease in the absorption coefficient with increasing photon energy is observed. Obviously it is so common, that everyone mentions it, but nobody cares to explain. For example: ![absorbtion spectra graph](http:\/\/i.stack.imgur.com\/ns5sv.gif) As I understand, in EXAFS (Extended X-Ray Absorption Fine Structure) case, a core electron is excited to the conduction band, and photons with higher energy should just excite it to the higher energy level in the conduction band. So why does absorption decrease?"}
{"id":"28508","title":"Impervious nature of solid matter due to quantum degeneracy pressure","text":"On Wikipedia the following statement is made without reference: > Freeman Dyson showed that the imperviousness of solid matter is due to > quantum degeneracy pressure rather than electrostatic repulsion as had been > previously assumed. Can anyone find the appropriate reference(s)?"}
{"id":"20077","title":"Simulating Gravity in 3D Game?","text":"Alright, I am writing a space simulator for a 3D game and I would like to implement gravity of objects into it. Is there a nice way to find a velocity vector which can be added to my engine output vector to create the effect of gravity. In addition, how would I be able to find the velocities required to get object A into orbit around object B at a certain distance?"}
{"id":"64277","title":"How to keep the clock of a spaceship synchronised to the clock of an observer?","text":"I read that the clocks of GPS satellites seem to run slower than the clock of stationary observer, because of their speed (special relativity) and seem to run faster than the clock of stationary observer because of their altitude (general relativity) . http:\/\/osg.informatik.tu- chemnitz.de\/lehre\/old\/ws0809\/sem\/online\/GPS.pdf From this, can you model the speed $\\upsilon(t)$ and the altitude $r(t)$ of a **spaceship** launched from the ground vertically at $t=0$ such that the clock of the **spaceship** will always seem to be synchronised to the clock of the observer on the ground (from the observer's frame of reference) ?"}
{"id":"28438","title":"Helium-4 superfluidity and gauge symmetry breaking","text":"Is there an accessible account of superfluidity in Helium-4 as a manifestation of \"global gauge symmetry\" breaking? And what is meant by \"global gauge symmetry\"? I was taught that gauge symmetries were by definition local. Is it just a different terminology in condensed matter?"}
{"id":"129099","title":"How powerful would an explosion have to be for its effects to be felt worldwide?","text":"If an explosion were to occur at any point on earth, how powerful would that explosion have to be for it to be audible or otherwise detectable by every person on the planet? Detection could mean either seeing or hearing the blast or feeling the tremors created by the shock wave. Bonus question: is any such explosion possible without it destroying the planet, the atmosphere or wiping out all life on earth? * * * A rough estimate puts the average distance between most antipodes on land at just shy of 20 000 kilometres. The largest nuclear bomb ever detonated, Tsar Bomba, had a yield of 50-58 megatons of TNT and was detectable almost a 1000 km away, according to Wikipedia: > The heat from the explosion could have caused third-degree burns 100 km (62 > mi) away from ground zero. A shock wave was observed in the air at Dikson > settlement 700 kilometres (430 mi) away; windowpanes were partially broken > to distances of 900 kilometres (560 mi). Atmospheric focusing caused blast > damage at even greater distances, breaking windows in Norway and Finland. > The seismic shock created by the detonation was measurable even on its third > passage around the Earth. The most powerful volcanic eruption known was that of Mount Tambora in 1815. Classified as Volcanic Explosivity Index 7 (note that it goes up to 8) with an estimated yield of 800 Mt, it was heard about 2 600 km away. > On 5 April 1815, a moderate-sized eruption occurred, followed by thunderous > detonation sounds, heard in Makassar on Sulawesi 380 km (240 mi) away, > Batavia (now Jakarta) on Java 1,260 km (780 mi) away, and Ternate on the > Maluku Islands 1,400 km (870 mi) away. On the morning of 6 April, volcanic > ash began to fall in East Java with faint detonation sounds lasting until 10 > April. What was first thought to be the sound of firing guns was heard on 10 > April on Sumatra island more than 2,600 km (1,600 mi) away. The Wikipedia page on TNT equivalents list some interesting events but branches off into seismic and cosmic events after the entry for the Tsar Bomba. So, is it possible to go bigger?"}
{"id":"129098","title":"Lagrangian Systems","text":"Given a manifold $M$, Arnold's \"Mathematical Methods of Classical Mechanics\" defines a Lagrangian system as a pair $(M,L)$ where $L$ is some smooth function on the tangent bundle $TM$. The function $L$ is called the Lagrangian. In the case when $M$ is a Riemannian manifold and a particle in $M$ is moving under some conservative force field, taking the Lagrangian to be the kinetic minus the potential energy we recover Newton's second law. I know that one of the main advantages to the Euler-Lagrange equations over Newtons is the way in which they simplify constrained systems. I know another is the coordinate independence of the equations. However, in all applications I've seen the manifold is always Riemannian and the Lagrangian is always $K-U$. My questions are:   1. Why do we have this abstract definition of a Lagrangian system and of an abstract Lagrangian?    2. What are some of the cases in which $L$ is not $K-U$ that gives interesting results?    3. Or cases in which the manifold is not Riemannian?"}
{"id":"67765","title":"How is a Hamiltonian constructed from a Lagrangian with a Legendre transform","text":"many textbooks tell me that Hamiltonians are constructed from Lagrangians like $$L=L(q,\\dot{q})$$ with a Legendre transformation to obtain the Hamiltonian as $$H=\\dot{q}\\frac{\\partial L}{\\partial \\dot{q}}-L$$ but none of the textbooks explain how this is done. My specific problem is that I have Lagrangians that do not depend on $\\dot{q}$ and therefore should have $\\frac{\\partial L}{\\partial \\dot{q}}=0$, hence $H=-L$. But my impression from the clues I have is that it is not that simple. Let's say the Lagrangian is $$L(q)=\\ln(q)-(2q-10)\\lambda$$ Now as far as I know the Legendre transformation should give a function $f^*(p)=\\sup(pq-L(q))$ (this implies $p=\\frac{\\partial L}{\\partial q}$) which is obtained by substituting the stationary point $q_s$ of $\\sup(pq-L(q))$ into $pq-L(q)$ thus getting $f^*(p)=pq_s-L(q_s)$ (for instance wikipedia's Legendre Transformation page explains this). Doing this for the example above: $$\\frac{\\partial (pq-L(q))}{\\partial q}=\\frac{\\partial (pq-\\ln(q)+(2q-10)\\lambda)}{\\partial q}=p-\\frac{1}{q}+2\\lambda$$ must be 0 for a stationary point, thus $q_s=1\/(p+2\\lambda)$. And hence the transformation should be $$f^*(p)=p\\frac{1}{p+2\\lambda}-\\ln(\\frac{1}{p+2\\lambda})+(2\\frac{1}{p+2\\lambda}-10)\\lambda$$ and this should be the Hamiltonian. But this equation does obviously have nothing to do with the textbook Hamiltonian. Rather, an answer to another question has in a similar case treated $L(q)$ as being dependent on $\\dot{q}$ implicitly (Writing $\\dot{q}$ in terms of $p$ in the Hamiltonian formulation ... answer by Qmechanic). It mentions using the Dirac-Bergmann method for obtaining the Legendre transform. Trying something along the lines of this other question the above example seems to give $$p=\\frac{\\partial L}{\\partial \\dot{q}}=0$$ and $$p \\approx 0$$ (an equality modulo constraint as the answer to the question linked above says). And then $H=\\dot{q}p-L$. The difference seems to be that the Legendre transform is done with respect to two different variables, $q$ and $\\dot{q}$ - but it was my understanding that it had to be done with respect to all variables the Lagrangian depends on. So how does the $qp_q$ term vanish if we have only the $\\dot{q}p_{\\dot{q}}$ term left? Thanks. edit: changed ln to \\ln as Plane Waves suggested, and sup to \\sup. And yes, sup is the supremum over all q, as Vibert said, sorry for forgetting to mention that."}
{"id":"67762","title":"Effect of slits and a lens","text":"We have a screen with two slits (Young style) separated by a distance $d$, one of them receives a planar wave of $600nm$, the other receives a planar wave of $400nm$. Behind the slits there's a screen we observe. Only with this, as the waves have different wavelengths, I guess there can't be any interference, we will only see the difraction pattern, the two functions of the form $\\sin^2(x)\/x^2$, with the principal maximums separated a distance $d$. Am I right here? Now, we put a convex lens behind the slits so the screen in which we observe is in the focal plane. Ok, I have three configurations: 1.- In the first one, the system is configured such that the slits are far away from the lens. Here, we can approximate the wave that arrives as a planar wave, and therefore the lens will perform the Fourier transform in the focal plane of the screen. The diffraction of the slits also performs the Fourier transform, so this configuration should lead to having only two bars of light in the screen, centered in the focus. Am I right? 2.- The slits are in the focal plane on the lens, such that the lens is in the middle of slits-screen. Here, the same thing should happen, right? As the light comes from the focal plane, the lens must do the fourier transform with no extra things, and we should get the two bars, again both of them in the same line (center of the screen). Am I right here? 3.- The last one, I can't see... the lens is just behind the slits so the distance between slits and lens is $\\approx 0$. My guess here is that the two centers of the intesity distributions $\\sin^2(x)\/x^2$ will go to the center of the screen (the focus) because they go perpendicular to the lens, but the rest of the pattern will just be compressed a little. Again the wave don't interfere, so the intensities just sum up, and the resulting will be: $$I=I_1\\frac{\\sin^2(\\alpha x)}{(\\alpha x)^2}+I_2\\frac{\\sin^2(\\beta x')}{(\\beta x')^2}$$ Being $\\alpha$ and $\\beta$ the some factor of compression due to the lens, that could actually be a function of $x$. Am I right here? Am I completely wrong? What would happen?"}
{"id":"89836","title":"$\\hat{\\imath}$ component of force exerted on an electron by a magnetic field?","text":"> The magnetic field over a certain range is given by $\\vec{B} = > B_x\\hat{\\imath} + B_y\\hat{\\jmath}$, where $B_x= 4\\: \\mathrm{T}$ and $B_y= > 2\\: \\mathrm{T}$. An electron moves into the field with a velocity $\\vec{v} = > v_x\\hat{\\imath}+v_y\\hat{\\jmath}+v_z\\hat{k}$, where $v_x= 5\\: \\mathrm{m\/s}$, > $v_y= 8\\: \\mathrm{m\/s}$ and $v_z= 9\\: \\mathrm{m\/s}$. The charge on the > electron is $-1.602 \\times 10^{-19}\\: \\mathrm{C}$. What is the > $\\hat{\\imath}$ component of the force exerted on the electron by the > magnetic field? Answer in units of $\\mathrm{N}$. I know that $\\vec{F}=q\\vec{v} \\times \\vec{B}$, so plugging in I have: $$\\vec{F}=(-1.602 \\times 10^{-19})<5,8,9> \\times <4,2,0>$$ My confusion is to whether or not multiply my velocity vector components by charge (the scalar) or if I take the cross product between $\\vec{B}$ and $\\vec{v}$ first? I'd also like to know why whichever operation comes first does in fact come first."}
{"id":"23676","title":"Is there a general physics simulator for learning purposes?","text":"Is there a complete physics simulator that I can use to do general simulations for learning purposes? For example:   1. Create a sandbox.   2. Fill with a gas.   3. Load a 3d solid model like this (but 3d).   4. Fill it with a dense liquid.   5. Load gravity.   6. Watch, measure and understand how a barymeter works. It doesn't need to be precise, just usable, so I guess it is not impossible. The point would be to simulate and visualize any kind of exercise you would find in your physics book. It would be the mother of the learning tools. If it doesn't exist, is anybody interested in programming it?"}
{"id":"66412","title":"Can the vanishing of the Riemann tensor be determined from causal relations?","text":"Given a _Lorentzian manifold and metric tensor_ , \"$( M, g )$\", the corresponding _causal relations_ between its elements ( _events_ ) may be derived; i.e. for every pair (in general) of distinct events in set $M$ an assignment is obtained whether it is _timelike_ separated, or _lightlike_ separated, or neither ( _spacelike_ separated). In turn, I'd like to better understand whether causal separation relations, given abstractly as \"$( M, s )$\", allow to characterize the corresponding Lorentzian manifold\/metric. As an exemplary and surely relevant characteristic (cmp. answer here) let's consider whether _the Riemann curvature tensor vanishes_ , or not, at each event of the whole set $M$ (or perhaps suitable subsets of $M$). Are there particular causal separation relations which would be indicative, or counter-indicative, of the Riemann curvature tensor vanishing at all events of set $M$ (or if this may simplify considerations: at all events of a _chart_ of the manifold); or on some subset of $M$? To put my question still more concretely, consider as possible illustration of \"counter-indication\": (a)   Can ~~any _chart of a 3+1 dimensional Lorentzian manifold with everywhere vanishing Riemann curvature tensor_ (or, at least, a whole such manifold) contain ~~ [Edit in consideration of 1st comment (by twistor59): -- _the Riemann curvature tensor vanish at least in one event of a 3+1 dimensional Lorentzian manifold_ if each of its _charts_ contains -- ]   * fifteen events (conveniently organized as five triples): $A, B, C$; $\\,\\,\\,\\, F, G, H$; $\\,\\,\\,\\, J, K, L$; $\\,\\,\\,\\, N, P, Q\\,\\,\\,\\,$, and $\\,\\,\\,\\, U, V, W$,   * where (to specify the causal separation relations among all corresponding one-hundred-and-five event pairs): $s[ A, B ]$ and $s[ A, C ]$ and $s[ B, C ]$ are _timelike_ ,   $s[ F, G ]$ and $s[ F, H ]$ and $s[ G, H ]$ are _timelike_ ,   $s[ J, K ]$ and $s[ J, L ]$ and $s[ K, L ]$ are _timelike_ ,   $s[ N, P ]$ and $s[ N, Q ]$ and $s[ P, Q ]$ are _timelike_ ,   $s[ U, V ]$ and $s[ U, W ]$ and $s[ V, W ]$ are _timelike_ , $s[ A, G ]$ and $s[ G, C ]$ and $s[ A, K ]$ and $s[ K, C ]$ and   $s[ A, P ]$ and $s[ P, C ]$ and $s[ A, V ]$ and $s[ V, C ]$ are _lightlike_ , $s[ F, B ]$ and $s[ B, H ]$ and $s[ F, K ]$ and $s[ K, H ]$ and   $s[ F, P ]$ and $s[ P, H ]$ and $s[ F, V ]$ and $s[ V, H ]$ are _lightlike_ , $s[ J, B ]$ and $s[ B, L ]$ and $s[ J, G ]$ and $s[ G, L ]$ and   $s[ J, P ]$ and $s[ P, L ]$ and $s[ J, V ]$ and $s[ V, L ]$ are _lightlike_ , $s[ N, B ]$ and $s[ B, Q ]$ and $s[ N, G ]$ and $s[ G, Q ]$ and   $s[ N, K ]$ and $s[ K, Q ]$ and $s[ N, V ]$ and $s[ V, Q ]$ are _lightlike_ , $s[ U, B ]$ and $s[ B, W ]$ and $s[ U, G ]$ and $s[ G, W ]$ and   $s[ U, K ]$ and $s[ K, W ]$ and $s[ U, P ]$ and $s[ P, W ]$ are _lightlike_ , the separations of all ten pairs among the events $A, F, J, N, U$ are _spacelike_ ,   the separations of all ten pairs among the events $B, G, K, P, V$ are _spacelike_ ,   the separations of all ten pairs among the events $C, H, L, Q, W$ are _spacelike_ , and finally the separations of all twenty remaining event pairs are _timelike_   ? Conversely, consider as possible illustration of \"indication\": (b)   Is there a _3+1 dimensional Lorentzian manifold with ~~everywhere vanishing Riemann curvature tensor_ (or, at least, one of its _charts_ ) which doesn't ~~ [Edit in consideration of 1st comment (by twistor59): -- _nowhere vanishing Riemann curvature tensor_ such that all of its _charts_ \\-- ] contain   * twenty-four events, conveniently organized as four triples ($A, B, C$; $\\,\\,\\,\\, F, G, H$; $\\,\\,\\,\\, J, K, L$; $\\,\\,\\,\\, N, P, Q$) and six pairs ($D, E$; $\\,\\,\\,\\, S, T$; $\\,\\,\\,\\, U, V$; $\\,\\,\\,\\, W, X$; $\\,\\,\\,\\, Y, Z$; $\\,\\,\\,\\, {\\it\\unicode{xA3}}, {\\it\\unicode{x20AC}\\,}$),   * where (again explicitly, please bear with me$\\, \\\\!^*$): the sixty-six separations among the twelve events belonging to the four triples are exactly as in question part (a), each of the six pairs is _timelike_ separated, the separations of all fifteen pairs among the events $D, S, U, W, Y, {\\it\\unicode{xA3}}$ are _spacelike_ ,   the separations of all fifteen pairs among the events $E, T, V, X, Z, {\\it\\unicode{x20AC}\\,}$ are _spacelike_ , $s[ D, {\\it\\unicode{x20AC}\\,} ]$ and $s[ S, Z ]$ and $s[ U, X ]$ are _spacelike_ ,   $s[ E, {\\it\\unicode{xA3}} ]$ and $s[ T, Y ]$ and $s[ V, W ]$ are _spacelike_ , $s[ A, {\\it\\unicode{xA3}} ]$ and $s[ A, Y ]$ and $s[ A, W ]$ are _spacelike_ ,   $s[ A, {\\it\\unicode{x20AC}\\,} ]$ and $s[ A, Z ]$ and $s[ A, X ]$ are _timelike_ ,   $s[ A, E ]$ and $s[ A, T ]$ and $s[ A, V ]$ are _timelike_ , $s[ C, {\\it\\unicode{x20AC}\\,} ]$ and $s[ C, Z ]$ and $s[ C, X ]$ are _spacelike_ ,   $s[ C, {\\it\\unicode{xA3}} ]$ and $s[ C, Y ]$ and $s[ C, W ]$ are _timelike_ ,   $s[ C, D ]$ and $s[ C, S ]$ and $s[ C, U ]$ are _timelike_ , $s[ F, {\\it\\unicode{xA3}} ]$ and $s[ F, D ]$ and $s[ F, S ]$ are _spacelike_ ,   $s[ F, {\\it\\unicode{x20AC}\\,} ]$ and $s[ F, E ]$ and $s[ F, T ]$ are _timelike_ ,   $s[ F, V ]$ and $s[ F, X ]$ and $s[ F, Z ]$ are _timelike_ , $s[ H, {\\it\\unicode{x20AC}\\,} ]$ and $s[ H, E ]$ and $s[ H, T ]$ are _spacelike_ ,   $s[ H, {\\it\\unicode{xA3}} ]$ and $s[ H, D ]$ and $s[ H, S ]$ are _timelike_ ,   $s[ H, U ]$ and $s[ H, W ]$ and $s[ H, Y ]$ are _timelike_ , $s[ J, D ]$ and $s[ J, U ]$ and $s[ J, Y ]$ are _spacelike_ ,   $s[ J, E ]$ and $s[ J, V ]$ and $s[ J, Z ]$ are _timelike_ ,   $s[ J, T ]$ and $s[ J, X ]$ and $s[ J, {\\it\\unicode{x20AC}\\,} ]$ are _timelike_ , $s[ L, E ]$ and $s[ L, V ]$ and $s[ L, Z ]$ are _spacelike_ ,   $s[ L, D ]$ and $s[ L, U ]$ and $s[ L, Y ]$ are _timelike_ ,   $s[ L, S ]$ and $s[ L, W ]$ and $s[ L, {\\it\\unicode{xA3}} ]$ are _timelike_ , $s[ N, D ]$ and $s[ N, S ]$ and $s[ N, W ]$ are _spacelike_ ,   $s[ N, E ]$ and $s[ N, T ]$ and $s[ N, X ]$ are _timelike_ ,   $s[ N, V ]$ and $s[ N, Z ]$ and $s[ N, {\\it\\unicode{x20AC}\\,} ]$ are _timelike_ , $s[ Q, E ]$ and $s[ Q, T ]$ and $s[ Q, X ]$ are _spacelike_ ,   $s[ Q, D ]$ and $s[ Q, S ]$ and $s[ Q, W ]$ are _timelike_ ,   $s[ Q, U ]$ and $s[ Q, Y ]$ and $s[ Q, {\\it\\unicode{xA3}} ]$ are _timelike_ , and finally the separations of all ninety-six remaining event pairs are _lightlike_   ? (*: The two sets of causal separation relations stated explicitly in question part (a) and part (b) are of course not arbitrary, but have motivations that are somewhat outside the immediate scope of my question -- considering Lorentzian manifolds -- itself. It may nevertheless be helpful, if not overly suggestive, to attribute the relations of part (a) to \"five participants, each finding coincident pings from the four others\", and the relations of part (b) to \"ten participants -- four as vertices of a regular tetrahedron and six as middles between these vertices -- pinging among each other\".)"}
{"id":"126241","title":"Electrostatics with Yukawa mass","text":"I seem to have forgotten how to solve Laplace's equation when there is a Yukawa mass $\\mu$. I want to find the Yukawa potential due to a homogeneously charged sphere of radius $R$ and charge density $\\rho$. **My attempt** I need to solve $(\\nabla^2-\\mu^2)\\phi = -\\rho$ inside and outside the sphere. Since there is spherical symmetry, I can convert to spherical coordinates, and assume the potential $\\phi$ has no angular dependence: $$ \\frac{1}{r}\\frac{\\partial^2}{\\partial r^2} (r\\phi) - \\mu^2 \\phi = -\\rho $$ Outside the sphere, there is no charge, so I solve the homogenous equation giving the solution: $$\\phi_\\text{out}(r) = A \\frac{e^{-\\mu r}}{r}+B \\frac{e^{\\mu r}}{r}$$ and inside the sphere, where there is charge density, I solve the inhomogenous equation, giving the solution: $$\\phi_\\text{in}(r) = \\frac{\\rho}{\\mu^2}+C \\frac{e^{-\\mu r}}{r}+D \\frac{e^{\\mu r}}{r}$$ **Problem** I have four constants of integration ($A$, $B$, $C$, $D$) but only three pieces of information to fix them.   1. The potential should not blow up as $r\\rightarrow\\infty$: I must have $B=0$   2. It should not be singular as $r\\rightarrow 0$: I must have $D=-C$. And therefore, $$\\phi_\\text{in}(r) = \\frac{\\rho}{\\mu^2} + 2 C \\frac{\\sinh( \\mu r)}{r}$$   3. Also, I must have continuity at the surface of the sphere $r=R$: Then I have $$A=\\big(\\frac{\\rho R}{\\mu^2}+ 2 C \\sinh \\mu R\\big)e^{\\mu R}$$ My potential has an undetermined constant $C$; What is the final piece of information that will fix $C$?"}
{"id":"59302","title":"How to charge a field?","text":"In a previous post [ Noether theorem, gauge symmetry and conservation of charge ] we were discussing the different ways to demonstrate the current conservation: via the first Noether theorem applied to a global $U(1)$ gauge symmetry, or via the covariant (or minimal, or Weyl, or ...) substitution with a $U(1)$ local gauge symmetry and antisymmetry of the electro-magnetic field applied to the equations of motion. I nevertheless still get confused about something: what is a charge ? More precisely, how to charge a field ? Indeed, if one believes that only the global gauge construction is able to demonstrate the conservation of the charge, one has to admit that the charge is not defined that way ! The same demonstration applies for the conservation of the particle number. Indeed, I used in [ Noether theorem, gauge symmetry and conservation of charge ] a Lagrangien for _uncharged_ particles to show the conservation of a particular current through Noether theorem. **So the question becomes: how to charge this particle ?** In particular: how to charge this particle and still conserve the global gauge ? NB: In the local gauge construction, it seems to be easier to give a charge to the system, but then only the second Noether theorem applies, and people get annoyed by that. **To conclude, a** (possibly important) **remark about my point of view:** I would like to understand the problem of \"charging the field\" in a condensed matter perspective, when I believe we have no a priori idea of the charge of the quasi-particles appearing in our effective theories. Can we overcome this problem ? Of course, any point of view (especially the opinion of QFT physicists who might have already tackled this problem) is warmly welcome :-) And the question \"how to charge a field ?\" is just a warm-up for this more complicated one \"how to define the charge of an effective, emergent field ?\" which is entirely subsidiary question for the moment. **IMPORTANT EDIT** I just became aware of this question [ How does non-Abelian gauge symmetry imply the quantization of the corresponding charges? ] which is strongly related, and well answered. I nevertheless think one can continue to discuss here about the condensed matter problem. Indeed, there are some people who believe that a superconductor (among other emergent phases of matter) have charge neutral excitations [see e.g. http:\/\/dx.doi.org\/10.1103\/PhysRevB.41.11693 which is too old to be on arXiv, but this one: http:\/\/arxiv.org\/abs\/cond-mat\/0404327 rephrases it in a different way.]. So the question follows: how can these excitations at low temperatures become again charged at higher temperature ? How can we measure charge current from an uncharged excitation ? How can we \"charge\" a field after all ?"}
{"id":"105973","title":"Is a scaled-up aircraft carrier 'ski-ramp' a viable system to impart enough velocity to significantly assist a spacecraft to orbit?","text":"Not a boost straight up to escape velocity, just sufficient added momentum to make a significant economic saving on fuel cost, mass and complexity of a standard launch. It seems ludicrous to me how much fuel in a standard launch appears to be wasted pushing thousands of tonnes of rocket fuel in a vertical direction from a standing start. The system I envisage would have a horizontal track of possibly 2-3km (distance to be debated), acceleration via external systems (electromag or JATO?), continue up a ramp of slowly increasing gradient, with craft's own engines now running, and exit the ramp at between Mach 1 or 2. Safety run-off track if craft's own engines fail to ignite properly. Winged style craft for stability from ramp & shuttle-style landing. Similar I know to the Phys.SE 'Rail gun' question here, but acceleration could be more survivable. Had a longer script but it appeared to over-run the site limit. Acknowledge AdamRedwine's railgun question -would have added my contribution to that but my Reputation today is only 1. Acknowledge Gerry Anderson's Fireball XL5 puppet show on British TV in the '60s - the launch concept has always stayed with me! Submitted for criticism."}
{"id":"73701","title":"What causes stress concentration (aka stress risers\/raisers) at corners?","text":"I've read a few explanations about why stress concentration occurs at sharp corners but I don't find the explanations intuitive. Can anyone explain it perhaps using an analogy such as atoms \"holding hands\" with neighbor atoms or a similar easy to understand analogy? Most explanations I've shown just show the result of a finite-element-analysis based stress analysis. This shows the emergent property but doesn't explain the underlying mechanism that causes the stress to concentrate at corners. Better understanding the cause of the crashes of the De Havilland Comet jet aircraft due to metal fatigue at the corners of their rectangular windows is the context for this question. See: http:\/\/www.cracked.com\/article_19623_6-small-math-errors-that-caused-huge- disasters.html The answer I'm looking for should not contain any math."}
{"id":"34317","title":"Estimate quarter mile time","text":"I need to estimate a drag race quarter mile time given the car's weight, bhp and preferably the drive (FWD, RWD, 4WD). I know $v(t) = ds\/dt$ and $a(t) = dv\/dt = d^2s\/dt^2$, but how can I get the function $s(t)$ to calculate $v(t)$ and $a(t)$? I thought maybe Power to Weight ratio would help for solving the acceleration but I have no idea what to do with it and maybe the drive would just be a constant which is then subtracted from the calculated time. Can you help me?"}
{"id":"32481","title":"Why does vibration loosen screws?","text":"I am trying to figure out why vibrations (say, from an engine) loosen screws. It seems to me that there is evident symmetry between loosening and tightening a screw. I am wondering what breaks this symmetry."}
{"id":"78866","title":"Uniform constant magnetic field and traditional attractive force","text":"Why uniform constant magnetic fields can not exert net force on a piece of iron whatever strong it might get?"}
{"id":"8456","title":"Derivation of relativistic energy","text":"The concept of relativistic energy comes from it's conservation in relativistic mechanics for an elastic collision. It seems to me that another possible derivation could equate the energy of a single particle before, with the kinetic energy after of N particles after the energy is distributed amongst them in the form of the Boltzman distribution, as well as individually being of the form $mv^2$. Are there any problems in doing this?"}
{"id":"92336","title":"physically modelling the Saltatory nerve impulse transmission?","text":"The nerve impulse transmission is specifically a biophysical process. Under a _resting_ stage, the membrane is already polarised (presence of charge on either side leading to a potential difference across it, due to its finite capacitance). Changes in this polarisations mediated by several agents, cause some part of this membrane to be depolarised (inversion of the charges in that localised area). This changes the potential difference across that region and starts ion flow from the depolarised region of the membrane to the adjacent polarised region, inducing the same depolarisation there. Myelinating a neuron is similar to lowering the capacitance of the neuron. Myelinating the neuron causes subsequent action potentials to be spatially separated, and the conduction of voltage between them occurs primarily through ion flow along the neuron. Here are some nerve conduction basics. > Now my actual question. **Modelling a neuron as a simple one dimensional > _membrane\/cable_ , how can a lowered capacitance lead to a faster conduction > of the voltage perturbation i.e a change in voltage (depolarisation), which > is initially limited to a small localised region? This conduction of the > voltage change can occur through the ion flow along the inner side of the > membrane and also due to long distance changes in potential due to the > altered distribution of charges along the membrane**. A very good physical > modelling of the neuron is given here and here, but because of the quite > complicated mathematical nature of the modified telegrapher's equation which > appears as the final answer, I am unable to understand how lowered > capacitance increases the speed of voltage conduction? Tell me if this question is off-topic or too biological to be answered here."}
{"id":"114191","title":"Help with a solid sphere sliding and then rolling","text":"Initially at time $t=0$, a solid sphere slides with velocity $v$ along a horizontal surface. The coefficient of friction is u. Find the required time for the sphere to stop sliding (the sphere will then be rolling). Ive tried using Newtons equations of motion but I'm not sure what the final velocity will be? At first I thought it may be zero because it will be rolling but obviously it will still have some translational velocity. I've also found the work done by the friction on the sphere and taking this figure away from the initial kinetic energy leaving (what I thought) would be the rotational kinetic energy that the ball has. However I can't get to the answer. I'm almost certain I'm just missing a small trick but I have no idea where to go with it."}
{"id":"35684","title":"Force applied perpendicular to direction of motion, static or kinetic friction?","text":"Let's assume we have a 1 kg cube on a flat uniform surface. The coefficients of friction between them are $\\mu_k = 0.25$ and $\\mu_s = 0.50$. This cube is moving at 1 m\/s in the y direction, with whatever force is needed to overcome friction being applied. There is no force or motion in the x direction. Assume $g = 10\\, {\\rm m \/ \\rm s^2}$ in the z direction. Now, if a force is applied in the x direction which type of friction applies, static or kinetic? Edit: To clarify, if the cube is moving forward, and gravity is acting down, the new force is being applied to the right. My rudimentary model of friction is that it acts like two pieces of sandpaper where the bumps can interlock when there is not motion, but when they are moving over top of each other they don't have time to settle in. In that model, I would think kinetic friction would apply, as the direction of movement shouldn't matter. However, I realize that in reality friction is a result of molecular attractive forces, and probably a lot of other more complicated things I'm unaware of. Because of this, I suspect the answer may lie somewhere between static and kinetic friction."}
{"id":"68111","title":"What made up light photons?","text":"mass is energy per c square $m=E\/c^2$ energy is made up of photons but what made up photon itself? what made up a single photon? * * * **Replay to comment:** but as we can see in history early phyisicists though that atom in non-fissionable (thats why they called it atom), now we call photon elementary but it may be a composite particle!, why not?"}
{"id":"52908","title":"How do I intuit viscosity in a rotating fluid?","text":"Suppose I have two plates with a viscous fluid in between. I slide them in the same direction (a direction in their own plane), one at $5 \\,\\text{m\/s}$ and the other at $6 \\,\\text{m\/s}$. Due to the relative velocity, there will be a viscous damping force tending to equalize the velocities of the plates. Now suppose I have two concentric cylindrical shells, the second with a radius $6\/5$ that of the first. I put viscous fluid between them and rotate the entire system such that the cylinders have tangential velocities $5 \\,\\text{m\/s}$ and $6 \\, \\text{m\/s}$. Because the system is executing rigid rotation, there will be no viscous damping. Suppose I take a small cube of the viscous fluid in either scenario. In both cases, there's a similar velocity gradient across the cube, but only in one case does the cube dissipate energy. Why? I can work this out by writing the Navier-Stokes equation and transforming to cylindrical coordinates, but I am still having a hard time visualizing the kinematics such that this becomes clear to me."}
{"id":"60810","title":"Earth's rotation isn't that orbit?","text":"If the earth is rotating at some $465~\\text{m}\/\\text{s}$ at the equator and that's really fast.   1. Shouldn't we in that case be in orbit with the earth just not fast enough?    2. How fast do we need to move?   3. If the earth stopped rotating (gradually), shouldn't gravity increase significantly?   4. Do satellites need extra speed when they are released or does it just depend on their inertia from the earth's rotation?   5. If the satellite decelerated against the earth direction of rotation will it fall?    6. If the earth rotated extremely fast, shouldn't the earth's crust along with everything else get ejected away by the centrifugal force? I'm sorry if the questions cover many matters but I preferred to ask them all at once instead of asking each separately."}
{"id":"60813","title":"Maximizing Multiplicity of Einstein Solid == (Temperature = $\\infty$)?","text":"If I have a system consisting of 2 Einstein solids (`A` and `B`) is it equivalent to say that maximizing the multiplicity of the system is the same as setting the temperature to $\\infty$? I have two reasons to support this Idea. 1) The total multiplicity of the system can be expressed as: $$\\Omega_{tot} = \\Omega_A(q_A,N_A) \\Omega_B(q_B,N_B)$$ Which is the product of the individual multiplicities, and $q = q_A + q_B = $ Total quanta of energy shared between the system, and $N = N_A + N_B =$ total number of quantum harmonic oscillators. I wish to maximize the multiplicity with respect to oscillator `A`'s total internal energy ($U_A)$ $$\\left( \\frac{ \\partial \\Omega_{tot} }{ \\partial U_A } \\right) = 0 \\implies \\left( \\frac{ \\partial Ln( \\Omega_{tot} )}{ \\partial U_A } \\right) = \\frac{ \\Omega_{tot}'}{ \\Omega_{tot}} = 0$$ This is true because Ln(x) is a monotonic function. But it should be noted that the definition of temperature is such that: $$\\frac{1}{kT} = \\left( \\frac{ \\partial Ln( \\Omega_{tot} )}{ \\partial U_A } \\right) = 0 \\implies T= \\infty$$ `k` here is the boltzman constant. 2) This makes some intuitive sense. On the Gaussian distribution as a function of Internal energy, the maximum multiplicity lies on the peak. The change in temperature gives a quantification of the direction of heat flow. If the maximum multiplicity is attained, then heat cannot flow any higher, it can only flow downward. This corresponds to infinite temperature. I hope my intuition here is correct."}
{"id":"52901","title":"Time-reversal symmery and topological insulators","text":"I have a very naive question about the notion of time-reversal symmetry applied to topological insulators that are studied in experiments. If I understand correctly, the exsistance of time-reversal symmetry is crucial for the occurence of topologically protected surface states in topological insulators. At least this is what the theory of topological insulators tells us. But I have a problem with applying this theory to real materials. In real world there is always dissipation and therefore there is an \"arrow of time\" and time-reversal symmetry does not hold (right?). So how can we talk about time-reversal symmetry in real materials that we call topological insulators? Please help me find where I went wrong in my argument. I realize that there must be something wrong but can't figure out what exactly."}
{"id":"56685","title":"What happens to this potential energy?","text":"Let's say I turn on a Van de Graaff which creates a large positive charge. Now let's say I have an object with a positive charge in my hand and I start walking toward the Van de Graaff from $x$ meters away. If I walk an arbitrary distance towards the Van de Graaff, I am doing work and the charge in my hand is gaining potential energy. Now somebody standing next to the Van de Graaff suddenly turns it off. What happens to that potential energy of the charge? By the conservation of energy it should be conserved somehow, but it appears not to be. Any help is appreciated."}
{"id":"106361","title":"Non-uniform lens flares","text":"I assume -- and maybe wrong -- that lens flares, in photography, are caused by light getting \"trapped\" inside the curved lens and bouncing around (due to refraction and non-perfect total internal reflection) causing bright spots at periodic intervals, when they partially escape on the imaging side. This would give the classic circular(ish) spots when a camera pans up towards, say, the sun, where the shape of the spots mimics that of the lens\/lenses and aperture. However, I was looking at the Hubble Deep Field image and noticed that a few of the brighter stars have _only_ horizontal and vertical flares. Similarly, stars (as in the light source) are often drawn as _n_ -pointed polygons, implying distinct flares at regular angles. What is it about the lens, in these cases, that causes flares to propagate seemingly non-uniformly or directionally?"}
{"id":"65334","title":"Calculating vector components from other vectors","text":"I'm busy studying for my Physics exam this evening and I've come across a vector problem that I cant quite solve. Any help would be appreciated! We are given the following vectors:               A = 5i - 6.5j     B = -3.5i + 7j      We are also told that a vector C lies in the xy-plane and is perpendicular to A. The scalar product of B and C is equal to 15. With this information we have to calculate the components of the vector C. I have tried calculating the magnitude of B and then trying to solve the equation:               |B||C|cos{theta} = 15      But I didn't get very far. I have a feeling I'm just not seeing something and making a careless error. Thanks in advance!"}
{"id":"60492","title":"Kinematics with non constant acceleration","text":"A particle experiences an acceleration described by $$ a=kx^{-2} $$ where x is the displacement from the origin and k is an arbitrary constant. To what value does the velocity v of the particle converge to as x approaches infinity if the particle starts at some point x0? If I approach this problem with energy, then $$ W = \\int F \\mathrm{d}x $$ $$ = \\int_{x_0}^\\infty mkx^{-2} \\mathrm{d}x $$ $$ = mk(-\\infty^{-1}+{x_0}^{-1}) $$ $$ W = K = mk{x_0}^{-1} $$ $$ \\frac{1}{2}mv^2 = mk{x_0}^{-1} $$ $$ v = \\sqrt{2k{x_0}^{-1}} $$ How would I solve this problem with pure kinematics? (there appears to be some sort of cyclical dependency where acceleration affects velocity, velocity affects displacement, and displacement affects acceleration) Likewise, two particles experience accelerations described by $$ a_1=k_1x^{-2} $$ $$and $$ $$ a_2=-k_2x^{-2} $$ where x is the distance between the two particles What two velocities do the particles reach as x approaches infinity if the two particles are initially separated by some x0?"}
{"id":"96057","title":"How two photons interfere in a double slit experiment","text":"I have a very basic question about photons in double-slit experiment. I am not good at math, and have some Quantum mechanics knowledge. Math free explanation would be very good to me. When we do double slit experiment with single photon at a time, it will be detected at some location of screen. Over the time period the accumulation of detected locations show interference pattern. The reason for the interference pattern is that photon's location in not well defined, the presence of photon in particular place can be determined by prbability and this probability is presented as a wave. that probability wave is splitted in to two while passing the two slits, collide with each other and causes interference pattern. Question 1: Is this reason for single photon interference correct? Question 2: How is the probability wave behaving while two-photons passing double-slit at a time? Are 4 probability waves (2 for each) interfering?"}
{"id":"116772","title":"What will happen after escaping earth's gravitational field?","text":"Suppose that I escaped the gravitational field of earth. Then: am I going to be pulled by Sun's gravity?"}
{"id":"116779","title":"Entropy is constant. How to express this equation in terms of pressure and density?","text":"In hydrodynamics of an ideal, non-compressive flow we use 5 variables: pressure $p$, density $\\rho$ and velocity field $\\mathbf{v}$. So we need 5 equations. Landau's \"Hydrodynamics\" states that the equations are:   1. The mass continuity equation $\\frac{\\partial \\rho}{\\partial t}+\\nabla (\\rho \\mathbf{v}) = 0$   2. The Euler equation (3 components) $\\frac{\\partial \\mathbf{v}}{\\partial t}+(\\mathbf{v}\\cdot \\nabla)\\mathbf{v} = -\\frac{1}{\\rho}\\nabla p$   3. A statement of the fact, that there is no dissipation of energy $\\frac{d s}{d t} = 0$ ($s$ is entropy per unit mass) My question: how to express the last equation in terms of the actual variables we are using? We have to assume some form of the equation of state for the fluid in order to do it right? Landau in his typical fashion glance over it, assuming the readers' perfect understanding of thermodynamics, which is not my case unfortunately. PS. The question is a little related to a more general one I posted yesterday: Explicit form of the entropy production in hydrodynamics"}
{"id":"25094","title":"What frame(s) of reference are used to measure the rotation of the Sun around the galaxy ?","text":"I can find various speeds and estimated durations listed at numerous places but none specifically describe the frame of reference. Possible options as example of kind of answer I expect.   * Local Galactic cluster    * Distance quasars    * The cosmic background radiation? **\\--------- UPDATE ---------** Thanks AIB and voithos. Lot of reading for me. Though technically, I still don't have an answer that meets the following criteria.   1. rotational velocity(average preferably) of sol around best estimate of center of Milky-way galaxy.    2. publicly available reference(I don't have immediate access to some of the books given)    3. frame of reference external to Milky-way galaxy. As I note below the only reference (wmap5basic_reprint.pdf) I can read that uses an external frame of reference doesn't specifically state the vector is rotational (despite wikipedia article assuming such). The topic is barely touched on in that paper. I realised the speeds are variable. What I had not realised is that the whole idea of a (relatively) clearly defined x orbiting y system doesn't really scale up well from the local solar system to the galactic scale. The galaxy is more like a whirlpool or tornado compared to the \"clockwork\" appearance of the solar system. Although both the solar system and galaxy are constantly(very slowly) changing \"fluid\" rotational systems, the galaxy is obviously far more fluid than the solar system. Also we have not yet been able to observe anything about it's center. Or in other words, we are not \"orbiting\" the galaxy, we are part of the galaxy. I suspect the topic is more in the realm of \"fluid dynamics\" than \"orbital mechanics\" I've accepted AIB's answer as the most enlightening to me personally. Also, it would appear I have wiki-sidebar blindness. Apologies for that. FYI   The paper referencing the speed relative to the CMB, as mentioned in wikipedia article can be found here http:\/\/cmbdata.gsfc.nasa.gov\/product\/map\/dr3\/pub_papers\/fiveyear\/basic_results\/wmap5basic_reprint.pdf The relevant section appears to be 7.3.1. \"... implies a Solar System peculiar velocity of 369.0 ± 0.9kms-1 with respect to the CMB rest frame.\" Although it's not obvious to me what vector that velocity is along. Though Dipole Anisotropy in the COBE DMR First-Year Sky Maps gives a specific velocity(including vector) for the local galactic group in relation to the CMB rest frame \"implied velocity of the Local Group with respect to the CMB rest frame is 627 +\/- 22 km\/s toward (l,b) = (276 +\/- 3 deg, 30 +\/- 3 deg).\" FYI   Other reference frames that are external to the local galaxy are \"The Supergalactic coordinate system\""}
{"id":"105505","title":"Proton gas density","text":"As far as I know the lightest gas is hydrogen due to low mass of its nucleus, but what if we were to somehow strip hydrogen atoms of electrons and enclose protons in a container made of teflon (high electronegativity). Would such a proton-only gas be stable? And what would be its density under atmospheric pressure (assume teflon envelope around the gas)? Would electrostatic effects between positively charged protons contribute to lowering density of such gas?"}
{"id":"105503","title":"Would gravity on the surface of a planet which is being consumed by a black hole change?","text":"Assuming that the black hole starts grow in the exact center of the planet and that the general structure of the planet does not degrade as it is eaten from the inside, would the gravity on the surface of the planet be affected by the black hole growing. My uneducated guess is that as the matter becomes more compacted the gravity on the surface would actually decrease due to the fact that the distance to the mass, with respect to the surface, is increasing. Also would the evaporation of the black hole actually decrease the total amount of mass in the center of the planet?"}
{"id":"134958","title":"Why do we need 2. Quantization of the Dirac Equation","text":"As a Mathematician reading about the Dirac equation on the internet, leaves me with a great deal of confusion, about it. So let me start with its definition: The **Dirac equation** , is given by $ i \\hbar \\gamma^\\mu \\partial_\\mu \\psi = m c\\cdot \\psi $ where the Dirac matrices $\\gamma^\\mu$ are defined by $\\gamma^\\mu\\gamma^\\nu + \\gamma^\\nu\\gamma^\\mu = \\eta^{\\mu\\nu}$ and where $\\psi$ is a \"solution\". Ok, the first deal of confusion already starts with the $\\psi$'s. It seems that people freely see them as spinor valued functios or as \"operator fields\". But if I understand this correctly, seeing them as operators, is not part of the original picture, but was later added as the so called **second quantization**. Right? Now my question is the following: Why do we need this second quantization of the Dirac equation? What experiments can not be described by the original Dirac equation? Maybe there is a list somewhere or such."}
{"id":"134955","title":"Rotational symmetry in integration","text":"Can someone please tell me why $$4\\int d^4x \\, x^\\mu x^\\nu ~=~\\int d^4x \\, g^{\\mu\\nu}x^2 $$ by some rotational symmetry argument?"}
{"id":"63896","title":"Purcell and the Magnetic Field","text":"When showing that the magnetic field derived from the curl being proportional to current and the divergence being zero is unique, Purcell says that at some sufficiently remote enclosing boundary the difference of two magnetic fields (two possible fields from the curl), D, would take a constant value. Why is this true?"}
{"id":"99140","title":"If the quarks in a neutron are (up,down,down), why isn't it negatively charged?","text":"If the quarks in a neutron are (up,down,down), why isn't it negatively charged? Excuse the silly question, just wondering."}
{"id":"99148","title":"spread of fock state distribution and infinite revival time of rabi oscillation in spontaneous emission","text":"In cavity QED for a 2-level atom, the revival time for oscillation b\/w the states $\\left|\\ e\\ 0\\right\\rangle$ and $\\left|\\ g\\ 1\\right\\rangle$ (absorbing the same photon that is emitted) is said to be infinity because in general, Rabi oscillation revival time is proportional to the square root of the mean of the occupancies of Fock states. Why should the spread of the Fock states distribution go to large values or close to infinity in the spontaneous emission case, to make the revival time infinity ? Refer :- http:\/\/prl.aps.org\/abstract\/PRL\/v94\/i1\/e010401 for a description of Rabi Oscillations Revival"}
{"id":"62761","title":"Lecture Notes confusion: Constructing the Einstein Equation","text":"This question is on the construction of the Einstein Field Equation. In my notes, it is said that > The most general form of the Ricci tensor $R_{ab}$ is > $$R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}$$ where $R$ is the Ricci scalar. Why is this the most general form (involving up to the second derivative of the metric --- by definition of $R_{ab}$)? I suppose there are symmetry and degrees of freedom arguments. But I can only see why the LHS is a _possible_ form. I can't se why it is the most general form... > Taking the covariant derivative $\\nabla_a$ of the expression above gives > $$C=\\frac{1}{2}$$. This I understand. > Compare the resulting expression with the Poisson equation gives > $$A=\\frac{8\\pi G}{c^4}$$. This I _don't_ understand --- perhaps I am being silly again... but still. I assume the \"Poisson equation\" referred to here is $$E^i{}_i=4\\pi\\rho G$$ where $E^i{}_i$ is the tidal tensor and may be expressed as $$E^i{}_i=R^i{}_{aib}T^aT^b$$ where $T^a$ is the tangent vector of the geodesic. So contracting $$R_{ab}=AT_{ab}+Bg_{ab}+CRg_{ab}$$ with $T^aT^b$ gives $$4\\pi\\rho G=AT_{ab}T^aT^b$$. But then what? Or perhaps I have already made a mistake? I only know the energy-momentum tensor $T_{ab}$ to be of the form $$\\begin{pmatrix}H&\\pi_i\\\\\\\\\\frac{s_i}{c}&T_{ij}\\end{pmatrix}$$ where $H$ is the energy density, $\\pi_i$ is the momentum density, $s_i$ is the energy flux. But I don't understand how it leads to $$A=\\frac{8\\pi G}{c^4}$$."}
{"id":"71618","title":"Hollow gold bar","text":"A scammer got a hollow gold bar and fills it with a combination of lead and air, with the same average density as gold. What's the simplest way of discovering the fraud? I know that x-rays will see it, but are there simple means for analyzing it? (without destroying it)"}
{"id":"18423","title":"Conservation of momentum leading to damage","text":"What would be an intuitive way to damage objects in a physics game using impulses? Since momentum is conserved, so is impulse (the change in momentum for any two time periods) in a closed system. So if I have the impulse of a collision (that is, the change in momentum during a time interval for one of the two bodies) what would be the intuitive way to damage both the objects. Should each object receive the same damage based on the impulse of the overall system, or should the damage incurred be split unevenly based on the relative mass of each object? My intuition is that objects with larger mass would have higher hit-points, and so would be less affected if the same damage was incurred to both the objects in the collision."}
{"id":"15615","title":"Equation that tells me the rpm and mass of a spinning disk needed to keep a second large mass stable using gyroscopic effects","text":"I am trying to figure out how large of a mass and how quickly I need to spin said mass to keep a two-wheeled robot stable. Ideally, I am looking for a formula that relates _m_ 1=mass of robot, _m_ 2=mass of spinning disk, _v_ =rotational speed of disk (rpm), and some sort of stability factor _s_ (or amount of mass _m_ 1 that it can counteract). Is there any such formula, and what would it be? Thanks!"}
{"id":"76344","title":"Making an equal amount of positive and negative energy?","text":"I was wondering if it would be possible to create an amount of positive energy out of a vacuum, in addition to an equal amount of negative energy, thus not violating the first law of thermodynamics, since the total amount of energy in the universe would remain the same. It could be like an \"energy loan,\" with the positive energy being like the \"loan money,\" and the negative energy being like the \"debt.\" Is this theoretically (or even practically) possible? For example, I am asking if you could create 200 units of positive energy out of a vacuum, as long as you also created 200 units of negative energy too, so as not to violate the second law of thermodynamics. By negative energy, I mean energy that is below 0 degrees Kelvin. I came across the idea as I was reading about the zero-energy universe theory. It suggested that this mechanism was what made the universe, and that it could have been started by an unusually large quantum fluctuation. Does this idea conflict with the second law of thermodynamics in any way? Would entropy have to be considered, since negative energy reduces entropy, violating the second law? If so, is there a way of changing the idea a bit so there as to avoid the problem? Could \"quantum interest\" account for the missing entropy? Do any restrictions apply on the negative and positive energy, such as quantum inequalities? Can the positive and negative energy be separated? Is there a way to get around the restrictions, if there are any? Finally, if all of this is possible, could it be done? Would a possible method be to recreate an unusually large quantum fluctuation, as described in the second paragraph? Are there other ways of making the \"energy loan\"?"}
{"id":"113859","title":"Is Ehrenfest theorem equivalent to Bohr's Correspondence Principle?","text":"Ehrenfest theorem is usually dubbed as the quantum mechanical equivalent of Newton's law and Griffiths states, in the first chapter of his textbook, that Ehrenfests theorem enables us to work with expectation values in quantum mechanics as _Ehrenfest theorem tells us that expectation values obey classical laws._ On the other hand, the correspondence principle states that _the behavior of systems described by the theory of quantum mechanics reproduces classical physics in the limit of large quantum numbers._ Are the two logically equivalent? References: http:\/\/farside.ph.utexas.edu\/teaching\/qm\/lectures\/node31.html"}
{"id":"107613","title":"Functional integral aproach for Feynman rules","text":"I am familiar with the basic ideas of quantum field theory but I feel uncomfortable when I have to derive Feynman rules by myself for a given action (for example in non-linear sigma models or electrodynamics). Could you provide a list of good books, articles, lecture notes and\/or any other sources (except Peskin), where Wick's theorem and the Feynman rules are derived from the functional integral **in detail** , preferably with examples?"}
{"id":"107885","title":"Gravity force and dark energy","text":"If gravity is a fundamental force which bends spacetime and dark energy is energy which stretches spacetime, what is the difference between the terms force and energy?"}
{"id":"107617","title":"dual variables for lattice fermions","text":"I am quite familiar with duality transformations for lattice spin systems (i.e. systems with global $O(n)$ symmetry) and pure gauge systems (i.e. local $SU(n)$). However, after searching for a bit, I can't seem to find literature on dual variables for lattice fermions (systems with inner products of grassmann fields). For the spin and gauge cases we do harmonic analysis and fourier expanded the exponential of the action (or a character expansion for the gauge case). Is there an analogous process for lattice actions with grassmann fields? Thanks."}
{"id":"78038","title":"Why does Joule heating not occur when no current flows through a conductor?","text":"Joule heating happens every time when the conduction electrons transfer kinetic energy to the conductor's atoms through collisions, causing these conductor's atoms to increase their kinetic and vibrational energy which manifests as heat. Then, why wouldn't it happen when no current is flowing through the conductor? When there is no current, the electrons are still moving randomly at a speed of $\\mathrm{~10^5\\ m\/s}$, but at a zero average velocity. Then, why don't these electrons collide with the atomic ions making up the system and transfer energy to them causing them to heat up even when no net current is flowing?"}
{"id":"112103","title":"String vibration","text":"In many textbooks an illustration of a vibrating string at a fundamental mode shown (and wikipedia) shown like this http:\/\/en.wikipedia.org\/wiki\/File:Standing_waves_on_a_string.gif). However if you watch youtube videos of string vibrating in slow motion you see something like this (http:\/\/projects.kmi.open.ac.uk\/role\/moodle\/pluginfile.php\/1239\/mod_page\/content\/1\/ta212_2_008i.jpg) So which one is this? Is the first one just a schematic version of the second? Also the notion of a wave propagation in a string sort of relevant only to the second pictire only? Or am I wrong? Please clarify, I think I am missing something fundamental here. Thanks"}
{"id":"18390","title":"Focusing laser & off-axis illumination & diffraction limit","text":"I've thought I had a good understanding how resolution enhancement tricks works for projection lithography, until I tried to understand if it's possible to get sub-diffraction performance for focused laser beam: Let's assume we have a laser with nice and shiny perfect aspherical focusing lens (with performance greatly exceeding diffraction limit for it's NA). This system moves around and draw stuff. The question is - can we get enhanced resolution if we apply usual lithography trick of annular\/quadruple illumination? Obviously, we won't reduce aberrations thanks to off-axis illumination, as there are none."}
{"id":"112109","title":"Casimir Forces and its associated Feynman Propagator","text":"This is a continuation to my previous question, in which I began an attempt solve the Casimir Force problem using path integrals. As one of the answers there suggest I solve the Feynman propagator subject to the boundary conditions $x=0$ and $x=L$ at the plate boundaries. The equation for Feynman propagator is $$ (\\Box^2+m^2)\\Delta_F(x-x') = -\\delta(x-x') $$ The solution to the free field is $$ \\Delta_F(x-x') = \\lim_{\\epsilon\\rightarrow0}\\int \\frac{d^4p}{(2\\pi)^4}\\frac{e^{ip_{\\mu}(x^{\\mu}-x'^{\\mu})}}{p^2-m^2+i\\epsilon} $$ What would be the boundary conditions that I have to exactly impose ? Imposing a boundary condition would mean, I think we might have to introduce the a new function (I don't if am right, but this is in general true for Green's function I guess) $$ \\Delta_F(x-x') \\rightarrow \\Delta_F(x-x') + F(x-x') $$ where $F(x-x')$ is such that it satisfies the Boundary condition. Now my question is in case I have boundary condition (like below) how do I solve the differential equation for the boundary conditions like, (take plates to be at $z=0$ and $z=L$) $$ \\Delta_F(x-x')\\bigg|_{z=0} = \\Delta_F(x-x')\\bigg|_{z=L} = 0 $$ EDIT 1: It just occurred to me that there might be short route to this problem with some conceptual reasoning, I gave this a try.. Considering the region between the plates, I know the momentum is quantised in the z-direction, so I have (which is some sense imposed by the boundary condtions) $$ p_z = \\frac{n\\pi}{L} $$ Now using the Feynman propagator in momentum representation, which is $$\\widetilde\\Delta_F(p) = \\frac{1}{(p^0)^2-(\\textbf p^2+m^2)+i\\epsilon}$$ In this I can substitute for, $p_z$, which will give me $$ \\widetilde\\Delta_F(p) = \\frac{1}{(p^0)^2-(p_x^2+p_y^2+\\Big(\\frac{n\\pi}{L}\\Big)^2)+m^2)+i\\epsilon} $$ Now can I get back to position representation, but with integral on $p_z$ replaced by a sum over $n$. Am I right in doing this procedure ? EDIT 2 : Following the procedure that I have mentioned, for a simple (1+1) case of the Feynman propagator in position representation, I have $$ \\Delta_F(x-x') = \\sum_{n=1}^\\infty\\int\\frac{dp_0}{(2\\pi)^2}\\frac{e^{ip_0(x^0-x'^0)}e^{i\\frac{n\\pi}{L}(z-z')}}{(p^0)^2-\\big(\\big(\\frac{n\\pi}{L}\\big)^2+m^2\\big)} $$ EDIT 3 : $$ \\text{Tr}\\log{\\Delta} = - \\sum_n \\int dp_0 \\log{\\bigg(p_0^2 - \\bigg(\\frac{n\\pi}{L}\\bigg)^2 + m^2\\bigg)} $$ But this term seems to diverge, how does one obtain a cutoff in the context of this problem. (A cutoff for $p_0$ integral is also needed I guess)."}
{"id":"18396","title":"Who is right? Force exerted on a body. Me or my friend?","text":"> Three masses $m_1 = 3\\text{ kg}$, $m_2 = 9\\text{ kg}$ and $m_3 = 6\\text{ > kg}$ hang from three identical springs in a motionless elevator. The > elevator is moving downward with a velocity of $v = -2\\text{ m\/s}$ but > accelerating upward with an acceleration of $a = 5\\ \\mathrm{m\/s^2}$. (Note: > an upward acceleration when the elevator is moving down means the elevator > is slowing down.) What is the force the bottom spring exerts on the bottom > mass? Take $g = 10\\ \\mathrm{m\/s^2}$. **My argument** The elevator is steadily slowing down and it get an upward acceleration. So as it goes down the net force is $$\\begin{gather}F_s - mg = ma \\\\\\ F_s = 6(10 + 5) = 6(15) = 90\\text{ N}\\end{gather}$$ But this is the force the spring has. It is the same as the force EXERTED on the body attached? Is there a Newton's third law involved? **Friend's argument** He says that BECAUSE it has an upward acceleration and it is going down that we need to be concern with $$Fs = mg - ma = 90 - 60 = 30\\text{ N}$$"}
{"id":"39363","title":"Using Quantum Teleportation in a way to have the effect of matter teleportation","text":"So, I understand that quantum teleportation is the transfer of a quantum state from one particle or system of particles and its correlations to another receiving system consisting of particle(s). Can't this be used in a way to achieve teleportation as depicted in sci-fi? You start out with a translationally entangled particle pair (position and momentum are correlated) as described by this paper: http:\/\/pra.aps.org\/abstract\/PRA\/v61\/i5\/e052104 Now according to these papers, you can theoretically teleport the position and momentum information of particle. Atomic teleportation of the external degrees of freedom, (Their position and momentum) http:\/\/pra.aps.org\/abstract\/PRA\/v49\/i2\/p1473_1 http:\/\/prl.aps.org\/abstract\/PRL\/v86\/i14\/p3180_1 http:\/\/iopscience.iop.org\/0295-5075\/75\/6\/847 Firstly, I need someone to help me understand **exactly what is being teleported in regards to the papers I've listed**. I understand completely the discrete case of quantum teleportation with spin or polarization, I'm not asking for that. This is what I know: 1) Start out with a translationally entangled pair. 2) Interact an \"input\" particle with one of the pairs. 3) Make a measurement of the input particle with one of the pairs' position and momentum. (This is the step I don't really understand) But I know that this is analogous to the Bell state measurement in discrete quantum teleportation that is widely described everywhere. 4) This result is communicated to the other laboratory where appropriate \"shifts\" of position and momenta are done to the entangled pair just like in the discrete case again. So, if position and momentum information is teleported, then does this mean that if the input particle was propagating in the X-direction like a wave- packet, then after teleportation, the receiving particle will now move in the X-direction relative to its original position? This is quite confusing to me. I want to understand this because I want to ask the trillion dollar question, \"What if you are to replace the input particle with a input MOLECULE?\" For a diatomic molecule, its simply two atoms of the same type. According to this wikipedia article: http:\/\/en.wikipedia.org\/wiki\/Linear_combination_of_atomic_orbitals I know there are vibrational states in molecules which ARE essentially position and momenta information right? If we are to set up TWO translationally entangled pairs of atoms of the same type... And have them simultaneously interact with the molecule in step 3), _then can't we do the appropriate SHIFTS of the entangled pairs of atoms to make them turn into the input molecule?!_ So, the overall effect is that if you start out with H2 in Lab A, and had a translationally entangled pair for each atom, then you'll end up with H2 in Lab B after teleportation which is exactly what we want if we want to stick close to the sci-fi sense. If what I am saying makes sense, then can't this be in principle be extrapolated to larger molecules and eventually cells, and organs, to an entire human? I know that this is a large jump... However, what I'm trying to say is that according to this kinda scheme I'm asking about, its sticking to real physics, and its essentially accomplishing teleportation in the sense people know about. Do I make sense?"}
{"id":"68251","title":"A few questions related to frame dragging","text":"I am trying to get my head around a few concepts related to frame dragging and related physics. In regards to black holes that have no charge and all their mass is tied up in rotational kinetic energy then the ergosphere is a maximum size. By definition, as I see it defined, the outer boundary of the ergosphere is the point at which a photon becomes stationary with respect to outside observation if it is **orbiting** retrograde. The radius at the **equator** of this maximal case is $2GM$ making it twice the inner event horizon radius if I am understanding correctly. The innermost stable orbit for massive particles is at GM in a pro-grade direction. If any of this is incorrect please let me know. So my questions are: >   1. Pretty much by definition it is impossible for a massive particle to go > slower than the speed of light relative to outside observers if it is within > the ergosphere - true? What is the limit of this speed relative to outside > observers? I'm looking for the limit before it passes into the inner > horizon. > >   2. Since the frame dragging extends outward (presumably to infinity with > extremely negligible effect) what would happen in the case where you had a > ring of orbiting black holes equally spaced with spin aligned such that the > axis of rotation was tangential to the circle? Without touching event > horizons it would appear to be able to pull massive particles through the > center of the circle in a straight line rather than a curved path as above - > correct? And if the holes were close enough would that allow the ergospheres > to merge (temporarily as I'd assume everything would merge eventually) would > that allow faster than light travel for massive particles in a straight > line? I'm neglecting that the orbits of the holes are unstable and would > radiate gravitational waves and that they are massive enough so tidal forces > are 'small' compared to the particle or mass of particles. > >   3. If the holes above are arranged as stated would this extend the > ergosphere inward toward the center if they are closely spaced at some > distance? I've seen cases for general non-maximal spin cases of two holes > and am unable to determine how this would play out. > > Thanks for taking the time to read and if I put too much in a single question I apologize but didn't want to spam several closely related questions."}
{"id":"102483","title":"Where is the Higgs Field?","text":"Where is the Higgs Field? I get the idea of how it works but I couldn't find anything explaining where it is."}
{"id":"108275","title":"what kind of system respects $SU(N)$ symmetry?","text":"I read this post, Is the symmetry group of two spin 1\/2 particles $SU(2) \\times SU(2)$ or $SU(4)$? If the picked answer is correct, can I believe that an $N$-degenerate system respects $SU(N)$ symmetry? But if so, I am confused in this way: can we understand that a spin $j$ particle contains $2j+1$ internal degenerate states? Then the symmetry of this particle is $SU(2j+1)$, right? If $j>2$, then it is larger than $SU(2)$! I must be wrong somehow since I know little about group theory. So, basically, what kind of system respects $SU(N)$? * * * **Update 1** Since I have not enough reputation, I could not add comment on the linked post, which I am still confused, so I may also post here(related to this issue): There are $M\\geq 2$ particles, if each particle respects $SU(N)$ symmetry, and if   * A. there is no interaction between them, then what is the symmetry group, if   *     * A1.they are non-identical particles?     *     * A2. they are identical particles?     * B. there is pairwise interaction between them, and the interaction is $SU(N)$ invariant,   *     * B0. what do I mean by $SU(N)$ invariant?     *     * B1. symmetry group for non-identical case?     *     * B2. for identical case?"}
{"id":"79348","title":"$t_1$, $t_2$, $t_3$ Hermitian generators of $SU(2)$","text":"What is the exact $SU(2)$ representation to which these Hermitian generators belong? \\begin{equation} t_a=\\\\{t_1,t_2,t_3\\\\}=\\left\\\\{\\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 & 1 & 0 \\\\\\ 1 & 0 & 1 \\\\\\ 0 & 1 & 0 \\end{pmatrix},\\,\\frac{1}{\\sqrt{2}}\\begin{pmatrix} 0 & -i & 0 \\\\\\ i & 0 & -i \\\\\\ 0 & i & 0 \\end{pmatrix},\\,\\begin{pmatrix} 1 & 0 & 0 \\\\\\ 0 & 0 & 0 \\\\\\ 0 & 0 & -1 \\end{pmatrix} \\right\\\\} \\end{equation} I'm a bit puzzled with this, these are not the generators of the triplet representation of $SU(2)$ (the triplet in $SU(2)$ is the adjoint rep which is real and constructed with the structure constants), nevertheless they are used as if they were in much literature. What are really these generators? They seem like the extension to 3 dimensions of the Pauli matrices. The 3 dimensional fundamental rep (this does not make sense to me)? some kind of non-irreducible representation?"}
{"id":"77126","title":"Help with calculating resistance at a given temperature","text":"> The resistance of a bulb filament is $100\\Omega$ at a temperature of > $100^\\circ \\text{C}$. If its temperature coefficient of resistance be $0.005 > \\space \\text{per} ^\\circ \\text{C}$, its resistance will become $200\\Omega$ > at a temperature of > >   1. $300^\\circ\\text{C}$ >   2. $400^\\circ\\text{C}$ >   3. $500^\\circ\\text{C}$ >   4. $200^\\circ\\text{C}$ > Now the linear approximation is $R_t=R_0(1+\\alpha T)$ Therefore \\begin{align*} 100\\Omega &= R_0(1+0.005\\times 100) \\\\\\ \\therefore R_0 &= \\frac{100}{1.5} \\\\\\ \\text{Now } 200 &= \\frac{100}{1.5}(1+0.005\\times t_2) \\\\\\ 0.005\\times t_2 &= 2 \\\\\\ t_2 &=400^\\circ\\text{C} \\end{align*} But there is also this formula $\\alpha=\\frac{R_2-R_1}{R_1(t_2-t_1)}$ \\begin{align*} 0.005 &=\\frac{200-100}{100(t_2-100)} \\\\\\ t_2 &= 1\/0.005 + 100 \\\\\\ t_2 &= 300^\\circ\\text{C} \\end{align*} Why are these answers differing?"}
{"id":"77129","title":"Why do smaller objects become harder to break?","text":"When grabbing a typical tree branch of at least two feet, it's so easy to snap with a less than one inch circumference that even a toddler can do it. However, after breaking it, the smaller halves with multiply from the center, going like this: 1 - BB (before break) 2 - AB (after break) 4 - AB (half of second break equals four) 8 - AB (continuing multiplication factors of two) And so on.... But what I want to know is why this is ... in quantum physics. I want to know why the more you break it, the harder it is to keep breaking it from the center of each half. This applies to tearing as well ... if you tear a cardboard box in half down the middle it's easy, but if you turn it over and continue on it keeps getting harder as the pieces get smaller."}
{"id":"132444","title":"Why is optical orbital angular momentum (OAM) called \"topological charge\"?","text":"The terminology \"topological charge\" is frequent in lots of research papers related to optical vortex or optical OAM, it is used to represent the optical OAM. Why? How to comprehend it?"}
{"id":"506","title":"Why doesn't a bike\/bicycle fall if going with a high speed?","text":"Why does a bike\/bicycle fall when its speed is very low or close to zero and is balanced when going with a high speed?"}
{"id":"82172","title":"Why does a bicycle (without any support of stand) falls down being at rest, but not under motion?","text":"I have always seen a bicycle not standing without any support, it either falls down to the right or to the left, may even to some other direction. But,the same two wheeler when under motion,moves balanced, without falling to either side,even it not falls, when the two wheeler is bent to left or right by the driver, like the one which we could see in the bike races. **So, what makes the bicycle at rest to fall? And what makes it not to fall when under motion?** _MY VIEW ON THE CONCEPT_   If the bicycle is at rest, it will be acted upon by downward force ($mg$,where $m$ is mass and $g$ is acceleration due to gravity). If it is affected just by this force, it should have stood without falling, because there is no force which pulls it to either side to make it fall. I think, it might fall, either due to the direction of wind, or due to the nonuniform distribution of mass, or due to other external forces. If we consider the same bicycle to be under motion, I don't know what makes it not to fall,even affected by nonuniform distribution of force,or direction of wind. I think there might be something related with vectors, angular momentum,and forces like centripetal and centrifugal. (What ever I said, is just my opinion about the concept, I don't claim it to be true. Correct me,if I am wrong anywhere.)"}
{"id":"14079","title":"Why do we fall when we ride bicycle slowly but we don't when riding it fastly","text":"> **Possible Duplicate:**   >  Why doesn't the bike fall if going with a high speed? Why is it that, when we ride bicycle at good speed, it is perfectly balanced and we don't fall. But we lose balance when we ride bicycle at lower speeds."}
{"id":"20234","title":"How do bikes stay upright?","text":"> **Possible Duplicate:**   >  Why doesn't the bike fall if going with a high speed? How does a bike stay upright? I know there are various theories involving castor effects, and gyroscopic motion but as far as I'm aware no one knows for certain how a bike stays up right. What is the best theory right now?"}
{"id":"91528","title":"Is Conformal Symmetry Local or Global?","text":"I'm just brushing up on a bit of CFT, and I'm trying to understand whether conformal symmetry is local or global in the physics sense. Obviously when the metric is viewed as dynamical then the symmetry is local, because essentially then we're dealing with a change of variables under which the metric transforms with a local scale factor $\\Omega = \\Omega(x)$. Usually, however, we think of the metric as fixed. David Tong's excellent notes suggest that in this case the symmetry should be thought of as global. But I'm not sure I agree. Say we work in 2D and have general conformal transformation given by holomorphic function $f(z)$. Under a conformal transformation $z\\to w$ say, viewed actively, a general field $\\Phi$ will transform to $$(\\frac{dw}{dz})^h \\Phi$$ where the prefactor is clearly dependent on the spacetime point. This would suggest that the transformation is local in the physics sense. Perhaps the distinction he is trying to make is between physical and gauge transformations. But then again I might be wrong because I thought that only global transformations had nonzero conserved quantities, and there's definitely a conserved current for conformal symmetry. Could anyone help to clarify this for me?"}
{"id":"103618","title":"What is $\\langle \\sigma_\\mu \\rangle$ $\\langle \\sigma_\\mu \\rangle$ for the Pauli Matrices?","text":"What is \\begin{align} \\sum_{\\mu=0}^{3} \\langle \\sigma_{\\mu} \\rangle^2 = ? \\end{align} $\\sigma_{\\mu}$ are the Pauli matrices. The Bra-Ket notation is used in this question: \\begin{align} \\langle \\sigma_{\\mu} \\rangle = \\langle \\Psi \\lvert \\sigma_{\\mu} \\lvert \\Psi \\rangle , \\end{align} where $\\Psi$ is the Pauli spinor of two complex components."}
{"id":"91520","title":"Derivation of normal shear stress","text":"I am self-studying this note and I am stuck in the derivation of the normal shear stress. Specifically I can't see how the relations (23) and (24) come about. Specifically, what I don't understand is $$ \\tau'_{xx} = \\frac{\\tau_{xx}+\\tau_{yy}}{2}+\\tau_{yx} \\tag{23} $$ and $$ \\tau'_{yy} = \\frac{\\tau_{xx}+\\tau_{yy}}{2}-\\tau_{yx}\\tag{24} $$ Can someone elaborate on the note to make it clearer?"}
{"id":"102480","title":"Are \"confinement\" and \"asymptotic freedom\" two sides of the same coin?","text":"On Wikipedia it says that the two peculiar properties of quantum chromodynamics (QCD) are: confinement and asymptotic freedom. Asymptotic freedom is the idea that at low energies we _cannot_ use perturbation theory because the coupling becomes extremely strong. One the other hand, at sufficiently high energies, the coupling becomes very small and so we can use perturbation theory. This can all be determined by investigating the $\\beta$-function. My question is: > Why is the asymptotic freedom of quarks not considered to be a sufficient > explanation for confinement? My logic: According to asymptotic freedom: at low energies the coupling becomes so strong that quarks cannot be isolated which explains confinement. If I am right (which I'm probably not), then why is confinement still considered to be such a mystery?"}
{"id":"135129","title":"How to do linear stability analysis on this system of ODEs?","text":"I was trying to do linear stability analysis of spring pendulum. I arrived at the differential equations which describe the system. But I am unable to proceed to linear stability analysis. Is it possible to do linear stability analysis on 2nd order differential equations by finding eigen values of Jacobian matrix? The equations are as below : $mr'' - m(L+r)(\\theta')^2-mgcos(\\theta)+kr' = 0$ and $(L+r)\\theta''+2r'\\theta'+gsin(\\theta)=0$ * * * UPDATE : The fixed points are $(mg\/k,0)$ and $(-mg\/k,\\pi)$"}
{"id":"101612","title":"Why in the relativistic quantum mechanics $ \\gamma_4$ name is not used instead of $ \\gamma_5$?","text":"I have seen in the in the Dirac equation $$\\gamma_0,\\gamma_1,\\gamma_2,\\gamma_3.$$ Then I have seen the definition of a new matrix $$\\gamma_5=i\\gamma_0\\gamma_1\\gamma_2\\gamma_3.$$ Now my question is why the name of the new matrix has not been given as $$\\gamma_4.$$ Is there any historical reason behind this or it is simply taken as $$\\gamma_5$$ without any special reason skipping $$\\gamma_4~?$$"}
{"id":"103599","title":"Electron propagator in a strong background B field","text":"Near a magnetar with $B>B_\\rm{QED}$, the strong B field will suppress the Compton scattering cross section of photons with a specific polarization (E-mode). Some references I know deal this problem using time-dependent perturbation in quantum mechanics. My question is there an intuitive way in QED to see how this takes place and why polarization is important in this case. The electron propagator will be modified. How can we include the background field in the filed theory calculation and derive it?"}
{"id":"76838","title":"How many oxygen molecules touch you in your lifetime?","text":"Following up on a comment by BlueRaja to this beautiful answer of Ilmari Karonen, I would like to phrase this follow up question: How many air molecules hit your average human's skin during their lifetime? Actually, make that **how many oxygen molecules do you breathe in your lifetime?** (In case you want some motivation: Ilmari calculates that the chance for a random oxygen molecule in the atmosphere (though I note that local effects could actually play a role for this one) to have existed in that form since the time of _Homo erectus_ is about one in 1014. However, there's a lot of them molecules, so maybe you _do_ brush up with quite a lot of them often.) Any takers?"}
{"id":"103556","title":"If a paper disc is cut into a spiral, does its moment of inertia change?","text":"It is obvious that there is no change in the mass of it and its radius. But the shape of the object does change. Does it mean its moment of inertia will also change?"}
{"id":"43944","title":"outward pressure in an confined electron gas","text":"Suppose i have some electrons stored in a empty shell container with a negative ion layer in the inner surface so the electrons keep bouncing inside without being able to leave the inner cavity. I want to compute the electrostatic pressure that the electrons exert on the shell container. So i assume the electrons have a uniform volumetric charge density $\\rho$, and the container has a radius $R_0$. My derivation is as follows: electric field at radius $r$ is $$\\frac{1}{3 \\epsilon_0} \\frac{\\rho r^3}{r^2} = \\frac{1}{3 \\epsilon_0} \\rho r$$ force on an element of charge $dQ = \\rho dr dA$ is $$F = p dA = \\frac{1}{3 \\epsilon_0} \\rho^2 r dr dA $$ which means that the pressure satisfies $$ \\frac{dp}{dr} = \\frac{1}{3 \\epsilon_0} \\rho^2 r $$ which by simple integration leads to $$ p = \\frac{1}{6 \\epsilon_0} \\rho^2 r^2 $$ evaluated at the shell this gives $$ p(R_0) = \\frac{1}{6 \\epsilon_0} \\rho^2 R_0^2 $$ Which looks reasonable. But the part that i'm not sure is if this result is sensitive to the assumption that the charge density is uniform inside the volume. The relaxed density of the electrons will tend to be maximum near $R_0$. > How can i estimate the relaxed density radial function and\/or how will this > change my result?"}
{"id":"74475","title":"What is physics of Collision between solid and liquid (or gas)?","text":"My mind has been busy recently by this question: What is physics of Collision between solid and liquid (or gas)? What is Conservation law's of such Collision? Why, when we drop a solid object in water, water shows an Strange behavior (infact what is Conservation law's of energy and momentum when we drop a solid object in water )? Figure 1: Strange behavior of water after impact ![Strange behavior of water after impact](http:\/\/wp.streetwise.co\/wp- content\/uploads\/2012\/05\/water-drop.jpg)"}
{"id":"42018","title":"Jupiter Core is Solid Hydrogen","text":"I was watching a show on the science channel about gas giants; there is something I do not understand. I am not a scientist, so this may be obvious to some. I learned that there a three states of an given physical object; solid, liquid, and gas depending on how cold or hot the object is. An easy example is ice, water and steam from coldest to hottest. So the theory is that Jupiter has a super-heated solid and very dense core that is made up of hydrogen. How does a gas like hydrogen become a solid while being super-heated? Is it that the pressure is so much that the gas is compressed into a solid? If so how much pressure does it take to compress hydrogen into a solid? How does the heat play into the equation?"}
{"id":"135140","title":"Elastic Collision","text":"A cube has a side length of 20 cm. An atom in the gas moves around the cube as shown. It continually bounces off the four lateral walls of the cube. The atom has a mass of 6.6×10−27 kg. Because of the elastic collisions with the walls, the atom maintains its speed of 300 m\/s as it moves around. The figure below shows the view from above looking down into the cube at the path of the atom. The repeated hits cause there to be a force applied to each of the four lateral faces. Find the force (in Newtons) that it applies to Face 1. Find the force it applies to Face 2.![enter image description here](http:\/\/i.stack.imgur.com\/mrgk6.jpg) I have tried the thing as F=change in momentum w.r.t time for face 1 chnge inn momentum is 2mv sin(60') how to calculate change in time? or is there any other approach to the problem??"}
{"id":"42015","title":"Huygens Principle and principal of rectilinear propagation of light","text":"Suppose I have an wave source and light waves are radiating from it. If I have a point source, then after a time t, with a radius of ct I will have a circular wave front.By Huygens principle each point on the wave front acts like a secondary source. From this point again light can travel in all directions. So isn't the principle of rectilinear propagation of light violated?![enter image description here](http:\/\/i.stack.imgur.com\/ipOCi.png) Consider the light following the specified path shown with the arrow head."}
{"id":"42017","title":"Is the density operator a mathematical convenience or a 'fundamental' aspect of quantum mechanics?","text":"In quantum mechanics, one makes the distinction between mixed states and pure states. A classic example of a mixed state is a beam of photons in which 50% have spin in the positive $z$-direction and 50% have spin in the positive $x$-direction. Note that this is not the same as a beam of photons, 100% of which are in the state $$ \\frac{1}{\\sqrt{2}}[\\lvert z,+\\rangle + \\lvert x,+\\rangle]. $$ It seems, however, that at least in principle, we could describe this beam of particles as a single pure state in a 'very large' Hilbert space, namely the Hilbert space that is the tensor product of all the $\\sim 10^{23}$ particles (I think that's the proper order of magnitude at least). So then, is the density operator a mathematical convenience, or are there other aspects of quantum mechanics that truly require the density operator to be a 'fundamental' object of the theory? (If what I mean by this is at all unclear, please let me know in the comments, and I will do my best to clarify.)"}
{"id":"134296","title":"How much energy is necessary to set a year to exactly 360 days","text":"How much energy would be necessary to slow down Earth rotation such that a year was 360 days long? In the same spirit: how much energy would be necessary to make the Earth rotate faster around the Sun such that a year was 360 days long? _(Just for fun: how would you do it practically?)_"}
{"id":"20165","title":"How to calculate the electric field at a point in space","text":"Let's say I have a uniformly-charged wire bent into a semi-circle around the origin. How can I find the electric field (magnitude and direction) I'm not even sure if I should use Coulomb's or Gauss' law either. Kind of stuck setting up a solution and starting somewhere."}
{"id":"129758","title":"What is the airspeed velocity of an unladen swallow?","text":"I got asked this question for my physics course at college, i'm not the best at airspeed velocities... > What is the airspeed velocity of an unladen swallow? I still can't find anything to help me, i'm quite stuck and have been googling and asking about on a few forums. Best help I found was this page on the subject. Thank you."}
{"id":"121362","title":"Does the Bohr van Leeuwen Theorem also apply to ferromagnetism?","text":"I know that the Bohr-van Leeuwen theorem shows that there could be not consistent pure classical explanation of dia- and paramagnetism. Does the same theorem also rule out a consistent classical theory of ferromagnetism? Do you have any realiable references for this?"}
{"id":"45245","title":"Does closing curtains 'make your home warmer'?","text":"I mean, in the sense that the act of closing curtains would somehow reduce the amount of heat loss of the house to the outside, thus making it warmer _for a given supply of heating_."}
{"id":"134233","title":"Symmetry breaking under isothermal expansion","text":"Is there any example of a symmetry breaking phase transition in a system of particles under isothermal expansion?"}
{"id":"44116","title":"How to reconstruct information from a graph of an oscillation?","text":"We are given a graph of the position of a wave (amplitude). How can we calculate the wavelength, frequency and the maximum speed of a particle attached to that wave? We have _Speed = wave length $\\times$ frequency_ , _$W=2 \\pi \\times$ frequency_ , $V_{max}=A\\times W$. So how to calculate A? ![enter image description here](http:\/\/i.stack.imgur.com\/9vkUA.png)"}
{"id":"129732","title":"Electrons and energy","text":"I'm an a-level student. While studying quantum physics I came across this statement:\"According to electromagnetic theory an accelerating charge emits energy\". The bottom-line was that an electron (which is an accelerating charge in an atom) should, according to this, emit electromagnetic radiation, but it does not. Can someone please explain the electromagmetic theory and why the electron does not emit energy? Please keep in mind I'm only an a-level student."}
{"id":"70205","title":"Meaning of terms and interpretation in the electric multipole expansion","text":"In section 3.4.1 of Griffiths' _Introduction to Electrodynamics_ , he discusses electric multipole expansion. He derives the formula or the electric potential of a dipole, which I follow, but right after, he begins talking about the electric potential at a large distance, which is as follows: $$V( \\vec{r}) = \\frac{1}{4 \\pi \\epsilon _{0}} \\int \\frac{\\rho (\\vec{r}')}{ℛ} dV$$ $ℛ$ is from some point inside the charge distribution to the point P. **What exactly does $\\vec{r}'$ denote?** Is it the distance from the center of the charge distribution? Next, he uses law of cosines to find the expression for $ℛ^{2} = r^{2}+(r')^{2}-2rr'\\cos\\theta'$. This gives the same question as above, what does the symbol $r'$ mean? Then he defines $ℛ = r(1+ \\epsilon)^{1\/2}$, where $\\epsilon \\equiv \\left(\\frac{r'}{r}\\right)^{2}\\left(\\frac{r'}{r}-2\\cos\\theta^\\prime\\right)$. The proceeding part is where I really get lost: > For points well outside the charge distribution, $\\epsilon$ is much less > than 1, and this invites a binomial expansion. $$\\frac{1}{ℛ} = > \\frac{1}{r}\\left[1- (1\/2) \\epsilon+ (3\/8) \\epsilon ^{2} - (5\/16) \\epsilon > ^{3}+\\ldots\\right]$$ **What is going on in this last step?** And finally, we eventually derive the formula $$V(\\vec{r}) = \\frac{1}{4 \\pi \\epsilon _{0}} \\sum ^{\\infty}_{n=0}\\frac{1}{r^{n+1}} \\int(r')^n\\,P_{n}(\\cos \\theta)\\,\\rho( \\vec{r}')dV $$ Why did we go for all this trouble? If this is supposed to be the electric potential at a large distance, **couldn't we have just used $V( \\vec{r}) = \\frac{1}{4 \\pi \\epsilon _{0}} \\int \\frac{\\rho (\\vec{r}')}{ℛ} dV$**? I don't understand what this equation _actually means_ in physical terms. And in regards to the actual equation, **what is $P_{n}$**? Any help is much appreciated."}
{"id":"70209","title":"How is energy extracted from fusion?","text":"I understand that combining deuterium and tritium will form helium and a neutron. There are three methods to do this (1) tokamak (2) lasers and (3) cold fusion. I would like to know after helium is formed. How is that energy extracted from tokamak and stored?"}
{"id":"18985","title":"What is wrong in this representation of relative reference systems?","text":"Up until now I've explained relative time to myself as looking at the 3D world from different four-dimensional perspectives, analogous to how looking at a 2D-ish object (eg. a sheet of paper) from different angles makes it appear to change shape. Also, I've used this representation below to explain how the time at which an event occurs could change depending on the body's reference system. This representation uses two bodies that move relative to each other, $B1$ and $B2$, to represent the time of occurrence of events $E1$ and $E2$. ![](http:\/\/i.stack.imgur.com\/KsZ2r.png) ( _Not very precise -- blame MS Paint :P -- but my intent should be clear._) I am not sure, but as far as I know $t(E1) \\le t(E2)$ should hold true for any reference system (order should be preserved), which means that the drawing is correct. But I just realized that in certain cases, it is possible for $E2$ to occur before $E1$ (according to my representation): ![](http:\/\/i.stack.imgur.com\/8X5n9.png) Here you can see that $B3$ and $B2$ have entirely different perceptions on the order of the two events, but that would break causality, so it cannot be true. What is wrong in my representation of relative systems of reference?"}
{"id":"122100","title":"Amount of fluid that moves along a path","text":"Let suppose that $\\vec{F}$ is the velocity vector field of of a fluid in space, and $\\vec{s}$ is a straight path of arbitrary length. Let's suppose that $\\vec{F}$ and $\\vec{s}$ are parallels and point in the same direction, so that $\\vec{F} \\cdot \\vec{s} = F s$. How can we account for the total amount of fluid that moves along the path (per unit of time). **Related question**"}
{"id":"5492","title":"Astronomical detection significance from magnitude error","text":"At this website: http:\/\/heasarc.gsfc.nasa.gov\/docs\/swift\/analysis\/threads\/uvot_thread_afterglows.html The passage at the bottom states that a V-band magnitude of 17.62, with an error $\\pm$0.02 is a 49.4-$\\sigma$ detection significance. How is this value calculated? Could you provide the working? I have a similar problem, with a set of magnitudes and errors, and would like to know at what magnitude limit I can claim a statistically significant (6-sigma) detection. Thanks"}
{"id":"83893","title":"Selection rules in Stark effect","text":"The energy level of an electron could be shifted by an electric field. $\\langle n, l,m|[L_z,z]|n^{\\prime},l^{\\prime},m^{\\prime}\\rangle=(m-m^{\\prime})\\hbar \\langle n, l,m|z|n^{\\prime},l^{\\prime},m^{\\prime}\\rangle$. Because $[L_z,z]=0$, we must have $m= m^{\\prime}$ for non-vanishing $\\langle n, l,m|z|n^{\\prime},l^{\\prime},m^{\\prime}\\rangle$. This is one selection rule in Stark effect. Then how to rigorously prove $l^{\\prime}=l\\pm1$ is the other selection rule for $\\langle n, l,m|z|n^{\\prime},l^{\\prime},m^{\\prime}\\rangle\\neq0$?"}
{"id":"83891","title":"Mass-spring system on an incline","text":"![block-spring system](http:\/\/i.stack.imgur.com\/oCVEr.jpg) I am reviewing for an exam next week, and this is one of the questions I am stuck on. I have the mass-spring system above with spring constant $k$ on a frictionless incline. I would like to find the total energy of the system at any time $t$. I know that the total energy of the system is going to be the sum $E = K + U_g + U_s $ My first question is that I am confused about how to approach the problem. If I take my axes along the ramp, then the problem becomes one dimensional. My idea was set up my coordinate system like that so I could describe the displacement of the mass around it's equilibrium position on the ramp so I could find $K + U_s$, and then switch back to a more traditional coordinate system where I take my axes along the perpendicular sides of the ramp to find $U_g$. Does this approach work? If not what should I be doing?"}
{"id":"41639","title":"What characterizes a metallic sound, and why do metals have a metallic sound?","text":"We know that when we strike a metal, it usually has a characteristic \"sharp\" sound, unlike when we strike wood, say. What characterizes this \"metallic sound\"? Does it have a well-defined power spectrum? What are its generic properties? Also, why do metals have a metallic sound? My best guess would be that metals have high Young's \/ Bulk modulus and so the resonances for typically sized metals should be at fairly high frequencies, and since they are very elastic, they wouldn't dissipate energy all that easily, so the resonances would have small FWHM and hence be sharp. So my guess is that a \"metallic sound\" has distinct resonance peaks in its power spectrum, whereas for a piece of wood, for example, these peaks merge into each other to create a continuum."}
{"id":"47672","title":"Element of area in 4-dimensional space-time","text":"How would you proof that $$ \\mathrm {Tr} (\\mathbf{S\\cdot \\bar S })=0$$ where $\\mathbf S$ is an element of area delimited for the 4-vectors $\\mathbf u$ and $\\mathbf v$ given by $$S^{\\alpha \\beta}\\equiv u^\\alpha v^\\beta-u^\\beta v^\\alpha$$ and $$\\bar S^{\\alpha \\beta}\\equiv \\frac{1}{2}\\epsilon^{\\alpha \\beta \\gamma \\delta} S_{\\gamma \\delta}$$ is the dual of $\\mathbf S$. I used an analogy with the Maxwell field tensor $\\mathbf T$. I know that $\\mathrm {Tr} (\\mathbf{T\\cdot \\bar T })=\\frac{4}{c}\\mathbf E \\cdot \\mathbf B$. Building the analog vectors $\\mathbf E$ and $\\mathbf B$ but with $\\mathbf S$ I get that $ \\mathrm {Tr} (\\mathbf{S\\cdot \\bar S })=0$. But I'm looking for a more ilustrative solution to this problem. Any ideas?"}
{"id":"47678","title":"What is the Hubbard-Holstein model?","text":"Please explain as simply as possible what the Hubbard-Holtstein model is and what it is used for."}
{"id":"8295","title":"Einstein's box - unclear about Bohr's retort","text":"I was reading a book on the history of Quantum Mechanics and I got intrigued by the _gendankenexperiment_ proposed by Einstein to Bohr at the 6th Solvay conference in 1930. For context, the thought experiment is a failed attempt by Einstein to disprove Heisenberg's Uncertainty Principle. ![Einstein's box as drawn by Bohr](http:\/\/upload.wikimedia.org\/wikipedia\/en\/6\/65\/Ebohr4.gif) > Einstein considers a box (called Einstein's box; see figure) containing > electromagnetic radiation and a clock which controls the opening of a > shutter which covers a hole made in one of the walls of the box. The shutter > uncovers the hole for a time Δt which can be chosen arbitrarily. During the > opening, we are to suppose that a photon, from among those inside the box, > escapes through the hole. In this way a wave of limited spatial extension > has been created, following the explanation given above. In order to > challenge the indeterminacy relation between time and energy, it is > necessary to find a way to determine with adequate precision the energy that > the photon has brought with it. At this point, Einstein turns to his > celebrated relation between mass and energy of special relativity: $E = > mc^2$. From this it follows that knowledge of the mass of an object provides > a precise indication about its energy.   >  \\--source Bohr's response was quite surprising: there was uncertainty in the time because the clock changed position in a gravitational field and thus it's rate could not be measured precisely. > Bohr showed that [...] the box would have to be suspended on a spring in the > middle of a gravitational field. [...] After the release of a photon, > weights could be added to the box to restore it to its original position and > this would allow us to determine the weight. [...] The inevitable > uncertainty of the position of the box translates into an uncertainty in the > position of the pointer and of the determination of weight and therefore of > energy. On the other hand, since the system is immersed in a gravitational > field which varies with the position, according to the principle of > equivalence the uncertainty in the position of the clock implies an > uncertainty with respect to its measurement of time and therefore of the > value of the interval Δt. **Question:** How can Bohr invoke a General Relativity concept when Quantum Mechanics is notoriously incompatible with it? Shouldn't HUP hold up with only the support of (relativistic) quantum mechanics? Clarifying a bit what my doubt is\/was: I thought that HUP was intrinsic to QM, a derived principle from operator non-commutability. QM shouldn't need GR concepts to be self consistent. In other words - if GR did not exist, relativistic QM would be a perfectly happy theory. I was surprised it's not the case."}
{"id":"8294","title":"How much of a star falls into a black hole?","text":"http:\/\/blogs.discovermagazine.com\/badastronomy\/2011\/04\/05\/astronomers-may- have-witnessed-a-star-torn-apart-by-a-black-hole\/ A lot of the star in the disc, a lot of the star in the jets, precisely how much of the star actually falls into the black hole?"}
{"id":"102283","title":"How would you go about finding the natural frequencies (resonance frequency)","text":"How would you go about finding the natural frequencies of solid materials like wood (e.g., teak, pine), stone (e.g., marble, granite) liquid, etc?"}
{"id":"82381","title":"Interpretation of Free Damped Vibrations","text":"I'm studying vibrations; so I'm using Beer-Johnston-Cornwell Dynamics book. I am worry about the equation for Underdamped Vibration, which in the book it is: $$x_{(t)}=x_0e^{-\\lambda t}\\sin(\\omega_d+\\phi)$$; where $$\\omega_d=\\sqrt{\\omega_n^2+c^2\/4m^2}$$. I think that $x_0$ would be replaced by the result vector of constants $c_1$ and $c_2$ affected by the factor $e^{-\\lambda t}$. It means, an $x_m$ or an amplitud, but not the initial position, because it could be 0, with an inicial velocity. Also the graphic, it depicts the boundary equation with $x_0$. I attached a picture. Can you help me and comment? How I should interpret $x_0$? Maybe I misunderstood this topic. ![damp](http:\/\/i.stack.imgur.com\/obyol.jpg)"}
{"id":"110169","title":"What does \"optical conductivity\" mean?","text":"Does it just mean \"AC electric conductivity\"? If so, why have a special name for it, and why mention optical specifically? The wikipedia page on it is very sparse. This (warning, PDF) document just says: > The term “optical conductivity” means the electrical conductivity in the > presence of an alternating electric field. Which sounds exactly like AC conductivity to me. Is it different? If not, then why the special name?"}
{"id":"110162","title":"What Happens When A Gravitational Wave Interacts With Another One?","text":"If two gravitational waves came in contact with each other what would happen? In another question entirely, what happens when a higher gravitational field interacts with a weaker one."}
{"id":"126739","title":"Determine the number of days with North-East wind direction from the number of days with North and East wind direction?","text":"I have some data like: Wind flow from north direction = “X” Numbers of days. Wind flow from east direction = “Y” Numbers of days. Then is there any formula to know numbers of days wind flows from North-East direction?"}
{"id":"88671","title":"Will larger balls make a Newton's cradle swing more stable?","text":"I got a rather small Netwon's cradle and when I start it, the effect is not very good since all the balls start to swing where the effect we want is obvious. The balls are small and I wonder if larger balls would make the swing more ideal, like only the outer balls appear to move and the 3 inner balls should be still? I know product recommendations are off-topic, but I'm leaning towards buying a cradle with larger balls and perhaps you can advice which alternative to get that best replicates the original ideal cradle where only the outer balls move and the inner balls appear still?"}
{"id":"13896","title":"Spin up, spin down and superposition","text":"I'm just starting to study quantum mechanics. Please explain the error in this thinking: You set up decay of two $\\pi$ mesons and get $2\\mathrm{e}^-$ on Mars and $2\\mathrm{e}^+$ on Earth. On Earth you may or may not measure the spin of those positrons, with 50% probability that they are the same spin. On Mars, your buddy \"immediately afterwards\" takes a $\\mathrm{He}^{2+}$ ion and adds the electrons. If you made the measurement, the $\\mathrm{He}$ has a 50%+ chance of being in a higher-then-base energy level due to Pauli exclusion. If you didn't make the measurement, the $\\mathrm{He}$ has a much lower chance of being in a higher-than-base energy level. Your buddy measures the energy state. Result: You just superluminally transmitted $\\approx 0.8$ bits of information."}
{"id":"64804","title":"What could cause a 30 k g acceleration?","text":"I'm reading the specs of an IC (Cypress 5LP SoC) and it says it's available in 30 k g shock resistance configuration. The fastest acceleration I heard of so far was hitting a golf ball hard, which would be around 1000 g. Does anyone have an example of a mechanical impact of 30 000 g (300 000 m$\\cdot$s$^2$), applicable to electronics devices?"}
{"id":"64803","title":"A rod of length $L$ & mass $M$ is rotating in a circle about one end then calculate tension in the rod at a distance $x$ from the support","text":"A rod of length L & mass M is rotating in a circle about one end then calculate tension in the rod at a distance 'x' from the support ? For its solution why should we take mass of L-x portion of rod instead of taking mass upto x distance from support as we have the formula T = m w2 x I m in confusion...."}
{"id":"116579","title":"What's the advantage of NASA's flying saucer over traditional aerodynamic models?","text":"NASA has recently tested a saucer-shaped spaceship. What's the advantage of this new design over traditional aerodynamic designs? The test launch was performed from within atmosphere which would offer higher air drag. What's the positive side of this that this big thing has been ignored?"}
{"id":"108833","title":"$SU(2)$ gauge symmetry","text":"Take the Lagrangian with one fermion: $$ \\mathcal{L} = -\\frac{1}{4}F^{\\mu\\nu}_aF^a_{\\mu\\nu} + \\bar{\\psi}(i\\gamma^\\mu D_\\mu - m)\\psi$$ where the gauge covariant derivative $D_\\mu = \\partial_\\mu+i\\frac{g}{2}t^aW^a_\\mu$. The Lagrangian is invariant under a local $SU(2)$ transformation: $$ \\psi(x) \\rightarrow \\exp \\left[-i\\theta^a(x)t^a \\right]\\psi(x) $$ $$W^a_\\mu(x) \\rightarrow W^a_\\mu(x) +\\frac{1}{g}\\partial_\\mu\\theta^a(x) + \\epsilon^{abc}\\theta^b(x)W^c_\\mu(x)$$ Often, we say that $W_\\mu^a$ transforms according to the adjoint representation of $SU(2)$ but how can we say that based on the previous equation?"}
{"id":"119567","title":"Does Hall Effect increases the resistance of a wire?","text":"If a current in a wire is flowing perpendicular to a magnetic field, the Hall effect will be observed, which is caused by the forces from magnetic fields 'pushing' the electrons to one side or the wire. So, does it increases the resistance of the wire, as there is now less area for current to flow through due to one side of the wire being occupied by these (assumed)immobile electrons?"}
{"id":"116570","title":"Radioactive stability of some nuclei","text":"While studying radioactivity I found that even the most radioactive substances i.e substances with the shortest half lives do not completely degenerate. Suppose there is a 1 mole sample of an element **X** whose half life is **1 day**. But as the _degeneration equation_ is _exponential_ in nature even after 1 million years some amount of **X** would remain in the sample. What i want to know is how come certain **_atoms of an element possess such stability that they do not generate after a million years_** , **_while some atoms taken form the same sample degenerate after 1 day._** Is it because they have some different particles in their nucleus or something ? What is the role of **quantum tunnelling** ?"}
{"id":"2239","title":"Why does an airplane have more lift near the ground?","text":"I've noticed that an airplane appears to have more lift when it's almost touching the ground then it has 100 feet or more in the air. What causes this to occur?"}
{"id":"69780","title":"An atomic bomb explodes inside of an \"unbreakable\" container which is on a scale. Does the \"weight\" of the container change?","text":"This may or may not be an incredibly stupid thought experiment, but a short time ago I read that most of the \"mass\" in the proton was actually energy from the quarks and gluons, as opposed to the actual mass which was coupled to the Higgs field. This made me start thinking about objects on macroscopic scales and the effects of energy and mass in a closed macroscopic system. I have only a superficial highschool level understanding of relativistic mass, and I have read extremely vigorous debates and arguments about the significance of the mass-energy equivalence, and what it means in terms of the \"weight\" of an object. So my question is that if said atomic bomb were to explode inside of a completely impenetrable container of some sort, would the measured weight of the container change, assuming no radiation was able to escape the container itself?"}
{"id":"51920","title":"Amount of thermal energy in the Earth?","text":"Does anyone know the amount of thermal energy that the Earth's mantle and core possess? I don't mean the maximum limit of electrical power we could generate with geothermal plants, but rather: if you took the Earth and magically cooled it down so that its temperature became homogeneously 0K, what would the change in the Earth's internal energy be? (ignoring the Sun) If we don't have that data, does anyone have an idea how to perform an order of magnitude estimate?"}
{"id":"51922","title":"Uncertainty relation and Energy-Position interference","text":"How would you prove that the simultaneous measurements of position and energy are not subject to interference? I was thinking in calculate the commutation relation between $x$ and $H$ (Because $\\Delta E=\\Delta H$), but I realized that $[H,x]\\neq0$, so I tried to use a more general expression for the Uncertainty Principle that says that if $H_1$ and $H_2$ are Hermitian operators then $\\Delta H_1 \\Delta H_2\\geq\\frac{1}{2}|\\langle [H_1,H_2]\\rangle|$, but again, $[H,x]\\neq0$. Can you suggest me a way to do this? Thanks."}
{"id":"5930","title":"Are collaborations needed to be productive in physics?","text":"If you look at most published articles, or those on the preprint server, you find the vast majority of articles have more than one authors. Are collaborations needed to be productive in physics? In other fields like math, single author articles are the norm."}
{"id":"32062","title":"A wobbly pan on an induction cooker, is it less effective?","text":"According to the Wikipedia, one of the limitations of the induction cooker is that the bottom surface of the pot should be flat. Accordingly, I commented on a question on Seasoned Advise, but I'd like to know whether this is actually true (My username is BaffledCook there). For a wok, it seems reasonable to believe the induction cooker will not be effective, but how about a wobbly pan? How wobbly should it be before it really affects efficiency?"}
{"id":"121203","title":"Calculating size\/weight of the base of a banner to withstand wind load","text":"I want to build what is essentially a banner - a rectangular piece of fabric that will be mounted to a stand and displayed vertically. This object will be outdoors, so the most important factor in it staying upright is that it be able to withstand wind load. I know how to calculate wind load on an object, and I understand that the force of that wind load can be considered to act on the center of mass of the banner, but I'm not sure how to figure the size or weight of the base into the system to determine whether the object will fall over under the wind load. For example, if I were to make a banner 3 feet wide and 6 feet tall, With a base also 3 feet wide, and 3 feet deep, how much would the base need to weigh to withstand a 30 mph gust of wind? (Of course, the ideal answer would show me how to calculate this for myself, to try different scenarios)"}
{"id":"90927","title":"How to find a hyperbolic escape orbit with a starting point and a desired velocity after sphere exit","text":"How would you find an escape orbit which includes a given starting point (not assuming a circular starting orbit) and escapes the sphere of influence of the current body with a given remaining velocity? A problem perhaps similar to Lambert's Problem, but instead of an arrival location and time, it is a final velocity. I have a solution that someone suggested to me, that takes the final velocity as a hyperbolic excess velocity, uses that to calculate specific energy and semi major axis of the departure orbit. However they then assume that you're departure burn will occur at the periapsis of the new orbit and derive the eccentricity, and determine other elements of the departure orbit using that. The approach would work, though you have to use various guesses as the starting position until you find one that results in an orbit departing in the correct direction and the start is always at the periapsis of the departure orbit. I am hoping that there is an answer to this problem that doesn't require starting at the periapsis of the escape orbit, as I think that would broaden the range of potential departure directions from a given point. However I don't know how I would begin to calculate that."}
{"id":"121829","title":"Kac-Moody algebras in 5 dimensional Kaluza-Klein theory","text":"I am trying to make sense to the issue of how does the Kac-Moody algebra encode the symmetries of the non-truncated theory. Let's contextualize a little bit. Ok, so in the 5 dimensional Kaluza-Klein we start with pure (5 dimensional) gravity. Then we assume a $M_4×S^1$ ground state where we have Minkowski space with a circle. The topology of this ground state breaks the 5D diffeomorphism invariance because now only periodic transformations are allowed. So considering a coordinate change $$x\\to{}x'=x+\\xi,$$ we can Fourier expand $$\\xi(x,\\theta)=\\sum{}\\xi(x)e^{in\\theta}.$$ It is known that if we truncate at $n=0$ we get a theory of gravitation in 4 dimensional space-time, electromagnetism and a scalar field (sometimes called the Dilaton). So, the remnants of the initial 5 dimensional diffeo invariance are 4 dimensinal diffeo invariance, $U(1)$ gauge and a scale invariance for the dilaton. In the non truncated theory we will keep all the modes in the previous Fourier expansion and we will have more symmetries. The 4d diffeo and $U(1)$ gauge do survive in the non truncated version but not the scale invariance of the dilaton. So far so good. But, how to know the symmetries of the full theory? well, with Kac-Moody generalization of Poincaré algebras. For example in http:\/\/arxiv.org\/abs\/hep- th\/9410046 says this. But HOW exactly? I would like a clear explanation of how this Kac-Moody algebra encodes the remnant symmetries of the original diffeo 5 and also how the zero modes of this algebra correspond to diffeo4 and $U(1)$ EDIT:: this is the algebra I am talking about $[P_{\\mu}^{(n)},P_{\\nu}^{(m)}]=0$ $[M_{\\mu\\nu}^{(m)},P_{\\lambda}^{(n)}]=i(\\eta_{\\lambda\\nu}P_{\\mu}^{(m+n)}-\\eta_{\\lambda\\mu}P_{\\nu}^{(m+n)})$ $[M_{\\mu\\nu}^{(n)},M_{\\rho\\sigma}^{(m)}]=i(\\eta_{\\nu\\rho}M_{\\mu\\sigma}^{(m+n)}+\\eta_{\\mu\\sigma}M_{\\nu\\rho}^{(m+n)}-\\eta_{\\mu\\rho}M_{\\nu\\sigma}^{(m+n)}-\\eta_{\\nu\\sigma}M_{\\mu\\rho}^{(m+n)})$ $[Q^{(n)},Q^{(m)}]=(n-m)Q^{(n+m)}$ $[Q^{(n)},P^{(m)}_{\\mu}]=-mP^{(n+m)}_{\\mu}$ $[Q^{(n)},M^{(m)}_{\\mu\\nu}]=-mM^{(n+m)}_{\\mu\\nu}$"}
{"id":"121826","title":"Problem with shell model and magnetic moment of Lithium-6","text":"I have a problem with the calculus of magnetic moment of Li-6. The configuration of protons is $1p_{3\/2}$, and the neutrons' one is the same. I have to add the magnetic moment of uncoupled proton and uncoupled neutron. I use the following formula for $J=l+\\frac{1}{2}$ (J is the particle spin): $$ \\frac{\\mu}{\\mu_N}=g_lJ+\\frac{g_s-g_l}{2}$$ For the proton I have: $g_l=1; g_s=5.58 \\rightarrow \\frac{\\mu}{\\mu_N}=J+2.29=3.79$ For the neutron I have: $g_l=0; g_s=-3.82 \\rightarrow \\frac{\\mu}{\\mu_N}=-1.91$ So the total $\\frac{\\mu}{\\mu_N}=3.79-1.91=1.88$, exactly 1 more than the correct value, 0.88! What's wrong?"}
{"id":"121827","title":"Poles for a particle scattered in a delta potential","text":"I am working on problem a professor gave me to get an idea for the research he does, and have hit a point where I'm having a difficult time seeing where I need to go from where I'm at. I would also like to go ahead and apologize for not knowing how to format correctly. I was given that a particle is scattered with the given Hamiltonian: $$ H = P^2 - g\\delta(x) $$ Where $\\delta(x)$ is the Dirac delta function. I was able to find the states in momentum representation, and was supposed to Fourier transform them to get the position representation states. Doing this leads to an integral with simple poles on the real axis, and can be solved by moving the poles above or below the real axis by some constant, applying the residue theorem, and taking the constant to zero. This leads to three different solutions, depending on if I move one pole up and one down (two possibilities), or both poles into a contour. While I was talking to my professor, he mentioned that the solutions only work for certain values of $x$, and that the range of $x$ is given by what makes the arc in the contour go to zero. I'm actually having trouble seeing this, since that introduces an ambiguity in the solutions. If I shift the left pole up and the right pole down and close the contour in the upper plane, this means that x has to be positive, in order to get a decaying exponential in the integral. However there is nothing stopping me from moving the poles in the opposite way, and closing above to obtain a different solution for $x>0$. Is there something wrong in the math, or is the way I move the poles governed by the physical situation I'm interested in? (Which would be no plane waves moving the left for $x>0$ d.) I should mention that I am assuming: $$ |\\psi> = |p> \\+ |\\psi_{sc}> $$ Where p is the incoming momentum and $|\\psi_{sc}>$ is the scattered portion of the wave function. Working in momentum representation, I obtain: $$ \\psi_{sc}(k) = \\frac{g - \\frac{g^2 i}{2p+ g i}}{2\\pi(k^2-p^2)} $$ Where k is the momentum variable, and p is the fixed momentum of the incoming particle. The transform I obtain is: $$ \\psi_{sc}(x) = \\eta \\int_{-\\infty}^{\\infty} \\frac{e^{i k x}}{k^2-p^2} dk $$ This is where I run into the problem with the poles. I know there shouldn't be any waves traveling to the left for $x>0$. My final goal is to check my states by checking the reflection and transmission coefficients and confirming they add up to 1."}
{"id":"27922","title":"black hole no-hair theorems vs. entropy and surface area","text":"I was revisiting some old popular science books a while ago and two statements struck me as incompatible.   * No-hair theorems: a black hole is fully-described by just a few numbers (mass, spin etc) irrespective of the type or configuration of the matter\/energy within the event horizon.   * Surface area measures entropy: you can't reduce the total entropy of the universe by throwing a box of hot gas into a black hole, its size\/entropy will increase by the necessary amount. Suppose I have two black holes, A and B. They are identical and interchangeable, same mass, spin, charge... I have two boxes, BoxA and BoxB. BoxA contains a kilogram of salt in the form of a single crystal and BoxB contains a kilogram of salt as loose powder. **BoxA has less entropy than BoxB** but is otherwise identical. I throw BoxA into A and BoxB into B. The no-hair theorems seem to imply that A and B will increase in size by the same amount. The entropy theorems seem to imply that B will end up larger than A. What am I missing here?"}
{"id":"27924","title":"Particles for all forces: how do they know where to go, and what to avoid?","text":"Here's an intuitive problem which I can't get around, can someone please explain it? Consider a proton P and an electron E moving through the electromagnetic field (or other particles for other forces, same argument). They exert a force upon one another. In classical mechanics this is expressed as their contributing to the field and the field exerts a force back upon them in turn. In quantum mechanics the model is the exchange of a particle. Let's say one such particle X is emitted from P and heads towards E. In the basic scenario, E absorbs it and changes its momentum accordingly. Fine. How does X know where E is going to be by the time it arrives? What's to stop E dodging it, or having some other particle intercept X en route? Are P and E emitting a constant stream of force-carrying particles towards every other non-force-carrying particle in the universe? Doesn't this imply a vast amount of radiation all over the place? I am tempted to shrug of the entire particle exchange as a mere numerical convenience; a discretization of the Maxwell equations perhaps. I am reluctant to say \"virtual particle\" because I suspect that term means something different to what I think it means. Or is it a kind of observer effect: E \"observes\" X in the act of absorbing it, all non-intercepting paths have zero probability when the waveform collapses? Or have I missed the point entirely?"}
{"id":"52784","title":"Is there a maximum amount of photons that can exist in a certain amount of space?","text":"If you have a set amount of space, lets say 10 cubic centimeters, and you would be able to trap photons in there.   If you would then add more and more photons to that space, could you then go on infinitely or would you eventually run into a maximum amount of photons that can be in that space?"}
{"id":"60698","title":"Observer effect, do this mean literally someone or just any interaction with other matter?","text":"I am a layman and was wondering, the quantum observer effect. The regular notion to laymen seems to be literally \"if you look at it\", but as I am coming to understand the world I live in better I feel it means just coming in contact with something. Is this an ongoing question? Whether a particle that is not interacting with anything undergoes changes in state? Though now that confuses me too, If it is in two states at once. What denote say spin to the left from spin to the right, what denotes the origins of the calculations we make?"}
{"id":"99389","title":"Self-adjoint and nonpositive differential operators","text":"I recently tumbled over a statement in a geophysics paper (PDF here). They have a wave equation which they formulate as > $$ \\frac{1}{v_0}\\frac{\\partial^2}{\\partial t^2} \\begin{pmatrix}p \\\\\\ > r\\end{pmatrix} = \\begin{pmatrix} > 1+2\\epsilon&\\sqrt{1+2\\delta}\\\\\\\\\\sqrt{1+2\\delta} &1\\end{pmatrix} > \\begin{pmatrix}G_{\\bar x \\bar x}+G_{\\bar y \\bar y}&0\\\\\\0&G_{\\bar z \\bar z} > \\end{pmatrix} \\begin{pmatrix} p\\\\\\r \\end{pmatrix}\\tag{20} $$ and they claim that > To achieve stability, the rotated differential operators $G_{\\bar x \\bar > x}$, $G_{\\bar y \\bar y}$, and $G_{\\bar z \\bar z}$ should be self-adjoint and > nonpositive definite as are the second-order derivative operators > ($\\tfrac{\\partial^2}{\\partial x^2}$, $\\tfrac{\\partial^2}{\\partial y^2}$ and > $\\tfrac{\\partial^2}{\\partial z^2}$). (see eq. 14 and statement under eq. 20). They also claim that self-adjointness and nonpositiveness of the differential operators of the wave equation are necessary to conserve the energy in this system, and that if they were not self-adjoint, numerical instabilities occur. > We have solved the problem by introducing the self-adjointness to the > operator matrices in equation 20 to make sure that energy is conserved > during the wave propagation to avoid amplitude blowup in the modeling. In this case the wave equation consists of two coupled elliptical PDEs. What happens in general, when some operators are not bounded and linear? Unfortunately I don't have the mathematical background to understand this statement."}
{"id":"80525","title":"How to derive the Bethe stopping power formula","text":"I need the derivation of Bethe formula for stopping power, but I can't see the corresponding paper to this matter. > Application of Ordinary Space-Time Concepts in Collision Problems and > Relation of Classical Theory to Born's Approximation. E. J. Williams. _Rev. > Mod. Phys._ **17** no. 2-3 (1945) pp. 217-226. Can anyone help me about this paper please?"}
{"id":"45767","title":"What makes an equation an 'equation of motion'?","text":"Every now and then, I find myself reading papers\/text talking about how _this_ equation is a constraint but _that_ equation is an equation of motion which satisfies _this_ constraint. For example, in the Hamiltonian formulation of Maxwell's theory, Gauss' law $\\nabla\\cdot\\mathbf{E}=0$ is a constraint, whereas $\\partial_\\mu F^{\\mu\\nu}=0$ is an equation of motion. But why then isn't $\\partial_\\mu j^\\mu=0$, the charge-conservation\/continuity equation, called an equation of motion. Instead it is just a 'conservation law'. Maybe first-order differentials aren't allowed to be equations of motion? Then what of the Dirac equation $(i\\gamma^\\mu\\partial_\\mu-m)\\psi=0$? This is a first-order differential, isn't it? Or perhaps when there is an $i$, all bets are off... So, what counts as an equation of motion, and what doesn't? How can I tell if I am looking at a constraint? or some conservation law?"}
{"id":"17335","title":"Vertical component of moving weight at a 45 degree angle","text":"Here's an easier one. I use the leg press machine at the gym so I don't have to worrying about hurting myself while lifting heavier weight. The weight glides on a track that looks to be 45 degrees. What's the equation to figure out how much weight I would be able to squat normally. IE the vertical component of moving 400lbs at a 45 degree angle."}
{"id":"45768","title":"How to express $ds$?(when we know expression for $ds^{2}$)","text":"We know that $$ds^2 = g_{\\mu\\nu}dx^{\\mu}dx^{\\nu},$$ Can you say how to calculate $ds$?"}
{"id":"78234","title":"How to calculate the energy required to generate a vortex?","text":"I am looking for the formula to calculate how much energy it requires to create a vortex. Can anyone assist please?"}
{"id":"120150","title":"Problems book recommendation on supersymmetry, supergravity and superstring theory","text":"I'm learning supersymmetry, supergravity and superstring. I want some problems books to have some idea in this area. Is there this kind of books? Or are there some papers that have many solved model?"}
{"id":"95726","title":"Pulley problem with two masses","text":"Assume a pulley and a massless rope. On one end there is a 10.0kg mass. On the other there is a 34.5kg mass. What is the acceleration of the 34.5kg mass? I created a free-body diagram with $m_1g_1$ pointing up and $m_2g_2$ pointing down, so the total tension is $m_1g_1 - m_2g_2$. This results in 240 N downwards, so I divide by 34.5kg to get $6.96 m\/s^2$. However that is incorrect. What is my mistake?"}
{"id":"54835","title":"Double Pendulum","text":"The equations of motions for the double pendulum is given by $$\\dot{\\theta_1} = \\frac{6}{ml^2}\\frac{2p_{\\theta1} - 3\\cos(\\theta_1 - \\theta_2)p_{\\theta2}}{16 - 9\\cos^2(\\theta_1 - \\theta_2)}$$ and similarly for the other pendulum. In respect to what does the change in angle for the first pendulum refer to? Is it with respect to time? So that $\\dot{\\theta_1} = \\frac{d\\theta}{dt}$?"}
{"id":"10229","title":"Historical background of wave function collapse","text":"I wonder what were the main experiments that led people to develop the concept of wave function collapse? (I think I am correct in including the Born Rule within the general umbrella of the collapse paradigm.) Are there any instances where cases once thought to be examples of collapse have since been explained as the normal time-evolution of the wave function? EDIT: I'm going to have to make an objection to Ron Maimon's very excellent answer about particle tracks as evidence of collapse. I've been waiting for someone to suggest what I personally have always considered the prototype of the wave function collapse, namely the appearance of flecks of silver on a photographic plate when exposed to the light of a distant star. This has the essential elements of collapse in a way that ordinary photographic exposures do not. The mere appearance of dots on a photographic plate does not signal the collapse of anything: it is readily explainable as a consequence of the rate of silver-bromide reduction being proportional to light intensity. It is only when the intensity becomes so very low that the time taken to accumulate enough energy for a single conversion becomes unreasonable that we must consider the explanation of wave function collapse. The tracks in the cloud chamber do not demonstrate this phenomenon since the energy needed for the creation of the tracks is already available in the supersaturated gas. It is not necessary for the incoming particle to supply energy for the creation of the track, so there is no need to collapse its wave function. The straightness of the tracks is explained by Mott as an ordinary consequence of time-evolution of the wave function. There is no experimental proof that a single \"particle\" cannot be responsible for multiple tracks in the cloud chamber, because the tracks are not tagged according to which particle created them."}
{"id":"10225","title":"How to calculate the projected area at different angles\/vectors?","text":"Please help me with the following. I want to know if there is an equation\/set of equations to find out the projected area of a (3-D) cube when it is oriented at different angles of attack to the fluid flow, rendering the velocity vector of the impinging fluid on the cube surface to change with each angle of attack. In particular, what is the projected area when the cube is oriented at such an angle that the flow velocity vector passes through one of the vertex and the geometric centre of the cube? Thanks."}
{"id":"10222","title":"direction of Pockels Effect. Refractive index eigendirections","text":"There is a linear electro-optic effect called Pockels Effect The brief is that refractive index changes due to electric field. If there is an anisotropy (like birefringence) and electric field is in perpendicular to optical axis then refractive index changes for different polarization of ordinary rays. It gets eigendirections 45 degrees to electric field. Why so? Why it gets min 45 degrees but not exactly towards direction of E?"}
{"id":"49846","title":"Isometry group from information about the center of the group","text":"I am reading this paper on Dyons and Duality in $\\mathcal{N}=4$ super- symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to the Bogolomony equation for bps monopoles. He finds out that when the simple root used for embedding the SU(2) monopole is simple then the bosonic zero modes which I believe is the solution for the higgs field transform as $1 \\oplus 0$, and when it is not a simple root it transforms as two doublets. Hope this is not required for my question. **Then he says that the zero modes are SENSITIVE to the center of SU(2), and hence the isometry group is SU(2)? What does sensitive to the center mean? Is center the subset which commutates with all elements of SU(2)? How can it tell me about the isometry group?**"}
{"id":"63341","title":"Evolution principle of the physical laws","text":"I wanted to know if there is a physical theory that considers that the laws of physics undergo an evolutionary process. That see the law of physics or the absence of them, as something dynamic, and that with time they slowly converge to something we know today. A kind of simulated annealing of the physical laws."}
{"id":"78586","title":"Can Helium Disappear from Earth?","text":"Can Helium disappear? As we know Helium is lighter than air, so basically Helium fly off from Earth. Is it possible that in the future we will run out Helium?"}
{"id":"78584","title":"Where do electrical charges go, on a nonconductor dielectric when we make it charged?","text":"My question is about electrically nonconductor dielectrics. We know such materials don't possess free charges.They have atoms bound together and every atom has specific numbers of electrons turning around its nucleus. When we make a dielectric charged: where does this charge go? what keeps the charge fixed on the structure? Are they bound to specific atoms? If your answer is \"yes\",so What makes that atom specific to possess the charge? (same questions about positively charging.)"}
{"id":"68490","title":"Applications to the Van der Pol equation?","text":"What are some applications to the Van der Pol equation? Are there any physical examples?"}
{"id":"54781","title":"Understanding Quantum Physics","text":"I have very little background in physics, and none in quantum physics, but I've been reading about how sub-atomic particles behave probabilistically, so I was wondering, is it possible (even though the probability would be unimaginably small) that all the particles which make up my body are located somewhere 100 light years away? Or have I misunderstood the concept?"}
{"id":"62699","title":"Domain wall and kink solutions from solitions equations","text":"A general solition equation can be obtaion from scalar field theory $$\\varphi(x) = v\\tanh\\Bigl(\\tfrac{1}{2}m(x - x_0)\\Bigr),\\tag{92.6}$$ where $x_0$ is a constant of integration when we drived this solution from Lagrangian. **Comparing to the sine-Gordon theorywith this article , can we convert our equation to the sine-Gordon equation?** Because this equation represented for 2 dimensions (1 space dimension and 1 time dimension) then how will I get Kink solutions and domain wall equation from the above equation. Any elaboration would be helpful for me. This is a continuing post of Solving the soliton equation without energy. **ADDITION: If I want to visualize the solition equation (92.6) by graph then what would be the best process?**"}
{"id":"105980","title":"Does Light Ever Cease Once Created","text":"I was reading an article on Huffington Post (link to article) about the Multiverse theory. In the article it said this: > The big news last week came from the Background Imaging of Cosmic > Extragalactic Polarization 2 (BICEP2) experiment at the South Pole, which > saw imprints in the cosmic microwave background— **the oldest light in the > universe** , dating from shortly after the big bang... I am far from scientific, actually just a web dev, but I have an interest in learning and reading about science, so please explain your answer in a way I can understand it. My question is in regards to the bold type in the quote. Does light ever actually end, go away, or just cease to exist once it is created? Will this \"oldest light in the universe\" ever completely cease to exist, or does light stay around indefinitely."}
{"id":"115124","title":"What are the prerequisites to study topological quantum computation\/topological phases of the matter?","text":"I am an undergraduate student and I would like to approach the subject of topological order with focus on topological quantum computation, I know (very) little QFT and basic algebraic topology (if that matters...). To give a reference frame, I'd like to understand papers like http:\/\/arxiv.org\/abs\/quant-ph\/9707021v1 or Preskill's great presentations: http:\/\/online.kitp.ucsb.edu\/online\/exotic_c04\/preskill\/, and later to take it as my field of research for a PhD. So my questions are: 1) What kind of studies are useful\/needed for this field? I assume Quantum Computation\/Information are, but what else? Quantum Field Theories, Topology, Quantum Optics...? 2) Which are univerities or research centers active in the field (preferably in Europe)? 3) Any reference suggested? (Better if beginner-focused)"}
{"id":"105984","title":"Friction influencing the motion of a mass and non-inertial frame of reference","text":"So... recently a Newtonian-mechanics related exercise has raised in me many questions basically ragarding which forces influence a given system and how, I know they might sound bit too \"noobish\" but I just had to get them out of my chest (yes it really gave me something to think about in the last day). perhaps you should take a look at the above mentioned exercise to better understand what I am trying to figure out ![image of the system](http:\/\/s21.postimg.org\/ek3mmimw7\/Nuova_immagine_bitmap.png) So, I'm trying to figure out how this system works... the exercise says (please forgive me for all the grammatical errors you will eventually find): > # Problem > > a cube with a mass $\\ M=50Kg$ is standing on a plain surface and can move on > it without friction. On that cube is standing another cube with a mass $\\ > m=10Kg$, at a given distance $\\ d=0.5 m$ from the upper-left corner of the > cube. Initially, when everything is stationary, a force $\\ F=100N$ is > applied to the bigger cube horizontally (I assume that the force $\\ F$ is > constant); at the moment $\\ t=2s$ the smaller cube falls. Calculate the > friction coefficient between the two cubes. # Variables used: > $\\ μ$= friction coefficient > > $\\ g$= gravitational acceleration on earth =$\\ 9.81 m\/s^2$ > > $\\ a_M$= acceleration of the bigger cube ($\\ M$) > > $\\ a_m$= acceleration of the smaller cube ($\\ m$) > > $\\ F_f$= friction force So, I assume that the bigger cube carries the smaller one which moves with a constant, negative acceleration until it falls (please correct me if I'm wrong), with equation of motion $\\ x(t)= x_0 + v_0t + 1\/2 at^2$ so $\\ 0.5m = - 1\/2 at^2$ ------> $\\ a= -0.25 m\/s^2 $ now... $\\ F- μ mg = Ma_M$ in other terms (and again please correct me if I'm wrong): $\\ F$ is partially dissipated by the friction between the cubes, and the bigger cube moves under the influence of a force which equals to the applied force $\\ F$ minus the friction force $\\ F_f= μ mg$ now the universe collapses: the solution says: > $\\ μmg = ma_m$ > > $\\ a_r = a_M - a_m = [ μg (m + M) -F] \/ M$ which leads to $\\ μ=0.15 $ But... what leads to those last two equations?"}
{"id":"120108","title":"How can I derive this Hamiltonian?","text":"I have a Lagrangian $L$, a momentum $p$ and a Hamiltonian $H$: $$L=\\frac m 2(\\dot z + A\\omega\\cos\\omega t)^2 - \\frac k 2 z^2$$ $$p=m\\dot z + mA\\omega\\cos\\omega t$$ $$H=p\\dot z - L=\\frac m 2 \\dot z^2 - \\frac m 2 (A\\omega\\cos\\omega t)^2 - \\frac k 2z^2$$ And I want to calculate $\\frac {\\partial H} {\\partial z}$ and $\\frac {\\partial H} {\\partial p}$. I understand from this question that I need to algebraically manipulate $H$ to express it in terms of $p$ and $z$. The answers there suggested trying to express $\\dot z$ in terms of $z$ and $p$, and presumably I need to express $t$ in terms of $z$ and $p$ as well. But that seems like its going to lead into very nasty territory... first of all, I've got quadratics, so that's not an invertible function. Second of all, for $t$, if I have to use inverse trigonometric functions, then it'll only be valid over a particular range of the variable. Could I get some pointers on how to tackle this calculation? Is inverting the various relations really the way to go?"}
{"id":"86335","title":"About the geodesics in general relativity","text":"I'm learning general relativity from the book \" Einstein's General Theory of Relativity - Øyvind Grøn and Sigbjorn Hervik\". The field equations are derived by the Hilbert - Einstein action and are written in the form $$ R_{\\mu\\nu}-\\frac{1}{2}R\\,g_{\\mu\\nu}+\\Lambda\\,g_{\\mu\\nu}=kT_{\\mu\\nu}$$   where $k$ is acostant that should be determined. The authors find this $k=\\frac{8\\pi G}{c^4}$ by imposing the well known conditions of \"Newtonian Limit\". The first requirement is the following > The particles in free fall moving along geodesics induced by $g$. This is a postulate? By the Einstein equivalence principle we know that locally we can eliminate the effects of gravity and so living in a (flat) Minkowskian space-time but there is not mention of the above statement. I agree on the fact that once we have a curved space-time, in presence of gravitational fields, it is natural to require that the free falling particles move along the geodesics, but this can be proved or it is postulated?"}
{"id":"112087","title":"Question about Bose-Hubbard Model","text":"By using an optical lattice, how can one change the interaction term $U$? And how is the superfluid phase achieved in the hard-core boson regime? Why are these phases identified as superfluid or mott insulator?"}
{"id":"91729","title":"Step by step algorithm to solve Einstein's equations","text":"I cannot completely understand what is a _regular_ method to solve Einstein's equations in GR when there are no handy hints like spherical symmetry or time- independence. E.g. how can one derive Schwarzschild metric starting from arbitrary coordinates $x^0, x^1, x^2, x^3$? I don't even understand the stress-energy tensor form in such a case - obviously it sould be proportional to $\\delta(x - x_0(s))$, where $x_0(s)$ is a parametrized particle's world-line, but if the metric is unknown _in advance_ how do I get $x_0(s)$ without any a priory assumptions?"}
{"id":"72663","title":"Transmutation with cosmic radiation possible?","text":"Can cosmic radiation (alpha radiation) transmute the material of a space craft, particular carbon, titanium and aluminum? Where can i find transmutation tables or formulas to calculate the possibility of a transmutation and the outcome?"}
{"id":"72664","title":"How many seconds is a temporal meter?","text":"Is there a proof that time is a 4th dimension? If it is, then why not measure it in units of the previous three? Logical right? How many seconds is a temporal meter?"}
{"id":"91723","title":"When to use ideal gas law in fluid mechanics?","text":"The ideal gas law (aka the equation of state) is given by $$ p\/\\rho_N = k_BT, $$ where $\\rho_N$ is number density. When am I allowed to use this to describe a fluid?"}
{"id":"77498","title":"A few questions about the Fermi Level\/Energy","text":"My first question is, how is the Fermi Energy for a material actually determined? I know this derivation, but it seems to say that the Fermi Energy is just based on the electron density (and maybe some effective mass) of the material. Is that really all that determines it? Secondly, I'm trying to figure out how the interfaces of various materials work in terms of their bands, but it's not clear to me exactly what _must_ be true in all cases (vs what is often but not necessarily true, or what is theoretically but rarely practically true). For example, Anderson's Rule starts by aligning the \"vacuum levels\" of the two materials, but then this article says that it's not a great idea to use the vacuum level, and the Anderson's Rule article says it's just not that accurate a rule, anyway. Similarly, it seems like the Schottky-Mott Rule isn't very successful either. Additionally, I've read somewhere that the Fermi Level (the electrochemical potential, the sum of the chemical potential and electric potential) has to be continuous everywhere in both of the materials, so that results in the _chemical potentials_ (i.e., the $T \\neq 0$ Fermi Energies, which were normally different in the two materials) lining up, and that happens by having an _electric potential_ difference across them. But this picture from wikipedia then seems to suggest that either what I just said is wrong, or the label should really be \"Fermi _energy_ \" (or chemical potential) in their definitions. Which is it? So, what can I always depend on and know is true in these situations? Thank you!"}
{"id":"73138","title":"Peierls Argument for Absence of Long Range Order","text":"I'm really confused about the argument in Cardy's book for why there can't be long range order in 1D for discrete models. Let me just copy it out, and hopefully someone can explain it to me. He takes an Ising-like system as an example. We start with the ground state with all spins up, and we want to see if this state is stable against flipping the spins in some chain of length $l$. This chain has two domain walls at the endpoints, so we get an energy change of $4J$. Then the claim is that there is an entropy of $\\log l$ associated with this chain, since \"each wall may occupy $O(l)$ positions.\" If this were true, we would get a free energy change of $4J-\\beta^{-1} \\log l$, and this would imply that the ground state is unstable to flipping very long chains. The only part I'm not on board with is the claim about the entropy. I would say that if $L$ is the length of the system, then we have $L$ places to put the chain, so we get an entropy of $\\log L$. Certainly as $L\\to \\infty$ this gives no long range order, as expected. So, is the entropy $\\log l$ or $\\log L$? (Incidentally, I'm perfectly happy with his argument in 2D...)"}
{"id":"66592","title":"List of the basic quantum mechanical variables","text":"Is there a list of basic quantum variables\/attributes that all quantum particles have? Ex. An electron has charge, position, speed, momentum, etc. Is there a complete list of these variables? I would figure not all quantum particles share the same set of variables? A photon has position and an electron has position but a photon does not have a charge and an electron does, though I guess it is said a photon has neutral charge."}
{"id":"32857","title":"How to calculate Temperature Humidity Wind Index?","text":"I would like to know how to calculate Temperature Humidity Wind Index (THW Index)? I know how to calculate Heat Index and Wind Chill. I am asking this because my weather station Davis Vantage Pro2 calculates THW index but I could not find any information on NOAA. This is what it says under Help for my weather station: > The THW Index uses humidity, temperature and wind to calculate an apparent > temperature that incorporates the cooling effects of wind on our perception > of temperature."}
{"id":"51554","title":"Why are magnetic lines of force invisible?","text":"We can very well feel the magnetic field around a magnet, but we can't see it. Why is that so? Also, can we cut a portion of the field and use it?"}
{"id":"51556","title":"Kinetic energy of a photon and Schwarzschild radius","text":"I have read here, that $\\frac{1}{2}mv^2$ must not be applied on a photon ever. If i want to calculate escape velocity $v_e$ i need to use $\\frac{1}{2}mv^2$ because we say that kinetic energy _(positive)_ must be same or larger than gravitational potential _(which is negative)_ in order for an object to escape. It is done like this: $$ \\begin{split} W_k + W_p &= 0\\\\\\ \\frac{1}{2}mv_e^2 + \\left(-\\frac{GMm}{r}\\right) &= 0\\\\\\ v_e &= \\sqrt{\\frac{2GM}{r}} \\end{split} $$ Than we say if there is a black hole inside certain radius we call Schwarzschield radius $r=R_{sch}$ not even light can escape because its escape velocity is smaller than needed to escape. At the border of the sphere with $R_{sch}$ light can barely escape, so it must hold that $c$ equals escape velocity $v_e$. So we write down the equation below and derive $R_{sch}$. $$ \\begin{split} c &= \\sqrt{\\frac{2GM}{R_{sch}}}\\\\\\ R_{sch} &= \\frac{2GM}{c^2} \\end{split} $$ This is a well known equation, but it is derived allso using $\\frac{1}{2}mv^2$ for light (photons). This is in contradiction with 1st statement in this post. So is the last equation even valid???"}
{"id":"111358","title":"What exactly is an anti-neutrino?","text":"According to the the definition of anti-particles, they are particles with same mass but opposite charge. Neutrinos by definition have no charge. So, how can it have an anti-particle?"}
{"id":"19499","title":"Should you run when under rain?","text":"When it's raining, would you get less wet if you run or more wet? I think you will get less wet, because rain is coming down in a constant volume and if you run you will have a shorter amount of time you will get less volume I'm not too sure though, some people say when you run you are running into a more volume of rain because rain is not all coming up to down, it will hit you more horizontally"}
{"id":"61640","title":"The role of the affine connection the geodesic equation","text":"I apologise in advance that my knowledge of differential geometry and GR is very limited. In general relativity the equation of motion for a particle moving only under the influence of gravity is given by the geodesic equation: $$ \\ddot{x}^\\lambda + \\Gamma^\\lambda_{\\mu\\nu}\\dot{x}^\\mu\\dot{x}^\\nu =0. $$ I am looking for a conceptual description of the role of the affine connection, $\\Gamma^\\lambda_{\\mu\\nu}$ in this equation. I understand that it is something to do with notion of a straight line in curved space. Comparing it to the equation for a free particle according to Newtonian gravity: $$ \\ddot{x}_i = -\\nabla\\Phi, $$ Then it kind of looks like the affine connection is our equivalent of how to differentiate, except that our scalar field is now some kind of velocity?"}
{"id":"113331","title":"Bullet entering target","text":"I've been given this question: > A bullet entering a target with an initial velocity of $ u $ loses $\\frac > {u}{n}$ of its velocity after penetrating a distance $ a $ into the target; > how much further will it penetrate? My question is, doesn't some information about the nature of the bullet's acceleration need to given? How otherwise would I formulate the kinematic equations?"}
{"id":"130358","title":"Is the energy of a photon continuous\/discrete?","text":"I was struggling today with this question: does a free photon have a continuous energy spectra? Free means in no context of any energy system (eg. an atom, em field). Although I'm asking myself if the quantization of the electromagnetic field is omnipresent and will always make the energy discrete? Edit: This leads me also to the question: if we have 2 energy levels (like in hydrogen: ground state and first excited) the uncertainty principle tells us, that the energy isn't quite exact defined: $\\Delta E\\Delta t \\ge \\hbar$ . Therefore the final energy of the emitted photon won't have a discrete energy, since it would be sth. like $E_{photon} = E_{0} + \\Delta E$ ?!"}
{"id":"93493","title":"Where does a spinning figure skater's energy go when she slows down?","text":"Today in physics class we were talking about angular momentum and rotational kinetic energy. My teacher used the classic example of a figure skater spinning on ice - when she pulls her arms in, her angular momentum is conserved and her angular velocity increases, meaning that her rotational kinetic energy also increases. Of course, this increase in energy must come from somewhere - in this case, it comes from the figure skater doing work on her arms and pulling them in toward her body. Then I started wondering - if the figure skater slows her rotation by extending her arms, she decreases her rotational KE. Where is her energy going? Or to put it another way, what force is doing work on the figure skater in order to decrease her energy?"}
{"id":"74674","title":"What is Maupertuis' principle good for?","text":"The strength of Hamilton's principle is obvious to me and I see the advantage. Now, for conservative systems we also have Maupertuis' principle that says: $$ \\delta \\int p dq =0$$ and I am not sure how to derive an equation of motion from this? Is this of any use in practical computations? So, can one apply this principle for example to the harmonic oscillator?- I have never seen anybody using it. Further, I read in Goldstein's classical Mechanics that the variation in Maupertuis' principle is not the one in Hamilton's principle, since we have constant Hamiltonian and changing time, whereas Hamilton's principle has constant time and varying Hamiltonian (in general). I am a little bit wondering about this, since you could easily get Maupertuis' principle from Hamilton's principle: $$ \\delta \\int L dt = \\delta \\int p \\dot{q} - H dt = \\delta \\int p \\dot{q} dt = \\delta \\int p dq =0,$$ if $H$ is constant. Can anybody here explain to me, why we have to use a different variation and how one can use this principle?"}
{"id":"131288","title":"Band-limited double-step Fresnel diffraction for images with radio waves?","text":"Is it possible to use band-limited double-step Fresnel diffraction to assemble a holographic image with radio waves? If not is there a simular principle?"}
{"id":"74672","title":"Question on expression in \"J.S.Bell : On the Einstein Podolsky Rosen paradox\"","text":"I have a question on the article `J. S. Bell, On the Einstein Podolsky Rosen paradox, Physics 1, 195, 1964.` (link) My question concerns the expression (3) of the article, at page 196. I don't understand what is the reasoning that leads to this expression of the expectation value... I think I miss something but I don't know what. This is what I understood from now on : $\\vec{\\sigma_1}$ and $\\vec{\\sigma_2}$ are the spins of the two particles that move apart and must be exactly opposite according to quantum mechanics when measured in a direction of the component $\\vec{a}$. First, did I understand well and do we really have $$A(\\vec{a},\\lambda) = \\vec{\\sigma_1}.\\vec{a} = \\pm 1 \\\\\\ B(\\vec{b},\\lambda) = \\vec{\\sigma_2}.\\vec{b} = \\pm 1$$ then ? If not, what does $A(\\vec{a},\\lambda)$ and $B(\\vec{b},\\lambda)$ correspond to ? A sort of $sign$ function or something like in the next section? Secondly, why $$ <\\vec{\\sigma_1}.\\vec{a}\\; \\vec{\\sigma_2}.\\vec{b}> = -\\vec{a}.\\vec{b}$$ Is it because $\\vec{\\sigma_1}$ and $\\vec{\\sigma_2}$ are opposite ? Thanks !"}
{"id":"27118","title":"Asymptotic Completeness, generalized free fields, and the relationship of thermodynamics with infinity","text":"Asymptotic completeness is a strong constraint on quantum field theories that rules out generalized free fields, which otherwise satisfy the Wightman axioms. If we were to take a limit of a list of continuous mass distributions $\\rho_n(k^2)$ that approaches a distribution in some topological sense, however, is there anywhere an analysis of how the behavior of the $\\rho_n(k^2)$ would approach the behavior of the free field? The following statement seems too bald (from the review \"Outline of axiomatic relativistic quantum field theory\" by R F Streater, Rep. Prog. Phys. 38, 771-846 (1975)): \"If the Källén-Lehmann weight function is continuous, there are no particles associated with the corresponding generalized free field; the interpretation in terms of unstable particles is not adequate\". Surely as we take the support of a generalized Lorentz invariant free field to be arbitrarily small we could expect that the behavior, at least as characterized by the VEVs, which constitute complete knowledge of a Wightman field, would eventually be arbitrarily close to the behavior we would expect from a free field? Classical thermodynamics has a complicated relationship with infinity, in that the analytic behavior of phase transitions does not emerge unless we take an infinite number of particles, but the behavior of very large numbers of particles nonetheless can approximate thermodynamic behavior rather well. By this elementary analogy, it seems premature to rule out generalized free fields. It also seems telling, although weakly, that the Källén-Lehmann weight function of an interacting field is nontrivial in quasiparticle approaches. Being able to derive an S-matrix requires that a theory must be asymptotically complete, however real measurements are always at finite time-like separation from state preparations, with the interaction presumably not adiabatically switched off at the times of the preparation and measurement, so that something less than analytically perfect asymptotic completeness ought to be adequate. EDIT: To make this more concrete, the imaginary component of the mass $1$ propagator in real space at time-like separation is $I(t)=\\frac{J_1(t)}{8\\pi t}$. If we take a smooth unit weight mass distribution $$w_\\beta(m)=\\frac{\\exp(-\\frac{\\beta}{m}-\\beta m)}{2m^2K_1(2\\beta)}\\ \\mathrm{for}\\ m>0,\\ \\mathrm{zero\\ for}\\ m\\le 0,$$ for large $\\beta$ this weight function is concentrated near $m=1$, with maximum value $\\sqrt{\\frac{\\beta}{\\pi}}$. For this weight function, the imaginary component of the propagator in real space at time-like separation is (using Gradshteyn&Ryzhik 6.635.3) $$I_\\beta(t)=\\int\\limits_0^\\infty w_\\beta(m)\\frac{mJ_1(mt)}{8\\pi t}\\mathrm{d}m= \\frac{J_1\\left(\\sqrt{2\\beta(\\sqrt{\\beta^2+t^2}-\\beta)}\\right) K_1\\left(\\sqrt{2\\beta(\\sqrt{\\beta^2+t^2}+\\beta)}\\right)}{8\\pi tK_1(2\\beta)}.$$ Asymptotically, this expression decreases faster than any polynomial for large $t$ (because the weight function is smooth), which is completely different from the asymptotic behavior of $I(t)$, $-\\frac{\\cos(t+\\pi\/4)}{4\\sqrt{2\\pi^3t^3}}$, however by choosing $\\beta$ very large, we can ensure that $I_\\beta(t)$ is close to $I(t)$ out to a large time- like separation that is approximately proportional to $\\sqrt{\\beta}$. Graphing $I(t)$ and $I_\\beta(t)$ near $t=1000$, for example, and for $\\beta=2^{20},2^{21},2^{22},2^{23},2^{24}$, we obtain ![Graph of $I\\(t\\)$ and $I_\\\\beta\\(t\\)$ near $t=1000$](http:\/\/i.stack.imgur.com\/Pk3pG.png) $I(t)$ and $I_\\beta(t)$ are very closely in phase, as seen here, until $t$ is of the order of $\\beta^{2\/3}$ in wavelength units. We can take $\\beta$ to be such that this approximation is very close out to billions of years (for which, taking an inverse mass of $10^{-15}m$, $\\sqrt{\\beta}\\approx \\frac{10^{25}m}{10^{-15}m}=10^{40}$), or to whatever distance is necessary not to be in conflict with experiment (perhaps more _or_ less than $10^{40}$). This is of course quite finely tuned, however something on the order of the age of the universe would seem necessary for what is essentially a stability parameter, and the alternative is to take the remarkably idealized distribution-equivalent choice $\\beta=\\infty$ as usual. ~~[I would like to be able to give the real component of the propagator at time-like and space-like separations for this weight function, however Gradshteyn &Ryzhik does not offer the necessary integrals, and nor does my version of Maple.]~~ EDIT(2): Turns out that by transforming Gradshteyn&Ryzhik 6.653.2 we obtain $$R_\\beta(r)\\\\!=\\\\!\\int\\limits_0^\\infty\\\\!\\\\!w_\\beta(m)\\frac{mK_1(mr)}{4\\pi^2 r}\\mathrm{d}m= \\frac{K_1\\left(\\\\!\\sqrt{2\\beta(\\beta-\\sqrt{\\beta^2-r^2})}\\right) K_1\\left(\\\\!\\sqrt{2\\beta(\\beta+\\sqrt{\\beta^2-r^2})}\\right)}{4\\pi^2 rK_1(2\\beta)},$$ which _is_ real valued for $r>\\beta$. As for $I_\\beta(t)$, the approximation to the mass $1$ propagator at space-like separation $r$, $R(r)=\\frac{K_1(r)}{4\\pi^2 r}$, is close for $r$ less than approximately $\\sqrt{\\beta}$. For the real component at time-like separation, ~~it is almost certain that~~ one ~~simply~~ replaces the Bessel function $J_1(...)$ by $Y_1(...)$."}
{"id":"27119","title":"Physical interpretation of different selfadjoint extensions","text":"Given a symmetric (densely defined) operator in a Hilbert space, there might be quite a lot of selfadjoint extensions to it. This might be the case for a Schrödinger operator with a \"bad\" potential. There is a \"smallest\" one (Friedrichs) and a largest one (Krein), and all others are in some sense in between. Considering the corresponding Schrödinger equations, to each of these extensions there is a (completely different) unitary group solving it. My question is: what is the physical meaning of these extensions? How do you distinguish between the different unitary groups? Is there one which is physically \"relevant\"? Why is the Friedrichs extension chosen so often?"}
{"id":"55699","title":"How can I add an acceleration vector to a velocity with a different direction?","text":"I am in my last year of high school and am struggling with some homework. I'm sorry if this question is incredibly stupid, but I simply can't find the answer in my notes. If I have an object with a velocity of 10ms-1 travelling at a bearing of 090 degrees (to the right) with no acceleration, and then all of a sudden it spontaneously gains an acceleration of 2ms-2 at a bearing of 180 degrees (downwards), how do I calculate the velocity (magnitude and direction) of the object for every second after the object gains this acceleration? Thank you very much for your help, I really want to be able to understand this!"}
{"id":"103041","title":"How are symmetries defined mathematically?","text":"I have started working on differential geometry very recently. I am little bit familiar with mathematical concepts such as manifolds, differential forms and associated concepts. As I was speeding through I caught interest in Lie groups and whose invaluable connection with the symmetries. I would be happy if anybody answers to these few questions.   1. I want to know what is a symmetry from the mathematical point of view?    2. How are $su2$, $su3$, and other symmetries first formulated?    3. What was the motivation?    4. How do we identify internal symmetries and corresponding conserved physical quantities?"}
{"id":"113818","title":"What is the relative speed of two near-light speed particles headed towards each other?","text":"I understand that nothing can move faster than light due to time dilation. I want to build upon my understanding of Einstein's theory of Special Relativity, so I came up with this hypothetical problem for myself: > If both particles are moving at 0.9c towards each other (according to an > observer on Earth), what speed does each particle have relative to the > other? I'm stumped since I am only familiar with Newtonian classical mechanics to solve this (which would be wrong). I am aware of Lorentz' factor $\\gamma =\\frac{1}{\\sqrt{1 - \\frac{v^2}{c^2}}}$ and the time dilation equation $t_m = \\frac{t_o}{\\sqrt{1 - \\frac{v^2}{c^2}}}$, but I'm not sure how to apply it in this instance. I'm just looking for a gentle push in the right direction. Edit: My prior research:   1. This article, https:\/\/what-if.xkcd.com\/1\/, talks about the effect of a baseball thrown at 0.9c would have as it traveled towards a batter. This is not what I'm looking for since it doesn't explain the math behind finding the collision speed between two high velocity entities.   2. This question on Yahoo, https:\/\/answers.yahoo.com\/question\/index?qid=20110529201519AAxbxvm, asks a similar question to mine, but no math is involved to demonstrate the calculations behind the reasoning.   3. This question on Physics.SE, Collision between a photon and a massive particle, does not answer my question either. I have not been able to find a similar Physics.SE question which can provide me to clues to my own hypothetical question.    4. I have also Googled the following phrases: > i. \"collision faster than the speed of light\" > > ii. \"collision of high speed particles\" > > iii. \"lorentz transformation in a collision\" > > iv. \"collision analysis at light speed\""}
{"id":"56520","title":"Spontaneous symmetry breaking: How can the vacuum be infinitly degenerate?","text":"In classical field theories, it is with no difficulty to imagine a system to have a continuum of ground states, but how can this be in the quantum case?   Suppose a continuous symmetry with charge $Q$ is spontaneously broken, that would means $Q|0\\rangle\\ne0$, and hence the symmetry transformation transforms continuously $|0\\rangle$ into anther vacuum, but **how can a separable Hilbert space have a continuum of vacuums deferent from each other?**      I saw somewhere that says the quantum states are built upon one vacuum, and others simply doesn't belong to it, what does this mean? and **then how could $Q$ be a well defined operator** which acting on a state (the vacuum) actually gives a state (another \"vacuum\") out of the space considered?"}
{"id":"104574","title":"Angular momentum of 2d harmonic oscillator","text":"So, I have the problem of determining the spectrum of H and L, in terms of creation and annihilation operators of angular momentum... The problem goes along with what is happening on this page. However, my professor is asking us for the commutators of H and L with a quantity A, which is defined as $$A=\\hat a_x^2+\\hat a_y^2$$ When I do this out, writing H as $1+\\hat a_x^{\\dagger}\\hat a_x+\\hat a_y^{\\dagger}\\hat a_y$, I find that $$[H,A]=\\hbar\\omega(a_x^3+a_y^3)$$ I have no problem with this, except that when I want to construct an eigenket representation of $E_{nm}$in terms of the $\\hat a_L^{\\dagger}\\text{ and } \\hat a_R^{\\dagger}$ and $A^{\\dagger}$, I am not sure how to use this result to find this. I know that the result should be $E_{n,m}=\\hbar\\omega[2n+1+|m|]$ Thanks"}
{"id":"104572","title":"What the heck is negative effective mass?","text":"I am reading this book:Solid State Electronic Devices by Ben G Streetman and Sanjay Kumar Banerjee.   I have some doubts in the article **3.2.2** Effective mass.   In this the aythors say that $E=\\dfrac{1}{2}mv^2=\\dfrac{1}{2}\\dfrac{P^2}{m}=\\dfrac{{\\hbar} ^2}{2m}k^2$. >   * Electrons usually have thermal velocity of order $10^7$m\/s. So Shouldn't > we use $E=mc^2$ rather than $E=\\frac{1}{2}mv^2$. > The author further says that electron mass is related to curvature of (E,k) relationship as $\\dfrac{d^2E}{dk^2}=\\dfrac{\\hbar^2}{m}\\tag{3.2(d)}$. then the author says that the effective mass is given by $m^{*}={{\\hbar^2}\/{{\\dfrac{d^2E}{dk^2}}}}\\tag{3.3}$ >   * Is equation $3.3$ derived from equation $3.2(b)$. In equation $3.2(d)$ > we have $m$ the original mass and in eqn $3.3$ we have $m^{*}$ the effective > mass which is completely different from $m$. > In the end of the article the author says that the total force on the electron is given by $F_{tot}=ma$. After this the author says the external force applied on an electron is related to effective mass as: $F_{ext}=m^{*}a$. >   * From where this equation comes? Sometimes the effective mass $m^{*}$ is > negative then the equation implies that if we apply certain external force > on electrons in the crystal they will accelerate in the opposite direction, > how this is possible? What's going on? >"}
{"id":"57694","title":"What breaks the symmetry between the electromagnetic and weak nuclear force?","text":"I know the electromagnetic force is mediated by a photon and the weak nuclear force is mediated by two massive bosons. Are there any other insights into why the masses are so different?"}
{"id":"135212","title":"Virtual particles\/quantum tunneling - conservation of energy?","text":"I'm confused as to how the above phenomena can take place since arent they breaking the law of conservation of energy (even, if temporarily)?"}
{"id":"64950","title":"Question on Type HO\/HE string theory","text":"The Heterotic string state is a tensoring of the bosonic string left-moving state and the Type II string right-moving state. Therefore, I expect the spectrum to be: $$\\begin{array}{*{20}{c}} \\hline & {{\\rm{Sector}}}&{{\\rm{Spectrum}}}&{{\\rm{Massless fields}}}& \\\\\\ \\hline & {{\\rm{Bosonic}} - {\\rm{R}}}&{{\\bf{1}}{{\\bf{6}}_v} \\otimes {{\\bf{8}}_s} = {{\\bf{8}}_v} \\otimes {{\\bf{8}}_v} \\otimes {{\\bf{8}}_s}}&?\\\\\\ \\hline & {{\\rm{Bosonic}} - {\\rm{NS}}}&{{{\\bf{8}}_v} \\otimes {{\\bf{8}}_v} \\otimes {{\\bf{8}}_v}}&?& \\hline \\end{array}$$   1. However, how does one calculate the massless fields of the Type I string theory using the spectrum of the type I string theory?   2. Furthermore, to calculate the mass spectrum of the Heterotic string, does one simply add the number operator of the bosonic string to that of the type II string and the same for the normal ordering constant? i.e. is it true that $$\\begin{array}{l} m = \\sqrt {\\frac{{2\\pi T}}{{{c_0}}}\\left( {B + {{\\tilde N}_{II}} - {a_B} - {{\\tilde a}_{II}}} \\right)} \\\\\\ {\\rm{ }} = \\sqrt {\\frac{{2\\pi T}}{{{c_0}}}\\left( {B + {{\\tilde N}_{II}} - 1 - {{\\tilde a}_{II}}} \\right)} \\end{array}$$ **Edit: I found the answer to Question 2. Check the answers section. I have answered my own question.**"}
{"id":"133219","title":"Relationship between Connection and Material Derivative","text":"Suppose $D\\subset \\Bbb R^3$ contains a fluid and that $f : D\\times \\mathbb{R}\\to \\mathbb{R}$ is a time dependent function defined on the fluid region. In that case, the material derivative is defined by $$\\dfrac{D}{Dt}f(a, t) = \\dfrac{\\partial f}{\\partial t}(a, t) + (\\mathbf{u}\\cdot \\nabla)f(a, t)$$ Where $\\mathbf{u}\\cdot \\nabla$ is the operator defined on scalar function $g$ by $$(\\mathbf{u}\\cdot \\nabla )g = \\mathbf{u}\\cdot (\\nabla g) = D_{\\mathbf{u}} g$$ That is the directional derivative along $\\mathbf{u}$ of the function $g$. On vector fields it is defined componentwise, that is, if $\\mathbf{v} = (v_1,v_2,v_3)$ then $$(\\mathbf{u}\\cdot \\nabla)\\mathbf{v} = ((\\mathbf{u}\\cdot \\nabla)v_1, (\\mathbf{u}\\cdot \\nabla)v_2, (\\mathbf{u}\\cdot \\nabla)v_3) = (D_{\\mathbf{u}}v_1, D_{\\mathbf{u}}v_2, D_{\\mathbf{u}}v_3)$$ But that latter thing is clearly the Covariant Derivative of $\\mathbf{v}$ along $\\mathbf{u}$ when we consider the Levi-Civita Connection on $\\mathbb{R}^3$ with the usual flat metric tensor, that is $$(\\mathbf{u}\\cdot \\nabla)\\mathbf{v} = \\nabla_{\\mathbf{u}}\\mathbf{v}$$ Now, is this conclusion right? Can we really write the material derivative as $$\\dfrac{D}{Dt}\\mathbf{v}(a, t) = \\dfrac{\\partial \\mathbf{v}}{\\partial t}(a, t) + \\nabla_{\\mathbf{u}}\\mathbf{v}(a, t)$$ and if it's right is there some usefullness in this relationship? I mean, I don't know that much of connections and how they can be used on Physics, but I know they are usefull. In that case, writing the material derivative in terms of a connection gives some advantage? Would it make sense if $\\nabla$ were another connection other than the Levi-Civita connection?"}
{"id":"32147","title":"From where does the force\/energy of action-reaction comes from? (Newton's Third Law of Motion)","text":"I was wondering, from where does the opposite force and thus energy comes from when we apply force. For example, lets say there are two persons (P1 and P2) on the universe, and no force is applied to them. P1 applies a force to P2. P2 will start to move. P1 lost some energy to do that. Now, according to the third law of newton, P1 will also get a force from P2 as a reaction. But from who does the energy of reaction comes from? I mean, I found it hard to believe that P2 is the one who losses energy, just for the sake of reaction. So, from where does this force, and thus energy, comes from? I am not an expert, I just know the basics. So please, be tolerant :P"}
{"id":"101892","title":"How to derive the spin-orbit term in Kane Mele model?","text":"The Kane mele model is a famous quantum spin hall model on honeycomb lattice (C.L. Kane and E.J. Mele, Phys. Rev. Lett. 95, 226801 (2005)). The Hamiltonian is $$H = - t\\sum\\limits_{\\left\\langle {i,j} \\right\\rangle \\alpha } {c_{i\\alpha }^\\dagger {c_{j\\alpha }}} + i{\\lambda _{SO}}\\sum\\limits_{\\left\\langle {\\left\\langle {i,j} \\right\\rangle } \\right\\rangle \\alpha \\beta } {{\\nu _{ij}}c_{i\\alpha }^\\dagger \\sigma _{\\alpha \\beta }^z{c_{j\\beta }}} $$ The first term is just the nearest hopping term for graphene, while the second term represents the spin-orbit interaction, which seems quite peculiar to me.   1. Must it be in the z direction? Why?   2. Why does the spin-orbit term only have the second nearest hopping term, but no nearest hopping term? How does it relate to the symmetry of honeycomb lattice?   3. Is there an intuitive way to understand this term? **Can somebody tell me the derivation for the second term?**"}
{"id":"90801","title":"Black hole entangled with the cosmological horizon","text":"Maldacena and Susskind recently proposed a interesting and very suggestive duality between entanglement and topological identification: http:\/\/arxiv.org\/abs\/1306.0533 But are such ideas applicable to horizons in general? Can I entangle a black hole horizon, with, say, a Rindler horizon as seen by an accelerated observer? Can I entangle a black hole with our cosmological horizon? Are these in principle, physically possible operations? **Note:** this question might be part of a multi-part question"}
{"id":"79960","title":"How do I prevent shattering of glass?","text":"I am not an expert of physics, instead I am more good at chemistry. I just wanted to ask that how do I prevent shattering of glasses on sudden large temperature changes? Sometimes, when I have to cool hot test tubes rapidly, the test tubes break down when I place them in water for cooling. Is there any way to prevent it?"}
{"id":"75236","title":"About units and plural form","text":"The plural form of physical units are always confusing me. I asked some senior students, some of them said we need to use plural form of units but some said units are always singular. For example, 1 meter is 1 meter. But it is 3 times, should it be 3 meters? I know that if we use the abbreviation form, no plural form, i.e. 3m. But if we use 'meter', for the case like 0.04 meter, should it be 0.04 meter or 0.04 meters? I think we should use 0.04 meter because it is less than or equal to 1. The last question I have is about foot and feet, when the magnitude is less than or equal to 1, I should use foot, otherwise, use feet instead, is that right?"}
{"id":"2172","title":"How does an electron microscope work?","text":"I am a physics novice. Google tells me that electron microscopes work much like their optical counterparts -- but the analogy falls apart for me when I think about what I'm \"viewing.\" Obviously, you can _see_ light through the lenses, but what is the \"image\" analog for electron microscopes? Is it at-all like spraying an invisible shape with bullets and examining where collisions took place? Like if you shot at an invisible car with a tommy gun and were able to make out bullet holes -- so that the more bullets you shoot the better your image? And, just for completeness, I suspect this implies that the best resolution you can get is the bullet-size, or in this case the size of the electron. How do you map \"objects\" or whatever they are considered on that scale if they are smaller than an electron? Is our perception of how _small_ we can see limited by this cap?"}
{"id":"2179","title":"Electron\/Photon Scattering","text":"Hey guys, I have a final tomorrow and I am going over some assignments. One of the questions from the assignment was: > A photon having 37 keV scatters from a free electron at rest. What is the > maximum energy that the electron can obtain? Now, I got the right answer (4.67 KeV) but I can't seem to figure it out now. Maybe I am just too tired. Any hints to where I should begin?"}
{"id":"17137","title":"superconductor levitating in earth's magnetic field?","text":"> **Possible Duplicate:**   >  Can superconducting magnets fly (or repel the earth's core)? I've seen superconductors levitating on magnets. But is it possible for superconductors to levitate on Earth from Earth's magnetic field?"}
{"id":"14040","title":"Is there a stable numerical algorithm for FWHM that isn't 2.35*sigma?","text":"This is a question that should have a simple answer, but which I can find no proper discussion of in the literature or on the internet. I start from the assumption that I have a noisy numerical signal with a peak in it - for example amplitude as a function of $x$, i.e. $A(x)$ - and I wish to determine its full-width at half maximum (FWHM). I am aware of two basic algorithms that may be used on numerical data:   1. Assume it's Gaussian, determine the r.m.s. value, and multiply by 2.35 ( $2\\sqrt{2 \\log{2}} $).   2. Determine the FWHM using a peak-finding algorithm that locates the peak $A_{max} = A(x_{peak}$), then locates the _first_ positions either side where $A(x)$ falls to $1\/2 A_{max}$. Method 1 is numerically robust as you don't have to bin (i.e. smooth) the data that determines $A(x)$. However, if the peak is significantly non-Gaussian in shape you get a systematically wrong answer. Method 2 is direct, but there are plenty of numerical cases where there are multiple points in $x$ either side of the peak where $A(x)$ crosses $1\/2 A_{max}$. And of course the number of crossing points depends sensitively on whether I smooth A(x) or not. My question is therefore whether there is a mathematically justifiable algorithm for determining FWHM numerically that doesn't assume the peak is Gaussian,"}
{"id":"14043","title":"Plotting a wave function that represents a particle","text":"The problem is this: > A particle is represented by the wave function $\\psi = > e^{-(x-x_{0})^2\/2\\alpha}\\sin kx$. Plot the wave function $\\psi$ and the > probability distribution $|\\psi(x)|^2$. This the problem 2.1 in the book _Fundamental University Physics Volume III_ by Marcelo Alonso and Edward Finn. The thing is I don't know what values $k, \\alpha$ and $x_{0}$ should have. Probably I don't know what $\\psi$ really represents in this case."}
{"id":"3846","title":"Basic mechanics problems, unsolvable by brute-force numerical integration","text":"I'm looking for simple problems in theoretical mechanics that are impossible or unreasonably difficult to solve by means of \"brute-force\" numerical integration of Newton or Euler-lagrange equations. I'm interested in these beacuse I noticed that kind of \"computer-nihilism\" point of view is getting popular (at least among some students): A person says, that \" _in the end we anyway doing real stuff by computer simulation_. _And for the numerical values of parameters we are usually able to numerically obtain the result with a given precision. So we just need to know how to write down the equations_ \".   And, therefore, \" _there is no need to learn all that complicated stuff in theoretical mechanics_ \". Apart from obvious counter-arguments for this, I'd like to show that there are basic problems you are unable to solve without \"the complicated stuff\".   Let me give an example of such a problem:   **Given:**   1. A center, that creates some strange field with the potential $U(r)=-\\frac{\\alpha}{r^3}$. (Mysterious planet)   2. A body with mass $m$ scattering off this center. (Our space ship.)   3. A radius R, at which we want to stay as long as possible. **Find:** the impact parameter $\\rho$ and the energy $E_0$ for our body, so it will stay in the \"ring\" $R<r<2R$ for as long as possible. The problem is easily formulated. And it is easy to solve even for \"newbies\" in theoretical mechanics. The specific feature of the problem -- there is no reasonable way of solving it by doing straightforward computer simulation. Can you propose other examples of problems with these properties?"}
{"id":"122161","title":"How can one calculate the phase difference between two quantum harmonic oscillator (Hermite-Gauss) states?","text":"The analytic solutions of a quantum harmonic oscillator are given by Hermite- Gauss states, which differ in the order $n$ of the Hermite polynomials. If two such states are plotted, there will be a relative phase shift. How can that be calculated analytically?"}
{"id":"53849","title":"Is my electric power cord creating magnetic field when coiled?","text":"Simple question. But I was wondering, does my mac power cord create a magnetic field when its all coiled?"}
{"id":"71423","title":"Hydrogen cloud at the universe's beginning?","text":"What prevented all of the hydrogen at the universe's start from coalescing into one gigantic star?"}
{"id":"71427","title":"Trouble justifying calculation of force on a sphere due to radiation pressure","text":"I know very little about electromagnetic waves and light in general so you are going to have to bear with me. I am attempting to calculate the force on a sphere in a plane wave of light where the light has intensity I. My calculations and research have concluded that, regardless of whether or not the sphere absorbs or reflects the light, that the force can be modeled as: $\\frac{\\pi{r}^2{I}}{c}$. This pdf document outlines what I am looking for (with $I$ replaced with $\\langle{S}\\rangle$) but in equation 24 it seems like a factor of $2\\pi\\text{r}$ appears out of nowhere. Another solution I have tried is a double integral in spherical coordinates and exploiting symmetry to find that $\\text{Force} = \\iint\\limits_A \\\\! p \\,\\,{dA} $ where p is a function describing the pressure on an area element. Every time I calculate this pressure I find a function of one angle rather than of two angles. I find that my integral looks like: $$\\int_{0}^{\\pi}\\\\!\\int_{0}^{\\frac{\\pi}{2}}\\\\!\\frac{4\\pi{r}^2{I}}{{c}}cos(\\theta)^3sin(\\theta) \\, d\\theta d\\phi$$ which evaluates to the correct expression for force (up to the sign) but I have trouble justifying that the pressure does not depend on $\\phi$ and only depends on $\\theta$. In this instance, I am integrating over half of a hemisphere where $\\theta$ is the polar angle and $\\phi$ is the azimuthal angle or as described in this image. I feel like these calculations are incorrect and that the double integral over ${dA}$ should include an element reflecting the influence of $\\phi$ as I cannot see any reason why the calculation would be independant of it."}
{"id":"104991","title":"If electromagnetic waves have magnetic fields, why beam of flashlight is not disrupted by a Magnet?","text":"Wikipedia article about Electromagnetic Radiation says \"As an electromagnetic wave, it has both electric and magnetic field components\". And this discussion also confirms Light is EM wave. Since we know that light travels through electromagnetic waves. Does that mean when we hold a Magnet close to the beam of a flashlight, it should interact with the waves and cause funny things to happen ?"}
{"id":"30547","title":"Do composite particles that are electrically neutral but have charged constituents radiate?","text":"For example an electron radiates when accelerated. So does a positron. But is the radiation emitted by accelerated positronium the sum of the radiation emitted by each separately? If not, why not? If so, does this provide a way of testing whether a given neutral particle is composite? For example, does a neutron bremsstrahlung when decelerated?"}
{"id":"30545","title":"polynomial fit and uncertainty","text":"I am trying to fit a polynomial function by least square method and I want to know the uncertainty on my fitting parameters. In which book can I find a clear answer?"}
{"id":"83562","title":"Fundamental Constants in a theory of everything (TOE)","text":"Do physicists ever expect to be able to derive the fundamental constants of nature from theory? For example, if string theory or some other theory unites the four forces, would the theory be considered complete if it relies on these measured constants, or would a true theory of everything (TOE) require that these constants come out of the theory itself?"}
{"id":"133078","title":"\"Delayed choice\" quantum imaging experiment - why wouldn't it work?","text":"I was just reading a Nature news article (Entangled photons make a picture from a paradox, 27 August 2014) about a way to form an image of an object out of photons that have not interacted with the object, but which are entangled with photons that have. The article has a photograph of the image, which is of a cardboard cutout of a cat. If I understand correctly, the technique is that   1. a photon beam is split, e.g. via a semi-silvered mirror, resulting in two beams, call them A and B;    2. in each beam there is some device that turns each photon into two entangled photons. At this point there are four beams, call them A1, B1, A2 and B2.    3. The photons in beam A1 interact with the cardboard cat   4. The photons in beam A1 are allowed to interact with the photons from beam B1 in some fashion, _and then discarded_   5. The photons in beams B2 and A2 (neither of which has interacted with a photon that's interacted with the object) are combined somehow and used to produce the image. This is pretty neat - the article talks about using to image an object with very low-energy photons, while still being able to make an image on a screen using high-energy ones. However, the following occurred to me while reading about it: presumably, as with all experiments of this kind, it shouldn't in principle make any difference if we change the length of the beams. What would happen if we keep the length of beams A2 and B2 short, but we make beams A1 and B1 really long, say a few light hours? In this thought experiment the cardboard cat, along with the device for combining the photons from beams A1 and B1, is on Pluto, and the rest of the equipment is in a lab on Earth. It seems as if it should work just as well in this case as it does with everything in the same lab. But now we're seeing the image of the cat five hours before the photons interact with it - and by swapping the cat out for a different object, someone on Pluto could send us a message from the future. Presumably this can't really work, since it would be possible to construct a paradox if it did. But where is the issue? Have I misunderstood something about the setup (entirely possible since I'm getting this from the news article rather than the actual paper), or is there some fundamental reason why it would stop working if the A1\/B1 beam length was much longer than the A2\/B2 one? This seems similar to delayed choice quantum eraser experiments. However (if I remember correctly from looking into them years ago) in those experiments you have to combine all the beams back again to make the image in the end, so there's no way to construct a paradox. In this case the Nature news article seems quite clear that the photons are thrown away after interacting with the cardboard cat, and the image is produced _only_ from those that haven't - this seems to be a major difference."}
{"id":"113468","title":"Dirac Delta function","text":"Although I find it mathematically dubious, we said that $\\Delta \\frac{1}{r} = -4\\pi \\delta(r)$. Now, I was wondering is there a similar relation to the delta function if we look at $\\Delta \\frac{1}{r^n} =?$."}
{"id":"112737","title":"Is the viscous sublayer size universal?","text":"We know the non-dimensionalized distance of the viscous sub-layer is 5 wall units for a turbulent boundary layer. It is common to see this in Fluid Mechanics books but seems somewhat arbitrary. I want to know if this true for all conditions or does it depend on the Reynolds number (high viscosity, high speed, etc)."}
{"id":"47475","title":"How does an electrical field really work?","text":"A little bit of background information: I'm planning to write a little booklet or web page about CPU\/computer architecture, basically for my own education, because we didn't cover it in depth in college. I feel like I should be learning a lot more about the fundamentals about how computers work, if I want to be a better programmer. The idea is to present the workings of a CPU by describing it from the ground up. Starting at simple electrical circuits, then semiconductors, transistors etc. From my school physics class I remember that I was very often left with a lot of gaps about how different concepts are linked together. I want to avoid this. So here's my question: How does an electrical field really work? How come that an electron can exert a force on another electron without physical contact? What is it in an electron that creates the field, where does the energy for doing the work come from?"}
{"id":"22790","title":"Are a measured object always part of the theory?","text":"Is there a notion of measurement, which doesn't correspond to a yes\/no question or with the idea of the comparison of two real world objects, which produces a real number? And does at least one of the object which are compared to produce the measurement result have to be modeled within the theory? (One of my motivations for that question is that I wonder if a human can make a measurement involving light without a length measurement and it also seems to me that almost all measurements involve looking at data, which therefore effectively makes them length measurements. And then I speculate if there are even other mearuements than these. For example, I can't test the quantum wave values unless I collapse it - its reality is hidden (this question is certainly not only about QM though.) and therefore the specific configuration is not real is a strict sense. Similarly there is no time mearuesment, is there? The configuration of the clock is just infered by a mearuement of light.)"}
{"id":"6887","title":"effective theory of graphene","text":"This is a question about deriving effective mass theory for graphene. For the two sub-lattice atoms, the wave equation can be written as the massless Dirac equation: $ \\displaystyle -i\\hbar v_F \\begin{pmatrix} 0 & \\partial_x -i\\partial_y \\\\\\\\\\partial_x +i\\partial_y & 0 \\end{pmatrix} \\left(\\begin{array}{c} \\Psi_A \\\\\\ \\Psi_B \\end{array}\\right)=E \\left(\\begin{array}{c} \\Psi_A \\\\\\ \\Psi_B \\end{array}\\right) \\ \\ \\ \\ \\ (1)$ where ${A,B}$ are two subatoms. The derivation of the equation went back to 1984, which is the paper I am trying to understand. In the article, they argued that at first order of ${\\vec{\\kappa}\\cdot \\vec{p}}$ expansion, the momentum matrix can be written in the form, under group-theoretic arguments (equation (3) in the paper): $ \\displaystyle \\bar{p}\\begin{pmatrix} 0 & \\hat{x} -i\\hat{y} \\\\\\ \\hat{x} +i\\hat{y} & 0 \\end{pmatrix} . \\ \\ \\ \\ \\ (2)$ What is the argument behind it? Is there anyone read the paper or know the answer?"}
{"id":"15791","title":"Why is the sun so dense?","text":"Looking at the radius and mass of stars on Wikipedia, I see that the Sun is the densest of all, often many times denser than other stars. Is that because only non-dense starts are easily seen from a distance? Are there any stars of comparable luminosity to the sun that can be seen with the naked eye, and do they have a similar density as the sun? If needed I can copy-paste the mass and radius of other stars here for reference."}
{"id":"39441","title":"Angular acceleration of stone disk","text":"I have a conceputal question regarding the following problem: A round massive stone disk with diameter $0.600 m$ has a mass of $50.0 kg$. The stone rotates at an angular velocity of $115.2 rad\/s$, around an axis located at the center of the disk. An axe pressed against the disk applies a tangential frictional force, $F_f$ at $50.0 N$. Assume that we turn off the rotating power of the disk, so that the frictional force is the only force acting on the disk. What is the magnitude of the angular acceleration, $\\alpha$, of the disk? OK, so I have calculated the torque ($-15 Nm$), and the moment of inertia ($I = 2.25 kg \\cdot m^2$). By using the formula: $$\\tau = I \\alpha$$ It is easy to show that $\\alpha = -6.67 rad\/s^2$, which is also the correct answer. However, say I want to try to solve this in another way, without using the torque and moment of inertia. Since the frictional force is the only force acting on the disk, I would assume that we would have: $$F_f = ma_T$$ $$-50 = ma_T$$ $$-50 = 50 \\cdot a_T$$ $$a_T = - 1 m\/s^2$$ And since we have: $$a_T = r \\alpha$$ This gives: $$\\alpha = \\frac{-1}{0.3} = -3.33 rad\/s^2$$ Which is not the same answer. So why doesn't this second approach work? If anyone can explain this to me, I would really appreciate it!"}
{"id":"24548","title":"Is the Portal feasible in real life?","text":"Firstly, I assume prior knowledge\/experience with the game Portal. Should it be possible that two openings (circles\/unidentified-constructions) would work as interchangeable gates? With one opening analyzing the molecular structure of the object passing through and the other reconstructing it accordingly? Maybe as a replica? The object doesn't need to be alive or biological at all."}
{"id":"31638","title":"What equations govern the formation of droplets on a surface?","text":"When some smooth surface (like that of a steel or glass plate) is brought in contact with steam (over e.g. boiling milk) then water is usually seen to condense on that surface not uniformly but as droplets. What are the equations which govern the formation and growth of these droplets ? In particular what role does the geometry of the surface plays in it? Also it is possible to prepare experimental conditions where no droplets are formed but water condenses uniformly ?"}
{"id":"88474","title":"Retarded (Advanced) GreenFunction","text":"I need to show that Green Function: $\\displaystyle G(\\vec{r},t;\\vec{r}^{\\prime},t^{\\prime}) = \\frac{\\delta(t-t^{\\prime}\\pm |\\vec{r}-\\vec{r}^{\\prime}|\\,\/c)}{|\\vec{r}-\\vec{r}^{\\prime}|}$ Obeys $\\displaystyle \\nabla^2 G -\\frac{1}{c^2}\\frac{\\partial^2 G}{\\partial t^2} = -4\\pi \\delta(\\vec{r}-\\vec{r}^{\\prime})\\delta(t-t^{\\prime})$ I tried to use the identity for Laplacian : $\\displaystyle \\nabla^2 (\\psi\\phi) = \\phi\\nabla^2\\psi + \\psi\\nabla^2\\phi + 2\\vec{\\nabla}\\phi\\cdot\\vec{\\nabla}\\psi $ As follows: ![My work](http:\/\/i.stack.imgur.com\/9DJKa.jpg) But I couldn't get nice results. Any suggestions?"}
{"id":"103798","title":"Rate of probability loss from absorbing boundary","text":"The following is the solution to the 1D diffusion equation with diffusion coefficient D, initial particle position $x_0$, and a perfectly absorbing boundary at $x=0$ (s.t. $P(x=0)=0$). $$ P(x;t)=\\frac{1}{\\sqrt{4 \\pi D t}}e^{-\\frac{(x-x_0)^2}{4 D t}} - \\frac{1}{\\sqrt{4 \\pi D t}}e^{-\\frac{(x+x_0)^2}{4 D t}} $$ If I understand correctly, for an $x_0>0$. $$ P(\\text{no collision at time $t$})=\\int_0^\\infty P(x;t) dx $$ In other words, the total probability in allowed space at time $t$ is exactly the probability that the particle _never_ contacted the absorbing boundary at $x=0$ up to time $t$. What I want to compute is the rate of probability loss $k(t)$. From above, it seems that would be: $$ k(t) = \\frac{d}{dt} \\int_0^\\infty P(x;t) dx $$ evaluating with mathematica reveals: $$ k(t)=-\\frac{D x_0}{2 \\sqrt{\\pi}} \\left(\\frac{1}{D t}\\right)^{3\/2} e^{-\\frac{x_0^2}{4 D t}} $$ which seems reasonable. It seems that there should be a way to compute $k(t)$ _without_ computing the spatial integration across $x$, perhaps some computation involving only the boundary. I thought since all the loss occurs at $x=0$, the time derivative of $P(x;t)$ evaluated at $x=0$ should be k(t). However, the result of that calculation is 0. _Question: is there a way to compute k(t) without computing the spatial integral over the $x$ domain?_"}
{"id":"35823","title":"Is there a deep reason why springs combine like capacitors?","text":"I was solving a practice Physics GRE and there was a question about springs connected in series and parallel. I was too lazy to derive the way the spring constants add in each case. But I knew how capacitances and resistances add when they are connected in series\/parallel. So I reasoned that spring constants should behave as capcitances because both springs and capacitors store energy. This line reasoning did give me the correct answer for how spring constants add, but I was just curious if this analogy makes sense, and if it does, how far one can take it. That is, knowing just that two things store energy, what all can you say will be similar for the two things."}
{"id":"100031","title":"Shrinking and Expanding objects in a CFT","text":"In a conformal field theory, is it possible to construct a machine that shrinks or expands objects?"}
{"id":"103867","title":"Including special relativistic effects in momentum in Heisenberg's Uncertainty Principle","text":"I've been told that an electron is somewhere within the space of $10^{-10}m$ and am supposed to find the uncertainty in its velocity. Simply applying $m\\Delta x \\Delta v \\geq \\frac{h}{4\\pi}$ results in $\\Delta v \\geq 5.79\\times10^{10} m\/s$. Now, this velocity is not approaching the speed of light, but I wanted to know if I could use the Lorentz factor to make this calculation more accurate (of course in this situation it would not make much of a difference. So I redefined momentum uncertainty as $\\Delta p = \\gamma m_0 \\Delta v$ and solved the inequality... but my final answer states $\\Delta v \\leq 3\\times10^8$... this relation is obvious... but why has it popped out of this? What if the space was even smaller when special relativistic effects actually would come into play, how would I then tackle the question?"}
{"id":"112927","title":"What is the criterion of stability of thick-walled spherical shell?","text":"Is there the formula (if someone already has discovered it) or what is the algorithm (if a particular formula was not deduced), to calculate the critical pressure of thick-walled spherical shell $−$ the pressure which provokes buckling?"}
{"id":"103422","title":"Visualising scalar and vector fields","text":"As I have been studying electromagnetic theory, I have always been lead to confusion when trying to visualise the fields. Fortunately, the electric and magnetic fields are vector fields and also along the propagation direction they provide a 3-Dimensional Coordinate system. However, in general to visualise a N-dimensional field (I mean field depending on N coordinates), N+1 dimensions are necessary. For instance, in the case of 1D field $\\phi(x)$, we can plot a graph $x$ vs $\\phi(x)$. But I am wondering if it is also possible to describe the fields using contours. To elaborate, if I have 2D field $\\lambda(x,y)$ will it possible to visualise the field in 2D itself using contours (without actually plotting the variation of $\\lambda(x,y)$ vs $x$ and $y$)."}
{"id":"132298","title":"How can Excel be used to model the flight of a rocket?","text":"Firstly, sorry for any silly mistakes or improper use of this website - this is my first time using it! I am trying to use Excel to model the flight of the Apollo 11 Saturn V rocket using the real values from the mission evaluation report. However, I can't seem to get the right acceleration. It is always too low. My initial acceleration at t=0.1s is 2.06 m\/s^2 while the values in the evaluation report show the acceleration to be close to 10 m\/s^2. My final goal is to calculate the acceleration at the S-IC OECO event (at t=161s). My spreadsheet showed 28m\/s^2 while the real value is 39m\/s^2. Obviously I have done something terribly wrong since the difference between the values is quite large, but I've been looking over my spreadsheet for days now and I just can't find my problem. What _should_ the columns be? This is what I have so far:   * Range time (s) - in increments of 0.1s down the column   * Mass (kg) - the mass at that time calculated by previous mass - mass flow rate*0.1s   * g (m\/s^2) - the acceleration due to gravity at that altitude   * Weight (N) - mass*g   * Thrust (N) - the thrust at that time using real values from the graphs in the evaluation report   * Drag (N) - using the drag equation and estimated coefficients    * Total force (N) - weight + thrust + drag    * Acceleration (m\/s^2) - using F=ma so I did total force\/mass    * Velocity (m\/s) - previous velocity + acceleration*0.1s    * Altitude (m) - previous altitude + velocity*0.1s I've taken into account changing thrust, mass flow rate, acceleration due to gravity and drag as well. The values in the report are given as 'inertial acceleration' but I haven't learnt this yet (still in high school). So is inertial acceleration different to the acceleration I have calculated using F=ma? Have I misused Newton's second law? Was I wrong to consider the weight of the propellants in this case? I'll appreciate any and all helpful feedback and I can provide more information if needed. Thank you so much! :)"}
{"id":"5961","title":"Find drop-off rate of magnetic interference from a mass of pure iron on a magnetic compass","text":"How can I find the magnetic interference a stationary 35000 kg block of 100% pure iron would have on a magnetic compass and what the drop off rate of the interference would be. So if said 35000 kg block of iron was 1 meter away from the compass, 100 meters away, or 1000 kilometers way I would like to calculate the rate of drop off of the interference. * * * This may seem absurd, but it is very important for a conceptual project I am working on. For the context of this question, assume everything is perfect, and that we are basically operating in a vacuum and there is no interference from anything else and that all instruments are 100% accurate and infinitely precise. And that I have only a very very basic understanding of physics, mathematics and magnetism."}
{"id":"74888","title":"What is the lifetime of an induced magnetization in a para\/diamagnetic material?","text":"To the best of my knowledge thermal fluctuations are responsible for washing out any effective magnetization, once the external field is switched off. Since thermal fluctuations need some time to drive the system back to random orientation, what would be the time scale for this process? Is it of the order of 200 ps?"}
{"id":"74886","title":"Speed of a falling pencil","text":"If you balance a pencil of length $d$ on its tip, and let it fall, how do you compute the final velocity of its other end just before it touches the ground? (Assume the pencil is a uniform one dimensional rod)"}
{"id":"17552","title":"What are some interesting calculus of variation problems?","text":"That I could create as a classical mechanics class project? Other than the classical examples that we see in textbooks (catenary, brachistochrone, Fermat, etc..)"}
{"id":"13428","title":"How are Monte Carlo simulations used in experimental high energy physics?","text":"How are Monte Carlo simulations used in experimental high energy physics? Particularly in studying detectors limitations (efficiencies?) and data analysis. I will appreciate giving a simple example to clarify how MC is used if the question is too technical to have a simplified answer"}
{"id":"101428","title":"\"String\" infinity paradox","text":"The smallest particles of the universe (or the smallest part of the smallest particles of the universe) must have infinity density. But if it have infinity density, you can calculate mass of one of these particles as infinity - so if any thing would composed of these particles, would it have infinity mass. How is it possible?"}
{"id":"81906","title":"Should particles in Smoothed Particle Hydrodynamics (SPH) always move during a simulation?","text":"Or can they just be used as an interpolation points and use some other \"transported property\" which are just evolved and propagated from boundary conditions like for eg. heat conduction through a solid metal bar. It seems to be possible, but isn't clearly elaborated. The force computation step is always used for fluid flow to move particles forward according to Newton's laws, but what if heat is transported purely by conduction in a solid? Here, nothing moves but still, heat flows and destroys temperature gradients. What equations are to be considered here (apart from the heat diffusion equation) to adapt this problem for this kind of discretization?"}
{"id":"81904","title":"Does a diatomic molecule falling into a black hole dissociate?","text":"I've just answered Dipping a Dyson Ring below the event horizon, and while I'm confident my answer is correct I'm less certain about the exact consequences. To simplify the situation consider a diatomic molecule falling into a Schwarzschild black hole with its long axis in a radial direction. ![Molecule](http:\/\/i.stack.imgur.com\/KZea4.gif) The inner atom cannot interact with the outer atom because no radially outwards motion is possible, not even for electromagnetic fields travelling at the speed of light. However I find I'm uncertain exactly what the outer atom would experience. We know that the outer atom feels the gravitational force of the black hole even though gravitational waves cannot propagate outwards from the singularity. That's because it's experiencing the curvature _left behind_ as the black hole collapsed. Would the same be true for the interaction of the outer atom with the inner atom? Would it still feel an electromagnetic interaction with the inner atom because it's interacting with the field (or virtual photons if that's a better description) left behind by the inner atom? Or would the inner atom effectively have disappeared? If the latter, presumably the fanciful accounts of observers falling into the black hole (large enough to avoid tidal destruction) are indeed fancy since it's hard to see how any large scale organisation could persist inside the event horizon. **Later:** I realise I didn't ask what I originally intended to. In the above question my molecule is freely falling and the question arose from a situation where the object within the event horizon is attempting to resist the inwards motion. I'll have to go away and re-think my question, but thanks to Dmitry for answering what I asked even if it wasn't what I meant :-)"}
{"id":"45162","title":"Spectrum of quantum fluctuations in a harmonic oscillator","text":"If we have a harmonic oscillator and look at it on small scale the energy is quantized and we can calculate the different eigenstates. In general the energy eigenvalues are given by $$E_n = \\left(\\frac{1}{2}+n\\right) \\hbar \\omega$$ Even if we can bring this system at $T=0$ into it's ground state, there will be zero point motion or quantum fluctuations remaining. Now if we heat a system, depending on its excitations we need Maxwell–Boltzmann, Bose-Einstein or Fermi statistics to calculate the occupation of each state. The resulting spectra are due to thermal excitations from the ground state. Now if we remain at about absolute zero the system still has quantum fluctuations. From Heisenberg's principle the energy is uncertain but that does not tell me which is the current energy or eigenstate. How can one calculate the 'quantum' spectrum of an harmonic oscillator, or bluntly, how can one calculate the probabilities for observing the system at the state $n=1,2,...$ at $T=0$ if we prepare the system to be in $n=0$?"}
{"id":"80057","title":"Racing balls question","text":"My question is related to simulation of racing ball demonstration. http:\/\/www.phy.ntnu.edu.tw\/ntnujava\/index.php?topic=142 One ball goes on a straight path, while another one goes on a curved path. On the simulation the second ball going on the blue part down hill which is parabola has a constant component of the velocity in the x direction. Why is that the case when the slope changes?"}
{"id":"80059","title":"Speed of approach between image and object","text":"A plane mirror image approaches the object at the same rate the object approaches the mirror. So the speed of approach is twice the speed at which the object approaches the mirror. If the object approaches the mirror at a speed greater than half the speed of light (c), does that mean that the speed of approach between object and image will exceed the speed of light?"}
{"id":"6075","title":"Can I calculate the height of a cliff from weight of falling object and time taken?","text":"I'm sure a simple question. I have a video of me jumping off a cliff into a river. I want to calculate how high it is. I know my weight, acceleration due to gravity of course, and I can get the time it took to hit the water. Can I calculate (albeit roughly) how high the cliff was? Cheers, Tim."}
{"id":"134008","title":"No photon interaction in free space","text":"How can the Hanbury Brown and Twiss effect (photon bunching) be explained if photons don't interact in free space? To explain it with the influence of the two photons on the two detectors simultaneously would seem strange to me, because of the fact that \"if the source consists of a single atom which can only emit one photon at a time, simultaneous detection in two closely spaced detectors is clearly impossible\" (same Wiki under the \"Quantum interpretation\" heading). Comment: The question is not about uniting or dividing two photons, which is possible only on the atomic level. The question is about synchronizing two photons from different sources like the sun."}
{"id":"7340","title":"Water rocket physics","text":"According to this page, the current record for greatest altitude achieved by a water and air propelled rocket is 2044 feet (623 meters).   In this connection, i had one question: How the greatest altitude depends of the density of the liquid used in the rocket?   Suppose that we have the opportunity to choose a liquid with an arbitrary density and with the same hydrodynamic properties as water. Then there must be a density at which the greatest altitude has its absolute maximum, i think. To be more specific let's consider a simpler problem: Let a two-liter soda bottle rocket, filled with some amount of a liquid mentioned above and air at the atmospheric pressure, is launched in deep space($g=0$)(say an astronaut is experimenting). What is the absolute maximum of the speed the rocket can achieve? For simplicity let's assume the expansion of the air to be isothermal (which it is not). ![enter image description here](http:\/\/i.stack.imgur.com\/RVXX7.png)"}
{"id":"10024","title":"Figuring out North and South Magnetic Poles for Earth","text":"I currently make my living as an electrophysicist, so am a bit embarassed at having not thought this one through, before. Magnetically, the North pole is.... North. We get out our compass, and wait for the North-seeking needle to line up. I've always thought of this end of the needle as the north end. In a current carrying loop, if we use the right-hand rule and point our thumb in the direction of conventional current, the fingers show the direction of the mag field lines around the wire. The direction the mag field lines point \"north\" when they pass through the interior of the loop. By stacking many such loops we can create a cylindrical electromagnet, for which it is easy to see which end is the \"north\" end. I resume permanent physical magnets follow the same naming convention: that is, mag field lines emanate from the north end and cycle around to the south. We know that like poles of a magnet repel and opposites attract. So, now, hanging a magnet from a string (or going back and using our trusty compass), if the north pole of the magnet points towards teh magnetic north pole of Earth, doesn't that mean that the pole near the Arctic circle is really magnetically-speaking the south pole? Or conversely, when we say \"north-seeking\" pole of a magnet, is that really the south pole of the magnet? Which is it?"}
{"id":"70431","title":"Is there an equivalent of a scalar potential for torques?","text":"For a given scalar potential $V$, it is known that the corresponding force field $\\mathbf{F}$ can be computed from $$ \\mathbf{F} = -\\nabla V $$ Suppose a rigid body is placed inside this potential. The torque on the body $\\mathbf{T}$ exerted by the scalar field will be $$ \\begin{align} \\mathbf{T} &= \\int_M \\left( \\mathbf{r} \\times \\mathbf{F}(\\mathbf{r}) \\right) dm \\\\\\ &= \\int_M \\left( \\mathbf{r} \\times -\\nabla V(\\mathbf{r}) \\right) dm \\\\\\ &= \\int_M \\left( \\nabla V(\\mathbf{r}) \\times \\mathbf{r} \\right) dm \\end{align} $$ with $\\mathbf{r}$ the position vector to the mass element $dm$, and the integration carried out over the entire body $M$. So, being not too familiar with rigid body dynamics, I was wondering -- does something like a (vector\/scalar) potential $P$ exist, such that the local torque induced by the potential $V$ can be expressed as $$ \\matrix{ \\mathbf{T} = I \\cdot \\nabla P & & &\\text{(or some similar form)} } $$ with $I$ the moment of inertia tensor of the rigid body? If such a thing exists:   * what is its name?   * where should I start reading?   * What is the proper expression for the torque $\\mathbf{T}$?   * how does $P$ relate to $V$? If such a thing _doesn't_ exist:   * why not? :)"}
{"id":"8227","title":"A mirror flips left and right, but not up and down","text":"Why is it that when you look in the mirror left and right directions appear flipped, but not the up and down?"}
{"id":"93964","title":"Axis of reflection in a mirror","text":"When I look at my reflection in a planar mirror, the image I see is reflected about a vertical axis. Why is it this axis and not the horizontal axis?"}
{"id":"70438","title":"How far would a radio signal propagate in free space as compared to Earth's atmosphere?","text":"A curiosity question. Radio Propagation within Earth's atmosphere is via atmosphere. Broadly speaking a signal loses strength between bouncing off the ionized layer, and absorption by various earthy elements. If an identical signal were to be produced in free space, how far would it travel as compared to it's range within Earth's atmosphere? EDIT: A commercial FM transmitter on Earth probably has a range no more than 25 miles where the listener uses a broadcast receiver. For the purpose of elaboration assume the the receiver being placed in static orbit - how far out could the same signal be generated, and still be intelligible?"}
{"id":"29880","title":"Propagator of the Klein-Gorden equation","text":"## Does this integral converge? My question is related to this one: Free particle propagation amplitude calculation I am reading the book of Peskin and Schroesder. In the second page of their chapter 2, they estimated the propagator for the Klein-Gorden equation. The final step of the calculation is to calculate an integral of the form $$\\int_0^\\infty \\mathrm{d}p\\;p\\;\\sin(px)e^{-\\mathrm{i}t\\sqrt{p^2+m^2}}.$$ My question is very simple: does this integral converge in the first place? Note that the integrand is an oscillating function whose amplitude grows without bound. If we take $m=0$, then for sure the integral does not converge. In fact, I can reproduce their result very easily (namely, the $e^{-m\\sqrt{x^2-t^2}}$ behavior). But I am not sure whether my calculation makes sense because I don't know whether the integral has a finite value. I know this integral can be converted into a loop integral in the complex plane, and then be estimated using stationary-phase method. But since the amplitude of the integrand does not go to zero uniformly at infinity, I feel such a procedure may not be valid. We may multiply a factor $e^{-\\epsilon p}$ ($\\epsilon>0$) to the integrand to make it converge, but then we are not calculating what we wanted to calculate in the beginning, right? Then, if the integral does not converge by itself, how should I make sense of the result in the Peskin and Schroesder book? I am quite confused by this. Thanks for any hints!"}
{"id":"29889","title":"How much thrust would be needed to turn a hobbyist weather balloon into a deep space probe?","text":"I was reading the article _Weather Balloon Space Probes_ that says you can put your own balloon probe at 65,000 ft temporarily. Is it even remotely possible to raise the probe high enough using balloons, and then put it in orbit or escape the field using an on-board propulsion system? Can it be done with easily purchased materials? How much mass can be added for instrumentation? How much fuel of is required? What kind? I know it's kind of a broad question, but I just want to know a rough estimate."}
{"id":"119340","title":"Magnetic moment with external magnetic field on a lattice?","text":"Consider a system in which atoms are located in a regular lattice, each atom having a spin $1\/2$ and an associated intrinsic magnetic moment $\\mu_0>0$. Assume that each atom interacts only weakly with other atoms so that all the other atoms act as a heat reservior. If the system is placed in an external magnetic field $H_0 \\hat{z}$, then the Hamiltonian is given by: $$\\hat{H}=-\\mu_0H_0\\sum_i\\sigma^z_i$$ where $\\sigma_i^z$ are (the $z$ components of) the Pauli matrices, at the $i$th site. For the magnetic moment $\\mu= \\pm\\mu_0$, the corresponding magnetic energy of the atom is $E=\\mp\\mu_0 H_0$. What would be the mean magnetic moment?"}
{"id":"119342","title":"Why didn't electroweak symmetry breaking happen earlier than it did?","text":"I've read that electroweak symmetry breaking occurred \"at a picosecond or so\" after the Big Bang (in a source I found). I can't help wondering why it took so long to get started. For instance, was something missing up until then, which was needed by the Higgs mechanism? What feature(s) of the Higgs mechanism, or perhaps what other theoretical consideration(s), was\/were seemingly responsible for this very brief delay?"}
{"id":"843","title":"Maxwell's Demon Constant (Information-Energy equivalence)","text":"New Scientist article: Summon a 'demon' to turn information into energy The speed of light c converts between space and time and also appears in e=mc^2. Maxwell's Demon can turn information supplied into a system into energy, which suggests there is a constant 'd' such that e=i*f(d) where i is information and f is a function of d. If d appears in the formula relating energy and information like c relates energy and mass, but c is a conversion factor between space and time, then what is d a conversion factor between? What dimensions does d have? Other than \"Maxwell's demon constant\", is there another name for d? Or is there not a 'd' after all, instead 'c' appears once again in the information-energy equivalence fomula e=i*f(c)? Or is there not a constant here at all and the situation is more complicated? Living things have evolved to exploit all kinds of opportunities - even quantum effects in the form of photosynthesis. Living things have senses to extract information from the environment, but do living cells utilise any demon effects to turn any of this information into energy ?"}
{"id":"1058","title":"Distinguishing mechanical systems from general dynamical systems","text":"[Remark: I admit that my first attempt on What makes a space a real space? was rather ill-posed and led to some confusion. Sorry for that, but please give me a second try. Part of the confusion arose from the ununsual term \"real space\". The far better term would have been \"physical space\" which I hope to be unambiguous.] In the following let a \"mechanical system\" be a system of $n$ spatial objects moving in physical space. Consider you are given a function $q:\\mathbb{R} \\rightarrow \\mathcal{M}^n$ with $\\mathcal{M}$ a mathematical space. **Question 1** > When would you assume that $q$ describes the evolution of a mechanical > system? I would, if I found or was given a function $\\mathcal{L}(x,\\dot{x},t)$ such that the Lagrange equations hold for all times $t$ when inserting $q(t)$ for $x$ and $\\dot{q}(t)$ for $\\dot{x}$, and which I can consider physically plausible. In this case I would interpret $\\mathcal{M}$ as physical space, maybe distorted by generalized coordinates. I would _not_ , if there is provably no such function. In this case $q$ describes the evolution of a general dynamical system, but not a mechanical one. **Question 2** > Are there mathematically definable conditions on $\\mathcal{L}$ for being > physically plausible, or possible, let's say? Besides, maybe, that $\\mathcal{L}$ must be of the form $\\mathcal{T} - \\mathcal{V}$. **Question 3** > Can anyone give an example of a non-mechanical system that _behaves_ like a > mechanical one (in the narrow sense above)? One answer to this question comes immediately in mind: a system that is _designed_ to behave like a mechanical one, e.g. a computer running a simulation. I'd like to exclude such cases if I could, and restrict the question to \"natural\" systems. **Question 4** > What about possible worlds with a physical space like ours but in which the > Lagrange equations do not hold?"}
{"id":"135327","title":"Does the image forced barrier lowering appear wnen the device is in Forward bias?","text":"Because of the image forced barrier lowering the barrier potential reduces in reverse bias . What happens to barrier height (potential) in forward bias??"}
{"id":"122317","title":"Why is supercritical fluid not considered a separate state of matter?","text":"As given on this link, supercritical fluids are viewed more as a continuum which has both liquid and gas properties. This continuum is obtained when a gas is brought to a pressure and a temperature higher than its critical values. The intermolecular distances of a pure supercritical fluid are in between that of a liquid and gas. With so many qualitative differences between a supercritical fluid and a gas and a liquid. Why is it not considered a separate state of matter?"}
{"id":"87824","title":"Is there some special cutoff density after which spacetime \"collapses\" and forms a black hole?","text":"With crude calculations following densities can be approximated: Given that radius of proton is $1.75×10^{−15} m$ and it's mass is $1.67 × 10^{-27}kg$, this gives density of proton to be $\\dfrac {1.67 × 10^{-27}kg} {\\frac{4}{3}(1.75×10^{−15} m)^3}=\\dfrac {3}{4} \\dfrac{1.67 × 10^{-27}}{5.36×10^{−45}} \\dfrac{kg}{m^3}=2.34 \\times 10^{15} \\dfrac{kg}{m^3}$ Density of the heaviest naturally occurring solid $\\mathrm U^{238}$: 1 cubic meter of Uranium is about $2 \\times 10^4 kg$, and since there are 238 protons+neutrons in Uranium, this gives the density $8 \\times 10^{-3} \\times 10^4 \\dfrac{kg}{m^3}=8\\times10 \\dfrac{kg}{m^3}$ This means the protons are dispersed by a factor of $2.92\\times10^{13}$, in other words there are $2.92\\times10^{13}$ empty space units to each unit of space filled with proton. Density of water $=\\frac{1}{20}\\times10 \\dfrac{kg}{m^3}=5\\times10^{-1}\\dfrac{kg}{m^3}$ The density in the core of neutron star is $8×10^{17} kg\/m^3$, and the density of a black hole is supposedly infinite according to Wikipedia, but then again before transitioning to a black hole there must have been some finite mass distributed over a non zero volume of space therefore having a finite density. So far on logarithmic scale (base 10) followings look obvious for $\\operatorname {log}_{10}(density \\dfrac {kg}{m^3}) = \\operatorname {log}_{10} \\dfrac{kg}{m^3} + \\operatorname {log}_{10}density$ Let $ d =\\operatorname {log}_{10}density,\\quad \\operatorname {log}_{10} \\dfrac{kg}{m^3}= \\text{what is one to make of this}?$ Then following crude observations can be made: $-1\\leq\\mathcal O(d)\\leq1$: Order of density of life as we know it. $\\mathcal O(d)\\approx 15$: Order of density of matter packed space $\\mathcal O(d)\\approx 17$:Order of density at the core of a Neutron star $\\mathcal O(d)\\geq x$:Order of density of black holes, (collapse of space to contain matter?) what is $x$? My question is: does these calculations make sense and is there anything more to the order of densities than these naïve observations?"}
{"id":"122319","title":"Bending of light - photon's inertia instead of mass","text":"Using classical mechanics, the formula for gravitational attraction is $$F = G\\frac{m_1m_2}{r^2}.$$ This formula does not work for photons, and we need to use Einstein's theory of gravity to account for that, as the photon is massless. What if, instead of using the mass of the photon, we rather used the inertia of it? The momentum of a photon is $$p=\\frac{h}{\\lambda}.$$ Thus, using that, we can calculate its supposed 'mass' (inertia) as momentum is also equal to mass times velocity. So 'mass' of a photon is $$m=\\frac{p}{v}=\\frac{h}{\\lambda c}.$$ If we use this mass as the mass of a photon, can we use Newton's equation for gravitational attraction?"}
{"id":"122318","title":"Simple Parallel Electric Circuit","text":"Ammeter (A1) and voltmeter (V) in series have parallel connection to another ammeter (A2). Currents in A1 and A2 are respectively 0.2 and 1.7 amp. Voltmeter's voltage is 6 volts. ![enter image description here](http:\/\/i.stack.imgur.com\/w9oZl.png) How should I find resistances of all elements?"}
{"id":"8640","title":"Can a disk like object (like UFO's) really fly?","text":"UFOs as shown in movies are shown as disk like objects with raised centers that emit some sort of light from bottom. Can such a thing fly? My very limited knowledge in physics tell me that a disk like object may not be able to maneuver unless it has thrusters on sides and simple light can not be enough to make any object go up in the air. Is it possible?"}
{"id":"64344","title":"Chain of balls on an inclined plane","text":"Suppose we have some inclined plane, and there is some chain of balls of length $l$ and mass $m$ lying on it. No friction at all in the system. 1) What is $x_0$ (the vertical hanging part of the chain) needed for the chain to be in equilibrium? _Answer_ : $l\/3$ 2) If we push the chain a little bit to the right, what is the acceleration of the chain, at instant when the vertical part of the chain is $x$ long ($x > x_0$)? _Answer_ : $a(x)=\\frac{g}{l}(\\frac{3}{2}x-\\frac{l}{2})$ 3) What is the velocity of the chain at instant when all of it is vertical (e.g. all the left part on the inclined plane \"became vertical\")? ( _no answer_ ) ![enter image description here](http:\/\/i.stack.imgur.com\/jydqH.png) 1) I do not understand, why the component of the weight of the left part should be equal to the weight of the right hanging part. Their vectors are not along the same line, so why should this condition be necessary for equilibrium? Also, the left part is being pulled by the right, so there should be another force along the incline. 2) Based on the first question, this one should be: $(1-\\frac{x_0}{l})mg \\sin 30^{\\circ}-\\frac{x_0}{l}mg=ma$ However, this leads to a wrong $a$, according to the answer. 3) Based on the book's answer of 2), is this right? From Torricelli's equation: $v^2=v_0^2+2a \\Delta x$ $v(x)=\\sqrt{v_0^2+2 \\int_{l\/3}^{x} a(x) dx}$ Therefore: $v(l)=\\sqrt{\\frac{20}{3}l}$"}
{"id":"89466","title":"Why does a superfluid climb up the wall of container?","text":"This has been asked  but I couldn't find a good answer why does super-fluid climb up the walls of container? What if the walls were too high? What forces are involved? ![enter image description here](http:\/\/i.stack.imgur.com\/rgYNn.jpg)"}
{"id":"89460","title":"Method of determining peculiar speed of the galaxy which moves on celestial sphere and emits the light","text":"This question is the continuation of this one. I came up with solution, but I'm not sure that it is correct. Can someone check it? Let's introduce transverse (to the observer) proper speed $v_{\\perp}$ of the galaxy. The galaxy travels distance $\\Delta D$ in a time $\\Delta t_{0}$: $$ \\Delta D = v_{\\perp}\\Delta t = v_{\\perp}\\Delta t_{0}\\frac{a(t)}{a_{0}}, $$ where $dt_{0}$ refers to the time interval of observation of the galaxy, $dt$ refers to the time interval of light emission. So $$ d\\theta = \\frac{dD}{a(t) r(t)} = \\frac{v_{\\perp}dt_{0}}{a_{0}r(t)}, $$ where $r(t)$ is the galaxy position, and finally $$ \\frac{d \\theta }{dt_{0}} = \\frac{v_{\\perp}}{a_{0}r(t)}. $$ Is this method correct? Also, can I identify the left side of this equation with the observing angular velocity? And how to find $r(t)$? To consider the FLRW interval for the light? $$ ds^2 = dt^2 - a(t)^2\\frac{dr^2}{1 - kr^2} = 0 \\Rightarrow f(r) = \\int \\limits_{0}^{z} \\frac{dz}{H(z)} \\Rightarrow r = f^{-1}\\left[ \\int \\limits_{0}^{z} \\frac{dz}{H(z)}\\right]? $$"}
{"id":"66866","title":"Maximum angular velocity to stop in one rotation with a known torque","text":"I have an object I can rotate with a given torque. I would like to stop applying torque once I've reached a defined maximum rotational speed. The maximum rotational speed should be defined so that applying maximum torque will stop the rotation of the object within one rotation. If I know my torque and moment of inertia, how can I find the maximum rotational velocity to allow me to stop the object in one rotation? Time is whatever is needed. I've tried finding the angular acceleration required to stop the object, but that leaves me with the time variable. Of the equations I've tried, I'm left with a time variable as well as the maximum angular velocity."}
{"id":"110399","title":"How Long does it take for a roller coaster cart to reach point $x_1$?","text":"Assume that we have a particle with weight m and initial speed v0, that follows a roller coaster path y(x). ![Roller Coaster Path](http:\/\/i.stack.imgur.com\/kyUeA.png) Assuming no energy loss to friction or air resistance, the total energy in the system is $$E0 = m g y(x0) + 0.5 m vo^2$$ then the speed at each point $x_f$ is ![enter image description here](http:\/\/i.stack.imgur.com\/jMCyQ.png) So, using the energy conservation principle, we can derive the speed at each point of the curve. It gets tricky though, to try to figure out at what time the particle reaches a specific point of the curve. Turning our attention to dynamics, should help us achieve the goal. $$dS\/dt = v(x)$$ Is the approach followed correct? ![enter image description here](http:\/\/i.stack.imgur.com\/hz3U7.png)"}
{"id":"28827","title":"Does wind speed things up or accelerate things?","text":"This question may seem odd, but I can't think of anything better. So I'll go straight to the point. Let's say there's a projectile in air going east, shot at a certain angle, with a certain speed. Then there's the wind blowing, for let's say 5 meters per second in the same direction as the projectile. So how will the wind affect the projectile? Will it accelerate for 5 meters per second every second it lasts in the air, or will the speed just increase by 5 meters per second all through out? And what if the wind goes exactly the opposite direction? Will it have a negative effect? Should I also take into consideration the drag the object makes, the mass, the terminal velocity, or if there are other things I've missed? I don't mind if you give formulas. It would also help. But I just wanted to know, in simple terms because I'm not really that inclined with terms used in physics, the effects of a blowing wind to a projectile motion. Thanks in advance and sorry if I made some errors or misunderstanding with this post or if I used the wrong tags. Feel free to edit it. Thanks again,   Vincent :)"}
{"id":"129159","title":"\"Dictionnary\" between QFTs in D and D-1 dimensions?","text":"Considering Einstein equations, suppose, for instance, that the RHS, the stress-energy tensor, is uniquely due to the electromagnetic field. Now, if we imagine a quantized version of these Einstein equations, we have a quantum field theory in $D$ dimensions (here the quantum electromagnetic field) as \"source\" of a \"quantum gravity\" theory in $D$ dimensions, so we may consider these equations as a kind of \"dictionnary\" between these two theories. On an other way, there are some holographic dualities, which relates some quantum theory of gravity in a $D$ dimensional space with boundary, with some quantum field theory on the boundary ($D-1$ dimensions) So, the question is, if, \"skipping gravity\", it is possible or not, and how, to directly relate the QFT in $D$ dimensions (source of gravity), and the other QFT (on the boundary) in $D-1$ dimensions."}
{"id":"82021","title":"Any FLOSS Monte-Carlo package for reactor physics?","text":"Any FLOSS Monte-Carlo package for reactor physics? Are there any Free and Open Source software packages for nuclear reactor processes simulation? Maybe, something similar to MCNP?"}
{"id":"96482","title":"No hair theorem and black hole entropy","text":"The _no hair_ theorem says that black holes rapidly converge to a state that is completely described just by their mass, spin and charge. Black hole thermodynamics says that the black hole entropy is proportional to the surface area of the event horizon. As I understand it in information theoretic terms the black hole entropy is 1 bit per planck unit of area, which should then be the amount of information required to fully describe a black hole. These two statements seem to be incompatible. It seems that the thermodynamics says that if we fix the macroscopic no hair parameters there still remain all the surface bits to be fixed. Is there any real conflict here or is somehow illusory? Is it a quantum vs classical issue? Does the entropic information not count in some sense? If not why, can 2 equal mass Schwartzchild black holes be distinguished by their \"surface information\"? Is it that the surface information is just not observable because the surface is considered _in_ the black hole? Or is this apparent discrepancy part of the black hole information paradox? There is a very similar question with a detailed answer here, but I believe I am actually asking something different. The question and excellent answer there focus on the objects as they fall into a black hole. I am more interested in how to think about the total information in a steady state black hole and the current status of the no hair theorem wrt black hole entropy and the information paradox."}
{"id":"96488","title":"How do you represent an equation with four variables?","text":"As the title asks, how do you represent an equation with four variables? Is there a practical way of representing them? (I've thought of one, but I don't think I can explain it clearly. This question has got me really curious.) Additionally, is there any practical way of solving these equations?"}
{"id":"16091","title":"What is the fundamental probabilistic interpretation of Quantum Fields?","text":"In quantum mechanics, particles are described by wave functions, which describe probability amplitudes. In quantum field theory, particles are described by excitations of quantum fields. What is the analog of the quantum mechanical wave function? Is it a spectrum of field configurations (in analogy with QM wave functions' spectrum of particle observables), where each field configuration can be associated with a probability amplitude? Or is the field just essentially a superposition of infinitely many wave functions for each point along the field (as if you quantized a continuous mattress of infinitesimal coupled particles)?"}
{"id":"16092","title":"Depalma Free Energy fields","text":"Few years ago I read some papers about Free Energy, written by Bruce de Palma, a physicist who is said to be the inventor of the N-Machine, which is an device that works with free energy latent in the space around us. There are several informal references to him and his work. However, I tried to find papers of him several times in Scielo, Science Direct, IEEE, Physical Letters, Nature and several other scientific databases and found no references to Dr. Bruce de Palma. Personally I find quite strange that a research of such importance do not have a formal paper in an journal of big impact. Does his work make sense?"}
{"id":"16096","title":"Synthesizing elements (Nuclear Physics\/Alchemy?)","text":"Based on my limited knowledge of nuclear physics, it seems that one day it may\/will be possible to synthesize whatever elements we may need, given enough energy. Is this accurate? Is there a table that lists the most efficient ways to create certain elements from more abundant ones via fusion or fission? I daresay this has rather large implications for precious metals such as gold\/silver which are used as stores of wealth, provided the energy requirements are not exorbitant."}
{"id":"13958","title":"How to bend 3d space embededd in a 4d enviornment?","text":"> **Possible Duplicate:**   >  Calculating position in space assuming general relativity I recently started to program a 4 dimensional visualization program. I have the 4 dimension space perspectively projected in the 3d space. Now I describe 3d sphere inside this space. I want to describe the movement of those 3d spheres. What I thought it will be cool is to apply the general relativity theory to it. As far as I know from the general relativity theory, is that masses bends the space around it. What I think it means: let's imagine a 2d plane. if I put a mass on this 2d plane, the plane would be actually pinched in the 3d space. Parctically what I have to do is to find a function that describe this pinch, in function of the mass. like a Gauss distribution under the mass. Now this transformation will be applied to the moving objects, which I think would bend the trajectory of the objet to a curve. What I want to do is to apply this to the 3d space: if you put a mass in the 3d space it will cause a pinch in the 4th dimension. So what I want is to find the transformation function which describe this pinch. I think that what Zaslavsky is the actual answer, I din't got it at the beginning because I didn't tought it was actually what I was looking for but anyway: $$\\frac{\\mathrm{d}^2x^\\lambda}{\\mathrm{d}t^2} + \\Gamma^{\\lambda}_{\\mu\\nu}\\frac{\\mathrm{d}x^\\mu}{\\mathrm{d}t}\\frac{\\mathrm{d}x^\\nu}{\\mathrm{d}t} = 0$$ The solution to this differential equation should give the pinch I mean, am I right? PS: found by surfing random on the internet: Lorentz transformations are just hyperbolic rotations. edit: I've edited the question, I hope, in a way is comprehensible."}
{"id":"19515","title":"Could gravity hold electron charge together?","text":"Could the gravitational force be what holds the charge of the electron together? It seems to be the only obvious possibility; what other ideas have been proposed besides side-stepping the issue and assuming a \"point charge\"? How would this affect the electron \"self-energy\" problem? The question is related to the idea of geons."}
{"id":"63992","title":"Ginzburg-Landau model for superconductivity","text":"Could someone kindly elaborate more on the Simple Interpretation section from this Wikipedia Article? I refer to the part on the natures of $\\alpha , \\beta$. Why can one assume that $\\alpha(T)=\\alpha_0(T-T_c)$, for example?"}
{"id":"16323","title":"Pull Force of an electromagnet","text":"How do I calculate the pull force of a cylindrical electromagnet to Iron as a function of distance? Is there any difference betweeen magnets and electromagnets?"}
{"id":"19519","title":"Kinetic energy absorbing in order to avoid damages?","text":"is this possible? To absorb kinetic energy and disable inertia force? http:\/\/www.youtube.com\/watch?v=KnKPbAbJI0w http:\/\/www.youtube.com\/watch?v=4_5oseSVUc4"}
{"id":"16329","title":"Net work done on the body when we lift it and put it on the table is zero?","text":"I'm little confused here. Work done on the body when we lift it and put it on the table is zero, because according to work energy theorem, change in kinetic energy of the body is zero. So, the net work done is zero. Fine, but now the object has $mgh$ (where $m=$mass of body; $h=$height of body) amount of potential energy stored in it. If the net work done is zero then who has increased its energy? I'm confused and unable to think it through kindly help."}
{"id":"65740","title":"Is glass a liquid?","text":"I was told by a condensed matter physicist that glass is a liquid with a very high viscosity (it would be more precise to say that it is a supercooled liquid). The example given was that in cathedral windows the glass flows very, very slowly, so they are thicker at the bottom. I recently read this article in Scientific American, which contradicts the previous statement. Are there any serious studies about this aspect of glass? Is it only a problem of definition?"}
{"id":"29641","title":"How positively charged protons remain glued to each other while they should repel each other out of nucleus?","text":"> **Possible Duplicate:**   >  Protons' repulsion within a nucleus How positively charged protons remain glued to each other while they should repel each other out of nucleus?"}
{"id":"52082","title":"Order of phase transitions","text":"I got to read things like > _In case of a first order phase transition, the volume and temperature > change in a discontinuous manner. However for phase transitions of higher > order the change in volume and temperature can be continuous._ I could not exactly understand the statement and thats why I thought what exactly it means by the order of a phase transition. May I know what exactly it means by the order of phase transition?"}
{"id":"52086","title":"What are the differences between indistinguishable and identical?","text":"What is the difference between indistinguishable particles and identical particles?"}
{"id":"69664","title":"Chemical Potential of Ideal Fermi Gas","text":"In Wikipedia's article on Fermi Gases, they have the following equation for the chemical potential: > $$\\mu = E_0 + E_F \\left[ 1- \\frac{\\pi ^2}{12} \\left(\\frac{kT}{E_F}\\right) ^2 > - \\frac{\\pi^4}{80} \\left(\\frac{kT}{E_F}\\right)^4 + \\cdots \\right]$$ where > $E_0$ is the potential energy per particle, $k$ is the Boltzmann constant > and $T$ is the temperature. I don't understand how they get the third term, particularly the 1\/80 factor. I've often seen this equation expressed just to the tau^2 term, and I understand how to get the 1\/12 factor from a binomial expansion of $$\\left( 1 + \\frac{\\pi^{2}}{8} \\left(\\frac{\\tau}{E}\\right)^{2}\\right)^{-2\/3} $$ However, I've tried continuing the binomial expansion and cannot figure out why the factor is 1\/80."}
{"id":"2372","title":"How long it will take for a upright rigid body to fall on a ground","text":"Let's suppose there is a straight rigid bar with height $h$ and center of mass at the middle of height $h\/2$. Now if the bar is vertically upright from ground, how long will it take to fall on the ground and what is the equation of motion of the center of mass (Lagrangian)?"}
{"id":"21234","title":"Cerenkov light - a practical calculation","text":"I need to calculate the approximate total amount of energy radiated, via Cerenkov, by a muon as it traverses $10\\:\\rm{cm}$ of quartz glass. Unfortunately I've spent so much time fiddling with electronics in the lab recently that I've completely forgotten how to do maths. The formula to use is the Frank–Tamm Equation: $$\\frac{dE}{dx} = \\frac{q^2}{4\\pi} \\int_{v>c\/n(\\omega)} \\mu(\\omega)\\omega\\biggl(1-\\frac{c^2}{v^2n^2(\\omega)}\\biggr)d\\omega$$ Where $\\mu(\\omega)$ and $n(\\omega)$ are the permeability and refractive index of the medium being traversed by the charged particle. For a $5\\:\\rm{keV}$ muon travelling through quartz glass I have looked up the following numbers (but I'm really not sure over what frequency range to consider the last two): $$x=0.1\\:\\rm{m},$$ $$q=1.6\\times10^{-19}\\:\\rm{C},$$ $$v\\approx c,$$ $$n(\\omega)=1.54-1.46 (\\lambda=200-700\\:\\rm{nm})\\approx1.5,$$ $$\\mu(\\omega)\\approx1NA^2.$$"}
{"id":"46370","title":"Efficiently calculating the gravitational force from spherical shell","text":"I am reading Kolenkow and Kleppner's _Classical Mechanics_ and they have tried to calculate the gravitational force between a uniform thin spherical shell of mass $M$ and a particle of mass $m$ located at a distance $r$ from the center. The shell has been divided into narrow rings.$R$ has been assumed to be the radius of the shell with thickness $t$ ($t<<R$). The ring at angle $\\theta$ which subtends angle $d\\theta$ has circumference $2\\pi R\\sin\\theta$. The volume is $dV=2\\pi R^2t\\sin \\theta d\\theta$ and its mass is $pdV=2\\pi R^2t\\rho\\sin\\theta d\\theta$. If $\\alpha$ be the angle between the force vector and the line of centers, $$dF=\\frac{Gm\\rho dV}{r'^2}\\cos\\alpha$$ where $r'$ is the distance of each part of the ring from $m$. Next, an integration has been carried out using $\\cos\\alpha=\\frac{r-R\\cos\\theta}{r'}$ and $r'=\\sqrt{r'^2+R^2-2\\pi R\\cos\\theta}$. Question: I would like to avoid these calculations and I was wondering if there exists a better solution."}
{"id":"1873","title":"The Concepts of Path Integral in Quantitative Finance","text":"I realize that path integral techniques can be applied to quantitative finance such as option value calculation. But I don't quite understand how this is done. Is it possible to explain this to me in qualitative and quantitative terms?"}
{"id":"90888","title":"Derivation of Lagrangian density for an infinite classical dielectric in interaction with the EM field","text":"I am tasked with reading and reproducing all the steps in J.J. Hopfield's 1958 paper \"Theory of the Contribution of Excitons to the Complex Dielectric Constant of Crystals\". Embarrassingly I am stuck on equation (3). Hopfield takes the Lagrangian _density_ for an infinite classical dielectric in interaction with the electromagnetic field to be ![enter image description here](http:\/\/i.stack.imgur.com\/0byQD.png) Instead of taking this equation at face value, I would like to fully derive this Lagrangian density, but am unsure how to approach the problem. I have found many articles and sites tackling a similar problem, but never ending up with an expression as here. Any pointers on how to approach the problem or links to material that could be useful would be very much appreciated."}
{"id":"57103","title":"Where is a particle bound in a delta potential?","text":"I can picture a bound state in a harmonic oscillator, or in an infinite square well, but where is a particle bound in a delta potential?"}
{"id":"98994","title":"How do I measure the total weight of a container with scales on four corners?","text":"I have a very large movable trash container, and I have mounted four scales, one on each corner. How do I get the **total weight** of the entire container? Is it simply a sum of the four weights, or is it more complex than that? The contents of the container are not evenly distributed, one corner will be heavier than the others. Also, there is a possibility that one corner will not be completely resting on the ground, and thus the scale on that corner will read zero, but the other three corners will bear it's weight."}
{"id":"98993","title":"Sunrise\/Sunset on a planet with an A-Class or B-Class Star","text":"I'm a sci-fi writer, and I have a question regarding a hypothetical planet that orbits an A-class or B-class Star. **Assuming Rayleigh Scattering occurs roughly the same way on this planet as it does on Earth, what colors would we see at Sunrise and Sunset?** I understand that what colours we see from our sun comes from the Rayleigh Scattering, but seeing as it's sci-fi I'm writing, I need a plausible colour scheme to use for my new planet. The Planet has a breathable atmosphere not too different to Earth, so for the sake of the argument, I'm assuming that the Rayleigh Scattering would occur roughly the same way, but my physics knowledge is sadly lacking in this area, and the Wikipedia article only refers to our own G-class Sun. On that note, would the Sky still appear Blue, from an A-class or B-class Star? Or would the Rayleigh scattering trigger a completely different colour of the sky? Is there no 'right answer' when it comes to what sunrise\/sunset\/sky would look like on a distant planet?"}
{"id":"102226","title":"What path will an object in a cockpit take?","text":"Situation: A fly is in the cockpit of a fighter jet, and flying directly in front of the pilot. The pilot becomes annoyed and decides to get rid of the fly by executing a sudden, high speed turn. The cockpit is so dark the fly cannot see any of the cockpit. will the fly continue to be in front of the pilot, or will it hit the side of the cockpit? If the pilot executes a slow turn will the result be different?"}
{"id":"102223","title":"Can a long pipe stop the flow of a liquid from a pump?","text":"If a pump is discharging a fluid into a pipe, is it possible to add a long enough pipe such that the flow of the liquid will be stopped and none will come out of the open end of the pipe?"}
{"id":"60718","title":"Concerning drag on a flow past a cylinder","text":"I am wondering about the drag coefficient for a flow past a cylinder. I am reading this article. I understand why the drag is high to begin with (point 2), when the boundary layer separates and the recirculations appear, the pressure behind the cylinder (where the wakes are located) will drop, creating a drag. This is similar to what happens when vortices appear at the end of an airplane wing, the result is a drag force. However, I am wondering about the following:   1) Why does the drag decrease as the Re goes up?   2) Why does the drag coefficient take a dive at point 5   3) Why does it rise after point 5? The text doesn't really explain why the drag coefficient is increasing or decreasing."}
{"id":"126675","title":"Singularity in Newton's gravitational law","text":"If $r=0$ in the well know equation $F= G\\dfrac{m_1\\cdot m_2}{r^2}$, it will not follow that the force will be infinite? May someone please clarify it to me?"}
{"id":"90779","title":"Separation of variables in various PDEs, physical meaning","text":"The method of separation of variables produces an undetermined separation constant and a family of solutions indexed by the values of this constant. For instance, in the case of an infinitely long rod along the positive x axis (with suitable boundary conditions, like a fixed temperature at the beginning, and a given temp. distribution at initial time), the separation constant, $-k^2$, can vary continuously, and the solutions are of the form $$T_k(x,t)=C(k)\\cdot \\sin{kx} \\cdot e^{-k^2\\alpha t}$$ One is then supposed to integrate over $k$ to obtain the answer. In the case of a finitely long rod, the values of the constant are quantized and can be summed, rather than integrated. But when we're dealing with Schroinger's equation for a particle in a box, we are only interested in the eigenfunctions corresponding to the values of the separation constant, which we associate with energy. But the general solution would be a superposition of these wavefunctions. Suppose we're dealing with a 3D box of length $L$, then the eigenfunctions are: $$\\psi_{k,\\ell,m}(x,y,z)=\\sqrt{\\frac{8}{L^3}} \\sin{\\frac{\\pi}{L}kx}\\cdot\\sin{\\frac{\\pi}{L}\\ell y}\\cdot\\sin{\\frac{\\pi}{L}mz}, \\quad k,\\ell,m\\in\\mathbb{N}$$ My question: is there any physical significance to the function: $$\\psi=\\sum_k\\sum_\\ell\\sum_m c_{k,\\ell,m}\\psi_{k,\\ell,m}, $$ where $c_{k,\\ell,m}$ are constant coefficients. Clearly this function is a solution to the Schrodinger equation, by the superposition principle. But does this function describe anything at all? Or is it that, unlike in the case of heat transfer, only the eigenfunctions matter?"}
{"id":"113664","title":"Mass,Energy,Matter","text":"In $E=mc^2$ what type of energy is $E$? For instance, you can find the kinetic energy of an electron by using $\\frac{1}{2}mv^2$ but you can also find the energy from $E=mc^2$. How are those two different besides the values?"}
{"id":"109043","title":"What's the total Cosmic Microwave Background energy?","text":"This is a part of my wider attempts at accounting matter in the observable universe. The total light ever emitted by stars and other space objects is capped around 1% of baryonic mass (in energy equivalent). How much light is there in space and how heavy is it? How does the CMB compare to that? How would we calculate it?"}
{"id":"104970","title":"Can relativistic momentum (photons) be used as propulsion for 'free' after the initial generation?","text":"In discussing this question about propelling a spacecraft with photons and their relativistic momentum, the author asked that I restate my comment as another question. If photons can really be used for propulsion, could two mirrors be affixed between two objects, focused well and continually adjusting their respective 'where they will be at the time of light arrival', and use the same photons bouncing back and forth between the two objects for perpetual thrust after the initial photon generation? The initial question was on an assignment problem: \"A rocket of mass $m_0$ is propelled by a giant monochromatic laser mounted on the back of the rocket. The laser emits a beam with a power of $P_0$ watts and a frequency $f_0$, both measured in the rest frame of the rocket. When the beam is turned on, the rocket is driven in the opposite direction by the recoil.\""}
{"id":"26704","title":"What happens to light and mass in the center of a black hole?","text":"I know that black holes are \"black\" because nothing can escape it due to the massive gravity, but I am wondering if there are any theories as to what happens to the light or mass that enters a black hole and cannot escape."}
{"id":"62712","title":"Whis is the difference between charge fractionalization in 1D and 2D?","text":"Both 1D Polyacetelene and 2D fractional quantum Hall state can support fractional excitations. But as I can see, there are some differences: the ground state of Polyacetelene breaks translational invariance but FQHE does not. In general a symmetry breaking state in 2D are considered as conventional state and does not support fractional excitations. I wonder why is there such a difference between 1D and 2D fractionalizations?   Is there any 1D model can support fractional excitations from a symmetry preserved ground state ?"}
{"id":"92382","title":"Can two spools shield you effectively against radiation","text":"a friend of mine thinks that by buying a product called rayguard he can effectively shield against electrosmog. On the website, there are multiple evidences that this might work. However the product seems to be just created of one plastic chasi with two spools inside. The functionality of the product is described as following: > With the RayGuards, the solution therefore is not based upon electronics but > on physics. In a radius depending on the model of device used, all > frequencies are absorbed. > > This spectrum of frequencies is being put through a dividing and analysing > system. There the compen- sation of the harmful waveparts is made. > > To explain it better, we will use the example of a camera. Compare the > effect of the RayGuard to the polarising filter. Disturbing effects of > reflection are being cleared by the filter and the foto looks clear and > colourful again. I do not claim that I am a physicist , however after some research my personal opinion on this product is that it is not so simple to shield against radiation. My questions to you are:   * Is it possible with the rayguard to shield effectively against radiation? If it is a scam? Why \"not\"   * From a physical standpoint, how to shield properly against radiation? What device would be needed to shield effectively against for example mobile phone radiation   * Is the everyday radiation from a physical standpoint \"unhealthy\"? I really appreciate for your reply! PS.: I know that this question is not the typical physician question which is asked here, however it would be great if you could give me some hard facts\/theories to prevent my friend from wasting a lot of money!"}
{"id":"132236","title":"Electric flux through nonspherical surphace","text":"Here's an explanation from a book that I read: ![enter image description here](http:\/\/i.stack.imgur.com\/9q7I7.png) Now, I have a few problems with this explanation as well as with the \"cone method\". I guess that all my problems boil down to the concept of infinitesimal.   1. The book says \"a normal to $dA$ makes an angle $\\phi$ with a **radial** line from $q$\". Now, the problem is that I can take another radial line intersecting $dA$ which will make a different angle with the area.   2. The strength of the electric field passing through the projected area ($dA \\cos \\phi$) is different from $\\vec{E}$ because the projected area is closer to the charge, yet the book seems to ignore that. The same goes for the \"cone method\": ![enter image description here](http:\/\/i.stack.imgur.com\/mXMH1.png) Books calculate proportion between the areas and the radii using the same radius for $dA \\cos \\phi$ and $dA$ although the distance of the projected area from the charge is different than from the original area to the charge. So how can I solve these contradictions? Is there a strict mathematical proof that an electric flux through any area inside a cone (regardless of its orientation) is the same?"}
{"id":"48722","title":"What's wrong with this reactionless drive?","text":"I think this is analogues for the Woodward effect, but macroscopic: We assume a spacecraft consisting of a broomstick, a donut and lots of gear for storing and transfering mechanical energy. Take the donut, spin it till the outer edge moves at relativistic speed (We assume a tough glazing), and push it along the broomstick, brake it's spin and then the donut, move donut forward along broomstick, repeat. Since the donut is heavier when it's being accelerated backwards then when it's braked\/accelerated forward, there should be a net momentum. Since nothing leaves the broomstick, this has to be wrong. But where?"}
{"id":"57813","title":"What will happen if we place salt water on a induction cooker?","text":"As we know that induction cooker works on the principal of induction of current in a conducting plate. So I just wanted to know what will happen if we place salt water in a plastic container on the induction cooker will it get hot as it is also a conductor or nothing will happen?"}
{"id":"109353","title":"Is the molecule of hot water heavier than that of cold water?","text":"We know that the molecule of hot water($H_2O$) has more energy than that of cold water (temperature = energy) and according to Einstein relation $E=mc^2$ ,this extra energy of the hot molecule has a mass. Does that make the hot molecule heavier?"}
{"id":"46724","title":"Correcting for bad eyesight on display monitors","text":"I think it would be neat if one could configure one's eyesight parameters (astigmatism and myopia in my case), viewing distance, and perhaps age into a special display driver, such that a computer would present its user interface distorted (from the view of a normal-sighted person) in such a way that the bad eye, by applying its own \"distortion\", would essentially un-distort it so that the brain would receive a sharp view without the need for physical glasses or lenses. I suspect though that this is not possible partly on grounds of limited information. If the error in eyesight would be linear (e.g. shift to the left or enlargement) an appropriate distortion seems trivial. But a realistic error (hence a realistic distortion) would lead to a situation where several image points get mixed into the same \"pixel\" (in the physical eye this would mean: hit the same receptor), and disentangling those would require a-priori knowledge about original image, which is not always available (could be available for regular window shapes, or moving images -- this may lead to headache :) It's strange though, because my glasses are simple physical devices that a computer should be able to simulate, but perhaps not in the limited confines of the 2D surface of its display. Now all of this is just a hunch and my question is this: is there a straightforward answer for what kinds of \"invertible\" distortions (in a mathematical sense) the imagined apparatus could work, and has the problem ever been formalized in a better way?"}
{"id":"79443","title":"A problematic integral in calculating the entanglement entropy in 1+1 D free massive bosonic field theory","text":"I encountered a curious integration identity when I was reading the paper by Pasquale Calabrese and John Cardy on the entanglement entropy of 1+1D quantum field theory (arXiv). The identity is given below: $$ \\int^\\infty_0 x I_{\\alpha}(x)K_{\\alpha}(x) dx \\,\\, ``=\" \\frac{\\alpha}{2}. $$ Here $I_\\alpha(x)$ and $K_{\\alpha}(x)$ are the modified Bessel functions of the fist and second kind. The above identity is implicitly used in Eq.4.19 and Eq.4.22 of the said paper. However, the above identity looks false to me as the left hand side is apparently divergent. Recall that we have the following asymptotic expansion: $$ I_\\alpha (x) \\sim \\frac{e^{x}}{\\sqrt{2\\pi x}};\\\\\\ K_\\alpha (x) \\sim \\sqrt{\\frac{\\pi}{2 x}}e^{-x}, $$ from which we obtain $$ xI_\\alpha(x)K_\\alpha(x) \\sim \\frac{1}{2}. $$ I believe that some sort of regularization has been done on the integral or I simply miss something. I hope someone here could help me understanding this identity. EDIT: I think the following might be a useful hint. The above integral rises from the expression of $\\int^\\infty_0 rG(r,r)dr$, where $r$ is the radial coordinate and $G$ is the usual Green's function (see Eq.4.18). Now let us consider $\\int^\\infty_0 G((1+\\epsilon)r,r)rdr$, motiviated by the short distance expansion. The corresponding integral is: $$ \\int^\\infty_0 x I_{\\alpha}((1+\\epsilon)x)K_{\\alpha}(x) dx = \\frac{(1+\\epsilon)^{-\\alpha}} {(1+\\epsilon)^2-1} = \\frac{1}{2\\epsilon}-\\frac{1}{2}(\\alpha+\\frac{1}{2})+O(\\epsilon). $$ The above identity can be found in Gradshteyn & Ryzhik's table, 7th Edition, entries (6.521.2), (6.521.8), and (6.521.9). Now we take the limit $\\epsilon\\to 0$. The first term of the above is divergent, which could be interpreted as the usual UV divergence part. The second term, however, is finite. If somehow we can argue that it is meaningful to drop the first term and take the $\\alpha$-dependent as the \"physical\" answer of the integral, then we are done."}
{"id":"131768","title":"The states of the adjoint representation correspond to the generators","text":"![enter image description here](http:\/\/i.stack.imgur.com\/3pHzS.png) I understand the definition of the adjoint representation. It uses structure constants as matrix components of generators, but I can't understand meaning of the states $|X_{a}\\rangle$. What does \"correspond\" mean? What is the exact definition of $|X_{a}\\rangle$? Thanks in advance for any help."}
{"id":"67877","title":"Is this a valid derivation of the Legendre transformation from the Euler-Lagrange condition","text":"E-L condition: $$\\frac{d p}{dt}=\\frac{\\partial L}{\\partial q}$$ Where $p=\\frac{\\partial L}{\\partial \\dot{q}}$ Are the following steps valid: $$\\frac{\\partial q}{dt} dp=\\partial L$$ $$\\dot{q} \\: dp = \\partial L$$ $$ \\int \\dot{q} \\: dp = L+C $$ By integration by parts the LHS becomes: $$ \\int \\dot{q} \\: dp = \\dot{q} p-\\int p \\: d \\dot{q} = \\dot{q} p- \\int \\frac{\\partial L}{\\partial \\dot{q}} \\:d \\dot{q}=\\dot{q} p-L+C_1$$ Substituting this back into the LHS: $$\\dot{q} p-L=L+C_2$$ If steps are valid, then this indicates that the Legendre transformation of the Lagrangian is just the Lagrangian plus some constant $C_2$, and that the Hamiltonian is thus $H=L+C_2$. Seems pretty fishy. If it is not fishy (if above steps are valid), then this question: Since it is derived from the E-L condition, does this result imply that the action of all functions that are their own Legendre transformation (plus a constant) is stationary?"}
{"id":"131761","title":"How to understand \"the speed of propagation of the condition of constant phase\"?","text":"I still can't understand that the phase can be a constant until now. If the phase is constant, from the $$y(x,t) = a \\times \\sin(phase)$$ the shape of wave will be a line parallel to x-axis.But I know that I am wrong,how to explain it?"}
{"id":"92389","title":"At what point is the spin determined in a Stern-Gerlach Apparatus","text":"Consider a particle with spin that travels through a Stern Gerlach box (SGB), which projects the particle’s spin onto one of the eigenstates in the $z$-direction. The SGB defines separate trajectories for the particles that travel through it depending on their spin. My Question: At what point is the spin determined when it is in superposition? When the particle starts to feel the magnetic field? Or only when the trajectory is measured in the detector? This is a similar question, however it does not answer my question."}
{"id":"23161","title":"Why don’t photons interact with the Higgs field?","text":"Why don’t photons interact with the Higgs field and hence remain massless?"}
{"id":"30772","title":"Why do electrons around nucleus radiate light according to classical physics","text":"As I navigate through physics stackexchange, I noticed Electron model under Maxwell's theory. Electrons radiate light when revolving around nucleus? Why is it so obvious? Note that I do not know anything about quantum mechanics - and I just want classical stories."}
{"id":"61435","title":"Power radiated by the sun at different locations","text":"_I am wondering can someone help to solve second part which extends first part;_ The power radiated by the sun is ${3.9*10^{26}}_{watt}$. The earth orbits the sun in a nearly circular orbit of radius $1.5 *10^{11}m$. The earth’s axis of rotation is tilted by $27^o$ relative to the plane of the orbit , so sunlight does not strike the equator perpendicularly. What power strikes a $0.75$ $m^2$patch of flat land at the **_equator_**? $P_0 = I*4*π*r^2$ $P = I*A*cos(\\alpha)$ then $P=\\frac{P_0 * A *cos(\\alpha)}{ 4*π *r^2}$ then $P=\\frac{(3.9*10^{26})(0.75)(cos(27))}{(4*π*(1.5*10^{11})^2)}=920_{watt}$ ~ How much power strikes a patch of the same area located at latitude (at the same time of year) of 43° S (like Christchurch, New Zealand) or 43° N?"}
{"id":"61434","title":"How do I quantize a classical field theory","text":"I have not been able to find any information about this on the Internet. I am a middle-schooler, 14, who self-studies physics, and I know up to and including ODEs, and some of the calculus of variations, enough so that I can take a variation of the action. However, I have absolutely zero budget, and no ability to get textbooks. So, I was wondering how a physicist would quantize a classical field theory, such as Gauss's law for gravity, which I'm currently trying to create a quantum field theory for, as part of my plan to create a possible theory of quantum gravity, by adding in the relativistic corrections later."}
{"id":"28126","title":"Why the color motion picture film appears as black?","text":"Why the color motion picture film, or a color photographic slide, appears as black? The film appears black but when the light from the projector lamp hit it then you can see all the colors."}
{"id":"38932","title":"An Ideal Transformer","text":"In a transformer assumed to be transformer, power in the primary is equal to power in the secondary. So in a sense, the power in the secondary is 'fixed'. Output voltage in the secondary is also fixed by the turns ratio and the voltage in the primary. Then, The current tapped from the secondary coil should also be fixed $I=\\frac{P}{V}$. Doesn't the resistance in the circuit connected to the secondary coil also affect the current tapped from the secondary coil? So if a load resistance in the secondary coil circuit is varied, would the current still be fixed the same $I=\\frac{P}{V}$ or would it change accordingly with $I=\\frac{V}{R}$"}
{"id":"98692","title":"Can the apparent velocity of an approaching spacecraft be faster than light?","text":"Consider watching an alien space ship at Alpha Centauri (at 4.5 light years away) through a telescope from earth. This space ship turns towards us, and starts travelling toward us at what appears to be a velocity of exactly 2 million kilometers per second, when measuring the distance every hour. The space ship will arrive here in about 8 months. How fast is the space ship really travelling?"}
{"id":"35516","title":"Why is the Heisenberg uncertainty principle stated the way it is?","text":"I spent a long time being confused by the Heisenberg uncertainty principle in my quantum chemistry class. It is frequently stated that the \"position and momentum of a particle cannot be simultaneously known to arbitrary precision\" (or any other observables $[A, B] \\neq 0$). This made no sense to me -- why can't you measure both of these? Is my instrument just going to stop working at a certain length scale? The Internet was of little help; Wikipedia describes it this way as well and gets into philosophical arguments on what \"position\" and \"momentum\" mean and whether they really exist (in my opinion, irrelevant nonsense that has no effect on our ability to predict things). Eventually it was the equation itself that gave me the most insight: $$\\sigma_x \\sigma_p \\geq \\frac{\\hbar}{2}$$ Look at that – there's two standard deviations in there! It is _impossible_ by definition to have a standard deviation of one measurement. It requires multiple measurements to have any meaning at all. After some probing and asking around I figured out what this really means: Multiple repeated measurements of identically prepared systems don't give identical results. The distribution of these results is limited by that formula. Wow! So much clearer. Thus $\\hat{r}(t)$ and $\\hat{p}(t)$ _can_ be known for the same values of $t$ to as much precision as your measuring equipment will allow. But if you repeat the experiment, you won't get identical data. Why doesn't everyone just state it that way? I feel like that would eliminate many a student's confusion. (Unless, of course, I'm still missing something – feel free to enlighten me should that be the case). EDIT: This post was at +1. Who downvoted me? I took a while to write out my question clearly and made sure it followed the guidelines on here."}
{"id":"31879","title":"Why are galactic centers always brighter than the edges?","text":"As you can see this image http:\/\/i.stack.imgur.com\/YOt8C.jpg and other galaxy images, the centers generally much brighter. Why is that? Is there a very big star? A very big gravitational field?"}
{"id":"114029","title":"Read-off particle from (projected) Dynkin labels","text":"In the review of Slanksy \"Group theory for unified model building\" in chapter 6: How do one relate the projected Dynkin diagrams from for example $\\overline{5}+10$ of $su(5)$ to the corresponding representation in of the subgroup $su(2)\\times su(3)$? An example is $P( 0 0 1)=(0)(01)$, how do one see that this must be a charge $1\/3$ antiquark singlet? Reference: http:\/\/cds.cern.ch\/record\/134739\/files\/198109187.pdf?origin=publication_detail"}
{"id":"101226","title":"Confused about Impulse","text":"Encountered a problem that involves impulse while studying for my exam and I'm not sure how to even approach it. I know that momentum is conserved, but I'm not sure how to relate that to avg force. Maybe someone can help point me in the right direction? I know that it's in quadrant III, through intuition, but I can't come up with a provable explanation Relevant equation: $J=F_{avg}\\Delta T$ ![Couldn't copy and paste the problem, description in link](http:\/\/i.stack.imgur.com\/dDlRS.png)"}
{"id":"101227","title":"Why is no EMF being induced in this ring passing through a magnetic field?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/B1KNO.png) The book's logic is that there is no induced EMF because flux is constant as it passes through the magnetic field. Which makes sense, but this seems counter-intuitive to what I previously learned. If i'm not mistaken, if an electron is relatively moving perpendicular to a magnetic field, it WILL experience a force. The electrons in the ring are moving perpendicularly relative to the stationary field, so why aren't they experiencing a force?"}
{"id":"130490","title":"How would one begin to try to identify the properties of the mathematical structure corresponding to physical reality?","text":"I was thinking about the Mathematical universe hypothesis and a natural question popped into my mind: _Assuming that the universe (by universe I mean the complete physical reality) is really isomorphic to some conceivable, mathematically constructible structure, how would one begin to narrow down the possibilities? How would one identify its properties without necessarily finding the structure itself?_ My first guess is that one should look at QFT and GR and assume that the mathematical structure we want would have to be **consistent** with those two theories at least in the appropriate approximations\/limits and that we could somehow find all the familiar symmetries in some form on that structure. But those are just words, I don't understand what would one have to do rigorously to rule out some of the structures, I would really appreciate some non-handwavy guidelines if it's possible. **P.S. I'll gladly elaborate further and edit my question if something is particularly unclear.** **EDIT 1:** I'm not asking how theoretical physics should proceed in general and how should the scientific method be used in order to understand the world. I'm interested in how much can we say about the **\"final, true and complete\"** theory (assuming it exists) without actually having it. I'm interested in the mathematical structure associated with that theory, what are the most general statements about it that are almost certainly true?"}
{"id":"27096","title":"ATLAS Higgs Interpretation","text":"I came across this abstract, and I am curious as to what the ATLAS Team has actually discovered: > Abstract Motivated by the result of the Higgs boson candidates at LEP with a > mass of about 115~GeV\/c2, the observation given in ATLAS note ATL-COM- > PHYS-2010-935 (November 18, 2010) and the publication “Production of > isolated Higgs particle at the Large Hadron Collider Physics” (Letters B 683 > 2010 354-357), we studied the γγ invariant mass distribution over the range > of 80 to 150 GeV\/c2. With 37.5~pb−1 data from 2010 and 26.0~pb−1 from 2011, > we observe a γγ resonance around 115~GeV\/c2 with a significance of 4σ. The > event rate for this resonance is about thirty times larger than the > expectation from Higgs to γγ in the standard model. This channel H→γγ is of > great importance because the presence of new heavy particles can enhance > strongly both the Higgs production cross section and the decay branching > ratio. This large enhancement over the standard model rate implies that the > present result is the first definitive observation of physics beyond the > standard model. Exciting new physics, including new particles, may be > expected to be found in the very near future. The abstract seems to be from a restricted web site (CERN Log-in required), however I have been able to track down a PDF that seems to be discussing the same phenomenon, however the abstracts are not the same. Currently, as I understand it, there is a lot of skepticism about the initial Higgs candidate. If this isn't the Higgs, then what part of the standard model is actually being represented? Or is this an entirely new phenomenon?"}
{"id":"27091","title":"Sampling typical clusters between distant points in subcritical percolation","text":"I have on several occasions wondered how one might proceed in order to sample large subcritical Bernoulli bond-percolation clusters, say on the square lattice. More precisely, let's consider the lattice $\\mathbb{Z}^2$, and open each edge independently with probability $p<p_c=0.5$. I am interested in the event that the vertices $(0,0)$ and $(N,0)$ are connected by a path of open edges, when $N$ some large number (\"large\" meaning here \"many times the correlation length\"). It is well-known (rigorously) that this probability decays exponentially fast with $N$, and one has a lot of information about the geometry of the corresponding cluster (e.g., the corresponding cluster converges to a brownian bridge as $N\\to\\infty$ under diffusive scaling, it has a maximal \"width\" of order $\\log N$, etc.). **Question:** Are there (not too inefficient) algorithms that would sample typical clusters contributing to this event? **Edit:** googling a bit, I have stumbled upon this paper. I have yet to read it (and to decide whether it is relevant to my question). What the author seems to be doing is growing his objects in a biased way, which helps create the very unlikely configurations he's after, but would result in the wrong final distribution. So he's at the same time correcting the bias (in a way I haven't yet understood by just browsing through the paper). As an example, he's sampling 18 disjoint crossing clusters for critical percolation on $\\mathbb{Z}^3$ in a box of size $128\\times128\\times2000$, an event of probability of order $10^{-300}$. So: **Alternative question:** does anybody know how this works? (I can also simply read the paper, but it might be useful for me and others if a nice self- contained description was given here by someone knowledgeable). Apparently, looking more thoroughly, there is a lot of material that could be useful. It seems that relevant keywords here are: rare events simulation, importance sampling, splitting, etc."}
{"id":"51746","title":"Word for the star around which an exoplanet orbits:","text":"Is there accepted nomenclature for the star around which a particular exoplanet orbits? Meaning, if I were to say \"The exoplanet blah blah blah's (noun)\" what noun would I put there? Sun? Star? Designated Gravity Buddy? Something else? As I understand it, this would not come up often currently in the real world (because Exoplanets are named by their star), and it more in the realm of science fiction and fantasy at the moment. However, just because something seems currently useless (to me) from a scientific standpoint doesn't mean that someone hasn't thought up a name. Would this name change if you're actually on the planet or in the system? (ie, you hear about Tatooine having \"two suns\" not \"two stars\")."}
{"id":"27099","title":"Some questions on a version of the O'Raifeartaigh model","text":"This form is taken from a talk by Seiberg to which I was listening to, Take the Kahler potential ($K$) and the supersymmetric potential ($W$) as, $K = \\vert X\\vert ^2 + \\vert \\phi _1 \\vert ^2 + \\vert \\phi_2\\vert ^2 $ $W = fX + m\\phi_1 \\phi_2 + \\frac{h}{2}X\\phi_1 ^2 $   * This notation looks a bit confusing to me. Are the fields $X$, $\\phi_1$ and $\\phi_2$ real or complex? The form of $K$ seems to suggest that they are complex - since I would be inclined to read $\\vert \\psi \\vert ^2$ as $\\psi ^* \\psi$ - but then the form of $W$ looks misleading - it seems that $W$ could be complex. Is that okay?  Now he looks at the potential $V$ defined as $V = \\frac{\\partial ^2 K}{\\partial \\psi_m \\partial \\psi_n} \\left ( \\frac {\\partial W}{\\partial \\psi_m} \\right )^* \\frac {\\partial W}{\\partial \\psi_n}$ (..where $\\psi_n$ and $\\psi_m$ sums over all fields in the theory..) For this case this will give, $V = \\vert \\frac{h}{2}\\phi_1^2 + f\\vert ^2 + \\vert m\\phi_1 \\vert ^2 + \\vert hX\\phi_1 + m\\phi_2 \\vert ^2 $   * Though for the last term Seiberg seemed to have a \"-\" sign as $\\vert hX\\phi_1 - m\\phi_2 \\vert ^2 $ - which I could not understand.  I think the first point he was making is that it is clear by looking at the above expression for $V$ that it can't go to $0$ anywhere and hence supersymmetry is not broken at any value of the fields.   * I would like to hear of some discussion as to why this particular function $V$ is important for the analysis - after all this is one among several terms that will appear in the Lagrangian with this Kahler potential and the supersymmetry potential.    * He seemed to say that if *``$\\phi_1$ and $\\phi_2$ are integrated out then in terms of the massless field $X$ the potential is just $f^2$\"* - I would be glad if someone can elaborate the calculation that he is referring to - I would naively think that in the limit of $h$ and $m$ going to $0$ the potential is looking like just $f^2$.    * With reference to the above case when the potential is just $f^2$ he seemed to be referring to the case when $\\phi_2 = -\\frac{hX\\phi_1}{m}$. I could not get the significance of this. The equations of motion from this $V$ are clearly much more complicated.    * He said that one can work out the spectrum of the field theory by _\"diagonalizing the small fluctuations\"_ \\- what did he mean? Was he meaning to drop all terms cubic or higher in the fields $\\phi_1, \\phi_2, X$ ? In this what would the \"mass matrix\" be defined as?  The confusion arises because of the initial doubt about whether the fields are real or complex. It seems that $V$ will have terms like $\\phi^*\\phi^*$ and $\\phi \\phi$ and also a constant term $f^2$ - these features are confusing me as to what diagonalizing will mean. Normally with complex fields say $\\psi_i$ the \"mass-matrix\" would be defined the $M$ in the terms $\\psi_i ^* M_{ij}\\psi_j$ But here I can't see that structure!   * The point he wanted to make is that once the mass-matrix is diagonalized it will have the same number of bosonic and fermionic masses and also the super-trace of its square will be $0$ - I can't see from where will fermionic masses come here!    * If the mass-matrix is $M$ then he seemed to claim - almost magically out of the top of his hat! - that the 1-loop effective action is $\\frac{1}{64\\pi^2} STr \\left ( M^4 log \\frac{M^2}{M_{cut_off}^2} \\right ) $ - he seemed to be saying that it follows from something else and he didn't need to do any loop calculation for that!  I would be glad if someone can help with these."}
{"id":"10277","title":"Does decoherence explain all instances of wave function collapse?","text":"Specifically, how can decoherence explain the appearance of flecks of metallic silver on a photographic plate when exposed to the very weak light of a distant star? EDIT: Perhaps the advocates of decoherence need some context for this question. There is a certain definite quantity of energy on the order of one or two eV to drive the chemical reaction 2AgBr -> 2Ag + Br2 This is the reaction responsible for the fleck of silver on the photographic plate. The amount of energy is far greater than can be accounted for in any realistic time frame by the classical e-m wave energy of the light of a distant star. Any explanation must explain where this energy comes from. How does \"decoherence\" claim to do this? I have heard over and over again that there is a matrix which is diagonalized, but no one has so much as volunteered to say just what matrix they are talking about. Is it, for example, the matrix of position states of the photon? Or perhaps it is the oxidation states of the silver atom? And I would really like a better explanation of how the matrix is \"diagonalized\" than to simply repeat that it is in \"thermal contact with the environment.\" EDIT: I have reviewed the comments again and I find that no one has come close to dealing with the question. I cannot find anything wrong in the way I have asked it so far, so I am posting this edit as my only means to prompt people to attempt an answer."}
{"id":"10275","title":"Imaginary time in quantum and thermodynamics","text":"The following question is about chapter 2 of Sakurai's _Modern Quantum Mechanics_. I wish I could link to the Google book, but it doesn't seem to have a satisfactory preview to be able to read the section I'm talking about, so I'll do my best to write out the part I'm talking about... In the section about propagators and Feynman path integrals (p. 113 in my edition) he gives the following example: $G(t) \\equiv \\int d^3 x' K(x', t; x',0) $ $=\\int d^3 x' \\sum_{a'} |\\langle x'|a'\\rangle|^2 \\textrm{exp} \\left(\\frac{-iE_{a'}t}{\\hbar}\\right)$ $=\\sum_{a'}\\textrm{exp} \\left( \\frac{-iE_{a'}t}{\\hbar} \\right)$ He goes on to say that this is equivalent to taking the trace of the time evolution operator in the $\\\\{|a'\\rangle\\\\}$ basis, or a \"sum over states\", reminiscent of the partition function in statistical mechanics. He then writes $\\beta$ defined by $\\beta=\\frac{it}{\\hbar}$ real and positive, but with with $t$ purely imaginary, rewriting the last line of the previous example as the partition function itself: $Z=\\sum_{a'} \\textrm{exp} \\left( -\\beta E_{a'} \\right)$. So my question is this: What is the physical significance (if any) of representing time as purely imaginary? What does this say about the connection between thermodynamics and quantum? The fact that you get the partition function exactly, save for the imaginary time, here seems too perfect to be just a trick. Can someone explain this to me?"}
{"id":"126904","title":"Writing Schrodinger equation with central potential in Atomic unit","text":"I'm struggling to write Schrodinger equation with a central potential in Atomic unit. $$ \\frac{\\hbar^2}{2\\mu}\\Big(\\frac{d^2}{dr^2}+\\frac{2}{r}\\frac{d}{dr}-\\frac{l(l+1)}{r^2}\\Big)R(r)+\\Big(E-V-\\frac{e^2}{4\\pi \\epsilon_0 \\epsilon r}\\Big)R(r)=0 $$ Taking Effective Bohr radius $a_0$ as length scale and effective Rydberg as energy scale $$Ry=\\frac{\\mu}{m_e \\epsilon^2}\\big(\\frac{hc}{e}\\big)\\frac{m_e e^4}{8\\epsilon_0^2 \\hbar^3c}$$ in MKSA or $Ry=\\frac{\\mu e^4}{2\\epsilon^2 \\hbar^2}$ in CGS. What I tried is: By substitution of $r'=r\/a_0$ : $$ \\Big(\\frac{d^2}{dr'^2}+\\frac{2}{r'}\\frac{d}{dr'}-\\frac{l(l+1)}{r'^2}\\Big)R(r')+\\frac{2\\mu a_0^2}{\\hbar^2}\\Big(E-V-\\frac{e^2}{4\\pi \\epsilon_0 \\epsilon a_0 r'}\\Big)R(r)=0 $$ where $\\mu$ is effective mass $\\mu=.06m_e$. Then we can write $m_e=\\hbar=e=1$. The potential is for example $V=4 \\ Ry$. Know I don't know what can I write instead of   $$ \\frac{2\\mu a_0^2}{\\hbar^2}\\Big(-\\frac{e^2}{4\\pi \\epsilon_0 \\epsilon a_0 r'}\\Big) $$ Any help appreciated."}
{"id":"17850","title":"Uniqueness of eigenvector representation in a complete set of compatible observables","text":"Sakurai states that if we have a complete, maximal set of compatible observables, say $A,B,C...$ Then, an eigenvector represented by $|a,b,c....>$, where $a,b,c...$ are respective eigenvalues, is unique. Why is it so? Why can't there be two eigenvectors with same eigenvalues for each observable? Does maximality of the set has some role to play in it?"}
{"id":"27324","title":"What is the connection between extra dimensions in Kaluza-Klein type theories and those in string theories?","text":"This follows to some extent from a question I asked previously about the flaws of Kaluza-Klein theories. It appears to me that Kaluza-Klein theories attach additional dimensions to spacetime that are related to the gauge freedoms of field theories. I believe the original model was to attach a $U(1)$ dimension to the usual 4-dimensional spacetime to reproduce electromagnetism. But, as explained in the answers to my previous question, these extra dimensions have all sorts of problems. String theories also (famously?) require extra dimensions. So, is there a connection between the higher-dimensional descriptions? What do they have in common and how do they differ? For example, the $U(1)\\times SU(2)\\times SU(3)$ group is 7-dimensional, which, when attached to the 4 dimensions of spacetime, gives 11 dimensions. I hear the same number is bandied about in string theory although there's no obvious reason they should be related at all."}
{"id":"25258","title":"What size aperature would I need to find and see the supernova in M51?","text":"On June 2, 2011, a new supernova was identified by an amateur French astronomer in M51. What size telescope aperture would one need to have any chance of seeing this? Given the right sized scope, how would I best identify which point of light is the supernova?"}
{"id":"70637","title":"Stress in a thick-walled pressure vessel","text":"I can find many references that give the stress in the walls of a pressure vessel for spheres and tubes, but they all seem to be limited to a thin-wall approximation. I'll limit my writing here to spherical vessels (I don't think it should be very different to find the answer for any structure with symmetry). My main trouble is in setting up the integral. $$ \\sigma = \\frac{P R }{2 t}$$ So my question is: how do you find material stress for a pressure vessel with a wall that goes from $R_i$ to $R_o$, and you want a solution that is still perfectly valid when $R_i \\ll R_o$? My guess is that you would take the equation for the thin wall case, and recast it into an integral. $$ P = \\frac{2 \\sigma t }{ R }$$ In the more general case, would we write something like this? $$ P = \\int_{R_i}^{R_o} \\frac{2 \\sigma }{r} dr$$ This would only be valid if the stress was constant for all differential shells. I'm wondering if that's a bad assumption, but if the material was highly elastic I think it would be decent assumption. My intuition is that the material would have to be much _more_ elastic for it to hold with the inner radius many times smaller than the outer radius. But regardless of that detail, is the above integral correct? Is there some good intuitive logic to justify it? And is there any other reference that gives the expression for stress in this case? It should be a logarithm if the above is correct, but I haven't seen this before."}
{"id":"101913","title":"Why does a remote car key work when held to your head\/body?","text":"I was trying to unlock my car, but I was out of range. A friend of mine said that I have to hold the transmitter next to my head. It worked, so I tried the following later that day:   * Walked away from the car until I was out of range   * Put key next to my head (it worked)   * Put key on my chest (it worked)   * Put key on my leg (didn't work) So first I thought it has to do with height. But I am out of range if I use the key at the same height as my head but not next to it, and normally my key is at the same height as my chest. So it has nothing to do with height. Or my whole body is acting like an antenna, but how is that possible if I am holding the key? Why would it be an antenna if I hold it against my head and not in my hand? Here's a vid of Top Gear demonstrating it."}
{"id":"33117","title":"What would happen if a hydrogen bomb were to explode in Saturn's atmosphere?","text":"Purely hypothetical since any kind of testing in atmosphere\/space is banned by international legislation\/agreement. The humans have already bombed Luna so ... what could be expected to happen on Saturn if a hydrogen bomb were to explode in it's atmosphere? Would the explosion set the planet's atmosphere ablaze?"}
{"id":"113346","title":"How can the microstates be measured with zero energy expenditure?","text":"_James P. Sethna. Statistical Mechanics. Exercise 5.2:_ > What prevents a Maxwellian demon from using an atom in an unknown state to > extract work? The demon must first measure which side of the box the atom is > on. Early workers suggested that there must be a minimum energy cost to take > this measurement, equal to the energy gain extractable from the bit. Bennett > showed that no energy need be expended in the measurement process. Why does > this not violate the second law of thermodynamics? Reference to Bennett's paper didn't help me much. Here the relevant model is a tape consisting of single atoms in pistons, where knowing which side an atom is in a piston counts 1 bit of information, which can be used to extract useful work by expanding the piston, as shown below: ![enter image description here](http:\/\/i.stack.imgur.com\/aKzX4.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/2MbfM.jpg) **My understanding is that after the measurement uncertainty of position is reduced by half and entropy decreases. But without energy expenditure, it seems that this decrease comes free. How can the second law hold if there's no corresponding increase of entropy elsewhere (which I can't identify)?** Something like an explanation is given at the end of the exercise: > The demon can extract an unlimited amount of useful work from a tape with an > unknown bit sequence if it has enough internal states to store the > sequence—basically it can copy the information onto a second, internal tape. > But the same work must be expended to re-zero this internal tape, preparing > it to be used again. **Does this mean after measurement, the reduced entropy in the first tape goes to the second \"internal tape\" which stores the information? How can such measurement take place?**"}
{"id":"113348","title":"Questions (intuitive) about circuits and the flow of electrons","text":"Why don't electrons accelerate in a circuit? If there is a potential difference and the electron moves in the circuit a force is exerted on it. Why does that force produce no acceleration, keeping the current constant? If we add a resistor to a circuit why is the resistance 'spread' over the circuit? Is it because electrons moving faster in the areas of lower resistance would create charge imbalances at places in the conductor?"}
{"id":"127836","title":"Fock matrix elements for RHF formalism","text":"Here I refer to a particular book _Molecular Quantum Mechanics_ by Peter W. Atkins and Ronald S. Friedman, but similar derivation could be found in many other texts. So, when obtaining the explicit form of the Fock matrix elements for RHF formalism (p. 295 in 4th edition), authors go from ![enter image description here](http:\/\/i.stack.imgur.com\/GtcVE.png) to ![enter image description here](http:\/\/i.stack.imgur.com\/kGKFJ.png) just by mentioning that $\\psi_u$ is expanded as a linear combination of basis functions $\\theta$. The only way I could go from the first equation to the second one is by expanding the same spatial orbital $\\psi_u$ on the left and on the right sides of integrand expression, i.e. before and after $1\/r_{12}$, _differently_ , $$ \\psi_u = \\sum_l c_{lu} \\theta_{l} \\quad \\text{\"on the left side\"} \\, , \\\\\\ \\psi_u = \\sum_m c_{mu} \\theta_{m} \\quad \\text{\"on the right side\"} \\, . $$ Is it true? And if yes, why on earth this should be done this way?"}
{"id":"763","title":"Amount of 2 amperage","text":"I have 2 equation describing the alternating amperage $I_1$ and $I_2$. I need to get amount of these amperages. My equations: $$I_1=10\\sin(\\omega t+30)$$ $$I_2=20\\sin(\\omega t-50)$$ How can i make it? I forgot. Thank you."}
{"id":"71587","title":"Exact summation of a sub-class of diagram: do we know the exact solved problem?","text":"In quantum field theories (to be relativistic, (non-)relativistic statistical or whatever), we have the powerful diagrammatic approach at our disposal. Most of the time we can not sum up all the diagrams, only a sub-class of them. I was thinking of the Hartree-Fock ones to give an example, but they are several classes we can re-sum exactly (skeleton, ladder, ...). Somehow this re- summation of diagrams is _exact_. I will keep italics to refer to this _exactness_. Clearly the solution we find that way is not the full solution of the problem we considered at the beginning. By \"problem\" I obviously mean the eigenstates\/eigenvalues problem of a secular equation given by a Hamiltonian\/Lagrangian. My question is the following: **do we know the problem we _exactly_ solved summing only partial class(es) of diagram ?** I mean: can we write out an effective Hamiltonian\/Lagrangian that our _exact_ summation is the _exact_ solution of ? Can we at least know the _exact_ sub-space (eigenstates and\/or eigenvalues), the solutions we found _exactly_ belong to ?"}
{"id":"38697","title":"Is the Fock state a superposition of product states?","text":"I know that in principle, the Fock state is not the same as a product state because if it were, the photons would be uncorrelated. What I don't understand is, can it be expanded in product states? For example, if two photons share the same mode, is the state $|2,0\\rangle$ the same as the product $|1,0\\rangle|1,0\\rangle$? If they are in different modes, is the Fock state $|1,1\\rangle$ the same as superposition of two Fock states, $\\frac{1}{\\sqrt{2}}\\left(|1_a,1_b\\rangle+|1_b,1_a\\rangle\\right)$ (with $a$ and $b$ labeling the photons)? If yes, then is this superposition, in turn, the same as $\\frac{1}{\\sqrt{2}}\\left(|1_a,1_b\\rangle+|1_b,1_a\\rangle\\right)$ ?"}
{"id":"127833","title":"Can all canonical transformation be obtained through generation function approaches?","text":"The question can be formulated as following: Suppose $$\\delta \\int_{t_1}^{t_2}{[p\\cdot \\dot{q} - H(p,q,t) ]dt} = 0$$ $$\\delta \\int_{t_1}^{t_2}{[P\\cdot \\dot{Q} - K(P,Q,t) ]dt} = 0$$ in which $$P = P(p,q,t), Q = Q(p,q,t)$$ is an invertible transformation. Can we prove that there must exist a $\\lambda$ and function $G(p,q,t)$ (or $G(p,Q,t)$, $G(P,Q,t)$, $G(P,q,t)$), such that $$\\lambda[p\\cdot \\dot{q} - H(p,q,t) ] = [P\\cdot \\dot{Q} - K(P,Q,t) ] + \\frac{dG}{dt}~?$$"}
{"id":"77988","title":"What are the strongest sources of collimated neutrons and protons?","text":"I am imagining an unusual experiment which will require intense beams of either protons or neutrons. The experiment would work better with neutrons, but neutron sources are much ~~weaker~~ messier so I am considering both. I am interested in beams of almost any non-relativistic energy. I am not sure yet about how focused I will need the beam to be. I know this is a beginner question, but what are the basic options here for the strongest source (as measured by neucleons per second)? * * * Neutron apparently come from two sources: reactors and accelerators. (As Sir Patrick Stewert can attest.) In the latter case (\"spallation\"), a proton beam is just slammed into a heavy-metal target and the neutrons that fly out are caught. It appears that the highest intensity protons beams are often used to produce the highest intensity neutron beams in this manner. Saclay High Intensity Proton Injector Project (IPHI): $6 \\times 10^{17}$ protons\/second. Low-Energy Demonstration Accelerator (LEDA): $6 \\times 10^{17}$ protons\/second. (These agree with Andre's chart below showing 100 mA. Note also that the LHC current is about 500 mA, although the protons are grouped into bunches. And as Andre pointed out, the LHC circulates the same protons and so could not provide a continuous 500 mA beam into a target\/experiment that consumed them.) Neutron sources, being the result of either fission or spallation, are messy. I still don't have a good handle on what kind of neutron intensities are possible for what momentum ranges. The Lujan Neutron Scattering Center at LANCE is a neutron source for many experiments. * * * In spallation, the number of neutrons produced per incident proton is usually on the order of 40, but the phase space distribution is much wider than the incident protons. (Protons beams can be directly focused using magnets.) According to a CERN report by Lengeler, reactors typically can produce thermal neutrons (~0.025 eV) fluxes of order $10^{14}$ neutrons\/cm${}^2$s, whereas spallation sources can exceed $10^{17}$\/cm${}^2$s. * * * It is probably best, then, to think of continuous neutron sources as phase- space-density limited, with the above numbers giving a rough estimate of the currently achievable densities. Apparently, spallation sources are limited by the need to dissipate the heat from the incident proton beam. If high fluxes are needed only briefly, laser pulses can produce $4 \\times 10^{9}$\/cm${}^2$ over a nanosecond, i.e. an instantaneous flux of $4 \\times 10^{18}$\/cm${}^2$s. As for protons, it's worth noting that the high-intensity 50 mA beam at the European Spallation Source is limited by \"space charge effects at low energy, by the power that can be delivered to the beam in each cavity at medium and high energies, and by beam losses.\""}
{"id":"89664","title":"TKNN invariant changes due to continuous deformation of parameter space","text":"Naively, I would assume that a topological invariant remains invariant under continuous deformations of whatever space the invariant belongs to. In the case of topological insulators, this space is the space of parameters characterizing the material and its environment. If we consider the magnitude of a magnetic field incident onto a material supporting metallic edge states a parameter, then why does the Chern number (TKNN invariant) change when $|\\textbf{B}|$ is increased smoothly and slowly? Since the Hall conductivity experiences sharp increases due to slow increase of magnetic field, the Chern number must be increasing sharply (by integer jumps), signifying sharp changes in the topology of the parameter space. What parameter is changing so drastically in response to the increasing magnetic field?"}
{"id":"108566","title":"Probability distribution of two particle types system","text":"Suppose that particles of two different species, A and B, can be chosen with probability $p_A$ and $p_B$, respectively. What would be the probability (and distribution) $p(N_A;N)$ that $N_A$ out of $N$ particles are of type A? I'm trying to apply the Binomial distribution here but am bothered by the fact that it applies for N trials (whereas here we only do 1)."}
{"id":"72802","title":"Why the free energy is called 'free'?","text":"The free energy, $F$ of a thermodynamic system at a given temperature $T$, is defined as, \\begin{equation} e^{-\\beta F} = \\mathcal{Z} = \\sum_{\\\\{configuration\\\\}} e^{-\\beta E(configuration) } \\end{equation} where $\\beta = 1\/k_BT$ and $E$ is the energy of a certain configuration. Now my question is: **Why the free energy is called 'free'? I mean it is free in which sense?**"}
{"id":"31405","title":"Remaining Potential Experimental Particle Physics Discoveries at the TeV Scale?","text":"With the discovery of the Higgs Boson, some have been calling it the end of experimental particle physics for our generation, due to the fact that all of the particles predicted by the standard model have been found (the W and Z Bosons in 1983 and the Top Quark in 1995) and no new physics is expected at the energy ranges we can probe. I have also generally heard from my friends who work in experimental particle physics (or, at the very least, accelerator physics) a similar story, that there isn't too much left to be discovered. Is this an accurate assessment? What about dark matter particle models, like the axion\/axino? Can the LHC (or SLAC, etc) probe enough of a range to perhaps shed light on this cosmological problem? Can we hope for any more insights into beyond the standard model theories from current accelerators?"}
{"id":"5769","title":"Efficiency of Metropolis algorithm","text":"Context is 1D Ising model. Metropolis algorithm is used for simulate that model. Among all possible spins configurations (states) that algorithm generates only states with the desired Boltzmann probability. Algorithm chooses spin at random and makes a trial flip. If trial satisfies certain conditions related to Boltzmann probability, flip is accepted. Otherwise flip is rejected and system is unchanged. Define \"acceptance ratio\" as a percentage of accepted trials. Simulation shows that acceptance ratio is higher on higher temperature (behave as increasing function of temperature). Questions:   1. Why Metropolis algorithm is not efficient at low temperatures?    2. Is efficiency of algorithm related to acceptance ratio?"}
{"id":"15184","title":"Why is moonlight less colourful than yellow light?","text":"Just what the title states; and this isn't an original question ... On full-moon, moonlight is almost as bright as mid-dawn ; yet there is hardly any colour visible. This is not the case with sunlight. I'm curious to know why this is the case"}
{"id":"15183","title":"Where did the idea of a neutrino come from?","text":"Exactly what paper by W. Pauli introduced the idea for the existence of a neutrino and how was its existence confirmed experimentally (who did that and in what paper)?"}
{"id":"47159","title":"Planetary Gravity and its effects","text":"This is my first question on the Physics portion of Stack Exchange. I was hoping to get some light on the topic of gravity. I don't have much background knowledge of physics so I might as well start here. I have a few planet and gravity related questions: **1).** Does planetary gravity depend on size or density? Or both? **2).** Is it plausible for a large (very large), terrestrial planet to have tall mountains like Mount Everest or even like on Mars? **3).** Is there a way, given an average density per cubic meter along with the average radius of the planet, to calculate the pull of gravity? I've been scouring the internet for these things with no luck so far. The idea behind this is to make a planet rendering program which takes all of this into effect. If this forum doesn't answer this sort of thing, feel free to point me in the right direction. Thanks!"}
{"id":"47157","title":"is this generalized Hawking radiation formula right?","text":"Look at equation 11.2.17 in this page. The expression is: $$ T = 10^{-5} \\text{K m} \\frac{\\xi}{\\frac{GM}{c^2} \\lbrace \\frac{GM}{c^2} + \\xi \\rbrace - e^2 }$$ where $$ \\xi = (r_s^2 - a^2 - e^2)^{1\/2}$$ and the usual parameters $$ r_s = \\frac{GM}{c^2} $$ $$ a^2 = \\frac{L^2}{M^2 c^2}$$ $$ e^2 = \\frac{Q^2 G}{4 \\pi \\epsilon_0 c^4}$$ This formula is supposed to describe the temperature of a black hole with angular momentum $L$, charge $Q$ and mass $M$ > **Question:** is the above formula correct? I'm trying to find the limit temperature for $a=0$ and $e=r_s$ The temperature expression could be simplified as $$ T = 10^{-5} \\text{K m} \\frac{\\xi}{\\xi^2 + a^2 + \\frac{GM}{c^2} \\xi }$$ if $a=0$, $$ T = 10^{-5} \\text{K m} \\frac{1}{\\xi + \\frac{GM}{c^2} } $$ so when the black hole has extremal charge, $\\xi=0$ and the temperature looks like the normal black hole temperature for the Schwarzschild black hole, which looks very wrong Any idea where is the mistake? I was expecting the temperature of the charged extremal black hole with $a=0$ to be infinite"}
{"id":"81635","title":"why does water do that?","text":"I found recently that water fills a bottle quicker through only observation if I hold the tube away from the filling water than if I leave the tube inside the bottle. why does that happen? does potential energy have anything to do with this? ![enter image description here](http:\/\/i.stack.imgur.com\/VUXJU.jpg)"}
{"id":"15236","title":"Is traveling at the speed of light the same thing as teleportation?","text":"If I were on one side of the room and moved at the speed of light to the other side of the room, to an observer it would appear that I teleported. If time stops at that speed, it would be instantaneous but we know that light is not instantaneous (see light years). So I'm curious what the difference between traveling at the speed of light and teleportation is. I think teleportation has to do with deconstructing objects and then reconstructing them which allows them to travel through things etc., but is there any other noticeable difference?"}
{"id":"87527","title":"Question about exterior derivatives","text":"I know from Carroll that the integration in GR is basically a mapping from n-form to the real number. And it's given that $$d^nx=dx^0\\wedge\\ldots\\wedge dx^{n-1}=\\frac{1}{n!}\\epsilon_{\\mu_1\\ldots\\mu_n}dx^{\\mu_1}\\ldots dx^{\\mu_n}$$ Now, I have an expression that is given in spherical coordinate system, where I have $$\\int_\\Sigma f(\\theta,\\phi) d\\theta\\wedge d\\phi$$ when I want to integrate this (the epsilon part is already computed), do I just have $\\int\\int d\\theta d\\phi$ to integrate, or do I need to put the part from integration in spherical coordinate system $\\int\\int\\sin\\theta d\\theta d\\phi$? I haven't done much integration that involved forms before, so any help is appreciated :) EDIT: From the article I have: $$k_\\xi[h,\\bar{g}]=k^{[\\nu\\mu]}_\\xi[h,\\bar{g}](d^{n-2}x)_{\\nu\\mu}$$ $$(d^{n-p}x)_{\\mu_1\\ldots\\mu_p}:=\\frac{1}{p!(n-p)!}\\epsilon_{\\mu_1\\ldots\\mu_n}dx^{\\mu_p+1}\\ldots dx^{\\mu_n}$$ $$k_\\xi^{[\\nu\\mu]}[h,\\bar{g}]=-\\frac{\\sqrt{-\\bar{g}}}{16\\pi}\\ldots$$ where $\\ldots$ is an expression. Does this mean, since I have an $n-2$ form and I'm in 4 dimensional space, I need to include this $\\sin\\theta$ after all? EDIT2: I need to add the $\\sin\\theta$. I got it. Thanks :D"}
{"id":"109491","title":"2 Event Horizons come into contact","text":"Just a question out of curiosity, what would happen if the event horizons of 2 black holes of the same mass were to come into contact? Would both gravitational accelerations be canceled where the event horizons overlap?"}
{"id":"88207","title":"Air insulation cavity depth: rule of thumb to avoid convection","text":"I want to know more about optimising an air gap between two surfaces, such as window panes or a building's walls. I was taught that as a general rule, a 6cm air gap is the optimum. Any larger and convection counters the effectiveness. But something tells me this is hardly an accurate figure! Can anyone improve on this rule of thumb? What are the most significant variables which would alter the optimum separation distance? Examples of what I would want to apply this to are:   * A building's window panes of different sizes: glass, perspex or combinations   * Oven \/ furnace surfaces of different sizes: between two ceramic refractory walls, or refractory walls and steel. (Temperature differentials and absolutes of perhaps up to 1000 degrees C) Perhaps there are some approximated formulas to calculate optimum separation distance? And if the effective thermal conductivity of the air changes, a formula for that too. Thanks! For interest to other readers, this question is related."}
{"id":"100639","title":"Eigenfunction associated with the $\\hat{x}$ operator","text":"Consider the following operator $\\hat{x}=i\\hbar \\frac{\\partial}{\\partial p}$. I am trying to show that the eigenfunctions of $\\hat{x}$ are not square- normalizable. I am interested in doing so since theoretically, we notice that the eigenfunction of the momentum operator $\\hat{p}=\\frac{\\hbar}{i}\\frac{\\partial}{\\partial x}$ is not square- normalizable. How do we set up the equation for the $\\hat{x}$ operator?"}
{"id":"69468","title":"Fourier series of single tone modulated wave","text":"When a single-tone continuous modulating signal modulates a sinusoidal carrier, isn't the modulated wave periodic? If so, can't we apply fourier series and determine the harmonic frequency components instead of finding the frequency components using fourier transform?"}
{"id":"21030","title":"How does a speaker produce sound?","text":"What I read is that a speaker produces sound by the movement of a coil attached to a cone which moves back and forth. So, If I try to move the coil by hand, would it produce sound? If not, why? or Why would it produce sound only when the current moves the coil?"}
{"id":"12189","title":"What is the Jost function in scattering theory?","text":"What is the Jost function in scattering theory? Is it an operator or some kind of determinant? How is it obtained?"}
{"id":"62391","title":"Zero-zero (lower indicies) term for affine connection ($\\Gamma_{00}^\\lambda$), why do some terms dissapear?","text":"More simply a tensor algebra question, but in General relativity I have the following when I calculate $\\Gamma_{00}^\\lambda$:- $$ \\Gamma_{00}^\\lambda = \\frac{1}{2}g^{\\nu\\lambda}\\left( \\frac{\\partial g_{0\\nu}}{\\partial x^0} + \\frac{\\partial g_{0\\nu}}{\\partial x^0} - \\frac{\\partial g_{00}}{\\partial x^\\nu} \\right) = -\\frac{1}{2}g^{\\nu\\lambda} \\frac{\\partial g_{00}}{\\partial x^\\nu} $$ Why are the $\\frac{\\partial g_{0\\nu}}{\\partial x^0}$ terms zero?"}
{"id":"81200","title":"Semiclassical description of EM waves reflection from metallic surfaces","text":"Imagine an EM wave impinging on a metal. Fresnel's formulas tell us that no wave can propagate through the metal, or that the transmitted field is an evascent wave with some penetration depth dependent on the refraction index of the metal. If we now would like to zoom into the microscopic physics of the reflection, we would have to take into account the crystalline structure of the metal, which imposes a certain electronic band-structure, which in turn determine the way the electrons respond to the external EM perturbation. We should calculate the reflected field from the back action of the electron motion on the incident field. Do you know of a semiclassical way of solving this problem? Is it feasable to compute numerically a set of coupled Maxwell-Schroedinger equations?"}
{"id":"119932","title":"What's The Minimum distance?","text":"My Friend gave me a question today. The question was.:: > We have a point A. At a distance of $x_0$ from the point. There is a > particle$(P_1)$. Also, a particle is present at the point A $(P_2)$.The > Particle $P_1$ moves with a velocity of $u_1$ towards Point A. At the same > instant the particle $P_2$ moves making a angle $\\theta$ with $P_1A$ with a > velocity $u_2$. So what will be the minimum distance between the two > particles when both of them start at the same time. So can anybody help me with this problem. I tried decomposing it into vectors, used $\\tan\\theta$, but it did not give me a correct answer."}
{"id":"29094","title":"Why can't helicopters reach mount everest?","text":"Is there a reason why people can't just take the helicopter to mount Everest? Why is it that helicopters can't reach that high?"}
{"id":"43437","title":"Wheatstone bridge galvanometer error","text":"We had to measure the resistance of $R_x$, we balanced the Wheatstone bridge and did calculations. My question is: we didn't include galvanometer error into calculations. Why is that? I read that it's very precise, but that doesn't seem like a good enough explanation in exact sciences :\/ Edit: The precision is not the case as I was told, I need to go into more detail. When I measured, I set a value on the potentiometer, then I adjusted the adjustable resistor, so that the galvanometer would show zero. Is the error somehow compensated? ![enter image description here](http:\/\/i.stack.imgur.com\/yODpD.png) ɛ - electromotive force. K - switch. R - adjustable resistor. C - potentiometer."}
{"id":"46605","title":"When can photon field amplitudes be written as field operators?","text":"Suppose I have some classical field equation for two photon fields with amplitudes $A_1(z),A_2(z)$ (plane waves) given as ${A}_1=\\alpha f(A_1,A_2) \\\\\\ {{A}_2}=\\beta g(A_1,A_2) $ Under what conditions can I make the replacement $A\\rightarrow \\hat{A}$ ? I am uncomfortable with the Bohr correspondence principle because the correspondence principle seems to be a very weak argument. Edit: $z$ is position, $\\alpha$,$\\beta$ are real numbers. $f$ and $g$ are functions of the amplitudes."}
{"id":"135259","title":"How can I make a dry dilution refrigerator quiet?","text":"The pulse tubes are really loud. We've wrapped the helium lines in sound absorbing foam, and we have a cage around the top of the cryostat also lined with foam, but it's still pretty loud. Are there any really good and practical ways to damp the sound?"}
{"id":"21782","title":"Normal distribution of x, xdot","text":"I have some real measurements from a process and I happened to look at the mutual distribution of (x(t), xdot(t)). I found that they seem to follow 2d normal distribution around (mu, 0). See image, the y-axis is xdot. ![x-xdot-histogram](http:\/\/i.stack.imgur.com\/rQfWk.png) Is there some well known process which exhibit this property? What kind of dynamic equations on x(t) cause this?"}
{"id":"13581","title":"How do we know that the nucleus isn't a quark-gluon plasma?","text":"The standard picture of the nucleus of atom is that is several distinct nucleons, which themselves are composed of quarks. However, it seems to me like a much simpler picture is that the nucleus is directly made out of quarks, without having nucleons as substructures. That is, that the nucleus is the ground state of a quark-gluon plasma. I can't find any evidence that this is false. For instance, if the distances between the nucleons was significantly larger than the size of the nucleons, there would be a difference in size between hydrogen and other atoms which can't be explained by this new framework, but according to wikipedia the diameter of all of the nuclei is the same order of magnitude. What is the evidence that large nuclei have nucleons as substructures?"}
{"id":"90570","title":"Small angle approximation for coupled pendulum","text":"I am looking at a coupled plane pendulum problem where the pendula are identical with mass $m$ and length $r$, and connected by a spring with spring constant $k$ and an equilibrium seperation $d_0$. If $\\theta_1$ and $\\theta_2$ are the angular displacement with the vertical then in these coordinates then the Lagrangian (I believe) is $$L=\\frac{1}{2}mr^2 \\left( \\dot{\\theta_1^2} + \\dot{\\theta_2^2} \\right) -mgr \\left( \\cos \\theta_1 + \\cos \\theta_2 \\right) -\\frac{1}{2}k \\left[ \\sqrt{\\left(r \\sin \\theta_1 - r \\sin \\theta_2 \\right)^2 + \\left(r \\cos \\theta_1 - r \\cos \\theta_2 \\right)^2} -d_0\\right]^2.$$ I'm trying to rewrite the approximate Lagrangian for small oscillations about the bottom in terms of $\\eta_1$ and $\\eta_2$ where $\\eta_{\\alpha}$ represents the horizontal displacements. I'm not sure how to go about doing this. Am I supposed to assume that $\\theta_1=\\theta_2=0$ is an equilibrium point and then expand the potential to second degree? In order to do this, can I assume that my $-mgr\\left(\\cos \\theta_1 - \\cos \\theta_2 \\right)$ can be approximated as $$-mgr\\left(\\cos \\theta_1 - \\cos \\theta_2 \\right) \\approx mgr\\left(\\frac{\\theta_1^2 + \\theta_2^2}{2} \\right)$$ and then for the spring terms write $$\\frac{1}{2}k \\left[ \\sqrt{\\left(r \\sin \\theta_1 - r \\sin \\theta_2 \\right)^2 + \\left(r \\cos \\theta_1 - r \\cos \\theta_2 \\right)^2} -d_0\\right]^2=P(\\theta_1, \\theta_2)$$ $$\\frac{1}{2}k \\left[ \\sqrt{\\left(r \\sin \\theta_1 - r \\sin \\theta_2 \\right)^2 + \\left(r \\cos \\theta_1 - r \\cos \\theta_2 \\right)^2} -d_0\\right]^2 \\approx \\frac{\\partial^2 P}{\\partial \\theta_{\\alpha} \\partial \\theta_{\\beta}}\\bigg|_{\\theta_1=\\theta_2=0} \\theta_{\\alpha} \\theta_{\\beta} $$ Then finally combine all like terms? Or before doing this can I simplify the root by writing $\\sin \\theta_{\\alpha} = 0$ before doing the partial derivatives? I'm not sure which process is correct here. Any help would be appreciated. Thanks. **\\------------------------EDIT----------------------------** If I write $$V(\\theta_1, \\theta_2) = mgr \\cos \\theta_1 + mgr \\cos \\theta_2 + \\frac{1}{2}k \\left[ \\sqrt{\\left(r \\sin \\theta_1 - r \\sin \\theta_2 \\right)^2 + \\left(r \\cos \\theta_1 - r \\cos \\theta_2 \\right)^2} -d_0\\right]^2$$ then according to my calculations $$\\frac{\\partial^2 V}{\\partial \\theta_1^2} = -\\frac{1}{2}kr(2r-d)-mgr = \\frac{\\partial^2 V}{\\partial \\theta_2^2}$$ $$\\frac{\\partial^2 V}{\\partial \\theta_1 \\partial \\theta_2} = -\\frac{1}{2}kr(2r-d_0)$$ Can I therefore write the Lagrangian as $$L \\approx \\frac{1}{2}mr^2 \\left( \\dot{\\theta_1^2} + \\dot{\\theta_2^2} \\right) + \\frac{1}{2} \\left[ \\frac{1}{2}kr(2r-d_0) + mgr \\right] \\left(\\theta_1^2 + \\theta_2^2 \\right)+\\frac{1}{2}kr(2r-d_0) \\theta_1 \\theta_2$$ and then proceed to find the vibrational frequencies and normal modes using the K-M matrix procedure (after making the $\\eta$ substitutions)? I'm wondering if there are any simplifications I can make to the potential (i.e. treat the approximation of the potential in multiple separate parts where I take derivatives of each individually) before I start taking partial derivatives to simplify the problem. Or does a multidimensional potential such as this always have to be treated as one massive beast?"}
{"id":"41877","title":"Finding Max Radius of Propeller (angular velocity)","text":"I am looking at a question from University Physics ![enter image description here](http:\/\/i.stack.imgur.com\/loQo7.png) The given answer ![enter image description here](http:\/\/i.stack.imgur.com\/cgP7p.png) Whats the intuition behind using the below diagram ![enter image description here](http:\/\/i.stack.imgur.com\/BX14V.png) of finding $v_{tip}$? I was looking the the $v_{tan}$ not knowing how to continue. Also $v_{tip}$ will rotate and turn positive and negative"}
{"id":"53118","title":"Andrew's experiment","text":"In Thomas Andrew's experiment, consider the dome shaped saturation region. If we increase the pressure at constant volume until we reach the critical point, why does the density of vapours rise and the density of liquid fall? Moreover, why will the meniscus or the separation boundary of two phase slowly become less distinct and finally at critical point disappear?  ![enter image description here](http:\/\/i.stack.imgur.com\/LXo52.gif) Will vapour pressure increase or decrease as we move vertically up in the P-V plane in the saturation region?"}
{"id":"54467","title":"Online magnetic field solver","text":"Is there a site that allows you to give the approximate specs of a set of magnets and then it produces the graph of the magnetic field for you? Here is a situation I try to understand: I have a set of rectangular magnets about 1*0.75*.25 inches with north and south on large faces. The magnets are glued in two long lines (like dominoes) about 1 inch apart, all north sides are up. In the area between the magnets there is a strong magnetic field, as if a magnet with opposite polarity is in that places (as if a south pole is there). Why does that happen?"}
{"id":"54465","title":"Gravity Question Can this be reopened?","text":"Anyway what I am trying to figure out is something I have been thinking about for many years. The dinosaurs didn’t die from a big explosion or anything just that the earths gravity increased and made them less competitive to smaller creatures. I have heard bone mass decreases for astronauts so I am think it must increase if gravity increased. Also I am not looking for a number but an understanding of what a creature would weigh with the increase of matter that has been hitting the earth over time. The 37,000# number is what I found on the internet as a theory to what is being added to the earth ever year so over 60 million years there must by an increaseHope someone can help"}
{"id":"535","title":"Why does kinetic energy increase quadratically, not linearly, with speed?","text":"As Wikipedia says: > _[...] the kinetic energy of a non-rotating object of mass $m$ traveling at > a speed $v$ is $mv^2\/2$._ Why does this not increase linearly with speed? Why does it take so much more energy to go from 1 m\/s to 2 m\/s than it does to go from 0 m\/s to 1 m\/s? My intuition is wrong here, please help it out!"}
{"id":"123365","title":"Is acceleration quadratic or linear outside of a gravity field?","text":"So as I understand it: If you hit a wall at twice the speed you hit four times as hard? I often ask people this and everyone says \"at twice the speed you hit twice as hard\". $K=\\frac{1}{2}mv^2$ is very counter intuitive. I have read an explanation that seems to say gravity is acceleration and that's where the squared comes from. Do we have anyway of knowing if the kinetic energy of an object outside of a gravity field is linear or quadratic? Could the gravity field somehow match the acceleration to give the $v^2$ Bonus question: If you get pulled over for speeding should you ever ask the policeman why kinetic energy is quadratic instead of linear? $K=\\frac{1}{2}mv^2=$($300) dollar fine where I live."}
{"id":"69169","title":"How - The force of a 60 mph crash is not just twice as great as a 30 mph crash; it’s four times as great!","text":"The DMV manual says that > _The faster you go, the less time you have to avoid a hazard or collision. > The force of a 60 mph crash is not just twice as great as a 30 mph crash; > it’s four times as great!_ My physics is quite rusty, so I could not figure it out. I guess the above statement is correct, but how do we prove it? **Edit** I figured this out myself, but alternative methods or new ways of understanding are still welcome."}
{"id":"58121","title":"Newton's Third Law of Motion","text":"I want to start off by saying that I've looked around for other explanations, but I've not really found any satisfying ones. My question is basically the whole \"Why can anything move at all?\" question, with a little twist. Refer to this picture: ![](http:\/\/i.stack.imgur.com\/bmhkv.png) Now I understand that opposite forces act on different objects, so in the top one, the rocket would have a net force of a 100 N to the right, and accelerate, as per Newton's Second Law. However, in the bottom example, wouldn't the box push back on the rocket, equal and opposite, as the rocket pushes on the box, therefore cancelling the forward push from the exhaust, making the rocket not move at all, whilst the box gets a net force of a 100 N? This isn't true obviously, but why not? Also, in this earlier Phys.SE post there is a great answer depicting a finger and a matchbox. Now this post is very related to mine. I'm wondering why in that picture, the force of the finger pushing on the matches doesn't equal the force of the muscles pushing forwards in the finger? Surely when I'm having a force pushing the finger forwards, that same force applies to the matches?"}
{"id":"20783","title":"How to calculate the Gibbs free energy out of a given Entropy?","text":"I'm stuck in a small calculation. I try to solve a small exercise. You should calculate for a given entropy the Gibbs free energy, where the entropy is given as $S\\left(U,V\\right) = \\frac{4}{3}\\left(\\alpha V U^3\\right)^{1\/4} \\qquad\\dots\\qquad U$ is the internal energy, $V$ is the volume and $\\alpha > 0, \\text{const}$ The exercise say something like \"you get a surprisingly expression for the Gibbs free energy\". * * * Basically I have to use the definition of the Gibbs free energy: $$ \\begin{equation} G\\left(T,p\\right) = U\\left(S,V\\right) - T\\cdot S + p \\cdot V \\label{eqn:gibbs} \\end{equation} $$ The question is now, how can we exchange the volume $V$ and the entropy $S$ in the equation above. My idea was to calculate $p=p\\left(T,V\\right)$ and transpose it to $V=V\\left(p,T\\right)$. The thing is, that I get for the pressure $p$ something like: $p = \\frac{\\alpha}{3}T^4$, which is not $p = p\\left(T,V\\right)$ and it's not possible to find a relation for $V=V\\left(p,T\\right)$. I thing, I'm making some mistake. Can someone help me?"}
{"id":"20786","title":"Calculating the force of a mass bearing down on an object","text":"I need to calculate the force of a weight bearing down on a smaller object, constraining it from expansion. The weight bearing down has a much bigger surface area than the smaller object. I know the mass of the weight. I know the surface area of the smaller object. What else do I need to know in order to calculate the force bearing down on the small object? What I mean is, what do I need to know in order to calculate what the force bearing down on that smaller object is? I think I might need to know about the acceleration of gravity (9.8 m s^-2), but I'm not sure how that applies if I don't know the dimensions of the larger object. I know that F=ma, but I can't visualise how that applies to an object bearing down on an object with a smaller surface area."}
{"id":"62554","title":"Is Dirac's description of a photon in a split beam still seen as correct today?","text":"This comes from the _Interference of Photons_ section in the book _The Principles of Quantum Mechanics_ by P Dirac: > We shall discuss the description which quantum mechanics provides of the > interference. Suppose we have a beam of light which is passed through some > kind of interferometer, so that it gets split up into two components and the > two components are subsequently made to interfere. We may, as in the > preceding section, take an incident beam consisting of only a single photon > and inquire what will happen to it as it goes through the apparatus. This > will present to us the difficulty of the conflict between the wave and > corpuscular theories of light in an acute form. > > Corresponding to the description that he had in the case of the > polarization, we must now describe the photon as going partly into each of > the two components into which the incident beam is split. The photon is > then, as we may say, in a translational state given by the superposition of > the two translational states associated with the two components. We are thus > lead to a generalization of the term \"translational state\" applied to a > photon. For a photon to be in a definite translational state it need not be > associated with one single beam of light, but may be associated with two or > more beams of light which are the components into which one original beam > has been split. In the accurate mathematical theory each translational state > is associated with one of the wave functions of ordinary wave optics, which > wave function may describe either a single beam or two or more beams into > which one original beam has been split. Translational states are thus > superposable in a similar way to wave functions.   1. Is this the view held by leading physicists today?   2. Can we really talk of the incident and exiting photon in the beam splitter being the same photon?   3. Is Dirac saying that the photon is partly in the two split beams, but no where else in space?"}
{"id":"38403","title":"Is an electric lamp a transducer?","text":"Silly thought. A transducer, by definition, is a device that converts variations in one form of energy to another. An electric lamp converts electricity into visible light - the brightness may vary depending upon the electric potential applied. Is it correct to state an Electric Lamp is a transducer?"}
{"id":"122435","title":"what property of steam makes it the preferable motive fluid in jet ejectors? why not air or nitrogen?","text":"I want to know what properties of steam make it the fluid of choice in   1. steam turbines   2. jet ejectors. I want to understand the mechanism of energy conversion in these equipment and hence understand why steam is the preferred choice. Please leave out economic factors for now and keep your responses rooted only in physics\/thermodynamics."}
{"id":"122432","title":"Dual particle-wave behavior","text":"If electrons and photons, and possibly more particles, exhibit dual character, why don't physicists create a new classification for them? Why describe them as both waves and particles. Why not rather create a new classification, which accurately describes their behavior?"}
{"id":"48522","title":"Term for \"atmospheric ricochet\" due to wrong \"angle-of-attack\"","text":"I watched \"Apollo 13\" yesterday, and they had the \"angle-of-attack\" problem that had to be manually solved, to prevent the ship from \" _ricochet[ing] off the atmosphere like a rock skipping off a pond_ \". I found and read this, but couldn't manage the term to further learn about this \"ricochet\". **Is there a term that reflects this \"ricochet\"?**"}
{"id":"41125","title":"Effect of Current on spring","text":"When Current passes thru a spring , some books mention that it gets compressed. However, I think due to the heating effect of current, molecules will increase kinetic energy and the spring should get elongated. What is the physical mechanism for the compression instead of elongation?"}
{"id":"129012","title":"What is the difference between negative acceleration and decreasing acceleration?","text":"According to Reisnick and Halliday, retardation is the acceleration in the negative direction of time axis ie. opposite to the motion. Decreasing accleretion is the decrease in the magnitude of time rate of change of velocity. Am I correct or not? What is the $v$-$t$ graph with retardation and decreasing acceleration? [I don't think they are same.]"}
{"id":"59652","title":"Accelerating an elevator","text":"A person is standing on a weighing scale in an elevator in upward acceleration. Let $N$ be normal reaction force exerted by the weighing scale to the person (upward). It is known that the person will experience a normal reaction force ($N$) larger than his own weight (larger reading on the weighing scale). While $N$ is equal, but in opposite direction, to the force that the person exerts to the weighing scale due to action-reaction pair (the person and the weighing scale). That means other than the person's weight, there should be an extra force exerting to the weighing scale downward ($A$). So $-N$ (upward)$ = mg + A$ (downward). I can't figure out where does $A$ come from?"}
{"id":"40832","title":"What are relativistic and radiative effects (in quantum simulation)?","text":"I'm reading about Quantum Monte Carlo, and I see that some people are trying to calculate hydrogen and helium energies as accurately as possible. QMC with Green's function or Diffusion QMC seem to be the best ways to converge on the \"exact\" solution to Schrodinger's equation. However, if one wants to be very exact, then the Born-Oppenheimer approximation must be removed. A lot of papers mention that the results are still not exact enough, and must be corrected for relativistic and radiative effects. I'm pretty sure I know what relativistic effects are -- the non-relative Schrodinger's equation cannot account for GR as particles approach the speed of light (or even small but measurable effects at lower speeds). But what are radiative effects? And I would think you would include these two things into your QMC calculation instead of applying a post-simulation correction factor if you wanted to be ultra-precise (e.g. use the Dirac equation instead for relativistic effects). So why don't most researchers do this? Does it raise the calculation time by orders of magnitude for an additional 4th decimal place of accuracy? Finally, is there anything at a \"deeper\" level than relativistic and radiative effects? In other words, if I left a supercomputer running for years to compute helium's energies without the BO approximation, and with relativistic and radiative effects included in the MC calculations, would this converge on the _exact_ experimental values? (Actually, I just thought of one such left-out factor -- gravity ... and might you have to simulate the quarks within the protons individually? Anything else?)"}
{"id":"102453","title":"Find Angular Momentum about any point","text":"How do I find the angular momentum of a body about any point?   We know that $L=I\\omega$ for a body rotating in space, where $L$ denotes the angular momentum, $I$ denotes the moment of inertia and $\\omega$ denotes the angular velocity. However, this is **only applicable** for **Fixed Axis of Rotation, Instantaneous Axis of Rotation and Center of Mass**.   Can somebody state and prove the value of the angular momentum of a body about any point? (If a formula for that exists).   Thanks!"}
{"id":"29454","title":"Does an uneven and\/or moving surface affect quality when using a solar filter?","text":"I have chosen to make my own solar filter using Baader as opposed to Mylar or anything else based on these: http:\/\/physics.stackexchange.com\/a\/25204\/6805 and http:\/\/irwincur.tripod.com\/mylar_vs__baader.htm and others. My last solar filter was the product which is no longer available direct from Celestron (I'm not even sure what it was made with). **My questions** : Does \"flapping in the wind\" baader effect image quality? See the images below to understand what I mean... Is it possible to achieve a better taut fitting when making my own? As seen here ![enter image description here](http:\/\/i.stack.imgur.com\/BcsMd.jpg) it is not flat or tight like this glass one is ![enter image description here](http:\/\/i.stack.imgur.com\/EHj3t.jpg)"}
{"id":"11608","title":"What happens when a lightning rod is not earthed?","text":"A Lightning rod is fixed to the building and it is not given earthing then what happens? How does the lightning reach the ground through the building?"}
{"id":"94989","title":"Moving conducting bar in (changing) magnetic field","text":"In my exam today I've been given this problem, yet even with the results at hand I simply can't warp my head around it; Given the picture below, a bar is placed on two conducting rails with a resistor (in the form of a single thin wire) at the left. Now at t=0 the magnetic field changes which causes the bar to move. (and a current flows through the \"circuit\"). The current can easily be found by Lenz's law: $$ \\mathscr{E} = -\\frac{\\mathrm{d}\\Phi_b}{\\mathrm{d}t} = -\\ell\\frac{\\mathrm{d}\\left( Bx \\right )}{\\mathrm{d}t} = -\\ell x\\frac{\\mathrm{d} B }{\\mathrm{d}t} -\\ell B v $$ ![enter image description here](http:\/\/i.stack.imgur.com\/TnAlF.png) Now this is follow up text for the problem > It may be assumed that the conducting bar **has a negligible mass and > friction** , so it will immediately move to a position $x$ in which there is > no force exerted on it. > > b) Explain why the relation between the actual position $x$ and the actual > value of the magnetic field $B$ is given by $Bx = B_0x_0$. What did the prof try to explain here? No force means constant speed, so not a \"position\" at all right?   Well the result sheet shows: > No force means $\\frac{\\mathrm{d}(Bx)}{\\mathrm{d}t} = 0$ Hence $Bx$ is > constant ..... what???? Sorry but maybe it stems from the fact that I already didn't understand what \"no force\" meant in the text but I can't understand this."}
{"id":"59382","title":"Dispersion relation and Heat Capacity","text":"I have seen case studies of the 3D Debye model where the vibrational modes of a solid is taken to be harmonic with dispersion relation $\\omega = c_sk$. It is said that for temperatures much less than the Debye temperature, the heat capacity at constant volume $$C_V\\sim T^3$$. Now I want to show that for bosons with dispersion relation $\\omega\\sim A\\sqrt k$ has heat capacity $$C_V\\sim T^4$$ for $T\\ll T_{Debye}$. In the case studies I have read, I can't find where the dispersion relation comes into play. I have no idea how to see this. Please help! * * * Perhaps I could get a response if I change the question to: How does the dispersion relation come into the calculations for the Debye model? Links to notes or an answer would be very much appreciated!! * * * @MarkMitchison kindly pointed out that the density of state depends on the dispersion relation. What are the general definitions of $$E, p$$ in terms of $$\\omega, k$$? E.g. for the first case $E=\\hbar \\omega$ and $p=\\hbar k$. So the question is: what are the respective values for $E,p$ in general?"}
{"id":"59383","title":"Is normalization consistent with Schrodinger's Equation?","text":"Schrodinger's Equation does not set a limit on the size of wave functions but to normalize a wave function a limit must be set. How is this consistent physically and mathematically with Schrodinger's Equation."}
{"id":"114434","title":"Specific problems with the quadrupole formulation of gravitational radiation","text":"the quadrupole formula has some counterintuitive consequences, when analysing the power output averaged over a period $$ P = \\langle \\frac{d^3 Q_{ij}}{dt^3} \\frac{d^3 Q_{ij}}{dt^3} \\rangle $$ Specifically, if two separated sources are considered inside the radiation zone, $\\rho_1$ and $\\rho_2$, and assume that each orbit around separate centers $C_1$ and $C_2$ by a length $R$, the quadrupole moment of the pair is going to be (around the common center $C_1 + \\frac{R}{2} = C_2 - \\frac{R}{2}$) $$ \\int ( \\rho_1(x) + \\rho_2(x) ) x^j x^k d^3x = \\int \\rho_1(x) (x_1 - \\frac{R}{2})^j (x_1 - \\frac{R}{2})^k d^3x + \\int \\rho_2(x) (x_2 + \\frac{R}{2})^j (x_2 + \\frac{R}{2})^k d^3x = \\int \\rho_1(x) x_1^j x_1^k d^3x - \\frac{R^j}{2} \\int \\rho_1(x) x_1^k d^3x - \\frac{R^k}{2} \\int \\rho_1(x) x_1^j d^3x \\+ \\int \\rho_2(x) x_2^j x_2^k d^3x + \\frac{R^j}{2} \\int \\rho_2(x) x_2^k d^3x + \\frac{R^k}{2} \\int \\rho_2(x) x_2^j d^3x$$ $$ = \\int \\rho_1(x) x_1^j x_1^k d^3x + \\int \\rho_2(x) x_2^j x_2^k d^3x + \\frac{R^j}{2}( \\int \\rho_2(x) x_2^k d^3x - \\int \\rho_1(x) x_1^k d^3x ) + \\frac{R^k}{2}( \\int \\rho_2(x) x_2^j d^3x - \\int \\rho_1(x) x_1^j d^3x )$$ plus some other terms that are constant over time that are not needed However, the terms $\\frac{R^k}{2} \\int \\rho_c(x) x_c^j d^3x$ will have nonzero third time derivatives in general, so they will contribute to the overall power, even though they depend on an arbitrary choice of a center over which the quadrupole moments are calculated? What is wrong with this expression, and how to obtain a power output from the quadrupolar moment that doesn't depend over what origin of coordinates is chosen for the calculation?"}
{"id":"9842","title":"Where is the particle during a tunneling event?","text":"If, say, a particle with energy $E<V_0$, approaches a finite potential barrier with height $V_0$, and happens to tunnel through, where would the particle be during the time period when it is to the left of the potential barrier and to the right of the potential barrier? Surely there must be a finite amount of time for it to travel through to the other side, unless it simply teleports there? If it travels through with energy less than $V_0$, however, doesn't that mean it cannot enter in the region of the potential barrier?"}
{"id":"9840","title":"Detection of W and Z bosons","text":"What specific behaviour confirmed the existence of the W and Z bosons at the UA1 and UA2 experiments? Thanks!"}
{"id":"52721","title":"How should I simulate the electric potential field from a wavefunction?","text":"I was interested in making what I thought would be a simple simulation of an electron encountering a positron by numerically solving the Schrodinger equation over several time steps, but I've run into 1 problem: the potential used in most textbooks (to solve the hydrogen atom for example) are still of a classical point particle, rather than a wavefunction. How do I generate the potential field from two real, moving and interacting wavefunctions?"}
{"id":"104800","title":"Experiments proving\/disproving extra dimensions","text":"I have been recently reading quite a little about Kaluza Klein theories. I am still far from mastering this but I am curious if any experiment that may disprove or give hints of the existence of extra spatial dimensions is to be expected to be performed relatively soon. This page of CERN, for example, says that the LHC might give some clues, but it doesn't really dive into much detail. So, in some detail, which experiments might say something about this, and when might they be expected?"}
{"id":"32402","title":"Why does laser light not affect glowing materials?","text":"I have this childrens rubber ball which glows in the dark after it's exposed to light. I \"charge\" it with a flash light then play with my dogs at night. I thought to try a very intense green laser, and see how the ball reacted. The laser light had no effect on the balls ability to glow. So I'm left wondering, why does laser light not allow luminescence (maybe not the right word) materials to glow? EDIT In Response to Answer. So I tried a little modification. I tried exciting the ball with three different light sources; a \"super bright\" Red LED, a very very \"super bright\" white LED and a blue LED of unknown specs (no package, bottom of my kit). I held the ball to each light source (driven with the same current) for the same approximate amount of time and compared the results. The red LED had no effect. The white had a bit of an effect, enough to see dimly in normal room lighting. The blue led had a significant effect, causing a bright glow. This was interesting as the blue LED was the least bright visually. Yay science!"}
{"id":"55404","title":"Mathematically, how do we deduce that angular momentum is bounded?","text":"So, how do we know $J_{+}|j,(m=j)\\rangle =|0\\rangle$? I.e. that m is bounded by j. We know that $J_{+}|j,(m=j)\\rangle =C|j, j+1\\rangle$, but how do I know that gives zero? Is it by looking at its norm-square?"}
{"id":"92910","title":"Exotic coupling","text":"I have encountered the minimal coupling between a field and charges before $$H = \\frac{1}{2m}(p-qA)^2,$$ whereby I am considering the classical case. The description minimal leads me to ask if there exist exotic couplings (non- minimal)? Could some examples of such couplings be discussed?"}
{"id":"92911","title":"How did gunsmiths create revolver cylinders in 1850's without the use of electrical drill?","text":"How did gunsmiths create revolver cylinders holes in 1850's without the use of electrical drill? Which referces\/ books specialize providing knowledge in similar molding method?"}
{"id":"110057","title":"$E=mc^2$ and particles","text":"This is a simple question. Does the Energy create both Fermion and Boson particles ? or just only the Fermion particles?"}
{"id":"104327","title":"Polarization vector and bound charge","text":"Why is it that the bound charge is $Q_b = - \\oint_S{\\mathbf{P} \\cdot d\\mathbf{S}}$? In particular, why is there a negative sign? Hayt's book on electromagnetism describes this as the \"net increase in bound charge within the closed surface\". Compared to Gauss' law $Q_{f} = \\oint_S{\\mathbf{D} \\cdot d\\mathbf{S}}$, the free charge is just the electric flux, so I don't see why the bound charge has a negative sign. Also, if the electric flux is independent of bound charge, how come the D-field changes across material boundaries? I thought the D-field was defined as $\\mathbf{D} = \\epsilon_0 \\mathbf{E} + \\mathbf{P}$ to cancel out the effect of the bound charge (since the effect of polarization is already incorporated into the E-field). Is the D-field dependent on bound charge? Just to add to the second part of my question. I know that for a dielectric- dielectric boundary the tangential components $\\frac{D_{t1}}{\\epsilon_1} = \\frac{D_{t2}}{\\epsilon_2}$ and if $Q_{free} = 0$ on the boundary, then the normal components $D_{n1} = D_{n2}$. The tangential component will not contribute to the flux integral so for any $\\epsilon_1$ and $\\epsilon_2$ across a boundary the electric flux integral will evaluate to 0. It's clear that the flux is independent of the bound charge, but the the tangential component changes so the D-field is not?"}
{"id":"51238","title":"What aspects of QFT do mathematicians find troublesome?","text":"What aspects of \"conventional\" quantum field theory (i.e. what's used by most practicing physicists) are considered to be lacking mathematical rigor?"}
{"id":"127117","title":"Argument for symmetry of potential of a spherical capacitor filled with dielectric","text":"A spherical capacitor with inner radius $r_1$ and outer radius $r_2$ is filled with dielectric material with permittivity $\\epsilon=\\epsilon_0+\\epsilon_1\\cos^2\\theta $.   $\\theta$ is the polar angle. Furthermore there is a charge $Q$ on the inner surface at $r_1$ and a charge $-Q$ on the surface at $r_2$. Give reasons why the potential $\\Phi(r,\\theta,\\varphi)$ is for all $r$ independent of $\\theta$ and $\\varphi$. The independence of $\\varphi$ is clear as our problem is invariant under $\\varphi$-rotations. I'm not able to find a valid argument why the potential is independent of $\\theta$. Thanks in advance for help !"}
{"id":"25019","title":"What does f\/4.6 mean in a telescope and how important is this value?","text":"In some specifications for telescopes, I saw a value marked as f\/4.6. What does it mean exactly, and how important is when it comes to choosing a telescope?"}
{"id":"110229","title":"What is the lenght of the rope?","text":"The $E$ string of a violin has a linear density of $0.5 g \/ m$ and is subjected to a $80\\text{N}$ of tension, tuned for a frequency $u = 660 \\text{Hz}$:   1. What is the length of the rope? I know that in order to answer the question I have manipulate the equation: $$ \\nu = \\sqrt\\frac{\\tau}{\\mu} $$ Getting this equation: $$ L = \\frac{m v^2}{T} $$ I have tried to find the mass but my attempts was unsucessufull. I think that i could use the equation: $$ f_1 = \\frac{\\sqrt\\frac{T}{m\/L}}{2L} $$ In order to find the lenght, but I need help to find the mass of the rope."}
{"id":"98329","title":"What is the constant appearing in the low energy action?","text":"Usually one finds this expression for the low energy action $$S = \\frac{1}{2\\kappa_0^2}\\int d^D X\\; \\sqrt{-G}\\; \\mathrm{e}^{-2\\Phi}\\,(R-\\frac{1}{12}H_{\\mu\\nu\\lambda}H^{\\mu\\nu\\lambda}+4 \\partial_{\\mu}\\Phi\\partial^{\\mu}\\Phi).$$ On dimensional grounds one can deduce $\\kappa_0^2$ near $\\ell_s^{24}$ (for $D=26$) or $\\ell_s^8$ (for $D=10$). Some authors use (obvious for the superstring, not for $D=26$ string): $$2\\kappa_0^2=(2\\pi)^7\\alpha'^4$$ with $\\ell_s=2\\pi\\sqrt{\\alpha'}$. Btw. this also confuses me, is $\\ell_s=\\sqrt{\\alpha'}$? So my question is how is $2\\kappa_0^2$ determined?"}
{"id":"82276","title":"Amateur moon laser ranging","text":"Questions first, then my rough estimations: 1) Is it possible to perform moon laser ranging with amateur motorized 114mm telescope? My calculations suggest that for 1mJ laser it should receive ~2 photons per source 1mJ laser pulse. 2) Given that we already talking about individual photons, how it was possible to perform moon laser ranging BEFORE retro-reflectors were deployed to the moon? Retroreflector sends back light in ~1arcsecond angle, while bare lunar sufrace - in ~6 archours, which means we supposed to receive signal ~(6*60*60)^2 = 4.5*10^8 weaker, i.e. even with 2.5 meter telescopes we are talking 1 photon per 250 1J pulses. **My rough estimations:** Given that atmosphere turbulence limits telescope resolution to ~1 arcsecond (adaptive optics was not available when laser ranging experiments started, nor it is available now for amateurs), if we use telescope with diameter larger than ~150mm (so that we are limited by atmosphere, not diffraction) to expand the laser beam we will get ~1939x1939 meter illuminated area on the moon surface (tan(1arcsec)*400'000km). Which means only 1\/(1939*1939) part of our energy will reach reflector. Retroreflector is ~1x1 meter in size. It will reflect the light with same beam divergence - 1arcsecond. Too sad, as diffraction limit for retro-reflector of such size is ~0.2 arcsecond. So, if our receiving telescope has area of ~1 meter^2, we will receive again 1\/(1939*1939) part of what reached the moon, so total attenuation is ~ 1.4*10^13. If we use 532nm pulse laser with 1mJ pulse energy, it will emit 2.67*10^15 photons, which means we are going to receive ~190 photons per pulse. Sounds realistic. These calculations suggests that 114mm amateur telescope should be able to detect 2 photon per pulse - again should be detectable statistically."}
{"id":"117209","title":"scattering of light","text":"How does the path of light become visible due to scaterring of light? According to me it will only tell the presence of colloidal particle but the book give the example of cinema hall projector and says that you could tell path of light due to scattering of light. Nevertheless, without scaterring of light also we can tell its path."}
{"id":"131204","title":"Are galaxies, stars, planets, and ultimately life in violation of the second law of thermodynamics?","text":"How can we think about entropy in these situations? To my knowledge all of these structures are born out of gravitational interaction. However, it would seem that the formation of these more organized structures conceptually violates the second law of thermodynamics. Is this dilemma arisen from my flawed understanding of thermodynamics or are these structures actually in a way defying an increase in entropy?"}
{"id":"113585","title":"Zitterbewegung for massless particle","text":"Is it possible for a massless particle to undergo zitterbewegung? In massive Dirac theory the Zitterbewegung frequency comes out to be $2mc^2\/\\hbar$. It looks like the effect will vanish for a massless particle. But the effect is due to the interference between particle and antiparticle state which is also there for a massless particle. So what would be the outcome?"}
{"id":"57672","title":"Force inversely proportional to the squared distance","text":"**Newton's law of universal gravitation:** \"Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and **inversely proportional to the square of the distance** between them.\" **Coulomb's law:** \"The magnitude of the Electrostatics force of interaction between two point charges is directly proportional to the scalar multiplication of the magnitudes of charges and **inversely proportional to the square of the distances** between them.\" How did Sir Isaac Newton and Sir Charles Augustine De Coulomb come to know that the force, gravitational or coulomb's, is _inversely proportional to the square of the distance between two point masses or charges_ , why didn't they just say that _the force is inversely proportional to the distance of two bodies from each other or two charges_? There must have been something that made them formulate these **inverse-square laws**."}
{"id":"105589","title":"Rotational Velocity and Rotational Frequency","text":"What is the difference between rotational velocity & rotational frequency? Their units seem to be the same, and I've read that one is a 'scalar' and the other is a 'vector,' but how do they differ?"}
{"id":"9069","title":"Bell Test statistics","text":"I have some questions about the statistics of Bell Tests. For convenience I'm going to refer to the test described in this paper: http:\/\/arxiv.org\/abs\/quant-ph\/0205171 They show a coincidence rate of 300 counts with polarizers aligned. I'm wondering how many counts there would be if you took out both polarizers altogether, and how many of those would be coincidence counts. I'm interested in the ideal case of 100% detector efficiency and 100% downcoversion efficiency, but I also wonder how close the ideal case can be approached in practise."}
{"id":"112411","title":"Ten-ping bowling: Can a ping pong ball knock over a bowling pin?","text":"In this video we see a rather unorthodox bowling technique on display. The gentleman appears to knock over all of the pins using only a ping pong ball. It's a fake, of course: you could never knock over a bowling pin with a ping pong ball. Or could you? How fast would a ping pong ball need to travel in order to knock over a heavy object like a bowling pin? Naturally, you are free to make any simplifying assumptions and order-of-magnitude estimates you like."}
{"id":"122289","title":"\"Vectors\" (i.e. 1-tensors) their definition and motivation for relativity","text":"I'm reading Einstein Gravity in a Nutshell (by Zee) and here he defines a vector as an object which is invariant under coordinate representation; concretely, if in one coordinate representation, $V$, $p_V=(p^1,p^2)$ then when we transform it via a rotation $p_W = R(\\theta)p_V$ then we do not violate any physical laws. To motivate the question further, I understand that individual terms after a transformation are not preserved but the law as a whole is: For example if $ma_V = F_V$ then $ma_W=F_W$ and although $F_W \\neq F_V$ it is the case that the scalar (tensor) is preserved $$ma_W-F_w=0=ma_V-F_V$$ The question that motivated this post is the following: Prove that if $p_V$ is a vector then $p'=(ap^1,bp^2)$ cannot be a vector unless $a \\equiv b$. This seems easy enough to show $$Rp'= (p_1a\\cos(\\theta)-p_2b\\sin(\\theta), p_1 a\\sin(\\theta) + p_2b\\cos(\\theta))^t$$ And clearly, we cannot just factor out the $a$ and $b$ unless $a\\equiv b$. But what does this really mean? The converse must be true, namely if $p'$ is a vector then surely $p_V$ cannot be a vector. I suspect if I keep reading there will be ample examples and it will click better (for example the force-mass-acceleration equation makes sense in how it transforms). I suppose my confusion rests in the idea that \"well of course a vector is just a tuple and if rotate it that's another tuple and that old-new tuple pair was prescribed by a rotation so of course all tuples are vectors!\" EDIT: Here's another example to make the point (this is an actual exercise): Suppose we are given two vectors ${p}$ and $q$ in ordinary 3-dimensional space. Consider this array of three numbers: $(p^2q^3,p^3q^1,p^1q^2)^t$. Prove that is not a vector, even though it looks like a vector. (Check how it transforms under rotation!) In contrast, $(p^2q^3-p^3q^2, p^3q^1-p^1q^3,p^1q^2-p^2q^1)^t$ does transform like a vector (so it's a vector). It is in fact the vector cross product $p \\times q$."}
{"id":"62678","title":"Contraction of the metric tensor","text":"This is perhaps a simple tensor calculus problem -- but I just can't see why... I have notes (in GR) that contains a proof of the statement > _In space of constant sectional curvature, $K$ is independent of position._ Here > _$$R_{abcd}\\equiv K(x)(g_{bd}g_{ac}-g_{ad}g_{bc})$$ where $R_{abcd}$ is the > Riemann curvature tensor and $g_{ab}$ is the metric of the spacetime._ The proof goes like this: > _Contract the defining equation with $g^{ac}$, giving $$R_{bd}=3Kg_{bd}.$$ > and so on._ Problem is I don't understand why the contraction gives $$R_{bd}=3Kg_{bd}.$$ I can see the first term gives $$g^{ac}g_{bd}g_{ac}=4g_{bd}$$ since it's 4D spacetime. But as far as I can tell, the second term gives $g^{ac}g_{ad}g_{bc}=\\delta_{bd}$ which is not necessarily $g_{bd}$. Where have I gone wrong?"}
{"id":"71876","title":"Electric field of uniformly charged spherical cap?","text":"I was wondering what the electric field of a uniformly charged spherical cap is? Thereby I am referring to a spherical shell that was sliced into two pieces and we are only looking at one part of it. So in spherical coordinates it would mean that you would have a shell with radius $R$, it contains a full revolution of $2\\pi$, but the polar angle does not go from $0$ to $\\pi$ but rather from $0$ to some $\\theta \\in (0,\\pi)$. And now I am looking for an equation that gives me the electric field for a given charge $Q$ on the shell. It would be sufficient if somebody could give me the integral, that I have to evaluate!"}
{"id":"108762","title":"Velocity required for orbit of satellite","text":"I am very confused about orbital velocity for satellite launched from earth. It is states that: $$ v=\\sqrt{\\frac{GM}{r}} $$ Some says that it is equal to 8000 m\/s and some say 3.1 km\/s. Please remove my doubt."}
{"id":"62671","title":"Heisenberg's uncertainty and $0 K$ temperature","text":"when a body is subjected to $0 K$ temperature, it becomes rigid. hence if we see in terms of quantum the lattice vibration decreases, resulting in no change in the direction of the Random velocity, then can we increase the degree of probability of finding of an electron in an given sys"}
{"id":"102056","title":"Are particles solid?","text":"My intuitive understanding is that although atoms are mostly empty space, the nucleus is solid. Thus a neutron or a neutrino can collide with it and do things like deflect or cause a nuclear reaction. But does this intuition reflect the underlying reality? Since we usually think of a solid in reference to substances comprised of atoms, how does the concept of solid extend into the subatomic realm? Thanks"}
{"id":"78610","title":"Why do comets have tails?","text":"There is no atmosphere for comets to get hot and burn and show tails but they still have tails. Why do they? Edit: Isn't the answer \"acceleration\"? ![enter image description here](http:\/\/i.stack.imgur.com\/bdR0F.gif)"}
{"id":"31242","title":"What does the discovery of the Higgs Boson mean for physics?","text":"Will this unite some theories, or cause some other change in physics, and perhaps our undertanding of the universe?"}
{"id":"31246","title":"What does the discovery of Higgs boson mean for steady state theory?","text":"The relationship between the steady state theory and the notion of Higgs boson is not clear to me. What does the discovery of Higgs boson mean for steady state theory? Or are the two ideas purely orthogonal?"}
{"id":"3151","title":"What's the problem with solar energy?","text":"What \"breakthrough\" from a theoretical point of view is needed for solar energy to become feasible energy alternative?"}
{"id":"3157","title":"Photon absorption probability for a given molecule in gas phase","text":"So I'm pretty sure I'm approaching this problem in the wrong way and I need some guidance (my first hint is that I think I'm thinking about a quantum mechanical problem too classically) Suppose there is an isolated molecule in the gas phase with an average cross- sectional area to be exposed to radiation of $A$. (For my specific problem, the molecule is trapped in a superfluid Helium droplet, but I think the calculation should be roughly the same). If the radiation source has a flux $f$ (in units of photons\/second\/square area\/0.1% BW) at energy $E$, what is the probability of the molecule absorbing a photon within a given interaction time $t$ if the absorption probability at a given energy is $P(E)$? This is pretty easy to calculate if I treat the whole problem classically, i.e. like a ball and a target model. For some reason, though, I get numbers that seem to be way too low if I do this. I know it has to be more complicated than that, since light is a wave also. What am I missing? I understand that transition probabilities are related to wavefunction overlap, etc. Also, I should note that the radiation in my specific problem is in the hard x-ray region, though I don't think that should change the answer. Thanks in advance for your help."}
{"id":"129259","title":"Quantum computing and ambiguity","text":"I do a bit of hobby programming and I often search the internet for little oddities that are fun to ponder over. I have read a few passages that try to explain quantum computing to the layman like myself. I have read of the Qubit, the more 'power' version of the bit, and its bad habit of being in superposition. This, to me, sounds as if it sits halfway between 1 and 0. So, I reason that one can create a qu-binary number with these; something resembling a ternary number, made from 0's, 1's and 1\/2's (or Q's). I have read that a quantum computer has more 'power' when it comes to computation because one qu-value is a possibility between at most n^2 regular values in n bits. I have constructed a little problem with this value when you try to store a specific set of regular values in a qu-value. Imagine a value is a superposition between 2 and 3. In qu-binary, I would write \"10 or 11 -> 1Q\", as the last bit is \"both\". OK, so this works. But what about real values 2, 3 and 4 in superposition? in my ternary notation \"QQQ\" is potentially any of the possibilities 0 through 7, and so actually represents a whole lot more values than I want!? My question is, how does it really work? Am I thinking about it all wrong? Because this is how the whole subject of Quantum computing looks like from the outside. Or is this an example of quantum computing's non-determinism? I assume all bits are completely isolated from one another and have no qu- knowledge of any other. Maybe something obscure like quantum gates sharing information between bits could explain the problem; or if the bits represent continuous probabilities. I don't know. Could someone explain it for me?"}
{"id":"106771","title":"Ice cube on a scale","text":"Lets put an ice cube on a fairly precise scale and take readings over a period of several hours. The ice cube is situated in a saucer so that the water is retained after melting. **What will the recorded curve look like?** Suppose ambients conditions: 25°C, RH40% and 1013 mbar."}
{"id":"66283","title":"Field At Magnetic Dipole","text":"Suppose I have a pure magnetic dipole $\\mathbf{\\vec m} = m\\hat z$ located at the origin. What is the magnitude of the field $|\\vec B|$ as $r\\to 0$? In other words, what is $\\lim_{r\\to 0}\\frac{\\hat{r}\\cdot \\vec{p}}{4\\pi\\varepsilon_0r^2}$? Is it just zero? $\\infty$? Do I have to use some sort of quadripole term?"}
{"id":"23082","title":"What is the physical or mathematical meaning of the Gibbs-Duhem equation?","text":"The Gibbs-Duhem equation states $$0~=~SdT-VdP+\\sum(N_i d\\mu_i),$$ where $\\mu$ is the chemical potential. Does it have any mathematical (about intensive parameters) or physical meaning?"}
{"id":"23083","title":"Why is the Higgs boson spin 0?","text":"Why is the Higgs boson spin 0? Detailed equation-form answers would be great, but if possible, some explanation of the original logic behind this feature of the Higgs mechanism (e.g., \"to provide particles with mass without altering their spin states\") would also be appreciated."}
{"id":"23086","title":"Moment of inertia of a coin","text":"I have a a coin infinitely thin, rotating along the diameter. How to derive the formula for it's moment of inertia passing through the diameter. I was suggested to use the surface density and infinitely small part of the surface area, equidistant from the axis of rotation (marked as $dS$ on the picture). ![](http:\/\/i.stack.imgur.com\/q6KEX.jpg) I've already figured out that: $$I~=~\\int r^2dm~=~\\int \\rho r^2 dS. $$ And now I'm stuck. Any help would be much appreciated. Greg edit: My coin is infinitely thin, so it's two-dimensional object (only $X$ any $Y$ axis). So let's assume that my rotating axis is equal to Y axis. So I have to integrate from $R$ to $-R$ on the $X$ axis. And every $dS$ will have different surface area. But I know, that total area is $S=\\pi r^2$. From Pythagorean theorem I know, that $ r^2 + h^2 = R^2 $. And my integrate is $$I ~=~ \\int\\limits_{-R}^{R} 2\\rho r^2 \\sqrt {R^2 - r^2} dr $$ ~~But now I'm confused how to solve that - every time I get different solution than inWolfram Alpha calculator ~~ Ok, I've solved that - final answer is $ I = \\frac {1}{4}MR^2 $"}
{"id":"109494","title":"Could an orbiting mass generate power?","text":"If an heavy object (e.g. 10 tons) orbiting around Earth at 370 miles high, is connected with a cable back to Earth, we assumed either Earth is going to pull the mass or vice versa (or it will fall back to Earth). Assuming correct pressure\/release\/pull can be applied from the ground to prevent the fall, or it naturally starts spinning around in a circular motion - what do you think about its power generation capability? This was a fun chat between friends, so just wanted to run it by."}
{"id":"130363","title":"Discrepancy ( or Confusion ) in the mass of photon","text":"$E$ = $mc^2$ And also $E$ = $hf$ (f - frequency) And hence Einstein said $m$ = $hf\\over c^2$ And so photons have mass But later he also said $M$ = $M_0\\over \\sqrt {1-v^2\/c^2}$ Where if we put $v = c$ we get $M = M_0\/0 \\leftrightarrow M=\\infty$ And so photon travelling at the speed of light ($c$) have undefined mass (or $\\infty$) And so in one equation it is said photons have mass and in one it is said they have $\\infty$ mass. So do i have a problem in understanding or is there really any discrepancy ?"}
{"id":"25869","title":"Stellar evolution simulation engine or software","text":"Is there any general purpose stellar evolution simulation engine or software? Something to throw in properties of the star and to watch how (and why) they change along the timeline - with or without visualization (but preferably with). The best of what I have found on the net is this site: http:\/\/outreach.atnf.csiro.au\/education\/senior\/astrophysics\/stellarevolution_links.html however, there is no good simulation software behind the links there (or at least I don't see one)."}
{"id":"67484","title":"Bose gas with $T = 0$ and $\\mu < 0$","text":"Is it possible to have a Bose gas with $T = 0$ and $\\mu < 0$ ? I think that there is a problem, because all the states $k$ are such as $$\\langle n_k \\rangle = \\dfrac{1}{e^{\\beta \\\\{\\epsilon_k - \\mu\\\\}}-1} = 0$$"}
{"id":"74960","title":"Why aren't there any arrays of visible light telescope?","text":"There have been many radio telescope arrays which are capable of resolving stars significantly fainter than can be resolved with a single \"small\" radio telescope. Why don't they do the same with light telescopes?"}
{"id":"66757","title":"Pion production in proton-proton collision","text":"Why is $\\pi^0$ created in the high-energy collision $p+p\\to p+p+\\pi^0$?"}
{"id":"55366","title":"Why does electromagnetic waves travel slower in well isolated conductor?","text":"Wikipedia writes, that > Propagation speed is affected by insulation, so that in an unshielded copper > conductor ranges 95 to 97% that of the speed of light, while in a typical > coaxial cable it is about 66% of the speed of light Why does insulation affect this speed?"}
{"id":"6624","title":"Are the solutions in radicals of cubic and quartic of any use in physics?","text":"We all know that there are analytic formulae to solve quadratic, cubic and quartic polynomial equations. But it seems to me that the only solution that widely used is physics is the solution of quadratic we all learnt at school. Is it true that when physicist faces quartic or even cubic equation -- he turns to numerical methods to solve them? Have anyone seen the analytic analysis of some real problems in physics, where these formulae were used?"}
{"id":"131138","title":"Deformation of light-cone","text":"In the paper _The geometry of free fall and light propagation_ by Ehlers and his colleagues ( _Gen. Relativ. Gravit._ **44** no. 6, pp. 1587–1609 (2012)), when the authors introduce the differentiable structure on page 70, they claim: > It should be realized that the representation of light in special relativity > by means of ordinary cones rather than by hypersurfaces of hour-glass shape > depends partly on a particular choice of differential structure. The question is: What do they mean by saying **particular choice of differential structure**? In what condition may the light cone be deformed and become like hour-glass as shown in this figure? Thank you in advance for any help. ![enter image description here](http:\/\/i.stack.imgur.com\/XoQWY.jpg)"}
{"id":"92075","title":"What are differences of different telecommunication wavebands?","text":"I'm just wondering what are the differences (advantage and disadvantage) of different telecommunication bands ? For example:   * Why does satellite TV use Ku band?    * Is it true the rule is that greater band frequencies (10 ghz vs 1 ghz vs 800 mhz) can transport more data?   * Why mobile phone bands use (for example) 800 MHz or 1800 MHz?    * Is there any other \"rules\" related to band type and distance covered ? Yes, i know, i'm a little confused. I just would like to understand the differences between \"radio frequencies\", particularly their advantages and disadvantages. Are there references that explain those types of arguments?"}
{"id":"100344","title":"Ampere's Law and Circulation?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/V2GdM.png) The net circulation should be zero if the net current is zero. However, I don't see how the circulation in the Amperian loop above can be zero. All the lines interact the loop only once and all their circulation contributions are positive or negative depending on the assumed direction of integration. Could someone explain how the net circulation would be zero? Thanks!"}
{"id":"92070","title":"Electrode potential","text":"If I connect a zinc half cell (Left hand side) and a copper half cell (right hand side), I get an E cell value of 1.1V taken from the voltmeter. If I replace the voltmeter with a light bulb, I will get light and heat energy of the bulb. My question: Will the voltage across the cell remain the same? I mean, the potential difference arises due to the difference in the amount of charge build up on the electrodes... so an electron flows from the zinc electrode to the copper electrode, will the voltage across the bulb decrease?"}
{"id":"92079","title":"Why can't matter pass the speed of the light?","text":"I have the following questions.   1. Why can't matter pass the speed of the light?   2. What happens to an object when it comes close to the speed of the light?   3. What makes the object unable to surpass this speed limit? And finally, I need to know whether this speed limit has been experimentally proven?"}
{"id":"60505","title":"Whats the anti-torque mechanism in horizontal take-off aircraft?","text":"In most helicopters there is the anti-torque tail rotor to prevent the body from spinning in the opposite direction to the main rotor. What's the equivalent mechanism in horizontal takeoff single engine propeller, and jet aircrafts, where the air or the jet coming out back from the turbine or propeller is spinning and will cause such aircraft to roll?"}
{"id":"110936","title":"An equivalent for the Bernoulli equation for viscous liquids","text":"I was wondering whether there is any equation for viscous liquids (probably derived from energy and mass conservation principles) relating the pressure, rate of volume flow, area of cross section and height of the liquid (or any other parameters if required). NOTE : I am aware of Poiseuille's law, but that is not what I am looking for as it relates the pressure _gradient_ to the rate of flow."}
{"id":"22067","title":"Why are there 4 Dimensions and 4 Fundamental Forces?","text":"Is it a coincidence that there are four fundamental forces and four spacetime dimensions ? Does a universe with three spacetime dimension contain four fundamental forces? Can magnetism be realized in three dimensional spacetime? Edit 1: Thanks alot for the answer. I'm a first year undergrad student so this may seem naive. If there is a symmetry that mixes spacetime symmetries and internal symmetries of the theory (e.g. gauge symmetry) then there might be ways that spacetime geometry have effects on the particle spectrum and interactions of the theory so you can describe things in geometrical manner . The mechanisms that lead to the structure of spacetime may explain why there are four forces."}
{"id":"31935","title":"geometry inside the event horizon","text":"I'm trying to understand intuitively the geometry as it would look to an observer entering the event horizon of a schwarszchild black hole. I would appreciate any insights or corrections to the above. inmediately after your enter the event horizon, if you look back and try to reach again for the horizon, it will seem to be expanding away quicker than the speed of light. Near this region, the apparent shape of the horizon is a sphere expanding away, and we are inside the sphere Near the singularity, we really don't know what happens. I've heard that spaghettification is not a necessarily occurrence, since the metric field diagonal components are shrinking as the curvature grows, so it could very well be the case that a infinite length hyper-cylinder $S^3 \\times R^+$ of constant _physical_ radius is being conformally mapped to the $S^3 - \\\\{0\\\\}$ region around the singularity, or that in general a region around the singularity can be mapped to anything in the other end, which is basically because the degrees of freedom of curvature and stress-energy in our end of the spacetime cannot really predict what sort of topology endpoint will connect to the matter in the other end. Since the metric components are tending to zero at the singularity, this argument sounds pretty interesting, since it would seem to imply that observers will \"shrink\" relative to kruskal coordinates, because the local physics would always be that physical observers will stay fixed relative to their local metric, since the metric is covariantly constant!. However, i'm not expert on how to describe the asymptotic physics in the neighbourhood of the schwarszchild singularity. (which is why i'm asking on this site, after all!). **Question:** _does this argument hold any water?_"}
{"id":"77707","title":"Waveguides and Maxwell's Equations","text":"**Background** In many textbooks on electromagnetism, the sequence in which topics are introduced are generally as follows -- electrostatics, magnetostatics, electrodynamics, wave propagation in free space, and wave propagation in confined space, in that order. As an example, consider Griffiths' 3rd edition of Introduction to Electrodynamics. Similarly, consider Orfanidis' Electromagnetic Waves and Antennas. By using the discussion on wave propagation in free space as a launchpad, the discussion on wave propagation in a confined space usually introduces some additional boundary conditions that the solutions to Maxwell's equations for wave propagation in free space did not have. If a proposed solution can be plugged into Maxwell's equations and not produce any contradictions, then they are valid. The goal is then to maintain the monochromatic wave solutions that worked for free space wave propagation with some \"patches\" to account for the fact that the propagating EM energy will be confined by the hollow waveguide. The reason monochromatic (one-wavelength) wave solutions are desirable is that, by superposition, any signal can be expressed as a sum of one or more monochromatic waves, and so any signal's behavior in a hollow waveguide can be understood. One of these \"patches\" is the boundary condition that $E_{\\parallel}=0$. This is because the net electric field in an ideal conductor, such as the material that makes up the confining hollow waveguide, must be $0$ because any externally applied electric field will influence electrons to rush to counter the externally applied electric field. If there is no net electric field (meaning $\\vec{E}=\\vec{0}$), then there is no net electric field in the tangential direction ($E_{\\parallel}=0$) along and inside the metal. If there is no net electric field along the metal, then, by Faraday's law, there must be no net changing magnetic field $\\frac{\\partial \\vec{H}}{\\partial t}$. This means that each of the components $\\frac{\\partial H_x}{\\partial t}$, $\\frac{\\partial H_y}{\\partial t}$, and $\\frac{\\partial H_z}{\\partial t}$ must all be $0$ too. If there is no initial magnetic field (meaning $\\vec{H}=\\vec{0}$, then there must be no magnetic field at any later point. Therefore $\\vec{H}=\\vec{0}$ and $H_{\\perp}=0$. So these are the two additional boundary conditions that must be considered. An implicit assumption is that the material on the interior of the waveguide is isotropic, homogeneous, and linear. This means that $\\vec{B}=\\mu\\vec{H}$ and therefore $\\vec{B}=\\vec{0}$ and $B_{\\perp}=0$ at the boundary of the metal + hollow interior interface as well. **Problem** Frequently, the monochromatic wave solution to Maxwell's equations given the boundary conditions of a hollow waveguide are presented as $$ \\vec{F}\\left(x,y,z,t\\right)=\\left[\\text{spatial component}\\right]\\left[\\text{temporal component}\\right] $$ For example from Orfanidis, $$ \\vec{E}\\left(x,y,z,t\\right)=\\vec{E}\\left(x,y\\right)e^{j\\omega t - jkz} $$ and from Griffiths, $$ \\vec{E}\\left(x,y,z,t\\right)=\\widetilde{\\vec{E}}_0\\left(x,y\\right)e^{i\\left(kz - \\omega t\\right)} $$ Further, $\\vec{E}$ is a 3-dimensional vector with components $E_x\\left(x,y\\right)$, $E_y\\left(x,y\\right)$, and $E_z\\left(x,y\\right)$. **Bottom line** : Why is the spatial component only a function of $x$ and $y$? Further, why does spatial component have $x$, $y$, $z$ components [e.g. $E_x$, $E_y$, and $E_z$] (where $E_x$, $E_y$, and $E_z$ are each functions of only $x$ and $y$)?"}
{"id":"77700","title":"Quantization surface in QFT","text":"![enter image description here](http:\/\/i.stack.imgur.com\/RKL4O.png) What does the Quantization Surface mean here? Reference: H. Latal W. Schweiger (Eds.) - Methods of Quantization"}
{"id":"31938","title":"Please explain me how the Higgs boson gives mass to other particles, more detail?","text":"As we know, the Higgs boson gives mass to other particles. But here is onething which is not clear for me. I mean, I do not understand how the Higgs boson gives mass to other particles? Does anyone could explain me more explicitly how procedure goes that the Higgs boson gives mass to other particles."}
{"id":"111992","title":"Neutrino mass difference and mixing angles","text":"What are the most recent values of the mixing angles associated with the 3 neutrinos flavors and what are their mass differences? Have they been measured in the first place?"}
{"id":"87513","title":"References requested on departments with recitation sessions","text":"What physics departments offer recitation sessions for their physics courses on the undergraduate level? If there is a site which has such info it will be greatly appreciated."}
{"id":"75709","title":"View from a helicopter rotor: why is the horizon distorted?","text":"This video (\"rotor panorama\") was captured by a camera attached to the rotor head of a radio-controlled helicopter, with the frame rate set to the rotor's frequency. During a long segment of the video, the horizon looks distorted: ![enter image description here](http:\/\/i.stack.imgur.com\/Bl6c0.jpg) What causes this distortion?"}
{"id":"100960","title":"Physical Interpretation of the Bloch vector","text":"In the expression of the density matrix of a (Electron-Spin) Qubit $$ \\rho=\\frac{1}{2}(I + x \\sigma_x + y \\sigma_y + z \\sigma_z) $$ where $\\tau=(x,y,z)$ is unit vector in the Bloch sphere, which is the physical Interpretation of $\\tau$? Can it be interpreted as the direction of spin axis of the electron?"}
{"id":"130268","title":"What are the main algorithms the LHC particle detectors use to reconstruct decay pathways?","text":"I am just starting to look into how we understand the data from particle collisions. My question is, what are the algorithms or ways that these detectors interpret the data? Are there standard approaches? Or if not, what are some good papers or places to look to get started in learning more about the implementation and\/or details of how this works? So far I haven't dug into any textbooks, but many articles on the web and this was somewhat helpful in pointing to where to look: http:\/\/arstechnica.com\/science\/2010\/03\/all-about-particle-smashers-part-ii\/ So from my understanding so far, there are a few different LCH \"experiments\", which are physical structures that are optimized to focus on specific aspects of data from a collision event. The detector measures all kinds of particle emissions and changes in electrical fields, and then seems to try to backtrack and figure out all the emission\/decay events that might have taken place in that split second. From my understanding so far, basically the computer programs used to compute these possible \"decay pathways\" must be using some standard algorithms or something, and must have built into them all possible particle emission pathways (like all possible Feynman diagrams if there is such a thing). Are there any good resources or standard algorithms\/approaches to understanding how particle detectors analyze their data?"}
{"id":"93303","title":"Gravity change as a function of time only","text":"Is it possible to find how gravity changes with time? That is, height is not part of the equation. For example, $g$ can be expressed as a function of height that does not involve time with this relation: $g_i=g_0 (r \/(r+h))^2$ where $g_i$ is $g$ at any height and $g_0$ is on the Earth's surface. I also found through the above relation that $g_i = g_{i-1} ((r+h_{i-1})\/(r+h_i))^2$ By the same token, there should be a relation between $g_0, g_i, t$ that does not involve distances. An object in free fall will be going through changes in $g$ as time progresses. Is it possible to create such an expression?"}
{"id":"95292","title":"How to measure distances to stars by means of spectroscopic parallaxes?","text":"How to measure distances to stars by means of spectroscopic parallaxes on practice? What is the accuracy of measuring distances using this method compared with distances based on HIPPARCOS trigonometric parallaxes?"}
{"id":"122631","title":"Why don't bubble lights explode?","text":"As a child, I loved to watch the bubble lights at Christmas time. The colorful bubbling was hypnotic. However, there was always something about them that bothered me. With all of that boiling happening within the glass vial, why didn't the entire thing explode? I knew how explosive steam boiling could be, why was the bubble light able to withstand the constant boiling, and pressure changes that were going on inside it? Also, how was the fluid able to boil within an enclosed space (where the volume could not change?)"}
{"id":"123","title":"How does one measure the mass of a galaxy? And other such large quantities","text":"Trained as a pure mathematician, I see claims about the mass of a galaxy and other such huge measurements that are arrived at experimentally, and I just have to scratch my head. I know this is a bit of a vague question--but does anyone have a good resource for something like \"measurement in astrophysics\" or an introductory history of how astrophysicists figured out how to make measurements of this kind?"}
{"id":"70280","title":"The Observance of Electromagnetic Phenomena","text":"As far as I've seen, E&M waves are sinusoidal. Our eyes observe these patterns and give us colors, but in a situation like such: Why do electrons emit radiation? There is still a wave, or at least a pulse, but it's not necessarily sinusoidal, and I'm not sure what 'wavelength' would be used. Nonetheless, radiation produced in this way does have a wavelength-it is used in synchrotrons. So what am I missing?"}
{"id":"45825","title":"What is the energy of interaction between a point charge and an infinite cylinder?","text":"I don't remember enough from my electromagnetism course and I can't find any simple, full example on this subject. I know, that I can consider the cylinder as a wire with the same charge density (when outside of the cylinder), so I consider the wire case with the same charge density, $\\lambda$. I set the $y$ direction to pass through the wire, and the $x$ direction to pass through the point charge and perpendicular to the wire. Everything is in the $z=0$ plane. Hence, the particle is at $x=r$ position. From Gauss law, I learn that the field at a point at distance $r$ from the wire I have a field which scales like $\\lambda\/r$. To find the potential, I need to solve the integral $V = -\\int_b^r E dl = - k \\int_b^r \\lambda\/x dx$ Where $k$ holds the constants and $b$ being a point where the potential is 0. The solution of this integral is $V = - k \\lambda \\log (1\/r) + C$ When $b=1$ the potential is 0 so the potential is $V = k \\lambda \\log (1\/r)$ and the energy is $U = k q \\lambda \\log (1\/r)$ Here are my questions:   1. Is any of this true?   2. I tried to derive this by starting with coulomb law and calculating the energy for a segment of the wire, I get an answer which scales like $1\/r$, can you derive the answer for this?"}
{"id":"113841","title":"UCM unidentified force","text":"> _A train rounds a curve of radius 235 m (turning right). The train track is > flat, and the train is traveling at a constant speed. A lamp in the train > makes an angle to the vertical of 17.5 degrees. How do you calculate the > speed of the train?_ The first thing I need is to draw a free-body diagram of the lamp.![free-body diagram of lamp](http:\/\/i.stack.imgur.com\/SQTWH.png) Sorry about the bad drawing. What confuses me is what force drives the lamp to the left. It seems to have something to do with inertia, I think it is probably equal to the centripetal force that the train exerts on anything in it. Can someone identify this force and explain the problem?"}
{"id":"123586","title":"How does the Alcubierre drive warp space in the vicinity of it's destination?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/kWreF.gif) I don't understand how this could work. So lets say the drive is set to go 30,000,000 meters in a straight line to get to a distant planet. Now lets say the ship size is 20 meters. Now let's make the assumption that the warp field in the photo is 30 meters (10 meters bigger than the ship). So the warp bubble is applied. Now the 30 meters of space the ship occupied is cut. But it only could apply the warp bubble to this confined region which means 30 meters are cut instead of 30 million and now 29,999,970 meters are left and the ship is not even close to the planet it wants to go to (barely moves). Are there any solutions to this problem?"}
{"id":"70866","title":"What provides the centrifugal force for planets orbiting a star?","text":"this is a question I had when I was a kid. I'm a bit ashamed because I think I am missing out on something very obvious since I have the same question despite almost being an engineer now! From Newtonian physics, I understand how although the gravitational force pulls the planet towards a star, and the planet 'falls' towards the star, due the angular momentum, it also moves laterally. In case of a planetary orbit, this is just enough to keep it moving around it in an elliptical orbit. (With speed and radius being such that angular momentum and energy is conserved) But let's consider a single planet and star system where the planet moves around the star in a circular path for simplicity. If gravitational force provides centripetal force, what can account for the centrifugal force? I have a feeling it is related to some inaccurate view I have, of centrifugal force. The xkcd comic and wikipedia article talk about two concepts: fictitious and reactive centrifugal force. But it would be safe to assume the star(EDIT: sorry, I meant to choose a frame of reference such that the centripetal force on the planet has to be cancelled by a centrifugal force on the planet in the direction: star to planet) as a stationary frame of reference for the purpose of this question, right? Which means that it is not a fictitious force, right? Or is it not considered a centrifugal force if you take the star as the reference frame?"}
{"id":"123588","title":"Why do books write $X(f)$ when they mean actually mean $\\lvert X(f)\\rvert$?","text":"All books write $X(f)$ in plots - the Fourier transform of $x(t)$ - when they actually mean $\\lvert X(f)\\rvert$, without even mentioning in passing that they are dropping the mod sign. And also they consider only the mod of FT in the analysis, completely ignoring the phase term - especially communications analysis like AM and FM. Why? It is well known that in image processing phase also place an important role? Is there any proof or valid reason that phase is unimportant in communication engineering?"}
{"id":"18909","title":"Simple applications of group theory which can be understood by a senior undergrad","text":"I am looking for references (books or web links) which have \"simple\" examples on the use of group theory in physics or science in general. I have looked at many books on the subject unfortunately they usually require extensive technical coverage of the basics, i.e. the 1st 100 pages or something, to be able to start discussing applications. I believe that there is an easy way to explain anything (it's just hard to find it)."}
{"id":"18901","title":"What will happen if we use a speed greater than light speed and find a body'motion and energy relative to it?","text":"In Einstein's papers, he used light speed as a reference speed. What if we use a greater finite speed and do the same calculations. Won't this greater speed then be the limit."}
{"id":"30652","title":"what is the 2D gravity potential?","text":"In 3D, I can calculate the total force due to gravity acting on a point on the surface of the unit sphere of constant density, where I choose units so that all physical constants (as well as the density of the sphere) is 1: $$F = 4\\int_{-1}^1 \\int_{-\\sqrt{1-x^2}}^{\\sqrt{1-x^2}} \\int_{-\\sqrt{1-x^2-y^2}}^{\\sqrt{1-x^2-y^2}} \\tfrac{x+1}{\\left[(x+1)^2+y^2+z^2\\right]^{3\/2}} dz\\, dy\\, dx = \\frac{16\\pi}{3}.$$ This force agrees with what we get if we treat the sphere as a point particle at the sphere's center of gravity with lumped mass $\\frac{4\\pi}{3}$. So far so good. But now if I try to calculate the force of gravity of a point on the boundary of the unit disk in 2D, I get infinity: $$F = 2\\int_{-1}^1 \\int_{-\\sqrt{1-x^2}}^{\\sqrt{1-x^2}} \\tfrac{x+1}{\\left[(x+1)^2+y^2\\right]^{3\/2}} dy\\, dx = \\infty?$$ Intuitively, if I think of the total force as the sum of contributions from nested circular rings around the point of interest, the circumference of the rings scales like $r$, the force density like $\\frac{1}{r^2}$, so the contribution of each ring scales like $\\frac{1}{r}$ which diverges as $r\\to 0$. What is going on here? How is the gravity potential derived? Is it an accident that in 3D, $\\frac{G m_1 m_2}{r}$ is harmonic on the punctured space $\\mathbb{R}^3 \\backslash\\\\{0\\\\}$? Is the \"right\" gravity potential in 2D something like $G m_1 m_2 \\log r$? If so, why, and isn't it a paradox if point masses in 2D orbit according to a different law than coplanar point masses in 3D?"}
{"id":"114496","title":"Which mathematical operation does the right hand rule for current come from?","text":"I am currently wondering about this famous rule: ![picture](http:\/\/ffden-2.phys.uaf.edu\/212_fall2003.web.dir\/Daniel_Lenord\/righthand.jpg) Where does it come from mathematically that when you point with your thumb in the direction of the current, your curved fingers will point in the direction of the $B$-field? In other words, which mathematical operation does it come from? Probably it has to do with some vector product or Stokes theorem application, but I am not quite sure about it."}
{"id":"76424","title":"Is wearing 3D glasses from the cinema as sunglasses hurtful to the eyes?","text":"I heard a few times that using them as sunglasses is hurting the eye since UV light is not filtered, but the pupil is wider than it would be w\/o wearing them because the visible light is dimmed. I have always assumed since atmospheric (i.e. not reflected from flat surfaces) sunlight is unpolarised and since the glasses used in 3D cinema only allow circular polarised light to pass they do indeed filter out a large portion of the intensity of the incoming (unpolarised) light. Does this not work for the UV part of the spectrum or is my initial assumption wrong?"}
{"id":"76425","title":"Expectation value of the stress-energy tensor in 2-D CFT","text":"Due to a previous question, I am confused with the expectation value of the stress-energy tensor in a 2-D conformal field theory. Let's take the example of string theory, to sketch the problem. Defining the action by $S = \\frac{1}{2 \\pi \\alpha'}\\int d^2z \\partial X^\\mu \\bar \\partial X_\\mu$, the (euclidean) expectation value of an operator is : $$\\langle\\mathcal F[X]\\rangle=\\int [dX]~ exp(-S)~F[X]\\tag{0}$$ My questions are : 1) In string theory, what is : $$\\langle \\partial X^\\mu \\partial X_\\mu\\rangle \\tag{1}$$ 2) In string theory, what is : $$\\langle :\\partial X^\\mu \\partial X_\\mu:\\rangle \\tag{2}$$ 3) In a general 2-D conformal field theory, what is : $$\\langle T^{\\alpha \\beta}\\rangle \\tag{3}$$"}
{"id":"7654","title":"Brachistochrone problem in general relativity","text":"This question Brachistochrone Problem for Inhomogeneous Potential has the obvious extension. Namely the same question, when gravity is treated according to general relativity. To make it specific let's consider the case of Schwarzschild metric. Given to points, outside the event horizon, what is the trajectory (or the worldline) of a test body, which goes from the first point to the second for least time. All considered from the point of view of a distant observer at rest (with respect to the star\/back hole). Of course other metrics or the general case are also interesting. I couldn't find much by searching on the net. There is one article with the tittle suggesting relevance (actually more but this one seems very relevant), but it is over 60 pages of technicalities. So it would be nice to see a shorter answer."}
{"id":"133183","title":"Glauber's Formula","text":"In the Cohen-Tannoudji Quantum Physics book, Complement BII, says: [...] two operators $A$ and $B$ with both commute with their commutator. An argument modeled on the preceding one shows that, if we have: \\begin{align} [A,C]=[B,C]=0 \\end{align} with $C=[A,B]$, then: \\begin{align} [A,F(B)]=[A,B]F'(B) \\end{align} Then this last property, is used to proof Glauber's Formula. \\begin{align*} e^Ae^B=e^{A+B}e^{\\frac{1}{2}[A,B]} \\end{align*} I understand this proof, but I couldn't find a way to demonstrate, \\begin{align} [A,C]=[B,C]=0 \\end{align} with $C=[A,B]$, then: \\begin{align} [A,F(B)]=[A,B]F'(B) \\end{align} I would like to know to do this, so I can understand better the Glauber's Formula proof."}
{"id":"78314","title":"How to demonstrate that there are just 14 types of Bravais lattice?","text":"Is it possible to demonstrate that there are just 14 types of Bravais lattice without the knowledge of group theory?"}
{"id":"93693","title":"Expansion of $L(v^2 + 2\\vec{v}\\cdot\\vec{\\epsilon}+\\epsilon^2)$","text":"How can I find the expansion of the Lagragian (it it only dependent on $v^2$) $L(v^2 + 2\\vec{v}\\cdot\\vec{\\epsilon}+\\epsilon^2)$ in powers of $\\vec{\\epsilon}$ ? (From L.Landau, E. Lifshitz, Mechanics , page 7)"}
{"id":"122929","title":"Lagrangian depends on second derivative of field","text":"In case of the gauge-fixed Faddeev-Popov Lagrangian: $$ \\mathcal{L}=-\\frac{1}{4}F_{\\mu\\nu}\\,^{a}F^{\\mu\\nu a}+\\bar{\\psi}\\left(i\\gamma^{\\mu}D_{\\mu}-m\\right)\\psi-\\frac{\\xi}{2}B^{a}B^{a}+B^{a}\\partial^{\\mu}A_{\\mu}\\,^{a}+\\bar{c}^{a}\\left(-\\partial^{\\mu}D_{\\mu}\\,^{ac}\\right)c^{c} $$ (for example in Peskin and Schröder equation 16.44) If you expand the last term (for the ghost fields) you get: $$ \\bar{c}^{a}\\left(-\\partial^{\\mu}D_{\\mu}\\,^{ac}\\right)c^{c} = -\\bar{c}^{a}\\partial^{2}c^{a}-gf^{abc}\\bar{c}^{a}\\left(\\partial^{\\mu}A_{\\mu}\\,^{b}\\right)c^{c}-gf^{abc}\\bar{c}^{a}A_{\\mu}\\,^{b}\\partial^{\\mu}c^{c} $$ And so, the Lagrangian has a term proportional to the second derivative of $c^a$. In this case, how does one find the classical equations of motion for the various ghost fields and their adjoints? I found the following equations of motion so far: $$ D_{\\beta}\\,^{dc}F^{\\beta\\sigma}\\,^{c}=-g\\bar{\\psi}\\gamma^{\\sigma}t^{d}\\psi+\\partial^{\\sigma}B^{d}+gf^{dac}\\left(\\partial^{\\sigma}\\bar{c}^{a}\\right)c^{c} = 0 $$ $$ \\sum_{j}\\partial_{\\sigma}\\bar{\\psi}_{\\alpha,\\, j}i\\gamma^{\\sigma}\\,_{ji}-\\sum_{\\beta}\\sum_{j}\\bar{\\psi}_{\\beta,\\, j}\\left(gA_{\\mu}\\,^{a}\\gamma^{\\mu}\\,_{ji}t^{a}\\,_{\\beta\\alpha}-m\\delta_{ji}\\delta_{\\beta\\alpha}\\right)=0 $$ $$ \\left(i\\gamma^{\\mu}D_{\\mu}-m\\right)\\psi=0 $$ $$ B^{b}=\\frac{1}{\\xi}\\partial^{\\mu}A_{\\mu}\\,^{b} $$ $$ \\partial^{\\mu}\\left(D_{\\mu}\\,^{dc}c^{c}\\right)=0 $$ $$ f^{abd}\\left(\\partial_{\\sigma}\\bar{c}^{a}\\right)A^{\\sigma}\\,^{b}=0 $$ But it is the last equation that I suspect is false (I saw the equation $ D_\\mu\\,^{ad} \\partial^\\mu \\bar{c}^d = 0 $ in some exercise sheet (http:\/\/www.itp.phys.ethz.ch\/education\/fs14\/qftII\/Series7-3.pdf Exercise 3) and I also saw the equation $D^\\mu\\,^{ad}\\partial_\\mu B^d = igf^{dbc}(\\partial^\\mu\\bar{c}^b)D_\\mu\\,^{dc} c^c$ which I don't understand how they were derived.) EDIT: Thanks to Qmechanic's answer I was able to derive the correct equations of motions (as noted in the comment to that answer) but I still don't know where to \"obtain\" the last equation I mentioned which connects the auxiliary field with the ghost fields."}
{"id":"14422","title":"Inelastic nucleon-nucleon cross section at LHC energies","text":"I am trying to reproduce the number of participants $N_{part}$ in Pb-Pb Collisions at LHC using a Glauber Monte Carlo simulation, specifically aiming to reproduce the values given in Phys. Rev. Lett. 106, 032301 (2011) by K. Aamodt et al. (the ALICE Collaboration). To do this, I am looking for the inelastic nucleon-nucleon cross section. A paper on the software written for the PHOBOS experiment (available on arXiv here) suggest that it is (at LHC energies) \" _expected to be around $\\sigma_{NN} = 72$ mb (with large uncertainty from the unknown elastic cross section)_ \". I have been using this web-interface provided by the GSI to get an estimate of $N_{part}$. The website suggests to use $60$mb at $\\sqrt{s_{NN}} = 5500 \\text{GeV}$. However, even using the 72mb suggested above, for 0%-5% centrality I only get about $376$ participant nucleons, compared with $N_{part} = 382.8$ for $2.76$ TeV published in the ALICE paper. I looked at the PDG website as well, but since (as hinted above) the plots do not show data for the elastic part of the cross-section, no value is provided for the energy I'm looking at. The fit funciton provided does not seem to give a reasonable value for LHC energies. > Am I right in assuming that we simply don't know the value of $\\sigma_{NN}$ > yet and that it must thus be estimated for such calculations? If that is the > case, is there a common way to do so?"}
{"id":"38083","title":"Eigen modes in a room","text":"In a rectangular room 10*8*5 how can the eigen modes belonging to octave and third octave bands centered at 50Hz be found ? I have the formula to calculate the number of modes but don't have the idea how to find the modes. Do I need to use Matlab for it ?"}
{"id":"38089","title":"Capacitance and Light Bulb And Dielectric Slab","text":"An electric lamp having coil of negligible inductance connected in series with a capacitor and an AC source is glowing with certain brightness. How does the brightness of the lamp change on reducing the (i) capacitance, and (ii) the frequency ? What if the diaelectric slab is introduced between the capacitor."}
{"id":"109789","title":"Virgo SuperCluster and space time","text":"Is the Virgo Supercluster affected at all by the expansion of space? What about the larger filament that Virgo resides in?I don't quite understand how gravity affects space time locally in groups of galaxies, and am wondering if the expansion of space is lumpy. It would be nice to know we are not all being stretched... :)"}
{"id":"131850","title":"How to find the electric field of a charged tube?","text":"Let's say there is a charged tube(cylinder with no top or bottom) with radius $a$, length $l$ and charge $q$ and a point which is collinear with the centre of the charged tube. Anyway, since we can think the tube as infinite charged rings we first should calculate the electric field of a single ring.(the distance between ring and point charge is $r$ and the distance between the centre of the ring and point charge is $d$) $$ dE= (k.dq)\/r^2 $$ where $dq$ is infinitesimal charge. $$ dE_x=(k.dq)\/r^2cos(\\theta) $$ $$ dE_x=(k.d.dq)\/(a^2+d^2)^{(3⁄2)} $$ $$ E_x=∫(k.d.dq)\/(a^2+d^2)^{(3⁄2)} =(k.d)\/(a^2+d^2)^{(3⁄2)}∫dq=kdq\/(a^2+d^2)^{(3⁄2)} $$ Finally, we found the electric field of a single ring. However, I am stuck at how to integrate the equation to find the total electric field of the charged tube. How should I integrate it? P.S: It may seem like the other question I asked but they are different in concept of what the question is."}
{"id":"131851","title":"Speed of light and distance","text":"Our measure of distance (the meter) is defined in terms of how far light in a vacuum travels in a specific time. When light travels through another medium, we say it travels at a different speed. Why not say it travels at the same speed, and the distance in the medium is different that the distance outside? Is it only because the 2nd explanation feels less intuitive than the first, or is there more to it?"}
{"id":"103086","title":"energy of electrons outside an atom","text":"The higher the quantum number(energy levels)m the higher the energy. What does the energy refers to? Kinetic energy, potential energy, or the total mechanic energy?"}
{"id":"88909","title":"Stresses on an inclined plane","text":"# Background This issue has been bothering me for a few days now. It's actually part of my homework, but I've already finished them and in a sense they're not part of the question here. What I would like to know is more of a theoretical question about this formulation. Now my professor's book and many online resources (sample) maintain the same mathematical relation, hence my dilemma. # Problem Consider the following: ![Problem](http:\/\/i.imgur.com\/bTaKei0.png) Under axial loading, the stresses can now be computed for any section. The question posed is what if we make an imaginary section at an _inclined_ , its normal defined by an angle $\\theta$. I will not write every detail here, but its clear that we can use $A$ as a placeholder for the area and we can define $\\sigma_x$ to be $\\sigma_x = \\frac{F}{A}$ and therefore in the diagram below we will use only stress. ![Problem](http:\/\/i.imgur.com\/u4PU86j.png?1) Now, I will refrain working with vector quantities, I will only use scalar relations. If you take a look at this triangle, its easy to see that the area is projected to the section and we can get the relationship: $$ A' = \\frac{A}{\\cos(\\theta)} $$ Now the rest of the equations go by the books: $$ T = \\frac{F}{A'} = \\frac{F}{A}\\,\\cos(\\theta) = \\sigma_x\\,\\cos(\\theta) $$ $$ T_n = T\\,\\cos(\\theta) $$ $$ T_s = T\\,\\sin(\\theta) $$ My dilemma starts here. First of all _notice_ how in each of the above expressions the value for the length of the object is not taken into consideration. Also, we're working with areas and forces hence the length itself is not part of the equation at all even as a _contained_ information. Now that would be all fine, if it were not for the following section(to simplify, I've taken the origin to be at the center of the object). ![Section C](http:\/\/i.imgur.com\/CZYfjZA.png) Now as you can see this section of the object gets very interesting. Clearly the formula for $A'$ does not hold anymore because the area itself is shrinking on the left side. To make things even worse, notice how the area on the right appears. The two areas are clearly linked, in this case, the sum of the two should be equal to the original area - from the fact that the height appearing on the right is equal to the height subtracting from the left. Calculating the angle at which the breaking point is made, that is when the section plane cuts the corner is easily calculated using: $$ \\tan(\\theta_k) = \\frac{l}{h} $$ Where $ \\theta_k $ stands for the critical angle, just so we can refer to it. Now the question that is on my mind is what is the distribution of the stresses on this section. Clearly there is stress on the left, there's stress on the right. Now I am confused as to the distribution of stress at this angle. # My Work Now, I've attempted a solution here. First, I am in a tight spot between assumptions. Clearly the original force $F$ must still be present here. But the force distribution is confusing me at this point. The best idea I could come up with, is the following. I would be grateful for your review. ## FBD ![Problem 3](http:\/\/i.imgur.com\/0DIhQMU.png) I've only written here $F_1$ and $F_2$ for visualization purposes, because personally I don't think that we will need the $F_2$. Knowing that $F_1 = F$ it's obvious that they will cancel each other out, and they will also cancel the moments they create against the newly formed centroid on the trapezoid. Hence I think the only remaining task is to compute the areas of each individual piece and compute stresses. ## Areas Computing the individual areas for the cuts, since we won't need the area at where supposedly $F_2$ was located, we will ignore that for the time being. Though it will come naturally as we try to find the left area. The inclined plane area is the easiest to find, clearly, if we project the length $l$ to the top of the smaller rectangle we can see that using trigonometric relations we can find: $$l\\,'=\\frac{l}{\\sin(\\theta)}$$ Multiply that by the base and you've got the area $A'=b l\\,'$. To find the areas of the left and right sections one must notice that the amount we deviate from the original height we add to the right side. So if we're to project the length like we did last time we get this: ![Area2](http:\/\/i.imgur.com\/9m1MbQ0.png) From the figure above it's clear that the following relation holds true: $$ h = h' + 2\\,h_2 $$ From trig we know $h'=l\\cot(\\theta)$. From this, $h_2$ is defined by: $$ h_2 = \\frac{1}{2}(h - l\\cot(\\theta)) $$ Now $ h_1 = h - h_2 $ and after some mathmagic we should get the following: $$ h_{1\/2} = \\frac{1}{2}(h\\pm l\\,\\cot(\\theta)) $$ Now let's multiply this by the base and we've got ourselves $A_1 = h_1\\,b$. ## Stress Now we have two stresses, namely $\\sigma_{x_1}$ and $T$. Let's analyze the relations. The forces clearly have an eccentric deviation from the centerline. Now, is this going to be a problem? I seem to think that it won't be, and I base my logic on the fact that they will cancel the bending moments as we said above. The formulas are: $$\\sigma_{x_1} = \\frac{F}{A_1}$$ $$T = \\frac{F}{A'} $$ Now obviously these are general equations, one could always dig deeper and I believe we could even express these stresses as a function of elementary stresses, namely $\\sigma_x$ and some form of this other _tangential_ stress $F\/(l\\,b)$. ## Appendix To be completely accurate, the formula for $\\theta_k$ seems to be a bit more complex, since there are four corners on the cut (in our case). Knowing that paired corners are symmetric to the x-axis we can use this equation: $$\\theta_k(n) = |\\arctan(\\frac{l}{h}) + \\pi\\,n|$$ where $n\\in(-2,-1,0,1)$. Is this ugly or is it just me? # Closing I hope to get some review on the ideas here, I'd really love to know how would this get solved correctly. And please someone, help me fix the appendix formula."}
{"id":"89362","title":"A Puzzle about $SO(3)$","text":"Lie algebra of nonabelian group is $[T^a,T^b]=if^{abc}T^c$. For $SO(3)$ case, is the representation $T^a_{ij}=-i\\epsilon^{aij}$ fundamental or adjoint? The fundamental representation is defined as identifying $T^a$ the original generator. The adjoint representation is defined as identifying $T^a$ as structure constant. For $SO(3)$ case, generators seem to be same as structure contant. Reference:   1. Srednicki QFT"}
{"id":"44599","title":"what is the basic form of the 'fire'?","text":"> **Possible Duplicate:**   >  Is fire matter or energy? What is the basic form of fire? physics defines every entity by a basic form either solid or liquid or as a gas, example: water is liquid, ice as solid, water vapor: gas so what is fire? when I queried is 'Plasma a fire?' I landed in below link, where it is said that fire isn't plasma, but not defined which basic form it belongs to! the link: Is fire plasma? What I understand is, it's a form of energy caused by reaction of gases(?) so it's basically a gas! is it true?"}
{"id":"127469","title":"How does weight add up to press on things?","text":"I think I understand how pressure works with gases. More molecules bouncing around -> more random impacts -> stronger force. But I realized to my embarrassment that I don't understand how solid things press on each other, on a molecular level. Say I put a block of iron on my head. If I put another one on top of it, I feel twice the weight. The two blocks together can tear through thin paper where one block can't. But the contact between my head (or paper) and the blocks is just a very thin layer of atoms of the lower block's structure. If the lower block doesn't move when I put the upper one on it, what causes this thin layer to \"press\" on my head (or paper) more? When the two blocks together tear through thin paper, where does the force come from that acts on the paper molecules - it can't be gravity from the upper block, right? And how come that whatever this source of pressure is only depends on the weight of the upper block, and not on what it's made out of, iron or wood?"}
{"id":"65130","title":"What is the result of applying a fourier transform n times to a distribution?","text":"For a function applying the fourier transform twice is equivalent to the parity transformation, applying it three times is the same as applying the inverse of the fourier transform, and applying four times, is the identity transformation. Wikipedia has a good description of this. What do we get when we apply the transform multiple times to a distribution? For example applying it twice would look like this: $ \\mathscr F \\mathscr F T(f) = T(\\mathscr F \\mathscr F f) = T( f_-) = ? $ Where $f_- = f(-x)$. Here $\\mathscr F$ is the fourier transform $T$ is a distribution, and $f$ is a function."}
{"id":"65131","title":"Solution of multiple materials","text":"## Introduction I'm trying to make some strawberry liqueur. The recipe I got says I should put alcohol, strawberries and sugar in a bowl and leave it soaking for 4-5 weeks. I read that making liqueur is not a chemical reaction, but only a solvent (mostly alcohol) acting on some fruit, herb or nut. So, intuitively, I would add the sugar only after the soaking time. I wonder if that makes a difference. ## Question If I dissolve material A in alcohol until it reaches saturation, how much of material B can I still dissolve in the solution?   * Nothing   * As much as with pure alcohol   * Something in between None of the materials are gaseous and both are soluble in alcohol."}
{"id":"64660","title":"Speed of air through valve","text":"Right now I'm working on a hot air balloon in Physics formulas. At the moment it's flying up like it should, but once it is at a certain height we want it to go back down. After some research about hot air balloons we found out that they drop down by opening a valve at the top. So what we want to do is once the balloon gets at a certain height it will open the valve and drop down again. We can get the size of the valve and that it opens at a certain height. How can we calculate the speed of cold air going into the valve? We know the speed of the balloon, the size, temperature of the air in and outside of the balloon, air pressure etc. Is there some kind of formula we can use to calculate how fast the cold air will flow in and then cool of the air in the balloon making it drop down."}
{"id":"19857","title":"Earth's stationary iron core","text":"Why is the earth's iron core stationary, while the liquid matal circles around it creating the magnetic shield. Don't understand how can the entire planet rotate where as the planet's center is stationary. Anyone could explain this?"}
{"id":"19947","title":"Why are bubbles formed in the empty space of soda when turned upside down?","text":"I know this may actually be a chemistry question, but I don't know where else to ask. When I have a soda plastic bottle, I turn it upside down so the soda fills the empty space, and bubbles start growing on the walls of the plastic. So, why does this happen? I can think of something like a film forming around the empty space, and it reacts with the liquid, if so, what is it? why does it react that way?"}
{"id":"19945","title":"Sources to learn about Greens functions","text":"For a physics major, what are the best books\/references on Greens functions for self-studying? My mathematical background is on the level of Mathematical Methods in the physical sciences by Mary Boas."}
{"id":"268","title":"What physical forces pull\/press water upwards in vegetation?","text":"Each spring enormous amounts of water rise up in trees and other vegetation. What causes this stream upwards? Edit: I was under the impression that capillary action is a key factor: the original question therefore was: what are the fundamental forces involved in capillary action? Can perhaps anybody make an estimation of the amount of gravity energy that is involved in this process earth-wide each season?"}
{"id":"19948","title":"Expressing a particle's matter wave in terms of its momentum","text":"I'm following Zettili's QM book and on p. 39 the following manipulation is done, Given a localized wave function (called a wave packet), it can be expressed as $$\\psi(x,t) = \\frac{1}{\\sqrt{ 2 \\pi}} \\int_{-\\infty}^{\\infty} \\phi(k) e^{i(kx-\\omega t)} dk$$ Now use the de broglie relations: $p = \\hbar k$ and $E = \\hbar \\omega$ and define $\\tilde{\\phi}(p) = \\phi(\\frac{k}{\\hbar})$. This should yield $$\\psi(x,t) = \\frac{1}{\\sqrt{ 2 \\pi \\hbar}}\\int_{-\\infty}^{\\infty} \\tilde{\\phi}(p) e^{i(px-E t)\/ \\hbar} dp$$ but I get $$\\psi(x,t) = \\frac{1}{\\sqrt{ 2 \\pi} \\hbar}\\int_{-\\infty}^{\\infty} \\phi\\biggl(\\frac{p}{h}\\biggr) e^{i(px-E t)\/ \\hbar} dp$$ when I make the change-of-variable $k=\\frac{p}{\\hbar}$. What am I missing?"}
{"id":"88013","title":"What is the condition for accelerating charge to radiate?","text":"I was always taught that any accelerating charge produces radiation, but I don't think this condition is sufficient condition. For instance, any free charge on Earth is accelerated due to Earth orbiting the Sun but it doesn't produce radiation."}
{"id":"88018","title":"A question about verifying the transverse of electric field","text":"I came accross a question about verifying the transverse of electric field in Peskin and Schroeder's QFT p179. Given > $$ \\mathcal{A}^{\\mu}(\\mathbf{k}) = \\frac{ -e}{| \\mathbf{k} | } \\left( > \\frac{p'^{\\mu}}{k \\cdot p'} - \\frac{p^{\\mu}}{k \\cdot p} \\right) \\tag{6.7} $$ and > $$ \\mathcal{E} = -i \\mathbf{k} \\mathcal{A}^0 (\\mathbf{k}) + i k^0 > \\mathcal{A}(\\mathbf{k}) \\tag{6.9} $$ it is said > you can easily check that the electric field is transverse: $\\mathbf{k} > \\cdot \\mathcal{E}(\\mathbf{k}) =0 $ I tried to vertify this almost trivial relation and I didn't get it... Here is my attempt $$ \\mathbf{k} \\cdot \\mathcal{E}(\\mathbf{k}) = \\left( \\frac{ -e}{| \\mathbf{k} | } \\right) \\left[ - i \\mathbf{k} \\frac{ \\mathbf{k} p'^0 - k^0 \\mathbf{p}'}{ k \\cdot p'} + i \\mathbf{k} \\frac{ \\mathbf{k} p^0 - k^0 \\mathbf{p}}{ k \\cdot p} \\right] \\tag{1}$$ I may use $p^{\\mu}=(p^0, \\mathbf{0})$ (since the particle is initially at rest) and $k^0=|\\mathbf{k}|$. It seems that Eq. (1) is still not necessarily zero. Would you provide, at least, any hint how to show Eq. (1) is zero?"}
{"id":"52479","title":"Resonant inductive coupling and Schumann resonances","text":"I was reading about WiTricity (http:\/\/en.wikipedia.org\/wiki\/WiTricity) a technology developed by MIT to wirelessly transmit electricity through resonance, and I have this question: Given the phenomenon of resonant inductive coupling which wikipedia defines as: > the near field wireless transmission of electrical energy between two coils > that are tuned to resonate at the same frequency. > http:\/\/en.wikipedia.org\/wiki\/Resonant_inductive_coupling And the Schumann resonances of the earth ( ~7.83Hz, see wikipedia), would it be theoretically possible to create a coil that resonates at the same frequency or one of it's harmonics (7.83, 14.3, 20.8, 27.3 and 33.8 Hz) to generate electricity? I have a feeling that these wavelengths may be too big to capture via resonance (they are as large as the circumference of the earth if I understand it correctly), so alternatively would it be possible to create a coil that resonates with one of the EM waves that the sun sends our way?"}
{"id":"9262","title":"Why do we take the first derivative of EPR\/ESR spectra?","text":"Apologies if this question is a bit too chemistry-flavoured. In electron paramagnetic resonance spectroscopy, there's a practically ubiquitous convention of plotting the first derivative of the absorption with respect to field, rather than the simple absorption. Why is this?"}
{"id":"9261","title":"Reversible Stirling Engine","text":"So it has been asserted that a Stirling engine with proper regeneration can be made reversible. It will consist of two isothermal quasi-static processes connected by two constant-volume processes. The working gas is supposed to be heated\/cooled through infinitesimal temperature differences (thus preventing the creation of entropy). I'd like to know an explanation on the working gas properties during each step of the cycle. What are the state conditions for the gas at each of the points in the cycle (including inside the regenerator)? I've been looking for a the past week and found nothing."}
{"id":"53707","title":"Additional mass of block on inclined plane","text":"I would like some help in solving this problem. ![Height is 15 cm , base is 88 cm](http:\/\/i.stack.imgur.com\/fjVLX.png) Height is 15 cm , base is 88 cm My attempt: $$(119.3 + m)(\\sin(\\theta)) = (88*9.8)$$ $$ 15^2 + 88^2 = 89.269^2 $$ $$ \\sin^{-1} = 15\/89.269 $$ $$ (119.3 + m)(9.6733) = (88*9.8) $$ $m = -30.14$ ....Am I missing something? I feel like it has something to do with the force of friction..."}
{"id":"10732","title":"History of the use of the concept of phase space in engineering","text":"Engineering textbooks constantly use the concept of 'phase space' (see e.g. http:\/\/www.cs.cmu.edu\/~baraff\/sigcourse\/notesc.pdf). That is, they think of the state of a mechanical system as a high-dimensional geometric space. For example, for a system of particles, the phase space has $6n$ dimensions, where $n$ is the number of particles. My understanding is that the notion of phase space initially developed among theoretical physicists (possible in conjunction with quantum mechanics? or possibly before that point?). Therefore, I'm wondering when and how engineers began to adopt this terminology and concept from physicists. It might have something to do with differential equations, since the concept of representing the state of a mechanical system as a point in a space becomes obvious when you view ODE's in terms of flows on euclidean spaces. NOTE: The phase space concept was invented by theoretical physicist and mathematician Willard Gibbs."}
{"id":"10733","title":"Is a weak measurement the same as an unsharp measurement or POVM?","text":"This is prompted by the strong claims made in Science 332, 1170 (2011) to have observed trajectories of photons, \"something all of our textbooks and professors had always told us was impossible\". I'm suspicious of this claim because they work entirely within quantum theory, and it seems to me that in making weak measurements at two points on what they say is a trajectory, they do nothing else than make unsharp measurements of incompatible observables, a concept that was first introduced, to my knowledge, by Busch and Lahti in Phys. Rev. D 29, 1634–1646 (1984), and that is now much used in quantum information. Certainly observables that are at time-like separation are in general incompatible, so an unsharp measurement approach would seem applicable. A brief account of the Science paper mentioned above can be found here, amongst other places, and a few quotes can be found here, but, this being Science, I couldn't find a preprint. I was first alerted to this paper by a page at the BBC. The claim highlighted by ZapperZ, \"Single-particle trajectories measured in this fashion reproduce those predicted by the Bohm–de Broglie interpretation of quantum mechanics (8), although the reconstruction is in no way dependent on a choice of interpretation\", seems to me particularly egregious, insofar as both their choice of what experiment to do and of what to do with their raw data seem partly driven by the form they want to present their results in. It seems entirely possible to present the data in terms of unsharp measurements and POVMs, which would give relatively less support to the de Broglie-Bohm interpretation."}
{"id":"53700","title":"Car parking special-relavity puzzle","text":"Hi I read the following puzzle from an old text book long time ago. However it doesn't provide the answer. So what is the solution? Let's suppose a car is going to park to a garage and the garage is exactly the same size of the the static size of the car. Now the car is running in a speed that in its point of view the garage is just 1\/10 of its static size. So it will consider the garage is too small to fit. On the other hand, the garage manager consider the car is only 1\/10 of the garage size so it definitely fit. The question is to explain what will ACTUALLY happen if the car try to park in the garage, and what the garage manager and the driver will see."}
{"id":"63039","title":"Are the hypotheses of the Bernoulli equation satisfied for a bird or airplane wing at low Mach number?","text":"A previous question by David Zaslavsky was a request for a broad, \"how things work\" type of explanation of the lift of an airfoil. The answers given there are enlightening, but don't address a more specific question I have. First a summary of my current understanding. There is no logical reason to assume that parcels of air take equal time to travel above and below the wing. If we did make such an assumption, Bernoulli's principle would make all kinds of incorrect predictions. \"How things work\" explanations (e.g., this one) often speak as though Bernoulli's principle is one effect, and then there are some other effects, such as conservation of momentum; it's implied that these effects are additive, and it's often stated that the Bernoulli effect is negligible. But in fact Bernoulli's principle is, under certain hypotheses, equivalent to conservation of energy. Conservation of energy and conservation of momentum are not effects to be added. They're two physical laws, both of which need to be obeyed. It's only under certain hypotheses that Bernoulli's principle is equivalent to conservation of energy. It requires nonviscous flow, and the version given in freshman physics textbooks also assumes incompressible flow, although there is also a version for compressible flow. The WP article on Bernoulli's principle states that it's usually valid for Mach numbers lower than about 0.3, which I guess might be valid for birds and some small planes (cruise speed for a Cessna 172 is Mach 0.25), but not passenger jets. Based on this, I can envision two logical possibilities: (1) The hypotheses of Bernoulli's principle hold for birds and some small planes. The equation gives at least an approximately correct result for the net force on the wing, but there is no obvious, simple way to know what velocities to assume. (2) The hypotheses of Bernoulli's principle fail in these cases. Which of these is correct? If #2, which hypothesis is it that fails? Is the incompressible-flow version inapplicable but the compressible-flow version OK?"}
{"id":"69166","title":"Quantum Cyclotron Frequency - Why is it off by a factor of 2?","text":"Say you have a magnetic field $\\vec{B}=(0,0,B_0)$. Then the Schrodinger Equation Hamiltonian for a spin-2 particle of charge $e$ moving in this field is: $$H = \\frac{1}{2m}[\\vec{p}-e\\vec{A}]^2-\\vec{\\mu}\\cdot\\vec{B},$$ where $\\vec{A}=(-\\tfrac{1}{2}y,\\tfrac{1}{2}x,0)$ is the magnetic vector potential. You can find the speed by finding this commutator: $\\frac{d\\vec{r}}{dt}=i[H,\\vec{r}]$. When I did out this calculation, I ended up with $\\frac{\\vec{p}-e\\vec{A}}{m}$. However, I'm told that I should have gotten $\\frac{\\vec{p}}{m}+\\vec{\\omega}_c\\times\\vec{r}$, where $\\vec{\\omega}_c$ is the cyclotron frequency. This has the value $\\vec{\\omega}_c=-\\frac{e\\vec{B}}{m}$, but if you plug that in, you get $\\vec{\\omega}_c\\times\\vec{r}=-\\frac{2e\\vec{A}}{m}$. Why is this off by a factor of 2?"}
{"id":"45451","title":"Spring coupled platforms & conservation of momentum - can it be solved with freshman physics?","text":"This question came up as an exercise in a first year undergraduate course I was a TA for. It turned out to be a lot more difficult (impossible?) than anticipated... > Two platforms of mass $M_1$ and $M_2$ ($M_1\\neq M_2$) are connected by a > spring of constant $k$, and are initially at rest with the spring > unstretched, sitting on a frictionless surface. A man of mass $m$ stands on > one platform and begins to run, always with a constant speed $v$ measured > relative to the platform he is running on. What is the maximum speed reached > by the other platform, relative to the ground? My intuition is telling me that I need to know something about how the man gets from rest to his constant speed (instantaneously? very slowly? with some smooth acceleration?) to solve this, but I haven't been able to prove to myself that this is required. If this is a requirement, I think the most reasonable assumption would be that he reaches his full speed instantaneously. What I'm most interested in is whether this problem can be tackled with a typical freshman's toolbox - simple arguments around conservation of energy\/momentum, no\/very limited differential equations, basic calculus. I can see an easy way to get an upper bound on the maximum speed from energy\/momentum considerations, but I don't see a way to check if this speed is ever reached."}
{"id":"45450","title":"What is the relation between the Higgs field and chirality?","text":"Wikipedia states that the spontaneous breaking of chiral symmetry _\"is responsible for the bulk of the mass (over 99%) of the nucleons\"_. How do the nucleons gain mass from the spontaneous breaking of chiral symmetry? Why don't leptons gains mass from it? What is the role of the Higgs field in this all?"}
{"id":"118830","title":"Finding the components of the tensor for potential and kinetic energy","text":"I have a rather poor understanding of what a tensor is, but enough to apply it to the biggest part of the classical mechanics I'm studying.   However, I've run into a small problem while studying \"Free vibrations of a linear triatomic molecule\". (We're assuming all 3 atoms are in 1 straight line and the forces they exert on each other are represented by springs with constant values for $k$). My potential energy is described as: $V=\\frac{k}{2}(\\eta_1^2+2\\eta_2^2+\\eta_3^2-2\\eta_1\\eta_2-2\\eta_2\\eta_3)$   Where $\\eta$ is a coordinate relative to the equilibrium position. And 'hence' the tensor has the form: $\\vec V=\\begin{bmatrix} k & -k & 0\\\\\\ -k & 2k & -k\\\\\\ 0 & -k & k \\end{bmatrix}$ How do I go from 1 to the other and vice versa?   I've tried googling terms like \"Equation to tensor\", \"Potential energy tensor\", \"Absolute value of tensor\" etc. but they didn't yield anything usable for me."}
{"id":"68410","title":"Question on static friction","text":"![enter image description here](http:\/\/i.stack.imgur.com\/fzIUA.png) This is the question, and the answer is given as 2). Now my basic doubt is, are they asking us the direction of the total contact force on the object by the floor? Or, are they asking the direction of the static friction on the block by the floor? What is it? Because, the normal reaction between any two surfaces in contact is ALWAYS perpendicular to the two surfaces. Please help me, and please answer this before closing it as being too specific."}
{"id":"90307","title":"Collection of histories vs. collection of momentary configurations","text":"For a given Hamiltonian, is the space of histories of a classical system the same as the symplectic manifold? Do I have to take care of gauge equivalences and if so, is this only an issue for fields (not for trajectories)?"}
{"id":"90303","title":"Car on a frictionless banked curve","text":"> _A car is on a banked curve, following a path which is part of a circle with > radius $R$. The curve is banked at angle $\\theta$ with the horizontal, and > is a frictionless surface. What is the speed the car must go to accomplish > this?_ What I don't understand about this problem is why we assume there is only the normal force and the gravitational force on the vehicle. From that point onwards, I have no trouble following the solution. The way I see it, if we're considering only that there exists a normal force and a gravitational force, something the car is doing (accelerating?) must be \"adding\" to the normal force. Or is it possible that the car could just be coasting, and it could keep a constant height along the banked curve?"}
{"id":"90309","title":"The expansion of a function in powers of a parameter","text":"In the perturbation theory for non-degenerate levels, the energy $E_n(\\lambda)$ of an eigenstate $|\\psi_n(\\lambda)\\rangle$ of the hamiltonian $\\mathcal{H}=\\mathcal{H}_0+\\lambda \\mathcal{H}_1$ (where $\\mathcal{H}_0$ is the unperturbed hamiltonian with eigenstates $E_n^0$ and $\\lambda\\mathcal{H}_1$ is the perturbation) is described by the equation $$E_n(\\lambda)=E_n^0+\\dfrac{\\lambda\\langle\\phi_n|\\mathcal{H}_1|\\phi_n\\rangle+\\lambda^2\\langle\\phi_n|\\mathcal{H}_1|\\psi_n^1\\rangle+\\lambda^3\\langle\\phi_n|\\mathcal{H}_1|\\psi_n^2\\rangle+\\dots}{1+\\lambda a_n^{(1)}+\\lambda^2 a_n^{(2)}+\\dots}.$$ Here, $|\\phi_n\\rangle$ is the $n$'th eigenstate of the unperturbed hamiltonian, and $a_n^{(p)}=\\langle\\phi_n|\\psi_n^p\\rangle$, with $|\\psi_n^p\\rangle$ being the $p$'th correction to the $n$'th eigenstate. The zeroth correction of course equals the eigenstate of the unperturbed hamiltonian: $|\\psi_n^0\\rangle=|\\phi_n\\rangle$. **My question is the following** : how does one go from the above equation to this? $$\\begin{array}{r l} E_n(\\lambda)&=E_n^0+\\lambda\\langle\\phi_n|\\mathcal{H}_1|\\phi_n\\rangle\\\\\\ &+\\lambda^2\\left[\\langle\\phi_n|\\mathcal{H}_1|\\psi_n^1\\rangle-\\langle\\phi_n|\\mathcal{H}_1|\\phi_n\\rangle a_n^{(1)}\\right]\\\\\\ &+\\lambda^3\\left[\\langle\\phi_n|\\mathcal{H}_1|\\psi_n^2\\rangle-\\langle\\phi_n|\\mathcal{H}_1|\\psi_n^1\\rangle a_n^{(1)}-\\langle\\phi_n|\\mathcal{H}_1|\\phi_n\\rangle a_n^{(2)}\\right]+\\dots \\end{array}$$ The book I'm working with tells me this is done by _\"Expanding this expression in powers of $\\lambda$\"_ , but I can't recognize the procedure. Of course the first term $E_n^0$ is of order $\\lambda^0$, and the second term $\\lambda\\langle\\phi_n|\\mathcal{H}_1|\\phi_n\\rangle$ is of order $\\lambda^1$, and these can be taken from the original equation right away, but I can't find out how the subtractions in the third and fourth term got there."}
{"id":"77389","title":"Hamiltonian formulation of single particle in an electromagnetic field","text":"Consider a charged particle in a static electromagnetic field. Suppose that the domain is simply connected so that the second law of Newton's dynamics reads: $$ m\\frac{\\mathrm{d}^2\\vec{x}}{\\mathrm{d}t^2}=-e\\left(\\vec{\\nabla}\\Phi+\\frac{\\mathrm{d}\\vec{x}}{\\mathrm{d}t}\\times(\\vec{\\nabla}\\times\\vec{A})\\right) $$ where:   * $m$ is the mass of the particle,   * $\\vec{x}$ is position vector,   * $\\Phi$ the (scalar) electric potential,   * $\\vec{A}$ the (vector) magnetic potential. My question is how to derive the Hamiltonian formulation from here. The Hamiltonian is: $$ H(\\vec{x},\\vec{p})=\\frac{1}{2m}\\left\\|\\vec{p}-e\\vec{A}\\right\\|+e\\Phi(\\vec{x}) $$ and the Hamilton equations look like: $$ \\begin{cases} \\frac{\\mathrm{d}x_i}{\\mathrm{d}t}=\\frac{1}{m}(p_i-eA_i)\\\\\\ \\frac{\\mathrm{d}p_i}{\\mathrm{d}t}=\\frac{e}{m}\\sum_{j=1}^3(p_j- eA_j)\\frac{\\partial A_j}{\\partial x^i}-e\\frac{\\partial\\Phi}{\\partial x^i} \\end{cases} $$ Another question related to this one is why the magnetic potential is \"included\" in the \"generalized\" moment $\\vec{p}-e\\vec{A}$. I undertand why from the Lagrangian of the system but perhaps there is a physical explanation. **EDIT:** Consider a single particle in a conservative force field $F=-\\vec{\\nabla}V$. Then the Newton's equations are: $$ m\\frac{\\mathrm{d}^2\\vec{x}}{\\mathrm{d}t^2}=-\\vec{\\nabla}V $$ This is a second order differential equation, which we can transform into a system of first order differential equations, by adding a new variable which is usually the speed $v$ in mathematics but here we will take the momentum $p=mv$. The system I am talking about is: $$ \\begin{cases} \\frac{\\mathrm{d}\\vec{x}}{\\mathrm{d}t}=\\frac{\\vec{p}}{m}\\\\\\ \\frac{\\mathrm{d}\\vec{p}}{\\mathrm{d}t}=-\\vec{\\nabla}V(x) \\end{cases} $$ which is just the hamiltonian form of the Newton's equations. Can we do the same kind of trick in the case of the system above?"}
{"id":"94537","title":"Help wanted achieving cryogenic temperatures","text":"My son's science project involves liquifying air. I found the Linde process which is the way it was done first. We are trying and not getting very far so I am looking for advice. I have an air compressor good to 150psi. We are slowly ramping up the pressure in the system. We are currently testing at 80psi. We pressurize 20 feet of copper tube bent in a coil running through an ice bath. The tube exits and goes into the bottom of a thermos. The end of the tube is capped, and we allow a small leak so that pressure drops from 80psi to 1atm. We insulated the few inches between the ice bath and the thermos. We do not yet have a double walled tube, but we taped the thermos so all the air coming out is exiting through the insulation. When we run, the thermos drops from room temperature (about 68F in the basement) to our best of 45F. I'm quoting in Fahrenheit right now because the boys are very obnoxious about not wanting to use a temperature system they aren't comfortable with, and I just haven't been able to convince them that they are being unreasonable. The point is, we are not even achieving the temperature of the ice bath. Incidentally, as measured with my multimeter, the ice bath is always 3C. I'm curious about that too, but that will be a separate question. Needless to say, we are not achieving any refrigeration. The next step we will increase the leak so that the flow through the system is much higher. And we will start trying to use a Peltier cooler so we can have an intermediate stage that is 20C or so cooler than the ice bath. Obviously we can up the pressure, but I would think we should see something more than we are getting. Any suggestions? Should I take a picture of our setup?"}
{"id":"94538","title":"Ice bath is always 3C, why?","text":"I've always learned that a mixture of ice and water should reach equilibrium at approximately 0C. I've actually tried to create that a number of times in different contexts and always fail. First, an ice bath in a plastic container. I get 3C, occasionally as low as 2C with a multimeter thermocouple. Granted those are inaccurate, so I've tried a digital thermometer from Vernier with claimed accuracy of 0.5C. I confirmed with an IR thermometer. Same story. Thinking that perhaps it is due to warming from the outside, I have put the ice bath in a thermos and chilled it a while. Same story. I am not using distilled water, so there are some minerals in the water, but if anything I thought that would lower the temperature of the ice bath."}
{"id":"129191","title":"Adding matter to make galaxies bigger","text":"What are the actual factors that play a role in the accretion of matter into galaxies? I read about Accretion Disks but I don't quite understand how they work yet."}
{"id":"66519","title":"Theoretic Mass Energy Conversion?","text":"say you had a large quantity of a heavy element such as uranium... AND you put a massive amount of energy into it, so that it began to undergo nuclear fission and transformed to a significantly smaller element(s). Would it be possible to use this energy to now begin a process of fusion, to output energy? And then once it's fused beyond a certain size, back to fission? Basically the idea is to extract back and forth as much energy as possible from the original sample."}
{"id":"97948","title":"Is there a better approximation of drag (other than quadratic drag)?","text":"This is a student inquiry sparked by sheer curiosity. Wikipedia states the drag equation, $F = 1\/2v^2pC_dA_c$. (p = mass density of fluid\/gas, v = velocity, c_d = drag coefficient, a_c = cross sectional area perpendicular to velocity.) Wikipedia claims that this equation is only accurate under certain conditions: > _The objects must have a blunt form factor and the fluid must have a large > enough Reynolds number to produce turbulence behind the object._ **Does there exist a more general equation that can accurately measure the drag of all objects, at all speeds\/Reynolds numbers and considering all physical properties affecting the drag?**"}
{"id":"97949","title":"What happens when electron degeneracy pressure is being overcome (right before an object turns into a black hole)?","text":"I was wondering about hydrostatic equilibrium (the balance of radially inward and outward forces) in large (i.e. above the Chandrasekhar limit) stars. It is often said that when the gravitational force exceeds any outward forces or pressures, mainly the electron degeneracy pressure I'm thinking, the star collapses into a black hole. But how can this happen without the Pauli exclusion principle being violated? A similar question: Does black hole formation contradict the Pauli exclusion principle? The answer by @Siva in that question makes some sense to me, but I don't understand at what point we start our counting of states from the object being a star to a black hole. What happens in the middle? Is there a sharp change?"}
{"id":"8550","title":"Escape Velocity of Asteriod 243 Ida","text":"I was reading about this asteroid (apparently, it has a moon, isn't that awesome?) and I started thinking about if I was on this asteroid, and I jumped, would I fall off? It's been a while since I did something like this, so I went to Wikipedia to get some formulas. There I found $$v_e = \\sqrt{\\frac{2GM}{r}}$$ Where   * $G$ is the gravitational constant ($6.673 × 10^{-11}\\ \\mathrm{m}^3\\mathrm{kg}^{-1}\\mathrm{s}^{-2}$)   * $M$ is mass of the body (found to be $4.2 × 10^{16}\\ \\mathrm{kg}$)   * $r$ distance from center of mass (found to be $53.6\\ \\mathrm{km}$) $$v_e = \\sqrt{\\frac{2(6.673\\times 10^{-11})(4.2\\times 10^{16})}{53600}} = \\sqrt{104} = 10.22\\,\\frac{\\mathrm{m}}{\\mathrm{s}}$$ I'm a fairly fit guy, about 73kg and .6m vertical leap here on earth. I found a formula for the speed of a free falling object to be $$v = \\sqrt{2gd}$$ If I assume that my speed when I hit the ground is the same as when I jump up, that means my jump speed is about 3.4 m\/s, which is significantly less than the required 10.4 m\/s. So it would seem I would not be able to jump off of this asteroid. Are my formulas, assumptions, and calculations correct here? Also, how would I use this information to determine how high I would jump (would it be enough to freak me out? Well I'd be jumping off an asteroid, so it would freak me out anyway... although I imagine I would be going slow enough for (my ship?) to pick me back up.)"}
{"id":"8557","title":"Is there a maximum number of types of elementary particles?","text":"Doing a Google search i found a paper called The maximum number of elementary particles in a super symmetric extension of the standard model. It claims in the abstract that the upper bound is 84 (i don't have access to the article) My question is: Is there a max number of types of elementary particles predicted in advanced physics theories such as string theory? What are the reasons for this?Are the arguments purely mathematical?"}
{"id":"111683","title":"Earnshaw's theorem for extended conducting bodies","text":"I read on Wiki that Earnshaw's theorem has been proven for extended conducting bodies. If we consider the case of a positive charge at the centre of a symmetric metal cavity- the positive charge and the induced negative charge constitute a system which violates Earnshaw's law. Where am I wrong in thinking this way?"}
{"id":"27193","title":"Consideration of static atomic displacements in electronic structure calculations","text":"I am hoping to discuss some details of electronic structure calculations. I am not an expert on this topic, so please forgive any abuse of terminology. It is my understanding that first principles electronic structure calculations are based on perfect translational symmetry of a crystalline lattice. Further, that the first round of calculations traditionally treat the atomic nuclei as fixed points due to the large disparity in nuclei and electron velocities, and after the initial electronic structure is solved, one can then turn on the nuclear motions and understand the interaction between the nuclear and electronic motions. The heart of my question is: is it possible to investigate local distortions of structures using existing techniques? By a local distortion, I mean a distortion which does not disrupt the long range crystalline periodicity, but results in a very different local atomic environment?"}
{"id":"52864","title":"Chiral edge state as topological properity of bulk state","text":"As far as I know, quantum hall effect and quantum spin hall effect has chiral edge state. Chiral edge state is usually closely related with delocalization, since back scattering is forbidden. However, some topological nontrivial state does not have chiral state, such as majorana bound state in topological superconductor. I am quite interested whether the existance of chiral edge state is determined by bulk topological properity? Thanks in advance"}
{"id":"52865","title":"Indicators on how even the heat is distributed?","text":"I'm wondering if there are any good indicators on how even the heat is distributed on an object (for simplicity, a flat object maybe)? What are the possibly reasonable ways to maximize the evenness if varying the shape of the object is allowed."}
{"id":"22664","title":"What are the physycal meaning of universal constants such as the magnetic, electric and gravitational constants?","text":"I know that: $G=6.67300 × 10^{-11}\\dfrac{Nm^2}{kg^2}$(Gravitational constant) $K_e=9×10^9\\dfrac{Nm^2}{C^2}$(Electric constant or Coloumb's constant) $k_m=1×10^{-7}\\dfrac{Ns^2}{C^2}$(Magnetic constant) * * * But what are the meaning of this values? Would the universe be much different with a little change on this values? What if we hadn't known them?"}
{"id":"68318","title":"Question about entangled states","text":"I have a question about entangled state. Suppose I consider the entangled state $\\frac{1}{\\sqrt{2}}(|00\\rangle + |11\\rangle)$. I saw an argument for how measurement of the first bit is affected by whether or not the second bit is measured because this is an entangled state in an article I am reading. The argument is clear to me and I describe it below, so please keep reading: If I measure the first bit without first measuring the 2nd bit, then the probability that I get $|0\\rangle$ is 1\/2. This is clear to me. If I measure the 1st bit after measuring the 2nd bit, the probability that it is $|0\\rangle$ is 0 or 1 depending on whether the measured value of the 2nd bit was $|0\\rangle$ or $|1\\rangle$. This is clear to me too. But what I am confused about is this: If I measure the first bit after the second bit, this is what I would calculate the probability of measuring the first bit being $|0\\rangle$ to be: $\\frac{1}{2}\\times 1 + \\frac{1}{2}\\times 0 = \\frac{1}{2}$. The first term is the prob. of measuring the second bit as $|0\\rangle$ and then the 1st bit as 0 and the 2nd term is the prob of measuring the 2nd bit as 1 and then measuring the 1st bit as zero. If the probability of measuring the first bit as 0 is to same regardless of whether the 2nd bit was measured or not is unchanged, what is the effect of the entangled state? Or am I missing something?"}
{"id":"52868","title":"Is spacetime an illusion?","text":"In consistent histories, for gauge theories, can the projection operators used in the chains be not gauge invariant? In quantum gravity, for a projection operator to be gauge invariant means it has to be diffeomorphism invariant. There are no localized diffeomorphism invariant operators, not even localized in time. No localized observables. So, what are the permissible chains in quantum gravity? Do they have to be delocalized, or localized on the boundaries of spacetime? Are space and time illusions, as Andrew Strominger claimed? A Cosmic Hologram projecting out the illusion that is spacetime from the future boundary of time?"}
{"id":"776","title":"Can Jupiter be ignited?","text":"Our solar system itself contains two candidate \"Earths\" One is Jupiter's moon Europa and another is Saturn's moon Titan. Both of them have the problem of having at low temperature as Sun's heat cannot reach them. Jupiter is made up mostly of Hydrogen so is our Sun. Can we ignite Jupiter so that it will produce enough heat to warm these two earth like planets\/satellites? What can be consequences of any of such attempt?"}
{"id":"107887","title":"Why aren't there more natural satellites orbiting the Earth than just the Moon?","text":"Considering that there's a lot of debris in space and that impacts fling out rocks into space all the time, why do we only have one large natural satellite - the Moon? Shouldn't there be all kinds of rocks in all shapes and sizes orbiting the Earth?"}
{"id":"107880","title":"Is spectrum of Discrete-Time Fourier Transform (DTFT) periodic or not","text":"I can't think of any better title. Here is the content that I got question http:\/\/cnx.org\/content\/m10247\/2.31\/ As it state the nature of DTFT's spectrum is periodic as it show in figure 1 ![enter image description here](http:\/\/i.stack.imgur.com\/RHr9S.png) ![enter image description here](http:\/\/i.stack.imgur.com\/892ct.png) However, in figure 3, spectrum is non-periodic ![enter image description here](http:\/\/i.stack.imgur.com\/lO3dP.png) So, the spectrum of DTFT is periodic or not or it is dependent on each case of $s(t)$"}
{"id":"110488","title":"\"Consciousness causes collapse\"-interpretation or free will excluded by quantum zeno effect?","text":"This question is if the von Neumann–Wigner interpretation, aka \"consciousness cause collapse\" (see e.g. John von Neumann's 1932 book The Mathematical Foundations of Quantum Mechanics) is compatible with the quantum Zeno effect (see Sudarshan, E. C. G.; Misra, B. (1977) \"The Zeno's paradox in quantum theory\", Journal of Mathematical Physics 18 (4): 756–763) respectively its empirical observation (such as by Wineland 1989 or by Raizen 2002). Or if the Quantum Zeno effect falsies or created the possibility to empirically verify the von Neumann-Wigner interpretation. Assume measuring the state of a system every second causes a zeno effect and measuring every three seconds causes an anti-zeno effect. Now let's assume the state is measured every second and recorded. A proponent of the \"consciousness causes collapse\" interpretation would argue that the wave function only collapses when a conscious observer actually looks at the measurement results. But what if the conscious observer first looks at only every third result and decides only afterwards on whether to look at the other results too. Will she observe the zeno effect depending on her future decision? Wouldn't such an experiment show that either (1) the consciousness of the observer has no impact and there is a zeno effect even if two out of three measurement results are destroyed before a conscious observer looks at them or that (2) the observer has no free will and whenever she observes a zeno effect looking at every third measurement she can't help but looking at the other results too, and conversely if the first set of results she looks at shows an anti-zeno effect she has an irresistible urge to destroy the other results? Well the first option looks more plausible to me, but it seems like something that could be experimentally tested."}
{"id":"126471","title":"Can an object reverse its direction of acceleration even though it continues to move in the same direction?","text":"Can anyone please explain me on this matter along with day to day examples?"}
{"id":"65279","title":"Scalar-fermion bound state","text":"Is it possible to have a bound state between a scalar and a fermion? For example, a squark--anti-squark bound state, provided that the decay width is sufficiently small compared to the binding energy?"}
{"id":"92893","title":"what is the mechanism of water conductivity?","text":"I'm trying to wrap my head around the kelvin water drop generator and more specifically what happens in the water before it drops. If on one side the drops of water are positive and on the other side negative, that means there is current flowing through the water _before it drops_. What I dont understand even more specifically is how electricity is carried by water, and why water conductivity increases with ions content (and other impurities perhaps). Are there electrons flowing ? Or are there ions flowing ? Let's imagine that we are below the potentials at whitch electrolysis occurs (about 1 volt). What carries the charges ? I first thought that ions were responsible for the charges in the kelvin generator, but when a spark occurs, because the water has dropped into separate boxes it's impossible for them to move, so there must be some kind of redox reaction occuring. But if table salt is in the water to raise it's conductivity, would that mean that Na+ is reduced to sodium and Cl- oxydized to chlorine ?"}
{"id":"114477","title":"plasma pressure initial condition","text":"The authors in the paper below did a simulation of plasma plume evolution from a laser heated metal target. They used compressible transport equations and ideal gas EOS for Ar. I am having difficulty understanding their initial condition. They say \", the initial internal pressure of plasma is assumed constant, 210 MPa.\". How is this value determined or assumed. It is based on the temperature of the surface? So, im thinking about the problem some more, and it seems the density is what should be known because when ablation occurs, phase transition occurs in the metal surface which generates a vapor. Thanks S. Harilal, G. V. Miloshevsky, P. K. Diwakar, N. L. LaHaye, and A. Hassanein, Phys. Plasmas 19(8), 083504 (2012)."}
{"id":"114473","title":"calculating electrodynamic momentum of a dumbbell (consisting of two point charges) in longitudinal motion","text":"I'm working through a paper on momentum in electrodynamics that requires the integration below and would greatly appreciate any help. I'm pretty sure it evaluates to $2\/d$ but I can't quite figure how. $$ \\int_{0}^{\\infty}\\int_{0}^{\\pi} {r^{4}\\sin^{3}\\left(\\theta\\right) \\over \\left\\\\{(r^{2} + d^{2}\/4)^2 - \\left[rd\\cos\\left(\\theta\\right)\\right]^{2}\\right\\\\}^{3\/2}} \\,{\\rm d}r\\,{\\rm d}\\theta \\space-[1] $$ If I focus on the integration w.r.t. r first, I get [2]. If I focus on the integration w.r.t. $\\theta$ first, I get [3]. A,B,C are just constants w.r.t. the variable of integration. I believe I can make a series of substitutions but they don't seem to get me closer to an answer. $$ \\int_{0}^{\\infty} {Ar^{4} \\over \\left[(r^{2} + B)^2 -\\left(rC\\right)^{2}\\right]^{3\/2}}\\,{\\rm d}r \\space-[2] $$ $$ \\int_{0}^{\\pi} {A\\sin^{3}\\left(\\theta\\right) \\over \\left\\\\{(C)^2 - \\left[B\\cos\\left(\\theta\\right)\\right]^{2}\\right\\\\}^{3\/2}} \\,{\\rm \\rm d}\\theta \\space-[3] $$"}
{"id":"104285","title":"Does measuring the exact position of a ball destroy the ball","text":"If you have a macroscopic ball (say, a tennis ball) and you (hypothetically) try to measure the exact position of the center of that ball by measuring the exact positions of the atoms making up the ball, will this measurement destroy the ball (i.e. is there a point where, as a result of the uncertainty principle, the more accurately you measure the position of each of the atoms, the more likely the atoms are to scatter off in some random direction, destroying the ball)?"}
{"id":"105626","title":"What is the capacitance of parallel plate capacitor with different electrode material?","text":"What is the capacitance of parallel plate capacitor with different electrode material? The capacitance of parallel plate capacitor depends on dielectric material, surface area, and separation between two plates. $$C = ε_r * ε_0 * \\frac{A}{D} $$ But how about electrode materials between dielectric, will the capacitance be different? Thanks."}
{"id":"102934","title":"Signal strength drop-off from a geosynchronous satellite as a function of angular pointing error","text":"If we have a compact antenna used to communicate with a satellite in geosynchronous orbit somewhere between the $K_u$ and $K_a$ bands of the spectrum (for high broadband applications where we can't use something like the Iridium network), we presumably need to properly orient the dipole of the antenna with respect to the position of the geosynchronous satellite. Let $\\alpha$ be the angular pointing error of the satellite (some fraction of a degree). How does the signal strength fall off as a function of $\\alpha$? Or, if it makes a difference, we can uniquely define an azimuth angle $\\theta$ and elevation angle $\\phi$. Now, I'm guessing that the answer to this question has something to do with the satellite beam configuration \/ geometry. So is it possible for there to be a \"typical\" answer to this question? Also, to be clear, here I'm asking about the \"form\" of the expression for the drop-off in signal intensity as a function of $\\alpha$ (or azimuth and elevation angles $\\theta$ and $\\phi$), as the exact parameters will likely depend on the precise antenna being used. Where I'm coming from with respect to this question: I heard recently that is indeed possible possible to send data packets back and forth to a geosynchronous satellite from a sufficiently large boat (or aircraft) with proper antenna equipment (indeed to establish a continuous connection), and I was wondering how in the world this worked? In other words, in the context of something like a small vessel or an airplane, to what extent is sophisticated inertial navigation equipment required for real-time compensation of rotational (pitch \/ roll \/ etc.) and translational transformations on the orientation and position of the antenna? [3-11-2014] Is it safe to assume that these vessel-to-GEO antennas use circular polarization to avoid issues associated with the relative orientation of the transmitting and (primarily) receiving antenna? It seems then safe to assume that only the angular error $\\alpha$ should be sufficient to estimate signal strength?"}
{"id":"60008","title":"Shear Flow corresponding to Eccentric Shear Force of a Closed-section Beam (Structural Analysis - Mechanics)","text":"Been stumped with this question for way too long... its a beam with a thin- walled rectangular cross-section, and a shear force is acting at a distance from the shear center. I know my decomposition of that force into the torsional moment and shear-center acting shear force is correct, but for some reason the shear flows don't add up.... rather than causing a higher shear to the left of the cross-section like I'd expect from my diagram, they cause a higher shear on the right side of the cross-section, sort of the mirror image of what I would expect. I've tried finding my mistake looking at books, websites, notes but none of them address this topic directly, leaving me with no other choice. :( Could anyone please tell me what I'm doing wrong? Also sorry for the blurry image and messy diagrams... I'll draw a new one if it's useless. ![enter image description here](http:\/\/i.stack.imgur.com\/U31Qw.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/zxXJH.jpg)"}
{"id":"14336","title":"total intensity measurement (of the whole visible light spectrum; from 400 - 800 nm) with a powermeter PM100?","text":"I was using a power meter PM100, with a Si-photodiode sensor head, to measure the intensity of the white light source (250 W quartz halogen lamp). I have recorded the power, shown by the sensor, by setting the wavelength equal to 633 nm (i.e. the maximum intensity is supposed to be at this wavelength, in the spectrum); however, this value for the intensity (power divided by area of the sensor) is only true for the monochromatic light (supposed to use for laser light?) and want to estimate the intensity (of the white light source) from the measured value. I have the responsivity curve for the senor, I used, and the intensity at a wavelength. How can I estimate the total power\/intensity of the light source? Your help will be much appreciated."}
{"id":"89170","title":"Explain Momentum","text":"It's a shame for me to be in Year 12 and still haven't understood the concept of Momentum. This is what I think, and I know I'm wrong. But, it is a good place to start for you to explain: \"Since the unit of the momentum is kg\/ms^-1, it is a measure how much mass is being moved at the speed of 1ms^-1. By the way, this is not a homework or anything. I'm really eager to learn about this concept. Thanks In Advance."}
{"id":"21925","title":"what is the proper way to connect two light bulbs in a circuit? in series or parallel?","text":"What is the proper way to connect two light bulbs in a circuit? in series or in parallel and why? My thought is that it's better to hook them in parallel, since if we take into account Ohm's Law, the sum of their resistance will be less than when connected in series therefore the battery will last longer. What do you think?"}
{"id":"21924","title":"Hawking radiation: direct matter -> energy conversion?","text":"When a black hole evaporates, does it turn all the matter that has fallen in directly to energy, or will it somehow throw back out the same kind of matter (normal or anti) that went in?"}
{"id":"94681","title":"If the universe could emerge from nothing, what about physical laws?","text":"I've watched Lawrence Krauss's lecture and read his book. I think I got what he was saying, and I don't have any problems with that; however, what I can't get is how the laws of physics that makes this phenomenon possible came to be in the first place?"}
{"id":"122508","title":"From where randomness comes from and why it exists?","text":"I recently began to study statistics and probability and I have two questions:   1. Where does randomness come from? What is the source of randomness?    2. Why does the randomness exist? Is it possible to answer my question that non mathematician or non physicist can understand it? I searched Internet for the answer but there is very little information that non mathematician or the information is for professional physicists and mathematicians. Also please suggest books concerning my questions that is written so that non mathematician or non physicist can understand it."}
{"id":"15717","title":"can radiocarbon dating be used on living things?","text":"I have been going through a wiki article about worlds oldest living creature. As a matter of fact its a plant, a shrub to be precise. Wiki says that the plant age was determined by carbon dating. But what i know is that carbon dating can only be done for dead tissues i.e which no longer assimilates atmospheric carbon onto itself. Can someone clarify this?? regards,"}
{"id":"15716","title":"Optics: finding total power of a syste, including wavelength","text":"Specifically I need to create an ABCD matrix given 2 radii of curvatures, the thickness of the lens, type of glass (BK7) and a given wavelength. I will have to figure this out eventually, so help would be appreciated with this as well, but my main question is, how do I correlate the wavelength of the incident light to the total power? If I have the wavelength of light, how do I decide what the power of the lens is if nothing in the lens changes, just the wavelength of the beam of light (for example, how would the power differ if a yellow helium line was shone, or a blue hydrogen line)? Thank you!"}
{"id":"113797","title":"Quantum entropy in term of density matrix","text":"Why in von Neumann expression of quantum entropy we have trace of density matrix expression? Why don't off diagonal term play a role?"}
{"id":"113791","title":"Simple question regarding the Green's function for the diffusion equation","text":"The differential operator for diffusion in three dimensions is given by $\\partial_t - k \\nabla^2$ where $k$ is a constant. The Green's function is (according to Wikipedia) $$\\theta(t)\\left( \\frac{1}{4\\pi k t} \\right)^{3\/2} e^{-r^2\/4kt}$$ where $\\theta$ is the Heaviside step function and $r = |\\mathbf{r}|$. Hence, we should have $$ (\\partial_t - k \\nabla^2)\\theta(t)\\left( \\frac{1}{4\\pi k t} \\right)^{3\/2} e^{-r^2\/4kt} = \\delta(t)\\delta^3(\\mathbf{r}).$$ I've played around with this for a while, but I simply can't see how evaluating the left-hand side in the above expression gives the right-hand side. Where do the delta functions even come from?"}
{"id":"107711","title":"Describing a motion of gyroscope with gimbal","text":"Can you tell be how to set the equations to describe the motion of this machine in movie \"Contact\": https:\/\/www.youtube.com\/watch?v=TSaO9VGjLXc This is gyroscope with gimbal, am I right?"}
{"id":"24155","title":"Infinitely charged wire and Differential form of Gauss' Law","text":"I have tried calculating the potential of a charged wire the direct way. If lambda is the charge density of the wire, then I get $$\\phi(r) = \\frac{\\lambda}{4 \\pi \\epsilon_0 r} \\int_{-\\infty}^\\infty \\frac{1}{(1+z^2\/r^2)^{1\/2}} dz.$$ But this comes to $+\\infty$ unless I am doing the calculation wrong. Why doesn't this work the direct way? Also, is it possible to calculate the potential of a charged wire using Gauss' differential law? What about in the case of an infinite charged sheet? Or does Gauss' differential law only apply to charged volumes?"}
{"id":"33679","title":"As the universe expands, why do some things stretch but not others?","text":"I got into watching a video on Olbers' Paradox a few days ago, and from there read about the origins of the universe, its expansion, and so on... it's always fascinated me, but now something about it bothers me. I've heard many analogies about it (the dot-balloon, the raisin bread loaf, and others), but none really seem to explain this question. (This comes close, but dances around the answer more than explain it.) At the beginning of the universe as we know it, the universe itself was very small, so all the stars giving off light would have made it very bright (16:29 in this video). Since that time, the wavelength of that light has been stretched (17:01, same video). I found a few explanations saying that space itself stretched (here; described as \"ether\" in the article), which would stretch out the wavelengths. But here's what bothers me: If space is stretching out, redshifting all the light soaring around our universe, why are we not stretching? Theoretically, the universe is expanding an incredible amount faster than the speed of light, and the edge of the universe is an unimaginably large number of megaparsecs away from us. But should we not notice some of the stretch here, too? That is to say, if the light in space (the \"ether\", though I'm not fond of that term) is stretching out, why is everything on Earth still the same size as it was a hundred years ago? Is it stretching uniformly, but we are just unable to notice such a small stretch? Or does mass have some property that space and light do not, that prevents it from stretching out? I've also heard about time stretching, too; does this have an impact on it?"}
{"id":"93784","title":"Flux through a conduting cylinder?","text":"> _A point charge of magnitude $Q$ is placed inside a conducting cylinder of > length $L$ and radius $R$ at its centre. What is the flux through the > cylinder?_ I know that I have to use Gauss Law here but my concern is: > _Since it is a conducting cylinder, there would be equal and opposite > charges induced on its surface. How do I account for these charges?_ Help would be appreciated."}
{"id":"20790","title":"Which limit for matsubara frequency sum?","text":"in the context of a simple toy problem for Feynman path integrals, I consider a two-site Hubbard model for spinless fermions. I expand the path integral to first order in the interaction $V$, which means that I have to compute an \"average\" of the type $$\\left\\langle \\bar \\psi_1(\\omega_n) \\bar \\psi_2(\\omega_m) \\psi_2(\\omega_p) \\psi_1(\\omega_q) \\right\\rangle$$ with respect to the non-interacting action. Since the Grassmann numbers for sites $1$ and $2$ are different numbers, this average immediately factors, and I'm left with two-variable averages, which give me just the non-interacting Matsubara Green's functions. But now I have to calculate a Matsubara sum of the type $$\\frac{1}{\\beta} \\sum_n \\frac{1}{i\\omega_n - \\epsilon}$$ where $\\epsilon$ is just a constant. From searching in the literature, I know that contour integration is the way to go, but unfortunately this sum is ambiguous and I have to introduce a factor $e^{-i\\omega_n \\tau}$ with $\\tau \\rightarrow 0^+$ or $\\tau \\rightarrow 0^-$, and the result of the contour integration will be quite different depending on which limit I take. My question now is, how do I know what limit to take? And is there an easier way of carrying out the sum over frequency?"}
{"id":"116412","title":"What's best shape for linear concentrated solar panel?","text":"I found 3 shapes which are suitable for concentrated solar panel.      1\\. **Semicircle** \\- first thing that comes to ones mind.   It's a simple shape, the position of the focal point is obvious.      2\\. **Parabolic mirror**   A bit more difficult shape.   ![Parabolic mirror](http:\/\/i.stack.imgur.com\/qWq30.png)      3\\. **Compound parabolic mirror**   ![Compound Parabolic mirror](http:\/\/i.stack.imgur.com\/VUupy.jpg)   Compound parabolic mirror has an advantage over the parabolic mirror in acceptance angle. Collector doesn't necessarily need to track sun, because it's able to receive sunlight from a range of angles.   **Is there any other shape, that might have interesting properties for the purpose of solar collector?**   I found many DIY designs with semicircle mirrors. And many scientific papers with parabolic mirrors.   **What is the advantage of parabolic mirrors are over semicircle mirrors?**"}
{"id":"56586","title":"POVM advantage in state discrimination","text":"Suppose we are given the task of discriminating, with minimum error, between a set of states $\\\\{|\\psi_1\\rangle,|\\psi_2\\rangle,\\ldots,|\\psi_N\\rangle\\\\}$. In other words, we are given an unknown state form this set and our task is to output which of the states we were given. Let $P_{err}$ be the probability of incorrectly identifying the state. Under what circumstances is this probability minimized by using a projective measurement?"}
{"id":"92549","title":"The most general procedure for quantization","text":"I recently read the following passage on page 137 in volume I of 'Quantum Fields and Strings: A course for Mathematicians' by Pierre Deligne and others (note that I am no mathematician and have not gotten too far into reading the book, so bear with me): > _A physical system is usually described in terms of states and observable. > In the Hamiltonian framework of classical mechanics, the states form a > symplectic manifold $(M,\\omega)$ and the observables are functions on $M$. > The dynamics of a (time invariant) system is a one parameter group of > symplectic diffeomorphisms; the generating function is the energy or > Hamiltonian. The system is said to be free if $(M,\\omega)$ is an affine > symplectic space and the motion is by a one-parameter group of symplectic > transformations. This general descriptions applies to any system that > includes classical particles, fields, strings and other types of objects._ The last sentence, in particular, has really intrigued me. It implies a most general procedure for quantizing all systems encountered in physics. I haven't understood the part on symplectic diffeomorphisms or free systems. Here are my questions:   1. Given a constraint-free phase-space, equipped with the symplectic 2-form, we can construct a Hilbert space of states and a set of observables and start calculating expectation values and probability amplitudes. Since the passage says that this applies to point particles, fields and strings, I assume this is all there is to quantization of any system. Is this true?    2. What is the general procedure for such a construction, given $M$ and $\\omega$?    3. For classical fields and strings what does this symplectic 2-form look like? (isn't it of infinite dimension?)   4. Also I assume for constrained systems like in loop quantum gravity, one needs to solve for the constraints and cast the system as a constraint-free before constructing the phase, am I correct?   5. I don't know what 'the one-parameter group of symplectic diffeomorphisms' are. How are the different from ordinary diffeomorphisms on a manifold? Since diffeomorphisms may be looked at as a tiny co-ordinate changes, are these diffeomorphisms canonical transformations? (is time or its equivalent the parameter mentioned above?)   6. What is meant by a 'free' system as given above?   7. By 'affine' I assume they mean that the connection on $M$ is flat and torsion free, what would this physically mean in the case of a one dimensional-oscillator or in the case of systems with strings and fields?   8. In systems that do not permit a Lagrangian description, how exactly do we define the cotangent bundle necessary for the conjugate momenta? If we can't, then how do we construct the symplectic 2-form? If we can't construct the symplectic 2-form, then how do we quantize the system? I have asked a lot of long questions, so please answer as many as you can and link relevant articles."}
{"id":"68352","title":"Potentials in Helmoholtz Theorem","text":"I read about Helmholtz theorem on the Griffiths (Introduction to electrodynamics) and in the appendix B it shows how to use the Helmholtz theorem to determine the field **F** starting from the potential. The book says that: Suppose we are told that the divergence of a vector function **F** (r) is a specified scalar function D(r): $\\nabla \\cdot \\textbf{F}=D(r)$ and the curl of **F** (r) is a specified vector function **C** (r): $\\nabla \\times \\textbf{F}=\\textbf{C}$ For consistency, **C** must be divergenceless, $\\nabla \\cdot \\textbf{C}=0$ because the divergence of a curl is always zero. Question: can we, on the basis of this information, determine the function **F**? If D(r) and **C** (r) go to zero sufficiently rapidly at infinity, the answer is yes, as I will show by explicit construction. I claim that $\\textbf{F}=-\\nabla U + \\nabla \\times \\textbf{W}$ where $U(\\textbf{r})=\\frac{1}{4\\pi}\\int \\frac{D(\\textbf{r'})}{\\textbf{r-r'}}d\\tau'$ and $\\textbf{W(r)}=\\frac{1}{4\\pi}\\int \\frac{\\textbf{C(r'})}{\\textbf{r-r'}}d\\tau'$ Can someone show me how to obtain the potentials formulas written above?"}
{"id":"68358","title":"Correlation between speed, distance, mass, etc between a planet and a star (or moon and planet)?","text":"While playing around with some little scripts, generating solar systems, I started to wonder about the correlation between the different objects. Up until now, distance, mass, size and speed are randomly generated. Are there any 'common rules' on how some of theese objects correlate? Are planets close to a star always smaller and faster, and planets far away usually larger and slower? Is the rotation speed affected? And how about the correlations between moons and their planet? At the end of the day, I'm triyng to make the the generation a bit more accurate \/ realistic. Thanks in advance! Sorry for the english, it's a second language for me."}
{"id":"108131","title":"Very short decay: A bare quark?","text":"I keep reading the same phrase about the very short life time of the top quark: > > Because the t-quark decays on a shorter than the characteristic QCD > interaction-time it cannot hadronize. Therefore it give to possibility to be > seen as a bare quark. I cannot find any more information beyond this simple phrase. There are however a few things that I would like to seen cleared out (sorry for the naive questions) :   * The first question is simple: I don't understand why this offers to opportunity to see a bare quark, since this quark decays that fast it must be very difficult te see it...   * A bare quark?? I thought this was ruled to see colorful particles?"}
{"id":"108134","title":"How do projectors work?","text":"So, objects are certain colors because they are absorbing every color except for that one. So why is it that if I take a projector and project a blue image on a red wall, the red wall still reflects the blue image rather than absorbing the blue light? I don't know if this is considered a dumb question, but I was thinking about it and it got me curious (also, I haven't taken any kinds of physics)."}
{"id":"108136","title":"What can I say about compatibility between predictions and results?","text":"If I have these theoretical predictions: \\begin{align} \\omega_{p_1} = 4.5132 \\pm 0.0003~\\text{rad\/s} && \\omega_{p_2} = 4.5145 \\pm 0.0002~\\text{rad\/s}\\\\\\ \\omega_{b_1} = 0.0707 \\pm 0.0003~\\text{rad\/s} && \\omega_{b_2} = 0.0700 \\pm 0.0002~\\text{rad\/s} \\end{align} And I got these experimental results: \\begin{align} \\omega_{p_1} = 4.5148 \\pm 0.0001~\\text{rad\/s} && \\omega_{p_2} = 4.5147 \\pm 0.0001~\\text{rad\/s}\\\\\\ \\omega_{b_1} = 0.0702 \\pm 0.0001~\\text{rad\/s} && \\omega_{b_2} = 0.0707 \\pm 0.0001~\\text{rad\/s} \\end{align} Can I only say that the experimental measurements are incompatible with the theoretical predictions even if the discrepancy is in the order of $\\sim 10^{-3}\\text{ s}$? Or can I say something else more?"}
{"id":"109847","title":"What happens in a car crash?","text":"My mom and I were in a car accident. We are ok, but I want to know how fast the car that hit us was going. We were stopped at a light. The car that hit us from behind was a big GMC SUV. Our car was a Cadillac srx and mom looked it up it weighs 4,277 pounds. Then we hit a little car in front of us at 12mph and the air bag went off! Mom says the car behind us must have going faster than that because she says our car absorbed some kinetic energy. I don't know what that means but I think it was going slower and the hit pushed us to go faster. How can we figure out what actually happened?"}
{"id":"10400","title":"Universe Expansion as an absolute time reference","text":"Why we call \"constant\" to the Hubble constant?, if the universe were really expanding then the Hubble \"constant\" should change, being variable, smaller and smaller..with \"time\". Other example\/view of this question: if Hubble constant were constant then the universe \"age\" would be constant too, so let's say 13.7 billion years, and stop growing. That's not an expanding universe. On the other hand: If the Hubble constant and universe age were not so constant then: There is some property that is changing like a \"time\" that -everyone in universe- could get same number, so there would be an absolute time.. I see the Hubble Constant, and even the age of the universe, a really too speculative theory (perhaps it is as speculative as the posibility of measuring), having more accuracy will lead to better theories... Is it because some theoric impossibility in measuring that kind of variation? I see a lot of inconsistency. Any light at this respect is welcome, thanks"}
{"id":"78558","title":"Calculating the age of the Universe now and 30 billion years from now","text":"I am confused about the age of the Universe. If you calculate it by $1 \/ H_0$, won't the answer be roughly the same today as it will be 30 billion years from now? I know $H_0$ is a parameter, not a constant, but it doesn't change that much, does it? And if the expansion is accelerating then $H_0$ is going up, implying the age of the universe $1 \/ H_0 $ was higher in the past than it is today."}
{"id":"81982","title":"Calculate age of the Universe at 7 billion and 28 billion years old","text":"If $1 \/ H_0 $ is about 14 billion years, then what happened when the universe was half its current age? Is the empirically determined $H_0$ supposed to have been twice its current value? And when the universe is twice its current age is Hubble's parameter half its current value? That would predict expansion is slowing. But expansion is actually speeding up. That implies that $ H_0 $ is getting larger, and because $1 \/ H_0 $ gets smaller in that case, the universe is growing younger. I asked this question already here, but it was marked a duplicate. I know that Hubble's Parameter is not a constant. That doesn't actually clarify the answers to these questions."}
{"id":"69050","title":"Is Hubble's constant really constant?","text":"How does Hubble's constant resembles age of universe? Isn't universe getting old each day? How can a constant be a reciprocal of age of universe? Hubble's value must be variable, isn't it?"}
{"id":"114889","title":"Electromagnetic energy with P and M","text":"In many books, for a linear dielectric medium in which we have Maxwell's equations and the relationships $${\\mathbf D}=\\varepsilon({\\mathbf x}){\\mathbf E}$$ and $${\\mathbf B}=\\mu({\\mathbf x}){\\mathbf H}$$ we find formulae for the electromagnetic energy as $$U=\\int d{\\mathbf x}\\frac{1}{2}({\\mathbf E}\\cdot{\\mathbf D}+{\\mathbf B}\\cdot{\\mathbf H})$$ in e.g. J.D. Jackson's book. Suppose, instead, we have Maxwell's equations and the ${\\it general}$ linear relationship $${\\mathbf D}=\\varepsilon({\\mathbf x}){\\mathbf E}+{\\mathbf P}$$ and $${\\mathbf B}=\\mu({\\mathbf x})({\\mathbf H}+{\\mathbf M}).$$ We are willing to suppose that we know what the energy for making the polarization $\\mathbf P$ and magnetization $\\mathbf M$ appear are - but want to know what the electromagnetic energy is. Think for example that we are only moving a bunch of permanent magnets around and want to know how the energy changes when we do. We don't care about the condensed matter cost of making the magnets and the medium has no hysteresis and is linear as above. What then is the formula for the electromagnetic energy? Do you have a reference for this (likely) more than century old result?"}
{"id":"72387","title":"Can classical systems exhibit \"strong coupling\"?","text":"Does the concept of strong coupling mean anything in a classical setting? If strong coupling means just an inability to apply perturbative methods to the Hamiltonian, then obviously yes, we can provide examples of \"classical\" systems that cannot be handled perturbatively, such as granular materials, dusty plasmas, etc. I have only rudimentary QFT knowledge (and strong coupling is originally a QFT concept), but my intuition from my background in spectroscopy tells me that there is more to the term strong coupling than just the need for nonperturbative methods. The example I am most deeply familiar with is Forster energy transfer or FRET in biophysics. You're in weak coupling if the non- radiative rate is small compared to the radiative rate. But you're in strong coupling if the non-radiative rate is so large that you can get energy migration, where multiple energy transfer processes occur in series. In that case, the excitation is delocalized and you have to treat the network of interacting atoms as a single entity. I was under the understanding that delocalization was an essential feature of the term \"strong coupling\". Now, back to the original question, to what degree is it appropriate to use the term strong-coupling in many-body classical systems, such as granular materials, dusty plasmas, and dense colloidal glasses? These systems are classical in the sense that there is complete decoherence, and hard sphere potentials are the rule. In one sense, I want to say yes, the kinetic energy terms carry little information and most of the energy is stored in potential energy terms, but I am still hung up on the issue of delocalization. Can anyone help set me straight?"}
{"id":"73471","title":"In famous Einsteins Photoelectric effect, Why does intensity of light doesn't raise the kinetic energy of the emitting electrons?","text":"The work function of any metal is no doubt constant for it is related to electromagnetic attraction between electrons and protons. However on increasing the intensity of any light source the kinetic energy of the emitting electrons must increase, mustn't it? Let us assume there is only 1 electron in a metal surface. Let hf be the energy required to expel it out with a velocity 'v'. Again let us increase the intensity of the source keeping frequency constant. Now ' hf ' will change to 'nhf' where n = no. of photons striking on electron at a same time. Since work function is constant the only variable must be 'v' letting it increase its K.E. This clearly shows Kinetic energy of emitting electrons is directly proportional to the intensity of light source. If kinetic energy depends upon the intensity, stopping potential for a particular frequency of light for a particular metal is a variable quantity. It is true that on increasing the intensity the no.of photo electrons will increase. But what if the no.of electrons in a metallic plate is constant. Suppose I have only two electrons in a metal, on increasing intensity i.e on increasing the no.of photons , the no.of photons colliding from different sides simultaneously may increase . So, K.E of emitted electron must increase on increasing intensity, mustn't it? But experimental data doesn't show this. What is wrong? Help me out"}
{"id":"23272","title":"Has Bose-Einstein theory been considered for dark matter?","text":"Has Bose-Einstein theory been considered for dark matter? The theory would explain why no measurable radiation is emitted due to zero temperature--its lack of interaction with other matter and its gravitational lensing. Yet I haven't seen it considered. Am I missing something or have I totally misunderstood dark matter and\/or Bose-Einstein theory?"}
{"id":"113124","title":"How was it possible to beat a soldier with a full plate armor?","text":"From my point of view, it seems that a soldier armed with a **full metal plate armor** was almost invulnerable at the time their opponents yielded swords, spears or bows. I understand that it couldn't be the case, but I'm not sure about the physics behind it. More specifically:   1. How could an archer beat this soldier? Would it matter whether the archer aimed at more vulnerable spots at the soldier, or any arrow shooted with full power would do?   2. How could a swordsman or a spearman beat this soldier? Would the impact of these weapons suffice?   3. Just out of curiosity, a modern gun would penetrate into the plates that easily? One headshot would do? I'm thinking on a soldier wearing an armor like this one below, or even more bulkier. ![Italian suit of Armor, c.1450](http:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/34\/Italian_- _Sallet_-_Walters_51580.jpg\/378px-Italian_-_Sallet_-_Walters_51580.jpg)[1] [1] Courtesy of Wikipedia"}
{"id":"130188","title":"Static Friction on a Circular Road","text":"I was recently brushing up on my dynamics, when I saw the interpretation of how a car moves in a circular road. Apparently, the car can stay in a circle due to static friction acting inwards. But I don't seem to understand exactly why this force exists and where it originates. What kind of motion causes it to exist?"}
{"id":"132652","title":"Why isn't jumping against a wall an elastic collision?","text":"According to this calculator http:\/\/www.abecedarical.com\/javascript\/script_collision1d.html when low mass object hits high mass object it is reflected gaining opposite velocity almost the same as initial velocity. If I jump onto the wall why my body is not reflected? I know that collision is not fully elastic but it should be at least similar."}
{"id":"130654","title":"Inertia and momentum","text":"Question #1: Why does speed have nothing to do with inertia? Question #2: If a car hits a steel wall and stops, where did the momentum go?"}
{"id":"43860","title":"If wave packets spread, why don't objects disappear?","text":"If you have an electron moving in empty space, it will be represented by a wave packet. But packets can spread over time, that is, their width increases, with it's uncertainty in position increasing. Now, if I throw a basketball, why doesn't the basketball's packet spread as well? Wouldn't that cause its uncertainty in position to increase so much to the point it disappears? EDIT: I realize I wasn't clear what I meant by disappear. Basically, suppose the wave packet is spread over the entire Solar System. Your field of vision covers only an extremely tiny part of the Solar System. Therefore, the probability that you will find the basketball that you threw in your field of vision is very small."}
{"id":"128867","title":"What does temperature look like at the subatomic level?","text":"I am trying to get a better understanding of the definition of temperature at the subatomic level. I have a background in molecular biology with some college physics, but no deep quantum mechanics background. Everything I've found on the web (Wikipedia, Google Scholar) seems to use 'temperature' very loosely as just \" _agitation of particles_ \": more movement\/agitation of particles equals higher temperature. But what exactly does this mean? The reason I'm asking is because the use of \"particles\" in relation to temperature seems to just mean **atoms**. The increase in agitation of _atoms_ is equal to an increase in temperature. But I am asking because I don't know if this is true. So atoms are made out of protons\/neutrons\/electrons. Protons and neutrons are _composite particles_ , each made up of 3 _elementary particles_ : quarks. Also, each of these examples I've mentioned are _matter_ particles, but other particles like photons are massless. So how do they fit into temperature? Basically, how do the different subatomic particles (both composite and elementary) relate to temperature?"}
{"id":"128868","title":"Is Moon really rotating about its axis?","text":"Wikipedia says yes but on Newtonian arguments. From general relativistic point of view Moon is not rotating but moving along geodesic trajectory. And like parallel transport of a vector (pointing to the direction of motion) on a surface of a sphere along equator, from outside it seems like rotation. So, would Moon still be rotating if we remove the Earth? Would Foucault pendulum on Moon detect its rotation?"}
{"id":"6789","title":"Find true anomaly given period, eccentricity and time","text":"Is it possible to find the true anomaly of an object in a Kepler orbit given the orbital period of the object, the orbital eccentricity and the time? Assuming a two body system and the mass of the orbiting body is negligible. I'm doing this computationally so I'd like to be able to place an object on an orbital ellipse with as few evaluations as possible. I ask about these orbital elements specifically because at the time in the program that I need to calculate the true anomaly, I have these values on hand already. ## Edit I've been searching extensively for an answer to this, and found these four steps to solving position as a function of time. The thing I'm having most trouble with is solving Kepler's equation -- it seems like there should be a simpler way to do this -- I can find the velocity at each apsis, and the orbital period relatively easily. As the eccentricity approaches 1 the orbiting object would spend more time near apoapsis and less time near periapsis -- but I can't seem to put that into numbers."}
{"id":"131305","title":"What makes General Relativity conformal variant?","text":"I have a question regarding the well known fact that General Relativity is not a conformal invariant theory or to put it in other words about the fact that it is conformal variant: What are the physical assumptions in General Relativity about the gravitation that make the resulting formulation of gravity to be conformal variant? I am not asking about a mathematical reasoning because that is obvious: One performs a conformal transformation and he can check that the action and field equations are not invariant under this transformation. This mathematical part is obvious for me. I am asking what is the physical assumption that make this happen? What physical property of gravity (which is absent in Electromagnetic interaction) makes it conformal variant (at least in GR)? is it weak equivalence principle? is it the requirement of the Corresponding principle? or what? there should be some physics behind this, it can't be a mere mathematical coincidence ..."}
{"id":"55581","title":"A puzzle from 'The origin of the hidden supersymmetry'","text":"In the paper arXiv:1004.5489 The origin of the hidden supersymmetry, the author use {Qa,Qa}={Qb,Qb}=2H, {Qa,Qb}=0 for N=2 hidden SUSY, which is different from what I was taught: {Qa,Qa}={Qb,Qb}=0, {Qa,Qb}=2H. I think they are controversially different. Is any of them wrong? If not, could anyone help by explaining some more?"}
{"id":"121555","title":"in an organic semiconductor, what is the average distance travelled by an exciton?","text":"In an organic semiconductor, what is the average distance travelled by an exciton up to recombination? How is this value related to the morfology \/ structure of the organic semiconductor?"}
{"id":"128288","title":"Where does energy go when performing a useless effort?","text":"I went to school one day, so I thought I was able to get this simple one.. but it looks like I'm not anymore. :( One lonely little spaceship is resting into space. It has a small fuel capacity that it suddently burns to run away and finally reaches a speed $v$. Its tank is now empty, its kinetic energy $E = \\frac{m v^2}{2}$ is exactly what it had in reserve, and it is equal to the work performed by the constant force $F$ it has just applied to itself over the run distance $d$ with $W = Fd = E$.. Now, instead of being free, it is tied with a rope to a wall. It does try to run away, burns its fuel.. But there is nothing to do, the rope is too strong. And it ends up with an empty tank and a null speed, a null kinetic energy, and the work $F$ has performed has been null all the time, for the rope's $-F$ has prevented any acceleration.. Where is its energy gone? The entire universe had energy $E$ all the time, the spaceship had $E$ initially (chemical potential) now $0$.. what is the system that had energy $0$ initially and now $E$, and why? what happened? If it is only friction and heat within the rope and the wall, does it mean there is no way modeling a \"perfect rope\" without assuming that some energy is lost?"}
{"id":"128139","title":"Landau's derivation of a free particle's kinetic energy- expansion of a function?","text":"I was reading a bit of Landau and Lifshitz's Mechanics the other day and ran into the following part, where the authors are about to derive the kinetic energy of a free particle. They use the fact that the Lagrangian of this particle must be the same (or at most, differ by the total time derivative of a function of co-ordinates and time) in different inertial frames. > _We have > $L'=L(v'^2)=L(v^2+2\\mathbf{v}\\cdot\\boldsymbol{\\epsilon}+\\boldsymbol{\\epsilon}^2)$. > Expanding this expression in powers of $\\boldsymbol{\\epsilon}$ and > neglecting terms above the first order, we obtain_ > $$L(v'^2)=L(v^2)+\\frac{\\partial L(v^2)}{\\partial > (v^2)}2\\mathbf{v}\\cdot\\boldsymbol{\\epsilon}.$$ I think I'm ok with all the physics in this section. What I don't get is just the part I quoted above (so maybe this post is better suited for the math site, but since this is book is so physics-y I thought I'd post it here). My math is pretty rusty, so I'm not really sure- how do the authors expand the function to arrive at the above expression? It reminds me a bit of a Taylor expansion, but not very much. What's the process used to arrive at it?"}
{"id":"128135","title":"What is the physical interpretation of the fermion field Hamilton?","text":"I am still unclear what the hamilitonian of a quantized field is, but what I do know is the hamilitonian of the boson field is defined as \\begin{align} H_{\\text{boson}} &=& \\frac{1}{2}\\sum_{\\mathbf{k},\\mu=-1,1} \\hbar \\omega_{\\mathbf{k},\\mu} \\Big({a^\\dagger}^{(\\mu)}(\\mathbf{k})\\,a^{(\\mu)}(\\mathbf{k}) + a^{(\\mu)}(\\mathbf{k})\\,{a^\\dagger}^{(\\mu)}(\\mathbf{k})\\Big) \\\\\\ &=& \\sum_{\\mathbf{k},\\mu} \\hbar \\omega_{\\mathbf{k},\\mu} \\Big({a^\\dagger}^{(\\mu)}(\\mathbf{k})a^{(\\mu)}(\\mathbf{k}) + \\frac{1}{2}\\Big) \\\\\\ &=& \\sum_{\\mathbf{k},\\mu} \\hbar \\omega_{\\mathbf{k},\\mu} \\Big( N_{\\mathbf{k},\\mu} + \\frac{1}{2}\\Big) , \\end{align} however something that appears scarcely mentioned in the literature is the Hamiltonian of the fermion field. Interpolating from the boson field Hamilton, my guess is the Hamiltonian of the fermion field would be \\begin{align} H_{\\text{fermion}} &=& \\frac{1}{2}\\sum_{\\mathbf{k},\\mu=-1,1} \\hbar \\omega_{\\mathbf{k},\\mu} \\Big({b^\\dagger}^{(\\mu)}(\\mathbf{k})\\,b^{(\\mu)}(\\mathbf{k}) + b^{(\\mu)}(\\mathbf{k})\\,{b^\\dagger}^{(\\mu)}(\\mathbf{k})\\Big) \\\\\\ &=& \\frac{1}{2}\\sum_{\\mathbf{k},\\mu=-1,1} \\hbar \\omega_{\\mathbf{k},\\mu} \\Big({b^\\dagger}^{(\\mu)}(\\mathbf{k})\\,b^{(\\mu)}(\\mathbf{k}) - {b^\\dagger}^{(\\mu)}(\\mathbf{k})\\,b^{(\\mu)}(\\mathbf{k})+1\\Big) \\\\\\ &=& \\sum_{\\mathbf{k},\\mu} \\hbar \\omega_{\\mathbf{k},\\mu} \\Big( 0_{\\mathbf{k},\\mu} + \\frac{1}{2}\\Big) . \\end{align} I found $H_{\\text{fermion}}$ to be very different from $H_{\\text{boson}}$, because $H_{\\text{boson}}$ depends on the number of bosons $N_{\\mu,\\boldsymbol{k}}$, however $H_{\\text{fermion}}$ is independent of the number of fermions in each of its states, but rather seams to equal the vacuum energy only. Is $H_{\\text{fermion}}$ real and what is its physical interpretation?"}
{"id":"134085","title":"Can induced-emf be divided?","text":"A conductor's area is reduced by half or possibly more. I calculated the induced-emf based on the original area unreduced, does it make sense to divide the induced-emf value to the reduction factor? Say the area was reduced by half,therefore, diving the induced-emf value by \/2? I assume by reducing the area the induced-emf should be less?"}
{"id":"38513","title":"what happens when I roll a gyroscope along its axis of spin","text":"Say:   1. I have a gyroscope that is spinning in the xy plane along the z axis.   2. I then roll its spinning axis by some angle theta Now I know the gyroscope will resist my attempting to change its axis of rotation. The question is:   1. When I perform work to change the axis of rotation.   2. Where will that work be transferred ? ... will the rotation speed of the disk slow down? To clarify: see the diagram below, the axis of rotation is around the Z axis - the disk is in the x-y plan. Now say the gyroscope is in a frame that allows it the disk's rotation axis to rotate in x-z plane -- which means the pivot is in the yaxis -- so assume there is a frame around the disk and there is a rod along the y axis on wither side of the disk attached to the frame. So, if I apply a fore in the -x direction at point 'a' the gyroscope will rotate and I will have to move the force a distance delta-x so clearly work is being done (lets assume no friction). The question is where does this work go?                                     \/|\\ z-axis                    | 'a'   |                    |       |                   \/ | \\     |______\\ x-axis                  \\ _ \/      \\     \/                    |         \\                    |         _\\| y-axis      Thanks."}
{"id":"102877","title":"Why do floating objects attract?","text":"Objects floating on the surface of calm water seem to be attracted to one another and tend to come together in a drift. Is this attraction like gravity?"}
{"id":"102920","title":"Calculating the Potential from the E-Field","text":"I find that often times I'll be tripped up by questioning whether or not I can do something mathematically, and be unable to come up with a satisfying answer. This is, unfortunately, one of those times. I'm told: > A uniform electric field, $\\vec{E} = E_0\\hat{x}$. What is the potential, > expressed using cylindrical coordinates, $V(s,\\phi,z)$? My first course of action is: We know... $$|r| = \\sqrt{x^2 + y^2 + z^2} = \\sqrt{x^2} = x = E_0$$ $$\\tan^{-1}{\\frac{y}{x}} = \\theta = 0$$ $$z = z = 0$$ So the electric field only has a component in the $\\hat{r}$ direction. Now, we know that $\\vec{E} = -\\nabla V(r, \\phi, z) = - \\frac{\\partial V}{r} - \\frac{1}{r}\\frac{\\partial V}{\\theta} - \\frac{\\partial V}{z}$ So, I think \"Oh. I just have to integrate.\" ... but over what? Do I integrate three times, once w.r.t. radius, then phi, then z? I'm pretty sure that won't give me the right answer. If I decide to express $\\nabla V$ in terms of cartesian coordinates, I get $- \\nabla V(x,y,z) = E_0 \\hat{x}$ ... but the question still remains. I feel like this is definitely the _easy_ part of the problem, and I can often do the more complicated parts—it's just small things like this often throw me off. How would I go about extracting the potential from either of those equations? I know I have to integrate, but... where?"}
{"id":"109281","title":"Who was the R. Dolen behind Dolen-Horn-Schmid duality?","text":"Can you provide biographical information for R. Dolen, known for Dolen-Horn- Schmid duality?"}
{"id":"36165","title":"Electric scooter acceleration","text":"I tested two electric scooters with almost same power (1500 W), but very different performances: A \"Zem Star 45\" (1500 W, 60 Nm) can reach 50 km\/h in \"a few seconds\" (not measured actually), An \"Ingaeta Ingo\" (1500 W, ??? Nm) took more than 10 seconds to reach 50 km\/h. How much does torque affect acceleration performance? Given two identical scooters (same weight, same engine power, same wheels size), rgardless of wheels and air friction, how can I calculate acceleration given the torque on a flat street?"}
{"id":"33534","title":"Is eternal inflation Lorentz invariant?","text":"Start without general relativity. Consider a metastable vacuum over good ol'-fashioned Minkowski space. It decays. A bubble forms and the domain wall expands. The domain wall is timelike, and accelerates, driven by the difference in pressure between both phases. Its speed approaches the speed of light asymptotically. Far away, it looks like an expanding light cone, but it stops looking so around the moment of decay. Roughly speaking, this breaks Lorentz covariance, and this picks out an approximate frame of reference. It might be argued that the metastable phase prior to decay is Lorentz invariant. If so, there is only one Lorentz invariant measure over the space of future- oriented unit 4-velocity vectors up to a multiplicative factor. This space has infinite measure, and the typical decay velocity of a decay would have to be infinite. Nonsense. The point to note is we have to specify the initial conditions of the metastable vacuum. That can't be infinitely in the past, as the lifetime, while exponentially long, is still finite. A Lorentz invariant initial boundary can only be an expanding light cone, or a hyperboloid. However, both have infinite volumes, and to regulate, we need to cut off these surfaces at some finite radius. This radius has to be less than the decay lifetime. This cutoff roughly picks out a frame of reference. At any rate, affine parameters on light cones are tricky. Boost them enough, and you can get an exponential change in the affine parameter, and in this boosted frame, the likelihood of crossing a bubble nucleation approaches one in probability. Turn on general relativity. Does eternal inflation break Lorentz invariance?"}
{"id":"89975","title":"If time isn't continuous, what is the best-known upper bound on the length of time intervals?","text":"There have been several questions about whether time is continuous or not and it seems like the answer isn't currently known. I know quantum mechanics treats time as continuous and any mathematics that involves integrating over some time interval treats time as continuous too. Surely though there are experiments that are quite sensitive to discrete time with large intervals. It seems the shortest laser pulse so far is only 67 attoseconds ($67 \\times 10^{-18}\\: \\mathrm{s}$) but wouldn't this experiment actually constrain time intervals to much less than that? Are short laser pulses even a good experiment to determine if time is discrete or not? **So, assuming time isn't continuous, what is the best-known upper-bound on time intervals? Also, which experiments have done the best to constrain how non-continuous time could be?**"}
{"id":"108855","title":"Ricci tensor of direct product of manifolds","text":"Imagine I have a (Lorentzian) manifold with a metric $\\left[ {\\begin{array}{cc} g_{\\mu\\nu} &0\\\\\\ 0&g_{mn}\\\\\\ \\end{array} } \\right]$ Will the Ricci tensor be also block diagonal without any mixing of the two manifolds?"}
{"id":"41578","title":"Is Force equal to components in different dimensions of Force or distance of those components","text":"I'm trying to understand to find my homework problem. Its a simple concept of finding force. I wonder if you have two formulas then is the total Force = force from x + force from y. Or $\\sqrt{forceX^2+forceY^2}$. And why. It would make sense to me for it to be the sum of two forces if its not so why not?? I have the actual example: 3.00-kg object is moving in a plane, with its x and y coordinates given by $x=5t^2 - 1$ and $y = 3t^3+2$, where x and y are in meters and t is in seconds. Find the magnitude of the net force acting on this object at $t = 2.00 s$ so I either get 66 by addition of the forces. Or 46.861 by distance formula.. Which one is correct and why. (feel free to use the example i provided or just explain the concept without it. I just wonder which way of finding force is correct and why)"}
{"id":"129427","title":"What physical phenomenon occured in this experiment?","text":"I noticed a strange behavior of my experiment. I have a glass container with a fully saturated solution of CuSO4, which I use to grow CuSO4 crystals, by waiting for the water to evaporate. Recently I noticed that crystals were build above water surface and outside of the container. After some time I noticed that half of my solution ended up on the table and floor creating a big blue pool. It looked almost like the behavior of a super-fluid, which can creep out of its container. I removed the crystals using a paper towel, but after some time the same thing happened. Once the crystals go over the edge and below the water line on the other side, the fluid goes outside quite quickly. Just like if you put an inverted U-pipe into a container. Once you force the liquid to start going, it will continue to flow by itself. ![CuSo4 creeping out of a container](http:\/\/i.stack.imgur.com\/juqYQ.jpg) Can you explain what physical phenomenon(s) is\/are happening here? Is it some sort of capillary action?"}
{"id":"76210","title":"Deriving the Hamiltonian density for a free scalar field","text":"I'm working through my old notes on QFT (cf. Ref 1) and I'm not quite sure how to approach the derivation of the Hamiltonian density for a free scalar field (question 2.3 on page 19) and the subsequent derivation of the components of the energy-momentum four-vector in terms of the new field operators. Could you point me in the right direction? $${\\mathcal{H}~=~\\frac{1}{2}\\left[\\left(\\partial_t\\phi\\right)^{2}+\\left(\\nabla\\phi\\right)^{2}+m^{2}\\phi^{2}\\right]\\textrm{.}}$$ References:   1. M Dasgupta, _AN INTRODUCTION TO QUANTUM FIELD THEORY,_ Lecture presented at the School for Experimental High Energy Physics Students Somerville College, Oxford, September 2009."}
{"id":"95979","title":"Equation 7.22 in Peskin & Schroeder: writing the Fourier transform of a two-point function as a series of 1PI diagrams","text":"In Peskin and Schroeder's QFT book, on page 219, there is the following equation:![enter image description here](http:\/\/i.stack.imgur.com\/nC46a.png) The heading to the equation is: \"The Fourier transform of the two-point function can now be written as\". Could someone help me formally prove this formula? I don't really get why the perturbation expansion decomposes into 1PI's."}
{"id":"13991","title":"two distant galaxies seen from earth","text":"From the Earth If we observe two galaxies that are diametrically opposed and each 1000ly far from the earth. the separation distance between the galaxies will be 20000ly? really question is: if the galaxies were separated from Earth 10Gly. then the separation distance between galaxies would 20Gyr?"}
{"id":"13996","title":"Finite difference formulation of the heat equation with thermal conductivity in 1D","text":"This may seem trivial, but I'm having some trouble deriving the finite difference form of the heat equation with a thermal conductivity function $a(x)$ depending on $x$: $$\\frac{\\partial u(x, t)}{\\partial t} = \\frac{\\partial }{\\partial x}[a(x)\\frac{\\partial u(x, t)}{\\partial x}]$$ I find some trouble deriving the right hand side. Let $h$ be the space step, and $x_{i−1}=x_i−h$ and $x_{i+1}=x_i+h$. I want to evaluate the right hand side on node $i$. If I take centered differences I get: $$\\begin{align*}\\left\\\\{\\frac{\\partial}{\\partial x}\\left[ a(x)\\frac{\\partial \\, u(x, t)}{\\partial x}\\right]\\right\\\\}_i &= \\frac{\\left[a(x)\\frac{\\partial\\, u(x, t)}{\\partial x}\\right]_{i+1\/2} - \\left[a(x)\\frac{\\partial \\, u(x, t)}{\\partial x}\\right]_{i-1\/2}}{h} \\\\\\ &= \\frac{a_{i+1\/2}\\left[\\frac{u_{i+1}-u_{i}}{h}\\right] - a_{i-1\/2}\\left[\\frac{u_{i}-u_{i-1}}{h}\\right]}{h} \\end{align*}$$ Is this ok? The conductivities should be in between the nodes I'm calculating? (intuitively it seems this is reasonable). Is there a way, for example by using forward or backward differences, to use conductivity nodes that overlap with the nodes I'm calculating (I calculate only the interger indexed nodes)? Can I mix a centered difference approach for one derivative and a backward or forward approach for the other one? I would really appreciate any hint. Thanks in advance, Federico"}
{"id":"119443","title":"Modelling of nuclear motions (Classification) after invoking the BO approximation","text":"I know that after invoking the Born-Oppenheimer approximation, the nuclei will move on the adiabatic potential provided by the electronic energy (also called potential energy surface (PES)). Such nuclear motions, can be described following quantum-mechanical, classical or hybrids methods on a fitted potential energy surface (PES) or when we want to avoid any analytical representation or interpolation of the PES, we can resort in “on the fly” calculation of the potential. My question is, what are those quantum, classical or hybrid theories\/methods that are in connection with an statistical description? and how can i make a general classification o each of them?, at least the most important methods."}
{"id":"119444","title":"Blackbody and standing waves","text":"I'm reading articles about black body radiation and why classical mechanics fails to explain it. My question is: Why do EM waves have to be standing wave in a cavity?"}
{"id":"43326","title":"A Derivation of Ehrenfest's Theorem in a particular case","text":"What are the missing lines in the integration? $$\\frac{\\text d \\langle {p} \\rangle}{ \\text{d} t} $$ $$= \\frac{\\text d}{\\text d t} \\int\\limits_{-\\infty}^{\\infty} \\Psi^* \\left( \\frac{\\hbar}{i}\\frac{\\partial}{\\partial x} \\right) \\Psi ~\\text d x$$ $$= \\frac{\\hbar}{i}\\int\\frac{\\partial }{\\partial t} \\left(\\Psi^*\\frac{\\partial\\Psi}{\\partial x}\\right)~\\text d x $$ $$= \\frac{\\hbar}{i}\\int \\frac{\\partial \\Psi^*}{\\partial t}\\frac{\\partial \\Psi}{\\partial x} +\\Psi^*\\frac{\\partial }{\\partial x}\\frac{\\partial \\Psi }{ \\partial t} ~\\text {d} x$$ $$= \\frac{\\hbar}{i}\\int\\left( -\\frac{i\\hbar}{2m}\\frac{\\partial^2 \\Psi^*}{\\partial x^2}+\\frac{i}{\\hbar} V\\Psi^*\\right)\\frac{\\partial \\Psi}{\\partial x}+\\Psi^* \\frac{\\partial}{\\partial x} \\left(\\frac{i\\hbar}{2m} \\frac{\\partial^2 \\Psi }{\\partial x^2} -\\frac{i}{\\hbar}V\\Psi \\right)~\\text d x$$ $$=\\int\\left(V\\Psi^*-\\frac{\\hbar^2}{2m}\\frac{\\partial^2 \\Psi^*}{\\partial x^2}\\right) \\frac{\\partial \\Psi}{\\partial x}+\\Psi^*\\frac{\\partial}{\\partial x}\\left(\\frac{\\hbar^2}{2m}\\frac{\\partial^2 \\Psi}{\\partial x^2}-V\\Psi\\right)\\text d x$$ $$=\\left. \\left(V\\Psi^*-\\frac{\\hbar^2}{2m}\\frac{\\partial^2 \\Psi^*}{ \\partial x^2} \\right) \\Psi \\right|_{-\\infty}^{\\infty}-\\int\\left(\\frac{\\partial}{\\partial x} (V\\Psi^*)-\\frac{\\hbar^2}{2m}\\frac{\\partial^3 \\Psi^*}{ \\partial x^3}\\right)\\Psi \\text d x+ \\left.\\Psi^*\\left(\\frac{\\hbar^2}{2m}\\frac{\\partial^2 \\Psi}{\\partial x^2} - V\\Psi\\right)\\right|_{-\\infty}^\\infty-\\int\\frac{\\partial\\Psi^*}{\\partial x} \\left( \\frac{\\hbar^2}{2m}\\frac{\\partial ^2 \\Psi}{\\partial x^2} - V \\Psi \\right)\\text d x$$ $$=0 + \\int\\left(\\frac{\\hbar^2}{2m}\\frac{\\partial^3 \\Psi^*}{ \\partial x^3}-\\frac{\\partial}{\\partial x} (V\\Psi^*)\\right) \\Psi \\text d x+0+\\int\\frac{\\partial\\Psi^*}{\\partial x}\\left(V\\Psi - \\frac{\\hbar^2}{2m}\\frac{\\partial^2 \\Psi}{\\partial x^2}\\right) \\text d x$$ $$=\\int\\frac{\\hbar^2}{2m} \\left(\\frac{\\partial^3\\Psi^*}{\\partial x^3}\\Psi -\\frac{\\partial\\Psi^*}{\\partial x} \\frac{\\partial^2\\Psi}{\\partial x^2}\\right)+\\frac{\\partial\\Psi^*}{\\partial x}(V\\Psi )-\\frac{\\partial}{\\partial x}(V\\Psi^*)\\Psi\\text d x $$ $$\\vdots$$ $$=\\int \\frac{\\hbar^2}{2m}\\left( \\Psi^* \\frac{\\partial^3 \\Psi}{\\partial x^3} -\\frac {\\partial^2 \\Psi^*}{\\partial x^2} \\frac{\\partial \\Psi}{\\partial x} \\right)+\\left( V \\Psi^* \\frac{\\partial \\Psi}{\\partial x}-\\Psi^* \\frac{\\partial}{\\partial x} (V \\Psi) \\right)~\\text d x$$ $$\\vdots $$ $$= \\int \\left( V \\Psi^* \\frac{\\partial \\Psi}{\\partial x}- \\Psi^* \\frac{\\partial V}{\\partial x} \\Psi-\\Psi^*V \\frac{\\partial \\Psi}{\\partial x} \\right)~\\text d x$$ $$= \\int\\limits_{-\\infty}^{\\infty} -\\Psi^* \\frac{\\partial V}{\\partial x} \\Psi ~\\text {d} x$$ $$=\\left\\langle - \\frac{ \\partial V }{\\partial x} \\right\\rangle $$"}
{"id":"92388","title":"2 Time Dimensions","text":"Our universe has 4 dimensions, 3 spacial and 1 temporal. We can add another spacial dimension and use math to explore it. I cannot think of any reason why we cannot add add another temporal dimension in the same way. So, what would a universe with 2 temporal dimensions be like? These are some thought I had about physics in 2 temporal dimensions. With one spacial dimension, forwards is the only movement possible. With two spacial dimensions, movement in many different ways is possible. It would follow then that with two temporal dimensions time travel in many different 'directions' would be possible. The formula for velocity is displacement\/time. With two times, how would velocity be effected? What would happen to light, which must travel at a constant speed? Frankly, I have no idea a universe with 2 times is like, but I very much would like to know."}
{"id":"16015","title":"Laser frequency tripling","text":"I've heard that common way to get 354nm laser is to triple 1064nm Nd:YAG one. But how to make non-linear crystal to do tripling instead of doubling? What are the best crystals for tripling and what's the efficiency?"}
{"id":"16014","title":"Defining a Riemannian manifold - made easy?","text":"In the context of GTR spacetime, I'm trying to get the basic idea of a Riemannian manifold clear in my mind. Apologies for the longwindedness. Question 1. Is this a reasonable, simplified summary of the steps needed to define such a manifold?   1. Start with an amorphous collection of points.   2. Find the minimum number of parameters needed to locate each point in space. That number will be the dimension of the manifold.   3. Construct a coordinate system to locate each point in space. I may need more than one coordinate system to to cover the whole manifold. I need a “good” coordinate system so that each point is uniquely described.   4. Decide on a metric that gives the infinitesimal distance between two adjacent point. The metric need not be the same all over the manifold (as $g_{\\mu\\upsilon}$ isn't in spacetime). In that case I would have a metric tensor field defined over the manifold.   5. The manifold has to be continuous and differentiable (not quite sure what this means, but I assume it's the obvious - I have to be able to differentiate it everywhere).   6. I can now start constructing scalar fields, covariant and contravariant vectors, tensors etc. Question 2. Assuming I do all the above with a flat plane (a table top for example), using Cartesian coordinates. Can I impose any kind of metric to my new “table top” manifold? Could I just make a weird one up: $$\\left(\\begin{array}{cc} x^{2} & \\sin xy\\\\\\ x+y & 13y \\end{array}\\right)$$ for example, instead of the Euclidean $$\\left(\\begin{array}{cc} 1 & 0\\\\\\ 0 & 1 \\end{array}\\right)$$ Thank you."}
{"id":"110209","title":"Easy proof of Noether's theorem?","text":"Where could I find an easy proof of Noether's theorem? I mean I know that the variation must be $ 0=\\delta S = (EULER-LAGRANGE)+ (CONSERVED\\, \\, \\, CURRENT) $ for the case of a particle $q(t)$. I know how to obtain it by I have doubts for the case of fields $ \\phi (x) $, any hint? I have checked several books but I cannot find any easy proof of Noether's theorem anywhere; they use too complicated methods."}
{"id":"94318","title":"Higgs field in space around us?","text":"Is the higgs field in space around us? I understand it as that the higgs field has a constant value on every space time point, is that right? And this value is the vacuum expectation value. This value is responsible for giving mass to the elementary particles, like the fermions, right? In the Lagrangian for e fermion there is a term $g \\phi_0 \\bar \\psi \\psi$. $g$ is the Yukawa coupling constant and $\\phi_0$ is the vacuum expectation value, right?"}
{"id":"90804","title":"Does brightness of the light remains constant with the variation of the distance with respect to the source in vaccum?","text":"I have this question going in my mind from many days, i.e why brightness of light emitted from any light source around us decreases with distance? The brightness of light from tube light, streetlight, etc or any light from any source around us, decreases with the distance. As my teacher has taught me that light is a form of radiation, I thought this decrease in brightness may be due to absorption of radiation in the medium. I don't know whether it is correct or wrong. If it is wrong, please explain _why there will be decrease in brightness of light with increase in distance from the light source?_   If we assume that decrease in brightness of light is due to absorption of radiation in the medium, then _in vaccum (where we can assume no energy dissipation), would there be no decrease in brightness of light with respect to distance from the light source?_   ![enter image description here](http:\/\/i.stack.imgur.com\/x6nhs.jpg)"}
{"id":"60244","title":"Higher order covariant Lagrangian","text":"I'm in search of examples of Lagrangian, which are at least second order in the derivatives and are covariant, preferable for field theories. Up to now I could only find first-order (such at Klein-Gordon-Lagrangian) or non-covariant (e.g. KdV) ones. Also some pointers to the literature about general properties of such systems are welcome. Thanks"}
{"id":"11747","title":"What is a simple physics application of the reciprocal of a square root?","text":"I learned yesterday that the inverse of square roots is used to calculate the vectors of surface normals in 3d graphics. It seems like such a mind- bogglingly simple idea, and it leads me to wonder if it hasn't found use elsewhere. Is there anywhere in physics where one might need to calculate the reciprocal of a number's square root?"}
{"id":"64030","title":"What is the Lorentz tensor with a superscript and subscript index?","text":"I have been reading about symmetries of systems' actions, e.g. the Polyakov action, and I have encountered Lorentz transformations of the form: $\\Lambda^{\\mu}_{\\nu} X^{\\nu}$. I am moderately familiar with $\\Lambda$, the Lorentz matrix. If the indices are in superscript then it is the inverse of $\\Lambda_{\\mu \\nu}$. However, what is $\\Lambda^{\\mu}_{\\nu}$ in terms of the Lorentz matrix? I have chosen mathematical physics as the tag, as I do not think any discipline of pure mathematics is appropriate based on the context. Please correct me if I am wrong."}
{"id":"64035","title":"Cosmological triangle with PLANCK results","text":"Is there an updated version of the cosmological triangle with recent PLANCK results included? ![](http:\/\/i.stack.imgur.com\/Q31Pq.png)"}
{"id":"96372","title":"Maximum potential of a van de graaff generator","text":"I know that we can calculate the maximum potential of a van de graaff generator by ( radius* electric field in which corona discharge begin to form in the surrounding gas( according to wikipedia 30 kv\/cm)). What if i covered the sphere with Mica with a dielectric strength ( according to wikipedia) 1.18 Mv\/cm , will that increase the maximum potential of the sphere? I know it will be hard to discharge the sphere in this case , but is it possible?"}
{"id":"45642","title":"Calculating partition function of ultra-relativistic 1D gas","text":"This is a problem (Problem 3.16) from the book Statistical Mechanics 2nd Ed. by Pathria. In the problem I have to calculate the partition function of an ultra-relativistic 1D gas ($E_i=cp_i$) consisting in $3N$ particles moving in one dimension. I know that the partition function is given by $$Q_{3N}=\\frac{1}{(3N)!h^{3N}} \\int e^{-\\beta H(q,p)}d\\omega,$$ where $d\\omega$ denotes a volume element of the phase space. In this case $$d\\omega=dq_1dp_1\\cdots dq_{3N}dp_{3N},$$ and $$H(q,p)=\\sum_{i=1}^{3N}cp_i.$$ Then, making the substitution I find that $$Q_{3N}=\\frac{L^{3N}}{(3N)!h^{3N}} \\left[\\int_{-\\infty}^{+\\infty} e^{-\\beta c p_j}dp_j\\right]^{3N}.$$ $L$ being the \"length\" of the space available. But I'm pretty sure that this integral does not converge. Where am I wrong?"}
{"id":"99536","title":"How much power would a space craft's magnetic shield require?","text":"I've read over the decades that a magnetic shield might protect a spacecraft from cosmic radiation. Its a fascinating idea that might only be theory or science fiction at the moment. In regards to that here is an article just in case someone isn't sure what I mean: http:\/\/physicsworld.com\/cws\/article\/news\/2008\/nov\/06\/magnetic-shield-could- protect-spacecraft The article even mentions simulations about midway into the article. Does anyone know if research has continued? My big question though, is how much electrical power would be required to protect a space vessel in this way? How large would the field be in order to be useful? Better yet, please use some practical objects that anyone can wrap their head around: What if you were using similar technology to protect the Apollo Command Module? Or the Discovery One from 2001 (140 meters long). _In laymans terms, how much electrical power would it take to create a magnetic shield?_"}
{"id":"99468","title":"how do compression and expansion of air transfer energy(sound) and why it is adiabatic not isothermal?","text":"Suppose a vibrating fork exerts force to the air particles to compress which leads to the increase of internal energy (heat).This heat or energy given by the fork is given to the next layer of air and in this way, this energy makes sound. But if it was so, it will be isothermal. And Laplace told it was adiabatic. How can it be? I am confused. The heat energy infact is the energy given by the fork. If it is adiabatic, how can the air pass this energy ?"}
{"id":"99813","title":"propagation of sound energy","text":"In my previous questions like 'how do compressions and relaxation carry energy...' and 'how are they not isothermal...' and 'where does the heat come from in the compressed gas...',i've made enquiries reagarding the phenomenon of sound energy propagation and got answers from experts.So i am now writting my concept:when a vibrating fork compresses the air,it does work on it increasing the K.E. of the layer.When the fork goes opp. side,it works on the compressed layer and relaxes it increasing theP.E.Meanwhile,during expansion,the layer works on the next layer and compresses it using the excess kinetic energy.And the next layer does the same to its next layer by compressing using that K.E. given to it by the fork via the first layer.Ultimately,the last layer works on our ear drum to produce sound using the kinetic energy.The question is am i right in my assumption?Plz help..."}
{"id":"16901","title":"Equation of motion for the reduced density matrix","text":"The equation of motion for the density matrix of a many body isolated quantum system is the von Neumann's equation: $\\dot{\\rho }(t)=i[\\rho (t),H]$. How about the equation of motion for the reduced density matrix of one particle, is there a general procedure to obtain it?"}
{"id":"80399","title":"Is this possible? LCD right in front of the eye","text":"I came across these swimming goggles with a LCD display integrated right on the goggle itself. http:\/\/www.psfk.com\/2011\/09\/smart-goggles-enable-swimmers-to-track-and-record- swim-pace.html\/my_pace_goggles2 Does this look possible as shown in the design? Would I need a lens over the LCD to make it easier to focus on? How thick and\/or curved would that lens need to be? Any advice appreciated!"}
{"id":"16904","title":"How do I find equations for the distance of the planets relative to one another?","text":"I know I should be able to piece together some basic Newtonian equations for this, but I'm not sure where to start. I want to be able to choose one planet as the center and calculate its distance from another planet over time as they both rotate the sun. First I need to even find equations for the motion of the planets, then... magic? Thanks for any guidance!"}
{"id":"80390","title":"Are neutrino stars theoretically possible?","text":"Since neutrinos   1. have a small mass and   2. are affected by gravity, wouldn't it be theoretically possible to have such a large quantity of them so close to each other, that they would form a _kind_ of a stellar object, i.e. one that would keep itself from dissolving due to the large gravity. If such objects were possible, how would they interact with rest of the world? Would they be invisible (dark matter?) because of neutrinos' lack of electromagnetic charge? Would this lack also make ordinary matter pass through them, or would the Pauli exclusion principle prevent this passing through due to the high density of neutrinos?"}
{"id":"12211","title":"Physics Equivalent of Physical Chemistry (P-Chem)","text":"Before I take P-Chem I would like to understand how physicists view the same material. What Physics course(s) should I take and\/or books should I read to learn the same material from a physicist's perspective? Would thermodynamics be equivalent? Background: I have done the math: Calculus 1-3 and differential equations and have already taken general physics 1+2 and 2 semesters of modern physics (relativity, QM...)."}
{"id":"95199","title":"Homework: A parachutist jumps from a plane","text":"I need help with the following problem. I think I almost have it . . . A parachutist bails out and freely falls a distance of $y_1$. Then the parachute opens and thereafter, the parachutist decelerates at a rate of $a_2$. She reaches the ground with a speed of $v_2$. Find her average speed for the fall. (The answer should be $18.5\\ \\text{m\/s}$.) Note: In the diagram, replace \"x\" with \"y\". :) ![http:\/\/i.imgur.com\/OPTZfWI.png](http:\/\/i.stack.imgur.com\/5lnSe.png) $ \\text{Let upwards be the positive y direction.}\\\\\\ \\text{Givens:}\\\\\\ y_1 = -59.7\\ \\text{m}\\\\\\ v_0 = 0\\ \\text{m\/s}\\\\\\ v_2 = -3.41\\ \\text{m\/s}\\\\\\ a_1 = -9.8\\ \\text{m\/s$^2$}\\\\\\ a_2 = 1.60\\ \\text{m\/s$^2$}\\\\\\ \\text{Solve for $t_1$:}\\\\\\ y_1 - y_0 = \\frac{1}{2}a_1t^2+v_0t\\qquad\\text{for}\\ \\ y_1 = 0; v_0 = 0; t=t_1.\\\\\\ t_1 = \\pm\\sqrt{\\frac{2y_1}{a_1}}\\\\\\ t_1 \\approx \\pm 3.4905\\ \\text{s}.\\\\\\ \\text{Plug $t_1$ into the velocity equation for $v_1$ to find $v_1$:}\\\\\\ v_1 = a_1t_1 + v_0\\\\\\ v_1 \\approx -34.207.\\\\\\ \\text{Plug $v_1$ into the velocity equation for $v_2$ to find $t_2$:}\\\\\\ v_2 = a_2t_2 + v_1\\\\\\ t_2 = \\frac{v_2 - v_1}{a_2}\\\\\\ t_2 \\approx 19.248\\ \\text{s}.\\\\\\ \\text{Plug $t_2$ into the position equation for $y_2$ to find $y_2$.}\\\\\\ y_2 = \\frac{1}{2}a_2t_2^2+v_1t_2 + x_1\\\\\\ y_2 \\approx -422\\ \\text{m}.\\\\\\ \\text{Now solve for $\\lvert\\overline{v}\\rvert$:}\\\\\\ \\lvert\\overline{v}\\rvert = \\lvert\\frac{x_2 - x_0}{t_2 - t_0}\\rvert\\\\\\ \\lvert\\overline{v}\\rvert \\approx 21.9\\ \\text{m\/s} \\neq 18.5\\ \\text{m\/s}. $ My solution is incorrect. What am I doing wrong?"}
{"id":"63264","title":"Independent systems and Lagrangians","text":"**Definition 1:** The notion of independent systems has a precise meaning in probabilities. It states that the (joint) probability or finding the system ($S_1S_2$) in the configuration ($C_1C_2$) is equal to the probability of finding the system ($S_1$) in the configuration ($C_1$) times the probability of finding the system ($S_2$) in the configuration ($C_2$). **Definition 2:** However, in we consider fields systems, the practical tool is Lagrangians. So I should say that 2 systems are independent if : $$ Lagrangian (S_1S_2) = Lagrangian (S_1) + Lagrangian (S_2)$$ **The question:** Now, what is the relationship between these 2 definitions? They could be only compatible, or they could be equivalent in the field domain. Is there a way to \"demonstrate\" the latter from the former ?"}
{"id":"62795","title":"Where do high-energy neutrinos come from?","text":"Last week the IceCube South Pole Neutrino Observatory published a press release reporting the possible discovery of two neutrinos with energies of over 1 PeV. Would anyone here be willing to help me understand the significance of this announcement? Although it was reported in the press that these neutrinos originated from outside our galaxy, the original press release seemed to say that this might possibly be the first sighting of 'astrophysical' neutrinos. Are there other observations of extragalactic neutrinos? Are there any conjectures as to processes which might generate? How would a neutrino even be accelerated if it is unaffected by magnetic fields?"}
{"id":"62794","title":"Mathematical explanation of quantum teleportation","text":"I am now studying quantum teleportation. I get what the process is like but I'm wondering why it happens this way. You've got two entangled particles A and B whose wavefunctions are entangled. You also have a third particle C, which is the one you want to teleport. You get C entangled with either of the two and final result is that the wavefuction of one of the two A and B (depending on which you entangled C with) becomes the same as the C's wavefunction is.   1. Why does it occur this way (mathematically explained, if possible)? Why does the wavefunction of A or B becomes that of C?   2. What is the equation explaining this process (addition of wavefunctions, maybe)?"}
{"id":"62796","title":"Electric Field in Dieletric","text":"I have a simple doubt. The textbook I'm working with says that if $E_0$ is the electric field in some region without diletric, then if we introduce a dieletric of constant $k$ the new field will simply be $E=k^{-1}E_0$. Is it always true? I mean, suppose we have concentric spheres of radii $a$ and $b$ being $a$ the radius of the inner sphere and $b$ the radius of the outer sphere and charge $Q_a$ in the inner and $Q_b$ in the outer. If there's no dieletric between them the field in the space between at a distance $r$ from the center will be given by Gauss' Law as: $$E_0=\\frac{1}{4\\pi\\epsilon_0} \\frac{Q_a}{r^2}$$ Now if we insert a dieletric on the space between is it true that the field on the dieletric will _simply_ be: $$E=\\frac{1}{4\\pi k \\epsilon_0}\\frac{Q_a}{r^2}$$ Thanks very much in advance."}
{"id":"8852","title":"How can I model buckyballs sticking together? (Tiny spherical magnets)","text":"Buckyballs are a desktoy that consist of 216 tiny spherical magnets. You can makes all kinds of interesting shapes and structures out of them (do a Google search). I want to model them in a physics simulation, but I'm not sure how to do it. Right now, I'm thinking calculate the sum of the forces between each of the buckyball's poles and each of the other buckyballs' poles, and update its linear momentum. Then calculate the torque that each of those forces causes about the magnet's center, and use that to update angular momentum. The problem I'm having is how to model the magnets touching. It's easy in real life to make a long chain or a hexagon. How should I model the magnets actually running into each other to mimic this behavior?"}
{"id":"8850","title":"Alcubierre Drive - Clarification on relativistic effects","text":"On the Wikipedia article on the Alcubierre drive, it says: > Since the ship is not moving within this bubble, but carried along as the > region itself moves, conventional relativistic effects such as time dilation > do not apply in the way they would in the case of a ship moving at high > velocity through flat spacetime relative to other objects. And... > Also, this method of travel does not actually involve moving faster than > light in a local sense, since a light beam within the bubble would still > always move faster than the ship; it is only \"faster than light\" in the > sense that, thanks to the contraction of the space in front of it, the ship > could reach its destination faster than a light beam restricted to > travelling outside the warp bubble. I'm confused about the statement _\"conventional relativistic effects such as time dilation do not apply\"_. Say Bob lives on Earth, and Jill lives on a planet in Andromeda, and we'll say for the sake of argument that they're stationary. If I were to travel from Bob to Jill using an Alcubierre drive such that the journey would take me, say, 1 week from my reference frame... how long would Jill have to wait from her reference frame? Do the time dilation effects cancel out altogether? Would she only wait 1 week?"}
{"id":"4972","title":"Why (in relatively non-technical terms) are Calabi-Yau manifolds favored for compactified dimensions in string theory?","text":"I was hoping for an answer in general terms avoiding things like holonomy, Chern classes, Kahler manifolds, fibre bundles and terms of similar ilk. Simply, what are the compelling reasons for restricting the landscape to admittedly bizarre Calabi-Yau manifolds? I have Yau's semi-popular book but haven't read it yet, nor, obviously, String Theory Demystified :)"}
{"id":"34312","title":"Is dark matter really present around the sun?","text":"Recently I read an article that there is dark matter around the sun but if it is so, than why can we see it clearly. If it is called matter than it shall show some hindrance in radiation we receive and the solar flares must bounce back against them though they are coming straight to us. To stay close to sun it has to be rotating at a very very high speed, so does it? Does it pose some kind of gravitational effect on the rest of the system? Is it constant or does it increase or decrease in size?"}
{"id":"91445","title":"How is Momenergy frame independent?","text":"momenergy = mass * spacetime displacement\/proper time for that displacement What I dont understand is how can the momenergy be frame independent? The unit-4 vector always points in the direction of the worldline? Agreed, the magnitude of momenergy will be invariant wrt to any frame. But wont the worldline of a particle be different in different frames? So wont the direction of the spacetime interval be frame dependent?"}
{"id":"91444","title":"Does electromagnetic radiation make sense in one dimension?","text":"I'm trying to do a simple simulation of a 1D charged quantum particle, which gets irradiated by an electromagnetic wave — in context of non-relativistic QM. The Schrödinger equation for such a particle would be $$i\\hbar\\frac{\\partial \\Psi(x,t)}{\\partial t}=\\frac1{2m}\\left(-i\\hbar\\frac{\\partial}{\\partial x}-qA(x,t)\\right)^2\\Psi(x,t)+q\\left(\\phi_0(x)+\\phi(x,t)\\right)\\Psi(x,t),$$ where $A(x,t)$ is (what I believe to be) an $x$ component of vector potential $\\vec A$ of radiation, $\\phi_0(x)$ is static scalar potential of non- irradiated system, and $\\phi(x,t)$ is scalar potential of radiation. What I can't seem to understand is how electromagnetic radiation should be incorporated in this equation. I suppose it shouldn't depend on coordinate, i.e. in non-relativistic context speed of light should be considered infinite, thus wavelength should be infinite. From basic knowledge of electromagnetism I suppose that such a wave would be represented by: $$E=E_0 \\cos(\\omega t)$$ $$B=B_0 \\cos(\\omega t)$$ But simulation is to be 1D, while electromagnetic wave has three dimensions: one for direction of propagation (which I guess is irrelevant because of infinite wavelength), second for $\\vec E$ and third one for $\\vec B$. To add the complexity, I need to determine how vector potential $\\vec A$ and scalar potential $\\phi$ would look in such case, which I might understand once I realize how to orient the radiation in terms of $\\vec B$ and $\\vec E$. So, my question is: does it even make sense to talk about external electromagnetic radiation in 1D case? If yes, how should the field be oriented in this case to make a sensible quantum simulation? What would its vector potential $\\vec A$ and scalar potential $\\phi$ look like?"}
{"id":"96678","title":"Considerations and questions for a DIY heat pipe; how to choose a working fluid?","text":"I plan to build a DIY heat pipe! For more info on what a heat pipe is, see Wikipedia. I would like to use it to cool a desktop graphics card I have. I am having trouble deciding how to choose the working fluid for the pipe. I plan to build it based on this design. Using gravity to return the fluid to the hot end. Working fluid considerations My graphics card consumes 250 watts at 100% load and 15-20 watts at idle, the maximum temperature is 90º Celsius and the temperature at idle is 30º Celsius.   * How does this translate to a choice of working fluid? I would assume the fluid has to have a phase transition temperature just lower than that of the lowest operating temperature of the card, which can be changed by based on the PV=nRT law? So just whatever substance will boil between 30–90ºC at the given pressure in the pipe?   * How can I decide how much fluid to use? As long as the pipe is filled with only the vapor of the working fluid at the proper pressure, then would the only amount of fluid you need be just enough to cover the surface area of the boiling point on the hot end? Would it be a good idea to add a little more?"}
{"id":"21528","title":"Raising a toolbox with a rope","text":"Suppose you use a rope to hoist a box of tools vertically at constant speed v. The rope exerts a constant upward force of magnitude F_up on the box, and gravity exerts a constant downward force (the weight of the box). No other forces act on the box. To raise the box twice as fast, the force of the rope on the box would have to have:- A) the same magnitude $F_{up}$ B) a magnitude of $2F_{up}$ The answer seems to be (A), and I don't understand why! It says, if it's constant velocity, both upward and downward force = 0. Since the downward force is always the same, the upward force must be the same, too, regardless of the box's speed. My take on that: But how are you gonna change the speed of the box without exerting force? If the force is the same, while there is one in the other direction, there would be no change of speed. So, can you explain to me the correct answer? EDIT: \"Consider ... (1) starting the motion, (2) maintaining the motion, and (3) slowing the toolbox to a stop ...\" (1) Starting. The object needs force greater than weight to go upward. (2) Maintaining. The object needs force equal to the weight to maintain motion upward. (3) Slowing. The object needs force less than weight for it to stop. Ummm.. I don't know what to conclude with that? I understand one important thing: It can't be $F_{up} = W$, or in other words $F_{up} = F_{down}$, because there would be no acceleration in any direction. If there is no acceleration at any direction, I can't get the velocity to be twice as fast, or any amount faster. EDIT: Thank you all. @Manishearth I could imagine this, but the book doesn't mention the \"jerk\", nothing about it in the question or in the explanation for the answer. (This is an example in the book). That's why I got confused. Anyway, again, thank you all for the clarification."}
{"id":"35041","title":"supressing certain decay paths and enhancing others with interference","text":"In a scattering reaction, there are many possible final states for the products, each with different production rates. > **Question:** Is there a way in which we could in general supress certain > rates of products of the reaction and enhance others? can we use coherence > and interference of the initial states to modulate such rates? As an application of such possibility, i'm wondering if we could enhance the production rate of antimatter (for the purpose of the question, let's consider antihydrogen) by a few orders of magnitude by such manipulations?"}
{"id":"32735","title":"How do Anamorphic images work?","text":"I found the following anamorphic image on this site ![enter image description here](http:\/\/i.stack.imgur.com\/c2f3G.jpg) Can anyone explain the physics behind the working of such images? Can we create our own anamorphic images? If so how? Thanks in advance."}
{"id":"23708","title":"Why is Mosely screening in heavy atom K-shell 1 unit?","text":"The Wikipedia article on Moseley's law seems to show that the screening of heavy atoms is by 1 electron charge _exactly_ (in the limit of large Z, experimental precision, within nonrelativistic limits, and). But why is this exactly one unit? The other K-shell electron is not screening exactly one unit, and this seems to be a conspiracy of other electrons. I suspect it is because of an unappreciated hole-picture of deep holes in heavy atoms (electrons missing in deep shells), and I will describe this theory briefly. If you remove an electron from close to the nucleus, the electron-hole behaves as an object with _positive charge_ and _negative mass_ (this is why it orbits the nucleus that it is repelled by). The state is not a vacuum quite, because of the presence of other electrons, but the rigidity of the Fermi liquid near the nucleus of a heavy atom means that the hole behaves as a single particle. This single-particle behavior is in the potential background of the nucleus and the other electrons, and it is possible that the result can give an exact 1 unit screening. I developed the formalism a little bit to see what the form should be, but I did not see any reason for 1 unit screening. Perhaps there is none, but it looks to be more than a coincidence."}
{"id":"93596","title":"Predicting the direction of sound incorrectly?","text":"When i was sitting outside my house yesterday, I heard a strange sharp sound. I could not point out it's exact direction but apparently looked like coming from left but later my friend told me that an accident had occurred (due to which the strange sound came) and it totally came from different direction? How did this happen? Why can not our ears point out the exact direction of sound sometimes?"}
{"id":"27017","title":"Covariant derivatives","text":"I need correctly define covariant derivatives on the coset space $G\/H$, where a group $G \\equiv \\\\{X_i, Y_a\\\\}$ ($X$ and $Y$ are generators) have a subrgroup $H \\equiv \\\\{X_i\\\\}$ Lie algebra of $G$ has the following structure: $$[X, X] \\sim X, \\quad [X, Y] \\sim Y, \\quad [Y, Y] \\sim X + Y.$$ The coset space $G\/H$ is parametrized by coordinates $\\xi^a$ corresponding to the generators $Y_a$. In the exponential parametrization an element of this coset has the form $$G\/H: \\quad \\Omega = exp\\\\{i\\xi^a Y_a\\\\}$$ The ordinary derivatives $\\frac{\\partial}{\\partial\\xi^a}$ of $\\psi(\\xi)$ ($\\psi(\\xi)$ - field defined on $G\/H$) do not transform covariantly. I was told once that it is quite easy to define covariant derivatives using the Cartan form $\\Omega^{-1}d\\Omega$. I looked through the bunch of the mathematical books about Lie groups and Lie algebras but I can't find a clear description of the procedure. Can anyone advise me a proper book (paper) or, if it's not take much time, tell it here? Thank you. Sorry for my English)"}
{"id":"73670","title":"Is the symmetrisation postulate unnecessary according to Landau Lifshitz?","text":"The symmetrisation postulate is known for stating that, in nature, particles have either completely symmetric or completely antisymmetric wave functions. According to these postulate, these states are thought to be sufficient do describe all possible systems of identical particles. However, in Landau Lifshitz Quantum Mechanics, in the first page of Chapter IX - Identity of Particles, he comes to the same conclusion without needing to state any ad-hoc postulate. It goes like this: Let $\\psi(\\xi_1,\\xi_2)$ be the wave function of the system, $\\xi_1$ and $\\xi_2$ denoting the three coordinates and spin projection for each particle. As a result of interchanging the two particles, the wave function can change only by an unimportant phase factor: $$ \\psi(\\xi_1,\\xi_2)=e^{i\\alpha}\\psi(\\xi_2,\\xi_1) $$ By repeating the interchange, we return to the original state, while the function $\\psi$ is multiplied by $e^{2i\\alpha}$. Hence it follows that $e^{2i\\alpha}=1$ or $e^{i\\alpha}=\\pm1$. Thus $$ \\psi(\\xi_1,\\xi_2)=\\pm\\psi(\\xi_2,\\xi_1) $$ Thus there are only two possibilities: the wave function is either symmetrical or antisymmetrical. It goes on by explaining how to generalize this concept to systems with any number of identical particles, etc. Im summary, no symmetrisation postulate was ever stated in this rationale. Is \"shifting by an unimportant phase factor\" a too strong requirement for ensuring identity of particles?"}
{"id":"73672","title":"Centrifugal Force","text":"When a body is in circular motion, realistically, it experiences only the centripetal force, created by gravitational pull, tension etc. , which gives it acceleration towards the centre. Now, assuming we were to fill a bucket with water, and whirl it in the vertical plane. There will be a certain velocity when the water doesn't fall out even when the bucket is upside down. At that instant, if we were to write down the forces that act on the water, they would be the Normal reaction by the bucket, and it's weight. This means that the net force is downwards. Why doesn't the water fall down, since there's no force in the upward direction? Centrifugal force is a psuedo force, so it shouldn't be applicable here, as the water won't wall down even if I watch from a non inertial frame. How can one explain this phenomenon? Also, say we were to fit a ring in a rod, and rotate the rod, the ring will eventually slip and fall off. Why does this happen if there is no outward force?"}
{"id":"37557","title":"Crystal magnetic response only skin deep?","text":"The Hamiltonian for a single electron in a magnetic field reads $$H=\\left(\\frac{{\\bf p}^{2}}{2m_{e}}+q_{e}\\phi\\right)+\\mu_{B}\\left({\\bf \\hat{L}}+g{\\bf \\hat{S}}\\right)\\cdot{\\bf B}+\\frac{e^{2}}{8m_{e}}\\left({\\bf B}\\times{\\bf r}\\right)^{2}$$ where symmetric gauge ${\\bf A}=\\frac{1}{2}{\\bf B \\times r}$ is used and where the last two term are responsible for paramagnetism and diamagnetism respectively. The many-body Hamiltonian, weak electron-electron interaction considered, reads $${\\cal H} = \\sum_{\\sigma}\\int d{\\bf r}\\,\\left[-\\left(\\psi_{\\sigma}^{\\dagger}({\\bf r})\\frac{\\hbar^{2}}{2m^{*}}{\\nabla}^{2}\\psi_{\\sigma}({\\bf r})\\right)+\\boldsymbol{\\mu}({\\bf r})\\cdot({\\nabla}\\times{\\bf A})+\\hat{n}_{e}({\\bf r})\\frac{e^{2}}{2m^*}{\\bf A}({\\bf r})^{2}\\right]$$ where $\\boldsymbol\\mu({\\bf r})=\\sum_\\sigma \\mu_B \\psi^\\dagger({\\bf L}+g{\\bf S})\\psi $ is local magnetic moment density of electron of total electron angular momentum and define $\\mathbf M_\\text{para} \\equiv \\langle \\boldsymbol \\mu\\rangle$ Now that we vary $\\bf A$, we should obtain the response current $${\\bf j}={\\nabla}\\times{\\mathbf M_\\text{para}}+\\frac{n_{e}e^{2}}{m^*}{\\bf A}$$ It is rather strange to me that (i) the last term is essentially the same as the one in London's equation, which gives a penetration depth $\\lambda$, and that (ii) the first term is non-zero only on the crystal surface if $M$ is uniform deep inside, which I think is just the magnetization current we saw in textbooks. If (i) is true, one may conclude that magnetic response for all crystals (neglect interactions) happens in a thin layer near to the surface (a few nanometer) since magnetic field can only penetrate that far. The length scale that electron density drops to zero is a few angstroms, so we are able to neglect the first term in an appropriate region."}
{"id":"128983","title":"Why does earth cool?","text":"This question is boggling me for some time. We know that heat can be transferred from matter to matter and heat is nothing more than tiny atoms vibration intensity (correct me if I'm wrong). But space is a vacuum, and so arises the question: Can you heat vacuum? :)) How can it be that earth loses it's heat to space itself? Thanks"}
{"id":"25702","title":"Did I see two Milky Ways next to each other?","text":"I am from the northern hemisphere and as far as I remember, the Milky Way was \"single lane\", just one stripe. At least, this is what I recall the Milky Way to look like when seeing it from near Irkutsk, Siberia, in late August 1997. Yesterday I went to Blackbutt, Australia (somewhere near 27° South, 152° East) and the Milky Way seemed to have a large black stripe in the middle. I was wondering, if these were two spiral arms of the galaxy that we are part of."}
{"id":"55796","title":"How \"big\" objects can WISE and NEOWISE detect?","text":"I mean WISE is monitoring near-Earth objects, but cannot see the latest Russian meteor and others. Why can it not detect small objects? What is the limit of it's infrared detectors?"}
{"id":"98610","title":"Question on the hydrostatic paradox","text":"The hydrostatic paradox, states that the > height of the water in any container is independent of the shape of the > container. This does make sense since it is only the height of the fluid relative to ground which determines the pressure, and it is the pressure equilibrium that decides the shape of the fluid. But then consider the (B) container in the figure below.   ![enter image description here](http:\/\/i.stack.imgur.com\/DoLKb.png) The pressure at a particular depth say, $h$ is same as in the case of the cylindrical container. But in the cylindrical case (A), any element of fluid is in equilibrium because the difference in pressure of the fluid element above it and that below it exactly balances the weight of the element. Extending this argument to (B), for any element at the edge, the pressure of the fluid above is the same as in the case of the cylindrical container, but below it is the surface of the container. This container, is in equilibrium with the atmospheric pressure outside, and therefore, would be exerting pressure equal to the atmospheric pressure onto the fluid within, which would not be enough to maintain the equilibrium of the fluid element. **How then can the equilibrium of the fluid element at the edge be explained?**"}
{"id":"88428","title":"What is the logic behind the Fahrenheit scale?","text":"The Fahrenheit scale is defined by fixed points on the scale. What interests me is the apparent arbitrary chosen numbers in these fix-points. First wikipedia wites from 32 to 212. -and later in the _History_ part we have from 0 to 32 to 96(based on a quadrupling of 7.5 to 22.5 to 60). I am stille confused. Does any of the numbers make sense in an other numeral system? Did Rømer or Fahrenheit use their lucky numbers or what is going on? **Edit:** I am not interested in what solutions or objects, that is used for the fix-points. I am asking what the logic is behind the chosen numbers?"}
{"id":"9975","title":"Wien's fifth power Law and Stephan Boltzmann's fourth power laws of emissive power","text":"Wien's fifth power law says that emissive power is proportional to the temperature raised to the fifth power. On the other hand, the Stefan–Boltzmann law says emissive power is proportional to the temperature raised to the fourth power. How can both of these be true?"}
{"id":"57792","title":"What is the frequency of occurrence of stellar classifications off the HR main-sequence?","text":"An alternative version of this question would be: \"if was to pick a star from the $10^{11}$ or so in our galaxy at random, what are the probabilities of it being various kinds of star?\" (and I do mean \"in our galaxy\" and not \"visible in the night sky\"). There is a nice table I've seen in various places (e.g wikipedia) which goes a long way towards answering this. It tells me that 0.00003% are 'O' types and 0.13% 'B' and 0.6% 'A' through to the 76% rump of 'M types. Unfortunately there is some small print associated with that table which is that it's only for stars **on the main sequence** (hence its 'M' types are all low-luminosity dwarfs - red giants aren't included - and its 'A's and 'B's are all young giants - white dwarfs aren't included). However, I've no idea what proportion of the stellar population are **off** the main sequence. One in a hundred? One in a million ? What I'd really like to find (or gain enough information to compile) is a 2D version of that table with the same axes as a HR diagram where I can look up the frequency of white dwarfs by looking in the cells in the A-B spectral and the luminosity 0.001 - 0.1 ranges, or the frequency of stars like Betelgeuse by looking in the M, $10^5$ cell. (Of course another issue with the cited table is that it claims to be frequencies in \"solar neighbourhood\". For example it'd be nice to have different versions specific to stellar populations of, say, globular clusters, the galactic disk and central \"bulge\". But I'll settle for some pan-galactic numbers initially)."}
{"id":"57794","title":"Calculating engine starter’s energy use","text":"During a discussion on start-and-stop vehicle technology some bloke began pushing the point that re-starting the car uses stored energy from the battery, which needs to be replenished by increased fuel usage once the engine is moving. Well, this is obvious, the question is that of significance of such additional usage, especially in comparison with savings from reduced idle. Common sense tells me that this is splitting hairs: if ICE efficiency outlays put all accessory usage at 2 to 3% of overall energy consumption, which includes AC, and all electricals, then it can’t be that much. More over, this dissertation explores idle-reduction technologies for long- haul commercial trucks, and one of the systems exampled is battery-powered (p. 14) that stores enough energy to power AC or heater overnight, and takes about six hours of charge while driving. Author acknowledges increased fuel consumption due to necessity of higher-amperage alternator, although there are no specific numbers provided. However, the mere existence of such commercial application available on the market leads me to believe that such added load is still better than idling. But, in the interest of science, I need some hard numbers (besides, the fellow just wouldn’t go away). I have some ideas on what to consider, but I’m not well versed in electrical and mechanical engineering, so I do not think I can account for most major factors. Energy use of the starter motor can be calculated by using the amount of current used per start (which would be 2 to 3 seconds) by the motor itself and the solenoid that engages the starter gear onto the flywheel. Both current and power demand can be found in the starter’s specifications, but I am not sure as to how reliable those numbers would be in the real-world application. Then there is fuel consumption of the start itself which is estimated at 10 to 15 seconds worth of idling (and Florida chapter of ASME even calculated six seconds for 6-cylinder engine (in a simple, non-rigorous field experiment), but the original link is broken). Now, how to calculate the increased consumption due to the charging of depleted battery from the starter motor itself, and, additionally, to account for all the accessories that were running while the engine was stopped? Is it a simple matter of using the same number calculated for energy usage of starter motor, and arriving at it by calculating the amount of fuel needed to produce that much extra energy given the losses in the engine itself and in the charging circuit? And, ultimately, how significant are those considerations in the bigger picture?"}
{"id":"32067","title":"Is the potential energy in a compressed spring a Lorentz invariant?","text":"The total energy of an object comes from the time part of the four-momentum, and so isn't a Lorentz invariant. On the other hand, is the potential energy of a compressed spring a Lorentz invariant?"}
{"id":"26053","title":"What is the position and orientation of the Earth as a function of time","text":"I would like to know the position and orientation of the Earth, relative to the sun, as a function of time. I.e. select any coordinate system with the Sun at the origin so that the earth lies in the x-y coordinate plane. Provide a function which takes as input the time and produces   1. The x-y position of the center of the Earth   2. The unit vector from the center of the Earth to the North Pole.   3. The orientation of the Earth around this vector This function is chaotic in general. I would be happy with a decent approximation. This problem is solved as part of the \"what is the position of the sun in the sky\" problem. Solutions to this problem tend not to be well separable (at least I have trouble separating my question out of the complete position-of- sun-in-sky solution.)"}
{"id":"26055","title":"Relationship between the cosmic microwave background (CMB) radiation and the visible universe","text":"I know that the cosmic microwave background radiation (CMBR) is the leftover radiation from the Big Bang. Have we been able to make a concrete map of the radiation and the objects in the visible universe? (Something like \"this point in the CMBR map evolved into this particular region of the visible universe\".)"}
{"id":"130691","title":"Pressure at a bend in a pipe","text":"This is a question from a study guide for a military officer exam: I have fluid flowing through an elbow in a pipe (the elbow is in the shape of an upside down and backwards \"J\") and the flow goes from south to north to east. If holes are drilled at the upper left outside part of the bend and the lower right inside part of the bend, at which point will pressure be lower? Please see attached image for a picture of the question. When I have thought about this problem, I keep coming back to pressure being transmitted equally to all parts of a fluid, so I don't understand how the pressure can be different at one of the holes. The book gives the answer as the inside bend which I can force myself to accept since I can picture the fluid attempting to go straight to the outside bend hole and thus hitting that part harder, but I don't really understand it."}
{"id":"75089","title":"Fermions in the same state","text":"I need some clarification of what is meant when someone says \"fermions cannot occupy the same quantum state\". Consider two bosons: $$\\psi(\\vec{r_1}, s_1, \\vec{r_2}, s_2) = \\frac{1}{\\sqrt{2}} \\left( \\phi_A(\\vec{r_1}, s_1)\\phi_B(\\vec{r_2}, s_2) + \\phi_A(\\vec{r_2}, s_2)\\phi_B(\\vec{r_1}, s_1) \\right)$$ This is one wavefunction of two particles. A wavefunction directly corresponds to a state, and since this is only one wavefunction, it seems there is only one state -- that two bosons occupy. But now consider two fermions: $$\\psi(\\vec{r_1}, s_1, \\vec{r_2}, s_2) = \\frac{1}{\\sqrt{2}} \\left( \\phi_A(\\vec{r_1}, s_1)\\phi_B(\\vec{r_2}, s_2) - \\phi_A(\\vec{r_2}, s_2)\\phi_B(\\vec{r_1}, s_1) \\right)$$ Again, one wavefunction (=> one state) and two particles that occupy it. Yeah, $\\psi(\\vec{r_1}, s_1, \\vec{r_2}, s_2) = -\\psi(\\vec{r_2}, s_2, \\vec{r_1}, s_1)$, but it's still just one state -- occupied by two fermions. Could someone clarify?"}
{"id":"75087","title":"Boundary between supercritical fluid and gaseous phase","text":"As far as I understand, going around the critical point, it is possible to transform a liquid into a gas without seeing a phase transition. Does this also mean that there is no clear separation between a gaseous phase and a supercritical fluid ? This is more or less what can be understood for the dotted line between the two that usually appears in a phase diagram such as the following: ![Phase diagram](http:\/\/i.stack.imgur.com\/lbHET.png) So my question is: Is there a standard physical measure for deciding whether a given fluid is in gaseous phase or in supercritical state? Alternatively, is the boundary between the two clear-cut or somewhat fuzzy?"}
{"id":"53960","title":"Deriving the Sommerfeld expansion by contour integration (Le Bellac p. 277)","text":"In Le Bellac's statistical physics book he derives the Sommerfeld expansion by a contour integral. The idea is to expand integrals of the type $I(\\beta)\\equiv \\int_{0}^{\\infty}d\\epsilon\\, \\frac{\\varphi'(\\epsilon)}{e^{\\beta(\\epsilon-\\mu)}+1}$. Integrate by parts and expand $\\varphi(\\epsilon)$ $$\\varphi(\\epsilon)=\\sum_{m=0}^{\\infty}\\frac{(\\epsilon-\\mu)^{m}}{m!}\\left(\\frac{d ^{m}\\varphi}{d \\epsilon}\\Bigg |_{\\epsilon=\\mu}\\right)$$ Make a change of variables to write $I(\\beta)=\\sum_{m=0}^{\\infty}\\frac{\\beta^{-m}\\varphi^{(m)(\\mu)}}{m!}I_{m}$ where $I_{m}=\\int_{-\\infty}^{\\infty}dx\\, \\frac{x^{m}e^{x}}{(e^{x}+1)^{2}}$. The idea then to evaluate the $I_m$ is to consider the following integral $$J(p)=\\int_{-\\infty}^{\\infty}dx\\, \\frac{e^{ipx}}{(e^{x}+1)(e^{-x}+1)}=\\sum_{m=0}^{\\infty}\\frac{(ip)^{m}}{m!}\\int_{-\\infty}^{\\infty}\\, dx \\frac{x^{m}}{(e^{x}+1)(e^{-x}+1)}= \\sum_{m=0}^{\\infty}\\frac{(ip)^{m}}{m!}I_{m}$$ Integrating $J(p)$ gives $J(p)=\\frac{2\\pi pe^{-\\pi p}}{1-e^{-2\\pi p}}=1-\\frac{1}{6}(\\pi p)^2+\\ldots$ hence, $$I(\\beta)= \\varphi(\\mu) + \\frac{\\pi^{2}}{6}(k_{B}T)^{2}\\varphi^{(2)}(\\mu)+O(k_{B}T^{4})$$ So far so good, my problem comes when he writes the expansion of the Fermi- Dirac distribution itself as $$\\frac{1}{e^{\\beta(\\epsilon-\\mu)}+1}\\simeq \\Theta_{H}(\\mu-\\epsilon)-\\frac{\\pi^{2}}{6}(k_{B}T)^{2}\\delta'(\\epsilon-\\mu)+O(k_{B}T^{4})$$ which is meant to be valid under an integral. It appears to mimic the expansion of the generating function, at least where the signs are concerned. If we consider the definition of the Sommerfeld expansion from Wikipedia, we have a formula in the form $$\\int_{-\\infty}^\\infty \\frac{H(\\varepsilon)}{e^{\\beta(\\varepsilon - \\mu)} + 1}\\,\\mathrm{d}\\varepsilon = \\int_{-\\infty}^\\mu H(\\varepsilon)\\,\\mathrm{d}\\varepsilon + \\frac{\\pi^2}{6}\\left(\\frac{1}{\\beta}\\right)^2H^\\prime(\\mu) + \\mathrm{O} \\left(\\frac{1}{\\beta\\mu}\\right)^4$$ If we give Le Bellac's form a try, $$I(\\beta)\\simeq\\int_{0}^{\\infty}d\\epsilon\\, \\varphi'(\\epsilon)\\left[\\Theta_{H}(\\mu-\\epsilon)-\\frac{\\pi^{2}}{6}(k_{B}T)^{2}\\delta'(\\epsilon-\\mu)+O(k_{B}T^{4}) \\right] \\\\\\ =\\varphi(\\mu)-\\frac{\\pi^{2}}{6}(k_{B}T)^{2}\\underbrace{\\int_{0}^{\\infty}d\\epsilon\\, \\varphi'(\\epsilon)\\delta'(\\epsilon-\\mu)}_{K}$$ but integrating $K$ by parts gives $$K=\\left[\\delta(\\epsilon-\\mu)\\varphi'(\\epsilon) \\right]_{0}^{\\infty}- \\int_{0}^{\\infty}d\\epsilon\\, \\varphi''(\\epsilon)\\delta(\\epsilon-\\mu)=-\\varphi''(\\mu)$$ So the signs work out, and the negative sign is probably not a typo. On the other hand, the next term in the expansion would be $+\\frac{7\\pi^4}{360}(k_{B}T)^{4}\\delta^{(3)}(\\epsilon-\\mu)$ which would take a negative sign when integrated. The alternating behavior of the signs, as far as I can tell, is due to the imaginary unit that was included in the exponential for the contour integral. A very similar approach is done on p. 255 of the following notes except there is no imaginary unit in the exponential. Also, the expansion is meant to operate only on $\\varphi$ instead of actually correcting the Fermi-Dirac distribution for small $T$. My questions are as follows:   1. Are the corrections to the Fermi-Dirac distribution given by Le Bellac correct to all orders? They seem to work for the 0th and 1st order terms.   2. Assuming (1) is correct, how do we \"know\", or what's the reasoning, that the series expansion of the generating function would give the corrections to the Fermi-Dirac distribution? Because it seemed to me that the generating function was only to determine the $I_m$ coefficients."}
{"id":"106678","title":"What does (001) Silicon mean?","text":"If someone gives me a thin film of Si, and they tell me it's (001) Si, does that mean that the (001) planes of Si are the ones making up the surface of the film?"}
{"id":"102744","title":"Why do some sunsets have a green flash?","text":"Some sunsets exhibit a phenomenon called \"green flash,\" where there is a quick flash of green light right as the sun is setting. I have seen this myself several times, and sort of understand it. I am a bit frustrated at my inability to explain it better to my family. I understand how Rayleigh scattering would make the dominant color shift towards the blue just after the sun passes below the horizon. How does mirage cause the green light to refract more than the blue? Is there an easy way to use the two effects to estimate the wavelength of the flash?"}
{"id":"102745","title":"Particle\/antiparticle annihilation and entanglement","text":"This is a basic question. Suppose that A and B are completely entangled particles and so are C and D. If B and C are antiparticles that annihilate each other will A and D be entangled as a result. It seems they should be in order to satisfy conservation laws. Also if the initial pairs were the results of pair creation events then the Feynman diagram of the entire can also be interpreted as a single particle starting as A emitting and absorbing photons and emerging as D. More generally if the state of the 4 particle system just before annihilation was represented as $$\\alpha_{ij}\\, |A_i\\rangle\\, |B_j\\rangle\\; \\beta_{kl}\\, |C_k\\rangle\\, |D_l\\rangle$$ would the state of A and D just the after the B-C annihilation be $$\\sum_j\\alpha_{ij}\\beta_{jk}\\,|A_i\\rangle\\, |D_k\\rangle$$ in a product state with the state of the emitted photons. Is this correct?"}
{"id":"83449","title":"Can a planet form before the parent star ignites?","text":"I'm unable to find an answer to my questions via my searches. This questions pertains to the timeline of a star system creation. My question: During the creation of a star system, can satellite planets form before the star has ignited and begun to burn? In more detail, I understand the basic creation of a star, with the gravitational collapse of a large cloud, leading eventually to fusion. Is this transition from protostar to maturity (or to where \"ignition\" occurs) long enough that planets have already started to form, even in a primitive state (just a big rock)? Or has the star long since been burning when planets finally start to form?"}
{"id":"133331","title":"Equivalent Dose absorption from Na22 source","text":"No idea what exactly the rules are or how I should ask this question, but here's the verbatim quesiton: \"A Na$^{22}$ source has an activity of 100 $ \\mu $C. If you handle the source, how big is the equivalent dose on your hands, if all the radiation were absorbed? Assume the volume and density to be 10 x 5 x 1 $cm^3$ and ~1 $g\/cm^3$, respectively.\" I'm using the SI-system mostly since I live in Norway, but seem to be using $\\mu$C for medical radiation. I presume the volume and density apply to the tissue of the hands, and Sodium 22 has mostly positron decay with (I think) energies of 546keV, which makes it have a quality factor of 1. And as far as I've gathered the equation I'm supposed to use is $$D_q = qD_E=q\\frac{1}{\\rho}\\frac{W}{V}$$ where $D_q$=equivalent dose, q=quality factor, $\\rho$=density, $W$=absorbed energy, and $V$=volume, but I honestly don't know how I'm supposed to find $W$. If it is as simple as to multiply the activity with the energy value of the positrons, or if I need to find something else entirely. But if it's the first, then when everything is inserted gives me a dosage of: $2.32*10^{-7} mSv \/ hour$, seems a bit too low dosage, I feel like it should be higher. I'm also wondering if this says that I have to consider both the positron decay, the gamma decay AND the two annihilation photons when I calculate my total energy? With the different weightfactors on the decay with the % listed there? Namely: positron decay=0.546MeV (89,9%), gamma=1.275MeV (100%), annihilation photon=0.511MeV (180%) Haven't had any classes about this so far so I'm sorta stuck trying to find things on the internet. So any help would be much appreciated!"}
{"id":"133330","title":"Can we represent Simple Harmonic function as triangular waves?","text":"Having studied the topic recently I found out that simple harmonic motion can represented well with sine and cosine functions.Take for example a pendulum swing which could look like : ![enter image description here](http:\/\/i.stack.imgur.com\/iGqpV.gif) and the equations governing the motion would be ![enter image description here](http:\/\/penguinphysic.files.wordpress.com\/2009\/11\/image0151.png) So I've been wondering why can't simple harmonic motion be represented in form of triangular waves.Although the equations above involve angular momentum so I may be contradicting myself but fundamentally the velocity time is sine function : $$-\\sin(x)$$ and the gradient represents the acceleration is non-uniformly increasing and decreasing. What if instead of that you use $$-(\\arcsin(\\sin(x))$$ Which would represent a triangular wave whose gradient would depict that the acceleration is uniformly accelerating and decelerating.So would this represent harmonic motion or is it fundamentally incorrect."}
{"id":"133338","title":"Work to Create a Vacuum","text":"I'm trying to determine the amount of work required (in Joules) to create a vacuum of a certain pressure in a tube. A pneumatic cylinder will be used to remove the air from the tube to create the vacuum. I understand the standard formula for the work to create a vacuum is: $\\mathrm{pressure \\; difference} \\times \\mathrm{volume}$. However, I've also read that when considering the use of a cylinder and piston there is actually some work that can be reclaimed since the piston and cylinder provides additional volume for the exhausted gas to expand into. Therefore, the correct formula to calculate work under this scenario is: $$W = V \\left( P_{\\mathrm{ambient}} - P_{\\mathrm{new}} \\right) + V P_{\\mathrm{new}} \\ln(P_{\\mathrm{new}}\/P_{\\mathrm{ambient}})$$ Could someone validate which is the correct work formula to use for this scenario?"}
{"id":"33880","title":"Is momentum conservation for the classical Schrödinger equation due to non-relativistic or due to some more exotic invariance?","text":"I had no problem appliying the Neothers theorem for translations to the non- relativistic Schrödinger equation $\\mathrm i\\hbar\\frac{\\partial}{\\partial t}\\psi(\\mathbf{r},t) \\;=\\; \\left(- \\frac{\\hbar^2}{2m}\\Delta + V(\\mathbf{r},t)\\right)\\psi(\\mathbf{r},t)$ $\\Longrightarrow\\ \\mathcal{L}\\left(\\psi, \\mathbf{\\nabla}\\psi, \\dot{\\psi}\\right) = \\mathrm i\\hbar\\, \\frac{1}{2} (\\psi^{*}\\dot{\\psi}-\\dot{\\psi^{*}}\\psi) - \\frac{\\hbar^2}{2m} \\mathbf{\\nabla}\\psi^{*} \\mathbf{\\nabla}\\psi - V( \\mathbf{r},t)\\,\\psi^{*}\\psi$ $\\Longrightarrow\\ \\pi=\\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\psi}} \\propto \\psi^{*}$ $T[\\psi]\\propto \\mathbf{\\nabla} \\psi$ $\\Longrightarrow\\ I_{\\ \\psi,{\\ T_\\text{(translation)}}}=\\int\\text d^3x\\ \\pi\\ T[\\psi]\\propto \\int\\text d^3x\\ \\psi^{*} \\mathbf{\\nabla} \\psi = \\langle P \\rangle_\\psi$ But I actually wonder why that works out, given that the Schrödigner equation is not invariant under Galileian transformations. It might well be that the Schrödinger group, which I'm not familiar with, is close enough to the Galileian group, that the fourth line $T[\\psi]\\propto \\mathbf{\\nabla} \\psi$ is just the same and that's the reason. I'd like to know if the evaluation of the infinitesimal transformation is the only point at which one has to know the transformations one is actually dealing with. Is my guess right? * * * Also, regrding the \"trick\" to establish Galilei-invariance after the conventional transformation via multiplication of the Schrödinger field by a phase (a phase which, among other things, is mass dependend): Some authors change $\\psi(r,t)$ to $\\psi(r',t')=\\psi(r-vt,t)$, like here in the paper referenced on wikipedia (there is also a two year old version of it online (google)), but other authors, like the writers of the page in the first link, also transform $p$ to $p+mv$ in $\\phi$ (which doesn't change the fact that they still have to add a phase). This is all before the phase multiplication. So what is the \"right way\" here? If I do this transformations involving a multiplication of the phase, do I only transform the actual arguments of the scalar field $\\psi(r,t)$ or do I also transform the objects like $p$, which classically transform too, but are really just parameters (and the Eigenvalues) or the field - and not arguments?"}
{"id":"72887","title":"relativity - equivalent frames do not have equivalent clocks and rods","text":"In problems about relativity we have a stationary frame S and a moving frame M and two observers initially at the same reference point. Then the two frames pass each other and we measure the distance and time with a second reference point. The two reference points are at rest in the S frame. The conclusion is that the observer in S reference point measured a longer time and distance than the observer. But as S and M are equivalent (Galilean relativity) these two observers must when they compare each other measurements have the same results. How can you explain this difference in measurements and the equivalence of both frames at the same time."}
{"id":"89882","title":"How do you calculate the change in temperature of an adiabatic system?","text":"I really don't even know how to start doing this problem. I understand PV=nRT and I understand that if either T, V, or P is kept constant, then the other 2 change at the same rate, and I understand that if either T, V, or P is kept constant while the other two change, then the system experiences a change in heat. I also understand that the internal energy is kept constant and that internal energy = heat - work. However I do not know how to relate the change in temperature to a change in heat. Can somebody please walk me through this? Here is a diagram to better visualize the problem. ![enter image description here](http:\/\/i.stack.imgur.com\/Q7Qu2.jpg) My homework question asks to find the temperature at point c I have already calculated that n=0.41 and that the pressure at point b is 2.01966 atm You do not need to solve for c, just please tell me how. Okay so I couldn't handle it anymore and I just looked at the answer which happened to be 810J. This make me even more confused. Where did the extra 110J of energy just appear from?"}
{"id":"71509","title":"Why were the fathers of quantum mechanics so sure radioactive decay was indeterministic?","text":"The classic example of an indeterministic system is a radioactive isotope, e.g. the one that kills Schrödinger's cat. I get there are arguments against hidden variables in quantum mechanics, but how could they be so sure, back in the twenties, that the strong nuclear forces involved in radioactivity were not governed by hidden variables rather than true randomness? Einstein was very unhappy about the indeterminism of quantum mechanics regarding even well understood effects like Young's slit experiments, but it seems kind of ideological and brash on behalf of Heisenberg & Co to extend the indeterminism over to phenomena they hadn't even begun to understand, like alpha decay. Is there a reason for this early self-assuredness in postulating indeterminsm?"}
{"id":"15109","title":"Could the contents of a bottle affect the surface temperature of that bottle?","text":"e.g. If I had two bottles one with Vodka in and one with Water in (both the same glass bottle spec\/size) If I then put these two bottles in a freezer would the surface temperature of the bottle be the same or different? Thanks"}
{"id":"30284","title":"Sources for new experimental limits on susy models?","text":"I know the LHC people are publishing new limits every now and then, but as a non-expert in reading experimental papers (yet), I was wondering if there's a friendly website that collects and presents the limits in a readable and accessible way for non-experts?"}
{"id":"77907","title":"Magnetic field of solenoid in cone shape","text":"I have done some drawings of the magnetic fields of cone shaped solenoid electromagnets. I based this on the work of Tesla and some explanations of Bashar, known as the Bashar Cones. I found very strange patterns. ![enter image description here](http:\/\/i.stack.imgur.com\/jMAun.png) ![enter image description here](http:\/\/i.stack.imgur.com\/dRcF7.png) ![enter image description here](http:\/\/i.stack.imgur.com\/XIO5p.png) ![enter image description here](http:\/\/i.stack.imgur.com\/l62x5.png) ![enter image description here](http:\/\/i.stack.imgur.com\/DPQLO.png) ![enter image description here](http:\/\/i.stack.imgur.com\/yxexk.png) ![enter image description here](http:\/\/i.stack.imgur.com\/UA9o6.png) Can someone study and analyze them. I think some of those fields can be used to generate rotation as others do have strong fields to generate electricity. http:\/\/s1316.photobucket.com\/user\/amtriorix\/library\/Bashar%20Cones I do see circular magnetic fields, as I do see on some strong fields to do more experiments on it. http:\/\/www.youtube.com\/watch?v=7Ii_Bb6YYQU Walter Russell Cones ?"}
{"id":"78102","title":"Coulomb gauge and two degrees of freedom of EM field","text":"The EM field has two possible polarizations, which is caused by spin-one nature of field (leads to the Lorenz gauge) and massless of the field. Really, the Klein-Gordon equations for the EM field $$ \\square A^{\\mu} = 0 $$ does not leave a $A^{\\mu}$ uniquely determined: $$ A^{\\mu} \\to A^{\\mu}{'} = A^{\\mu} + \\partial^{\\mu}\\varphi , \\quad \\square \\varphi = 0 \\Rightarrow \\varphi^{0}{'} = 0, \\quad \\partial_{\\mu}A^{\\mu} = (\\nabla \\mathbf A ) = 0, $$ which is called Coulomb gauge. So EM field has two components. But after using Lorentz- transformations the new $A^{0}$ component isn't equal to zero. So I'm not imagine, what does it mean. It looks like that EM field begin to have three independently components in the new frame. Of course, this is not true, but I want to know, can the Coulomb gauge be an argument of existence only two independently components. Can you help me?"}
{"id":"18080","title":"Some questions about the logics of the principles of independence of motion and composition of motion","text":"In high-school level textbooks* one encounters often the principles of independence of motion and that of composition (or superpositions) of motions. In this context this is used as \"independence of _velocities_ \" and superposition of _velocities_ (not of forces). This is often illustrated by the example of the motion of a projectile, where the vertical and horizontal motions are said to be independent and that the velocities add like vectors. Now, if $x\\colon \\mathbb{R} \\to \\mathbb{R^3}$ describes the motion of the of the considered object, it is clear that one can decompose the velocity $v = \\dot x$ arbitrarily by $v = v_1 + v_2$ where $v_1$ is arbitrary and $v_2 := v - v_1$. This leads me to my first question: Am I correct that this is purely trivial math and contains no physics at all? If so, it would not deserve to be called \"principle of composition of motions\" or something like that and said to be fundamental. However it seems that one could interpret the decomposition above such that $v_1$ is the velocity of the object with reference to a frame of reference moving with $v_2$. If so, how can one see, that this goes wrong in the relativistic case? Now suppose you have two forces $F_1$ and $F_2$ which you can switch on and off, suppose that $F_i$ alone would result in a motion $x_i$ ($i=1,2$). Newtonian Mechanics tells us the principle of superposition of forces, i.e. if you turn both forces $F_1$ and $F_2$ on, the resulting motion $x$ is the solution of the differential equation $\\ddot x = \\frac1m F(x, \\dot x, t)$ (where m is the Mass of our object) with $F = F_1 + F_2$. One might interpret the principle of composition of motions such that always $\\dot x = \\dot x_1 + \\dot x_2$ holds true. This is clearly the case if $F_i$ depends linearily on $(x,\\dot x)$. However I think that it doesn't need to be true for nonlinear forces. This leads me the my third question: Is there any simple mechanical experiment where such nonlinear forces occur, which shows that in this case the \"principle of composition of motions\" doesn't hold? *I have found this in some (older) german textbooks, for example: Kuhn Physik IIA Mechanik, p. 107, Grimsehl Physik II p.16,17 compare also   http:\/\/sirius.ucsc.edu\/demoweb\/cgi-bin\/?mechan-no_rot-2nd_law and Arons"}
{"id":"78106","title":"Which magnetic force is stronger? Electron or proton?","text":"Assume the particles generates the magnetic field with their own spin. No motions except spin.(they have their intrinsic angular momentum.) Then which one generates stronger magnetic force? I've seen the formula which represents the magnetic moment of the 1\/2 spin particles with charge $q$, mass $m$, $\\vec{\\mu}=\\frac{g_sq}{2m}\\vec{S}$ where $g_s$ called g-factor. But I can't understand the formula. Let me ask a question, which magnetic force is stronger between electron and proton?"}
{"id":"18083","title":"Finding coefficient of proportionality","text":"Recently in my AP Physics class I did a lab in which I measured k for a spring by setting up an oscillating system with it, and timing the period, repeating for different masses. Since $T=2\\pi\\sqrt{\\frac mk}$, we were supposed to relate $T^2=\\frac{4\\pi^2}km=Cm$ for some $C$. This is a proportionality equation in $T^2$ and $m$, which we were to plot on a graph and find the slope $C=\\frac{4\\pi^2}k$, from which we can find $k=\\frac C{4\\pi^2}$. Now my lab partner and I found the $k$ in two different ways; while I simply averaged $k_i={C_i\\over4\\pi^2}$ for each trial, he used statistics software to find the equation $\\hat{T}^2\\approx C_am+\\beta$, where $\\beta\\approx0$ and $\\bar{k}={C_a\\over4\\pi^2}$. He claimed that the $\\beta$ term helped to eliminate error, leading to a better $k$. We recalculated $k$ by direct use of Hooke's Law and found that the percent difference was lower in my partner's case, leading me to believe that he may be correct in his method. So, my question is, physics.stackexchange: How does using the linreg analysis on the data produce better results? Or, if this is a fluke of data and my results should have been better, why shouldn't it? Thank you in advance."}
{"id":"34055","title":"First class and second class constraints","text":"Hello I am working on a project that involves the constraints. I checkout the paper of Dirac about the constraints as well as some other resources. But still confuse about the first class and second class constraints. Suppose, from the Lagrangian, I found two primary constraints $\\Phi_a$ and $\\Phi_b$. Let $\\dot\\Phi_a$ and $\\dot\\Phi_b$ both leads to the secondary constraints $\\Sigma_a$ and $\\Sigma_b$ respectively. From the consistency condition $\\dot\\Sigma_a$ leads to a tertiary constraints $\\Theta_a$ but $\\dot\\Sigma_b$ become zero. Now How can I check which one of them are first-class and which one of them are second-class?"}
{"id":"34057","title":"What's the role of field equation in QFT?","text":"For free field theory, it seems the solutions of a field equation are used to give a representation of Poincare group, and the field equation is still satisfied after quantization. However for an interacting theory, such as QED, the field equation: $$\\partial_\\nu F^{\\nu \\mu} ~=~ e \\bar{\\psi} \\gamma^\\mu \\psi.$$ I don't see any possibility of satisfying this equation using operators since LHS will only act nontrivially on bosonic sector of the Hilbert space and RHS only on fermionic sector, and I don't think perturbation technique can fix this since it's qualitatively not satisfied. Yet nobody seems to be bothered by this, so I'm wondering what's the role of field equation in QFT, especially for interacting ones."}
{"id":"34056","title":"How does one prove:$\\nabla(\\vec{\\mu_m}\\cdot\\vec{B})\\cdot\\vec{dr}=0$?","text":"Work done by a magnetic force(even over an infinitesimally short displacement)=0 _Net Force_ on a current loop in an external magnetic field is given by: $$\\vec{F}=\\nabla(\\vec{\\mu_m } \\cdot \\vec{B})$$ How does one prove: $$dW=\\nabla(\\vec{\\mu_m} \\cdot \\vec{B})\\cdot\\vec{dr}=0$$ $\\vec{\\mu_m}$: Magnetic Moment of the current loop."}
{"id":"46801","title":"How do electrons jump orbitals?","text":"My question isn't how they receive the energy to jump, but why. When someone views an element's emission spectrum, we see a line spectrum which proves that they don't exist outside of their orbitals (else we would see a continuous spectrum). Electrons can be released in the form of beta decay, thus proving that they are capable of traveling outside of orbitals contrary to the statement my teacher said that they stay within orbitals. Then, to add to the confusion, the older model of rings floating around a nucleus has, from what I can tell, been outdated, which would support this model. My teacher's explanation was that the electrons made a quantum jump of some kind. **_How_** do electrons move between orbitals or do we know how they jump, excluding the reason that energy causes them to jump, and why are positrons formed sometimes instead of electrons in Beta decay? When I'm asking \"how do electrons jump\" I would like to know how an electron can jump between each orbital such as how it moves and how it knows where to jump since it appears to be a jump where the electron doesn't slow into a orbital position. Specifically how they jump what is this Atomic electron transition, I understand that they jump and that they do this through absorbing and releasing energy but what is this Atomic electron transition other than what is already on the wikipedia article http:\/\/en.wikipedia.org\/wiki\/Atomic_electron_transition."}
{"id":"133808","title":"Finding directions of the internal forces within a freely jointed rigid object?","text":"Consider the the following problem: ![enter image description here](http:\/\/i.stack.imgur.com\/utK3P.png) Here is a force diagram showing the situation: ![enter image description here](http:\/\/i.stack.imgur.com\/eHwGx.png) For part (i) Taking moments about $B$ for $BC$ gives $84.5Lcos\\beta=2LT$ So $T=39$N For part (ii) Resolving vertically upwards on $BC$ gives $39\\cos\\beta -84.5=-r$ which results in $r=48.5$N in the direction shown on diagram (downwards). I'm aware that the question asks for the forces acting _on_ $BC$ so I deduced that by newtons third law $Y=48.5$N (upwards). Everything so far is correct but here's the problem: To find the horizontal component I resolved leftward on $BC$ to obtain $p=39\\sin\\beta$ giving $p=15$N direction as shown in diagram (leftward). Again realizing that the question asks for forces _on_ $BC$, by newtons third law $X=15$N ( **rightward** ): Here's the official answer: ![enter image description here](http:\/\/i.stack.imgur.com\/fvK3u.png) The marks-scheme is always correct. So could someone please kindly explain what I did wrong?"}
{"id":"133806","title":"Can we explicitly solve the Hamilton–Jacobi equation for a particle in a uniform magnetic field?","text":"HJE for nonrelativistic charged particle in an electromagnetic field is $$\\frac{1}{2m}\\left(\\nabla S - q\\mathbf{A}\\right)^2 + q\\phi + \\frac{\\partial S}{\\partial t} = 0.$$ For a uniform magnetic field $\\mathbf{B} = B_0 \\hat{\\mathbf{z}}$ and a particular choice of gauge this becomes something like: $$\\left(\\frac{\\partial S}{\\partial x}\\right)^2 + \\left(\\frac{\\partial S}{\\partial y} - qB_0 x\\right)^2 + \\left(\\frac{\\partial S}{\\partial z}\\right)^2 = -2m \\frac{\\partial S}{\\partial t}.$$ Can we solve this explicitly? It seems to me we can start by separating the variables $t, z$ by letting $S = f(x, y) + p_z z - Et$, giving $$\\left(\\frac{\\partial f}{\\partial x}\\right)^2 + \\left(\\frac{\\partial f}{\\partial y} - qB_0 x\\right)^2 = 2mE - p_z^2,$$ but I'm really bad at solving differential equations, so I don't know how to proceed. I also have no intuition for what the solution $S$ is supposed to look like, even though I already know how a particle moves in a uniform magnetic field, so I don't have any idea how to guess a form for $S$."}
{"id":"103901","title":"How does $\\bar{r}\\times(\\bar{\\nabla}\\times) - \\bar{\\nabla}\\times(\\bar{r}\\times)$ relate to the orbital angular momentum operator?","text":"When I attempted to calculate the following by hand $$\\bar{r}\\times(\\bar{\\nabla}\\times\\bar{F}) - \\bar{\\nabla}\\times(\\bar{r}\\times\\bar{F}),$$ I noticed some of the terms I extracted looked similar to the terms that appear in the oribtal anglar momentum operator $$\\bar{L} = -i\\hbar(\\bar{r}\\times\\bar{\\nabla}).$$ Is there a condensed expression that uses $\\bar{L}$? How does $$\\bar{r}\\times(\\bar{\\nabla}\\times) - \\bar{\\nabla}\\times(\\bar{r}\\times)$$ relate to the orbital angular momentum operator?"}
{"id":"103900","title":"Physical Consequence of Global Charge Conservation versus Local Charge Conservation?","text":"Gauge invariance of the electromagnetic field implies local charge conservation. If the gauge is no longer invariant, then the electromagnetic field must no longer conserve charge locally. Is it possible either physically or mathematically, to break local charge conservation, but still enforce global charge conservation? Charge can be created and destroyed at will, but the total amount of charge created at any given time is zero?"}
{"id":"88283","title":"If charged particles always attach to black hole event horizons, how can ordinary matter fall in?","text":"(A friend at work kindly loaned me loaned me his copy of Kip S Thorne's \"Black Holes & Time Warps\". This may have been ill-advised... :) BH&TW 1994 paperback p.410 Figure 11.5: > ... all positive charge that enters the horizon from the outside Universe > becomes attached to the horizon, and moves around on it, until it exits back > into the outside Universe (in the form of of negative charge falling inward > to neutralize the positive charge. Whoa there buckaroos, QED this ain't. Protons are made of quarks, and quarks are _individually_ charged at fractions of 1e. So, if I'm reading the above very straightforward statement the way it seem to be saying, _every charge_ should be individually trapped by the horizon. So how does that nasty lumpy $-1e$ of a charge cancel out each of the fractional charges of the proton quarks, some of which (the _d_ quarks) are negative BTW? So while I'm mostly OK with protons and antiprotons doing this little dance and falling in, because in that case all you are really doing is shooting gammas into the interior of the horizon. Even for that case, wow, that one statement above would be skipping over a plethora of particle complexity for any real event. E.g. you really need to look at the _quarks_ cancelling to get a complete picture, not just two particles with actual spatial size. The Thorne book is nine years and intended to be a popular read, so am I safe to assume that someone has subsequently looked in gruesome detail at the troublesome issue how exactly you cancel the mismatched charges of ordinary matter in a way that allows out-of-balance all-matter (or all-antimatter) lumps and atoms to fall into the interior of an event horizon? I note that the conservation principles in play for this question are the same ones that keep hydrogen atoms from turning into gammas. So while the interior of a black whole is a thing of beauty and pure energy and simplicity, messy matter first has to _get_ there by getting, more or less, converted 100% into gamma radiation. Once you have converted matter into all gammas and shot those into the event horizon interior, you can spit back whatever matter-antimatter or radiation combo you want. But if horizon charge attachment is real and fully self-consistent, I fully confess that I flatly do not see how you can ever get _into_ that interior. Also, I assume the membrane model has not been dramatically modified or abandoned since that time? After all, the idea that horizons are conductive seems so relevant to how quasars are powered that I find it a bit hard to believe that the conductive- horizon idea could ever be abandoned, since in effect quasars qualify as decent experimental evidence for the idea. Also, for whatever it's worth, I found the arguments about misbehaving field lines leading to charge capture to be nicely self-consistent and persuasive. So, more crisply: > If charge cancellation is required for particles to fall past an event > horizon, how can ordinary matter with its odd mix of spatially separated > fractional charges ever reach the 100% self-cancellation of charge needed > for it to pass through the event horizon? If appropriate, a reference to some widely accepted paper that adequately explains the nature of the whole-fractional charge cancellation mechanism would be fine."}
{"id":"88282","title":"Muons Internal Clocks and Time Dilation?","text":"I don't get how time dilation affects the half-life of muons? Time is just a tool that is used by humans how does that affect the internal clocks of elementary particles?"}
{"id":"131403","title":"Massive ropes and tension","text":"Consider the diagram bellow, a massive string hanging in equilibrium ![enter image description here](http:\/\/i.stack.imgur.com\/Vdwh2.png) My question is this: Can we take any 2 random point on the rope, say A and B and then use $\\sum F=0$ on the rope between them: i.e. say that $$T_0 \\sin(\\theta)+ T_0 \\sin(\\theta)=mg.$$ Where $\\theta$ is the angle the two forces $T_0$ make with the horizontal and m is the total mass of the rope between these two points. If this is the case would is work with the points in green so that in resolving horizontally would get us say $$T_1 \\cos(\\theta_1) = T_2 \\cos(\\theta_2)$$ where $\\theta_1$ and $\\theta_2$ are the angles they make with the horizontal. If you need any more information please just ask, thanks."}
{"id":"108307","title":"Coordinate System vs. Angular Properties vs. Centroid","text":"Please help me check my understanding related to the rotational motion of a 3D rigid body after reading some Physics textbooks and googling for some more materials (e.g., Wikipedia's Torque, Wikipedia's Moment of Inertia). The following points are to be checked against a situation where the rotational motion of a 3D rigid body happens about a single rotation axis in an inertial Cartesian coordinate system where the z-axis is taken to be the rotation axis but the origin $O$ of the coordinate system is **not** necessarily on the centroid of the 3D rigid body and can even be **outside** the 3D rigid body:   1. Each particle in the rigid body has $\\vec{v} = \\vec{\\omega} \\times \\vec{r}$ where $\\vec{r}$ is the position vector of the particle with respect to the origin $O$, and therefore, the speed of each particle depends on the choice of origin $O$.   2. With respect to $O$, the moment of force $\\vec{\\tau}$ and the angular momentum change $d\\vec{L}$ about the z-axis have the same directions.   3. With respect to $O$, the angular acceleration $\\vec{\\alpha}$ and the angular velocity change $d\\vec{\\omega}$ about the z-axis have the same directions.   4. Moment of inertia $I$ is always calculated with respect to z-axis, not the centroid of the rigid body, such that the $s$ in $\\int s^2 dm$ is always the distance between a point mass in the rigid body and z-axis.   5. Angular momentum of each particle in the rigid body $\\vec{L} = \\vec{r}\\times\\vec{p}$ where $\\vec{r}$ is the position vector of $\\vec{p}$ with respect to the origin $O$ (i.e., $||\\vec{r}||$ is **not** the distance between $\\vec{p}$ and z-axis), and therefore, the direction of $\\vec{L}$ may be different from $\\vec{\\omega}$.   6. Total angular momentum of the rigid body is then $\\sum \\vec{L} = I \\vec{\\omega}$, and therefore, the direction of $\\sum \\vec{L}$ is the same as $\\vec{\\omega}$.   7. The torque experienced by each particle in the rigid body $\\vec{\\tau} = \\vec{r}\\times\\vec{F}$ where $\\vec{r}$ is the position vector of $\\vec{F}$ with respect to the origin $O$ (i.e., $||\\vec{r}||$ is **not** the distance between $\\vec{F}$ and z-axis), and therefore, the direction of $\\vec{\\tau}$ may be different from $\\vec{\\omega}$.   8. The total torque experienced by the rigid body is then $\\sum \\vec{\\tau} = I \\vec{\\alpha}$, and therefore, the direction of $\\sum \\vec{\\tau}$ is the same as $\\vec{\\alpha}$.   9. The torque experienced by each particle in the rigid body is related to each particle's angular momentum rate of change $\\vec{r}\\times\\vec{F} = \\vec{\\tau} = \\frac{d\\vec{L}}{dt} = \\frac{d(\\vec{r} \\times \\vec{p})}{dt} = m\\frac{d(\\vec{r} \\times \\vec{v})}{dt} = m\\frac{d(\\vec{r} \\times (\\vec{\\omega}\\times\\vec{r}))}{dt} = m\\frac{d((\\vec{r}\\cdot\\vec{r}) \\vec{\\omega} - (\\vec{r}\\cdot\\vec{\\omega})\\vec{r})}{dt}$, and therefore, for each particle the directions of $\\vec{\\tau}$ and $d\\vec{L}$ are the same but may be different from the direction of $\\vec{\\omega}$.   10. The total torque experienced by the rigid body is related to the total angular momentum's rate of change $I \\vec{\\alpha} = \\sum \\vec{\\tau} = \\frac{d\\left(\\sum\\vec{L}\\right)}{dt} = \\frac{d\\left(\\sum I\\;\\vec{\\omega}\\right)}{dt} = \\sum I \\frac{d\\vec{\\omega}}{dt}$, and therefore, the directions of $\\vec{\\alpha}$, $\\sum \\vec{\\tau}$, $\\sum d\\vec{L}$, and $d\\vec{\\omega}$ are the same. Any mistake? Because the rotation axis can be outside the 3D rigid body, I feel uneasy especially for Point 5 to 10."}
{"id":"86699","title":"Confused about elasticity and collisions","text":"I was solving the following problem and the explanation to it confused me. > _There are two objects with mass $m$ and $M$, respectively. The object with > mass $m$ has a velocity of $\\sqrt{2gl}$ and collides with the other object > that is initially at rest. If the collision is elastic, what is the velocity > of the object with mass $M$ right after the collision? Friction is > negligible._ My understanding is, for these kind of collision problems momentum is conserved so, $$mv_{m} +Mv_{M} =mv'_{m}+Mv'_{M}$$ or in other words, $$mv_m=mv'_{m}+Mv'_{M}$$ Since the initial velocity is 0 for the object with mass $M$. In addition, elastic collision means that kinetic energy is conserved and intuitively, I am imagining that the smaller object (say, $m$) will \"bounce\" to the opposite direction as its initial velocity while the bigger object starts moving in the direction that is as same as the initial velocity of $m$, just a little slower. However, the answer explanation tells me that right after the collision the momentum is preserved, so $$mv_m=(m+M)v'_{m+M}$$. I thought that this equation represents the situation where the collision is completely INELASTIC and the objects stick to each other. Does it have to do with the wording \"right after the collision\" ? Or, is the explanation not making sense ?"}
{"id":"43175","title":"Uncertainty Principle for Information?","text":"I'm not familiar (yet) on how Information theory can be emerged\/used in QM\/QFT but I was thinking about this question: While we have Heisenberg uncertainty principle on measuring coupled observables, can we express it using somehow a more fundamental\/abstract concept like Information uncertainty (especially that we have Information conservation principle like for energy and momentum, and associated symmetry, i.e CPT)? in sense that because there is always some details that we can't measure\/know, can we express that as some information uncertainty? And despite of the answer please explain why there is or there isn't connection\/relation between the two concepts."}
{"id":"54372","title":"What are the biggest unanswered questions in physics today?","text":"Are there fundamental big questions in physics that most scientists encounter no matter what they're working on, or are most questions details of smaller subtopics? What are the great unknowns of our day? Sorry in advance if my question is too general. I told my kid that there are some things science still doesn't have theories for, and she asked \"like what\" and I said I wasn't sure, so I'm asking here. So much science has been done since I was in school that I don't know what all has changed. I think what she's after is some real mysteries to wonder about, ones that don't have generally accepted theories yet."}
{"id":"123628","title":"Ultrafast photon detectors","text":"Are there already photon detectors that can resolve the shape of photons in the time domaine if it looks like below where the two humps are separated by a time of the order of femtoseconds? What I know is that there are photon detectors that can resolve in the picosecond regime (see here). But this article is 6 years old... What is the best resolution that is possible today? ![enter image description here](http:\/\/i.stack.imgur.com\/wYDlb.png)"}
{"id":"116418","title":"Is it possible to tell whether the space ship is moving or not?","text":"Consider a space ship which is not under any force. Being inside the space ship, I will make a robotic fly from the platform to crawl inside using my remote. For simplicity assume that space ship will be drifting at uniform speed only under forward direction with respect to me, or else it will be under rest. The following experiment is assumed to be done in vaccum. From the top, the first image is of space ship. Assume that I am at point A and the robotic fly to be beside me as shown. In the second image, I will make the insect just to lift up and I will stop its engine, as the robot is not under any force it will not fall down, it will stay where it was. As the fly will continue to be under same motion as the space ship, I will not be able to tell whether my ship is moving or not. In the third image and fourth image, I will apply force to the robotic fly to move little back and I will stop its engine.   * (Third image) If the ship was under forward motion, the fly, because of the applied force backward i.e against its acquired motion from the spaceship, the forward motion of fly will not be synchronzed with the forward motion of the ship, so it will crash the wall with in an interval of time. This ensures that space ship was under uniform velocity.    * (Fourth image) If the ship was at absolute rest, the fly, even though there was an force applied in the backward direction, the fly will not crash with the backside of the space ship. This ensures that space ship was at absolute rest.  ![enter image description here](http:\/\/i.imgur.com\/S97AJfl.jpg) But, according to Feynman: > The principle of relativity was first stated by newton, in one of his > corollaries to the laws of motion: \"the motions of bodies included in a > given space are the same among themselves, whether that space is at rest or > moves uniformly forward in a straight line.\" This means, for example, _that > if a space ship is drifting along at a uniform speed, all experiments > performed in the space ship and all the phenomena in the space ship will > appear the same as if the ship were not moving, provided, of course, that > one does not look outside_...........in a moving space ship the electrical > and optical phenomena should be different from those in a stationary ship. Hoping that my above experiment is neither optical nor electrical, my experiment is in contradiction with the Feyman's explanation. From my experiment, I am able to say whether the space ship is moving or not. How is it possible?"}
{"id":"101500","title":"Horizontal projectile motion, finding the height of the object at a given time.","text":"A canon is fired Horizontal to the ground 80 meters above ground The canon ball is fired a $T=0$ and hits the ground at $T_g$. Calculate the height at the time $T_g\/2$ So far I have calculated time. which gives me the equation $$y=y_i+v_{iy}(4.05) - (0.5)(9.82)(4.05^2)$$ Now my problem is that when I try to calculate $v_{iy}$ I am getting $-37\\text{ m\/s}$ which I know is not the case; it wouldn't make any sense. $Y$ initial is equal to $80\\ \\ V_{iy}$ I don't know $4.05$ is the time it takes the projectile to hit the ground. $9.82$ is gravity which is my acceleration.   So my question is two parts. First, is this the best equation to find what I'm trying to find? Second, how do I find velocity in the $y$ direction when $\\theta$ is $180$ degrees? This is an edited question I do apologize for being unclear. Thank you for responding and requesting clarification. I was using the incorrect equation a simpler way to find the answer would be $y=.75\\times H$. Because change in $y = -.5\\times g\\times t^2$ sense initial velocity in the $y$ direction equals $0$. Thank you for the help in structuring my question better."}
{"id":"20702","title":"Statistical Mechanics treatment of the reaction process?","text":"I'm searching for an at least semi-rigorous Statistical Mechanics description\/treatment of a (spatially resolved) chemical reaction process of a macroscopic portion of at least two different species along the lines of $$A_1+A_2\\longrightarrow A_3,$$ (with $A_3=A_1A_2$, necessarily) or $$A_3\\longrightarrow A_1+A_2.$$ Might be a description here or a reference. I'm aware of the reaction kinetics description of combustion processes, where the idea is to modify diffusion equations with a reaction rate term $$''\\frac{\\partial}{\\partial t}[A_i]=D\\ \\Delta [A_i]+\\Sigma_j\\ R_{ij}(T,[A_k]),''$$ where the last term in the end is coming from a Boltzmann'esque theory, but I'm interested in \"more modern\" approaches or a self contained treatment from first principles, which might, for example, model the whole thing in therms of ensembles, obviously as linkage between thermodynamics and non-equilibrium physics."}
{"id":"80159","title":"Why has Earth's core not become solid?","text":"The Earth is billions of years old, yet its core has not yet cooled down and become solid. Will this happen in the foreseeable future?"}
{"id":"94125","title":"Angular momentum and spin","text":"I am having problems with this excercise. We look at a system where the total angular momentum is given by an electron with $l=1$ and $s=\\frac{1}{2}$. Now I am supposed to calculate the eigenfunctions $|\\frac{3}{2},m_j\\rangle, |\\frac{1}{2},m_j\\rangle$ with respect to $J^2 $ and $J_z$, where $J=L^{(1)}+S^{(2)}.$ In part a) I derived the equation $J^2=L^{(1)2}+S^{(2)2}+2L_3^{(1)2}S_3^{(2)2}+L_+^{(1)2}S_-^{(2)2}+L_-^{(1)2}S_+^{(2)2}$. Maybe this one is useful here. Also a hint was given: I should check that $|1,1\\rangle \\otimes |\\frac{1}{2},\\frac{1}{2}\\rangle $and $ \\sqrt{2\/3}|1,1\\rangle \\otimes |1\/2,-1\/2\\rangle-\\sqrt{1\/3}|1,0\\rangle \\times |1\/2,1\/2\\rangle $ are eigenfunctions with respect to $J_z$. This was clear to me how to do this and I checked it, but I don't know what this has to do with the actual excercise. Is there anybody who could help me with this excercise?"}
{"id":"94120","title":"Difference between mirror reflected light rays and rays of a screen","text":"What is the difference between the light rays reflected from a mirror and rays directly emmited from a screen if the screen shows the same image as the mirror would."}
{"id":"79787","title":"The reference frame of $c$","text":"I don't have a lot of knowledge of special relativity and associated topics; some of the few things I know are that \"all motion is relative\" (that is, there is no 'stationary reference frame'), and the speed of light ($c \\simeq 3 \\cdot 10^8~\\mathrm{m~s^{-1}}$) is the absolute asymptotic speed limit ( _asymptotic_ meaning that you can never equal it, only get arbitrarily close). What escapes me is how those concepts work together - to my naive understanding, an object would never move in its own reference frame (and so would never reach $c$). What reference frame is $c$ measured against? ( _Is_ it measured against a reference frame?) Or am I looking at this the wrong way?"}
{"id":"121299","title":"Why does my infrared thermometer say the sky is at -2 °C?","text":"I just got myself an infrared thermometer. I wouldn't have been able to predict what temperature it would give me when pointing at the sky, but it turned out to be -2 °C the first time I measured, and slightly going up and down with the temperature on the ground (say from -6 °C to 2 °C in the course of a day in which the ground temperature goes from 20 °C at night to 30 °C during daytime). Some details: this was for pointing straight up (as long as its not pointing too closely to the sun), a clear sky and a humidity of 63% according to internet. The range in which it measures radiation is from 8-14 μm and the specified range of temperatures is -50 to 550 °C. What is it that causes this reading?"}
{"id":"121298","title":"Computing fine structure of terms in LS coupling,","text":"In order to compute the fine structure of the terms in LS coupling (Russell- Saunders coupling), we must treat the hamiltonian $$H_2 = \\sum_{\\mbox{open subshells}} \\xi(r_i) \\vec{l_i}\\cdot\\vec{s_i}$$ as a perturbation to the perturbation given by the hamiltonian $$H_1 = \\sum_{\\mbox{open subshells}}\\frac{1}{r_{ij}}$$ which splits in differents terms labeled by $^{2S+1}L$ the energies obtained in the central field approximation. $H_2$ can be written as $$ H_2 = \\xi(\\vec{L}\\cdot\\vec{S}) \\vec{L}\\cdot\\vec{S}$$ and then the correction given from $H_2$ is then readily obtained. **My question is, how can this form of $H_2$ be derived from Wigner-Eckart's theorem?**"}
{"id":"53515","title":"Force exerted on a cone","text":"This is probably a pretty elemental question, but I can't figure out what the concept is called or how to calculate it. Say you have a solid cone shaped piece of ice, you hold it in your hand and squeeze the cone and it slips up as you continue the same force until it slips all the way up and out of your grip. What is this called and how do you calculate the speed of the cones movement with a given amount of force?"}
{"id":"6244","title":"Gribov ambiguities for splitting strings in the BRST conformal gauge","text":"The standard textbook treatment of the conformal gauge for a free string with BRST ghosts is fine, but when a string splits into two, we have Gribov ambiguities for the conformal gauge. We can impose the conformal gauge for the initial string before the split, and also the final two strings after the split, but not for the interpolating region. How do we deal with this?"}
{"id":"6246","title":"What role does gravity have on flame composition?","text":"In 2000, Nasa conducted an experiment and found that gravity plays an indirect role in flame formation and composition (see the wikipedia article). What role does gravity play? Why does a flame extend in the opposite direction of gravity? Assuming the experiment could be re-created, are there any other forces that would cause a flame to extend beyond a spherical shape?"}
{"id":"121742","title":"Why is the Taub-NUT instanton singular at $\\theta=\\pi$?","text":"Consider the following metric $$ds^2=V(dx+4m(1-\\cos\\theta)d\\phi)^2+\\frac{1}{V}(dr+r^2d\\theta^2+r^2\\sin^2\\theta{}d\\phi^2),$$ where $$V=1+\\frac{4m}{r}.$$ That is the Taub-NUT instanton. I have been told that it is singular at $\\theta=\\pi$ but I don't really see anything wrong with it. So, why is it singular at $\\theta=\\pi$? EDIT:: I have just found in this paper that the metric is singular \"since the $(1-\\cos\\theta)$ term in the metric means that a small loop about this axis does not shrink to zero at lenght at $\\theta=\\pi$\" but this is still too obscure for me, any clarification would be much appreciated."}
{"id":"121740","title":"Vapor-compression refrigeration","text":"In a vapor-compression refrigeration system I would like to understand the following :   1. Why is there a need for the compressor ? Can't the saturated vapor from the evaporator be cooled directly in the condenser ? Why does it need to be a superheated vapor in order to be condensed ?   2. How an expansion valve works exactly (physically speaking) ? I understand it uses flash evaporation to transform part of the liquid to vapor without heating it. But I don't understand how is this mixture temperature lowered below the temperature of the environment to be cooled. There is some kind of auto-refrigeration that is mentioned in the books I have read that I don't grasp. Can anyone explain to me how this happens ?"}
{"id":"45331","title":"How to calculate the supporting force for a cantilever beam?","text":"How do I find the supporting force $R_A$ and the moment $M_A$? Should I use the sum of moments or?"}
{"id":"114745","title":"Meaning of the Reduced Density Operator","text":"I am confused about what it is exactly that a reduced density operator describes. To illustrate, I came across the following seemingly paradoxical argument. Consider a biparte system $AB$, described by the pure state: $$| \\psi \\rangle = \\sum_{a,b} \\psi(a,b) |ab \\rangle$$ Its density matrix is defined as: $$\\rho \\equiv | \\psi \\rangle \\langle \\psi |$$ With reduced density operators: $$\\rho_A = tr_B(\\rho) \\qquad \\rho_B = tr_A(\\rho)$$ We can construct a new operator: $$\\rho_{AB} = \\rho_A \\otimes \\rho_B$$ Which is only equal to $\\rho$ if $|\\psi\\rangle$ is a seperable state (no correlations between $A$ and $B$). Another way to say this is: $$0 = S(\\rho) \\leq S(\\rho_A) + S(\\rho_B) = S(\\rho_A \\otimes \\rho_B) = S(\\rho_{AB})$$ Because if $A$ and $B$ are entangled, then $\\rho$ contains information about these correlations, but $\\rho_{AB}$ does not (is this wrong?). So say $A$ and $B$ are entangled, then $\\rho_{AB}$ does not contain that information, but I was under the impression that $\\rho_A$ ánd $\\rho_B$ do contain (some, all?) of this correlation information; otherwise both $A$ and $B$ should be described by pure states. I realise that $\\rho_{AB}$ has no physical meaning in the case of entanglement, but even as a purely mathematical construct I don't see how that information can just vanish. I guess my question is this: what information exactly does the reduced density operator encompass? And, if any of it is related to the correlations, how can this be reconciled with the argument in the paragraph above?"}
{"id":"80843","title":"How can I simulate a rigid bounced from a wall?","text":"How can I simulate a rigid sword bounced from a wall and hit the ground(like in physical world)? I want to simulate a simple animation. The sword is controlled by a center\/mass point.(Actually finally it't in a game engine). I can detect the figure and the size of the sword(maybe needed in doing bounce). And the position is $\\\\{x,y,z\\\\}$ rotation can be controled by quaternions\/matrix\/euler angles.[In mathematica, by default is matrix] # Step1 * * *               point1 = {0, 0, 500};point2 = {500, 0, 500};      Fly 500 miles. (Here I can add an acceleration)               Graphics3D[{Red, Cuboid[{600, -100, 400}, {510.1, 200, 600}],   GeometricTransformation[swordX[[1]],    TranslationTransform \/@ {point1, point2}]}, ViewPoint -> Front]      ![enter image description here](http:\/\/i.stack.imgur.com\/tYdjg.png) As you see in the picture, my sword inserted into the `Red` wall. # step2 * * * sword bounced from the wall, It should turn the head and do rotations and fly to the ground. Something like a projectile How can I simulate this physical process? Maybe what I need is an equation and some parameters? Any references? I've played with this _Mathematica_ Demonstration model, and I've made a video before. And I need make it more physical, especially how the sword hit and bounced from the wall and fly like a hammer in this model... video[03:07] # step3 * * * hit the ground, (maybe can bounce from the ground and hit again...) Some model maybe the same as that in step2. * * * I have no physical engines in Mathematica, if you know, welcome to tell me something related."}
{"id":"104311","title":"Hamilton formalism for Dirac spinors","text":"Let's have the Dirac free lagrangian: $$ L = \\bar {\\Psi} (i\\gamma^{\\mu}\\partial_{\\mu} - m) \\Psi . $$ I can rewrite it as $$ L = i\\Psi^{\\dagger}\\partial_{0}\\Psi - H_{d}, \\quad H_{d} = \\Psi^{\\dagger}(-i(\\hat {\\alpha} \\cdot \\nabla ) + \\beta m)\\Psi , \\quad \\hat {\\alpha} = \\gamma_{0}\\gamma . $$ Here $H_{d}$ is the Hamiltonian density. Is it possible to build Hamilton equations and Poisson brackets for spinors (for example, $\\Psi$ may be the canonical coordinate and $\\Psi^{\\dagger}$ be the canonical impulse)?"}
{"id":"13944","title":"How colored light can be explained if light is considered as emission of photon?","text":"If light is considered as wave, then different colored light can be explained as waves of different wavelength\/frequency. How colored light can be explained if light is considered as emission of photon?"}
{"id":"14337","title":"Circuit Simulation","text":"I am making a circuit simulation program, and I am not sure how to simulate the behavior of Diodes. Take this circuit:                                                          \\------100 Resistor -----LED------   Positive 5V-----470 Resistor-----| |---------Negative 5V                                                          \\-----------200 Resistor----------   How would you calculate the voltage and current through all of these components? Also, is it true that with a simple circuit like this: Positive 5V------470 Resistor------LED--------Negative 5V That you would subtract the forward voltage of the LED from the Battery's voltage, then use Ohm's law to calculate the current and resistance through the circuit? voltage = 5;   newVoltage = voltage - forwardVoltage;   current = newVoltage \/ 470;   resistance = voltage \/ current;   Thanks for the help."}
{"id":"11128","title":"Do Bell inequality violations appear instantly when the source is turned on, or do they increase over time?","text":"This experimental Question is a result of reading a particular article on Bell violations. I addressed the e-mail below to the corresponding authors —because who knows, they might reply— but it is not specific to them, it's just that their article crystallized the Question, and perhaps someone on Physics SE can tell me what the results of the experiment would be without doing it (although I think these time-dependent properties are not part of the usual data sheets for off the shelf components). There is a subsidiary question: have such experiments been done, and I haven't seen them in the literature? > Dear Johannes and Anton, > >   I've read your PNAS article \"Violation of local realism with freedom of > choice\" as a result of a blog posting by Sabine Hossenfelder at > http:\/\/backreaction.blogspot.com\/2011\/06\/nonlocal-correlations-between- > canary.html _[Physics SE readers can find PNAS and arXiv links there]_. I > was impressed by your three key remarks on the second page, which seem to me > nicely balanced. > >   Reading your article I wondered as I have many times whether Bell > inequality violations appear _instantly_ when the source is turned on, or to > what extent they increase over time. I am partly led to the questions below > because it seems that Bell violating experiments run continuously, whereas > it seems that technological applications may well be intermittent. > >   I imagine, specifically, an experiment _based on_ an experiment in which > we observe Bell violations routinely (presumably in CHSH form). The crucial > physical modification is to block both paths from the source to the two > detectors physically, at a point near the source, on a time scale of, say, > about a second, so that light travels from the source only for half a second > at a time. A steady state condition of the source, the quantized > electromagnetic field, and the detectors will presumably not be instantly > established, but, I suppose, would be established in half a second, so that > if we considered data only from the last quarter second of the on-phase we > would see the usual violation of Bell inequalities. > >   The crucial modification of the analysis is to consider the way Bell > violation changes over time, microsecond by microsecond (or more or less > finely time-sliced as experience indicates), after the moment that the > physical block is removed. Clearly this is a statistical analysis, since we > would expect to see approximately one photon pair every four microseconds > (at a local production rate comparable to the 250,000 photon pairs per > second that you report in your PNAS article). > >   I suppose that a number of questions emerge, some of which may be > characteristics of the source and detectors; others might be characteristic > of the experimental apparatus more generally considered. Firstly, most > elementarily, how does the rate of single photon detections change over time > in the microseconds after the physical block is removed? We would expect > that there would be a rapid approach to the steady state, but not > instantaneous. > >   Secondly, how does the rate of photon pair detections change over time in > the microseconds after the physical block is removed? Does the approach to > the steady state have the same relaxation rate as the relaxation rate for > the single photon approach to the steady state? > >   Thirdly, how does the violation of Bell inequalities vary over time in the > microseconds after the physical block is removed? Again, does this have the > same relaxation rate for the approach to the steady state? > >   It seems a matter of technological interest as well as of foundational > interest whether relaxation of Bell inequality violations towards the steady > state immediately after a light source is exposed is the same as the > relaxation rates of single photon detections and\/or double photon > detections, or whether we have to wait longer for usable Bell violations to > emerge. > >   Because I suppose these questions to be of general interest, I have also > posted them at http:\/\/physics.stackexchange.com. I don't imagine the results > of such an experiment will go against quantum theoretical expectations, but > I'm curious whether characterizing these time-dependent properties of the > sources and detectors might give some surprises."}
{"id":"95233","title":"Contracting Indices in General relativity","text":"I was reading a book about general relativity and I came across these two equations $$ \\begin{align} \\mathrm{g}^{\\mu\\nu}_{,\\rho}+ \\mathrm{g}^{\\sigma\\nu}{\\Gamma}^{\\mu}_{\\sigma\\rho}+ \\mathrm{g}^{\\mu\\sigma}{\\Gamma}^{\\nu}_{\\rho\\sigma} -\\frac{1}{2}( {\\Gamma}^{\\sigma}_{\\sigma}+{{\\Gamma}^{\\sigma}_{\\sigma\\rho}} ) \\mathrm{g}^{\\mu\\nu} &=0, \\tag1 \\\\\\ \\mathrm{g}^{[\\mu\\nu]}_{,\\nu} +\\frac{1}{2}( {\\Gamma}^{\\rho}_{\\rho\\nu}-{\\Gamma}^{\\rho}_{\\nu\\rho} ) \\mathrm{g}^{(\\mu\\nu)} &=0, \\tag2 \\end{align} $$ Where the connection is not symmetric with respect to the lower Indices. and it says that by contracting equation (1) once with respect to ($\\mu,\\rho$), then with respect to ($\\nu,\\rho$), then subtracting the resulting equations we can get equation (2), however I can't see how that's possible."}
{"id":"18149","title":"When has the speed of light been measured, recently?","text":"Yes, it is weird, absurd, but I can't stop thinking that the would-be superluminal neutrino speed has been computed by an arithmetic operation (space\/time) and not by direct comparison with a simultaneous light ray running in \"parallel\". So the \"unspeakable\", outrageous question: is the speed of light increasing in the last months? When it has been measured the last time? Also: is the speed of light measured in a one-way or two-ways (forward and back) method? It is a \"politically incorrect\" question, but its logic is rock-solid, I think."}
{"id":"122214","title":"George Green's definition of Green's function","text":"This is a curious question about the way George Green could have defined his Green's function. All the definitions I see have only Dirac-delta $\\delta(x−x′)$ function as their source on the RHS. But the Dirac-delta was defined about a century after Green's work. I just want to know, how George Green have defined his function (in case he defined it)? In case he didn't define his Green's function, how did he give a hint about this method (without Dirac-delta function) ?"}
{"id":"17971","title":"Is the Energy Sharply or Fuzzily Defined in Quantum Mechanics?","text":"According to quantum mechanics, energy of a state is uncertain within a small range in hydrogen atom. But we also know that energy of a state is quantized which is contradictory to the first. Which one is correct?"}
{"id":"111019","title":"Confusion with curl of Lorentz magnetic force","text":"Since the magnetic force is a no work force, $dW=\\vec F\\cdot d\\vec r=0$ for $\\vec F(\\vec r)=q(\\vec v(\\vec r) \\times \\vec B(\\vec r))$, therefore $\\oint \\vec F \\cdot d\\vec r=0$ by Stoke's theorem. Therefore, $\\vec\\nabla \\times \\vec F=0$. If this conclusion is true then expanding $\\vec\\nabla\\times[\\vec v(\\vec r) \\times \\vec B(\\vec r)]$ explicitly using the relevant vector identity one must also obtain zero. But it seems to be non-zero in general. So my question is whether $\\vec\\nabla\\times\\vec F=0$ or not?"}
{"id":"111017","title":"Liouville's theorem and conservation of phase space volume","text":"It can be proved that the size of an initial volume element in phase space remain constant in time **even for time-dependent Hamiltonians**. So I was wondering whether it is still true even when the system is dissipative like a damped harmonic oscillator?"}
{"id":"111010","title":"High Current at first and then steady from a battery","text":"I just came up with a question, When a battery is connected to a circuit, the current first suddenly peaks up and slowly goes down to a steady value. Why is it so? I have not actually observed this, but this question has been asked in exams, so it should be that way I hope."}
{"id":"8727","title":"Are valence electrons located solely in the s and p subshells?","text":"Or are they in all subshells??"}
{"id":"114227","title":"Interesting relationship between diffraction and Heisenberg's Uncertainty Principle?","text":"I recently came across an interesting explanation of diffraction through an aperture which does not use Huygens' Construction but instead relies on Heisenberg's Uncertainty Principle: > The Uncertainty Principle states that trying to pin a particle down to a > definite position will create uncertainty in its momentum, and vice-versa. > Therefore when you are confining a particle to go through a narrow aperture, > one is very certain about its position (the aperture is very narrow and > hence the particle must have been somewhere in that extremely small gap) So, > by Heisenberg's Principle, the particle will now have a crazy momentum... so > it can go in any direction. However, it seems we cannot use the same principle to explain the case where diffraction occurs when light passes around an obstacle. We are not locking the photons in a small position in that case, are we? Keeping the Uncertainty Principle in mind, diffraction through an aperture makes sense but in the case of a solid obstacle, my intuition says the wave should just be deflected. Is there a way to relate the Uncertainty Principle to diffraction around an obstacle? Besides, given that the Uncertainty Principle is defined in terms of particles, can it be used to explain diffraction in mechanical waves such as sound which are not composed of photons? _Do note that my question has nothing to do with diffraction patterns or the process of interference, and is rather about the relationship between Heisenberg's Principle and diffraction._"}
{"id":"118576","title":"Neel order and O(3) model","text":"The coarse grained fluctuations of the Neel order parameter in the half integer spin anti-ferromagnetic Heisenberg model is described by the O(3) non- linear sigma model with a strange berry phase term. See for example : Eq-2 in http:\/\/arxiv.org\/pdf\/cond-mat\/0311326.pdf I was wondering if someone could explain to me where the Berry phase term comes from. Apart from the origin of the term, it is also not clear to me how such a Berry phase term is allowed in a Lagrangian without any periodicity in the time direction. Thanks in advance for any inputs !"}
{"id":"118681","title":"determinism, and quantum computing","text":"It might seem logical to presume that the exact state of the world around us could have been predicted given a big enough computer to analyse all the particle's interactions and what not. Determinism seems to be logical. Then comes quantum computing. Since a collection of entangled qubits are only coherent when isolated from their environment, is this new information being fed into our known universe system? Whats going on here? Since you could hypothetically have hardware that starts manipulating particles as a result of these isolated quantum wave collapses, does this put the nail in the coffin of the deterministic universe? Or are we just awaiting a mechanism by which these wave function's chose the value to collapse to (quantum foam geometry or something)? As far as I understand we don't yet know how the wave function choses its collapsed spin state except that statistically it follows a predictable probability spread."}
{"id":"8189","title":"Knowing when wavefunction collapses","text":"So I learned that after a measurement of, lets say the position of the wavefunction of a particle is made, if another measurement of the position of the particle is made right away, you should get the same result because the wavefunction has collapsed. How can you tell the difference between the observer measuring the wavefunction which causes the collapse vs. something else that measured the wavefunction and collapsed it already and you are measuring it the second time around?"}
{"id":"12475","title":"Two paths having the same phase in the path integral approach","text":"In the path integral approach to Quantum Mechanics, can two distinctly different paths of the possible infinite paths have the same phase, i.e can there be a bimodal distribution of the phases associated with each path? And Why?"}
{"id":"116251","title":"What is the math\/physics behind maximizing tinker's damage?","text":"Tinker is a hero in the popular game Dota 2. He has a spell called 'March of the Machines' that creates a stampede of little robots in a rectangular area. The robots do damage on impact of an enemy unit. Basically this can be considered a vector field and enemy units are surface area. We want the maximum flux in order to optimize damage. If you're really interested in what this looks like: https:\/\/www.youtube.com\/watch?v=AmXRvkaYhIE Anyways, you often end up in static situation where enemy units aren't moving in this common pattern. It makes sense that a diagonal march will provide the most damage: ![enter image description here](http:\/\/i.imgur.com\/8hqUH4D.png) A vertical march would have one enemy 'shielding' damage from the one behind it (depending on if it were downwards or upwards). A horizontal march does negligable damage to the bottom due to way less surface area being hit. What kind of confuses me is once things start moving. The common advice is(artwork not mine): ![http:\/\/cloud-2.steampowered.com\/ugc\/596998708777054720\/3ED611DEA2B3245C834C28C12DE9D0188A72C1C0\/](http:\/\/i.stack.imgur.com\/0RpEH.jpg) Things tend to run roughly single file so you can assume the stuff running through the march is a wall of surface area. My intuition always made me think that when things are running through march, going fully perpendicular would maximize damage. The sadder part is I've taken vector calculus before, so I've encountered a great deal of this stuff. But this is certainly the area where my brain had a melt down! I'd love to see a mathy justification for this angled approach being optimal."}
{"id":"92998","title":"Negative sign in the Dirac term from the SUSY Kahler potential","text":"I want to calculate the Dirac term from the canonical Kahler potential, \\begin{equation} K = \\Phi ^\\ast \\Phi \\tag{1} \\end{equation} but I'm coming across a pesky negative sign in the final result. I am finding (see derivation below), \\begin{equation} \\- i \\bar{\\psi} \\bar{\\sigma} ^\\mu \\partial _\\mu \\psi \\tag{2} \\end{equation} This agrees with lecture notes by Matteo Bertolini (pg. 87) but is off by sign from lecture notes by Fernando Quevedo (pg. 50), while they both supposedly use the same conventions. I would be okay with deciding Quevedo's notes have an error but my result also seems to contradict the regular Dirac Lagrangian, which in four-vector notation is \\begin{equation} {\\cal L} _D = + i\\bar{\\Psi} \\gamma ^\\mu \\partial _\\mu \\Psi \\tag{3} \\end{equation} Any ideas where this problem is coming from? Here are my steps: The chiral and antichiral superfields take the form (I am using the (+---) metric), \\begin{align} & \\Phi = \\phi + \\sqrt{2} \\theta \\psi + \\theta ^2 F + i \\theta \\sigma ^\\mu \\bar{\\theta} \\partial _\\mu \\phi - \\frac{ i }{ \\sqrt{2} } \\theta ^2 \\partial _\\mu \\psi \\sigma ^\\mu \\bar{\\theta} - \\frac{1}{4} \\theta ^2 \\bar{\\theta} ^2 \\Box \\phi \\tag{4}\\\\\\ & \\Phi ^\\ast = \\phi ^\\ast + \\sqrt{2} \\bar{\\psi} \\bar{\\theta} + \\bar{\\theta} ^2 F ^\\ast - i \\theta \\sigma ^\\mu \\bar{\\theta} \\partial _\\mu \\phi ^\\ast + \\frac{ i }{ \\sqrt{2} } \\bar{\\theta} ^2 \\theta \\sigma ^\\mu \\partial _\\mu \\bar{\\psi} - \\frac{1}{4} \\theta ^2 \\bar{\\theta}^2 \\Box \\phi ^\\ast \\tag{5} \\end{align} Calculating the Dirac term involves the product of the second term of $ \\Phi ^\\ast $ and the fifth term of $ \\Phi $ and vice versa. I find: \\begin{align} \\- i \\bar{\\psi} \\bar{\\theta} \\theta ^2 \\partial _mu\\psi \\sigma ^\\mu \\bar{\\theta} & = - i \\partial _\\mu \\psi ^\\alpha \\sigma ^\\mu _{ \\alpha \\dot{\\alpha} } \\bar{\\psi} _{\\dot{\\beta}} \\bar{\\theta} ^{\\dot{\\beta}} \\bar{\\theta} ^{\\dot{\\alpha}} \\theta ^2 \\tag{6}\\\\\\ & = \\frac{ i }{ 2} \\partial _\\mu \\psi ^\\alpha \\sigma ^\\mu _{ \\alpha \\dot{\\alpha} } \\bar{\\psi} ^{\\dot{\\alpha}} \\bar{\\theta} ^2 \\theta ^2 \\tag{7}\\\\\\ & = - \\frac{ i }{ 2} \\bar{\\psi} \\bar{\\sigma} ^\\mu \\partial _\\mu \\psi \\bar{\\theta} ^2 \\theta ^2 \\tag{8} \\end{align} where in the last step I used the spinor identity, $ \\psi \\sigma ^\\mu \\bar{\\chi} = - \\bar{\\chi} \\bar{\\sigma} ^\\mu \\psi $. Repeating the calculation for the product of the fifth term of $ \\Phi ^\\ast $ and the second term of $ \\Phi $ and summing the two results gives: \\begin{equation} \\- i \\bar{\\psi} \\bar{\\sigma} ^\\mu \\partial _\\mu \\psi \\bar{\\theta} ^2 \\theta ^2 \\tag{9} \\end{equation} which after stripping off the $\\theta,\\bar{\\theta}$ is the result I quote above."}
{"id":"19254","title":"Violation of conservation of energy and potential energy between objects","text":"I would like to clarify my question. I have numbered them to be independent questions   1. For any conservative fields, $\\vec{F} = -\\nabla U$. Which means the restoring force is opposite to the increasing values of the potential energy. If it so happens that $F = \\nabla U$, does that mean the restoring force (not even called that anymore I guess) would be in the same direction as the increasing values of energy? Is this a violation to the first law in thermodynamics.   2. For attracting bodies, the closer the objects come together, the potential energy drops as $r \\to 0^-$ and $U \\to -\\infty$, but for repelling bodies we get $r \\to 0^-$ as $U \\to \\infty$ which means the potential energy actually increase. Am I right?"}
{"id":"51814","title":"Wave functions for three identical fermions","text":"I would like to express the wave functions for three identical particles, each with orbital angular momentum $L=1$ and spin angular momentum $S=1\/2$, in terms of single-particle wave functions. In other words, I would like to obtain the Clebsch-Gordan coefficients for this problem. The problem is discussed in Sakurai's Quantum Mechanics textbook around p. 375 and in Greiner and Muller's \"Quantum Mechanics: Symmetries, 2nd Edition\" on p. 300. I know I have to find the spin wave functions and the orbital angular momentum wave functions separately, and then combine them to get fully antisymmetric wave functions. I have the spin wave functions (four symmetric, 2 mixed symmetric under exchange of particles 1 and 2, and 2 mixed antisymmetric under exchange of 1 and 2), but I haven't been able to get a small enough number of angular momentum wave functions to get just 20 fully- antisymmetric total wave functions. In Sakurai's book, p 376, eq'n (6.5.20), we see that the 20 states can be decomposed into 2 states with total angular momentum $j=1\/2$, 4*3=12 with $j=3\/2$, and 6 with $j=5\/2$. Could anyone fill in _how_ Sakurai got 6 for $j=5\/2$, 12 for $j=3\/2$, and 2 for $j=1\/2$? Most importantly, referring to my comments on Y Macdisi's answer below, could anyone answer the following question: Is the orbital angular momentum state with single particle wave functions $\\alpha=1$, $\\beta=0$, and $\\gamma=0$ related in some way to the wave function with $\\alpha=-1$, $\\beta=0$, and $\\gamma=0$, or $\\alpha=1$, $\\beta=-1$, and $\\gamma=-1$, and so on? I would love it if I could just keep the first set of values for $\\alpha$, $\\beta$, and $\\gamma$, but I see no good reason to do that. If anyone happens to know where this problem is discussed more fully, I would appreciate a reference. Or, if anyone knows how to do this, I would appreciate help in knowing what the $j=3\/2$ and $j=5\/2$ states could be in terms of single-particle orbital ang. momentum states and spin states. I think I actually have the normalization factors already, I just need to know what the single-particle states are."}
{"id":"51815","title":"Where inside a large uniformly dense, symmetrical sphere would its gravity toward the center be the strongest?","text":"Imagine a sphere of uniform density with similar volume and average density of our Earth. There is a bore leading to the center of the sphere from the surface with a scale at regular intervals. At what part of your journey towards the center would the scale read your greatest weight? There are some caveats to consider. The sphere is in a very isolated region of space and has no angular motion. * * * I wrote this question down last night while I was having trouble sleeping. My intuition tells me that the surface of the sphere will have the greatest force of gravity. Any distance beyond the surface and the mass of the sphere above you will be reducing your weight. Is this correct? * * * If however the core is much more massive similar to Earth, one would feel a greater strain to stand upright as they descend."}
{"id":"54415","title":"Higgs-Boson\/Graviton","text":"The Higgs boson gives particles mass. And the graviton is the theoretical force-carrier of gravity. Gravity depends on mass. So if the Higgs Boson gives things mass, it therefore gives them gravity. Is the Higgs Boson the same thing as a Graviton? Or is there a difference? The only thing I know is that the Higgs Field is something very different from the Gravitational Field. Yet, I'm not satisfied with that fact. I want to know why the Higgs Boson is not the Graviton."}
{"id":"96677","title":"Nomenclature clarification concerning solitons","text":"My experience with solitons is restricted to the classical setting, namely solutions to the quartic interaction $\\phi^4$, the Sine-Gordon equation, and Korteweg–de Vries equations. I was explicit to show the boundary on my solitonic knowledge so that I could ask for an interpretation of the recent paper http:\/\/prl.aps.org\/abstract\/PRL\/v112\/i2\/e025301 In particular I do not understand the difference between solitons and vortex rings. My confusion stems from the introduction to the video Baths and Quarks: http:\/\/www.youtube.com\/watch?v=Ederft9dkag In the intro David Tong is blowing toroidal shaped bubbles, which I thought were examples of solitons. A phenomena which dolphins exhibit for play. However, in the watered down discussion of the aforementioned paper http:\/\/www.washington.edu\/news\/2014\/02\/03\/solving-a-physics-mystery-those- solitons-are-really-vortex-rings\/ the nomenclature vortex ring is used. Moreover, the so called \"heavy soliton\" posited in the Fermi gas is shown to be a quantized vortex ring. My guess is that the difference in word usage stems from the language of superfluid density functional theory. However my lack of expertise in the subject allows me to not go further than guessing at this stage. Could someone explain the difference, if there exists any, between the usage of the words soliton and vortex ring?"}
{"id":"62470","title":"Is angular momentum conserved if you move off at a Tangent?","text":"Lets imagine a binary system of two astronauts in space connected to one another via light rope. The rope is taut and they're spinning round and round with their axis of rotation being the the axis perpendicular to the their centre of mass. Now, my question is this. Lets say they each let go of the rope; they move off at tangents. Is angular momentum conserved ? And what is their subsequent motion ? My hypothesis : They will move off at constant velocity at a tangent to their circular motion. I believe that because they now under the effects of no external force by the rotational analogue of newton's second law (dL\/dt = T) their angular momentum will be of equal magnitude but opposite sign. Does this make sense ?"}
{"id":"81726","title":"Defining simultaneity with a central light vs with clocks","text":"So there's the classic example of the relativity of simultaneity involving two people on a train, with a light source exactly between them. Moments after the lights turn on, observers on the train will say the light struck the passengers simultaneously, while folks on the ground looking into the train will see the light hit one of them first. Now let's say the passengers start out next to each other, synchronize their clocks, and then slowly proceed to their respective ends of a 2 lightsecond long table. At 2:59:59 PM according to their clocks, the light between them turns on. Now there are 4 events: 1) The light hits Passenger A (the one closer to the front of the train). 2) The light hits Passenger B. 3) Passenger A's clock ticks 3 PM. 4) Passenger B's clock ticks 3 PM. Passengers on the train should all agree the four events are simultaneous, but what will people outside the train see? Will the clocks stay simultaneous with each other, or will 2 and 4 happen simultaneously, followed by 1 and 3? And regardless of the answer, can you justify it in terms of the fixed speed of light?"}
{"id":"2521","title":"Flux through a Mobius strip","text":"I was sent here from mathoverflow, hoping for a complete answer to this: === A friend of mine asked me what is the flux of the electric field (or any vector field like $$ \\vec r=(x,y,z)\\mapsto \\frac{\\vec r}{|r|^3} $$ where $|r|=(x^2+y^2+z^2)^{1\/2}$) through a Mobius strip. It seems to me there are no way to compute it in the \"standard\" way because the strip is not orientable, but if I think about the fact that such a strip _can_ indeed be built (for example using a thin metal layer), I also think that an answer _must_ be mathematically expressible. Searching on wikipedia I found that http:\/\/en.wikipedia.org\/wiki\/Mobius_resistor A Möbius resistor is an electrical component made up of two conductive surfaces separated by a dielectric material, twisted 180° and connected to form a Möbius strip. _It provides a resistor which has no residual self- inductance, meaning that it can resist the flow of electricity without causing magnetic interference at the same time._ How can I relate the highlighted phrase to some known differential geometry (physics, analysis?) theorem? Thanks a lot! === Now, I'm convinced that there are no way to apply Gauss law since there are no ways to bound a portion of space with a Mobius strip. Nonetheless I would like to \"see\" some equations showing me that \"[the mobius resistor] can resist the flow of electricity without causing magnetic interference at the same time\"."}
{"id":"2523","title":"Is there a limit to loudness?","text":"Is there any reason to believe that any measure of loudness (e.g. sound pressure) might have an upper boundary, similar to upper limit (c) of the speed of mass?"}
{"id":"94293","title":"Trying to turn a nonlinear differential equation into a linear one","text":"The problem I have today is to determine the differential equation for a square fin protruding from a wall that experiences surface to ambient radiation and has an internal heat generation of $\\dot q$ per length. The fin does not experience convection. I performed an energy balance on a differential element of the fin and reduced the differential equation to this: $$\\frac{d}{dx}(kA_{c}\\frac{dT}{dx})+\\dot q=\\frac{P \\sigma}{kA_{c}}(T(x)^{4}-T_{sur}^{4})$$ Because of constant corss-sectional area and constant specific heat of the fin, the equation reduces to: $$\\frac{d^{2}T}{dx^{2}}-\\frac{P \\sigma}{kA_{c}}T(x)^{4}+\\frac{P \\sigma}{kA_{c}}T_{sur}^{4}+\\frac{\\dot q}{kA_{c}}=0$$ When a fin is normally subjected to convection, we can say that $\\theta(x)=T(x)-T_{\\infty}$ and $\\frac{d \\theta}{dx}=\\frac{dT}{dx}$ and $\\frac{d^{2}\\theta}{dx^{2}}=\\frac{d^{2}T}{dx^{2}}$ . In that scenario, the $h(T(x)-T_{\\infty})$ term reduces to $\\theta(x)$ and the differential equation becomes second-order and linear. In this case with radiation, the differential equation is second order and highly nonlinear. I saw that I could \"linearize\" the radiation heat loss term but it still looks like there is a $T^{2}$ dependence which suggests it is still nonlinear. I am confused on the next steps I should take. If the differential equation that I created is fine then that is also great. Any Help is appreciated."}
{"id":"94295","title":"Parity of annihilation and creation operator - Real Klein-Gordon field","text":"In calculating the total-momentum operator of the real Klein-Gordon field, I end up with an equation like $$ \\vec P = \\frac{1}{(2\\pi)^3}\\int d^3p \\mspace{9mu} \\vec p\\Big(a_pa_{-p} + a_pa^\\dagger_{p}-a_{-p}^\\dagger a_{-p} - a_{-p}^\\dagger a_p^\\dagger\\Big) $$ and it seems the first and the last integrals vanish because they are antisymmetric w.r.t $ \\vec p \\rightarrow -\\vec p $. I am unable to see this anti-symmetry. Also I am not able to see how to relate $ a_{-p} $ to $ a_p $ or $ a_p^\\dagger $ other than some implicit integral relations like(which also am not very sure how to prove), $$ \\int d^3p a_p^\\dagger e^{-i\\vec p.\\vec x} = \\int d^3p a_p e^{i\\vec p.\\vec x}$$ Thanks for any inputs."}
{"id":"94296","title":"How to choose Gaussian surfaces while solving problems?","text":"I have a doubt regarding this problem: > _Two large identical flat metal plates are placed parallel to one another, > seperated by a small distance compared to their linear size. One plate is > given a charge per unit area $-5.50 \\times 10^{-5} Cm^{-2}$ and the other a > charge per unit area $+5.50 \\times 10^{-5} Cm^{-2}$. Using Gauss's law find > the electric field at an arbritrary point between the two plates._ I did it with the plug and chug formula without much trouble. However, when I am trying to use Gauss' Law, I don't know what surface to choose. I drew a cylinder but I'm not sure of its orientation (should I set the circular faces perpendicular or parallel to the field?). As well, I'm not sure on which surface to use when confronted with new problems. Is there a foolproof way of doing it or any set forms recommended?"}
{"id":"11465","title":"The most price-efficient experimental setup involving SPDC, single-photon counting etc","text":"I would like a suggestion on a price-efficient way of experimenting with spontaneous down-conversion and single-photon counting. The simple dual-slit experiment could be one part of an application although there are many, but I'm mainly asking about the key components which are similar - the laser that pumps a SPDC-crystal, the crystal itself, and photon-counters (the other standard optical bench items I'm more familiar with). For example, is a 405 nm (blue) diode-laser module the way to go, with a BBO- crystal, and what to use for detectors - are avalance photo-diodes good enough? What kind of quality does the laser need to have (wavelength stability etc)? I know there is a Journal of american physics teachers or something which have articles covering these fundamental experimental setups now and then, but I thought I'd ask here in case anyone who works at a lab (with limited funding :) with a good setup could share some ideas. If this gets many answers, it should be turned into a community wiki entry I guess."}
{"id":"57223","title":"Two photons transition","text":"if an atom in its ground state is coupled to an electromagnetic field it can absorb a photon if the EM field contains one with the right frequency. These transitions depends on $⟨f|H_i|i⟩$ (from Fermi's golden rule) with $|i⟩$ the initial state, $|f⟩$ the final one and $H_i=d.E$ the interaction between the field and the atom (with d the dipolar momentum and E the electric field). If we use $_F$ for the field and $_A$ for the atom we have : $⟨f|Hi|i⟩=_F⟨f|E|i⟩_F⋅_A⟨f|d|i⟩_A$ If $|⟨f|H_i|i⟩|^2=0$ the transition is not possible, for example when $_A⟨f|d|i⟩_A=0$. But is it also possible that $_F⟨f|E|i⟩_F=0$ ? If we look at transition with one photon (with the right impulsion p) we have $_F⟨0|E|1,p⟩_F\\neq0$. Is $_F⟨0|E|2,p'⟩_F$ (with p'=p\/2) also different from zero? I have tried to make these calculations for a Klein-Gordon field : I find $_F⟨0|\\phi(x)|1,p⟩_F=e^{ipx}\\neq0$, but I'm not sure whether $_F⟨0|\\phi(x)|2,p⟩_F=0$ or if I've made a mistake. I think it's strange since I only heard that two photon linear absorption was forbidden by consideration on the atom states, but I know that non-linear two photons absorption is possible and it seems to make sense with $⟨0|\\phi(x)|2,p⟩=0$ and $⟨0|\\phi^n(x)|2,p⟩\\neq0$ So here is my question, am I right? What prevents linear two photons transition? EDIT : Thanks to Emilio Pisanty for making everything more clear, I'm not very good at notations."}
{"id":"126268","title":"Relative wind velocity explanation - understanding Irodov problem 1.6","text":"I am having trouble understanding the reasoning behind the solution in this Irodov General Physics problem. The problem is 1.6: > 1.6. A ship moves along the equator to the east with velocity vo = 30 > km\/hour. The southeastern wind blows at an angle Θ = 60° to the equator with > velocity v = 15 km\/hour. Find the wind velocity v' relative to the ship and > the angle Θ' between the equator and the wind direction in the reference > frame fixed to the ship. My calculations: ![My diagram](http:\/\/i.stack.imgur.com\/u4AlL.jpg) After drawing the initial diagram I consider that in order for the wind direction to be relative to the boat, I need to subtract the boat vector from the wind vector. In my diagram I have repositioned the boat vector to be subtracted from the wind vector and using trigonometry I solve as follows: First the cosine law to determine the magnitude The magnitude = $\\sqrt{30^2 + 15^2 - 2\\cdot 15\\cdot 30\\cdot \\cos{60}}$ = $\\sqrt{675} \\approx 25.98$ Then for the angle I can use sine law $\\frac{sin(\\Theta)}{30 km\/h} = \\frac{sin(60)}{\\sqrt{675}}$ $\\Theta= \\arcsin{30km\/h \\cdot\\frac{sin(60)}{\\sqrt{675}}}$ $\\Theta = 90°$ The answer I get is therefore that the wind, in reference to the boat (as felt by the people standing on the boat), is approx 26 km\/h at an angle of 60° South of West (in relation to the equator). However the answer given in the solutions is always that the approximate speed of the wind is 40km\/h and the angle is approx 19°. I have looked at the solutions and I think the best explanation is given in the following diagrams obtained from link #3: ![Diagram](http:\/\/i.stack.imgur.com\/9rxFQ.jpg) ![Formula](http:\/\/i.stack.imgur.com\/VFCDY.jpg) I do not understand, however, why he is using cos(180+$\\Theta$). I want to understand why my solution is incorrect and yields a different answer. What is my mistake and what is the proper intuition for understanding this problem if I have done so incorrectly? Solutions I have viewed:   1. This Youtube Video   2. This online blog   3. This online blog"}
{"id":"68120","title":"How is the sun's equatorial circumference measured?","text":"How is the sun's equatorial circumference measured by scientists?"}
{"id":"98896","title":"What is the constant $g_0$ used for calculating the specific impulse in seconds?","text":"Wikipedia defines specific impulse as: $$ I_\\mathrm{sp} = \\frac{F_\\mathrm{thrust}}{\\dot{m} \\cdot g_0} $$ The $g_0$ is said to be _the acceleration at the Earth's surface_. So is it actually the acceleration at Earth's surface, or is it a universal constant that just happens to be equal to that acceleration? In other words, is the specific impulse of the same rocket different on either of two planets which have different gravitational constants and no atmosphere? * * * Update: Let me restate the question in yet another way: If I was building a rocket on Mars, would I need to adjust my $I_{sp}$ values (eg. for delta-v calculations to make sense for combining engines)? Unless I am mistaken, I understand now that the answer is **_no_**. However, I don't think that any of the current answers state this clearly enough."}
{"id":"9665","title":"What happens if you try to freeze water in an water tight container","text":"If I have a container that is full of water and I attempt to freeze the water by freezing the container, what would happen if the container is strong enough to prevent the water expansion? Could the water be in liquid state even though it is at or below freezing point? This is probably a daft question but its something that's been bugging me."}
{"id":"102508","title":"Large-N critical NLSM (equation 13.115 of Peskin and Schroeder)","text":"Any opinions if the equation 13.115 of Peskin and Schroeder is true on arbitrary manifolds in arbitrary dimensions for the same Lagrangian? I a priori see no problem. The point I also want to ask is - if one evaluates that equation for spacetime >2 then one will miss the symmetry broken phase of the theory - right? - BUT isn't the saddle point obtained from it still the critical point of the corresponding theory on whatever is the (>2)-manifold? * * * Is this in general true that the large-N critical NLSM on a manifold $M$ is the same as the theory of $N$ non-interacting conformally coupled scalars on $M$? Or in otherwords - will the Lagrange multiplier field in NLSM have to have at the large-N saddle point an expectation value which is the same as the conformally coupled mass for scalars on that manifold?"}
{"id":"90638","title":"Relative linear velocity of a particle to a rotating object","text":"I am trying to calculate the \"relative linear velocity\" of a particle moving over a rotating object. According to this paper (section 2.2) I am reading the relative linear velocity is calculated by: $$ v'= v - \\omega r $$ Where $v$ is the velocity of the particle, $\\omega$ the angular velocity of the rotating object and $r$ the radius of the object. What I don't understand is **why subtract?** For high angular velocities with low particle velocities that will produce a negative velocity. The particle's surely not traveling backwards relatively, if the particle is traveling in the same direction as the rotation?"}
{"id":"112679","title":"Classical and Semi-classical treatments of the ideal gas","text":"In the semi-classical treatment of the ideal gas, we write the partition function for the system as $$Z = \\frac{Z(1)^N}{N!}$$ where $Z(1)$ is the single particle partition function and $N$ is the number of particles. It is semi-classical in the sense that we consider the indistinguishability of the particles, so we divide by $N!$. The resulting expression for the entropy of the system is $$S = Nk \\left(\\ln \\left[\\left(\\frac{V}{N}\\right) \\left(\\frac{2\\pi mkT}{h^2}\\right)^{3\/2} \\right] + \\frac{5}{2}\\right)$$ Now consider a fully classical analysis. There $Z(1) = \\sum_{E} \\exp (-\\frac{E}{kT})$, where $E = p^2\/2m$ (assuming no interaction potentials). The problem can be mapped to an integral over the phase space of the Hamiltonian to give $$Z(1) \\rightarrow \\int \\exp \\left(-\\frac{1}{2mkT} (p_x^2 + p_y^2 + p_z^2) \\right)\\text{d}^3 \\underline{p} \\,\\text{d}^3 \\underline{x}$$ This can then be rewritten like $$ \\int \\exp \\left(-\\frac{p_x^2}{2mkT}\\right) \\text{d}p_x \\int \\exp \\left(-\\frac{p_y^2}{2mkT}\\right) \\text{d}p_y \\int \\exp \\left(-\\frac{p_z^2}{2mkT} \\right) \\text{d}p_z \\cdot V$$ where $V$ is the volume of the container. Those are Gaussian integrals and so evaluation is immediate. The result is that $Z(1) = (2\\pi mkT)^{3\/2} V$ The corresponding entropy can be calculated and the result is that $$S = Nk \\left(\\frac{3}{2} + \\ln\\left(\\frac{(2\\pi mkT)^{3\/2}}{V}\\right)\\right)$$. My question is: What is the significance of the factors $5\/2$ in the semi- classical treatment and the factor $3\/2$ in the classical treatment and why are they different? They look like the number of degrees of freedom a monatomic and diatomic molecule would have at room temperature, but I think this is a coincidence."}
{"id":"112670","title":"Stationary and Static","text":"I have some confusion about the concept of stationary and static. A metric $g$ is called **stationary** if there is a time like Killing vector $K$. $g$ is called **static** is further HSO (hyper surface orthogonal), i.e. there exists a foliation of hyper surface orthogonal to the integral curve of $K$. The Schwarzschild solution requires $g$ to be static, and the Kerr solution only require $g$ to be stationary. Here comes my question. To my understanding, requiring $K$ to be HSO is to enabling us to slice the spacetime into pieces with fixed time. The Kerr solution is for the metric outside a rotating body. In the note I am reading, it says > Do not require $K$ to be HSO, after all we want a rotating body. Why a rotating body indicates that the metric cannot be HSO? I think I can also slice the spacetime into piece for every time."}
{"id":"125901","title":"Is a non-Newtonian gas possible?","text":"I was wondering if it was possible to have a Non-Newtonian Gas. I was planning a science fair experiment to try to create one if its possible."}
{"id":"125909","title":"What is the phase space for outgoing photons?","text":"For a scattering process for which $n$ fermions are scattered, (by some conventions) the cross section acquires a phase space factor of: $$d\\sigma \\sim \\prod_{i=1}^n\\frac{d^3p_i}{(2\\pi)^3 2E_i}$$   1. In this convention, what is the equivalent phase space factor for an outgoing photon?   2. How is it that different conventions differ by arbitrary factors of $2$ and $\\pi$? isn't the cross section a measurable which has to \"stay put\" irrespective of the convention used?"}
{"id":"68787","title":"A question about speed","text":"> A object with mass 0.5 kg slides in straight line for 200 m, where the > friction act on it is 0.4 times the normal reaction acting on it. Find its > initial speed. I let $u$ be the speed. $u^2=2as=2*0.4mg*200=2*0.4*0.5*9.81*200,u=28.01ms^{-1}$. But the answer should be $40ms^{-1}$, can someone tell me what's wrong with my solution? Thank you. EDIT: Correct answer: $u^2=-2as=-2(f\/m)s=-2(-0.4N\/m)s=-2(0.4mg\/m)s=2*0.4*10*200=1600$ $u=40ms^{-1}$"}
{"id":"27587","title":"Higgs mass and the hierarchy problem","text":"I was wondering what is the opinion about importance of the hierarchy problem in the hep community? I'm still a student and I don't really understand, why there is so much attention around this issue. 1 loop corrections to the Higgs mass are divergent - in the cut-off regularization proportional to $\\Lambda^2$ and therefore require large fine tuning between the parameters to make those corrections small. But this kind of problem do not appear in the dimensional regularization. People like the value of $\\Lambda$ to be very large, with an argument that it should correspond to same energy scale at which our theory breaks down. I don't think, that we should treat the scale $\\Lambda$, as some kind of a physical scale of our model cut-off, as it is just a parameter to regularize the integral. Just like the $4+\\epsilon$ dimension in the dimensional regularization is not a physical thing. Why do we apply a physical meaning to $\\Lambda$? Not to mention the troubles with the Lorentz invariance. Maybe the hierarchy problem is an argument that the cut-off regularization scheme is just not right to use?"}
{"id":"27586","title":"Partition Functions in (A)dS\/CFT","text":"I'm trying to understand some aspects of dS\/CFT, and I'm running into a little trouble. Any help would be much appreciated. In arXix:1104.2621, Harlow and Stanford showed that the late-time Hartle- Hawking state in de Sitter space, \\begin{align*} \\Psi[h]=\\int_{dS} Dg_{g|_{\\partial dS}=h} \\exp(iS(g)) \\end{align*} is the analytic continuation of the infrared wavefunction from anti-de Sitter space near the boundary, \\begin{align*} \\Psi_{\\text{IR}}[h]=\\lim_{z\\to \\partial AdS}\\int_{AdS}Dg_{g|_{z}=h}\\exp(-S(g)). \\end{align*} For simplicity I've taken the metric to be the only field in the theory. Maldacena's formulation of dS\/CFT in arXiv:astro-ph\/0210603 is that the partition function of the dual CFT is equal to the Hartle-Hawking state at late times, i.e. $Z[h]=\\Psi[h]$. Now, if I take $h$ to be the metric on a sphere cross a circle, say, then $\\Psi_{\\text{IR}}$ computes the finite temperature partition function of the CFT dual to the AdS theory. This seems to imply that the partition function of the dual to the dS theory is just the analytic continuation of the partition function of the dual to the AdS theory. However, this seems to contradict the only known example of dS\/CFT, which is the $SP(n)$ theory dual to Vasiliev theory on $dS_4$, conjectured in arXiv:1108.5735. In this case, the dS\/CFT partition function is that of fermions at finite temperature, while the AdS\/CFT partition function is that of bosons. These are definitely not analytic continuations of one another under $\\Lambda\\to -\\Lambda$. Any ideas?"}
{"id":"67158","title":"How many points are required to make a black box","text":"I have a black box with an arbitrary mass distribution inside it. I want to replace that object with n point masses without changing any mechanical properties of the box (center of mass, total mass, moment of inertia tensor). What is the fewest number of points I need and how can I find them? I count 10 constraints. 3 from center of mass, 1 from total mass, and 6 from the moment of inertia tensor. 3 point masses give me 12 degrees of freedom so at first glance it seems the answer should be 3. In practice however it does not seem possible to represent some objects (a sphere for instance) with only coplanar points. At least 4 are required."}
{"id":"67159","title":"Optimal way to heat liquid from the sun?","text":"I understand that one way to tell how much a liquid will be heated up by a remote heat source is by looking at its color. I assume clear and colorless heats up least, then white, other colors and last black liquids will heat up the most. ( Is this right and can more detail be added?) However I don't understand the physics well enough to answer this simple sounding question. What other factors except for _visible_ color affect how much the liquid will be heated? If it is possible to give quantitive figures that would be great."}
{"id":"57916","title":"Did the Feynman heuristic of \"simple effects have simple causes\" fail for spin statistics?","text":"Someone here recently noted that _\"The spin-statistics thing isn't a problem, it is a theorem (a demonstrably valid proposition), and it shouldn't be addressed, it should be understood and celebrated._ \" \"Spin statistics\" is of course the shorthand way of referencing a most curious fact about the universe, which is this: If a particle of any type has a \"spin\" measured in whole units of quantized angular momentum, it will be a boson, a group that includes energy-like force particles such as photons. However, if its spin is (rather strangely) off-set by half a unit, it will instead be a fermion, which includes the particles that occupy space and that make up most ordinary matter. The rule is very simple. The explanation of it is arguably a bit more complicated. I find it fascinating that Nobel Laureate Richard Feynman worried over this simple theorem for decades, yet he never seemed to find an explanation for it that truly satisfied him. It was _not_ a lack of mathematical explanations, I should note. It was because Feynman deeply believed in a rather simple search heuristic: very simple relationships should in general also have simple, easily-conveyed explanations. Alas, Feynman's last attempt to explain spin statistics, in his _Dirac Lecture_ , always seemed to me one of his least clear bits of exposition ever. I am fairly confident Feynman would have assessed his Lecture that way himself, as he tended to be quite brutal in self-critiques on anything related to clarity of explanation. (I think there is an interesting family insight in that observation, incidentally: Richard Feynman's scientifically inclined father always hoped that his son, who had received the education he was never able to have, would someday explain all those little physics mysteries to him. The young Richard took that duty very seriously, and never really abandoned it, even towards the end of his own life.) So, my question and challenge: How is everyone doing on Feynman's spin- statistics challenge these days? Do you, fair reader, have in your hands some truly simple explanation for why whole-spin particles always seems to be bosons, and ones with half-spin offsets always seem to be fermions? I am not asking for twisty belts and wine glasses (please, _no!_ ), nor am I asking for something math free... though I do think anyone trying to answer this question should first look at how Feynman handled even complex numbers in his book _QED_. What I _am_ asking for is insight, the kind of explanation that makes the reader stop and think _wow, of course that's it, why didn't I see it what way before?_ So, anyone? I probably will not put an explicit bonus on this one, but if someone can provide an explanation that knocks everyone's +1 socks off, I guarantee I'll contribute at least a couple of hundred points to that overall consensus."}
{"id":"9391","title":"Derivation of polytropic process equation","text":"I know that a polytropic process equation is often the result of an empirical fit of a P-V curve that allows one to derive some nice analytical equations for the process. Is there actually a derivation for the polytropic process equation?"}
{"id":"9390","title":"Casimir effect plate experiment","text":"i was a little confused about the casimir effect. my understanding is with the 2 plate experiment there was a force pushing the plates together because there were less virtual particles in between the plates than outside them. Why are there less particles in between the plates?"}
{"id":"59321","title":"Is weak lensing the statistical effect of microlensing?","text":"I am looking into the effects of gravitational lensing of gravitational waves. I know that gravitons travel along null geodesics, just as photons, and so they will suffer the same deflection angle by a massive close to their paths I think I know what _microlensing_ is, with the impact parameter, Einstein angle, and amplification factor. However, I have read that _weak_ lensing is likely to be a problem for advanced gravitational wave detectors. Is weak lensing just the statistical, overall effect of many different microlensing events, or is it something else?"}
{"id":"95658","title":"When can i separate spin from the wavefunction?","text":"I am currently working on a Tight-Binding model and for the derivation of nearest neighbor spin interactions I have terms like $$ \\left<x\\uparrow\\right|\\eta\\;\\mathbf{l}\\cdot\\mathbf{s}\\left|z\\downarrow\\right> $$ where $\\left|x\\uparrow\\right>$ is an atomic orbital.For the simplification it is very important that i can assume $$ \\left|x\\uparrow\\right> = \\left|x\\right>\\left|\\uparrow\\right> $$ My question is:   * is this always possible in a solid body or do i neglect some effects with that? (like spin polarization or something?)"}
{"id":"78341","title":"Adiabatic filling of a container","text":"Suppose a thermally insulated container is filled with atmospheric air until the pressure reaches 5000 psi. This could represent the filling of a diving cylinder, before thermal dissipation becomes significant. Initially, then, some mass of air has atmospheric temperature $T_1$ and $p_1$. When forced in to the tank adiabatically, we expect its final temperature to be \\begin{align} T_2 &= T_1\\left(\\frac{p_2}{p_1}\\right)^{1-1\/\\gamma} \\\\\\ &= (300\\,\\mathrm{K})\\left(\\frac{5000\\,\\mathrm{psi}}{14.5\\,\\mathrm{psi}}\\right)^{1-1\/1.4} \\\\\\ &\\approx 1590\\,\\mathrm{K} \\end{align} This final temperature seems far too hot. A diving cylinder would likely melt at this temperature. More likely, though, something is wrong with my reasoning. Is this not the correct use of the temperature-pressure relations, assuming an ideal gas? If not, what other information is needed to predict the final temperature of the container?"}
{"id":"34297","title":"Bigger anti-matter particles","text":"I have learned about the existence of positrons as a decay product from uranium fission - if I'm not mistaken. Is there any evidence for higher 'mass' anti-matter, or is that mere speculation or rationalisation? i.e. Is the argument that there anti-matter based planets in the universe, or just a soup of positrons?"}
{"id":"109412","title":"What would we observe from thompson's lamp thought experiment?","text":"The thought experiment goes like this: > Say there is some circuit which turns a lamp on\/off with just a flick of a > switch. Say its off; you flick it, it turns on; flick it again it turns off, > and so on. > > So say you are conducting the experiment for two minutes. When the remaining > time halves, you flick the switch; and when that remaining time halves, you > flick again...This process goes on infinitely, until two minutes is reached. Mathematics says that the sum of all those remaining half-times will eventually tend to two minutes. And the outcome i.e. the final state is always unpredictable. We can't really know if its on\/off. Lets not concern about the final state. What would we actually observe towards the end, few micro seconds towards the two minute mark? Wouldn't the lamp appear turned on? What would happen if the time between two flicks tend to zero?"}
{"id":"34291","title":"Special Relativity: What differential equation describes an accelerated object from a non-inertial reference frame?","text":"I am looking for a set of differential equations (to be solved numerically for an educational program) that would describe the position and apparent time of an accelerated clock relative to a non-inertial reference frame. For now, all movement is along a single spatial dimension. I am _not_ interested in considering the time light takes to travel between observers; assume that they are clever enough to account for this delay. As inputs to this system of equations, I would have:   * An arbitrary, time-dependent acceleration a(t) as experienced by an observer in the non-inertial frame. In practice, this would be a simple list such as a(t=0) = 0; a(t=1) = 0.2; ...   * The initial position x(t=0), clock time t'(t=0), and velocity v(t=0) of the accelerated clock as seen in the non-inertial reference frame   * Another time-dependent acceleration a'(t') which describes the acceleration experienced by an observer traveling with the clock. To make this (hopefully) more clear, I'll give an example: an observer A, initially at rest, sees a relativistic rocket pass by carrying observer B. When A's watch reads t=0, she notes that B's position is x(t=0), B's speed is v(t=0) and B's (prominently displayed) clock reads t'(t=0). A has a list of instructions a(t) which tell her to set her rocket engines to achieve a specific acceleration at specific times. B has a similar set of instructions a'(t'). Both A and B consult their _own_ watches when determining when to change their acceleration. I have seen this question: What is the displacement of an accelerated and relativistic object?, which tells me enough to model the situation for an _inertial_ observer, but I have not figured out how to reapply this for a non- inertial observer. Similarly, I can compute the worldlines of non-accelerated objects from the perspective of a non-inertial observer, but it seems something sneaky happens when both the observer and observed are accelerating. I have also tried computing both worldlines from an inertial reference frame and then using the Lorentz transformation [x' = g (x - vt); y' = g (t - vx\/c^2)] to ask what B's worldline would look like from A's perspective, but this did not work (I can describe more about this, but for now it should suffice to say that the worldlines were not remotely correct)."}
{"id":"72067","title":"Total scattering rate of light for an atomic transition given a laser field with an intensity distribution?","text":"I have a laser with an intensity distribution $I(\\lambda)$ over a broad wavelength range (arroun 40 nm), I want to calculate the total scattering rate due to the full spectrum of the laser; the the scattering rate is given by $\\gamma_p(\\lambda) = \\gamma \\rho_{ee}(\\lambda) = \\frac{\\gamma}{2}\\cdot\\frac{A(I(\\lambda))}{1+A(I(\\lambda))+B(\\lambda)}$ where $\\gamma$ is the linewidth of the atomic transition and $\\rho_{ee}$ is the population in the excited state. I was directly integrating $\\int\\gamma_p(\\lambda)d\\lambda$, but the result makes no sense. Any idea?"}
{"id":"68877","title":"How to compute the change in entropy of s system when the temperature is decreasing?","text":"How to compute the change in entropy of s system when the temperature is decreasing? Such a question may be too general,then we may ask a special case:How to compute the change in entropy of Annealing system when such a process may be regarded as Markov Chain ?"}
{"id":"59519","title":"Heisenberg evolution equation for $\\hat{\\phi}$","text":"Consider quantum Hamiltonian of free massive scalar particle: $$\\hat{H} = \\int d^3x \\left[\\frac{1}{2} \\hat{\\pi}^2 (t, \\vec{x}) + \\frac{1}{2} \\partial_i \\hat{\\phi}(t, \\vec{x}) \\partial_i \\hat{\\phi}(t, \\vec{x}) + \\frac{m_0^2}{2} \\hat{\\phi}^2(t, \\vec{x}) \\right].$$ I'd like to show that Heisenberg evolution equation for $\\hat{\\phi}$ holds, i.e.: $$\\partial_t \\hat{\\phi} (t, \\vec{x}) = i [\\hat{H}, \\hat{\\phi}(t, \\vec{x})].$$ My problem is to compute the following term: $$\\int d^3 y \\left[\\partial_i \\hat{\\phi}(t, \\vec{y}) \\partial_i \\hat{\\phi}(t, \\vec{y}), \\hat{\\phi}(t, \\vec{x})\\right].$$ I would appreciate some help on how should I rearrange this commutator to see it vanishes."}
{"id":"59515","title":"Conservation of Linear Momentum at the point of collision","text":"This is a pretty basic conceptual question about the conservation of linear momentum. Consider an isolated system of 2 fixed-mass particles of masses $m_1$ and $m_2$ moving toward each other with velocities $v_1(t)$ and $v_2(t)$ respectively. Now conservation of momentum says that at any point during the particles' motion the quantity $$m_1v_1(t) + m_2v_2(t) =constant$$ With non-zero velocities and non-zero masses this constant will be **non- zero**. Let us say the particles collide at time $t_0$. At the point of collision, both particles have velocity **zero**. which would mean that the constant above will be zero. Contradiction. I realize I might be going wrong in my reasoning at the point of collision. In fact, I feel defining velocity at that point would not even make sense, since if one considers the displacement functions $x_i(t)$ $i=1,2$ of the particles, then $t_0$ would represent a point of **non-differentiability** of $x_i(t)$ for $i=1,2$. So **assuming** there are no collisions, by following the text-book derivation I can see why $$m_1v_1(t) + m_2v_2(t) = C_1$$ before the collision and $$m_1v_1(t) + m_2v_2(t) = C_2$$ after the collision would hold true, but not why $C_1=C_2$ Can someone help me in clearing this up?"}
{"id":"59514","title":"Lorentz invariance of the action for free relativistic particle","text":"I tried to check the Lorentz invariance of the standard special relativity action for free particle **directly** : ($c=1$) $$ S=\\int L dt=-m\\int\\sqrt{1-v^{2}}dt $$ Lorentz boost: $$ dt=\\frac{dt^{'}+udx^{'}}{\\sqrt{1-u^{2}}}$$, $$dx=\\frac{udt^{'}+dx^{'}}{\\sqrt{1-u^{2}}}$$, $$v=\\frac{dx}{dt}=\\frac{u+v^{'}}{1+uv^{'}}$$ Substitute these expressions into the action: $$ S^{'}=-m\\int \\sqrt{1-\\left(\\frac{u+v^{'}}{1+uv^{'}} \\right)^{2}} dt=-\\int \\sqrt{1-\\left(\\frac{u+v^{'}}{1+uv^{'}} \\right)^{2}} \\left(\\frac{dt^{'}+udx^{'}}{\\sqrt{1-u^{2}}}\\right) $$ $$ S^{'}=-m\\int \\left(\\frac{\\sqrt{1-\\left(\\frac{u+v^{'}}{1+uv^{'}} \\right)^{2}}}{\\sqrt{1-u^{2}}} \\right)dt^{'}-m\\int \\left(\\frac{u\\sqrt{1-\\left(\\frac{u+v^{'}}{1+uv^{'}} \\right)^{2}}}{\\sqrt{1-u^{2}}}\\right)dx^{'} $$ But I know that action of this type is invariant under lorentz transformations, so we want to get that $$ S^{'}=-m\\int \\sqrt{(1-v'^{2})} dt^{'} $$ But it doesnt seem to me that these expressions for action are equal. So I'm confused. Maybe there is a mistake in my calculations?"}
{"id":"78697","title":"Hydrogen atom in partial differential equations, show it is is independent of time.","text":"For the hydrogen atom, if $\\int|u|^2\\ \\mbox{d}x=1$ show that the same is true at all later times. Hint: Differentiate the integral with respect to t, taking care about the solution being complex valued. assume that $u$ and $\\nabla u\\rightarrow \\infty$ fast enough as $|x|\\rightarrow\\infty$."}
{"id":"78696","title":"Angle needed for object A to intercept with object B","text":"Object B is 15 degrees East of North at a distance of 20km\/h. Object B is moving at an average speed of 30km\/h in the direction 40 degrees East of North. If object A is capable of moving at 100km\/h, at what angle does it need to move to intercept object B? ![obligatory sketch](http:\/\/i.stack.imgur.com\/O2qUC.jpg) I've tried calculating it as vectors, but got stuck. In the end I just tried using good old geomatry, where $$ \\frac{\\sin A}{a} = \\frac{\\sin B}{b} $$ So I called the angle I'm searching for A, and the angle from the line between A and B, and the direction of object B, as angle B. 'a' and 'b' are the average speeds of object a and b. $$ \\frac{\\sin A}{30 \\times t} = \\frac{\\sin(155°)}{100\\times t} $$ $$ A = arcsin\\left(\\frac{30\\times t\\times \\sin(155°)}{100\\times t} \\right) = 7.284° $$ So the final angle should be $15°+7.284°=\\mathbf{22.284°}$ Is this correct? Isn't there a better way of doing this, like using vector products? PS. I've found a very similar question here, but didn't quite understand how i.e. $\\alpha + \\beta = 40° $, because my calculations show $ =35° $. And trying to read through comments and answers, confuses me more due to updates being made inconsistent to comments about the updates done. And due to me not having enough reputation, I'm not able to ask counter-questions. BONUS Question: I have been searching high and low on how to make a simple graph of my problem. Like this here question, with a graph of the problem hosted at stack.imgur.com which I guess was generated by LaTeX on stackexchange.com, right? EDIT: Added obligatory sketch!"}
{"id":"79562","title":"Hamiltonian form of Noether's Theorem","text":"I understand that Noether's Theorem has a Hamiltonian form, whereby {X, H} = 0 iff {H, X} = 0. The proof of this is trivial, as it follows from the antisymmetry of the Poisson Brackets. First question: It remains a question on my mind whether every symmetry, which transforms q(t) into q(t) + k(q(t)) truly has a representation as X(q,p) on the phase space. In the Butterfield paper, they rely on recitfication to make q a cyclic coordinate, in which case the problem is trivial. Is there an easier way to see it in the general case? Second question: What about general non-symmetry transformations? Do they always have a representation X(q,p) on phase space? In other words, suppose I have a know L(q, q'), and p(q, q'). Suppose I define a transformation that changes q(t) into q(t) + k(q(t)), and therefore changes q'(t) into q'(t) + q'(t)dk\/dq. Does there exist a corresponding X(q, p) on phase space such that dX\/dq = -k(q) and dX\/dp = q'(t) dk\/dq (after changing variables from q' to p)"}
{"id":"106993","title":"Different solutions for floating a DIY torii gate deeper than it's original floating level","text":"I'm facing a dead end with my physics experience on a new build I am working on. Any help would be appreciated. Basic problem: I am trying to build a floating wooden Torii gate to be placed on water in a way that the structure's original floating level is pulled\/pushed downwards in some way - Maybe 15-30 cm below surface. Reasons for a build like this: 1) It can't be there all year due to weather conditions 2) I want to go through it with my kayak 3) It's flipping cool! Questions: A) How can I achieve a STABLEish structure floating under water surface? I guess the best way would be to somehow pull it downwards with weights? How would YOU do it? B) How can this be calculated? C) Tips, tricks? Ask questions too. ![Mockup image](http:\/\/i.stack.imgur.com\/svOEo.png)"}
{"id":"57914","title":"Parallel capacitors without battery. Does charge flowing after a dielectric input?","text":"If I charge two capacitors which are connected parallel $[$the minus (-) of the one opposite to the minus (-) of the other and the plus(+) of the one opposite to the plus (+) of the other.$]$, will I get current (charge flowing) after adding a dielectric material to one of them ?"}
{"id":"2891","title":"Impedance formula for an edge coupled differential stripline in an asymmetrical stackup?","text":"I've been searching and have not been able to find a formula for calculating the impedance of differential lines on inner layers where the dielectric above and below it is not symmetrical. I've seen plenty of examples of symmetrical stackups, but that would not help in my case. Does anyone know what the formula would be for this? Thanks! Edited for more information: For those that don't know, a stackup consists of all the different layers of a PCB (Printed Circuit BOard). You can have, for example, a 4 layber PCB which has individual copper material & glass epoxy stacked on top of each other. This allows the board to have more layers for routing the nets of the circuits."}
{"id":"131657","title":"How much energy would the Human Torch need?","text":"I wasn't sure whether to put this in the SciFi section or here, but since I'm interested in a physical quantity,so I decided to ask it here. I was wondering about approximately how much energy would the Human Torch need to reach X temperature. I was wondering if $Q=mC\\Delta T$ would be useful, but I'm not sure if there's such thing as a $C$ for human skin?"}
{"id":"66201","title":"What are the theoretical prerequisites for experimental work with SEM and TEM?","text":"What level of knowledge and in what areas would be required to perform experimental work with an SEM (Scanning Electron Microscope) and a TEM (Transmission Electron Microscope) respectively? E.g. would a strong background in quantum mechanics be necessary for EELS (Electron Energy Loss Spectroscopy) using a TEM? Assume that both the SEM and TEM machines are modern (made within the last 3 years).   Samples to be observed in the microscope would be metal alloys for materials science research.   Imaging methods of particular interest are EBSD and Surface Imaging for the SEM, and BSE, EELS, and Electron Diffraction for the TEM."}
{"id":"66205","title":"Why is Helium-3 better than Deuterium","text":"I see that many websites and magazines with physics thematic are pretty excited about mining Helium 3 isotope on the Moon. But this seems to be a very hard-to-get resource. For more than one reason:   1. There is not too much of it on the moon. It actually seems to be 0.007% of the moon soil.   2. Transport to earth would be also very VERY complicated in my opinion.   3. People would be required to take care of the harvesters. Deuterium can be found in the seas as \"heavy water\". No spaceships needed. Now even when we get the Helium-3, I have a strong suspicion that it contains less energy than Deuterium but it will require more energy to start the fusion. ## Why I think it needs more initial energy to start the fusion?   * Deuterium contains one proton and one neutron. That makes a +1e core.   * Helium 3 has 2 protons and one neutron - That's +2 core. Now the worst problem with fusion reaction is, that once you pull the atoms so close, that the electron shells no longer shield the polarity of the protons they start repelling with electric force. Now if you're trying to pull +1 and +1 together it seems much easier than pulling +2 and +2 together. Just think about the standard equation: $F_e = k*\\frac{|Q_1*Q_2|}{r^2}$ where $k$ is the permittivity constant and $Q_{1,2}$ are the proton charges. Repelling force must be **4 times as large**. ![enter image description here](http:\/\/i.stack.imgur.com\/gORNB.png) ## Why I think it contains less energy? This is even simpler to explain. In the fusion process we merge smaller cores to create larger. We normally start with hydrogen and create helium. Now when we already have Helium fused, it means that we **missed one step** in the fusion. Why would we do that?   Can't actually Helium 3 core be one of cores that are created in Hydrogen fusion?"}
{"id":"22358","title":"How can an an ant lift 50 times its own weight and pull 30 times its weight?","text":"According to many sites like this one, an ant can apparently lift 50 times its own weight and pull 30 times its weight. Is it true? Can it be proved using physics? Though most sites agree that an ant can lift many times its own weight, not all agree with exactly _how many_ times its weight. The explanations provided if any are usually vague and do not use specific numbers. Can a specific numerical value be calculated? Secondly, how is it possible for the ant to pull 30 times its weight? I find it unbelievable. Can anyone explain this?"}
{"id":"10462","title":"What makes an Antenna special?","text":"The most important property of an antenna, beyond its conductivity(?) seems to be it's length (or more generally, its shape). But the antenna device is connected to the receiver by an arbitrarily-long connector. What is the fundamental difference between an antenna and the connector? Why doesn't the connector's length, straying from some 1\/2, 1\/4 length of the targeted band, interfere with the reception?"}
{"id":"22350","title":"Boundary conditions for static electric field","text":"Consider a surface that carries surface charge density. In electrostatics, boundary conditions are studied by showing that there is a discontinuity in the normal component of the electric field across the charged surface and that there is no discontinuity in the horizontal component. My question is aren't the electric fields at the surface always perpendicular to the surface? I mean where does the horizontal component come from if wherever we look locally, close enough to the surface, it looks like a flat infinite patch and there is no horizontal components!"}
{"id":"22353","title":"The nature of time, according to quantum field theory","text":"I will try my best to ask the question that best fits something I have been pondering on for a few days. > **Are virtual particles really constantly popping in and out of existence? > Or are they merely a mathematical bookkeeping device forquantum field > theory?** > > Gordon Kane, director of the Michigan Center for Theoretical Physics at the > University of Michigan at Ann Arbor, provides this answer. > > Virtual particles are indeed real particles. Quantum theory predicts that > every particle spends some time as a combination of other particles in all > possible ways. These predictions are very well understood and tested. > http:\/\/www.scientificamerican.com\/article.cfm?id=are-virtual-particles-rea We define a second by the \" _duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom_ \" (from wikipedia). Or in other words, we define a second by movement of a certain amount of waves\/particles. Since virtual particles pop in and out of existence, could you still claim that these things have a movement on the quantum level? If not, then how do you explain time\/spacetime in the eyes of quantum field theory? And how does quantum entanglement fit in to this picture? I am not a scientist, so please forgive me if my understandings of these concepts are flawed."}
{"id":"92386","title":"Why do I still get static shocks when hands are wet?","text":"Quite a few times now I have washed my hands, and while my hands are still wet I will touch something metal and get a static shock. I thought static electricity doesn't really build up when there's a lot of moisture, so that's why you don't get static shocks in the summer when there's more humidity. Even though it's winter now, I figured I wouldn't feel any static shocks if my hands are wet, but I still do."}
{"id":"73000","title":"seemingly impossible query","text":"While on an expedition to a lake, it was noticed that the wind speeds gusted violently in seemingly periodic wakes and the proposed explanation was Von Karman vortex shedding off of nearby mountains that the lake was in the lee of. Therefore I am attempting to calculate the amount of influence that a Karman vortex street would have on some wind speed readings obtained during that time in order to see if I can detect the influence in wind speed readings collected during that time that the observation of intense gusts was made. From some lengthy research into this topic I have found a cadre of equations which I think give me the desired results of the frequency and maximum wind speed within the Karman vortexes all based solely upon wind speed readings from the lake.I however have come across an issue in the calculations which reduces all parameters to either constants or to be exactly directly proportional to the presumed oncoming wind speed, a highly unlikely and illogical occurrence. The problem rests in the fact that the laboratory calculated kinematic viscosity of air doesn't hold up to large scale observations of vortex street formation, thus necessitation a new parameter as stated in this article. This new parameter is β and is defined as $$β=St\/Re$$ with $Re$ being the Reynolds number and $St$ being the Strouhal number. Due to the fact that I don't have the shedding frequency but need to calculate it, I cannot calculate St except through the formula $$St = .212(1-\\frac{21.2}{Re})$$ as found in this article. However, doing this and substituting into the parameter of $β$ yields a relationship between the constant $β$ and $Re$, which makes $Re$ constant and in turn $St$ constant as $St$ is defined in terms of $Re$ only. This causes all the parameters that I have to be either linear with respect to any oncoming flow velocity or constant, something obviously wrong. Although I have everything that I need after this point in order to obtain maximum wind speed and shedding frequency, this initial step has stopped me dead in my tracks due to the fact that every author whom I have seen on this subject had a picture to work with to calculate the distances between the vortices and use that and an assumed wind speed to calculate shedding frequency, plug that in to find Strouhal number and then use the $β$ parameter to calculate Reynolds number. All those working in lab tests similarly calculated shedding frequency and recorded it by video taping or counting the eddy shedding process. I, however, only have the wind speed to work with and so have problems by reducing out too many variables. My question therefore becomes how do I define a either a variable $β$ parameter or redefine either my $St$ or $Re$ number so that one can be calculated independently with the limited information at my disposal of only the wind speeds and the ability to assume other parameters such as landmass dimensions etc. Thank you for your time reading this."}
{"id":"77788","title":"Stable equilibrium given force","text":"If a particle moves under the influence of a resistive force proportianal to velocity and a potential $U$, $$F(x,\\dot x)=-b\\dot x-\\frac {\\partial U}{\\partial x}$$ Where b>0 and $U(x)=(x^2-a^2)^2$ My thoughts were to make $F=0$, which would result in: $$\\dot x= \\frac 1b(4x^3-4a^2x)$$ This means that the points of equilibrium are a function of time? EDIT:: So I set up as suggested, using $F=m\\ddot x$ to get $$0=m\\ddot x+b\\dot x+4x(x^2-a^2)$$ I am trying to solve for $x(t)$. I know how to solve the equation that has only x, but I do not know how to solve for the x^3 term. I am curious how to combine these to get x(t)"}
{"id":"35651","title":"Inductance of loop to a square wire? What is the induced current and the induced $\\bar E$?","text":"_I am practising for the elementary-physics -course Tfy-0.1064, this puzzle is a simplified version of the problem 3Ahere (sorry not in English). My basic idea here is to understand a scenario with one loop and then understand the solenoid case._ Suppose a circle around the square of 20-cm-side and $\\frac{dB}{dt}=35\\frac{mT}{s}$ in the circle, please, see the picture below. What is the induced current and the induced $\\bar E$ to the square by the circle? ![enter image description here](http:\/\/i.stack.imgur.com\/CoIpB.jpg)"}
{"id":"32923","title":"vector-scalar gravity","text":"Has anyone developed a vector-scalar theory of gravity using an acceleration field equation that is based on a ratio in which the numerator is the familiar Newtonian vector sum of all masses times the inverse square of their respective distances, and the denominator is the scalar (not vector) sum of the very same formula? As a scalar sum, the denominator would always be larger than the numerator because there is no cancellation from opposite directions, and the quotient could be interpreted as the proportion of \"directionality\" of the combined contributions. Or has anyone developed a relativistic theory based on the same ratio of vector to scalar sum of the same contributions?"}
{"id":"77965","title":"An Equality in The Calculus of Many Instantons","text":"I am reading the review on instantons. When I tried to derive formula (2.27) on page 17, I always get the different coefficient of $gF_{mn}$ term. My calculation is just directly expanding the first term $-\\frac{1}{2} \\int d^4x {\\rm{tr_N}} F^2_{mn}$ in action to quadratic form, and leaving the second term $i\\theta k$ invariant. Under the fluctuation $A_n \\to A_n +\\delta A_n $, the first term $-\\frac{1}{2} \\int d^4x {\\rm{tr_N}} F^2_{mn}$ becomes $$ -\\frac{1}{2} \\int d^4x {\\rm{tr_N}} F^2_{mn} \\\\\\\\-\\frac{1}{2} \\int d^4x {\\rm{tr}_N }\\left( 2(D_n\\delta A_m)^2 -2D_n\\delta A_mD_m\\delta A_n +2g F_{mn} [\\delta A_m,\\delta A_n] \\right) +...$$ Integrating by parts and disregarding the surface term, the quadratic term becomes $$ \\int d^4x {\\rm{tr}_N }\\left( \\delta A_m D^2 \\delta_{mn}\\delta A_n -\\delta A_mD_nD_m\\delta A_n+2g F_{mn} \\delta A_n \\delta A_m \\right)$$ Using the equalities $D_n D_m=D_mD_n-gF_{mn}$, $D_n\\delta A_n=0$, the quadratic form becomes $$ \\int d^4x {\\rm{tr}_N }\\delta A_m\\left( D^2 \\delta_{mn} +g F_{mn}+2g F_{mn} \\right)\\delta A_n $$ So the coefficient of $gF_{mn}$ term is 3. What is wrong with the above calculation?"}
{"id":"23914","title":"\"Conceptualizing\" neutrinos","text":"Layman here. EE and BS physics. I am \"content\" in viewing photons\/electromagnetic radiation as an \"emergent\" property of spacetime? due to the electrons (\"particles..?\") and all their activity jumping around in atoms. [or is this likely too simple?] ... in the \"same\" manner that sound and water waves are an \"emergent\" property of air and water particles. OK. So WHAT do neutrinos represent? Can I visualize them as an emergent property of neutrons and protons [and quarks?] and all their activity in the nucleus? Neutrinos are a real mystery to me, as if everything else weren't!"}
{"id":"74783","title":"Centripetal acceleration for particle that moves in a circle at non-constant speed","text":"How is $a_{R}=v_T^{2}\/r$ derived for changing tangential speed? ($a_R$-centripetal component $r$-distance from axis $v_T$- tangential speed)"}
{"id":"65963","title":"Right-angle lever paradox in special relativity","text":"I remember to have read somewhere an interesting special relativity \"paradox\" considering two perpendicular rods $A$ and $B$ of equal proper length $L$ fixed at point $O$. ![rest frame](http:\/\/i.stack.imgur.com\/ULxzd.png) In the \"rest\" frame equal (in magnitude) forces are applied on the edges of the rods, perpendicular to them, thus giving zero net angular force: $$-F_AL_A + F_BL_B = -FL + FL = 0$$ (assuming $z$-axis is towards us). In the frame moving parallel to rod $B$ with the velocity $v$ it's length is diminished due to Lorentz-contraction $$L_B' = L \\sqrt{1-v^2\/c^2}$$ and perpendicular component of the force is diminished also $$F_B' = F\\sqrt{1-v^2\/c^2}$$ while $L_A' = L$ and $F_A'=F$. So the net angular force in this frame is not zero: $$-F_A'L_A' + F_B'L_B' = -FL + FL (1-v^2\/c^2) = -\\frac{v^2}{c^2} FL$$ and the system should be rotating clockwise with angular acceleration. I don't remember the explaination of the paradox as I did not understand it at that time, but now it seems to be very simple: even in Newtonian Mechanics total angular force would not be zero in \"moving\" frame given the fact that the net force is not zero. If we take into account that rods are _fixed_ relative to $O$, we should assume that in the junction point $O$ there is an additional force which compensates the forces on the edges of rods. The same argument seems to be valid in the relativistic case too. However, I remember that the explaination was not that simple. Am I wrong with my explaination, or the statement of the paradox is wrong? Does anyone know similar \"paradox\"?"}
{"id":"27209","title":"What do theoretical physicists need from computer scientists?","text":"I recently co-authored a paper (not online yet unfortunately) with some chemists that essentially provided answers to the question, \"What do chemists need from computer scientists?\" This included the solution of theoretical problems, like combinatorial enumeration and the sampling of certain classes of graphs; and practical programming problems, like open-source implementations of algorithms that are currently only implemented in expensive software packages. This motivates me to ask: what about this field? Are there theoretical issues of combinatorics, algorithm analysis, that physics needs a theoretical computer scientist to solve? Or how about creation of practical tools that would allow a theoretical physicist to do a better job: \"If only I had a program that would solve this type of problem for me!\" Intended as community wiki."}
{"id":"68873","title":"Particle in a 1D box (momentum representation)","text":"I have this problem. I want to find the wave function in the momentum space for the particle in a 1D box. We know that the wave function in the position space is: $$Y_n(x) = A\\sin{(n\\pi x\/L)}$$ Well, if I write the Schrödinger equation in the moment space I have: $$\\frac{p^2}{2m}Y_n(p) = E_nY_n(p)$$ So, this equation doesn't give me any information about the wave function $Y_n(p)$ I know that I can solve this problem just using the Fourier tranformation, but I'm asking myself if there is another posibility to solve this problem. PD. If I use the Fourier transformation, do I have to integrate just from $O$ to $L$?"}
{"id":"25516","title":"Balmer lines and their positions","text":"I have two stars, both of which have Balmer lines at the same wavelength positions. The depths of the Balmer lines on each star are different, and the stars are different as well. Why does this pattern exist?"}
{"id":"98409","title":"Compute an aerodynamic shape","text":"I'm interested in building a Fairing for a recumbent bicycle. A fairing is just a rigid curved surface (usually plastic or fiberglass) that surrounds the bike in order to reduce wind drag. Apparently it can increase your speed dramatically (as much as 300% with zero wind relative to the ground). Questions of building the thing aside (which looks feasible actually), how can I compute the \"most\" aerodynamic surfaces to enclose a given space? In other words, if I have a given three dimensional \"blob\" (my body's range on motion + the bike itself) and a given direction of motion, what formulae and\/or software can I use to give me some highly-aerodynamic shapes that are just slightly larger than the blob?"}
{"id":"70577","title":"Deriving the transformation under Weyl rescaling in Polchinski eq. (1.2.31)","text":"I have another question in Polchinski's string theory book volume 1, namely how to derive Eq. (1.2.32)? $$(-\\gamma')^{1\/2} R'=(-\\gamma)^{1\/2} (R-2 \\nabla^2 \\omega) \\tag{1.2.32}$$ I have awared his Eq. (1.2.21) for the transformation of world-line metric $$\\gamma_{ab}'(\\tau,\\sigma)=\\exp(2 \\omega(\\tau,\\sigma )) \\gamma_{ab} (\\tau,\\sigma)$$ , the definitions of Ricci scalar, $R:=R_{\\mu \\nu} g^{\\mu \\nu}$, Ricci, and Riemann tensors. It is still far from obvious how to construct the transformation of $R'$ from metric->Christoffel symbols->Riemann->Ricci-> (1.2.32) My question is, how to derive Eq. (1.2.32)?"}
{"id":"105311","title":"Conversion of QCD cross section formula","text":"I'm writing a program to calculate NLO cross sections for semi-inclusive high-$p_T$ pion production in proton-proton-collisions for my bachelor thesis. I've got a paper describing the production cross section as function of the differential cross sections of the partons $\\frac{d\\sigma}{dv}$ and $\\frac{d\\sigma}{dv dw}$ where $v = 1+\\frac t s$ and $w = - \\frac{u}{s+t}$, with $s$,$t$,$u$ being the conventional mandelstam variables. And I found a paper tabulating all the partonic cross sections. But unfortunately they use the formula $$d\\sigma = \\frac{1}{32 \\pi^2 s} \\frac{d^{n-1}p_3}{p_3^0} \\frac{d^{n-1}p_4}{p_4^0} \\sum M_{fi} \\delta \\left(p_1+p_2-p_3-p_4\\right)$$ How do I calculate the conversion from $\\frac{d\\sigma}{d^{n-1}p_3 d^{n-1}p_4}$ to $\\frac{d\\sigma}{dv}$? I know in principle how to deal with differentials, and that the conversion should somehow begin with $\\frac{d\\sigma}{dv} \\frac{dv}{d^{n-1}p_3 d^{n-1}p_4}$ but I don't have a good ansatz how to deal with the posterior part..."}
{"id":"30366","title":"Why wet is dark?","text":"When something gets wet, it usually appears darker. This can be observed with soil, sand, cloth, paper, concrete, bricks ... What is the reason for this? How does water soaking into the material change its optical properties?"}
{"id":"113265","title":"Time travel to the past in binary black hole system","text":"I just saw this YouTube video (only 3 mins long) of Neil deGrasse Tyson explaining what happens when two black holes collide. He says that when the black holes are rotating around each other, there exists a path that you could follow that would cause you to travel back in time. Any chance that someone could explain this in an intuïtive way to me? I just finished my 1st bachelor physics, to give you an idea of my knowledge. I also understand the basics of special relativity."}
{"id":"131788","title":"Are temperature increases discrete?","text":"Are temperature increases ever discrete in nature, or is it a continuous variable? If a discrete case exists, is there any material that exhibits particularly strange behavior?"}
{"id":"100830","title":"Enthalpy Change in Reversible, Isothermal Expansion of Ideal Gas","text":"For the reversible isothermal expansion of an ideal gas: $${∆H}={∆U}=0 \\tag1$$ This is obvious for the case of internal energy because $${∆U} = \\frac {3}{2} n R {∆T} = 0 \\tag2$$ and $${∆U} = -C_P n {∆T} = 0 \\tag3$$ For the case of enthalpy it is easy to see that $${∆H} = -C_v n {∆T} = 0 \\tag4$$ I've also seen $${∆H} = ∆U + ∆(PV) = ∆U + nR{∆T} = 0 \\tag5$$ Now for the part I don't understand. $$dH = dU + PdV \\tag6$$ $$dH = dU + nRT \\frac {dV}{V} \\tag7$$ $${∆H} = {∆U} + nRT \\ln\\frac {V_2}{V_1} \\tag8$$ $${∆H} = 0 + nRT \\ln\\frac {V_2}{V_1} = \\ln\\frac {V_2}{V_1} ≠ 0\\tag9$$ Clearly, it is incorrect to make the substitution $ P = nRT\/V$ in going from $(6)$ to $(7)$. Why is that? I thought equation $(6)$ was always valid, and integrating such a substitution should account for any change in the variables throughout the process. Why does this not yield the same answer as $(4)$ and $(5)$?"}
{"id":"75123","title":"Can parallel universes constitute the missing mass aka dark matter?","text":"I was pondering about this question for a while now. And with my limited knowledge of physics I can answer myself - if parallel universe theory is true and if indeed there is gravitation leakage from 1 universe to the other, there would be infinite number of universes and thus unimaginable amount leakage would occur making life or formation of any matter impossible. Does that sound like a plausible answer? :)"}
{"id":"107579","title":"How can I see this equation describes advection?","text":"My notes say that this is the equation for the \"advection of a quantity **A** at speed **v** \": $$ \\frac{\\partial \\mathbf{A}}{\\partial t} = \\nabla\\times(\\mathbf{v}\\times\\mathbf{A}) .$$ Is this true? How can I see (mathematically) that this equation corresponds to advection?"}
{"id":"33383","title":"Relation for Dirac-spinors of different helicities","text":"Assume that we have massless spin-1\/2 particles. The Dirac-spinor is the solution of the Dirac equation: $$ p^\\mu \\gamma_\\mu u_\\pm(p) = 0, \\quad p^2 = 0$$ The subscripts $\\pm$ denote two different solutions, belonging to two different helicities. Is it possible to find a representation for the $\\gamma_\\mu$ so that the following relation holds true? $$ u_+ = \\left( u_- \\right)^* $$ I know this possible for (polarization) vectors, i.e. one may choose $\\epsilon^\\mu_+ = \\left( \\epsilon^\\mu_- \\right)^* $, but I guess it is not for spinors (in the weyl representation it obviously is not). So here is my **Question:** Is it possible to find a representation of the Dirac-Gamma- matrices so that the spinors of different helicities are related by complex conjugation?"}
{"id":"41329","title":"Lorentz transformation matrix and its meaning in Minkowski diagram","text":"I have sucessfully derived the Lorentz matrix for the boost in $x$ direction and its inverse. So I know how to get these two matrices: $$ \\Lambda = \\begin{bmatrix} \\gamma & 0 & 0 & -\\beta \\gamma\\\\\\ 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 \\\\\\ -\\beta \\gamma & 0 & 0 & \\gamma \\end{bmatrix} $$ $$ \\Lambda^{-1} = \\begin{bmatrix} \\gamma & 0 & 0 & \\beta \\gamma\\\\\\ 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 \\\\\\ \\beta \\gamma & 0 & 0 & \\gamma \\end{bmatrix} $$ What I lack is a geometrical meaning of those two. I have read, that it is some sort of a rotation, so I have already drawn the Minkowski diagram for steady _observer 1_ and one for the moving _observer 2_ as _observer 1_ sees it. Examples are drawn for $\\beta = 0.5$. I have noticed that:   * _base cell_ is a square (picture 1) for a steady observer and a parallelogram (picture 2) for a moving observer   * light travels only diagonals of _base cells_   * axis $x'$ has coefficient $\\beta$   * axis $ct'$ has coefficient $1\/\\beta$ **question 1:** Is there a way I can help myself with what I have figured out so far to show, how the Lorentz matrix deforms a square into a parallelogram and vice versa? **question 2:** I have heard of the invariant quantity $\\Delta s$ in relativity for which I have stumbled upon an equation $\\Delta s^2 = \\Delta x^2 - c \\Delta t^2$. Is it possible $\\Delta s$ is an area of a base cell? **question 3:** Why is it so necessary for a scalar product to be defined with a minus sign for Minkowski spacetime? **picture 1:** ![Minkowski diagram for steady *observer 1*](http:\/\/i.stack.imgur.com\/nmqKd.png) **picture 2:** ![enter image description here](http:\/\/i.stack.imgur.com\/9KD3S.png) Thank you."}
{"id":"55934","title":"Differences of behaviour of a particle in a box in quantum theory between that in classic physics","text":"Can anyone help me enlist 3 major differences between the quantum and classical physics of the behaviour of a particle in a box? I would like some insight into the differences without solving PDEs because I am not a physics student. Thank you very much"}
{"id":"37656","title":"Does cosmological horizon grow or decrease as it radiates?","text":"Ron Maimon in many posts claimed that cosmological horizon is like a big black hole. Black holes decrease as they evaporate and their radius decreases as well. So what is with a cosmological horizon? If cosmological horizon is just a black hole centered at the opposite side of the universe, we should see the radius of the external space groving as the radius of the BH decreases. But if the radiation decreases the horizon entropy we should see the area of the horizon decreasing. What is the correct conclusion?"}
{"id":"15809","title":"Why does light in a room not form constructive and destructive interference patterns?","text":"This is something that I have wondered for a long time. How come when I walk around why do I not see random black spots where light has collided destructively and bright spots where it has collided constructively?"}
{"id":"15803","title":"LQG Demystified Book?","text":"Does a similar easy to read introduction like David Mc Mohan's Demystified books exist for LQG? I'd like something easier to start with than a paper but deeper than a popular book."}
{"id":"114255","title":"Has the existence of anyons been experimentally verified?","text":"I've been wondering whether there has been any experimental evidence for the existence of anyons or are they just objects of purely mathematical interest?"}
{"id":"18322","title":"Why are noble gases used for lights?","text":"I know that neon is used in advert signs due to its inertness. However, I am not entirely sure how the inertness is exploited. I think it is because Ne being inert means that after electricity frees an electron we can count on it returning to the ion left behind to emit a photon, whereas for more active elements, the gas may just exist as a plasma?"}
{"id":"38009","title":"Pion Decay and Fractional electric Charge disappeared, why?","text":"![Pion Plus Decay](http:\/\/hyperphysics.phy- astr.gsu.edu\/hbase\/particles\/imgpar\/pidec1.gif) Since the charged pions decay into two particles, a muon and a muon neutrino Fractional electric Charge disappeared, why? The decay proceeds by the weak interaction $W^{+}$ and can be visualized in terms of Feynman diagrams. my suggestion is that the Fractional electric Charge disappeared because it does not exist at all in the Quarks mechanism."}
{"id":"38006","title":"How 'Faster Than Light' Neutrino Was Product of Loose Cable at CERN?","text":"A \"faster than light\" neutrino discovery was actually the result of a loose cable. A fiber-optic cable in a GPS receiver at the European Center for Particle Physics ( CERN) near Geneva Can someone show me the light here, What's Going On ? how long was fiber-optic cable?"}
{"id":"89050","title":"Is a particular force different in different frames","text":"Can a particular real force have different magnitude in different frames?"}
{"id":"47581","title":"Entropy and the principle of least action","text":"Is there any link between the law of maximum entropy and the principle of least action. Is it possible to derive one from the other ?"}
{"id":"5642","title":"What is an \"inclined magnetic field\"?","text":"What is meant by an \"inclined magnetic field\"? How is it different from the usual magnetic field?"}
{"id":"103006","title":"How organic impurities in the air affects laser?","text":"We have a Nd YAG laser pumped Optical Parameter Oscillator in our lab. A technician came to our lab (my professor called him for its maintenance) and while he was working on it, he told me that because of the organic impurities in the air, the crystals got contaminated. He showed me a layer formation on the crystal where the laser hits. He explained this is due to the organics in the air which interacts with laser and get deposited. I am not clear what kind of organics (aromatic or some other?) which creates this and how it is get deposited there? Can anyone please explain me in detail how the atmospheric impurities (especially organics) can cause damage on lasers?"}
{"id":"103007","title":"Oscillations and Majorana Neutrino","text":"In neutrino oscillations, neutrinos can convert from one flavor to another. This implies individual lepton number is not conserved. Doesn't it also imply that, if the neutrinos have mass, the mass should be of Majorana type? But I'm confused. Can I define lepton number (and the states like $\\nu_e ,\\nu_\\mu$ and $\\nu_\\tau$) if neutrinos are Majorana particles? Or in other words, can Dirac neutrinos exhibit oscillations? Is the total lepton number conserved?"}
{"id":"47056","title":"Interpretation of $e|\\psi|^2$ as electron density","text":"In solid state physics the electron density is often equated to $e|\\psi|^2$. However, the Sakurai says (Chapter 2.4, Interpretation of the Wave Function, p. 101) that adopting such a view leads \"to some bizarre consequences\", and that Born's statistical interpretation of $|\\psi|^2$ as a probability density is more satisfactory. I am aware that a position measurement of an electron leads to (in the Copenhagen interpretation) a collapse of the wave function into a position eigenstate $x$ with the probability given by $|\\psi(x)|^2$, and that it is known from some scattering experiments that the electron behaves as a point- like particle. However, the electron density is experimentally observable, e.g. by X-ray scattering. One could argue that X-ray scattering is done with a large ensemble of atoms, so that what we actually observe is the average over many position measurements. I am wondering if this the only \"bizarre consequence\", since this distinction seems to be a minor difference to me. My questions aims at clarifying the difference between the probabilistic interpretation of $|\\psi(x)|^2$ and the interpretation as an electron density."}
{"id":"47055","title":"Conservation laws in mSUGRA model","text":"Can somebody list all the quantum numbers (beside R-parity) that are conserved in vertex for SUSY particles in mSUGRA model?"}
{"id":"87400","title":"How many mirrors placed perpendicular to each other shall make a light ray completely reverse its path?","text":"The answer is 3. But according to me 2 shall also work in 2 Dimensions? Does anyone has a theoretical proof?"}
{"id":"23441","title":"How to combine measurement error with statistic error","text":"We have to measure a period of an oscillation. We are to take the time it takes for 50 oscillations multiple times. I know that I will have a $\\Delta t = 0.1 \\, \\mathrm s$ because of my reaction time. If I now measure, say 40, 41 and 39 seconds in three runs, I will also have standard deviation of 1. What is the total error then? Do I add them up, like so? $$\\sqrt{1^2 + 0.1^2}$$ Or is it just the 1 and I discard the (systematic?) error of my reaction time? I wonder if I measure a huge number of times, the standard deviation should become tiny compared to my reaction time. Is the lower bound 0 or is it my reaction time with 0.1?"}
{"id":"94869","title":"How to estimate a final error when you have N experiments with N errors?","text":"if we have Xo±e1, X1±e2, X2±e3, ..., Xn±en when we know these, what is going to be our estimation for X, and it's error? for example if we have 344±1, 350±2, 345±5, 338 ± 2 estimation? error of our estimation?"}
{"id":"15374","title":"Does the heat death of the universe really imply a maximum entropy state *all* of the time? Or most of the time?","text":"Statistically speaking, you're going to still encounter deviations from equilibrium, even though the expected value is equilibrium. But these rare deviations from equilibrium - which are inevitable - might have the power to do work. So does the universe inevitably descend towards a maximum-entropy state? Or is it only probabilistically destined towards a maximum-entropy state - that is - it will be in that state more than any other state. After all, someone has even hypothesized a Poincare recurrence time, as described below: http:\/\/en.wikipedia.org\/wiki\/Timeline_of_the_far_future > Scale of an estimated Poincaré recurrence time for the quantum state of a > hypothetical box containing an isolated black hole of stellar mass.[47] This > time assumes a statistical model subject to Poincaré recurrence. A much > simplified way of thinking about this time is that in a model in which > history repeats itself arbitrarily many times due to properties of > statistical mechanics, this is the time scale when it will first be somewhat > similar (for a reasonable choice of \"similar\") to its current state again."}
{"id":"107682","title":"Kolmogorov-Smirnov test vs Chi-squared test","text":"What is the difference between the Kolmogorov-Smirnov test and the Chi-squared test? When should we use one instead of the other? I was reading this article, and I got confused a lot. It is hard to get a physics-related answer in the Cross Validated SE section, so this is why I am asking here."}
{"id":"114253","title":"How do you calculate the energy output if you know the volts, amps and time?","text":"How do you calculate the energy output if you know the volts, amps and time?"}
{"id":"79535","title":"How do you derive Lagrange's equation of motion from a Routhian?","text":"Given a Routhian $R(r,\\dot{r},\\phi,p_{\\phi})$, how do you derive Lagrange's equation for $r$? Do you just solve the following for $r$? $$\\frac{d}{dt}\\frac{∂R}{∂\\dot{\\phi}}-\\frac{∂R}{∂\\phi}=0$$ And as a related question, what is the motivation for using a Routhian?"}
{"id":"127977","title":"How to compute observables from the boson field operator?","text":"I think I understand that if given the two boson wavefunction of two different states \\begin{align} \\Psi(\\boldsymbol{r}_1,\\boldsymbol{r}_2) = \\dfrac{ \\psi_1(\\boldsymbol{r}_1)\\psi_2(\\boldsymbol{r}_2) \\+ \\psi_1(\\boldsymbol{r}_2)\\psi_2(\\boldsymbol{r}_1) } {\\sqrt{2}} \\end{align} the expectation value of an operator $\\hat{Q}$ for $\\Psi(\\boldsymbol{r}_1,\\boldsymbol{r}_2)$ is \\begin{align} \\langle \\Psi | \\hat{Q} | \\Psi \\rangle = \\langle \\psi_1(\\boldsymbol{r}_1) | \\hat{Q}_1 | \\psi_1(\\boldsymbol{r}_1) \\rangle \\+ \\langle \\psi_2(\\boldsymbol{r}_2) | \\hat{Q}_2 | \\psi_2(\\boldsymbol{r}_2) \\rangle \\end{align} If I add one more boson to state $\\psi_1$, then the total wavefunction changes to \\begin{align} \\Psi(\\boldsymbol{r}_1,\\boldsymbol{r}_2,\\boldsymbol{r}_3) = \\dfrac{ \\psi_1(\\boldsymbol{r}_1)\\psi_1(\\boldsymbol{r}_2)\\psi_2(\\boldsymbol{r}_3) \\+ \\psi_1(\\boldsymbol{r}_2)\\psi_1(\\boldsymbol{r}_3)\\psi_2(\\boldsymbol{r}_1) \\+ \\psi_1(\\boldsymbol{r}_3)\\psi_1(\\boldsymbol{r}_1)\\psi_2(\\boldsymbol{r}_2) } {\\sqrt{6}} \\end{align} The expectation value of $\\hat{Q}$ for $\\Psi(\\boldsymbol{r}_1,\\boldsymbol{r}_2,\\boldsymbol{r}_3)$ becomes \\begin{align} \\langle \\Psi | \\hat{Q} | \\Psi \\rangle = 2\\langle \\psi_1(\\boldsymbol{r}_1) | \\hat{Q}_1 | \\psi_1(\\boldsymbol{r}_1) \\rangle \\+ \\langle \\psi_2(\\boldsymbol{r}_2) | \\hat{Q}_2 | \\psi_2(\\boldsymbol{r}_2) \\rangle \\end{align} I am stuck with understanding how to compute the expectation values of $\\hat{Q}$ in terms of the boson field operator \\begin{align} \\Psi(\\boldsymbol{r})=\\sum_{\\nu} \\psi_{\\nu} \\left( \\boldsymbol{r} \\right) b_{\\nu} \\end{align} where \\begin{align} \\left[\\Psi(\\boldsymbol{r}_1),\\Psi^\\dagger(\\boldsymbol{r}_2)\\right]=\\delta (\\boldsymbol{r}_1-\\boldsymbol{r}_2) \\\\\\ [b_\\alpha^\\dagger,b_\\beta^\\dagger]=[b_\\alpha,b_\\beta]=0,\\quad [b_\\alpha,b_\\beta^\\dagger]=\\delta_{\\alpha\\beta} \\end{align} I am assuming that \\begin{align} \\langle \\psi_{\\mu}^\\dagger \\left( \\boldsymbol{r} \\right) | \\psi_{\\nu} \\left( \\boldsymbol{r} \\right) \\rangle = \\int \\psi_{\\mu}^\\dagger \\left( \\boldsymbol{r} \\right) \\psi_{\\nu} \\left( \\boldsymbol{r} \\right) d^3\\boldsymbol{r} = \\delta_{\\mu\\nu} \\end{align}"}
{"id":"132061","title":"How long does it take plasma to radiate its heat?","text":"Lets say we have 1 gram of plasma (Argon) at 1 million kelvin confined in a vacuum with electromagnets. If we keep the magnets on but shut down the device that heated the plasma, how long will it take for it to cool off?"}
{"id":"39060","title":"Neutrino mass with Dirac and Majorana","text":"Why both Dirac mass and Majorana mass terms are needed to explain the mass of a neutrino?"}
{"id":"112867","title":"Does mechanical resonance of an object changes in different surrounding medium?","text":"When comparing mechanical resonance of an object (for example string) in air and in water, does the resonance frequency changes? My guess is that it does change because the surrounding medium will slow it down. Thank you."}
{"id":"73942","title":"How do we know photons have spin 1?","text":"Electrons have spin 1\/2, and as they are charged, they also have an associated magnetic moment, which can be measured by an electron beam splitting up in an inhomogeneous magnetic field or through the interaction of the electrons's magnetic moment with an external magnetic field in spectroscopic measurements. On the other hand, a photon is neutral - how can one measure its spin if there's no magnetic moment? How do we know it has spin 1?"}
{"id":"67826","title":"How can Magnets be used to pick up pieces of metal when the force from a magnetic field does no work?","text":"I learned that the force from magnetic fields does no work. However I was wondering how magnets can be used to pick up pieces of metal like small paperclips and stuff. I also was wondering how magnets can stick to pieces of metal list a refrigerator?"}
{"id":"16770","title":"What is deep Fresnel region?","text":"If I understand correctly, it has something to do with autocorrelation function, but can someone give me a definition or exact explanation? In case of scattering, if you wish to analyze pattern with Fourier transform, you can only do so within deep Fresnel region. Apparently (according to the text I've been reading) in practical experiment this would mean that detector has to be close to the sample. Say we want to observe scattering of light on some soft material and use laser to do so. Could this simply mean that if we don't go too far from source, light waves remain coherent? This article contains an example of the use of this terminology."}
{"id":"16772","title":"Definition regarding percolation","text":"in a homework sheet studying bond-percolation on the Bethe lattice, a function $g(r)$ is introduced as \"the probability of finding two nodes separated by a distance $r$ on the same cluster\". Now just to see if I get this definition right, for the simple one-dimensional case (Bethe lattice with coordination number 2), would this mean that $$g(r) = p^r$$ if $p$ is the probability that a given bond exists, or would it be $$g(r) = 2p^r - p^{2r}$$ The former would mean: Given two nodes of distance $r$, what's the probability that they are connected, whereas the other would mean: Given one node, what's the probability that it's connected to _some_ node a distance $r$ away. And since in the one-dimensional case there are 2 nodes of distance $r$, we get add the probability for having a bond to each of them and then subtract the joint probability... I'm mainly asking because it will change the answer for the percolation threshold at coordination numbers larger than $2$. If the former definition is correct, I'd guess the correlation function would still be $p^r$ since there's exactly one path between two nodes. Whereas for the latter definition, I'd get something like $$g(r) = 1 - (1-p^r)^{N(r)}$$ where $N(r)$ counts how many nodes exist a distance $r$ away from a chosen \"center\" node, which is $N(r) = z(z-1)^{n-1}$ if $z$ is the coordination number."}
{"id":"12064","title":"Right Angle Circuitry","text":"This question concerns the article Right Angle Circuitry. Having passed a grad E&M course, I feel like this should make perfect sense to me and I should be able to answer all the questions it raises. Sadly that is not yet the case. What are the answers to all the unanswered questions here? In particular:   * Why does the circuit in Figure 8 not act as a transformer?   * Is it possible to extract energy from a toroid with capacitor plates as shown in Figure 9? If not, why not?   * Have structures like that in Figure 10 ever been used in practice as transmission lines?   * Last question in the article: Is it possible to create \"insulating electric circuitry which conducts only displacement current\"?"}
{"id":"60834","title":"How does one write eigenstates of field operators in terms of particle states in scalar field theory?","text":"I am reading the first paper in Schwinger's QED anthology, where he discusses his action principle. In this, he writes down states that are simultaneous eigenkets of the field operators at all points of a space-like surface $\\sigma $, $\\lvert\\Phi', \\sigma\\rangle$, which obey $\\Phi(x)|\\Phi', \\sigma\\rangle = \\Phi'(x)|\\Phi', \\sigma\\rangle$ for any $x \\in \\sigma$. I am aware that $\\Phi(x)$ will commute with $\\Phi(y)$ if the separation between x and y is spacelike, and hence what he does is doable, at least in principle. But how does one write down such a state in terms of particle states? I have also seen this construction somewhere else (I think Peskin), where functional integrals are first introduced to deal with amplitudes of the form $\\langle\\Phi'_1, t_1\\vert\\Phi''_2,t_2\\rangle$, i.e. the propagation amplitude between a simultaneous eigenket of all $\\Phi(x,t_1)$ and a simultaneous eigenket of all $\\Phi(x,t_2)$, and the formalism is then extended to deal with the vacuum in far past to vacuum in far future amplitude."}
{"id":"69253","title":"Why is the Pegasus launched from a subsonic airplane?","text":"Considering that the reason typically given for launching spacecraft from sea- level as opposed to mountains is that the limiting factor is _velocity_ , not _altitude_ , then why isn't the Pegasus rocket launched from a supersonic airplane? At ~50,000 lbs, a B-1 could take two of them to Mach 2, even if the plane had to be modified as is the case with the shuttle carriers. Other planes could do, especially the easily-available F-111 or some old Russian hardware such as the Tu-22 or Tu-160. Considering that NASA now launches humans on Russian hardware this is not entirely infeasible."}
{"id":"100065","title":"What does the $\\Delta$ notation mean? E.g. for potential energy: $\\Delta U$ vs just $U$? What is the difference?","text":"I've seen in this article that potential energy is defined like this: $U=-\\int _{\\text{ref}}^r\\overset{\\rightharpoonup }{F}\\cdot d\\overset{\\rightharpoonup }{r}$. However I've seen in other text books that potential energy is: $\\text{$\\Delta $U}$. What is the difference? Why is \"Delta\" used sometimes and sometimes not? How should I interpret the \"Delta\" form?"}
{"id":"135198","title":"Definition of transmission and reflection probability","text":"This is a basic question, but it does not seem to be well defined anywhere. Generally, two terms are mixed somewhat randomly: transmission PROBABILITY and transmission coefficient. So to be clear, generally for a scattering problem, one defines the transmission coefficient as: $T = \\frac{C}{A}$ where $C$ is the amplitude of the \"outgoing solution\" and A of the \"incoming one\" and similarly for R, the reflection coefficient. But the square of these quantities are not probabilities they do not add up to 1. I found somewhere without explanation that the transmission probability, is: $ T^2 k_2\/k_1 $ where $k_2$ is the outgoing wavenumber. So to sum up, in scattering problems: what is the definition of the transmission\/reflection PROBABILITY?"}
{"id":"46721","title":"Why can't a superconductor make a DC motor self sustaining?","text":"Superconducting wire can host a low current magnetic field. I do not know if it supports a corresponding electrical field. Can a superconducting wire that sustains a current accelerate a DC motor? Where is the resistance in a superconducting homopolar motor? Please explain if I am way off of target. What am I getting wrong?"}
{"id":"17423","title":"Island of Stability","text":"When I was much younger, I remember being fascinated by the thought of an Island of Stability at very high atomic numbers. However, I have not heard much on this and I was wondering   * Did this idea ever have any meaningful support in the scientific community, or was it a fringe theory?   * Is there any validity to this considering how much more we actually know about atomic construction?"}
{"id":"38823","title":"atmospheric phenomenon? What causes condensation trails to converge?","text":"This air plane just caught my eye. Two contrails apparently are flowing backward, slightly off-centered and then ultimately converge, giving the overall shape of a very narrow rhomboid parallelogram, with one diagonal being on the order of about four plane-width's. Please explain what I am seeing here as detailed as possible. Some of you may even know what class of plane this may be and thus infer the scales involved, to give some hints of the scale of the contrail under investigation? I looked up other pictures, and if I am not mistaken, most contrails are slightly angled outward. Is this by design or due to expansion and convection of the gases involved? Can someone confirm this? ![enter image description here](http:\/\/i.stack.imgur.com\/yzKW3.jpg) **Edit** :   * I did observe the described contrail-shape, with quite some consistency, over varying angles of ascension and declination.   * I have never heard of conspiracies regarding contrails (e.g. chemtrails) until after writing this post. It shows that good, hard explanations for _atmospheric contrail phenomena_ are in demand.   * Given the right circumstances (large distances) some apparent contrail convergence can be attributed to perspective alone. **Links** :   * Contrail Science (impressive resource, which I just peeked at so far)   * Contrail gallery   * Contrails - clouds (ppt)   * http:\/\/www.es.lancs.ac.uk\/hazelrigg\/amy\/Introduction\/The%20Science.htm   * http:\/\/www.epa.gov\/oms\/aviation.htm   * http:\/\/www-pm.larc.nasa.gov\/newcontrail.html   * http:\/\/www.epa.gov\/oar\/oap.html   * http:\/\/www.history.com\/shows.do?episodeId=464914&action=detail   * http:\/\/mynasadata.larc.nasa.gov\/P4.html"}
{"id":"13330","title":"Galilean invariance proof","text":"I'm studying for a physics test, but I think I don't really understand Galilean invariance. In my textbook there is an example in which they prove that if you consider 2 frames S and S' in standard configuration that the second law of Newton is Galilean invariant by proving that if $x' = x - Vt$ than $F_x = F'_x$, so this law holds in both frames. So far I understand this. However, in the book there is one assignment in which they ask me to verify that the relationship between kinetic energy and momentum, $E = p^2\/2m$, is Galilean invariant. I couldn't really figure it out by myself so I looked at the answers. The answer is as followed, according to my text book: > In S: > > $E = \\frac{1}{2}m\\dot{x}^2;$      $p=m\\dot{x}.$ > > Substitute $\\dot{x} = p\/m$ in the equation for the energy: > > $E = \\frac{1}{2}m(\\frac{p}{m})^2=p^2\/2m$ > > In S': > > > $E'=\\frac{1}{2}m\\dot{x}'^2-\\frac{1}{2}m(\\dot{x}-V)^2=\\frac{1}{2}m\\dot{x}^2-m\\dot{x}V^2$ > > $p'=m\\dot{x}'$ > > Assume the relationship holds: i.e., > > > $E'=\\frac{p'^2}{2m}=\\frac{1}{2m}(m\\dot{x}-mV)^2=\\frac{1}{2}m(\\dot{x}^2-2\\dot{x}V+V^2)=\\frac{1}{2}m\\dot{x}^2-m\\dot{x}V+\\frac{1}{2}mV^2,$ > > in agreement with the Galilean transformation of the kinetic energy Source: McComb, W. D., 1999. _Dynamics and Relativity._ New York: Oxford University Press Inc. I understand all of the equations, however I just don't understand why this verifies that this relationship is Galilean invariant. Can someone explain this to me?"}
{"id":"80294","title":"Is the $SU(2)$ flux defined in the context of Projective Symmetry Group(PSG) an observable quantity?","text":"The $SU(2)$ flux defined in the context of PSG is as follows: Consider the mean-field Hamiltonian $H_{MF}=\\sum(\\psi_i^\\dagger\\chi_{ij}\\psi_j+H.c.)$ description of a 2D lattice spin-model, the definition of $SU(2)$ flux $P_C$ for a loop $C=i\\rightarrow j_1\\rightarrow j_2\\rightarrow ...\\rightarrow j_n\\rightarrow i$ with the base point $i$ is $P_C=\\chi_{ij_1}\\chi_{j_1j_2}...\\chi_{j_ni}$. On the other hand, the two $SU(2)$ gauge-equivalent mean-field ansatz $\\chi_{ij}$ and $\\chi_{ij}'=G_i\\chi_{ij}G_j^\\dagger$ describe the same projected spin- wavefunction. And the $SU(2)$ flux $P_C'$ for the same loop $C$ is given by $P_C'=\\chi_{ij_1}'\\chi_{j_1j_2}'...\\chi_{j_ni}'=G_iP_CG_i^\\dagger$, in general $P_C' \\neq P_C$, but an observable quantity should be invariant under the $SU(2)$ gauge transformation $\\chi_{ij}\\rightarrow\\chi_{ij}'=G_i\\chi_{ij}G_j^\\dagger$,. Thus, does this mean that the $SU(2)$ flux $P_C$ is not an observable quantity? Thanks in advance."}
{"id":"62343","title":"Can an electron interact with itself to create interference?","text":"I have been recently brushing up my elementary physics concepts, specifically quantum physics. So, if I set up a single photon emitter in the double slit experiment, it is possible for me to see interference (I am assuming), when I thought about this I realized that a photon is mass-less and travels at the speed of light, so time does not affect a photon, therefore a photon can interact with another instance of itself (which in our frame of reference exists in another time). My question is, is the same effect observable when a single electron is fired at speeds much lower than the speed of light? If so, how? because for this to happen, the electron (which has mass and experiences time the way we do) has to physically be present at two locations at the same time (both the slits)?"}
{"id":"46895","title":"Why is the Hubble Space telescope able to see farthest in the universe?","text":"What determines how far a telescope can see in the universe? How does recording data for a very long time (~10 years) help? If we could build a telescope which work at microwave region, will it be a better telescope (so that we could see even farther)?"}
{"id":"46899","title":"Why does glass break at the line where you score it?","text":"Why does it take such a small incision for the glass to break at that spot? Why is the structural strength of the material influenced by such a small imperfection?"}
{"id":"41044","title":"How can I estimate relative humidity ($e\/e_s$) from temperature and dewpoint?","text":"## Background my goal is to estimate vapor pressure deficit ($vpd$) from Relative Humidity ($rh = \\frac{e}{e_s}$): $$vpd = \\frac{100 - \\frac{e}{e_s}}{100} e_s$$ (from Hartmann \"Global Physical Climatology) But it is surprisingly difficult to find values of climatological mean $rh$ (e.g. see motivating question at gis.SE. It turns out that WeatherUnderground provides values of dewpoint. ## Problem At first I thought it would be trivial to convert from dewpoint to $e$ or $rh$, but according to Lawrence 2005, there is no direct conversion, only a \"highly accurate conversion\" to get from water vapor pressure ($e$) to the dewpoint ($t_d$) (equation 7): $$t_d = \\frac{B (ln\\frac{e}{C})}{A - ln (\\frac{e}{C})}$$ Where $[A,B,C]$ are empirical constants$[17.625, 243.04^oC,610.94Pa]$ ## Questions   1. why is there no exact relationship between $e$ and $t_d$?   2. how can I solve the last equation for $e = f(t_d)$ (or $rh$)?"}
{"id":"134710","title":"Calculate temperature at isobaric or isochoric process","text":"I've had some trouble in regards to calculate the temperatures in the different states. Considering the following pV-diagram: ![enter image description here](http:\/\/i.stack.imgur.com\/j14ca.jpg) Here is the data I have: The adiabatic constant $\\gamma$ is 1.4 $C_{mv}=20.8\\frac{J}{mol\\cdot K}$ $V_{2}=100\\cdot 10^{-6}\\ m^{3}$ $T_{2}=300\\ K$ $p_{2}=100000\\ Pa$ $V_{4}=4\\cdot V_{3}$ Now my question is: How do I calculate the temperatures at the states 1, 3, 4 and 5. And how do I calculate the temperature or pressure at any given state in general? And how do I determine if for instance the initial state 1 is isobaric, isochoric or even both??"}
{"id":"59755","title":"Hydrostatic pressure at lateral directions","text":"I just read that, with respect to a stationary tiny cube, suspended in a fluid, that has a negligible weight and dimensions: > Pressure is the same in every direction in a fluid at a given depth, if it > weren't the fluid would start to move. This made sense to me the first time, but then I realised that the condition for the fluid not to move is that the net force is zero. However, the net force being zero doesn't necessarily require that forces in all directions are equal, just that two forces pointing in opposing directions are equal? So say, the vertical forces on a cube, pointing up and down, are 1N each, but then all the other forces pointing towards the sides of the cube, may as well be 10N each, and yet all the forces would cancel each other out, resulting in a net force of zero. Am I missing something here?"}
{"id":"44492","title":"Are the intersections of past and future light cones spacelike?","text":"Given a timelike reference worldline (not necessarily geodesic), we can define light-cone coordinates $\\tau^+$ and $\\tau^-$ so that the 3-D hypersurfaces of constant $\\tau^+$ are past light cones of events on the reference worldline, the value of $\\tau^+$ being reference proper time. Similarly $\\tau^-$ for future light cones.   * Is there a common name for the 2-D surfaces of constant $\\tau^+$ and $\\tau^-$ induced by a timelike worldline?   * Is it correct that these are \"locally spacelike\" in the sense that any curve entirely within a surface is spacelike?   * Under what assumptions are they also \"globaly spacelike\" in the sense that there are no causal (timelike or null) curves between any two points on a surface?  From this we can also do a simple coordinate change to $r^\\star = \\frac{\\tau^+ - \\tau^-}{2c}$ and time $\\tau^\\star = \\frac{\\tau^+ + \\tau^-}{2}$. In flat spacetime with a geodesic reference worldline, the 3-D hypersurfaces of constant $\\tau^\\star$ are just the usual orthogonal simultaneous spaces. [EDIT: should have been $r^\\star = \\frac{\\tau^+ - \\tau^-}{2}c$ ]   * Is there a common name for the hypersurfaces of constant $\\tau^\\star$ ?   * Are they also locally spacelike?   * Under what (very strong I expect) assumptions are they globally spacelike? ### Addendum In response to the request to clarify the co-ordinates: To illustrate, suppose spacetime is flat, and suppose the chosen reference worldline is geodesic. It can then be taken to be stationary at the spatial origin of Minkowski co-ordinates WLOG. In that case, $\\tau^+$ is $t + r\/c$ and $\\tau^-$ is $t - r\/c$, since an event at time $t$ and distance $r$ would meet the past and future lightcones originating from the reference worldline at those times, respectively. As a result $\\tau^\\star = t$ and $r^\\star = r$. Of course this is not the case I am asking about. It is a general definition, not restricted to a geodesic reference worldline, nor to flat spacetime. Summary: if I am the reference worldline, then $\\tau^+$ is what my watch is reading when I see the event. $\\tau^-$ is what my watch is reading when the event sees me."}
{"id":"95264","title":"Tricky operator identity: $[L^2,[L^2,\\vec{r}]]=2 \\hbar ^2 \\{ L^2, \\vec{r}\\}$?","text":"This operator identity showed up in a course I was taking, and it was given without proof. $$[L^2,[L^2,\\vec{r}]]=2 \\hbar ^2 \\\\{ L^2, \\vec{r}\\\\}$$ The curly brackets denote the anticommutator, $AB+BA$. The $\\vec{r}$ operator is the position operator. The $L^2$ operator is given by: $$L^2 = -\\hbar ^2 \\left( \\frac{1}{\\sin \\theta} {\\partial\\over\\partial\\theta} (\\sin \\theta {\\partial\\over\\partial\\theta}) + \\frac{1}{\\sin^2 \\theta} {\\partial^2\\over\\partial\\phi^2}\\right)$$ Is there a way of proving this identity without tediously expanding all the commutators? I've been trying to find one but was unable to."}
{"id":"28043","title":"Kinematics - concept question","text":"A child tosses a ball directly upward. Its total time in the air is T. Its maximum height is H. What is its height after it has been in the air a time T\/4? Neglect air resistance. Ok so I know that there is no x-component to the velocity: $$D=v_0 t+\\frac{1}{2}at^2$$ $$\\frac{1}{2}D=H$$ so, $$\\frac{1}{2}H=v_0(\\frac{1}{4}t)+[(4.9)(\\frac{1}{4}t)^2]$$ I am confused as to where to go from here? I am overcomplicating this, missing the underlying concept. I don't want the answer, but just some guidance."}
{"id":"129807","title":"How are electromagnetic waves differentiated?","text":"I would like to know how the signals for remote controlled cars, radios, etc.. That use radio waves are told apart from each other. I know that the radio waves are modulated to encode data and the frequency or amplitude are changed, so then the waves are propagated through the air and received at another location via a receiver that is tuned to a certain frequency that the waves were emitted, but I'm sure in most places in the world by now there are numerous amounts of waves traversing at any point, why doesn't the receiver of this device happen to catch another wave of the same frequency instead of the one that was intended? or is there anything that stops me from having a device that emits a wide range of frequencies or amplitudes that would manipulate nearby electronics?"}
{"id":"114387","title":"Which planet in the solar system is it?","text":"> It is known that the period of revolution of the earth about the sun is > $365.3$ days and that the semimajor axis of the earth's orbit is $1.495 > \\times 10^{11}$ m. An astronomer notes that the period of a distant planet > is $29.7$ earth years. What is the semimajor exis of the distant planet's > orbit? Check in a reference source (encyclopedia,etc.) what planet of the > solar system this might be. This problem is from Anderson's _Introduction to Flight_ , 4th edition, Exercise 8.4. I was presented a formula $$\\left( \\frac{\\tau_2}{\\tau_1} \\right)^2=\\left(\\frac{a_2}{a_1} \\right)^2$$ where $\\tau_1, \\tau_2$ denote the periods of earth and the other planet, respectively, and $a_1,a_2$ denote the periods of earth and the other planet, also respectively. When I plug in $a_1=1.495 \\times 10^{11}$ m, $\\tau_1=365.3$ days, and $\\tau_2=29.1 \\cdot 365.3$ days, I see the equation like this: $$\\left( \\frac{29.1 \\cdot 365.3}{365.3} \\right)^2=\\left(\\frac{a_2}{1.495 \\times 10^{11}} \\right)^2$$ Solving for $a_2$, I get $a_2 = 4.44 \\times 10^{12}$. I looked up for the planet with this semi-ajor axis on Wikipedia, and it turns out to be Neptune. Am I right on this?"}
{"id":"135347","title":"A question on quantum computing and de Broglie's pilot wave theory","text":"I don't know much about quantum computing except what I have read about on wiki and popsci. I have been reading about the de Broglie-Bohm pilot wave theory and how they describe quantum mechanics in terms of discrete particle traveling a pilot wave and it seems produce the same quantum mechanical statistics as the Copenhagen version. some experiments by fluid dynamicists from MIT also seem to support this alternate view. My question is if this alternate view is the correct view of reality would that effect quantum computing in any way? ie. dealing with discrete particles instead of probabilistic \"particles\". both theories explain the superposition of waves and both produce the same stats so I would guess not."}
{"id":"95845","title":"Electrons, spins, and degeneracy","text":"In an atom, two electrons can have the same set of $n,\\ell,m$ quantum numbers as long as they have opposite spins. My introductory physics and chemistry courses have all introduced this as two electrons occupying the same energy state; however, the spins of the electrons cause splitting of the energy levels they occupy. This leads me to think two electrons with opposite spins are only able to share the same set of $n,\\ell,m$ quantum numbers because they actually occupy slightly different energy levels. My question is: if there were no splitting of energy levels due to spin, would two electrons still be able to occupy the now-exactly the same energy level or do they need to have different energies, no matter how small the difference?"}
{"id":"92811","title":"Conservation of mass in fluid flow","text":"When deriving the continuity equation in physics class for two immiscible fluids flowing in succession we used the principle of conservation of mass. My question is, shouldn't volume be conserved instead of the mass? To further elaborate, assume section 1 of a pipe has cross-section area $A_1$ and section 2 has area $A_2$. At the point where liquid 1 with density $r_1$ is completely in section 2 and liquid 2 with density $r_2$ is just about to enter section 2. Shouldn't the volume of liquid in section 2 be conserved, i.e., $V_\\mathrm{in}$ = $V_\\mathrm{out}$ instead of the mass? (Assume ideal fluids and ideal flow)"}
{"id":"92812","title":"Mechanics Landau Galilean Principle","text":"I started reading Landau's Mechanics book and was having some trouble understanding the Galilean Relativity Principle. What does Landau mean by saying space to be homogenous and isotropic and time is homogenous for an inertial observer ? Also, how does he conclude using this that, the Lagrangian must be independent of $\\vec{r}$, time and direction of $\\vec{v}$ . I am not understand what he wants to convey. Also a side note, are there any lecture notes\/books\/videos etc. that explain in a bit more detail what Landau wishes to convey to the reader at times in his book."}
{"id":"126781","title":"Eigenstates of an observable","text":"Can we use eigenstates of ANY observable as base of the Hilbert space? If we can, is this equal to the statement that those eigenstates are orthogonal to each other and normalizable?"}
{"id":"118729","title":"how to calculate acceleration?","text":"The problem is as follows. A searchlight casts a spot of light along the face of a horizontal wall located 100m above from the searchlight. It rotates at a constant $\\omega$ of 4rads\/s. Determine the magnitude of the velocity and acceleration at which the spot appears to travel across the wall at the instant $\\theta$ =45 degrees. ($\\theta$ = the angle of the spot from the vertical) I managed to find the velocity as 800ms\/s but I can't figure out the way to find the acceleration. Any help?"}
{"id":"51375","title":"Why is the spring constant $W_p''(0)$?","text":"According to my physics book, the spring constant can be calculated from knowing the potential energy, with the formula $k = W_p''(0)$. I don't really understand why, and the book doesn't explain it any further. How do I know the two are equal?"}
{"id":"61068","title":"Sommerfeld results & van Hove singularities","text":"According to Sommerfeld the derivative of the density of states $g'(\\varepsilon)$ apears in several thermodynamic quantities. Will this also be the case if one use the correct dispersion relation of crystal? If yes don't we encounter the divergence in these quantities when we pass a van Hove singularity (where $g'(\\varepsilon)\\rightarrow \\infty $)?"}
{"id":"25350","title":"What physical interactions actually make single stars leave their binary companions at formation?","text":"From an interesting ScienceDaily article, I read this > Before the groups of stars disperse, binary stars move through their birth > sites and the group studied how they interact with other stars > gravitationally. \"In many cases the pairs are torn apart into two single > stars, in the same way that a pair of dancers might be separated after > colliding with another couple on a crowded dance floor,\" explains Michael > Marks, a PhD student and member of the International Max-Planck Research > School for Astronomy and Astrophysics. The population of binaries is > therefore diminished before the stars spread out into the wider Galaxy. > > The stellar nurseries do not all look the same and are crowded to different > extents, something described by the density of the group. The more binaries > form within the same space (higher density groups), the more interaction > will take place between them and the more binary systems will be split up > into single stars. This means that every group has a different composition > of single and binary stars when the group disperses, depending on the > initial density of stars. Here's the question: what are the types of gravitational interactions that make binary systems split up into single stars?"}
{"id":"122259","title":"Can $M_3$ move upwards and $M_2$ move to the right?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/vxYJd.png) Can M3 move upwards and M2 move to the right? in the solutions they assume this, but I find it impossible, it doesn't make sense physically. I get the answer wrong if I assume M3 moving down and M2 moving right in the next question, which is the one that makes sense."}
{"id":"77783","title":"Finding the equation of motion of anharmonic potential","text":"If I have a potential given by: $$U=U_0\\left[2\\left(\\frac xa\\right)^2-\\left(\\frac xa\\right)^4\\right]$$ It says that at $t=0$, the particle is at the origin ($x=0$) and the velocity is positive and equal to the escape velocity, which I found to be $\\sqrt {2U_0\/m}$ I have the differential equation: $$m\\ddot x=-\\nabla U=-U_0\\left[\\frac {4x}{a^2}-\\frac {4x^3}{a^4}\\right]$$ EDIT:: So I have the initial $(x,\\dot x)$. Now to find $x(t)$, I use conservation of energy. $$E=K+U=\\frac 12mv^2+U_0\\left[2\\left(\\frac xa\\right)^2-\\left(\\frac xa\\right)^4\\right]$$ The energy of the system is $U_0$, so I can change the above equation to: $$U_0=\\frac 12 m \\left (\\frac {dx}{dt}\\right)^2+U_0\\left[2\\left(\\frac xa\\right)^2-\\left(\\frac xa\\right)^4\\right]$$ $$\\left(\\frac{dx}{dt}\\right)^2=\\frac 2m U_0\\left[1-2\\left( \\frac xa \\right)^2+\\left( \\frac xa\\right)^4 \\right]$$ I found that the value in brackets reduces to $\\frac{1}{a^2} (x^2-a^2)^2$ So: $$\\frac {dx}{dt}=\\sqrt{\\frac{2U_0}{m}} \\frac {x^2-a^2}{a^2}$$ So I want $$\\int^{x}_{x_0} \\frac{dx}{x^2-a^2}=\\int^t_0 \\sqrt{\\frac {2U_0}{m}}\\frac{dt}{a^2}$$ This ends up being a hyperbolic arctan function, which could potentially make sense, but am I going in the right direction?"}
{"id":"127789","title":"CPT invariance of Dirac equation","text":"We know that Dirac equation is \\begin{equation} ( i \\partial _\\mu \\gamma ^\\mu - m ) \\psi ~=~0. \\end{equation} How can we show that Dirac equation is invariant under CPT transformation?"}
{"id":"129589","title":"Uniformly charged sphere's electric field","text":"I am facing this topic for the umpteenth time in my college career and, of course, every teacher has explained it in a different way. In this course, to find the expression of the electric field of a uniformly charged sphere, the professor started from this equation. $$ \\frac{1}{r^2}\\ \\frac{\\partial}{\\partial r}\\ \\lgroup r^2 \\frac{\\partial u}{\\partial r}\\ \\rgroup = -\\frac{q}{\\epsilon}\\ \\delta(r) $$ some idea of ​​what he is doing?"}
{"id":"60087","title":"What Speed Would an object need to leave the earth at to reach L1?","text":"Let's say the Earth is an airless sphere. What speed would an object weighing 1 kg need to leave the surface at in order to get to and be motionless at L1, where the Moon's gravity becomes stronger than the Earth's gravity?"}
{"id":"64504","title":"Bleaching groundstate","text":"I'm reading an article about two-dimensional infrared spectroscopy and I don't understand the following sentence. > Bleach or stimulated emission contributions yield negative signals. What are \"bleach contributions\"? I have never heard of it and cannot find a suitable explanation on the Internet. Can somebody explain that to me? The article I'm referring to is J. Chem. Phys. 121, 5935 (2004)"}
{"id":"91373","title":"What if the Earth stops in its orbit and instead falls straight towards the Sun?","text":"The Earth orbits around the Sun because it has angular momentum. If we stopped the Earth in orbit and then let it fall straight towards the Sun, how long would it take to reach the Sun in seconds? Details and assumptions               The mass of the sun is 2*10^30 Kg.     The mass of the earth is 6*10^24 Kg.     The earth is 149,600,000 Km from the Sun.     You may treat the Earth and Sun as point masses."}
{"id":"15249","title":"What does it take to understand Maxwell's equations?","text":"Assume I want to learn math and physics enough to reach a level where I understand Maxwell's equations (The terms and reasoning in the equations I.e. why they \"work\"). What would I have to learn in order to have the tools I need to make sense out of it? I'm kindof looking for a road map which I can use to get started and know what to focus on. The fields are pretty big so some pruning would be very helpful (if possible)"}
{"id":"116832","title":"Finding the equation of motion regarding a block pulling a uniform disk","text":"![enter image description here](http:\/\/i.stack.imgur.com\/QkMLC.png) My problem is finding the equation of motion for the block. Using Newton's Second Law, I did: $$4g-T-R=4\\frac{dw}{dt}$$ But the mark scheme says: $$4g-T-R=0.2*4\\frac{dw}{dt}$$ And I don't understand why the radius has anything to do with this. Can somebody please explain?"}
{"id":"56347","title":"energy of deflection of a moving object","text":"Imagine an object moving in a straight line in space, perhaps shaped like a yo-yo (to make rotational effects negligible). Suddenly a constant force is exerted at right angles to the direction of motion, such that the object is now deflected into a circular path. What is the energy required (or work done) to maintain the constant deflection of this circular path? Start over, and imagine a second identical object moving in the opposite direction, but offset so that when a similar force is applied to it, it travels in the same circular path, but offset by 180 degrees. Clearly, the energy required to keep both objects moving in the path is twice that for just one of them. Now, start again, but this time have the two objects instead link up with a cable and snap fastener, as they go by each other. The cable tension causes both objects to travel in the same circular path as before. Does the cable do the same work that was done before to keep the two objects in the circular path? If not (and besides, how can a cable do work?), then was the energy wasted before, to do what a simple cable can do without energy?"}
{"id":"10908","title":"intensifying a light source","text":"When trying to intensify a beam of light by refracting it through a lens, (as in a lighthouse fresnel system or similar railroad style switch lamp from years past), is the beam intensity increased by moving the lens closer to the beam? That is, refracting it as close as possible to the light source? Or of a similar condition, would an old kerosene barn lantern be brighter if the globe was an inch away from the flame instead of 3 to 4 inches? TE"}
{"id":"16430","title":"Numeric method to calculate the charge distribution on a conducting surface?","text":"If I have an arbitrary (closed?) conducting surface and a nearby charge density, is there a simple numeric way of computing the induced charge distribution on the surface?"}
{"id":"93852","title":"about field gradient","text":"I read the term field gradient in most of the article about magnetic field. I search it online but most of the explanation is about the math. I wonder in physics, what the gradient field really mean? To my understanding, I think a gradient field means the magnitude of field is position dependent, is that right? I read an article online introducing the strength of the magnetic field of some objects. The earth's magnetic field strength is about 0.5Gauss and a small magnetic bar (toy) produce about 5Gauss field. Let take the magnetic toy as example, I wonder how to estimate the gradient of the field? From the math, it seems that we should estimate how does the field changed in a unit of distance, so for the bar producing field of 5Gauss, roughly how big is the gradient of that field?"}
{"id":"113715","title":"Apparent dimensional mismatch after taking derivative","text":"Suppose I have a variable $x$ and a constant $a$, each having the dimension of length. That is $[x]=[a]=[L]$ where square brackets denote the dimension of the physical quantity contained within them. Now, we wish to take the derivative of $u = log (\\frac{x^2}{a^2})-log (\\frac{a^2}{x^2})$. Here, we have taken the natural logarithm. It is clear that $u$ is a dimensionless function. $$\\frac{du}{dx} = \\frac{a^2}{x^2}.\\frac{2x}{a^2} - \\frac{x^2}{a^2}.(-2a^2).\\frac{2x}{x^3} \\\\\\ = \\frac{1}{x} - 4. $$ Here, the dimensions of the two terms on the right do not match. The dimension of the first term is what I expected. Where am I going wrong?"}
{"id":"53262","title":"Do I need to have an airtight container to siphon boiling water?","text":"I'm attempting to siphon boiling water into another container for a school project. I have researched the physics of a siphon but I can't seem to make my boiling water siphon out of its initial pot. Does the pot need to be airtight to increase the pressure? Is 3\/8\" tubing too thick for this to work?"}
{"id":"76207","title":"Simple argument for unexpected behavior in SUSY model","text":"Consider a supersymmetric theory with 3 chiral superfields, $X, \\Phi_1$ and $\\Phi_2,$ with canonical Kahler potential and superpotential $$ W= \\frac12 h_1 X\\Phi_1^2 +\\frac12 h_2 \\Phi_2\\Phi_1^2 + fX.$$ One can show, by doing calculations, that (i) supersymmetry is spontaneously broken, but (ii) one- loop corrections do not lift the classical pseudo-moduli space. QUESTION: is it possible to say (ii) without looking at the explicit form of Coleman-Weinberg potential, e.g. making some field redefinition which shows that this is not an interacting theory and it is very close to Polonyi model?"}
{"id":"63576","title":"Is there a momentum for charge?","text":"Since mass and charge behave similarly, so, just like center of mass, I define a point center of charge, that is defined by $$\\vec r_{qm} = \\frac {\\sum{q_i \\vec r_i}} {\\sum{q_i}}$$ where $\\vec r_i$ is a position vector w.r.t. the origin. Now suppose just like momentum,there is a quantity we call _charge-momentum_ $q \\vec v$ for a system which is changed iff there is an external quantity denoted by $q \\vec a$ where $\\vec a= \\frac{d\\vec v}{dt}$ . Let us name this new quantity $ q\\vec a$ as _charge-force_. Now I conjecture, for every _charge force_ , there is an equal and opposite _charge force_. And that charge-momentum of an isolated system will be conserved. Just like momentum, can this be a conserved quantity of the universe? Does this make any sense ?"}
{"id":"132383","title":"Variable area manometer duct","text":"How does one measure the pressure at the end of the manometer tube when the manometer tube itself is having a variable area? ![manometer with variable area tube](http:\/\/i.stack.imgur.com\/ZMAcK.gif)"}
{"id":"47084","title":"Why are so many forces explainable using inverse squares when space is three dimensional?","text":"It seems paradoxical that the strength of so many phenomena (Newtonian gravity, Coulomb force) are calculable by the **inverse square** of distance. However, since volume is determined by **three** dimensions and presumably these phenomena have to travel through all three, how is it possible that their strengths are governed by the inverse of the distance **squared**? The gravitational force and intensity of light is merely **4 times** weaker at **2 times** the distance, but the volume of a sphere between the two is **8 times larger**. Since presumably these phenomena would affect all objects in a spherical shell surrounding the source with equal intensity, they travel in all three dimensions. How come these laws do not obey an **inverse-cube** relationship while traveling through space?"}
{"id":"22967","title":"can one introduce magnetic monopoles without Dirac strings?","text":"To introduce magnetic monopoles in Maxwell equations, Dirac uses special strings, that are singularities in space, allowing potentials to be gauge potentials. A consequence of this is the quantization of charge. Okay, it looks great. But is this the only way to introduce magnetic monopoles?"}
{"id":"53475","title":"A fly in an accelerating car","text":"A fly is flying around in a car, the fly never touches any surface in the car only fly’s around in the air inside the car. The car accelerates. does the fly slam in to the rear window. or does the fly continue to fly uninterrupted?"}
{"id":"94210","title":"Mosquito in a Bus","text":"My Question goes like this: If a mosquito inside the bus wants to move from back to front, will there be any difference if the bus is stationary or moving (ie extra effort in the latter case?)? Consider it as a closed bus & mosquito at a fixed point in bus, ie the air column will be constant throughout the process, & the mosquito will always be in that air column & never touch any part of the bus. Here's another question which has the similar answer as above question : Suppose the bus is at rest, & mosquito is also at rest (wrt to both ground & bus) and is in the air column, & not touching any part of bus. If bus starts moving, will the mosquito stay at rest wrt ground, ie go to back, or will it stay at rest wrt bus? For both questions, consider that when bus starts,it instaneously starts moving in constant velocity.(no acceleration) Thanks in advance to everyone who tries to answer it."}
{"id":"100768","title":"Butterfly inside the accelerated train","text":"Suppose there is a train moving right with constant acceleration and there is a butterfly inside the train which moves opposite to direction of motion of train i.e. towards left. The butterfly always moves with constant velocity outside the train. Does the butterfly retard or accelerate inside the train due to acceleration of train or does it move with constant velocity easily? Please explain."}
{"id":"71938","title":"Does the Milky Way have dark matter satellite galaxies?","text":"This recent paper by Weinberg et al. discusses that one potential problem with our current model of Cold Dark Matter (CDM) is that is predicts a greater number of satellite galaxies for the Milky Way than are actually observed, and satellite galaxies with larger masses than those the Milky Way has (although the paper does say that many of the Milky Way's satellite galaxies may be too dim for us to currently observe). I know that at least dark matter galaxies are believed to have been observed (see this article). Is there any observational or other evidence for the existence of dark matter satellite galaxies to the Milky Way? And how have they been found? Edit: As has been pointed out, this is a question we likely don't have the answer to yet (otherwise there wouldn't be recent papers posing the question still), so perhaps a better question would be how could\/would you find dark matter satellite galaxies of the Milky Way?"}
{"id":"89701","title":"Speed of Magnetic Signal over Large Distance","text":"If I had a very strong magnet on Earth and a very sensitive compass on Mars (just using planets to illustrate large distance), how long would the compass take to notice if I turned the magnet 180deg? I assume it can't beat the speed of light. Do the outer reaches of the magnetic field move as a rigid body with the turning magnet or do they lag in motion?"}
{"id":"102312","title":"What are the limits of validity for the magnetic field of a solenoid?","text":"1) The field outside a solenoid is approximated to be zero, because of opposites points \"cancelling out\". Does this approximation of the field being almost zero become worse as the diameter of the solenoid increases? (since the opposite sides are further apart) I'm looking for an answer for two cases: 1) Only the diameter increases, but all other dimensions remain the same. 2) All dimensions increase proportionally, keeping the aspect ratio the same. 2) Are solenoids approximated as a series of perfect circles (since opposite sides cancel only in a circle)? And consequently shouldn't a stretched out solenoid's field be completely different since its nothing like a series of circles? **EDIT:** ![enter image description here](http:\/\/i.stack.imgur.com\/IHvMQ.png) The field at P is the superposition of the fields from the opposite \"points\". If the diameter of the solenoid increases, the bottom \"points\" are further away from $P$, so their contribution to the field decreases. Therefore the assumption that the fields from the top points cancel with the bottom points becomes falser\/less valid."}
{"id":"102310","title":"How do inductors produce current?","text":"This is how I understand batteries and capacitors. The terminals produce an electric field which causes the electrons to move in random directions. The surface charge builds up causing new electric fields to be created, which forces the electrons to move in the direction of the wire, creating current. But with inductors, there is no negative or positive terminal (as in an end with a surplus or deficit of charge), so, on a microscopic level, how does it produce current?"}
{"id":"103253","title":"If the wave function can be collapsed, can we collapse the particle function?","text":"No doubt it's been questioned before so what is the theory or experiment? If there is an experiment."}
{"id":"3059","title":"Symmetry breaking","text":"What is a good place to learn the details of symmetry breaking? What I am looking for is a more serious exposition than the wiki-article, which explains the details, especially the mathematical part, but at the same time less detail than the typical quantum field theory book. Preferably I would like to see a clean made up example with all the details, but not necessarily a real example, what one can see in most books, which naturally has a lot of details that are there because that's what the world is, and not to illustrate the idea. I hope it is clear what I am looking for. EDIT: Just to clarify. I am looking for a reference, but if somebody is willing to write an example here, I would be more than happy."}
{"id":"91178","title":"Orbifold CFT of SU(2)\/G and SO(3)\/G","text":"In this paper by Dijkgraaf, Vafa, Verlinde, Verlinde, orbifold CFT is discussed. In that paper, it outlined that orbifold CFT provides a way to generate the new theories from the old known ones. (i.e. \"... it would be very interesting to devise operations that act on the space of RCFT's and generate new theories from a given one. As an example one can think of the construction of nondiagonal modular invariant combinations of characters, coset models, etc. Another operation of that kind is the concept of **an orbifold**.\") In p.521, Table 6 of this paper, they provide an example of SU(2)\/H$_8$ $\\simeq$ SU(2)\/($Z_2 \\times Z_2$) orbifold. (here H$_8$ is a quaternion group.) **Question** : $\\bullet$ Here I am interested in **orbifold CFT** of **SU(2)\/G and SO(3)\/G** kinds (with G as its normal subgroup) . Can someone provide some more examples and data (i.e. S and T matrix, fusions of quasi-particles) of **SU(2)\/G and SO(3)\/G** **orbifold CFT**? $\\bullet$ What are their **central charges** $c$ of this 1+1D CFT? (e.g. what is the central charge of this SU(2)\/D$_2$ model?) What is their **ground state degeneracy** of 2+1D bulk TQFT on the spatial $T^2$ torus? $\\bullet$ If there is also information on its corresponding **twisted quantum doubles** , in the form of $D^\\omega(G')$ of some certain group $G'$. It will be nicer. Reference and books are welcome, but please **offer summaries** instead of just giving Reference. (ps. Francesco yellow-book CFT Sec 17.8.4 has some discussions, but I could not find orbifold CFT of $SO(3)\/D_n$ of $D_n$ dihedral group or $SU(2)\/H$ of $H$ as a quaternion.)"}
{"id":"27468","title":"Renormalization Group for non-equilibrium","text":"For equilibrium\/ground state systems, a (Wilson) renormalization group transformation produces a series of systems (flow of Hamiltonians\/couplings $H_{\\Lambda}$ where $\\Lambda$ is the cut-off) such that long-wave\/asymptotic behaviour of $H_{\\Lambda}$ is the same as of $H_{\\Lambda'}$ after rescaling by $\\Lambda\/\\Lambda'$. The idea of this definition implies an exact starting point for RG formalisms, with technical details varying between the fields and approximation methods. (For examples, see arXiv:1012.5604 and Wikipedia article). Now, for **non-equilibrium condensed-matter systems** there is research direction aiming at generalization of the RG approach to a steady state, e.g., a voltage-biased strongly interacting quatum dot (or Kondo impuryity). For examples, see arXiv:0902.1446 and related references. I would like to understand the conceptional foundations for the non-equlibrium RG. > What is the definition of an RG transofrmation in a non-equilibirum, steady > state ensemble? I see a problem in the fact that the non-equilibirum desnity matrix which is used to define the problem is not a function of the Hamiltonian alone, thus it is not clear to me how is the effect of the change in the cut-off is split between the Hamiltonian (running couplings) and the density matrix (renormaltizaton of the boundary\/external conditions?)"}
{"id":"91170","title":"Which microscopy technique is for wet condition?","text":"I am searching for a microscopy technique in wet conditions. It must be possible to see through the specimen. ESEM is good but can't see through. TEM needs high vacuum which will boil away water. The specimen is embedded inside a gel. The size of the specimen is between 0.5 µm and 20 µm."}
{"id":"67054","title":"Does spin of the earth vary at different locations?","text":"Does the spin of the earth around its axis vary at different distance from the sun?,if so how does it correspond to the length of day and night,?"}
{"id":"106542","title":"Conjugate of an operator applied to a function","text":"In section 6.3.1 of the following MIT Open Course Ware (PDF) (22-02, Introduction to Applied Nuclear Engineering), the author, Prof Paola Cappellaro, derives Heisenberg's equation using the definition of expectation values, $$ \\frac{d}{dt}\\langle\\hat{A}\\rangle=\\frac{d}{dt}\\int d^3x\\psi^*(x,t)\\hat{A}\\psi(x,t)\\\\\\ =\\int d^3x\\left(\\frac{\\partial\\psi^*}{\\partial t}\\hat{A}\\psi+\\psi^*\\frac{\\partial\\hat{A}}{\\partial t}\\psi+\\psi^*\\hat{A}\\frac{\\partial\\psi}{\\partial t}\\right) $$ and Schrodinger's equation, $$ \\frac{\\partial\\psi}{\\partial t}=-\\frac{i}{\\hbar}\\hat{H}\\psi,\\qquad \\frac{\\partial\\psi^*}{\\partial t}=\\frac{i}{\\hbar}\\left(\\hat{H}\\psi\\right)^* $$ Then the author uses $\\left(\\hat{H}\\psi\\right)^*=\\psi^*\\hat{H}^*=\\psi^*\\hat{H}$ to get $$ \\frac{d}{dt}\\langle\\hat{A}\\rangle=\\int d^3x\\left(\\frac{i}{\\hbar}\\psi^*\\left[\\hat{H}\\hat{A}-\\hat{A}\\hat{H}\\right]\\psi+\\psi^*\\frac{\\partial\\hat{A}}{\\partial t}\\psi\\right)\\\\\\ =\\frac{i}{\\hbar}\\langle\\left[\\hat{H},\\,\\hat{A}\\right]\\rangle+\\langle\\frac{\\partial\\hat{A}}{\\partial t}\\rangle $$ There are two things I don't understand in this derivation. The first is that the total time derivative $\\frac{d}{dt}$ turns into a partial time derivative $\\frac{\\partial}{\\partial t}$ when inside the integral. The second is that the author states $$(\\hat{H}\\psi)^*=\\psi^*\\hat{H}$$ This confuses me as $\\hat H$ is the Hamiltonian operator, and contains an $x$ derivative. Thus the left hand side of the above equation is another state $\\phi^*$ such that $\\phi=\\hat H\\psi$, but the right hand side is an operator: the derivative is yet to be applied to anything. This equation by itself just makes no sense to me. Any help with either of these two issues would be greatly appreciated!"}
{"id":"26427","title":"What is the simplest way to prove the Earth is round?","text":"Assume you've come in contact with a tribe of people cut off from the rest of the world, or you've gone back in time several thousand years, or (more likely) you've got a numbskull cousin. How would you prove that the Earth is, in fact, round?"}
{"id":"73962","title":"Proofs that Earth is not flat?","text":"Well, I do not believe that Earth is flat, but I met some conspirationistas believing that all physics we learn in school is twisted, all the Earth's pictures from space are photoshopped by NASA, etc, etc. So those guys wanted to put a satellite into orbit, take some pics, and see for themselves. Of course, a satellite won't stay above a flat Earth, but what other very simple means to prove\/check that earth is round (well, geoid) do we have? I am looking for simple means, available to anyone, that **do not need any external tool or knowledge** that a naysayer would reject. Some things I've considered:   * Meteorological balloon with camera and a phone with GPS, with a parachute to recover it. It has been done by amateurs, and in good conditions, you can see the Earth's curvature.   * Measure the Sun's height above horizon at noon at two different latitudes. The difference, along with some simple geometry would give you Earth's diameter."}
{"id":"91283","title":"What if Correspondence principle in QM happening in classical sense?","text":"I assume a simple set up (hope to generalize it later)... suppose there is a position pdf (need not be position but any) (probability density function) which is the magnitude of a $\\Psi(x)$ predicted by QM (lets call it P_quantum) and let there be a classical prediction pdf (P_classic), according to CP, for large quantum numbers, P_quantum agree with P_classic in an average sense, but my question is for suppose, what if it agress to converge to P_classic(x) in pointwise convergence, while still explaining all the microscopic things like hydrogen atom..blah blah, what would that mean philosophically? for example see Particle in a 1-D box and the correspondence principle For supose What if classical mechanics originate from QM in a classical sense ( it currently does in only statistical sense) but still explaining all the microscopic things like Hydrogen atoms, atomic spectra etc..., what would that mean? Right now they converge only in statistical sense, hence they say you need not converge pointwise as anyway you lack measuring device at that microscopic level for macroscopic things. What if they did converge pointwise, would that mean the lack of measuring device argument disappear from the QM scene?"}
{"id":"123697","title":"How to calculate the charge on a capacitor","text":"![enter image description here](http:\/\/i.stack.imgur.com\/YiGA9.png)I have a network in which $C_1$ ($8.73 \\;\\rm \\mu F$) and $C_2$ ($2.46 \\;\\rm \\mu F$) are connected in parallel and $C_3$ ($7.45 \\;\\rm \\mu F$) and $C_4$ ($2.22 \\;\\rm \\mu F$) are connected in parallel. $C_{12}$ and $C_{34}$ are connected in series. The network is connected to a $21.9\\;\\rm V$ battery. I calculated the total capacitance to be $5.187 \\rm \\;\\mu F$. How do I find the charge on $C_1$? I use the equation $Q=CV$ right? So I did $5.187\\;\\mu F$ multiplied by $21.9\\;\\rm V$ and got $113.6\\;\\rm\\mu C$ but that is not the right answer? What am I doing wrong?"}
{"id":"29056","title":"Loudspeaker frequency response dip","text":"I'm trying to determine the reason for a dip in the frequency response curve of my loudspeaker. I have a loudspeaker mounted on the inside of an enclosure, and a 100mm ID tube mounted in-front of the speaker on the outside of the enclosure. The measured frequency response is: ![enter image description here](http:\/\/i.stack.imgur.com\/AsZbD.png) At roughly 4100 Hz, a large dip in the response is noted. This dip corresponds to the cut-off frequency of the (0, 1) cross mode. This leads me to believe that it has something to do with the higher-order mode propagation, but I don't understand what. Could someone help me understand this problem please."}
{"id":"129496","title":"Interpretation of the wave function in quantum mechanics","text":"I just started watching the coursera lectures on the basics of quantum mechanics and one of the first lectures were on deriving Schrodinger's equation and its interpretation it under Born's interpretation. What I want to ask is what the wave function \\begin{equation} \\psi({\\bf r},t) \\end{equation} return and represent. Now I know that \\begin{equation} |\\psi({\\bf r},t)|^2dxdydz \\end{equation} is the probability of finding the quantum particle described by \\begin{equation} \\psi({\\bf r},t) \\end{equation} in the volument element \\begin{equation} dV = dxdydz \\end{equation} at time T. But im not sure what the wave function $\\psi$ returns. Could someone please explain in laymans term the return type of the $\\psi$ function and what the lone $\\psi$ function represents."}
{"id":"66011","title":"Why does water have several different solid phase but only one liquid and gas phase","text":"Why does water have several different solid phase but only one liquid and gas phase? Is there any meaning? or any reason behind it? Or is it just the way the nature behaves?"}
{"id":"34486","title":"Reflection positivity in general","text":"In the Euclidean QFT obtained by \"Wick-rotating\" a unitary QFT, the correlation functions satisfy a property called reflection positivity, see e.g. this Wikipedia article for the case of a scalar field. What's the correct formulation if you have chiral fermions and\/or terms like the QCD theta angle? Could someone give the references?"}
{"id":"126814","title":"Is there a simulator for observing the force between two charges comprehensively?","text":"I basically want to observe the electrostatic force between two charges. I want to know if is there a simulator which could both help me to observe and calculate the force between two charges. I found this simulator which is pretty good I think. However, that would be great if there is a more comprehensive simulator with greater distances, more parameters etc. I know this is a duplicate but nobody answered the other question and I really want to know if there is a simulator like that. I am sorry if I broke the rules. Thank you."}
{"id":"34489","title":"Can all quantum superpositions be realized experimentally?","text":"When textbooks in QM give example of finite dimensional Hilbert spaces they give examples of photon polarizations or of 2-states systems and sometimes they mention how one can achieve superposition in such cases experimentally. On the other hand when they talk about simple potentials like particle in an infinite potential well, and talk about superposition of the stationary states of this problem they never mention how such superposition can be achieved experimentally (despite the fact that we are in the era of nanotechnology and scientists can \"engineer\" effectively such potentials and many others). Typical example that may appear in textbooks can be as simple as $\\Psi(x)=\\frac{4}{5}\\phi_1(x)+\\frac{3}{5}\\phi_5(x)$, where {$\\phi_n(x)$} are the normalized eigenfunctions of the particle in a box Hamiltonian. Other more general superpositions could be between infinite numbers of stationary states via $\\Psi=\\sum a_n\\phi_n(x)$. That makes me wonder, is there a fundamental reason that prevents us from engineering such superposition in case of particle in a box and the like, or that we just do not know how to do it yet? why is it possible with spin and seem to be hard with particle in a box? or is it something related to the energy eignestates in the position representation? It is kind of frustrating to study for long hours\/read\/solve problems\/HW on all kinds of potentials and on superpositions without knowing how\/if they can be realized in experiment. If someone knows references in which this issue is discussed it would be greatly appreciated."}
{"id":"105930","title":"Radiation pressure at high temperature","text":"I am reading a book on the history of the hydrogen bomb. The author describes the Teller-Ulam design and explains that the liquid deuterium charge was compressed by the radiation pressure generated by a vaporized plasma of polyethylene. This would allow the generation of pressures orders of magnitude higher than with high explosives. Is it possible to quickly estimate this pressure? If I assume a temperature of $T=1,000,000K$ and use $P=\\frac{4\\sigma}{3c}T^4$ I calculate a pressure of about $2.5$ million bars. Is this realistic? How does it compare with explosives?"}
{"id":"92488","title":"How can the Lorentz transformation cause two objects' masses to both increase?","text":"Suppose there are two people **A** and **B**. A is standing on the earth and **B** is on a rocket in space. Now **B** passes by the earth at a speed of 0.9c and **A** sees him. According to **A** , **B's** mass would have increased. But since the lorentz factor considers only relative velocity, we can say that even **A** is moving at a speed of 0.9c relative to **B**. So will **A's** mass also increase according to **B** after applying the the lorentz transformation formula? If yes then my doubt is how can the mass of both increase?"}
{"id":"92489","title":"How strong is electron degeneracy pressure?","text":"I'm trying to get some specific numbers for electron degeneracy that I can understand, using a concrete example. Take for example this portion of carbon crystal: ![carbon crystal lattice](http:\/\/i.stack.imgur.com\/WODZS.gif)   1. Exactly how much energy would be required to overcome the electron degeneracy pressure of these 18 carbon atoms, in a theoretical laboratory capable of such things.   2. I believe that neutrinos would be emitted as a byproduct of the collapse, is that correct? What other reactions if any would be needed to fully account for conservation of energy\/momentum etc?"}
{"id":"31056","title":"Group rings in Physics","text":"In my institute a few mathematicians work on Group Ring. Since it has close connection with representation theory, I thought that there must be some interesting connections of it with Physics. However I did not manage to find any information from our local experts; perhaps for one reason, they are more interested on finite groups and its structure. Hence I am asking the same question to the community. Does there exists any connection between group rings and physics? (I am having a feeling, that it can be used as a tool for solving representation theoretic problems. But does its usefulness (if there is any) end there only?) Advanced thanks for any reply. Also please provide some references (paper or books), which describes such connection(s), if exists,"}
{"id":"31057","title":"How to calculate specific heat capacity of an alloy","text":"I have done an experiment in which I measured the specific heat capacity of a coin. I would like to compare the results to the actual value. I have found out that the coin is made of Cupronickel alloy (for simplicity let's assume it's just 75% copper and 25% nickel). Knowing the SHC values of the two elements (copper and nickel) how would I go about calculating the theoretical value of the SHC of the Cupronickel alloy, so that I can compare it to my experiment's results?"}
{"id":"92484","title":"Equation for Hubble Value as a function of time","text":"I am trying to write the equation for the situation where the Hubble parameter $H$ would be changing over time. In other words, it would represent an accelerated expansion of the Universe. That is, $H$ can no longer be the simple $H=1\/t$. In the new equation, I should be able to plug a future time and see what the Hubble Value will be in that future. I think I got most of the concepts right. First of all I understand that the key to the problem is in $H=a˙(t)\/a(t)$. Where $a(t)$ is the scale factor from the Friedman equations. I also understand that if $H$ is changing then $a¨(t)> 0$ and also $H'(a)>0$. But I'm still very uncomfortable when my pencil meets the paper. The Friedman equations are not stated as a function of $t$, but as a function of $a$ where $a$ is the time scale, and frankly, I don't know how to use the time scale factor. In any event, here is my poor attempt to do it. According to Wikipedia, one of the solutions of Friedman equation is (assume flat space k=0): $a(t) = a_0 t ^{2\/3(w+1)}$ Therefore: $a'(t) = d(a_0 t ^{2\/3(w+1)}) \/ dt$ $a'(t) = (2a_0\/3(w+1)) t^{ -(1+3w)\/3(w+1)}$ And I suppose that we could now substitute: $H=a˙(t)\/a(t)$ with the above: $H = (2a_0\/3(w+1)) t^{ -(1+3w)\/3(w+1)} \/ a_0 t ^{2\/3(w+1)}$ Simplified: $H = (2\/3(w+1)) t^{-(3w+2)\/3(w+1)}$ And $w$ is typically known from observation. I will appreciate if someone can let me know if I am in the right path or totally derailed. I have a feeling that $a(t) = a_0 t ^{2\/3(w+1)}$ was not the right place to start because if $w=-1$, then everything goes down the drain. But then again, in an accelerated expansion, $w$ would not equal -1. It would always be less than -1. Also, in the final equation, if $w<-1$ then H<0, which could not be right. So I'm not sure what to think. Many thanks in advance, Luis"}
{"id":"32882","title":"Rotation, cats landing on their feet, and conservation of angular momentum","text":"Let _θ_ be the orientation (angle) of a body (such as a cat), and let _ω_ be its angular velocity. It is well-known that _θ_ can change even when the body is not rotating, using the conservation of angular momentum; that is, even when _ω_ = d _θ_ \/d _t_ = 0. That's how cats land on their feet so well. But how can _θ_ possibly ever change, when its derivative is zero?! What's wrong with the math?"}
{"id":"100265","title":"Determining $y$ coordinate of charged particle motion","text":"I've tried several different ways, with a tutor, and with other colleagues. One of my colleagues has even gone to the professor with whom he worked out an equation for the component, and it was later confirmed to be incorrect. Here is the question: > A proton ($q = 1.6 \\times 10^{-19}$ C, $m = 1.67 \\times 10^{-27}$ kg) moving > with constant velocity enters a region containing a constant magnetic field > that is directed along the $z$-axis at $(x,y) = (0,0)$ as shown. The > magnetic field extends for a distance $D = 0.55 m$ in the $x$-direction. The > proton leaves the field having a velocity vector $(v_x, v_y) = (4.8 \\times > 10^5, 1.8 \\times 10^5)m\/s$. This is the diagram: ![enter image description here](http:\/\/i.stack.imgur.com\/EDPe3.png) The question we have is how to determine $h$, the $y$ coordinate of the proton as it leaves the magnetic field. I've already determined that the velocity $v$ is equal to 512640.2247 m\/s, and that the radius $R$ is 1.533 m. The trajectory of the proton is curved on the inside, so we can't use anything involving any derivation of similar triangles. I've tried solving for $v$ as a function of time, but then realized that $v_y$ is always changing anyway, which lead me to calculate the acceleration so the time term that I used incorporated the change, but that didn't work either. Edit: Really sorry folks, I wrote this in a hurry to get to work on time. Yes, the radius I mentioned is indeed the radius of the circular trajectory of the proton after it enters the magnetic field. Also, I do understand to keep things variable, and I mentioned the radius and velocity only as to sort of show what I have available to work with. I've drawn the imagined circle suggested below, and I've made the realization that it would be helpful to relate the radius, the angle, and the velocity, but I'm not sure how to do so accurately."}
{"id":"110411","title":"What areas of physics depend on the sum $1 + 2 + 3 + 4 + 5 + 6+ 7+\\ldots= -1\/12$?","text":"This youtube video from Numberphile, http:\/\/youtu.be\/w-I6XTVZXww shows how the value is derived. In the video, one interviewee claims that \"this result is used in many areas of physics\". In the video, only string theory is mentioned. Which areas of physics use or depend on the sum $$1 + 2 + 3 + 4 + 5 + 6+ 7+\\ldots= -1\/12?$$"}
{"id":"50106","title":"Pair production - mathematically?","text":"Allover the web i am only seeing a statement similar to this: > Pair production is not possible in vaccum, 3rd particle is needed so that > conservation of momentum holds. Well noone out of many writers shows, how to prove this matematically. So this is what interests me here. * * * **First i wanted to know if pair production really cannot happen in vacuum** , so i drew a picture and used equations for conservation of energy and conservation of momentum to calculate energy of a photon $(h \\nu)$ needed for pair production. It turns out $h\\nu$ is different if i calculate it out of conservation of energy or conservation of momentum. And even more! It can never be the same because equallity would mean parts $v_1 \\cos \\alpha$ and $v_2 \\cos \\beta$ should equall speed of light $c$. Well that cannot happen. Below is my derivation. * * * ![pair production](http:\/\/i.stack.imgur.com\/wGF1v.png) CONSERVATION OF ENERGY: $$ \\scriptsize \\begin{split} W_{1} &= W_{2}\\\\\\ W_f &= W_{e^-} + W_{e^+}\\\\\\ h\\nu &= W_{ke^-} + W_{0e^-} + W_{ke^+} + W_{0e^+}\\\\\\ h\\nu &=\\left[m_ec^2 \\gamma(v_1) - m_ec^2\\right] + m_ec^2 + \\left[m_ec^2 \\gamma(v_2) - m_ec^2\\right] + m_ec^2\\\\\\ h\\nu &=m_ec^2 \\gamma(v_1) + m_ec^2 \\gamma(v_2)\\\\\\ h\\nu &=m_ec^2 \\left[\\gamma(v_1) + \\gamma(v_2) \\right]\\\\\\ \\end{split} $$ * * * CONSERVATION OF MOMENTUM: $y$ direction: $$ \\scriptsize \\begin{split} p_{1} &= p_{2}\\\\\\ 0 &= p_{e^-} \\sin \\alpha - p_{e^+} \\sin \\beta \\\\\\ 0 &= m_e v_{1} \\gamma(v_{1}) \\sin \\alpha - m_e v_{2} \\gamma(v_{2}) \\sin \\beta\\\\\\ &\\text{If $\\boxed{\\alpha = \\beta} \\Longrightarrow \\boxed{v_1 = v_2}$ and:} \\end{split} $$ $$ \\begin{split} \\scriptsize 0 = 0 \\end{split} $$ $x$ direction: $$ \\scriptsize \\begin{split} p_{1} &= p_{2}\\\\\\ \\frac{h}{\\lambda} &= p_{e^-} \\cos \\alpha + p_{e^+} \\cos \\beta \\\\\\ \\frac{h \\nu}{c} &= m_e v_{1} \\gamma(v_{1}) \\cos \\alpha + m_e v_{2} \\gamma(v_{2}) \\cos \\beta\\\\\\ h \\nu &= m_e c \\Big[ \\gamma(v_1) \\underbrace{v_{1} \\cos \\alpha}_{\\neq c} + \\gamma(v_{2}) \\underbrace{v_{2} \\cos \\beta}_{\\neq c} \\Big] \\end{split} $$ Alltogether: Because momentum in $y$ direction equals 0 (holds for some combinations of $\\alpha, \\beta, v_1, v_2$) whole momentum equals just the momentum in $x$ direction. So if i add them i get: $$ \\scriptsize h \\nu = m_e c \\Big[ \\gamma(v_1) \\underbrace{v_{1} \\cos \\alpha}_{\\neq c} + \\gamma(v_{2}) \\underbrace{v_{2} \\cos \\beta}_{\\neq c} \\Big] $$ * * * From this i can conclude only that i cannot sucessfully apply conservation of energy and conservation of momentum at the same time and therefore pair production in vacuum cannot happen. **QUESTION1:** Why do writers state that 3rd particle is needed so that conservation of momentum holds? What if conservation of momentum holds and conservation of energy doesn't? How can we say which one holds and which one doesnt? **QUESTION2:** Do writters actually mean that if a 3rd particle is included we can achieve $h \\nu$ to match in both cases? **QUESTION3:** Can someone show me mathematically how this is done? I mean it should right?"}
{"id":"61540","title":"Induced charge on sphere","text":"I have a conducting sphere ($radius = a$) at potential $V_0$. It is enclosed by another thin shell ($radius = b, b > a$) which has a charge density $\\sigma (\\theta) = \\sigma_0 \\cos(\\theta)$ for the polar angle. I'm supposed to find the potential everywhere, noting the field discontinuities at $r = a$ and $r = b$ (this probably involves Poisson's equation), as well as the surface charge density on the inner conducting sphere. So inside the conductor (r less than a), the potential should be constant. Additionally, integrating $\\sigma (\\theta)$ over all $\\theta$ gives me $0$ total charge (very positive at the top of the sphere and negative at the bottom of the sphere), so the potential for $r > b$ should be constant as well because there is zero total charge inside (using Gauss's law with $Q_{enclosed} = 0$). I'm really stuck how to do the potential in the in-between shell, as well as finding the surface charge densities (though once I know the former, the latter mat be doable via the discontinuity of the field $E_{in} - E_{out} = \\sigma \/ 2$). Any ideas?"}
{"id":"61542","title":"How do forces work","text":"Is there a mechanistic-type explanation for how forces work? For example, two electrons repel each other. How does that happen? Other than saying that there are force fields that exert forces, how does the electromagnetic force accomplish its effects. What is the interface\/link\/connection between the force (field) and the objects on which it acts. Or is all we can say is that it just happens: it's a physics primitive? A similar question was asked here, but I'd like something more intuitive if possible."}
{"id":"61549","title":"Relativity of simultaneity - An example","text":"I am trying to understand the relativity of simultaneity in different frames, and I am trying to work out an example. Suppose along the x-axis there are two points 2000m apart. Event A happens at t=0 and event B happens at $t=2\\cdot10^{-6}s$ after event A happens in the rest frame. If there is an observer moving along the positive x-axis, how fast must he move in order to see the two events happen simultaneously? So a diagram might look like this: `A------------------------------B`   `Observer ->` My understanding is that the observer should \"see\" the two events simultaneously if the information reaches him at the same time. Thus, if he stands somewhere between A and B such that after A happens, the information of A and B happening reach him at the same time, then he would think that A and B are simultaneous events. If my logic is correct, then he should stand somewhere to the right of the middle point between A and B. But the example is asking for the velocity of the observer. I am not sure how to do this since I think the initial position of the observer should matter as well (if it does matter, can we assume he starts at the left point?)."}
{"id":"55554","title":"Does a photon have a north and south pole?","text":"A photon has an oscillating magnetic and electric field. Is the magnetic field a dipole?"}
{"id":"32774","title":"Average Neighbouring Impurity Separation in a Random 1D chain","text":"![](http:\/\/i.imgur.com\/MwtfT.jpg) I have a finite and discrete 1D chain (edit: linear chain, i.e. a straight line) of atoms, with unit separation, with a set number of impurities randomly distributed in the place of these atoms in the system. What I would like to do is describe the separation between neighbouring impurities (call it \"D\" which will always be an integer) statistically, and also to work out the average separation . For example, the plot below was calculated from several thousand simulations of a chain of length 200 atoms and 10 impurities where the y-axis is the probability $P(D)$ of finding an impurity at distance D, and the x-axis is nearest impurity distance $D$. It kind of looks like a Poisson distributon, which one would expect since the system is discrete and random and a kind of counting exercise, but it doesn't work to well as a fit to the data points. It has been a long time since I did any statistics so I'm not sure how to start expressing what I found mathematically. Since I know the system length ($L = 200$) and the number of impurities ($N_i$) is a fair starting point the impurity density $\\rho = N_i\/L$ ? _EDIT: The chain isn't allowed to self-intersect, it's a straight line in each case. The system I'm using above is a straight line of 200 evenly spaced atoms, and I'm distributing 10 impurities in the place of random atoms (e.g. at sites 4, 11, 54,...so there are still discrete steps between sites). The graph above is the result of finding the spacings between these impurity sites._ _EDIT 2: Attached a picture at the top_ ![](http:\/\/i.imgur.com\/Rqgn6.jpg) EDIT 3: Okay so it seems it could be my PRNG code causing problems. I'm using Fortran 95, here is the code:               `CALL RANDOM_SEED(size = n)     ALLOCATE(seed(n))          CALL SYSTEM_CLOCK(COUNT=clock) !!! intentionally slows it down to prevent succesive calls from returning the same number          do i = 1 , 1000000          end do     seed = clock + 37 * (\/ (i - 1, i = 1, n) \/)     CALL RANDOM_SEED(PUT = seed)          call random_number(x)`      EDIT 4!: Repeated this for a system size of 50 with 3 impurities (10,000 iterations), fitted a geometric distribution to it, as one can see immediately there is a huge variation. Is this explained by the fact for small n and L when we go from our first impurity (at site j) the probabilities to find the next impurity change drastically since effectively we are now looking for a system with n=2 and L= 50-j, which would be really sensitive to the location of the first impurity hence the huge variation. When we go to a huge system we can essentially treat the remainder of the chain as still being really long, and n and L are pretty much unchanged so this effect is masked and we see a geometric distribution arise? EDIT 5: Just did a bit of reading around Bernoulli Trials, is what I'm seeing and describing a binomial distribution hence the peak I see with smaller systems? Still that wouldn't explain why my 200 system is different from Ilmari's below... ![](http:\/\/i.imgur.com\/ybsJa.jpg)"}
{"id":"22143","title":"Finding the correct units for the energy-momentum tensor?","text":"I'm trying to understand the energy-momentum tensor $T^{\\mu\\nu}$ but I'm confused about the units. My textbook says the components of $T^{\\mu\\nu}$ are $\\mathrm{Jm^{-3}}$. Four-momentum is is given by$$P^{\\mu}=\\left(E\/c,\\mathbf{p}\\right)=\\left(E\/c,p_{x},p_{y},p_{z}\\right)$$ The $E\/c$ component of $P^{\\mu}$ has units $\\mathrm{Jsm^{-1}}$. The definition of $T^{\\mu\\nu}$ is “the rate of flow of the $\\mu$ component of four-momentum across a surface of constant $\\nu$.” Using this definition, how do you get the rate of flow of the $E\/c$ component of four-momentum across a surface of constant time (ie the $T^{00}$ component) to have the correct units of $\\mathrm{Jm^{-3}}$? Surely you need to multiply $E\/c$ by something with $\\mathrm{s^{-1}m^{-2}}$ units, but what exactly? Thank you"}
{"id":"73216","title":"How do we measure a quantum system properties? (length, mass and time)","text":"How do we measure a quantum particle properties? (length, mass and time)"}
{"id":"101074","title":"Modelling multiple heat sources?","text":"I am working on a project to model heat transfer in a data center. I have basic models for a one dimensional heat transfer using conduction and convection. Such as, taking a server and modelling the temperature decrease over time and models taking distance into account based Fourier's equations. However I am now looking to step the model into 2 dimensions and this involves considering a stack of servers with multiple heat sources stacked on each other (such as a server rack). I am struggling to find any useful journals that detail the interactions of multiple heat sources, would people mind pointing me in the direction of some useful sources or explain the basics to me so I can research and expand on them?"}
{"id":"130382","title":"Penetration and skin depth in 'metallic' magnetic induction tomography (classical electromagnetism)","text":"Does anyone know how AC magnetic fields behave in 'metallic' magnetic induction tomography (MIT). Do they form a magnetic circuit around a metallic specimen or penetrate through it, or both? This would be in respect of the skin effect limiting the penetration of the AC magnetic driving field (primary). MIT works by an AC magnetic driving field inducing eddy currents in a metallic specimen (to be imaged) and sensors (coils) on the other side of the specimen picking up the secondary magnetic field due to the eddy currents induced in it. It actually senses or picks up the total field (secondary and primary magnetic fields). Sensor coils measure the spatially distributed total fields to be tomographically represented as an image."}
{"id":"130388","title":"Dissipating light pollution","text":"If all the lights in a city, and the area surrounding it were shut off. How long would it take for the light pollution to clear, so the light from stars could be seen?"}
{"id":"98231","title":"How far we can see in the thickest possible fog?","text":"There is quite foggy day today here, I cannot see end of the street. This made me think about how much worse can it be. So here is my question: how thick a fog can be? What's the smallest visibility distance possible? By visibility distance I mean how far I can see a black object against a white background in the fog. UPDATE: There were some nitpicks, so let me clarify a bit:   * I think of water\/ice fog.   * It's said that angular size matters, so let's say an object whose angular size as big as the full moon.    * Whiteout doesn't count, assume the soil remains dark."}
{"id":"98236","title":"Show that the circumference of the Bohr orbit of hydrogen is an integral multiple of the wavelength of the electron in it","text":"We have an electron revolving in a the $n$th Bohr orbit of an atom of hydrogen. The problem is to prove that the circumference of the orbit is an integral multiple of wavelength of this electron $\\lambda_e$. $$2\\pi r_n = k\\lambda_e, \\ k \\in \\mathbb{N}$$ where $r_n$ is the radius of the $n$th Bohr orbit. I started out by figuring out by expanding out the different terms in the equation: The $n$th Bohr radius for hydrogen is given by: $$r_n = 5.29n^2 \\rm{pm}$$ $$\\implies 2\\pi r_n = 10.58n^2\\pi\\tag1$$ The wavelength of an electron is given by: $$\\lambda_e = \\frac{h}{p_e}$$ The angular momentum of an electron is given by: $$p_e = k_1\\hbar, \\ k_1 \\in \\mathbb{N}$$ $$\\implies \\lambda_e = \\frac{h}{k_1\\hbar} = \\frac{2\\pi}{k_1}\\tag2$$ How do I show that $(1)$ is an integral multiple of $(2)$?"}
{"id":"122062","title":"Do metals have their distinctive look because of the electron sea which surrounds the metal atoms?","text":"are metals shiny because of the electron sea which surrounds the atomic lattice of the metal sample. are metals more shiny because the electron are more evenly distributed on the surface?"}
{"id":"133602","title":"Will I be able to push a small object in front of me in the outer space?","text":"Imagine I am standing on Earth, and pushing a tennis ball away from me. The ball moves. If it is very heavy, I will move back instead of the ball. Now consider the same scenario in outer space, where both the ball and I are hanging without any third body to support\/assert gravity on us. What happens in this case? Will the ball move or will it make me move backwards, despite being very small? Or will both of us move equally away from each other? How exactly does inertia come into picture here?"}
{"id":"87197","title":"Why do arrows point backwards in time for Feynman Diagrams?","text":"I've been looking into Feynman Diagram for quite some time, and the fact that anti-particles point backwards in time after an interaction has been puzzling me. I understand this to be some convention regarding charges and conservation, yet I wish to know what does this all mean. Does the fact that the pair of particles have lines at, say t=0, and an interaction occurs at t=5 mean that both particles existed at t=0 and we are just measuring the effects of a future interaction? Thanks!"}
{"id":"39643","title":"How to handle D-brane particles in the S-matrix using string worldsheets?","text":"In string theory, it's possible to collide a number of strings only in the asymptotic past, and get a final asymptotic state which contains a D0-brane, an anti D0-brane, and a number of strings. If space is compactified, the D0-brane can be replaced by some wrapped D-brane particle, or a stable bound state of a number of them. How is this handled using string worldsheets only? It's true if the string coupling is weak, many strings have to collide to produce a pair of D-branes, but the string coupling is always nonzero, even if tiny. Does this mean a worldsheet formalism only is incomplete? It's true this is a nonperturbative effect, but does this mean perturbation theory is incomplete?"}
{"id":"71214","title":"splitting spin representation when reducible","text":"I'm looking at the spin representation of the orthogonal algebra $so(2m,C)$. This representation is $2^m$ dimensional and an explicit construction with gamma matrices is well known (see for example http:\/\/hitoshi.berkeley.edu\/230A\/clifford.pdf). It is also well known that this is reducible into two $2^{m\/2}$ dimensional irredible reps. I'd like to get an explicit construction of these two irreps as matrices... similar to the construction in the link above. Does anyone know where that's give."}
{"id":"123338","title":"transfer of electricity through electromagnetic waves","text":"Is it possible to transfer electricity from one place to another without the help of physical wires."}
{"id":"8817","title":"What is wrong with the De Broglie–Bohm theory a.k.a \"Causal Interpretation\" of quantum theory?","text":"> **Possible Duplicate:**   >  Why do people still talk about bohmian mechanics\/hidden variables I've heard of De Broglie–Bohm theory a.k.a causal interpretation of quantum theory. The predictions match accurately with with the nondeterministic quantum theory. As a philosophy buff this one seems just like the classical universe. No funny new-age religion style gimmicks. So why don't people use this interpretation instead of the Copenhagen \"magic\" interpretation? P.S. I'm not an expert. I possess only superficial knowledge of quantum physics."}
{"id":"123331","title":"Quantum mechanics with non-cartesian coordinates","text":"Let say we have the classical hamiltonian of a harmonic oscillator: $$H=\\frac{p_x^2+p_y^2+p_z^2}{2m}+\\frac{k_1x^2+k_2y^2+k_3z^2}{2}$$ and we want to find the hamiltonian operator in quantum mechanics, we use the transformations $x\\rightarrow x$, $y\\rightarrow y$, $z\\rightarrow z$, $p_x\\rightarrow -i\\hbar\\frac{\\partial}{\\partial x}$, $p_y\\rightarrow -i\\hbar\\frac{\\partial}{\\partial y}$ and $p_z\\rightarrow -i\\hbar\\frac{\\partial}{\\partial z}$ to obtain: $$\\hat{H}=\\frac{-\\hbar^2}{2m}\\Delta+\\frac{k_1x^2+k_2y^2+k_3z^2}{2}=\\frac{-\\hbar^2}{2m}\\left(\\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2}+\\frac{\\partial^2}{\\partial z^2}\\right)+\\frac{k_1x^2+k_2y^2+k_3z^2}{2}$$ Now suppose I want to treat this system in sphercal coordinates, should I transform the classical hamiltonian in spherical coordinates ($p_r\\rightarrow \\frac{\\partial}{\\partial r}$, $p_{\\theta}\\rightarrow \\frac{\\partial}{\\partial \\theta}$, $p_{\\varphi}\\rightarrow \\frac{\\partial}{\\partial \\varphi}$) or should I transform the hamiltonian operator to spherical coordinates using $$\\Delta=\\frac{1}{r^2}\\frac{\\partial}{\\partial r}\\left(r^2\\frac{\\partial}{\\partial r}\\right)+\\frac{1}{r^2\\sin\\theta}\\frac{\\partial}{\\partial\\theta}\\left(\\sin\\theta\\frac{\\partial}{\\partial\\theta}\\right)+\\frac{1}{r^2\\sin^2\\theta}\\frac{\\partial^2}{\\partial\\varphi^2}$$ and why?"}
{"id":"135369","title":"Numerical tests of Titius–Bode formula for artificial planetary orbits","text":"Titius–Bode law is an empirical formula stating that positions of planets in Solar system is described by: $$a=0.4+0.3 \\times 2^m [AU],$$ where $m$ are natural numbers. All planets, with exception of Neptune, fall within a few percent of it. Moreover, it predicted the existence of Uranus (so, at least to me, it does not sound like a coincidence). There are simulators of solar system formation (even some online). Are there numerical simulations generating such orbits (as expected from Titius–Bode law). Or are they showing that there is no preference for such orbits? See also:   * Why do the planets' orbital distances fall on an exponential curve?"}
{"id":"132957","title":"\"Rocket in a box\" thought experiment","text":"Here's a thought experiment that I came up with a while ago. It might be hard to visualize, so I'll describe it as best I can. Take a rocket in a vacuum, in outer space. Attach a large \"box\" to the end, so that the exhaust gases from the nozzle will shoot straight into it\/ Let's make the box large. Really large - for our purposes, about a mile long, wide and tall, with the nozzle stuck on in the center of one side. When the rocket is turned on, the exhaust gases will shoot out the back end, and the rocket - and thus the box - will move forward. But what about when the gases hit the other side of the box? For the first scenario, imagine that the exhaust gases are in a single, quick pulse (all effects involving sideways motion can be ignored). In an (admittedly poor) attempt to model it, we can imagine it is a small object - like an oddly shaped cube. It will go in the opposite direction as the rocket. Now, when it hits the other side of the box, it will collide. A few things could happen here: 1) The collision is a gaseous \"elastic\" collision - in other words, the box will rebound in the other direction, and the box\/rocket will go _backward_ while the gases go _forward_. When the gases hit the front of the box, the rocket will go forward and the box will go backward. And so on. 2) In an inelastic collision, some energy will be lost during each impact, and the oscillations will grow smaller and smaller. 3) The gases and the box will undergo an \"inelastic\" collision, and the rocket and gases will stop moving. In both these scenarios, the gases are in a small pulse. I have an idea of what will happen (thought using a small cube is a very poor analogy). But what if the gases continue to emanate from the nozzle (and thus the mission had been better planned!)? So my question) are this: 1) In the first scenario, am I correct in saying that the rocket and gases will oscillate back and forth, or will something completely different happen? 2) What will happen in the second scenario?"}
{"id":"83028","title":"Vertical and horizontal components of forces and vectors","text":"I'm getting a bit confused when finding components of vectors and forces. In problems for vectors, I've always known that if you want to get the components of a vector, you would use the following: ![enter image description here](http:\/\/i.stack.imgur.com\/UWNQM.png) * * * Recently we've started working with Forces in homework problems. This diagram is from one of my homework problems, which they want me to draw the free body diagram: ![enter image description here](http:\/\/i.stack.imgur.com\/4zGQN.gif) Here is the free body diagram given by the homework solution: ![enter image description here](http:\/\/i.stack.imgur.com\/KGMsO.gif) * * * What I don't understand is why would I be using $m_{2}gsin(\\theta)$ instead of $m_{2}gcos(\\theta)$ when finding the X-component of the gravitational force of $m_{2}$. Force _is_ a vector, so why wouldn't you use $m_{2}gcos(\\theta)$? Are the rules different when dealing with forces?"}
{"id":"83029","title":"Creating small polarised scientific lenses?","text":"How would I go about creating small polarised scientific lenses? Thinking 5mm by 5mm, with a light-source behind it for the purposes of reducing (polarised) surface reflection (assisted with a cross-polarised viewer). Is the idea that I just need to apply magnesium fluoride; and if so, at what consistency and against what type of lens?"}
{"id":"47934","title":"Don't understand the integral over the square of the Dirac delta function","text":"In Griffiths' _Intro to QM_ [1] he gives the eigenfunctions of the Hermitian operator $\\hat{x}=x$ as being $$g_{\\lambda}\\left(x\\right)~=~B_{\\lambda}\\delta\\left(x-\\lambda\\right)$$ (cf. last formula on p. 101). He then says that these eigenfunctions are not square integrable because $$\\int_{-\\infty}^{\\infty}g_{\\lambda}\\left(x\\right)^{*}g_{\\lambda}\\left(x\\right)dx ~=~\\left|B_{\\lambda}\\right|^{2}\\int_{-\\infty}^{\\infty}\\delta\\left(x-\\lambda\\right)\\delta\\left(x-\\lambda\\right)dx ~=~\\left|B_{\\lambda}\\right|^{2}\\delta\\left(\\lambda-\\lambda\\right) ~\\rightarrow~\\infty$$ (cf. second formula on p. 102). My question is, how does he arrive at the final term, more specifically, where does the $\\delta\\left(\\lambda-\\lambda\\right)$ bit come from? My total knowledge of the Dirac delta function was gleaned earlier on in Griffiths and extends to just about understanding $$\\tag{2.95}\\int_{-\\infty}^{\\infty}f\\left(x\\right)\\delta\\left(x-a\\right)dx~=~f\\left(a\\right)$$ (cf. second formula on p. 53). References:   1. D.J. Griffiths, _Introduction to Quantum Mechanics,_ (1995) p. 101-102."}
{"id":"131048","title":"How accurate are constants in cgs units?","text":"I am just curious that if constants in cgs units change the answer of an equation. For example, Coulomb's constant, in SI units it equals to $8.98...\\times 10^9 \\,\\mathrm{N\\,m^2\\,C^{-2}}$. However in cgs units it equals to 1. I think the difference between an answer calculated with Coulomb's constant in SI units and Coulomb's constant in cgs units would be a lot. I don't know if my logic is right."}
{"id":"12788","title":"How does electron gun accelerates electrons?","text":"I know that in electron guns we see in TV's and lots of other places, we have electron emitter (cold\/hot W needle in the simplest case) and electrons are accelerated using lattice with high-voltage potential. But the question is why doesn't this lattice slow down electrons once they passes through it? PS. Please correct if lattice is not the correct word here :-D"}
{"id":"5567","title":"Quantum shot-noise and the fluctuation dissipation theorem","text":"Classically, shot noise observed in the signal generated by a laser incident on a photodiode is explained as being due to the quantization of light into photons, giving rise to a Poisson process. In quantum optics, on the other hand, the shot noise is said to arise from interference with the vacuum field, which leaks in at points of optical loss (absorption). Meanwhile, mechanical oscillators are subject to the fluctuation-dissipation theorem, which says, roughly, that the thermal excitement of the various mechanical modes of the oscillator are proportional to the dissipation in that mode. In the context of electronics, Johnson noise is an example of this effect. I recently read a thought-provoking statement (in this paper) that these effects are really \"the same\"; just as thermal motion arises at points of mechanical dissipation, shot noise arises at points of optical dissipation. **Is this analogy physically\/theoretically significant?**"}
{"id":"5561","title":"Best way to keep food cool indoors without using a fridge\/freezer","text":"I've just moved into my first place, and for reasons I do not condone or recommend I don't have any method of refrigeration right now. Luckily I now live very close to shops, so it isn't a big deal, but it made me think: if I really had to, what would be the most efficient way to keep food cool without using any electrical (or any other means which would be impractical) method of refrigeration?"}
{"id":"72979","title":"How to prove that we are living in a 3D world?","text":"Is there any scientific experiment that can lead us to conclude we live in 3 dimensions without the premise of the conception of limited dimensions? Thank you all who helped in the improvement of this question (which was not clear at first)."}
{"id":"76294","title":"Can classical orders coexist with quantum orders?","text":"For example, the ground state of the antiferromagnetic(AFM) Heisenberg model $H=J\\sum_{<ij>}\\mathbf{S}_i \\cdot \\mathbf{S}_j(J>0)$ on a 2D square lattice is a Neel state, which is a classical order described by conventional order parameters $\\left \\langle \\mathbf{S}_i \\right \\rangle$. On the other hand, if we use the Schwinger-fermion mean-field theory to study $H$, we will get a mean-field Hamiltonian $H_{MF}=\\sum (f_i^\\dagger \\chi_{ij} f_j+f_i^\\dagger \\eta_{ij} f_j^\\dagger+H.c.)$ which is studied in Wen's papers of PSG, and now we can study the PSG of the mean-field ansatz $(\\chi_{ij},\\eta_{ij})$ and associated quantum order. So according to the above example, can we say that the AFM SDW phase(classical order) _possesses_ the quantum order(PSG)? Thank you very much."}
{"id":"76297","title":"Tycho's stellar parallax measurements","text":"How, exactly, did Tycho Brahe do his measurements of (no) stellar parallax? All the descriptions of parallax that I have been able to find seem to talk about the change in position of a nearby star compared to background stars farther away, but as far as I know the Ptolemaic system assumed that the fixed stars were all at the same distance? I guess I am looking for a description of how parallax measurements where actually historically done at Tycho's and Copernicus's time."}
{"id":"38277","title":"Breaking of conformal symmetry","text":"I am wondering something about the breaking of conformal symmetry: I know that it can be broken at the quantum level, anomalously, but I never encountered or heard about a model where it is broken \"à la Higgs\" with a potential whose true minimum would spoil this invariance (e.g. making appear a particular energy scale). I guess we would then get some Goldstone bosons, would there be something special about them?"}
{"id":"89240","title":"Yearly onshore wind turbine energy production","text":"I am trying to work out the yearly energy production of an onshore Siemens SWT-3.0-101, hub height 94m, total height 144,5. Here is the power curve at each wind speed level worked out through a software: ![power curve](http:\/\/i.stack.imgur.com\/s9yri.jpg) I am trying to work out the average yearly production at wind speed= 8 m\/s. My calculations are the following: 1138(=Kw at 8 m\/s wind speed)*3600(=sec in 1hr)=4096800 KW\/h; 4096800Kw\/h*6765,81(hours in one year)=27718170408Kw\/h in one year=27718,17GW\/h in one year. This is my result, but everywhere is said that an yearly yeld of an onshore wind turbine stays around 5.5-6 GW\/h, which is very far from my result. Where am I wrong? Thank you very much in advance for your help"}
{"id":"11216","title":"the search for calabicles","text":"Pop-sci ST books say, \"At every point of space, there is a Calabi-Yau space...\" or similar. But uncertainty p. and noncommgeom say, no points in space, only planckons (not my coinage). Is 'point of space' (meaning real, not abstract\/mathematical space) inexact terminology, or a category error, like \"an atom of mayonnaise'- only worse? So I speculate that C-Y space can not be a continuum either, but must consist of discrete calabicles (my coinage)."}
{"id":"89246","title":"What are the current experimental restrictions of the possible speeds of gravitation?","text":"Somewhere I read that the Hulse-taylor binary pulsar can not differentiate between competing theories assuming different speeds of gravity. Is it mathematically true in general, that the orbital decay of two orbiting bodies cannot possibly favor certain speeds of gravity over others? If so, why is it the case? Any references would be much appreciated! Edit: I found the page where I read it: > The speed of gravitational radiation ($C_{gw}$) depends upon the specific > model of Gravitation that you use. There are quite a few competing models > (all consistent with all experiments to date[~1992]) including of course > Einstein's but also Brans-Dicke and several families of others. All metric > models can support gravity waves. But not all predict radiation travelling > at $C_{gw} = C_{em}$. ($C_{em}$ is the speed of electromagnetic waves.) So, **is it possible to construct a metric theory of gravitation with arbitrarily chosen $C_{gw}$** , so that it will be consistent with orbital decay experiments to date? If not, what is the possible range. I believe it must have been discussed in scientific literature, but I can't find anything. I know that it can be a very hard question to answer with mathematic rigor, but a non-technical overview should be sufficient."}
{"id":"132719","title":"A force opposing Gravity","text":"_Every Action Has An Equal and Opposite Reaction_ (Newton's Third Law.) If this is the case, does gravity have an equal-opposing force? From asking around I still haven't got a very clear answer; those who I've talked to seem to believe there isn't one - that gravity is actually a singularity [a one way force] which somehow \"just works\", others think it differently - believing there **is** an opposing force of which prevents gravity from compressing masses more than it already does. So which one is the right answer? (if either!)"}
{"id":"102859","title":"Why are there no true oneway mirrors?","text":"\"One way mirrors\" are used in interrogation rooms etc. The wikipedia article states that: > A true one way mirror does not, and cannot, exist. Light always passes > exactly equally in both directions. However, when one side is brightly lit > and the other kept dark, the darker side becomes difficult to see from the > brightly lit side because it is masked by the much brighter reflection of > the lit side. Which I think I understand. What I find strange is that this surely can't be any deep principle of physics, as it's easily \"violated\"; you could just place a camera on one side and a TV on the other side, and it would be perfectly \"see-through\" from one side but not the other. Couldn't you imagine a sort of microscopic version of the above \"violation\"?"}
{"id":"57543","title":"Trapping EM radiation","text":"Is there a material which can allow light (or any other EM radiation) to pass through from one side as if it is transparent but its other side reflects light like a mirror? ![material](http:\/\/i.stack.imgur.com\/SSOC7.png)"}
{"id":"12784","title":"Physics-oriented books on fractals","text":"I'm looking for some good books on fractals, with a spin to applications in physics. Specifically, applications of fractal geometry to differential equations and dynamical systems, but with emphasis on the physics, even at the expense of mathematical rigor. Hope that was clear and specific enough."}
{"id":"127541","title":"What is the physical significance of the fermi-field asymmetric stress-energy tensor?","text":"Using the ideas from a previous question here it can be shown that if one takes the boson spin 1 stress-energy tensor of the form \\begin{align} T^{\\mu\\nu}_{\\text{spin one}} = \\begin{bmatrix} \\frac{1}{8\\pi}(E^2+B^2) && S_x\/c && S_y\/c && S_z\/c \\\\\\ S_x\/c && -\\sigma_{xx} && -\\sigma_{xy} && -\\sigma_{xz} \\\\\\ S_y\/c && -\\sigma_{yx} && -\\sigma_{yy} && -\\sigma_{yz} \\\\\\ S_z\/c && -\\sigma_{zx} && -\\sigma_{zy} && -\\sigma_{zz} \\end{bmatrix} \\end{align} created from a boson spin 1 electromagnetic field $F=E+iB$ and changes the spin of $F$ from spin one to spin half, $\\\\{F,A\\\\} \\rightarrow \\\\{ \\phi_+ , \\phi_- \\\\} $ using here, then $T^{\\mu\\nu}_{\\text{spin one}}$ changes to \\begin{align} T^{\\mu\\nu}_{\\text{spin half}} = mc^2 \\begin{bmatrix} I && 0 && 0 && i\\gamma^{0123} \\\\\\ \\gamma^{01} && 0 && 0 && i\\gamma^{23} \\\\\\ \\gamma^{02} && 0 && 0 && i\\gamma^{31} \\\\\\ \\gamma^{03} && 0 && 0 && i\\gamma^{12} \\end{bmatrix} , \\end{align} which I can only understand $T^{\\mu\\nu}_{\\text{spin half}} $ as a fermion spin half stres-energy tensor. What is the physical significance of $T^{\\mu\\nu}_{\\text{spin half}} $ compared to $T^{\\mu\\nu}_{\\text{spin one}}$? I am struggling to understand how to interpret $T^{\\mu\\nu}_{\\text{spin half}} $. This fermi-field stress-energy tensor is asymmetric. The divergence of $T^{\\mu\\nu}_{\\text{spin half}} $ is zero."}
{"id":"103212","title":"Can someone please explain the \"infrared catastrophe\"?","text":"In my readings I've run into this idea of an \"infrared catastrophe\" associated with 1\/f noise. As far as I can tell it is because when you graph the periodogram of the 1\/f signal you see the PSD goes to infinity as frequency goes to 0. Not sure what that means practically though. If we are talking about a sound wave, does that mean the sound becomes infinitely loud at low frequencies? What is the \"catastrophe\"?"}
{"id":"120018","title":"Does the Night Mode of the screen display (LCD) save more energy?","text":"In some cases, we can enable the Night Mode (reversing the bright and dark color of the display; such as White Text, Black Background) for the screen display. LCD(Liquid-crystal display) seems to be a common monitor display devise nowadays, so let us consider the LCD at this moment. > (1) Will the Night Mode save more energy? > > (2) How can we evaluate the energy efficiency ratio between, for example > **the Day Mode and the Night Mode**? > > (3) A bonus question: whether the energy efficiency (of Day Mode and Night > Mode) varies for CRT(Cathode ray tube), LCD, Plasma, and OLED(organic light- > emitting diode)? ![enter image description here](http:\/\/i.stack.imgur.com\/SuwPE.jpg)"}
{"id":"69592","title":"What does this feynman diagram represent?","text":"This feynman diagram represents the elastic proton scattering of electrons. ![enter image description here](http:\/\/i.stack.imgur.com\/gKizX.png) The $e^-$ has an initial momentum $k$ and a final $k^\\prime$. The circle represents that the target(the proton that is) has a structure and $q$ is the momentum transfer between the electron and the proton, which is mediated by a photon. What does the $p$ and $p^\\prime$ lines mean?"}
{"id":"82642","title":"Reliability of time dependent measurements","text":"How is a \"baseline\" established. When an experiment is performed to verify the reliability of something like carbon dating. How do we know the results we get over the period of the test is still valid for an item older than one billion years?"}
{"id":"82648","title":"Changing Momentum","text":"![](http:\/\/i.stack.imgur.com\/ohx7t.jpg)![](http:\/\/i.stack.imgur.com\/yLA6b.jpg) Okay, so I am having a hard time wrapping my head around this. Before looking at the answer, I worked the example and got the answer only because I used dimensional analysis to get me to the right units for force. However, to be honest, I don't think I even understand the concept here. I know mass is changing and velocity is constant, but what is throwing me off is where it says \"The water strikes the window at $32\\, m\/s$ so each kilogram of water loses $32\\, kgm\/s$ of momentum. Water strikes the window at the rate of $45\\,kg\/s$, so the rate at which it loses momentum to the window is $1400\\, kgm\/s^2$\" How is it losing $32\\, kgm\/s$ of momentum and then at the same time also losing it at a rate of $1400\\,kg m\/s^2$? Why does it even mention \"The water strikes the window at $32\\,m\/s$ so each kilogram of water loses $32\\,kgm\/s$ of momentum\" what is the purpose of that sentence. That is really confusing me. **EDIT:** I added this simple diagram to show how I see this system. Where is $32\\,kgm\/s$ of momentum come in? (I am aware that the window is now showing)"}
{"id":"80647","title":"What temperature will create amorphous TiSi?","text":"At what temperature will create amorphous TiSi? I think it's around 500 Celsius but I'm not sure."}
{"id":"88117","title":"Problem in applying Takens correlation dimension","text":"I am trying to implement Takens' correlation dimension for noisy time series formula given in Eq (8) in (Estimation of the dimension of a noisy attractor. Schouten JC, Takens F, van den Bleek CM Physical Review E, vol 50, number 3, 1994) . The actual derivation and explanation is given in Takens' maximum likelihood estimation (F. Takens, On the numerical determination of the dimension of an attractor, in: D. Rand, L.S. Yong (Eds.), Dynamical Systems and Turbulence, Warwick, 1980, Lecture Notes in Mathematics, Vol. 898, Springer, Berlin, 1981.) and summarized below according to my understanding. My time series are noisy with additive white Gaussian noise, `Z = x+noise` Step1 = Perform phase space embedding of the 1D time series. This becomes U. Step2: Find euclidean distances or maximum norm distances of the higher dimensional space. Find a scaling distance, $l_0$ (or $r_0$) called as the maximum scaling distance. Step3: Then find those distances, $l_z$, which are $\\le l_0$. This step converts distances into $l_z$ that now belongs to range $[0,1]$. Sometimes, the maximum scaling length $l_0$ is taken as the $l_0$ = average absolute deviation of the time series by the formula $l_0 = 1\/{\\rm total\\ elements}[ {\\sum (u_i - {\\tt mean}(u)}]$. Step 4: In Takens' paper, it is mentioned under Section2, that the distances can be made independent of $l_0$ by dividing all distances $l_z$ by $l_0$. Step5: Take the logarithm of each element so found by Step 4. ${\\rm Takens\\ dimension} = -(\\frac{2}{n(n-1)} \\sum [\\log (||X_i- X_j||\/r_o)]^{-1}$ (Formula for Takens) $n$ = number of samples. **Questions:** (A) It is not clear to me what should be $l_0$ - the upper bound of the scaling distances. Due to the term \"distance\", I initially thought them to be the average standard deviation of the euclidean distances calculated. But, then this doe snot follow what is mentioned under Eq (12). When I took $l_0$ as the average absolute deviation of the time series and divided the distances by it, the result of correlation dimension is incorrect. So, if I follow what is mentioned in the paper1, then the value of the dimension is coming out to be incorrect. What should be $l_0$ ? (B) How do I formulate the dimension for noisy case , I mean will the final formula change and do I have to include the noise effects as done by Ellner http:\/\/rullf2.xs4all.nl\/msct\/node28.html I really do not understand what should be $l_0$, the upper bound of scaling distances. Should be the maximum distance, or the maximum standard deviation of the distance vector or the average standard deviation of the time series ? Please help, I am not sure if I understood correctly."}
{"id":"17328","title":"Consequences for causality if superluminal neutrinos were explained by extra dimensions","text":"One suggestion for explaining superluminal neutrinos (assuming for the sake of argument that the OPERA results should hold up) is that the neutrinos have taken a route through extra dimensions, with an appropriate metric of the form $\\mathrm{d}s^2=-\\mathrm{d}t^2 + f(r)\\mathrm{d}x^2 + \\mathrm{d}r^2$. This would be consistent with ordinary physics, but we would still have faster-than-light travel in our 4 dimensions, so what would be the consequences for causality?"}
{"id":"64745","title":"Boundary conditions on wave equation","text":"![wave equation question](http:\/\/i.stack.imgur.com\/cvSGd.png) I am having trouble understanding the boundary conditions. From the solutions, the first is that $D_1(0, t) = D_2(0, t)$ because the rope can't break at the junction. The second is that $\\dfrac{\\partial D_1}{\\partial x} D_1(0, t) = \\dfrac{\\partial D_2}{\\partial x}(0, t)$. How can I interpret this physically? I'm not quite sure how to think about $\\partial D \/\\partial x$."}
{"id":"64746","title":"Energy conserved... or not? Confused!","text":"I am confused. Could someone kindly explain what's going on in this question? > A particle of mass $m$ and charge $e$ moves in the $x,y-$ plane. There is a > constant magnetic field $B$ that points in the $z$ direction. Initially the > particle has energy $\\gamma m$ in the rest frame of the observer. After a > time elapse of $\\Delta t$, the observed energy is $\\gamma' m$ (again in the > rest frame of the observer). How are $\\gamma,\\gamma',t$ related? Some thoughts: Since we have an accelerated charge, energy is radiated. Since the question does not specify that the motion is non-relativistic, I shall use the relativistic equations. Larmor's formula: $$P=\\frac{\\mu_0 q^2}{6\\pi m^2}\\frac{dp^a}{d\\tau}\\frac{dp_a}{d\\tau}$$ To get $\\frac{dp^a}{d\\tau}$ I need the equation of motion: $$\\frac{dp^a}{d\\tau}=eF^a{}_b\\frac{dx^b}{d\\tau}=\\frac{e}{m}F^a{}_bp^b$$ where $$F^a{}_b=\\begin{pmatrix}0&0&0&0\\\\\\ 0 &0& B&0\\\\\\0 &-B& 0&0\\\\\\0&0&0&0\\end{pmatrix}$$ This gives 4 coupled ODEs. Actually the first 2 are trivial such that $$p_0=p_{0, init}\\\\\\p_4=p_{4,init} $$ and $$\\frac{dp_1}{d\\tau}=\\frac{qB}{m}p_2\\\\\\ \\frac{dp_2}{d\\tau}=-\\frac{qB}{m}p_1$$ But then doesn't $$p_0=p_{0, init}=\\gamma m$$ say that the energy is conserved? It is also not physically surprising if it were since the magnetic field induces a force perpendicular to the velocity of the particle, therefore does no work on it. What's gone wrong? * * * **[Update 1]** With Mike's and Michael's help, I have proceeded to figured out how to do it for the non-relativistic case (but I'm not very sure how to extend it to the relativistic case :-\/ ) -- if I'm not mistaken, the NR case should be something of the following: $$F=e \\vec v\\times \\vec B$$ Assuming circular motion and the magnetic field being perpendicular to the plane of motion, $$r=\\frac{mv_{init}}{eB}$$ Therefore acceleration is $$a=\\frac{v^2}{r}$$ Then I can substitute it into Larmor's formula and so on. How do I generalize it to the relativistic case? **[Update 2]** Do I simply add a $\\gamma$ factor in front of the $v$'s?"}
{"id":"52337","title":"Question about Classical Transport Theory","text":"With a distribution function of the form $f=f_{0} + \\vec{v} \\cdot \\vec{g}$, one can obtain the current density. My question is about $\\vec{g}$; we assume a general solution to $\\vec{g}$ of the form $\\vec{g}= \\alpha \\vec{E}+\\beta \\vec{B} + \\gamma(\\vec{B} \\times \\vec{E}) $. By substitution into $$\\frac{\\mu}{c}(\\vec{v} \\times \\vec{B})\\cdot \\vec{g} + \\vec{v} \\cdot \\vec{g} = e \\tau \\frac{\\partial f_{0}}{\\partial \\epsilon} \\vec{E} \\cdot \\vec{v}$$ and by equating both sides and equating the coefficients of $\\vec{v} \\cdot \\vec{E}$, $\\vec{v} \\cdot \\vec{B}$, and $\\vec{v} \\cdot (\\vec{B} \\times \\vec{E})$ we are supposed to get $$\\vec{g} = \\frac{\\partial f_{0}}{\\partial \\epsilon} \\frac{e \\tau}{1+ (\\omega_{c}\\tau)^{2}}[\\vec{E}+(\\omega_{c}\\tau)^{2}(\\hat{z}\\cdot\\vec{E})\\hat{z}+\\omega_{c}\\tau(\\vec{E} \\times \\hat{z})]$$ My question is how? I'm afraid this isn't apparent to me. I don't know where to start to try and prove this!"}
{"id":"10674","title":"Focusing Light with Flat Mirrors","text":"What is the best way to focus (sun)light using flat mirrors? My goal is to start a fire. Cutting the mirrors is easy."}
{"id":"16987","title":"The time component is $\\gamma m c$, so shouldn't $E=mc$?","text":"Basically, the book is Brian Cox's _Why Does $E=mc^2$?: (And Why Should We Care?)_. I just finished Chapter 5, where we derived the spacetime momentum vector (energy-momentum four vector, as he establishes the physics jargon). Let $\\gamma=\\frac{1}{\\sqrt{1-\\frac{v^2}{c^2}}}$ So, as we found out, the vector's spatial component is $\\gamma m v$, leaving the time component as $\\gamma m c$. He went on, under the guise of making the outcome more intellgible, to saying we could happily multiply the time component by $c$ without changing it's meaning. Cool, I thought, no problem. Next, he pointed out $\\gamma\\approx1+\\frac{1}{2}\\frac{v^2}{c^2}$, so $\\gamma m c^2 \\approx mc^2 + \\frac{1}{2}mv^2$. Et voilà, $mc^2$. Granted, he's obviously trying to simplify things so I can reach an intuitive understanding, but from that point onwards, he uses $mc^2$ as the conversion value. I'm confused. Could someone explain why we use $mc^2$ and not the version scaled down by a factor of $c$?"}
{"id":"21142","title":"Can we solve the particle in an infinite well in QM using creation and annihilation operators?","text":"The particle in an infinite potential well in QM is usually solved by easily solving Schrodinger differential equation. On the other hand particle in the harmonic oscillator oscillator potential can be solved elegantly algebraically using the creation and annihilation operators to find its spectrum. Is it possible to do the particle in a box problem using creation and annihilation operator and how?"}
{"id":"69040","title":"What is Schrödinger's wave equation?","text":"What is Schrödinger's wave equation? We recently studied about the Schrödinger wave equation in school. I understand that because of Heisenberg's Uncertainty principle, and De Broglie's Hypothesis, it was not possible to consider an electron as entirely a particle, and also know about it's velocity and position accurately. I also know the bare basics of the Schrödinger equation, and that it tells us what the probability of finding an electron within an atom. I really need to know how the quantum numbers come into the picture, and what the mathematical aspect to the equation is. I'm not looking for derivations, and just want to understand the background of the equation. Lastly, i know that Psi is the amplitude of the electron wave, which is essentially a length. Length should ideally be a positive rational number, however small. Why then, is Psi, a complex number? Also, it would be much appreciated if the explanation was a bit simple. Thanks a lot. I really appreciate any help you can provide to me."}
{"id":"28519","title":"Kepler problem in time: how do two gravitationally attracting particles move?","text":"Two particles with initial positions and velocities $r_1,v_1$ and $r_2,v_2$ are interacting by the inverse square law (with G=1), so that $$ {d^2r_1\\over dt^2} = - { m_2(r_1-r_2)\\over |r_1-r_2|^3} $$ $$ {d^2r_2\\over dt^2} = - { m_1(r_2-r_1)\\over |r_1-r_2|^3} $$ (the inverse square law along the line of separation). What is the complete solution of these differential equations? What is the position of the two objects as a function of time? After reading a lot on Wikipedia, I've come to the defintion of center of mass and relative coordinates: $$R(t) ~=~ \\frac{m_1 r_1 + m_2 r_2}{m_1+m_2}$$ $$\\ddot{r}(t) r(t)^2 ~=~ (m_1+m_2)G$$ Where $R$ is the center of mass, and $r$ is the displacement between the particles... Is this correct? How do I proceed to solve the differential equation?"}
{"id":"30153","title":"Nature of orbit due to central force!","text":"> **Possible Duplicate:**   >  Kepler problem in time: how do two gravitationally attracting particles > move? How do we get the shape of orbit under the condition that   1. Force is centrally directed   2. Force follows inverse square law   3. Angular momentum is conserved I read somewhere under these three constraint the orbit becomes elliptical. I tried to put together these conditions but couldn't find the solution! Any help in derivation of elliptical orbit would be appreciated. Thanks"}
{"id":"20813","title":"How would a black hole power plant work?","text":"A black hole power plant (BHPP) is something I'll define here as a machine that uses a black hole to convert mass into energy for useful work. As such, it constitutes the 3rd kind of matter-energy power (formerly \"nuclear power\") humans have entertained, the first two being fission and fusion. Putting aside the fact that the level of technological advancement needed for this is far beyond modern day humans, it seems to be pretty well established that this is possible (and maybe someday inevitable) among physicists. Personally, I feel like I don't understand the proposal because no one has really created a coherent picture of how it would work. To keep this focused, here are my objective questions: Related to the conversion itself:   * **What mechanism of energy release are we talking about?** Possibilities include radiation from accretion of material, Hawking radiation, magnetic conversion of rotational energy, and maybe others (EDIT: Manishearth's answer revealed that I don't know the possibilities myself). Can we rule some of these out pragmatically? How well are these mechanisms understood? Do we really have the physics\/observations to back them up, or is it still on the forefront of physics?   * **How would the energy be harvested?** Some possibilities include a thermal cycle (like a Rankine cycle), radiation to electricity conversion (like photovoltaics), magnetically induced currents, and charge movements. Could the particles of emission be _too_ high energy to harness with known materials as a part of the machine structure? That would certainly apply to Hawking radiation from small BHs, do other options have the problem of overly-energetic emissions? Related to application:   * **Is there _anything_ we could use to change the Hawking radiation power output other than the BH mass?** Would we be dancing on a knife's edge by using Hawking radiation as a power source? As you should know, a BH evaporation would be a fairly major cosmic event, and to get more power out, one would have to use a smaller BH, which presents a greater risk of catastrophe. This problem could be mitigated by using several BHs so a lower energy throughput would be needed for each, but then that risks BH collision which is also a catastrophic event. Does the spin, charge, or local gravitational potential affect the output? Is there any way for Hawking radiation to be a reasonably safe method of matter to energy conversion (even for an advanced civilization)?   * **What are the parameters for the total mass and energy output?** Specifically, what would the total energy output divided by mass be, and what would the mass-to-energy conversion efficiency of new in-falling material be? For accretion, I've heard numbers like 30% of the in-falling material can be converted to energy and ejected back into space at the poles... but could we change and control that? How would a BHPP's parameters compare to our sun and current Earth-based power plants? Sorry I couldn't wrap this up into a single question. I tried. To address the \"on topic\" concern, I hear people mention that tokomak _fusion_ power research still has many unknowns that are in the domain of physics, as opposed to engineering, which will come later in its development. I tried to edit my question above to be actually physics in nature. I think I speak for most people when I say that I don't understand the vast majority of the _physics_ of BHs. My question here is generally how those physics would play out in a machine that does useful matter to energy conversion."}
{"id":"20816","title":"If a measured value is oscillating so fast between two numbers, what does the detector read? ","text":"There is a picture usually depicted in introductory quantum mechanics about how the probability pattern appears in the classical limit, that the oscillation is too fast and that it cannot be resolved by the experimental apparatus, hence the detector will detect only the average value. ![Image of oscillation probability](http:\/\/i.stack.imgur.com\/SXnFl.gif) I do not see how the detector \"decides\" to give the average value if the observed value oscillate too fast between 2 numbers!"}
{"id":"78010","title":"Circular motion question","text":"> _(Q) Find the maximum speed at which a truck can safely travel without > toppling over, on a curve of radius 250m. The height of the centre of > gravity of the truck above the ground is 1.5m and the distance between the > wheels is 1.5m, the truck being horizontal._ I actually can't figure out which force acts as the centripetal force in the question. Also since it is a truck so it can't lean, so I can't apply the method we generally use in leaning of cycle. Also no friction is described here between the road and wheels. ![My progress till now](http:\/\/i.stack.imgur.com\/A19c4.jpg)"}
{"id":"21491","title":"Could a real-life X-Wing fly in Earth's atmosphere?","text":"From an aerodynamic point of view, could a full-size aircraft of X-Wing design fly in Earth's atmosphere? Assuming you were free to add control surfaces here and there, could the wings in open configuration create lift ? ![X-Wing](http:\/\/i.stack.imgur.com\/GS4Mt.jpg)"}
{"id":"46914","title":"What is the color of Water?","text":"What is the color of Water? I am not talking about large water bodies like Ocean or lake but a glass of water. If it has no color how can we see it?"}
{"id":"17589","title":"Quantum systems with real structure","text":"A lot has been said as to why quantum mechanics needs complex numbers. However, all measurements produce real values. Expectation values are real, the observables form a real Lie-algebra (use $-\\mathrm i[*,*]$ as Lie bracket instead of the raw commutator) and any state $\\mathbf z$ is uniquely determined by the real-valued function $P_{\\mathbf z}(*) = |\\langle \\mathbf z,*\\rangle|^2$ on the space of states. The space of states itself, however, is a complex projective space, and even though phase factors are irrelevant when taking expectation values, they do matter when combining states - linear superposition $α\\mathbf z + β\\mathbf z'$ depends on the ration $|α|\/|β|$ as well as the phase difference $\\arg(α) - \\arg(β)$. But there is at least one important quantum system which can be modeled as a real system: the qubit, ie the space of spin states of a single electron. In addition to the standard representation as $\\mathbf z=(z_1,z_2)\\in\\mathbb C^2,\\langle \\mathbf z,\\mathbf z\\rangle =1$, states can also be represented as $\\vec z\\in\\mathbb R^3,\\vec z^2=1$ via $$ \\vec z = \\begin{pmatrix} \\langle\\mathbf z,\\pmb σ_1\\mathbf z\\rangle\\\\\\ \\langle\\mathbf z,\\pmb σ_2\\mathbf z\\rangle\\\\\\ \\langle\\mathbf z,\\pmb σ_3\\mathbf z\\rangle \\end{pmatrix} = \\begin{pmatrix} 2\\Re(\\bar z_1z_2)\\\\\\ 2\\Im(\\bar z_1z_2)\\\\\\ |z_1|^2 - |z_2|^2 \\end{pmatrix} $$ where $\\pmb σ_1, \\pmb σ_2, \\pmb σ_3$ are the Pauli-matrices. This is the well-known Hopf fibration. What makes this useful is the fact that $P_z$ takes a particular simple form: $$ P_{\\mathbf z}(\\mathbf z') = \\frac 12 (1 + \\vec z*\\vec z') = \\frac 12 (1 + \\cos\\measuredangle(\\vec z,\\vec z')) $$ This also means that states $\\mathbf z,\\mathbf z'$ are orthogonal in $\\mathbb C^2$ exactly when $\\measuredangle(\\vec z,\\vec z')=180°$ in $\\mathbb R^3$, ie $\\vec z'=-\\vec z$ The expection value of any observable $\\mathbf A$ can be expanded in terms of $P_{\\mathbf a_i}$ after a choice of orthonormal eigenvectors $\\\\{\\mathbf a_i\\\\}$, ie $$ \\langle\\mathbf z,\\mathbf{Az}\\rangle = \\sum_i α_iP_{\\mathbf a_i}(\\mathbf z) $$ where $\\\\{α_i\\\\}$ denote the eigenvalues. Substituting our definition of $P_{\\mathbf a_i}$ and using the fact that $\\mathbf a_1=-\\mathbf a_2$, this is simply $$ \\langle\\mathbf z,\\mathbf{Az}\\rangle = \\frac 12(α_1 + α_2) + \\frac 12(α_1-α_2)\\,\\vec a_1*\\vec z $$ Restricting ourselves to $\\mathfrak{su}(2)$, ie the traceless matrices with $α_2=-α_1$, yields $$ \\langle\\mathbf z,\\mathbf{Az}\\rangle = \\vec A*\\vec z $$ where $\\vec A=α_1\\,\\vec a_1$ is the representation of our observable. We arrived at a quantum system which can be described with nothing more than school math. After this lengthy exposition, finally my question: Are there any other quantum system with a similar real structure? If not, is there a particular reason why? ### Update To get a better understanding of the problem, I read up on complex manifolds, and it seems that the qubit is indeed special. In particular, the projective spaces $P_n\\mathbb C$ _are_ complex manifolds, but spheres $S^k$ do not even permit almost-complex structures for $k\\not= 2,6$ (Borel, Serre 1951) and are out as state spaces in the general case. The finite-dimensional complex projective spaces can be realized as different homogeneous spaces. In particular, we have the obvious $$P_n\\mathbb C \\cong \\mathbb C^{n+1}\\setminus\\\\{0\\\\}\\:\\big\/\\:\\mathbb C^*$$ the not-so-obvious $$P_n\\mathbb C \\cong U(n+1)\\:\\big\/\\:U(n)\\times U(1)$$ and the one which is probably the most clear-cut $$P_n\\mathbb C \\cong S^{2n+1}\\:\\big\/\\:S^1$$ but does not convey that we are dealing with complex spaces. This quotient also only ever results in a sphere for $n=1$, ie our qubit case. The Wikipedia page on Hopf fibration links to this paper, where the fibration $$S^7\\xrightarrow{S^3}S^4$$ is used to model the state space $P_3\\mathbb C$ of two qubits. Even though there _might_ be some insight into the structure of the state space in this fibration, it is less natural (and less useful, for that matter) than the single qubit Hopf fibration $$S^3\\xrightarrow{S^1}S^2$$ where actually $S^2 \\cong P_1\\mathbb C$, whereas obviously $S^4\\not\\cong P_3\\mathbb C$ as they do not even have the same dimensions when considered as real manifolds."}
{"id":"54599","title":"Factors that make beam divergence worse after refocusing","text":"A beam of light of width $W$ and wavelength $\\lambda$ with divergence that is diffraction-limited is refocused with an optical element placed at a distance $D$ from the beam source. Will the refocused beam have worse divergence than the one imposed by diffraction limits? how much worse? Is there an optical way to improve the beam divergence of the refocused beam? If a 2nd refocusing element, very similar to the 1st one, is placed on front of it, will the divergence get progressively worse? by how much? Finally, does it matter if the beam is a Gaussian or Fresnel beam?"}
{"id":"11986","title":"How can we get this formula for Gravitational potential?","text":"I just read the wikipedia page http:\/\/en.wikipedia.org\/wiki\/Gravitational_potential But I don't understand how to get this formula: $$\\rho(\\mathbf{x}) = \\frac{1}{4\\pi G}\\Delta V(\\mathbf{x})$$ Can anyone tell me why?"}
{"id":"20855","title":"Show that the energy levels of a particle in a specific potential are $E_n=(n+\\frac{1}{2})h\\omega-\\frac{1}{2}\\frac{F^2}{m\\omega^2}$","text":"A particle of mass m moves on the x-axis under the influence of the potential $$V(x)=\\frac{1}{2}m\\omega^2x^2+Fx$$ Can anyone help me, using Schrödinger's equation in one dimension that the energy levels are: $$E_n=(n+\\frac{1}{2})h\\omega-\\frac{1}{2}\\frac{F^2}{m\\omega^2}$$ Where n is a non negative integer?"}
{"id":"20363","title":"Statistical interpretation of Entropy","text":"I'm preparing my statistical physics course, and while writing the lecture notes it says that a system with non distinguishable particles has much less microstates asociated with a particular macrostate. Hence, a system with distinguishable particles has much more microstates asociated with a particular microstate. Entropy $S$ is related with the number of microstates $\\Omega$ via: $$S=k \\ln (\\Omega) $$ where $k$ is Boltzmann's constant. Then my question is **Given the same macrostate for both systems, the entropy of the first is much lower than the entropy of the second?** Thanks for your time."}
{"id":"44515","title":"Electric field and electric potential of a point charge in 2D and 1D","text":"in 3D, electric field of a piont charge is inversely proportional to the square of distance while the potential is inversely proportional to distance. We can derive it from Coulomb's law. however, I don't known how to derive the formula in 2D and 1D. I read in a book that electric potential of a point charge in 2D is proportional to the logarithm of distance. How to prove it?"}
{"id":"112909","title":"Is there an atomic level explanation for diffraction of light around opaque aperture?","text":"It has always been a hard time understanding the phenomenon of diffraction of light around opaque objects. Hoe does it happen? Why does transparent objects do not diffract light? What quantum mechanics operate that bends light? I used to have an intuitive understanding that surface atoms of the aperture edges reflect the light that interferes and forms the fringes."}
{"id":"126085","title":"Drake's equation","text":"Some years ago i read about this equation that helps us estimate the number of active, communicative extraterrestrial civilizations in the Milky Way galaxy. ![enter image description here](http:\/\/i.stack.imgur.com\/yqMHa.png) Now, i am wondering if there are results from Drake's equation (given that some parameters are extremely hard to be calculated) and if do these reflect the real world?"}
{"id":"123509","title":"Optical devices for micro meter photons?","text":"Are there all the common optical devices like beam splitters, mirrors, polarizers, fibers, etc for photons with a wavelenght in the range of micro meters? If so, how broad can the bandwidth be for the optical devices to work properly? Consider for example a photon with a mean frequency of $10^{12}\\,\\text{Hz}$ (i.e. a wavelength of $300\\,\\mu\\text{m}$). Will problems occur if the bandwidth of the photon is also in the range of $10^{11}\\,\\text{Hz}$ or $10^{12}\\,\\text{Hz}$?"}
{"id":"129307","title":"Neil deGrasse Tyson says that electrons \"teleport\" between energy levels?","text":"This page: https:\/\/blog.afach.de\/?p=62 Discusses the error Neil deGrasse Tyson made when talking about electronic transitions (video included there). Tyson clearly said in his Cosmos series that electrons disappear from one level and appear in the other one. The guy in the page discusses that this is wrong, because there is no proof of that, and because it breaks simplest rules of relativity, and because it contradicts the simplest quantum model. Is his argument correct? And why would Tyson do such a horrible mistake? For more information please visit the link."}
{"id":"59271","title":"Does the possibility of large scale entanglement mean 2-Body Problems are also unsolvable?","text":"Experiments are showing that larger and larger objects can be entangled whereby proving that this quantum feature has no upper limit. Assuming this is true, does entangled celestial bodies mean even the 2-Body and not just N-body problems are also unsolvable?"}
{"id":"59272","title":"How do you tell what forces do no work?","text":"> The total mass of the children and the toboggan is 66 kg. The force the > parent exerts is 58 N (18 degrees above the horizontal). What 3 forces\/ > components do no work on the toboggan? I said the vertical component of the force being applied, since the displacement and the force will be perpendicular. I also said gravity and the $x$ component of the force being applied since it will also be perpendicular. I don't know any other things that will make work $0$."}
{"id":"119394","title":"How can the Earth keep spinning with a liquid core?","text":"In regards to the 'conservation of angular momentum' being the explanation of why celestial objects spin... If you fill a ball or any other container with a liquid and try to spin it, you will not see any more than 5 or 6 revolutions because of the frictional losses of the liquid inside the container. I first discovered this during lunch in grade school when I tried to spin my milk carton by throwing it up into the air while giving it a 'spin'. At best, I could get only 3 rotations out of it. This same principle can be seen when you spin a raw egg and a hard boiled egg on a table top. The cooked 'solid' egg will spin the raw one won't. Because of my early experiences with milk cartons and eggs, somewhere in the intervening years I found it hard to believe some of the accepted 'facts' about our planet: So here is my question: **If the age of the earth 4.5 BILLION years, how can it be spinning freely in space with a liquid or semi liquid core for that length of time?** Combine the effects of the liquid core with the effects of liquid oceans and a gaseous atmosphere, all of which are creating resistance to rotation, these frictional losses would have stopped any rotation long ago. If the earth had a solid core, I could understand it... If the earth was less that 4.5 billion years old I might understand it... But given the accepted age of 4.5 billion years with a liquid core and a fluid outer shell I say there is a fly in the ointment somewhere!"}
{"id":"119393","title":"Local fermionic symmetry and GS action","text":"I have a trouble understanding an argument which I think has a simple answer but I am not getting it. The question is that if you don't impose local fermionic symmetry the GS action has only one term and one of the equation of motion looks as $Γ\\cdot p\\partial_0\\psi=0$, where $\\psi$ is a Majorana-Weyl spinor in 10 dimensions and $Γ$ are just generalized gamma matrices. Now $(Γ\\cdot p)^2 = p^2= 0$ for the on-shell condition, but now they say that since the square is zero hence half of the eigenvalues of $Γ\\cdot p$ is zero and this is where I am not getting it. If the square of a matrix is zero then how can we say that half of it's eigenvalues are zero. Since the eigenvalues of a squared matrix is square of the eigenvalues of the matrix whose square is taken, so how can we say that half of the eigenvalues are zero? May be it can happen when spinor contain complex component but here the spinor is Majorana too ( thinking in some different way that how it will impose constraint on half of the component of $\\psi$ but this is not working either). (For reference, see chapter 5 of GSW Vol.1.)"}
{"id":"66653","title":"Derivation of Universal Variables","text":"Does anyone know of link to the derivation of the Universal Variables? This is in regards to Kepler's equations. The Wikipedia link didn't have enough information for me, and the Colorado page doesn't explain how they came up with $S$ and $\\chi$. A universal variable is the universal anomaly in the universal Kepler equation. It is traditionally marked as chi. There is a derivation in Battin books `An Introduction to the Mathematics and Methods of Astrodynamics`."}
{"id":"110870","title":"Could a hard drive actually have been erased as described in Cryptonomicon?","text":"In chapter 80, _The Primary_ , Neal Stephenson has one of his character describe how the drives of a computer carried through a door (by the police, who where raiding the facility) will have been erased.   Basically, there's an enormous electromagnet around the door frame: > Cantrell is now drawing an elaborate diagram, and has even slowed down, > almost to a stop, the better to draw it. It begins with a tall rectangle. > Set within that is a parallelogram, the same size, but skewed a little bit > downwards, and with a little circle drawn in the middle of one edge. Randy > realizes he’s looking at a perspective view of a door-frame with its door > hanging slightly ajar, the little circle being its knob. _STEEL FRAME_ , > Cantrell writes, _hollow metal channels_. Quick meandering scribbles suggest > the matrix of wall surrounding it, and the floor underneath. Where the > uprights of the doorframe are planted in the floor, Cantrell draws small, > carefully foreshortened circles. _Holes in the floor_. Then he encircles the > doorframe in a continuous hoop, beginning at one of those circles and > climbing up one side of the doorframe, across the top, down the other side, > through the other hole in the floor, and then horizontally beneath the door, > then up through the first hole again, completing the loop. He draws one or > two careful iterations of this and then numerous sloppy ones until the whole > thing is surrounded in a vague, elongated tornado. _Many turns of fine > wire_. Finally he draws two leads away from this huge door-sized coil and > connects them to a sandwich of alternating long and short horizontal lines, > which Randy recognizes as the symbol for a battery. The diagram is completed > with a huge arrow drawn vigorously through the center of the doorway, like > an airborne battering ram, labeled B which means a magnetic field. _Ordo > computer room door_. > > \"Wow,\" Randy says. Cantrell has drawn a classic elementary-school > electromagnet, the kind of thing young Randy made by winding a wire around a > nail and hooking it up to a lantern battery. Except that this one is wound > around the outside of a doorframe and, Randy guesses, hidden inside the > walls and beneath the floor so that no one would know it was there unless > they tore the building apart. Magnetic fields are the styli of the modern > world, they are what writes bits onto disks, or wipes them away. The > read\/write heads of Tombstone’s hard drive are exactly the same thing, but a > lot smaller. If they are fine-pointed draftsman’s pens, then what Cantrell’s > drawn here is a firehose spraying India ink. It probably would have no > effect on a disk drive that was a few meters away from it, but anything that > was actually carried through that doorway would be wiped clean. Between the > pulse-gun fired into the building from outside (destroying every chip within > range) and this doorframe hack (losing every bit on every disk) the Ordo > raid must have been purely a scrap-hauling run for whoever organized it - > Andrew Loeb or (according to the Secret Admirers) Attorney General > Comstock’s sinister Fed forces who were using Andy as a cat’s paw. The only > thing that would have made it through that doorway intact would have been > information stored on CD-ROM or other nonmagnetic media, and Tombstone had > none of that. Would this actually work?"}
{"id":"67358","title":"Interpretation of theoretical Schrodinger's cat experiment","text":"I do appreciate how to interpret the famous theoretical experiment of Schrodinger's cat experiment in the following situation: > _A Lab has a closed box with a dead or live cat and and a person who have no > idea what has inside that box open it for the first time and nobody has > opened it before. How its conscious cood make a choice - dead or live cat - > if he even don't know that there is a cat inside it?_"}
{"id":"61384","title":"Proton scattering off electrons","text":"I'm looking at a question which starts off as: A proton is travelling through a material and scattering the electrons in the material. Express the scattering angle in terms of the impact parameter, $b$, ther reduced mass $\\mu$, the relative speed $v$, and the scattering angle in the ZMF. Hence show that the momentum transfer is $q=\\frac{2\\mu{v}}{\\sqrt{1+z^2}}$ where $z=b\\mu{v^2}\/\\alpha$. My attempt: Somethings are confusing me straight off, I'm not sure whether this is inelastic scattering or elastic scattering and I'm not sure whether $v$ is the relative speed in the LAB or ZMF frame. Also, assuming that $\\alpha$ is meant to be the fine structure constant, this means that $z$ has units which seems a bit of an issue to me... I've also assumed that this is all none relativistic. By looking at Rutherford scattering, we have that (in the electron rest\/LAB frame) the scattering angle $\\chi$ is given by $cot(\\frac{\\chi}{2})=\\frac{mv^2b}{\\alpha\\hbar{c}}$ I have assume that $v$ is the relative speed in the LAB frame (not sure how correct that is). From here I'm really not sure where I am meant to be going, I am pretty sure that this is pretty basic mechanics but I can't see how I should be going about this so any tips\/ advice would be much appreciated!"}
{"id":"61380","title":"Speed and transparency of light","text":"I have been puzzled with a fact that as an object moves faster, it ceases its property of opacity. I mean to say that as an object moves faster we can see right through it (more clearly than in a situation when his speed is low). Now with the fact that the speed of light has been controlled upto 61kmph in Bose- Einstein condensate, does it imply that light also becomes more and more opaque as its speed slower down?"}
{"id":"78951","title":"Magnetic field strength of a hydrogen electron","text":"Does the rotation of the electron around the hydrogen nucleus cause a resultant magnetic field in the atom? If so, the current density must be very high and the resulting field should affect the stability of the atom. How is this field counteracted so that the atom remains stable? Thanks"}
{"id":"78953","title":"Elastic collision of photon","text":"Consider an elastic collision of a photon with 100 eV energy hitting a mirror. How much momentum is exchanged in the collision? Also, can one model the reflection of an elastically colliding photon with a single electron in the mirror? It would seem that the photon would be absorbed by the electron, so that the collision is necessarily inelastic."}
{"id":"8340","title":"How do horseshoe orbits work?","text":"An asteroid was recently discovered that is in a horseshoe orbit with respect to the earth. Is there an intuitive explanation for these orbits? It seems that the earth acts as a repulsive force where one would expect gravity to be attractive. From the above linked article: Accepted in MNRAS, Apostolos A. Christou, David J. Asher, 1104.0036, _A long- lived horseshoe companion to the Earth_ ![enter image description here](http:\/\/i.stack.imgur.com\/lesXN.png) A better description of the horseshoe orbits, along with the tadpole orbits and Lagrange points:   ![enter image description here](http:\/\/i.stack.imgur.com\/FrWhl.jpg)"}
{"id":"12417","title":"What happens to light in a perfect reflective sphere?","text":"Let's say you have the ability to shine some light into a perfectly round sphere and the sphere's interior surface was perfectly smooth and reflective and there was no way for the light to escape. If you could observe the inside of the sphere, what would you observe? A glow? And would temperature affect the outcome? Seems silly, it's just something I've always thought about but never spent enough time (until now) to actually find an answer."}
{"id":"86677","title":"How can there be a Current and an Electric field in an idealized wire with no voltage drop?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/GEnoN.gif) In an ideal circuit, How can there be a current b\/w points a & b, when there is no potential difference and thus no electric field between a & b? If there is no current, then where does current across the resistor come from, because that means no charge is coming from the battery (?)."}
{"id":"127062","title":"How are the distances to the most distant $\\gamma$-ray bursts measured?","text":"This question came to my mind when I read on NASA's website that an explosion (gamma ray burst, GRB 080913) took place 12.8 billion light years away from us. How do they measure such large distances?"}
{"id":"127066","title":"If there were fundamental forces weaker than gravity, would we know about it?","text":"We know that gravity is a very weak force compared to electromagnetic forces and the nuclear forces. We know about the other forces because they're necessary to explain atoms, and we can detect gravity easily, because unlike the other forces it is always attractive, so the weak gravitational force from every particle in a planet can add up to a measurable effect. However, is it possible that there are other fundamental forces that are much weaker than gravity, or even of a similar magnitude but with \"charges\" that attract their opposite and therefore cancel out over large scales? It seems that we wouldn't necessarily have detected such a force if it did exist, and it seems different from the kind of thing you can probe using a particle accelerator. I realise this question is kind of naïve. I know a little quantum mechanics but never studied quantum field theory or particle physics. I'm curious about whether those formalisms provide a way to rule out such additional weak forces."}
{"id":"129956","title":"Why does this model fall apart when angular velocity is small?","text":"I'm doing a physics problem in which a marble spins around a spinning bowl and both have angular velocity $\\omega$. It rotates with radius $r$ around the central axis and the hemispherical bowl has radius $R$. I've solved for the radius $r$ in terms of $\\omega$: $$r=\\sqrt{R^2-\\frac{g^2}{\\omega^4}}$$ But I can't figure out what this means when $\\omega$ is really small ($<\\sqrt{\\frac{g}{R}}$ ). Some hypotheses:   * Something to do with friction   * It falls through the center of the bowl (if it were hollow)   * We're missing something in this model Are any of these right? Is this more complex than it seems?"}
{"id":"3635","title":"Radiation from a pair of charged objects orbiting each other","text":"This question on binary black hole solutions, led to me think about the similar question from the perspective of what we know about the Hydrogen atom. Prior to quantum mechanics, it was not understood what led to the stability of the Hydrogen atom against collapse of the electron orbits due to Bremsstrahlung, i.e. the emission of radiation due to the fact that it is in a non-inertial (accelerated) frame of reference. Bohr and Sommerfeld came up with a somewhat _ad-hoc_ procedure \\- the first ever instance of quantization - according to which in the quantum theory only those classical orbits are allowed the value of whose action is quantized in units of $\\hbar$. $$ \\int_i p dq = 2 \\pi n_i $$ where the integral is over the $i^{th}$ (closed) classical orbit. Now what I'm thinking of next has probably been thought of before but I haven't done a literature review to find out. Classically, we expect the accelerating electron to radiate resulting in the catastrophic collapse of its orbit. However, in a _complete_ description we must also take the proton into consideration. It is also a charged object and as is well known from the two-body, inverse square law, central force problem (see eg. Goldstein) both the proton and electron orbit each other. Therefore the proton, being a charged object, must also radiate if we don't ignore its (accelerated) motion around the electron. An observer sitting at a distance $d \\gg r$, where $r$ is the mean size of the two-body system, will measure radiation which is a superposition of that coming from both the electron _and_ the proton. The question is this: **a).** What is the phase difference between the two contributions ($\\mathbf{E}_e$ and $\\mathbf{E}_p$) to the net electric field $\\mathbf{E}$ as seen by this observer?, **b).** What is the value of the total energy $E$ emitted by the electron-proton system, given by the integral of the Poynting vector across a closed surface enclosing the system, as seen by this observer? [ My motivation is to see if we can learn more about the Bohr-Sommerfeld quantization condition by considering the classical electrodynamics of the **full** electron-proton system. The quantity $E$ will depend on the size and shape of the classical orbits of the two charged objects or more simply of their mean separation $E:=E(r)$. As we vary $r$ from $0$ to some value $r_{max} \\ll d$ we would expect $E(r)$ to oscillate and have local minima for some classical orbits. If these classical minima occur for orbits which satisfy the Bohr-Sommerfeld condition then we would have established a connection between the full classical problem and its quantization. ]"}
{"id":"99920","title":"Does $\\langle\\Omega|\\mathrm{e}^{\\mathrm{i}Px}=\\langle\\Omega|\\mathrm{e}^{\\mathrm{i}0x}$? $(\\langle\\Omega| =$ ground state of the interacting theory)","text":"Let $\\langle\\Omega|$ be the ground state of an interacting theory, just as Peskin & Schroeder(PS) describes on page 82 and page 213. On page 213 PS do the following $$\\tag{1}\\langle\\Omega|\\phi(x)|\\lambda_{\\vec{p}}\\rangle = \\langle\\Omega|\\exp(+\\mathrm{i}Px)\\phi(0)\\exp(-\\mathrm{i}Px)|\\lambda_{\\vec{p}}\\rangle \\\\\\ =\\langle\\Omega|\\phi(0)|\\lambda_{0}\\rangle\\exp(-\\mathrm{i}px)\\bigg|_{p^0=E_{\\vec{p}}} $$ In the above $|\\lambda_0\\rangle$ is an eigenstate of the full interacting hamiltonian $H$. The ket $|\\lambda_{\\vec{p}}\\rangle$ is the boost of $|\\lambda_0\\rangle$ with momentum $\\vec{p}.$ Furthermore $E_{\\vec{p}} = \\sqrt{p^2+m_\\lambda^2}$ and $\\vec{P}|\\lambda_0\\rangle = 0.$ I assume PS used $$\\tag{2}\\langle\\Omega|\\mathrm{e}^{\\mathrm{i}Px}=\\langle\\Omega|\\mathrm{e}^{\\mathrm{i}0x}$$ in $(1)$. Is $(2)$ true? I thought $\\langle0|\\mathrm{e}^{\\mathrm{i}Px}=\\langle0|\\mathrm{e}^{\\mathrm{i}0x}$ where $\\langle0|$ is the ground state of the free theory. I hope I have provided enough info for the question to make sense."}
{"id":"99923","title":"Do we need inertial frames in Lagrangian mechanics?","text":"Do Euler-Lagrange equations hold only for inertial systems? If yes, where is the point in the variational derivation from Hamilton's principle where we made that restriction? My question arose because I was thinking how could you describe the motion of pendulum with an oscillating suspension point, something like this: ![enter image description here](http:\/\/i.stack.imgur.com\/g5LT3m.gif) If you take your coordinate frame fixed in the suspension point, you would get the motion of a simple pendulum, which is false (because it's a non-inertial frame and you should have virtual forces). But if it were moving at constant speed, a moving coordinate frame should give the same equations of motion. **So where does the restriction to non-inertial frames come from in the Lagrangian formalism?** I've read that in non-inertial frames the lagrangian would be $L(q,\\dot q, \\ddot q, \\dddot q..., t)$, but I didn't see a detailed explanation. And I'm not sure if the situation would be similar to Kepler's problem in polar coordinates."}
{"id":"48271","title":"Is there a limit on how hot you can superheat water at 1 atm?","text":"Is there a limit on how hot you can superheat water at 1 atm in a perfect container? What about in a microwave?"}
{"id":"106366","title":"Deriving Rabi rotation matrix","text":"I want to understand where the matrix: $$ \\left|\\psi(t)\\right> = \\binom{a(t)}{b(t)} = \\begin{bmatrix} cos(\\Omega t\/2)&-ie^{i\\phi_L t}sin(\\Omega t\/2) \\\\\\ -ie^{-i\\phi_L t}sin(\\Omega t\/2) & cos(\\Omega t\/2) \\end{bmatrix}\\binom{a(0)}{b(0)}$$ comes from. I've already derived $$\\dot{a} = \\frac{\\Omega}{2}e^{i\\phi_L t} b$$ and its counterpart $$\\dot{b} = \\frac{\\Omega}{2}e^{-i\\phi_L t} a$$ where $\\Omega = \\frac{1}{\\hbar} \\left<1\\right|-d\\cdot\\varepsilon\\left|0\\right>$ and, I think, $\\phi_L$ is the frequency of the incident radiation. But how can I get from the above two relations to the matrix?"}
{"id":"48274","title":"Is the inducing signal weakened to the extent it induces a signal in the receiving antenna?","text":"Silly question ... An antenna is necessary to receive a transmitted signal from atmosphere, or otherwise. The transmitted signal weakens as it travels through the atmosphere. When the signal encounters an antenna (subject to antenna gain) a corresponding signal is induced in the antenna. Is the inducing signal weakened to the extent it induces a signal in the receiving antenna?"}
{"id":"102875","title":"Mirror Transportation","text":"Why do people transport mirrors in inclined position? I frequently find people carrying mirrors in slant position on mini trucks.I guess it has some advantage of carrying them without breaking,and I am now so curious to know the reason behind it ![Either sides of the wooden piece are aligned by mirrors](http:\/\/i.stack.imgur.com\/J1S63.jpg)"}
{"id":"113037","title":"Which experiment gave scientists reason to believe nuclear fission\/fusion produced energy?","text":"Every piece of knowledge in science has a beginning lying in someone's experiment. I would like to know which experiment gave scientists the reason to believe nuclear fission\/fusion existed and was instrumental in the development of the field of nuclear energy. I would also accept a thought experiment as an acceptable answer, as long as it answers the question."}
{"id":"24873","title":"What is the most distant object from the Earth that a spacecraft has visited to date?","text":"What is the most distant object from the earth that a spacecraft visited has visited so far? What was the mission and when did it happen?"}
{"id":"24876","title":"For observing planets what kind of telescope offers the best view?","text":"I've heard that using a refractor is better than a reflector when it comes to planets to best reproduce their colors. But perhaps other factors can weight in too? For example, do you want a slow or fast f-stop to get the best view of a planet? What about eye-pieces, what style of eyepiece construction will give the best views?"}
{"id":"134828","title":"Radioactive decay - What mechanism decides when an unstable nucleus decays?","text":"My first question on Stackexchange (if it is formatted wrong or something please tell me so I know in future) - here it is: **Given an unstable nucleus (exactly which nucleus is not particularly pertinent) - what decides _precisely_ when it will decay?** I am somewhat familiar with the concept- a nucleus becomes 'unstable' when the mutual repulsion between the protons exceeds the binding force of the strong nuclear force. That being said, if we were to theoretically isolate a single unstable atom, there are a few possibilities (correct me if I am wrong):   1. atom immediately decays   2. atom decays x seconds later   3. atom doesn't decay There is an unpredictable nature (which I assume arises from something to do with quantum uncertainty). But getting back to my question, what is it that suddenly makes an 'unstable nucleus' decay? Any further reading would be appreciated, although nothing too complicated (university level is my limit I think). UPDATE Thanks all, my first time using this forum and I was not let down :) With regards to the question itself, I'm a bit disappointed that there is no precise mechanism, but I guess Einstein was wrong when he quoted: \"God does not play dice\""}
{"id":"134820","title":"When to pull out a negative sign from a variable","text":"I get confused about when there should be a negative sign in certain equations or not. I will give three short examples (that I will make long with explanation) that show my confusion. Example 1: Rockets. Consider a rocket of mass $m$ traveling in the positive x direction with velocity $v$. The ejected fuel has a velocity relative to the rocket $v_{ex}$. At time $t$, the rocket's momentum is $P(t) = mv$. In a short time later, $t + dt$, the rocket's mass is $m + dm$ (where $dm$) is negative and momentum $P(t + dt) = (m + dm)(v + dv)$. The fuel ejected in a time $dt$ has mass (-$dm$) and velocity $v - v_{ex}$ relative to ground. The total momentum at time $t + dt$ is $$P(t+dt) = (m+dm)(v+dv) - dm(v-v_{ex})$$ $$ = mv + mdv + dmv_{ex}$$ 1a) Why is the negative contained in the $dm$ of the rocket mass while the negative is pulled out of the $dm$ for the fuel? To reconcile this, I thought of something like, if ejecting mass is a positive, then use positive $dm$ as they did in the text from which I'm reading out of. So ejecting means positive, while the fuel gains mass in a way. So I can define gaining mass with a $-dm$ as they did. However, if I was doing this problem, I would have made ejecting mass negative and gaining mass positive. But if I change the sign on all of the $dm$'s I don't get the same result in the text. Example 2: I'm not reading this one out of a text but I can explain the question. Often times you'll see something like $$\\frac{d(Q)}{dx} = ax$$ where $Q$ is some function. The book or whatever will then say, but as $x$ increases $Q$ decreases so the expression should really be $$\\frac{d(Q)}{dx} = - ax$$ 2a) Why can't the negative sign be contained in the constant of proportionality? Does it change the result? All it does by putting a minus sign there is force $a$ to be positive. Example 3: Projectile motion with quadratic drag. The equation of motion for this is given by $$m\\ddot{\\vec{r}} = m\\vec{g} - cv^2\\hat{v} = m\\vec{g} - cv\\vec{v}$$ where $v$ is the magnitude of the velocity of the projectile. If we only look in the vertical direction, with y measured vertically upwards, my text gives $$m\\ddot{v_y} = -mg - c\\sqrt{v^2_x + v^2_y}v_y$$ 3a) So $\\vec{g} = -g\\hat y$ and they pulled negative out of the $g$. So it makes sense that its $-mg$ since y is pointed upwards. Why isn't it $+ c\\sqrt{...}$? Is the reason because $v_y$ is negative, so a negative times a negative is a positive? If so, why did the negative come out of the $g$ while the negative was kept inside the $v_y$?"}
{"id":"102292","title":"Electron and photon interaction","text":"Assuming I shoot a beam of light to an electron, the electron will take the energy and reemit EMR at all direction randomly? If it will, is it happening at the same time?"}
{"id":"113690","title":"Superluminal travel without time travel","text":"Can superluminal travel or communication be possible without leading to the possibility of closed timelike curves (CTCs) and causality violations? I have seen conflicting opinions regarding this issue, and whether all possible methods of superluminal travel (e.g. wormholes, Kraskinov tubes, and warp drives like the Alcubierre drive) necessarily open the door to causality violations. I understand that wormholes may lead to CTCs in certain conditions (i.e. through time dilation or Roman rings, as well as Kraskinov tubes). My question is basically this: Are there possible (hypothetical) methods of superluminal travel that do not open Pandora's box of causality violation?"}
{"id":"113693","title":"Can an electromagnet repel a strong magnet?","text":"When the direction of current is reversed, the polarity of an electromagnet should be reversed. I am trying to understand why this did not happen in an experiment I conducted today. An electromagnet was connected to a 12v battery and I verified that it was working. I then touched the electromagnet to a strong regular magnet (rated to hold over 50 kg). There was a strong force between the magnets. I then disconnected the electromagnet from power, reversed the wires, and turned it back on. I expected to feel the electromagnet repel the other magnet, but instead they two magnets got stuck together again. It seems like reversing the wires had no affect on the polarity of the electromagnet. Here is a picture of the electromagnet I used for reference: ![enter image description here](http:\/\/i.stack.imgur.com\/9mea3.jpg) And here is the strong regular magnet."}
{"id":"20047","title":"Is the stability of the nucleus due to pions or gluons?","text":"> **Possible Duplicate:**   >  Is it pions or gluons that mediate the strong force between nucleons? (I'm currently studying particle physics in my physics classes at high school so simple language would be appreciated.) We have learnt that the proton is the only stable hadron and the neutron has a mean lifetime of approximately 20 minutes. This leads to the question of how nuclei are stable. As I currently understand it, pions are exchanged between the neutrons and protons which turn the proton into a neutron and the neutron into a proton. My question is how gluons are involved in this process and what simplifications have been made in my understanding? My initial guess is something to do with quark color."}
{"id":"99073","title":"Optics Brewster's Angle Reflection Intensity","text":"An incident unpolarised light beam of intensity $I_{0}$ strikes glass plate B at Brewster's Angle. The reflected light travels vertically and strikes a second glass plate A, again at Brewster's Angle. (We ignore the light transmitted by the glass plates.) Plate A is then rotated about the z-axis as shown. Briefly explain how the intensity of the light reflected by the apparatus varies with the angle of plate A. Illustrate with a qualitative sketch of intensity vs. angle of plate A. ![Diagram](http:\/\/i.imgur.com\/2AZstU1.png?1) We have an unpolarised beam incident on plate B at Brewster's Angle which upon reflection should become s-polarised or polarised perpendicular to the incident plane. My understanding of Brewster's Angle then says that upon the second reflection at plate A if the plate is oriented such that the incident plane is the same as for plate B there will be no reflection. Otherwise as it rotates there will be light reflected, is this correct? Also does the intensity of the light reflected off of A increase gradually to a maximum when the beam is parallel to the incident plane at A? Any help would be much appreciated."}
{"id":"112348","title":"Domain walls intersection","text":"I was reading this article( _On domain shapes and processes in supersymmetric theories_ ). In the paragraph about domain walls intersection (paragraph $4$, page $7$) the authors say: > In a one-field theory it is known that an intersection of domain walls is > unstable They explain it as follows: > Indeed the configuration shown in Fig. 2a has translational zero modes, > whose shape is given by the gradient of the field. Since there are > directions in the plane, where the field approaches the same values at both > innities, the component of the gradient in such direction necessarily has > zero. Thus the zero mode cannot be the lowest in the spectrum and a negative > mode exists, leading to a separation of the walls. So, i have the theory of one scalar field in $2+1$ with mexican hat potential: $V=\\frac{\\lambda}{4}\\left( \\phi^{2}-v^{2}\\right)^{2} $. I consider configuration (domain walls intersection) with boundary conditions: $$ \\phi_{d}(+\\infty, +\\infty)=-v $$ $$ \\phi_{d}(+\\infty, -\\infty)=v $$ $$ \\phi_{d}(-\\infty, +\\infty)=v $$ $$ \\phi_{d}(-\\infty, -\\infty)=-v $$ Here $\\phi_{d}$ satisfies the stationary equation of motion (recall that $\\dot{\\phi}=0$): $$ \\Delta \\phi_{d}-\\frac{\\partial V}{\\partial \\phi}(\\phi_{d})=0. $$ I consider the small excitation over $\\phi_d$: $\\tilde{\\phi}=\\phi_{d}+\\phi$. After linearizing, I obtain the following equation ($\\phi=e^{i\\omega t}f_{\\omega}(x_1,x_2)$): $$\\left[-\\Delta + \\frac{\\partial^{2}V}{\\partial \\phi^{2}}(\\phi_{d})\\right]f=\\omega^{2}f \\tag{1}$$. I have found **zero modes** : its directional derivative : $f_{0}(x_{1},x_{2})=(\\nabla\\phi, \\mathbf{n})$. It satisfies the equation $(1)$ with $\\omega=0$. Now I can consider the directions along \"diagonals\", at their ends the field takes the same value $v$ or $-v$ respectively (see figure 2 at page 10 in the file). So at this diagonals there is a point where $f_{0}=0$, as authors said. But I cant understand the last sentence of their explanation: **How the existance of such point (where $f_{0}=0$) implies that we also have negative modes?** Can you explain me this? Thanks in advance. _PS. I will accept another approaches to show the existance of negative modes (and therefore instability of such configuration) too._"}
{"id":"112343","title":"$E=mc^2$ resembles kinetic energy formula?","text":"The simplest equation expressing mass–energy equivalence is the famous $E=mc^2$ where $c$ represents the speed of light. Compare this with $E_K = \\frac{1}{2}mv^2$. Since $E=mc^2$ can be applied to rest mass ($m_0$) and rest energy ($E_0$) to show their proportionality as $E_0=m_0c^2$, I ask whether this resemblance is just a coincidence created by the need of any equation to be homogeneous for units, or are the two equations fundamentally related? I know about Mass-Energy Equivalency but I could not find the answer I'm looking for there."}
{"id":"102120","title":"Equation of motion for a falling rod (with one end touching a frictionless surface)","text":"I have a quick question about the equation of motion for a falling rod (with one end touching a frictionless surface). The end touching the surface is not fixed. I am given the moment of inertia about the center of mass. Only gravity is acting on the rod. The rod begins at an angle $\\theta$ above the ground. I know that the rod will rotate about the center of mass, and the point touching the frictionless surface will slide along the surface, but I am having trouble calculating the torque. For reference, the left end of the rod is touching the surface, and the right end is in air. I have calculated the torque, $\\tau$, from the right moment arm as $\\tau=\\frac{mg\\cos{\\theta}}{4I_G}$ because the right half of the rod contains half of the mass and half of the length. I don't really understand how to calculate the total net torque, though. Any hints would be appreciated."}
{"id":"102123","title":"Huggins Displacement Theory and Retrocausality","text":"I was looking at the Wikipedia entries on Time Travel and the Grandfather paradox and noticed a paragraph on the so-called Huggins Displacement Theory. I haven't been able to find the source although I assume it's originator was the Victorian astronomer Sir William Huggins. Does anyone know of the precise reference? Normally sending information into one's past can lead to contradictions. For example based on some condition now I could send a message into my past that prevents that very condition from occurring in my present. Thus sending information into the past, and therefore arbitrarily changing it, seems to be forbidden physically. However the Huggins Displacement Theory says that there is a loop-hole in this argument (or more accurately a \"corner case\"!) I could send information along my past lightcone so that if I influence something a year in my past then it must be a light-year away from me. Thus no change in the past has a enough time to travel back to me to cause a contradiction in my present. For example imagine that I see a stock price going up today. The Huggins Theory says that I can beam knowledge of this stock price rise back in time provided that the message is also displaced by a comparable distance. Thus if I send it back a year in time then it must also travel away from me by a light-year in distance. If my twin gets the message then he might try to profit from the information by sending back a message to me to buy the stock. Unfortunately the message will only get back to me at the instant that the price rises so that neither myself nor my twin will be able to profit from this situation. Maybe physics does allow information to be sent back in time along a past lightcone so that this effect could be measured. My hunch is that perhaps if one charged up an electrode then electrostatic influences from it might travel back along its lightcone to cause measureable repulsion or attraction on distant electrodes. I don't think electromagnetic radiation itself could go backwards in time as in principle it could be reflected back onto an observer's past worldline and therefore trigger contradictions in his\/her present."}
{"id":"114220","title":"How would one compute the angle of deflection, in a relativistic collision - underspecified system?","text":"Consider the simplistic case of two identical mass particles colliding elastically with the second particle initially stationary and the first particle travelling with energy $E$. By conservation of 4-momentum we have: $$p_{1}^{\\mu}+p_{2}^{\\mu}=p_{1}'^{\\mu}+p_{2}'^{\\mu}$$ Taking the inner product of this with itself: $$\\left\\langle p_{1}^{\\mu} \\middle| p_{1}^{\\mu}\\right\\rangle + 2\\left\\langle p_{1}^{\\mu} \\middle| p_{2}^{\\mu} \\right\\rangle+\\left\\langle p_{2}^{\\mu} \\middle| p_{2}^{\\mu} \\right\\rangle = \\left\\langle p_{1}'^{\\mu} \\middle| p_{1}'^{\\mu}\\right\\rangle + 2\\left\\langle p_{1}'^{\\mu} \\middle| p_{2}'^{\\mu}\\right\\rangle +\\left\\langle p_{2}'^{\\mu} \\middle| p_{2}'^{\\mu}\\right\\rangle $$ Using the fact that $\\left\\langle p_{i}^{\\mu} \\middle| p_{i}^{\\mu}\\right\\rangle = m_{0}^{2}c^{2}$, we can simplify this: $$2m_{0}c^{2}+2m_{0}E= 2m_{0}c^{2}+2\\left(\\frac{E_{1}'E_{2}'}{c^{2}}-\\vec{p}_{1}'\\cdot\\vec{p}_{2}'\\right) \\implies m_{0}E=\\frac{E_{1}'E_{2}'}{c^{2}}-\\vec{p}_{1}'\\cdot\\vec{p}_{2}'$$ We note that $\\vec{p}_{1}'\\cdot\\vec{p}_{2}'=\\|\\vec{p}_{1}'\\|\\|\\vec{p}_{2}'\\|\\cos(\\theta)$, where $\\theta$ is the inner angle between $\\vec{p}_{1}'$ and $\\vec{p}_{2}'$. However, this results in a system with $\\|p_{1}'\\|$,$\\|p_{2}'\\|$ and $E_{1,2}$ unspecified and I cannot see how we could thus extract $\\theta$ from the initial conditions; what have I misunderstood or misapplied here?"}
{"id":"107185","title":"Are gravitons bound by the event horizon?","text":"I understand that photons, even when traveling at the speed of light, cannot escape the event horizon of a black hole. Are gravitons and other virtual particles traveling at the speed of light also confined by event horizons? If so, it seems that the gravitational field created by the black hole would result only from the mass of the black hole beyond the event horizon, where gravitons are capable of escaping. As a result, would there would be a disparity between the apparent mass of the black hole due to its gravitational field on other celestial bodies and the total amount of matter contained within the black hole? Also, I was reading this question: Nature of gravity: gravitons, curvature of space-time or both?, which suggests that gravitons and curved space may be indistinguishable. However, if gravitons are bound by the event horizon it seems that a black hole would act differently based on whether gravity results from gravitons or curved space-time. The existence of bound gravitons would negate the gravitational field of mass within the event horizon, resulting in a significantly lower gravitational field outside of the black hole. Would this occur, or am I neglecting some effect of relativity upon the gravitational field?"}
{"id":"107186","title":"Will an anti-neutron annihilate a regular neutron even though they have no charge","text":"Since neutrons have no electric charge will an anti-neutron annihilate when it comes into contact with a regular neutron as protons and anti-protons do? if so what causes it to annihilate?"}
{"id":"78186","title":"Locality in the scattering amplitude","text":"Early in this talk by Nima Arkani-Hamed, he describes what locality means for scattering amplitudes. \"Locality tells you that the only poles in the scattering amplitude occur when the sum of a subset of the momenta of the particles goes on shell.\" The talk then goes on to describe his recent research, but right now I'm not worrying about understanding that. I'm just trying to understand the statement quoted above. I know what locality looks like in the Lagrangian density (the terms of the Lagrangian density contain field operators which are all functions of the same spacetime coordinate), but it's not at all obvious to me how that is equivalent to the above statement about scattering amplitudes. Can you explain it (preferably at about the level of a first graduate course in QFT)?"}
{"id":"78185","title":"S-matrix and it's exponential form","text":"By using Dyson series for the representation of the $S$-matrix, it's expression can be written in a form $$ \\hat {S}(\\infty , -\\infty) = \\sum_{n = 0}^{\\infty}\\frac{(-i)^{n}}{n!}\\int \\limits_{-\\infty}^{\\infty}d^{4}x_{1}...\\int \\limits_{-\\infty}^{\\infty}d^{4}x_{n}\\hat {P}(\\hat {\\Gamma }_{i}(x_{1}))...\\hat {\\Gamma}_{i}(x_{n})), \\qquad (.1) $$ where $$ \\hat {N}(\\hat {H}_{k}(t_{i}), \\hat {H}_{k}(t_{j})) = \\left\\\\{ \\begin{matrix} \\hat {H}_{k}(t_{i}) \\hat {H}_{k}(t_{j}), t_{i} > t_{j} \\\\\\ \\hat {H}_{k}(t_{j}) \\hat {H}_{k}(t_{i}), t_{j} > t_{i} \\end{matrix}\\right. . $$ How to rewrite $(.1)$ into exponential form?"}
{"id":"133889","title":"How is voltage increased from a battery?","text":"I have recently watched a video about a powerful Octa-copter that uses 30,000 Volt brushless motors to propell, the batteries however look quite small about a 19v laptop battery. How is the voltage increased to suffice the voltage requirements? where does the extra electrical energy come from or what is the tradeoff for stepping up the voltage? and if the batteries are at most about 19V how does it obey ohms law?"}
{"id":"35983","title":"Anybody have example of two-qubit non-Pauli and non-Clifford quantum gate?","text":"A lot of known quantum gates are in the Pauli group (I,X,Z,Y) or in the Clifford group (H,P,Cnot). I need examples of the quantum gates that aren't in this groups. Also, are there are matlab functions to check if a quantum gate (2x2 or 4x4) is in any of these groups? Or maybe there are matlab routines that generate quantum gates outside of this groups? Thanks.."}
{"id":"104982","title":"Speed of light that is traveling away from the observer","text":"The second postulate of Special Relativity states: > The speed of light in a vacuum is the same for all observers, regardless of > their motion relative to the source. Now imagine the observer that is looking at the light which is traveling away from him in a direction he is looking. In this case, the light should look somewhat like a \"dot\" to such observer - not really moving at any direction for him. However according to postulate above the light should move at the speed of c (if we are in vacuum to be precise). What is wrong with this reasoning? Or is it just some kind of exception from the rule?"}
{"id":"3394","title":"What would the electromagnetic field of a massless electron look like","text":"The Standard Model gives non-zero mass to the electron via the coupling to the Higgs field. Issues of renormalizability aside, this is fundamentally unrelated to the fact that the electron couples to the EM field. However, if the Higgs mechanism did not operate - that is, if there were no vacuum symmetry breaking, the electron field would have no effective mass term. In QFT perturbation theory, this model offers no special difficulty. My question is, what is the classical limit of this theory, if it has one? Does the electron acquire a purely EM mass? If the low energy renormalized mass is set to zero (is there an obstacle to doing that?) what do the classical field configurations look like? My puzzlement is related to the classical description of the EM field of a relativistic charged particle - namely, that it becomes \"flattened\" in the direction of motion, just like a Lorentz-contracted sphere. That is, the field is weaker than for a motionless particle in the longitudinal direction, and stronger in the transverse directions. The limiting case is that the field becomes concentrated on a 2-d surface transverse to the particle location, where it has infinite strength - obviously an unphysical situation. So what actually happens?"}
{"id":"68541","title":"Heat equation with Neumann boundary conditions","text":"I would like to explore some ideas with different geometries exchanging energy mainly by radiation. This means solving Laplace equation for the steady state. However the boundary conditions are always Neumann's because the only constraints are fluxes. After many questions and attempts, I realized that Mathematica cannot (yet ?) treat PDEs with Neumann boundary conditions. To my great surprise I have also found no literature describing the treatment of the heat equation with the derivative at boundary proportional to $T^4$ even if I thought that similar problems would play a paramount role in engineering. What are some good books\/papers to treat the Heat equation with Neumann boundary conditions and more specifically when the flux at boundary is proportional to $T^4$ ?"}
{"id":"68546","title":"Equations of motion for bob-on-a-string -- am I missing some terms?","text":"The dynamics of a type of physical system I am currently working on are modeled in most of the literature by replacing the moving parts of that system with an equivalent set of pendulums. Parameters and initial conditions for this approximate pendulum model are chosen such that the total mass and the center of mass of these pendulums describe those of the real system as accurately as possible. For one particular system however, I need to take this one step further because that system is losing mass _rapidly_. (Also, it is spinning about an axis that lies outside the system, but let's leave that out of consideration here). Therefore, I ended up modeling this particular system with a series of \"specialized\" pendulums:   * the pendulum masses are free to move in any direction (i.e., not constrained to move in any single plane)    * the point masses are attached to the pendulum hinge points by means of massless strings of infinite strength but finite elasticity (spring constant $k$)   * The pendulum mass $m$, the height of the hinge point $h$, the restoring force $m\\mathbf{g}$ and the nominal string length $L$ are all functions of time. ![enter image description here](http:\/\/i.stack.imgur.com\/RJv7C.png) In this figure, $h(t) = \\left|\\mathbf{h}(t)\\right|$ and $\\left|\\mathbf{s}(t)\\right| = L(t)$ when the string is _just_ stretched. $\\dot{\\mathbf{h}}$ and $\\ddot{\\mathbf{h}}$ are constrained to only the pure vertical direction. Leaving the fictitious forces out of consideration, I have come up with the following equation of motion for this kind of pendulum: $$ m\\ddot{\\mathbf{r}} = -m\\mathbf{g} + \\begin{cases} 0 & \\text{if } |\\mathbf{s}| < L\\\\\\ m\\left(\\frac{\\mathbf{s}\\times\\mathbf{\\dot{r}}}{|\\mathbf{s}|^2}\\times\\mathbf{\\dot{r}}\\right) - \\gamma_1\\left(\\frac{\\mathbf{s}\\times\\mathbf{\\dot{r}}}{|\\mathbf{s}|^2}\\times\\mathbf{r}\\right) & \\\\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ - k\\left(|\\mathbf{s}|-L\\right)\\frac{\\mathbf{s}}{|\\mathbf{s}|} - \\gamma_2\\left(\\frac{\\mathbf{s}}{|\\mathbf{s}|}\\cdot\\mathbf{\\dot{r}} + \\dot{L}\\right)\\frac{\\mathbf{s}}{|\\mathbf{s}|}& \\text{if } |\\mathbf{s}| \\geq L \\end{cases} $$ where $\\gamma_1$ is the torsional damping ratio, $k$ the string constant of the (flexible) string and $\\gamma_2$ its linear damping ratio. The first two terms of the second case lie in the current plane of rotation of the mass $m$ about the hinge point. They describe the centripetal force towards the hinge point, and torsional damping, respectively. The last two terms are in the radial direction, and describe the stretching and damping of the stretched string, respectively. Numerical results of this equation are encouraging. However, this is all a bit messy (especially with the terms for the ficticious forces included), so I'd really appreciate if someone else took a look at it. Why I ask this is because I can't figure out why the $\\dot{\\mathrm{h}}$, $\\ddot{\\mathrm{h}}$, $\\ddot{L}$ and $\\dot{m}$ do not show up in this equation... Am I missing any terms anywhere?"}
{"id":"68544","title":"Integral related to particle diffusion","text":"In the context of particle diffusion, I am trying to understand the equations that describe Brownian motion as a macroscopic process. Assume $N(x,t)$ is a number concentration and $D$ is a diffusion constant, then we can write: $$\\frac{\\partial N(x,t)}{\\partial t} = D \\frac{\\partial^2}{\\partial x^2} N(x,t)$$ next, we can multiply by the mean square displacement $x^2$ and integrate over $x$ from $-\\infty$ to $\\infty$: $$\\int x^2 \\frac{\\partial N(x,t)}{\\partial t} dx = \\int x^2 D \\frac{\\partial^2}{\\partial x^2} N(x,t) dx $$ Now the script that I'm reading writes that the left-hand side can be written as: $$\\int x^2 \\frac{\\partial N(x,t)}{\\partial t} dx = N_0 \\frac{\\partial \\langle x^2 \\rangle}{\\partial t} $$ where I assume that $N_0$ is the initial number concentration of particles and $\\langle \\rangle$ is the mean square displacement. I do not understand the last equation. Could somebody explain why I can express the integral as the partial derivative wrt to time of the mean square displacement (multiplied by $N_0$). **EDIT: $\\langle x^2 \\rangle$ is the mean square displacement (contrarily to what I have written earlier)**"}
{"id":"108384","title":"Static friction","text":"Just a quick question, When a force on an object is equal to its static coefficient * normal force , would the object move or would it stay stationary? Or does the force always need to be greater than this value for the object to move? My guess is that it need to be greater but im not completely certain."}
{"id":"103454","title":"Stress-Energy Tensor","text":"As of recent, I've been doing a bit of self education in GR, equipped with a working knowledge of the key elements of the differential geometry in GR, and in looking at the Einstein-Rosen bridge, ![Einstein-Rosen Bridge](http:\/\/i.stack.imgur.com\/bMT6L.jpg) I see that geometrically it is a hyperboloid of one sheet. Now when using this manifold to calculate things like curvature and geodesics, we need a metric, which we can somewhat easily derive from the equation of the aforementioned surface. Now per $ R_{\\mu\\nu} - \\frac{1}{2}g_{\\mu\\nu}R = \\frac{8\\pi G}{c^{4}} T_{\\mu\\nu} $ ( setting $ \\Lambda = 0 $ for the simple case ), one should be able to solve for the stress-energy tensor, albeit the grotesque mathematics. By this logic, based on the predetermined shape of the local spacetime, we should be able to calculate the mass and geometry of the object required to create such a metric. My questions then are: To what extent is this a valid statement if at all? Given the fact that there are going to be initial value problems embedded in the PDE's with conditions that require the input of some properties of the geometry of the aforementioned object: How is it that one goes about finding physically viable objects that satisfy the EFE under the constraints that we've applied if any exists. How does one test to see if there are even any solutions that work? For an attempt at the second questions, I was thinking that if we have to apply constraints to get rid of the unknown constants that resulted from the PDE, we just put constraints on what the solution can be, to ensure that it fits with what we want. Ie I want to make sure my sphere of matter is of this size, but can't exceed this mass. Am I at least on the right track with this?"}
{"id":"103458","title":"Definition of a non-pertubative Quantum field theory","text":"How do you define a non-perturbative Quantum field theory. What does it mean? I was just digging around some math about the meaning of $Z_1$, and such in terms of probabilities. It turns out these and associated counter terms are derived via perturbation theory. So the question is , if you associate meaning to things like probability of self interaction and such, how do you make these things make sense if you are working non-perturbatively. What is a self interaction outside of perturbation theory? I mean considering some probable sketch, for some arbitrary calculations , something looking like the lines below might appear, then some cleaver physics is said. $1 + \\Sigma_a$ $\\Gamma^{n} \\rightarrow Z_{b} \\bar{\\Gamma^{m}}$ ok so what is a self interaction in non-pertubative models like. Is a non-pertubative model real? Can it make sense. . . . I don't know .. . . . ."}
{"id":"79272","title":"Question regarding waves squeezed","text":"in one MOOC I am taking, the professor had a slide stating \"when squeezed into a narrow space wave is amplified\". I am not sure I understand what he meant by that, and I am trying to think into terms of wave equations. Could anyone explain it to me? Thank you."}
{"id":"132754","title":"How was the formula for kinetic energy found, and who found it?","text":"My questions mostly concern the history of physics. Who found the formula for kinetic energy $E_k =\\frac{1}{2}mv^{2}$ and how was this formula actually discovered? I've recently watched Leonard Susskind's lecture where he proves that if you define kinetic and potential energy in this way, then you can show that the total energy is conserved. But that makes me wonder how anyone came to define kinetic energy in that way. My guess is that someone thought along the following lines: > Energy is conserved, in the sense that when you lift something up you've > done work, but when you let it go back down you're basically back where you > started. So it seems that my work and the work of gravity just traded off. > But how do I make the concept mathematically rigorous? I suppose I need > functions $U$ and $V$, so that the total energy is their sum $E=U+V$, and > the time derivative is always zero, $\\frac{dE}{dt}=0$. But where do I go from here? How do I leap to either   * a) $U=\\frac{1}{2}mv^{2}$   * b) $F=-\\frac{dV}{dt}$?  It seems to me that if you could get to either (a) or (b), then the rest is just algebra, but I do not see how to get to either of these without being told by a physics professor."}
{"id":"132753","title":"Length of the missing part","text":"I have got a very interesting question. A car is driving this way on the picture down. In the looping is a missing part. The radius of the looping is r = 40cm. The car starts at the height h = 3r. If the car jumps from the interruption of the looping it flys and lands an the other side of the interruption and goes ahead to drive. Now I have to find out, how long the missing part of the looping is. phi has no value. I don't have to calculate the friction. ![enter image description here](http:\/\/i.stack.imgur.com\/mAVxR.png) The car has got a velocity of 17,5 km\/h in B, right?"}
{"id":"132759","title":"A beam of light, traveling around a planet","text":"Some time ago, I happened to have found one problem: > The refractive index of the planet's atmosphere decreases with height above > the surface under the following law: > > $$n=n_0-\\alpha h$$ > > with condition $\\frac{\\alpha h}{n_0}\\ll1$ > > Find the altitude $H$ above the surface of the planet, where a ray, emitted > horizontally, will bypass the planet, remaining at this elevation. Radius of > the planet is $R$. Since optics is not my field I decided to ask here. What would be the height $H$ for the Earth? (approximately)"}
{"id":"87831","title":"Application of Ampère's law in low $Re_m$ MHD","text":"Ampère's law states that $$ \\nabla \\times \\mathbf{B} = \\mu_0 \\mathbf{J} \\tag{1}$$ Simultaneously, from Ohm's law, we know that $$\\mathbf{J}=\\sigma\\left(\\mathbf{E}+\\mathbf{u}\\times\\mathbf{B}\\right)\\tag{2}$$ When equating both currents $\\mathbf{J}$, and applying Faraday's law, and knowing that the magnetic field $\\mathbf{B}$ is solenoidal, one arrives at the transport equation of the magnetic field $$ \\frac{\\partial\\mathbf{B}}{\\partial t}=\\nabla\\times(\\mathbf{u}\\times\\mathbf{B)}+\\frac{1}{\\sigma\\mu_0}\\nabla^2\\mathbf{B}\\tag{3}$$ In the low magnetic Reynolds numbers limit $Re_m=\\mu_0\\sigma u L \\ll 1$, the externally imposed magnetic field does not change and equation $(3)$ is not explicitly solved, because the diffusional term becomes negligible. In this so-called one-way coupled magnetohydrodynamics (MHD) (valid for most liquid metals), the electric field $\\mathbf{E}$ has a potential $\\Psi$, which is calculated from the divergence free ($\\nabla\\cdot\\mathbf{J}=0$) condition, using equation $(2)$, with a Lorentz force $\\mathbf{J}\\times\\mathbf{B}$. My question is how should one look at Ampère's law in the context of one-way coupled, or low $Re_m$ MHD? Specifically considering the following two examples **Example 1** Suppose a situation with an insulating boundary (with $\\mathbf{n}=\\mathbf{\\hat{y}}$). This means that $J_y=0$. Furthermore, the imposed magnetic field is constant in time, but varies in space. From $(1)$ $$J_y = \\frac{dB_x}{dz}-\\frac{dB_z}{dx},$$ which, for an arbitrary, non-uniform, steady magnetic field, would be non- zero. **Example 2** Suppose the externally imposed magnetic field $\\mathbf{B}(x,y,z,t)=\\mathbf{B}_0$. According to $(1)$, $\\mathbf{J}=\\mathbf{0}$, and thus there would be no Lorentz force? What is exactly happening at the interface, and am I applying Ampère's law correctly. What am I overlooking?"}
{"id":"23428","title":"Subjectivity of decoherence","text":"I read that quantum decoherence is subjective, in the sense that two observers may not have the same \"environment\" and after each one has traced over those degrees of freedom they will end up with a different density matrix for the couple system-apparatus. I kind of understand the idea, but has an example of this calculation already been done? With the same system looked at by two different observers, and both density matrices calculated\/compared."}
{"id":"23426","title":"Maximum speed of a water skier","text":"Water skiing is a sport where an individual is pulled behind a boat or a cable ski installation on a body of water, skimming the surface. Consider an idealized case where the boat is moving at a constant velocity $\\overrightarrow v_0=\\text{const}$ (relative to the water), independently of the skier. The boat and skier are connected by a massless, unstretchable rope. Surface of the water is assumed to be smooth. Then, what is the maximum possible (instantaneous)speed of the skier $ v_\\text{max}$ (relative to the water)? My first guess, based on intuition, is that $v_\\text{max}=2v_0$. But I'm not at all sure. **Edit:** Skier's velocity may vary. Consider vector projections of velocity vectors $\\overrightarrow v_1 $ and $\\overrightarrow v_2 $ of the boat and skier in the direction of the rope. Because the rope is unstretchable, these projections must be equal. That means, must hold the equality: $$v_1\\cos{\\alpha}=v_2\\cos{\\beta}$$ $\\alpha$ is an angle between $\\overrightarrow v_1 $ and the direction of the rope   $\\beta$ is an angle between $\\overrightarrow v_2 $ and the direction of the rope So for example, if $\\alpha<\\beta$ then $v_2>v_1$. I.e. the skier is moving faster than the boat relative to the water surface."}
{"id":"76945","title":"Physics textbooks that distinguish between laws and definitions?","text":"Often when I am learning physics I start to think about whether the laws I'm learning are mere definitions or experimentally determined, and usually the textbook does not make this clear. As Thomas Kuhn writes in _The Structure of Scientific Revolutions_ , > These generalizations look like laws of nature, but their function for group > members is not often that alone. Sometimes it is: for ex­ample the Joule- > Lenz Law, $H = RI^2$. When that law was discovered, community members > already knew what $H$, $R$, and $I$ stood for, and these generalizations > simply told them something about the behavior of heat, current, and > resistance that they had not known before. But more often, as discussion > earlier in the book indicates, symbolic gen­eralizations simultaneously > serve a second function, one that is ordi­narily sharply separated in > analyses by philosophers of science. Like $f = ma$ or $I = V\/R$, they > function in part as laws but also in part as definitions of some of the > symbols they deploy. Furthermore, the bal­ance between their inseparable > legislative and definitional force shifts over time. In another context > these points would repay detailed anal­ysis, for the nature of the > commitment to a law is very different from that of commitment to a > definition. Laws are often corrigible piecemeal, but definitions, being > tautologies, are not. For example, part of what the acceptance of Ohm's Law > demanded was a redefinition of both 'current' and 'resistance'; if those > terms had continued to mean what they had meant before, Ohm's Law could not > have been right; that is why it was so strenuously opposed as, say, the > Joule-Lenz Law was not. Another example is conservation of momentum; is momentum conserved because it is defined as a quantity that is conserved or because it has been observed to be conserved? Are there any physics textbooks (maybe for classical mechanics) that take a more axiomatic approach, clearly distinguishing definitions from laws determined by experiment?"}
{"id":"73532","title":"Tensor Operators","text":"**Motivation.** I was recently reviewing the section 3.10 in Sakurai's quantum mechanics in which he discusses tensor operators, and I was left desiring a more mathematically general\/precise discussion. I then skimmed the Wikipedia page on tensor operators, and felt similarly dissatisfied. Here's why In these discussions, one essentially defines an indexed set of operators $T_{i_1\\cdots i_k}$ to be a \"cartesian\" tensor operator of rank $k$ provided $$ U(R)\\, T_{i_1\\cdots i_k}\\, U^\\dagger(R) = R_{i_1}^{\\phantom{i_1}j_1}\\cdots R_{i_1}^{\\phantom{i_1}j_1}T_{j_1\\cdots j_k} $$ for each rotation $R\\in\\mathrm{SO}(3)$ where $U$ is some unitary representation of $\\mathrm{SO}(3)$ acting on a Hilbert space (usually that of some physical system whose behavior under rotations we with to study). Similarly one defines a \"spherical\" tensor operator of rank $n$ as an indexed set of operators $T^{(n)}_{q}$ with $-n<q,q'<n$ for which $$ U(R)\\,T_q^{(n)}\\,U^\\dagger(R) = \\sum_{q'=-n}^n D_{q'q}^{(n)}(R)T_{q'}^{(n)} $$ where $D^{(n)}$ is the irreducible representation of $\\mathrm{SO}(3)$ of dimension $n$. Based on these standard definitions, I would think that one could define something less \"coordinate-dependent\" and extended to representations of any group, not just $\\mathrm{SO}(3)$, as follows. > **Candidate Definition**. Let a group $G$ be given. Let $U$ be a unitary > representation of $G$ on a Hilbert space $\\mathcal H$, and let $\\rho$ be a > representation of $G$ on a finite-dimensional, real or complex vector space > $V$. A $k$-multilinear, linear operator-valued function $T:V^k\\to > \\mathrm{Lin}(\\mathcal H)$ is called a **tensor operator** relative to the > pair of representations $U$ and $\\rho$ provided \\begin{align} U(g) T(v_1, > \\dots, v_k) U(g)^\\dagger = T(\\rho(g)v_1, \\dots, \\rho(g)v_k) \\end{align} for > all $g\\in G$ and for all $v_1, \\dots, v_k\\in V$. Notice that if a basis $u_1, \\dots, u_N$ for $V$ is given, and if we define the components $T_{i_1,\\dots i_k}$ of $T$ in this basis by \\begin{align} T_{i_1 \\dots i_k} = T(u_{i_1}, \\dots, u_{i_k}) \\end{align} and if $\\rho(g)_i^{\\phantom ij}$ denotes the matrix representation of $\\rho(g)$ in this basis, then by using multilinearity the defining property of a tensor operator can be written as follows \\begin{align} U(g) T_{i_1\\cdots i_k} U^\\dagger(g) = \\rho(g)_{i_1}^{\\phantom {i_1}j_1}\\cdots \\rho(g)_{i_k}^{\\phantom {i_k}j_k} T_{j_1\\cdots j_k} \\end{align} So this definition immediately reproduces the cartesian tensor definition above if we take, $V =\\mathbb R^3$, $G=\\mathrm{SO}(3)$, and $\\rho(R) = R$, and similarly for the spherical tensor definition if we take $V=\\mathbb C^{2n+1}$, $G=\\mathrm{SO}(3)$, $\\rho = D^{(n)}$ and $k=1$. **Question.** Is the sort of object I just defined the \"proper\" formalization\/generalization of the notion of tensor operators used in physics; it seems to contain the notion of tensor operator used in the physics literature? Is there any literature on the sort of object I define here? I would think that the answer would be yes since this sort of thing seems to me like a natural generalization a mathematically-minded physicist might like to study."}
{"id":"73531","title":"Is squared motor gearbox ratio proportional to inertia ratio?","text":"I read an interesting article http:\/\/m.machinedesign.com\/news\/motor-sizing-made-easy It is very interesting, but I can not follow the 2nd last paragraph. I don't understand why it is true. _**Gearboxes offer a significant benefit in that they affect the inertia ratio by a factor of the gearbox ratio squared._** Please help. Thanks"}
{"id":"73530","title":"What would earth look like if traveling at 77% the speed of light?","text":"I've been having a bit of trouble wrapping my mind around this. If I were in a ship going 77% the speed of light (enough to experience reasonable time dilation) would I see earth going in fast forward? Would people on earth see my ship moving very slowly? I have this question because I was reading Physics of the Impossible (Micho Kaku) and he stated that if we were to travel this fast, we would get to a nearby star in about 10 years, but on earth it would have been many more years. This leaves me to suspect that if many years passed on earth even though I would experience time as normal, that I would see people in fast forward. Or would they be moving slowly because relative to me _they_ are moving the speed of light?"}
{"id":"74547","title":"Acceleration of bouncing ball with air resistance","text":"Here is a strange test question I really don't get: > A ball falls freely from a certain height, and due to air resistance it will > reach a certain terminal velocity (assuming there is a normal gravitational > acceleration going on all the time). Then it hits the ground, and bounces > back without loss of energy. > > Question : what is the acceleration on the ball immediately after it > bounced? The answer is $2g$ (2 times gravitational acceleration), but the explanation is not included. _Note: This is a question from the Physics Olympiad!_"}
{"id":"74546","title":"Is reparameterization invariance some kind of gauge symmetry?","text":"On page 116 of this book it is said, that reparameterization invariance of the string action is analogous to the gauge invariance in electrodynamices. Whereas Maxwell's equations are symmetric under a gauge transformation which allows to describe the same $\\vec{E}$ and $\\vec{B}$ fields by different potentials $A_{\\mu}$, reparameterization invariance makes it possible to use different grids on the world sheet to describe the same physical motion of a string. So is reparameterization invariance some kind of gauge invariance too? What would take the role of the gauge potential $A_{\\mu}$ and what would the corresponding \"gauge invariant\" action look like?"}
{"id":"74543","title":"Which twin produces more widgets?","text":"Two twins are identical in every way, including their widget production. One twin goes on a long space trip at near the speed of light to produce widgets during the trip. When space-twin returns he has aged less than earth-twin. But, who produced more widgets?"}
{"id":"93238","title":"Delta to Star\/Y Conversions and vice versa in Electric Ciruits","text":"We all know the basic rules for conversion of $\"Delta\"$ circuits to $\"Star\"$ circuits and vice versa. We also know that this is needed for simplification of circuits in complex cases. Can anyone please explain HOW the concept of such conversions came about? To be clearer, can anyone show the derivation of the Conversions? for both Capacitors and Resistors?"}
{"id":"81863","title":"Is it possible to extract nitrogen from the air via a centrifuge","text":"I was thinking about how they have cans of compressed nitrogen for filling paint cans, and also that they sometimes will blow nitrogen in to food containers before sealing them, and that got me interested in extracting nitrogen for a nitrogen filled refrigerator. I read up on it and it appears the main way that nitrogen is extracted from the air now is via liquidification of the air and then relying on the different boiling points. I was curious why they didn't use something like a centrifuge to separate the nitrogen out. I'm guessing from what little I know of centrifuges use in nuclear refinement is that it's not energy efficient since the weights are not different enough to cause easy separation. But I was curious if that was the real reason, or if there were other forces that prevented a centrifuge from being used in this kind of gas separation."}
{"id":"75630","title":"Why does one of the extra dimensions of F-Theory have to be a temporal dimension?","text":"F-Theory, as I understand it, is a realisation of Type IIB String Theory as a 12-dimensional theory in such a way that the $SL(2,\\mathbb Z)$ symmetry becomes natural because Type IIB String Theory is F-Theory compactified on a torus. _Question 1:_ However, this explains only why F-Theory needs to be 12-dimensional. Why does one of these extra infinitesimal dimensions need to be temporal, though. _Question 2:_ I've also heard that F-Theory isn't exactly a 2-time theory. However, that doesn't make sense to me, as, since the metric signature needs to be $(-1,-1,1,1,1,1,1,1,1,1,1,1)$, i.e. mostly plus, with 2 temporals and 10 spatials. Or am I missing something obvious, here?"}
{"id":"122648","title":"Compressilibility and the form of Newton's second law in fluid mechanics","text":"In deriving Euler's equations for fluid mechanics, in particular $$f=\\rho \\partial_t v +\\rho v\\cdot \\nabla v$$ for some body force $f$ (e.g Landau & Lifschitz 2.3) one assumes the continuity equation $\\partial_t\\rho=-\\nabla\\cdot\\left(\\rho v\\right)$ and Newton's second law in the form $F=ma$ so $$f=\\rho D_t v$$ where $D_t=\\partial_t+v\\cdot \\nabla$ is the total time derivative (e.g Landau & Lifschitz 2.1). If one instead uses $F=\\dot p$ so that $$f = D_t\\left(\\rho v\\right)$$ then in order to get to get our initial equation back one must assume $D_t \\rho=0$ but of course $D_t \\rho=\\partial_t\\rho+v\\cdot \\nabla\\rho=\\nabla\\cdot\\rho$ and so we have assumed the flow is incompressible. Landau implies that he has not assumed incompressibility at this point so how is it that one can choose the first form for Newton's second law without loss of generality? _(If your answer would appeal to stress-energy tensors or Navier-Stokes etc., if possible please show how the assumption isn't implicitly made.)_ To be more clear, as I have shown above, if you say that $f = D_t\\left(\\rho v\\right)$ and also that $f=\\rho \\partial_t v +\\rho v\\cdot \\nabla v$ then it follows that the flow is incompressible. I don't think that this does imply incompressibility so why is the assumption false?"}
{"id":"70040","title":"Why do we use spinors for describing fermions?","text":"I.e., what properties of the spinors gives us a reason for using them for describing of wavefunctions of fermions?"}
{"id":"131888","title":"Potential energy graphs of chemical systems","text":"![enter image description here](http:\/\/i.stack.imgur.com\/ggTmN.png) I really wish someone could dissect this for me in the language of physics. The system is that of two atoms that approximate each other along 'r' (or distance themselves apart). My concerns are these:   * What does negative potential energy mean here?   * Where is the \"zero\" reference defined?   * Why is there reference to repulsion and attraction in the PE graph?   * If $F=-dU\/dr$, shouldn't the graph of the attractive force be flipped upside down? (Specifically they say the Energy of attraction is $-1.436\/r$ and the energy of repulsion is $7.32x10^-6\/r^8$.)   * My textbook makes reference to $U=\\int F\\,dr$. Why isn't it defined as the negative work in this system?"}
{"id":"122745","title":"Asymmetry of induced e.m.f due to a bar magnet falling through a coil","text":"It is well known that as a bar magnet falls freely through a coil, an electromotive force is induced in the coil by Lenz's Law. However, it appears that the peak e.m.f induced (ignoring the direction) is higher after the passing through a coil, than before it passes through the coil. Why so? My explanation was that after passing through the coil, gravity causes it to accelerate after the coil in the same direction in which the magnetic flux linkage through the coil was changing. By Faraday's Law, e.m.f induced will be greater. However, under this hypothesis, gravity would also have the same effect on the bar magnet before it passes through the coil. Is there any theory that can explain this asymmetry well?"}
{"id":"122742","title":"Hot water freezing faster than cold water","text":"This question has puzzled me for a long time. There is already a question like this on Physics.SE. John's answer to the question seems quite satisfying. But when I googled the cause I found this and this explanation. They maybe wrong but I think How Stuff Works is a reliable source. And here's the original paper. I am quite confused now reading the different explanations. Can anyone please shed some light on the issue?"}
{"id":"122741","title":"Capillary action meniscus height in a tube fitted inside another tube?","text":"I was thinking about how would capillary action change in a tube (classic example) and in a tube fitted inside another tube (considering water as the liquid involved). Height of liquid column: ![enter image description here](http:\/\/i.stack.imgur.com\/P5s6S.png) where: $\\gamma$ = liquid-air surface tension $\\theta$ = contact angle $\\rho$ = density of liquid $g$ = gravity acceleration $r$ = radius I tried my best to draw the examples I'm interested in order to help my explanation. I didn't consider the capillarity inside the smaller tube in both example #2 and #3 because I'd like to assume that $a\/2$ in example #1 is close to $c$ in example #2 and #3 (drawings not to scale). ![enter image description here](http:\/\/i.stack.imgur.com\/Au1cp.png) Since from what I understand the column height is given, among other things (most of which can't be changed, like liquid-air surface tension, contact angle, density of liquid and gravity acceleration), by the tube radius, I'd like to know if $c$ in example #2 can be considered as $a\/2$ in example #1 to calculate column height using above formula. Also I'd like to know how having beads of slightly smaller diameter than $c$ between the two tubes (example #3) would affect the column height. If said beads were less dense than water, could they still improve column height or would they just form a floating mat on top of 1 unit thickness? What'd be the column height of example #2 and #3 assuming $c$ as 1mm? I'm quite sure that given the same reached height $h$ in example #2 and #3, $c$ of #2 has to be smaller than $c$ in #3."}
{"id":"18617","title":"Do interaction-free measurements require a physical collapse or splitting in order to be truly interaction free?","text":"Interaction-free quantum experiments like Renninger's experiment or the Elitzur-Vaidman bomb tester are often taken to be examples of interaction-free measurements of a system. Unfortunately, such assumptions presuppose the ability to post-select in the future just to make sense. Interpretationally speaking, it is hard to see how post-selection can possibly be made without some form of physical collapse of the wave function or a preferred physical splitting of the wave function into branches. Without either a physical collapse or splitting, is it possible to gain information about a system of which we are totally ignorant about the preparation of its properties we are interested in without an actual interaction with it? Basically, does the idea of interaction-free measurements only make sense within some interpretations of quantum mechanics and not others? Is there a philosophical reading of the two-state formalism which does not presuppose a collapse or a splitting? In the two-state formalism, does the necessity of normalizing the overall probability factor to 1 entail the ontological reality of the other outcomes because we have to sum up over their probabilities to get the rescaling factor? The other outcomes where the bomb did in fact go off?"}
{"id":"122290","title":"How do two rigid bodies with different 3rd moment of inertia rotate differently?","text":"If rigid bodies $R_1$ and $R_2$ has exactly same total mass $M$, central of mass, and rotational inertia $I$, but different third moment of inertia $M_3$, how would they move\/rotate differently? What's the observable phenomenon of the difference in their third moment of inertia?"}
{"id":"122293","title":"What are some publications which continue Schrödinger's \"What is life?\" discussion?","text":"I'm looking for modern publications on the physical nature of life in which the primary reference is to the discussion started by Schrödinger in 1944 in his book \"What is life\". For example, publications that suggest amendments (which have been generally accepted) to his ideas, similar books (not text books) that take into consideration new discoveries and offer a modern follow up, compiled works, reviews, and the like."}
{"id":"44066","title":"Find Inductance in a Coil","text":"Question: An inductive coil has an internal resistance of 20 $\\Omega$. When an AC Voltage source with a frequency of 100 Hz is connected to the coil, the current lags the voltage by 30 degrees. What is the value of the inductance L. My Work: I am unsure where to begin here. If we know that $\\omega_d=\\frac{1}{\\sqrt{LC}}$ and we also know that $f=2\\pi \\omega_d$, then we can just find the capacitance and solve for L in the first equation and we will have our answer. However, I am not sure how to find capacitance. Any help is appreciated."}
{"id":"41748","title":"Is synchrotron radiation considered incandescent?","text":"Synchrotron radiation is produced via the acceleration of charged particles, much like incandescence. However, all information I've seen state that incandescent light is produced exclusively from the thermal motion of the charges. Is there therefore a better term to describe the light produced via synchrotrons? Thanks!"}
{"id":"71023","title":"What force does bathroom scale measure?","text":"When you stand on a bathroom scale, I know that the force displayed is the normal force. Since it is the normal force, then technically the force displayed would actually be Fg= (mass)(acceleration) right? If this is true, then why is it that your normal force changes in an elevator? (i.e. if you travel down an elevator, why would the normal force decrease?) Please explain this to me, as I am really confused! It would be best if you could relate this to the Newton laws. Thanks!"}
{"id":"9296","title":"How does carbon dioxide or water vapour absorb thermal infra red radiation from the sun?","text":"We are all told at school water vapour and carbon dioxide are the top two greenhouse gases, and that they absorb thermal infra red radiation, trap heat and warm up the Earth. My question is how do they do that? Why can't Oxygen or Nitrogen or any other gas not absorb infra red radiation as well as water vapour or CO2?"}
{"id":"126536","title":"Trying to combine red, green and blue to produce white","text":"I tried to mimic the mechanism of typical screens to produce white color out of red, green and blue. What I did is displayed the attached image on the screen, and moved far away as to let the diffraction effects take place, so that the three colors appear as if they're coming from the same point. Nonetheless, quite paradoxically, what I have seen was black instead of white. I don't know if this question fits this place, so excuse me. ![enter image description here](http:\/\/i.stack.imgur.com\/EHzCq.png)"}
{"id":"47741","title":"Why does motion help you balance on ice skates?","text":"It's almost impossible to balance on a single ice skate if you're standing still. But give yourself just a little forward motion—it doesn't take very much—and it suddenly becomes easy. You can stand there on one leg and glide effortlessly half way across the rink. Friction will cause you to gradually slow down, and it's only when you've slowed almost to a complete stop that balancing becomes hard again. Why? This seems suspiciously similar to the question of why it's easier to balance on a moving bicycle. But the standard answer to that involves the angular momentum of the wheels. There's nothing rotating here. In fact, other than the very slight deceleration from friction, moving should make no difference. In both cases, you're standing still in a nearly inertial reference frame. Does that slight deceleration somehow matter? Or does the interaction between the ice and a moving skate somehow help you to balance?"}
{"id":"127037","title":"Can you explain Fermat's Principle to me?","text":"The textbook(F.A.Jenkins and H.E.White **_Fundamentals of Optics_** ) states that the Fermat's principle is that > _the path taken by a light ray in going from one point to another through > any set of media is such as to render its optical path equal, in the first > approximation, to other paths closely adjacent to the actual one._ To be honest, I can't understand that description. But I know that > _the actual ray path between two points is the one for which the optical > path length is stationary with respect to variations of the path._ Are the two descriptions same?"}
{"id":"69943","title":"What is the physics behind explosion of Stars?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/mjhTo.jpg) Looking at the above photo, you will soon find it is impossible to explain what actually happens (with today's physics), in one word star is like a nuclear fusion bomb which its pressure is in balance against gravitational force.   1. A supernova (or a black hole) is triggered by an imbalance between them, so isn't supernova a natural pure fusion bomb?    2. Then what equations are able to simulate a virtual supernova explosion?"}
{"id":"123190","title":"Why are interactions always mediated via bosons (as opposed to fermions)?","text":"Weak, Strong and electromagnetic force are mediated via bosons. I've come across the Walecka-model where the exchange particles are $\\sigma$ and $\\omega$ mesons; bosons again. why are interactions always mediated via bosons (not fermions)?"}
{"id":"69949","title":"How do spiral arms form?","text":"Why aren't all spinning galaxies shaped as discs as my young mind would expect? I understand how the innermost parts of a galaxy spin faster than the outer parts, and that could explain why some galaxies are more spiraled than others based on age. Though, this doesn't explain how the arms came into existence in the first place. Might it have something to do with an imperfect distribution of mass, and therefore an imperfect distribution of gravity, causing a split in the disc from which point gravity, centripetal force, and inertia could take over? Or does something happen earlier in a galaxy's life?"}
{"id":"69948","title":"Question about Komar integral derivation in Wald","text":"I have a question about derivation 11.2.10 in Wald (page 289). Here is a screenshot of the relevant passage: ![Wald p289](http:\/\/i.stack.imgur.com\/UI69w.png) I don't get the step $$-\\frac{1}{4\\pi}\\int _{\\Sigma}R^{d}{}{}_{f}\\xi^{f}\\epsilon_{deab} = \\frac{1}{4\\pi}\\int _{\\Sigma}R_{ab}n^{a}\\xi^{b}dV.$$ Could someone explain to me how he gets that equality? Thank you very much in advance."}
{"id":"95526","title":"Dimensional aspects of the imaginary unit $i$ in physics","text":"From a real world perspective each dimension in the 3-D Cartesian System can be represented by an axis that is perpendicular to 2 other axes. I read somewhere else that the effect of ${\\sqrt {-1}}$ is to reorient data 90 degrees on the \"imaginary\" axis. I guess my question is this: What role ( if any) does ${\\sqrt {-1}}$ serve in everyday or even quantum physics? ${Edit}$ To clarify, I am more concerned with the dimensional aspects of ${\\sqrt {-1}}$. If for instance you were to consider a 3-D location to be a \"point\" with length width and height in a complex graph representation with three additional mutually perpendicular axes? PS. I know this thread is getting a bit hair brained. But in my defense, I did look for a \"specualation\" tag before I posted this thread here."}
{"id":"44959","title":"In classical mechanics, are complex numbers unphysical?","text":"> **Possible Duplicate:**   >  Physics math without $\\sqrt{-1}$ When I produce a complex final solution to a problem that began without complex coefficients at all, I have so far (with my limited expertise) tended to discard it as unphysical, unless I had to tinker some more, in which case the $i$s may square and real-ity be restored. Is this acceptable? I understand that the Wessel plane is useful in modelling planetary motion, and many other things, but the fact (?) that you could do it all (albeit more awkwardly) without them seems to be indicative that they are a little contrived. In short, is there a counter-example?; a physical theory ( _not_ a particular, neat, mathematical model of a physical theory) in classical mechanics that could not be formulated without complex numbers?"}
{"id":"105516","title":"Complex variables in classical mechanics","text":"In quantum mechanics complex numbers are absolutely essential because of the relation $$[\\hat q_i,\\hat p_j]=i\\hbar\\delta_{ij}.$$ But is complex number also essential anywhere in the formalism of classical physics, except for some convenience of mathematical calculations? Is there any example from classical physics where there is no other way out except to introduce complex variables? Is there any strong reason by which I can say that complex number are unavoidable in classical physics too?"}
{"id":"89161","title":"Gauge fields in 2d spacetime","text":"I believe it is only a technical question. However I cannot realize it. It is said in 2d spacetime the gauge fields $A_\\mu$ can be rewritten in lightcone coordinates as $A_+=ig\\partial_+g^{-1}$ and $A_-=ih\\partial_-h^{-1}$, where $g$ and $h$ are the elements of the gauge group. Anyone could give me a simple and transparent explanation on it? Thanks!"}
{"id":"13702","title":"What is a quark condensate?","text":"1. What is a quark condensate? is it a bound state between 2 quarks? can we have 3(or more)-quarks condensate?   2. What mediates the interaction between the constituents of the condensate?   3. Are the constituents of the condensate always fermions?"}
{"id":"41398","title":"Calculate acceleration and time given initial speed, final speed, and travelling distance?","text":"A motorcycle is known to accelerate from rest to 190km\/h in 402m. Considering the rate of acceleration is constant, how should I go about calculating the acceleration rate and the time it took the bike to complete the distance?"}
{"id":"134040","title":"Motion with damping involving both $-v^2$ and $-v$ terms","text":"I'm trying to solve for the motion of a particle, accounting for both viscous and drag forces. (There is no potential) The total resistance by medium is modeled by: $\\vec{F}=-(\\mu_1v + \\mu_2v^2)\\hat{v} $ Where $v$ is the velocity; There is no potential. I need to set up the Lagrangian and describe motion under the following conditions: $\\frac{\\mu_1}{\\mu_2} = 10^{-5}$ $\\frac{\\mu_1}{\\mu_2} = 1$ $\\frac{\\mu_1}{\\mu_2} = 500$ I have set up the Lagrangian as follows: $L = \\frac{1}{2}m(\\dot{x}^2+\\dot{y}^2+\\dot{z}^2)$ $\\frac{d}{dt} \\frac{\\partial L}{\\partial \\dot{x}} -\\frac{\\partial L}{\\partial x} =- (\\mu_1 \\dot{x} + \\mu_2\\dot{x}\\sqrt{\\dot{x}^2+\\dot{y}^2+\\dot{z}^2})$ and similarly for z and y. Equations of motion turn out to be: $m\\ddot{x}= - (\\mu_1 \\dot{x} + \\mu_2\\dot{x}\\sqrt{\\dot{x}^2+\\dot{y}^2+\\dot{z}^2})$ Similarly for z and y. I'm at a total loss to how these should be solved. Is there even a way of analytically solving such coupled equations? However, If I orient the x axis along initial direction of velocity, I should get: $m\\ddot{x}= - (\\mu_1 \\dot{x} + \\mu_2\\dot{x}^2)$ $y=0$ $z=0$ I'm stuck at this point. Wolfram Alpha suggests: ![](http:\/\/puu.sh\/bmj5K\/8d1edff9ab.png) Where $\\frac{\\mu_1}{m}=a$ and $\\frac{\\mu_2}{m}=b$ I'm not aware of the techniques used to get to this. Would appreciate some help\/pointing to resources on that. After this, I'm not really sure of how to apply limits (Since we have the additional constraint of the domain of Log functions), and how to interpret the motion in the 3 regimes mentioned above. Can someone give me some pointers? Also what really happens when there is a potential (Say, gravity)? (which really is the real world scenario)"}
{"id":"76819","title":"How do we deal with irrationals in Physics?","text":"This question is really very basic: how do we deal with irrationals in Physics? If for instance, in some meaningful calculation we get a length of $\\pi$ meters, or a force of $\\pi$ Newtons, how should we understand this? One could say: \"well, we understand it as a rational approximation\", but then there comes another question: which rational approximation? We can approximate $\\pi$ by $3,14$ or $3,1415$ and so forth, increasing the precision, but yet never getting really to $\\pi$. Another problem is: well, the equations gave us $\\pi$, and this is not a physically meaningful thing, so is there something wrong with the equation? Because, in nature I think there really isn't irrationals lurking arround, so that this becomes a little complicated. So: how do we deal with irrationals in Physics?"}
{"id":"41397","title":"The earth's magnetic field","text":"This might sound like a silly question. Is it possible for the earth's magnetic field to actually destroy or harm earth? (implosion, crushing etc.)"}
{"id":"65621","title":"How the value of permitivity of free space is determined?","text":"when electric charges are placed on the medium then force of interaction between them decreases. We say this factor permitivity of the medium. The force of interaction between the charges is max if the charges are at space or if there is absence of medium. We know the electric permittivity of free space is $8.85 * 10^{-12} F\/m$. My question is how this value is determined. Is there any experiment held?"}
{"id":"51891","title":"Do multiple electrons exist during superposition?","text":"Wikipedia says: > _Quantum superposition is a fundamental principle of quantum mechanics that > holds that a physical system—such as an electron—exists partly in all its > particular, theoretically possible states (or, configuration of its > properties) simultaneously;[...]_ What interests me is what happens when location is the state that is superposed. If an electron exists in two locations at once, as a result of quantum superposition, does that mean that the mass of the universe has increased so long as the superposition exists? So in other words, does it follow that effectively two electrons exist during superposition? Why or why not?"}
{"id":"16282","title":"Critics of Mannheim's Conformal Gravity Theory?","text":"I'm looking for more articles\/reactions\/critiques\/support for Philip Mannheim's recent conformal gravity theory. See here: http:\/\/arxiv.org\/abs\/1101.2186v1 Any ideas on where to start?"}
{"id":"41395","title":"Peak deconvolution","text":"I need some help understanding how much information I can pull out of this data. I have a sample made up of two materials. Materials A and material B. Then I took a picture of the sample. The two materials mix quite well, but not perfectly, so on my image I can see that some areas are mostly material A, some are mostly material B, and most of the areas are a mix between. I also took an image of a sample consisting only of material A. On my images material A will look white, and material B will look black. If I make a histogram of the image of material A (and only A) it has the center around 6 [A.U] which fits my theory. The histogram also has a FWHM of 0.55 [A.U.] The same histogram of material A & B is centered around 7 [A.U] which also fit my theory since it's a 50:50 mix of A & B and A is centered at 6 [A.U] and B should be centered at 8 [A.U]. The FWHM of this peak 0.64 [A.U] - thus only slightly larger than the image with pure A. Here's my problem, since the peak with the mix only is slightly wider than the peak for the pure material it means that my resolution i not good enough to distinguish the two materials from each other (if I could see areas with pure material A or B it would be a camel\/double-peak, if I could see areas with mostly material A or B I would have a very wide peak - here I only have a slightly wider peak). However, it is still wider, so there must be some kind of information I can subtract. I'm not really sure what - if any. EDIT: If anything is unclear, please ask!"}
{"id":"116957","title":"A strange audio phenomenon, could there be a physical interpretation to it?","text":"http:\/\/mathoverflow.net\/q\/165038\/14414 Motivation : Here is a motivation as to why this problem is so important. Let $f(t)$ be an audio signal. We can safely asume it to be bandlimited to 0-20kHz as we cannot hear anything above that. Capture this signal in digital computer with appropriate sampling frequency and denote it as $f[n]$. Now take Discrete Hilbert transform of $f[n]$ to get $f_h[n]$, (using the code $f_h$ = imag(hilbert(f)); in Matlab). Compute the signal $f_{\\theta}[n] = f[n]\\cos\\theta + f_h[n]\\sin\\theta$ for any value of $\\theta$, then listen to the signal with different values for $\\theta$. They all sound **exactly identical**. Similarly our $MI_{\\omega_0,\\omega_1}(t)$ is same for all $f_{\\theta} = f\\cos\\theta + f_h\\sin\\theta$, for any value of $\\theta$. **Question :** just try it. $<f,f_h> = 0$, they why do they produce same effect in the listner? Is it some quantum mechanical effect gone wrong? **Added :** Also see this metric space : metric space I've recently filed a patent using this metric with a slight change, instead of arccos i used sqrt(2(1-cos(theta))), which makes it a Hilbertian metric. I had then embedded this metric space into an Hilbert space isometrically, to model using vectors. **MATLAB code :** [f,fs] = wavread('audio_file.wav'); fh = imag(hilbert(f)); theta = pi\/4; f_tht = f*cos(theta) + fh*sin(theta); wavplay(f,fs); wavplay(f_tht,fs);"}
{"id":"69363","title":"Why is the summer, in the temperate latitudes, in average, hotter that the spring?","text":"It is common knowledge that the transition from the Spring to the Summer season occurs in the Summer Solstice when the \"Sun reaches its highest excursion relative to the celestial equator on the celestial sphere\" (as stated in Wikipedia). It is also stated in Wikipedia' Summer page: \"Days continue to lengthen from equinox to solstice and summer days progressively shorten after the solstice, so meteorological summer encompasses the build-up to the longest day and a diminishing thereafter, with summer having many more hours of daylight than spring.\" My question is: why is the summer, in the temperate latitudes, in average, hotter that the spring?"}
{"id":"119987","title":"Based on note, how fast is a stringed instrument's string oscillating?","text":"to be quite honest, I have no idea where to start on this problem mathematically. However, it struck my curiosity and would love to know how it works on a mathematical level. # the problem You have a guitar. All of the strings start at a resting position. When you play a string, it is now oscillating between two points (as far as I can tell). The speed at which it oscillates is correlating to the frequency, and the distance between the points is how hard the note was struck or the volume of the note. You play the E string, open, and let it ring. Then you play the 7th fret of the same string, and let it ring. Now how does one determine how quickly the string is oscillating between the two points in this scenario, in a unit of measurable distance? All I really have so far is that E and B are a perfect fifth apart, making the ratio of their frequencies 3:2. If one were to play the 12th fret of the same string, that would be a 1:2 relationship. Therefore it follows that the speed they oscillate should follow the same proportions. I, however, am not sure how to get an initial distance. ### Edit for some clarity; the application to put this question in context, I am trying to programmatically work with a three-dimensional model of a guitar as it is played by a guitar player and make it visually convincing. As such, when he plays an open string, it would move more slowly. If he put his finger on a fret on that string and played it, nothing from his finger to the nut would vibrate, but the rest would vibrate FASTER. I'm trying to discern a reasonable level of accuracy of this visual effect. Thank you"}
{"id":"82854","title":"Alternative derivation for the capacitor energy equation","text":"I hope this is the right place for this kind of post. A friend is trying to derive the equation for the energy stored in a capacitor by analysing the change in potential on one plate when the capacitor is shorted, however I don't think this valid. Could anyone explain why this is right or wrong? He says that when the capacitor is shorted the potential at each plate changes by V\/2 (from 0 to V\/2 on one side and from V to V\/2 on the other), so by substituting in V\/2 into the equation he gets E = 1\/2 QV, the capacitor energy equation. ![Diagram](http:\/\/i.imgur.com\/Q7H7oXk.png)"}
{"id":"46034","title":"Characteristics of bloch electron in a priodic potential","text":"Effective mass of a Bloch electron in a periodic potential is negative why ?"}
{"id":"126269","title":"Can two single particles interfere with each other?","text":"Groups of particles can interfere with one another; In the double slit experiment when measuring single photons at the screen each one arrives at the screen in a random manner and they only show the interference pattern once several particles are detected. Obviously two waves can obviously interfere with one another, but can two single particles interfere with one another? Cohen-Tannoudji writes that > light simultaneously behaves like a wave and like a flux of particles But do two particles constitute a flux of particles? I doubt that this could be tested experimentally but if it were so would this constitute a measurement for each particle? **Here was my thought** (Disclaimer:I do not have a good idea of what the interference of two single particle states is): To have interference of two single particles you would have to know something about there position to be able to describe their interference pattern hence the measurement."}
{"id":"64289","title":"Self Inductance and Mutual Inductance","text":"Systems have mutual inductance and self inductance-however won't the effects of these two effect each other as well? Do systems undergo current and magnetic field behavior that are, in this way, dictated by differential equations?"}
{"id":"64280","title":"What's the physical significance of using fourier transform for diffraction?","text":"I am studying some basic idea of diffraction and there mention in far field, the diffraction pattern could be understood by Fourier transform. But I just don't understand what's the physical fact for that. And why cannot use Fourier transform for the the near-field case? Also, when I am trying to understand the theory of diffraction, it ends up with some complicate math (integrals). I want to learn that but the books I am reading are not easy to understand. Anyone recommends some good books or video lectures (more theoretical but to explain most of the math in plain way)?"}
{"id":"133837","title":"Change in time period of a simple pendulum","text":"There's a question in my book which goes as: A pendulum has a bob of a hollow sphere filled with a liquid and having a hole at the base. If the liquid is allowed to flow out, what would happen to the time period of the pendulum? My thought: Since the time period is independent of the mass of the bob, the time period would remain unchanged. What my books says: The time period would initially decrease, and then gets restored. This is because as the liquid gets lowered, the center of gravity of the bob gets lowered, which increases the effective length of the pendulum string slightly. When the liquid is completely drained out, the center of gravity again comes to the center of the sphere, restoring the time period. I don't understand why the 'effective length' would increase due to the shift of cog. And would the change in time period would be so significant?"}
{"id":"80287","title":"Time period of simple pendulum with varying mass","text":"How do you find time period as a function of time for a simple pendulum that is in the form of a hollow sphere that is filled with mercury and there is a hole in the bottom through which the mercury is constantly falling at a fixed rate? I tried creating a function for the time period to check how it varies with the mass of mercury, but I had too many variables. EDIT: This question was a concept based multiple choice question in my homework book. The option were time period becomes erratic, time period increases, time period decreases, time period first increases then decreases."}
{"id":"16864","title":"Where to find experimental\/theoretical value of (charge) radius of particles?","text":"Where to find experimental\/theoretical value of (charge) radius of particles like proton, neutron, $pi^+$, etc. ?"}
{"id":"16866","title":"Resources for current thought on time\/spacetime?","text":"Are any of the big-name physicists associated with the time in the same way that Hawking and Penrose are associated with black holes? I'm interested in some good books that focus on the topic."}
{"id":"125980","title":"At what temperature does air become liquid at a pressure of X bar?","text":"... with X = 6 in my particular case. I was trying to get a phase diagram for air, only got one for nitrogen and was slightly puzzled. Here's why: I was wondering about a particular detail in industrial air liquification. The original Linde process used 200 bar pressure and cycled the air though several cycles to condense it, as one decompression only gives around 40 K reduction in temperature. This process was replaced rather early by a modified process (called the Linde-Fränkle process) that compresses the air to only 6 bar and then runs it through a pre-cooled regenerator, where it liquifies in the first pass, proceeding to the distillation column immediately. That process was in use until the 1990s, so it probably had decent performance. Now, I was wondering to what temperature the regenerator has been precooled. From the phase diagram of nitrogen, I'm estimating to somewhere not abive 150 K, most probably lower. Since the pre cooling is done with the gaseous $N_2$ and $O_2$ boiling off the distillation column, it sounds like quite close to the liquification temperature at atmospheric pressure. This puzzles me because it makes me wonder why such little compression is sufficient to make the whole process economic. Also, I can't quite see at what point the Joule-Thompson effect comes into play here. The air is liquefied entirely without decompression, so where dies the cooling come from? I was also wondering if I might be confusing liquid and supercritical here, since I'm told pressurized gases in typical containers at room temperature are really supercritical already, even though laymen like me still call them liquid."}
{"id":"16863","title":"A physical understanding of fractionalization","text":"all! Is there a physical understanding of fractionalization in condensed matter physics? The textbook approach is theoretical, not physical. I'm thinking of spin-charge separation for electrons, the fractional quantum hall effect, and things like that. The theoretical approach is to introduce an auxiliary gauge field with no kinetic term at the bare level so that it is apparently confining and nondynamical at the bare level, but somehow, dynamics intervenes, and it becomes deconfining, and somehow, there's some mixture between the emergent gauge symmetry and the original symmetries, and somehow, fractionalization comes in in the diagonalization. What is the physical interpretation without introducing theoretical auxiliary gauge symmetries right from the onset?"}
{"id":"12154","title":"What exactly is a mole?","text":"I have some trouble understanding exactly what a mole represents. As I understand, one unit mole is 1\/12 of the mass of an atom of carbon-12 (thus it is the mass of one nucleon?). What is a mole, then?"}
{"id":"90012","title":"Spring system - 3 DoF system and its properties while changing stiffness","text":"I was given the attached 3 degree of freedom spring system with the purpose of analyzing it. ![system](http:\/\/i.imgur.com\/pK0p9kG.jpg) I came up with the following equation of motion ![equation](http:\/\/i.imgur.com\/vr2QWQr.jpg) and then I ran Matlab to calculate the corresponding natural frequencies and mode shapes using eigenvalues and eigenvectors; I was asked to see what happens when value of stiffness $k_{12}$ is changed. This is the plot of the value $k_{12}$ against the natural frequencies. ![plot](http:\/\/i.imgur.com\/fdwweRr.jpg) The problem is that I do not know WHY values of natural frequencies are insensitive at low values of $k_{12}$ and why both the 1st and 2nd natural frequencies are insensitive to changes in $k_{12}$ when values are large (first two level off and the third one seems to go to infinity). I assume it has something to do with the equation for force due to a spring between 2 masses but I cannot figure it out. That is why I ask for your help - thanks in advance."}
{"id":"59842","title":"Showing Lorenz gauge is satisfied in retarded potential - vector calculus","text":"I am trying to show that $\\nabla\\cdot \\vec{A}=-\\mu_0 \\epsilon_0 \\frac{\\partial V}{\\partial t}$ $V=\\frac{1}{4\\pi\\epsilon_0}\\int \\frac{\\rho(\\vec{r}',t_r)}{r}d\\tau'$ $\\vec{A}=\\frac{\\mu_0}{4\\pi}\\int \\frac{J(\\vec{r}',t_r)}{r}d\\tau'$ Where $d\\tau'=dxdydz$, $t_r=t-\\frac{r}{c}$ is the retarded time, $\\vec{r}$ is the position vector of the charge and $\\vec{r}'$ is the position of the observer (us) and $r=|\\vec{r}-\\vec{r}'|$ the distance from us to the source. Probably best to start with $\\nabla\\cdot \\vec{A}$: $\\nabla\\cdot \\vec{A} =\\frac{1}{4\\pi \\epsilon_0}\\int \\nabla \\cdot \\left(\\frac{\\vec{J}(\\vec{r}',t_r)}{r}\\right)d\\tau'$ Apparently, $\\nabla\\cdot ({\\vec{J}\\over r})=\\frac{1}{r}(\\nabla\\cdot \\vec{J})+\\frac{1}{r}(\\nabla'\\cdot \\vec{J})-\\nabla'\\cdot ({\\vec{J}\\over r})$ This is the first thing I need to do to get any further but I cannot do this. $\\nabla$ is differentiation with respect to $r$ and $\\nabla'$ is with respect to $r'$. I thought that $\\nabla\\cdot ({\\vec{J}\\over r})=\\frac{1}{r}(\\nabla\\cdot\\vec{J})+\\vec{J}\\cdot (\\nabla(\\frac{1}{r}))$"}
{"id":"63107","title":"Problem involving Dirac's equation","text":"I'm stuck in an equation derivation of Ryder's QFT book. Starting with Dirac's equation: $$(i\\gamma^\\mu\\partial_\\mu-m)\\psi=0$$ If I multiply by $i\\gamma^\\nu\\partial_\\nu$, I get: $$((\\gamma^\\nu\\partial_\\nu)(\\gamma^\\mu\\partial_\\mu)+i\\gamma^\\nu\\partial_\\nu m)\\psi=0$$ I should get: $$(\\gamma^\\nu\\gamma^\\mu\\partial_\\nu\\partial_\\mu+m^2)\\psi=0$$ I suppose that this means: $$i\\gamma^\\nu\\partial_\\nu=m$$ But I don't know why. ¿Could anyone show me the way to prove this property? Thanks."}
{"id":"9317","title":"Is it possible to orbit the sun next to Earth?","text":"With Geostationary orbit your limited to the equator and ~36000km. Would it be possible to orbit the sun at the same speed as Earth, and then in basically the same place so that the distance to the Sun is slightly less or more than Earth, and the inclination is slightly less or more, in order to achieve a geostationary effect over any point at any height. Wouldn't the Earth's gravity interfere. (The reason I ask is, would this be a way of creating a space elevator?)"}
{"id":"29225","title":"What do scientists believe about existence in dimensions?","text":"I couldn't really think of a suitable question title, I'm not sure if it's completely related or not. But this is as far as I know (well, I thought it all up last night and it seemed extremely logical so I think it's probably what scientists believe): There are 11 possible dimensions, the 1st, 2nd, 3rd, 4th, 5th and so on and each dimension consists of all the dimensions below it, e.g. the 6th dimension consists of the 1st, 2nd, 3rd, 4th, 5th and 6th, the 4th of which is time, which allows for existence because of course there must be time for existence as you need time to forward to exist. So this means that the space-time (the 4th dimension) must be the lowest dimension with possible existence as that is the lowest dimension to include a dimension of time. So we, as humans, must exist in the 4th dimension because we need time to exist and because all the gravity, space and space curvature around us is governed by the extra dimension of time, creating space-time\/4th dimension, but time isn't a spacial dimension so our view of the universe doesn't change because of the extra dimension of time, which is why we only see it as 3-dimensional, because they are the only 3 spacial dimensions that we can actually see\/visualise. Are my views correct and a physicist would agree with me or are they completely wrong? If they are wrong, could someone explain what I have explained, but correctly please."}
{"id":"57995","title":"2D quantum well energy spectrum (analytical vs numerical)","text":"I am trying to understand the energy spectrum difference between the analytical and the approximated solution for a quantum well. The particle is inside a box with domain $\\Omega=(0,0)$X$(1,1)$. For this I have $\\hbar = m = 1$ and the energy is given analytically by $E_{m,n}=\\frac{\\pi^2}{2}(n^2 + m^2)$ My approximation is done using finite differences with a grid of $60$X$60$. The eigenvalues that I'm getting with the exact solution are always positive whereas the eigenvalues I'm getting with the approximated solution are always negative. I am not familiar with quantum mechanics. That said, can you help me understand this? What can be happening, is it a big error due to the approximation? Where or what can I think of to try to understand these results? I hope the question and the problem are well stated if not just tell me, probably I missed some important data and\/or assumption. **UPDATE** Here I paste the code used to do this little simulation               clc;     xO = 0;     xL = 1;     N = 60;     %First build the matrix for a 1D-mesh     h = (xL - xO) \/ (N-1);     H = diag( (-2\/h^2)*ones(1,N), 0 ) + ...         diag( (1\/h^2)*ones(1,N-1), 1 ) + ...         diag( (1\/h^2)*ones(1,N-1), -1 );     %Using the tensor product build the matrix for the 2D-mesh     H = kron(H, eye(N)) + kron(eye(N), H);     H = (-1\/2) .* H;          %Compute the eigenvalues E and eigenvectors Psi     [Pcomp, Ecomp] = eig(H);          fEexac = @(m,n) ((pi^2)\/2) * (n^2 + m^2);     Eexac = [];     for n=1:N         for m=1:N             Eexac((n-1)*N+m,(n-1)*N+m) = fEexac(n,m);         end     end          %Plot energy from analytical vs computed     figure(1); hold on;     plot(1:N^2, sort(diag(Eexac)), 'b', 1:N^2, sort(diag(Ecomp)), 'r');          %Plot discretized energy spectrum     figure(2); hold on;     plot(1:40, diag(Ecomp(1:40,1:40)), 'r');          %Compare lowest 300 eigenvalues (exact and computed)     figure(3); hold on;     x = 1:300;     plot(1:300, sort(diag(Eexac(1:300,1:300))), 'b', ...         1:300, sort(diag(Ecomp(1:300,1:300))), 'r');"}
{"id":"51788","title":"What does the quantum state of a system tell us about itself?","text":"> _In quantum mechanics, quantum state refers to the state of a quantum > system. A quantum state is given as a vector in a vector space, called the > state vector. The state vector theoretically contains statistical > information about the quantum system._ The article in Wikipedia about quantum state suggests that the quantum state of a system is indeed its state vector which contains statistical information about it. But what are those statistical information? The particle's location, momentum, wavefunction and.. energy level? When I measure the momentum and position of an electron, the quantum superposition principle suggests that I'm only getting a result corresponding to one of its possible state because of it exhibits both wave and partile behaviors. But what exactly is a possible state?"}
{"id":"54942","title":"Similarity between the Coulomb force and Newton's gravitational force","text":"Coulomb force and gravitational force has the same governing equation. So they should be same in nature. A moving electric charge creates magnetic field, so a moving mass should create some force which will be analogous to magnetic force."}
{"id":"8951","title":"What are the ideal centers of mass and rotation for a steadicam?","text":"For those who don't know what a steadicam is, I'll explain it in two points to give a _basic_ idea. If you know what a steadicam is, skip to point 3. 1) A steadicam is a stabilizing mount for a motion picture camera, which mechanically isolates the operator's movement from the camera, allowing a very smooth shot even when the operator is moving quickly over an uneven surface. 2) How does it work (image courtesy of wsclater): ![Steadicam](http:\/\/i.stack.imgur.com\/jMrJJ.jpg) Red dots = control masses, Blue dot = center of rotation The idea behind a steadicam is that the center of the mass is taken **out of the camera** itself. Therefore, when moving \/ panning the whole system (steadicam + camera), the motion of the camera itself is more fluid. This can be seen in action in this amazing youtube example. 3) The question: How do we determine the ideal distance of the center of rotation from the camera? How do we determine the ideal distance of the center of mass from the center of rotation? What's the ideal length of the handle (the point where we hold the system), i.e. how far should we hold the system from its center of rotation? _Disclaimer: This is neither a matter of profit, nor a matter of homework. I'm just trying to build some of those for my own personal use, especially to find out how it really works, as I'm kind of amazed from this._"}
{"id":"21756","title":"Irreducible representation in physics","text":"I am confused about something. Group theory books written for physicists say that any reducible representation can be decomposed in terms of irreducible representations (so correct me if I am wrong, to me irreducible representations are like the unit vectors i j k in terms of which any 3D vector can be expanded, or they are like sines and cosines in terms of which any periodic function can be Fourier expanded.) Now at the same time they say that any bigger representation of a group can be built out of irreducible ones. What is unclear to me is the physical motivation for each direction. Of course those books contain physical applications but the big picture is never obvious (they get to applications after 200-300 pages of abstract details). If someone could answer the following questions I would be really appreciated: 1-what is the physical motivation to write a representation in terms of irreducible ones? 2-what is the physical motivation to build bigger representations using irreducible ones?"}
{"id":"62858","title":"Derrick’s theorem","text":"Consider a theory in D spatial dimensions involving one or more scalar fields $\\phi_a$, with a Lagrangian density of the form $$L= \\frac{1}{2} G_{ab}(\\phi) \\partial_\\mu \\phi_a \\partial^\\mu \\phi_b- V(\\phi)$$ where the eigenvalues of G are all positive definite for any value of $\\phi$, and V = 0 at its minima. Any finite energy static solution of the field equations is a stationary point of the potential energy $$E = I_K + I_V ,$$ where $$I_K[\\phi]= \\frac{1}{2} \\int d^Dx G_{ab}(\\phi) \\partial_j \\phi_a \\partial_j\\phi_b$$ and $$I_V = \\int d^Dx V(\\phi)$$ are both positive. Since the solution is a stationary point among all configura- tions, it must, a fortiori, also be a stationary point among any subset of these ̄ configurations to which it belongs. Therefore, given a solution $\\phi(x)$, consider the one-parameter family of configurations, $$f_\\lambda(x)= \\bar{\\phi}(\\lambda x)$$ that are obtained from the solution by rescaling lengths. The potential energy of these configurations is given by \\begin{align} E_\\lambda &= I_K(f_\\lambda) + I_V(f_\\lambda)\\\\\\ &=\\lambda^{2-D} I_K[\\bar\\phi]+\\lambda^{-D} I_V[\\bar\\phi]\\tag{1} \\end{align} > $\\lambda = 1$ must be a stationary point of $E(\\lambda)$, which implies that > $$0=(D-2) I_K[\\bar\\phi]+D I_V[\\bar\\phi] \\tag{2}$$ **My problem is, how they got the equation(2) from (1)?**"}
{"id":"26900","title":"What videos should everyone watch?","text":"Similar to this cstheory question \"What videos should everyone watch?\" What conferences, workshops or lectures have recordings that researchers in theoretical physics should watch?"}
{"id":"26903","title":"Lorentz transformation in light cone coordinates in string theory","text":"What is the explicit form of the Lorentz transformation changing the light cone coordinates in the light cone gauge in string theory? The extended nature of the strings complicate matters, especially with interactions turned on. String modes tend to be mixed up."}
{"id":"68705","title":"Schrödinger equation in position representation","text":"$$ \\DeclareMathOperator{\\dif}{d \\\\!} \\newcommand{\\ramuno}{\\mathrm{i}} \\newcommand{\\exponent}{\\mathrm{e}} \\newcommand{\\ket}[1]{|{#1}\\rangle} \\newcommand{\\bra}[1]{\\langle{#1}|} \\newcommand{\\braket}[2]{\\langle{#1}|{#2}\\rangle} \\newcommand{\\bracket}[3]{\\langle{#1}|{#2}|{#3}\\rangle} \\newcommand{\\linop}[1]{\\hat{#1}} \\newcommand{\\dpd}[2]{\\frac{\\partial #1}{\\partial #2}} \\newcommand{\\dod}[2]{\\frac{\\dif{#1}}{\\dif{#2}}} $$ We start from an abstract state vector $\\ket{\\Psi}$ as a description of a state of a system and the Schrödinger equation in the following form $$ \\ramuno \\hbar \\dod{}{t} \\ket{\\Psi(t)} = \\hat{H} \\ket{\\Psi(t)} \\, . \\quad (1) $$ Now if we move to position representation of the state vector what will happen to the Schrödinger equation? In _Mathematics for Quantum Mechanics: An Introductory Survey of Operators, Eigenvalues And Linear Vector Spaces_ by John David Jackson I found the following information (pp. 77-78). By taking an inner product of both sides of (1) with $\\ket{x}$ and using the resolution of identity $\\linop{I} = \\int\\nolimits_{-\\infty}^{+\\infty} \\ket{x'} \\bra{x'} \\dif{x'}$ on the right side we obtain $$ \\ramuno \\hbar \\dod{}{t} \\braket{x}{\\Psi(t)} = \\int\\limits_{-\\infty}^{+\\infty} \\bracket{ x }{ \\linop{H} }{ x' } \\braket{ x' }{ \\Psi(t) } \\dif{x'} \\, . $$ We then introduce a wave function $\\Psi(x,t) \\equiv \\braket{x}{\\Psi(t)}$ and if I understand correctly $\\bracket{ x }{ \\linop{H} }{ x' }$ is also replaced by a function $h(x, x')$ which will lead us to $$ \\ramuno \\hbar \\dod{}{t} \\Psi(x, t) = \\int\\limits_{-\\infty}^{+\\infty} h(x, x') \\Psi(x', t) \\dif{x'} \\, . $$ Now we are one step away from the familiar Schrödinger equation in position representation: we need Hamiltonian operator in position representation $\\linop{H}(x, \\frac{\\hbar}{\\ramuno} \\dod{}{x})$ to be given by $$ \\linop{H}(x, \\frac{\\hbar}{\\ramuno} \\dod{}{x}) \\Psi(x, t) = \\int\\limits_{-\\infty}^{+\\infty} h(x, x') \\Psi(x', t) \\dif{x'} \\, . $$ Author claims (p. 44) that > For our purposes the general linear operator $K$ can be written in the > explicit form $$ g = K f \\rightarrow g(x) = \\int\\limits_{a}^{b} k(x,x') > f(x') \\dif{x'} \\quad(2) $$ The function $k(x,x')$ is called the _kernel of > the operator_ $K$. It is not that I do not trust the author, but since my knowledge of mathematics is not great and I've never seen something like (2) before, I'm confused with this \"for our purposes\". What does it actually mean? Is (2) true for _any_ linear operator or for a particular kind of linear operators, say, for self-adjoint linear operators on Hilbert spaces?"}
{"id":"68706","title":"Can massive fermions have zero modes?","text":"I'm confused about fermion zero modes in relation to instantons. I understand that instantons can create fermion zero modes, but it's not clear to me when a fermion has a zero mode. For example, consider QCD with nonzero fermion masses: i.e. there are explicit mass terms in the Lagrangian. Do these fermions have zero modes? Are they created by the QCD instanton? Is it generically possible to have a zero mode when the field has an explicit mass term, and if so, do those zero modes behave differently in an instanton background than the zero modes of massless fermions? Thanks."}
{"id":"45077","title":"Direct nuclear reaction in nuclear physics","text":"Time taken to occur a direct nuclear reaction is very low $10^{-22}$sec . I want to know the Importance of direct nuclear reactions."}
{"id":"48085","title":"NOT Universal Operator and Computational basis","text":"This is the relationship between density operator and Bloch vector: $$\\rho= \\frac{1}{2}({\\bf{\\hat{1}}}+{\\bf{b}}.\\boldsymbol{\\hat{\\sigma}})$$ We define the NOT Universal Operator in the following way: $$U: {\\bf{b}}\\to -{\\bf{b}}\/3$$ My question is - How does NOT Universal Operator act on the elements of the computational basis: $|0\\rangle, |1\\rangle$?"}
{"id":"29489","title":"Gravity and Magnetism on Stars","text":"I am familiar with the fact that Magnetars have a really strong magnetic field and an ordinary star such as the Sun has a very strong gravitational field. But what about the opposite? What is the strength of the _gravitational field_ of a Magnetar and how strong is the _magnetic field_ of an ordinary star?"}
{"id":"24220","title":"Current string-theory version of Big Bang","text":"I always wondered what the description of Big Bang would be like in string theory. How is it different from QFT version of Big Bang? Thanks."}
{"id":"79074","title":"What is meant by boiling off electrons in a heater coil?","text":"In my electricity and magnetism course, we used a Thompson tube to produce an electron beam. There is a heating element at the back of the tube and the lab manual claims that \"electrons are boiled off.\" Can someone explain what is meant by this? Boiled off seems an odd term to use but maybe if I understand it meaning, it won't seem so odd."}
{"id":"62277","title":"Smallest force to move a brick","text":"Having a brick lying on a table, I can exert horizontal force equal to $\\mu m g$ to a middle of it's side, and it will start moving (assume $\\mu$ is the friction coefficient). However, can I make the brick moving with less horizontal force? May be, applying it not to a middle of a side can help? I have no idea of how to calculate or estimate it, but it would be interesting to know."}
{"id":"123595","title":"How does the friedmon solution to Einstein's equations resolve paradox of bounded infinities?","text":"This article talks about a potential explanation of dark matter based on something called the \"friedmon.\" I have no interest in the dark matter question, but the article has made me curious about this friedmon thing. Googling \"friedmon\" turns up very little. The article says the friedmon is a solution to Einstein's equations of GR wherein an infinite expanse (field or space-time, I guess you would call it) sits as an object within another infinite expanse. Well, let me just quote the article: \"To an external observer, a friedmon looks like a micro blackhole with an electric charge the same as that of the electron. However, the friedmon's interior can be macroscopically large — up to the size of the known universe.\" I believe the notion of an infinite universe sitting as a finite object in another infinite field is also central to the multiverse theory. I'm just curious as to how you express this idea mathematically. At face value there appears to be a contradiction in the idea of an infinite universe which is also a finite object situated in some broader expanse. I would also appreciate if someone can point out a paper or something which shows the details of the friedmon solution to Einstein's equations of GR."}
{"id":"126273","title":"Does speed in space really matter?","text":"This is about validating the science from a science fiction novel. My understanding is that speed is all relative. So while I am in a car on the freeway my speed relative to another passenger is zero, relative to a street sign is 75, relative to a car travelling the opposite direction is 150, and relative to some point off the planet it is much greater. Now two cars traveling at the same speed and direction would have a relative speed of 0, but because of imperfect roads and wind resistance, the speed relative to earth would still impact the car's interactions. For example, walking between a board connecting the two cars would be much easier at 10 km\/h than at 100 km\/h. But the relative speed of the cars compared to a distant star has no measable impact. So how does this translate to two spaceships? Is a space ship traveling at 1000 km\/h relative to a star in the solar system any more likely to strike debris at a faster speed, than one that is stationary relative to the star? Likewise, is it more difficult for two ships traveling at 1000 km\/h relative to a star (both with the same heading, speed, etc.) to dock, than two ships stationary relative to that star? It would seem like acceleration would be the only influence, unless solar winds have a bigger impact then I imagine. Without resistance, it would require just as much effort from the manuvering thrusters to maintain a specific speed as it would be to hold still. Theoretically, no mater how fast, or in what direction I am traveling, there is some point, at some distance, in space, to which I am stationary (relatively)."}
{"id":"126278","title":"Does a spinning part affect the moment of inertia of a composite object?","text":"I have been going back through some Kleppner problems and have a doubt concerning problem 6.18. It states: > Find the period of a pendulum consisting of a disk of mass $M$ and radius > $R$ fixed to the end of a rod of length $l$ and mass $m$. How does the > period change if the disk is mounted to the rod by a frictionless bearing so > that it is perfectly free to spin? The first part (with the disk not free to spin) was reasonably straightforward. My only doubt was that I assumed the moments of inertia of the disk and rod could be added; this seems reasonable, but I don't quite know how to justify it rigorously (side-note: if anybody could give a hint on this, it would be highly appreciated). My result was: $$T=2\\pi\\sqrt{\\frac{MR^2\/2+Ml^2+ml^2\/3}{gl(M+m\/2)}}$$ For the second part, the issue I had was mainly conceptual... So when the disk is free to spin, it's no longer part of the rigid body; so it won't contribute to the moment of inertia, right? The problem comes here: earlier, to calculate the torque on the rigid body about the pivot, I said that: $$\\tau=R_{CM}\\times W$$ Where $R_{CM}$ is the center of mass of the rigid body. This formula comes from a summation over the torques on every small mass in the rigid body, so I figured that the torque on the rigid body, once the disk was no longer a part of it, would depend only on the center of mass of the rod (the disk would no longer affect the 'effective' center of mass that the torque acts on). This gives $$T=2\\pi\\sqrt{\\frac{2l}{3g}}$$ I checked my answer with this website (pages 5-7) afterwards, and the first part agreed but the second part was in disagreement; the problem was that in that site, $R_{CM}$ was still 'affected' by the spinning disk. Why is this so? (I explained above why I think that $R_{CM}$ should not contribute.)"}
{"id":"51465","title":"proof of radius of convergence of perturbation series in quantum electrodynamics zero","text":"Can anyone show detailed proof of why radius of convergence of perturbation series in quantum electrodynamics is zero? And how is perturbation series constructed? So, as this argument requires imaginary electric charge of bispinor field (negative coupling constant), how is this argument considered valid?"}
{"id":"23624","title":"why  water molecules in vapor contributes to the green house effect and water condensed in drops not?","text":"I am a teacher planning a unit on climate change. I came across this statement: \"Clouds are water vapor (green house gas) and water molecules will reflect heat into space… actually clouds are condensed water and the water molecule absorbs the heat, lowering the temperature of the atmosphere around the cloud.\" I don't understand why water vapor is a green house gas whereas the water molecule is not. I also don't understand why the water molecule absorbs heat and does not reemit it.... Please explain!"}
{"id":"47619","title":"Calculating the speed of an object moving relative to another moving Object (Frame of reference)","text":"> **Possible Duplicate:**   >  How to deduce the theorem of addition of velocities? Let's say that you are in a rocket speeding at 90% the speed of light away from Earth. Now fire a bullet inside the rocket that is also going at 90% the speed of light. According to Newtonian physics, we add both velocities. Thus, the bullet should be going at 180% the speed of light. But we **now** know that according to Einstein the sum of these velocities is actually close to 99% the speed of light, because length contracts and time slows down. But how can you determine mathematically, that bullet is going at 99% the speed of light?   Additionally, can you also calculate, how much time is slowed down, if you know the velocity? Is there any formula?"}
{"id":"90865","title":"Speed of light and relativity","text":"Suppose there are two observers $A$ & $B$ both are in motion, now $A$ sees $B$ is moving with speed $'u'$. A says that another object $'d'$ is moving with speed $c+u$ w.r.t. A in the same direction as $B$. What stops him from saying this ? The formula $\\frac {u+v}{1+\\frac{uv}{c^2}}$ is such that $'v'$ is speed of the object $'d'$ w.r.t.$ B$ and not $A.$ and $u$ is the speed of $B$ w.r.t. $A$, now even though $B$ also sees that the object $'d'$ moves faster than the speed of light. He can still always see light to travel with the speed of light and also he can be an inertial observer? What is the problem here ?"}
{"id":"93655","title":"Can a revolving body self-gravitate?","text":"If a body is revolving around a point at radius R with tangential velocity V, does General Relativity predict that at some tangential speed, the body will revolve around the point without any external forces acting on it? Since General Relativity models gravity as a curvature of spacetime, I was thinking that an observer at the center of rotation of the revolving body would measure some curvature of space as a result of the revolution. This is because the body would be contracted in the tangential direction relative to the observer, such that C<2*PI*R where 'C' is the circumference of the revolving body measured by the observer (the revolving body is thought of as a ring of particles). This implies a curvature of space, which would be equivalent to a gravitational force."}
{"id":"24042","title":"Baryon asymmetry","text":"Baryon asymmetry refers to the observation that apparently there is matter in the Universe but not much antimatter. We don't see galaxies made of antimatter or observe gamma rays that would be produced if large chunks of antimatter would annihilate with matter. Hence at early times, when both were present, there must have been a little bit more matter than antimatter. This is quantified using the asymmetry parameter $\\eta = \\frac{n_{baryon} - n_{antibaryon}}{n_{photon}}$ From cosmological measurements such as WMAP, $\\eta \\approx (6 \\pm 0.25) \\times 10^{-10}$ However, the source of baryon asymmetry is said to be one of the Big Problems of Physics. What is currently the state of the art regarding this puzzle? What's the best fit we can get from the Standard Model? What do we get from lattice simulations?"}
{"id":"129530","title":"Why does the second law of thermodynmics only occur in the time dimension?","text":"If I break an egg, how broken it is depends on \"when\" I am not \"where\" I am. Why is the time dimension special?"}
{"id":"73736","title":"Does gas spread out equally everywhere?","text":"An excerpt from this page: > Gases can fill a container of any size or shape. It doesn't even matter how > big the container is. The molecules still spread out to fill the whole space > equally. That is one of their physical characteristics. If a fixed quantity of gas is let out in a limited space, will it spread out equally and maintain a fixed gas distribution throughout the space? Or, does it go where the gravitational pull is maximum? And what factors affect the distribution of gas in a given area?"}
{"id":"129538","title":"Why does magnetic field increase when the number of coil turns increase?","text":"Since we all know that to increase the magnetic field we can do these of the following things: (i) Increase the number of turns in a coil, (ii) Increase the current following through conductor, and there would be more but I know only these (upto my standard - Xth ) now if I use the first method to increase the magnetic field then the length of conductor will also increase because I am increasing the number of turns; say I am increasing the length of conductor by $x$ meters then as we all know that: $R = \\rho l\/a$ you know resistance is equal to resistivity multiplied by length of conductor and then divided by the area of cross section. According to this $R$ is directly propotional to length of conductor and by Ohm's law $R$ is inverously proportional to current Therefore if I increase the length by $x$ m of conductor then the resistance will also increase at a rate of $x \\Omega $ and current will decrease at $x$ Ampere, then how will the magnetic field increase if the current is being reduced?"}
{"id":"129539","title":"Is there any difference between a proper photon and a photon of thermal radiation?","text":"Does a thermal photon radiate spherically whereas a photon radiates in a circle\/beam? How do you tell a thermal photon of, say 3 eV, from a photon of light?"}
{"id":"23996","title":"Simple Projectile Motion Question","text":"A volcano erupts 50m below the sea level. A rock leaves the crater at 20 m\/s at an angle 30 deg with the vertical line. The rock has a mass of 15kg. IGNORE WATER RESISTANCE. It gets out of the water, reaches a max height and falls back it. When does it reach 30 below sea for the first and second times? $a = <0, -9.8>$ $v = <20sin(30), -9.8t+20cos(30)> = <10, -9.8t+17.32>$ $s = <10t, -4.9t^2+17.32t>$ For the object to be 30m below sea level, y=20m, so $s_y=20 \\implies -4.9t^2+17.32t-20=0$. However, no real solutions exist to this. This is so simple, I apologize for asking, but what am I overlooking?"}
{"id":"92305","title":"Inflation of universe and Information stored therein","text":"In an inflationary model of our universe, the space-time expands within a finite boundary. Some other models are also proposed where the universe is not explained to have been expanding rather the masses of the objects are presented to be expanding with time. If so, what happens to the information scattered in space? Does it inflate too? And in literal sense, what is the meaning of inflation of information?"}
{"id":"112503","title":"Why doesn't amplitude affect the speed of sound?","text":"I understand why amplitude doesn't affect the speed of the sound AFTER the 'leading compression'. The extra force provided on one stage of the cycle is countered on the other stage. But shouldn't the 'leading compression' travel faster if amplitude is greater? Because wave motion hasn't been set up yet shouldn't all compressions face the same amount of resistance while moving forward?"}
{"id":"73082","title":"Observed angular velocity and motion parallax","text":"I would be highly obliged if someone would guide me in the right direction regarding this. I am trying to understand and mathematically explain the relation between angular velocities as observed by a moving observer of an otherwise, static object. For example- A tree passes by, as we travel in a bus. So what is the relation of the angular velocity of the tree as observed by an observer in the bus with the velocity of the bus and the distance, so involved? Also please point me towards any published material on this topic. Thanks."}
{"id":"73081","title":"A question about Poincare invariance of Polyakov action","text":"I have a question the variation of the Polyakov action, related to this Phys.SE post. For Polyakov action $$ S_p[X,\\gamma]=-\\frac{1}{4 \\pi \\alpha'} \\int_{-\\infty}^{\\infty} d \\tau \\int_0^l d \\sigma (-\\gamma)^{1\/2} \\gamma^{ab} \\partial_a X^{\\mu} \\partial_b X_{\\mu}.$$ Consider the variation (p. 14 in Polchinski's string theory book) $$ \\delta S_p~=~ \\frac{1}{2\\pi \\alpha'} \\int_{-\\infty}^{\\infty} d \\tau \\int_0^l d \\sigma (-\\gamma)^{1\/2} \\delta X_{\\mu} \\nabla^2X^{\\mu}$$ $$ - \\frac{1}{2\\pi \\alpha'} \\int_{-\\infty}^{\\infty} d \\tau (-\\gamma)^{1\/2} \\delta X_{\\mu} \\partial^{\\sigma} X^{\\mu} |^{\\sigma=l}_{\\sigma=0} \\tag{1.2.27} $$ The open-string and closed-string boundary conditions $$\\partial^{\\sigma} X^{\\mu}(\\tau,0)=\\partial^{\\sigma} X^{\\mu}(\\tau,l)=0 \\tag{1.2.28} $$ $$ X^{\\mu}(\\tau,l) = X^{\\mu}(\\tau,0), \\partial^{\\sigma} X^{\\mu}(\\tau,0)=\\partial^{\\sigma} X^{\\mu}(\\tau,l), \\gamma_{ab}(\\tau,l)= \\gamma_{ab}(\\tau,0) \\tag{1.2.30} $$ will make the \"surface term\" (the second term in the RHS of Eq. (1.2.27)) vanishes. It is said (p. 14-15 of Polchinski) > _The open-string boundary condition (1.2.28) and closed string boundary > condition (1.2.30) are the only possibilities consistent with > $D-$dimensional Poincare invariance and the equation of motion._ The boundary condition (1.2.28)\/(1.2.30) will make the surface term vanish. The resulting equation of motion is tensor equation.   $$(-\\gamma)^{1\/2} \\nabla^2 X^{\\mu}=0$$ My question is, why non-vanishing surface term will break the Poincare invariance? E.g. a Lorentz transformation $X_{\\mu} \\rightarrow \\Lambda_{\\mu}^{\\nu} X_{\\nu}$ will not affect $\\sigma,\\tau$ space"}
{"id":"55908","title":"Significance of the the Lagrange multipliers in statistical mechanics","text":"In classic thermodynamics one can derive the Maxwell Boltzmann statistics by solving a Lagrange multipliers equation. In this process a new parameter $\\beta$ is introduced to take account of the total energy constraint, $$L = \\beta(E-\\sum_i p_i \\epsilon_i)+...$$ and it is later connected to $\\frac{1}{k_B T}$ by inserting our knowledge about perfect gases. On a side note, about every book around seems to skip the passage with a frustrating \"it can be shown that\". My problem is that, given similar Lagrange multipliers equation, I'm not sure whether one can in general relate $E$ and $\\beta$: it is evident that $\\beta$ has no physical significance by itself, and it's always bounded to some parameter in the constraints that it is multiplying. For example, say that I have an array of cells that can be either occupied by 0, 1 or 2 dots. I have some constraints, one of which is the density of the dots $\\rho$. The Lagrange multipliers equation has the form $$L = \\gamma(\\rho-p_1-2p_2)+...$$ where $p_i$ is the probability of having a cell with $i$ dots. Is there some way to eliminate the $\\gamma$ parameter in the final explicit formula for $p_1$, and use $\\rho$ instead?"}
{"id":"55905","title":"A Book about the Bohr-Einstein debate?","text":"A book about the Bohr-Einstein debate? Is there any book that details the correspondence between the two? The only books I could find are popular science books, I wonder if there is a book that lists the correspondence in a more 'raw' form."}
{"id":"98485","title":"Calculating the Center of Mass","text":"We have a homogeneous body that looks like this: ![Homogeneous body](http:\/\/i.stack.imgur.com\/j9fuR.jpg) I have tried dividing the body into different parts using the following definition: **R** g * **A** = **R** 1 * **A** 1 + ... **R** n * **A** n I was thinking I could first divide it into two parts using the x axis, and then add those two together using the definition above. But how can I calculate the coordinates for the center of mass for the lower part?"}
{"id":"26482","title":"What does velocity dispersion (sigma) reveal about a galaxy?","text":"I'm getting hung up on this term. In studying SMBHs, I see that velocity dispersion strongly correlates with mass. Just what is the velocity dispersion? How can the velocity dispersion of the galaxy be expressed in one figure (sigma) if it has to be measured all over the galaxy? I can imagine the velocity dispersion changes with radius, so \"which\" point is used? Why exactly is it that higher velocity dispersion is correlated with higher Mass? And is all this referring to the bulge only or to the entire galaxy? So to get velocity dispersion, is \"one slit\" at the center enough or do you scan across? I hope someone can answer this without relying heavily on the maths. I am really just trying to understand the concept, because I think that velocity dispersion is real, not just a mathematical construct."}
{"id":"104667","title":"Why such hypersurface orthogonal vector leading to $g_{0i}=0$ for $i=1,2,3$?","text":"Suppose that the hypersurface orthogonal co-vector $W$ us perpendicular to the family of hypersurface defined by a function $\\varphi$ with $\\varphi=constant$. If we choose a coordinate in which $x^0=\\varphi$, how to prove that the metric components $g_{0i}=0$ for $i=1,2,3$? My try is as follows. Because $W$ is perpendicular to the surface $x^0=c$, let $U=(c,0,0,0)$, then for every vector $V=(c,x^1,x^2,x^3)$ inside $x^0=c$, we must have $$ W_a(V^a-U^a)=0 $$ which leads to $$ W_i=0,\\;\\;i=1,2,3 $$ Now because $W$ is HSO, we must have $$ W_{[a}\\nabla_bW_{c]}=0 $$ Then I am stuck. Since when $W_i=0,i=1,2,3$, the above equation holds automatically, thus can give me nothing. Could anyone help?"}
{"id":"26151","title":"Galaxies seen from Earth","text":"If we observe two galaxies from the Earth that are diametrically opposed and each 10,000 light-years from the Earth, will the separation distance between the galaxies be 20,000 light-years? Really, the question is: if the galaxies instead were separated from Earth by 10 billion light-years, would the separation distance between the galaxies then be 20 billion light-years?"}
{"id":"25598","title":"Are telescopes with a concave lens useful for astronomy?","text":"Are telescopes with a concave lens (instead of convex ones) ever useful for astronomy? And if so, where are they used? Do they ever affect resolving power?"}
{"id":"63845","title":"Will changing amplitude change the frequency?","text":"Will changing the amplitude change the frequency of a wave, or is it possible for a specific frequency (50 Hz. for example) to generate from shifting amplitude patterns?"}
{"id":"4812","title":"Time evolution of wave spectrum","text":"A useful way of thinking (not only) oceanic waves is to consider them as a superimposition of linear modes: the elevation η of the sea surface is given by: 1: $\\eta({\\bf x}, t) = \\int_{-\\infty}^{\\infty} \\hat{\\eta}({\\bf k}) e^{i({\\bf k \\cdot x} - \\omega t)} d{\\bf k}$ where $\\omega$ and ${\\bf k}$ are related by a dispersion relation, and $\\hat{\\eta}({\\bf k})$ is a sort of time independent Fourier transform. $\\hat{\\eta}$ is given by: 2: $\\hat{\\eta}({\\bf k}) = const \\int_{-\\infty}^{\\infty} e^{-i{\\bf k \\cdot x}}(\\eta({\\bf x}, 0) + \\frac{i}{\\omega({\\bf k})} \\frac{\\partial \\eta({\\bf x}, 0)}{\\partial t}) d{\\bf x}$ meaning that if we know $\\eta({\\bf x}, 0)$ and its time derivative we can determine the amplitude of all the normal modes, and neglecting non linear interactions between the modes we know the evolution of the wave. Clear and smart. What's the simples way to obtain relation 2 from relation 1? Thanks!"}
{"id":"2045","title":"Ring theory in physics","text":"Surely group theory is a very handy tool in the problems dealing with symmetry. But is there any application for ring theory in physics? If not, what's this that makes rings not applicable in physics problems?"}
{"id":"6428","title":"Snell's law starting from qft?","text":"Can one \"interpret\" Snell's law in terms of QED and the photon picture? How would one justifiy this interpretation with some degree of mathematical rigour? At the end I would like to have a direct path from qft to snell's law as an approximation which is mathematically exact to some degree and gives a deeper physical insight (i.e. from a microscopic = qft perspective) to Snell's law."}
{"id":"79813","title":"Vapor Pressure and Container","text":"The following comes from my text and concerns vapor pressure: Consider a liquid in a closed, cylindrical container that is half full. The molecules at the surface of the container and along the edges have the fewest neighbors; therefore, they will evaporate more quickly than other molecules. This intuitively makes sense. The text also says that vapor pressure is independent of the shape and volume of a container. Now consider the above situation in two cylinders of the same volume. The first has a much smaller radius than the second. Since the smaller radius cylinder has a greater proportion of its molecules along the edges, I would think that it would have a slightly higher vapor pressure. Is this correct? Perhaps I'm just over-complicating things. Any help is appreciated!"}
{"id":"79816","title":"Why does an armature of a generator rotate?","text":"Why does an armature of a generator rotate? a) like poles repel b) unlike poles attract c) current flows up d) an outside force turns it I am thinking that (d) is the correct answer but still not sure of it"}
{"id":"112250","title":"If the universe has a shape, what is the surface?","text":"Since the idea of the universe was that it was expanding, is there a way to know the surface area of the universe like the growing 3D object it is illustrated? The idea was that the universe was expanding, and then I think that the surface would expand, but my concept is limited about such a metric. I think I read about the mass of the universe, but is there an agreed measure of the surface of the (bounds of the) universe? Since the universe is assumed to have a shape, then what the surface be?"}
{"id":"57502","title":"Rate of Cooling","text":"I was given the following problem: > Two objects have a temperature difference of 20 degrees C at time 0. Two > minutes later, they have a temperature difference of 18 degrees. How long > does it take for them to have a difference of 10 degrees. Initially, I tried to approach the problem with Newton's Law of Cooling. However, I believe my solution is flawed: $$\\frac{dT}{dt} = -kT$$ letting $T$ be the temperature difference between the two objects. The solution to the above equation is $T(t) = ce^{-kt}$. Inserting the provided values, I found that $c = T(0) = 20$ and $T(2) = 18 = 20e^{-2k}$, so $k = 0.05268$. Setting $20e^{-0.05268t} = 10$, I found that $t = 13.2$ minutes. **Is Newton's Law of Cooling applicable if I simply let $T$ be the difference in temperatures between two objects?** Also, the problem implies that both objects are changing temperature. **What do I need to do to accommodate for both objects changing temperatures?**"}
{"id":"123198","title":"Tensors in special relativity","text":"I'm trying to understand tensors, but I've come across the following question: >   1. Let $T^{\\mu\\nu}$ by a $(2,0)$ tensor. Give the definitions of > $T_\\mu^{\\,\\nu}$, $T_{\\mu\\nu}$, and $T^{\\mu}_{\\,\\nu}$. Then give explicit > formulas for the following components in terms of components of > $T^{\\mu\\nu}$: > $T_{00},\\,T_{10},\\,T_{12},\\,T_0^{\\,1},\\,T_1^{\\,1},\\,T_0^{\\,0},\\,T^1_{\\,1}$   >  $$T_\\mu^{\\,\\nu}\\equiv\\eta_{\\mu\\sigma}T^{\\sigma\\nu} \\\\\\ > T_{\\mu\\nu}\\equiv\\eta_{\\sigma\\mu}\\eta_{\\rho\\nu}T^{\\sigma\\rho}\\\\\\ > T^\\mu_{\\,\\nu}\\equiv\\eta_{\\sigma\\nu}T^{\\mu\\sigma}$$ > I'm finding it hard to understand these definitions. I really don't know what these tensors represent or what it means when both the indices are raised? Also are these standard definitions in special relativity or is it more likely that they are just made up by my professor for this assignment? I've always thought that things with two indices are matrices in general."}
{"id":"107465","title":"Torque and angular acceleration with bicycle wheel","text":"This might be a simple problem for many of you. However, please help me understand it too. I have been looking trough a lot of materials online, and I still have the following questions, that would help me a lot. I saw this video on youtube: see wheel in action If we consider the following model: ![enter image description here](http:\/\/i.stack.imgur.com\/bGYsu.gif) Please clarify for me the following questions:   1. I believe the gravity is not pulling the wheel down because the torque resulting from multiplying gravity by the l vector is pointing sideways. In other word, the down-pointing gravity vector is transformed into a vector pointing sideways and spinning the wheel around the string. (Is this how force multiplication works? do the multiplied factors transform into the resulting force?)   2. If the wheel would keep a constant angular velocity(assuming no friction) the wheel would spin indefinitely without rising or falling. What would happen if the angular velocity would be increased while spinning? would the wheel rise (ie the free floating end would rise)?   3. Mathematically what causes the wheel to fall when the angular velocity decreases? The only changing quantity is L, which depends on the speed of the spinning wheel. But the direction of L is not changing. And from (1) above, the gravity should not be pointing downwards. So why does the wheel fall when L gets smaller(by wheel falling I mean the free floating end goes down, until it is under the string)"}
{"id":"112132","title":"Energy fluctuations in quantum canonical ensemble","text":"How would you go about showing that in the quantum canonical ensemble (that is, in the density matrix and operator formulation), the energy fluctuations, namely $\\langle H^2\\rangle - \\langle H\\rangle^2$ is the same as in the classical case, i.e. $k_b T^2 C_V$? I've seen a lot of books asserting this, but can't find a formal derivation similar to the classical one, as given here, for example."}
{"id":"70224","title":"What do I need to know to calculate the price per km for plug-in hybrid or electric cars?","text":"Knowing the per kWh price of electricity, what other factors are needed to calculate how much would one pay per km traveled on electric power compared to gasoline power?   EDIT: For example, the Chevrolet Volt reports a 35 kWh per 100 miles. Supposing the price for electricity is 11 cents per kWh, so this mean one can drive 100 miles for $3.85, or a mile for 3.8 cents?"}
{"id":"83279","title":"Why does the fluid inside a cup not spin when the cup is spinning, but starts to spin when the cup stops spinning?","text":"How come when I spin a cup with water in it, the water does not spin, but the moment I stop spinning the cup, the water starts spinning the other way?"}
{"id":"83275","title":"Dark matter detection","text":"In the detection of weakly interacting massive particles (WIMPs), which is the basis of dark matter, what is the use of the tank filled with liquid xenon? I mean, how does the releasing of photons contribute in the detection of WIMPs?   To relate to what I'm talking about, you can follow this link:   http:\/\/www.universetoday.com\/105943\/new-dark-matter-detector-draws-a-blank-in- first-test-round\/"}
{"id":"83270","title":"Does coffee cool faster than tap water?","text":"A friend of mine, based on casual observation, believes that coffee will cool faster than ordinary hot tap water. Being curious about this I have tried to investigate it myself, but I'm not well versed in physics and thermodnamics. Also, I know that there are many variables to this, so I will attempt to ask the question as specifically as possible: I have 8oz of water (we can assume that it is impure, but we can not know what those impurities are) that is heated to 200 degrees F in a microwave (let's say we have a really precise microwave that can do that easily) and that is then placed in a standard, throw-away styrofoam coffee cup. I have an exact duplicate scenario, except that this time we will also brew coffee in the water for several minutes, filter out the coffee using a standard coffee filter (we can assume some coffee particles remain in the water, but we can not know anything about those coffee particles) and then re- heat the water to 200 degrees F before adding it to the foam coffee cup. If each of these coffee cups are then placed on separate tables in a room- temperature room (let's say 70 degrees F) and allowed to sit, undisturbed, will the liquids cool at roughly equivalent speeds (i.e., will the water and the coffee reach 180 degrees F at roughly the same time) or will one cool faster than the other? For either scenario, why? If there's any clarification or wording choices that should be changed, please let me know as I am genuinely curious. My guess is that there's something to do with the specific heats begin different, and that the surface tension of the coffee mixture will be disrupted due to the coffee particles. But, again, I don't really know :-)"}
{"id":"53169","title":"Room temperature and fan orientation","text":"So I'm in a tiny dorm room and I normally point my fan blowing outside the window to cool my room off. I've been in some debates on blowing air out or in is more effective, so I'm hoping to get some empirical data to back up my claim. Things I have   * A Heater with only low, medium, high settings.   * The Fan.   * I do _not_ have a thermometer. So the best I could think of was melting ice cubes. I don't care about the exact temperature, just which method is more effective.   I've outline the basis of my experiment: Shut the windows and turn the heat on high, 15 minutes Open the windows, turn off the heat, turn on the fan on high, 15 minutes Leave $n$ ice cubes out and take the average of how long it takes them to melt. All the ice cubes are from the same tray. Can anyone devise a better experiment with no tools..."}
{"id":"107662","title":"Current when the slide wire attains a constant velocity","text":"In a slide wire generator, when the slide wire attains a final constant velocity, is the current in the circuit 0? It does not make sense if it is 0 as the area keeps increasing, allowing more flux to pass, meaning more current would need to be generated to counteract the increasing flux."}
{"id":"107665","title":"Does the cell phone make dipole particles in our body dance the same way Microwave oven does in food","text":"I am a non-physics person trying to wrap my head around some EM radiation facts. Please help me with this. I have read about Dielectric Heating on Wikipedia. And watched quite a few videos on internet on how Microwave oven works. It all boils down to the following facts. Please correct me if I am wrong on any of them.   1. The Microwaves are absorbed by Water molecule and the Negative charged Hydrogen atoms want to align with negative radiation and Positive Oxygen atom wants to face positive radiation which makes the water molecule dance.    2. Microwave oven Magnetron changes the negative-positive polarity in charge as often as 2.4 billion times in a second which makes Water molecules dance really really fast.   3. This dance makes the water molecules heated. Now coming to my question, I learned that Cell Phones use UHF frequencies such as 800 MHz and 400 MHz. Does that mean water molecules in our body are dancing at the rate of 800\/400 steps per seconds ? Why don't they heat up then ?"}
{"id":"107668","title":"Would electromagnetic radiation impart a pressure on a surface of neutrons?","text":"In my physics textbook, it says that a qualitative way to envision pressure from EM waves is as follows: the electric field drives charges in the $x$ direction, and the magnetic field then exerts on them a force $q\\vec{v}\\times\\vec{B}$ in the $z$ direction. The net force on all the charges in the surface produces the pressure. But if this is the case, then does that mean that EM waves wouldn't impart any pressure on a surface of neutrons, given there are no charges to move? Let me know if this is just an analogy that doesn't hold... but in this case, is there another intuitive way to understand why there's a pressure associated with the waves? I understand mathematically, but not physically."}
{"id":"39357","title":"Can cos(x) or sin(x) be the function of stationary action?","text":"Is there a way to express $\\cos(x(t))$ (or $\\sin(x(t))$) as the solution to the Euler-Lagrange equation, in other words is there a sense in which this function is the path of stationary action?"}
{"id":"47503","title":"Shouldn't the change in kinetic energy be more in a moving elevator from a stationary frame of reference?","text":"Consider an elevator moving down with uniform velocity. A person standing inside watches an object fall from the ceiling of the elevator to the floor. Say the height of the elevator is $h$. Then the work done by gravity in that frame of reference should be $mgh$. But consider this same event being watched by someone else in the stationary frame of reference. In his reference frame, the object travels a larger distance as it falls from the ceiling to the floor of the elevator because the floor itself is moving downwards (one can calculate this extra distance covered to be $u \\sqrt{\\frac{2h}{g}} $) and hence the change in kinetic energy should be more in that frame than in the moving frame! I just can't seem to figure out where I'm going wrong here. I'm probably missing something very obvious. So I would be very grateful if anyone could explain this to me. **Edit:** Okay, let's say the object is a clay ball and it collides with the floor inelastically such that it's kinetic energy is converted into heat. In the moving frame of reference the heat would be simply equal to $\\frac{1}{2}mv^2$ which is equal to $mgh$. In the stationary frame of reference it would be equal to $\\frac{1}{2}mv^2-\\frac{1}{2}mu^2$ since the ball after colliding is moving with speed $u$. This can be calculated to be equal to $mgu\\sqrt{\\frac{2h}{g}} + mgh$ which is clearly greater than the heat produced in the frame attached to the elevator and this is a contradiction because the heat measured in any frame should be the same."}
{"id":"47502","title":"Does black hole evaporation respect quantum superpositions?","text":"I've confused myself about the following scenario: Suppose you make a black hole out of states with spin aligned into one direction, say the positive x-direction, and let's call this \"up\". Then the so-created black hole has an non-zero angular momentum because the expectation value of the spin is nonzero. (Let's assume there's no other net contribution to the angular momentum, just for simplicity. I don't think this matters.) The so-created black hole evaporates in the semi-classical limit with an energy spectrum that depends on this expectation value of the total angular momentum. (Forget about what happens in the final stage where the semi-classical approximation breaks down, I don't think this matters.) Next you make a black hole out of the same initial state with the only difference that all spins are down. Same thing: the emission spectrum - the distribution of outgoing particles over energy - depends on the expectation value of the angular momentum, at least in the semi-classical limit we're pretty sure of that. Finally, you make a black hole out of a superposition of the two initial states. The expectation value of the total angular momentum is now zero, correspondingly the angular momentum in the semi-classical Kerr-background is zero. What is the decay spectrum (in the early radiation) in this case? Is the outgoing radiation a) a superposition of the radiation that one obtained from the purely up and purely down case? But then it would depend on the angular momentum which the black hole doesn't have. Or does it b) not depend on the angular momentum of the superposed states (note that what is and what isn't a superposition is ambiguous anyway) but then the evolution doesn't seem to be linear? (In the sense that |psi_in> + |psi'_in> doesn't result in |psi_out> + |psi'_out> when |psi_in> resulted in |psi_out> and same for the dashed case)."}
{"id":"114696","title":"The $U(1)$ charge of a representation","text":"My question is about the reduction of a representation of a group $SU(5)$ to irreps of the subgroup $SU(3)\\times SU(2) \\times U(1)$. For example the weights of the 10 dimensional representation of SU(5) are ![enter image description here](http:\/\/i.stack.imgur.com\/SlEAK.png) One can identify the irreps of the subgroup by regrouping the dynkin labels into $((a_3 a_4) ,(a_1), a_2)$ such that (denoting $-1$ by $\\bar{1}$): $$ (1,1)_{Y} \\rightarrow \\left\\\\{ \\begin{array}{l l} (0 0,0,1 ) \\end{array} \\right. $$ $$ (\\overline{3},1)_{Y} \\rightarrow \\left\\\\{ \\begin{array}{l l} (0 1,(0),\\bar{1}) \\\\\\ (1 \\bar{1},(0),\\bar{1})\\\\\\ (\\bar{1}0,(0),0) \\end{array} \\right. $$ $$ (3,2)_{Y} \\rightarrow \\left\\\\{ \\begin{array}{l l} (1 0,1,\\bar{1}) \\\\\\ (\\bar{1} 1,\\bar{1},1)\\\\\\ (0\\bar{1},\\bar{1},1)\\\\\\ (1 0,\\bar{1},0)\\\\\\ (\\bar{1}1,1,0)\\\\\\ (0\\bar{1},1,0) \\end{array} \\right. $$ My problem is: how can I derive the $Y$ charge of the $U(1)$ factor for each of these from the Dynkin labels? * * * **Edit** The metrictensor for SU(5) is thus $$G= \\frac{1}{5}\\left( \\begin{array}{cccc} 4 & 3 & 2 & 1 \\\\\\ 3 & 6 & 4 & 2 \\\\\\ 2 & 4 & 6 & 3 \\\\\\ 1 & 2 & 3 & 4 \\end{array} \\right). $$ However in the reference, Slansky, on page 84 the same exercise is done but the axis have negative values... $$\\tilde{Y}^W = \\frac{1}{3} [-2 \\;1\\, -1\\; 2]. $$ How come they do not agree?"}
{"id":"88369","title":"Why must the final state be stationary?","text":"I faced the following sentences: > We consider a gravitational collapse taking place in this spacetime. The > singularity theorems assure us that a singularity will form. The assumption > that the cosmic censorship conjecture is valid will then imply that the > collapse must produce a black hole which will settle down to a stationary > final state. Can anyone explain to me why **The assumption that the cosmic censorship conjecture is valid will then imply that the collapse must produce a black hole which will settle down to a stationary final state**? Why must the final state be stationary?"}
{"id":"88364","title":"4 element super conductor without oxygen and iron","text":"I am trying to find a list with super conductors. In particular I am searching for a superconductor, that consists of 4 elements none of which is Oxygen. The reason I rejected Oxygen, is because(from the kinematic factor) the maximum energy from scattering with the Oxygen is about $730 keV$. Therefore-unfortunately-I have to reject _YBCO_ superconductors. In particular I have to analyse an RBS spectrum of a supposed 4-element well known super conductor. I can't find any 4-element-superconductor without oxygen. Is there anyone, or it might be the case where the sample is bad described? Just for reference the RBS spectrum is ![enter image description here](http:\/\/i.stack.imgur.com\/eEndF.png) **EDIT** I also tested iron based superconductors, but it seems that I need a heavier element. For instance a superconductor containing $Ba, K, Fe, As$ has the following RBS spectrum ![enter image description here](http:\/\/i.stack.imgur.com\/P5ssM.png) Please ignore the concentrations... The analysis indicates that the superconductors consists of 4 elements which are   * Bi(28.5%)   * Sr(28.4%)   * Cu(28.2%)   * K(14.8%) Does a superconductor like that really exists? I mean OK about Bismuth or Copper or even Strontium...How about Potassium? The combination of those doesn't seem right to me..."}
{"id":"88367","title":"Radio vs optical telescope imaging","text":"As I understand, the visible light from an optical telescope is focused on a sensor which correlates light exposure to an electrical voltage, which is then converted to an image. A single antenna radio telescope's signal is focused on some sort a sensor but this signal represents only a single 'pixel.' To generate a composite image of an object, a time intensive scan pattern needs to be carried out. Why does this difference exist? Why does a radio telescope see a single pixel, while optical telescopes see an entire image? Are radio telescope sensors only a single pixel, or is a radio telescope itself, regardless of the sensor, only capable of seeing a single pixel?"}
{"id":"11732","title":"How do I calculate the power consumed by a lightbulb?","text":"I'm studying a lightbulb and its variable resistance, given by the expression: $R(T) = Ro[1 + α(T-T_0)]$, where $R_0$ is the resistance of the lamp at $T_0$. In this case, $R$ is not given by Ohm's law ($V=Ri$). So, which expression can I use to calculate the power consumed by a lightbulb?   * $P = R\\cdot i^2$   * $P = \\frac{V^2}{R}$   * $P = V\\cdot i$ (I don't think I can use this one) I have measured the current and the e.p.d. at the bulb using instruments."}
{"id":"130197","title":"In KK theory, is proper time defined using the 5 dimensional or the 4 dimensional line element?","text":"Let's consider five dimensional KK theory. This is Klein's metric $\\hat{g}_{AB}= \\begin{pmatrix} g_{00}+A_{0}A_{0}&g_{01}+A_{0}A_{1}&g_{02}+A_{0}A_{2}&g_{03}+A_{0}A_{3}&A_ 0\\\\\\ g_{10}+A_{1}A_{0}&g_{11}+A_{1}A_{1}&g_{12}+A_{1}A_{2}&g_{13}+A_{1}A_{3}&A_ 1\\\\\\ g_{20}+A_{2}A_{0}&g_{21}+A_{2}A_{1}&g_{22}+A_{2}A_{2}&g_{23}+A_{2}A_{3}&A_ 2\\\\\\ g_{30}+A_{3}A_{0}&g_{31}+A_{3}A_{1}&g_{32}+A_{3}A_{2}&g_{33}+A_{3}A_{3}&A_3\\\\\\ A_{0}&A_{1}&A_{2}&A_{3}&1 \\end{pmatrix}$ my question is the following, is proper time defined using the five dimensional line element? $d\\tau^2=\\sqrt{-\\hat{g}_{AB}dx^{A}dx^{B}}$ or is it defined as usual using the 4 dimensional line element? $d\\tau^2=\\sqrt{-g_{\\mu\\nu}dx^{\\mu}dx^{\\nu}}$ in either case, why?"}
{"id":"100061","title":"A paradox with spin: Is spin a physical degree of freedom?","text":"Suppose I want to calculate the state associated with a spin particle under a magnetic field. I suppose the particle interacts via the Zeeman effect, and only through it. Then I want to resolve the Schrödinger equation: $$\\mathbf{i}\\hbar\\dfrac{\\partial\\Psi}{\\partial t}+\\dfrac{\\hbar^{2}}{2m}\\dfrac{\\partial^{2}\\Psi}{\\partial x^{2}}=h_{Z}\\sigma_{z}\\Psi\\left(x,t\\right)$$ with $h_{Z}$ the Zeeman term. All other terms are usual. To resolve, I apply the transformation $$\\Psi=e^{-\\mathbf{i}h_{Z}\\sigma_{z}t\/\\hbar}e^{-\\mathbf{i}Et\/\\hbar}e^{\\mathbf{i}kx}\\Psi_{0}$$ with $\\Psi_{0}$ a constant taken to be real for commodity, and the dispersion relation reads $$E=\\dfrac{\\left(\\hbar k\\right)^{2}}{2m}$$ as usual for a **free** particle. There is no mark of the spin splitting we are used to, like the energy doublet $E_{\\pm}=E_{0}\\pm h_{Z}$ due to the Zeeman term. Of course this is just a redefinition of the energy origin, done in two different ways for spin up and spin down (or something like that). If I calculate observable like the density $\\rho\\propto\\left|\\Psi\\right|^{2}=\\Psi_{0}^{2}$ or the current $j\\propto\\Im\\left\\\\{ \\Psi\\partial_{x}\\Psi^{\\ast}\\right\\\\} =k\\Psi_{0}^{2}$ again I find no _measurable_ signature of the spin degree of freedom... So where is the spin degree of freedom gone ? Do we need to discuss only space and\/or time dependent spin degree of freedom to make any sense of it ? And then the spin is only a _relative_ degree of freedom (in other word, I know what is a spin _mismatch_ , but not what is a spin, a bit like a phase: only phase difference matters). Or anyone gets a better idea of what a spin is? Thanks in advance."}
{"id":"4037","title":"Existence and uniqueness of solutions for Einstein equations","text":"Now that an equivalence of Navier Stokes and Einstein equations has been established, and it is known solutions to Einstein-Maxwell-Boltzmann exist and are unique, and it is known that Einstein equations in a wave guage with electromagnetic source with a lorentz gauge also have unique solutions that exist, what are the next steps? What are the good references on research for existence and uniqueness for einsten solutions without electromagnetic sources? Also, what does it mean physically that there are unique solutions to Einstein Field equations when electromagnetic sources are included? For those struggling with the concept of existance and uniqueness, I think this post provides some good insight."}
{"id":"86353","title":"Euclidean derivation of the black hole temperature; conical singularities","text":"I am studying the derivation of the black hole temperature by means of the Euclidean approach, i.e. by Wick rotating, compactifying the Euclidean time and identifying the period with the inverse temperature. Consider the Schwarzschild case as an example. The Euclidean Schwarzschild metric is of course $ds^2=\\left(1-\\frac{2M}{r}\\right)d\\tau^2+\\left(1-\\frac{2M}{r}\\right)^{-1}dr^2$ where $\\tau=it$ the Euclidean time. Here $\\tau\\in[0,\\beta]$ with $\\beta=T^{-1}$ the inverse temperature, where the points $\\tau=0$ and $\\tau=\\beta$ are equivalent. (I ignore the 2-sphere part of the metric.) Outside but close to the event horizon $r=2M$, we can (after simple some steps) write this as $ds^2=\\frac{\\sigma^2}{16M^2}d\\tau^2+d\\sigma^2$. Here $\\sigma^2 \\equiv 8M(r-2M) $, though this is not really relevant for my question. Now the next step in all the literature is to require that $\\tau\/4M$ is periodic with period $2\\pi$ to prevent conical singularities. I have trouble understanding this.   * A conical singularity basically means that the point at $\\sigma=0$ looks like the tip of a cone, right? So we have a singularity at $\\sigma=0$. But don't we still have a singularity there for polar coordinates $(r,\\theta)$, since the coordinate chart $\\theta$ is not continuous there?    * If so, removing the point $\\sigma=0$ would erase the conical singularity, right? Then why do we want to get rid of the conical singularity, why is it any worse than the polar coordinate singularity? Of course polar coordinates just describe flat space without the origin, is this not the case for the conical coordinates? Clearly I have not understood the concept of a conical singularity... Last question: suppose I did understand it, and continued the derivation to get the temperature $T=1\/8\\pi M$. Apparently this is the temperature as measured by an observer at infinity, how can I see this? I know the temperature gets redshifted, like frequency, but I don't see where the derivation identifies $T$ with the one measured at infinity."}
{"id":"91513","title":"Does thermal energy change within a chamber depending on what's in it, where it is, what it is?","text":"I have noticed that some people will say that, when you put more mass in an oven, it takes longer for both to cook than if just were one. Some others will say that timings are the same. Assume we have an oven that's heated to 350 degrees fahrenheit, and it's 10 cubic feet(it should hold about 340 liters of liquid, volume-wise). If we stick a 1 lb, 6 ounce cake in the oven in a 9x9 aluminum pan, let's say that it cooks until succinct in about 25 minutes. If we were to add the exact same mass, but in another cake and equivalent pan (with the exact same ingredients to the nearest degree possible on the atomic scale), and we place that pan right next to it in the center of the one-rack oven, will cooking times differ? Basically, does thermal energy become affected when you place more mass over an area of heat within a chamber? It takes different cooking times for different things(molecule-wise, etc.), and mass is always a factor(e.g. a chicken drumstick will cook at 350F in less than 15 minutes, but a whole 5 lb chicken, say, will most definitely take quite longer). So my question is, in an enclosure of 10 cubic feet (or the like), if you add more mass\/volume to the chamber, will thermal energy change, will cooking times change with varying mass and volume, distributions and placement inside the chamber, and why or why not? PS: I know this is broad, but we can narrow it down by just taking my aforementioned examples _as just examples_ , and provide answers relevant to the idea within any scope of measurements desired."}
{"id":"66243","title":"Path integral formulation understanding","text":"I have done basic quantum mechanics and now I want to do the path integral formulation. I find Feynman's book _Path Integrals in Quantum Mechanics_ difficult. Is there an easier alternative?"}
{"id":"7327","title":"Is it possible to restart formal higher education in physics at a later age?","text":"I would begin by apologizing for a soft question. Reading the FAQ, I have tried to formulate my question precisely in the last paragraph and you may skip the following paragraph where I make subjective observations, which I think may be relevant and any comments would be more than helpful. By formal education I mean registering in an academic department in a university as a grad\/Phd Student. Subjective belief: I feel I am personally not upto the mark of high-quality physics students. I need to learn a lot more, and do a lot more problems to be any good. I've noticed that Physics particularly is a very selective field dominated by hard-working and primarily young people who accomplish a lot at an early age (olympiads\/contests, early phd's from very good places, etc). Even among this bunch a lot get filtered out and wouldn't get faculty positions. Theoretical Physicists tend to be outliers in a field almost entirely composed of outliers. And to learn from such a community, I feel an academic environment is essential, and not just reading books in isolation. So I beleive academia is the only legitimate environment that functions to produce sane physicists who know their field. However, if one has been set- back for some reason but carries the self-beleif that he can be smart enough to do it, would ageing be a parameter. I read a statement in Kleppner's undergrad mechancis book that \"by finishing all this at the age of 26, Newton set the trend which follows today, that great advances are made by young physicists\". I haven't found a counterexample so I would trust academic departments have the same criterion. Q: Do people who missed out all the action in their 13-20 ages miss out on having a good career? What are the possible reasons to take a hiatus of \"self- study\" for a 22-year old so that he may be a good candiate for a respectable physics career (Phd\/postdoc etc) later."}
{"id":"43994","title":"Aharonov-Bohm Effect and Integer Quantum Hall Effect","text":"What is the relationship between Aharonov-Bohm effect and Integer Quantum Hall effect?"}
{"id":"135112","title":"A few questions on passive vs active Lorentz transformations","text":"1.) How do we physically interpret an active Lorentz transformation? The passive transformation seems simple enough: you view a fixed object from the perspective of a new observer. When we actively Lorentz transform a vector are we interpreting this as moving the vector to a new point in spacetime considered from the perspective of a single observer? 2.) I am reading David Tong's QFT notes, and am having a hard time interpreting what he means by active transformations. The notes in question can be found here: http:\/\/www.damtp.cam.ac.uk\/user\/tong\/qft\/qft.pdf The notes in question are on pg 11-12 as labeled by the book or pages 17-18 as labeled by the PDF. In his notes, Tong states that we can transform a scalar field as follows: $$\\phi (x) \\rightarrow \\phi'(x) = \\phi \\left(\\Lambda^{-1} x \\right).$$ When he does this, I'm interpreting this as $$\\phi (x) \\rightarrow \\phi'(x') = \\phi \\left(\\Lambda^{-1} x' \\right),$$ where $x'=\\Lambda x$. From what I understand, the advantage of using the inverse Lorentz transformation on the primed system is that we can use the same functional form of $\\phi$. However, when moving to the primed system we have still used $\\Lambda$, not its inverse. Can anyone tell me if I'm correct in my understanding up to this point? If my understanding is correct up to this point then I really don't understand the next section in his notes. He states that under this transformation derivatives transform as $$(\\partial_\\mu \\phi) \\rightarrow \\left( \\Lambda^{-1} \\right)^\\nu_{\\phantom{\\nu} \\mu} (\\partial_\\nu \\phi)(y),$$ where $y=\\Lambda^{-1}x$ (where this $x$ is primed, right?). But we've still gone from $x \\rightarrow x'$ where $x'=\\Lambda x$ (again based on my understanding which may be terribly wrong). Using $\\Lambda^{-1}x'$ was simply a mathematical trick to allow us to use the same functional form of $\\phi$. If that's the case, why are the derivatives transforming as $\\Lambda^{-1}$ instead of just $\\Lambda$? I'm sorry -- I know this is a little convoluted, but I'm having a really hard time getting my head around this, especially with his notation. I really wish he would have used primes or something... Am I completely lost? Someone please rescue me."}
{"id":"80210","title":"Deriving the rocket equation","text":"Recently I decided to brush on the those parts of classical mechanics that always gave me trouble. One of those parts was the rocket equation so I figured that I would try to derive the appropriate equations from scratch. Evidently I ran into trouble somewhere, so I am hoping that someone can point out my errors and steer me in the right direction. Anyway here goes ... Suppose a rocket expels exhaust behind it with constant velocity (relative to the rocket) $\\mathbf{u}$. Fix an inertial reference frame and consider the rocket + exhaust system. If $m_R$ denotes the instantaneous mass of the rocket, then the momentum of the rocket can be written as $m_R\\mathbf{v}$, where $\\mathbf{v}$ is the instantaneous velocity of the rocket. The momentum of the exhaust ought to be given by the integral: $$\\int (\\mathbf{v}+\\mathbf{u})\\frac{\\mathrm{d}m_E}{\\mathrm{d}t} \\; \\mathrm{d}t$$ where $m_E$ denotes the instantaneous mass of the exhaust. Combining these expressions for the momentum, we arrive at the total momentum for the system: $$\\mathbf{p} = m_R\\mathbf{v} + \\int (\\mathbf{v}+\\mathbf{u})\\frac{\\mathrm{d}m_E}{\\mathrm{d}t} \\; \\mathrm{d}t$$ Now according to the 2nd Law of Motion, the force acting on the rocket + exhaust system is obtained from $\\mathbf{p}$ by taking time derivatives. In other words: $$\\mathbf{F} = \\dot{\\mathbf{p}} = \\frac{\\mathrm{d}m_R}{\\mathrm{d}t}\\mathbf{v}+m_R\\frac{\\mathrm{d}\\mathbf{v}}{\\mathrm{d}t}+(\\mathbf{v}+\\mathbf{u})\\frac{\\mathrm{d}m_E}{\\mathrm{d}t} = m_R\\frac{\\mathrm{d}\\mathbf{v}}{\\mathrm{d}t} - \\frac{\\mathrm{d}m_R}{\\mathrm{d}t}\\mathbf{u}$$ This expression superficially matches the ones in my college mechanics books, but there is one big difference. In this expression $\\mathbf{F}$ is the total force acting on the rocket + exhaust system, while in my texts this $\\mathbf{F}$ is just the force acting on the rocket alone. So based on this I have made some fundamental error somewhere in my argument. As I asked before, if anyone could help me identify this error and correct it, I would be much obliged."}
{"id":"80212","title":"Reference for the polar parameterization of quaternions","text":"I would like to find the original reference in which the polar parameterization of quaternions was given (i.e. the relationship between the components of a unit quaternion and the polar angles of an axis and rotation angle of a three-dimensional rotation). I assume it would have been written by Hamilton, but the physics literature from the 60's doesn't really point back any further (See, e.g. Biedenharn LC. J Math Phys 1961;2:433, equation 10). Any suggestions?"}
{"id":"10150","title":"Calculation of a Gravity Resonance Keyhole","text":"Can anyone describe the mathematics behind the calculation of a resonance keyhole (for a two-body model)? It seems like the size and position of the keyhole should be a function only of mass and relative velocity, but I'm having difficulty deriving a formulaic solution."}
{"id":"10151","title":"Radio waves and frequency of photon","text":"How radio waves create the current in antenna in terms of photons? If it is Compton scattering then why is not changed the freuency of photons?"}
{"id":"7713","title":"Predicting Decay Rates via the Standard Model","text":"Question 7584 illustrated a procedure to forecast the decay rates of isotopes with known long average lifetimes. Lifetimes of the many U isotopes vary from micoseconds to gigayears. F has only one stable isotope while Sn has 10. Can Standard Model principles be used to predict the stability of isotopes and the average lifetimes for unstable isotopes, or can this only be done by measurement?"}
{"id":"46720","title":"Is equivalent resistance always lower if we add a resistor to a passive electronic circuit?","text":"How to prove that equivalent resistance of any passive network is always lower if we add a resistor between arbitrary two nodes? Note that this is not necessarily a parallel circuit, 2 nodes that we connect with a resistor are not the same 2 nodes between we want to measure equivalent resistance but completely arbitrary 2 nodes in passive network. So imagine a circuit with 4 access points: A, B, C and D. We want to measure equivalent resistance between A and B. How can you prove that Rab will be less if we add a resistor between C and D? I tried to search the web but didn't have any success."}
{"id":"98938","title":"Why Transmission of electrical energy without wires was not practical even after Tesla's proposal","text":"Why we couldn't transmit Electricity wirelessly even after so many proposals available proposed by Tesla ? > Tesla tried to generate more interest in Wardenclyffe by revealing its > ability to transmit wireless electricity, but Morgan was not interested, and > the 1903 \"rich man's panic\" on Wall Street dried up any further > investment.[8][9][10] By July 1904, Morgan (and the other investors) finally > decided they would not provide any additional financing. In May 1905, > Tesla's patents on alternating current motors and other methods of power > transmission expired, halting royalty payments and causing a severe > reduction of funding to the Wardenclyffe Tower. In an attempt to find > alternative funding Tesla advertised the services of the Wardenclyffe > facility but he was met with little success. By this time Tesla had also > designed the Tesla turbine at Wardenclyffe and produced Tesla coils for sale > to various businesses. Source > > In 1891 and 1892, Tesla demonstrated the oscillatory transformer that bears > his name in lectures delivered before meetings of the American Institute of > Electrical Engineers (AIEE) in New York City\" and the Institute of > Electrical Engineers (IEE) in London, and in a later presentation titled \"On > Light and Other High Frequency Phenomena\" (Philadelphia\/St. Louis; Franklin > Institute in 1893), where he put forward his ideas on the wireless > transmission of electrical energy. Source ![enter image description here](http:\/\/i.stack.imgur.com\/O4iFF.png) The Tesla coil wireless transmitter"}
{"id":"135190","title":"Non-stationary capacitor","text":"A common exercise in non-stationary electromagnetism is to find the electric and the magnetic field generated by a capacitor with round plates, if the potential difference between the plates varies in time (typically $V=V_0 sin( \\omega t)$ or $V= \\alpha t$). While finding the fields internal to the capacitor is pretty easy, I was struggling with those on the outside. $B$ is given instant per instant by Biot-Savart law and varies in time, so exists an electric field following the third Maxwell equation; However, I couldn't find an expression for $E$ without evident absurdities. I thought that maybe with some approximations (quasi-stationarity?) such fields could be ignored. Some hints?"}
{"id":"14892","title":"Does space have to be filled with charged particles to carry electromagnetic waves?","text":"I'm a newbie here so have mercy. I'm studying electromagnetic waves. This is the propagation of energy via the vibration of charged particles, as I understand it. A charged particle could be like a proton, which has a positive charge. So I'm imagining a simplified \"universe\" that just exists along a two dimensional line, a long string of protons. You can send energy from one end of this space to the other by vibrating this string of charged particles. Maybe this could be done by \"lifting\" one proton at one end of space, and that would effect the next proton along the string in a similar way, only less so, to create a wave. You can move your end of space up and down as fast as you'd like. If you move it slowly, you create long waves (from crest to crest). If you move your proton at your end quickly, you produce short waves. But no matter how quickly or slowly you lift your proton up and down, the wave you create (either longish or shortish) will always reach the other end of space at the same time. In other words, you can't control how fast your wave travels through space, you can only control the shape of the wave itself. The energy transmitted always travels the same speed (C, the speed of light). Lift your proton up and down slowly, and you produce a radio wave. Lift it up and down really quickly, and you'll produce an x-ray wave. Lift it up and down somewhere in the middle, and you get a light wave. Here's one problem I'm struggling with. If we can see light emitted from stars that are light years away from us, that must mean that every part of space along that line of sight is full of charged particles right? If that's so, what are these charged particles? Is space full of particles in every direction, in every place? They aren't protons and electrons right, maybe more elementary charged particles like quarks? So space isn't empty, it's absolutely jam packed with charged particles. The \"fabric\" of space time is \"made of something,\" and that something must be charged either positive or negative. Is this a correct line of thinking, or am I missing some fundamental idea here? Thanks for your help."}
{"id":"123829","title":"Atomic Physics - Bohr's model of atom","text":"Well I'm learning about the models that have been proposed for the atom, and the Bohr model came up. My teacher told me that the one of the main postulate of the theory is that when an atom is in ground state, it can neither absorb nor emit energy. This is what gives the atom its stability. However, it can obviously absorb energy to jump to a higher energy level. I don't understand, how can it jump to a higher energy level, from the ground state obviously, without absorbing energy? Some problem here?"}
{"id":"123822","title":"What does this question about entanglement and classical geometry mean?","text":"Below is the question from Andy Strominger's presentation at the String 2014 conference. The question was asked by credible physicist Ashoke Sen as an important question. > \"What is the precise relation between quantum entanglement and classical > geometry?\" Could somebody describe the question and explain why it is an important question and its implications?"}
{"id":"43782","title":"current in series resistors and voltage drop in parallel resistors","text":"when we have resistors in series, the current through all the resistors is same and the voltage drop (or simply voltage) at each resistor is different. **Question 1:** it is fine that voltage drop (potential drop) is different because each resistor offers different resistance (suppose). but how is the current through each resistor same? if we have resistors of different resistance, shouldn't the current be different through each resistor? similarly, when we have resistors in parallel, the current through each resistor is different but the voltage drop at each resistor is same. **Question 2:** current is through each resistor is different because resistance of each resistor is different (suppose). but how is the voltage drop at each resistor same here? shouldn't the voltage drop at each resistor be different because each resistor offers different resistor?"}
{"id":"5195","title":"Resistance between two points on a conducting surface","text":"Suppose we have a cylindrical resistor, with resistance given by $R=\\rho\\cdot l\/(\\pi r^2)$ Let $d$ be the distance between two points in the interior of the resistor and let $r\\gg d\\gg l$. Ie. it is approximately a 2D-surface (a rather thin disk). What is the resistance between these two points? Let $r,l\\gg d$, (ie a 3D volume), is the resistance $0$ ? * * * Clarification: A voltage difference is applied between two points a distance $d$ apart, inside a material with resistivity $\\rho$, and the current is measured, the proportionality constant $V\/I$ is called $R$. The material is a cylinder of height $l$ and radius $r$, and the two points are situated close to the center, we can write $R$ as a function of $l$, $r$ and $d$, $R(l,r,d)$, for small $d$. The questions are then:   What is $$ \\lim_{r \\rightarrow \\infty} \\lim_{l \\rightarrow \\infty} R(l,r,d) $$   What is $$ \\lim_{r \\rightarrow \\infty} \\lim_{l \\rightarrow 0} R(l,r,d) $$"}
{"id":"134790","title":"Does the real part of the inverse dielectric function have to be negative at some point for Cooper pairs to form?","text":"Electrons naturally repel one another. However, in a superconductor, a phonon- mediated interaction causes the electrons to have a weak attractive interaction. Suppose that the interaction between two electrons is $V(\\textbf{q})$ (the bare Coulomb interaction, which is $>0$) and that the effective interaction between two electrons is $V(\\textbf{q})\/\\epsilon(\\textbf q, \\omega)$. My question is threefold: (1) Is the only way to have an attractive interaction between electrons is for the $\\Re[\\frac{1}{\\epsilon(\\textbf{q},\\omega)}] $ to become negative at a certain value of $\\textbf{q}$ and $\\omega$? (2) If so, does $\\Re[\\frac{1}{\\epsilon(\\textbf{q},\\omega)}] $ have to become negative at the values of $\\textbf{q}$ and $\\omega$ that correspond to a phonon? (3) (If yes to Question (1)): Since high-temperature superconductors form Cooper pairs, does this mean that in the space of $\\textbf{q}$ and $\\omega$ such that $\\Re[\\frac{1}{\\epsilon(\\textbf{q},\\omega)}] < 0 $, there are clues lurking about high-temperature superconductivity yet unbeknownst to us?"}
{"id":"114604","title":"Quantum Field Theory with (2,2) metric","text":"Does someone know some reference which treats QFT in a space with the following non-Lorentzian signature: $g_{\\mu\\nu}=\\text{diag}(-1,-1,1,1)$. I'm interested in basic stuff like the shape of the scalar propagator and Feynman rules in such a space."}
{"id":"14272","title":"Two-state system problem","text":"Given a 2-state system with (complete set) orthonormal eigenstates $u_1, u_2$ with eigenvalues $E_1, E_2$ respectively, where $E_2>E_1$, and there exists a linear operator $\\hat{L}$ with eigenvalues $\\pm1$,   1. would the normalized eigenfunctions (in terms of the given eigenstates) just be $$u_1\\over \\sqrt{\\int u_1^*u_1}$$ ? Since I am not given an argument\/coordinate system for the eigenstates, perhaps I should first project them into one? But I don't have any info on the nature of the quantum system...   2. A second question asks for the expectation values of the energy in the respective states. But isn't that just $E_1$ and $E_2$ respectively?? Grateful for any enlightenment."}
{"id":"14279","title":"Wave \/ particle duality question for sound and light","text":"Sound is usually referred to as just \"sound waves\" - we do not talk about a \"sound particle\" and only as a wave or \"matter wave.\" Could something similar apply to light i.e. that there really is no \"light particle\" (photon) and that the wave respresentation is what applies both to sound and light? We say light is an electromagnetic movement, could it be that light is an electromagnetic movement with just movement and similarly to sound like there is no \"sound particle\" (only a movement that makes some sound from vibration and movement of particles that are \"ordinary\" particles in the air and not in fact any particular \"sound particle\" so similarly the photon could be just a name or just a concept in fact used to describe what is similar to sound and just a wave and not a special particle? I know this seems controversial like saying that there is no photon and I say it might be not that controversial since there is no particular particle for sound and a waveform only. Thanks for any insight!"}
{"id":"45898","title":"View of the Sun from Voyager 1","text":"Today Voyager 1 is approximately 122 AU distance from the Sun. What does the sun look like from this distance? How much brighter does the sun appear to the naked eye compared to other stars in the same field of view? Are any of the other planets from our solar system visible to the naked eye at this distance? Would the sun produce enough light for a normal person to read a book, or would the light be too dark for reading (just trying to get a practical feel of the amount of light)?"}
{"id":"118744","title":"Universal central charge in higher dimensional AdS\/CFT?","text":"In the $AdS_3\/CFT_2$ correspondence, the central charge of the dual CFT2 is universally given by $$ c = \\frac{3\\ell}{2G} $$ This is independent of the matter in the bulk of AdS3. Is it also universal in the higher dimensional analogues $AdS_d\/CFT_{d-1}$ or does it depend on details of the matter in the bulk, etc.? In particular, what is the central charge for $d=4$?"}
{"id":"108020","title":"can interaction between a massless fermion and external sourse exist?","text":"For example, let's consider the electromagnetic interaction between a massless fermion and a electromagnetic externel sourse $A^\\mu$, then the lagrangian is $$\\mathcal{L}=\\bar{\\psi}\\gamma^\\mu\\partial_\\mu\\psi+ieA_\\mu\\bar{\\psi}\\gamma^\\mu\\psi$$ and how can we calculate the Feynman diagram and $\\mathcal{M}$ of it? But according to relativity, the equation of motion of a particle with charge is $$e F^a\\ _bU^b=U^b\\partial_bP^a$$ where $F_{ab}$ is the electromagnetic tensor, $U^a$ is the 4-velocity of the particle and $P^a$ is the 4-momentum of the particle. If the particle is massless, then the speed will be $1$ and this equation fails. So can a massless particle have charge and interaction with externel sourse? If could, for example, how can I calculate the force and equation of motion? If couldn't, why?"}
{"id":"89558","title":"Is it possible to harness energy from the exothermic reaction of changes of state of matter?","text":"Since a substance emits heat as it transitions from a high energy state (gas) to a lower energy state (solid), is it possible to devise a method to capture this heat and convert it to usable energy? I'm not sure how economical this would be, but consider the exothermic reaction of water freezing. The heat of fusion for water is $3.33 *10^5 J\/kg$ So if a 1 cubic meter of water was frozen, there would be: $3.33 * 10^5 J\/kg * 1000kg = 3.33 * 10^8 J$ of heat energy produced. The heat value for the vaporization of water is $2.36 * 10^6 J\/kg$ which would be even more efficient, though it is more difficult to capture 1000kg of water vapor. I imagine this might be akin to how an air conditioner works, but I cannot imagine how this may be done or if it is already being attempted. So I pose the question here, would it be possible to harness the heat energy from the air and water concentrate it into a usable form of energy--a clean form of energy at that. Why, or why not?"}
{"id":"129888","title":"Earthing of conductors","text":"Can anyone please explain me the process of earthing of a spherical conductor in detail. I am aware about the basic idea of earthing but I am not able to understand and visualize it properly."}
{"id":"110157","title":"Spectroscopy and the current state of our ability to determine the composition of extra-solar planets","text":"I was interested in learning more about the status of our scientific understanding and technological instruments regarding extra solar spectroscopy. I am motivated by this almost 3 year old question\/answer here at physics SE. Specifically: > The composition of extra solar planet should be taken with a pinch of salt, > the science is very new and uses instruments not designed for the task. Keep > an eye out for new missions such as ECHO which might fly in the next 5 to 10 > years... Since it has been almost 3 years, can someone give an update on our current understanding on this subject? I am curious as to how accurate our measurements of the composition of planets and stars outside our solar system are. Also, what is the impact of our atmosphere? Are we able to tease out the effects of absorption by our own atmosphere in order to get a clean signal?"}
{"id":"110155","title":"Mass-Energy Equivalence theory energy or momentum is not conserved?","text":"The famous equation for mass energy equivalence: $E=mc^2$ It cannot conserve energy or momentum without some loss in one way or another. To elaborate further if I take $1kg$ of mass and I also take electromagnetic radiation with same energy as the $1kg$ mass and ensuring its accuracy using the above equation. Then use this radiation and I use it to smash it into a \"crash\" mat I would get this: $$E=mc^2$$ $$E=1.c^2$$ $$E=89,401,000,000,000,000$$ **Approx.** That being the energy we then use this to calculate the momentum of the electromagnetic radiation transferred onto the \"crash\" mat which would be: $$p=mv$$ $$p=mc$$ $$p=e\/c^2 . c $$ $$p=e\/c$$ $$p=89,401,000,000,000,000\/c$$ $$p=299,000,000$$ **Approx.** where as if I now take $1kg$ of matter and accelerate it to high velocity (say, 10,000 kilometers per hour ) to measure its momentum we get roughly $10,000 kg.m\/sec$ That being said, momentum cannot be conserved, why is that? Next, Energy cannot be conserved. For example if I used the same $1kg$ and I shoot it into space at a $x$ velocity and after say 1000 years I recieve it and measure mass of the object it would weigh exactly the same as 1000 years before therefore I can conclude it has same energy in its mass however if I use the mass-equivalent of electromagnetic radiation and do the same but my conclusion would be vastly different as first due to travelling in long fabric of space it would get slightly or even highly redshifted due to doppler effect. That being said the observer would get different energy reading to that of 1000 years ago. That in mind where did that energy go?"}
{"id":"19352","title":"Why doesn't phase space contain acceleration\/forces?","text":"I'm watching some Physics lectures on the internet by Leonard Susskind:   http:\/\/www.youtube.com\/watch?v=pyX8kQ- JzHI&feature=BFa&list=PL189C0DCE90CB6D81&lf=plpp_video In this lecture, and also in Wikipedia and other places phase space is described as the space of all the states we need to know to determine the configuration of the system infinitely into the future. But I don't understand how is position and velocity enough to determine this, what about forces? Let's say we have a particle and we want to know where it will be in 10 seconds, we obviously need it's starting position, it's starting velocity **and also** all the forces acting on it, or it's acceleration. Where is my misunderstanding?"}
{"id":"96538","title":"How should the implicit sum $C_{ijkl}u_{i,j}u_{k,l}$ be interpreted?","text":"$C$ is a 3x3x3x3 tensor. How should the expression $C_{ijkl}u_{i,j}u_{k,l}$ be interpreted? This is my guess: $$ \\sum_{i=1}^3\\sum_{j=1}^3 \\sum_{k=1}^3\\sum_{l=1}^3 C_{ijkl}u_{i,j}u_{k,l} $$"}
{"id":"19358","title":"How to calculate pressure at altitude when there is 0 temperature lapse?","text":"To calculate pressure at a given altitude, the following formula is used: $p = p_0 \\cdot \\left(1 - \\frac{L \\cdot h}{T_0} \\right)^\\frac{g \\cdot M}{R \\cdot L}$ _(the values can be foundhere)_ This formula is great, but only works completely with the US Standard Atmosphere. I really want to use ISA (International Standard Atmosphere) more, because it accounts for the dynamics of the atmosphere much better. The problem is that the Tropopause has a Lapse rate ($L$) of 0. I don't know how to get that to work in this formula because that would involve dividing by zero. Any ideas?"}
{"id":"51910","title":"Transformators (primary and secondary windings)","text":"Consider the following problem: _Robby wants to put a light in the shed so he puts a cable between his house and the shed. In the cable there are 2 wires with a combined resistance of $0,92 \\space \\Omega$. Robby connects his transformator at his house on an electricity grid of $230 V$. He connects the secondary inductor($14.3V$) to the cable. A $12 \\space V$ \/ $30 \\space W$ lamp in the shed works perfectly._ There are a couple of questions about this piece of text, which have been bothering me:   * **Calculate the heat per second which is created in the cable.** My problem with questions like this is that I have an overwhelming amount of data and it is hard for me to distinguish which is needed and which is not. So my initial thought was that I'd calculate $I$ like this: $ I = 230 : 0,92 = 250 \\space A$, and then use the formula $P = I^2 \\times R$ which would give me a ridiculous amount of $57500 \\space J\/s$. I knew I did something wrong when I saw this, so I looked at the correction model and found out that they found the $I$ of the lamp, and afterwards used that in he formula I gave in combination with $R= 0,92 \\Omega$. Why is it done like this? I first thought that you could you $I_{lamp}$ because this is in series and $I$ is the same everywhere in series, but I have 2 problems with that idea:   * How do we know it is in series?   * Also, if we do $ I = 230 : 0,92 = 250 \\space A$ this isn't equal to $I_{lamp}$, so I dismissed that idea and now I am back at square one.. Then we get the following passage which confused me even more: _Robby decides the put the transformator in the shed. The lamp functions perfectly when he puts the loop ratio to 958:50._ What?! Does this mean that he didn't use the transformator in the first passage? But if he didn't why is the secondary inductor mentioned in the first passage then?! The following 2 questions are easier (if you made the first one correctly):   * **Calculate the primary voltage on the transformator** I just did $(958\/50) \\times 12 \\approx 230V$   * **Explain without using calculations that he loses much less now** $I_{secondary}$ is much smaller now, and because of the formula $P=I^2 \\times R$ this leads to less loss of energy. So the bulk of my problem lies with 'analyzing' the first text. How can you conclude from that test that the transformator isn't used?"}
{"id":"16123","title":"What's the big deal with Nd:YAG rods?","text":"I see the prices of Nd:YAG rods for lasers, and could someone explain why they are so expensive? Do they need to be mono-crystalline or something? Will Nd:YAG laser work with amorphous\/poly-crystalline rod?"}
{"id":"96535","title":"How come the universe is made of matter and not antimatter?","text":"Antimatter is like matter on opposite day: it has the same properties as the stuff that makes up planets, stars and galaxies, but one vital piece is different—its charge. The universe supposedly started off with equal parts matter and antimatter, but somehow, matter won out. Most of both substances annihilated each other shortly after the big bang, and left a small surplus of matter remaining."}
{"id":"65183","title":"Why the magnetic flux is not zero?","text":"If $\\vec{\\mathbf B}=B\\vec{\\mathbf a}_z$, compute the magnetic flux passing through a hemisphere of radius $R$ centered at the origin and bounded by the plane $z=0$. ![Figure](http:\/\/i.stack.imgur.com\/WavK2.png) * * * **Solution** The hemisphere and the circular disc of radius $R$ form a closed surface, as illustrated in the figure; therefore, the flux passing through the hemisphere must be exactly equal to the flux passing through the disc. The flux passing through the disc is $$\\Phi=\\int_S\\vec{\\mathbf B}\\cdot\\mathrm d\\vec{\\mathbf s}= \\int\\limits_0^R\\int\\limits_0^{2\\pi}B\\rho\\,\\mathrm d\\rho\\,\\mathrm d\\phi =\\pi R^2B$$ The reader is encouraged to verify this result by integrating over the surface of the hemisphere. * * * According to Maxwell's equations the magnetic flux over a closed surface must be zero, why in this case does not happen?"}
{"id":"65186","title":"Some Dirac notation unclarities","text":"**Q1:** Ok so i have come to a point where i know that $\\Psi(r,t)$ which we denote only by $\\Psi$ can be represented in a Hilbert space by a vector which we denote $\\left|\\Psi\\right\\rangle$. Does this mean $\\left| \\Psi(r,t) \\right\\rangle$? **Q2:** I know that equation $\\Psi = \\psi e^{- iWt\/\\hbar}$ represents a link between a stationary Schrödinger equation and a time dependant Schrödinger equation (we denote $\\Psi(r)$ as $\\psi$). I want to know now if we denote $\\psi$ in a Dirac notation as $\\left|\\Psi(r)\\right\\rangle$ or any differently? **Q3:** How do we write a Schrödinger equation and a time independant Schrödinger equation using a Dirac notation?"}
{"id":"82951","title":"Heisenberg uncertainty principle - question","text":"> _A beam of particles each having mass $m$ and velocity $v$ in the incident > on a circular hole of radius $b$ located on a screen. If another screen is > placed at a distance $D$ from the hole, determine the minimum size of the > pattern formed on the screen by the beam._"}
{"id":"122126","title":"Is there a depth at which a human body, with lungs full of air, will no longer float?","text":"I was watching a special on divers who dove down hundreds of feet without tanks or gear. The show referred to a depth at which the diver was no longer buoyant and would actually sink, even with lungs full of air. Is there such a point and if so, why? What is happening there?"}
{"id":"119534","title":"What to work on?","text":"(I am not exactly sure if this question is supposed to go here, but it seems like the appropriate place for it.) Hello Scientific Community, I am soon graduating from high school and, as everyone does, I am searching for something to study at college. I have always wanted to study straight forward physics, but recently I have been having problems finding the application (possible jobs) of a physicist. My parents are trying to push me towards mechanical engineering, but I am trying to convince them I can get a decent job studying physics. They hate the idea of having me become a teacher, so that is out of question. I know many of the members in the community have studied physics, and I would like all of you out there to please help me making a short description of your job and it`s availability. If you know of any other jobs apart from the one you are on, feel free to list them. Thanks a lot for the help P.S: I live in a country were not many resources are spent on scientific investigation. If you work in some kind of job on applied physics, please answer, at least a short response. Still make an answer if you are some kind of investigator for anyone else that reads the thread in the future."}
{"id":"119531","title":"Why does water freeze from top to bottom","text":"I was wondering why water freezes from up to bottom and not the other way around. The reason I question this is because in a Video I came across, I discovered water freezes from top but surely with the current knowledge I would say it should be opposite because we know cool air drifts down while hot air drifts up so similarly the heat should fall down and this that the freezing should start from bottom and not up? Unless my knowledge applied is wrong and potentially stupid so I want to ask: Why does water freeze from top to bottom?"}
{"id":"81609","title":"Is there an official name for \"Lorentz Pairs\" like energy and momentum?","text":"In learning about relativity I've noticed that in the construction of Lorentz invariants (specifically four-vectors) two physical quantities that were previously considered distinct are instead treated as a single object. Examples include position and time (spacetime), electric and magnetic fields, electric and magnetic potential, energy and momentum, charge and current density, and so on. I was interested in learning more about this I tried googling my best guess, \"Lorentz Pairs\", to no avail. My question is whether there is more specific terminology referring to these pairs of physical quantities outside of the general term \"Lorentz Invariant\". Additionally, why do we so often see two physical quantities being combined instead of, say, three or some other number?"}
{"id":"11670","title":"Significance of isolation of radium?","text":"This is an extension of my previous question here. Marie Curie isolated radium in 1903, which paved the way for the development of the theory of radioactivity. In regards to the techniques she used however, what was the impact of the establishment of those techniques? Were the techniques used by Curie new? Are they used today? What impact did they have on modern physics? I can find a lot of information online about the actual isolation of radium, but not regarding the significance of the techniques used."}
{"id":"46208","title":"Constant volume specific heat","text":"* A monatomic ideal gas is confined to move in two dimensions. What is the constant volume specific heat for this gas?    * Consider a system of N independent harmonic oscillators moving in two dimensions. What is $C_v$ for the system. For the first case, I guess this one is $C_v = \\frac{3}{2}R$. Am I correct? However for the second case, I was confused a little bit. Any help on this?"}
{"id":"46200","title":"Why is energy-momentum 4-vector so much easier to explore\/observe than spacetime 4-vector","text":"I have read that spacetime 4-vector is quite difficult to observe\/explore and that energy-momentum 4-vector is much more appropriate for CERN etc. **Why is that?** Could anyone give me a brief explanation of the differences regarding exploration and observation?"}
{"id":"46204","title":"How do I calculate the probability that the oscillator is in a certain state using partition function?","text":"So let's say I have a single ($N=1$) quantum harmonic oscillator and the energy is determined by $E_n = (n + 1\/2) \\cdot \\hbar \\omega$ (where $n$ is the quantum number and $n$ = $0, 1, 2, \\ldots$) What's the probability that the oscillator is in the state labeled $n$ at temperature $T$? So according to my calculation, $Z$, the partition function, is $Z = 1 \/ (1 - x)$ where $x = e^{-\\beta \\hbar \\omega} \\Rightarrow P = x ^ n (1 - x)$. Is it correct? Also, how do I go about calculating the probability of finding the oscillator in a state with odd quantum number?"}
{"id":"94001","title":"How do bicycle spokes work?","text":"As you know, it is quite obvious that Bicycle spokes attach the hub in the centre to the rim. What else do they do? If you compare the wheels today with the ones from ancient times, there are more spokes now on motor bikes and bicycles than of a wheel of a chariot. Why is that? What effect does it have on the vehicle if there is a higher number of spokes on the wheels?"}
{"id":"17293","title":"How would a large a mass be stable at the Earth Sun L4 or L5 point?","text":"I've heard about the Trojan asteroids and there is the famous idea of putting a space colony at one of these points, but the explanations I see for how something is stable at those points it they are 'insignificant mass.' What does that mean? Insignificant to Earth could be pretty big, insignificant to the sun could be downright huge. What are we talking about here, the mass of a large asteroid? The moon? The Earth? Surely it couldn't be bigger than one of the two bodies in the system?"}
{"id":"67118","title":"Static as opposed to Kinetic Friction in Rolling Motion","text":"During analysis of rolling motion, why do we consider coefficient of friction as that of _static_ friction and not kinetic friction?"}
{"id":"57078","title":"Does universal wave function exist?","text":"Does universal wave function exist? What the science tells us?"}
{"id":"11636","title":"Hydrogen as a fuel in Sun","text":"The source of Sun's incessant energy is hydrogen; which is continuously converting to helium through nuclear fusion reaction releasing energy. Why does not all hydrogen convert into helium in one big explosion just like a man-made bomb or cracker explodes?"}
{"id":"57074","title":"What is the current induced in a straight wire when a magnetic field is turned from on to off?","text":"> Consider a straight wire which runs along the y-axis. A B-field exists which > is uniform and only in the x-direction: > > $$\\vec{B}=(B(t),0,0)$$ > > The B-field at time $t_0$ is 'switched off' in a sharp step, which will be > modelled by the tanh function: > > $$ B(t)= \\begin{cases} B_0 & t\\leq t_0 \\\\\\ 0 & t>t_0 \\end{cases} \\approx > \\frac{B_0}{2}\\left [ \\tanh(100(t-t_0)) +1 \\right ] $$ > > **What is the current induced in the wire due to the change in magnetic > field as a function of time?** What I have tried is to work out the E-Field y-component using Maxwell's 3rd law (using the notation $\\partial_{yz} x \\equiv \\frac{\\partial^2 x}{\\partial y \\partial z}$): $$\\nabla\\times\\vec{E} = -\\partial_t \\vec{B}$$ But the B-field only has an x-component, so: $$\\begin{align} \\partial_y E_z - \\partial_z E_y &= -\\partial_t B(t)\\tag{i} \\\\\\ \\partial_z E_x - \\partial_x E_z &= 0\\tag{ii} \\\\\\ \\partial_x E_y - \\partial_y E_x &= 0\\tag{iii} \\end{align}$$ So if you differentiate (i) by x, (ii) by y and (iii) by z: $$\\begin{align} \\partial_{xy} E_z - \\partial_{xz} E_y &= -\\partial_{xt} B(t) \\\\\\ \\partial_{yz} E_x - \\partial_{xy} E_z &= 0 \\\\\\ \\partial_{xz} E_y - \\partial_{yz} E_x &= 0 \\end{align}$$ But then if you sub them all in together you get: $$\\partial_{xt} B(t) = 0$$ Which isn't very useful.   * If I want to work out the current induced in the wire which runs along the y-axis do I want to work out: $E_y$? And If so how do I do so?   * Would current be induced in any other direction?"}
{"id":"9430","title":"Max rocket speed in interstellar space?","text":"Interstellar space propulsion...if a spaceship were to get beyond our Sun's gravitational pull and since there is no atmosphere\/wind\/friction in space...does that mean, if an engine was constantly thrusting, that the spaceship would continue to accelerate to the speed of light, as long as the engine was propulsing forward? In other words, does speed constantly increase in a vacuum? or are the two ideas (Space\/Vacuum) separate ideas?"}
{"id":"64326","title":"Absolute Viscosity of Water at certain temperatures","text":"I just started my class on fluid mechanics. There's a problem that requires absolute viscosity $u$ of water at 25C. I looked up my table in the book and I only have it for water at 20C and 30C. I figured we had to manually calculate it in this case. The school notes only have Andrade's equation $u = D \\times e^{\\ B\/T}$ So I've got the absolute temperature which is 25C. There is no information of what D and B are(Defined as constants). So how do I figure out the absolute viscosity at any temperature? I've looked up a table online with the value for water at 25C and it gives me the right answer, is it okay to interpolate the values of my table of 20C and 30C to find the value for 25C?"}
{"id":"122830","title":"Why is the second the SI base unit for time?","text":"Specifically, during the moves towards _Le Système international d'unités_ in the 18th and 19th centuries, why didn't anyone attempt to move us away from the definition of there being 24 hours in a day? Wikipedia informed me that the name **second** comes from the _second division_ of an hour. It also claims that the concept of having 24 hours in a day dates back at least 6000 years to the earliest Egyptian civilization. It also says the word second (as a measurement of time) didn't make it into English until the 1500's and couldn't be measured with any accuracy for 100 years. So by one way of looking at things, the base unit of time (or perhaps the grand-pappy unit of time) is the hour. Is there some convenient property of **24 hours = 1 day** that caused it to hold on over other units? I understand the significance of a degree being 1\/360th of a circle (360 = 60 * 60 = (3 * 4 * 5)^2), and 24 is closely related to those factors, but that seems a flimsy reason. The math for converting earth's rotation to elapsed time doesn't even work out very conveniently (IMO) with these units. I expected to find a reason in time being one of the basic properties of the universe, being immutable (in 18th century eyes) and therefore used as a pivot point for other unit definitions. But the dependency graph of SI units mostly dispelled me of that: both temperature and mass were found to be as independent as time, and both units were redefined during this period (to Kelvins and kilograms, respectively). ![The seven SI base units and the interdependency of their definitions. Clockwise from top: kelvin \\(temperature\\), second \\(time\\), metre \\(length\\), kilogram \\(mass\\), candela \\(luminous intensity\\), mole \\(amount of substance\\) and ampere \\(electric current\\).](http:\/\/i.stack.imgur.com\/jnGwh.png) During the same period, other units were attempted to be re-measured from new standards:   * Length was adopted to use the meter at one ten-millionth from the North Pole to the Equator in 1793   * Mass was adopted to use the gram at the cube of the hundredth part of the metre (at STP) in 1795   * Temperature was adopted to use degrees Celsius, one hundredth of the separation between boiling and freezing of water. And yet, efforts to define the second seem to center on more perfectly measuring the existing definition of 1\/86,400th of the Earth's rotation. Given the predilection for base 10 (and the gradual move away from units depending on our earthly frame of reference), why was the second left at such a cumbersome, ancient interval? Why didn't we redefine a new measure of time to be (say) 1\/100,000th of a day, put 10 hours in a day, etc. etc.? Did no one think to challenge the convention of 24 hours in a day? Why not? That came out a little more jumbled than it was in my head, so let me sum up: I'm mainly interested in answers addressing attempts to redefine units of time and why they didn't achieve wide adoption in the scientific community."}
{"id":"48128","title":"Surface UV exposure with cooler star","text":"If the sun's surface was ~ 4000K (and earth closer to compensate), the UV component of the radiation would be less. However, UV makes ozone via photolysis of oxygen. Also, the stratosphere would probably start up higher (and contain less mass) because UV heating is what's responsible for the inversion. So what is the relationship between surface UV exposure (measured in DNA damage) and star temperature, which effect \"wins\"?"}
{"id":"99409","title":"Does fluorescence produce most of the colors we see day-to-day?","text":"I read the following on Wikipedia regarding fluorescence, > _Fluorescence is the emission of light by a substance that has absorbed > light or other electromagnetic radiation... The most striking examples of > fluorescence occur when the absorbed radiation is in the ultraviolet region > of the spectrum, and thus invisible to the human eye, and the emitted light > is in the visible region._ Recalling my grade-school knowledge of color perception: The colors I perceive of an object that does not produce light are those that the object does not absorb. Both those notions seem the same to me; however, I suspect that I've missed something. Is there a difference between the way that the colors we perceive of most objects is produced and the way that flourescent color is produced?"}
{"id":"109518","title":"Derivation of Euler-Lagrange equations for Lagrangian with dependence on second order derivatives","text":"Suppose we have a Lagrangian that depends on second-order derivatives: $$L = L(q, \\dot{q}, \\ddot{q})$$ If we're working on the variational problem for this Lagrangian, then I know that we'll wind up with the following Euler-Lagrange equation: $$\\frac{\\partial L}{\\partial q} - \\frac{d}{dt} \\frac{\\partial L}{\\partial \\dot{q}} + \\frac{d^2}{dt^2} \\frac{\\partial L}{\\partial \\ddot{q}} = 0.$$ However, I can't see how to derive this equation. Obviously, the final term is supposed to come from integrating by parts the contribution of the $\\ddot{q}$ dependence to the variation in the Lagrangian; doing that yields (writing $T$ for the period of time over which we're extremising the action): $$\\int_T \\frac{\\partial L}{\\partial \\ddot{q}} \\delta \\ddot{q} = \\left[ \\frac{\\partial L}{\\partial \\ddot{q}} \\delta \\dot{q} - \\frac{\\partial L}{\\partial \\ddot{q}} \\delta q \\right]_{\\partial T} + \\int_T \\frac{d^2}{dt^2} \\frac{\\partial L}{\\partial \\ddot{q}}.$$ Now, the term in square brackets presumably has to vanish. The right-hand term will do so, since $\\delta q$ vanishes on the boundary of $T$; but why should the left-hand term vanish? Is it just a condition of dealing with the variations for such a Lagrangian that we consider only variations for which $\\delta \\dot{q}$ vanishes on the boundary of integration as well? Or is there something I'm missing?"}
{"id":"53407","title":"Data For Quantum Entanglement","text":"Is there any publicly accessible data that shows quantum entanglement empirically. I want to see what these researches are seeing that is showing them that indeed this phenomenon is real. Also, any explanation of such data would be greatly appreciated, ie. what instruments are used to measure what aspects, how to interpret the data, and what specific elements prove the phenomenon beyond any doubt or what things would have to be different in the data to falsify the claim of quantum entanglement."}
{"id":"63318","title":"Phase space in quantum mechanics and Heisenberg uncertainty principle","text":"In my book about quantum mechanics they give a derivation that for one particle an area of $h$ in $2D$ phase space contains exactly one quantum mechanical state. In my book about statistical physics they do exactly the same, but now for $6D$ phase space (now one quantum mechanical state covers a volume of $h^3$). I'm not sure that I interpret this well. Does this mean that in this $6D$ volume of phase space (of $h^3$), the particle can only be in exactly one location with only one possible set of momentum vectors $p_x$, $p_y$, $p_z$ (this is how I interpret \"one possible state\")? If this interpretation is right, I have the following problem. Both books say that this is conform the uncertainty principle of Heisenberg, but I don't see why. It seems to me that because one can measure (f.i. in $2D$ phase space) $p_x$ and $x$ so that $∆x∆p_x = h$ and because this is exactly the surface of phase space in which there is only one state, that one has now determined exactly the position and momentum of the particle (because there is only one state in this area). Because of this I guess something is wrong with my interpretation (although it seems to match with the derivation)."}
{"id":"63319","title":"What is a sudden singularity?","text":"I've seen references to some sort of black hole (or something) referred to as a sudden singularity, but I haven't seen a short clear definition of what this is for the layman."}
{"id":"54750","title":"Unclear how heat interacts with Navier Stokes","text":"I am playing around with an Navier stokes solver and I'm having trouble introducing heat. Am I right in thinking this would be introduced in the ${\\bf f}$ term of ${\\partial{\\bf u}\\over\\partial t} = -({\\bf u}\\cdot\\nabla){\\bf u}+v\\nabla^2{\\bf u}+{\\bf f}$? I find the lack of mass term disturbing. Please reassure me! Also, how do I calculate the force vector, given that I have computed the scalar $dQ\\over dt$ from Newton's law of cooling? Or is that the wrong approach?"}
{"id":"112440","title":"Why does the alcubierre drive get roasted by hawking radiation?","text":"I read that an alcubierre drive would get destroyed by hawking radiation inside the warp bubble. Where does this hawking radiation come from? And why does it only happen to objects inside the bubble?"}
{"id":"37976","title":"Pulsed Spherical Wave","text":"Can somebody help show me how a pulsed spherical wave has a wavefunction of the form U(r,t) = (1\/r)a(t-r\/c), where a(t) is an arbitrary function, r is the radius of the spherical wave, t is time, and c is the speed of light?"}
{"id":"37975","title":"do Higgs Bosons happen in nature all the time? Rarely? Or do they only happen when the Higgs field is excited in a particle accelerator?","text":"I'm trying to reconcile an apparent contradiction between explanations given by Dr. Cox in 2009 and 2012, and those given by a panel of Berkeley professors. I'm not a physicist, and so I realize this contradiction is likely my layman misunderstanding. I appreciate your time in teaching me. Dr. Cox seems to describe, I think, the Higgs Boson as though it were just another gauge boson, i.e. one of the fundamental \"force carrier\" particles by which force is transmitted to other particles. Except in this case, the H.B. is not transmitting force but imparting mass. He goes on to say explicitly that Higgs Bosons exist in vast numbers all around us and are constantly bombarding all other particles at all times. It is through this bombardment that mass is gained. Is my summary and interpretation correct? If so, is his explanation accurate? My understanding was the H.B. is more or less a byproduct of the Higgs field itself, the result of its excitation, and it is interaction with the field itself, not the H.B., that endows other particles with mass. This slide from the Berkeley lecture lists the H.B. as neither a gauge boson particle, nor a matter particle, but an entirely different, third type of particle. Does the Higgs field exist because there's an abundance of H.B. particles just moving around, themselves giving rise to the Higgs field? Or does the Higgs field simply exist everywhere, and by virtue of its existence, occasionally an H.B. pops out? (Of this last point, I am certain. Several sources have directly stated that what makes the Higgs field unique is that its default energy state is nonzero, i.e. unlike the \"electron field\", it exists everywhere, even in a vacuum). **Do Higgs Bosons happen on their own in nature all the time?** Rarely? Or do they only happen when the field is excited under deliberate conditions, e.g. within a man-made particle accelerator? Or is my question malformed and running afoul of the wave\/particle duality? A final summary of my running understanding: Consider an electron and a positron, some distance apart. They each experience an attractive force to one another, via the electromagnetic field they produce. The actual means by which this force is transmitted, the means by which the electron \"knows about\" the positron, is by way of a gauge boson, in this case the photon. \"The photon is the mediating particle of the EM field\". **You quite literally have photons moving back and forth between the electron and positron**. They serve as the actual means of transmission of force\/energy\/momentum transfer and exchange between the two particles. (I'm referencing some illustration I saw in a video that quite literally had an animation of photons bouncing back and forth from, in their example, nuclear protons and their orbiting electrons). I think I'm just at a loss to explain the relationship between and difference among the Higgs field and the H.B. itself. Explanations in lectures and the media seem to vary from \"With respect to endowing mass, the H.B. does it all\" to \"The Higgs field does it all; the H.B. is just a theoretical consequence of the existence of the Higgs field; finding the H.B. is strong confirmation that the Higgs field itself exists\". Thanks!"}
{"id":"34405","title":"ADM Hamiltonian formalism and Quantum gravity","text":"is there a Hamiltonian reformultion of gravity ?=? if so if we use the usual Quantization scheme we can not we quantizy the gravity ?? in terms of a Gauge Theory with the potential $ A_{\\mu}^{i} $ how can we get the Schroedinguer equation ??"}
{"id":"26731","title":"What is this shadow of the Sun on the Moon?","text":"I was reading the article _Moon Phases_ on HowStuffWorks. In the picture, each moon has a dark green area which represents the shadow of the Sun. How is this shadow formed and why is this important?"}
{"id":"26730","title":"Energy and Matter","text":"I was watching a show about the big bang theory. They were saying that in the beginning all that existed was energy. After the big bang that energy transformed into matter which then started forming into different elements. How exactly does energy change into matter and at what point does this happen?"}
{"id":"68937","title":"What is a photopeak?","text":"I'm currently studying for an exam, where I'm looking into decays and gamma radiation. I then have a diagram where there are stuff like \"Compton scattering\", \"Single Escape Peaks\", \"Double Escape Peaks\" which all are calculated if I know the value of the photopeak. And all these three are actually quite well explained But, nothing in my notes actually tells me what the photopeak is :\/ So could anyone explain, just briefly, what the photopeak is\/means, at least in this regard. Thanks in advance :)"}
{"id":"103520","title":"What does it mean to \"wrap\" a D-brane around some manifold?","text":"I am getting quite confused with this terminology when I read the papers. Like while constructing the near horizon $AdS_3$ in the $D1-D5$ system one considers $IIB$ on $R^{1,4}\\times M^4 \\times S^1$ and one \"wraps\" $N_1$ D1 branes on the $S^1$ and $N_5$ D5-branes on $M^4 \\times S^1$. What does it exactly mean? Coming from reading how D-branes were introduced in Polchinksi's book I would think that in a $9+1$ spacetime Dp branes are some planar streched out stuff with a $p+1$ worldvolume and whose $p$ spatial dimensions are transverse to the $9-p$ spatial dimensions which have been compactified and T-dualized. So are we now saying that its possible that instead of imagining the Dp branes as some set of periodically arranged planes on the T-dual torus we can also think of their spatial world being compactified on some arbitrary p-manifold? If \"wrapping\" is really a choice of topology for the p-spatial dimensions of the Dp-brane then what determines this choice? Is this something put in by hand or does this happen naturally?"}
{"id":"67947","title":"The force arrow for centripetal force points in the opposite direction of my calculation?","text":"I need someone to clarify a conceptual problem I can't seem to surpass. Image there is a rollercoaster loop and a rollerbcoaster car enters the loop at high speed. Once the car completes the full loop and is then at the last point at the bottom of the loop, I am trying to understanding the forces at play. I can see that there is gravitational force: $F_g$ = $mg$ and centripetal force: $F_c = $ $v^2\/r$, and then there is the normal force $F_N$. Let's say $F_g = 10$ ,and $F_c = 25$, wouldn't $F_N = 35$? And if so, I am very confused about the force diagram. We have $F_N$ pointing up and $F_g$ pointing down, but what about $F_c$? My calculation has it pointing down, but in reality I thought centripetal force always points to the center of a circle? Thanks for the help!"}
{"id":"131074","title":"How do you measure neutron energy?","text":"I think you can detect a neutron event with a scintillator (neutron to photon), photomultiplier tube (photon to electric signal) system, but how would a fusion neutron's energy (2.45 MeV) be typically measured?"}
{"id":"67949","title":"Outflow for fluid simulation based on \"Stable Fluids\"","text":"I've implemented a fluid simulation based on the paper Stable Fluids. It works quite well, except I would like to have the velocity at the \"upper\" edge just to outflow and not to re-enter on the lower(opposite) edge. Apparently that's a side-effect of the periodic boundaries. Is there an easy method to still achieve this? How would I need to modify the velocity? **Edit:** The fluid may just exit, since I'm just interested in how the velocity evolves (and since I'm adding new tracing particles all the time anyway). Actually I use an (3d-)extended version of the CUDA fluid example. I guess it's mesh-full and yes, it involves the FFT."}
{"id":"77354","title":"How do I split a vector into components parallel and perpendicular to a known line?","text":"> Given $\\mathbf{F}=\\langle 7.20,−12.0,28.2\\rangle\\text{ Newtons}$, find the > component of $\\mathbf{F}$ that acts perpendicular to member DA such that the > vector addition of the perpendicular and parallel components of $\\mathbf{F}$ > ($\\mathbf{F}=\\mathbf{F}_\\perp + \\mathbf{F}_\\parallel$) with respect to DA > equals $\\mathbf{F}$. Express your answer in component form. $\\mathbf{A} = (−5.60,3.68,5.76)$ and $\\mathbf{D} = (0,2.72,3.00)$ By finding the position vector of DA, which was $\\langle -5.6,0.96,2.76\\rangle$, then getting the unit vector of DA, which was $\\langle -0.886,0.152,0.437\\rangle$ and dotting it with the force vector I got the magnitude of the force along DA, which was $4.11\\text{ Newtons}$. Now I am not sure how to continue this problem. I know that the magnitude of $\\mathbf{F}$ minus the magnitude of the force along DA equals the magnitude of $\\mathbf{F}_\\perp$, but how do I get it in components? ![Diagram](http:\/\/i.stack.imgur.com\/cAY6w.jpg)"}
{"id":"83703","title":"Flux increase due to velocity?","text":"I have a book question I'm trying to understand: ![http:\/\/i.imgur.com\/IqmS9OB.png](http:\/\/i.stack.imgur.com\/LJCeE.png) The first bit is easy enough. The second part I was confused so I checked the solution: > B is into the page. The flux increases as the bar moves to the right, so the > magnetic field of the induced current is out of the page inside the circuit. > To produce magnetic field in this direction the induced current must be > counterclockwise, so from b to a in the rod. I thought I had a pretty good grip on increasing\/decreasing magnetic field inducing negative\/positive current(respectively). Maybe I'm missing something real trivial but I don't understand how flux increases as the bar moves. Why?"}
{"id":"56081","title":"Why there is no operator for time in QM?","text":"Is there one central reason why there is no \"Time\" operator in QM? I know this question has been asked before, but I thought I would try to stimulate some fresh thinking."}
{"id":"34243","title":"Is there an observable of time?","text":"In Quantum Mechanics, position is an observable, but time may be not. I think that time is simply a classical parameter associated with the act of measurement, but is there an observable of time? And if the observable will exist, what is an operator of time?"}
{"id":"6584","title":"Time as a Hermitian operator in QM?","text":"In non-relativistic QM, on one hand we have the following relations: $$\\langle x | P | \\psi \\rangle ~=~ -i \\hbar \\frac{\\partial}{\\partial x} \\psi(x),$$ $$\\langle p | X | \\psi \\rangle ~=~ i \\hbar \\frac{\\partial}{\\partial p} \\psi(p).$$ On the other hand, despite the similarities, the relations cannot be directly applied to energy and time: $$\\langle t | H | \\psi \\rangle ~=~ i \\hbar \\frac{\\partial}{\\partial t} \\psi(t),$$ $$\\langle E | T | \\psi \\rangle ~=~ -i \\hbar \\frac{\\partial}{\\partial E} \\psi(E).$$ Just wondering, how can one mathematically prove that the \"classical time\" (which means no QFT or relativity involved), unlike its close relative \"position\", is not a Hermitian operator? I ask your pardon if you feel the question clumsy or scattered. But to be honest, if I can clearly point out where the core issue of the problem is, I may have already answered it by myself :\/"}
{"id":"64656","title":"How can we detect X-rays?","text":"I know that X-rays can be detected by various ways, like ionizing of air particles. Is there a way to detect X-rays,which are photons, by detecting? Can something absorb the energy of the X-rays and detect if it is there?"}
{"id":"72652","title":"How to measure the arbitrariness of a quantum state?","text":"An arbitrary qubit is represented as $\\alpha|0\\rangle+\\beta|1\\rangle$ with $|\\alpha|^2+|\\beta|^2=1$. If we know either $\\alpha$ or $\\beta$, the state can be completely identified. The 'arbitrariness' of an arbitrary qubit thus can be said to be '1'. For a known state this is '0'. Now if we consider an arbitrary two qubit state, the total number of variables are 3 (actually 4 but 1 less due to normalization). If any two of these variables are equal or one of them is known, the state is still arbitrary but from a given restricted class. We can say that the 'arbitrariness' of this quantum state is something between 0 and 1. Can anyone suggest a measure for the arbitrariness of this quantum state? I want to extend this measure to higher order states. Can anyone suggest any reference for this?"}
{"id":"92492","title":"How does the Planck length hold true?","text":"If there is a length which no object can breach in terms of the smallest of size, how can that length hold true? Our earth is undoubtedly considered the Planck length to some monolithic planet out there (relatively speaking of course), yet a number was derived stating the smallest possible space occupied? I'm not saying this theory is wrong, I would just like it explained to me."}
{"id":"98131","title":"Which is smallest distance measurement which makes sense talking about","text":"This might seem a silly question, but here it goes: Of all possible values for a measure of distance, which is the smallest that makes sense talking about? I mean, I could talk about $10^{-100000}$ meters but in real life this value is smaller than anything we know (i think), so it wouldn't make any sense talking about this value, now would it? Initially I thought about Planck length, but I'm not sure. Please not I'm not referring to mathematics, any number can be of use in mathematics. But in real life, small values like $10^{-100000}$ meters don't make sense. So, what is smallest distance value that makes sense talking about? Thanks."}
{"id":"62003","title":"Why is the Planck length the shortest measurable length?","text":"I quote from the Wikipedia article on Planck length: > _According to the generalized uncertainty principle, the Planck length is in > principle, within a factor of order unity, the shortest measurable length – > and no improvements in measurement instruments could change that._ I also link to an answer provided to this Phys.SE question: > _The reason there is only a finite number of ways is that we assume > separations smaller than the Planck length can't be distinguished._ I have hear this claimed every once in a while, but I never learned this formally at university. Can it be proven using quantum mechanics\/uncertainty principle that the Planck length is the smallest measurable quantity, or is this still an area of debate and opinion in modern physics? If you have a simple proof, could you please provide."}
{"id":"32802","title":"Need help understanding dynamic and static contact angles","text":"I've run into a conceptual road block. I'm coming to you guys because I think my adviser is getting annoyed with me. The concept involves a meniscus being pulled up a cylinder. I understand that the main force \"pulling\" on the meniscus at the contact line is due to the difference between the static and dynamic contact angles. This force per unit length takes the form: $F=\\gamma (Cos(\\theta_{static})-Cos(\\theta_{dynamic}))$ The concept that describes determining the static and dynamic contact angles have been given to me in terms of minimizing the interface between the liquid and air when the meniscus has reached a known height. In the words of someone much more knowledgable than me: \"The fluid interface only knows about the equilibrium contact angle as a boundary condition. For any circle and any height there is a minimum energy configuration of the fluid surface. All of these solutions have a different angle. You find the equilibrium by finding the minimum energy solution that also intersects the solid at the contact angle.\" For some reason, I am unable to see where I can explore these concepts to determine a relationship between the dynamic and contact angles. Can anyone offer elucidation as to these concepts? Thanks"}
{"id":"55008","title":"A basic question on AdS\/CFT","text":"Previously I asked a question Question on dimensions of CFT operators (ref: MAGOO, hep-th\/9905111) here and it was (correctly of course) answered by Motl. I realized I didn't understand a part of it even though I thought I did. It was about dimensionlessness of bulk scalars from the perpective of boundary conformal transformation. If we have a bulk scalar $\\phi$ it is dimensionless from the point of boundary transformations as (as answered by Motl) ``it is an actual field in the AdS space (the bulk), unlike $\\phi_0$ that is defined on the boundary. They mean that it is dimensionless under the conformal transformations of the boundary - and that's true because the conformal transformations are realized as simple isometries in the bulk, so they can't rescale the AdS field $\\phi$ - at most, they move it to another point. '' But the way we talk about conformal dimension of a boundary operator is through it's transformation under the diffeomorphism $x\\to\\lambda x$ and then if a boundary field $f(x)\\to f^\\prime(x)=\\lambda^hf(\\lambda x)$ (or the relation written later by Motl in that answer), then $h$ is it's conformal dimension. I hope what I said so far is right. But, isn't conformal transformation at the boundary also act as a diffeomorphism in the bulk (at least for boundary part of the coordinates of the bulk field). Then why will it not transform in such a way that it has a dimension? I believe, e.g. spin-1 gauge field in the bulk has (confomal\/scaling\/mass) dimension 1. So, how does it have a dimension? Sorry for the novice question and thanks in advance!"}
{"id":"51583","title":"Difference between $\\partial$ and $\\nabla$ in general relativity","text":"I read a lot in _Road to Reality_ , so I think I might use some general relativity terms where I should only special ones. In our lectures we just had $\\partial_\\mu$ which would have the plain partial differentials. On a problem set, the Bianchi identity for the Maxwell field tensor is given as: $$ \\partial^\\alpha F^{\\beta\\gamma} + \\partial^\\beta F^{\\gamma\\beta} + \\partial^\\gamma F^{\\alpha\\beta} = 0. $$ In Penrose's book, this identity is given as $\\nabla_{[a} R_{bc]d}{}^e = 0$ where the square brackets denote the antisymmetrisation like in the previous form. **Are those square brackets standard notation in Physics?** Since $\\partial_\\mu$ is basically the $(\\partial_t, \\nabla)$ it is a covector or covariant vector. Penrose calls this $\\nabla_a$ the covariant derivative (something with a connection and curved manifolds as far as I understood). If I am in a non-curved $\\mathbb M$ Minkowski $(1, 3)$ space where I have no curvature (since $\\eta_{\\mu\\nu}$ is $\\mathop{diag}(1, -1, -1, -1)$?), I was thingking that $\\partial_\\mu = \\nabla_\\mu$. **Can I write $\\nabla_\\mu$ instead of my partial derivatives or do they mean something different?**"}
{"id":"51587","title":"Spinning spheres colliding","text":"In an ideal environment with no friction, in a vacuum, what happens to the velocity of the spin of two spheres spinning in perfect parity at two different velocities when they come into contact?"}
{"id":"35467","title":"Why a star commonly exhibits 6 rays?","text":"Have you realized, that often the photo of a star shows 6 rays spreaded symmetrically around it, independently of the camera chosen? Do you have idea of what kind of optical phenomena is behind it?"}
{"id":"35463","title":"Oscillating Dipole: Principle of Receiving and Transmitting Electromagnetic wave?","text":"I am practising to Tfy-0.1064 -elementary-physics-exam and doing this practise -exam here. The problem in Finnish goes like this: > _\"Selosta lyhyesti sähkömagneettisen säteilyn lähettämisen ja > vastaanottamisen periaate.\"_ and its translation is: > **\"Outline the principle of transmitting and receiving Electromagnetic > wave.\"**"}
{"id":"73297","title":"A question for the generalization of gauge transformation with two antisymmetric indices","text":"I have a question about the generalization of gauge transformation with two antisymmetric indices. Starting from Eq. (3.7.6) in Polchinski's string theory book p. 108. $$S_{\\sigma} = \\frac{1}{4 \\pi \\alpha'} \\int_M d^2 \\sigma g^{1\/2} \\left[ \\left( g^{ab} G_{\\mu \\nu}(X) + i \\epsilon^{ab} B_{\\mu \\nu} (X) \\right) \\partial_a X^{\\mu} \\partial_b X^{\\nu} + \\alpha' R \\Phi(X) \\right] \\tag{3.7.6} $$ where $B_{\\mu \\nu}(X)$ is the antisymmetric tensor. It is said variation $$ \\delta B_{\\mu \\nu} (X) = \\partial_{\\mu} \\zeta_{\\nu}(X) - \\partial_{\\nu} \\zeta_{\\mu}(X) \\tag{3.7.7} $$ can add a total derivative to the Lagrangian density. I tried to do some integration by part for $\\partial_{\\mu}$, $\\partial_{\\nu}$, $\\partial_a$, and\/or $\\partial_b$ in (3.7.7), miserably I didn't get a total derivative. My question is, how to see Eq. (3.7.7) gives a total derivative?"}
{"id":"107850","title":"Possible to calculate carrier lifetime with doping concentrations?","text":"I'm working on a problem involving a silicon semiconductor. A PN junction is created with $N_a = 5\\cdot10^{14}$ throughout the entire semiconductor, and is doped with $N_d = 2\\cdot10^{15}$ on the N side. I'm asked to calculate the Diffusion Constant $D$, carrier mobility, and carrier lifetime for both sides of the semiconductor. I think I've properly calculated the carrier mobility and the diffusion constant, but I can't figure out how to get the carrier lifetime (or recombination time)."}
{"id":"23814","title":"Is it circular reasoning to derive Newton's laws from action minimization?","text":"Usually, a typical example of the use of the action principle that I've read a lot is the derivation of Newton's equation (generalized to coordinate $q(t)$). However, in the classical mechanics interpretation, isn't this like tripping oneself up? Because we've already identified $T$ and $U$ as the kinetic energy and the potential energy, respectively which are derived, in fact, from Newton's equations."}
{"id":"65840","title":"Quantum master equation in the Batalin-Vilkovisky formalism","text":"I am reading the Section 15.9 of Weinberg's book \"The Quantum Theory of Fields, vol. 2\". Under a shift $\\delta\\Psi[\\chi]$ in $\\Psi[\\chi]$, we have $$ \\begin{split} \\delta Z&=i\\int[d\\chi]\\,\\exp\\big(iI_{\\Psi}[\\chi]\\big) \\left(\\frac{\\delta_RS[\\chi,\\chi^{\\ddagger}]}{\\delta\\chi_n^{\\ddagger}} \\right)_{\\chi^{\\ddagger}=\\delta\\Psi\/\\delta\\chi}\\left( \\frac{\\delta(\\delta\\Psi[\\chi])}{\\delta\\chi^n}\\right) \\\\\\ &=i\\int[d\\chi]\\,\\exp\\big(iI_{\\Psi}[\\chi]\\big)\\left\\\\{\\frac{\\delta_L} {\\delta\\chi^n}\\left(\\frac{\\delta_RS}{\\delta\\chi_n^{\\ddagger}} \\delta\\Psi\\right)-\\frac{\\delta_R}{\\delta\\chi_n^{\\ddagger}} \\frac{\\delta_LS}{\\delta\\chi^n}\\delta\\Psi\\right\\\\}_{\\chi^{\\ddagger} =\\delta\\Psi\/\\delta\\chi} \\\\\\ &=\\int[d\\chi]\\,\\exp\\big(iI_{\\Psi}[\\chi]\\big)\\left\\\\{ \\frac{\\delta_RS[\\chi,\\chi^{\\ddagger}]}{\\delta\\chi_n^{\\ddagger}} \\frac{\\delta_LI_{\\Psi}[\\chi]}{\\delta\\chi^n}-i\\Delta S[\\chi,\\chi^{\\ddagger}]\\right\\\\}_{\\chi^{\\ddagger}=\\delta\\Psi\/ \\delta\\chi}\\delta\\Psi[\\chi] \\end{split} $$ The last line is exactly the same to Eq. (15.9.33). Referring to the definition of antibracket $$ (F,G)=\\frac{\\delta_RF}{\\delta\\chi^n}\\frac{\\delta_LG}{\\delta\\chi_n^{\\ddagger}}- \\frac{\\delta_RF}{\\delta\\chi_n^{\\ddagger}}\\frac{\\delta_LG}{\\delta\\chi^n} $$ we can see that the quantum master equation reads $$ -(S,S)-2i\\Delta S=0 $$ which has an extra minus sign. I are not sure whether this is a typo or not. Could someone help me to check this derivation? Besides, I am also confused by $\\delta_L$ and $\\delta_R$. Any clarifications will be appreciated. Many thanks in advance!"}
{"id":"74114","title":"Heisenberg picture of QM as a result of Hamilton formalism","text":"Let's have formula of full time-derivative of physical value in Poisson's formalism: $$\\tag{1} \\frac{df}{dt} = -[H, f]_{P. br.} + \\frac{\\partial f}{\\partial t}, $$ where $[A, B]_{P. br.}$ is Poisson's bracket, and $H$ the Hamiltonian. By using quantum mechanics the equation can be rewritten in form $$\\tag{2} \\hat {\\frac{d f}{d t}} = \\hat {\\dot {f}} = \\frac{i}{\\hbar }[H, f] + \\hat {\\frac{\\partial f}{\\partial t}}. $$ It is a bit like Heisenberg picture, but in $(2)$ in the left side there is operator of full time-derivative of function (also, in the right side - operator of partial time-derivative). In Heisenberg picture in the left side is full time-derivative of operator, which represents function. So my question is: can I get Heisenberg formalism from $(1)$?"}
{"id":"75868","title":"Voltage and parallel-plate capacitors","text":"Suppose we have two parallel plates (each of area $A$) spaced with distance $d_1$ acting as a parallel-plate capacitor within a circuit. If the capacitor is connected to a power source with constant voltage $V_s$, positive charge will accumulate on the parallel plate connected to the positive terminal and negative charge will accumulate on the plate connected to the negative terminal. This will create an electrical potential between the two plates. Does this process continue until the voltage $V_t$ between the plates is equal to $V_s$, and if not, how can we calculate the terminal voltage? Suppose we move the plates closer to distance $d_2$. Will the terminal voltage be identical to that of $d_1$?"}
{"id":"99890","title":"Energy of a damped oscillator","text":"$$ E=\\frac{1}{2}m\\left(\\frac{dx}{dt}\\right)^2+\\frac{1}{2}m\\omega_0^2x^2. $$ This is the equation for the energy of a oscillator. The second term is the potential energy. Now, my question is, will this second term remain the same if we are to consider the oscillator as a damped oscillator. Because doesn't the second term arise from the consideration of a conservative force field? And when talking of damped oscillator, isn't the force field non- conservative?"}
{"id":"32118","title":"Is there a physical reason steel balls are not suitable to play billiards with?","text":"Steel is an elastic matter and is often used when demonstrating elastic collisions. We also make springs from it. Due to these properties and for its relative low price, it seems obvious to me to use them to play billiards. But I don't find any information on the internet about this. Maybe I miss something very obvious?"}
{"id":"57335","title":"A partial differential equation for kinetic energy","text":"The kinetic energy of a point particle of mass $m$ and speed $v$ is $K = \\frac{1}{2}mv^2$. An elementary mathematics textbook I saw asked one to show that $$ \\frac{\\partial K}{\\partial m}\\frac{\\partial^2 K}{\\partial v^2} = K.$$ While this is a straightforward exercise in partial differentiation, is there supposed to be any physical meaning behind this formula? For example, is there a significance to quantities that satisfy this nonlinear PDE?"}
{"id":"101869","title":"multiple frequency trapped power signal","text":"is there any possibility to generate a waveform that consists of multiple power signals with different frequencies such that these signals travel together like they constitute [to form a knot(waveform) from a group of threads(signals)] without losing any power while propagation?"}
{"id":"134719","title":"Am I missing a trick to solving this differential equation?","text":"I was playing around with a 3-D potential $V$ such that $V_{(r)} = 0$ for $r<a$, and $V_{(r)} = V_0$ otherwise. By using the Schrödinger Equation, I showed that: $$\\frac{-\\hbar}{2m}\\frac{1}{r^2}\\frac{d}{dr}\\bigl( r^2\\frac{d}{dr}\\bigr)\\psi = E\\psi$$ I then used the substitution $\\psi_{(r)}=f_{(r)}\/r$ and $k=\\sqrt{2mE}\/\\hbar$ to get: $$\\frac{1}{r}\\frac{d^2f_{(r)}}{dr^2}=-\\frac{k^2}{r}f_{(r)} \\tag{I}$$ which describes the wavefunction $\\psi_{(r)}=f_{(r)}\/r$ inside the sphere. Hence, the differential equation has the domain $0\\leq r<a$, and I cannot multiply both sides by $r$. This is unfortunate, because there is a similar equation for the outside of the sphere: $$\\frac{1}{r}\\frac{d^2f_{(r)}}{dr^2}=\\frac{k'^2}{r}f_{(r)}$$ As this is outside the sphere, I can multiply both sides by $r$ to get a familiar differential equation that can be solved easily: $$\\frac{d^2f_{(r)}}{dr^2}=k'^2f_{(r)}$$ If I do the same thing to $(I)$, I obtain the equation for simple harmonic motion, but substituting the solution back into $(I)$ as a sanity check gives a division by zero when evalutating for $r=0$. After that, I tried a number of substitutions to make $(I)$ have a more recognisable form - to no avail. Then I had the idea of multiplying my trial solution by some other function of $r$ so that upon substitution into $(I)$, the evaluation of $r=0$ doesn't give an infinity... but I don't know quite how to do that... Long story short..... my question is: what trick do I need to get a meaningful solution to $(I)$?"}
{"id":"4990","title":"Chaos theory and determinism","text":"My professor in class went a little over chaos theory, and basically said that Newtonian determinism no longer applies, since as time goes to infinity, no matter how close together two initial points are, the distance between them will increase greatly. But why isn't this merely a matter of the imprecision of our measuring instruments? If we can somehow know our initial conditions _exactly_ , wouldn't we still be able to calculate what the system will be like at some time _t_ in the future?"}
{"id":"37699","title":"Calculate time to go from one point to another point","text":"in a 2D football game I want to calculate cycle(time) until ball go from one point to another point. in this link exist all fomulas that we need for my problem but anyone does not have any parameter that shows the effect of decay on ball speed."}
{"id":"106077","title":"A real gas with gravitation-like interaction","text":"Consider a system (a gas) of point-like particles with a gravitation-like interaction (potential) $V(r) \\sim \\frac{1}{r}$ between pairs of them. One can rule out statistically that two particles will approach each other exactly along their line, so two particles will never collide directly - thus giving rise to infinite (kinetic) energies. So the particles will always whirl around each other in one way or the other. What can be said about such a system in the framework of statistical mechanics? How does its partition function look like? What are its thermodynamical equilibrium states? What happens when cooling such a system down (e.g. from very high temperatures)?"}
{"id":"70741","title":"Calculating Mass for a Reduced Trajectory while Keeping Projectile Force Constant","text":"Apologies if this is not a research-specific question, but applied mechanics. A firefighter has to pass a test in which he is to throw a medicine ball a certain distance. The ball is thrown at a 45 degree angle (Fig. 1) and has to travel a ground distance of 7.5 meters. For the test, a medicine ball is a rough sphere with a mass of 4 kg and ~250 mm in diameter. In preparing for the test, however, the firefighter has only access to a dumbbell, which he does not intend to throw. Given that a certain force is needed to throw the medicine ball with a preparation phase of 1 m and then released for a 7.5 meter throw, how heavy should the training substitute dumbbell be in order to achieve the equivalent force? Assume that the dumbbell only covers a 1 m distance at 45 degrees and that the firefighter intends to train at a 1 m\/s velocity. * * * Figure 1 (Source: http:\/\/www.bochum.de\/C125708500379A31\/vwContentByKey\/W27LXNMW860BOLDDE#par4)"}
{"id":"70742","title":"An Electromagnetic Paradox?","text":"![Two masses](http:\/\/i.stack.imgur.com\/VfRrV.jpg) The above diagram represents an isolated system with two masses $M$, at position $X$, and $m$, at position $x$, connected together by an extended spring. Each mass is connected by rigid rods to charges $Q$ and $q$ (of negligible mass) separated by a vertical distance $r$. The masses and charges are constrained to move in a horizontal direction so only horizontal forces affect the dynamics of the system. The system is released and the two masses are pulled together by the spring. As there is no external force, $F_{ext}$, acting on the system then we have $$F_{ext} = M_{tot} \\ddot{X}_{com} = 0,$$ where $M_{tot}=M+m$ is the total mass of the system and $X_{com}=(MX+mx)\/(M+m)$ is the center of mass of the system. Thus we have: $$M \\ddot{X} + m \\ddot{x} = 0.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (1)$$ But let us now consider the masses $M$ and $m$ separately together with the internal forces acting on them. The force on $M$ consists of the force $F$ from the spring together with a horizontal electromagnetic radiation force on charge $Q$ due to the acceleration of charge $q$. Therefore we have: $$M \\ddot{X} = F - \\frac{qQ}{4\\pi\\epsilon_0 c^2 r}\\ddot{x}.$$ Similarly the force on $m$ consists of the force $-F$ from the spring together with a radiation force on charge $q$ due to the acceleration of charge $Q$. $$m \\ddot{x} = -F - \\frac{qQ}{4\\pi\\epsilon_0 c^2 r}\\ddot{X}.$$ Adding these two equations together we obtain: $$M \\ddot{X} + m \\ddot{x} = -\\frac{qQ}{4\\pi\\epsilon_0 c^2 r}(\\ddot{X} + \\ddot{x}).\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (2)$$ Equation (2) is not consistent with Equation (1). There seems to be a paradox here. One could answer that I have not taken into account the electromagnetic reaction force on the charges due to their emission of electromagnetic radiation. In fact the electromagnetic reaction force on a moving charge $q$ is given by $$F_{reaction} = -\\frac{2}{3}\\frac{q^2}{4\\pi\\epsilon_0 c^3}\\dot{a}.$$ Therefore $F_{reaction}$ depends on the rate of change of the acceleration, $\\dot{a}$, rather than $a$ itself. As the reaction force term has an extra factor of $c$ in the denominator and the rate of change of acceleration is small (the length of the spring doesn't change very fast if the masses are reasonably large) then we can ignore the possibility of the emission of electromagnetic radiation and the subsequent reaction forces on the charges. Therefore this is not an answer to the paradox."}
{"id":"102982","title":"Difference between dielectric constant and dielectric function","text":"I'm confused by the definition of dielectric constant. We all know that dielectric is function of wavelength, because materials respond in different ways depending on the energy of the light. But what does it mean when it says > _Dielectric constant of a material is = 5._ ?"}
{"id":"63961","title":"Does relativistic mass have weight?","text":"If an object was sliding on an infinitely long friction-less floor on Earth with relativistic speeds (ignoring air resistance), would it exert more vertical weight force on the floor than when it's at rest?"}
{"id":"38767","title":"Relation between quarks and strings","text":"In string theory(s), are quarks just individual strings, or are they made of multiple strings? Are the heavier quarks made of heavier or longer strings? Are there red, blue, and green strings corresponding to the color charges of qcd?"}
{"id":"26936","title":"Measurement of Tangential Momentum Accomodation?","text":"> **Possible Duplicate:**   >  Measurement of Tangential Momentum Accomodation?   * I am using atomic force microscopy (AFM) for characterizing pores of the size of nanometers for application in gas flow. For this, knowing the value of Tangential Momentum Accommodation (TMA) coefficient is an important task. Following is a formalism that describes measurements done by AFM and the formula to compute TMA: Sneddon demonstrated that for any axisymmetric tip of an AFM, the relation between indentation force and displacement can be written as: > $F=\\mathrm{TMA} (Zp-Zc)^n$ Where, > TMA = Tangential Momentum Accommodation for a spherical tip, $\\mathrm{TMA} = \\frac43 E_r R_t^{\\frac12}$ > $E_r$ = reduced elastic modulus = $\\frac{E}{1-\\nu^2}$ > > $n$ = constant between 1 and 2 that depends on the shape of the tip > > $Z_p$ = piezoelectric displacement of the chip during indentation > > $Z_c$ = cantilever deflection > > $R_t$ = tip radius > > $E$ = Young's elastic modulus > > $\\nu$ = Poisson ratio of the sample (for rock minerals it lies between 0.1 > and 0.3) AFM can be used to measure $F$, $Z_p$, $Z_c$ and **Bulk modulus of elasticity of materials**. To obtain the exponent $n$, a log-log plot of $F$ versus $(Zp- Zc)$ is plotted, although TMA is not calculated from this plot because doing so would require accurate knowledge of $(Zp-Zc)$ and, thus, the contact point.   * Based on the above information, I would like to know how we can get TMA from force measurement of molecule and pore wall. **ABOUT THE EXPERIMENT:**   1. The goal is to measure TMA by AFM. We measure interactive forces between AFM tip (hemisphere) and planar surface. Using the models for hemisphere and plane surface we can extract necessary data. We extract coefficients that are independent of the shapes to be able to use it in a pore.   2. The tip measures interactive forces.   3. In addition to hydrodynamic forces, electrostatic and van der Waals forces are additional forces."}
{"id":"18705","title":"the causality and the anti-particles","text":"How can I quantitatively and qualitatively understand the fact that there is a relevence between the existence of anti-particles and the causality?"}
{"id":"18513","title":"Motion is relative, right? And most pop sci relativity explanations are somewhat incorrect?","text":"On page 20 of _A Brief History of Time_ : > . . . all observers should measure the same speed of light, no matter how > fast they are moving. But in an observer's frame of reference, they're actually not moving at all. So the sentence \"should\" read, \". . . they are moving, _relative to each other_.\" BTW, I do realize that this kind of writing is intended to be easy to read and thus omits the obligatory caveats to every claim. I'm just trying to deeply, intuitively understand the material. From a UVA lecture, Conserving Momentum: the Relativistic Mass Increase: > Mass Really _Does_ Increase with Speed . . . particles become heavier and > heavier as the speed of light is approached . . . And here, isn't \". . . heavier and heavier _relative to an observer_ . . .\" actually correct?"}
{"id":"132048","title":"How to tell whether photons are entangled?","text":"Suppose you have some sort of a \"black box\" system - you know nothing of its inner workings. The system has two outputs, let's call them A and B, and it occasionally emits photons - one photon from each output, with both the A photon and B photon emitted simultaneously. You observe the system for a while, and notice that when you measure the polarization of the two photons, they always have the opposite polarization - if one is horizontal, the other is vertical, and vice versa. Now, is there any experiment you can do which will show you whether the photons are truly entangled in the quantum sense, or whether it is some purely classical process within the box that's causing the polarization to be opposite? If so, what is that experiment? An answer that won't assume any more than undergrad knowledge of physics would be appreciated!"}
{"id":"109894","title":"Modified gravity and dark energy","text":"Is there a model of _modified gravity_ that either was close to or have succeeded in modelling the problem of dark energy, and suffers from no instabilities and explains the observational data. Also has the approach to use modified gravity to explain the problem of dark energy been abandoned? Needed to write a review about the challenges facing modified gravity, Any insight would be great. Cheers"}
{"id":"87723","title":"The formation of ice from water properties?","text":"> The formation of ice from water is accompanied by: > > A. absorption of energy as heat > > B. temperature increase > > C. decrease in volume > > D. an evolution of heat > > E. temperature decrease For the first one I think it's correct but I don't know the explanation For second one it is wrong since ice is cold For the third one it is wrong since ice has more volume than water for same quantity For the fourth one I'm confused what is heat? It is colder but doesn't mean heat evolute? For fifth one I think it is correct since ice is cold"}
{"id":"21831","title":"Why does separating plastic from a cough drop create flashes?","text":"I was tearing open individually-wrapped cough drops in the dark, and noticed that when I pulled apart two adhesive-joined pieces of silvery plastic film, small flashes of light were produced. It was not a reflection--it was pitch dark. What causes this? The product is Bee M.D. Organic Honey Throat Drops--kind of gooey, old ones (laugh). They contain menthol and eucalyptus oil. I did some more experimenting and think the the flash could actually be separating the plastic from the cough drop itself rather than pulling apart the adhesive. The item is small, the room is dark, my finger has a wound, and all in all it's hard to tell for sure. I don't know if this is a physics or chemistry question--it depends on the answer."}
{"id":"103294","title":"What is the reason behind why a quantum particle cannot be at rest?","text":"So I've seen different reasonings for this; which is correct, or are they both corollaries of each other? 1) For a particle to be at rest, we would know its momentum and therefore by Heisenberg's uncertainty principle, the uncertainty in its position would be infinite. Is this acceptable? 2) $\\lambda = h \/ p = h \/ 0 \\implies \\lambda \\rightarrow \\infty$ which is not possible. EDIT: Another reason: 3) Due to the zero-point energy, even in its ground state there will always be non-zero kinetic energy."}
{"id":"103296","title":"Function for curved line in log-log-plot","text":"I've done experiments on the thermo-viscoelasticity of liver tissue and the following results               temp shift     1   20     4     2   30     8     3   40    20     4   45    49     5   50   300     6   55  8000     7   60 60000      ![shift vs temp plot](http:\/\/file2.npage.de\/013648\/02\/bilder\/curved_loglog.jpg). show a curved line in a loglog-plot, which means I can't use a power-law function to approximate my data. Do you have any idea what kind of function might work for my data? I've tried superlinear functions of the type y(x) = a^x^b but that doesn't seem to lead to a good fit. I'm adding a plot of the same data plus 2 additional temperatures (70 and 80). temp shift 1 20 4 2 30 8 3 40 20 4 45 49 5 50 300 6 55 80000 7 60 600000 8 70 9000000 9 80 9000000 ![shift vs temp plot](http:\/\/file2.npage.de\/013648\/02\/bilder\/curved_loglin.jpg)."}
{"id":"87729","title":"E&M question in example in my book","text":"So I still not sure how to apply like Right-hand rule (RHR) in this setup in problem like the one in the following so I tried to do RHR in order to get the direction but it didn't work out. This is an example from the Halliday and Resnick book. > Figure 32-24 shows a wire segment, placed in a uniform magnetic field B that > points out of the plane of the figure. If the segment carries a current $i$, > what resultant magnetic force acts on it ? Here is image of the problem at hand; $F_1$ and $F_3$ are the force on the straight wire segments ![enter image description here](http:\/\/i.stack.imgur.com\/WFe6M.jpg) Solution: $$F_1 = F_3 = -iLB $$ so its equal to $$iL = \\langle iL,0,0\\rangle \\quad B = \\langle0,0,B\\rangle$$ $$iL\\times B = \\langle0,-iLB,0\\rangle$$ So it points in the **negative y** direction I tried to do same thing now for circular part to get the directions: $$iL = i\\langle r \\cos \\theta,r \\sin \\theta ,0\\rangle \\quad B =\\langle0,0,B\\rangle$$ $$iL \\times B = \\langle r \\sin\\theta B,r \\cos\\theta,0\\rangle$$ So since its a circle, $\\cos \\theta$ will gets canceled by symmetry so it will be so it will have only a horizontal component equal to $r \\sin \\theta B$ of course this works out in the problem when you do the integral but its not right reasoning as the book have and book used RHR."}
{"id":"15038","title":"Why do I hear beats through headphones only at low frequencies?","text":"I was recently playing with this Wolfram Demonstrations applet, which demonstrates beats. At first I thought the app didn't work because I couldn't hear any beats. Then I realized that the applet doesn't analytically add two sine waves of nearby frequencies and play the result; it plays different sine waves to the left and right audio channels. Since I had headphones on, I only heard one frequency per ear, and only consciously detected a single tone. When I took the headphones off and held the ear buds close together, I could hear the beats distinctly. Messing around a little more, I found that when I lower the frequency to around 600 Hz or lower, I begin to be able to hear the beats even with headphones on. What's happening? Are my ears for some reason picking up relative phase information only at lower frequencies? Are only lower frequencies transmitted effectively through my skull? I would guess it's a coincidence that I begin to hear beats when the wavelength of the sound (in air) is on the order of the distance between my ears, but is that right?"}
{"id":"77837","title":"How do people work out the trajectories of planets and stars just by looking at them?","text":"I've been thinking about how astronomers can look at bright dots in the sky and deduce a whole bunch of things from their movements. I'm particularly interested in how people like Kepler and Galileo, 400 years ago, were able to deduce that planets move in ellipses and work out laws like equal times means equal areas, among others. It seems really remarkable to me that people have figured out a lot of very detailed things about the universe just looking from our planet. There is a similar question that asks how people can determine what the different celestial objects are made of. I'm more interested in the mechanics side of it; in class we've worked out Kepler's laws theoretically using Newtonian mechanics, but of course that's not how Kepler did it. I'd be interested in a more-or-less detailed explanation of how can one deduce so much about celestial objects just by looking at the sky. I realize this is probably not trivial, so references to books or websites are fine."}
{"id":"35878","title":"Is acceleration due to gravity constant?","text":"I was taught in school that acceleration due to gravity is constant. But recently, when I checked Physics textbook, I noted that $$F = \\dfrac{G \\cdot m_1 \\cdot m_2}{r^2} $$ So, as the body falls down, $r$ must be changing, so should acceleration due to gravity."}
{"id":"33334","title":"What would the universe be like if Electroweak symmetry were unbroken?","text":"> **Possible Duplicate:**   >  What happens to matter in a standard model with zero Higgs VEV? What if the Higgs did not have a \"Mexican hat\" potential and the therefore it's vacuum expectation value were zero. Then the elementary particle masses would be zero, so the electron, W+, W-, Z and quarks would all be massless. Would a proton with non-zero mass be possible since most of the proton mass comes from the binding energy of gluons which are already massless? The weak force would be long range and fall off like $1\/r^2$. I assume the Z would take the role of the photon. How many kinds of weak charge are there? Would like charges be repelled and unlike charges be attracted via the Z? Would there be something equivalent to magnetism? What would the massless W do? Would the long range weak \"W\" force change the electrical charge values of electrons\/neutrinos and quarks? How weird is that! I know, we wouldn't exist to see all of this, but I am curious about how this would all work."}
{"id":"119884","title":"To find the center of charge in current waveform","text":"I have a circuit for which output current waveform has both positive and negative points. Now I want to get a center of the charge for this waveform (means where my net charge is concentrated ?). Is there any formula or theorem to calculate this. My current waveform has time on $x$ axis and current values at $y$ axis."}
{"id":"88199","title":"Illegal gauge condition in electrodynamics","text":"Just a quick sanity check here: I'm preparing a tutorial for a class on classical electrodynamics and I wanted to show an example of a gauge condition which leads to a contradiction, so I simply thought of $$(A^\\mu)' = A^\\mu - \\partial^\\mu \\chi \\stackrel{!}{=} 0\\quad \\forall \\mu \\in \\\\{0,1,2,3\\\\}\\qquad (1)$$ For the 0th component this gives me the Weyl gauge ($\\phi = 0$), which is allowed since I will always find a $\\chi$ satisfying $$\\phi = \\frac1c \\partial_t\\chi.$$ I do, however, have a contradiction in the spatial components, since $$\\vec{A} = \\vec{\\nabla}\\chi$$ is only true for $\\vec{A}$ with $\\vec{\\nabla}\\times\\vec{A}=0$ (on a simply-connected region), so if a given vector potential has non-vanishing rotation, I won't be able to find a $\\chi$ so that the transformed field vanishes in all components. **Does anyone have some other examples?** * * * (1) - As mentioned by QMechanic: Here are actually four gauge-fixing conditions disguised as one; this is unusual since common gauges (e.g. $\\vec{\\nabla}\\cdot\\vec{A}=0$) generally eliminate only one degree of freedom."}
{"id":"4174","title":"High-spin neutron star","text":"Suppose a large neutron star were to be spun-up by a particular pattern of mass-accretion. The increased centrifugal force could presumably mitigate the increased gravity thus delaying gravitational collapse. How far could this process be taken?"}
{"id":"69510","title":"Why do some air-conditioned stores blast you with jets of air as you enter?","text":"I went to a grocery store on a hot day that was very well air-conditioned, and I noticed as I went through the open entrance that there seemed to be a very powerful downward air current right at the doorway. After crossing the invisible threshold, the temperature immediately dropped a good 15 degrees or so. How does this process work?"}
{"id":"86097","title":"Holes in a P-type semiconductor under external force E","text":"Basically in almost every semiconductor texts, there will be all these concepts concerning electrons, holes, dopants, fermi-levels. However, I have been always confused about the picture of hole transport in semiconductor device, say, a simple PN junction. With a specific acceptor level and dopant concentration, we have some \"holes\" in the valence bands, which are in fact the absence of some electrons having gone to the acceptor levels, then all theses books seem to assume in the valence bands of the p-type region, there will be only holes that conduct current? My questions:   * aren't there still many electrons in the valence band? though they have some negative effective masses, but still do contribute to the transport?   * under a certain external force(say,E), the electron and holes in the valence band are moving in the same directions since electrons have negative effective mass and holes have positive one, so the corresponding currents cancel with each other in a P-type semiconductor?"}
{"id":"135016","title":"What is going on in front of and behind a fan?","text":"Why is it that when you drop paper behind a fan, it drops, and is not blown\/sucked into the fan, whereas if you drop paper in front of a fan, it is blown away?"}
{"id":"81378","title":"Can all systems be put in equilibrium?","text":"I'm in a first year statics course. We have spent the whole semester solving for forces and moments so that the system is in equilibrium. When we are given a system, we immediately begin solving for all variables for the system to be in equilibrium. But how do we know that the system _can_ be put in equilibrium? I asked my prof about this and he said that all systems can be solved to be put in equilibrium, but some systems might possibly have multiple solutions. Why can all systems be put in equilibrium? If I misunderstood the prof and all systems _can't_ be put in equilibrium, is there a quick way to check if a system can be put in equilibrium? (Other than the obvious way of solving the system). Thanks."}
{"id":"17509","title":"What is the physical meaning of the affine parameter for null geodesic?","text":"For time-like geodesic, the affine parameter is the proper time $\\tau$ or its linear transform, and the geodesic equation is $$\\frac{d^{2}x^{\\mu}}{d\\tau^{2}}+\\Gamma_{\\rho\\sigma}^{\\mu}\\frac{dx^{\\rho}}{d\\tau}\\frac{dx^{\\sigma}}{d\\tau}=0. $$ But proper time $\\Delta\\tau=0$ for null paths, so what the physical meaning of is the affine parameter for null geodesic?"}
{"id":"41943","title":"Why does the law of increasing entropy, a law arising from statistics of many particles, underpin modern physics?","text":"As far as I interpret it, the law of ever increasing entropy states that \"a system will always move towards the most disordered state, never in the other direction\". Now, I understand why it would be virtually impossible for a system to decrease it's entropy, just as it is virtually impossible for me to solve a Rubik's cube by making random twists. However the (ever so small) probability remains. Why does this law underpin so much of modern physics? Why is a theory that breaks this law useless, and why was Maxwell's demon such a problem? Does this law not just describe what is _most likely_ to happen in complex systems, not what _has_ to happen in all systems?"}
{"id":"20670","title":"How can a human eye focus on a screen directly in front of it?","text":"I am asking this question here because I think the answer has something to do with the way light is bent as it's captured through the eye. I saw a show a while ago about tiny screens on contact lenses to pull up data on objects you see in the real world, I also just saw this article about Google testing the same idea with screens in the lenses of sunglasses. The part I do not understand is how your eye would focus on a screen that is so close. My eyes (and I believe most others) cannot focus on anything closer than a couple inches away. Yet the lenses on glasses are much closer to the eye than a couple inches. So I would guess they would have to use some special technology to separate the light rays in a way that your brain could make an image from it. If that's correct, how would it work? if not, how would they get your eye to focus on a screen so close?"}
{"id":"62426","title":"What are \"packets\"?","text":"Paraphrased from Wikipedia: > Infrared sensing in snakes depends on a kind of natural thermography, by > which tiny **packets** of cellular water are raised in temperature by the > infrared radiation. What is \"packets\" refering to in this is example?"}
{"id":"907","title":"Type of stationary point in Hamilton's principle","text":"In this question it is discussed why by Hamilton's principle the action integral must be stationary. Most examples deal with the case that the action integral is minimal: this makes sense - we all follow the path with the least resistance. However, a stationary point can be a maximal or minimal extremum or even a point of inflexion (rising or falling). Can anybody give examples from theory or practice where the action integral takes on a maximal extremum or a point of inflexion?"}
{"id":"46996","title":"Are different frequencies of light lensed differently during gravitational lensing a bit like refraction?","text":"So I was wondering about the event horizon on a black hole. And wondering if the point of no return for radio waves vs gamma rays would be different. I guess the logic being, since gamma rays have more energy than radio waves, their point of no return might be different. But I'm not sure if gravity has the same affect on all frequencies of electromagnetic radiation. I'm sure the answer for this would lie in gravitational lensing, ie are different colours of light lensed by the same amount? or do higher frequencies lens less?"}
{"id":"128761","title":"Historical analysis of light interference - difference frequencies","text":"It is well-known that light of two different frequencies illuminating a detector will produce an output with a component at the difference frequency. While such considerations are eminently useful (various heterodyne measurement techniques) it has not been practical until lasers came along. Regardless of practicality, the effect has been known for a long time. The question is: how long? And, specifically, can you provide a reference to an analysis which establishes it? A web-accessible reference, for preference. I'd expect it was determined shortly after Maxwell's equations were established, but I could be wrong. Ideally, this would be an acknowledged seminal paper, comparable to Skutt's explanation of the color of the sky."}
{"id":"128766","title":"Can a hydrogen atom sit still?","text":"Can we take a single lone hydrogen atom (proton and electron?) and set it in one place and observe it staying there? How about just a proton? A neutron? How about just an electron?"}
{"id":"128767","title":"Will an electron gun fire behind it?","text":"If an electron travels as a wave, can it therefore not be aimed with any precision? If you fire an electron gun, can't you aim it at a particular slit? If the electron travels as a wave, will an electron gun hit anything at a 180 degree angle from the barrel? 360 degrees?"}
{"id":"128764","title":"Effect of crystal growth on its scintillation","text":"How does crystal growth affect its scintillation properties? What are possible ways in which growth of CaWO crystals grown via Chochralski method could be modified to avoid non-linearity in its scintillation output with respect to incident Gamma rays? Apparently this is relevant for rare event searches (eg. CRESST) using these scintillating crystals as detectors."}
{"id":"134050","title":"Generator phase noise influence on its averaged signal","text":"I need to estimate averaged signal of a generator with known phase noise and the amplitude noise can be neglected. The averaged signal of the generator is expressed as: $$ \\langle A\\exp{i(\\Omega t + \\varphi)}\\rangle + \\mathrm{c.c.} = Ae^{i\\Omega t} \\langle \\exp{i\\varphi} \\rangle + \\mathrm{c.c.}, $$ with $\\Omega$ the generator carrier frequency and $\\varphi$ the phase noise, $\\langle\\varphi\\rangle = 0$. I suppose that phase noise is stationary. Its spectral density $ S_{\\varphi\\varphi} = \\frac 1{2\\pi} \\int_{-\\infty}^\\infty\\mathrm d\\omega \\, e^{-i\\omega t} \\langle \\varphi(0)\\varphi(t) \\rangle$ is given. To obtain $\\langle \\exp{i\\varphi} \\rangle$ I suppose $\\varphi$ to have Gaussian statistics. It is easy to show for that case that $$ \\langle \\exp{i\\varphi} \\rangle = e^{-\\frac{\\langle\\varphi^2\\rangle}2}. $$ Then, one finds $$\\langle\\varphi^2\\rangle = \\int_{-\\infty}^\\infty\\mathrm d\\omega \\, S_{\\varphi\\varphi}(\\omega). $$ I've taken $S_{\\varphi\\varphi}$ of the Agilent N5173B at 20GHz carrier, see the picture below and the generator's datasheet. (Note the two-sided phase noise spectral density is called $\\mathcal L$ in industry, see the IEEE standart.) ![Agilent N5173B phase noise ](http:\/\/i.stack.imgur.com\/qd2p6.png) I approximate the integral of the spectral density as the square under the connected points shown in the table: $$ \\begin{array}{|r|r|} \\hline f, \\mathrm{Hz}& 1& 10& 100& 10^3& 10^4& 10^5& 10^6& 1.1\\cdot10^6& 1.4\\cdot10^6\\\\\\ \\hline S_{\\varphi\\varphi}, \\mathrm{dBc}\/\\mathrm{Hz}& -40& -50& -80& -90& -95& -95& -125& -130& -135 \\\\\\ \\hline \\end{array} $$ First plotting the spectral density and approximating its integral by hand, than using Python scipy trapz integration, I obtain $$\\sqrt{\\langle\\varphi^2\\rangle} \\sim 10\\,\\mathrm{rad}$$ which is hard to believe for such a costly generator. And, according to the abovesaid, it will actually diminish the generator output, as $e^{-\\frac{\\langle\\varphi^2\\rangle}2} \\sim 0.01$. So, 0) What the hell? 1) Probably I've misunderstood the IEEE definitions? I guess there should be some narrower frequency cutoffs? 2) Is it valid to assume the phase noise of a generator to be gaussian? 3) Probably better approaches exist for estimating the averaged signal of a generator with phase noise?"}
{"id":"121586","title":"History of the names \"Feynman-gauge\" & \"Landau-gauge\". How arised & how settled?","text":"Warning: Students, stay away from antiquities. The aim to learn is to survive. Hi. Today the nomenclatures _Feynman gauge_ and _Landau gauge_ seem established, but could you explain the history? It's two-fold: 1. Who used first such gauges and how? 2. How did the terminology gain ground? It also inevitably asks some history of the Lorenz gauges in QFT. In particular, when was it realized that the freedom (and need) of the gauge parameter ξ lies in quantum theory? First I apologize that I have only limited access to literature on asking such questions. My preliminary studies revealed below. 1 **Terminology in literature** (Schweber 1961) is not gauge-parameter conscious, and \"Lorentz[sic] gauge\" (in Feynman gauge) is opposed to coulomb. But it also refers to today's Yennie gauge in sec 15g, saying (Fried and Yennie 1958) found it is possible to take gauge where the photon propagator ∝ $g + 2pp\/p^2$. In (Nakanishi 1966), the word \"Landau gauge\" is seen, and it cites several near-past articles regarding Landau gauge quantization. (It is an important paper in canonical quantization in Landau gauge, together with (Lautrup 1967). Nakanishi was a strong proponent of the Landau gauge.) In p. 74 of (Nakanashi 1972), it reads \"Feynman gauge or Fermi gauge\" and \"Landau-Khalatonikov[sic] gauge, or simply Landau gauge\". (Landau and Khalatnikov, 1955) is listed in the bibliography section, but I couldn't find which part of Nakanishi actually cites it. (Nakanishi 1972) is a review article, one of whose main topics is canonical quantization of EM field in arbitrary Lorenz gauge, i.e., for any gauge parameter. In p 134 of (Itzykoson & Zuber, 1980), the words \"Feynman gauge\" and \"Landau gauge\" are used. Were the names settled at that time? Hmm, in p 389 of (Siegel 1999), \"Fermi-Feynman gauge\" is introduced. (Srednicki 2007) uses the word \"$R_\\xi$\" for QED, remarking \"[it] has been historically used only in the context of spontaneous symmetry broken [...] but we will use it here as well.\" 2 **Gauge parameter symbol** ξ is now usual. Is it due to (Fujikawa, Lee and Sanda 1972)? For other symbols, I mention $\\alpha$. (Nakanishi 1972) uses it, and even after (Fujikawa, Lee and Sanda 1972) it is sometimes used, for example and in (Siegel 1999). 3 **Theory Timeline** 1930 - Fermi: P. 240 of (Schweber 1961) says Fermi proposed to add $-\\frac{1}{2} (\\partial A)^2$ to the Langrangian. (Fermi was the first to introduce a subsidiary condition, but it was not perfect. See also Gupta and Bleuler below.) Although I haven't checked Fermi's papers, it may be better to call \"Fermi-Feynman(-'t Hooft) gauge.\" 1948 - Feynman: Feynman simply justifies the use of Feynman gauge in the section 8 of (Feynman 1949). Before Feynman, it wasn't Lorentz covariant, and transverse photons were separated. Feynman says it's not necessary, and it's ok to do $\\gamma^\\mu$...$\\gamma_\\mu$. 1950 - Gupta & Bleuler: They say Gupta and Bleuler succeeds in covariant canonical quantization in Feynman gauge, by discovering the correct subsidiary condition. 1956 - Landau & Khalatnikov: See (Nakanishi 1972) above. 1958 - Yennie gauge: It is said (Fried and Yennie 1958) uses the \"Yennie gauge\" of today, $\\xi = 3$, in bound state problems. Early or mid 60's - Rise of interest in Landau gauge? See (Nakanishi 1966) above. 1966 - 67 Nakanishi & Lautrup: canonical quantization of EM field for any ξ. 1967 - Faddeev & Popov 1971 - 't Hooft: In 1971, 't Hooft used \"Feynman-'t Hooft gauge\" or simply \"'t Hooft gauge\" for broken gauge symmetry. (Fujikawa, Lee and Sanda 1972) generalized to any ξ. Its abstract uses the word \"Feynman-'t Hooft gauge\". (According to Weinberg. Haven't read both two.) 1972 - Still canonical quantization for arbitrary ξ is of interest, including massive vector field. See (Nakanishi 1972). 4 Loren't'z gauge (misspelling) You may know that in the 20th century, the common spelling was \"Lorentz gauge\", with extra \"t\". I couldn't find any exceptions at my hand. The turning point might be the errata of Peskin & Schroeder. Srednicki and Siegel spell it correctly. 5 **Bibliography**   * Fermi, E., Atti. Acad. Lincei. **9** (1929) 881, Atti. Acad. Lincei. **12** (1939) 431, Rev. Mod. Phys. **4** (1932) 87.   * Feynman, R., Phys. Rev. **76** (1949) 769.   * Fried H.M., Yennie, D.R., Phys. Rev. **112** (1958) 1391.   * Fujikawa, Lee and Sanda, PRD **6** (1972) 2923   * Itzykson & Zuber \"Quantum field theory\", 1980.   * Landau, L. D., Khalatnikov, I. M., J. Exper. Theor. Phys. USSR 29 (1955), 89 [English translation: Sov. Phys. J. E. T. P. 2 (1956), 69].   * Lautrup, B., Mat. Fys. Medel. Dan. Vid. Selsk. 35 (1967), No. 11.   * Nakanishi, N., Prog. Theor. Phys. **35** (1966) 1111 (Downloadable gratis)   * Nakanishi, N., Prog. Theor. Phys. Supple. **51** (1972) 1 (Downloadable gratis)   * Schweber, \"An introduction to relativistic quantum field theory\", 1961.   * Siegel, W., \"Fields\", arXiv:hep-th\/9912205   * Srednicki, (2007) _Quantum field theory_ 6 **Revisions of this question** 26 Jun: Added the post 't Hooft era and the symbol ξ."}
{"id":"7319","title":"Forces on a simple flat slope","text":"![enter image description here](https:\/\/docs.google.com\/drawings\/pub?id=1cm2nq5OKck4zKphMYgMIJuCREZDO9JlNriRA0_4pUTc&w=400) This diagram is of a block (b) moving down a ramp unaided. Taking gravity (g) as $9.8ms^{-2}$, and the mass of the block (b) as $0.2kg$, how can I find the force acting along the slope (f). BTW: the ramp is inclined 1.8 degrees, not radians. I thought I could use basic trigonometry, and $F=ma$: $$ Sin(1.8) = \\frac{0.2*9.8}{f} $$ $$ f = \\frac{0.2*9.8}{sin(1.8)} $$ However this results in: $f = 62.40$ Seeing as force is a vector I thought I could use trig to split into it's components but obviously I'm doing something wrong. (the answer should be ~0.06N) Sorry for the noob question, but it's got me clueless."}
{"id":"11902","title":"Is this simulation following real physics?","text":"I am trying to simulate a game in Box2D(Physics engine). The game that I am trying to simulate is very simple and can be found here: http:\/\/www.makaimedia.com\/#\/speartoss What I want to know is that, is the spear in that game following real physics principles? In other words, if I went out and threw a spear, will it behave like the spear in this game? Specifically, I am taking about the rotation of the spear. In the game, the spearhead is point up first half of the trajectory and as soon as it hits the vertex, the spear head starts point down. I am simulating a spear in Box2d physics engine. The density of the spear is 15x more than the density of the handle. I set an angle (between 0, 90 degrees) and give it a linear velocity like this:                   b2Vec2 force;         force.Set(cos(angle) * 22.0f , sin(angle) * 22.0f);         _spearBody->SetLinearVelocity(force);      What happens is, the spearhead never points downwards. It stays at the exact same angle throughout the trajectory. Is this more realistic or the physics in speartoss game? Thanks."}
{"id":"11904","title":"Why aren't there compression waves in electromagnetic fields?","text":"I just started learning about optics, and in the book I'm reading they explain how the electrical field caused by a single charged particle could be described by a series of field lines, and compare them to ropes, to provide an intuition of the concept. Then they say that and that if we wiggle the particle up and down, that would produce transversal waves in the horizontal field lines, but no waves in the vertical lines. I know that the physical analogy is not to be taken literally, but I don't understand why wouldn't that cause compression waves in the vertical lines. I mean, even though the direction of the field in the points directly above and below the particle doesn't change, the intensity does. And I assume it wouldn't instantly. So what am I missing?"}
{"id":"11905","title":"When is the Hamiltonian of a system not equal to its total energy?","text":"I thought the Hamiltonian was always equal to the total energy of a system but have read that this isn't always true. Is there an example of this and does the Hamiltonian have a physical interpretation in such a case?"}
{"id":"129380","title":"Is there an analogous Gauss' law which is applicable for a gravitational field?","text":"Consider the Earth to be a flat infinite plane having linear mass density equal to the mass density of the actual earth.   1. Can there be an analogous Gauss' law that can give the gravitational field for any point on a particular Gaussian surface.    2. If yes, what would be the substitute for the permittivity constant that appears in the Gauss' law for an electric field?"}
{"id":"121634","title":"Which areas of physics are related to the act of playing drums?","text":"I'm musician (drummer) and I'm trying to figure out what can I study (related to Physics) for better understanding of the drumsticks and wrist movements, the force applied and the better way to apply it and introduce some theory in my training. I'm really awful at physics, so I really don't know what areas are important (pressure, mechanics, acceleration?)."}
{"id":"111167","title":"Yang-Mills CP violation","text":"Why does a term proportional to $\\left(F,\\,\\tilde{F}\\right)\\propto Tr\\left[ F_{\\mu\\nu}\\tilde{F}^{\\mu\\nu}\\right]$ in the Lagrangian of the pure Yang-Mills theory violate CP?"}
{"id":"123673","title":"Gibbs equation and third law of thermodynamics","text":"I have seen a discussion of what would happen for an ideal gas expands irreversibly and adiabatically until absolute zero degree K. The entropy change is like that: Δs=Cv ln(T2\/T1)+R ln(V2\/V1) It is impossible for T2 to be zero K in the equation and so it becomes one justification of the third law. I wonder whether it is valid and whether an irreversible process can be represented by the above equation. For irreversible process the heat exchange is not equal to Tds so can the Gibbs equation be applied? The video is in https:\/\/www.youtube.com\/watch?v=r4fGG_7NQr8 , 46:12"}
{"id":"123672","title":"Problem understanding basic sail mechanics","text":"I am trying to understand the physics behind a single sailed boat. I have read the following questions Sailing against the wind. Is this a fair model? What exactly is the 'lift' of a sailboat as explained by Bernoulli principle I've also been following this explanation which seemed to be helping but I am still confused. Full disclosure: my physics is pretty weak and I've never sailed, but my maths should be sound. I am (now) aware of the two components that provide the force that moves the boat: The _\"Newtonian aspect\"_ , where the wind collides with the sail and this produces a force and the _\"Bernoulli aspect\"_ resulting from pressure differences resulting from the air passing around the sail. (If people think this would be better as two questions, happy to break it down, but I suspect one or both will be silly) **Update:** It has been said by @CarlWittoff that the Bernoulli aspect is pretty trivial, so I would like to focus on how the Newtonian part of the force can achieve a force that goes into the wind. **Newtonian Aspect** I understand this in the most simplest sense, a sail perpendicular to the wind will produce a force on the boat proportional to the relative wind speed. As the sail is angled in to the wind, you get more \"lift\" and less \"drag\" (in plane terms). In the sailing explanation, it implies that the Newtonian aspect can be used to explain going into the wind. This is also hinted at in the first question. However, I don't understand how this can be. Surely the force applied to the sail, and hence the ship, is always in the direction of the wind. Hence it can never provide a force into the wind. The article seems to suggest you can get away without the Bernoulli effect, but I don't see how. ![Diagram of the air flow](http:\/\/i.stack.imgur.com\/16B4F.png) This is my attempt at how this translates into sail. The air is pushed around the sail, which exerts a force (in purple) on the boat. However, I can see no way for this force to be into the wind. **Bernoulli Aspect** This is the area I am even more lost with. My understanding is that somehow, you can angle the sail in such a way that the air that goes past on one side knows to go faster which reduces the pressure, giving a force. However, I cannot seem to connect that intuition to any kind of applicable formula. Specifically, what determines the direction of the force and what is it proportional to? I'm trying to simulate a boat for a game and any help on which formulae I actually need to be plugging in would be hugely appreciated :)"}
{"id":"123674","title":"Why does a flat universe imply an infinite universe?","text":"This article claims that because the universe appears to be flat, it must be infinite. I've heard this idea mentioned in a few other places, but they never explain the reasoning at all."}
{"id":"17883","title":"What happens to a rotating rod that breaks in two?","text":"I know that the approximation for the moment of inertia of an infinitely thin rod of mass $m$ and length $L$ spinning around an axis perpendicular to its own axis at its center is $\\frac{mL^2}{3}$: ![enter image description here](http:\/\/i.stack.imgur.com\/LVgxB.png) What happens when this rod is instantaneously broken in the middle with no loss or gain of energy? Rather than some external event triggering the breakage, imagine some sort of latch that held it together before, and now it is simply released. Easier to pretend with than an infinitely thin rod with mass, at any rate. I'd like to know how I should determine the motion (absent gravity) of the two halves. Suppose I view the scene from directly above (viewing in the same the direction as the axis of rotation), the rod is rotating clockwise from that perspective (viewing from below, in the picture above), and the event occurs when the rod is vertically aligned from that view. Based on this I assume a coordinate system: The rod was rotating within the x-y plane with its center at the origin. The axis of rotation is the positive z axis. My camera is pointing in the positive z axis from some position $(0,0,-z)$. The top half of the rod travels to the right. Its center of mass is at its middle point, which is at position $(0,\\frac{L}{4},0)$ at $t=0$. The bottom half is going to travel to the left, in a symmetrical fashion. What are their angular velocities? Do they spin faster or slower, or at the same speed as the original rod? The length has halved, which means the moment of inertia has been quartered. But conservation of angular momentum states each rod carries half of the rotational energy (is this correct application of conservation of angular momentum? Is there such a thing as conservation of rotational kinetic energy?) Is part of the original angular kinetic energy now split between resultant angular kinetic energy and linear kinetic energy? It seems so, because the original rod had zero linear kinetic energy (and momentum) but after breaking, the parts now both have linear kinetic energy since they are flying apart. At this point I'm almost positive I can stumble my way to the solution for the split happening exactly at the middle. How might things change if the split is for example 1\/10th of the way down the rod? Surely the little piece would go off much faster but would it spin at the same rate also?"}
{"id":"110698","title":"Spin-Position Coupling\/Interaction","text":"Firstly, is there a general procedure\/algorithm for attacking Hamiltonians having spin-position coupling, e.g., $$\\frac{\\mathbf{\\sigma_1}\\cdot\\mathbf{\\sigma_2}}{|\\mathbf{r_1}-\\mathbf{r_2}|}$$ I do not know where to begin. Secondly, I would like you to recommend a textbook which discuss such a kind of systems. I looked out and found none."}
{"id":"89450","title":"Where to find detailed measured emission spectra of all chemical elements?","text":"I'd like to have something like this, but for single atoms and with more extended range of wavelengths. All I could find e.g. for hydrogen was lots of talks about Rydberg formula etc. and plots of spectrum in visible region. But I need _experimental_ data to look at, not something plotted from formulas like Rydberg's one. Similarly for other elements, I mostly found only plots of spectral lines in visible range. So, where can one find detailed measured spectra for all (or most) chemical elements, preferably with UV and IR parts? It'd be most useful to have it in numeric form (spreadsheet etc.) instead of pictures."}
{"id":"89452","title":"Does the strength of a smell depend on the volume or surface area of an agglomerate of material?","text":"I just switched from Timberland to Adidas shoes, and noticed a horrible trait of the Adidas tread: instead of being composed of externally-protruding bumps like the Timberland, it is composed of channels striated _internally_. This has the horrible trait of trapping and making difficult to remove small pebbles, mud, and other foreign objects that one might inadvertently step on. Assuming that one were to step on an unusually aromatic organic viscoplastic, **would the 'strength' of the smell depend on the surface area of the exposed agglomerate, or of its mass**? My instinct leads me to think that the surface area would be the major factor, but empirically I see that larger massed agglomerates, even if wedged deep within an Adidas tread with little exposed surface area, seem to be more easily detected by human olfactory organs. However I contend that my sensing apparatus and testng methodology are not scientifically sound, nor do I have access to facilities which would allow me to make more objective observations. Of course, the principle of \"a month in the lab could save an hour in the library\" applies, so I searched online but found surprisingly little (read: none) literature on the subject."}
{"id":"110696","title":"What Would Negative Mass Do To Spacetime?","text":"It's known that positive mass bends space-time to create a curvature. But if something had negative mass what would it do? Make it flat or like a crest?"}
{"id":"61305","title":"Static electrical attraction","text":"Coulomb's law is used to calculate the electrical attraction between 2 charged particles, what formula do I use to calculate an electrical attraction magnitude between 2 plates? Let's assume the first plate holds a charge of +3x10^-4 coulombs and the other one holds the same negative amount of charge (-3x10^-4), both plates area is 5 square centimetres and they are fully exposed and facing to each other. Also let me assume that the distance between them is 2 meters."}
{"id":"110340","title":"Vibration transmission using spring and damper model","text":"I want to simulate the vibration transmission between two colliding objects. I use a Finite Element Model to simulate the vibration on each object, but I haven't found any good mathematical model to transfer the vibration between two FEM models that collided and keep in contact. ![enter image description here](http:\/\/i.stack.imgur.com\/LcbS8.jpg) In the figure, a sphere is dropped into a plate. the change in color represent the displacement on the FEM models. Currently, I am using a spring\/damper model to transmit the vibration from one object to the other. But, I haven't found any good model that give any theoretical foundation to my simple spring\/damper model."}
{"id":"46805","title":"Rod slipping against block due to gravity?","text":"A uniform rod of mass $m$ and length $l$ is pivoted at point O. The rod is initially in vertical position and touching a block of mass M which is at rest on a horizontal surface. The rod is given a slight jerk and it starts rotating about point O. This causes the block to move forward. The rod loses contact with the block at $\\theta = 30^\\circ$. All surfaces are smooth.   1. The value of $M\/m$?   2. The velocity of block when the rod loses contact with the block?   3. The acceleration of center of mass of rod, when it loses contact with the block?    4. The hinge reaction at O on the rod when it loses contact with the block? ![enter image description here](http:\/\/i.stack.imgur.com\/uJWRv.jpg) Here is my attempt at the problem The component of angular speed of the tip of the rod in horizontal direction is same as that of the block. Let V be the speed of the block when rod loses contact with block. So,Horizontal component of angular speed of the tip of the rod = $\\omega l\\cos(60)=V$ So, $\\omega l\/2=V$ Thus, $\\omega = 2V\/l$ Applying Conservation of Energy, Loss in Potential Energy of Rod = Gain in Rotational Kinetic Energy of Rod + Gain in KE of Block $$\\frac{Mgl}{2}(1-\\sin60^\\circ) = \\frac{1}{2}\\frac{ml^2}{3}(ω ^2) +\\frac{1}{2}MV^2 $$ $$\\frac{Mgl}{2}(1-\\sin60^\\circ) = \\frac{1}{2}\\frac{ml^2}{3}\\frac{4V ^2}{l^2} +\\frac{1}{2}MV^2 $$ $$\\frac{Mgl}{4} = \\frac{2mV^2}{3} +\\frac{1}{2}MV^2 $$ The forces on the rod are Hinge force in x direction = $ H_{x} $ Hinge force in y direction = $ H_{y} $ Force due to gravity = mg Normal contact force from the block = $N$ The force on the block are Normal contact force from the rod =$ N$ Now, I am not able to proceed further."}
{"id":"28879","title":"Why do humans have bilateral symmetry?","text":"About the eyes I know that it requires for gauging distance as in Modern 3D cameras have two sensors. And two ears for sound source localization using differences in levels and timing (But not yet two microphones in mobile phones\/other devices). What about other organs? Even not all organs are in pair. If it is about about redundancy then why not more than two? Thanks for your answer."}
{"id":"78138","title":"Is there a proof from the first principle that the Lagrangian L = T - V?","text":"Is there a proof from the first principle that for the Lagrangian $L$, $$L = T\\text{(kinetic energy)} - V\\text{(potential energy)}$$ in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it? On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the answer, and if possible, the primary source (who published the answer first in the literature). * * * Notes added on Sept 22:   \\- Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of.   \\- We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider.   \\- But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion.   \\- Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future."}
{"id":"25162","title":"What is the current status of Pluto?","text":"Pluto has been designated a planet in our solar system for years (ever since it was discovered in the last century), but a couple of years ago it was demoted. What caused this decision? And is there a chance that it could be reversed?"}
{"id":"21881","title":"Decomposition of SU(N) adjoint representation under SU(2)","text":"I am having trouble in demonstrating that under SU(2) transformations the adjoint representation of SU(N) transforms as one spin 1, 2(N-2) spin $\\frac12$ and singlets. I am trying to demonstrate it from $N \\otimes \\bar{N} = 1 + A$ where $A$ is the adjoint representation; and the fact that an N vector of SU(N) decomposes as one $j=\\frac12$ plus two $j=0$. So basically: $(1\\oplus^{N-2}0)\\otimes(1\\oplus^{N-2}0)= ?$. But I end up with one spin 1, $2N-3$ spin $\\frac12$, and $(N-2)^2+1$ spin 0. Embarrassing. Is my equation with the question mark wrong?"}
{"id":"21882","title":"Gravitation in a space that is topologically toroidal","text":"In my scant spare time I'm building an Asteroids game. You know - a little ship equipped with a pea shooter and a bunch of asteroids floating around everywhere waiting to be to blown up. But, I wanted to add a little twist. Wouldn't it be cool if Newtonian gravity was in effect and you could do thinks like enter into an orbit around an asteroid and fire at it, or shoot gravity assisted bankshots around a massive asteroid so that you can shoot one behind it. But the problem is that in asteroids, space is topologically toroidal. If you fly off of the top of the screen, you reappear at the same x coordinate at the bottom of the screen (and similarly for the right of the screen). So how does one calculate the distance between two bodies in this space? Really, I realize that this question doesn't make sense because body A would pull upon body B from a variety of directions each with their corresponding distances. But anyway, the main questions: How would Newtonian gravity work in toroidal space? AND Is there any applicability to the answer to this question outside of my game?"}
{"id":"21887","title":"I think there is a mistake in solutions to this problem. plus and minus sign","text":"![](http:\/\/img526.imageshack.us\/img526\/2593\/49699289.jpg)! I pretty much had what the solutions had, but we disagreed with one thing, a minus sensitive sign. $\\sum W = \\Delta K$ $\\int_0^{-s} - ks^2 ds = \\frac{ks^3}{3} = \\frac{1}{2}m(v^2-4^2)$ Rearranging, I get $v^2 = 4^2 - \\frac{2k(-0.2)^3}{3m} = 19.2$ So v = 4.38m\/s But the key did $v^2 = 4^2 - \\frac{2k(0.2)^3}{3m} = 12.8$ where they let $\\int_0^{s} - ks^2 ds = \\frac{ks^3}{3}$ **EDIT** Apparently I am stupid and I made a question that didn't even need answering... I had this on paper and I typed up the wrong thing, that's why I was confused. $\\sum W = -\\int_{-s}^{0} ks^2 ds = \\frac{1}{2}m(v^2-4^2)$ Sorry everyone"}
{"id":"56260","title":"how to determine if a vortex is laminar or turbulent","text":"In a cylindrical chamber with a high diameter-to-height ratio; a fluid is tangentially injected. there is an axial exit to the cylinder. how do I determine if the vortex so formed is laminar or turbulent? what parameter will indicate this? (this is a simulation; so I can measure whichever velocity\/turbulent KE that is needed.)"}
{"id":"10823","title":"If 100% of the energy from the sun is reflected back into space","text":"100% of the energy from the sun is reflected back into space, it's just shifted from a low-entropy state to a high-entropy state, and from a high frequency (ultraviolet) to a low frequency (infrared). But does the total number of photons change as a result? Are there more low frequency photons, then, in order to make the same amount of energy?"}
{"id":"56265","title":"How to get the angle needed for a projectile to pass through a given point for trajectory plotting","text":"I am trying to find the angle needed for a projectile to pass-through a given point. Here is what I do know:   * Starting Point $(x_0,y_0)$   * Velocity   * Pass-through point $(x_1, y_1)$ I also need to incorporate gravity into the equation. Anyone have any ideas? I haven't had much luck so far, so any ideas\/suggestions would be great."}
{"id":"60037","title":"Do all the 4 forces of nature act at the same speed?","text":"It is believed that gravity, the weakest of the four forces propagates at the speed of light, cf. e.g. this Phys.SE post. One would expect (perhaps erroneously) that the other, stronger, forces acted at an even quicker speed. Of course, it is not like that. At least, it's not for electromagnetism. What do we know about the speed of the other two?"}
{"id":"93959","title":"Wall visible after light source removed","text":"Let's say you have a lamp pointed at a wall, in a dark room with no other light source, if you turned off that lamp #  Would the wall still be visible for any amount of time?"}
{"id":"61872","title":"How long does a supernova last?","text":"Just what the title indicates. Is a supernova over instantaneously? Or, does the (for want of a better word) explosion continue for a while? What is\/are the order of timescales involved? EDIT: Following the posted replies\/comments I figure what I mean to ask is the duration for which the supernova continues to release copious amounts of energy."}
{"id":"61874","title":"Energy stored in fields","text":"How to intuitively think of energy stored in a magnetic\/electric field? Kindly answer in a bit simple terms without referring to mathematics."}
{"id":"61876","title":"Can the laws of quantum mechanics be derived from a more fundamental theory?","text":"String theory takes quantum mechanics and tries to make it compatible with gravity. If it turns out to be a theory of everything then would it explain why our world is described by the laws of quantum mechanics? I want to understand if the laws of quantum mechanics are the only logical possibility for a low-energy theory potentially derivable from a better understanding of string theory. The other possibility is that they are just strange laws that are discovered through experiments that can't be reached by pure theoretical reasoning. Which of the two possibilities is more reasonable from a physics point of view and why?"}
{"id":"13233","title":"Vacuum and repulsive gravity","text":"How can one show from General Relativity that gravity is attractive force, and under which conditions it becomes repulsive, also why positive energy vacuum drives repulsive gravity?"}
{"id":"113611","title":"Measuring flow rate of gases","text":"Flow rates of liquids could be easily measured with a stopwatch and a known volume container. But gases, if entrapped in a container will create a back pressure, and may lead to decreased flow. I wonder how they measure the quantity of gases generated by a reaction or flow rate of fluids in turbines, etc., without creating back pressure. What\/how back pressure corrections are providedif this cannot be prevented.? Are there flow meters for gases at all pressures similar to fluids.?"}
{"id":"53300","title":"Chiral perturbation theory: what is the Quark Condensate? why expand in $U$ rather than Goldstone fields?","text":"I'm studying Chiral Perturbation Theory ($\\chi PT$) from Scherer's _Introduction to Chiral Perturbation Theory_. What I am currently having some trouble understanding are two things:   1. The quark condensate. What is this and why is it a sufficient condition for spontaneous breaking of chiral symmetry? What I do not really understand is where the operator $S_a=\\bar{q} \\lambda_a q$ comes from ($\\lambda_a$ are the Gell-Mann matrices) and why the expectation value of this (which I gather is zero) gives us this thing called the quark condensate.   2. The formulation of the effective Lagrangian. There is some stuff in Scherer about the coset $G\/H$ where in this case G is the full chiral group and $H$ is the vector subgroup which is left after spontaneous symmetry breaking, but I do not really follow how this discussion explains why the Lagrangian is given in terms of the SU(3) matrix $U=\\exp{\\frac{i}{F_0} \\Phi} = \\exp{\\frac{i}{F_0} \\phi_a\\lambda_a}$ for (individual) Goldstone fields $\\phi_a$? Why can't we write down the effective Lagrangian in terms of the actual degrees of freedom in the theory, i.e. the Goldstone fields? I've read something about them not transforming non-linearly (and the $U$ transforming linearly) but could not really follow so if someone could elaborate on this I would be very glad. A big thanks in advance for all help given! And another thing - if anyone has another tip for an introductory reference to $\\chi PT$, I would be very grateful. Scherer works decently but it's always good to read about things from a different viewpoint."}
{"id":"113615","title":"What does it mean to \"convert energy into time\"?","text":"In a recent article about creating electron-positron pairs by colliding photons in a laboratory, Andrei Seryi, director of the John Adams Institute at Oxford University, was quoted to said: > It's breathtaking to think that things we thought are not connected, can in > fact be converted to each other: matter and energy, particles and light. > Would we be able in the future to **convert energy into time** and vice > versa? Let's ignore for the moment that a physicist is amazed that $\\gamma\\gamma\\rightarrow e^-e^+$ process is possible in a laboratory. But what does he mean by converting energy into time? Is this just a poetic sentence to emphasize his amazement, or is there some deeper meaning behind it? Only thing that comes up in my mind is that, because our currently best theory of gravity - general relativity - is describing the dynamics of spacetime, \"creating time\" might mean inducing some change in the metric tensor $g_{\\mu\\nu}$, so he might be talking about the prospect of converting ordinary Standard Model particles into gravitational waves and vice versa. However, he might be talking about something else. Am I missing something here?"}
{"id":"53304","title":"Orientation in GaAs","text":"I can't find the precise definition of what is the orientation of a GaAs lattice. Being the superposition of two fcc lattices (one of Ga, the other of As), I would think that it is the direction of the closest Ga atom to any As atom of the structure. Is it right?"}
{"id":"56872","title":"What is the path integral exactly?","text":"I asked a question here about path integrals and QFT. I just want to confirm something. Is the path integral in quantum field theory a mathematical tool only? I thought the path integral meant that particles literally do take all paths as stated by Victor Stenger and Stephen Hawking in their interpretation of what happens in reality? If particles are field excitations, as QFT says, does that mean Stenger and Hawking's interpretation is right and in reality particles do take all paths (sum over histories)?"}
{"id":"81032","title":"Does heat kernel factorize on product spaces?","text":"I have a doubt regarding whether the trace of the vector heat kernel on a product space factorizes into the corresponding heat kernel traces on each manifold in the product space. I know this holds for the spin-0 as well as spin-1\/2 cases, but I am not sure about the spin-1 case. Can somebody please help me out in this regard?"}
{"id":"81035","title":"Canonical form representation of a Linear Gaussian CPD","text":"I'd like to know how a linear Gaussian conditional probability distribution can be represented to a canonical form. For example, let X and Y be two sets of continuous variables, with |X| = n and |Y| = m. Let p(Y | X) = _N_ (Y | _a_ \\+ BX; C) where _a_ is a vector of dimension m, B is an m by n matrix, and C is an m by m matrix."}
{"id":"81034","title":"Educational applications of a small Michelson interferometer?","text":"The community college where I teach has some nice old Michelson interferometers. There appear to be a bunch of versions of these that used to be sold (may still be sold?) which were all probably knock-offs of some manufacturer's original model. Ours are Cenco 71990-549, and are identical to the one in the photo here (but without the stepping motor). Here and here are manuals for a couple of other similar ones. (We no longer have the manuals for ours.) The arms are about 20 cm, the movable mirror can travel about 25 mm, and it has a micrometer that is accurate to about 2 um. What is a good educational application for these sweet old instruments? Here are the things I've tried, both by myself and using students as guinea pigs:   1. Crank the micrometer while counting fringes with a Na discharge tube. Determine the wavelength.   2. Like 1, but with laser light, projecting the fringes rather than squinting into the instrument.   3. Equalize the arms by finding the condition where fringes are visible with white light. Insert a sample of glass with an unknown index of refraction, and reacquire the condition of equal optical path length. Determine the unknown index of refraction.   4. The Na line is actually a close doublet, so as you move the micrometer you go back and forth from a condition of high-contrast fringes (the bull's eye patterns for the two wavelengths agree) to a condition of little or no contrast (the patterns disagree and counteract each other). If $\\Delta d$ is the distance traveled between two successive positions of maximum contrast, then the splitting of the doublet is $\\Delta \\lambda\\approx \\lambda^2\/2\\Delta d$. Comments on how these went:   1. Incredibly tedious, and nearly impossible to do with high precision because you pass through the conditions of low contrast described in #4. The same result could be obtained with much higher precision in five minutes with a diffraction grating.   2. Not as bad as #1, but still doesn't use the instrument for an application to which it's uniquely well suited.   3. This was very cool. It was amazing how accurately the equal-length condition could be detected using this technique (to within 1 um). However, it doesn't seem to be a high-precision way to determine an unknown index of refraction, because you have to use a thin sample like a microscope slide, and your precision ends up limited by the determination of the thickness of the slide using Vernier calipers (e.g., 1.8 mm, so only 2 sig figs).   4. This is sublimely tricky, but I doubt that most students understand it. Measuring the spacing of a close doublet, while impressive to a spectroscopist, is of little intrinsic interest to a student in a freshman survey course. Other than the Michelson-Morley experiment (which obviously can't be done in a 3-hour student lab with a tabletop instrument), the classic historical application seems to have been the determination of the meter in terms of atomic standards. (See Hardy and Perrin, p. 584.) This doesn't seem practical for us. I've seen references to gas cells for use with this setup, presumably using the technique of #3. A back-of-the-envelope calculation seems to suggest that you would get a low-precision measurement, so I don't really see the point. Is it possible to use a technique sort of like #3 to determine the ratio of two wavelengths from the same source with high precision? This would have some intrinsic interest for hydrogen, whose wavelengths are in the ratios of small integers."}
{"id":"98966","title":"Maximum theoretical efficiency of internal combustion engine","text":"This Wikipedia article states that: > Most steel engines have a thermodynamic limit of 37%. Is it correct? If so then it is not clear where this number comes from. Is it derived somehow from Carnot's theorem? I tried to google it but found nothing usefull. If this limit of 37% is wrong then what is correct limit?"}
{"id":"98961","title":"Bravais lattice with sublattices : why multiple bands?","text":"I have a very naive question : given a tight-binding model (with nearest- neighbor hoping) on a lattice defined by a Bravais lattice with a number of sublattices (for instance the honeycomb lattice is a triangular lattice with two sublattices), why is there a band associated with each sublattice ? For instance when a lattice is defined by a Bravais lattice with a 2 sublattice basis, the tight-binding model will have 2 bands.. But why is that ? There must be something very simple that I am not getting here... is it just because that to define a band structure, you need some translation invariance ?"}
{"id":"31199","title":"How and why can random matrices answer physical problems?","text":"_Random matrix theory_ pops up regularly in the context of dynamical systems. I was, however, so far not able to grasp the basic idea of this formalism. Could someone please provide an instructive example or a basic introduction to the topic? I would also appreciate hints on relevant literature."}
{"id":"98969","title":"Detail of deriving Berry Curvature From Berry Connection","text":"The Berry curvature of the $n^{\\mathrm{th}}$ eigenstate of Hamiltonian $H$ for the vector of external parameters $\\vec{R}$ can be derived in part by writing the following two lines: $B^n(\\vec{R}) \\equiv \\nabla \\times A^n(\\vec{R}) = -Im \\left[ \\left< \\nabla n(\\vec{R}) | \\nabla n (\\vec{R}) \\right> \\right]$ where $A^n(\\vec{R})\\equiv -Im \\left[ \\left< n(\\vec{R})| \\partial_{\\vec{R}} n(\\vec{R}) \\right> \\right]$ is the Berry connection. I don't know exactly how perform this step in the calculation in detail: $\\nabla \\times A^n(\\vec{R}) = -Im \\left[ \\left< \\nabla n(\\vec{R}) | \\nabla n (\\vec{R}) \\right> \\right]$ Help would be greatly appreciated. Thank you very much."}
{"id":"94633","title":"Spoon released out of a satelite moving with orbital velocity","text":"![enter image description here](http:\/\/i.stack.imgur.com\/VwmZU.png)Suppose we are in a satelite revolving around the earth with a velocity equal to orbital velocity. Now if we release a spoon from the satelite then will it reach the earth or will it just revolve around the earth like the satelite with a velocity equal to orbital velocity?"}
{"id":"90707","title":"Explanation of the classical coupling of the Higgs Field to Electromagnetism","text":"I'm interested in learning about the classical coupling of the Higgs Field to Electromagnetism. There are numerous sources explaining the Higgs mechanism quantum mechanically, i.e. How does the Higgs mechanism work? In addition to the question about classical coupling I'm also interested in how the Mexican hat describes charged particles?"}
{"id":"90708","title":"Neutrinos arrived before the photons (supernova)","text":"A while back I read about the super Kamiokande detector detected a large neutrino flux and then several hours later a supernova was seen. Anyone know of this with sources? I don't recall the source at the moment."}
{"id":"95211","title":"How do electrons get the energy to jump from one orbital to the next when in stationary orbits the electron does not radiate energy","text":"My Question is how do electrons get energy\/lose energy to jump up or down an orbital as in a stationary orbit they do not gain energy and their energy is finite?"}
{"id":"54655","title":"Point charge moving towards a conducting plane","text":"A point charge $q$ of mass $m$ is released from rest at a distance $d$ from an infinite grounded conducting plane. Show that the charge hits the plane after an amount of time given by: $ \\Delta t= \\frac{\\pi d} {q} \\sqrt{2\\pi\\epsilon_0md} $ I can't seem to start this one. I'm assuming I need to start by finding the force by the charge as a function of $z$, and setting it equal to $\\frac{md^2z}{dt} $ giving a differential equation. The diff eq. seems hard to solve, but a hint I recieved was to multiply both sides by $\\frac{dz}{dt}$, but I don't see how this simplifies the problem."}
{"id":"54650","title":"Once I have the eigenvalues and the eigenvectors, how do I find the eigenfunctions?","text":"I am using Mathematica to construct a matrix for the Hamiltonian of some system. I have built this matrix already, and I have found the eigenvalues and the eigenvectors, I am uncertain if what I did next is correct: I took the normalized eigenvectors, placed them in matrix form, and did matrix multiplication with the basis set of solutions. Let me try to be more precise since I am not sure I am using the right language when mentioning the basis solutions. In the problem we are using the set of solutions of the particle in a box model as our basis. I can increase the number of basis elements in the calculation of the matrix of the Hamiltonian (which amounts to doing $<\\psi_n|H \\psi_k>$ over a specified range of $n$ and $k$) in order for some of my smallest eigenvalues to begin to converge. Once I have this $H$ matrix built, and that I see that my eigenvalues are converging to some degree, I take the eigenvectors of the $H$ matrix, format them to be in matrix form, and multiply them by the set of basis solutions. I hope that makes things clearer."}
{"id":"108075","title":"Experimental evidence for the relic neutrinos","text":"What are the experimental (indirect) evidence for the cosmic neutrino background? Where can I read more about this? The discussion on the wikipedia page about the C$\\nu$B seems to me to be more about the evidence of the number of generation of neutrinos, than about the cosmic neutrino background..."}
{"id":"104901","title":"A question about Gell-Mann Low theorem?","text":"Let $|\\Psi_0\\rangle$ be an eigenstate of $H_0$ with energy $E_0$ and let the 'interacting' Hamiltonian be $H=H_0 + gV$, where $g$ is a coupling constant and $V$ the interaction term. We define a Hamiltonian $H_\\epsilon=H_0 + e^{-\\epsilon |t|}gV$ which effectively interpolates between $H$ and $H_0$ in the limit $\\epsilon \\rightarrow 0^+ $ and $|t|\\rightarrow\\infty$. Let $U_{\\epsilon I}$ denote the evolution operator in the interaction picture. The Gell-Mann and Low theorem asserts that if the limit as $\\epsilon\\rightarrow 0^+$ of $|\\Psi^{(\\pm)}_\\epsilon \\rangle = \\frac{ U_{\\epsilon I} (0,\\pm\\infty) |\\Psi_0 \\rangle}{\\langle \\Psi_0 | U_{\\epsilon I}(0,\\pm\\infty)|\\Psi_0\\rangle}$ exists, then $|\\Psi^{(\\pm)}_\\epsilon \\rangle$ are eigenstates of $H$. Here is my question: In interacting field theory, whether the infinite past state $|p_1p_2>_{in} = a^{\\dagger}_{in}(p_1)a^{\\dagger}_{in}(p_2)|0>$ is some kind of $|\\Psi_0\\rangle$ I think it may be wrong. Where is my fault? Thanks!"}
{"id":"98545","title":"Needle Floating on a Water Surface","text":"So, I'm trying to find the largest radius of a needle before it ends up sinking, and I know that it's a balance of needle weight vs surface tension. However, I'm not really sure how the surface tension force is defined. In a lot of resources, I see the surface tension force define as gamma, which is the force \/ length. Also, I am seeing surface tension defined in terms of curvature give by the Young-Laplace equation. Which one am I supposed to use? If I can use the gamma one, it would be straightforward, but I'm not clear on what length to use. Obviously, it's the portion of the circumference in contact with the needle, but this length can vary based upon a lot of factors. Aside from that, I'm thinking that I would have to set the contact angle equal to zero (for the maximum surface tension force), but does that mean that I use gamma*2*pi*r? Wouldn't that imply that the needle is completely submerged? Any help would be appreciated!"}
{"id":"48713","title":"Is the gravitational potential of a planet in orbit always equal to minus the squared velocity?","text":"Say a planet (mass $m$) is orbiting a star (mass $M$) in a perfect circle, so it is in circular motion. $F=ma$ and the gravitational force between two masses $F=\\frac{GMm}{r^2}$ so $\\frac{GMm}{r^2}=ma$ $\\frac{GM}{r^2}=a$ And in circular motion $a=\\frac{v^2}{r}$ so $\\frac{GM}{r^2}=\\frac{v^2}{r}$ $\\frac{GM}{r}=v^2$ And gravitational potential $V=-\\frac{GM}{r}$ So $v^2=-V$ Is there a (qualitative\/less math\/wordy) reason why this is the case? (or have I got this wrong?) and is this limited to the specific case of perfect circular motion?"}
{"id":"99616","title":"While holding an object, no work done but costs energy (in response to a similar question)","text":"I read the answer to Why does holding something up cost energy while no work is being done? and wanting to know more, I asked my teacher about it without telling him what I read here. Instead of referring to muscle cells and biophysics, he answered my question in terms of entropy. He told me that while my arm muscles are stretched when I hold the object, they are more ordered. When my arm is at rest and muscles are not contracted, the muscles are less ordered (more entropy). So his conclusion was that the energy is required to keep the system (my arm muscles) from going to a state of higher entropy. However, the answer in terms of muscle cells doing work on each other (i.e the answer to the hyper-linked question) made more sense to me. Could someone please give me some intuitive sense to my teacher's answer or explain the link between the two answers if there are any..."}
{"id":"79237","title":"Calculating the uncertainty of $r$, when $1\/r^2$ is used","text":"I have tried to search through the forums and the Internet, however I haven't been able to find a source I was able to understand fully. I have a distance $r$, with an uncertainty of $\\mathrm{±\\ 0.003\\ m}$, however in order to linearise my graph, I need to divide $1$ with $r$ squared ($1\/r^2$), what would the final uncertainty for $r$ be? - Furthermore, what would the unit be? Would that still be $m$? or $m^2$?"}
{"id":"91364","title":"Rijke tube: Why Sound is maximum when heated 1\/4th length from bottom of tube","text":"Why is it that the maximum sound coming from a Rijke tube happens only when heated wire gauze is placed at 1\/4th the length of the tube from the bottom. According to Rayleigh: > If heat be periodically communicated to, and abstracted from, a mass of air > vibrating in a cylinder bounded by a piston, the effect produced will depend > upon the phase of the vibration at which the transfer of heat takes place. > If heat be given to the air at the moment of greatest condensation, or be > taken from it at the moment of greatest rarefaction, the vibration is > encouraged. On the other hand, if heat be given at the moment of greatest > rarefaction, or abstracted at the moment of greatest condensation, the > vibration is discouraged. Wouldn't the sound be maximal if heated gauze is placed at the bottom, where there exists an antinode?"}
{"id":"29398","title":"The Synchrotron at CalTech","text":"In the Feynman Lectures Vol. 1, it says that the Synchrotron at the California Institute of Technology is capable of producing electromagnetic Radiations with a frequency of $10^{24}$ cycles per second. Since I'm not familiar with _cycles_ , I'd like to know where this lies in the UV Spectrum."}
{"id":"630","title":"Is it possible to observe interference from 2 independent optical lasers?","text":"It seems that if the coherence length of a laser is big enough, it is possible to observe a (moving) interference picture by combining them. Is it true? How fast should photo-detectors be for observing of the interference of beams from two of the \"best available\" lasers? What is the coherence length of the best- available laser? More specifically, does there exist any (optical single- wavelength) laser with coherence length exceeding 500 meters?"}
{"id":"6438","title":"What if the LHC doesn't see SUSY?","text":"A question in four parts.   1. What are the main problems which supersymmetry purports to solve?    2. What would constitute lack of evidence for SUSY at the proposed LHC energy scales (e.g. certain predicted superpartners are not in fact observed)?    3. Are there alternative theoretical approaches which would address the SUSY problem set and which would still be credible in such an LHC no-SUSY-scenario?   4. Where would LHC-disconfirmation of SUSY leave String Theory? I would like to think that these four points could be taken together as one question."}
{"id":"78762","title":"What is going on in the system? How are the formulas `mg sin(x)` and `mg cos(x)` derived?","text":"When a load is resting on an inclined plane, there is force $mg \\sin(x)$ that's vertical to the inclined plane and force $mg \\cos(x)$ horizontal to the plane acting on it like this: ![enter image description here](http:\/\/i.stack.imgur.com\/mfCsM.png) My textbook says the weight $mg$ has two components: $mg \\sin(x)$ and $mg \\cos(x)$. But when adding up $mg \\sin(x)$ to $mg cos(x)$ it does not equal to $mg$. If `mg` and N are the only forces in this system, which means the net force should be 0, how should I explain the existence of $mg \\sin(x)$ and $mg \\cos(x)$? Does $mg \\sin(x)$ and $mg \\cos(x)$ have their own reaction force to cancel them out and turn the net force to be 0 too or is it wrong the say the system's net force is 0? And how are the formulas $mg \\sin(x)$ and $mg \\cos(x)$ that are used to describe these two forces derived?"}
{"id":"78","title":"Why do neutrons repel each other?","text":"I can understand why 2 protons will repel each other, because they're both positive. But there isn't a neutral charge is there? So why do neutrons repel? (Do they, or have I been misinformed?) The reason why I'm asking this is because, I've just been learning about neutron stars and how the neutrons are forced (as in, they repel) together according to my teacher (he's a great teacher btw, though what I just said doesn't make it seem so). So I wondered, why they have to be forced by gravity and not just pushed?"}
{"id":"29647","title":"If a fundamental theory exibits e.g. a mirror symmetry, in what sense it the underlying geometry real?","text":"> Are the more recently discovered symmetries in string theory such that the > theories based on mirroring geometries are absolutely the same from an > observable point of view? I have mirror symmetry in mind, i.e. by symmetries I mean the dualities encountered in string theory. * * * (v2) Say you take e.g. take the Schwarzschild metric from general relativity and consider its quivalent in some theory of quantum gravity theory. There, if you don't look close enough, i.e. if you look at averages, the quantum- Schwarzschild space would look classical, namely like the general relativity Schwarzschild space. Now the target space in string theory is one manifold and it leads to a quantized theory. But if there are now actually two mirroring manifolds where this theory can come from, the I guess the classical version looks like neighter of these geometries. The following question comes from this observation, that there are totally different (mirroring) options with which Calabi-Yau you can start (considered in the Lagrangian) if you go in the quantization direction and the conclusion that neighter of will be the limit, or there are more limits. > If we assume that the world is described by such a theory, do we besically > live in two indestingushable geometries at once? To put it differently, I ask if string theory can be considered a quantum gravity in the sense that there is a classical limit down to classical differential geometry, especially regarding the extra dimensions. And if so, to what geometry in these higher dimensions does the limit lead? I do know that at least some of the geometry must come from the string excitation representing a quantizations of a metric\/graviton. But I suspect different metrics in the non-linear sigma model Lagrangian will also affect the classical picture in some essential way, directly or not."}
{"id":"72441","title":"Refraction of light in water","text":"Here's a question in a paper based on refraction of light. I can't seem to solve it for some reason. > A man looks down at a fish length of 20 cm. His eye is 2m above the surface > of the water ($\\mu = 4\/3$) an the fish is 2m below the surface shown in the > figure. The ratio of angular width $\\Delta\\theta_1$ of the fish as seen by > the man in presence of water to the $\\Delta\\theta_2$ in the absence of water > is:   >  _(A) 6\/5 $\\hspace{3cm}$ (B) 5\/6 $\\hspace{3cm}$ (C)7\/8 $\\hspace{3cm}$ > (D)8\/7_ ![fish](http:\/\/i.stack.imgur.com\/5zvpD.jpg) * * * _Here's what I've done:-_ When there is no water present: $$\\frac{\\Delta\\theta_{air}}{2} = \\sin^{-1}(10\/400)$$ $$\\therefore \\Delta\\theta_{air} = 2\\sin^{-1}(1\/40)$$ ...from the triangle obtained by the eye, and half the fish. When the water is present: ![diag](http:\/\/i.stack.imgur.com\/qCTOk.jpg)   ($P.S:$ the equation written below the diagram is wrong, I've just realised. I've written the correct version below) Let the light rays at angle $\\phi$ with the normal refract at the water surface and converge at the eye. Here the angle of refraction is $\\theta$. So we have: $$\\sin\\phi . \\mu_{water} = \\sin\\theta . \\mu_{air}$$ $$\\therefore\\frac{\\sin\\theta}{\\sin\\phi} = \\frac{\\mu_{water}}{\\mu_{air}}$$ $$\\therefore\\frac{\\sin\\theta}{\\sin\\phi}=\\frac{4}{3}$$ After which I am stuck and have no clue how to go about this. Can anyone help me? * * * _**Addendum_** : Will taking $\\sin x = x$ help? $x$ is very small anyway, and $\\sin x \\approx x$ is turning out to be true ($x = 0.025$, $\\sin(0.025) = 0.024997...$). _**Addendum2_** : Another thing I figured out is that the water will increase the observed angular width, so the ratio is definitely $>1$. So we can safely rule out (B) and (C) which are $<1$."}
{"id":"3863","title":"What happens to an astronaut who's floating in a spaceship (in space) when it begins to move?","text":"I feel this is somehow a stupid question, but I don't know the true answer. What happens to an astronaut who's floating in a spaceship in space when it begins to move? Will the astronaut not move until he smashes onto a wall in the spaceship? Or will he move with the spaceship due to gravity? Or does it depend on the speed\/acceleration of the spaceship; if it's too fast, the astronaut will smash onto a wall and if it's enough slow, he'll move with the spaceship? Thanks for helping me out!"}
{"id":"3862","title":"Is the \"consistent histories\" interpretation of QM a \"many worlds interpretation\" in disguise?","text":"The so called consistent histories interpretation is claimed to be a correction of the Copenhagen Interpretation. One of its aim, as much as I can see is to show that observers don't have any special role and a history is represented as a sequence of projection operators in the Heisenberg picture. It has been convincingly shown that due to environmental decoherence the density matrix will be almost diagonalized within a short time (expressed in terms of a certain basis of interest). It will smell just like a \"collapse\" of the wave function. However, if we want to see the whole picture, we see that superposition still exists in the entangled whole. My question is whether this interpretation is actually a \"many worlds interpretation\" in disguise? If not, then why not?"}
{"id":"50027","title":"Is it possible to write a Density Matrix in the following form?","text":"Is it possible to write an arbitrary density matrix $\\hat{\\rho}$ in the following form ? $$\\hat{\\rho} ~=~ \\frac{1}{N} \\sum_{\\ell=1}^N \\left|x_{\\ell}\\right\\rangle \\left\\langle x_{\\ell}\\right|,$$ where $\\left\\\\{\\left|x_{\\ell}\\right\\rangle\\right\\\\}_{\\ell = 1}^{N}$ are normalized states (but not necessarily orthogonal). If yes, how can one prove this ?"}
{"id":"66118","title":"Poynting vector direction","text":"Just a quick question if I may. The Poynting vector, or the energy flux density, is given by: $\\mathbf{S} = \\frac{1}{\\mu_{0}}(\\mathbf{E} \\times \\mathbf{B})$ So it's the cross product between the $\\mathbf{E}$-field and $\\mathbf{B}$-field. So depending on the direction of the fields, the Poynting vector will point in some direction. So lets say the $\\mathbf{E}$-field has the direction $\\mathbf{e}_{y}$ and the $\\mathbf{B}$-field has the direction $\\mathbf{e}_{z}$, then the resulting direction for $\\mathbf{S}$ will be $\\mathbf{e}_{z}$. So my question is, is that the direction of which the energy is flowing, or is there some fancy thing I need to know, like it's the opposite or something like that ? Thanks in advance :)"}
{"id":"61467","title":"Beta function of pure $SU(N_\\text{c})$ Yang-Mills theory","text":"What is the dependence of the beta function of pure $SU(N_\\text{c})$ Yang- Mills theory on the number of colors? I guess $$\\mu\\frac{dg_\\text{YM}}{d\\mu}=-\\beta_0N_\\text{c}g_\\text{YM}^3-\\beta_1N_\\text{c}^2g_\\text{YM}^5-\\beta_2N_\\text{c}^3g_\\text{YM}^7-\\cdots\\,,$$ with the appropriate constants $\\beta_0,\\beta_1,...$, as has been computed in QCD (including quarks) at four loops  [arXiv:hep-ph\/9701390]. Is this correct? How could it be proved (to all orders)? Then the 't Hooft coupling $\\lambda=g^2_\\text{YM}N_\\text{c}$ runs independently of $N_\\text{c}$: $$\\frac{\\mu}{2}\\frac{d\\lambda}{d\\mu}=-\\beta_0\\lambda^2-\\beta_1\\lambda^3-\\beta_2\\lambda^4-\\cdots\\,.$$ The goal is to assure that in the large $N_\\text{c}$ limit the coupling runs independently of $N_\\text{c}$, so that the confinement scale is held fixed. So the series in $\\lambda$ could be truncated for $N_\\text{c}\\rightarrow\\infty$ (there could be negative powers of $N_\\text{c}$)."}
{"id":"61462","title":"What happens to pipe outlet temperature here?","text":"I have a fluid flowing through a pipe in the ocean and there is heat transfer from the ocean to the fluid in the pipe. I prepared a simulation and the results show that if I increase the mass flow rate of the fluid through the pipe then the heat transfer rate to the fluid increases and the outlet temperature increases. These results don't seem right to me. I would think if the heat transfer rate increases, then the outlet temperature should also increase. Does anyone have any suggestions as to why my intuition might be wrong? It is also quite possible the simulation could be wrong."}
{"id":"28114","title":"Describing quantum intereference with only currents and densities","text":"I know about and believe to understand the general wave equation based Kirchhoff diffraction formula, which in the Fraunhofer limit leads to a farfield complex wave function by Fourier transforming the complex wave function of the incident beam in the aperture. I believe it should be possible to describe this phenomenon through only the density (wavefunction squared) and probability current (quantum mechanically). These two should be able to provide the phase and amplitude information necessary to interfere something. What I'm looking for is a \"hydrodynamic\" theory of diffraction and\/or interference, which I think should be very well possible. Has this been done before? Am I barking up a wrong tree?"}
{"id":"22771","title":"Does the universe follow a general physical and computational principle of evolution?","text":"Evolution is a principle in biology, whereby organisms evolve their ability to replicate and multiply in numbers over successive generations. From a computational point of view, the organisms employ a common programming (DNA, RNA, etc.) and runtime environment (cell biology). This evolution has eventually evolved a more powerful ((in a restricted sense) computational environment, the hierarchy of biological brains, with the currently most powerful class instance being the human brain. A collection of human brains is now working on evolving a quantum computer. Perhaps one day the the quantum computers will produce a.... From an anthropic point of view, we are here to think about these things, because our universe creates environments such as our planet earth, which support the evolution of life. So my question is whether there is a general physical and computational principle at work here, demonstrated by the following incomplete, in parts almost surely incorrect, and highly speculative hierarchical chain of evolution:   1. Within the Multiverse, universes are instantiated following particular physical laws originated and adapted from their embedding universe   2. Each instantiated embedded universe follows computational rules (the particular physical laws of this universe) and create a finite number of embedded universes (the only candidate seems to be for singularities of black holes to correspond with a white-hole, inflationary, embedded universe), each of which evolve in the same general manner   3. Each universe produces a finite number of computational schemes to build organisms that follow certain classes of program patters. These organism evolve according to the theory of biological evolution.   4. Biological organisms evolve biological brains, a higher class computational scheme   5. biological brains build computational devices, which at some point attain the property of replication   6. The replicating computational devices build higher classes of computational devices, following the general principle of evolution..."}
{"id":"77674","title":"How does one calculate where the \"surface\" of a gas-giant would be?","text":"Okay, so Jupiter, Saturn, et. al are gas giants. I understand that they have large gassy atmospheres, which, due to the pressure would eventually become more and more dense as one approaches the center of the planet. Falling into Saturn or Jupiter, would we pass through it until we hit the nucleus? hits my model about right - gas eventually compressed by gravity would form, I suppose, a liquid core, followed by a solid surface. Assuming that Jupiter and Saturn were made of pure hydrogen (It's actually about 90%, but I'll simplify it a bit, how would one go about calculating the distance at which there is a \"surface\"? Put another way, at some distance, gravity should compress the gas into a liquid, and at a point even lower, I would assume that the liquid would compress into a gas. Is there an equation that would tell me what those points are? And, for bonus points, assuming that Saturn has a diameter of 72,367 miles \/ 116,464 km, at what distances would the transition points from solid to liquid and liquid to gas be? Full disclosure: I do realize that the same surface pressures would actually implode a human being trying to step on said surface. Despite desires to attempt this in reality, I am not actually going to \"try this at home.\" This is a thought experiment more about how I calculate where gravity forces a compound into a different state of matter. I am curious more about how one models the effect of mass into gravity and compression. As a thought experiment, I suppose technically this is not a 'problem I face,' but I am curious. Thanks in advance :)"}
{"id":"73315","title":"$d=\\frac{1}{2}at^2$ not giving expected results","text":"Say I have an object that starts falling from rest at 2 m\/s2. Its total displacement and velocity at the end of every second should look something like this:               t    d    v     1    2    2     2    6    4     3    12   6     4    20   8      At the end of the first second, it should have accelerated to 2 m\/s and moved 2 m. At the end of 4 seconds, it should moved 20 m. But when I use the equation for getting displacement, $d=\\frac{1}{2}at^2$, I get 16 m. Shouldn't it be 20 m? What is going wrong here?"}
{"id":"31841","title":"Rotational speed of a coil in a uniform magnetic field at equilibrium","text":"I'm looking at the following problem from \"Physics 3\" by Halliday, Resnick and Krane (4th edition): > The armature of a motor has 97 turns each of area 190 cm² and rotates in a > uniform magnetic field of 0.33 T. A potential difference of 24 V is applied. > If no load is attached and friction is neglected, find the rotational speed > at equilibrium. My attempt is as follows: Initially, the potential difference is 24 V. Because of this potential difference, current starts flowing through the motor. Then, because of the 0.33 T magnetic field, the motor starts rotating. The rotational motion changes the magnetic flux through the motor's coil, which produces an induced EMF (electromotive force) that opposes the original EMF. If we call N the number of turns, A the area of the motor's coil, and θ the angle from the magnetic field vector B to the vector normal to the plane of the coil, then the magnetic flux through the coil is: $\\Phi=NAB\\cos{\\theta}$ So, the induced EMF is: $\\varepsilon_{ind}=-\\frac{\\mathrm{d} }{\\mathrm{d} t}(NAB\\cos{\\theta})=NAB\\frac{\\mathrm{d} \\theta }{\\mathrm{d} t}\\sin{\\theta}$ If we call ω the angular velocity as a function of time ($\\frac{\\mathrm{d} \\theta }{\\mathrm{d} t}$), then the net EMF as a function of time is: $\\varepsilon = 24 - \\varepsilon_{ind} = 24- NAB\\omega\\sin\\theta$ Is this correct so far? I'm not sure how to proceed from here. Anyway, the answer given in the back of the book ( **39.5 rad\/s** or **6.3 rev\/s** ) seems to suggest that the correct way of finding ω is by using $\\varepsilon=NBA\\omega$ and plugging in the values, that is, ε = 24 V, N = 97, B = 0.33 T and A = 0.0190 m². This gives the correct result of 6.3 revolutions per second. But I'm not sure how to obtain $\\varepsilon=NAB\\omega$ from the expression that I found above ($\\varepsilon = 24 - NAB\\omega\\sin\\theta$). * * * _**Edit**_ : Based on the suggestions given in the comments to the answer below, I understand that the \"equilibrium\" that this question is referring to is when the torque on the coil reaches the value zero; this happens when the total EMF ($\\varepsilon = 24 - NAB\\omega\\sin\\theta$) on the coil equals zero. That is: $\\omega = \\frac{24}{NAB\\sin\\theta}$ To reach the desired result, $\\sin\\theta$ should equal $1$; but I don't understand why that is the case here. Is this reasoning correct? How should I proceed from here?"}
{"id":"32635","title":"when is the stationary phase approximation exact?","text":"I am thinking about some topological field theories, and I am wondering when one can say that the stationary phase approximation (ie. a sum of the first- order variations about each vacuum) is exact. I am looking perhaps for conditions on how the space of vacua is embedded into the space of all field configurations. I suspect that when the action is a Morse function (and I suppose the space of field configurations is finite dimensional) that the exactness of the stationary phase approximation implies some very strict topological constraints on the configuration space... torsion-free and so on. Anyone have a good reference or some wisdom? Also I'd like to dedicate this question to the memory of theoreticalphysics.stackexchange"}
{"id":"48311","title":"Physical explanation for capacitive circuit","text":"In capacitive circuit, the voltage and current are out of phase. the current leads the voltage by 90 degrees. this can be explained by these two equations: $$V = V_{\\max} \\sin \\omega t$$ $$I = I_{\\max} \\cos \\omega t.$$ Now, my question is that what does this statement mean physically that \"current leads the voltage by 90 degrees? It is clear mathematically that if $V = 0$ then $I$ is maximum but how do we explain this physically? hope you understood my question"}
{"id":"93546","title":"Is kinetic energy in QM a state-property or is it distributed?","text":"Suppose we have a quantum mechanical system, which is well described by its wave function in r-representation $\\Psi$. We are interested in the properties of an observable, say the kinetic energy $T$. So we let the kinetic energy operator act on the wave function, multiply the result by Psi and integrate over spatial coordinates. Thus we obtain the expectation value $\\langle T \\rangle$. We note that this result is a property of the operator and also of the quantum state. Now in QM text books one will often find detailed discussions of the result, in order to provide physical meaning and context. For example, in discussing the kinetic energy of the ground state of a hydrogen atom, it is pointed out that the kinetic energy $T$ is intimately related --through the Schroedinger equation-- to the electrostatic potential $V$ and the total energy $E$. Both $V$ and $E$ are operators that are multiplicative in $r$-representation. So $T$ (which at its core is represented by the Laplace differential operator), when written as $E$-$V$, may be regarded as multiplicative. Thus we get (with energy in Rydberg energy units and distance in Bohr radius units): $$T = -1 + \\dfrac{2}{r}$$ This result is then discussed in terms of \"the classical turning point\" $r=2$. The region $r > 2$ is forbidden in classical physics, since the kinetic energy is here negative. Yet in QM there is a non-zero probability that the electron is found at $r > 2$. I wonder how much \"truth\" there is the above formula for $T$. It is suggestive to regard it as a proper function of the distance to the nucleus $r$. However, as soon as one digs a little deeper, conflicts arise. For example, one may compute the Fourier transform of the wave function. Now since the initial wave function was spherically symmetric, so is the FT. Hence in $\\Phi(k)$ there are only states with positive wave factor $k$. There is simply no room for negative kinetic energies! So immediately a paradox arises when we regard kinetic energy as a distribution in $r$- or $k$-representation. Furthermore, one may look at the higher moments of the kinetic energy operator. This leads to new problems. For example, the second moment $\\langle TT\\rangle$ can be computed as either $(\\Psi, TT \\Psi)$, or as $(T \\Psi, T \\Psi)$ [provided $T$ is self-adjoint when acting on these wave functions]. Of course it is satisfactory that these two integral representations lead to the same net result. However their integrands are obviously different! Therefore if the kinetic energy is distributed within the state, it can not be pin- pointed without some level of ambiguity. I would like to hear your comments on this issue !"}
{"id":"98577","title":"Superficial Degree of Divergence for Feynman Diagrams","text":"The superficial degree of divergence for a diagram is defined as the power of $k$ in the nominator minus the power of $k$ in the denominator. It is written to be equal to $4\\times$ (number of loops)$ - $(number of internal Fermion lines)$ - 2 \\times $(number of internal Boson lines). Why is it 4 and not 3? When you do a 4-dimensional integral in spherical coordinates, you get a factor of $k^3$ from your volume element. In particular, when you calculate the QED vertex, for instance, you get: $$ \\int_{\\mathbb{R}^4} \\frac{d^4k}{(2\\pi)^4} \\frac{\\gamma^\\rho (\\not\\\\!{p_1}-\\not\\\\!{k}+m)\\gamma^\\mu(-\\not\\\\!{p_2}-\\not\\\\!{k}+m)\\gamma_\\rho}{k^2\\left[\\left(p_1-k\\right)^2-m^2\\right]\\left[\\left(p_2+k\\right)^2-m^2\\right]} \\overset{k\\to\\infty}{\\propto} \\int_{\\mathbb{R}^4} \\frac{d^4k\\, k^2}{k^6} \\overset{\\text{spherical}}{\\underset{\\text{coordinates}}{\\propto}} \\int_0^\\infty\\frac{dk}{k} $$ Thus I would say that the superficial degree of divergence of this diagram is $3\\times1-2-2=-1$ and not $0$. What am I missing?"}
{"id":"132715","title":"Calculating new temperature of an object when air temperature changes","text":"I'm trying to calculate the new temperature of an object when the air temperature around it changes, given a period of time. Basically I get periodic readings from an air temperature sensor in a refrigerator. In some cases these readings are every 5 minutes, in others every 1 minute, so the time between readings is variable. For each reading I get, I'd like to also calculate the approximate temperature of food at its core; something like a chicken for example (I know that part is vague, but if there is a variable I can tweak then that is fine). The result should be a \"damped\" version of the actual air temperature, as obviously any objects will slowly change temperature to eventually meet the air temperature. Initially there used to be \"food simulant\" put around the sensor, so the temperature would automatically be damped, but this is no longer the case. I do not know much about thermodynamics. I'm not sure if I can just add a percentage of the temperature change to the previous damped value, or if I need a calculation based on the last few air temperature readings, or what. I guess I'm looking for a result a bit like:               10:00 2 degrees (air), 2 degrees (product)     10:05 2.5 degrees (air), 2.1 degrees (product)     10:10 2.5 degrees (air), 2.2 degrees (product)     10:20 2.7 degrees (air), 2.5 degrees (product)      I could do something really cheap like averaging the readings over the last 30 minutes but I don't think that will cut it! I'd appreciate any help - thanks very much."}
{"id":"43707","title":"Another faster-than-light question","text":"Imagine we have something very heavy (i.e supermassive black hole) and some object that we can throw with 0.999999 speed of light (i.e proton). We are throwing our particle in the direction of hole. The black hole is so heavy that we can assume that in some moment acceleration of gravity would be say 0.0001 speed of light\/s^2. So the question is what will be the speed of proton in few a seconds later, assuming we have such distances that it will not hit the black hole before."}
{"id":"134699","title":"How do I use MAUD for phase quantification of XRD data with AMCSD standards?","text":"I'm a biologist with some raw powder XRD data of biominerals, need to quantify different phases, can't access commercial software or databases at the moment. Does anyone have experience with using the cif standards from the AMCSD with MAUD? It seems to display the list of phases fine, but when I pick one it doesn't load. Alternatively, suggestions of other free\/trial software that works on Win 7, accepts the AMCSD and is not too impenetrable to noobs would be welcome. I have tried XPowder12 trial (which kept crashing), FullProf (which I couldn't figure out), and GSAS (which I couldn't even install). MAUD seemed like such a nice program until it didn't load the standards..."}
{"id":"25205","title":"Seeing cosmic activity now, really means it happens millions\/billions of years ago?","text":"A Recent report about a cosmic burst 3.8 billion light years away. It is written as though it is happening now. However, my question is, if the event is 3.8 billion light years away, doesn't that mean we are continuously looking at history, or is it possible to detect activity in \"realtime\" despite the distance?"}
{"id":"74320","title":"Planck satellite 2013 results impact on cosmological models","text":"I've recently watched a \"Through the Wormhole\" episode from 2010 named \"What Happened Before the Beginning?\", where cyclic and ekpyrotic cosmological models are explained. In this episode the creators of the ekpyrotic model, Neil Turok and Paul Steinhardt, say that the results from the Planck satellite about gravitational waves will either support one theory or the other. After doing a little bit of research about the Planck satellite I've come across some 2013 results on the Wikipedia page. However, this information is far too complex for me to understand. How does this information affect the cyclic and ekpyrotic models? Are the results leaning towards one of the models?"}
{"id":"74324","title":"Information Preservation and Burning Books","text":"I recently read an article in the NY Times called A Black Hole Mystery Wrapped in a Firewall Paradox. I really liked the article, but reading one quote immediately made me think of asking Physics.SE a question: > From the material in the smoke and flames of a burning book, for example, > one could figure out whether it was the Bible or the Kama Sutra; the same > should be true of the fizz and pop of black holes, these physicists argued. So, Physics.SE, _theoretically_ , how would you figure out what book was burned from the smoke and flames? I do not intend for this to be a silly question. If the idea of \"information loss\" inside of a black hole is so difficult for some physicists to come to terms with, I figure those same physicists should have a perfectly good answer to my question."}
{"id":"71315","title":"Which \"space-time coincidences\" are described by a \"co-ordinate system in which the gravitational field does not appear\"?","text":"In Einstein's exposition of the foundations of General Relativity (cmp. http:\/\/en.wikisource.org\/wiki\/The_Foundation_of_the_Generalised_Theory_of_Relativity , end of §3) there appears an emphasis on considerations of \" _coincidences_ \": > _All our well-substantiated space-time propositions amount to the > determination of space-time coincidences. [...] The results of our > measurements are nothing else than well-proved theorems about such > coincidences of material points_ and their relation to \" _co-ordinates_ \": > _The introduction of a system of co-ordinates serves no other purpose than > an easy description of totality of such coincidences._ Subsequently (in §4) one more or less specific \" _system of co-ordinates_ \" is suggested: > The nature of acceleration of an infinitely small (positional) co-ordinate > system is hereby to be so chosen, that the gravitational field does not > appear; this is possible for an infinitely small region. $X_{1}, X_{2}, > X_{3}$ are the spatial co-ordinates; [...] Question:   What, explicitly, is contained in the corresponding \" _totality of such coincidences_ \";   or in some specific subset of this \" _totality_ \"?,   of which the indicated \" _co-ordinate system in which the gravitational field does not appear_ \" is a suitably \" _easy description_ \". Does this \" _totality_ \" contain, for instance: (1)   for any three distinct \" _material points_ \" (${\\textbf A}$, ${\\textbf B}$, and ${\\textbf P}$, who are assigned the corresponding unequal constant \" _spatial co-ordinate_ \" triples $\\\\{ \\, X_{1}^{\\textbf A}, X_{2}^{\\textbf A}, X_{3}^{\\textbf A} \\, \\\\}$, $\\\\{ \\, X_{1}^{\\textbf B}, X_{2}^{\\textbf B}, X_{3}^{\\textbf B} \\, \\\\}$, and $\\\\{ \\, X_{1}^{\\textbf P}, X_{2}^{\\textbf P}, X_{3}^{\\textbf P} \\, \\\\}$, respectively),   and for any \" _coincidence_ \" ${\\textbf A}_{\\mathscr X}$ in which ${\\textbf A}$ had been participating,   the \" _coincidence_ \" that ${\\textbf A}\\\\!$'s indication of having seen that ${\\textbf P}$ saw ${\\textbf A}\\\\!$'s indication of having seen that ${\\textbf B}$ saw \" _coincidence_ \" ${\\textbf A}_{\\mathscr X}$   was coincident to   ${\\textbf A}\\\\!$'s indication of having seen that ${\\textbf B}$ saw ${\\textbf A}\\\\!$'s indication of having seen that ${\\textbf P}$ saw \" _coincidence_ \" ${\\textbf A}_{\\mathscr X}$   ? And is \" _an easy description_ \" of this \" _totality_ \" afforded by the indicated system for instance in the sense that: (2)   for any two distinct \" _material points_ \" (${\\textbf A}$ and ${\\textbf B}$, who are assigned the corresponding unequal constant \" _spatial co-ordinate_ \" triples $\\\\{ \\, X_{1}^{\\textbf A}, X_{2}^{\\textbf A}, X_{3}^{\\textbf A} \\, \\\\}$, and $\\\\{ \\, X_{1}^{\\textbf B}, X_{2}^{\\textbf B}, X_{3}^{\\textbf B} \\, \\\\}$, respectively) there is another triple, $\\\\{ \\, X_{1}^{\\textbf M}, X_{2}^{\\textbf M}, X_{3}^{\\textbf M} \\, \\\\}$, reserved as constant \" _spatial co-ordinates_ \" to denote a \" _material point_ \" ${\\textbf M}$, if it happens to exist, as \"middle between ${\\textbf A}$ and ${\\textbf B}$\", described by the \" _coincidences_ \"   * that for any \" _coincidence_ \" ${\\textbf A}_{\\mathscr X}$ in which ${\\textbf A}$ had been participating,  ${\\textbf A\\\\!}$'s indication of having seen that ${\\textbf M}$ saw ${\\textbf A\\\\!}$'s indication of having seen that ${\\textbf M}$ saw \" _coincidence_ \" ${\\textbf A}_{\\mathscr X}$   was coincident to   ${\\textbf A\\\\!}$'s indication of having seen that ${\\textbf B}$ saw \" _coincidence_ \" ${\\textbf A}_{\\mathscr X}$, and   * that for any \" _coincidence_ \" ${\\textbf B}_{\\mathscr Y}$ in which ${\\textbf B}$ had been participating, ${\\textbf B}$'s indication of having seen that ${\\textbf M}$ saw ${\\textbf B}$'s indication of having seen that ${\\textbf M}$ saw \" _coincidence_ \" ${\\textbf B}_{\\mathscr Y}$   was coincident to   ${\\textbf B}$'s indication of having seen that ${\\textbf A}$ saw \" _coincidence_ \" ${\\textbf B}_{\\mathscr Y}$, and   * that for any \" _coincidence_ \" ${\\textbf M}_{\\mathscr Z}$ in which ${\\textbf M}$ had been participating,  ${\\textbf M}$'s indication of having seen that ${\\textbf A}$ saw \" _coincidence_ \" ${\\textbf M}_{\\mathscr Z}$   was coincident to   ${\\textbf M}$'s indication of having seen that ${\\textbf B}$ saw \" _coincidence_ \" ${\\textbf M}_{\\mathscr Z}$   ?"}
{"id":"108195","title":"What is the principle of equivalence in thermodynamics?","text":"I've been searching for this for a while. There is a principle of equivalence in general relativity: http:\/\/en.wikipedia.org\/wiki\/Equivalence_principle But I need the principle of equivalence in thermodynamics. Is it the same as the second law of thermodynamics? http:\/\/en.wikipedia.org\/wiki\/Second_law_of_thermodynamics"}
{"id":"105223","title":"What transformation is the metric of general relativity invariant under?","text":"My limited understanding of metrics comes from Cartan. From there, I understand that a metric is something invariant under certain transformations, e.g. Lorentz in special relativity. But with the metric varying from point to point (event-to-event), what is the meaning of the metric?"}
{"id":"33143","title":"Can anyone explain this demonstration with a helium balloon?","text":"See the YouTube video (click) named \"Testing the Magnetic Nature of MagneGas.mp4\". A helium balloon is attracted to a metal bar."}
{"id":"123472","title":"Demonstration that vibrating basic particles constitute non-vibrating individuals","text":"I am a dilettante in physics; I ask for pardon for my confusion-causing (if any) terminology usage, and also for my imprecise choice of question tags. I know that basic particles of any individual stopping vibrating contradicts Heisenberg's uncertainty principle, so the particles must whenever vibrate. Then how could basic particles constitute a seemingly non-vibrating individual? Is it a natural limitation of human eyesight or ...? Is there any scientific explanation for it? A demonstration not using analogy or common sense is seeking after."}
{"id":"60678","title":"Questions about entanglement from a laymen\/quantum hobbyist","text":"Please note I am not a physicist I just read every article I can on it, I understand a good amount on it though. But no maths. (currently trying to learn the maths) By what means can we as humans entangle two particles? How does (just examples are fine) it happen in nature? And I would guess technically there could be a particle in me entangled with a particle in another person; maybe my future wife? I think the answer to the first is... We make them by shooting a photon at a half silvered mirror and somehow it splits and one bounces off the half silver while the other one leaves and we capture the one, and wait for the other and capture it too. Then measure one and measure the other and see they are the same? EDIT: One more question. From one of the answers two entangled particles have the same wave function. Does this mean every particle with the same wave function is entangled? Can an electron and photon or any two arbitrary particles or groups of particles(molecules) entangle?"}
{"id":"26933","title":"Topics in particle cosmology","text":"> **Possible Duplicate:**   >  Topics in particle cosmology I am interested in learning more about this interdisciplinary approach. 1) What are some of the top questions in particle cosmology (e.g nature of dark matter, inflationary structure, topological defects in the universe)? 2) Who are some of the top researchers in this field? 3) Recommended review papers (on arxiv, for instance) will be greatly appreciated."}
{"id":"37716","title":"What is the angular distance between Ptolemaic perigees of Mercury?","text":"In his excellent treatment of the history of the science of astronomical distances and sizes, Albert van Helden says (p.29) that > The complicated [Ptolemaic] model of Mercury has the curious property of > producing two perigees, each about 120° removed from the apogee. But when I try to confirm this using the 88 day period Mercury's epicycle, I get approximately half the expected value. In a given time (in days), $t$, Mercury will travel through an angle $\\varepsilon = 2\\pi t\/88$ along its epicycle which will traveled through an angle $\\delta = 2\\pi t\/365$ along its deferent. In transitioning from apogee to perigee, it must be the case (since Mercury must go half way round the epicycle, and then an additional $\\delta$ to \"catch up\" with the angle traveled by the center of the epicycle) that $$\\varepsilon=\\delta+\\pi$$ solving which yields $$t=[2\\cdot(1\/88-1\/365)]^{-1}\\approx58$$ which corresponds to about 57°, roughly half the number expected. What is missing from the above reasoning? Is there something about the definitions of epicyclic period I've missed, perhaps?"}
{"id":"18693","title":"Using the area element in derivation of geodesic","text":"In the derivation of the geodesic, one starts with the integral of the line element (arclength): $$L(C)=\\int_{\\tau_1}^{\\tau_2}d\\tau\\sqrt{g_{\\mu \\nu}\\dot{x}^{\\mu} \\dot{x}^{\\nu}}$$ The integrand is then substituted into the Euler-Lagrange equation, which simplifies to the geodesic equation. My question is would it be possible to start with the area element instead of the line element, and thereby arrive at a surface area geodesic equation, which I guess would be an equation for the surface area of a given patch of manifold?"}
{"id":"83340","title":"conservation of momentum when a bullet hits a block","text":"why momentum is conserved when a bullet hit a block horizontally even when force of bullet is acting on it and net external force is not zero ?"}
{"id":"83348","title":"Why does this cowbell appear to vibrate at the wrong frequencies?","text":"The cowbell in question: http:\/\/www.youtube.com\/watch?v=OuU0BzXm-qk In most cowbells, the larger the metal dome, the lower the frequency produced. What would cause a phenomenon where the opposite is true: the larger dome creates a higher-pitched sound, as the one in the video does."}
{"id":"113673","title":"Why is free neutron unstable but free proton is stable?","text":"Aren't they basically made up of the same stuffs (quarks) in almost the same configuration?"}
{"id":"47878","title":"Do stars appear to move with uniform motion?","text":"The Ancient Greek astronomers had quite an obsession over uniform circular motion; I was wondering if there was a logical reason for this. Did it develop through actual observations of the stars? Do they appear to move uniformly? If so, why? I've read a few things online that attributes this belief mainly to \"Pythagorean mysticism\", saying that the Ancient Greeks simply looked for an idealistically satisfying description of celestial motions. This too makes me wonder though: 1.) Why uniform? 2.) Why circular?"}
{"id":"108535","title":"Calculate speed of proton at infinity","text":"I have this problem to solve: Two protons are on the x axis at x=-1 and x=0. An alpha particle is placed at x=2, and the proton on the left is released, find its speed at infinity. I know that this involves the change in kinetic and potential energies of the particles, but I dont understand which particles are included in that calculation. Is the proton on the left neglected because it is the one released? An explanation in general would be appreciated. For the purposes of following the rules in asking a homework question properly, the specific physics topics I was asking about were: which particles are included in the kinetic and potential energy calculation for determining the speed at infinity of a proton, and why. My effort in solving this question was finding out that work is the change in kinetic energy + change in potential energy, but not knowing which particles to include in the calculation. This should clear up any confusion that this question is off topic."}
{"id":"72839","title":"Two-oscillator system coupled to reservoir","text":"I am implementing the Monte Carlo wave-function approach to dissipation problems. So far, I have simulated the quantum harmonic oscillator coupled to a finite temperature reservoir given in section 5A of the above paper. I must now implement the two-oscillator system, whose Hamiltonian I am told is of the form $$ H=\\hbar\\omega_aa^\\dagger a+\\hbar\\omega_bb^\\dagger b-\\hbar ga^\\dagger a(b^\\dagger+b). $$ My question is this:   1. Can I form my wave-function vector in such a way that, for example, every odd entry represents the first oscillator in the corresponding state and every even entry represents the second? As an example, if both oscillators were in the second state, the wave-function would be $$ \\left|\\psi\\right\\rangle=\\frac{1}{\\sqrt{2}}\\begin{pmatrix}0\\\\\\0\\\\\\1\\\\\\1\\\\\\0\\\\\\0\\\\\\\\\\vdots\\end{pmatrix} $$   2. If this is possible, what sort of form would the ladder operators $a,a^\\dagger,b,b^\\dagger$ take and would the corresponding number operators for each oscillator still be $a^\\dagger a$ and $b^\\dagger b$? Sorry if this is a dumb question, I have not yet taken a real QM course. Thanks for the help."}
{"id":"30725","title":"Compton scattering multiple wavelengths?","text":"The formula given for compton scattering shows that when x-ray of one specific wavelength hits carbon or some materials, emitted x-ray will be of one new specific wavelength. However, according to scattered x-ray intensity (y-axis) vs wavelength (x-axis) graph, it shows that there are multiple wavelengths for each scattering angle. So, what is the wrong part of my thoughts? How should compton scattering formula be interpreted? And how can one explain that scattered x-ray frequency can be higher than original x-ray frequency?"}
{"id":"31456","title":"Using Moments to influence the thickness of beams in structure?","text":"Are moments relevant to the thicknesses of beams in a structure (ie. the larger the moment the thicker the beam)?"}
{"id":"126192","title":"Examples of nearly perfect fluids and gases","text":"When learning for physics (hydrodynamics and gases) I wanted to know what would be examples of nearly perfect fluids (no surface effects and no friction) and perfect gases (only elastic collisions and a simplified interaction potential) that are used in experiments etc. When searching for that in the Internet I only found things like _ultra-hot quark-gluon plasma_ and _optically trapped 6Li atoms_ for perfect fluids. I rather thought of more practicable examples like fluid helium, is it near to a perfect fluid? For gases I think that would be hydrogen and light halogens (because the volume of the molecules is neglected), but at which temperatures?"}
{"id":"11311","title":"Using the Scalar Electrostatic Potential to Calculate Transition Probabilities","text":"**transition probabilites** of atomic systems prone to some time-varying electromagnetic field are very often calculated using **perturbation theory** leading to expressions including the vector potential $\\mathbf{A}$. This approach seems to be very complicated and I am wondering: > ### Is using the electrostatic potential $U$ to calculate \"electric > transitions\" (approximately) sufficient? This was the short version of my question; below you will find some further notes that (hopefully) contain all information needed to clearify it. In **relativistic quantum mechanics** , the action of an an **electrodynamic field** on an electron can be realized by minimal coupling using the replacement $\\partial_\\mu \\rightarrow \\partial_\\mu - \\mathrm{i}eA_\\mu$. Thus, in a certain sense, (quantum) electrodynamics can be derived from the Dirac equation using the invariance of the free-electron action with respect to a phase change. ### Electromagnetic Fields and the Schrödinger Equation Nevertheless, often one must ignore the beauty of gauge theory to be able to **practically calculate** something under certain simplifications using the non-relativistic Schrödinger-equation. There, one can incorporate an external electromagnetic field applying the replacements $$H\\left(\\mathbf{p},\\mathbf{x}\\right) \\rightarrow H\\left(\\mathbf{p}-e\\mathbf{A}\\left(t,\\mathbf{x}\\right),\\mathbf{x}\\right)+eU\\left(t,\\mathbf{x}\\right)$$ to the Hamiltonian of the system where $\\mathbf{A}$ is the known vector potential and $U$ is the potential corresponding to the electric field. In the Coulomb gauge $\\mathrm{div}\\mathbf{A}=0$ one can always achieve $U\\equiv 0$ if no sources are present and the overall Hamiltonian will look like $$H = H_0 -\\frac{e}{m}\\mathbf{A}\\left(t,\\mathbf{x}\\right)\\mathbf{p}$$ where $H_0 = \\frac{\\mathbf{p}^2}{2m} + V(\\mathbf{x})$ and the term in $\\mathbf{A}^2$, the ponderomotive potential has been neglected. ### Time-Dependent Perturbation Theory The last expression for $H$ has the form $$H = H_0(\\mathbf{p},\\mathbf{x}) + H_t(\\mathbf{p},\\mathbf{x},t)\\ .$$ Using the Dyson series, one can formally write down the time-dependent solution in the interaction picture as $$|\\psi_I(t)> = T\\exp\\left[\\frac{1}{\\mathrm{i}\\hbar}\\int_{t_{0}}^{t}H_{t,I}\\left(t^{\\prime}\\right)dt^{\\prime}\\right]|\\psi_I(t_0)> .$$ Then, one takes only the first two terms into account to find that **transition amplitudes** are approximately given by $$A_{n\\rightarrow m}\\approx\\frac{1}{\\mathrm{i}\\hbar}\\int_{t_{0}}^{t}\\left\\langle m\\right|H_{t,I}\\left(t^{\\prime}\\right)\\left|n\\right\\rangle dt^{\\prime}\\ \\mathrm{with}$$ $$H_{t,I}\\left(t\\right) = e^{\\mathrm{i}\\left(t-t_{0}\\right)H_{0}\/\\hbar}H_t\\left(t\\right)e^{-\\mathrm{i}\\left(t-t_{0}\\right)H_{0}\/\\hbar}\\ .$$ ### Relation to the Question In the given case we have $$H_t = -\\frac{e}{m}\\mathbf{A}\\left(t,\\mathbf{x}\\right)\\mathbf{p}$$ and assuming that $$\\mathbf{A}\\left(\\mathbf{x},t\\right) = \\mathbf{A}\\left(\\mathbf{x}\\right) T(t)\\ ,$$ and further using $$\\mathbf{p}=\\frac{\\mathrm{i}m}{\\hbar}\\left[H_{0},\\mathbf{x}\\right]\\ ,$$ one will have to solve $$A_{n\\rightarrow m}\\propto <m|\\mathbf{A}\\left(\\mathbf{x}\\right)\\cdot \\mathbf{x}|n>\\int_{t_0}^t e^{-\\mathrm{i}\\omega_{mn}\\tau}T(\\tau) d\\tau\\ +\\ \\mathrm{h.c.}$$ with $\\hbar \\omega_{mn} = E_m - E_n$, where the $E_i$ are energy levels of the unperturbed $H_0$. Using here the vector potential $\\mathbf{A}$ is the correct way to incorporate the field, of course. For me, it seems **extremely complicated** to do so since to calculate the transition probabilities for an **arbitrary field distribution** one will have to expand $\\mathbf{A}$ into vector spherical harmonics. So, the use of a scalar potential like $U$ would greatly simplify computations and my question can now be re-asked as > ### Why can't we just simply use $U(\\mathbf{x})\\cdot T(t)$ to calculate the > transition amplitudes $A_{n\\rightarrow m}$ for \"electric excitations\"? **Thank you very much** for insights, comments and corrections."}
{"id":"103315","title":"Why electrons get excited?","text":"Why and how are electrons get excited and what happen inside an atom when electrons get excited?"}
{"id":"103310","title":"Time Evolution Operator in Interaction Picture (Harmonic Oscillator with Time Dependent Perturbation)","text":"**1\\. The problem statement, all variables and given\/known data** _Consider a time-dependent harmonic oscillator with Hamiltonian_ $$\\hat{H}(t)=\\hat{H}_0+\\hat{V}(t)$$ $$\\hat{H}_0=\\hbar \\omega \\left( \\hat{a}^{\\dagger}\\hat{a}+\\frac{1}{2} \\right)$$ $$\\hat{V}(t)=\\lambda \\left( e^{i\\Omega t}\\hat{a}^{\\dagger}+e^{-i\\Omega t}\\hat{a} \\right)$$ *(i) Compute $\\hat{U}_S(t,0)$ using the interaction representation formula (Equation 1 in next section) to second order perturbation theory. (ii) Compute $\\hat{U}_S(t,0)$ using (Equation 2 in next section) to second order perturbation theory. **2\\. Relevant equations** EQUATION 1: $$U_I(t,0)=1-\\frac{i}{\\hbar}\\int_0^t dt' V_I(t')+\\left( \\frac{-i}{\\hbar} \\right)^2 \\int_0^t dt' \\int_0^{t'} V_I(t')V_I(t'') + \\dots$$ EQUATION 2: $$U(t,0)=1+\\sum_{n=1}^{∞}\\left( \\frac{-i}{\\hbar} \\right)^n\\int_0^t dt_1 \\int_0^{t_1} dt_2 \\dots \\int_0^{t_{n-1}}dt_n H(t_1)H(t_2)\\dots H(t_n)$$ **3\\. The attempt at a solution** So I know that for the interaction picture the transformation of the operator $\\hat{V}_I$ is.. $$\\hat{V}_I=e^{\\frac{i}{\\hbar}\\hat{H}_0 t} \\hat{V} e^{\\frac{-i}{\\hbar}\\hat{H}_0 t}$$ I also know that both operators and kets evolve in time. So I use the interaction picture equation of motion on the ladder operators so I can obtain an expression for them as a function of time. $$\\frac{d\\hat{a}}{dt}=\\frac{1}{i\\hbar}\\left[ \\hat{a},\\hbar \\omega \\left(\\hat{a}^{\\dagger}\\hat{a} + \\frac{1}{2} \\right) \\right]$$ $$\\frac{d\\hat{a}^{\\dagger}}{dt}=\\frac{1}{i\\hbar}\\left[ \\hat{a}^{\\dagger},\\hbar \\omega \\left( \\hat{a}^{\\dagger}\\hat{a} + \\frac{1}{2} \\right) \\right]$$ I then got.. $$\\hat{a}(t)=\\hat{a}(0)e^{-i\\omega t}$$ $$\\hat{a}^{\\dagger}(t)=\\hat{a}^{\\dagger}(0)e^{i\\omega t}$$ I plugged these into the expression for V to get, $$\\hat{V}=\\lambda \\left[ \\hat{a}^{\\dagger}(0)e^{i(\\Omega + \\omega)t} + \\hat{a}(0)e^{-i(\\Omega + \\omega)t} \\right]$$ So now what needs to be done, is to transform this into the interaction picture and then plug it into Equation 1 from above and integrate. But this seems very messy and I am having doubts if this is the correct way to I also know that both operators and kets evolve in time. So I use the interaction picture equation of motion on the ladder operators so I can obtain an expression for them as a function of time. go about this problem. If anyone can shed some light onto this I would really appreciate it!"}
{"id":"59696","title":"exponential potential quantization","text":"What are the energies $E_{n}$ of the Schroedinger operator $$ -\\frac{d^{2}}{dx^{2}}y(x)+ae^{bx}y(x)=E_{n}y(x) $$ for some real and positive 'a' and 'b' with the Boundary conditions $ y(0)=0=y(\\infty) $? Working a bit I get the quantization condition $$ J_{E_{n}\/4}(\\sqrt{-a})=0,$$ ($ J_{a}(x) $ is a Bessel function) but unfortunately this equation yields only imaginary roots."}
{"id":"20441","title":"Energy in an EM wave should depend on frequency","text":"I just finished reading Feynman's _Lectures on Physics_ vol.I, §34-9: \"The momentum of light\". The author explains that there is a relation between the wave 4-vector $k^{\\mu}$ and the energy-momentum 4-vector $p^{\\mu}$ of an EM wave, namely $$p^{\\mu}=\\hbar k^{\\mu}, $$ or equivalently $$\\tag{deB}W=\\hbar \\omega, \\mathbf{p}=\\hbar \\mathbf{k},$$ and those equations are called _de Broglie relations_. However, as I learned in my classical electromagnetism course, flux of energy in such a wave is quantified by Poynting's vector, yielding formulas such as the following: $$\\tag{1} I=\\frac{1}{2 \\mu_0 c} E_0^2, $$ where $I$ stands for \"average intensity\" of the wave and $E_0$ for \"maximum amplitude of electric field\". > **Question** Where is $\\omega$? It does not appear in formula (1) nor in any > other formula based on Poynting's vector. But as of equations (deB) it > should do so. Am I wrong? Thank you."}
{"id":"48110","title":"How can the surface of a liquid exist if there are no attracting molecules above it?","text":"Okay this is probably a stupid question but here goes: We know that molecules in a liquid are attracted from all the molecules around it so there is no net attraction. Well, then how do the molecules at the surface exist because there are only molecules below it and not above it? Won't they accelerate downwards? Also, does this have something to do with surface tension? If it does, then isn't surface tension a force that acts along the surface and not into or out of it?"}
{"id":"15208","title":"Radial Schrödinger equation","text":"I found a problem that says: > Show by direct substitution that $R_{10}$ is a solution of Schrödinger's > radial equation. AFAIK Schrödinger's radial equation is $\\frac{-\\hbar^2}{2m}[\\frac{d^2}{dr^2}+\\frac{2}{r}\\frac{d}{dr}-\\frac{l(l+1)}{r^2}]R+E_p(r)R=ER$ and $R_{10}$ in this case is: $2(\\frac{Z}{a_0})^{3\/2}e^{-\\rho\/2}$ where $\\rho$ is $\\rho = \\frac{2Zr}{na_0}$ and $E_p(r)$ is: $E_p(r) = -\\frac{Ze^2}{4\\pi\\epsilon_0r}$ The thing is I don't understand very well what $E$ is in this case and (I think) I need it to solve the problem. It's the total energy but in the book (Alonso y Finn) they never talk about the kinetic energy. I suppose it depends on $r$, $n$ and $l$ because once I do the substitution on the left side of the equation and I factorize it I get $R_{10}$ multiplied by a term that's full of those variables."}
{"id":"108933","title":"Formula for molar specific heat capacity in polytropic process","text":"I found this formula for a polytropic process, defined by $PV^n = {\\rm constant}$, in a book: $$C = \\frac R{\\gamma-1} + \\frac R{1-n} $$ where $C$ is molar specific heat and $\\gamma$ is adiabatic exponent. I do not know how it was derived, can someone guide me?"}
{"id":"16578","title":"How can you test to see if a dice is weighted?","text":"I was browsing Etsy today and came across this. What tests are there to see if the dice are usable, ie, if one side isn't favored over another, and if all sides are balanced? Would this just be to roll the dice a large nubmer of times and collect the data? Or are there other tests that could be done without doing that?"}
{"id":"16573","title":"What lepton neutrino would be in this reaction?","text":"If I have a reaction (shown below), and I'm supposed to work out what the products will be, what would the lepton neutrino $\\nu$ be? e.g electron neutrino: $$ \\nu + p \\to $$"}
{"id":"12668","title":"Converting Angular Momentum?","text":"I am aware of three apparently different forms of angular momentum: spin, orbital and macroscopic, like a spinning crystal or baseball. Can angular momentum be transferred or converted from one of these forms to another? If so, how?"}
{"id":"12669","title":"Are the rest masses of fundamental particles certainly constants?","text":"In particular I am curious if the values of the rest masses of the electron, up\/down quark, neutrino and the corresponding particles over the next two generations can be defined as constant or if there is an intrinsic uncertainty in the numbers. I suppose that since there is no (as of yet) mathematical theory that produces these masses we must instead rely on experimental results that will always be plagued by margins of error. But I wonder does it go deeper than this? In nature do such constant values exist or can they be \"smeared\" over a distribution like so many other observables prior to measurement? (Are we sampling a distribution albeit a very narrow one?) Does current theory say anything about this? (i\/e they must be constant with no wiggle room vs. no comment) I am somewhat familiar with on-shell and off-shell particles- but I must confess I'm not sure if this figures into my question. I'd like to say that as an example I'm talking about the rest mass of the electron as could be determined by charge and charge\/mass ratios. But perhaps this very number is itself influenced by off-shell electron masses? Perhaps this makes no sense. Needless to say I'd appreciate any clarification."}
{"id":"69495","title":"Source of potential energy in a capacitor being pulled apart","text":"When the plate spacing is large the capacitor system is storing more energy than when it is small. If one manually changes the spacing, where does the additional energy \"come from\" as the plate spacing increases?"}
{"id":"69492","title":"Can we see light as it 'interferes' with itself and produces the characteristic double-slit pattern?","text":"This TED talk suggests that we can now watch as a beam of light propagates through a bottle filled with water. My question is: can we use this new technology to perhaps 'see' the photon as it makes its way to the detector screen in the double slit experiment? What implications can femto-photography have on the single\/double-slit experiment and physics in general?"}
{"id":"69491","title":"Is a singularity a real thing?","text":"I've heard the work a few times now, the most recent in the star trek film. Is a singularity a real thing? If so what is it?"}
{"id":"103437","title":"Dispersion in a rectangular prism","text":"Why are the colours visible only at the fringes of the light refracted through a rectangular prism?"}
{"id":"63636","title":"Creating matrix Hamiltonian for Feynman's CCNOT","text":"I'm trying to read Quantum Mechanical Computer and to implement the CCNOT logical gate with Mathematica. Since i wish to use the SWITCH implementation of the CNOT [Fig.8] i've realized that i need to use 7 cubits (there's no difference with registry cubits and command line cubits). Also, i know that every operation is described by unitary matrix (we can only rotate ours cubit) i was thinking to use Pauli matrices, composed by tensorial product. As Feynman said, I'll obtain as results a $M=2^n * 2^n$. The vector will rapresent every possibile configuration of my system (it's composed by $2^n$) Also, using the observation that the upsite in the command line are invariant for my system (the expected falue of the number of the spinup commute with the hamiltonian of the spin) i can readuce the dimension of my matrix to $s*2^n$, and so the vector will be a unitary vector with only one $1$ in the corrispondence of the atom with the spin up. I have to create an algebra of operation that preserve the invariant of the clock, so not every possible operation is allowed on my chain (ex: NOT is not allowed because i will have 2 or 0 spin up that is an unacceptalbe substate). An example of a matrix that change my vector state from state $|1>$ to $|2>$ is the matrix $|2><1|$"}
{"id":"21064","title":"role of pure mathematics in fusion?","text":"Is there any way a pure math guy can get his rigorous math fix while simultaneously moving fusion research forward? Is fusion simply too complicated to take a pure math approach? What open math questions are there pertaining to fusion?"}
{"id":"21062","title":"Does inducing a current in a wire have any effect on the strength of a permanent magnet?","text":"Does inducing a current in a wire result in any changes in the strength of a permanent magnet? Specifically, what would the results of the following controlled experiment be? You set up two alternators that each simply consist of a permanent magnet located inside a stationary loop of wire as depicted here. With the first (control) alternator, you simply rotate the magnet at a certain velocity for a short time, once per day, and measure the amount of induced current on a daily basis. With the second alternator, you hook the magnet up to your favorite (external) source of rotational motion and spin it continuously and measure the amount of induced current at the same intervals as with the control alternator. Would the amount of current change over time (in either one)? If it would, in which direction and why? And if this is still not well-specified enough for an definite answer, what would it depend on? (how fast the magnets are rotating? what kind of circuit the wire is hooked up to? what the magnet is made out of? etc.)"}
{"id":"21060","title":"Riemann Tensor Calculation trick(number of element)","text":"When we calculate Riemann Tensor for different curvature we have lots of components. However, there are many components that are zero. How can we argue, based on the symmetry of connection , that those elements are zero? For example if I am calculating the Riemann Tensor of $S^2$ sphere, I get only one non zero component i.e. $R_{\\phi,r,\\phi}^{\\theta}$ = $sin^2 {\\theta}$ and other components are zero. So, How can I argue, without calculating that all other components are zero. Edit: (Dimension, No. of independent Riemann Components) = (2,1; 3,6 ; 4,20)"}
{"id":"81232","title":"The magnitude of spacecraft anomalous velocities?","text":"That Voyager and several other spacecraft have had anomalous changes in speed is known. According to the news reports, the differences are unsettling. But just how large are these anomalies really? Are they 'way down there' at the levels of computational precession? Or, if I understood how to calculate them, could I do so with my home computer in a reasonable period of time? (Say a few days calculation.)"}
{"id":"106733","title":"Why is the Legendre transformation an application of the duality relationship between points and lines?","text":"When I read the Wiki about Legendre transformation, there is a statement > The Legendre transformation is an application of the duality relationship > between points and lines. What's the meaning of this statement?"}
{"id":"63185","title":"Lethality of sounds and extreme \"loudness\"","text":"In theory, could pure sound be lethal? How loud would it have to be? Also, which events are the loudest in the universe, and how loud are they? I'm confining attention to events which occur regularly, e.g. starquakes."}
{"id":"45560","title":"Is there a mathematical derivation of potential energy that is *not* rooted in the conservation of energy?","text":"For simplicity I'll consider only gravity, but in general this question only applies to conservative forces. As per my understanding, the way one gets to the equation for gravitational potential energy is this: $(1)$ $\\ddot{r}=-\\frac{MG}{r^2}$ is given experimentally $(2)$ $KE=\\frac{1}{2}m\\dot{r}^2$ is derived theoretically and (thought) experimentally $(3)$ Energy is conserved $(4)$ For a set of generic initial conditions,   * Using $(1)$, find $r(t)$ and then derive $\\dot{r}(r)$   * Using $(2)$ find $KE(r)$   * Using $(3)$, $PE(r)=E_0-KE(r)$ **Is this a correct method?** If the answer is 'no' my question ends here, if not: surely $(3)$ is _by definition_ correct (circularly so), and so the equation for potential energy is arbitrary? I would be interested in seeing a method that does not do this (or is correct at all, if the answer _was_ 'no')."}
{"id":"45568","title":"Wire morphing shape?","text":"Can one morph a wire by electrical current into a shape different than static shape at beginning? As in the beginning shape being in a spiral, then when electrical energy is passed through the wire it becomes straight. When the current is shut off it would return to a spiral shape as in the beginning form. OR vise versa?"}
{"id":"54490","title":"Is it possible to calculate chromacity coordinates using absorbance values?","text":"I have absorbance values for blood plasma. Absorbance was measured in range of 380 – 600 nm every 5 nm ending up with absorbance values for 45 wave lengths. I was browsing through literature, and I came across chromacity coordinates (XYZ) and this seems to be a very convenient way of summarizing data about color in just two data points. I have been reading some literature about different colorimetric systems, but since my background is in genetics, it didn’t make much sense for me. Therefore, I would like to ask you if it is possible to calculate these chromacity coordinates using my absorbance values and if so does anyone now some kind of software for this or exact formulas on how to do it?"}
{"id":"1908","title":"Buckyballs in vacuum","text":"I've read about the idea that buckyballs and other nanostructures could be used to hold drugs and things until they reach certain places in the body and then get released. So I was wondering, if you created a buckyball in air, so that some molecules that are in air (such as oxygen and nitrogen) were inside the buckyball. if you then put the buckyball in a vacuum, would the air be able to escape, or would it be trapped in the buckyball. (And vice versa, a buckyball created in a vacuum put into the atmosphere, would any molecules fit through into the buckyball?)"}
{"id":"90090","title":"Between a solid and a hollow cylinders of the same mass, which one has the greater rotational kinetic energy?","text":"I know that rotational kinetic energy is $\\frac{1}{2}I\\omega^2$. Therefore, the rotational kinetic energy will depend on the moment of inertia. I came to the conclusion that since both have the same mass, both should have the same moment of inertia as   $$I = \\sum_\\text{over all mass} m r^2$$ But the answer says that since the hollow cylinder has greater moment of inertia, it has greater rotational kinetic energy. How can the hollow cylinder have greater moment of inertia even though the masses of both the bodies is same?"}
{"id":"45384","title":"Air density equation in stratosphere","text":"Does the density equation for air on Earth change in the outer atmospheres? I am generating a model for the Red Bull Stratos jump and am trying calculate a precise air density function."}
{"id":"45387","title":"Why can't missing mass be photons?","text":"After a star lives and dies, I assume virtually all of it's mass would be photons. If enough stars have already lived and died, couldn't there be enough photon energy out there to account for all the \"missing mass\" (=dark matter) in the universe? And if there were enough photons to account for all the missing mass, what would it look like to us?"}
{"id":"110246","title":"How to find the force of the compression at the core of a planet?","text":"For example the core of the planet Jupiter, how to find out the magnitude of the force that compresses its core? and can that be thought of as the total force that holds the planet together?"}
{"id":"29400","title":"Ground state degeneracy of a variation of Toric Code model","text":"We know that the ground state degeneracy of Toric Code model is 4. An easy way of seeing this is the following: Consider a 2D spin model where all the spins live on the links. The Hamiltonian is $H=-\\sum_v{A_v}-\\sum_p{B_p}$, where $A_v=\\sigma_{v1}^z\\sigma_{v2}^z\\sigma_{v3}^z\\sigma_{v4}^z$ is the product of 4 spins around a vertex $v$, and $B_p=\\sigma_{p1}^x\\sigma_{p2}^x\\sigma_{p3}^x\\sigma_{p4}^x$ is the product of 4 spins around a plaquette. All the $A_v$ and $B_p$'s commute with each other and they all square to one; thus each $A_v$ and $B_p$'s can take values $\\pm1$. To see the ground state degeneracy, we notice that $\\prod_vA_v=1$ and $\\prod_pB_p=1$, being two constraints on the $A_v$ and $B_p$'s. Thus different $A_v$ and $B_p$'s can now only label $2^{2N}\/4$ different configurations ($N$ being the total vertex number) because of the 2 constraints. But the total Hilbert space is still $2^{2N}$-dimensional, thus on average each configuration has a 4-fold degeneracy. Now suppose we add a third term in Toric Code model, making it $H=-\\sum_v{A_v}-\\sum_p{B_p}-\\sum_p{C_p}$, where $C_p=\\sigma_{p1}^y\\sigma_{p2}^y\\sigma_{p3}^y\\sigma_{p4}^y$ being the product of 4 $\\sigma_y$'s around a plaquette. What happens to the ground state degeneracy? Note that after adding the third term, the Hamiltonian is still a sum of commuting projectors, thus the counting method should still work; I'm just having trouble counting the number of constraints. My guess is that after adding the $C$ term, the ground state degeneracy reduces to 2. Anyone can prove\/disprove me?"}
{"id":"129028","title":"Do we hear sounds differently on the highest mountains?","text":"Some searching gives that above 6,000 meter altitude the air density is less than half of that at sea level. Speed of sound is about 15-20% slower and \"acoustic impedance\" seems to change too. Do humans notice these differences in sound? Does the pitch of tones change noticeable?"}
{"id":"44385","title":"Does quantum mechanics predict instantaneous action at a distance even without entanglement?","text":"The suggestion that quantum mechanics implies that instantaneous action at a distance occurs is normally based on the contention that this follows from the entanglement of particles that share a common origin. My question is: does quantum mechanics predict that this happens even when there is no entanglement? In particular, does this happen when two quantum systems interact in such a way that the wave function of at least one of them changes drastically? Take, for example, an interaction between a hydrogen atom and a hydrogen-oxide radical: H + HO = H2O. While the hydrogen atom is free, the wave function of its electron allows it to be anywhere in the universe. Or rather, it is everywhere in the universe. Instantly after the interaction has completed, the amplitude of the quantum wave of the same electron is smaller everywhere, except in the very immediate neighbourhood of the water molecule of which it has become a part. In the rest of the universe, the wave function has taken on other values, if not instantaneously – the reaction could take some time – then at least far quicker than light. One could object that the electron is never really at more than one point at any one time, only we don’t know which until we make a measurement. But that amounts to a refusal to accept quantum mechanics. The effect of a wave on its environment is determined by the totality of the wave, including even those places where its amplitude is partly or entirely imaginary. For example, the hydrogen atoms in a water molecule are positioned at an angle of 108 degrees from each other, as seen from the oxygen nucleus, due to some very particular properties of the wave functions which define the hydrogen-oxygen bonds. That results in water having some very peculiar properties, including the property that its solid form – ice – is less dense than the liquid form and therefore floats on it. Even the mere stability of water molecules is due to the wave function of the electrons which the hydrogen atoms share with the oxygen atom: in this wave function the electrons are as much as near the oxygen atom as the hydrogen atom."}
{"id":"40804","title":"Barrier in an infinite double well","text":"I am stuck on a QM homework problem. The setup is this: ![enter image description here](http:\/\/i.stack.imgur.com\/uRIu3.png) (To be clear, the potential in the left and rightmost regions is $0$ while the potential in the center region is $V_0$, and the wavefunction vanishes when $|x|>b+a\/2$.) I'm asked to write the Schrödinger equation for each region, find its solution, set up the BCs, and obtain the transcendental equations for the eigenvalues. **Where I'm at** : I understand the infinite potential well easily and I have done a free particle going over a finite barrier before (which I understood less well, but I can deal with it it).   * The problem asks me to make use of \"a symmetry\" in the problem, which is a vague hint. Are they trying to get me to make $\\psi$ an even function?   * I am supposed the condition for there to be one and only one bound state for $E<V_0$. How do I go about that?"}
{"id":"44380","title":"Problem with convergent geodesics at 2D sphere","text":"There is a chapter on general relativity in the book _Spacetime Physics Introduction To Special Relativity_ by Taylor and Wheeler, which qualitatively explains how attractive gravitational force can be understood in terms of space(time) curvature. They give (chapter 9.5, pp. 281-283 in 2nd edition) an example of two travelers on a 2d surface of sphere who start their motion some distance from each other at the equator straight north and move _along geodesic_ (great circle). Because of the surface curvature they _accelerate_ toward each other, and finally meet at the north pole (Fig. 9-4, attached). ![enter image description here](http:\/\/i.stack.imgur.com\/x2xDm.jpg) My problem begins once I (probably wrongly) extend the idea and imagine that the surface of the sphere represents the entire closed universe (in which case it would be a 3D sphere). In the case of flat Euclidean universe the two observers moving along straight lines are in inertial motion (they are in free fall though isotropic, homogeneous space) and their distance changes linearly with time, that is without accelaration. When the universe is closed and its curvature is positive and constant, they would observe acceleration although the universe is still homogeneous. What exactly is wrong in my speculation (I assume that in reality such two observers in Einstein 1917 spherical universe would not meet at a single point) and what makes the difference? The fact that real space has 3 dimensions, the fact that one has to consider 3D+1 space*time* (with Minkowski rather than Euclidean geometry in the flat case), the fact that Einstein universe is unstable, or something else?"}
{"id":"1279","title":"Why exactly does a boomerang return back to the thrower?","text":"I was always intrigued by the phenomena that govern the returning back of a boomerang to the thrower. Even if it is dependent on various factors such as velocity, aerodynamics, wind resistance and so on I was wondering, there must be some concise explanation to it?"}
{"id":"76692","title":"Is coherent light required for interference in Young's double slit experiment?","text":"In this Veritasium video, a home experiment is presented which appears to produce a very good double-slit interference pattern with normal sunlight. The experiment is an empty cardboard box with a visor and a placeholder for a microscope slide with two slits on one side. This is arranged with the slits and visor facing the Sun, so the interference forms on the bottom of the box. ![enter image description here](http:\/\/i.stack.imgur.com\/x2ssa.png) They claim to observe a good interference pattern from the two slits: ![enter image description here](http:\/\/i.stack.imgur.com\/4ZNX6.png) Discussions of interference in optics textbooks often stress that coherent light is needed to produce such patterns, and that sunlight and other thermal sources of light do not have such coherence. How, then, is this possible?"}
{"id":"116831","title":"Reasons for choosing $SU(3)$ as the color group vs. $SO(4)$","text":"What are the reasons that $SU(3)$ is used for QCD? Why wouldn't the simpler & smaller group $SO(4)$ make a better candidate?"}
{"id":"97836","title":"Neglecting mass at asymptotic spacelike momenta","text":"What is the rational\/reason for neglecting masses at asymptotic non- exceptional space-like momenta. I have come across this as a first fix for being able to extract information from the Callan-symanzik and Gell-Mann-Low methods. Is it purely physical, or is it both physical and mathematically sound reason? All responses are welcome."}
{"id":"95064","title":"Same partition functions, different theories","text":"I am reading the book \"Basic Concepts of String Theory\" by Blumenhagen, Lust and Theisen and in page 290 they say: \"It follows that the $E8\\times E8$ and the $SO(32)$ heterotic string theories have the same number of states at every mass level which are however differently organized under the internal gauge symmetries. So, even though the partition functions are identical, the theories are nevertheless different. The differences show up in correlation functions.\" I am trying to understand how these differences would show up in correlation functions, ie what would be an example of two correlation functions that are different in these two theories. Any references where this is done\/explained would be greatly appreciated!"}
{"id":"95061","title":"Influence of a Capacitance Matrix of a voltage measurment","text":"I am having a problem in understanding how in a multiconductor system the Capacitance Matrix influences the measure of voltage between two electrodes. In the case that interest me we have 4 electrodes, 2 emitting ones and 2 receiving ones. We transmit a known sinusoidal current into the the emitting electrodes and we measure a voltage on the receiving ones. We are looking for the phase and amplitude difference between the current emitted and the measured voltage. It is my understanding that the measured voltage is influenced by any conductor present in the medium close-by, therefore we have to determine the Capacitance Matrix to obtain the real voltage measured. $$ V_{real} = V_{measured} * C_{ij} $$ What I don't understand is the relation between the measured voltage ($V_{measured}$), the capacitance matrix ($C_{ij}$) and the real voltage ($V_{real}$). If any of you could help understand this I would be very grateful."}
{"id":"8425","title":"Laws of fluid flow in porous medium","text":"What are the equations that describe the flow of a fluid in a porous medium? Is there a variation of the Navier-Stokes equations? I would like to model the flow of air through a sponge-like structure. Instead of modelling the structure of the sponge, I would like to treat it like a homogenous area, where I solve appropriate equations for conservation of momentum, mass and energy."}
{"id":"38550","title":"is gravity always the weakest force","text":"> **Possible Duplicate:**   >  What does it mean to say \"Gravity is the weakest of the forces\"?   >  Strongest force in nature It is usually taught that the gravitational force is the weakest but I wanted to know if it will be always so. For example, On earth gravity is weaker than electromagnetic force or weak nuclear force, so is it that on black hole also if gravity is strongest than other forces shall be more stronger than it or it's that gravity becomes most strong suppressing others. Because if other forces increase in same proportion than it will have serious implications."}
{"id":"4243","title":"What does it mean to say \"Gravity is the weakest of the forces\"?","text":"I can understand that on small scales (within an atom\/molecule), the other forces are much stronger, but on larger scales, it seems that gravity is a far stronger force; e.g. planets are held to the sun by gravity. So what does it mean to say that \"gravity is the weakest of the forces\" when in some cases, it seems far stronger?"}
{"id":"128217","title":"Why do we say that the electric forces are stronger than gravitational forces?","text":"May be I am asking a stupid question but I am very confused...I understand both forces are inversely proportional to the distances squared, and that the gravitational constant is roughly $10^{-20}$ times the Coulomb's constant. But one involves charges, while the other involves mass. To me this makes as much sense as a saying a second is larger than a meter. Is it related to the energy densities of these fields? If yes, how? I think that it is not the duplicate of the mentioned one because the point asked in that question is why do we consider gravitational fields weaker even if they are more dominant in universe at macroscopic levels but I want to ask that on what criteria we define stronger or weaker fields....What do we exactly mean to say by declaring any one field stronger than the other ? Is it about their energy densities?"}
{"id":"53411","title":"Escape Velocity question","text":"I would like some help in understanding the derivation of the formula for escape velocity. Why does $t$ approach infinity? And why is $v(t)$ approaching zero? I know that $r(t)$ is infinite because the gravitational field is infinite. But I don't understand why $t$ (time) and velocity would be zero. I don't think I understand what $v(t)$ is. Is it velocity at a given point in time? And when time increases, you are farther away from the body producing the gravitational field and thus velocity needed for escape is lower? .... Here is a photo of the derivation: ![](http:\/\/i.stack.imgur.com\/KIH3X.png) From the wikipedia page: http:\/\/en.wikipedia.org\/wiki\/Escape_velocity"}
{"id":"111733","title":"Do Baryons have Quantized Vibrational and Rotational States?","text":"Given that baryons are not point particles but are composed of three quarks, are there quantized vibrational and rotational states analogous to those of molecules? If not why not, and if so are there spectroscopically observable transitions between such states?"}
{"id":"9830","title":"Tension in a curved charged wire (electrostatic force) - does wire thickness matter?","text":"Consider a conducting wire bent in a circle (alternatively, a perfectly smooth metal ring) with a positive (or negative) electric charge on it. Technically, this shape constitutes a torus. Assume that the radius of the wire is much smaller than the radius of the circle. **Find the tension in the wire due to the repulsive electrostatic force** (in Newtons). For this problem, here are example physical values, (may or may not be necessary to solve the problem) charge, $Q = 1 C$, circle radius, $R = 1 m$, wire radius, $r = 1 mm$. Use the most straightforward notations possible for Coulomb's law, $F=\\frac{k q_1 q_2}{r^2}$. This is a question about the mathematics of a set of inverse-square problems, not about theoretical E&M. Other notations that will be of useful to this discussion:   * $\\lambda = \\frac{Q }{ 2 \\pi R } $ = linear charge density    * $\\sigma = \\frac{\\lambda }{ 2 \\pi r }$ = surface charge density **What happens to the tension in the wire as the wire radius goes to zero** with a constant circle radius, assuming:   * A constant charge on the entire thing (also constant linear charge density)    * A constant surface charge density    * A constant voltage    * If it really bothers you, consider the tensile stress for the above as opposed to the wire tension, although my question is for the latter The following are some helping equations. Any of them could be wrong. Electric field from an infinite straight line charge $$F(r) = \\frac{ 2 \\lambda k }{ r }$$ Electric field face-on for a ring of charge, R is radius of ring and d is distance away $$F(d) = \\frac{2 \\pi \\lambda R k d }{ (d^2+r^2)^{3\/2} }$$ Electric potential in vicinity of ring of charge, $\\rho$ is distance from z-axis with torus in xy-plane, centered at origin. Electric field is simple knowing this.   * $l = \\sqrt{ (\\rho+R)^2 + z^2 }$ = distance to furthest point on ring    * $K(x) =$ Elliptic integral of the 2nd kind $$E(\\rho,z) = \\frac{ 4 R k \\lambda }{l} K\\left( \\frac{ 2 \\sqrt{ R \\rho } }{ l } \\right) $$ Now, I think that a valid approach would be to find the net effective field that a differential part of the ring experiences. _If_ you found this, I believe the tension could be written as follows, with T being tension, R the circle radius, and F the electric field. $$T = \\lambda R F$$ I have my own thoughts on the answer (and more equations than you would ever want), and I'll return to offer them later, but I certainly don't have a complete picture, and I want to see what approaches people here use. Maybe this could be answered very simply and maybe some people will know intuitively which cases limit to infinity."}
{"id":"9836","title":"How many boats does it take to find an acoustic buoy by Doppler shift?","text":"Inspired by this question on the Doppler shift, suppose there is buoy somewhere on the surface of the ocean emitting a pure frequency. You get to place some boats wherever you want on the surface of the ocean, moving whatever direction you want with whatever speed you want. The boats listen to the pure frequency, which will in general be Doppler shifted due to the relative motion between buoy and boat. If you do not know ahead of time the frequency the buoy is emitting, is it possible to deduce the buoy's location simply by observing the frequency measured by the boats? What is the minimum number of boats necessary, how should they be positioned, and what velocities should they have? This is a toy problem with the mathematics of the Doppler effect, so let's leave out attenuation of sound over distance and assume the surface of the ocean is a plane. Also, if the boats could continuously monitor the Doppler shift, they could collect extra information based on their own changes in position and velocity, so imagine locating the buoy based solely on a single frequency measurement from each boat."}
{"id":"9835","title":"Three-Dimensional Gravity","text":"Does anyone have any references that discuss gravity in three-dimensions? I'm trying to make my way through some papers by Witten relating $SL(2,\\mathbb{C})$ Chern-Simons theory and gravity in three spacetime dimensions and would like some explanation about the latter of the two topics. Thanks."}
{"id":"89869","title":"Is position fundamentally different from other observables?","text":"I apologize in advance for what might be a very naive question, and for its science-fictionesque flavor. It's still, I think, a real physics question. Suppose I have a (quantum) particle, whose state is represented by some element $x$ of some Hilbert space $H$. Given an observable $O$, I can express $x$ as a sum of eigenstates of $O$. For example, I could take $O$ to represent the particle's position --- but so far, at least, \"position\" --- as opposed to any other observable --- seems to play no special role. On the other hand, at the macro level, our brains seem to assign a very special role to position --- namely, when we're deciding whether to think of some collection of particles as an \"object\", one of our main criteria is (I think) whether these particles occupy nearby positions, and whether they continue to do so over time. Moreover, we ourselves are objects by this criterion. One can (perhaps barely) imagine some other sort of creature that perceives a collection of particles as an object when (and only when) those particles have, say, similar momenta (regardless of their positions) and continue to do so over time --- or similar values of some linear combination of position and momentum, or similar values of some other observable $O$ for which we don't even have a name. Presumably these creatures would themselves be collections of particles that are $O$-localized but not necessarily position-localized. But for us humans, position seems to play this unique role. **Question 1.** Is the unique role of position a matter of physics or of biology? In other words, is there anything in fundamental physics that rules out (or renders unlikely) the sort of $O$-localized (but not position- localized) creatures I'm imagining? **Question 1A.** Let me reword that. It seems clear that (spacetime) position plays a special role in relativity, which quite plausibly rules out my $O$-localized creatures. So let me ask more specifically whether there is anything in **quantum mechanics** (as opposed to all of physics) that explains these creatures' absence. **Question 2.** If the answer to Question 1A is yes, then how does one reconcile this with the fact that any basis of the Hilbert Space $H$ can be transformed via an isomorphism to any other basis, which seems to say, in essence, that there can be nothing special about the position basis, and hence nothing special about the position operator? **Question 3.** If relativity grants a special role to position and quantum mechanics does not, does that all by itself constitute a fundamental incompatibility between relativity and quantum mechanics? And can it be viewed as a significant failure of quantum mechanics that it cannot account for this aspect of the world? **Edited to add:** I'm glad for the answers I've gotten so far, but I infer from some of them that I need to be clearer about what I'm asking. The question is not just \"Why can't there be $O$-creatures?\". Instead, it is: \"If we have good reasons for thinking that $O$-creatures are unlikely, while knowing that position-creatures like us exist, how can we reconcile this with the fact that quantum mechanics sees no essential difference between $O$ and position? (Or am I wrong about the \"no essential difference\" part?)"}
{"id":"98361","title":"Is heat energy from electronics wasted in the winter?","text":"In the winter, are not all electronics heating the house - thus no energy is wasted? **Considering the energy I'm paying for:**   If I leave my computer running while I'm out, how much energy is actually wasted in the winter? Is all of the energy used in the computer expelled as heat into my home? To be even more specific:   **If my computer ran at a constant 500 watts and I had a 500 watt space heater, would they heat my home the same (thus cost the same)?** If so, wouldn't it make sense to leave the computer on in the winter, since the heat produced is not really wasted? That way I always have instant access. (Ignore variables like computer lifespans, gas furnaces, and cost- effectiveness for the sake of the question.)"}
{"id":"88780","title":"Is everything 100% efficient at heating?","text":"Everyone always talks about the efficiency of their appliances. I was wondering if everything was 100% efficient at heating its surroundings ?"}
{"id":"120172","title":"Are all means of heating equally efficient?","text":"In terms of cost of electricity, does it matter if I heat my room using a space heater, an open oven, a flood light, an ASIC? Assume no energy escapes the room, that the heat quickly becomes uniformly distributed, and ignore the fact that these produce heat at different rates. Consider only the amount of heat produced per unit of electricity."}
{"id":"26401","title":"Have scientists been able to find an example of a possible White Hole?","text":"We went to a planetarium last night last night and watched \"Black Holes\" narrated by Liam Neeson. If I recall correctly, he said that scientists believe that they have identified a number of Black Holes, and that they think there are White Holes at the bottom of Black Holes and instead of drawing matter\/light in, the White Holes spew matter out. I didn't get to ask this question after the session was over: Have scientists been able to find an example of a possible White Hole?"}
{"id":"52959","title":"Radiance equation","text":"I am trying to understand the equation of the radiance, but there is one thing i don't understand: $L_r = \\frac{d^2\\Phi}{d\\omega \\space dA\\cos{\\theta}}$ Why is that second exponent there in the numerator? Thanks ahead!"}
{"id":"55452","title":"Has hep-th\/0312070 forgotten to fix $s_{0} = 1\/2$ for the fermionic states in the second table on page 52?","text":"Link to the original paper: The Gauge\/String Correspondence Towards Realistic Gauge Theories (arXiv paper) On page 52 we see that, for a theory of Dp-branes placed at an orbifold (orbifold = $C_{2}$\/$Z2$) fixed point, we have a certain spectrum of open string NS and R states. The first thing to note is that the orbifold action has the effect of making the spinor labels behave as follows: $s_{3} = s_{4}$ or $s_{3} = -s_{4}$. This means the 3rd and 4th components are no longer independent. Since $s_{i}$ equals $1\/2$ or $-1\/2$, this means there are now 16 different combinations of the state $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$ corresponding to the different 16 permutations of the values of $s_{0}, s_{1}, s_{2},s_{3} = -1\/2,+1\/2$. We would have had 32 different permutations corresponding to $s_{0}, s_{1}, s_{2},s_{3}, s_{4} = -1\/2,+1\/2$ if $s_{4}$ wasn't fixed by the value of $s_{3}$. However, this isn't the end of it. We must also take a GSO projection which projects out half the states. This leaves 8 states (8 permutations) for each $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. So far so good; we now have 2 different types of Chan-Paton factor in front of our $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. Thus, we must consider two types of $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. This is why each row of the second table on page 52 has 2 states with different Pauli matrices in front. Each of these 2 states contributes 8 permutations so we have 16 fermions for each row of the table. But this is where I threw a spanner into the problem. For the _physical_ Ramond sector states, don't we normally fix $s_{0}$ to equal 1\/2. This is the case for flat spacetime and is explained on page 23 of Notes on D-branes. So now we have $s_{0}$ and $s_{4}$ fixed leaving permutations of $s_{1}, s_{2}, s_{3} = 1\/2, -1\/2$. This would give us 8 states per $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$ and after GSO projecting half the states out we'd be left with 4 per $|s_{0}, s_{1}, s_{2}, s_{3}, s_{4}>$. The result: I'd expect 8 fermions per row of the 2nd table on page 52 of our original source. My interpretation seems to be supported by page 30 of UV\/IR Mixing in Noncommutative Field Theories and Open Closed String Duality  which says we have 4 copies of each spinor. Note: It ignores $s_{0}$. I'm guessing they do this as it is fixed by the physicality condition. Sorry for the length of my description and thanks in advance."}
{"id":"98375","title":"What really makes a rainbow happen?","text":"Yes, a rainbow is formed when a raindrop refracts light, but why then does the whole sky not become a huge rainbow when it rains? Would the light not be dispersed into ordinary 'white' light? What forces it to look as if each end is nearly touching the ground?"}
{"id":"92921","title":"Is Information a potential or kinetic kind of energy?","text":"It is said that _the law of least action is that nature tries to convert potential energy into kinetic one as fast as possible._ Information can't be thought without a physical realisation, see here. It may be thought as a form of entropy. So information is physical and therefore it's connected to a certain kind of energy. But is it a potential or a kinetic kind of energy?"}
{"id":"92926","title":"Ionizing an atom with a proton","text":"What happens when a proton collide with an atom? How does the ionization occurs, assuming the proton has enough kinetic energy? How much energy is required for the proton to enter the atomic nucleus, resulting a higher chemical element ion?"}
{"id":"93038","title":"The inertia tensor is \"global\", how do I make it local?","text":"A solid cuboid has the following moment of inertia: $$I= \\begin{pmatrix} \\frac{1}{12}h^2+d^2 & 0 & 0 \\\\\\ 0 & \\frac{1}{12}w^2+d^2 & 0 \\\\\\ 0 & 0 & \\frac{1}{12}w^2+h^2 \\end{pmatrix} $$ When integrating the torque to the angular velocity, as far as I know, this holds: $$v_a=\\int I^{-1} T dt$$ Where $T$ is the torque and $t$ is the time. However, this seems to only hold when the orientation is the identity orientation. How do I extend this onto an arbitrary orientation? Do I need to make the inertia tensor local in some way? Note that the angular velocity $v_a$ is \"global\" (!)"}
{"id":"55983","title":"Dealing with experimental data","text":"I have some experimental data about a value $n$, now, I am supposed to give, in the ending, a single value with an error: $n=a\\pm b $. I have originally 6 values of $n$, each one comes as an indirect measurement from direct measurement, each one with it's systematic errors, so in the ending I have those 6 values, each one with an error. So what I guess I have to do is to mix the systematic error with the random error the way I've been taught $(E_{sys}^2+E_{rand}^2)^{1\/2}$. The systematic is already calculated, what do I use for the error? the mean of systematic errors? Another question is that those values, which are by the way moles of a quantity of a gas, have been got from different ways (basically from calculating different isothermic curves and getting the $n$ value that best fits each of them), so they're actually not from the same kind of measurement, but from different ones. This makes me doubt about how this would affect the calculation of the final error, if it does, or if I can just do it the way I said above."}
{"id":"51205","title":"Explanation of energy levels in molecules, atoms, nuclei and their relationship","text":"Why are the energy levels of molecules, the atoms that form them and the nuclei inside the atoms considered separately? Or phrased in a different way- what is it that makes their energy levels so different? What is the relationship between the energy levels of molecules, atoms and nuclei? I'd interested to learn different interpretations on this question."}
{"id":"51201","title":"Is 't Hooft's Determinism based on the holographic principle?","text":"Does 't Hooft's determinism work need the holographic principle in order to work or is it just an extension of his work?"}
{"id":"116722","title":"What are torque parameters?","text":"Torque is just the turning force on an object, i.e the force multiplied by the perpendicular distance from the force to the pivot. But when torque parameters are mentioned in mechanical engineering papers, what are they referring to? It's the plural \"parameters\" thats confusing me. Is it the collection of properties such as maximum torque, starting torque, etc? If so, can somebody please provide me with a list of the all, or some of the \"torque parameters\", along with what each means? Finally, if looking at the torque parameters of a crankshaft, what would be the important ones to pay attention to, and what are some typical values?"}
{"id":"82246","title":"How to write a noisy state as a separable operator","text":"Bipartite operators close enough to the identity are separable. But how does one compute the product operator terms of the separable expansion? In particular, if $\\left| \\Phi \\right> = m^{-1\/2} \\sum_{i=1}^n \\left| i \\right> \\otimes \\left| i \\right>$ is the $n$-dimensional Bell state then $I \\otimes I - \\left| \\Phi \\right>\\left< \\Phi \\right|$ is separable. But how do I write this as a sum of positive product operators $\\sum A_i \\otimes B_i$?"}
{"id":"65341","title":"fast quantum random number generator + limited decoherence rate => Schrödinger cat state?","text":"Suppose that fast quantum random number generator (QRNG, https:\/\/qrng.physik.hu-berlin.de\/) is placed in a subsystem which has limited interaction with the rest of the world. What would happen if decoherence rate for this subsystem were not fast enough to extract all the random numbers being created by QRNG? It seems that then the subsystem has to evolve into a state with macroscopic coherent properties (Schrödinger cat state). At least this occurs if one simulate QRNG using measurement of a rotated qubit and copying a result in a vector of qubits representing a (partly) closed subsystem. If so, why the question about feasibility of macroscopic coherent states is still not settled?"}
{"id":"65340","title":"Does the slip-stick phenomenon have any application?","text":"The slip-stick phenomenon is present all around us, be it the noise of car breaks or in earthquakes. But does it have any real-life application?"}
{"id":"111525","title":"Direct rocky planet formation without protostellar disk?","text":"Inspired by this question, which sort of asked what I'm asking, but emphasized a different point (and the answers accordingly addressed that). Can a rocky planet form directly from the collapse of an interstellar gas cloud? The picture I have of rocky planet formation is that dust and rocky material in a protostellar disk accumulates into a rocky planet. There must be some mechanism to prevent the accretion of significant amounts of (hydrogen\/helium) gas, I'm guessing this is something to do with the gas either being unable to cool, reheated if it does cool, or outright removed from some region around the protostar as the stellar wind and\/or radiation pressure kick in. My intuition is that even the most metal enriched (astronomer metal, i.e. everything heavier than $\\rm{He}$) cloud still by far hydrogen dominated, so that a cloud in isolation must _always_ form a primary star\/brown dwarf\/gas giant. But has anyone ever heard of a mechanism where a cloud in the right environment (some radiation field that efficiently heats the gas but not the dust\/metals perhaps?) could form a rocky body directly, with no associated star? This seems far fetched, but I'm curious to know whether anyone has ever looked at this; literature references appreciated."}
{"id":"2597","title":"Energy conservation in General Relativity","text":"I understand that energy conservation is not a rule in general relativity, but I'd like to know under what circumstances it can still be possible. In other words, when is it possible to associate a potential energy to the gravitational field, so that the energy is constant in the evolution of the system? Here are some examples, is there a convenient way to define energy in these scenarios?   * Just a system of gravitational waves.   * A point mass moving in a static (but otherwise arbitrary) space-time. Equivalent (if I'm not mistaken) to a test mass moving in the field of a second much larger mass, the larger mass wouldn't move.   * Two rotating bodies of similar mass. Overall, I'm trying to understand what keeps us from associating a potential energy to the metric. When we break the time translation symmetry of a system by introducing an electromagnetic field, we can still conserve energy by defining an electromagnetic potential energy. **Why can't we do the same when we break TT symmetry by making space-time curved?**"}
{"id":"134901","title":"Calculating coefficient of friction","text":"> Consider a body attached to a horizontal spring and resting on a surface, > inclined at an angle $\\theta$ from the ground.![Set- > up](http:\/\/i.stack.imgur.com\/F2G97.png) > > The spring constant is $k$. Initially the spring was kept in its natural > length while the body was held still by some external agent. When the > external agent was removed, the body slid $x$ units down the inclined plane > to achieve equilibrium.The coeffecient of (kinetic) frictional force acting > on the body is $\\mu$ To solve this problem, I can use two methods: 1.Work-Energy Theorem: This yields $$0=mg(x\\sin\\theta)-\\frac12kx^2-\\mu (mgx\\cos\\theta)$$ 2.Equating forces at equilibrium: This yields $$kx+\\mu mg\\cos\\theta- mg\\sin\\theta=0$$(along the inclined plane) On solving the equations the two methods give different answers, while they should give the same. Is there anything I am missing out here? Please help me solve this problem."}
{"id":"134900","title":"What are Killing spinors?","text":"What are Killing spinors? How can they be motivated? Are they directly related to Killing vectors and Killing tensors and is there an overarching motivation for all three objects? Any answer is greatly appreciated but a less formal one would be preferred."}
{"id":"2598","title":"Discussion of the Rovelli's paper on the black hole entropy in Loop Quantum Gravity","text":"In a recent discussion about black holes, space_cadet provided me with the following paper of Rovelli: Black Hole Entropy from Loop Quantum Gravity which claims to derive the Bekenstein-Hawking formula for the entropy of the black hole. Parts of his derivation seem strange to me, so I hope someone will able to clarify them. All of the computation hangs on the notion of _distinguishable_ (by an outside observer) states. It's not clear to me how does one decide which of the states are distinguishable and which are not. Indeed, Rovelli mentions a different paper that assumes different condition and derives an incorrect formula. It seems to me that the concept of Rovelli's distinctness was arrived at either accidentally or a posteriori to derive the correct entropy formula. > Is the concept of distinguishable states discussed somewhere more carefully? After this assumption is taken, the argument proceeds to count number of ordered partitions of a given number (representing the area of the black hole) and this can easily be seen exponential by combinatorial arguments, leading to the proportionality of the area and entropy. But it turns out that the constant of proportionality is wrong (roughly 12 times smaller than the correct B-H constant). Rovelli says that this is because number of issues were not addressed. The correct computation of area would also need to take the effect of nodes intersecting the horizon. It's not clear to me that addressing this would not spoils the proportionality even further (instead of correcting it). > Has a more proper derivation of the black hole entropy been carried out?"}
{"id":"134906","title":"Passing a Car With known speeds vs. Time","text":"While driving, it takes 22.0 s to pass a car. If my speed was 64.0 mph when I started to pass, and was 80.0 mph as I finished, how far (in feet) did I travel while passing?"}
{"id":"30805","title":"What am I doing wrong finding the eigenvectors of this rotation matrix?","text":"This question is from Principles of Quantum Mechanics by R. Shankar. > Given the operator (matrix) $\\Omega$ with eigenvalues $e^{i\\theta}$ and > $e^{-i\\theta}$ , I am told to find the corresponding eigenvectors. I think these equations are relevant: $$\\begin{align}\\Omega &= \\begin{bmatrix}\\cos{\\theta} & \\sin{\\theta} \\\\\\ -\\sin{\\theta} & \\cos{\\theta} \\end{bmatrix} \\\\\\ \\Omega \\begin{bmatrix} x_1 \\\\\\ x_2 \\end{bmatrix} &= \\begin{bmatrix} x_1 \\cos{\\theta} + x_2 \\sin{\\theta} \\\\\\ -x_1 \\sin{\\theta} + x_2 \\cos{\\theta} \\end{bmatrix} \\\\\\ e^{i\\theta} \\begin{bmatrix} x_1 \\\\\\ x_2 \\end{bmatrix} &= \\begin{bmatrix} x_1 \\cos{\\theta} + x_1 i\\sin{\\theta} \\\\\\ x_2 \\cos{\\theta} + x_2 i\\sin{\\theta} \\end{bmatrix}\\end{align}$$ I let the matrix operate on the generic vector $(x_1, x_2)^T$ and demand that the resulting vector is equal to $(e^{i\\theta}x_1, e^{i\\theta}x_2)^T$ . From this i get the condition that $x_2 = ix_2$ and $x_1 = -ix_2$ , which implies that $x_1 = x_2 = 0$ . Am i missing something crucial?"}
{"id":"2625","title":"Difference between thermodynamics and statistical mechanics","text":"I've just finished a class a few weeks ago which taught thermo and stat mech, and I still don't know the exact difference between the two. Can someone help clear this up for me? (Yeah it's sad, and even sadder that I got an A- in the course)"}
{"id":"20945","title":"Negative energies and a partition function","text":"I'm writing down the partition function for a system, for which I know the dispersion relation $$E \\left( \\mathbf{k} \\right) = \\sqrt{ \\left| \\mathbf{k} \\right|^2 + m^2 + \\cdots }$$ The exact form is not important, what matters is that technically, as the dispersion relation is the solution of a 2nd degree polynomial equation, there should be a $\\pm$ sign before the square root. When I write the partition function I get: $$ Z = \\sum_{\\mathbf{k}} \\exp \\left( - \\frac{E \\left( \\mathbf{k} \\right)}{\\beta} \\right) $$ but if I wanted to include the negative energies too, I would get: $$ Z = \\sum_{\\mathbf{k}} \\exp \\left( - \\frac{E \\left( \\mathbf{k} \\right)}{\\beta} + \\frac{E \\left( \\mathbf{k} \\right)}{\\beta} \\right) = \\sum_{\\mathbf{k}} 1$$ which is clearly absurd: my system has dynamics! ;-) Now my question is: are the negative really unphysical? Wouldn't it be more correct to keep track of the two-branch dispersion relation with something along these lines: $$ Z = \\sum_{\\mathbf{k}} \\exp \\left( - 2 \\frac{E \\left( \\mathbf{k} \\right)}{\\beta} \\right) $$"}
{"id":"9582","title":"Phenomena which are incorrectly declared as resonance phenomena?","text":"In standard college physics text books, high-school books and popular level physics books, the collapse of the Tacoma Narrows Bridge is often taken as an example of _resonance_. However a more detailed analysis shows that the collapse was not due to simple resonance (see the article by Billah and Scanlan and Wikipedia). Now my question is, whether there are further examples from mechanics, which are often incorrectly or oversimplified declared as examples of resonance phenomena (for example in physics text books)? **Edit:** I should add, that the definition usually given in textbooks is the following: Consider a mechanical system which can be described by a differential equation like $\\frac{d^2}{d t^2}x(t) + 2 \\rho \\frac{d}{d t} x(t) + \\omega_0^2 x(t) = f(t) $ where $f(t) = f_0 \\cos(\\omega t)$. Then for example the amplitude or the energy of the system as function of $\\omega$ has a maximum near $\\omega_0$. In this case one speaks of _resonance_ of the system."}
{"id":"9584","title":"stability of hypothetical lunar atmosphere","text":"Assume that by some means, the moon could be given an atmosphere, of the same density and pressure at the surface as the earth's. Obviously in a stable atmosphere there are temperature variations from pole to equator, day to night, and a lot of interesting dynamics follow from that, but for argument's sake assume an average of something like 0 to 5 C. How long would it last? Clearly there's no air there now (unless you count traces of potassium, or whatever it has), so such at artificial atmosphere would not be permanent, but presumably the moon could hang onto some air, for some time. With the moon's lower gravity, the density or pressure vs altitude curve will be different, and I'd guess that a significantly larger fraction of the mass would be at higher altitudes than on earth, and there's no protection from the solar wind. Someone with the right equations could probably work out how long it would take the pressure at the surface to drop by half."}
{"id":"107552","title":"Why are cgs units the norm in astrophysics?","text":"Other physics communities, e.g. the particle physics one, have their own set of units, custom-tailored to their own needs. Now, the astrophysics community is somewhat similar, in that a lot of measurements are often expressed in terms of corresponding numbers related to the Sun (solar mass, solar luminosity, solar radius, et cetera). However, when it comes to using down and dirty numbers, I've seen cgs units used way more often than SI units. Why is that? Is it just a matter of tradition or is there something more subtle at work?"}
{"id":"100604","title":"Observable which dependes on the cutoff","text":"In arXiv:0710.4330v1 Balitsky calculate the eikonal scattering of dipole composed of quark anti-quark, $Tr(U_{x}U^{\\dagger}_{y})$, to NLO accuracy. The result he found is: ![enter image description here](http:\/\/i.stack.imgur.com\/EGe1e.png) Where $\\mu$ is the normalization point in the MS bar scheme. I have already read the discussion Why do we expect our theories to be independent of cutoffs? and understood the observable cannot depends on the cutoff. While $Tr(U_{x}U^{\\dagger}_{y})$ is observable, taking the derivative according to $\\eta$ should produce an equation which is independent of the cutoff (at least naively). Why is it still there? any idea?"}
{"id":"52872","title":"Calculating (Glass) thermal conductivity","text":"I read that glass has a thermal coefficient of 0.8-1 W\/mK. Given that my window window thickness is around $5\\ mm$, then I would calculate my heat loss per area being $160-200\\ Wm^{-2}K^{-1}$. Yet standard glass has a U value of $5.6\\ Wm^{-2}K^{-1}$. What is going on here? I'm missing something huge..."}
{"id":"102009","title":"Question about units of mass, $M = (L^{3})(T^{-2})$?","text":"In section 5 of the \"Preliminary: On the measurement of quantities\" chapter (page 3) in \"A treatise on electricity and magnetism\" Maxwell uses, total length, $s=mt^{2}\/{2r^{2}}$to show that $m=2sr^{2}\/{t^2}$ is in units of Length cubed over Time squared. This is the first I have encountered this way of looking at mass and am wondering:   1. Is this just per-relativity nonsense?   2. Does mass being \"length cubed over time squared\" have any meaning?"}
{"id":"9059","title":"jumping into water","text":"Two questions:   1. Assuming you dive head first or fall straight with your legs first, what is the maximal height you can jump into water from and not get hurt?    In other words, an H meter fall into water is equivalent to how many meters concrete-pavement fall, force wise? (I'm assuming the damage caused will be mainly due to amount of force and not the duration)   2. Assume you jump head first and hold a sharp and strong long object that cuts the water before you arrive, will that make the entrance to the water more smooth and protect you?![enter image description here](http:\/\/i.stack.imgur.com\/RGzj6.png)"}
{"id":"67412","title":"What is the theoretical limit for farthest we can see back in time and distance?","text":"13.2 billion years ago the universe was rather small, having started only half a billion years ago. Today, with the help of Hubble Space Telescope, we are able to capture the light of galaxies emitted at that time. The point at which Earth exists now must have been quite close to those galaxies back then. If so, why is it that it is only now, after 13.2 billion years later the light from those galaxies has reached us? Or in other words, are we sure that the light we are seeing from those galaxies indeed travelled for 13.2 billion years? It looks like as if there was a race between our point running away from those galaxies (with the expansion of universe and space) and the light that was emitted at that time. And only now that light has reached and overtaken us. But if that is so, then wouldn't it put a limit on the oldest light we can see, no matter how powerful the telescope is (even it is more powerful than the James Webb Space Telescope)? This should be expected, because at the time just after the Big Bang, the light emitted by all objects must have already overtaken all other objects, including the location of earth. Therefore we will never see the light that old (close to the time of Big Bang) no matter how powerful the telescope. If this is so, what is the theoretical limit we can see far back in the past?"}
{"id":"22896","title":"Why is compressible flow near the choke point so efficient?","text":"Imagine a steady state, one-dimensional, compressible flow in a horizontal pipe of constant cross sectional area. This flow can be isothermal, adiabatic (Fanno), or diabatic (Rayleigh). As an example, the relevant macroscopic energy balance for Fanno flow is: $$\\Delta h=-\\Delta (KE)$$   In other words, Fanno flow converts enthalpy into kinetic energy. Below are Fanno lines for a real gas (steam) for different flow rates at the same pipe ID and inlet conditions. ![enter image description here](http:\/\/i.stack.imgur.com\/IFw8A.jpg) The lines clearly indicate the \"efficiency\" with which enthalpy is converted to kinetic energy. At low velocity (left upper portion of lines), almost no enthalpy is converted while the entropy generation is large (the line is nearly horizontal). Near Mach 1 (the rightmost maximum entropy point for each curve) the enthalpy conversion is high and entropy generation is nearly zero (the line is nearly vertical). This happens whether you are originally subsonic (the upper branch) or supersonic (the lower branch). Another way of saying no entropy generation occurs is that the viscous losses (friction) are zero. This would imply that the molecules no longer impart (as much?) force upon each other near the choke point. This seems absurd. In isothermal flow, you also find that as you near the choke point (interestingly, below Mach 1!), the flow approaches frictionless. As I understand it, these examples are physically realizable flow regimes (if obviously very short in length scale). Why is compressible flow so efficient near the choke point? Is something at the molecular scale really going on or is it simply a poor mathematical model at these velocities? UPDATE: Obviously, in the supersonic regime (lower branch), the reverse process takes place, i.e. kinetic energy is converted into enthalpy. Still, my question holds since this conversion is nearly perfect as you approach Mach 1."}
{"id":"68194","title":"Toward the establishment of non-equilibrium (quasi-equilibrium) magnon BEC theory","text":"In 2006, Demokritov et al have reported that they have achieved the observation of quasi-equilibrium magnon Bose-Einstein condensation (BEC) in YIG at finite (room) temperature by using the method called \"microwave pumping\" http:\/\/www.nature.com\/nature\/journal\/v443\/n7110\/abs\/nature05117.html. I know there are various (sometimes controversial) discussion on their experiments and the understanding of the microscopic mechanism; about spontaneous coherence (spontaneous symmetry breaking), spontaneous condensation[A] or driven condensation, thermodynamics properties, the novel dispersion relation of YIG, and thermalization effects. From the view point of theoretical issues (to the best of my knowledge), Prof. Bunkov and Volovik http:\/\/arxiv.org\/abs\/1003.4889 has generalized (extended) the idea of magnon condensation and have proposed a non-equilibrium magnon BEC theory[B], in which they have viewed the magnon annihilation operator associating with (you call) the off-diagonal long-range order, $a_0=\\sqrt{N_0} \\exp(i\\mu t-i\\alpha)$, as the macroscopic condensate order paramater. Taking the experimental result by Demokritov et al into account, we can easily expect that the magnitude of the condensate order parameter decreases due to thermalization effects[C] (i.e. The dissipation into environments, phonons) and it takes the following form; $a_0= \\sqrt{N_0} \\exp(i\\mu t-i\\alpha -t\/\\tau)$, [D] where the variable $\\tau$ reflects the dissipation. Thus although we can understand an aspect of the above experiment by Demokritov et al, I wonder whether we can regard it as Quasi-equilibrium or non-equilibrium magnon BEC (until $t<\\tau$) also in the case where the magnitude of the condensate order paramater is time-dependent; $a_0= \\sqrt{N_0} \\exp(i\\mu t-i\\alpha-t\/\\tau)$ ? [A] In the standard or traditional equilibrium magnon BEC theory treating only equilibrium situations, the idea of \"the spontaneous $U(1)$-symmetry breaking\" plays the key role. [B] Magnon BEC corresponds to coherent precessions in the language of original spins. [C] By tracing out the enviromental degrees of freedoms, the time-development of the system we focus on becomes a non-unitary form. As an eaxample, please see the famous Caldeira-Leggett model or quantum master equation. [D] When we neglect the thermalization effects or the dissipation, we can easily show and see that the magnitude of the macroscopic condensation order parameter takes a time-independent constant value after switching off microwaves because the $U(1)$-symmetry of the system is recovered. I guess although this theoretically means the stable magnon BEC, it does not reflect the experimental result by Demokritov et al."}
{"id":"113716","title":"Photobombing without app used by Harley Davidson","text":"Here we see a photobombing trick, used by Harley Davidson. They achieve to incrust a logo on a photo you take. The logo is invisible with naked eyes and only revealed on the photo you take. How does it work?"}
{"id":"71845","title":"Is a single particle transition allowed between two superconductors having Majorana modes?","text":"Sorry maybe I ask a very naive and stupid question. The motivation to ask it is as follows: I am considering the parity protection of Majorana qubit based on Kitaev's 1D model. As we know, in the Kitaev's chain model there are two free MFs on the end of the chain and the information in these two free MFs is protected by the parity of the superconductor substrate under the chains. My question is as follows: > How about the parity protection if we consider a Kitaev junction (A nanowire > hang on two superconductors and the middle of the nanowire is break up by a > gate voltage)? > > There are four MFs in the junction (from left to right named as 1,2,3,4), of > course the combined parity of 12+34 is protected if we ignore the > quasiparticle poisoning from environment, but how about the parity > protection if we only consider 23? > > Can 12 and 34 also have parity protection respectively?"}
{"id":"71840","title":"Where does the change in energy come from when trapping a photon between mirrors?","text":"You have a photon traveling with E=hf and you trap it between two perfectly reflecting mirrors (like a QM particle in a box). The photon has to make a standing wave between the mirrors and its spacial frequency is dependant on the distance between the mirrors, L. Its time frequency, and hence energy are derived from the spacial frequency; Therefore E is dependant on L. Where does the change in energy come from?"}
{"id":"109703","title":"Why does the heaviest point on a spinning ball tend to become the topmost pole?","text":"I have a mostly hollow, small clear plastic ball. Inside the ball is a weight stuck to the edge but the weight is smaller than the ball so the ball will always come to rest with the weight at the bottom when placed on a smooth surface. I've been spinning it on my desk with the weight at the top, it spins well despite it being top heavy, I assume like a gyroscope. If I spin it with the weight at the bottom however, it loses stability and it very quickly slows down until it regains stability with the weight at the top again, with further experimentation I found that wherever the weight is, once the ball starts spinning it stabilises with the weight at the top. What causes the rotational energy to transform into gravitational potential energy? Any why is the ball more stable with the weight at the top axis than with it at the bottom axis?"}
{"id":"30122","title":"Do wormhole exits require matter? If not, would they change the edge of the universe?","text":"As a layperson who watches way too many theoretical physics and astronomy documentaries, I've seen the folded-piece-of-paper analogy of wormholes a bajillion times. Usually they explain that black holes might connect to other parts of the universe, but they don't really explain what would have to be on the other end for any kind of information to travel through it. When they do, they usually suggest white holes -- but I'm not entirely convinced. I don't have too much trouble understanding how a black hole forms, and to some extent, how that could lead to a distortion of space-time sufficient to causing the \"fold\". But it's hard to imagine what would cause a white hole to form. Even if there were such thing as negative mass, there doesn't seem to be any way that an object can come to have a negative density sufficient to becoming a white hole, so there couldn't possibly be a process that causes them to form. So if black holes could be wormholes at all, it doesn't seem to be the case that the exit would require matter to be present before it forms. If anything, a black hole wormhole would have to \"perforate\" a white hole or some other kind of exit into existence -- somewhere. But there doesn't seem to be anything that determines where that exit would form, so one could exist outside the boundaries of the matter that defines the edges of the universe. That would mean that the universe is infinite in size, since white holes could (and probably would, if their position were completely random) appear very, very far away. So then either black holes can't be wormholes, the universe is infinite in size, or I'm missing something. Neither of the former two conclusions have been reached to the best of my knowledge -- so what am I missing?"}
{"id":"114852","title":"The position of a particle at any time $t$ is given by $S = V0\/a [1-e^{-at}]$. What are the dimensions of $a$ and $V_0$?","text":"To find the dimensions of and V0, I must know the dimension of S and e. So I want to know it."}
{"id":"37846","title":"Find Trace of Dirac Matrix","text":"The matrices present in the Dirac equation must have the following properties:   1. $\\\\{a^j,a^k\\\\}_{ab} = 2\\delta^{jk}\\delta_{ab}$      2. $\\\\{a^j,\\beta\\\\} = 0$      3. $(\\beta^2)_{ab} = \\delta_{ab}$ How can one show from this that the trace of $\\beta$ must be $0$?"}
{"id":"37844","title":"What was the first discovery of the delta baryon $\\Delta^{++}$?","text":"The delta baryons (also called delta resonances) are a family of subatomic hadron particles which have the symbols $\\Delta^{++}$, $\\Delta^{+}$, $\\Delta^{0}$, and $\\Delta^{−}$ and electric charges +2, +1, 0 and −1 elementary charge respectively.   1. i want first reference claims delta baryon discovery   2. why the mass of all delta particles are the same!?   3. proton and neutron quarks content are the same with $\\Delta^{+}$, $\\Delta^{0}$, So why they have so different mass!?"}
{"id":"79336","title":"How can electrostatic forces provide lift to the gossamer spider?","text":"I was reading this article entitled \"“Ballooning” Spiders Use Electrostatic Forces To Generate Lift\". What I would like to know is how can a electrostatically charged object generate lift within the atmosphere? If I have a charged piece of paper or silk, I don't notice that it can lift off my hand and fly up 100's of meters, yet a spider can do so to the hight of 4 kilometres according to the article. Clearly my silk or paper is much heavier, but therefore, shouldn't I be able to construct a suitably charged object which could do the same as the spider up, lift up to 4 kilometers? If we assume that spider is using electrostatic charges solely as the means of lift, how can it work? How could we develop a model which can do the same thing?"}
{"id":"79334","title":"Are the perpetual machines in this video really perpetual?","text":"This video lists a number of perpetual motion machines that have been proposed throughout history:   * Robert Boyle's Self-Flowing Flask   * A \"perpetual train\" using rolling cones   * F. G. Woodward's hoop wheel machine   * Jacob Leupold’s Overbalanced Wheel   * Bob Schadewald’s Wheel   * Villard's wheel These are all mechanical machines, which use gravity to operate, i.e, they are a way to transform the force of gravity into energy, and do not need batteries or motors. Since I think that those machines are not really perpetual motion ones, can they turn for hundreds of years or will it stop just after some minutes?"}
{"id":"79330","title":"Comparing sensitivity of flat spiral coils with regular solenoids","text":"I'm attempting to compare the sensitivity that can be gained by using flat spiral coils against regular solenoid coils. A nice paper here gives reasonably accurate expressions for the inductance of such coils. More precisely, for a given solenoid of N turns and area A, we can write (note that $d\/dt \\rightarrow j\\omega\\times$ when taking Fourier transforms) $V_{induced}(\\omega) = j\\omega NAB$ and we can define sensitivity as $\\left|\\frac{V_{induced}}{\\omega B}\\right| = NA$ to indicate just how much amplitude can be picked up by the coil at a given frequency. This is a good way of comparing solenoid-type coils. For the spiral coil, how would you define an equivalent sensitivity metric? I suppose what this really means is finding the expression for the flux of such a coil, or somehow using its inductance to define pickup sensitivity."}
{"id":"132417","title":"Kirchhoff current loop in Resistor Diode Ladder network","text":"I am looking for an approach on how to apply Kirchhoff current \/ voltage law in the infinitely long diode ladder network. Can anyone help me with this ? I am looking for 1D differential equation or an implicit equation.![enter image description here](http:\/\/i.stack.imgur.com\/qMNhd.jpg)"}
{"id":"132416","title":"QED Laser Reflection Mirror Expirment","text":"Feynman in his famous lectures in NZ, talked about an experiment with a reflection of a light from a \"back\" mirror with stripes on it I tried to find a video of such an experiment, but with no success. Can someone help me find such a video? or something very related to it that delivers the same point?"}
{"id":"132419","title":"Chern-Simons on a lattice and the framing anomaly","text":"Can someone make or refer me to the argument for why $U(1)$ Chern-Simons theory in three dimensions cannot be defined by a lattice action? (Unlike Dijkgraaf-Witten theories, which are defined on the lattice.) Possibly related: what is the \"framing anomaly\"?"}
{"id":"91599","title":"Red Shift Effects","text":"I dong know much about physics, but I was curious about the red shift effect. I think I have the basic idea of it down in that as an object 1 accelerates from object 2 the light from object 1 will appear more red to object 2 because it stretches the light between them. My question is how fast is the red shift effect? If object 1 and 2 are viewing each other and are stationary with say one light year between them and object 2 suddenly accelerates away from object 1, how soon could object 1 see the light from object 2 shift red? My thinking was that if there was no lag that you could potentially use it to transmit information over vast distances with a sort of red shift mores code."}
{"id":"91598","title":"Lorentz force helix","text":"I have a few questions about the Lorentz force.   1. Is it possible to create a Lorentz force helix in a cloud chamber and if so how would this be achieved?   2. How would the pressure inside the helix be measured in comparison to the pressure surrounding the helix?   3. Is there a limit on how much pressure the helix could produce? Ultimately what I am asking is, whether it is possible to create mini tornadoes and for those tornadoes to support a payload. Thank You"}
{"id":"103664","title":"What is the physical significance of the Dipole Transformation of Maxwell's Equations?","text":"# The Question Given Maxwell's equations of the form \\begin{align} \\bar{\\nabla}\\times \\bar{B} = \\dfrac{4\\pi}{c} \\bar{J} \\+ \\partial_0 \\bar{E} \\\\\\ \\bar{\\nabla}\\times \\bar{E} = -\\partial_0 \\bar{B} \\\\\\ \\bar{\\nabla} \\cdot \\bar{B}=0 \\\\\\ \\bar{\\nabla} \\cdot \\bar{E} = 4 \\pi \\rho, \\end{align} what is the physical significance of the following transformation of Maxwell's equations: \\begin{align} -i \\bar{\\nabla} \\times \\bar{G} \\+ \\bar{\\nabla} G_0 = \\dfrac{4\\pi}{c} \\bar{R} \\+ \\partial_{0}\\bar{G} \\tag{Amp-Far Dipole} \\\\\\ \\bar{\\nabla} \\cdot \\bar{G} \\- \\partial_{0}G_0 = 4\\pi R_0 , \\tag{Gauss Dipole} \\end{align} where $\\bar{G} =(i\\bar{r} \\times\\bar{F}-x_0 \\bar{F}) $, $G_0 = (\\bar{r} \\cdot \\bar{F})$, $\\bar{R} = (\\rho c \\bar{r}-x_0\\bar{J}+i \\bar{r} \\times\\bar{J})$, $R_0 = (\\bar{r}\\cdot\\bar{J} - x_0\\rho c)\/c$ and $\\bar{F} = \\bar{E} - i \\bar{B}$. # The Definitions In this post, I refer to (Amp-Far Dipole) and (Gauss Dipole) as the dipole equations, because I do not know what there actual names are or who first published these equations. I just stumbled upon them by accident on pencil and paper. $x_0 = ct$ is the time variable multiplied by the speed of light for condensed notation purposes. $\\bar{R}$ is the complex combination of the electric dipole field density $\\rho c \\bar{r}-x_0\\bar{J}$ and the magnetic dipole field density $\\bar{r} \\times\\bar{J}$. $\\bar{R}$ is interpreted as a fictitious current in the dipole equations (Amp- Far Dipole). The corresponding fictitious charge density of $\\bar{R}$ is $R_0$, which is equal to the Minkowski inner product of the four-position and the four-current. # Physical Consequences Though $R_0$ and $\\bar{R}$ are fictitious charge and current, they are conserved as a current when $G_0 = 0$. This implies that $G_0$ breaks charge conservation of the fictitious charge and current $R_0$ and $\\bar{R}$. An interesting consequence of the dipole equations is they are identical to Maxwell's equations when $G_0 = 0$. # Complex Formulation of Maxwell's Equations I first write Ampere's law, Faraday's law and Gauss' law in complex form \\begin{align} -i\\bar{\\nabla} \\times \\bar{F} = \\dfrac{4\\pi}{c} \\bar{J} \\+ \\partial_{0} \\bar{F} \\tag{Amp-Far} \\\\\\ \\bar{\\nabla} \\cdot \\bar{F} = 4\\pi \\rho \\tag{Gauss} , \\end{align} where $\\bar{F} = \\bar{E} + i \\bar{B}$. # Formulation of Ampere-Faraday Law in Dipole form I use the following differential vector calculus identity \\begin{align} \\bar{r} \\times (\\bar{\\nabla} \\times) \\+ \\bar{r} (\\bar{\\nabla} \\cdot) \\+ x_0(\\partial_{0}) = \\bar{\\nabla} \\times (\\bar{r} \\times) \\+ \\bar{\\nabla} (\\bar{r} \\cdot) \\+ \\partial_{0}(x_0) \\end{align} to transform (Amp-Far) into the following: \\begin{align} \\bar{r} \\times (\\bar{\\nabla} \\times \\bar{F}) \\+ \\bar{r} (\\bar{\\nabla} \\cdot \\bar{F}) \\+ x_0(\\partial_{0} \\bar{F}) =\\\\\\ \\bar{r} \\times \\left( i\\dfrac{4\\pi}{c} \\bar{J} \\+ i\\partial_{0} \\bar{F} \\right) \\+ \\bar{r} \\left(4\\pi \\rho\\right) \\+ x_0\\left(-i\\bar{\\nabla} \\times \\bar{F} \\- \\dfrac{4\\pi}{c} \\bar{J}\\right) =\\\\\\ \\dfrac{4\\pi}{c} (i\\bar{r} \\times\\bar{J}) \\+ \\partial_{0} (i\\bar{r} \\times\\bar{F}) \\+ 4\\pi \\left( \\rho \\bar{r} \\right) \\- i\\bar{\\nabla} \\times (x_0\\bar{F}) \\- \\dfrac{4\\pi}{c} (x_0\\bar{J}) =\\\\\\ \\bar{\\nabla} \\times (\\bar{r} \\times \\bar{F}) \\+ \\bar{\\nabla} (\\bar{r} \\cdot \\bar{F}) \\+ \\partial_{0}(x_0 \\bar{F}) , \\end{align} which reduces to the following expression \\begin{align} -i \\bar{\\nabla} \\times \\left( i\\bar{r} \\times \\bar{F} \\- x_0\\bar{F} \\right) \\+ \\bar{\\nabla} (\\bar{r} \\cdot \\bar{F}) =\\\\\\ \\dfrac{4\\pi}{c} \\left( \\rho c \\bar{r} \\- x_0\\bar{J} \\+ i \\bar{r} \\times\\bar{J} \\right) \\+ \\partial_{0} \\left( i\\bar{r} \\times\\bar{F} \\- x_0 \\bar{F} \\right) . \\end{align} One can perform the following substitutions $\\bar{G} =(i\\bar{r} \\times\\bar{F}-x_0 \\bar{F}) $, $G_0 = (\\bar{r} \\cdot \\bar{F})$, and $\\bar{R} = (\\rho c \\bar{r}-x_0\\bar{J}+i \\bar{r} \\times\\bar{J})$ to obtain \\begin{align} -i \\bar{\\nabla} \\times \\bar{G} \\+ \\bar{\\nabla} G_0 = \\dfrac{4\\pi}{c} \\bar{R} \\+ \\partial_{0}\\bar{G} . \\tag{Amp-Far Dipole} \\end{align} # Formulation of Gauss' Law in Dipole form I use the following differential vector calculus identity \\begin{align} -x_0 (\\nabla\\cdot) \\+ \\bar{r}\\cdot (-i\\bar{\\nabla}\\times) = \\bar{\\nabla} \\cdot (i(\\bar{r}\\times)- x_0) \\end{align} to transform (Gauss) into the following: \\begin{align} -x_0 (\\nabla\\cdot\\bar{F}) \\+ \\bar{r}\\cdot (-i\\bar{\\nabla}\\times\\bar{F}) =\\\\\\ -x_0 (4\\pi \\rho) \\+ \\bar{r}\\cdot \\left(\\dfrac{4\\pi}{c} \\bar{J} \\+ \\partial_{0} \\bar{F}\\right) =\\\\\\ \\dfrac{4\\pi}{c} (\\bar{r}\\cdot\\bar{J} - x_0\\rho c) \\+ \\partial_{0} (\\bar{r}\\cdot\\bar{F}) =\\\\\\ \\bar{\\nabla} \\cdot (i\\bar{r}\\times\\bar{F} - x_0\\bar{F}) , \\end{align} which reduces to the following expression \\begin{align} \\bar{\\nabla} \\cdot (i\\bar{r}\\times\\bar{F} - x_0\\bar{F}) \\- \\partial_{0} (\\bar{r}\\cdot\\bar{F}) = \\dfrac{4\\pi}{c} (\\bar{r}\\cdot\\bar{J} - x_0\\rho c) . \\end{align} One can perform the following substitutions $R_0 = (\\bar{r}\\cdot\\bar{J} - x_0\\rho c)\/c$ to obtain \\begin{align} \\bar{\\nabla} \\cdot \\bar{G} \\- \\partial_{0}G_0 = 4\\pi R_0 . \\tag{Gauss Dipole} \\end{align}"}
{"id":"91593","title":"Understanding the difference between co- and contra-variant vectors","text":"I am looking at the 4-vector treatment of special relativity, but I have had no formal training in Tensor algebra and thus am having difficulty understanding some of the concepts which appear. One such concept is the notion of contra- and co-variant vectors. Wikipedia describes a contra-variant vector as varying with the co-ordinates, i.e. the inverse of the reference axes. It gives the example of a change of basis matrix $\\mathbf{M}$, and a contra-variant vector $\\vec{v}$ and states that in the new set of co-ordinates, we have: $$\\vec{v}'=\\mathbf{M}\\vec{v}$$ It then states that a covariant vector varies with the co-ordinate axes. > Does this mean that for a covariant vector $\\vec{u}$ and change of basis > matrix $\\mathbf{M}$ we have: $$\\vec{u}'=\\mathbf{M}^{-1}\\vec{u}?$$ I am also having problems understanding the concept of raising and lowering indices. For instance, Wikipedia states that for two four-vectors $\\vec{A},\\vec{B}$ and using Einstein summation we have: $$\\left\\langle \\vec{A} \\middle|\\vec{B} \\right\\rangle=A^{\\mu}B_{\\mu}=A_{\\nu}B^{\\nu}$$ And I am having difficulty understanding how this relates to the other definition of the inner product: $$\\left\\langle \\vec{A} \\middle|\\vec{B} \\right\\rangle =\\mathbf{A}^{T}\\mathbf{\\eta}\\mathbf{B}$$ If anyone could help give me a more intuitive understanding of these concepts I would be grateful."}
{"id":"91592","title":"Experimental information: is momentum conserved?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/yGSEt.jpg) Could I say that it is conserved during the time, or instead there is a slight decrease?"}
{"id":"25811","title":"Are there any Earth-Meteorites on Earth?","text":"A follow-on from this question, if meteorites can come from Mars, it seems they could have come from Earth and ended up back here (though I'm guessing that to be less likely). Do we have any evidence of this having happened? Would there be a way of verifying that they left and came back, other than direct observation of them falling from space?"}
{"id":"67470","title":"Why doesn't Ehrenfest's theorem work for a particle in a an infinite square well?","text":"I'm reading Griffith's Intro to Quantum Mechanics, and he mentions that in an infinite potential well, a classical particle would simply bounce back and forth between the two walls indefinitely. He then proceeds to calculate the situation for a quantum particle, and gets a whole bunch of _stationary_ state solutions. My question is: Using Ehrenfest's theorem, can't we immediately say that the expectation values would mimic the classical particle and bounce back and forth as well? I know that this would give us the wrong answer(i.e. a non- stationary solution), but I'd like to know why this reasoning is wrong."}
{"id":"130811","title":"Is enthalpy also valid for liquids and solids?","text":"The definition of enthalpy is $$H=E+PV,$$ is it only valid for (ideal) gas? As this naturally comes from the first law of thermodynamics, $$dE=dQ+PdV$$ for a enclosed system of gas. Can we talk about the enthalpy of (perfect) fluid and even solid (the pressure P seems uneasy to specify.)?"}
{"id":"72760","title":"Would a small puddle of water evaporate faster if you spread a dry towel over it?","text":"Let's say you spill 10ml of water on the kitchen counter. It forms a small puddle that would evaporate after a while (assuming room temperature and sane humidity). Would spreading a large, dry towel over the puddle cause the water to evaporate quicker? What I mean is a towel that's much larger than the diameter of the puddle. Say 4x as large. On the one hand, the water by itself comes into direct contact with the air which causes it to evaporate. On the other hand, assuming the towel absorbs the water almost entirely, it increases the wet surface area significantly, but also \"locks\" away some of the water in the fabric where it has less contact with air."}
{"id":"105847","title":"Why is earth on an axial tilt of 23.4 degrees?","text":"What were the influences for this to occur? Did some things play a bigger part that others?"}
{"id":"77191","title":"Explaining Newton's Laws of motion to a 6 year old","text":"An old professor of mine once said that an effective means to get people interested in Physics is to get them started early. What would be an effective and meaningful (and fun) means to explain Newton's 3 laws of motion to a 6 year old?"}
{"id":"77196","title":"Weak force: attractive or repulsive?","text":"We are always told that there are the four fundamental forces or interactions of nature: gravitation, electromagnetism, and the weak and strong forces. We know that gravitation is attractive, that electromagnetism can be attractive or repulsive depending on the electric charge of the interacting particles, and that the strong force is attractive between quarks. But when the weak force is mentioned, the description is always something such as 'responsible for radioactive decay', but there is no mention of whether this force is attractive or repulsive. So my question is: is the weak force\/interaction attractive or repulsive?"}
{"id":"6675","title":"How to write units?","text":"In a scientific article, should I write \"3m\", \"3 m\", \"3 meters\", or \"3 [meters]\" ?"}
{"id":"6674","title":"How long would it take for electricity to flow from one terminal to other, via a 1 LY long wire?","text":"Basically, how long does it take for electricity to determine there is a closed circuit and how does it know that the circuit exists? I'm curious to know how it knows there is a closed circuit at any length **Edit:** For clarification of the thought experiment, picture an electrical source of some type with a 1 light year wire coming from both terminals and connected in the middle. The wires are of zero resistance and capacitance. If electrons were to start flowing immediately then information (one end of the circuit knowing the circuit is now closed at the opposite end) has obviously traveled faster than light, which is impossible. **Edit:** It has been pointed out that at least some capacitance is required. So if required, consider that as a possibility in the hypothetical setup. I'm not quite sure how that affects this situation due to my lack of knowledge. I'm trying to not get to bogged down in the setup and trying to focus more on \"How does the circuit know its complete & how long does it take to determine it (particularly when the distance is long)\""}
{"id":"130694","title":"Energy of gravitation","text":"EDIT: As some confusion has appeared, I want to make another clear question. If gravitational energy is meaningless in general relativity (since it is the geometry), how can one come up with the graviton, the quanta of the gravitational field? I would like to understand the concept of gravitation's energy (and maybe momentum) in term of Eintein's General Relativity and its non-quantum extension. Since the action of matter and metric is: $$S=\\int d^4x(\\sqrt{-g}R+\\mathcal{L}_{matter})$$ it's understood that the energy momentum of matter is $$T_{\\mu\\nu}\\sim\\frac{\\delta(\\sqrt{-g}\\mathcal{L}_\\text{matter})}{\\delta g^{\\mu\\nu}}$$ while $$G_{\\mu\\nu}\\sim\\frac{\\delta(\\sqrt{-g}R)}{\\delta g^{\\mu\\nu}}$$ where $G_{\\mu\\nu}$ denotes Einstein's tensor. How can we come up with a energy term of the metric?"}
{"id":"92027","title":"Harmonic oscillator modified by infinite well: are analytic solutions possible?","text":"I'm trying to find solutions to a harmonic oscillator that sits within an infinite square well. I haven't spent too much time yet, and I've had no success so far. I'm wondering how possible or complex an analytic solution would be? The potential of a simple harmonic oscillator is: $$V_1(x)=1\/2 m \\omega_0^2 \\, x^2$$ For simplicity, let us set $\\omega_0=1$ and work in unitless time. The potential for the infinite well $V_2(x)$ is infinity outside of the box, $|x|>L\/2$ and zero just about everywhere else. The potential for the modified problem is: $$V(x)=V_1+V_2 \\,=\\, 1\/2 m x^2 + V_2(x)$$ I would like to find the energy eigenstates of this modified problem, and the eigenvalues. What does your intuition say about how energy eigenvalues of the SHO would be affected by an added infinite well (assume a well width $L$ much larger than the \"wavelength\" of the ground state), and how does this compare to the actual analytical or a numerical solution?"}
{"id":"121316","title":"Masses and Pulley","text":"I have a two mass and a pulley system as shown here: ![enter image description here](http:\/\/i.stack.imgur.com\/leuTC.jpg) I have the following two questions: First, why does $\\dfrac{d^2}{dt^2} y_p$ is not zero, i I mean I don't understand why does $ y_p$ is going to change over time (it's a constant, right?). Second, and why does $\\dfrac{d^2}{dt^2} y_p = A$? (A is defined as the acceleration upward of the pulley)"}
{"id":"121311","title":"Confused about SI and CGS Units For E\/M Energy","text":"I've been looking at some papers on wakefields of electron beams and I am a little confuded about the units. They do not specify whether or not they are in cgs or SI units. Here is a sample equation: ![enter image description here](http:\/\/i.stack.imgur.com\/CbEpg.png) I am assuming that these are in cgs, am I correct? What units should electron charge and radius be in? What units is the change in energy in? Thank you!"}
{"id":"81251","title":"How much energy does it take to simply run forward?","text":"I'm interested in tracking as much data about my runs as I can in an effort to get faster, and while I can easily estimate energy expenditure during an uphill run due to the change in elevation, I can't estimate energy expenditure due to the requirements of just keeping in motion. I could estimate wind resistance, creating a simple cylindrical model and estimating drag, but obviously it would take energy to move forward on a flat surface in a vacuum as well. Where would I start for this? What would be my variables? I know for a bicycle it's pretty simple to account for rolling friction, but I'm not sure what I'd need to account for. The question: What makes running so much less energy- efficient than bicycling? seems to downplay the importance of vertical motion in energy dissipation. I guess to further boil down the question I'd ask: **where does the energy go when you run?**"}
{"id":"77739","title":"Commutation of abstract $O(3)$ generators and vectors","text":"I've been given the following problem, and I'm quite lost with it. Let $L_1$, $L_2$, and $L_3$ denote the abstract o(3) algebras. You are given that $\\vec{A} = (A_1, A_2, A_3)$ and $\\vec{B} = (B_1, B_2, B_3)$ transform as vector operators of o(3). Show that $[L_j, \\vec{A} \\cdot \\vec{B}] = 0$ I know that $L_j = \\varepsilon_{jlm} q_l p_m$, and I can obviously determine the dot product, but I'm not sure where to go from there. I do, however, know that $\\vec{A} = \\frac{1}{Ze^{2}\\mu}(\\vec{L} \\times \\vec{p}) + (\\frac{1}{r})\\vec{r}$, but I'm not sure how to integrate that into this problem."}
{"id":"2006","title":"Dark energy converted to electricity","text":"> **Possible Duplicate:**   >  Dark Energy saving the world In theory, dark energy be used to generate electricity. Given its apparent abundance, this could provide rather a lot (perhaps too much) clean energy. Has any research been conducted on this or any other practical applications for dark energy? (other than repelling galaxies ;)"}
{"id":"128330","title":"Equation describing the motion of an object attracted to magnet","text":"Imagine there is a strong magnet at `(0, 0)`. There is a horizontal rail on `y=1` on which a ferromagnetic object rides. The object can only move on the rail. No friction anywhere. The object starts at `(0, 1)` with some positive velocity. How will the graph of its movement look like?"}
{"id":"128337","title":"Phase space derivation of quantum harmonic oscillator partition function","text":"I would like to derive the partition function for the quantum Harmonic oscillator from scratch: $$\\tag{1} Z = \\int dp \\, dx\\, e^{-\\beta H}.$$ The free particle appears in many textbooks. $H = p^2$ so it is a Gaussian integral $$\\tag{2} Z = \\int dp \\, dx \\, e^{-\\beta p^2} = L \\int dp \\, e^{-\\beta p^2} = L \\sqrt{2\\pi\/\\beta}.$$ I wanted to do the same calculation for the Harmonic oscillator I get could take advantage that $$\\tag{3} Z = \\int dp \\, dx \\, e^{-\\beta (p^2 + x^2)} = \\int dx \\, e^{-\\beta x^2} \\cdot \\int dp \\, e^{-\\beta p^2 } = \\sqrt{2\\pi} \\cdot \\sqrt{2\\pi} = 2\\pi .$$ Or I could integrate over the circles $$\\tag{4} H = p^2 + x^2 = E$$ take advantage the Hamiltonian is symmetric on rotation in phase space $$\\tag{5} Z = \\int dp \\, dx \\, e^{-\\beta (p^2 + x^2)} = \\int dE \\, d\\theta \\, E \\, e^{-\\beta E} = 2\\pi. $$ It seems I have forgotten that energy levels are quantized, so I should integrate over the circles $$\\tag{6} H = E_n = \\hbar \\omega (n + \\tfrac{1}{2}).$$ * * * **Where are the $E_n$ wavefunctions $\\psi_n$ located in phase space?** By Heisenberg uncertainty, we can't specify both $(p,x)$ in the phase plane. Are they equidistributed on the circle $p^2 + x^2 = E$?"}
{"id":"75772","title":"Minimal Extension of Wave Equation to Include Dispersion","text":"Let's say you are modeling some process with the wave equation $\\frac{1}{c^{2}}\\frac{\\partial^{2}\\psi}{\\partial t^{2}} = \\nabla^{2}\\psi$. You wish to improve your model by including dispersive effects, but you want your model to be as simple as possible for computational tractability. What is the minimal appropriate model for phase velocity, say $c \\approx c_{0} + c_{2}\\omega^{2}$? and how should the wave equation be altered?"}
{"id":"14525","title":"Trying to understand the weak gravitational field metric (1)","text":"In Carroll's online Lecture Notes on General Relativity (p112) he says, \"the weakness of the gravitational field allows us to decompose the metric into the Minkowski form plus a small perturbation\" ![enter image description here](http:\/\/i.stack.imgur.com\/mnlOe.gif) He then says that, to first order in h ![enter image description here](http:\/\/i.stack.imgur.com\/PjVle.gif) How does that happen? I've tried juggling round with various inverse metrics but I just can't see where that minus sign comes from. Nice and simple please. Thank you."}
{"id":"134247","title":"Why is water vapor released when an empty water bottle is squeezed?","text":"I recently came across this tutorial and wondered why vapor will come out when the cap comes off. Heres the Tutorial ![enter image description here](http:\/\/i.stack.imgur.com\/Pwbg1.jpg)"}
{"id":"134242","title":"How is weight distributed when legs are astride?","text":"Is there a simple and quick formula to find the weight (the force acting) on each leg, especially when legs are astride and they are not equally stretched? ![enter image description here](http:\/\/i.stack.imgur.com\/lUesD.jpg) (or, better) not equally distant from the center of mass? Can you give a brief description of the principle and formulas involved? I couldn't find an appropriate picture, but suppose this man here below brings his back foot\/leg forward nearer the Center of Mass. ![enter image description here](http:\/\/i.stack.imgur.com\/25USG.jpg) Supposing he weighs 80Kg and is tall 170 cm, given that in a male the CoM is near the sternum, if the feet are equidistant from CM we would have two triangles   * 1-2) from CM to foot 1m, angle:45°, height (to CM) 71 cm, base 71 cm In this position can intuitively say that the weight is equally distributed (40-40) on the legs. (or are there other relevant factors to be considered?). But, if   * 2) the rear leg is moved further forward, then the triangles will not be equal anymore, how do you find out:   * the distribution of thw weight on each leg?   * the horizontal component of the force that must be counterbalanced by friction?   * can we apply the parallelogram to each triangle here? _Is it possible to find just the **percentage of the weight** distribution without examining the whole problem and doing a lot of operations?_ If you do not wish do use numbers just explain the principles. > You write down the equation that the total forces on all parts of the feet > from the floor are equal to the weight of the person; and the total torque > r⃗ i×F⃗ i relatively to the person's center of mass is zero Is this the _only or the simplest way to do it_? I asked if there is a **simple, quick formula** to get the weight on each leg, can you find one that i so simple that you can post it in a comment?"}
{"id":"5114","title":"Gravitational and gauge-gravitational anomalies in N=1 D=4 supergravity coupled to a SUSY gauge theory with chiral matter","text":"When people talk about the first superstring revolution they often mention the miraculous cancellation of anomalies via the Green Schwarz mechanism. My question is whether such a string-theoretic mechanism is also at work when the 4D gravitational and gauge-gravitational anomalies are tackled? In this context, would it be fair to say that a possible discovery of superpartners at the LHC, which automatically implies some version of N=1 D=4 supergravity, imply that stringy couplings (higher order in alpha') must be present in the corresponding lagrangian to cancel the anomalies? What type of coupling are those?"}
{"id":"72866","title":"Can flux pinning be used for propulsion in Earth's magnetic field?","text":"I was wondering if the following steps would result in a net displacement on a spacecraft operating in Earth's magnetic field:   1. Turn on a superconductor   2. Use a linear actuator to shift some mass   3. Turn off the superconductor   4. Shift the mass back to its original position   5. Repeat Would flux pinning allow pushing or pulling against the field?"}
{"id":"33330","title":"Matrix Representations of Quantum States and Hamiltonians","text":"I am a high school student trying to teach himself quantum mechanics just for fun, and I am a bit confused. As a fun test of my programming\/quantum mechanics skill, I decided to create a computer program to model an atom. Initially, I learned about the matrix representations of quantum states (the first example I came across was that of a polarized photon, with the base states [0, 1] and [1,0]). This was all fine and dandy, but then I learned about the Schrodinger Equation, which replaces these matrices with functions. The way I understood it, both matrices and functions are just concrete representations of an element in a vector space; is this correct? It is much easier to work with matrices in code, so how do I transform operators and functions (Hamiltonian operator and state functions, specifically) into their corresponding matrix forms?"}
{"id":"72645","title":"Why terminal velocity exists?","text":"Why the acceleration of free-falling bodies become zero after some time? My only idea would be that the closer to the core of Earth we are, the value of \"g\" the lower becomes (since the mass below the body is lower and consequently, its gravitational force is also lower) but it works on large scales, like hundreds and thousands of kilometres. (and also, I'm not sure in it) I don't count with air resistance, since an experiment with checking terminal velocity would work in vacuum too. EDIT: I thought about enery aspects too, but obviously a body that falls with constantly increasing speed wouldn't be a perpetuum mobile, because once it's gonna reach with the larger body, and then speed becomes zero."}
{"id":"62988","title":"$U(N)$ gauged quantum mechanics","text":"I'm studying the $U(N)$ gauge theory theory in 0+1 dimensions. The aim is to show that this is equivalent to a matrix model. Is there any literature on this topic? The action I am interested in is $$ S_E = \\int d\\tau \\text{Tr} \\left[ \\left( D_\\tau \\Phi \\right)^\\dagger \\left( D_\\tau \\Phi \\right) + m^2 \\Phi^\\dagger \\Phi\\right] $$ where $\\Phi$ is an adjoint matter field and $S_E$ is the Euclidean action. Also $$ D_\\tau \\Phi = \\partial_\\tau \\Phi + \\left[A_\\tau, \\Phi\\right] $$ We gauge fix using $\\partial_\\tau A_\\tau = 0$. I want to be able to describe this system in terms of a matrix model $U \\in U(N)$."}
{"id":"44149","title":"If something weighs 25 kg, how do I find the mass of the object?","text":"> An object is falling and it weighs 25 kg (on a scale, presumably). What is > its mass? I know that weight is measured in Newtons and mass in kilograms, but what if a problem states that something weighs in kilograms? Would I still use $\\text w=mg$?"}
{"id":"44143","title":"What does the velocity of a wave mean?","text":"I know that the velocity of a wave is given by $v=\\lambda f$ but what does this velocity represent in the physical sense. For instance, if I am told a car moves at a velocity of 5 $m\/s$ I know that the car itself will cover 5 meters in displacement every second. What part of the wave is moving at velocity $v$?"}
{"id":"133733","title":"Why group velocity represents energy or information transmission? What relation between phase velocity and special relativity","text":"I have two question,   1. why group velocity represents energy or information transmission?   2. what is the relation between phase velocity and Special relativity: **why can it exceed C without violation of SR**? I didn't take any classes in special relativity or information theory or quantum mechanics, my background only contains   1. general physics (basic knowledge about Special relativity)   2. electromagnetism   3. signals and systems (basic knowledge about signal transmission), so please reply in a simple and easy way."}
{"id":"41662","title":"Why does no physical energy-momentum tensor exist for the gravitational field?","text":"Starting with the Einstein-Hilbert Lagrangian $$ L_{EH} = -\\frac{1}{2}(R + 2\\Lambda)$$ one can formally calculate a gravitational energy-momentum tensor $$ T_{EH}^{\\mu\\nu} = -2 \\frac{\\delta L_{EH}}{\\delta g_{\\mu\\nu}}$$ leading to $$ T_{EH}^{\\mu\\nu} = -G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu} = -(R_{\\mu\\nu} - \\frac{1}{2}g_{\\mu\\nu}R) + \\Lambda g_{\\mu\\nu} $$ But then, in the paragraph below Eq(228) on page 62 of this paper, it is said that this quantity is not a physical quantity and that it is well known that for the gravitational field no (physical) energy-momentum tensor exists. To me personally, this fact is rather surprising than well known. So can somebody explain to me (mathematically and\/or \"intuitively\") why there is no energy-momentum tensor for the gravitational field?"}
{"id":"100872","title":"Comparing predictions and reality for the gravitational attraction due to light beams","text":"While doing some on-the-side reading, I stumbled across this question: Do two beams of light attract each other in general theory of relativity?. Great question and a great, easily understandable answer. In short, it explains that General Relativity allows photons\/beams of light to gravitationally attract other things. It mentions a very important and well known phenomenon; that the deflection of light passing through the gravitational field of a massive particle is twice that predicted by Newtonian Gravitation. It also mentions and links to a wonderful article that shows that light beams travelling anti- parallel are deflected by each other's gravitation by four times what is predicted by Newtonian methods. The Newtonian predictions were able to be made because of the commonly accepted gravitational mass for a photon, which effectively uses Einstein's $E=mc^2$ and Planck's $E=h f$ to get $m=h f\/c^2$. Not a bad strategy. My question is why we choose to equate the photon's gravitational mass with a hypothetical particle's rest mass? Given that the total energy of a photon (if you rescale the potential energy to 0) can be written as: $$Total~Energy=Kinetic~Energy+Rest~Energy$$ And given that it is nice to set the rest energy of our photon to $0$. Why then should we choose the mass on which to base the predictions using Newtonian Gravity to be a rest mass? Especially when Newtonian physics provides an adequate way of obtaining mass from kinetic energy (as long as that mass is used only in other Newtonian physics). I mean, why can we not say the following for purely Newtonian calculations: $$E=hf,~~K=\\frac{1}{2}mv^2,~~Rest~Energy=E_o=0$$ $$\\therefore hf=\\frac{1}{2}mv^2\\rightarrow m=2hf\/v^2=2hf\/c^2$$ This effectively doubles the gravitational mass of a light beam without altering the actual momentum of it. When predicting the deflection of a beam due to a massive particle, this would make the force of Newtonian gravitation twice as large and the fact that momentum didn't change means the deflection prediction would be twice as large. For the deflection of two antiparallel beams, since the gravitational masses of both are doubled, this would quadruple the force of attraction again without modifying each beam's momentum, making the Newtonian prediction four times that compared to using mass from the rest energy equation. Both of these new predictions are what is shown to actually happen. Understandably, if this were a good and valid idea, it would have been used or realized a long time ago. My question is centred around understanding why the rest mass equation must be used; I am not trying to say what has been done is wrong, just trying to understand why it makes sense."}
{"id":"71109","title":"Preons and 't Hooft condition","text":"If some fundamental particles, like leptons or quarks are composite (e.g.in preonic models), or the same with gauge bosons or the higgs particles. **How could it be possible that preons were more massive than the \"known\" elementary particles?** Remark: I think to remember there are some conditions called 't Hooft's arguments or similar that are related to this question. Can someone help me? Any good reference about how to understand this hard issue? I am interested in compositeness since I think that BSM theories should be simpler in the same they should have less \"elements\"\/\"constituents\"\/\"degrees of freedom\" than the SM itself... Remark (II): I am interested in as simple as possible answers AND in technical responses related to effective theories and their operators with mass dimension X..."}
{"id":"60278","title":"Calculate\/estimate power of a fission bomb","text":"I have some questions about the released energy and power of a nuclear fission bomb.   * What are the key dependencies of the power of a fission bomb?   * Is it true that the power of a fission bomb depends linearly on the mass of the bomb? If so, why?   * Is there a simple physical argument to estimate the magnitude of the total released energy of the bomb if you know the mass and the used isotope? Can you provide an example calculation?    * How can I estimate the power (released energy per time) from this information?   * Is there a closed formula to calculate the power of a fission bomb."}
{"id":"15508","title":"The Born-Oppenheimer approximation and muonic molecules","text":"Does the Born-Oppenheimer approximation fail for muonic molecules (i.e. molecules where one or more electrons are replaced with muons)?"}
{"id":"44852","title":"About interchange phase of identical particles in Weinberg's QFT book","text":"In Weinberg's textbook on QFT(google book preview), he discussed the phase acquired after interchanging particle labels in the last paragraph of page 171 and the footnote of page 172. It seems he's suggesting interchanging particles of same species but different spin states will only bring a phase determined by convention, that is, the phase does not have to be $\\pm1$. I'm having a hard time understanding this, because I was taught that interchange of identical particles must give a phase of $\\pm 1$ and took it for granted. Is Weinberg actually treating particles of same species but different spin states as distinguishable particles? If so can I take it further and conclude particles of same species, same spin states but different momenta are also distinguishable, so that the interchange phase is also conventional? EDIT: I did a calculation of the commutation relation of $a'_m$ and $a'_n$ as Lubos suggested and verified his conclusion. Recently I did the same calculation again but could not reproduce the result I had, here is my recent calculation: $a'_ma'_n=(-1)^{N_{m',m'<m}} a_{m'}(-1)^{N_{n',n'<n}}a_{n'}=\\pm (-1)^{N_{m',m'<m}}(-1)^{N_{n',n'<n}} a_{m'}a_{n'}$ where $+$ is taken when $m'\\neq n'$, $-$ is taken when $m'=n'$. Also $a'_na'_m=(-1)^{N_{n',n'<n}}a_{n'}(-1)^{N_{m',m'<m}} a_{m'}=\\pm (-1)^{N_{m',m'<m}}(-1)^{N_{n',n'<n}} a_{n'}a_{m'}$ again $+$ is taken when $m'\\neq n'$, $-$ is taken when $m'=n'$. So $a'_m$ and $a'_n$ should obey the same commutation relations as those of $a_m$ and $a_n$. From another persepctive, it seems just from Pauli exclusion principle we can deduce the antisymmetry of fermion wavefunctions: Exclusion principle tells us the square of a fermion creation or annihilation operator must be zero, so $a^2_m=0,\\ a^2_n=0,\\ (a_m+a_n)^2=0$(assuming no superselection rule on $m,\\ n$ so that the 3rd operator is well defined), and we can easily see this implies $a_ma_n+a_na_m=0$. So, I'm again puzzled by this question."}
{"id":"44856","title":"What does Metric Transitivity Mean?","text":"Jaynes In his paper \"Information theory and Statistical mechanics\" says \"Previously, one constructed a theory based on equations of motion, supplemented by additional hypothesis of ergodicity, Metric transitivity, or equal a priori probabilities, and the identification of entropy was made only at the end, by comparison of the resulting equation with the laws of phenomenological thermodynamics.\" In this context what does \"Metric Transitivity\" mean?"}
{"id":"47624","title":"Is there is any difference between Electric Flux ($\\Psi$) and Total number of Electric Field lines ($E\\times$Area)?","text":"Is there is any difference between Electric Flux ($\\Psi$) and Total number of Electric Field lines ($E\\times$Area)? $\\psi = \\Sigma Q$, and $\\phi$ = Electric field intensity $\\times$ Area where Area = Total surface integral over Gaussian surface"}
{"id":"101556","title":"de Broglie formula inconsistency","text":"I recently stumbled across a small peculiarity I don't understand: According to de Broglie, the frequency of a matterwave can be written as $f=\\frac{E}{h}$, and its wavelength as $\\lambda = \\frac{h}{p}$. However, relativistically for a particle with matter the following formulae are valid: $E = \\gamma m_{0} c^{2}$, and $p = \\gamma m_{0}v$, where $m_{0}$ is the particle's rest mass. So, with $v=\\lambda f$ (with the particle's velocity v, unequal the speed of light!), using de Broglie we would end up with $E=v p$, which is clearly not valid when inserting the relativistic energy and momentum equations: $v*p = \\gamma m_{0} v^{2} \\neq E = \\gamma m_{0} c^{2}$. Where am I mistaken?"}
{"id":"114095","title":"Logistic map and attractors","text":"Does the logistic map have a strange attractor for some \"chaotic\" values of the parameter?"}
{"id":"89088","title":"How do virtual-photons curve in a magetic field?","text":"From what I understand photons only move in straight lines unless reflected or refracted (other than influences from gravitational fields and their usual wavelike movement). And since they are a fundamental force carrier I am having difficulty understanding how they are \"pulled\" into a pole of a magnet after leaving the opposite end. How does this occur? ![Curves in magnetic field:](http:\/\/i.stack.imgur.com\/vZmTY.png)"}
{"id":"122993","title":"Do airlocks in space decompress violently as they do in movies?","text":"The standard scene follows: The good guys have finally captured the enemies and have them in an airlock chamber on a spaceship. A button is pressed and out they go, violently propelled by the blast of air moving from a pressurized zone to a zone of much less pressure (space). Here's my question: If people don't immediately decompress, then why do airlocks do this in movies? If the air moves due to expansion to reduce pressure, then why wouldn't the human body do the same?"}
{"id":"116525","title":"Calculating a spacecraft's deflection angle","text":"> There is a spacecraft with mass $M_1$ moving in a certain trajectory with a > constant speed $V_r$. To modify its trajectory, a engine is turned on and > starts ejecting gas at a speed $V_g$, which is constant relative to the > spacecraft and perpendicular to the spacecraft's trajectory. The engine is > turned off when the mass of the rocket is $K$ times lower. What is the > trajectory's deflection angle, in radians, caused by the engine? I guess I should use the rocket equations to discover the rocket's thrust and the thrust caused by the engine. So I would have both vectors and then I could calculate the \"total\" thrust. But I don't know how could I get the angle with it, maybe using some projectile motion equations? Can anybody help me?"}
{"id":"96334","title":"What is the world-sheet picture of high energy open string scattering?","text":"Following Gross-Manes paper (which came right after Gross-Mende), the planar one-loop amplitude (the annulus) is dominated by the region in moduli space where the radius of the annulus shrinks to a point, i.e. it looks like the tree amplitude. Does anyone know if this generalizes for to all-loops?"}
{"id":"17939","title":"What if we could give photons some mass?","text":"I was reading an article and these paragraphs got me wondering... > Before I list the replies, here is some background. The Higgs mechanism > describes an invisible field that, it is argued, split one force into two > soon after the birth of the universe. Specifically, it divided an ancient > \"electroweak\" force into the electromagnetic and weak forces we see at work > today. The latter is seen in some radioactive decay processes, and is > involved in creating sunshine. > > The Higgs field splits the electroweak force by giving mass to the particles > that carry the weak force (the W & Z bosons) and leaving the particle that > carries the electromagnetic force (the photon) massless. The Higgs boson is > the quantum particle associated with the Higgs field. What I'm wondering is... can anyone hazard a guess as to the possibility at some point in the future of humans being able to manipulate the Higgs mechanism so as to give photons some kind of mass? I'm thinking they obviously couldn't go at the speed of light anymore, so could we make 'slow light'?"}
{"id":"96336","title":"Double slit experiment","text":"I can't understand the relation between Schrödinger's Cat and this amazing experiment of the double slit. It seems like in the double slit we know if we observe the particle the wave function collapse and the electron acts like a normal particle, so doesn't that solve the problem in Schrödinger's Cat? When we open the box the atom that should help release the poison acts like a particle,the poison is in the bottle so the cat stays alive, don't know really?!"}
{"id":"127631","title":"Given eigenvalues of $\\vec l^2$ and $\\vec s^2$, calculate the eigenvalue for $\\vec j^2$","text":"There was an exam question that read approximatly: > _Let $\\vec j = \\vec l + \\vec s$. Given eigenvalues of $\\vec l^2$ and $\\vec > s^2$, calculate the eigenvalue for $\\vec j^2$._ We came up with $$\\vec j^2 = (\\vec l + \\vec s)^2 = \\vec l^2 + \\vec s^2 + 2 \\vec l \\cdot\\vec s$$ How can we get the eigenvalue for $\\vec j^2$ only with this information? I think that we would need $m_l$ and $m_s$ or so."}
{"id":"105659","title":"Why does the gas cloud of an underwater gunshot pulse?","text":"I have been watching some slow motion video and I was intrigued by the slow motion underwater gunshot. The first moments of the video go as expected. The gun fires and a cloud forms in front of the gun. Then the bullet rips through the water, leaving an open space (which I assume is a cavitation vacuum.) Soon after, the rip closes, and the gas bubble shrinks (I'm assuming this is due to the cooling and\/or compression of the hot high pressure gasses.) Then something funny happens. The bubble pulses a few times and almost appears to emit light. Each pulse is accompanied by a noise. The video can be seen here. What is this pulse effect and\/or what causes it? **EDIT:** Here is another great video for a possible explanation:"}
{"id":"39712","title":"What are some of the best books on complex systems?","text":"I'm rather interested in getting my feet wet at the interface of complex systems and emergence. Can anybody give me references to some good books on these topics? I'm looking for very introductory technical books."}
{"id":"89333","title":"Stellar Power(Luminosity) Flux","text":"So I was applying some mathematical techniques I learned to physics, and one thing that captured my interest, is the power or luminosity flux of a star. So modeling the situation, taking the scalar field of Luminosity($L$), and the vector surface(stellar surface), $S$, I began by setting up a Surface (Flux, in this case) integral of $$ \\iint_{S}L\\,dS $$ Where, $$ S=\\langle r\\sin\\theta \\cos\\phi,r\\sin\\theta \\sin\\phi,r\\cos\\theta \\rangle \\;,\\; \\theta \\, \\epsilon \\, [0,\\pi] \\wedge \\phi \\, \\epsilon \\, [0,2\\pi] $$ $$ L = 4\\pi r^{2}\\sigma T^{4} $$ $r$ in this model is the invariant stellar radius. Following logic, since $L$ has no explicit dependency on $ \\theta $ or $\\phi$, I rearranged as: $$ 4\\pi r^{2} \\sigma T^{4} \\iint_{S} dS $$ Since the integral of dS over S is merely surface area, and our surface is a sphere of radius $ r $, this expression reduces to: $$ 16\\pi^{2} r^{4} \\sigma T^{4} $$ The dimensional analysis for this breaks down to $ W \\cdot m^{2} $. Isn't flux supposed to be quantity per area?"}
{"id":"127435","title":"Spin wave theory for 1D spin systems","text":"In general, when one performs spin-wave theory, he has to assume some kind of long-range ordered state and perform spin-wave theory (using for instance Holstein-Primakoff bosons) around that state. Recall that in 1D there is no long-range order. Then it seems that spin-wave theory in 1D would be meaningless. **But yet, people still do spin-wave in 1D... why ?** (you can just type \"spin-wave for spin chain\" on google and you'll see plenty of examples)"}
{"id":"65104","title":"Easy to do experiments that clearly show outstanding properties of graphene","text":"By chance we received for free some monolayer graphene sheets (20 cm x 20 cm) and mixed coper-graphene wires at our University. I would like to prepare some very easy to do experiments for the chemistry students that show the awesome properties of graphene, and which could be carried out with very simple lab equipment or even home experiments, any ideas or references?"}
{"id":"69282","title":"Question regarding Drift velocity in general?","text":"The derivation of drift velocity in case of electrons is equivalent to the case of an charged ionic gas and therefore all the arguments also apply there. Now for an ideal \"ionic\" gas which interacts with each others only as perfect spheres would (for arguments sake), when we apply an external electric field, they would just show drift, that means that they \"favour\" one direction more than the others where drift velocity is given by $v_d=(eE\/m)\\tau$ where $\\tau$ is the average relaxation time. Hence the net velocities of the electrons wont vectorially add to zero, and moreover the speed as given by kinetic theory of matter would not depend only on the temperature but also depend on electric field. Does this not contradict two fundamental postulates of kinetic theory? Moreover in case of electrons in a circuit with many series resistors, the relaxation time would be different. The resistivity as defined by $E=\\rho J$ and $\\rho=m\/(ne^2\\tau)$ where j=current density, n=number of electrons per unit volume and e=electronic charge, should be different for all the series resistors. But then will not the drift velocity be different for all the resistors if the top equation is correct since the resistivities might defer, causing the relaxation time to differ. Doesn't this contradict the postulation of a steady current which requires constant drift velocity. Moreover suppose we use the same material of different cross section, since there is no area dependence in the drift velocity, it shouldn't change but then again we don't get a steady current as J remains a constant and I is simply J times A."}
{"id":"90979","title":"Gravitation law paradox for very close objects?","text":"We all know that gravitation force between two small (not heavenly) bodies is negligible. We give a reason that their mass is VERY small. But according to inverse square law, as $r\\to 0$, then $F\\to \\infty$. But in real life we observe that even if we bring two objects very close, no such force is seen. Why is this so?"}
{"id":"11521","title":"What is the meaning of the Fourier transform of Feynman propagator?","text":"I know $K(a,b,t)$ is the probability amplitude of find a particle that starts at point a in b in a time t later. There is also an expression that sometimes is called green function:   $$G(a,b,E)=(i\/\\hbar)\\int_{-\\infty}^\\infty\\;\\exp(iEt\/\\hbar)\\;K(a,b,t)\\;dt$$ or Fourier transform of Feynman propagator   See: Grosche. Handbook of Feynman path integrals page 149. Keller. the feynman integral page 461.   http:\/\/arxiv.org\/abs\/cond-mat\/0304290v1 I want to know if $G(a,b,E)$ could be the amplitude that a particle of energy $E$ at the initial point $a$ will appear at some (arbitrary) time at $b$. It seems that Martin Schaden and Larry Spruch use this interpretation in http:\/\/arxiv.org\/abs\/cond-mat\/0304290v1 but I have not found this in any book of quantum mechanics."}
{"id":"121876","title":"Temperature stability of permanent magnets","text":"Hello physicists\/engineers, I have a somehow very specific question I hope some expert can answer me: I'm working at university and try to build an experiment which requires comparable high magnetic field (2000Gauss or 0.2T if you prefer) at room temperature. This field is actually no problem using Neodymium iron boron magnets, the problem however is: I can see relatively slow drifts in the magnetic field in the order of several mG. Since this basically destroys my experiment, and definitely occurs due to temperature drifts of a few mK, i was wondering if there are permanent magnets that do not change their magnetization under such temperature drifts (SmCo?) and\/or if there's a procedure (annealing) to stabilize the magnets I have right now Thanks"}
{"id":"45484","title":"Laser Coherence Length\/Time","text":"Scenario: Imagine a collimated beam of white light falling on one refracting face of a prism. Let the light emerging from the second face be focused by a lens onto a screen. Suppose there is linear dispersion at the screen. Now by introducing a narrow \"exit slit\" (on the order of 0.01cm) in the screen, one has a type of monochromator that provides a nearly monochromatic beam of light. The white light has a mean wavelength of 500nm. Questions: What is meant by linear dispersion at the screen. Furthermore how does the slit effect the coherence length and coherence time of the light."}
{"id":"46289","title":"\"Inverted\" quantum oscillator","text":"I'm trying to understand the problem of the \"inverted\" oscillator, which has the following Hamiltonian: $$ \\hat{H}=\\frac{\\hat{p}^{2}}{2m}-\\frac{k\\hat{x}^{2}}{2} $$ Suppose that a particle at the initial time is in a state $$ \\Psi(x)=\\frac{1}{(\\pi b^{2})^{\\frac{1}{4}}}\\exp^{-x^{2}\/2b^{2}} $$ We want to find $\\langle\\Psi\\mid\\hat{x}(t)\\mid\\Psi\\rangle$ and $\\langle \\Psi \\mid \\hat{x}^{2}(t) \\mid \\Psi \\rangle$ and their asymptotic when $t \\rightarrow +\\infty$. Heisenberg equation for operator $\\hat{x}$ (we use that $\\hat{x}$ does not depend explicitly on time, $[\\hat{x},\\hat{x}]=0$, $[\\hat{x},\\hat{p}]=i\\hbar$ and Leibniz rule:$[A,BC]=[A,B]C+B[A,C]$): $$ i\\hbar \\dot{\\hat{x}}=\\left[\\hat{x},\\hat{H}\\right]=\\left[\\hat{x},\\frac{\\hat{p}^{2}}{2m}-\\frac{k\\hat{x}^{2}}{2}\\right]=\\left[\\hat{x},\\frac{\\hat{p}^{2}}{2m}\\right]=\\frac{\\hat{p}}{m}\\left[\\hat{x},\\hat{p}\\right]=i\\hbar\\frac{\\hat{p}}{m} $$ $$ \\dot{\\hat{x}}=\\hat{p}\/m $$ Similarly, Heisenberg equation for operator $\\hat{p}$: $$ i\\hbar\\dot{\\hat{p}}=\\left[\\hat{p},\\hat{H}\\right]=\\left[\\hat{p},\\frac{\\hat{p}^{2}}{2m}-\\frac{k\\hat{x}^{2}}{2}\\right]=-\\left[\\hat{p},\\frac{k\\hat{x}^{2}}{2}\\right]=-k\\hat{x}\\left[\\hat{p},\\hat{x}\\right]=i\\hbar k\\hat{x} $$ $$ \\dot{\\hat{p}}=k\\hat{x} $$ So, we have following set of equations: $\\dot{\\hat{x}}=\\hat{p}\/m$; $\\dot{\\hat{p}}=k\\hat{x}$. Using them, we can write: $$ m\\ddot{\\hat{x}}=k\\hat{x} $$ Solution of this differential equation is $\\hat{x}=A\\cosh(\\omega t)+B\\sinh(\\omega t)$, where $\\omega=\\sqrt{\\frac{k}{m}}$. Initial conditions: $A=\\hat{x}(0)$; $B=\\frac{\\dot{\\hat{x}}(0)}{\\omega}$. Remember that $\\dot{\\hat{x}}=\\hat{p}\/m$. Then $B=\\frac{\\hat{p}(0)}{m\\omega}$. So we have $$ \\hat{x}=\\hat{x}(0)\\cosh(\\omega t)+\\frac{\\hat{p}(0)}{m\\omega}\\sinh(\\omega t) $$ Then $$ \\hat{x}^{2}=\\hat{x}^{2}(0)\\cosh^{2}(0)+\\frac{\\hat{p}^{2}(0)}{(m\\omega)^{2}}\\sinh^{2}(0)+\\frac{1}{m\\omega}(\\hat{x}(0)\\hat{p}(0)+\\hat{p}(0)\\hat{x}(0))\\sinh(\\omega t)\\cosh(\\omega t) $$ Now we want to calculate $\\langle\\Psi\\mid\\hat{x}\\mid\\Psi\\rangle$ and $\\langle \\Psi \\mid \\hat{x}^{2} \\mid \\Psi \\rangle$: $$ \\langle\\Psi\\mid\\hat{x}(t)\\mid\\Psi\\rangle=\\langle\\Psi\\mid\\hat{x}(0)\\cosh(\\omega t)+\\frac{\\hat{p}(0)}{m\\omega}\\sinh(\\omega t)\\mid\\Psi\\rangle=\\langle\\Psi\\mid\\hat{x}(0)\\mid\\Psi\\rangle\\cosh(\\omega t)+\\langle\\Psi\\mid\\hat{p}(0)\\mid\\Psi\\rangle\\frac{\\sinh(\\omega t)}{m\\omega} $$ Similarly $$ \\langle \\Psi \\mid \\hat{x}^{2}(t) \\mid \\Psi \\rangle=\\langle\\Psi\\mid\\hat{x}^{2}(0)\\mid\\Psi\\rangle\\cosh^{2}(\\omega t)+\\langle\\Psi\\mid\\hat{p}^2(0)\\mid\\Psi\\rangle\\frac{\\sinh^{2}(\\omega t)}{(m\\omega)^{2}}+\\frac{1}{m\\omega}\\langle\\Psi\\mid(\\hat{x}(0)\\hat{p}(0)+\\hat{p}(0)\\hat{x}(0))\\mid\\Psi\\rangle\\sinh(\\omega t)\\cosh(\\omega t) $$ Lets introduce the notation:$x_{0}=\\langle\\Psi\\mid\\hat{x}(0)\\mid\\Psi\\rangle$, $p_{0}=\\langle\\Psi\\mid\\hat{p}(0)\\mid\\Psi\\rangle$,$x_{0}^{2}=\\langle\\Psi\\mid\\hat{x}^{2}(0)\\mid\\Psi\\rangle$, $p_{0}^{2}=\\langle\\Psi\\mid\\hat{p}^{2}(0)\\mid\\Psi\\rangle$. Then: $$ \\langle\\Psi\\mid\\hat{x}(t)\\mid\\Psi\\rangle=x_{0}\\cosh(\\omega t)+\\frac{p_{0}}{m\\omega}\\sinh(\\omega t) $$ $$ \\langle\\Psi\\mid\\hat{x^{2}}(t)\\mid\\Psi\\rangle=x_{0}^{2}\\cosh^{2}(\\omega t)+p_{0}^{2}\\frac{\\sinh^{2}(\\omega t)}{(m\\omega)^{2}}+\\frac{1}{m\\omega}\\langle\\Psi\\mid(\\hat{x}(0)\\hat{p}(0)+\\hat{p}(0)\\hat{x}(0))\\mid\\Psi\\rangle\\sinh(\\omega t)\\cosh(\\omega t) $$ **Here is the first big trouble** : i've found some lecture notes in the internet, where they say that $\\langle\\Psi\\mid(\\hat{x}(0)\\hat{p}(0)+\\hat{p}(0)\\hat{x}(0))\\mid\\Psi\\rangle=0$ for real $\\Psi(x)$. I tried to prove this statement, but i can't. **Can you explain why it is equal to zero?** Well, suppose that this statement is true. Then: $$ \\langle\\Psi\\mid\\hat{x}(t)\\mid\\Psi\\rangle=x_{0}\\cosh(\\omega t)+\\frac{p_{0}}{m\\omega}\\sinh(\\omega t) $$ $$ \\langle\\Psi\\mid\\hat{x^{2}}(t)\\mid\\Psi\\rangle=x_{0}^{2}\\cosh^{2}(\\omega t)+p_{0}^{2}\\frac{\\sinh^{2}(\\omega t)}{(m\\omega)^{2}} $$ Thus, we expressed values through their meanings at the initial time. So, lets find $x_{0}$, $p_{0}$, $x_{0}^{2}$, $p_{0}^{2}$. From this point we will work in the coordinate representation. $$ x_{0}=\\langle\\Psi\\mid\\hat{x}\\mid\\Psi\\rangle=\\int\\Psi^{\\ast}x\\Psi dx=\\int\\Psi x\\Psi dx=\\int\\Psi^{2}xdx $$ Remember that $\\Psi(x)=\\frac{1}{(\\pi b^{2})^{\\frac{1}{4}}}\\exp^{-x^{2}\/2b^{2}}$ : $$ x_{0}=\\frac{1}{(\\pi b^{2})^{\\frac{1}{2}}}\\int x\\exp^{-x^{2}\/b^{2}}dx $$ We integrate in domains from minus infinity to plus infinity. But this integral is equal to zero! So $$ x_{0}=0 $$ Similarly: $$ p_{0}=\\langle\\Psi\\mid\\hat{p}\\mid\\Psi\\rangle=-i\\hbar\\int\\Psi^{\\ast}\\frac{\\partial}{\\partial x}\\Psi dx=\\frac{2}{b^{2}}i\\hbar\\frac{1}{(\\pi b^{2})^{\\frac{1}{2}}}\\int x\\exp^{-x^{2}\/b^{2}}dx $$ The domains of integration are the same:from minus infinity to plus infinity. But this is also is equal to zero! So, we have : $$ \\langle\\Psi\\mid\\hat{x}(t)\\mid\\Psi\\rangle=0\\cosh(\\omega t)+\\frac{0}{m\\omega}\\sinh(\\omega t)=0 $$ **I'm confused by this result! Is there a mistake?** If not, can you explain me this result? Now we are going to calculate $x_{0}^{2}$ and $p_{0}^{2}$: $$ x_{0}^{2}=\\langle\\Psi\\mid\\hat{x}^{2}\\mid\\Psi\\rangle=\\int\\Psi^{\\ast}x\\Psi dx=\\frac{1}{(\\pi b^{2})^{\\frac{1}{2}}}\\int x^{2}\\exp^{-x^{2}\/b^{2}}dx=\\frac{1}{(\\pi b^{2})^{\\frac{1}{2}}}\\frac{\\sqrt{\\pi} (b^{2})^{\\frac{3}{2}}}{2}=\\frac{b^2}{2} $$ $$ p_{0}^{2}=\\langle\\Psi\\mid\\hat{p}^{2}\\mid\\Psi\\rangle=-\\hbar^{2}\\int\\Psi^{\\ast}\\frac{\\partial^{2}}{\\partial x^{2}}\\Psi dx=\\frac{\\hbar^{2}}{2b^{2}} $$ Finally: $$ \\langle\\Psi\\mid\\hat{x^{2}}(t)\\mid\\Psi\\rangle=\\frac{b^2}{2}\\cosh^{2}(\\omega t)+\\frac{\\hbar^2}{2b^{2}}\\frac{\\sinh^{2}(\\omega t)}{(m\\omega)^{2}} $$ Using the equation $\\cosh^{2}(x)-\\sinh^{2}(x)=1$ we get: $$ \\langle\\Psi\\mid\\hat{x^{2}}(t)\\mid\\Psi\\rangle=\\frac{1}{2}\\left(b^{2}+\\frac{\\hbar^{2}}{b^{2}(m\\omega)^{2}}\\right)\\cosh^{2}(\\omega t)-\\frac{\\hbar^{2}}{2b^{2}(m\\omega)^{2}} $$ Thus we see that when $t \\rightarrow +\\infty$ dispersion of coordinate $\\langle\\Psi\\mid\\hat{x^{2}}(t)\\mid\\Psi\\rangle$ tends to infinity. To sum up, I have these several difficulties:   1. why $\\langle\\Psi\\mid(\\hat{x}(0)\\hat{p}(0)+\\hat{p}(0)\\hat{x}(0))\\mid\\Psi\\rangle=0$ for real $\\Psi(x)$?   2. why $\\langle\\Psi\\mid\\hat{x}(t)\\mid\\Psi\\rangle=0$?   3. why when $t \\rightarrow +\\infty$ $\\langle\\Psi\\mid\\hat{x^{2}}(t)\\mid\\Psi\\rangle$ tends to infinity? Also i would be happy if you provide some articles(or books) where i can read about it."}
{"id":"52425","title":"Exact diagonalization of graphene's tight binding Hamiltonian","text":"While directly diagonalize graphene's tight binding Hamiltonian, which is numerical. We have to use a finite-sized graphene. So how to deal with boundary conditions? The usual solutions are zigzag or armchair condition, but to make our model more realistic to real infinite graphene plane, how about using periodic condition at boundaries while exactly diagonalizing the tight binding Hamiltonian?"}
{"id":"90352","title":"Why does a light wave invert at a boundary with greater index of refraction?","text":"Is there a reason why a EM wave reflects invertedly when it meets a boundary point with a greater index of refraction. In the case of ropes, if remember correctly, the reason why it inverts is to agree with Newtons third law. But, its unclear why for EM waves, upon reflection, a phase change of $\\pi$ occurs."}
{"id":"12361","title":"D'Alembert's Principle: Where does $-Q_j$ come from?","text":"This is a follow-up question to D'Alembert's Principle and the term containing the reversed effective force. From the second term of Eq. (1.45) $$\\begin{align*} \\sum_i{\\dot{\\mathbf{p}}_i \\cdot \\delta{\\mathbf{r}_i}} &= \\sum_i{m_i\\ddot{\\mathbf{r}}_i \\cdot \\frac{\\partial\\mathbf{r}_i}{\\partial q_j} \\delta q_j} \\end{align*}$$ Starting from Eq. (1.50), I was able to follow that $$\\begin{align*} \\sum_i{m_i\\ddot{\\mathbf{r}}_i \\cdot \\frac{\\partial\\mathbf{r}_i}{\\partial q_j}} &= \\sum_i{\\left[ \\frac{d}{dt} \\left( m_i\\mathbf{v}_i \\cdot \\frac{\\partial\\mathbf{v}_i}{\\partial\\dot{q}_j} \\right) - m_i\\mathbf{v}_i \\cdot \\frac{\\partial\\mathbf{v}_i}{\\partial q_j} \\right]} \\end{align*}$$ Goldstein substituted the above equation to (1.45) by saying: > ... and the second term on the left-hand side of Eq. (1.45) can be expanded > to By \"second term\", I understood it to be the very first equation I mentioned above. Therefore this is how I understood it: $$\\begin{align*} \\sum_i{\\dot{\\mathbf{p}}_i \\cdot \\delta{\\mathbf{r}_i}} &= \\sum_i{m_i\\ddot{\\mathbf{r}}_i \\cdot \\frac{\\partial\\mathbf{r}_i}{\\partial q_j} \\delta q_j}\\\\\\ &= \\sum_i\\left[ \\frac{d}{dt}\\left[ \\frac{\\partial}{\\partial\\dot{q}_j} \\left( \\frac{1}{2}\\sum_i{m_iv_i^2} \\right) \\right] - \\frac{\\partial}{\\partial q_j} \\left( \\frac{1}{2}\\sum_i{m_iv_i^2} \\right) - Q_j \\right]\\delta q_j\\\\\\ &= \\sum_i{\\left[ \\frac{d}{dt} \\left( \\frac{\\partial T}{\\partial\\dot{q}_j} \\right) - \\frac{\\partial T}{\\partial q_j} -Q_j \\right] \\delta q_j} \\end{align*}$$ I was able to follow $$\\begin{align*} T &= \\sum_i{m_i\\mathbf{v}_i} \\cdot \\partial\\mathbf{v}_i\\\\\\ T &=\\frac{1}{2} \\sum_i{m_iv_i^2} \\end{align*}$$ But I am at a loss: **Where does $-Q_j$ come from?**"}
{"id":"54865","title":"What's the principle used in infant breath monitor devices?","text":"Such a device consists of a sensor pad that is placed under a mattress: ![sesnor pad placing under a mattress](http:\/\/i.stack.imgur.com\/OMVui.gif) What's the principle it uses to detect breathing?"}
{"id":"12369","title":"Why atomic bomb makes a mushroom?","text":"Atomic bombs make mushroom. It's well-known and we've seen many images of atomic experiments around the world. My question is, why mushroom? Does TNT and Grenade also make mini-mushrooms? Is it a physical attribute of explosion, or something specific to atomic energy? Thanks"}
{"id":"3327","title":"Optical explanation of images of stars?","text":"Very often when viewing pictures of the cosmos taken by telescopes, one can observe that larger\/brighter stars do not appear precisely as points\/circles on the image. Indeed, the brighter the light from star, the more we see this effect of four perpendicular rays \"shooting\" out from the star. ![Stars1](http:\/\/i.stack.imgur.com\/LAA8M.jpg) (Taken from this page.) My question is: what are the optics responsible for this effect? I would suppose that both the hazy glow around the stars and the rays shooting outwards are optical effects created by the camera\/imaging device, but really can't surmise any more than that. (The fact that the rays are all aligned supports this, for one.) A proper justification, in terms of (geometrical?) optics for both the glow and the rays would be much appreciated. Here are a few other examples of such images:   * http:\/\/grizzlyhugs.files.wordpress.com\/2009\/12\/stars-sky-lg.jpg   * http:\/\/spacespin.org\/images\/articles\/hubble-large-small-stars-harmonious_3.jpg"}
{"id":"49835","title":"Is the normal force a conservative force?","text":"Most of the time the normal force doesn't do any work because it's perpendicular to the direction of motion but if it does do work, would it be conservative or non-conservative? For example, consider the following block-incline system where the incline is on a friction-less surface. Now as the block moves down, the incline itself starts moving in the opposite direction. Here, the normal force acting on the incline by the block does do work. ![Block-incline](http:\/\/i.stack.imgur.com\/35dSa.png) This question was on some website (it asked something about the final velocities of the block and the incline) and they solved this using mechanical energy and momentum conservation which confused me as I couldn't understand why the normal force here is conservative. So,   1. Is the normal force always conservative or is this only in some cases?    2. If yes, how do we deduce that like in the above example?   3. Also, if it is conservative, what would be its corresponding potential energy function?"}
{"id":"46823","title":"What is the difference between UT0, UT1 and GMT time?","text":"Every reference I find says that they are **_\"essentially\"_** the same, which we all know really means that they are not the same, but different only by a some small amount that someone else other than me decides is negligible. They also like to say that UT1 and UTC are the same, but if you look into it you find that they differ by as much as 0.9 seconds. This is, admittedly, insignificant to me. But the practicality of it is not my concern here. The technical reality of it is. I want specifics and exactness where applicable here."}
{"id":"63390","title":"Explanation for Negative $\\rho$ (radial distance) in Cylindrical Coordinates","text":"**My question :** What does it mean when we arrive at **negative** values for _distance_ variables like $\\rho$ in cylindrical coordinates? (after some discussion here,I revised the question, at the end of the post) **My specific example:** For the motion of an ion in a quadrupole Paul trap, we arrive at the same equation (mathieu equation) (below) for both $\\rho$ and $z$ components (the potential has only $\\rho$ and $z$ dependance ): $$\\frac{d^2u}{d\\tau^2} + [a_u-2q_u\\cos (2 \\tau) ]u=0 \\qquad\\qquad $$ here $u$ represents a coordinate variable and may be $r$ , $\\rho$ in cylindrical coordinates or $x , y ,z$ in Cartesian coordinates. **The problems is that , this equation has an oscillatory behavior and it's solution has negative values too. What is the meaning of negative $\\rho$ in cylindrical coordinates?** Here is a sample solution for $a_z=0$ and $q_z=0.4$ ![A sample solution for $a_z=0$ and $q_z=0.4$ ](http:\/\/i.stack.imgur.com\/eJSCb.jpg) **EDIT 1:** after some answers posted , I found out that the problem has something to do with our derivation of equations.Here is how _correct_ equations are. The potential is only a function of $r$ (or $\\rho$) and $z$ in cylindrical coordinates,so we don't have any force in $\\hat \\theta$ direction: $$\\mathbf F=-\\nabla V(r,z)=-(\\hat r{\\partial V \\over \\partial r} +\\hat \\theta \\frac{1}{r} {\\partial V \\over \\partial \\theta} +\\hat z {\\partial V \\over \\partial z})$$ so the equations of motion will be (I assumed the mass to be 1) $$\\ddot r - r\\dot{\\theta}^2=-{\\partial V \\over \\partial r} ***$$ $$r \\ddot{\\theta} +2 \\dot r \\dot \\theta=0 \\to \\dot{(r^2\\dot \\theta)} = 0$$ $$\\ddot z=-{\\partial V \\over \\partial z}$$ so the equations for $z$ and $r$ **will not** be the same. Now , **the problem is : how can we arrive at those equations (mathieu equation, above)(in the literature on Paul traps they use this equations) from these equations of motion ; eliminate $\\theta$ dependance and instead of that let $r$ get negative values and interpret it as a positive $r$ with $\\theta_0+\\pi$?**"}
{"id":"45730","title":"Theoretically speaking what are the limits for super-capacitors in terms of their storage capability?","text":"According to wikipedia, commercially available super-capacitors in 2010 can store 30Wh\/kg, with 85Wh\/kg achieved in a lab in 2011. Have any theoretical calculations been done to show what can be achieved as an upper limit to super-capacitors?"}
{"id":"45736","title":"Accelerating expansion of universe - entire universe a non-inertial frame of reference?","text":"If the expansion of the universe is accelerating, doesn't that mean that the entire universe is a non-inertial frame of reference? And if so, what implications does this have (if any)?"}
{"id":"2914","title":"Software to calculate forces between magnets","text":"I am working on a complex configuration of magnets and every time I make an experiment something unforseen happens. Now I believe I could speed up the development by sitting down and calculating the configuration in all scenarios. My question is however: Is there any software that can help me to calculate the forces between magnets. I am already working with AutoDesk as my CAD software."}
{"id":"29016","title":"Cyclist's electrical tingling under power lines","text":"It's been happening to me for years. I finally decided to ask users who are better with \"practical physics\" when I was told that my experience – that I am going to describe momentarily – prove that I am a diviner, a psychic, a \"sensibil\" as we call it. The right explanation clearly needs some electrodynamics although it's \"everyday electrodynamics\" and theoretical physicists are not trained to quickly answer such questions although each of us has probably solved many exercises that depend on the same principles. ![enter image description here](http:\/\/i.stack.imgur.com\/azt0a.jpg) When I am biking under the power lines – which probably have a high voltage in them – I feel a clear tingling shocks near my buttocks and related parts of the body for a second or so when I am under a critical point of the power lines. It is a strong feeling, not a marginal one: it feels like a dozen of ants that are stinging me at the same moment. It seems almost clear that some currents are running through my skins at 50 Hz. I would like to know the estimate (and calculation or justification) of the voltage, currents etc. that are going through my skin and some comparison with the shock one gets when he touches the power outlet. Now,   * my bike that makes this effect particularly strong is a mountain bike, Merida;   * the speed is about 20 km\/h and the velocity is perpendicular to the direction of the current in the power line;   * the seat has a hole in it and there is some metal – probably a conducting one – just a few centimeters away from the center of my buttocks. It's plausible that I am in touch with the metal – or near touch;   * my skin is kind of sweating during these events and the liquid isn't pure water so it's probably much more conductive than pure water;   * the temperature was 22 °C today, the humidity around 35%, clear skies, 10 km\/h wind;   * the power lines may be between 22 kV and 1 MV and at 50 Hz, the altitude is tens of meters but I don't really know exactly. What kind of approximation for the electromagnetic waves are relevant? What is the strength? How high currents one needs? Does one need some amplification from interference etc. (special places) to make the effect detectable? (I only remember experiencing this effect at two places around Pilsen; the most frequent place where I feel it is near Druztová, Greater Pilsen, Czechia.) Is the motion of the wheels or even its frequency important? Is there some resonance? Does the hole in the seat and the metal play any role? Just if you think that I am crazy, other people are experience the effect (although with different body parts), see e.g. here and here. This PDF file seems to suggest that the metals and electromagnetic induction is essential for the effect but the presentation looks neither particularly comprehensive nor impartial enough. An extra blog discussion on this topic is here: > http:\/\/motls.blogspot.com\/2012\/05\/electric-shocks-under-high-voltage.html"}
{"id":"13014","title":"Physical significane and context in which Dirac introduced the Dirac Delta function","text":"I'd like to know the exact context in which Paul Dirac introduced the Dirac delta function. What was the physical significance of the Dirac delta function when he first used it in Physics ?"}
{"id":"13015","title":"How to get statics out of a dynamic force concept?","text":"If one defines force as the time derivative of momentum, i.e. by $$ \\vec{F}= \\frac{d}{dt} \\vec{p} $$ how can this include static forces? Is there a generally accepted way to argue in detail how to get from this to the static case? If not, what different solutions are discussed? Edit: I shoud add, that by static forces I mean forces involved in problems where bodies don't move."}
{"id":"20213","title":"How to calculate the coefficient of restitution for 2 bodies?","text":"I have 2 rigid bodies (from different materials) in a collision. As you know I should have the coefficient of restitution value to get the velocities after collision. What is the information\/values should I know about each body to calculate the restitution? And how to calculate it?"}
{"id":"79688","title":"Why is distance between two successive nodes equal to $\\frac{1}{2} \\lambda$ in standing wave","text":"I'm reading my textbook and it says the distance between two successive nodes is equal to $\\frac{1}{2} \\lambda$ in standing wave. ![enter image description here](http:\/\/i.stack.imgur.com\/86Y0R.jpg) If $\\lambda$ here means the wavelength of the standing wave shouldn't it be: when there are two nodes: $L=\\lambda$ when there are three nodes: $L=\\frac{1}{2} \\lambda + \\frac{1}{2} \\lambda$ when there are three nodes: $L=\\frac{1}{3}\\lambda + \\frac{1}{3}\\lambda + \\frac{1}{3}\\lambda$ Does $\\lambda$ here mean anything?"}
{"id":"79687","title":"Magnetic induction in Relativity","text":"As we know magnetic phenomenon is a mere relativistic effect.My question is how to explain the magnetic induction in a relativistic manner?"}
{"id":"126839","title":"SInce heat is a vibration in solids, isn't it just small scale sound?","text":"On an atomic scale I think of electrons hitting positive ions in an electric lattice. Most energy there is heat energy. If I think of ball bearings flying through a large lattice, I think sound is produced. The only difference between the two interactions, seemingly, is size. Am I wrong here? In retrospect, I didn't realize there was a similar thread: If the source of sound is vibration, why can't we \"hear\" a object whose molecules are vibrating?"}
{"id":"62045","title":"Parabolic motion (experiment)","text":"We performed a laboratory, performing six releases of a sphere with angles $15^\\circ,30^\\circ,45^\\circ,60^\\circ,75^\\circ,40^\\circ$ a parabolic movement, took five distances for each angle, the initial velocity was calculated $3.025~\\text{m\/s}$. Then doing 5 tosses of the sphere with an angle of $90^\\circ$ times were recorded and calculated initial velocity was $3.33~\\text{m\/s}$, the question is: Why are these speeds almost the same?"}
{"id":"115175","title":"Quantum Bit Commitment, restricting ourselves to pure states","text":"I have been going through the Preskill lecture notes on quantum computation, and there is a question on Quantum Bit Commitment: > Alice wants to make a prediction, either $0$ or $1$, before an event. She > doesn't want Bob to know her prediction beforehand, but wants to be able to > prove to him afterwards that she was right all along. To do this she > prepares one of two distinguishable states represented by density operators > $\\rho_0$ or $\\rho_1$ of a bipartite system $A\\otimes B$, sending $B$ to Bob > and keeping $A$ for herself. Later to unveil her prediction she sends $A$ to > Bob who performs a measurement to determine if $A\\otimes B$ is in $\\rho_0$ > or $\\rho_1$. We say the protocol is binding if after commitment Alice is > unable to change her prediction, and concealing if after commitment and > before unveiling Bob is unable to discern the value of the bit. Show that if > this protocol is concealing it is not binding. I am asked to show that we may without loss of generality restrict ourselves to $\\rho_0$, $\\rho_1$ pure states, which is what I am unsure about. My guess is that we may do this because we can always regard a mixed state $\\rho$ on $A\\otimes B$ as a pure state $\\rho'$ on some $A\\otimes B\\otimes C$, where $\\rho=\\mathrm{Tr}_C\\\\{\\rho'\\\\}$. The tricky bit would be to show we can do this simultaneously for $\\rho_0$ and $\\rho_1$. To do this, if: $$\\rho_0=\\sum_{i=1}^np_i|\\psi_i\\rangle\\langle\\psi_i|$$ $$\\rho_1=\\sum_{j=1}^mq_j|\\phi_j\\rangle\\langle\\phi_j|$$ We can then define $$p'_k=\\begin{cases}p_k & k\\le n \\\\\\ 0 & k > n\\end{cases}$$ $$|\\psi'_k\\rangle=\\begin{cases}|\\psi_k\\rangle & k\\le n \\\\\\ 0 & k > n\\end{cases}$$ And similarly for $q'_k$, $|\\phi_k'\\rangle$. Then: $$\\rho_0=\\sum_{k=1}^{m+n}p'_k|\\psi'_k\\rangle\\langle\\psi'_k|$$ $$\\rho_1=\\sum_{k=1}^{m+n}q'_k|\\phi'_k\\rangle\\langle\\phi'_k|$$ Then if we pick $C$ such that $|\\alpha_k\\rangle\\in C$ are orthonormal for $k\\in\\\\{1,\\cdots,m+n\\\\}$ we have $\\rho_0'=|\\Psi\\rangle\\langle\\Psi|$ and $\\rho_1'=|\\Phi\\rangle\\langle\\Phi|$ pure for: $$|\\Psi\\rangle=\\sum_{k=1}^{m+n}\\sqrt{p'_k}|\\psi_k'\\rangle\\otimes|\\alpha_k\\rangle$$ $$|\\Phi\\rangle=\\sum_{k=1}^{m+n}\\sqrt{q'_k}|\\phi_k'\\rangle\\otimes|\\alpha_k\\rangle$$ Does this seem correct?"}
{"id":"18465","title":"Why is the angle of a pendulum as a function of time a sine wave?","text":"OK so I'm trying to understand why the angle of a pendulum as a function of time is a sine wave. I can't really find an explanation online and when I do find something partial there are certain symbols I don't understand. $$\\frac{d^2\\theta}{dt^2} + \\frac{g}{l}\\sin\\theta = 0$$ This is the equation I found on Wikipedia.   1. What I don't understand here is the part $d^2\\theta\\over dt^2$, from what I know $d$ means instantaneous delta, instantaneous rate of change, why does the upper part of the function has the square sign right after the $d$ ($d^2\\theta$) and the lower part of the fraction has the square sign after the $t$ and not after the $d$ ($dt^2$).   2. This part still doesn't really show the answer to my question cause the change of the angle and the time are squared so it doesn't mean much to, I'm hoping for an explanation that is simple as possible and intuitive as possible for why the angle as a function of time is a sine wave, hopefully as much mechanics as possible ans less math, a good reference is also good. EDIT: OK, I Understand the first part (the square sign notation), what I still don't understand is how we can get from the the equation I wrote above, or from the acceleration as function of the angle: $a = g\\sin\\ \\theta$, to an equation that shows the angle as a function of time. It's confusing for me since the angle itself changes all the time, in both eqaution (the one at the top and $a = g\\sin\\ \\theta$ we have $\\theta$, but $\\theta$ itself changes all the time! It seems it's not possible to use \"little math\" here, so use math where necessary. Thank you."}
{"id":"51505","title":"What are virtual excitations in the context of superconductors?","text":"In a superconductor I have read that the energy gap can be defined as the energy difference between the ground state and the virtual excitations of the system. Is this the energy required to form a cooper pair?"}
{"id":"23543","title":"Are quantum mechanics and quantum physics the same field?","text":"What is the difference between _quantum mechanics_ and _quantum physics_?"}
{"id":"23547","title":"Centripetal\/Gravitational Force","text":"Suppose a satellite is orbiting the Earth. The gravitational and centripetal force supposedly point towards the Earth. Therefore, the net force is towards the Earth. Since the satellite doesn't fall immediately towards the earth, what's the third force holding the satellite in orbit? First impression tells me its the elusive \"centrifugal force\", but I was told to use this with caution as it's often a misnomer. What's the correct line of reasoning behind this?"}
{"id":"97976","title":"Are Lifshitz and Berestetskii right in this case?","text":"In the Quantum electrodynamics book (look at the problem) its authors Lifshitz and Berestetskei claim that operator of charge conjugation $\\hat {C} = -\\alpha_{2}$ in Majorana basis transforms as $\\hat {C}^{M} = \\alpha_{2}$. Here (they started from the standart (Dirac) representation of the gamma-matrices) $$ \\alpha_{2} = \\begin{pmatrix} 0 & \\sigma_{y}\\\\\\ \\sigma_{y} & 0\\end{pmatrix}, \\quad \\beta = \\begin{pmatrix} 1 & 0\\\\\\ 0 & -1\\end{pmatrix}, $$ $$ \\hat {C}^{M} = \\hat {U}^{+} \\hat {C} \\hat {U}, \\quad \\hat {U} = \\frac{1}{\\sqrt{2}}(\\alpha_{2} + \\beta ) = \\frac{1}{\\sqrt{2}}\\begin{pmatrix} 1 & \\sigma_{y}\\\\\\ \\sigma_{y} & -1\\end{pmatrix} = \\hat {U}^{+}. $$ I tried to get their result, but I only got $$ \\hat {C}^{M} = \\frac{1}{2}\\begin{pmatrix} 1 & \\sigma_{y}\\\\\\ \\sigma_{y} & -1\\end{pmatrix}\\begin{pmatrix} 0 & \\sigma_{y}\\\\\\ \\sigma_{y} & 0\\end{pmatrix}\\begin{pmatrix} 1 & \\sigma_{y}\\\\\\ \\sigma_{y} & -1\\end{pmatrix} = \\begin{pmatrix} 1 & 0\\\\\\ 0 & -1\\end{pmatrix}, $$ because $\\sigma_{y}^{2} = 1$. Where is the mistake? **An edit**. I found the mistake. I used the wrong definition of transformation of charge conjugation operator under unitary spinor transformation. The correct one is $\\hat {C}^{'} = \\hat {U}\\hat {C}\\hat {U}^{T}$."}
{"id":"129275","title":"Where do flying gas molecules get their energy from?","text":"If you heat up water in a tank to boiling point, some of the water will turn to steam. Gas bubbles are created in the water. According to Archimedes principle, the gas bubbles will have a force equal to the weight of what they are occupying. Let's denote this force $F_g$. $$F_g = \\text{Bouyant force of a gas bubble}.$$ Let's denote the height of the tank $h$. We also assume the steam bubbles are created at the bottom of the tank and also that the viscosity of water is nearly zero. So the work done when a bubble reaches the surface is $$\\text{Energy} = (F_g-g m)h$$ $g$ is the gravitational acceleration and $m$ the mass of the bubble. Where did this energy come from? From the surrounding water? From the boiler? From the bubble itself? Can someone in detail explain to me this phenomena?"}
{"id":"110499","title":"Does the doppler shift have any effect on the energy that the atom absorbs?","text":"Does an atom moving towards the light that is falling on it, absorb more energy than when it is stationary? The atom should a see blue shift due to Doppler effect so the atom will see a light of greater frequency, causing it to absorb a greater amount of energy (or, equivalently, it will absorb less energy when it is moving away from the light due to red shift). Or will it absorb the same energy in all the cases? Does the atoms motion have any affect on the energy it absorbs?"}
{"id":"86974","title":"Why are high voltage lines “high voltage?”","text":"If I have two spheres of the same size and one sphere has a small amount of charge compared to the other that has a lot more charge, then clearly the sphere with the larger charge has a larger voltage (relative to the ground). My question is do high voltage power lines have a lot more charge that is placed on them? Is that what gives them the high voltage? I think I have a grasp of the step up stations that use transformers to kick up the voltage from power plants. This question seems almost silly to me but I have been struggling with this for a long time. I’ve done several searches online and I am not able to find answers. If there is a link that someone can provide, I much appreciate it."}
{"id":"74625","title":"Does alternating current (AC) require a complete circuit?","text":"This popular question about \"whether an AC circuit with one end grounded to Earth and the other end grounded to Mars would work _(ignoring resistance\/inductance of the wire)_ \" was recently asked on the Electronics SE. ![Picture edited from the one in the above link](http:\/\/i.stack.imgur.com\/fcz1i.png)   _(Picture edited from the one in the above link)_ Though I respect the AC\/DC experts there, I think _(with the exception of the top answer)_ they are all wrong. * * * My issue is that they all assume that AC requires a complete circuit in order to function. However, my understanding is that a complete circuit is necessary for DC, but not AC. My intuitive understanding is that AC is similar to two gas-filled rooms with a pump between them - the pump couldn't indefinitely pump gas from one room to another without a complete circuit _(DC)_ , but it could pump the gas back and forth indefinitely _(AC)_. In the latter case, not having a complete circuit just offers more resistance to the pump _(with smaller rooms causing a larger resistance)_. Is my understanding correct - **can AC circuits really function without a complete loop?**   More importantly, **what are the equations that govern this**?   If larger isolated conductors really offer less AC-resistance than smaller AC conductors, how is this resistance computed\/quantified? Would its \"cause\" be considered inductance, or something else?"}
{"id":"126420","title":"Maybe photon energy is constant as the Universe expands?","text":"This is a question following on from my previous post Time-like Killing vector in FRW metric? For simplicity I take the spatially flat FRW metric in cartesian co-ordinates given by: $$ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)$$ I transform the time co-ordinate $t$ to conformal time $\\tau$ using the relationship: $$d\\tau = \\frac{dt}{a(t)}$$ so that the metric now becomes: $$ds^2 = a^2(t)(-d\\tau^2 + dx^2 + dy^2 + dz^2)$$ We can introduce a function $\\alpha(\\tau)$ giving $$ds^2 = \\alpha^2(\\tau)(-d\\tau^2 + dx^2 + dy^2 + dz^2)$$ One can now see that the FRW metric has a time-like conformal Killing vector field.. Now electromagnetic waves are conformally (or scale) invariant. If one makes the transformation: $$x \\rightarrow \\lambda x$$ $$t \\rightarrow \\lambda t$$ the electromagnetic wave equations are invariant. Thus it seems to me that there is a symmetry here that Nature could obey. If it did so then the implication would be that the energy of electromagnetic waves is constant as the Universe expands.Therefore wave lengths that scale with the Universe are associated with constant energy. However atoms have a fixed wavelength so that their energy would increase as the Universe expands. Thus the cosmological redshift would be explained not by electromagnetic waves losing energy in an expanding Universe but rather by atomic energies increasing. In conventional cosmology one tacitly assumes that atomic energies are constant as the Universe expands. But what is the basis for such an assumption? At least with photons one can offer a symmetry argument, consistent with general relativity, as to why their energy should stay constant."}
{"id":"92229","title":"Problem with ring-method of du Nouy for calculating surface tension","text":"I need to calculate the surface tension of a liquid with the ring-method of du Nouy and the correct formula to use is: > $\\gamma = \\frac{F}{4\\pi R}$ I understand that the surface tension is the force that acts on a certain length which, in this case, is the circumference of the ring. Since the circumference of the ring is $2\\pi R$ where does this extra $\\times2$ come from? Thanks!"}
{"id":"77690","title":"Square bracket notation for dimensions and units: usage and conventions","text":"One of the most useful tools in dimensional analysis is the use of square brackets around some physical quantity $q$ to denote its dimension as $$[q].$$ However, the precise meaning of this symbol varies from source to source; there are a few possible interpretations and few strict guidelines. What conventions are there, who uses them, and when am I obliged to follow them?"}
{"id":"92223","title":"Is there a physical system that emulates mathematical cellular automata?","text":"Theoretical cellular automata have been proposed as models for many physical phenomena from music to quantum mechanics. My question concerns the reverse: **Is there a _simple_ physical system that emulates a theoretical cellular automaton?** A computer running CA software such as Hashlife can be considered an (extremely complex) physical system that behaves according to the rules of the programmed CA. I am however looking for something much simpler, such as a set of dominoes that topple each other in a fashion conforming to the rules of a certain CA, wave interference patterns that model a CA etc. The point is that there have been enormous advances in CA theory lately (such as the discovery of the first self-replicating pattern in Conway's Game of Life in November 2013) that it would be interesting if these patterns could simply be \"set up\" in the physical world somehow and then they would evolve according to the CA rules without requiring a computer to simulate the rules. In the long run, this might even lead to the creation self-replicating physical systems."}
{"id":"74198","title":"Why was the conversion factor of the metric unit bar chosen the way it was?","text":"The unit **bar** for pressure is clearly a metric unit, but its order of magnitude is a bit strange. In the centimeter–gram–second system of units we have:               1 bar = 1 000 000 baryes = 1 000 000 dyn\/cm²      so the bar is not \"coherent\" with this system (the factor is not one). Also in the meter–kilogram–second (and SI) system we get:               1 bar = 100 000 pascals = 100 000 N\/m²      while in the meter–tonne–second system:               1 bar = 100 pièzes = 100 sn\/m²      So my question is simply, where does the conversion factor for bar come from, since it seems to not fit into usual systems? According to the Wikipedia article _bar_ this unit was created already in 1909 by British meteorologist Shaw, but not much detail is provided. Maybe the factor was simply chosen as the power of ten making the unit closest to the atmospherical pressure at sea level (which is 1.01325 bar by convention, and close to 1.01 bar on average)?"}
{"id":"74196","title":"Is it possible to levitate a toy maglev train using only permanent magnets as long as the train stays in motion?","text":"I am fascinated with magnets, and specifically the idea of magnetic levitation. I recently purchased a Levitron toy maglev train and have enjoyed finding the perfect sweet spot at which it levitates, and I understand that the Levitron does not violate Earnshaw's theorem because it is in rotating motion. I would like to build a toy maglev train that makes no contact with the track, or any guide rails, using only permanent magnets and their repulsive force. I understand that if I attempt to build a flat track with no guide rails, the magnets will not be able to stably levitate, and will shunt the train car to one side or the other. However, what if I built a trough, or a V-shaped track, with the train car sitting inside the track, and a corresponding v-shaped bottom, with repulsive magnets running the length of the track, inside the V? I realize that it wouldn't be stable while sitting still, but if the train were in motion, would that enable it to levitate stably? In my mind there should be a sweet spot where the train's motion overcomes any sideways shunting effects as it travels above the magnetic fields of the magnets in the track, while gravity's pull on the train car into the V would be negated by the repulsive force of the magnets. I haven't been able to find anyone who has tried this approach. Every toy maglev train that I've seen has either used guide rails or a superconductor. My gut tells me it's not possible, but I want to believe. :) Is a configuration like this possible? Is it possible to overcome the instability through one-directional motion?"}
{"id":"61095","title":"Photon as the carrier of the electromagnetic force","text":"My physics background goes as \"far\" as reading popsci books on QM, Particle Physics, and Cosmology so pardon my ignorance in the below questions. I've read that the photon is the particle (quanta in QFT) exchanged between electrons, and that is how the electromagnetic force is explained in QFT.   1. How do we know about these virtual photon exchanges given the fact that we can't observe them directly? How did we arrive at \" ** _electrons exchange virtual photons and that's the cause of the electromagnetic force between them_** \" from merely observing electrons absorbing or emitting photons?    2. If electrons throw photons at each other doesn't that mean that they should only scatter (repel)? If that is so why do magnets and opposite charged particles attract ?   3. I've read that because photons don't posses electric charge therefore they can't exchange between themselves other photons so they can't collide (scatter) in the usual way, unlike the W+, W-, Z0, and gluons which posses the charge of the force they mediate. The only collision processes between photons that I've heard of are Delbruck scattering and the phenomenon of conjuring matter (1 electron , 1 positron) out of 2 photons that collide at a close angle between their trajectories. My third question is: Do these two rare phenomena of photon \"scattering\" happen here on earth naturally (no LHC or other particle accelerators), in the upper atmosphere, or only in the deep dark space? Thanks!"}
{"id":"61094","title":"A sphere, a simple object?","text":"In this video, the woman says that a sphere is a _pretty simple object_. What intrigues me is the use of a sphere for such a calculation. First of all, the sphere wouldn't be perfect as a perfect sphere doesn't exist in nature (first source of imprecision). Second, one will have to use π for calculating the volume. As π is irrational, one will have to use an approximation for it (second source imprecision). So why not using a simple cube? The shape would be closer to a cube due it's harder to make a sphere and it's volume would be calculated more precisely. Why they choose to do this calculation using a sphere?"}
{"id":"111508","title":"Kinetic energy of a gas molecule","text":"I am working through the derivation of the equation $$pV=\\frac13mNC_{rms}^2,$$ but relating to this formula my textbook states that the kinetic energy of an average molecule of gas is given by $$KE=\\frac12mC_{rms}^2$$ but how is this formula right since the mean squared speed ($C_{rms}^2$) is not the same as the average speed of the molecule squared, $v^2$, so why do we use the mean squared speed for kinetic energy and not the means speed squared? Is it just that it is easier to calculate?"}
{"id":"110457","title":"Magnetic Induction - How would different coil configurations affect the outcome?","text":"This is a branch question (from this question) related to magnetic induction: These deal with how different configurations of coiled layers might affect the outcome:   * If one were to bundle multiple layers of coiling to up the coils, would it be more effective to have each layer connected in parallel and running in the same direction? (example - 2 ins and 2 outs)   * If one were to coil 'loosely' such that the wire is wrapping around the coil at a 45 degree angle - would it generate less induction than the same number of coils tighter coiled at nearly a 90 degree angle (pretty sure this is a yes - but wondering if there is any silver lining in terms of resistance - perhaps with a deluge of parallel-connected layers))?   * In the event that two or more layers are connected in parallel - would the energy created across each layer likely be significantly less? In the event of (say) five layers of coiling connecting, each with differing currents (or voltages), how might the resultant current be combined once reunited (presuming a similar wire is receiving the energy)?  I am keenly interested in the variables at play being defined - as well as any information on how such might play out. Thank you for any guidance."}
{"id":"110454","title":"Lippmann-Schwinger solution","text":"What's wrong with this general solution of the Lippmann-Schwinger equation: $$ |\\psi_k \\rangle=|\\phi_k \\rangle+G_k V|\\psi_k \\rangle $$ Taking the inner product with $\\langle\\phi_{k'}|$ \\begin{align} \\langle \\phi_{k'}|\\psi_k \\rangle&=\\langle \\phi_{k'} |\\phi_k \\rangle+\\langle \\phi_{k'} |G_k V|\\psi_k \\rangle \\\\\\ &=\\delta(k'-k)+g(E_k-E_{k'})\\langle \\phi_{k'} |V|\\psi_k \\rangle\\\\\\ &=\\delta(k'-k)+g(E_k-E_{k'})\\langle \\phi_{k'} |H-H_0|\\psi_k \\rangle\\\\\\ &=\\delta(k'-k)+g(E_k-E_{k'})\\langle \\phi_{k'} |E_k-E_{k'}|\\psi_k \\rangle \\end{align} Basic algebra then gives us $$\\langle \\phi_{k'}|\\psi_k \\rangle[1-g(E_k-E_{k'})(E_k-E_{k'})]=\\delta(k'-k)$$ and isolating the inner product on the left hand side gives us $$\\langle \\phi_{k'}|\\psi_k \\rangle=\\delta(k'-k) [1-g(E_k-E_{k'})(E_k-E_{k'})]^{-1}$$ where $g(E_k- E_{k'})$ is the eigenvalue of $G_k$ resulting from hitting $\\langle \\phi_{k'} |$ on the left? This seems to suggest that the eigenstate $|\\psi_k \\rangle $ of the full Hamiltonian $H$ is the same as the eigenstate $|\\phi_k \\rangle$ of $H_0$, up to a proportionality factor, since $|\\psi_k \\rangle $ has no amplitude to be in $|\\phi_{k'\\neq k} \\rangle$."}
{"id":"32196","title":"Are there any well-known theories successfully unifying the inertial and gravitational mass?","text":"From what little I know of general relativity, the equality of inertial and gravitational mass is an axiom of the theory. I suspect that this precludes GR from unifying them in the same sense as electric, magnetic and weak interactions are unified as the electroweak force. But since the two appear to be equal under staggeringly precise measurements, there must be some interest in constructing a theory of mass and motion which fundamentally unifies the two, that is, makes the separation nonsensical \/ impossible. Are there any particularly outstanding attempts in this regard, even if the resulting theories are known to fail to correspond to our universe in some other important way?"}
{"id":"32197","title":"Inverse square law in 2+1 dimensional universe from a Yukawa coupling?","text":"There is a nice result that in 3+1 space time, a Yukawa coupling leads to an inverse square law force as the mass of the scalar field goes to zero. I was wondering what the corresponding force in a 2+1 dimensional Universe would be? Geometrically, I would guess that it should be $1\/r$ rather than $1\/r^2$, but my explicit calculation for a Yukawa coupling gives that it is $1\/r^2$ in 2+1 space as well. I was wondering if that was correct?"}
{"id":"26214","title":"Next crescent moon?","text":"How do I find the date of next crescent moon on wolframalpha or any other site which tells the appearance date time of next crescent on a specific location? _I am searching for it to look for 1st of Ramadan_"}
{"id":"25706","title":"Density of stars near the center of the Milky Way","text":"At night, I can look up and see the Milky Way across the sky. My question is, supposing our solar system was, instead of way out on an 'arm' of the galaxy, if we were near the galactic center. Would the night sky be much brighter?? If so, how close would we have to be to never have darkness of night?? Thanks!"}
{"id":"21975","title":"Does the \"Andromeda Paradox\" (Rietdijk–Putnam-Penrose) imply a completely deterministic universe?","text":"**Wikipedia article:** http:\/\/en.wikipedia.org\/wiki\/Rietdijk–Putnam_argument **Abstract of 1966 Rietdijk paper:** > A proof is given that there does not exist an event, that is not already in > the past for some possible distant observer at the (our) moment that the > latter is \"now\" for us. Such event is as \"legally\" past for that distant > observer as is the moment five minutes ago on the sun for us (irrespective > of the circumstance that the light of the sun cannot reach us in a period of > five minutes). Only an extreme positivism: \"that which cannot yet be > observed does not yet exist\", can possibly withstand the conclusion > concerned. Therefore, there is determinism, also in micro-physics. **Question:** I realize that this is a very philosophically-heavy topic, but at the same time, it is physical in the strictest sense (in that it's not a \"What if\" question, but more of a thought experiment, like the Twin Paradox, that follows the logical implications of relativity to some extreme scenario). So, setting aside matters of \"free will\" and whatnot, which I'm not interested in... If I understand this article correctly: the content of these papers implies a 100% deterministic universe — one in which the future of any object is \"set in stone\"; **but,** at the same time, this future is completely inaccessible to any observer that is in its past (i.e: impossible to predict even given initial conditions, because of small-scale uncertainty, chaos, etc.). Are the proofs in these articles are well-accepted? (Meaning that this \"everything has already happened depending on the observer\" universe is the only conclusion one can arrive at?). I was not aware of the Andromeda Paradox before. I can't imagine it being very useful, but it's interesting enough as a mind-bender that I'd've thought one would hear about it more often."}
{"id":"21973","title":"Factorization of fermionic scattering integral in 2d momentum rep","text":"the scattering integrals for fermions involves both momentum ($k$) and energy ($k^2$) conservation and a nonlinear phase space factor of a distribution function $f(k)$. $$\\begin{multline}I(k) = \\sum_{k_1, k_2, k_3} \\delta(k^2+k_1^2+k_2^2+k_3^2) \\delta(k+k_1+k_2+k_3)\\times \\\\\\ \\Bigl[f(k)f(k_1)\\bigl(1-f(k_2)\\bigr)\\bigl(1-f(k_3)\\bigr) - f(k_2)f(k_3)\\bigl(1-f(k)\\bigr)\\bigl(1-f(k_1)\\bigr)\\Bigr]\\end{multline}$$ In energy space, energy conservation is linear and powerful factorizations are possible to compute the integrals fast. In momentum space, the nonlinear energy conservation constraint complicates factorizations. Trivially, in 2 dim one can work in polar coordinates separating off the radial part. But has anybody seen a more efficient reduction of numerical complexity (e.g. mapping to FFT,...) Thanks for any hint!"}
{"id":"56390","title":"What libraries are available for solving problems at the college physics level?","text":"Are there any libraries available for solving problems at the level of college physics? In particular, I'm interested in libraries for general purpose programming languages that can calculate results both numerically _and_ symbolically. Libraries which could be used for implementing something like this are SymPy (Python), GiNaC (C++), or SymbolicC++ (C++). I'm asking this question because I've started cooking up a C# library for symbolic computation and have used it to solve a few projectile motion problems, but would like to review other work out there. Here's an example of a program which solves such a problem. Here's what the program outputs when run: ![photo](http:\/\/i.stack.imgur.com\/ZX14W.png) Any pointers or suggestions welcome. Thanks!"}
{"id":"3706","title":"Does any particle ever reach any singularity inside the black hole?","text":"I am not a professional physicist, so I may say something rubbish in here, but this question has always popped in my mind every time I read or hear anyone speak of particles hitting singularities and \"weird things happen\". Now to the question at hand, please follow my slow reasoning... As far as I've learned, to reach a black hole singularity, you must first cross an event horizon. The event horizon has this particular property of putting the external universe on an infinite speed to the falling observer. Now due to the Hawking radiation, and knowing that the cosmic background radiation is slowly dimming, sooner or later every black hole in this particular instance of inflation we are living in will evaporate, according to an external observer of said black holes. This means that every black hole has a finite timespan, as long as this universe survives that long. Now if we go back to the falling observer, we had already established that such an observer would see the outside universe \"speed up\" infinitely. This means that when the falling observer \"hits\" the event horizon, he will (or it if we speak about particles, which is clearer in this case), be immediately transported in time towards the final evaporation moment of the black hole. Either this or the particle gets some weird treatment. My point is, such a particle never gets to the singularity, because it has no time to get to it. The moment it crosses the event horizon, the black hole itself evaporates. Where am I wrong here?"}
{"id":"99814","title":"Does a muon detector on Earth's surface correctly measure the mean lifetime of a muon?","text":"Just a simple question. Does a muon detector on Earth's surface correctly measure the mean lifetime of a muon? I would think the answer is no because most muons detected are created about 15 km above Earth's surface, so on average they will have survived for some amount of time calculated using special relativity before reaching the detector. The detector I'm talking about is a scintillator, so it detects when the muon enters the detector with a flash, then if the muon decays it detects another flash. The time between flashes is said to be the muon's lifetime. Is this muon's lifetime the machine giving me the actual lifetime (2.2$\\mu s$), or just the lifetime of it on Earth's surface? here is a plot of my data showing a histogram of decay times ![enter image description here](http:\/\/i.stack.imgur.com\/Oc1yB.png)"}
{"id":"30907","title":"Concentration of fluid pumped through a tube?","text":"I have a bottle filled with a fluid A, and a tube of volume $V_0$ filled with a fluid B leading away from it. When I now start pumping fluid A out of the bottle, I am interested in the concentration $c(V)$ of fluid A in fluid B at the end of the tube after the volume $V$ exited the tube. In my second situation (I am afraid there are two), after having pumped $V_0$, a valve at position $V_1$ switches and the pump is reversed. The mixed fluids are now pumped into a second tube of volume $V_2$. Again, I'd like to know what drips out of there. **Situation 1**               |    A    |     |         |      V0     |      ====================    <- B     |         |     |         |     |_________|      **Situation 2**               |    A    |     |         |          V0     |      =========++==========    <- B     |         | V1  ||     |         |     || V2     |_________|     ||                           ^- B      I've tried using the Hagen-Poiseuille equation (or its proof as given in the Wikipedia), but that grows ugly fast. Is there a simple(r) way to approach this?"}
{"id":"82103","title":"Algorithm to simulate impulse response of a solid?","text":"There are well-documented methods of calculating the impulse response of a room (ex. image method, ray-tracing method); however, I have not been able to find anything similar for the impulse response of an arbitrary solid given its size, density, etc. Can anyone point me in the right direction?"}
{"id":"46048","title":"Tachyon vertex operator (Polchinski's book)","text":"* I would like to know how does Polchinski in his book \"derive\" what is the \"tachyon vertex operator\" (..as say stated in equation 3.6.25, 6.2.11..) I can't locate a \"derivation\" of the fact that $:e^{ikX}:$ is the tachyon vertex operator.  (..I understand that it follows from some application of the state-operator map but I can't put it together..)   * And then what is the meaning of the ``higher vertex operators\" - which are of the form of arbitrary number of either operators of the above kind or the derivatives of $X$ w.r.t either $z$ or $\\bar{z}$. (..like in equation 6.2.18..)"}
{"id":"57620","title":"What does it mean that an inspiral is 'adiabatic'?","text":"Binary systems emit gravitational radiation. This causes the system to lose energy, which results in a shrinking of the semi-major axis. I have read on countless occasions that this 'inspiral' is _adiabatic_ (here for example). What does this mean that the shrinking is adiabatic?"}
{"id":"86887","title":"Slow light and gravitational lensing","text":"It has been proposed that the Sun's gravitational lens be used to observe distant objects, but according to relativity a spacecraft would need to be 550 AU away in order to take advantage of the effect. Would it be possible to instead slow down the light while it is being deflected using, for example, a Bose-Einstein condensate such as a large quantity of very cold sodium placed into orbit around an asteroid? An answer to this question mentioned that it is useless to do the same thing using an ordinary material with a high refractive index because of dispersion, but as I understand it the mechanism for producing slow light changes the group velocity, which is the speed at which energy is carried, instead of the phase velocity."}
{"id":"105675","title":"Hydrogen ionization energy","text":"Proton and electron have a distance equal to the Bohr radius apart in the hydrogen atom. Knowing this, what's the ionization energy of the atom? So we know $U = -kq^2\/a_B$, but this is potential energy, not ionization. How should I proceed?"}
{"id":"59605","title":"What cosmic event would cause Sun rising from the west?","text":"I'm wondering what kind of astronomical or geological events would cause the reversal of Earth's rotation. For instance, Is a meteorite passing very close to Earth able to reverse its rotation? Can the presence of Earth inside extreme magnetic field cause such event?"}
{"id":"102905","title":"Phonon scattering process in raman spectroscopy","text":"I am trying to understand diagrams that involve showing phonon scattering processes that contribute to raman spectroscopy peaks. For example, I drew the one at the bottom of this post. This is supposed to the scattering process responsible for the D peak in graphene. I'm trying to understand exactly what the arrows represent. At initial glance it looks almost as if the phonon scatters and increases energy of magnitude $E_L$ and transitions to another branch, then scatters again, losing energy of magnitude $D$, which is somehow indicative of the D peak? ![enter image description here](http:\/\/i.stack.imgur.com\/dXi9j.png) Thanks!"}
{"id":"102903","title":"Getting nonphysical results when solving for the index of refraction of a slab?","text":"I'm trying to computationally find the refractive indices (real and imaginary) for a thin slab suspended in air (so the only indices to deal with are air and my material's). I've experimentally taken transmission and reflection measurements of intensity in a certain wavelength range. From a textbook chapter I read, $$r_{ij}[n_i, n_j]= (n_i - n_j)\/(n_i + n_j)$$ $$t_{ij}[n_i, n_j]= (2 n_i)\/(n_i + n_j)$$ $$r = (r_{ij}[n_1, n_2] + r_{ij}[n_2, n_3] e^{2i \\pi n_2 d\/\\lambda})\/(1 + r_{ij}[n_1, n_2] r_{ij}[n_2, n_3] e^{2i \\pi n_2 d\/\\lambda})$$ $$t = \\sqrt{n3\/n1} (t_{ij}[n_1, n_2] t_{ij}[n2, n3] e^{ 2i \\pi n_2 d\/\\lambda})\/(1 + r_{ij}[n_1, n_2] r_{ij}[n_2, n_3] e^{2i \\pi n_2 d\/\\lambda})$$ and $$T = |t|^2$$, $$R=|r|^2$$ for transmission and reflection of a plane wave through a 'slab' (though I may be missing a factor of 2 in a couple places. I've used Mathematica style brackets for the functions). From here, $T$ and $R$ are both functions of $n_1$ (the air, ~1) and $n_2$ my material ($n_3$ is air again, here). So mathematically I should be able to figure out $n_2$ for each $T$ and $R$ pair, at each wavelength. So I've done this. I have Mathematica (MM) find the $n_2$ that minimizes $$\\sqrt{(T-T_{exp})^2+(R-R_{exp})^2}$$ at each wavelength. Then, to see how close it is, I plug $n_2$ back into the original equation and plot it against my experimental results to see how close they match up. The problem is, they're matching up incredibly well -- but the results I'm getting for $n_2$ aren't realistic (more specifically, the real part of $n_2$ is negative, and this is not a negative refractive index material...). Here is a graphical example of what I mean (the horizontal axis is wavelength, in units of microns. The vertical axis is unitless for all): ![expT](http:\/\/i.stack.imgur.com\/ziXv4.png) ![ns](http:\/\/i.stack.imgur.com\/VhNxv.png) And here I've plotted the values of $T$ and $R$ from my computed $n_2$'s (green and black) over the first graph. As you can see, they're so similar you can't tell, except at the leftmost tails. ![exptheoryT](http:\/\/i.stack.imgur.com\/xNzzQ.png) What could be going on? One possibility is that though my solution for $n_2$ gives very close values for $T$ and $R$, there are very different values of $n_2$ that give even closer values. Someone I talked to told me that the plane wave model doesn't always apply at some scales -- which blew my mind because I have always seen it used. He said it was the far field solution to dipole radiation but that it might not apply at my length scale. Could anyone verify or refute this?"}
{"id":"113102","title":"Why does a perpendicular force not change speed?","text":"I have been taught that a perpendicular force acting on an object will only change the direction of its velocity, not its magnitude. The explanation that was provided to me is that because there no force in the direction of its velocity, there must be no work done and thus change in the object's kinetic energy. However, I have also studied the vectorial form of the equations of motion $$\\vec v_f = \\vec v_i + \\vec a\\cdot t$$ So I assume that the velocity after a constant force has acted on it for a period of time is simply the vector sum of $\\vec v_i $ and $\\vec a \\cdot t$, which I just assume is the acceleration vector (determined by the force) scaled up by a factor of the time the force acts.Diagrammatically, I took the sum as ![Vector Sum of initial and change in velocity](http:\/\/i.stack.imgur.com\/Mo9VJ.png) It is obviously clear from the diagram that the new speed (which is the length of the $\\vec v_f$ vector) is bigger than the previous speed. This can be confirmed by Pythagoras's theorem, the magnitude of the $\\vec v_f$ vector is $$ |\\vec v_f|=\\sqrt{|\\vec v_i|^2 + |\\vec a|^2 \\cdot t^2} $$ which is very clearly not equal to $|\\vec v_i| $ So, I conclude that the speed of the object has changed even though the force acting on it was perpendicular to its velocity. So my question is, what have I missed? Is my assumption that the final velocity is the vector sum of the initial velocity and the product of acceleration and time incorrect? Or have I made a mistake somewhere else? Thank you in advance for your help."}
{"id":"53817","title":"Muon production in particle accelerator","text":"PAMELA is a particle accelerator which have two concentric rings, protons are accelerated in the inside ring. At ISIS muons are produced when a 800 MeV proton beam collides with a graphite target(carbon 12) can the same be done with this device? The accelerated protons from inside may be sent to the outside ring which will have slow carbon atoms and the collision will produce pions which will decay into muons. Do pions have any magnetic properties which can help to contain them after production?."}
{"id":"53815","title":"The Lorenz gauge in electrodynamics","text":"What is the fundamental reason to fix the Lorenz gauge to $0$?"}
{"id":"34839","title":"How do NASA's Curiosity determine the elemental composition of Mars using spectrometer?","text":"From this article on hindustantimes.com - Curiosity sends data about Martian surface: > The resultant flash of glowing plasma is viewed by the system’s 4.3-inch > aperture telescope, which sends the light down an optical fiber to a > spectrometer located in the body of the rover. There, the colours of light > from the flash are recorded and then sent to Earth, enabling scientists to > determine the elemental composition of the vaporized material. How do NASA's Curiosity determine the elemental composition of Mars using spectrometer? Do just examining the colors of light enabled scientist to determine the elemental composition of any object?"}
{"id":"34835","title":"Having Trouble With The Principle Of Conservation Of Momentum For a Multiparticle System","text":"I'am reading John Taylor's Classical Mechanics chapter 1 page 20 where he proves the principle of conservation of momentum which states \"If the net external force $F^{ext}$ on an $N$-particle system is zero,the system's total momentum $P$ is constant. I'am not sure if i'am having trouble understanding the usage of summation notation or the entire derivation (which includes the concept) itself.Anyways here it is.A five particle system labelled by $\\alpha$ or $\\beta = 1,2,...,5$. The particle $\\alpha$ is subject to four internal forces,shown by small arrows and denoted $F_{\\alpha \\beta}$ and the net external force shown by the large arrow and denoted by $F^{ext}_{\\alpha}$. ![enter image description here](http:\/\/i.stack.imgur.com\/5bjnf.jpg) Thus the net force on particle $\\alpha$ is $$F_{\\alpha} = \\sum_{\\beta \\neq \\alpha}F_{\\alpha \\beta}+F^{ext}_{\\alpha}$$ According to Newton's second law,this is the same as the rate of change of $P_{\\alpha}$:$$\\dot{P}_{\\alpha} = \\sum_{\\beta\\neq\\alpha}F_{\\alpha\\beta}+F^{ext}_{\\alpha}$$ Then he considers the total momentum of the $N$-particle system,$$P=\\sum_{\\alpha}P_{\\alpha}$$ Differentiating wrt time we get $$\\dot{P}=\\sum_{\\alpha}\\dot{P}_{\\alpha}$$ or,substituting for $\\dot{P}_{\\alpha}$,$$\\dot{P}=\\sum_{\\alpha}\\sum_{\\beta\\neq\\alpha}F_{\\alpha\\beta}+\\sum_{\\alpha}F^{ext}_{\\alpha}$$ .The above expression is what is bothering me .I need someone to give me an intuition behind the double summation sign. How come it's $\\dot{P}=\\sum_{\\alpha}\\sum_{\\beta\\neq\\alpha}F_{\\alpha\\beta}+\\sum_{\\alpha}F^{ext}_{\\alpha}$ and not just $\\dot{P}=\\sum_{\\alpha}\\sum_{\\beta\\neq\\alpha}F_{\\alpha\\beta}+F^{ext}_{\\alpha}$. He eventually concludes that $$\\dot{P}=\\sum_{\\alpha}F^{ext}_{\\alpha}\\equiv F^{ext}$$."}
{"id":"106876","title":"Can all the systems have a Hamiltonian description?","text":"I have heard of mechanical systems that might not have a Hamiltonian dynamics, but I cannot figure out an example that supports it. Please help."}
{"id":"83536","title":"Why is lightning considered a plasma?","text":"I was wondering why a lightning bolt is coined as \"plasma\", or a \"spark\" from an electrical wire\/device is as well, yet flares, molten lava, and burning buildings are not(flares are pyrotechnic, exothermic, and very much like flames). I hear arguments saying that plasma is ionized gas, but how can a flare not be ionized at such high temperature(and why isn't ionized, very hot flames considered plasma)? This guy, the top answerer, says that fire is not plasma because it's not ionized: Is fire plasma? Also, some lava exceeds 3,000 Fahreheit. How is that not plasma, and what draws the fine line between plasma and fire itself(not as a state of matter, but as a relation with gas)? Fire is gas, but some say it's plasma if you can see it (check the link the poster added below in the first comment), so when does it become a full-on plasma if it's not normally one?"}
{"id":"98731","title":"Effects of microwaves on optical properties of human eye","text":"I have read a long series of paragraphs on wiki. Previously I thought that microwaves are not harmful to living beings but Wiki does not claim this explicitly. while doing an experiment in my college laboratory our teacher cautioned us not to see the face of Horn antenna. He said it will damage eyes so I am asking here to be sure about that. > What are the effects of microwaves emitted by Horn antenna upon the optical > properties of human eye(a natural optical device)?"}
{"id":"15742","title":"Derivative of the product of operators","text":"I'm asked to show that $\\frac{d(\\hat{A}\\hat{B})}{d\\lambda} = \\frac{d\\hat{A}}{d\\lambda}\\hat{B} + \\hat{A}\\frac{d\\hat{b}}{d\\lambda}$ With $\\lambda$ a continuous parameter Should I use the definition $\\frac{d\\hat{A}}{d\\lambda} = \\lim_{\\epsilon \\to 0} \\frac{\\hat{A}(\\lambda + \\epsilon) - \\hat{A}(\\lambda)}{\\epsilon}$ applied to $\\hat{A}\\hat{B}$ like $\\frac{d(\\hat{A}\\hat{B})}{d\\lambda} = \\lim_{\\epsilon \\to 0} \\frac{\\hat{A}(\\lambda + \\epsilon)\\hat{B}(\\lambda + \\epsilon) - \\hat{A}(\\lambda)\\hat{B}(\\lambda)}{\\epsilon}$ and do some algebra to get the RHS of the first equation, or I'm missing something? Thanks for your time Another interesting derivative to pay attention to is: $\\frac{d}{d\\lambda}\\exp(\\hat{A}(\\lambda) )$?"}
{"id":"29235","title":"Would it matter if the Earth rotated clockwise?","text":"In the Futurama episode _\"That Darn Katz!\"_ they save the world by rotating the Earth backwards saying it shouldn't matter (which direction Earth rotates). If Earth rotated clockwise and remained in it's current counter- clockwise orbit around the Sun, would it affect anything beyond the direction the constellations move across the sky?"}
{"id":"6583","title":"Can Gravity be described in terms of velocities and rotations of local Minkowski space?","text":"In the paper _The River Model of Black Holes_ :   Am.J.Phys.76:519-532,2008, Andrew J. S. Hamilton, Jason P. Lisle   http:\/\/arxiv.org\/abs\/gr-qc\/0411060   The authors give a way of describing the action of a rotating (and or charged) black hole through a collection of local Minkowski frames, that is, as a sort of collection of preferred reference frames, or more accurately, by the tidal effects arising from the movement from one frame to another. Each frame is defined by a \"river field\" $\\omega_{km}$ (See around equation 74). This field is composed as follows:   $\\omega_{0m} = \\beta_m$ is the \"velocity\" of the river, while   $\\mu^i = 0.5 \\epsilon^{ikm}\\omega_{km}$ gives the \"twist\" of the river. Then the motion of objects due to the black hole can be calculated from the tidal change $\\delta\\omega^k_m$ which is a local infinitesimal Lorentz transformation. My question is this: Can this description of a black hole be used to describe general relativity? Note that there is an obvious limitation: since this is based on a flat background metric you can't get worm holes and the like, but I mean, subject to the requirement of trivial topology, can every GR situation be described by a \"river field\"?"}
{"id":"59911","title":"Newton's second law?","text":"$F=ma$. A car strikes a wall at 60 mph. Its acceleration is zero at the time. The force of the car against the wall or vice versa is? To look at the car the force is not zero. Please explain."}
{"id":"59912","title":"Local minima in Ising model in a Monte Carlo simulation","text":"Is there any way to check whether in a Monte Carlo simulation using Ising model is stuck in any (false) local minima of energy or not, particularly in 3D system ?"}
{"id":"39413","title":"Where does the energy for repulsion in charges come from?","text":"If I bring together 2 objects with the same charge, they repel. But even after the repulsion has taken place, there is no loss of charge in the objects. Where does the energy come from then?"}
{"id":"39410","title":"Work Done in an Isobaric Process","text":"I am given the information that an air parcel undergoes isobaric heating from 0° C to 20° C, and that's all I'm given. I have to determine the work done by the parcel on its surroundings. I know that $dW = pdV$, and that $pV = RT$ ($V$ is the specific volume), but I don't know how to go about solving this in terms of the change in temperature and not the change in volume."}
{"id":"39419","title":"Proof of equality of the integral and differential form of Maxwell's equation","text":"Just curious, can anyone show how the integral and differential form of Maxwell's equation is equivalent? (While it is conceptually obvious, I am thinking rigorous mathematical proof may be useful in some occasions..)"}
{"id":"31649","title":"Cause of sea ice freezing during an upwell event","text":"One of the Episodes of BBC's documentary Frozen Planet describes an upwell phenomenon in the Antarctic Ocean. I am trying to understand the cause of seawater freezing when this upwell occurs. Liquid sea water can remain liquid below salt water's one-atmosphere freezing temperature when that salt water is maintained under pressure at deep ocean depths. The high pressure conditions deep in the ocean apparently prevent the salt water from freezing. According to the documentary, shifting ocean currents occasionally bring a deep mass of liquid antarctic sea water from near the bottom of the ocean all the way up to the surface very quickly. And much of that mass of liquid sea water quickly freezes due to the change in pressure. Pressure is reduces when the water moves from the bottom of the Antarctic Ocean and arrives near the surface. If temperature remains constant but the state changes from liquid to solid, I assume this state change is caused by the pressure allowing the sea water to expand and form a crystalline lattice - ice. Someone watching the BBC documentary with me believes the ice formed in such an upwell event is caused by the molecular movement of the water molecules slowing down with the reduction in pressure, and she argues the slowing of movement brought on by reduction in pressure causes the ice to form. She asserts the reduced molecular movement is the cause, not the physical expansion of the overall volume of the water as it rises. My understanding is that the extreme high-pressure water simply doesn't have enough physical space to expand into a volume which will allow the molecules to form a lattice, and this has little or nothing to do with the molecular movement. What is the correct way to understand this phenomenon?"}
{"id":"100430","title":"Why is $\\vert \\phi \\vert ^2$ infinite in QFT?","text":"I've read here¹ that for a scalar field $\\phi$, the square $\\vert \\phi \\vert ^2$ is infinite (which gives an infinite contribution to mass), more precisely: > the square of the field – a quantity which diverges in QFT as necessary > consequence of the commutation rules of the theory and unitarity. It seems that this is related to the fact that $\\phi$ is a distribution and the square would be the correlation function. > if we compute in quantum field theory a correlation function like $\\langle > A(x)A(y)\\rangle$ and let $x\\to y$, we find a divergent quantity. Is there a simple way to see why is this true, with little knowloedge of QFT?   1. Inertial Mass and Vacuum Fluctuations in Quantum Field Theory, Giovanni Modanese, PACS: 03.20.+i; 03.50.-k; 03.65.-w; 03.70.+k 95.30 Sf"}
{"id":"55084","title":"Entropy change relation to the number of lost bits","text":"Can we use entropy change value to define the (perhaps fractional) number of bits of information that are lost by and, or, xor gates?"}
{"id":"63660","title":"Irreducible Representations Of Lorentz Group","text":"In Weinberg's The Theory of Quantum Fields Volume 1, he considers classification one-particle states under inhomogeneous Lorentz group. My question only considers pages 62-64. He define states as $P^{\\mu} |p,\\sigma\\rangle = p^{\\mu} |p,\\sigma\\rangle $, where $\\sigma$ is any other label. Then he shows that, for a Lorentz Transformation : $$P^{\\mu}U(\\Lambda)|p,\\sigma\\rangle = \\Lambda^{\\mu}_{\\rho} p^{\\rho}U(\\Lambda)|p,\\sigma\\rangle $$ Therefore: $$U(\\Lambda)|p,\\sigma\\rangle = \\sum_{\\sigma'} C_{\\sigma' \\sigma}(\\Lambda,p)|\\Lambda p,\\sigma'\\rangle.$$ Then he wants to find $C$ in irreducible representations of the inhomogeneous Lorentz group. For any $m$ he chooses a $k$ such that $k^{\\mu}k_{\\mu} = - m^2$. Then defines express $p$'s with mass m, according to $p^{\\mu} = L^{\\mu}_{v}(p)k^v$. Then he defines $$|p,\\sigma\\rangle = N(p)U(L(p))|k,\\sigma\\rangle$$ (where $N(p)$ are normalization constants). I didn't understand this last statement. Is $\\sigma$ an eigenvalue of the corresponding operator, or just a label? I mean, if $J |k,\\sigma \\rangle = \\sigma |k,\\sigma\\rangle $ then is it true, $J |p,\\sigma\\rangle = \\sigma |p,\\sigma\\rangle$. If so how can we say that if $$U(\\Lambda)|k,\\sigma\\rangle = \\sum_{\\sigma'} C_{\\sigma' \\sigma}(\\Lambda,k)|\\Lambda k=p,\\sigma'\\rangle$$ Thanks for any help. First pages of these notes on General Relativity from Lorentz Invariance are very similar to Weinberg's book."}
{"id":"132597","title":"How to define tensor contraction without referring to summation?","text":"The textbook defines a tensor to be an element in $(T^*)^k×T^l→R$. It then expresses tensors as arrays of components with respect to a certain basis, and defines tensor contraction using summation convention. My question is: Can we define tensor contraction without referring to basis, components, and summation, but just the definition of tensor as a multilinear function? If components are inescapable, why bother define tensors as multilinear functions?"}
{"id":"18265","title":"Is there a theory which treats particles as classical point singularities?","text":"Is there a published theory that looks at all matter as occupying no space and only being felt because of its gravitational pull? We've been taught in school that matter has mass and occupies space. I was just wondering if anyone tried to look at matter as having no physical reality..."}
{"id":"72354","title":"Fourier transform of two pulses of light","text":"I have laser beam path that fires two pulses of light in a gaussian distribution, so the intensity graph over time is two identical gaussians separated by a distance $t_0$. In other words, a gaussian convolved with two $\\delta$ functions equidistant from the origin. When I take the Fourier transform of this, I get a graph with beats, but I'm having trouble figuring out what this graph means physically. When we feed this beam through a spectrometer, we are trying to find out if we should see fringes, and if so, what their frequency is. Any help would be appreciated."}
{"id":"91253","title":"Virial theorem and variational method: a question","text":"I have an hydrogenic atom, knowing that its ground-state wavefunction has the standard form $$ \\psi = A e^{-\\beta r} $$ with $A = \\frac{\\beta^3}{\\pi}$, I have to find the best value for $\\beta$ (using the variational method). After having included the Darwin correction $$ H_D = D\\delta(r) $$ with $D = \\frac{\\alpha^2\\pi Z}{2}$, into the Hamiltonian $$ H_0 = -\\frac12\\nabla^2-\\frac{Z}{r} $$ I calculated $$\\langle\\psi(\\beta)|-\\frac{\\nabla^2}{2}|\\psi(\\beta)\\rangle = \\langle \\psi(\\beta)|T|\\psi(\\beta)\\rangle = \\frac{\\beta^2}{2}$$ then I thought to use the virial theorem to calculate the part $\\langle V\\rangle $: $$ \\langle \\psi(\\beta)|-\\frac{Z}{r}|\\psi(\\beta)\\rangle = \\langle \\psi(\\beta)|V|\\psi(\\beta)\\rangle = -2\\langle \\psi(\\beta)|T|\\psi(\\beta)\\rangle = -\\beta^2 $$ with the Darwin part left to be easily calculated. Looking at the solutions that my professor wrote, I've just found a different result: > $$\\langle V\\rangle = \\langle \\psi(\\beta)|-\\frac{Z}{r}|\\psi(\\beta)\\rangle = > \\frac{Z}{\\beta}\\langle \\psi(\\beta)|-\\frac{\\beta}{r}|\\psi(\\beta)\\rangle = > -2\\frac{Z}{\\beta}\\frac{\\beta^2}{2} = -\\beta Z$$ Why's that? Is this somehow related to the fact that I'm going to use the variational method later? Please, help me understand."}
{"id":"130996","title":"A common definition of a scalar","text":"Some dictionaries define a scalar as follows: > A quantity, such as mass, length, or speed, that is completely specified by > its magnitude and has no direction. -- The Free Dictionary However, it is my impression that in many contexts scalars can be signed, in which case their magnitude (their absolute value) does not specify its value. This definition is even used on a test question here. Is it true that this definition is inaccurate?"}
{"id":"15093","title":"Best way to solve P-V-T Question","text":"Typically in an exam or problem set we have a problem that's like this: Determine the specified property at the indicated state of water. Locate the state on a sketch of the T-v and P-v diagram. p= 3 bar, T= 240°C. Find v, in $m^3\/kg$. I believe I can figure out $v,u,h$ when it ask to find this information, however I'm having trouble figuring out what's the best way to look at the steam table and figure out how to draw the correct information. For reference I've provided what the answer to this problem is. Here is the solution: ![Problem Set Answer](http:\/\/i.stack.imgur.com\/XDuc3.png) Thanks,"}
{"id":"130662","title":"EmDrive Cavity modes","text":"The EMDrive described in patent application GB 2493361 is a simple section of a sphere surrounded by a superconducting cooling system. I have attempted to solve the oscillation modes possible and find spherical Bessel functions for radial directions and Legendre polynomials for \"theta\". When I apply boundary conditions, $E_r$ must be zero at the side walls so $P_n^0$(cos($\\theta_w$)) = 0 can only happen for specific angles. If the wall is not at the right angle, $E_r$ must be zero every where. Same is true for $E_\\phi$, but that has zeros for $P_n^1$(cos($\\theta_w$)) so it would be a different mode. The claim in the patent is that putting RF into the cavity creates a force - but it does not say in what direction! The EmDrive web site talks about group velocity, but I don't see how that makes sense for a standing wave in a cavity. What am I missing? I can find $B$ from $\\nabla\\times E$), but is there an easy way to find energy density directly from the $E$ field solutions? Once I know energy density, finding forces should be straight forward. What I expect is the net total force to be zero, but it would be nice to prove it."}
{"id":"130663","title":"Why does $\\nabla \\to ik$ when you Fourier transform?","text":"I am reading a text that describes the scattering of light by a particle with dielectric constant $\\epsilon$ After a bit of maths starting from Maxwell's equations they obtain: $$\\nabla (\\nabla \\cdot E(r)) - \\nabla^2E(r)=\\mu_0\\omega^2\\epsilon(r)\\cdot E(r)$$ then say \"Fourier transforming with respect to $r$ gives (replace $\\nabla$ by $ik$):\" $$[k^2 \\hat{I}-kk]\\cdot E(k) = \\mu_0 \\omega^2 \\int \\epsilon(r) \\cdot E(r) \\exp(-ik \\cdot r) \\,\\mathrm{d}r$$ I don't understand why all $\\nabla$ have turned into $ik$? Is there any way to visualize why this can be done?"}
{"id":"135010","title":"Why can't angular momentum be used in flying vehicles?","text":"![Animation of relationship between torque and momentum](http:\/\/i.stack.imgur.com\/yIa9K.gif) If angular momentum ( **L** ) works like this animation from Wikipedia leads me to believe, why can't we put a large flywheel or several small ones on a chassis, all rotating counter-clockwise around their vertical axes and move the system upwards? It seems that there's nothing countering **L**. Am I missing something? I think the concept of such a vehicle would break Newton's first law, but right now to me that's like saying 'you can't because Newton said so'. Can anyone provide a more detailed explanation that someone with high-school- level knowledge of physics can understand?"}
{"id":"128810","title":"Bloch theorem, Energy, Free electron","text":"I'm trying to learn on my own a bit of solid physics to tackle semiconductors afterwards. I'm struggling with the Energy versus $k$ diagrams for a free electron which shows that for a single value of $k$ we can have many energies value although that there's only one band (in this example). Bloch's theorem tells us that we can label the energies the system can take with a vector $k$ and an integer $n$ (band index). And the theorem tells us also that for any vector $G$ in the reciprocal lattice we have: $E(n,k)=E(n,k+G)$ Let's apply this to the free electron. This tells us that (there's only one band index) $$\\frac{\\hbar^2}{2m}\\lvert k\\rvert^2 = \\frac{\\hbar^2}{2m}\\lvert k +G\\rvert^2 $$ for all $G$ in the reciprocal lattice$ \\lvert k\\rvert=\\lvert k+G\\rvert$ which is absurd. We find another contradiction by making the following reasoning: Since the potential is constant, we can consider that the potential is periodic with respect to a lattice with arbitrary vectors $a_1,a_2,a_3$. (In 1D, all that I'm saying is that a constant function can be seen as 1-periodic or 100-periodic or 0.0001 periodic..). So by using this fact we can prove that there is only one possible value for $E$ which is also absurd. Can someone please explain in detail what am I doing wrong?"}
{"id":"128814","title":"Does Free Will Theorem imply that quantum mechanics plays crucial role in our brain’s functioning (consciousness)?","text":"Is opposite statement of Free Will Theorem right: If elementary particles have a certain amount of free will, then so must we? Because to me elementary particles does have a bit of free will – quantum mechanics guarantees that nobody can predict what one is going to do, say in double slit experiment. So Penrose was right and origins of our consciousness lie in the laws of quantum mechanics? Is the only way our free will can come from is that of quantum mechanics?"}
{"id":"119270","title":"Can any gas be liquidified by lowering the temperature?","text":"Can any gas be liquidified by lowering the temperature? What happens with gases at absolute zero? Are there gases that remain gases at absolute zero? Do their molecules move at these temperatures? Does the gas acquire a crystallic structure at these temperatures while still remaining gaseous?"}
{"id":"123711","title":"Why is powdered sugar less dense than granulated sugar?","text":"The density of powdered sugar is about 20% lower than that of granulated sugar. There is small composition difference between the two - about 3% cornstarch and 0.5% water in powdered sugar, but that's it. It seems this is an issue of packing then. Why does powdered sugar pack worse than granulated sugar?"}
{"id":"7395","title":"Are regular light bulbs better for the eyes than CFLs or \"tube lights\"?","text":"I've heard that regular light bulbs with a filament are better for the eyes. Is the spectrum of one worse than the other? If so, are there any regulations for their use in industrial settings for worker safety?"}
{"id":"7391","title":"Difference between Higgs and anti-Higgs Fields","text":"I'm assuming the LHC can create a Higgs and an anti-Higgs boson. If so, would their fields be identical with respect to mass effects? How would LHC detectors distinguish between the two bosons?"}
{"id":"121503","title":"Angular momentum for 3D harmonic oscillator in two different bases","text":"I know that the energy eigenstates of the 3D quantum harmonic oscillator can be characterized by three quantum numbers: $$ | n_1,n_2,n_3\\rangle$$ or, if solved in the spherical coordinate system: $$|N,l,m\\rangle$$ The relationship between capital $N$ and the little $n_i$'s is straightforward: $N=n_1+n_2+n_3$, but this can't be said for the other quantum numbers. I want to find a way of relating the two representations, but I'm not sure how to go about doing this (my linear algebra background is pretty weak). Let's say I fix the energy to be $\\frac{5\\hbar \\omega}{2}$, which is equivalent to saying $N=1$. There are three states corresponding to this situation in the first representation: $| 1,0,0\\rangle, |0,1,0\\rangle,| 0,0,1\\rangle$. But what are the corresponding states in the second representation? If $N$ is fixed to be $1$, what are the allowed values of $l$ and $m$? I remember that $l=0,...,N-1$, and $-l\\leq m\\leq l$, but this makes no sense, since it would mean both $m$ and $l$ would have to be $0$..."}
{"id":"50107","title":"What is the difference between center of mass and center of gravity?","text":"What is the difference between center of mass and center of gravity? These terms seem to be used interchangeably. Is there a difference between them for non-moving object on Earth, or moving objects for that matter?"}
{"id":"45135","title":"A problem concerning the force between currents or moving electrons","text":"Concerning two identical wires carrying the same current (same direction, speed and magnitude), they will be attracted because of the Ampere force. But when I was in the frame moving with the same speed as the electron in the current, why will there be the same force between each wire (supposing that the electrons are moving at a speed far less than $c\/10$). Also, when two bundles of electrons are moving in the vacuum at the same speed, will they be attracted because of the Ampere force? Will they be dis- attracted because of the Coulomb force?"}
{"id":"133507","title":"Intense of liner polarized wave: with a polarizer vs. without a polarizer differences","text":"We had this experiment in which we measured the intense of **linear polarized** wave - with and without a polarizer. I noticed that **without the polarizer** the intense was slightly lower than **with the polarizer**. How is that possible? ![enter image description here](http:\/\/i.stack.imgur.com\/mTQR7.png) The wave length was about 2.8 [cm]. With polarizer I got 0.317 [Volt] . Without polarizer I got 0.2835 [Volt]."}
{"id":"133504","title":"Bulk-boundary cutoffs in AdS\/CFT","text":"I'm studying the holographic entanglement entropy (HEE) in this paper (Ryu- Takayanagi, 2006). In section 6.3 they compute the HEE for a segment in a 2D CFT. To do so, they obtain the corresponding geodesic in the bulk (in the Poincaré patch) and compute its length. I understand all that process, but I'm having some trouble when they introduce the cutoff. The metric diverges when $z\\to0$ so we introduce a cutoff $\\epsilon>0$, I understand that. But then they say > Since $e^\\rho\\sim x^i\/z$ near the boundary, we find $z\\sim a$ Here, $\\rho$ is the hiperbolic radial coordinate un the global coordinates for AdS, $$ ds^2 = R^2(-\\cosh^2\\rho\\ d\\tau^2 + d\\rho^2 + \\sinh^2\\rho\\ d\\Omega^2) $$ $x^i$ and $z$ are coordinates in the Poincaré patch, $$ ds^2 = \\frac{R^2}{z^2}(dz^2-dt^2+\\sum_i(dx^i)^2) $$ And $a$ is the inverse of the UV cutoff of the CFT in the boundary, that is, the spacing between sites. I have two problems: 1) First, I don't see why near the boundary $e^\\rho\\sim x^i\/z$. I made up the relations between both coordinate systems and I find more complicated relations than that (even setting $z\\sim0$). 2) Even assuming the previous point, I don't understand why we obtain that relation between the CFT and the $z$ cutoff."}
{"id":"75258","title":"Pseudo-randomness of observables","text":"There is a somewhat recent paper by Colbeck and Renner that, given the assumptions — 1) QM accurately predicts the correct statistical results in experiment (which so far has always been found to be true) and 2) the measurement can freely be chosen — that no extension to quantum mechanics can have improved predictive power. I've taken basic QM and I just started taking QFT so understanding the paper is somewhat beyond my skills; so perhaps someone can help me with this situation: Suppose you have a pseudo-random number generator (e.g. it could be any function really as long as its outputs appear random and the appropriate correlations [entanglement] between different positions exist) that is a function of position and time: $F(\\vec{x}, t)$. If you knew the algorithm for this generator, you could plug in the spatial coordinates and time of your experimental measurement*, and get information that allows you to predict the exact* eigenvalue of the linear operator that corresponds to your measurement. This hypothetical PRNG would be for the whole universe, so I suppose it would count as a non-local hidden variable. My question is: why does Colbeck and Renner's paper rule out this possibility? (*Okay, you can't plug in the exact coordinate and time of your measurement because of HUP. Perhaps you just get _improved_ predictive power over QM by giving close enough coordinates and time)."}
{"id":"4349","title":"Are W & Z bosons virtual or not?","text":"W and Z bosons are observed\/discovered. But as force carrying bosons they should be virtual particles, unobservable? And also they require to have mass, but if they are virtual they may be off-shell, so are they virtual or not."}
{"id":"71735","title":"What is the optical illusion called where our sun seems to disappear from view and then rise on the third day? And where can it be viewed from?","text":"What is the optical illusion called where our sun seems to disappear for 3 days and then rise again into view on the third day? And where can it be viewed from?"}
{"id":"17801","title":"Dimensional Analysis: Buckingham Pi Theorem Using a Matrix","text":"I have the following problem - which is basically that $\\omega = f(g,h,l)$ now the book claims that the two $\\pi$ terms as follows $\\pi_1 = \\omega \\sqrt{\\frac{l}{g}}$ and $\\pi_2 = \\frac{h}{l}$. Now we know we can get different dimensional $\\pi$ terms because of the nature of the theorem. The ones I get however I obtain by solving $Ax=0$ for a matrix $A$ so my solution space looks like this $x = \\begin{pmatrix} a \\\\\\ b \\\\\\ c \\\\\\ d \\end{pmatrix} = c \\begin{pmatrix} 2 \\\\\\ -1 \\\\\\ 1 \\\\\\ 0 \\end{pmatrix} + d \\begin{pmatrix} 2 \\\\\\ -1 \\\\\\ 0 \\\\\\ 1 \\end{pmatrix} $ such that my $\\pi$ terms are then $\\pi_1 = \\frac{\\omega^2 h}{g}$ and $\\pi_2 = \\frac{\\omega^2 l}{g}$ However I am uncertain as to whether or not this is the correct answer. It seems right to me given the theorem, but I don't see any way to confirm that."}
{"id":"87160","title":"Flow of water through sand","text":"Imagine a set up with a column of sand with porosity $\\phi$ and permeability $k$ contained in a cuboid container. When one pours water evenly (let's say $V_w\/A$ cubic meters per square meter) on the top surface of the sand at the same time, how does the distribution of water (water content at a particular height) of the sand vary over time (until the water reaches the bottom, and after the water reaches the bottom)? The sand may not be saturated."}
{"id":"29830","title":"Helium Fusion and the Origin of Elements","text":"In the fifth of his Messenger Lectures Feynman mentions the fact that Carbon-12 has an energy level at 7.82 MeV. He then states that this is what enables the fusion of three Helium-4 ions into Carbon and ultimately enables the production of the other elements. Moreover one can apparently calculate that an energy level of 7.82 MeV is necessary in advance. Now unfortunately I have almost no knowledge of nuclear physics and therefore do not know how fusion is modelled theoretically. And I do not know if it is hard or easy to estimate that energy level (i.e. it could simply be a matter of if energy conservation allows the process). So my question are: What goes into this kind of calculation? and more generally: What are the current methods for a quantitative (qualitative) understanding of nuclear fusion? Naively I would model the fusion process by an estimate for the transition amplitude between a $\\lvert \\text 3{He}\\rangle$ state into a $\\lvert C\\rangle$ state maybe by Fermi's golden rule. This should give you approximately a delta function $\\delta(E_f - E_i)$, where $E_f$ and $E_i$ are the energies of the initial and final state. Then you would need to estimate the energy levels of the $3\\text{He}$ system and one of them would be approximately at 7.82 MeV. You can assume in your answers that I took graduate level courses on quantum mechanics and quantum field theory, if that helps."}
{"id":"100311","title":"Difference between horizon and flatness problems & how inflation solves flatness (w\/out math)","text":"Layman here so I'm hoping for an answer for my query that doesn't involve math. I'm reading about inflation and how it solves the flatness and horizon problems. I get that the horizon problem deals with the uniformity of matter\/energy in the universe. Flatness deals with the curvature. But isn't the universe flat because of the uniformity of matter\/energy? Or is it the uniformity plus also the density (i.e. how far apart everything is)? In this way, is it correct to say that the flatness problem encapsulates the horizon problem? Finally, is there any way to explain why inflation would have given this particular density without using math?"}
{"id":"118448","title":"What does it mean to \"solve an equation\"?","text":"I don't understand what is meant by there being a \"solution\" to an equation. For example, what does a solution to the wave or heat equation represent, and _what are we solving for_? Of course, we can use such equations to perform computations, but what can the \"solutions\" tell us, and how can we use them? (Please excuse my knowing almost nothing about differential equations, but I am curious)"}
{"id":"66833","title":"What is a \"lateral fringe displacement\"?","text":"I encountered the term \"lateral fridge displacement\" in my optics homework for a problem about inserting a thin plate of glass over one of Young's double splits. So what does \"lateral fringe displacement\" mean in this context? I Googled but found nothing quite useful. Most of the results are about how this phenomenon emerges, but not specific about the exact definition."}
{"id":"66834","title":"Problem counting spin states","text":"I can't figure out how many different spin states I can create with a four- electron system. I think I can create a spin-zero state, three spin-one states, and five spin-two states. That gives me nine possible states altogether. My problem is that I can specify a maximum of eight (complex) numbers to completely describe the spin states of the four electrons. But the nine spin- states I can seemingly create correspond to the spherical harmonic functions and I know for sure that they are linearly independent. It seems very wrong. This discrepancy in the counting doesn't appear until you get to four electrons. It gets worse as you add more electrons. Does anyone else have a problem with this? EDIT: Thanks for the excellent answers, especially from Lagerbeer. It turns out I was even more messed up than I thought I was. I don't run into trouble with _four_ electrons....I'm already in trouble with _three_. I didn't realize it because I was counting 2n electron-parameters instead of n^2, so I had 2, 4, 6, 8... (parameters to describe electron spin) as opposed to 2, 4, 8, 16... as people have pointed out. And this has to relate to (l,m) spin states of 2, 4, 6, 9, 12, 16... So there's already a problem with three electrons and it boils down to this: you have two different states with z-axis spin 1\/2: the are (3\/2, 1\/2) and (1\/2, 1\/2). To make them up from electrons you have at your disposal these three states: {A} = duu {B} = udu {C} = uud The obvious thing to do is add the three together (and of course normalize); and I believe when you do, you get the (3\/2, 1\/2) state. The question is: how do you create the (1\/2, 1\/2) state? I think I know the answer and I've posted it on my blog. Anyone want to take a stab at it?"}
{"id":"126622","title":"Quantum entanglement and spooky action at a distance","text":"When quantum entanglement is explained in \"layman's terms\", it seems (to me) that the first premise, that we have to accept on faith, is that a particle doesn't _have_ a certain property (the particle is not IN one state or another) until that property is measured. For example, I have read that a particle may have \"spin up\" or \"spin down\" but -- here's where I get lost -- it's not that we don't KNOW which spin the particle has until we measure it, but actually the particle has NEITHER spin until we measure it. Now, if we accept that premise, then we can see how spooky it is, when two particles are created with correlated spins -- if one has spin up, the other has spin down, but neither particle \"has\" either spin until one of them is measured. At that instant, the other particle \"assumes\" the other spin. At least that's how I interpret things I have read. Certainly, if we accept that a particle doesn't actually HAVE the property until it is measured, but rather the particle exists in a superposition of states until the measurement is made, then we \"regular citizens\" can understand what is meant by \"spooky action at a distance\". (As soon as one particle is measured, the other particle assumes the \"other\" spin, no matter how far away it is. How did the second particle know the first particle was measured? That, I think, is the spooky action at a distance (and the information transfer can be faster than light).) However, from a layman's perspective, I want to cry out \"but the particle does have a definite spin, we just don't KNOW what it is, until it is measured! Duh!\" Now, I can't believe that all of the physics community hasn't thought of that objection -- but -- here's my point -- although the spooky action at a distance can be explained to us, once we accept the premise of superposition -- Why isn't the fact of superposition explainable also, in terms mortals can understand? I have looked, but I haven't found a layman's explanation of why we should accept the fact that a particle doesn't have a particular spin, or other property, until it's measured. It sure seems that the particles \"have\" the properties, even though we haven't measured them yet. I know that much of quantum mechanics is not \"intuitive\". If anyone can explain why particles don't have a definite property, even before we measure them, I would be grateful. Thanks."}
{"id":"95926","title":"Does Galileo's Tower of Pisa argument contradict quantum mechanics?","text":"(My questions are at the end, but they may not mean much without explanation below.) Galileo argued that because the mass of a falling object can always be redistributed in ways that asymptotically approximate a set of smaller falling objects, e.g. linked by light strings, then the equations of motion for an object falling in a vacuum cannot depend on its mass. That is, because dependency on mass would lead to paradoxical motion predictions if in one case you used the total mass $m_\\Sigma$ and in another one of the asymptotically free submasses $m_i$ were used as the arguments to the equation of motion. Thus for example, the equation for the instantaneous velocity of any object falling in a vacuum does not include mass as a significant parameter: > $v_i=\\sqrt{2gd}$ Galileo's argument was insightful but largely tautological. The problem is that the equations of motion for objects falling though fluids, such as terminal velocity, provide an existence proof that self-consistent equations of motion with mass as a significant parameter _can_ be constructed. However, such self-consistency becomes possible only if the distribution of object mass in space also becomes a non-trivial parameter. This additional information allows paradoxes to be resolved by converting seemingly inconsistent state predictions into asymptotic limits of a larger continuum of predictions. For example, in fluid mechanics a mass $m_{\\Sigma}$ is correctly predicted to have a higher terminal velocity than the same mass configured as two smaller spheres $m_1$ and $m_2$ linked by a thin string. The latter form is an example of how the predicted behavior of a single falling sphere can be made to approach asymptotically the prediction for two smaller spheres by slowing \"morphing\" the original object into one that closely approximates the case of two smaller spheres. What Galileo really proved, then, was that since experimental evidence showed that mass was not a relevant parameter for the equations of motion of objects falling in a vacuum, then the specific distribution of mass in any one object must also be irrelevant to those equations. What Galileo could never have realized was that exists a domain of physics for which his argument is very relevant: quantum physics. The de Broglie wavelength $\\lambda=h\/p $ of an object falling at modest speeds in a vacuum is a function of its instantaneous momentum $p_i=mv_i$, so: > $\\lambda_i=\\frac{h}{mv_i}=\\frac{h}{m > \\sqrt{2gd}}=\\frac{h}{m}\\sqrt{\\frac{1}{2gd}}$ While not an equation of motion, this equation violates Galileo's argument by predicting that the de Broglie frequency of an object falling in a vacuum _is_ a function of its mass. As in the case of fluid dynamics, the inclusion of mass as a significant parameter results in paradoxical predictions unless and until the configuration of the mass in the object is also taken into account. In other words, the de Broglie wavelength of an object must necessarily depend upon its shape and how mass is distributed within that object. The resulting more closely resembles solid state acoustic resonances in complexity than it does the simple singular mass parameter of pointlike particles isolated in space. For example, if $m_1=m_2=m$ and $m_\\Sigma=m_1+m_2=2m$, three state equations are relevant: > $\\lambda_{\\Sigma}=\\lambda_{2m}=\\frac{h}{2m}\\sqrt{\\frac{1}{2gd}}$ > > $\\lambda_1= \\lambda_m=\\frac{h}{m}\\sqrt{\\frac{1}{2gd}}$ > > $\\lambda_2= \\lambda_m=\\frac{h}{m}\\sqrt{\\frac{1}{2gd}}$ Self-consistency alone argues that if these equations are even approximately correct, then $\\lambda_{2m}$ should dominate when the $2m$ mass takes the form of a compact spherical object. Conversely, reconfiguring the $2m$ sphere into two smaller $m$ spheres linked by a slender thread should asymptotically approach the case of two separate $m$ spheres, and so should be dominated by the $\\lambda_m$ wavelength. However, any single wavelength cannot be accurate in any of the cases. There should instead be a spectrum of wavelengths and intensities that is highly dependent on the particulars of how the mass is distributed and connected in the object. That is why I mentioned earlier that an accurate equation model for predicting de Broglie wavelengths for non-point object will necessarily be comparable in complexity and subtly to resonance models in solid state physics. I do not readily see a way out of this. Galileo's argument, while largely specious in the case of the equations of motion for objects falling in a vacuum, is extraordinarily relevant to quantum mechanics, where it appears to say that the entire concept of \"mass\" requires revisiting if it is to be used accurately and meaningfully. A correct model should instead have more kinship with the equations of motion for objects falling though a fluid, in the sense that both must define very carefully how the distribution and \"connection\" of mass impacts the predictions. * * * So, my two main questions are:   1. Do there currently exist mathematical models of de Broglie wavelengths that pass the Galilean self-consistency test? For example, are they perhaps hidden away in the mathematical details of phenomena such as quantum models of molecular vibration? (My impression is \"no\", but I could certainly be wrong.)   2. What are the particle physics implications, if any, of quantum mass being dependent on form?"}
{"id":"89276","title":"Relative motion question","text":"I'm sitting on my seat in a train and the train is moving at a very high speed, let's say 600 mph. If I throw a ball vertically up in the air (while still sitting on my seat), will the ball fall back in my hands or will it fall in the seat behind me since my position would have changed from the time I threw the ball, but the balls vertical trajectory will remain the same?"}
{"id":"108079","title":"Ball thrown in a moving train","text":"> A ball is thrown upward in a train moving with a constant velocity. Where > will it land? My intuition tells me that the ball will fall at my back. But my book says that it will return back to the thrower."}
{"id":"110744","title":"What is $k$ in Newtons Law of Cooling?","text":"I am writing the definition of quantities for Newton's Law of Cooling and am not sure what to actually call k$,$ or its units. This is what I have so far $$\\frac{dT}{dt}=-k(T-T_s)$$ Where $\\frac{dT}{dt}$ is the rate of cooling measured in degrees Celsius per second $\\left[°C\\cdot{}s^{-1}\\right]$ $T$ is the temperature of the object measured in degrees Celsius $\\left[°C\\right]$ $T_s$ is the ambient temperature of the objects surrounding measured in degrees Celsius $\\left[°C\\right]$ $k$ is a constant and has no units? I don't think that's a good enough description"}
{"id":"19097","title":"Practical personal aircraft today?","text":"Is it possible to build, today, a personal aircraft that not use an impracticable amount of fuel for everyday use? What are the physical concepts that could be used to build it?"}
{"id":"16042","title":"Momentum and indirect bandgap","text":"When an electron is excited from valence band to conduction band, it has to have a finite momentum in the case of indirect bandgap. Does that mean that the electron cannot be created at rest ? does it have a non zero \"speed\" ?"}
{"id":"97787","title":"Relation between complex dielectric constant and resistance","text":"I am having a real hard time understanding **how** I reach the expression that relates resistance and $\\varepsilon''$ as the complex part of the dielectric constant. I was trying to follow a clear and eloquent way of describing this. I understand how the idealized concept of a capacitor requires a $\\pi\/2$ phase shift between polarization and electric field which is related by the dielectric constant. If it is not $\\pi\/2$ , then expressing the the dielectric constant as a complex number will allow us to express that phase shift. I know that not having a phase shift equal to $\\pi\/2$ means that I do not have a ideal capacitor, but also a related resistance. My professors keep telling me that the following expression is a _definition_ , not a deduction: $$\\varepsilon''=\\sigma\/{\\varepsilon_0\\omega} $$ i.e. $$\\varepsilon''=d\/{A\\varepsilon_0\\omega R} $$ where _d_ and _A_ are thickness and contact area of the resistance _R_ But it seems to me that I should be able to explain it better... How?"}
{"id":"103584","title":"Why is the direction of a flame is always upward?","text":"why a candle's flame direction is always upward? What is the coposition of fire? Don't say it's energy. There is something else."}
{"id":"97902","title":"Why is general relativity considered to be a gauge theory?","text":"I have studied the first five chapters of Carroll's book (up to the Schwarzschild solution). I see similarities to the Yang-Mill theories such as the covariant derivative to account for curvature in GR which is similar to the covariant derivative to account for local gauge transformations in Yang- Mill theories. I have heard that some consider GR to be a gauge theory, but I do not understand what the redundant degrees of freedom are required to make it gauge theory."}
{"id":"94320","title":"Is Lagrangian a scalar?","text":"I may be wrong: Lagrangian are scalars. They are NOT invariant under coordinate transformations. The simplest example is when you have a gravitational potential ($V=mgz$) and you translate $z$ by $a$ (some number). $$L=\\frac12m\\left(\\frac{dz}{dt}\\right)^2-mgz\\to L=\\frac12m\\left(\\frac{dz}{dt}\\right)^2-mgz-mga$$ thus the Lagrangian changed under this coordinate transformation. However Euler-Lagrange equations ARE invariant under coordinate transformations. **So some scalars do vary under coordinate transformation!** Thus components of vectors are scalars, thus time is. Again, try to say the following sentence: > the vector $a$ is the first component of the vector $b$. Can you please clarify?"}
{"id":"52386","title":"Time travel in the past","text":"The past weeks I watched some episodes from \"Through the Wormhole\" with Morgan Freeman. When talking about traveling back in time, they said you can only travel back up to the date when the first time travel occured. So, if for instance time travel will be discovered in 2045, all time travelers(from any given time) won't be able to travel in time earlier than 2045. Why is this? Can anyone explain this please? I know this is just theoretical but still I would like to know the concept behind it."}
{"id":"43663","title":"Gas kinetic representation of trans-critical conditions","text":"From a molecular point of view, can we think of the super-critical conditions as conditions where T and p are large enough that the collisions of gas molecules are frequent and powerful enough to prevent the molecular bonds that would form a liquid cannot be sustained. So if we consider a single fluid in a constant pressure system and with a temperature gradient from sub-critical temperatures to super-critical temperatures, how accurate is this diagram? ![Trans-critical event from a molecular dynamics point of view](http:\/\/i.stack.imgur.com\/NgSSC.png) How can we represent the disappearance of surface tension on the right side? Are molecules being stripped from the surface by collisions? Because the molecules inside the compressed liquid are still bonded right? What would you change in this crude diagram to make it more representative of what really happens? To give a bit more context, I'm more of a fluid mechanics guy modeling inject. So I might not be using the exact terms that I should. The terminology I use is within the context of this diagram: ![O2 phase diagram](http:\/\/i.stack.imgur.com\/W1wRj.png) Phase diagram of oxygen in the $p-T$ space. The black dot represents the critical point at the intersection of the long-dash lines for the critical temperature and pressure. The short-dash line is the pseudo-boiling line (Oschwald_2006a) or Widom line (Simeoni_2010a) and corresponds approximately to the region of maximum specific heat at constant pressure. The density is represented by the colored contours in the background."}
{"id":"60292","title":"Does performing a measurement on a system change its internal energy?","text":"I'm studying Quantum Mechanics in my spare time from a general point of view (no technical details) so some fundamental question came into my mind: How is it possible to detect a single photon without making any change to it? How does observation is related to the total energy of a system and does the act of performing measurement alter the internal energy of the observed system as a general rule?"}
{"id":"90837","title":"Observations in the cathode ray tube experiement","text":"![enter image description here](http:\/\/i.stack.imgur.com\/6drQ1.png) 1.One of the observations I learned was that the glass tube begins to glow with a brilliant green light. Many websites I read through refer to a fluorescent material. However, as shown in the above diagram there was no fluorescent material in the experiment carried out first on the cathode ray tube. **So where does the green glow come from. Is this the color of the radiation itself?** 2.\" _Cathode rays travel in straight lines. That is why, cathode rays cast shadow of any solid object placed in their path. The path cathode rays travel is not affected by the position of the anode._ \" I just can't seem to understand this explanation of the one of the observations.Also, different websites analyses this observation differently. For example, \" _The cathode rays consist of material particles because they produced shadow of objects placed in the way_ \" 3.Two of the conditions of the experiment were air at very low pressure and secondly a very high potential difference. Could someone please tell me why these conditions were necessary? I know the questions are very silly but because different websites refer to different things, I am becoming confused with something that should be simple to understand."}
{"id":"90832","title":"Characteristic waves in a plasma","text":"What does \"characteristic wave\" mean in the context of plasma waves? For example, when propagating parallel to the plasma magnetic field, the characteristic wave is circularly polarized. Does this mean that only a circularly polarized wave can propagate in this direction? Could an elliptically polarized wave travel in this direction instead?"}
{"id":"63924","title":"Critical dimension in quantization of p-branes","text":"So I have what might be a fairly basic question, but my understanding that in the quantization of the the string, or the 1-brane, there are conditions on the number of spacetime dimensions to ensure that the Lorentz algebra still holds. For the bosonic string we have D=26 and for the superstring we have D=10. Now I know that an equivalent condition on the critical dimension comes up by requiring that the conformal anomaly vanish. This is the prelude to my question: Are there any similar or equivalent restrictions on the number of spacetime dimensions are come up in the quantizing more general branes? Is it only the string because only the worldsheet theory has conformal invariance that we want to ensure holds in the quantized theory? If this statement is equivalent to the Lorentz algebra closing, why for a general p-brane do we not need to impose any conditions on the dimensions for this to hold?"}
{"id":"64043","title":"Is it possible that universe might not be speeding up expansion?","text":"I'm not sure but I was thinking of galaxies shrinking with time while still moving apart from each other at almost a constant speed or less (i.e: uniform\/slightly decelerating expansion). This may need further evidence such as tracking the distance between stars inside our galaxy to check if they are getting closer. If so, would this make an alternative explanation for visualizing the universe as if it were expanding in an accelerating rate?"}
{"id":"64041","title":"What causes the collapse of a Magnetic Field?","text":"It's theorized that Mars once had a magnetic field which collapsed at some point. There's also lots of speculation that a similar process will eventually happen to Earth. More recently has been the information that the earth's magnetic field is slowly dropping, presumably preparing for its 'flip' that happens every few hundred thousand years. But what causes a magnetic field to eventually collapse to the point that it ceases to be?"}
{"id":"60748","title":"What is the physical meaning of charges at light-like infinity in asymptotically flat space-times?","text":"In the case of charges defined at space-like infinity, I can understand the physical meaning of them because they can be related to measurements made by a physical observer (that is an observer whose wordline is time-like). For example in four dimensions, for the Schwarschild solution, the ADM mass coincide with the mass measured by an observer that is motionless (in Schwarschild coordinates) and far from the center of the solution. My puzzle is then the following : since no physical observer can reach light- like infinity, how are physically interpreted the charges defined at light- like infinity? How are they related to any measurement? Many thanks for your help."}
{"id":"53381","title":"Statistical Mechanic","text":"One can define entropy as $$S=k\\log{\\omega(E)},$$ where $\\omega(E)$ is the numbers of states with energy equal $E$; and the canonical partition function for a set of N particles is defined as$$Z_N=\\sum_{\\phi}e^{-\\beta E[\\phi]}=e^{-\\beta F(\\beta,N)},$$ where the sum run on states $\\phi$ and the free energy is defined as $F(\\beta,N)=U-TS.$ The mean value of the internal energy and the entropy i learned that would be $$\\langle U\\rangle=\\frac{\\partial(\\beta F)}{\\partial\\beta}$$ $$\\langle S\\rangle=\\beta^2\\frac{\\partial(F)}{\\partial\\beta}.$$ For definition, the mean value of any physical observable is $$\\langle O\\rangle=Z_N^{-1}\\sum_{\\phi}O[\\phi]e^{-\\beta E[\\phi]}.$$ I'm quite sure that there will be a problem of amount $k$, if one verify the definition of $\\langle S\\rangle$. Am i wrong?"}
{"id":"53386","title":"Circular Polarization","text":"As one observes an clockwise (cw) circular polarized electromagnetic wave which is reflected off a denser medium or metal interface it is changed to a counter-clockwise (CCW) polarized em wave plus a phase of PI. A counter clockwise polarized wave which is reflected off a lower dense medium interface only get an additional phase of PI.   1. This means the reflected wave always has the opposite polarization?   2. But **ONLY** because the reference coordinate system is changed (since by definition of polarization the wave always travels away from the observer) and NOT because of some phase which is added only to one or the other linear polarization axis?"}
{"id":"53384","title":"In Noether's theorem, what is a \"classical solution of the equations of motion\"?","text":"I'm reading a book which states that: > for each generator of a global symmetry transformation, there is a current > $j^{\\mu}_{a}$ which, _when evaluated on a classical solution of the > equations of motion $\\phi_{cl}$_ , is conserved. I.e. > $\\partial_{\\mu}j^{\\mu}_{a}\\vert_{\\phi=\\phi_{cl}} =0 $ I get the general principle, but I'm uncertain about the bit I have made _italics_. Can anyone shed some light on this?"}
{"id":"80696","title":"Rabi oscillations and two level dynamics","text":"I'm currently looking at Rabi Oscillations, and not I have a look at the following equations: $$W = \\sqrt{\\Omega^2+\\delta^2}.$$ The amplitude: $$\\frac{|\\Omega|^{2}}{\\delta^{2}+|\\Omega|^{2}}$$ Now, I see that if the detuning $\\delta = (\\omega - \\omega_{0})$ is large, the two level approximation is bad, since the ratio between level 2 and 3 is getting closer to 1. But why does the two level system not work for high $\\Omega$?"}
{"id":"80691","title":"How does concave bowl amplifies the sound?","text":"I recently came across an article \/ answer in Quora see here that mentioned that if we place the iPhone in a bowl and play music, the volume is amplified. What is the physics principle behind this?"}
{"id":"20537","title":"Why is $c$ considered as the speed of the photons?","text":"Maxwell equations brought $\\ c_{o}=\\frac{1}{\\sqrt{\\mu_{o}\\varepsilon_{o}}}\\ $. Since this is a constant, it made all physicists at that time wonder where was the frame of reference? They ended up with an _aether_ answer until the Michelson–Morley experiment denied it. Now there was no frame of reference, but the genius of Einstein resolved it. However, although Einstein's theory is mathematically good, it does not imply there can not be other ones mathematically good. I just read an article written by someone not knowledgeable about physics who had this interesting idea: There is no need to consider c as the speed of the photons unless natural evidence tell us that, we could consider c as the difference between the speed of the photon that just appeared (c value) and the one at (0 value) that is going to speed up and the result is a constant speed as well, c, for all references. That is what he says, if I understood it correctly. Here is the article: http:\/\/jesusgonzalezllorente.com\/ http:\/\/vixra.org\/abs\/1202.0013 The pictures are horrible though. So, is it actually possible to consider light as he states or is there any logical, mathematical physical law that does not allow it?"}
{"id":"20531","title":"Pushing, inertia of a cart system","text":"I have a cart with another cart on top which gets pulled down by another cart on the side. There is no friction. The question is: > How strongly do I have to push with $F$ to keep the cart $m_1$ stable? ![](http:\/\/wstaw.org\/m\/2012\/02\/04\/m62.png) I work in the system of the cart $m_3$. This system accelerates with some $a$. The force of inertia pulls back on $m_1$ with $m_1 a$. That is opposed by the gravitational force of $m_2$, which is simply $m_2 g$. The acceleration that I need for this to be stable is: $$ a = \\frac{m_2}{m_1} g $$ Here comes the point I am not too sure about: The force is on $m_3$ and on $m_2$, so the driving force would be this: $$ F = (m_2 + m_3) \\frac{m_2}{m_1} g $$ On second thought, I think also need to accelerate $m_1$ some way or another, so that my total force would be a little higher: $$ F = (m_1 + m_2 + m_3) \\frac{m_2}{m_1} g $$ I prefer the latter, but what is the right solution?"}
{"id":"90048","title":"Relationship between mass preserving four-fources and proper acceleration","text":"**Background** I studied Rindler's book on Relativity. Relevant information from this book is available on-line here: > http:\/\/www.scholarpedia.org\/article\/Special_relativity:_mechanics Given that we work with four-vectors $\\mathbf{R} = \\left(\\vec{r},ct\\right)$ where the four-velocity is defined as $$ \\tag{1} \\mathbf{U}=\\frac{d\\mathbf{R}}{d\\tau}=\\gamma\\frac{d\\mathbf{R}}{dt}=\\gamma\\left(u\\right)\\left(\\vec{u},c\\right), $$ $\\vec{u}$ is the spacial \"three-velocity\" and the four-acceleration is defined as $$ \\tag{2} \\mathbf{A}=\\gamma\\frac{d\\mathbf{U}}{d\\tau}=\\gamma\\frac{d}{dt}\\left(\\gamma\\vec{u},\\gamma c\\right)=\\gamma\\left(\\dot{\\gamma}\\vec{u}+\\gamma\\vec{a},\\dot{\\gamma}c\\right), $$ we can define the four-force as $$ \\tag{3} \\mathbf{F}=\\frac{d}{ {d\\tau } } \\mathbf{P}=\\frac{d}{ {d\\tau}}(m_{0}\\mathbf{U})=m_{0}\\mathbf{A}+\\frac{ {dm}_{0} }{ {d\\tau} }\\mathbf{U}. $$ The four-force $\\mathbf{F}$ is said to be **mass-preserving** if $dm_0\/d\\tau$ equals zero, i.e. the force leaves the resting mass of the particle unchanged. Moreover $$ \\tag{4} \\mathbf{F}=\\frac{d}{d\\tau}\\mathbf{P}=\\gamma(u)\\frac{d}{ {dt} }( \\vec{p},mc)=\\gamma (u)\\left(\\vec{f},\\frac{1}{c}\\frac{dE}{dt}\\right). $$ We note that $\\mathbf{A}\\cdot\\mathbf{U}=0$ and therefore for a mass preserving four-force $\\mathbf{F}\\cdot\\mathbf{U}=0$ from Eq. (3) or from Eq. (4) $$ \\tag{5} \\mathbf{F}\\cdot\\mathbf{U}=\\gamma^2(u)\\left(\\frac{dE}{dt}-\\vec{f}\\cdot\\vec{u}\\right)=0 $$ which means that $$ \\frac{dE}{dt}=\\vec{f}\\cdot\\vec{u} $$ and $$ \\tag{6} \\mathbf{F}=\\gamma(u)\\left(\\vec{f},\\vec{f}\\cdot\\vec{u}\/c\\right). $$ We also have the relationship $$ \\tag{7} \\gamma(u)\\vec{f}=m_0\\frac{d^2\\vec{r}}{d\\tau^2}. $$ Moreover we have that the **proper acceleration** $\\alpha$ is defined as $$ \\mathbf{A}\\cdot\\mathbf{A}=\\mathbf{A}^2=-\\alpha^2. $$ I studied the questions formulated in > Why proper acceleration is $du\/dt$ and not $du\/d\\tau$? > > relativistic factor between coordinate acceleration and proper acceleration and realized that the spatial vectors can be broken down into a parallel component and an orthogonal component, i.e. $\\vec{v}=\\vec{v}_\\parallel+\\vec{v}_\\perp$ or $\\vec{a}=\\vec{a}_\\parallel+\\vec{a}_\\perp$ where one makes a distinction between the component of the velocity that is parallel with the acceleration and vice versa. **Question** Now it can be shown that for a force $\\vec{f}$ acting upon a particle traveling with a velocity vector $\\vec{u}$ where there is an angle $\\theta$ between $\\vec{f}$ and $\\vec{u}$ the relationship between the proper acceleration of the particle and the force $\\vec{f}$ is $$ \\tag{8} \\vec{f}^2=m_0^2\\frac{\\gamma_\\parallel^2}{\\gamma^2}\\alpha^2 $$ where $$ \\tag{9} \\gamma_\\parallel=\\frac{1}{\\sqrt{1-u_\\parallel^2\/c^2}}\\;\\;\\;\\text{and}\\;\\;\\;\\gamma=\\frac{1}{\\sqrt{1-u^2\/c^2}}. $$ Now how can I derive or prove equation (8)? I have tried every feasible combination of Eq. (1) to (7) without success. I've also tried the more basic variants of the Lorentz transformation > https:\/\/en.wikipedia.org\/wiki\/Lorentz_transformation but it gets very confusing when there are two components to consider when assessing the relativistic effects. Any ideas?"}
{"id":"95498","title":"Determine orbital position after change in velocity","text":"I am working on a satellite simulator for a project\/game and I am stuck on this one bit of physics. So far I have a satellite that revolves around earth on a 2D plane following Keplerian motion using Kepler's equation. Everything is fine and the satellite will orbit many times without problem. I can also change the velocity of the satellite to manipulate apo\/peri points as well as augment of periapsis. However, the problem occurs if I change the velocity once the satellite passes the `PI\/2` mark, the entire orbit gets flipped (augment of periapsis is offset by `PI`) and the true anomaly resets to 0. Here is how it is currently implemented: With the assumption that the satellite starts at the periapsis at t0, I find the eccentric and true anomaly given a time since periapsis $w$. Then I find the radius at that angle to get the final position. $$t_0 = 0$$ $$E_0 = 0$$ $$M(0) = E_0 - e \\sin E_0$$ $$n = \\sqrt{ \\frac{GM }{ a^3}}$$ $$E = {\\rm kepler}(e, M)$$ $$v = 2 \\tan^{-1}{(\\sqrt{\\frac{(1+e)}{(1-e)}}\\tan\\frac{E}{2})}$$ $$r = a\\frac{1-e^2}{1 + e\\cos{v}}$$ When I update the velocity of the satellite (flight path angle $\\phi$ is recomputed after change in velocity), I recalculate a new true anomaly using $\\phi$, I then find the difference between the old true anomaly and the new true anomaly, and add that to the augment of periapsis $w$. After that I recompute time since periapsis $t$ using the new true anomaly and recompute the orbital parameters as well. $$vel = \\sqrt{GM\\frac{2}{r} - \\frac{1}{a}}$$ $$\\phi = \\tan^{-1} ( \\frac{e\\sin{v}}{1 + e \\cos{v}})$$ $$v_2 = \\tan^{-1}{\\frac{r v g m \\cos{\\phi} * \\sin{\\phi}}{r v g m \\cos^2{\\phi} - 1}}$$ $$w \\mathrel{{+}{=}} v_2 - v$$ $$E = \\cos^{-1}{ \\frac{e + \\cos{v}}{1 + e \\cos{v}} }$$ $$M = E - e \\sin{E}$$ $$n = \\sqrt{ \\frac{GM}{a^3} }$$ $$t = \\frac{M}{n}$$ $${\\rm calculateOrbit}()$$ I noticed if I were to $v = v + \\pi$ if ever $v_2 < 0$ and $v > 0$, then the weird orbit\/position flip won't occur until the $v = \\pi$ Otherwise I do not know what the problem is. I would really appreciate it if someone can point me the way, as I have been scratching my head over this for a long time now."}
{"id":"29311","title":"What is spontaneous symmetry breaking in QUANTUM systems?","text":"Most descriptions of spontaneous symmetry breaking, even for spontaneous symmetry breaking in quantum systems, actually only give a classical picture. According to the classical picture, spontaneous symmetry breaking can only happen for non-linear systems. Classical linear systems, such as Harmonic oscillators, can never have spontaneous symmetry breaking. (Here \"linear\" means that the equation of motion is linear.) But the real QUANTUM systems are always linear, since the Schroedinger equation is alway linear. But how can a linear quantum system have spontaneous symmetry breaking? Do we have a simple intuitive understanding spontaneous symmetry breaking WITHIN QUANTUM mechnics? (ie without using the classical picture, such a Mexican hat -- the logo of physics.stackexchange) The Mexican hat does give us a intuitive and pictorial understanding of spontaneous symmetry breaking in classical systems. Do we have a intuitive and pictorial understanding of spontaneous symmetry breaking in quantum systems."}
{"id":"53520","title":"How is energy produced in Alpha emission (radioactivity)?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/pW1vD.gif) In the above Alpha decay reaction, we get thorium by the alpha decay of uranium with an alpha particle. But we also get energy of amount Q by this process. And according to my course books, this Q energy comes from E = mc2 i.e. some of the mass is converted into energy. My question is that which mass converted into energy? I mean if you add the nucleons of alpha particle (helium nucleus), you will get the total nucleons present in Uranium. But, where did the energy come from then?"}
{"id":"53528","title":"What is mass of free up and down Quark?","text":"1. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly observed or found in isolation.   2. Color confinement, often simply called confinement, is the physics phenomenon that color charged particles (such as quarks) cannot be isolated singularly, and therefore cannot be directly observed.   3. In theoretical physics, quantum chromodynamics (QCD) is a theory of the strong interaction (color force), a fundamental force describing the interactions between quarks and gluons which make up hadrons (such as the proton, neutron or pion).   4. Yang–Mills theory List of unsolved problems in physics Yang–Mills theory in the non- perturbative regime: The equations of Yang–Mills remain unsolved at energy scales relevant for describing atomic nuclei. How does Yang– Mills theory give rise to the physics of nuclei and nuclear constituents? When you read about quarks, All folders of Quark model have not been defined. I just want to know what is mass of free Quark? But in my calculations seem free quark have no mass. while Quark become free is like to melt ice."}
{"id":"78459","title":"Computing the efficiency of a heat engine","text":"With the heat engine ![enter image description here](http:\/\/i.stack.imgur.com\/ulLxt.png) we have an ideal gas working the substance. I want to show that the efficiency of this heat engine is $$\\eta = 1 - \\frac{1}{\\gamma}\\frac{\\bigg(\\frac{V_3}{V_1}\\bigg)^\\gamma - \\bigg(\\frac{V_2}{V_1}\\bigg)^\\gamma}{\\bigg(\\frac{V_3}{V_1} - \\frac{V_2}{V_1}\\bigg)} $$ where $\\gamma = \\dfrac{C_p}{C_v}$ is the adiabetic index. So we have that $Q_1$ enters during the expansion $B \\to C$ and $Q_2$ leaves during the compression $D \\to A$. Therefore, $$\\eta = \\frac{Q_1 - Q_2}{Q_1} = 1 - \\frac{Q_2}{Q_1}$$ I'm kind of stuck at this point. Any further help would be appreciated."}
{"id":"95143","title":"Can we observe diffraction even if the slit size is tending to zero?","text":"According to Huygens principle, if we consider any point of the primary wave fronts, each point is the source of secondary disturbance. If the point size we consider tends to zero, the number of points on that wave front which enters the slit tends to infinity. Thus, if we consider the slit how much ever small, we must see the diffraction pattern. Because we can consider many points on the wave front that enter slit, whose size is very very small-thus producing interference of waves emanating from them which can give diffraction pattern. But in reality as I know we don't see any diffraction pattern if we have the slit of very small in width compared to wavelength. Is the point size that we consider in diffraction is restricted to some value? If not as explained above, we can have many points of very very small size on the wave front that enter the slit, and we can expect diffraction even if slit size is tending to zero. I don't know whether I have misunderstood anywhere, if yes pardon me and explain. If not, is it that (according to huygens principle) we can observe diffraction pattern even if slit size is tending to zero? Sometimes I might not be communicating with you better with my problem, if so please comment on where there is problem."}
{"id":"8828","title":"What is the difference between 'running' and 'current' quark mass?","text":"When looking at the PDG, there is a difference between the 'running' and the 'current' quark masses. Does anyone know which is the difference between these two?"}
{"id":"30869","title":"Exercise QFT and CFT","text":"Consider the action functional $S[z;t_1,t_2]=\\int_{t_1}^{t^2}[g(z,\\bar{z})\\dot{z}\\dot{\\bar{z}}]^{\\frac{1}{2}}dt$ with $z(t)$ a complex path with end points $z_i=z(t_i),\\; i=1,2$. $g(z,\\bar{z})$ is a positive real function on $H=\\\\{z\\in C: Im(z)>0\\\\}$. After some easy questions the exercise asks me to determine (up to a multiplicative constant) the function $g(z,\\bar{z})$ such that the action is symmetric under the transformation $z\\rightarrow \\gamma(z)= \\frac{az+b}{cz+d}$, with $a,b,c,d \\in R$. Can somebody help me or give me some hints?"}
{"id":"115039","title":"Question regarding Schrödinger's equation","text":"Here is my understanding of Schrödinger's equation and wave functions: (I'm considering the time independent equation.) > The equation takes in values of energy as an input and, being a differential > equation, gives functions as outputs. These functions correspond to the > different orbitals. > > When we input coordinate values we get an answer that we can use to plot the > probability distribution of an electron around the nucleus. Can someone tell me if my understanding is correct, and correct me if I'm wrong?"}
{"id":"30861","title":"Isotropy and Noise","text":"If you have a field which value is just Gaussian noise plus a constant do you call it isotropic?   * there is no preferred direction   * however it is not \"the same\" in all directions if \"the same\" means \"constant\" The question is about a definition of a word, thus obviously conflicting answers can arise. However what definition do **you** use or have seen used: \"no preferred direction\" or \"constant wrt translation\"?"}
{"id":"104952","title":"Free fall from space","text":"If you leave a ball weighing 10 kg at a height of 500 km above sea level (neglecting air friction). How can calculate how long the ball hits the ground and what will be its speed? ![enter image description here](http:\/\/i.stack.imgur.com\/5Emdd.png) I know that: On earth I can calculate the time taken to the object to fall by this equivalent:               x = x0 + v0t + (at^2)\/2     x = 0     x0 = 500000     v0 = 0     a = -g = 10 m\/s^2     0 = 500000 - 5t^2     t = 316.227sec      Also I know how to calculate the force between objects:               g = GMm\/r^2"}
{"id":"31475","title":"Instability of a thrown tennis racquet","text":"Someone once mentioned to me that it's impossible to throw a tennis racquet (or similarly shaped object) into the air, perpendicularly to the string plane, in such a way that it won't turn. What is this effect, or was he talking rubbish?"}
{"id":"114927","title":"Does perpetual motion machine really exist?","text":"I recently saw a video on Youtube about a perpetual motion machine. Here is the link: http:\/\/www.youtube.com\/watch?v=OS1KXMsE2qk It's on Robert Boyle's flask. Where does the energy come from?"}
{"id":"114926","title":"Young's Double Slit Experiment : What would happen if the \"first slit\" was too wide?","text":"![A picture showing Young's Double Slit experiment](http:\/\/i.stack.imgur.com\/QTGeQ.gif) I was wondering what would happen to the fringe pattern displayed on the screen if the first slit (as shown in the picture), which is also known as \"single slit\", was made a bit wider. I read it in my book. I don't understand it. Anyway, it quotes > _If the single slit is too wide, each part of it produces a fringe pattern > which is displaced slightly from the pattern due to adjacent parts of the > single slit. As a result, the dark fringes of double slit pattern become > narrower than the bright fringes, and contrast is lost between the dark and > the bright fringes._ Please answer the question in your own words and try to explain to me what the quote is trying to say. Also, when it says \"parts\", what does it mean?"}
{"id":"131738","title":"Explanation for this type of (magic-trick) suspension?","text":"Well let's start off with that I'm not a physicist but I'd like some thoughts on something I came across in my hometown. This guy: ![enter image description here](http:\/\/i.stack.imgur.com\/G3U6R.jpg) Is it possible that due to the electrical charge of magnets this guy can make the illusion that he can float ? Or is this probably a cheap trick that fools the eye ? I was standing there for quite some time watching the guy and he keep moving his feet. The resistance that he appeared to have was from a magnet force keeping him afloat. So after I passed this guy I did some physics searches on the web and the first thing that caught my eye was the electrical charge of magnets. So the question is : Is this related to the electrical charge of a magnet or a cheap trick ?"}
{"id":"67828","title":"Tree level and loop level","text":"I'm trying to read through a paper which explains the following about Universal Extra Dimensions (UED) vs ADD models: > The new feature of the UED scenario compared to the brane world is that > since there is no brane to violate translation invariance along the extra > dimensions, momentum is conserved at tree level leading to degenerate KK > mode masses at each level and conservation of KK number in the interactions > of the four dimensional effective theory. This statement is broken at the > loop level, where the fact that the extra dimensions are compact leads to > (calculable) violations of the full Lorentz symmetry, and as a result shifts > the masses of the KK modes away from their tree level values. I'm unfamiliar with the concepts of tree-level and loop-level. Could someone perhaps explain this in context? If I would have to guess, I'm thinking that tree-level correspond to the scattering correlation functions of the classical theory and loop-level are a kind of correction?"}
{"id":"24532","title":"Flux if there were only one type of charge in the universe","text":"There was this question that i saw in a book and it also had an answer given. The Question was: > If there were only one type of charge in the universe, then: > >   * $\\phi = \\oint \\boldsymbol{E}\\cdot \\partial\\boldsymbol{A} \\neq 0 $ on any > surface. >   * $\\phi = \\oint \\boldsymbol{E}\\cdot \\partial\\boldsymbol{A} = 0$ is the > charge is outside a surface. >   * $\\phi = \\oint \\boldsymbol{E}\\cdot \\partial\\boldsymbol{A} $ will not be > defined. >   * $\\phi = \\oint \\boldsymbol{E}\\cdot \\partial\\boldsymbol{A} = > \\frac{q}{\\epsilon_o}$ if the charge is inside the surface. > The answers were given as option second and the last one. (Gauss' law statements) How?"}
{"id":"24531","title":"Do ~1THz oscillators output \"light\"?","text":"Upon looking at some radiation levels higher than Microwaves, I have come across \"Terahertz radiation\". According to the article, there are numerous ways to generate such radiation, and that brings questions to me. My confusion lays in how it is different (if it is!) from a heating fan emitting infrared radiation (simple), or an LED emitting visible green light, or a flashlight emitting various spectral frequencies. Are those examples emitting Terahertz radiation? I would hypothesize that is, but can see no reference to light being called \"Terahertz radiation\" even though it is radiating. Although I am unsure if the technologies listed in the article can go above infrared in to visible light, but assuming so, would it emit \"light\" (in the respective frequencies)? How would this be different than an ordinary visible light emitter? At that point is it no different from an \"ideal\" colour changing LED?"}
{"id":"54337","title":"Partial derivative potential energy of 'free' vibration","text":"I have this rather mathematical question about the calculation of the partial derivative of a potential energy function given by: $$U(x_i)=\\frac{1}{2}\\sum_{i,j}\\frac{\\partial^2U(0)}{\\partial x_i\\partial x_j} x_ix_j.$$ Or if we use $b_{ij}$ for the Hessian : $$U(x_i)=\\frac{1}{2}\\sum_{i,j}b_{ij}x_ix_j.$$ I want to calculate the force: $k_i=\\frac{-\\partial U}{\\partial x_i}$. This should be $-\\sum_{j} b_{ji}x_j$. My questions:   1. Why is the summation over the index $i$ gone?   2. Why are the indices of the hessian $b$ switched? I hope someone can give me an answer. I put this in the physics section because it's a physics problem, but my question is actually purely mathematical."}
{"id":"72112","title":"What is the friction between cylinder and wall (ground)?","text":"A hollow cylinder (radius $R$) is rolling against the wall at angular speed $\\omega$. The coefficient of friction between the cylinder and the wall(ground) is $\\mu$. After how many rotations the cylinder will stop rotating? So I figured I need to find the time taken for cylinder to stop moving, and that would be $$ \\beta=-\\omega\/t => t = -\\omega\/\\beta $$ Where $\\beta$ is angular acceleration, which is known from torques: $$ 2*F_f*R = I*\\beta $$ That's where I got stuck... How do I know the friction? I'm familiar with such equation: $$ F_f = \\mu*F_n $$ How do I find the normal force? Does it have anything to do with centripetal force?"}
{"id":"1566","title":"Which experiments prove atomic theory?","text":"Which experiments prove atomic theory? Sub-atomic theories:   1. atoms have: nuclei; electrons; protons; and neutrons.   2. That the number of electrons atoms have determines their relationship with other atoms.    3. That the atom is the smallest elemental unit of matter - that we can't continue to divide atoms into anything smaller and have them retain the characteristics of the parent element.   4. That everything is made of atoms.  These sub-theories might spur more thoughts of individual experiments that prove individual sub-atomic theories (my guess is more was able to be proven after more experiments followed)."}
{"id":"73649","title":"Transformation of $t=0$ line in moving frame of reference","text":"How does $t=0$ transform into $t - vx\/c^2 = 0$ if a frame of reference is moving as given in here? It seems that the relativistic transformation is given by $$ \\begin{bmatrix} x' \\\\\\ ct' \\end{bmatrix}= A\\begin{bmatrix} x \\\\\\ ct \\end{bmatrix} = \\left( \\begin{array}{cc} \\frac{1}{\\sqrt{1-\\frac{v^2}{c^2}}} & -\\frac{v}{c \\sqrt{1-\\frac{v^2}{c^2}}} \\\\\\ -\\frac{v}{c \\sqrt{1-\\frac{v^2}{c^2}}} & \\frac{1}{\\sqrt{1-\\frac{v^2}{c^2}}} \\\\\\ \\end{array} \\right).\\begin{bmatrix} x \\\\\\ ct \\end{bmatrix} = \\begin{bmatrix} \\frac{x-t v}{\\sqrt{1-\\frac{v^2}{c^2}}} \\\\\\ \\frac{c^2 t-v x}{c \\sqrt{1-\\frac{v^2}{c^2}}} \\end{bmatrix} $$ I was wondering if the transformation is equivalent to measuring in new basis $A\\begin{bmatrix} 1 \\\\\\ 0 \\end{bmatrix}$ and $A\\begin{bmatrix} 0 \\\\\\ 1 \\end{bmatrix}$ for an event?"}
{"id":"73644","title":"Could there be a correlation between the Island of Stability and dark matter?","text":"I can only boast of my passion for science and not of any credentials. I ask because I am curious, and it is not the type of question one can easily Google! What if the key to discovering and unlocking the secrets of dark matter and energy lie in the Island of Stability? Am I missing a vital reason that outright contradicts an attempted connection between the two?"}
{"id":"73641","title":"Definition: Coupling","text":"What does it mean to say that 2 fields are coupled? More generally, what does \"coupling\" mean?"}
{"id":"55163","title":"Wormholes and the illusion of time?","text":"I was watching a video on Youtube by Brian Greene, \"the illusion of time.\"http:\/\/www.youtube.com\/watch?v=j-u1aaltiq4 In that video, he introduces to me the idea of a \"brane,\" or a slice of the universe at specific points in time. And then, he talks about an alien that is tens of billions of lightyears away from Earth. At that distance, he says that even the slowest movements can angle his brane, or slice of time, such that he is angling it towards the Earth's past (if the alien is moving away from us) or future (if the alien moves towards us). So I got to wondering. What if you had a wormhole here on Earth, and walked through it to appear somewhere tens of billions of lightyears away from Earth. Depending on your speed of crossing through it, would you appear further into their past\/future (depending on which direction you cross)? And, if you had another wormhole there connecting back to the Earth, wouldn't you be able to travel back into Earth's past? Which would violate causality? I know lots of people don't like the idea of wormholes, but it doesn't seem to have been scientifically ruled out. So, my question is, despite the fact that wormholes seem to possibly violate causality, why can't it be ruled out? Why is it still a theoretical possibility? As a side question, if the past, present, and future exist at the same time -- what does that even mean? (If anyone could edit my post, that would be great. My internet is giving me a horrible screen to work with. I have totally no ability to see my own tags -- look at this interface: https:\/\/dl.dropbox.com\/u\/106018191\/phystackex.png) (The image above is going to take a while to upload because of the slow internet, possibly a few hours at worst.)"}
{"id":"48585","title":"Where and how is the entropy of a black hole stored?","text":"Where and how is the entropy of a black hole stored? Is it around the horizon? Most of the entanglement entropy across the event horizon lies within Planck distances of it and are short lived. Is it stored near the singularity? How can you pack so much information in such a small region? Can you please help me?"}
{"id":"79497","title":"Overlap integral and probability","text":"I have a question regarding how to extract probability from an overlap integral. Specifically, I am calculating the probability of a particle in a bound state in a delta potential $V=-\\alpha \\delta(x)$ ($\\alpha$ is positive) remaining in a bound state after $V$ changes strength to $V=-\\beta \\delta(x)$. I am able to calculate the overlap integral of the bound state wavefunctions of these two $V$'s just fine; I am just wondering if I need to square the value of the overlap integral to find the probability of remaining in a bound state or if we just keep the value of the overlap integral as our probability. And if we do have to square the value of the overlap integral, why is that?"}
{"id":"35541","title":"In Copenhagen, can this idea preserve locality for Bell inequalities?","text":"Generate an entangled pair of qubits. Send to Alice and Bob far away from each other. Both measure along basis in one of two possible orientations. The result is sent to Charlie at some later time, who compares the corellations and concludes the Bell inequalities have been violated. But in Copenhagen, Charlie can say, the pair of results didn't become real until I collected both observations from Alice and Bob. They weren't real when either of them measured them. They only became real once I observed them. Then, both results, when they materialized, materialized at the same place. So, no nonlocality?"}
{"id":"73398","title":"On the theoretical aspects of the development of the first nuclear bombs","text":"I've just read that 68 years ago _Little Boy_ was dropped on Hiroshima, which made me wonder about some rather historical facts about the development of the first nuclear bombs; they seem to be several questions, but they boil down to the same thing: the theoretical aspects of the development of the bombs. As far as I know, nuclear fission was already understood at the time, so was all the _competition_ between _the american team_ and _the german team_ to develop the bomb first, a mere matter of engineering? Or what was the role of theoretical physicists such as Richard Feynman? Also, I've heard things like that the _team_ led by Werner Heisenberg had _misconceptions_ about nuclear fission, so that his _team_ could not develop the bomb first. Can anyone put in perspective the theoretical aspects of the development of the bombs taking care of this historical issues?"}
{"id":"73397","title":"Can entanglement be explained as a consequence of conservation laws?","text":"This article at NewScientist magazine (subscription required) describes entangling photons by passing them through a half silvered mirror. http:\/\/www.newscientist.com\/article\/mg21929282.100-quantum-weirdness-the- battle-for-the-basis-of-reality.html It talks about the \"weirdness\" of entanglement and how it means that either information is traveling faster than the speed of light or other fundamental assumptions about reality are wrong because the spins of the emerging photons are always correlated when measured. My question is just what's so strange about it? I don't get it... It seems to me to be just another variation of conservation laws. It would be like saying that it's \"weird\" for a photon in pair production to always produce particles of opposite charge. That's a given... Is there something to the timing of it that ruffles everybody's feathers? I think I'm missing something."}
{"id":"73394","title":"Bounded and Unbounded (Scattering) States in Quantum Mechanics","text":"I understand that bounded states in quantum mechanics imply that the total energy of the state, $E$, is less than the potential $V_0$ at + or - spatial infinity. Similarly, the scattering state implies that $E > V_0$. But I do not understand why for bounded states, the complete set of wave- function solutions to the time-dependent Schrodinger equation is constructed from a **discrete** linear combination of solutions, whereas for scattering states, the complete set, or general solution, is instead written as an integral over the **continuous** value of $k$ (or more appropriately, as a Fourier-type integral). Could anyone clarify why this difference is true for bound and scattering (unbound) states?"}
{"id":"114079","title":"Are there reasons for the discrepancies in absolute temp units - Kelvin vs. kelvins vs. degrees Kelvin?","text":"Before 1968, the units for absolute temperature were described as \"degrees Kelvin\" or \"degrees absolute.\" After that, the SI system got rid of the idea of \"degree\" for absolute temperature, so the new unit should apparently be expressed as a \"kelvin\" (with lowercase k) and abbreviated simply \"K\" (without the degree sign). Also, official SI conventions suggest that not only should the unit name be lowercase, but it should be pluralized as other units would be: \"Il en résulte que la température thermodynamique du point triple de l’eau est égale à 273,16 kelvins exactement, Ttpw = 273,16 K.\" Or, in English: \"It follows that the thermodynamic temperature of the triple point of water is exactly 273.16 kelvins, Ttpw = 273.16 K.\" Despite the official SI usage, however, it seems that there are still a variety of conventions in use. Many of the questions on this forum, for example, use Kelvin (with a capital) instead of kelvin in referencing the unit. Also, it appears that the plural usage is somewhat mixed in the physics literature: something like \"200 kelvins\" occurs, but more rarely than \"200 kelvin\" or even \"200 Kelvin.\" The NIST guidelines do not list the kelvin as an exception to the normal pluralization rules: \"the following plurals are irregular: Singular — lux, hertz, siemens; Plural — lux, hertz, siemens.\" On the other hand CERN's writing guidelines suggest that there is an exception: \"And note that it is always kelvin, even when plural (not kelvins or degrees kelvin).\" Given all of this, here is my question: Is the SI standard actually to pluralize kelvins, as would be suggested from the quotations from the official SI guides above? Is this officially stated anywhere in some standards organization's guidelines? Or, is there some rationale given somewhere for the continued use of plural \"kelvin\" (as in the CERN guidelines) or even \"Kelvin\" (with an apparently anomalous capital)? Or, is it -- as I suspect -- just a failure to treat the kelvin as an actual SI unit, despite the redefinition from \"degrees Kelvin\" to \"kelvins\" that happened decades ago? (Perhaps we just dropped the \"degree\" but effectively still treat it the same way as Celsius or Fahrenheit?) **EDIT:** Just to be clear -- I did NOT intend for this to be only a question about linguistic convention. The SI redefinition of the absolute temperature scale and units was apparently meant to refine or change some physical conception about temperature. Resistance to this change may indicate some other elements about the underlying physics (perhaps including the fact that temperature is an intensive property, as suggested in one answer below). While I'd certainly be interested in other official standards and usage recommendations, I also wanted to know if there were other PHYSICAL rationales for the inconsistent units."}
{"id":"114070","title":"Pulse Reflection","text":"I wanted to ask why a pulse that reaches to a free end will reflect off and return with the same direction of displacement that it had before reflection. I am looking for explanation with forces."}
{"id":"55291","title":"Form of the Classical EM Lagrangian","text":"So I know that for an electromagnetic field in a vacuum the Lagrangian is $\\mathcal L=-\\frac 1 4 F^{\\mu\\nu} F_{\\mu\\nu}$, the standard model tells me this. What I want to know is if there is an elementary argument (based on symmetry perhaps) as to why it has this form. I have done some searching\/reading on this, but have only ever found authors jumping straight to the expression and sometimes saying something to the effect that it is the \"simplest possible\". Thanks for the great answers, I accepted the one that was exactly what I was looking for, but the other two long ones were equally interesting and useful."}
{"id":"98649","title":"Confused about relating spring force to maximum distance","text":"I keep working through this problem and getting the wrong answer: > A spring with spring constant k is positioned vertically and then compressed > from its equilibrium length by a distance Δy. A ball of mass m is placed on > top of the spring, and is launched into the air when the spring is released. > The ball travels to a maximum height hmax. Using the Work-Kinetic Energy > theorem, determine hmax in terms of the given variables and any other > constants you deem necessary. The thought process was ∆K = $K_f - K_i$ = W --> $\\frac{mv^2}{2}-\\frac{mv_0^2}{2}$ = W $W$ = $W_s - W_g$ where $W_s$ is a spring and $w_g$ is gravity The final velocity is also zero so $K_f$ = 0. Which gives us $\\frac{mv_0^2}{2} = -k∆y^2 - mgh_m$ (solve for $h_m$) $h_m = \\frac{-k∆y^2}{2mg}+\\frac{v_0^2}{2g}$ I might be completely off but I can't figure out what else to do. I would be very grateful is someone walked me through it ![enter image description here](http:\/\/i.stack.imgur.com\/19aGI.png)"}
{"id":"98642","title":"Modelling a matter dominated universe collapsing into a black hole","text":"With the FLRW equations we can get solutions for a matter dominated closed universe in which the finale is an ultimate collapse, but this is only in terms of $a$ (the scale factor) and $t$ (time) and some $q_0$ and $\\theta$: $a = {\\frac {q_0}{2 q_0 -1 } } (cosh (\\theta) -1) $, (1) $t = {\\frac {q_0}{2 q_0 -1 } } (sinh(\\theta) - \\theta)$ (2) where $dt=a d\\theta$ Unfortunately, the FLRW does not show that the model should collapse into a black hole. So it only tells you when things might end depending on $q_0$. So I will appreciate if someone can give me an opinion of whether the following solution is appropriate for a universe that has collapsed into a black hole. Please feel free to rip it apart if necessary. And secondly, how can the final black hole equation be factored into the FLRW solution to get a complete model? Here it goes. First the energy of the matter is simply: $E_{m}= M c^2$ (3) The gravitational potential energy $U$ of the universe is (see http:\/\/arxiv.org\/abs\/1004.1035): $U = - \\frac {3 G M^2} {5 R}$ (4) For a particle away from the event horizon the Newtonian gravity is simply: $g r^2 = G M $ (5) where $r$ is the radial coordinate of the observer. If I assume that $U = E$ once the universe has completely collapsed then I get, $Mc^2 = - \\frac {3 G M^2} {5 R}$ (6) Substituting (5), I get: $c^2 = - \\frac {3 r^2 g}{5R}$ (7) And finally, $R = - \\frac{3}{5} \\frac {r^2 g}{c^2}$ (8) Now it turns out that the formula for the Schwarzschild radius in Newtonian gravitational fields is remarkably similar http:\/\/en.wikipedia.org\/wiki\/Schwarzschild_radius): $R_s = 2 \\frac {r^2 g} {c^2}$ (9) The obvious issue here is that I got the constant $-3\/5$ instead $2$. So the real question now is, are these computations correct and there is a reason for the difference? (for example, a supermassive BH Universe is not the same as your conventional BH). Or is it this a completely wrong approach that yields a completely wrong answer? If wrong, what is the right answer? Note that I tried to avoid solutions with $M$ and $G$ because that results in absurd answers such as those arrived in If the universe were compressed into a super massive black hole, how big would it be? (i.e., the radius of the BH would turn out larger than the current radius of the Universe, etc, etc)."}
{"id":"98641","title":"Bra-ket notation, Bits, & Superposition","text":"I am a quantum computing enthusiast, and recently I stumbled upon this the following two propositions: $$ \\alpha|1\\rangle + \\beta|0\\rangle$$ What does this mean? My understanding of this is that: the two bits, 1 and 0 are represented in a state of superposition, hence the bra-ket notation (which is commonly used for quantum mechanics), i.e., this is a qubit. Or is there a more concise explanation of this? Also: $$(\\alpha|1\\rangle + \\beta|0\\rangle)^N$$ What does it mean to raise this quantity (of superimposed bits, i.e., qubit) to the $N$th degree? If we take $2^N$ where $N$ is the number of qubits then this tells us the number of bits in the desired number of qubits. Is what I have stated in this post, generally correct?"}
{"id":"26395","title":"Which way do spiral galaxies rotate?","text":"Is it known whether spiral galaxies typically (or exclusively?) rotate with the arms trailing or facing? Intuitively it feels weird to think of the arms as facing the direction of rotation, but that's silly, of course -- it seems to have its origin in an instinctive assumption that some sort of friction with the intergalactic vacuum would make it hard for the galaxy to rotate against the barbs, which is nonsense. I assume that for galaxies that we don't view head-on, we can measure rotation curves spectroscopically. But that's not enough; it must also be possible to determine which end of the minor axis is the far one. Dust in front of the bulge can help with that -- but in the images I have seen where it is clear which way the galaxy is tilted, it's not easy to see which way the spiral curls. Still, I suppose there must be some galaxies where both the spiral and the rotation can be determined. What's the verdict for them?"}
{"id":"26390","title":"Determining cloud cover from observer to near the horizon","text":"Does there exist a Clear Sky Chart with the following enhancements?: 1 - Actual Cloud Cover (Offered Visually and not just Colors with a Legend, Over Time\/Past & Predictive) 2 - Simulate\/Predict Cloud Cover taking into account the direction from Observer to Observed Object and Angle of view \\- Close to the horizon (May be helpful to know when you can reasonably start\/end tracking something you want to catch that night that is close to the horizon) The reason I'm curious is: A) I wonder if it's just not feasible for any\/many reasons. B) It would be great help to know this information. In general, does anyone know of other Earth Weather, Clear Sky Clocks and Charts or anything else that gives more information?...anything related will be helpful. EDIT: I would love to find this lecture \"You can do better than Clear Sky Chart\" mentioned: http:\/\/stjornuskodun.blog.is\/blog\/stjornuskodun\/entry\/966941\/"}
{"id":"101948","title":"Why is radiation for an ultrarelativistic charge zero on axis?","text":"I attribute it to the fact that for an ultrarelativistic charge the field is contracted and essentially there are only fields in the transverse direction and nothing longitudinally (wrt the charges velocity). Any thoughts?"}
{"id":"24881","title":"What is meant when it is said that the universe is homogeneous and isotropic?","text":"It is sometimes said that the universe is homogeneous and isotropic. What is meant by each of these descriptions? Are they mutually exclusive, or does one require the other? And what implications rise because of this?"}
{"id":"133348","title":"If electrons are magnets do they attract each other?","text":"It is said every electron is negatively charged and so they repel each other but if electrons are tiny magnets(which are responsible for atomic attraction and how solid magnet works) does it mean opposite electron poles attract each other?"}
{"id":"133342","title":"Geodesic curvature and Weyl transformations","text":"The geodesic curvature is given by $$k=\\pm t^a n_b\\nabla_a t^b,$$ where $t^a$ is a unit vector tangent to the boundary of the string worldsheet and $n_a$ is an outward vector orthogonal to $t^a$. I don't understand why under the Weyl transformation $$\\gamma_{ab}\\rightarrow e^{2\\omega}\\gamma_{ab},$$ $t^a$ and $n_b$ transform as $$t^a\\rightarrow e^{-\\omega}t^a,~~~~~n_b\\rightarrow e^{\\omega}n_a.$$ Is this really as trivial as a normalisation? Also, what do \"time-like boundary\"$(+)$ and \"space-like boundary\"$(-)$ mean? I appreciate any discussion related to this. The geodesic curvature was somehow never mentioned in my GR class."}
{"id":"113317","title":"Electric Field and Magnetic field of Guassian beam","text":"The center of the gaussian beam in glass has a beam power of 1 mW and a beam radius of 20 μm (micro meter). The wavelength of the light $\\lambda_0$ is 1550 nm. How can I find the electric field E and magnetic field H?"}
{"id":"19937","title":"Time dilation as an observer in special relativity","text":"I've been having a discussion regarding time dilation relating to special relativity and how it should be observed from the FoR (Frame of Reference) of \"the person moving\" : **I assert** \\- If we have a time is slower in the moving FoR than when I observe the stationary frame of reference (earth) then I should perceive events as occurring faster, i.e., if I observe 15 years passing and 90 years pass in the stationary frame then I should observe events on earth happening at a rate of 6:1. So if I observe a 1 min event on earth I watch that entire event occur in 10 seconds. My thought process is similar to the idea that if I was near a massive object and looked out I would observe events further away as moving faster. My teacher seems to believe this line of reasoning is incorrect but will not (or cannot) give me the correct thought process. Hence my question is, \"is my logic correct, and if not why?\""}
{"id":"103898","title":"What was the major discovery on gravitational waves made March 17th, 2014, in the BICEP2 experiment?","text":"The Harvard-Smithsonian Centre for Astrophysics held a press conference today to announce a major discovery relating to gravitational waves. What was their announcement, and what are the implications? Would this discovery be confirmation of gravitational waves as predicted by general relativity (even though Sean Carroll links to the Nobel website implying that G-waves was detected decades ago, while my book on GR (B. Schutz) says they're still looking...I'm confused) Also, regarding inflation theory, would this discovery confirm inflation or refute it? Or something else, a la string theory?"}
{"id":"37793","title":"Can a neutron be created from pure energy","text":"Is it possible to create a neutron out of pure energy, i.e. not by bringing a bunch of already-existing quarks together? (A quick calculation using E = mc2 shows the energy required would be about 1.5 e-10 J.) Would an equivalent antiparticle always need to be created?"}
{"id":"14623","title":"Does magnetic geometry determine the scaling of a Polywell fusor?","text":"Does magnetic geometry determine the scaling of a Polywell fusor? Forgive imprecise terminology here - by \"magnetic geometry\" I mean the configuration of the magnets, the configuration that creates the \"Christmas star\" geometry of the electron field in the fusor. On to my question... I've read that the rate of fusion scales at $r^5$, with $r$ being the radius of the reaction chamber. Is this because of the aforementioned magnetic geometry? Or is it because of something more essential, like Bremsstrahlung, or something else?"}
{"id":"71534","title":"What are the consequences in high-energy of the non-interaction of the Higgs Field?","text":"At high-energies when the Higgs field won't affect (interact with) particles, when the symmetry breaking won't occur, what would be $\\rm W\\pm$ or $\\rm Z^{0}$ bosons speed if they would then have a $0$ rest mass? What about Fermions?"}
{"id":"14628","title":"How can I explain why the weak nuclear interaction between individual nucleons is 'weak'?","text":"> By considering the energy-time uncertainty principle, estimate the range of > the weak nuclear interaction at low energies. Compare this range to the size > of a typical nucleon (for example, a proton) and therefore explain why the > weak nuclear interaction between individual nucleons is 'weak'. I'm a bit stuck with this question: to estimate the range of weak nuclear interactions do I need to put the energy levels of a quark into the energy- time principle? Beyond doing this I am a bit lost! Any comments would be much appreciated; this is an assignment so the work has to be original."}
{"id":"38649","title":"What is the curvature scalar $\\Psi_{4}$?","text":"What is the curvature scalar $\\Psi_{4}$? Is it related to the scalar curvature $R$? What does its real and imaginary parts represent?"}
{"id":"76621","title":"Magnetic field when moving into the influence of a ferrous object","text":"Suppose I have a magnetometer. I am on the Earth's surface, away from the influence of other magnetic sources. Assuming the magnetometer is aligned with the reference frame, it should read only the Earth's magnetic field vector at that point. Suppose now I walk towards a large ferrous object, for example, a large parked truck. As I approach the truck, what will happen to the magnetic field that I measure? Is the distortion to the magnetic field able to be modelled as additive as a function of position? If not, how could one mathematically describe the distortion? Are there any canonical references on the subject? * * * EDIT: In response to the answer(s) below, here are some additional details:   * Assume that I know what the magnetic field in the local geodetic frame _should_ be to within some desired tolerance, e.g. by using EMM2010 or similar.   * Ultimately, I do want to calculate attitude, but I have other sensors at my disposal (IMU and GPS) where heading is observable under motion. My desire to to \"learn\" the magnetic deviation on-the-fly to help stabilise the heading during near-stationary periods. My intuition says that the deviation should be observable when yaw is observable. But I need to start with some sort of model as to what the deviation looks like :)"}
{"id":"106624","title":"What is the gist of the equations on Alejandro Guijarro's \"2\/14\" \"Momentum\" blackboard?","text":"The artist Alejandro Guijarro has an exhibit of photographs of a variety of blackboards with physics equations, drawings and text on them, as covered at Quantum Chaos on Display in Top Physicists’ Chalk Scrawls | Raw File | WIRED What is the one in the first closeup photograph (numbered \"2\/14\") about? The circled text \"non SUSY\" suggests that at least one of the topics is Supersymmetry (Wikipedia), but what is the gist of it? For clarity, this question is about the blackboard in the first closeup image on the Momentum exhibit web site, labeled \"2\/14\". ![enter image description here](http:\/\/i.stack.imgur.com\/YhrRI.jpg) Caption: $\\uparrow$ Click image to enlarge it. For example, the equation in the upper left starts with > $T_{ds} \\sim 1\/R$ For extra credit - where is the blackboard \/ who wrote on it \/ etc? (Note that I am also interested in the other blackboards, and suggest separate questions here for each of those. There is some commentary on some of the others at The Beautiful Blackboards at Quantum Physics Labs | MetaFilter)"}
{"id":"106623","title":"Electric field for two adjacent infinite layers of width h charged + and − uniformly","text":"I'm trying to figure how to calculate the electric field created by two adjacent infinite layers of width h charged + and − uniformly. I don't understand why isn't the field outside of the layers isn't simply zero. Why aren't the 2 cancelling each other? And how can I calculate the field anywhere between the two using gauss law? Thanks"}
{"id":"80511","title":"How do I derive the Lorentz contraction from the invariant interval?","text":"Reviewing some basic special relativity, and I stumbled upon this problem: From the definition of the proper time: $$c^2d\\tau^2=c^2dt^2-dx^2$$ I was able to derive the time dilation formula by using $x=vt$: $$c^2d\\tau^2=c^2dt^2-v^2dt^2=c^2dt^2\\left(1-\\frac{v^2}{c^2}\\right)\\rightarrow d\\tau = dt\\sqrt{1-\\frac{v^2}{c^2}}=t\/\\gamma$$ Now, I would very much like to be able to derive the length contraction formula in a similar fashion, and feel strongly that this should be possible. The definition of the invariant interval is: $$ds^2=dx^2-c^2dt^2$$ using $t=\\frac{x}{v}$ I tried: $$ds^2=dx^2-\\frac{c^2}{v^2}dx^2=dx^2\\left(1-\\frac{c^2}{v^2}\\right)\\rightarrow ds=dx\\sqrt{1-\\frac{c^2}{v^2}}$$ This is where I'm stuck: I don't see how this can be converted to a Lorentz factor... Any help that will allow me to arrive at the desired result $ds=\\gamma dx$ would be very much appreciated."}
{"id":"102717","title":"Confusion With Neutrino Oscillation","text":"Neutrinos produced in the weak interaction are eigenstates of the weak interaction hamiltonian or flavor eigenstates. Right? If this is so, then $\\nu_e,\\nu_\\mu,\\nu_\\tau$ should be the stationary states of weak hamiltonian and weak hamiltonian is diagonal in this basis. Then, why do they oscillate during the propagation from one flavor to another? **_Is the hamiltonian different during propagation (i.e., no longer weak hamiltonian?) and $\\nu_e,\\nu_\\mu,\\nu_\\tau$ are no longer eigenstates of the new hamiltonian_**?"}
{"id":"83476","title":"Normal force of ball sliding on concave surface","text":"Imagine a ball is sliding along a surface shaped like $y=x^2$. Like ![y=sin\\(x\\)](http:\/\/i.stack.imgur.com\/7u0dr.png), but please ignore the fact that the center of the ball is on the surface instead of the edge. When the ball is stationary, I can calculate the normal force fairly easily. $F_\\text{net}$ in the direction normal to the surface is 0. The normal force is normal to the surface, so it is the force that is exactly strong enough to cancel the normal component of the force of gravity. The normal force comes out to $<\\frac{-a}{(1 + a^2)}, \\frac{1}{(1 + a^2)}>$, where $a=f_{\\text{surface}}'(x)$ (or in this specific case: $a=2x$) I initially expected this to be true even after the ball begins sliding down the surface, but I found that it was not. Intuitively, I understand that a velocity with a non-zero component in the normal direction affects the strength of the normal force. This seems like the same mechanism by which the normal force is able to stop a falling object as it hits the ground, for example. In the falling ball example, the relationship between velocity and normal force seems a bit sticky mathematically. Assuming neither the ground nor the object compress, the normal force would have to be infinite initially, right? It would only be infinite for an infinitesimal amount of time (so the integral would still be finite). However, in that case, the $v(t)$ curve is discontinuous, while it seems like $v(t)$ would be continuous for a ball sliding on $y=x^2$, so hopefully the relationship is less obnoxious in the continuous case. **Is there a mathematical formulation of this relation between normal force and velocity?** Can I add a velocity parameter to my equation for $F_\\text{net}(x)$ (making it $F_\\text{net}(x, v)$) to take this relationship into account? **Note** : I also see a conservation of energy solution that sidesteps the normal force issue altogether, but I'd like to understand the forces aspect."}
{"id":"33850","title":"Cellular automata rules for quantum mechanics","text":"My limited understanding of quantum theory is that a **quantum system is completely described by its wave function** , which **deterministically evolves according to Schrödinger's equation** until wave function collapse. Schrödinger's equation is essentially just a differential equation, which means (as far as I can see) that the **state at every point in space changes depending only on its surroundings** (i.e local rules). Correct me if I'm wrong but it's at the point of wave function collapse that the non-locality comes in, which is what cellular autonoma (CA) models can't deal with very well. I'm not hypothesizing or suggesting that the universe **IS** a cellular autonoma (as Wolfram has suggested), but **what would a CA _model_ of Schrödinger's equation be like**? (Ignoring wave function collapse for the minute) Providing my understanding described above isn't wrong, I want to know, for the CA model:   1. What values describe the state at a given point? (i.e. what information would be in each CA cell?)   2. What rules are followed every discrete time tick to progress these values? I ask because I've gained a good intuition (an \"understanding\" that I'm happy with at least!) of heat flow, wave propagation and the rules of fluid dynamics through **interpreting their differential equations as local cellular automata rules** , and writing little toy simulation programs to see how they behave and look. I'm a computer science\/mathematics undergraduate and I've always said that I don't understand something until I can write a program that simulates it, so being a little too ambitious for my own good I want to write a simulation for a QM system! My background is I've read a few Penrose\/Feynman books, and I've got the complete Feynman lectures on physics (not read them yet...), and physics has always been a side hobby, but I'm not too hot yet on the mathematical language that is used for QM."}
{"id":"89617","title":"Is the spin-singlet state also a Resonating-Valence-Bond(RVB) state?","text":"The _spin-singlet state_ of a lattice spin-1\/2 system is defined as $S_x\\Psi=S_y\\Psi=S_z\\Psi=0$, where $S_\\alpha=\\sum S_i^\\alpha(\\alpha=x,y,z)$ are the total spin operators, in other words, a _spin-singlet state_ is a spin state with _gobal $SU(2)$ spin-rotation symmetry_. On the other hand, a RVB state is the superposition of various configurations of singlet-product states (no need for equal weight superposition here), thus, a RVB state is of course a _spin-singlet state_ according to the above definition. For the reversed statement, is the spin-singlet state also a RVB state? How to prove it? For the extreme 2-spin system, it's direct to show the equivalence between the spin-singlet state and the RVB state."}
{"id":"89610","title":"What is the nucleon axial charge?","text":"Can someone point me to a short definition of what the nucleon axial charge is?"}
{"id":"130640","title":"Physical Quantities","text":"What are the physical quantities which most relate to their definition? Could you give some examples? The definition I found is: \" it is any property which can be measured\". In the case of mass the given example is \"when we go buy apples we purchase two kilograms of apples. Please give another example."}
{"id":"132125","title":"Quantifier problems of equations in physics","text":"Equations in physics are often written without quantifiers. For instance, from time to time we can see the equation $$E = mc^2$$ is casually written down. I am wondering shouldn't it be written rigorously as something like: For all $E$ in a given range of values and all $m$ in a given range of values, we have $E =mc^2$? If not, then logically the equation $E = mc^2$ is sometimes valid and sometimes invalid as $E, m$ run through their given ranges, so that one cannot use \"$E = mc^2$\" to express what it is intended to mean. Then how does one incorporate this phys-mathematics writing style, as illustrated in the beginning, with the authentic mathematical writing style, as illustrated in the second paragraph, in the sense of logical rigor?"}
{"id":"132124","title":"Are Both figures correct?","text":"Actually I am in high school I just want to confirm that whether two figures of this question possible: Question is : A car is moving along a straight line OP.It moves from O to P in 18 seconds and returns from P to Q in 6 seconds , where OP = 360 metres and OQ = 240 metres.What are the average velocity and average speed of the car is going (a) From O to P and back to Q? My physics teacher made this figure from the question: ![by My teacher](http:\/\/i.stack.imgur.com\/SCZjb.jpg) and I this figure is also right: ![by me](http:\/\/i.stack.imgur.com\/URVXD.jpg) But When I solve the above Question by first figure I got Average speed of 20 m\/s and 40 m\/s from 2nd figure. Which one is correct? (When I ask my Question to my physics teacher , He said \"First one is correct,Second one doesn't make any sense\")"}
{"id":"47217","title":"Where does the energy flow differ between a forward and reverse process?","text":"![PV diagram of cycle](http:\/\/i.stack.imgur.com\/Yn1d3.png) Mathematically ($W=\\int PdV$) and by the First Law, I understand that $1 \\rightarrow 4 \\rightarrow 3 \\rightarrow 2 \\rightarrow 1$ and $1 \\rightarrow 2 \\rightarrow 3 \\rightarrow 4 \\rightarrow 1$ are not symmetric; the first requires work, whilst the latter gives it to the environment. What I am having trouble with is bellyfeeling _why_. A physical example: imagine a machine that works thusly: > **(1-2)** A piston loaded with 3 bricks in a hot reservoir isothermally > expands when 1 brick is removed. > > **(2-3)** Then, removed from contact from any reservoir, it adiabatically > expands when the 2nd brick is removed, cooling to the temperature of the > cold reservoir, where > > **(3-4)** the 2nd brick is replaced, isothermal compression takes place, > then > > **(4-1)** away from reservoir the 3rd is added, the temperature rising to > the intial hot temperature. **Given that the two possible cycles are asymmetric, _at what point (1,2,3,4) does the internal energy of the gas or the potential energy of the bricks differ for the two different cycles?_**"}
{"id":"109993","title":"Angular momentum of a black hole","text":"I recently read this Phys.SE post and, since I didn't know that black holes had a spin, a question came to my mind: how can I calculate the spin velocity of a black hole? Does mass or radius affects it? I googled it but I couldn't understand much, all I found was about Orbital Velocity of a planet..."}
{"id":"109998","title":"what forces are present when slicing an object.","text":"I found here:What happens when we cut objects? that it is due to fracture wood splits open when using a wedge. Does it only apply when the wedge is already in the wood? How about the moment before a crack is made, do you need to overcome the compressive and shear stress of the wood to start a crack? I want to compute for the forces needed to cut the wood, along the grain, using a wedge. but Im confused whether shear and compressive strength or fracture initiation is involved to start the cut. and when the wedge in the the wood should I compute for fracture propagation or the cosine of the normal force on the wedge that is pushing the grains apart. I hope I'm making sense."}
{"id":"76087","title":"How do you calculate the position of the antinodes on a wave?","text":"if for example i have a waveform with the formula $y=\\sin(1.2x)+\\sin(1.8x)$. The first 6 antinodes on my graph come up at around: $$ x=\\begin{cases} 1.01\\\\\\ 2.97\\\\\\ 4.65\\\\\\ 5.80\\\\\\ 7.48\\\\\\ 9.46\\\\\\ \\end{cases} $$ but I can't find a way to find the exact positions."}
{"id":"39591","title":"Gaussian surface question","text":"There is an infinite slab of charge, and a (Gaussian surface) cylinder whose ends are both outside of the slab. $\\phi_A$ is the flux through this cylinder, by symmetry the component of the flux through the top end is $\\frac{ \\phi_A}{2}$. Now, lift bottom end of the cylinder until it is _inside_ the slab. The net flux is now $ \\phi_B$. Proposition 1: The component of the flux through the top end is unaffected by the position of the bottom end (even though the total flux _is_ ), as the electric field and area at and of the top end are both unaffected by the movement of the bottom end.Thus the flux through the top end is _still_ $\\frac{ \\phi_A}{2}$. Proposition 2: Therefore the flux through the bottom part is all the flux that is remaining (that is, $\\phi_B-\\frac{ \\phi_A}{2}$). This must be balderdash, as with by extension of my logic, the flux through the bottom in the second case is$\\phi_B-(\\phi_B-\\frac{ \\phi_A}{2})=\\frac{ \\phi_A}{2}$: that is, the net flux is unaffected as the top and bottom flux add up to $ \\phi_A $, in contradiction with $\\phi=\\frac{Q}{\\epsilon_0}$. What is incorrect with propostion 1, both physically and mathematically?"}
{"id":"77973","title":"Riemann tensor in 2d and 3d","text":"Ok so I seem to be missing something here. I know that the number of independent coefficients of the Riemann tensor is $\\frac{1}{12} n^2 (n^2-1)$, which means in 2d it's 1 (i.e. Riemann tensor given by Ricci Scalar) and in 3d it's 6 (i.e. Riemann Tensor given by Ricci tensor). But why does that constrain the Riemann tensor to only be a function of the metric? Why not a tensorial combination of derivatives of the metric? What I mean is why is the Riemann tensor in 2D of the form \\begin{align} R_{abcd} = \\frac{R}{2}(g_{ac}g_{bd} - g_{a d}g_{b c}) \\end{align} and in 3D, \\begin{align} R_{abcd} = f(R_{ac})g_{bd} - f(R_{ad})g_{bc} + f(R_{bd})g_{a c} - f(R_{bc})g_{ad} \\end{align} where $f(R_{ab}) = R_{ab} - \\frac{1}{4}R g_{ab}$? Wikipedia says something about the Bianchi identities but I can't work it out. A hint I got (for the 2d case at least) was to consider the RHS (the terms in parenthesis) and show that it satisfies all the required properties of the Riemann tensor (sraightforward) and proceed from there - but, I have not been able to come up with any argument as to why there must be a unique tensor satisfying those properties. Of course I could brute force it by computing $R_{abcd}$ from the Christoffel symbols etc., but surely there must be a more elegant method to prove the statements above. Help, anyone? I haven't been able to find any proofs online - maybe my Googling skills suck."}
{"id":"77976","title":"How can I calculate the pressure against the sides of a container?","text":"While answering this question on DIY.SE, I wanted to calculate the amount of pressure (in psi) that would be exerted on the walls of the container. However, I don't know much about how to calculate such pressures. # Given: **The container:**   ![Hot Tub Frame with Banding and Bracing](http:\/\/i.stack.imgur.com\/9Coxs.jpg) _Assume the interior of the container is solid (e.g. covered in plywood and a rubber liner)_   **Container outside dimensions:** 8' x 8' x 3'   **Container inside dimensions:** 89\" x 89\" x 36\"   **Container volume:** 165.020917824 cubic feet (`(89x89x36) x 0.000578704`)   **Amount of water held by container :** 1234.521486241344 gallons (`165.020917824 x 7.481`)   **Weight of water held by container :** 10,297.3052722176 pounds (`165.020917824 x 62.4`) How would I calculate the amount of pressure on the walls in pounds per square inch?"}
{"id":"77975","title":"Can bad guys really hide \"dirty bomb\" material via bananas?","text":"This one probably has some cross-over with Skeptics SE, but I thought it fits here a little better. The concept comes from the TV show NCIS:LA second season episode \"Empty Quiver\" Here, the villains steal some nuclear material from a hospital's radiology department (in itself dubious). Then in order to transport it across the country without detection, they surround it in a truck full of bananas, as if it's nothing more than a produce delivery to a grocery store. Ok, so let's accept that bananas contain potassium, that there exists an isotope of potassium that is radioactive and found normally in nature (including bananas) and other things as indicated here. Is this collectively, i.e. an entire truck load, enough to fool a trained inspector?"}
{"id":"77974","title":"How is a Lichtenberg figure created?","text":"Related: Breakdown voltage of a dielectric When I think of the dielectric breakdown of air, an electric field of $3×10^6$ V\/m is enough to accelerate initially ionized electrons created primarily from muons via cosmic radiation. These ionized electrons then collide with neutral air molecules, which in turn generate an avalanche of charge that produces lightning. Using Wikipedia’s language, lightning is a three-dimensional Lichtenberg figure. Reading Wikipedia’s section on Modern 3D Lichtenberg figures, a piece of acrylic (or glass) is impregnated with electrons from a linear accelerator. When this acrylic is made to discharge, a “beam tree” is formed that is absolutely beautiful. ![enter image description here](http:\/\/i.stack.imgur.com\/sy0sY.jpg) What is the role of the pointed conductive object (or nail) that initiates the dielectric breakdown of the acrylic? The electrons trapped inside the acrylic create a charged dielectric and create some potential. I assume that the nail grounds out this acrylic so that electrons burn a path towards the nail to form the beam tree. However, **I want to know what is happening at the atomic level**?"}
{"id":"72251","title":"Mass is a stuff or not?","text":"Einstein said, there can be no clear definition can be give for moving body mass – instead of giving its name as “rest mass”, it is better to mention, “momentum and energy”. Matter is a stuff, particles and fields are stuffs – but “mass” is not a stuff?"}
{"id":"128879","title":"How to restore the potential energy?","text":"How to restore the potential energy, of a particle with variable mass and variable coordinate, by сartesian coordinates and mass of the particle? ? Dimension - one dimensional. Data that is to compute, only the mass and Cartesian coordinate of the particle, like this:                x        m      100.0  2710988       99.8  1000395      100.42 1669902       99.74  927829       99.74  927829      101.3  2153945      etc.      x and m are time-dependent ! No other data to calculate potential energy."}
{"id":"72259","title":"Speed of birth of a star","text":"A star forms when enough material gets close enough. But how fast does the star start to shine? Is it gradually heating up (similar to an oven) slowly shining more and more? Or is it more like an explosion (similar to an atom bomb) that happens the second the critical mass is reached?"}
{"id":"87579","title":"Speed of light, lasers and mass","text":"Hopefully this isn't a bad question. Light travels at the speed it does and nothing else can travel that fast because things have mass, correct? Or at least correct on an elementary level. And if lasers, which are light as I understand it, have the ability to destroy other things. Doesn't that mean light has energy? But if light has energy shouldn't it have mass? I thought $E = mc^2$ and $E$ is energy, $m$ is mass. 0 mass times speed of light squared would equal zero energy, right? It's probably obvious that I'm no physicist but hopefully its also obvious that I enjoy science and physics."}
{"id":"15284","title":"There is absolute silence?","text":"Silence is a relative or total lack of sound. But it is possible for real? Is there any way to achieve this?"}
{"id":"15281","title":"Renormalizing Chaos: Transition in a Logistic Map","text":"I am currently trying to understand the analysis of a logistic-like map $$f_\\mu (x) = 1-\\mu x^2$$ after section 2.2 in \"Renormalization Methods\" by A. Lesne. As I understand it, the physical situation is that $f_\\mu$ has exactly $2^n$ attractors $x_{i\\dots 2^n}$ in a certain region $\\mu\\in [0,\\mu_c)$. Since one can order them such that $f_\\mu(x_i) = x_{i+1}$ and the number of attractors grows in some way always by a factor of two, it is called \"double-periodic scenario\". However, at $\\mu_c\\approx1.401$, this behavior cannot be found any more and one might interprete this as critical point since the system appears to be **chaotic** from there, or, in some sense, the number of iterations it takes until a point $x$ is reached again becomes infinity. Here is a scetch of the situation: ![Logistic-Like- Map](http:\/\/i.stack.imgur.com\/zNq95.png) At certain points $\\mu_i$, $i=[i,\\infty]$, the number of attractors doubles until $\\mu_c$ is reached where this periodicity cannot be found. $N$ has been chosen to be sufficiently large. To understand the transition from the periodic point of view, Lesne conjectures that some delta exists such that $$\\lim_{i\\rightarrow \\infty}\\delta^i (\\mu_c - \\mu_i)=A\\neq0\\ .$$ Then, it is stated that $\\delta \\approx 4.67$ is somehow universal and can be derived using a **renormalization** approach of the form $R\\left[ f\\right] \\propto f^k$ with $R$ beeing an operator that contains in the end all the informations about the system. Two things are unclear to me: > ### How can one use the renormalization operator $R$ to analyze the critical > behaviour, thus $\\delta$? and > ### Is there a systematic way to find an appropriate $R$ better than just > looking at self-similarities of some graphs? Thank you in advance"}
{"id":"87575","title":"Unitary Lorentz transformation on quantized Dirac spinor","text":"I am stuck again on page 59 of Peskin and Schroeder. In particular, I do not know how they get equation (3.110). Let me first give some background in the way that I understand it (but I might be completely wrong). A unitary operator $U(\\Lambda)$ acts on states as follows: \\begin{equation} |p,s\\rangle \\rightarrow U(\\Lambda)|p,s\\rangle \\end{equation} and therefore any operator, such as a Dirac field, transforms as: \\begin{equation} \\psi'(x) = U(\\Lambda)\\psi(x)U^{-1}(\\Lambda) \\end{equation} Now, from equation (3.109): \\begin{equation} U(\\Lambda) a_p^s U^{-1}(\\Lambda) = \\sqrt{\\frac{E_{\\Lambda p}}{E_p}}a^s_{\\Lambda p} \\end{equation} we can find the transformation of the positive frequency solution of $\\psi$: \\begin{equation} U(\\Lambda) \\psi U^{-1}(\\Lambda) = U(\\Lambda) \\int \\frac{d^3 p}{(2\\pi)^3} \\frac{1}{\\sqrt{2 E_p}} \\sum_s a_p^s u^s(p) e^{-ip\\cdot x} U^{-1}(\\Lambda) \\end{equation} \\begin{equation} \\Rightarrow U(\\Lambda) \\psi U^{-1}(\\Lambda) = \\int \\frac{d^3 p}{(2\\pi)^3} \\frac{1}{\\sqrt{2 E_p}} \\sum_s U(\\Lambda) a_p^s U^{-1} (\\Lambda) U(\\Lambda) u^s(p)U^{-1}(\\Lambda) e^{-ip\\cdot x} \\end{equation} and using equation (3.109) this becomes: \\begin{equation} \\Rightarrow U(\\Lambda) \\psi U^{-1}(\\Lambda) = \\int \\frac{d^3 p}{(2\\pi)^3} \\frac{1}{2 E_p}\\sqrt{2 E_{\\Lambda p}} \\sum_s a^s_{\\Lambda p} U(\\Lambda) u^s(p)U^{-1}(\\Lambda) e^{-ip\\cdot x} \\end{equation} and from this point I have no idea how to get to equation (3.110). If anybody could push me in the right direction, then this will be greatly appreciated. (I am aware that the integration measure is Lorentz invariant.) Another question: does anybody have any other references\/notes\/books where they discuss how the quantized Dirac _operator_ field transforms? I find P&S explanation thoroughly confusing (as may have become clear from the questions I have been asking recently on this forum :) ), but I cannot find any other book that treats this stuff."}
{"id":"133839","title":"Is negative mass for a bound system of two particles forbidden?","text":"Is there any theorem that forbids the bound system of two massive particles to have negative mass?"}
{"id":"89850","title":"What are the key properties of and differences between classical and quantum statistical mechanics?","text":"I'm studying different ensembles and different statistics (M-B, B-E, F-D), and I have some ambiguities about which of these models are applicable to quantum systems and which are usable for classical systems or both. (For example, the applicability of grand canonical ensemble to classical systems, M-B statistics to quantum mechanical systems, etc.) So I arrived at the question of what are the key properties of classical and quantum statistical mechanical systems, and what are their differences?"}
{"id":"35935","title":"What are these rays that appear in photograph of sun?","text":"In many images of light emitting objects we see such rays. Why do they appear ? What is the math behind their number and direction? ![enter image description here](http:\/\/i.stack.imgur.com\/eXMj0.jpg)"}
{"id":"127809","title":"Thoughts on the ice cube from orbit problem","text":"Let's say we have a really exquisite cocktail party somewhere in New Mexico, and we just ran out of ice cubes. To the rescue comes this new service provided by Orbital Glacier Inc. They provide ice cubes around the world within only 5 minutes (!). How do they do it? How big must an ice cube be when you drop it from the orbit, so that it will have the size of a typical ice cube when it hits your glass of scotch that you're holding in your hand. Assuming the ice cube in the orbit needs to be of an enourmous size, would Orbit Glacier Inc. inevitably impact the climate conditions on planet earth when celebrating several such cocktail parties? Disclaimer: We are in no way related to our competitor Moon Ice Now, Inc."}
{"id":"126224","title":"Anomalous Total Transmittance","text":"I am measuring with a Spectrometer UV-Vis-NIR Lambda 900 the total transmittance of a sample made by a sheet of Ethylene vinyl acetate (EVA) between two glasses of 3 mm thickness. I am obtaining an apparent total transmittance greater than 100% (up to 120%) for wavelengths 500-1000 nm. Can this measurement be ascribed to a physical phenomenon? Or there is something wrong in my apparatus?"}
{"id":"131473","title":"Can the Large Hadron Collider break the speed of light?","text":"Can Large Hadron Collider experiments break the speed of light at some point in the future? ![enter image description here](http:\/\/i.stack.imgur.com\/i438z.jpg)"}
{"id":"19702","title":"Problem based on Rotational Motion","text":"> A spool of mass $\\mathsf m$ and inner radius $\\mathsf r$ and outer radius > $\\mathsf{2r}$, having moment of inertia $\\Large\\mathsf{\\frac{mr^2}{2}}$ is > made to roll without sliding on a rough horizontal surface by the help of an > applied force $\\small\\mathsf{(F=mg)}$, on ideal string wrapped around the > inner cylinder (Shown in the figure). > > ![](http:\/\/i.stack.imgur.com\/qOKD2.png) > > Find the minimum Co-efficient of Friction $\\small\\mathsf{(\\mu_{k})}$ > required for Pure Rolling. My Work For Pure Rolling, $$\\mathsf{ω.r=v \\tag 1}$$ $$\\mathsf{α.r=a\\tag 2}$$ Two torques are acting on the Spool about it's center (Circle's center), one due to the applied force F and other due to the friction generated by the rolling of the spool on the rough horizontal surface. $$\\mathsf{∑τ=2r.f_k−F.r=Iα \\tag 3}$$ Translation motion's equation, $$\\mathsf{\\sum F_{ext} = F−f_k=Ma\\tag 4}$$ Solving equations $(2), (3)$ and $(4)$ and using $\\mathsf{μ_k=\\frac{f_k}{N}}$. I am getting $\\mathsf{μ=\\frac{3}{5}}$. While the options are $\\mathsf{\\frac{2}{9},\\frac{4}{9},\\frac{5}{9}}$and $\\mathsf{\\text{'none of these'}}$."}
{"id":"19703","title":"What makes up a resonator of radio?","text":"I was reading this article about resonators. **Quote** : > The sine wave that matches that particular frequency will get amplified by > the **resonator** , and all of the other frequencies will be ignored. As far as I know, in picture below, the capacitor and inductor below makes up an **oscillator circuit** which oscillates at some frequency so that it can be matched and amplified by resonator. So what is this resonator that the article is talking about? ![enter image description here](http:\/\/i.stack.imgur.com\/O0xxy.gif)"}
{"id":"17621","title":"Relationship between inductance and number of turns in a coil in loudspeaker","text":"I am in the process of designing a loudspeaker and have a question regarding the number of turns in the multilayered solenoidal coil and the speaker impedance of $8\\:\\Omega$. I understand that the Impedance is equal to the real part $R$ (resistance) added to the inductance and that the inductance is partly a function of the number of turn of the copper wire. I am having difficulty with understanding which equation describes this relationship between inductance and number of turns. Thanks."}
{"id":"17624","title":"What are some interesting coupled harmonic oscillators problems?","text":"That I could create as a classical mechanics class project? Other than the classical examples that we see in textbooks."}
{"id":"17628","title":"Quantum Computing, Qubit Creation\/Entanglement","text":"I am currently a high school student researching quantum computing. I was referred to this site by Google and a friend. Currently I am researching the qubit part of quantum computing. My question is exactly how are qubits created in the lab, and how are they entangled? I don't expect the answers to be incredibly specific but a general overview would be of a great deal of help."}
{"id":"13539","title":"parallel\/anti-parallel vs. triplet\/singlet description of two spins","text":"If we consider two spins, we can think of the spins as being either parallel (up|up or down|down)or anti-parallel (up|down or down|up). Or we can think of them as being in the triplet or singlet configuration. Is one description more correct than the other? Or is it just a matter of choice between two basis sets? It would seem to me that using T\/S is correct because it accurately reflects the symmetry needed in the wavefunction."}
{"id":"41826","title":"Does the magnitude of pressure on the sides (not the top and bottom) of a volume depend on depth?","text":"I understand that at a point in a fluid, such as water, the fluid will be in equilibrium as the sum of the pressure from all directions will equal to zero. For a volume of water, the reasoning should be the same otherwise unequal pressure would mean the volume would move. However, are the magnitudes of different points along the side of the volume the same for all depth or are they different? A picture would help illustrate my problem ![box](http:\/\/i.imgur.com\/ZOUm7.jpg) The box can be a solid object or just an imaginary boundary enclosing water (it shouldn't change the question I believe). Since the volume is at rest in the picture, I believe that Pressure - Side.1 and Side.2 will have the same magnitude and cancel out, and the same goes for Side.3 and Side.4. However, is there a difference in magnitude for Side.1 in comparison with Side.3 or Side.4? If so, is there an equation? It's been a long time since I read about it and I tried skimming an introductory physics book but didn't find it."}
{"id":"53672","title":"What is enstrophy?","text":"In turbulence, the enstrophy of a flow in a domain $\\mathcal{D} \\subset \\mathbb{R}^{D}$ $$ \\mathcal{E} = \\int_{\\mathcal{D}} |\\vec{\\nabla} \\times \\, \\vec{v}|^2 d^{D}x $$ appears sometimes, it's cool and has cute properties in 2d (see this answer about turbulent flows), etc. Now, what does it represent physically? Okay, it's the integral of the vorticity squared, but has it a physically understandable interpretation? Why is it interesting to study _that_?"}
{"id":"83687","title":"How algebraic geometry and motives appears in physics?","text":"First, I'm not a physicist so I have just a little background in physics. I have been reading some noncommutative geometry books and papers (Connes, Rosenberg, Kontsevich etc) and a lot of high machinery from algebraic geometry such as étale cohomology and motives appears in such books, however I could not guess where these structures arise in physical situations. How algebraic geometry and motives appears in physics? Why do physicists needs to use a projective scheme? When this scheme (or other structure) needs a noncommutative analog?"}
{"id":"62501","title":"Color of a Metal's Threshold Wavelength?","text":"How do I find the color of the threshold wavelength if the metal has a threshold wavelength of $\\mathrm{6.5\\times 10^{-7}m}$? I know that converts down to $\\mathrm{650\\ nm}$, but can I still use the visible color spectrum's wavelengths to find the color or is it different for metal? I'm sorry I'm new to this and am just making sure it still applies."}
{"id":"13281","title":"Scaling an electron microscope to fit within 1 cubic cm when running multiple parallel scans","text":"Yes, I'm serious.   * What components correspond to an individual scan vrs what components can be cascaded across all objects? Ex: The vacuum chamber can be cascaded ( colocate the sample chambers within a single vacuum enclosure). Whereas there will need to be individual detectors for each sample   * Can there be a singular electron source? Using multiple magnetic focal lenses to direct those electrons to the individual samples?   * Can this idea be extrapolated to significantly downsize the components of an electron microscope?"}
{"id":"80106","title":"How to find the wavefunction that solves an infinite square well with a delta function well in the middle?","text":"Solutions for the wavefunction in an infinite square well with a delta function _barrier_ in the middle are easily found online (see here for an example). I am wondering what the wavefunction is for an infinite square well with a delta function _well_ in the middle. The setup is the bottom of the infinite square well is defined to be zero energy. I realize that there will be two situations, one where the particle's energy is less than zero and will therefore be bound to the delta function well and one where the particle's energy is greater than zero and is bound to the infinite square well. What are the wavefunctions for these two situations?"}
{"id":"46655","title":"Why is boiling water the second time more quiet than boiling it the first time?","text":"First of all: This is a different question than Why is boiling water loud, then quiet?, although the answers might be similar. When I wake up, I boil some water for a cup of tea. It happens quite often that I do something else for 10-15 minutes and forget the water, so I boil it again. I noticed that boiling the water the first time is much louder than boiling it the second time. Why is this the case? Did the air dissolved in the water disappear and therefore it became more quiet? How long would it take until the water has again air (I'm sorry, I know this is linguistically wrong, but I don't know how to write it correct). Is there any simple way I could get air again into the water to check if it's only the air?"}
{"id":"45308","title":"Special Relativity","text":"Could someone explain to me how special relativity works? I know there are thousands of sources and databases of knowledge out there, but I find it difficult to understand, even after reading up on those sources. (Note: if you're an admin to close my question down, would you please be so nice as to point out something to help me with this question?)"}
{"id":"45300","title":"$E^2 = (mc^2)^2 + (pc)^2$: What units are used to measure $E$, $m$, $c$ and $p$?","text":"\\begin{equation} E^2 = (mc^2)^2 + (pc)^2 \\end{equation} If I am using this equation to figure out the energy of something, what units would I use? Would it be the metric system? I.e. kilograms for $m$, meters per second for $p$, kilometers per second for $c$? And what units of measurement are used for $E$?"}
{"id":"20080","title":"Why in professional spectroscopes diffraction gratings are used instead of prisms?","text":"Why in professional spectroscopes diffraction gratings are used instead of prisms? I guess transparency is not an issue, as we need optics anyway."}
{"id":"44657","title":"Gauss Law for Electric Fields","text":"What is the integral form for the **Gauss Law for Electric Fields**? ![a](http:\/\/i.stack.imgur.com\/to4Wa.png) or ![b](http:\/\/i.stack.imgur.com\/i0fa9.png) ?"}
{"id":"70344","title":"Why is a general Green's function invariant under translations?","text":"I am struggling to understand Green's functions, as used in Quantum Field Theory. One of my main problems is that the source I have been reading has a definition which is certainly correct, but involves enough integrals to obscure any intuition I initially had. Could someone explain what a Green function $G^{(n)}(x_1,x_2,\\dots,x_n)$ intuitively represents (or provide a reference to a readable explanation)? In particular, I am being told that the Green function enjoys the invariance $$G^{(n)}(x_1+y,x_2+y,\\dots,x_n+y) = G^{(n)}(x_1,x_2,\\dots,x_n)$$ Supposedly, this is very simple and intuitive, but I can't see it (other than going a few levels down in the definitions, which in not very enlightning). Could someone please explain this property? _(my usual disclaimer: I am not a physicist, but a mathematician who occasionally comes into contact with physics. Please forgive my lack of physical insight.)_"}
{"id":"41173","title":"Microsoft Excel not graphing $x = y^{1\/2}$","text":"The experiment was relating the period of one \"bounce\" when you hang a weight on a spring and let it bounce. I have this data here, one being mass and one being time. The time is an average of 5 trials, each one being and average of 20 bounces, to minimize human error.               t          0.3049s     0.3982s     0.4838s     0.5572s     0.6219s     0.6804s     0.7362s     0.7811s     0.8328s     0.869s      The mass is the mass that was used in each trial (they aren't going up in exact differences because each weight has a slight difference, nothing is perfect in the real world)               m          50.59g     100.43g     150.25g     200.19g     250.89g     301.16g     351.28g     400.79g     450.43g     499.71g      My problem is that I need to find the relationship between them, I know $m = \\frac{k}{4\\pi^2}\\times T^2$ so I can work out $k$ like that but we need to graph it. In this equation, $k$ is the spring constant, $m$ is the mass hanging on the spring and $T$ is the period. I can assume that the relationship is a `sqrt` relation, not sure on that one. But it appears to be the reverse of a square. Should it be $\\frac{1}{x^2}$ then? Either way my problem is still present, I have tried $\\frac{1}{x}$, $\\frac{1}{x^2}$, $\\sqrt{x}$, $x^2$, none of them produce a straight line. The problem for SU is that when I go to graph the data on Excel I set the y axis data (which is the weights) and then when I go to set the x axis (which is the time) it just replaces the y axis with what I want to be the x axis, this is only happening when I have the `sqrt` of $m$ as the y axis and I try to set the x axis as the time. The problem of math\/physics is that, am I even using the right thing? To get a straight line it would need to be $x = y^{1\/2}$ right? I thought I was doing the right thing, it is what we were told to do. I'm just not getting anything that looks right."}
{"id":"41171","title":"'t Hooft limit of coupling fundamental fermions to Chern-Simons theory","text":"This question is in reference to this paper: arXiv:1110.4386 [hep-th].   * I would like to know what is the derivation or a reference to the proof of their crucial equation 2.3 (page 12).    * In their analysis of fundamental fermions coupled to Chern-Simons theory, why have they been able to ignore the ghosts altogether? Does it have something to do with working in the light-cone gauge in which the self-interaction of the gauge field probably disappears and hence that removes the necessity to have ghosts?    * Their Lagrangian is 2.1 (page 11) is massless and they say that they can always tune the bare mass to be $0$ and they can always ignore the mass of the fermion. Can someone elaborate on this? Why was this possible? Isn't this tantamount to assumption of conformality which they want to prove in the 't Hooft limit? (Then isn't the argument becoming circular?)  Similarly if this were a scalar field theory then in the same strain one might want to say that the quadratic and the quartic scalar interactions can always be held at zero - but again wouldn't that be an assumption of conformality in the 't Hooft limit? What would be a genuine proof of conformality for this theory or its scalar version?   * Is the scalar version of this theory somehow uninteresting or known?    * In this paper whatever is claimed as the higher-spin currents seem to have their conservation laws broken in in $1\/k$ or $1\/N$, then isn't the theory interesting only when $k$ and $N$ are both infinite and then isn't that a trivial theory?  What sense does it make to say the Fermion $2$-point function depends on the 't Hooft coupling (as in equation 2.23 page 16)? Isn't that a non-physical quantity to talk of?"}
{"id":"40886","title":"Could someone jump from the international space station and live?","text":"Felix Baumgartner just completed his breathtaking free-fall skydiving jump from $120,000\\,\\text{feet} = 39\\,\\text{km}$ above the Earth, breaking the speed of sound during the process. I was wondering if the next step could be jumping from the international space station. The person would have to overcome the orbital velocity of the station, re-enter the Earth's atmosphere, and land on the Earth with their feet, assisted by a parachute. Would such a stunt be survivable by a human? Could the person wear a space suit like the one depicted in the movie \"Sunshine\"? Is such a space suit possible? ![enter image description here](http:\/\/i.stack.imgur.com\/W4fSO.jpg) **UPDATE** : I came across an interesting news article about a company that is working on a skydiving suite that would survive freefall from outer space. Read More Here ![enter image description here](http:\/\/i.stack.imgur.com\/tbqPi.jpg)"}
{"id":"40889","title":"How does Baumgartner slow his velocity so that the parachute can open up?","text":"1. How does Baumgartner slow his velocity so that the parachute can open up?   2. What is Baumgartner's speed just before his parachute opens?   3. What causes his speed to slow down just before the chute opens?   4. Is it air-resistance or is it altering his position from pencil to delta? The reason for this question is the supposition his body can endure only a maximum of \"anti-force`\" when the chute opens up."}
{"id":"121772","title":"Supersymmetric cancellation of loop contributions in a SUSY gauge theory","text":"It is known that in SUSY models, loop contributions are automatically zero leading to a **_technically natural_** solution of the Higgs mass hierarchy problem. In many SUSY books\/notes, it is often shown that, with a Lagrangian consisting chiral superfields, $$ {\\cal L} = \\int \\left\\\\{\\Phi_i^{\\dagger}\\Phi_i+\\left[\\frac{1}{2} m_{ij}\\Phi_i\\Phi_j+\\frac{1}{3}\\lambda_{ijk}\\Phi_i\\Phi_j\\Phi_k\\right]\\delta^2(\\bar{\\theta}) +h.c.\\right\\\\}d^2\\theta d^2\\bar{\\theta} $$ All the loop diagrams of the superfields are identically zero. Thus in terms of component fields, the loop contributions are canceled out. Now, the same kind of vanishing loop contributions should be true in SUSY gauge theory as well. Without any direct\/explicit calculation of the loop supergraphs, is there a simple argument\/way to see how they are zero? Is that the same reason as the chiral model above, each supergraph contains a factor of $\\delta^4(\\theta)$ or higher order, which is $0$ ? Even if you provide references, your own interpretation\/explanation is appreciated."}
{"id":"123730","title":"Calibrating the relative intensity of a spectrometer with a blackbody source?","text":"I am trying to compare the relative brightness of H-Beta and H-Alpha lines using a CCD spectrometer. In order to correct for the different grating efficiency at the two different wavelengths. I took the spectrum of a 100W bulb to use as a blackbody source. I also know the pixel location that correlates to the H-alpha and H-beta line. I am wondering how to translate the blackbody spectrum to a relative intensity calibration. I thought to try to approximate the peak and use Wien's law to get the temperate of the filament and use that temperature in the Planck's law equation. However, I was unsure if there was a standard method of obtaining a relative intensity calibration for a spectrometer."}
{"id":"80816","title":"Is $E=mc^2$ false?","text":"Recently I have found a Youtube video saying that $$E=mc^2$$ is false. It is said that the real one is $$ E^2=m^2 c^4 + p^2 c^2 $$ where $E$=energy, $m$=mass, $c$=speed of light, $p$=momentum. Is there anything true in it? It says that if it was $E=mc^2$ then as light had only momentum it would not have any energy. Youtube link Youtube Search result"}
{"id":"80815","title":"Calculating sum of heat transfered","text":"I have a formula of thermal conductance heat transfer rate. Here it is: $$ Q = \\lambda{S (T_1 - T_2) \\over L} \\Delta t $$ For my calculations I have got some constant values available $$ Q = 0.58{1 (T_1 - T_2) \\over 0.01} \\Delta t $$ I can express difference of $T_1$(90°C) and $T_2$(0°C) as (note: $c_1$ and $c_2$, $m_1$ and $m_2$ are equal for my values, so I join them) $$ T_1 - T_2 = {Q_1\\over c_1m_1} - {Q_2\\over c_2m_2} = {Q_1 - Q_2 \\over c_1m_1} $$ $$ 90 - 0 = {172431270 - 0\\over2090\\times 916,7} $$ for beginning values. Now the problem: $T_1 - T_2$ is not constant, because $Q$ is just a rate of energy in one moment. In next moment, the difference of $T_1$ and $T_2$ will be smaller by $Q$ $$ T_1 - T_2 = {(Q_1 - Q) - (Q_2 + Q)\\over c_1m_1} = {Q_1 - Q_2 - 2Q\\over c_1m_1} = {172431270 - 2Q\\over 2090\\times 916,7} $$ This happens every following moment. So, completing everything into one formula, $$ Q = {58{172431270 - 2Q_{moved} \\over 2090\\times 916,7} \\Delta t} $$ My question: Is there any way to do sum of all moments in 10 seconds to get heat transmitted ($Q$) between two solids? The only problem I have there is that $T_1$ - $T_2$ is not constant. I can't evaluate it without using $Q$. Please help."}
{"id":"96427","title":"Why does the blade height goes on decreasing in a multi-stage axial flow compressor?","text":"The axial compressors used in jet engines have their blade heights in decreasing order in the direction of the flow."}
{"id":"1768","title":"Why doesn't air freeze?","text":"I am in no way experienced in the Physics field so this question may seem a bit silly but i'd appreciate an answer :) Why doesn't air freeze?"}
{"id":"1766","title":"Fermi's Golden Rule","text":"It is well known that to calculate the probability of transition in the scattering processes, as a first approximation, we use the Fermi golden rule. This rule is obtained considering the initial system in an eigenstate of the unperturbed Hamiltonian $H_0$ and considering the time-dependent perturbation $V(t)=\\begin{cases} V(\\underline{r}),& \\mbox{if } t_i<t<t_f \\\\\\ 0, & \\mbox{otherwise} \\end{cases}$ where $t_f-t_i$ is the interval of interaction, and then considering the limit for $t_i \\rightarrow -\\infty$, $t_f \\rightarrow + \\infty$. Now, what is the range of validity of this approximation?"}
{"id":"87850","title":"Parity Transformation Behavior of (Weyl-)spinors","text":"Suppose $u^\\pm (p)$ are Weyl-spinors with helicity $h = \\pm$, i.e. solutions of the Dirac equation $p_\\mu \\gamma^\\mu u(p) = 0$. How do these spinors behave under parity transformation? In Peskin&Schroeder, chapter 3.6, \"Parity\", the authors write $$u(p) = \\gamma_0 u(\\tilde{p})$$ where the tilde is understood to implement the parity on the coordinates: $\\tilde{p} = (p_0,-p_1,-p_2,-p_3)$. The helicities are not explicity mentioned, but I guess this means that they are preserved. If I google for \"parity spinors\", however, the first result[1] suggest they do (scroll down the page). I also found references suggesting that the phase may change. My Question: Is the formula above correct? Is there an additional phase factor? Does it depend on the representation of the Gamma matrices? [1] http:\/\/www.ippp.dur.ac.uk\/~krauss\/Lectures\/QuarksLeptons\/Dirac\/RealDirac_6.html"}
{"id":"28355","title":"What does the Fermi Energy really signify in a Semiconductor?","text":"In understanding the behavior of semiconductors, I'm coming across a description of the Fermi Energy here and at Wikipedia's page (Fermi Energy, Fermi Level). If I understand correctly, the _Fermi Level_ refers to the energy state at which there's a 50% chance of finding an electron. This varies with temperature. The _Fermi Energy_ is the highest occupied energy state of fermions at absolute zero. I'm a little confused as to the relation of the two terms. Additionally, in a semiconductor, the _Fermi Energy_ falls between the valence band and the conduction band. However, my understanding is that electrons cannot exist between the two bands -- so why isn't the _Fermi Energy_ the top of the valence band?"}
{"id":"11112","title":"Stresses put through joints on a steel climbing wall","text":"Being a fan of stackoverflow, I thought I could try this question here, hope its not too off topic. I'm wanting to build a free standing climbing wall in my garden and dont know where to start working out its strength and so construction. It will be made from interlocking (using bolts) 50cmx50cmx50cm mild steel cubes stacked 3m tall with a base of 1mx1m. The 4 base cubes will then be bolted to a concrete foundation. If I then bolt other cubes to the side of the main stack to create an overhang how can I work out the stress put onto the joints between the overhang and the main stack and between the stack and the foundations? The bolts between the cubes will be 3cm long and 2cm diameter. I havent decided what fixings to use between the stack and concrete as I believe this is where the problem will lie. Thanks for any help its much appreciated!"}
{"id":"111046","title":"Is the cross section of a relativistic water hose or string always a perfect circle?","text":"Given is a very long tube, such as a water hose or a tubular string with finite thickness, that has a constant circular cross section of radius $r$ along the length and that is at rest in an inertial frame. The hose is either   1. straight or    2. curved with radius $R$, or    3. irregularly shaped. Is the shape of the cross section always a circle for a relativistically fast observer of velocity $v$? What is the observed cross-section radius $r(v)$ of the hose? How does $r(v)$ depend on the geometry of the situation, i.e., on the angles between v and the hose? Is it constant along the length of the hose? The overall shape of the hose will vary for each observer. But will its radius $r(v)$ vary or not? We suspected that the cross section remains circular in all cases, and that $r(v)=r$ in all cases, for all observers, along the full length of the hose. The answer is a bit unexpected, though. P.S. This is not homework. It is a question we discussed at lunch yesterday. P.P.S. If the \"hose\" would be a string with some radius, the result might also apply to string theory?"}
{"id":"8713","title":"Determining Average Tidal Effects","text":"Maximum tidal heights vary widely across the globe, from 16 m in the Bay of Fundy to mere centimeters elsewhere. These variations are due to coastline and shoreline differences. This makes it difficult to determine a global average ocean tidal maximum height. Let's assume the Earth is a landless planet covered throughout by an average one km of ocean. How can we calculate the maximum ocean tidal height from a new moon or full moon syzygy, assuming semi- major Moon & Sun distances?"}
{"id":"94953","title":"Clarifying constraint forces in Lagrangian dynamics","text":"In the Lagrangian formulation, the addition of constraint forces that are unknown can be done with Lagrange multipliers, which allows for the forces to be found. Taking $k$ constraints of the form $f_k(q,t)=0$, we have $$\\frac{d}{dt}\\frac{\\partial L}{\\partial \\dot{q}}-\\frac{\\partial L}{\\partial {q}}+\\sum{\\lambda_k \\frac{\\partial f_k}{\\partial {q}}}=0.$$ So, the questions:   1. Are the $\\lambda_k$ the constraint forces (and therefore equal for each $q_i$), or is $\\lambda_k \\frac{\\partial f_k}{\\partial {q_i}}$ the $i$th component of the $k$th constraint force?   2. Should the signs of the constraint forces be interpreted as positive along the $\\hat{q}$ direction and negative opposing? I have worked out the Lagrangian for a car going around a banked curve at constant speed, radius, etc., to see if I could derive the constraint forces. $$L = \\frac{m}{2}(\\dot{z}^2+R^2\\dot{\\theta}^2+\\dot{R}^2)-mgz$$ with constraint $f = z-R\\tan(\\alpha)=0$. I get, from the $z$ equation (after applying $\\ddot{z}=0$), $$\\lambda = mg$$ and, from the R equation (after applying $\\ddot{R}=0$), $$\\lambda=-\\frac{mR\\dot{\\theta}^2}{\\tan(\\alpha)}.$$ From the FBD, it's easily seen that $tan(\\alpha) = \\frac{m\\omega^2R}{mg}$, which is consistent with the idea that what I really had above were $\\lambda_z$ and $\\lambda_R$, leading to $F_z=\\lambda \\frac{\\partial f}{\\partial {z}}$ and $F_R=\\lambda \\frac{\\partial f}{\\partial {R}}$."}
{"id":"114274","title":"Taylor series: Epsilon not differentiated?","text":"Why isn't epsilon differentiated with respect to time? (see my question on the right) ![enter image description here](http:\/\/i.stack.imgur.com\/AlAgL.png)"}
{"id":"114270","title":"Can rain evaporate?","text":"Are there any atmospheric condition under which a visible raindrop can fail to hit the ground by evaporating first? I imagine this would require a large vertical temperature difference, and possibly the rain forming very high up. Has anything like this been observed experimentally, or if not - is it possible to perform a calculation to show whether this is plausible?"}
{"id":"96086","title":"Special Relativity moving in space","text":"Given that time is 'just another' dimension and people get hung up on the fact that we cannot go back and forth in time like the other dimensions. Is there any proof that the corner of 8th Avenue and 14th Street at 3pm is the same place as as 8th and 14th at 4pm? Consider that in a 3 dimensional space, coordinates (8, 14, 3) isn't the same spot as (8, 14, 4) why can we assume that in a 2D place with Time that (8, 14, 3pm) is the same place as (8, 14, 4pm) only that the time has changed? Could it be that we cannot move back and forth in the other dimensions either, i.e. if we walk forward 10 meters and back 10 meters, it isn't really the same place since the time component has changed."}
{"id":"56109","title":"distance of electron from proton","text":"An electron is projected, with an initial speed of $1.10 \\times 10^5 \\text{m\/s}$, directly towards a proton that is essentially at rest. If the electron is initially a great distance from the proton, at what distance from the proton is its speed instantaneously equal to twice its initial value? i know it can be solved by equating the total energy as K.E+P.E(electron)=K.E+P.E(Proton) kinetic energy is $ \\frac{1}{2}mv^2$. how can i find out potential energy of each particle?"}
{"id":"56106","title":"What was the first ionization radiation?","text":"While trying to make somewhat of a timeline of the history of ionizatig radiation, i am wondering about the following questions: The first photoelectrical effect was found 1839 by Alexandre Becquerell - the photoelectrical effect and therefore UV-light, would be the first ionization radiation. Is that right? While i am quite sure about it for electromagnetical radiation, i am not sure about particle radiation. Was there any ionizating particle radiation before the discovery of uran-radiation by Antoine Becquerell? A source for proof would be quite a nice extra, thanks!"}
{"id":"105028","title":"Metric to describe an expanding spacetime from coordinates reflecting the perspective of a local observer","text":"The FLRW metric describes the metric expansion of spacetime from the perspective of comoving coordinates. Given the way this metric is usually formulated, comoving distances stay constant, and the size of the unit ball \"shrinks\" as you translate it along the time axis (but not the spatial axes). I'm interested in seeing how to reformulate this metric given a choice of coordinates that lines up with the perspective of an observer who is in the space. From the observer's perspective, everything is falling along timelike geodesics that are moving away from it, and the further objects are from the observer, the faster they seem to be falling, and the size of the unit ball around the observer remains constant with time. To keep it simple, I only care about the case where curvature is 0 and we set $c=1$. So the FLRW metric in those circumstances is $$\\mathrm{d}s^2 = -\\mathrm{d}t^2 + a(t)^2 \\cdot (\\mathrm{d}x^2 + \\mathrm{d}y^2 + \\mathrm{d}z^2)$$ Where $a(t)$ is the time-dependent scale factor. I thought that to flip this on its head, you'd want a metric that looks like this: $$\\mathrm{d}s^2 = -b(x,y,z)^2 \\cdot \\mathrm{d}t^2 + \\mathrm{d}x^2 + \\mathrm{d}y^2 + \\mathrm{d}z^2$$ Where we now have the space-dependent scale factor of $b(x,y,z)$, and it affects the time coordinate. I was thinking that for some such suitable function $b$, which is presumably monotonic, you'd end up with a metric which stays constant in time rather than in space, and timelike geodesics that are all constantly moving away from one another. (They're also moving away from the line $(t,0,0,0)$, representing the observer I was talking about.) Am I on the right track here? I'm looking for something which is basically exactly the same as FLRW, but just reflects the different choice of coordinates."}
{"id":"69997","title":"Why don't planets have Circular orbits?","text":"This might be a completely wrong question, but this is bothering me since many days ago. Given the mass (Sun) curves the space around it, gravitation is the result of such curved space (Correct me if I am wrong, source: A Documentary film). Given any point on a circle with center same as center of the mass, curvature in the space should be equal (Intuition). Planets rotate around the Sun because of the curve in the space they should follow a circular path and the distance between planet and Sun should be at a distance. Given the fact that earth has a elliptical orbit around the sun, and the distance between Earth and Sun varies according to position of the earth. Why do we have a elliptical orbit not a circular orbit."}
{"id":"69990","title":"In spontaneous symmetry breaking, global symmetry broken by gauged subgroup?","text":"My question is simple. Given a group $G$ broken to a subgroup $H$, gauging a possibly different subgroup Hg breaks explicitly the global symmetry $G$, generating what is known as pseudo-Goldstone bosons. Why is this? The usual answer I get is that the gauging determines a specific direction in field space, but I really don't understand this statement, how having a subgroup $Hg$ gauged can break the global $G$ explicitly?. What I have in mind is that inside the gauge transformations $Hg$ there are included global ones as well, so this is what is confusing me."}
{"id":"69991","title":"What is the alternative of PMG (Permanent Magnet Generator) in a Horizontal Axis Wind Turbine?","text":"I need to make a report about a HAWT (Horizontal Axis Wind Turbine) and list the parts with their costs, but as a Permanent Magnet Generator is a costly thing so what can I use in alternative of this to make the turbine less expensive? P.S. Due to the short time, I didn't do any kind of research on this PMG part's alternative."}
{"id":"9506","title":"Tunneling Rate Constant","text":"I am trying to \"decode\"\/derive an expression for the macroscopic rate constant for the tunneling of protons through a potential energy barrier that I read in a journal article: $$ k_{\\rm tun}(T)=(2\\pi\\hbar)^{-1}\\int_0^{V_{\\rm max}} Q(V,T) P_{\\rm tun}(V)\\ dV. $$ So basically: the authors say work out the probability of tunneling at each point up the potential energy barrier (from 0 to $V_{\\rm max}$), multiply this by the Boltzmann factor ($Q(V,T)$), integrating over all energies (i.e. we are taking a Boltzmann average )and then convert from energy in J to rate by multiplying by $(2\\pi\\hbar)^{-1}$. The authors use $(2\\pi\\hbar)^{-1}$ to convert from an energy in J to a rate in per second. This implies that the relationship between energy and frequency being used is: $$ E=\\hbar\\omega $$ I thought this just applied to electromagnetic radiation and free particles not in a potential. Is it OK to use this relation for protons tunneling in a potential? Or should you use some info about the shape of the potential? Many thanks N26"}
{"id":"9505","title":"Have the correlation functions of the XY spin chain model been calculated using a functional partition function with source terms?","text":"Have the correlation functions of the XY spin chain model, \\begin{equation} H=-\\sum_l (J_x \\sigma_l^x \\sigma_{l+1}^x+J_x \\sigma_l^y \\sigma_{l+1}^y)-B\\sum_l \\sigma_l^z \\end{equation} been calculated using a functional partition function with source terms? By functional partition function I mean the partition function plus source terms (or generating functional) which allow correlation functions such as $\\langle \\sigma_l^x \\sigma_{l+1}^x \\rangle$ to be calculated analytically: \\begin{equation} Z[\\bar{\\omega},\\omega] = Tr \\left[Exp \\left[-\\int_0^\\beta d\\tau H(a^{\\dagger},a) - \\int_0^\\beta d\\tau \\sum_l (\\bar{\\omega} a + a^{\\dagger} \\omega) \\right] \\right] \\end{equation} where $H(a^{\\dagger},a)$ is the XY Hamiltonian expressed in terms of spinless fermions (i.e. after a Jordan-Wigner transformation). **Edit:** Anyone? An Ising chain would be fine too!"}
{"id":"132952","title":"Is Hodge star operation can be understood as contraction after tensor product of a $p$-form with the volume element?","text":"By defintion, the Hodge star of a $p$-form $\\omega_{a_1\\cdots a_p}$ on a $n$-dimensional manifold is given by $*\\omega_{b_1\\cdots b_{n-p}}=\\frac{1}{p!}\\omega^{a_1\\cdots a_p}\\epsilon_{a_1\\cdots a_pb_1\\cdots b_{n-p}}$ Here, I assume the manifold is equipped with a metric $g_{ab}$ and the volume element is compatible with the metric, ie., $\\epsilon_{a_1\\cdots a_n}=\\sqrt{|g|}\\tilde\\epsilon_{a_1\\cdots a_n}$ with $\\tilde\\epsilon_{a_1\\cdots a_n}=+1$ when $a_1,\\cdots,a_n$ is an even permutation of $1,2,\\cdots,n$; $=-1$ when old permutation; $=0$ when there are repeated indices. From this formula, I guess we can calculate the Hodge dual by first take the tensor product of $\\omega_{a_1\\cdots a_p}$ with $\\epsilon_{a_1\\cdots a_n}$ and then contract the first $p$ indices. Am I correct?"}
{"id":"43148","title":"Zero Point Fluctuations","text":"The total energy of a mode in a quantum mechanical resonator is given by $E_n ~=~ (n+ 1\/2)hf$ where $n$ is the number of modes. So when there are no modes or vibrations, i.e. $n=0$, the energy is called the zero-point energy. What I don't understand is, if there are no modes, then what is this energy associated with? So what exactly is a mode? Also, is there a way of measuring this zero-point fluctuations? Why is the 1\/2 photon introduced?"}
{"id":"46517","title":"Problem evaluating moles in a an isochor transformation","text":"I have a problem with an isochor transformation. Me and my group of study made an experiment that want to check Gay-Lussac’s law. We registered the equilibrium states and fitted the $P = nRT \/ V$, in the $P(T)$ graph. The moles found are $n_f=(5.63 \\times 10^{-3} \\pm 1.24 \\times 10^{-4})$ mol. The problem is that, initial moles are $n_i=(3.19 \\times 10^{-3} \\pm 6.97 \\times 10^{-5})$ mol. How can the initial moles be smaller? _Details of the experiment_ : We started stated with a certain number of moles of air inside a metal cylinder closed with a rubber top and connected, through it, to a temperature and pressure sensor. The starting gas conditions are $V, P_0, T_0$ assuming the volume to be constant (the container is, in a good approximation, hermetic). We heated an amount of water in a calorimeter with a resistance and a temperature $T' \\gg T_0$, then we put the cylinder in contact with the hot water plotting the trend of $\\Delta P-T$ on a graph. From a certain point the trend gets linear, and with a fit we calculated the slope $m = nR\/V$ and from the latter we calculated the number of moles $n$ assuming the air as an ideal gas because of the higher temperature. We checked the consistency between this value and the one calculated from the initial conditions $n' = P_0V\/RT_0$ and surprisingly this value is smaller than the one calculated from the fit. Now what is really happening? the difference $n'-n < 0$ suggests that the number of moles in the cylinder is growing? Or maybe it might be that one of our assumptions is incorrect."}
{"id":"98842","title":"A dielectric sphere in an initially uniform electric field and representation theory of SO(3)","text":"I learned recently that the highest order spherical harmonic required to represent the spatial distribution of decay products of a particle can be used to determine its spin, by using arguments involving the representation theory of SO(3) \/ addition of angular momentum. I seek a similar argument for the well-studied problem of a dielectric sphere in an otherwise uniform electric field. This is, for example, solved in Griffiths' Electrodynamics, section 4.7 on the Third Edition (I'm using a Low Price Edition from India, where it is on Page 205). He concludes at the end of that problem that \"the field inside is (surprisingly) uniform\". I have a feeling that this result is less surprising when one applies ideas from SO(3) representation theory, but I'm not sure how to formulate the argument precisely. Here is my loose train of thought -- the field is a vector field with a specific direction, so it has one unit of angular momentum (or is given by an $l = 1$ spherical harmonic). Therefore, the induced surface charge on the dielectric sphere must be given by $P_1(\\cos \\theta) = \\cos(\\theta)$. The field inside is a vector field originating from these charges, so it must again be uniform. I'm also trying to use this to determine the highest order (i.e. highest $l$) component of $Y_{lm}$ in the total electric field after accounting for the polarization of the sphere."}
{"id":"60467","title":"Why are *high redshift* measurements of supernovae required to measure dark energy?","text":"Why are _high redshift_ measurements of supernovae required... in order to measure the equation of state parameter of dark energy? The luminosity distance can be written as \\begin{equation} d_{L}(z) = \\frac{c}{H_{0}}\\left(z-\\frac{1}{2}(1+q_{0})z^{2}\\right), \\end{equation} where \\begin{equation} q_{0} = \\frac{1}{2}\\left(1+3w_{\\small{DE}}\\Omega_{\\small{DE}}\\right). \\end{equation} Here, $c$ is the speed of light, $H_{0}$ is the Hubble constant, $z$ is the redshift of the galaxy hosting the observed supernova, $w_{\\small{DE}}$ is the equation of state parameter of dark energy, and $\\Omega_{\\small{DE}}$ is its density parameter. I was taught that because $q_{0}$ doesn't enter until quadratic order, we need to measure _high_ -$z$ supernovae. However, there are a lot of practical issues in measuring _high_ -$z$ supernovae. Now consider this. We can also write \\begin{equation} d_{L}(z) = c(1+z)\\int_{0}^{z}\\frac{dz^{\\prime}}{H_{0}\\left[ \\Omega_{\\small{M}}(1+z')^{3}+\\Omega_{\\small{DE}}(1+z')^{3(1+w_{DE})}\\right]}. \\end{equation} Here, I cannot see any reason to prefer high-$z$ supernova of low-$z$ supernovae. It would seem, then, that we can avoid the practical difficulties by simply using a different formula. Where am I going wrong?"}
{"id":"94170","title":"Is the concept of a field necessary to electrodynamics?","text":"I've read (in Griffith's text) that it is \"possible, though cumbersome\" to dispense with the field concept in electrodynamics entirely and instead use an action-at-a-distance theory. What exactly is meant here? Since fields physically exist, why is this permissible? Edit: In light of the answers below, I'm curious about the apparent contradiction here - if fields physically exist, then how can one dispense with the field concept?"}
{"id":"94172","title":"What force creates ions out of neutral atoms?","text":"Consider the reaction between Na and Cl to form NaCl. Na loses an electron and \"gives\" it to Cl because this makes both atoms more stable. But what forces \"pulls\" the electron from Na to Cl? Both atoms are electrically neutral at the beginning, so the electron that is going to be transferred isn't acted on by electrostatic forces. Since its not electrostatic forces that pull the electron from Na to Cl (since both atoms are neutral) what force pulls the electron from Na to Cl to make both atoms more stable?"}
{"id":"90668","title":"How to smooth the spectrum of a light source?","text":"Could somebody please tell me if there's a reasonably cheap substance or device can I use to smooth the spectrum of a light source? For example, if the spectrum has spikes as in the blue graph below, is it possible to smear in terms of wavelength (not spatially, as in ordinary diffusers) so that it resembles the red graph more? ![smooth_spectrum.png](http:\/\/i.stack.imgur.com\/LFaeM.png) The actual shape of the smoothing (convolution kernel) is not really important, but it is preferable that most of the energy is transmitted. I've searched for \"spectrum flattening\" filters, but they instead attenuate certain wavelengths rather than spread them out. I suppose it's possible to rapidly move the light source forward and back so that the Doppler effect would cause a similar shift, but the speeds required would be tremendous!"}
{"id":"102554","title":"Tensor decomposition under $\\mathrm{SU(3)}$","text":"In Georgi's book (page 143), he calculates the tensor components of $3\\otimes 8$ under the $\\mathrm{SU(3)}$ explicitly using tensor components. Namely; $u^{i}$ (a $3$) times $v^{j}_k$ (an $8$, meaning that it is also traceless, $v^j_j =0$), equal $$ \\dfrac{1}{2}\\left(u^{i}v^{j}_{k}+u^{j}v^{i}_{k} - \\dfrac{1}{4}\\delta^{i}_{k}u^{l}v^{j}_{l} - \\dfrac{1}{4}\\delta^{j}_{k}u^{l}v^{i}_{l} \\right) + \\dfrac{1}{4}\\epsilon^{ijl}\\left(\\epsilon_{lmn}u^m v^{n}_{k} + \\epsilon_{kmn}u^m v^{n}_{l}\\right) +\\dfrac{1}{8}\\left(3\\delta^{i}_{k}u^l v^{j}_{l} - \\delta^{j}_{k}u^{l}v^{i}_{l} \\right). $$ I understand that initially (the first term in the above equation) he symmetrizes with respect to the two upper indices while at the same time he subtracts the traces. In the second term above, I can see that he lowers two indices with the invariant $\\epsilon_{lmn}$ but I don't understand why he then symmetrizes! ($\\epsilon_{lmn}u^m v^{n}_k \\propto A_{lk}$ and he symmetrizes $A_{lk}$). While at the last term he adds the traces he subtraced with a spurious 3 and signs I don't understand how he got. Can someone elaborate on how exactly one can generally decompose tensor products into irreducible parts? _Is there a connection with the Young Tableaux? With the Young tableaux one can easily conclude that $3\\otimes 8 = 15 \\oplus \\bar{6} \\oplus 3$, **however how does one translate the Young tableaux into tensor components to get the above expression?**_ Thanks in advance."}
{"id":"102557","title":"How to prove the relations between the variables in bispherical coordinates?","text":"When I was reading papers about the forces between two charged spherical conductors in a uniform electric field, I was confused with bispherical coordinates the author used to describe the spherical conductors. The schematic diagram is as follows: ![enter image description here](http:\/\/i.stack.imgur.com\/DW2JI.png) where $β= \\ln(\\text{PL}_1 \/ \\text{PL}_2) , \\text{a = OL}_1 = \\text{OL}_2$; Here is a link about the bispherical coordinates. I want to know why all the points on the sphere (for example sphere 2) satisfy $β=β2$. What's more, the relations between $\\text{xyz}$ and $\\text{aβα}$ are not entirely clear to me.Thank you!"}
{"id":"102556","title":"A paradox to Lenz's law","text":"I have read that in simple words, Lenz's law states that: > _The direction of current induced in a conductor is in such a fashion, that > it opposes its cause._ This validates law of conservation of mass-energy. I arranged the following thought experiment: Let there be a pendulum with its bob being a small bar magnet. The pendulum is oscillating in a direction parallel to the horizontal axis of the bar magnet on which the North and South poles lie. Also, the pendulum is in complete vacuum. (But gravity is there to make the pendulum oscillate.) At one of the extreme positions of the pendulum, we keep a solenoid, ends of which are connected to a load resistance. As the North pole of the bar magnet approaches the solenoid, current is induced in the solenoid in such a fashion that a North pole is formed at the end near to the bar magnet's North pole, and the bar magnet gets repelled towards the other side. The bar magnet then goes to the other end and then comes back (as a pendulum does) and again the same process is repeated. This _should_ go on forever, and current _should_ keep appearing across the load resistance. How does the law of conservation of energy hold here?"}
{"id":"54992","title":"Loop integral using Feynman's trick","text":"I am trying to show for the one-loop integral with three propagators with different internal masses $m_1$, $m_2$, $m_3$, and all off-shell external momenta $p_1$, $p_2$, $p_3$ the following formula appearing in 't Hooft(1979), Bardin (1999), Denner (2007): (unfortunate metric $-,+,+,+$) $$\\int d^d q\\frac{1}{(q^2+m_1^2)((q+p1)^2+m_2^2)((q+p_1+p_2)^2+m_3^2)} $$ $$=i\\pi^2\\int_0^1dx\\int_0^xdy\\frac{1}{ax^2+by^2+cxy+dx+ey+f}$$ where $a$, $b$, $c$, ... are coefficients depending on the momenta in the following way: $a=-p_2^2$, $b=-p_1^2$, $c=-2p_1.p_2$, $d=m_2^2-m_3^2+p_2^2$, $e=m_1^2-m_2^2+p_1^2+2(p_1.p_2)$, $f=m_3^2-i\\epsilon$. I don't really care about factors in fromt like $i\\pi^2$. My **simple problem** is: I am totally unable to reproduce coefficients $d$, $e$ and $f$. The problem is, when I integrate over the third Feynman parameter, $m_3$ appears in all three coefficients $d$, $e$ and $f$. How do I squeeze the denominators to reproduce this formula?"}
{"id":"112681","title":"Simple harmonic motion","text":"A uniform straight rod of length $L$ is hinged at one end. It is free to oscillate in vertical plane. Time period of oscillation with small angular amplitude when a point mass of mass equal to that of rod is fixed with lower end is : $1$. ![enter image description here](http:\/\/i.stack.imgur.com\/eqjPb.jpg) $2$. ![enter image description here](http:\/\/i.stack.imgur.com\/Qjk1K.jpg) $3$. ![enter image description here](http:\/\/i.stack.imgur.com\/w2WSl.jpg) $4$. ![enter image description here](http:\/\/i.stack.imgur.com\/Evpiy.jpg) Hey,please could you help me solve this question. Actually,this seems to be a torsional pendulum and the time period in such a case is $2\\pi\\sqrt{I\\over C}$ where C is the restoring torque.Now,my doubt is what should the restoring torque be?Should it be 2mg(mass of the rod as well as the one suspended)times the length..or something else because that is not giving me my answer. High school student,so please keep it simple."}
{"id":"104860","title":"metastable energy levels","text":"can someone give me a good definition of \"metastable energy levels\" in a helium plasma (glow discharge). Am I right when I say : \"by adding some neon in my helium plasma, I will obtain a molecular plasma of NeHe\" ? I guess the metastable energy levels are the reason of forming NeHe, but I don't know why, can someone explain to me ? If both my guesses are wrong, please give me some explanation, if you can. thx"}
{"id":"108731","title":"Center of mass of a cylinder from which liquid is flowing out","text":"A cylindrical container is of mass M, radius R and height H. The cylinder is filled with a liquid of density D. The base of the cylinder has an outlet tap of radius r through which the liquid can flow out of the cylinder. Initially the center of mass of the cylinder and the liquid in the cylinder system is at a height of H\/2 along the axis of the cylinder. At time t = 0, the outlet tap at the base of the cylinder is opened and the liquid starts flowing out. We want to study the locus of the center of mass of the system. As liquid flows out, the center of mass will start coming down till it reaches a minimum point ans then start rising up again until it returns back to the height of H\/2 along the axis when the cylinder is completely empty. **Assumptions** :   1. The upper surface of the cylinder is closed.   2. The experiment is done at an altitude A above sea level. **Questions** :   1. Where will the center of mass of the system be at time t = T?   2. Find T at which the center of mass of the system will be lowest.   3. How much liquid will be present in the cylinder when the center of mass of the system is lowest?"}
{"id":"108737","title":"Resonance and a tuning fork","text":"I carried out this experiment in class: I struck a tuning fork with a hammer. The sound lasted for some time. However, when I connected the tuning fork onto a wooden sounding box, the sound lasted for a few seconds (much shorter than the previous case) But why is that? I know the reason is because of resonance but can someone please explain the reasoning behind these observations with detail."}
{"id":"102082","title":"Does limit $\\hbar \\rightarrow 0$ in Quantum Mechanics mean anything?","text":"Assuming that I learn Quantum Mechanics first, and then I approach Classical Mechanics as a special case of Quantum Mechanics, I will definitely find the relationship between Quantum Mechanics and Classical Mechanics very confusing. I don't know how to make sense of what happens when $\\hbar \\rightarrow 0$. For one, you can't recover classical mechanics from quantum theory by setting $\\hbar \\rightarrow 0$. However, it is possible to recover classical mechanics from Schrodinger equations. So, does limit $\\hbar \\rightarrow 0$ in Quantum Mechanics mean _anything_? How should we interpret it? Or does the above contradiction reveal yet another flaw in the fundamentals of Quantum Mechanics?"}
{"id":"102089","title":"Refraction problem - limiting radius of observation","text":"I solved this problem but I don't know if the solution is correct. Please, correct me if I'm wrong. There are two mediums, 1 and 2, with absolute refraction indexes $n_1$ < $n_2$, and an observer in medium 1 with height H (measured from surface to his eyes) . What is the limiting radius R that the observer can look at the surface and see an object located in medium 2? (as a function of $n_1$, $n_2$ and H) Let $\\theta_c$ be the critical angle of refraction. Then light that comes from medium 2 leaves the surface only for incident angles $\\theta_2 < \\theta_c$. Call $\\theta_2 = \\theta_c - \\epsilon$, with $\\epsilon > 0$. The refrated angle will be, by Snell's Law: $$\\theta_1 = \\sin^{-1}(\\frac{n_2}{n_1}\\sin(\\theta_2))$$ Such that: $$R = \\frac{H}{\\tan(\\frac{\\pi}{2} - \\theta_1)} = H\\tan{\\theta_1} = H\\tan{(\\sin^{-1}(\\frac{n_2}{n_1}\\sin(\\theta_2)))}$$ But in the limit $\\epsilon \\rightarrow 0$: $$\\theta_2 \\rightarrow \\theta_c = \\sin^{-1}(\\frac{n_1}{n_2}\\sin(\\frac{\\pi}{2})) \\Rightarrow \\theta_2 \\rightarrow \\sin^{-1}(\\frac{n_1}{n_2})$$ And therefore: $$R \\rightarrow H\\tan(\\sin^{-1}(1)) = H\\tan(\\frac{\\pi}{2}) \\Rightarrow R \\rightarrow \\infty$$ But this conclusion seems absurd to me since you can only see the mirror of an lake at long distances."}
{"id":"89679","title":"Capacitance of an arbitrary electrode configuration","text":"I have to find the capacitance of a circuit element in three dimensions. I have the potential and charge in this cross sectional slice calculated using numerical methods. Does this value of charge have any useful physical meaning in 3D so that I can calculate the capacitance of a length of electrode? I will ignore edge effects in the third dimension for ease of calculation."}
{"id":"112645","title":"Which clock is the fastest inside an accelerating body?","text":"The picture shows an accelerating spaceship with two clocks inside it. It is so far away from all other bodys that gravity is of no importance. Will the bottommost clock be slower than the topmost one? Or will both clocks have the same speed? ![Spaceship with two clocks inside it.](http:\/\/i.stack.imgur.com\/yMaWm.png?editPost)"}
{"id":"27552","title":"Why aren't the spin-3\/2 fields in the (3\/2,0)+(0,3\/2) representation?","text":"Why is it that spin-$\\frac 32$ fields are usually described to be in the $(\\frac 12, \\frac 12)\\otimes[(\\frac 12,0)\\oplus(0,\\frac 12)]$ representation (Rarita-Schwinger) rather than the $(\\frac 32,0)\\oplus(0,\\frac 32)$ representation? Does the latter not describe a spin-$\\frac 32$ field? Why is the gravitino given by the Rarita-Schwinger-type representation rather than the $(\\frac 32,0)\\oplus(0,\\frac 32)$ representation? This is related to a recent question I asked on gauge invariance of the Rarita-Schwinger field. Thanks!"}
{"id":"27559","title":"Relevance of SIC-POVMs to quantum information","text":"What is the real relevance of SIC-POVMs (symmetric informationally complete POVMs) to concrete tasks in quantum information theory? A lot of work has been put into giving explicit constructions, and yet it seems that in many concrete applications that at first glance require SIC-POVM-like objects, one can do without them using e.g. suitably chosen set of random projectors. So, my question is - if the problem of constructing SIC-POVMs in arbitrary dimensions was solved one day, would that enable solving problems which can't be solved or approximated using different techniques now? SIC-POVMs seem to play a role in some approaches to quantum foundations (see e.g. Chris Fuchs, Bayesian approaches to QM that rely on the existence of SIC- POVMs), but it's unclear to me how \"mainstream\" such applications are."}
{"id":"67161","title":"Replacing an operator with its expectation value","text":"While dealing with a _circling particle in an spherical symetric potential_ our professor said that we can replace an operator of $z$ component of angular momentum $\\hat{L}_z$ with the expectation value - he denoted it just $L_z$ - of the angular momentum if $L_z$ is constant. Why is that? So we first had this equation: \\begin{align} \\underbrace{\\psi (r,\\varphi,\\vartheta)}_{\\llap{ \\text{wave function in spherical coordinates}}} &= \\exp\\left[\\hat{L}_z \\frac{i}{\\hbar}\\, \\varphi\\right] \\underbrace{\\psi (r,0,\\vartheta)}_{\\rlap{\\text{wave function in spherical coordinates at $\\varphi=0$}}} \\end{align} and we got this one (notice that there is no operator over an $L_z$): \\begin{align} \\psi (r,\\varphi,\\vartheta) &= \\exp\\left[L_z \\frac{i}{\\hbar}\\, \\varphi\\right] \\psi (r,0,\\vartheta) \\end{align} Anyway here is the spherical coordinate system we ve been using all the time (the blue spherical aure is supposed to be a spherical potential...): ![enter image description here](http:\/\/i.stack.imgur.com\/G8aUE.png)"}
{"id":"29293","title":"Does a tidally locked planet have seasons?","text":"Could a tidally locked planet have seasons? According to my understanding, a tidally locked planet rotates around itself exactly once per rotation around its sun. However, if the axis of rotation of the planet is significantly off from the axis of rotation around the sun, wouldn't it still have seasons? Or does tidal locking require that the axis of rotation of the planet about itself be perfectly orthogonal to the axis of rotation of the planet about its sun? If not, what would happen to a \"tidally locked\" planet with an axis of rotation \"pointing towards\" its sun (sort of like Uranus)?"}
{"id":"29297","title":"weird terms in W boson self energy","text":"I calculated the correction to the self energy of the W boson due to a fermion doublet (below I have n(e) for neutrino (electron), but it could be up and down quarks or just any 4th generation contribution). My result checks with the results from section 21.3 in Peskin, so I am fairly confident in the form below. I think of this as a fairly straight-forward calculation, but I am getting inconsistent results. The interaction comes from the term $$\\mathcal{L}= \\frac{g}{\\sqrt{2}}W^+_\\mu \\bar{n}P_R \\gamma^\\mu P_L e + \\frac{g}{\\sqrt{2}}W^-_\\mu \\bar{e}P_R \\gamma^\\mu P_L n $$ so we get a single diagram with a electron (neutrino) running in the top(bottom) leg of the loop. For the self energy correction I get (with $\\bar{MS}$ scheme): $$\\Pi^{\\mu \\nu}(q^2) \\sim \\eta^{\\mu \\nu} \\int^1_0 dx \\left( x(1-x)q^2 - \\frac{1}{2} ( x m_e^2 +(1-x)m_n^2)\\right) \\log \\left[ \\frac{(1-x)m_n^2 + x m_e^2 +(x-1)x q^2}{\\mu^2} \\right] $$ where I have dropped the $q^\\mu q^\\nu $ term.This seems innocuous enough however if I perform the x integration and then expand in the external momentum $q^2$ I get weird terms like $$\\Pi(q^2) \\approx(m_n^3-m_e^2)^3 \\log \\frac{m_e}{m_n}\\frac{1}{q^2}$$ which seems incorrect since this should correspond to a term in the effective action, but there shouldn't be $\\frac{1}{q^2}$ terms in the action. Furthermore the original integrand is not singular in $q^2$ so I could have just as well expanded in $q^2$ first and then performed the Feynman parameter integral and no such $\\frac{1}{q^2}$ terms would be present. Any idea why the Feynman parameter integral and the momentum expansion don't seem to commute, or why I get these weird terms if I integrate first? Thanks a lot."}
{"id":"68806","title":"Are multipole fields, multipole expansion, and multipole radiation the same thing?","text":"Interaction between electromagnetic radiation and nuclei can be written in terms of multipole radiation. Are multipole fields, multipole expansion and multipole radiation the same thing? I have found this topics in Nuclear physics book written by R.R. Roy and B.P. Nigam."}
{"id":"109469","title":"Understanding Poynting's Theorem intuitively","text":"I understand that, broadly speaking, Poynting's theorem is a statement of conservation of energy. Energy density of a volume of current and charge decreases proportionally to work done on charges within the volume and by energy flux through the surface bounding the volume. The energy flux term is the area integral of the Poynting vector. My question is this: is this energy flux that is described by the Poynting vector a form of electromagnetic radiation? If not, how is energy radiating out of the charge and current distribution if not by work done on charges?"}
{"id":"66746","title":"Origin of motion and relative speed of bodies in the universe","text":"Charged particles can hit the earth at relativistic speeds. But it seems that all large bodies have fairly low relative speed. Of course, speed can increase considerably when a body orbits close to a massive object, but then it will not travel very far, and it can be discounted by averaging the speed over a long enough time (maybe a year). The Earth cruises around the Sun at 30 km\/s, the sun cruises at 200 km\/s and the Milky Way at 600 km\/s. Not that much. I have two somewhat opposite questions:   * What do we know of the relative speeds of massive bodies in the universe, small or large (not particles)? I am not sure whether it was the proper way to state the question. Also, do relative speeds get very high if we correct out the part due to universe expansion? Are there braking phenomena?   * On the other hand, what initiated that motion? If the initial soup had been homogenous, the coalescence of randomly moving particles should have produced structures essentially at rest (ahem: where is the energy going?) with respect to each other. There were small variations in temperature, but how should that create speed differences? Or did it cause large scale streams that coalesced into moving bodies? Even if some speed is due to contraction of large scale rotating structures, it does require initial momentum to exist. To put it together, do we have measured speed statistics in conformance with universe evolution models? What does it say about speed? Sorry if the questions are not well stated; that is my best. A reference to a paper for non-specialists would do too. * * * **Added after 3 weeks, considering the answer by @BenCrowell and the comment of @ChrisWhite** (thank you both). Please, forget the part about cosmological speed comparisons. **This is my own answer** to my question; I leave it as the question, since it raises other issues. The first point is: does my answer make sense? It is only guesswork on my part. I wrote this answer because, though really helpful, Ben Crowell's answer did not really answer the heart of my initial question. Though I suggested it in my initial question, I realize even more now that the problem is only the origin of momentum, probably only angular momentum. This was confirmed by the answer of Ben Crowell regarding the fact that structures were essentially at rest to start with. Probably my main error was to think that there might be another \"source of speed and momentum\", and my awkward attempt to discount observed very high speeds as going nowhere because in tight orbit. I do not see how large scale momentum could come from some form of accretion of spontaneously emerging angular momentum quanta. I would think they would balance on average without any visible macroscopic effect, outside very special and anisotropic places like a black hole horizon (and even then). I feel the same regarding linear momentum quanta (if such a thing exists). My guess is that high scale momentum arises from momentum exchange between large collapsing structures. It is well known that celestial structure, such as planets and satellites, can exchange angular momentum. Though I no longer try to follow the mathematical analysis, I also read that much of the exchange can be mediated by tidal effects. But tidal effect should be even stronger when it is between structures that are not yet collapsed and are hence very deformable. Angular Momentum exchange is not necessarily one structure slowing while another is speeding up. It is vectorial and may be two structures both speeding up in opposite directions, provided extra energy gets in from somewhere, such as potential energy from collapse. So it would be the case that, as they are collapsing, structures are being deformed under the gravitational influence of other structures, so that instead of converging to some global center of gravity, some subparts collapse separately around their own centers of gravity, and rotate around the main center of gravity. Is it actually in that way that the originally small variations of temperature (density?) created the various structures of the universe? Observed momentum arising from such momentum exchanges, there is no reason it should have the same orientation in all subparts of a larger structure, and this is indeed observed in the solar system. (see Can the axis of rotation of a celestial body point in any arbitrary direction?) What about other structures in the universe? The variations in speed (earth in solar system, sun in galaxy, etc.) are obviously just the consequence of variations in mass and size of the structures, very massive structure permitting higher speed as observed. The speed formula being $v = \\sqrt{\\frac{G\\cdot M} r}$, and the mass most likely increasing like the cube of the (initial) structure size. Hence the \"average speed\" in a structure should grow more or less like the initial linear size of the structure (assuming an almost uniform initial density). As reported in the initial question above, this is not quite what is observed. Where do I err? Could it be that very large structures collapse less than smaller ones because of the angular momentum of the subpart? This could reduce the influence of the total mass on individual subparts, and increase the orbital radius, thus reducing the speed. Is there any way to measure the degree of collapse as a function of initial size of a structure, and some notion of reference speed of substructures? Licence CC BY-SA 3.0 from the author."}
{"id":"25890","title":"Convert ECI coordinates to latitude\/longitude?","text":"I have been given output in (what I believe to be) ECI format (from OrbitTools):               TSINCE            X                Y                Z             0.00     4733.08693007    5370.64476064       0.09227655      360.00    -3928.94737512   -4784.93071195   -3602.13095174      720.00     2078.23905091    2928.47218732    6181.06427083     1080.00      343.60889986    -292.76288815   -7150.74620799     1440.00    -2682.96778230   -2415.73481378    6170.18900818      and need to convert the x,y pairs to latitude\/longitude (to plot on a map where the object is above). I can't reason through the conversion. Is there a document that a laymen could get through or formula that would take a given time\/point and yield a terrestrial latitude and longitude?"}
{"id":"67786","title":"Terms allowed in Lagrangian density","text":"I never fully understood why the Lagrangian density (let's say for a scalar field) was restricted to have only first order derivatives of the field in time and space in QFT. One reason I can think of is to ensure locality. But that doesn't restrict one from having a finite number of higher-order derivatives of the field, does it? Thanks!"}
{"id":"67780","title":"Why does a moving electric charge have nonuniform electric field lines?","text":"The electric field of a stationary electron is uniform, radially symmetric with straight field lines. However, the electric field of a moving electron still has radial and straight field lines, but they are no longer uniformly distributed, and the effect becomes more pronounced at higher speeds (that is, the field lines become more and more perpendicular to the direction of motion). I found this effect in this applet. Can someone physically explain to me the cause of this nonuniformity in the field lines? Thanks in advance."}
{"id":"34532","title":"Relation between Higgs boson and force of gravitation","text":"I don't know and I can't handle it. I am so much curious about the recent discoveries in science. European Space Agency and NASA are doing great work. Hats up to these institutions. My nights are sleepless just thinking about the consequences of Higg's Boson. I am so excited for the thing that all the science fictions turning into reality will be happening in my generation. I am able to see it. Currently I am 28. So I am very optimistic about it when I went through the NASA and ESA's vision and missions down the years till 2025. I would be very much happy and grateful to know if someone can explain me a little about the relation between Higgs Boson and Force of Gravitation(Gravitons if exists). By any means does Higgs Boson supports String Theory?"}
{"id":"131674","title":"Are voltages discrete when we zoom in enough?","text":"Voltages are often thought of as continuous physical quantities. I was wondering whether by zooming in a lot, they are discrete. I feel like the answer to the above question is yes as voltages in the real world are generated by actions of electrons. Can someone give me a more formal proof or a disproof? Whether voltages are discrete of continuous can have some impact the correctness of devices such as the analog to digital converter. For example, if voltages in the real world are continuous, then the Buridan principle[1] says that there cannot be a perfect analog to digital converter because such a device makes a discrete decision from continuous inputs. [1] : Lamport, L. (2012). Buridan’s Principle. Found Phys 42, 1056–1066. http:\/\/link.springer.com\/article\/10.1007\/s10701-012-9647-7 (It would be great if someone could also answer a related question http:\/\/electronics.stackexchange.com\/questions\/126091\/is-there-an-adc-with-a- finite-bound-on-measurement-time)"}
{"id":"26603","title":"How can you calculate the exact position of the Earth taking into account the gravitational attraction of Mars and the Sun?","text":"I think there must be a formula to calculate the exact position of the Earth taking into account the gravitational attraction of Mars and the Sun. Is it possible with vectors?"}
{"id":"110967","title":"Significant Figures (rules of addition)","text":"It is a simple one but different teachers answer this question differently. $$34+1.4+0.2$$ what would be its answer with due regards to significant figures?"}
{"id":"23381","title":"semiclassical exact expression (in one dimension only)","text":"let be $ N(x)= \\sum_{n} H(x-E_{n}) $ the eingenvalue 'staircase' function and let be a system so $ V(x)=V(-x)$ and $ V^{-1}(x)=\\sqrt \\pi \\frac{d^{1\/2}}{dx^{1\/2}} N(x) $ then would it be true that the two function $\\sum_{n}exp(-tE_{n})=Z(t)= \\int_{0}^{\\infty}dN(x)exp(-tx) $ and $ \\int_{0}^{\\infty}dx\\int_{0}^{\\infty}dpexp(-tp^{2}-tV(x))=U(t) $ would be equal ?? i have just compared the two results $\\int_{0}^{\\infty}dnN(x)exp(-tx)=\\int_{0}^{\\infty}dx\\int_{0}^{\\infty}dpexp(-tp^{2}-tV(x)) $ i have taken the Laplace transform inside and get the desired result assuming that the potential V(x) is EVEN"}
{"id":"23382","title":"How did the energy\/entropy\/volume\/pressure\/temperature relationship exist at the Big Bang and how did it evolve thereafter?","text":"According to the current Big Bang with inflation cosmological model? I was under the mistaken impression that there was very low volume, very high temperature\/pressure, very low entropy and the Big Bang (perhaps prompted by the high pressure) resulted in an expansion of volume where energy didn't change much so temperature went down and entropy went up. Now I've learned about inflation and how there is no conservation of energy at cosmological scales and that the universe gained an exponential amount of energy during the inflationary period. How did this happen with a _decrease_ in temperature? Inflation was a period of supercooled expansion, was it not? Why was positive pressure not enough to cause inflation such that we need a negative pressure in the form of the inflaton field (or later dark energy) to explain the expansion of the universe? Thanks for your time"}
{"id":"23386","title":"Conceptual quantum field theory","text":"Often papers and books give some bold(deep physical insight) statements in quantum field theory which are not backed by mathematics, and seldom by citing papers. Being a student I don't grasp the real meaning of those statements, making me think that I don't really understand QFT. I have access to all the standard books in QFT, but is there a book\/lecture notes that really aims to explain QFT in a way in which the ideas emerged, it's philosophy with a physical insight at a level of a graduate student ? I am mighty interested in doing theoretical high energy physics, and crave for a better understanding of QFT. Could somebody help me ...."}
{"id":"24767","title":"Proportion of energy compared to mass in universe","text":"In Wikipedia (http:\/\/en.wikipedia.org\/wiki\/Dark_matter), there is a picture that shows us the proportion of matter and energy in the current universe - CMB. It seems to suggest there is nothing that exists as energy form except dark energy - which we are yet to see - and every observable exists as matter forms. Is this correct interpretation? Of course, I know that matter's mass can be converted into energy by Einstein's basic and famous equation. If every observable exists as matter, is energy always converted to matter after being converted into energy from matter? Thanks."}
{"id":"22030","title":"Can someone explain the science behind MIT's 230% efficient LEDs?","text":"I was reading Gizmodo the other day and I didn't quite understand the Physics behind this. Could anybody shed some light on how this effect actually works?"}
{"id":"35666","title":"How is an ideal mirrored box of photons distinguishable from massive particles?","text":"Suppose you have an ideal mirrored box that contains enough photons as to have a relativistic mass equivalent to the [rest mass + kinetic energy] of an electron. In other words, the two systems have the same total energy. How would I distinguish these two systems? They have equivalent gravitational pulls and I believe they have equivalent inertia. Are charge and spin the only distinguishing factors (I'm assuming the \"box\" itself has no properties of interest)?"}
{"id":"35662","title":"why nontrivially space-like connected event horizons do not respect unitarity?","text":"I want to understand the assertion that the gluing between distant event horizons is forbidden by unitarity. What is exactly the argument that unitarity will necessarily forbid topological nontrivial spacetimes?"}
{"id":"35669","title":"representation of SU(2)","text":"The question is regarding SU(2) group and SU(2) algebra. The SU(2) group can be generated by exponentiating the generators of SU(2) algebra $X_a$ as $exp(i t_a X_a )$ with $t_a$ being three parameters. Genreally we use half the Pauli matrices as $X_a$ when discussing SU(2) algebra as well as the group. But if we use the spin-1 representation of $X_a$ (which can surely be done at the Lie Alebra level) can they be exponentiated to give another rep. of the SU(2) group (which would be 3 dimensional)?"}
{"id":"13577","title":"photons in expanding space: how is energy conserved?","text":"If a photon (wave package) redshifts (streches) traveling in our expanding universe, is it's energy reduced? If so, where does it go?"}
{"id":"118590","title":"With redshift, energy is lost. Where does it go?","text":"A photon emitted by a distant source billions of light years away arrives here with a lower frequency hence less energy than it started with. What happened to the energy?"}
{"id":"56663","title":"What are hidden variables exactly?","text":"What are hidden variables in quantum mechanics? I am aware there are many types but what exactly do they mean or even \"do\" exactly? Do they mean that the quantum indeterminacy becomes hidden but still exists in reality somewhere but is hidden? Or does it mean that it is a hidden mechanism which means the reality underneath really is single world determinism or there abouts? I have done some searching but it really is not clear to me."}
{"id":"23927","title":"What's the physical meaning of the Fourier transform of magnetic flux density?","text":"I have here below the distribution of the magnetic flux density $B$ across a 1 pole pitch in the airgap of a synchronous machine. The horizontal axis represents the distance along the arc length formed by a pole pitch across the airgap. ![plot of flux density distribution](http:\/\/i.stack.imgur.com\/0NEXq.jpg) If I take the Fourier Transform of this flux density,how should I interpret the results? I am used to taking Fourier Transform with respect to time and not with respect to spatial data that I have here. Thanks a lot... **Update** I think it makes sense instead to use the Fourier Transform on a complete cycle (i.e. not the half cycle shown up) to look at the harmonics that we have in the magnetic flux density. This is what I get: ![enter image description here](http:\/\/i.stack.imgur.com\/GxmWI.jpg) I guess from the Fourier Transform, this is the only information that is relevant."}
{"id":"23921","title":"Why would it be true that people with longer legs walk faster than ones with shorter legs?","text":"When a person walks, the only force acting on him is the force of friction between him and the ground (neglecting air resistance and all). The magnitude of acceleration due to this force is independent of the mass of the object (longer legs have more mass). Hence the person should move with with a velocity independent of the length of his legs. But I have heard (also observed) that people with longer legs walk faster than ones with shorter legs. If that is true, then why? One can argue that the torque about the pivot due to friction is more in case of longer legs, But then the torque due to gravity (when one raises his leg to move), which opposes the frictional torque, is also more for longer legs. And why would these torques make a difference anyway, as they have no effect on the acceleration of the center of mass?"}
{"id":"104032","title":"Is Space conserved","text":"Can space be created or destroyed? Is space conserved? I am **not asking for matter,energy and time**. Its just a question about **conservation of space** PS: I am asking for **what was there before big bang**. Surely Big bang would require space in the first place..."}
{"id":"128448","title":"Negative energy\/mass bounds on de-Sitter spacetime","text":"There exists a Positive Energy theorem for General Relativity in Anti-de Sitter and asymptotically flat spacetimes, but there is no equivalent theorem for de Sitter spacetimes > **Question:** Is there a lower bound theorem on negative mass-energy density > on de Sitter spacetimes? The intuition says that the absence of a positive energy theorem in dS has to do with the fact that for small enough positive energy densities, the cosmological expansion beats the gravitational attraction, which means that positive energy densities need to exceed a threshold in order to behave attractively from far away. Is this intuition correct?"}
{"id":"129446","title":"Cause and properties of northern lights","text":"If outside on a dark night in the middle to high latitudes, one might be able to see an aurora. It may be several hundred kilometers high and several thousand kilometers long, stretching around Earth in an arc. However, it is less than 1 km thick. What produces this huge display and what makes it so thin?"}
{"id":"25560","title":"Quality loss when adding more components to a telescope?","text":"I would think this would be inherently true, but I'm curious: (Assuming the highest quality _amateur_ instruments are used.) Is quality lost when, say, adding a T-mount adapter, diagonal or any other instrument between your eyes (or camera) and the primary focus? Maybe not discernible to our eyes, but could this be something to avoid when approaching astrophotography?"}
{"id":"74865","title":"How can sunlight be white , but pictures of the sun be orange?","text":"I was thinking about elementary physics today and realised: We learned and proved that sunlight is white. But i don't understand why when you look at a picture of a satellite orbiting the sun , it(the sun) seems yellow-orrange. How could this be? It is like there is a red light bulb in the room but what you see is green."}
{"id":"134429","title":"Topological insulator vs. topological superconductors in any dimension","text":"My question today is simple. What is the difference between a topological insulator and a topological superconductor? How that difference depends on the dimensionality of space(time)? What is the difference in terms of the so- called edge states? PS: I only know that a topological insulator is some material that is conductor on the edge but insulator in the bulk...Is that definition right in ANY dimension?What about topological superconductors?"}
{"id":"70585","title":"Why some particles interact with the Higgs field and others don't?","text":"Why some particles interact with the Higgs field and others don't? Higgs doesn't explain that much: why some particles have mass and others don't? is like why some particles interact with the Higgs field and others don't?"}
{"id":"87351","title":"Absorption of radiation due to temperature","text":"I was wondering if the temperature of an object affects the amount of radiation it absorbs. For example, if I have a box that is hotter, will it absorb more energy as compared to the same cooler box?"}
{"id":"105347","title":"Is it foolish to distinguish between covariant and contravariant vectors?","text":"A vector space is a set whose elements satisfy certain axioms. Now there are physical entities that satisfy these properties, which may not be arrows. A co-ordinate transformation is linear map from a vector to itself with a change of basis. Now the transformation is an abstract concept, it is just a mapping. To calculate it we need basis and matrices and how a transformation ends up looking depends only on the basis we choose, a transformation can look like a diagonal matrix if an eigenbasis is used and so on. It has nothing to do with the vectors it is mapping, only the dimension of the vector spaces is important. So it is foolish to distinguish vectors on the way how their components change under a co-ordinate transformation, since it depends on the basis you used. So there is actually no difference between a contravariant and covariant vector, there is a difference between a contravariant and covariant basis as is shown here http:\/\/arxiv.org\/abs\/1002.3217. An inner product is between elements of the same vector space and not between two vector spaces, it is not how it is defined. Is this approach correct? Along with this approach mentioned, we can view covectors as members of the dual space of the contra-vector space. What advantage does this approach over the former mentioned in my post? _Addendum:_ So now there are contra variant vectors and their duals called covariant vectors. But the duals are defined only once the contravectors are set up because they are the maps from the space of contra vectors to $R$ and thus, it won't make sense of to talk of covectors alone. Then what does it mean that the gradient is a covector ? Now saying because it transforms in a certain way makes no sense."}
{"id":"105349","title":"How exactly is the Poisson bracket of the modes of a classical string defined?","text":"In the theory of a classical bosonic string, we have expressions like: $$ \\\\{\\alpha^\\mu_m,\\alpha^\\nu_n \\\\} = - i m \\delta_{m,-n} \\eta^{\\mu \\nu} $$ were $\\alpha^\\mu_n$ are the Fourier modes of the string. How is this Poisson bracket defined? The definition from analytical mechanics involves partial derivatives with respect to the generalized coordinates and momenta. Since the modes $ \\alpha^\\mu_n $ are constants, their partial derivatives are zero, so there must be something I'm missing here."}
{"id":"89790","title":"Why is static friction called self adjusting force?","text":"Explain what is limiting force of static friction."}
{"id":"18958","title":"Confusion concerning the use of wavenumber in exercise about Fabry-Perot etalon","text":"In the exercise we are given that the spectrum of a light source consists of two spectral lines, which both have wavelengths around $500 \\text{ nm}$ and the separation between them - given in wavenumbers - is close to $0.1 \\text{ cm}^{-1}$. One part asks us to calculate the distance between the mirrors if we want the free spectral range of our Fabry-Perot etalon to satisfy $FSR(\\text{in } \\overline \\nu) = 2.5(\\overline \\nu_2 - \\overline \\nu_1)$. Now, I know that the free spectral range ($\\theta = 0$, $n_{air} \\approx 1$) is given by $$FSR = \\frac{c}{2d}$$ where $d$ is the distance between the mirrors and we have $$\\Delta\\overline \\nu = \\frac{c}{\\lambda_2} - \\frac{c}{\\lambda_1} = \\frac{c}{2\\pi}\\Delta k $$ So shouldn't then $$d = \\frac{c}{2 FSR} = \\frac{c}{2 \\cdot 2.5c\/(2\\pi) \\Delta k} = \\frac{\\pi}{2.5\\Delta k} \\approx 12.566 \\text{ cm}$$ According to the solution this is wrong: They get $FSR = 0.25 \\text{ cm}^{-1}$ and $d = \\frac{1}{2 \\cdot FSR} = 20 \\text{ mm}$ I seem to be missing something. If they calculate the free spectral range in terms of the wavenumber, shouldn't they convert it to frequency by $$FSR(k) \\to FSR(\\nu) = \\frac{c}{2\\pi} FSR(k)$$ instead of $$FSR(k) \\to FSR(\\nu) = c \\cdot FSR(k)$$ Does anybody see what's going on here (maybe the solution is simply wrong, but I'd like to know for sure)?"}
{"id":"18959","title":"What's the difference between arc length and signed distance?","text":"Suppose I give you a position function $x(t)$ where t is time and x is position. Then i ask you to find the total distance travelled from $t = 0$ to $t = t_f$. Which formula would give you that distance? $$\\int_{0}^{t_f} \\sqrt{1 + [x'(t)]^2} dt$$ $$\\int_{0}^{t_f} |x(t)| dt$$ I know the first one gives you the arc length of the path (which is the total length of the path) and the second one gives me the signed distance, but why are they fundamentally different again?"}
{"id":"86409","title":"Commercial large scale production of graphene","text":"I am a third year undergraduate Physics student, and for my solid state physics course I am asked to give a short (10 minute) qualitative presentation on the current standings of graphene production, about what can be done right now, what is expected to be doable in the near future, what the current problems are that have to be solved, etcetera. However, I am having quite some trouble in finding suitable sources. It is of course such a hot topic that new papers come out basically every day, and it is hard to find 'objective' sources that don't just talk about how a specific company has found the new, best way to do it. I suppose this is a bit of a non-traditional question to ask, but I was wondering if someone could help me out by pointing me towards some reliable, recent sources. A review article would be optimal, but of course as it is such a swiftly changing field this might be difficult. What I plan on doing at this point is structuring my presentation along these rough lines To start with, talk about how the 'first' graphene was made, using the scotch tape. Everyone I will talk to knows this, but it is still nice to see, especially with good pictures. I will then move on to what can already be done: several techniques that exist and that are being implemented as we speak, and what exactly they produce (multilayer graphene instead of monolayer, for example), and what the problems are that 'we' are currently facing in producing good graphene. Then I would like to talk about what is on the horizon, and on what timescales we expect what. If anyone has any comments on this (improvements or things I should definitely include) I would also be very happy to read them."}
{"id":"86402","title":"Is antumbra part of shadow darker than penumbra part of shadow?","text":"I'm exploring different types of shadows casted by objects.   I want to know if antumbra part of a shadow is darker than penumbra part. I've found two misleading pictures on Wikipedia:   First: ![Antumbra light](http:\/\/i.stack.imgur.com\/c5pTC.png) Second: ![Antumbra darker](http:\/\/i.stack.imgur.com\/ApN9C.jpg) Which one is correct? This example outside astronomy shows that antumbra is lighter than penumbra part of the shadow: ![Antumbra outside astronomy](http:\/\/i.stack.imgur.com\/wABFH.jpg) btw. Are there any reliable online references on that topic?"}
{"id":"86401","title":"Neglecting second order differentials","text":"I am currently doing some Lorentz invariance exercises considering infinitesimal Lorentz transformations, and have been told to neglect second order differentials. It's not the first time I have come across seeing something like this, and I wonder which is the justification behind this, I mean, is this a justifiable exact procedure or just an approximation?"}
{"id":"14181","title":"Do stationary states with higher energy necessarily have higher position-momentum uncertainty?","text":"For simple potentials like square wells and harmonic oscillators, one can explicitly calculate the product $\\Delta x \\Delta p$ for stationary states. When you do this, it turns out that higher energy levels have higher values of $\\Delta x \\Delta p$. Is this true for all time-independent potentials? Certainly, it is possible to find two states $\\mid \\Psi_1 \\rangle$ and $\\mid \\Psi_2 \\rangle$ with $\\langle \\Psi_1 \\mid H \\mid \\Psi_1 \\rangle > \\langle \\Psi_2 \\mid H \\mid \\Psi_2 \\rangle$ and also $\\Delta x_1 \\Delta p_1 < \\Delta x_2 \\Delta p_2$. For example, choose a quadratic potential, let $\\mid \\Psi_2 \\rangle$ be the first state and let $\\mid \\Psi_1 \\rangle$ be a Gaussian coherent state (thus with minimum uncertainty) and fairly high energy. So I'm asking here just about the stationary states. As Ron pointed out in the comments, this question is most interesting if we consider potentials with only a single local minimum, and increasing potential to the right of it and decreasing to the left."}
{"id":"15858","title":"Relation between unitarity and conservation of probability","text":"In a seminar, I heard that the unitary aspect of representations was important physically, because in quantum mechanics unitarity is closely tied to the conservation of probability. Could someone explain why this is so? After a bit of thought, one vague connection I can think of is that a matrix generated by a Hermitian matrix, $$U =e^{iH},$$ where $H$ is Hermitian is bound to be unitary . As Hermitian matrices correspond to observables, this must have some signficiance. Though, as I said, I still have the faintest idea."}
{"id":"18374","title":"What do \"local\" and \"global\" mean when talking about standard deviations in experimental particle physics?","text":"I found the following sentence: The excess is most compatible with a SM Higgs hypothesis in the vicinity of 124 GeV and below, but the statistical significance (2.6σ local and 1.9σ global after correcting for the LEE in the low mass region) is not large enough to say anything conclusive. I am familiar with sigma, but not familiar with local and global. What do they mean?"}
{"id":"18372","title":"Time passing near a black hole","text":"Given distance $d$ from a normal black hole with mass $m$, how much seconds will pass at that distance while in infinite distance from the black hole $t$ seconds has passed?"}
{"id":"116716","title":"What is the correct term to describe matter converting into energy?","text":"Matter and energy are related; one can convert into the other. What is it called when this happens? For example, solids melt\/liquefy into liquid, and liquid vaporizes into gas. Gas condenses into liquid, and liquid solidifies into solid. So matter [verb] into energy, and energy [verb] into matter. What are the correct verbs to use here? I am thinking that they are \"energize\" and \"materialize\", but I'm not sure, especially about \"energize\"."}
{"id":"13861","title":"Lev Landau's \"Theoretical Minimum\"","text":"The great russian physicist Lev Landau developed a famous entry exam to test his students. This \"Theoretical Minimum\" contained everything he considered elementary for a young theoretical physicist. Despite its name, it was notoriously hard and comprehensive, and in Landau's time, only 43 students passed it. I wonder if anyone can provide the list of topics, or even a copy of the exam? (I'm sure I'd have no chance to pass, but I'd like to see it out of a sense of sportmanship ;-). Also, I think it would make quite a good curriculum of theoretical physics (at least pre 1960).)"}
{"id":"13860","title":"What are the differences between the terms flammable, inflammable, and non-flammable?","text":"This question should perhaps go in Chemistry, to which I am committed, but since the Chemistry Stack Exchange forum is not yet in beta, I am wondering if you can help me from a physics perspective. I am wondering about the definitions of the terms _flammable_ , _inflammable_ , and _non-flammable_. Wikipedia says that _flammable_ and _inflammable_ are **synonyms** , meaning \"capable of burning\". In contrast, _non-flammable_ means \"not capable of burning\". To avoid confusion, it is recommended that use of the term _inflammable_ be avoided, since the prefix _in-_ here does not mean \"not\". This generally makes sense and sounds like good advice, but as I was reading a journal paper today, I came across a paragraph using these terms. I am wondering if you can help me by telling me your interpretation of the intent of this paragraph: > During the last decade room temperature ionic liquids (RTIL), being salts of > low temperature melting points, have been studied extensively. They may be > used as solvents for many processes and as electrolytes in electrochemical > devices. Batteries filled with such a type of electrolytes do not contain > any volatile components and therefore, they are not flammable. Room > temperature ionic liquids, being usually quaternary ammonium salts, are > characterized by negligible vapor pressure, which makes them inflammable. In > addition, they show a broad electrochemical stability window, generally >4 > V, which is necessary for the application in lithium-ion batteries with > high-energy cathodes. This is from page 602 in \"Ionic liquids as electrolytes for Li-ion batteries-- An overview of electrochemical studies\", Lewandowski, A.; Swidereska-Mocek, A. _Journal of Power Sources_ **2009** , _194_ , 601-609, which is available to subscribers here. The first part, **\"...do not contain any volatile components and therefore, they are not flammable\"** , seems to make sense; room temperature ionic liquids are promising in part because they do not ignite or burn easily, so they can potentially improve the safety of batteries. However, the next mention seems to say just the opposite: **\"...are characterized by negligible vapor pressure, which makes them inflammable\"**. So, my question is, does it seem to you that the term _inflammable_ has been used incorrectly in this paragraph? Does negligible vapor pressure imply that a substance is likely to burn readily or not burn readily? Thank you very much."}
{"id":"20822","title":"What is the physical meaning of a \"complete\" Hilbert space in QM?","text":"What does the word \"complete\" means from the physical point of view? I do not understand what it physically means to say that a Hilbert space is a complete vector space."}
{"id":"105209","title":"Completeness of the state space and Hilbert space","text":"I am wondering just why is it supposed that quantum states lie in a Hilbert space, which mathematically requires completeness? In other words, what does completeness (defined in terms of Cauchy sequences) imply physically?"}
{"id":"33505","title":"How can a strong water current be cold","text":"This is a layman question. If heat is the motion of atoms, how can a fast moving water current be cold?"}
{"id":"47006","title":"What is the meaning of the gradient of a current vs time graph?","text":"What is the meaning of the gradient of a current vs time graph? In other words, what does the unit $As^{-1}$ or $Cs^{-2}$ represent?"}
{"id":"15325","title":"What variables does the action $S$ depend on?","text":"Action is defined as, $$S ~=~ \\int L(q, q', t) dt,$$ but my question is what variables does $S$ depend on? Is $S = S(q, t)$ or $S = S(q, q', t)$ where $q' := \\frac{dq}{dt}$? In Wikipedia I've read that $S = S(q(t))$ and I think that suppose, $q$ and $t$ are considered as independent coordinates. Then $S$ should depend on $q'$ also because, for the typical lagrangian $$L ~=~ \\frac{q'^2}{2} - V(q).$$ I think we cannot find a function $Z(q, t)$ such that $$L = \\frac{dZ(q, t)}{dt}?$$"}
{"id":"15320","title":"Why do neutrino oscillations imply nonzero neutrino masses?","text":"Neutrinos can pass from one family to another (that is, change in flavor) in a process known as neutrino oscillation. The oscillation between the different families occurs randomly, and the likelihood of change seems to be higher in a material medium than in a vacuum. Why does the oscillation of neutrinos directly imply that they must have a nonzero mass, since the passage of one flavor to another can only occur in massive particles? And What are the cosmological implications of these oscillations?"}
{"id":"82407","title":"x-coordinate of the ball when it's velocity is perpendicular to projection velocity","text":"We have a ball projected at 20 m\/s at an angle $\\pi\/6$ from the x-axis. Now we have to calculate x-coordinate of the ball when it's velocity is perpendicular to projection velocity. I am thinking about how to solve this problem. Is it related to the equation of trajectory? What I have figured out so far, is that it will be making angle $\\pi\/3$ from x-axis at this instant how to get the x-coordinate?"}
{"id":"82405","title":"Does our existence cost us energy?","text":"Anything when it needs to inform its presense such as electromagnetic presense of charged particles and gravitational presense of particles due to their mass does so by sending information of its existencs propogated in space via means of electromagnetic waves or hypothesised gravitational waves. So as a part of telling others of their presense to others, do the particles constantly lose their enegy ? I understand that they could also gain energy in same way, but if a particle was completely isolated would it lose its existence and turn into pure energy over time ? I understand that the mass and energy would together remain constant, but my question is that whether something would happen to a particle or would it remain a particle ?"}
{"id":"39382","title":"Large gauge transformations","text":"I would like to understand what is the importance of large gauge transformations. I read that these gauge transformation cannot be deformed to the identity, but why should we care about that?"}
{"id":"39383","title":"Hawking radiation and reversibility","text":"It's often said that, as long as the information that fell into a black hole comes out _eventually_ in the Hawking radiation (by whatever means), pure states remain pure rather than evolving into mixed states, and \"the universe is safe for quantum mechanics.\" But that can't be the whole story! For quantum-mechanical reversibility doesn't merely say that information must _eventually_ be retrievable, after 1070 years or whatever; but also that, if U is an admissible transformation of a physical system, then U-1 is also an admissible transformation. So, just like it must be consistent with reversible microlaws for smoke and ash to spontaneously reassemble into a book, it must also be consistent for a black hole to spontaneously \"uncollapse\" into a star, or into whatever configuration of ordinary matter could have collapsed to form the black hole in the first place. And this \"white-hole uncollapse process\" must be possible in exactly the same amount of time as the black-hole collapse process, rather than an astronomically longer time (as with Hawking radiation). In both cases, the explanation for why we never see these processes must be thermodynamic -- i.e., sure they're allowed, but they involve such a crazy decrease in entropy that they're exponentially suppressed. I get that. But I'm still confused about something, and here's my best attempt to crystallize my confusion: > In order to explain how it could even be _possible_ for information to come > out of a black hole, physicists typically appeal to Hawking radiation, which > provides a mechanism based on more-or-less understood quantum field theory > in curved spacetime. (Granted, QFT also predicts that the radiation should > be thermal! But because of AdS\/CFT and so forth, today people seem pretty > confident that the information, after hanging out near the event horizon, > _is_ carried away by the Hawking radiation in some not-yet-understood way.) > However, suppose it's objected that a Hawking radiation process seems > nothing whatsoever like the time-reverse of an ordinary black-hole formation > process. Then the only response I know would be along the lines of, \"well, > do you believe that QM will survive unaltered in a future quantum theory of > gravity, or don't you? If you do, then consider the unitary U corresponding > to a black-hole formation process, and invert it to get U-1!\" My question > is: why couldn't people have made that same straightforward argument even > _before_ they knew anything about Hawking radiation? (Or _did_ they make > it?) More generally, even if Hawking radiation _does_ carry away the > infalling information, that still seems extremely far from implying full > quantum-mechanical reversibility. So, how much does the existence of Hawking > radiation really have to do with the case for the compatibility between > quantum mechanics and black holes?"}
{"id":"131884","title":"Using bra-ket notation?","text":"Am I using Bra-ket notation correctly? I want to define a superposition of the states, $|0\\rangle$ and $|1\\rangle$. Is it simply? $|0\\rangle + |1\\rangle$ I can't find any non-technical information on it. I'm just a hobbyist."}
{"id":"108821","title":"What happens in a circuit, when the wire and the battery are superconducting. And shorted","text":"When a wire with no resistance is connected to the terminals of an ideal battery, will a current exist in the circuit? If a capacitor is added to the circuit, will it be charged by the battery or will it remain uncharged? In all cases there is absolutely no resistance through the circuits."}
{"id":"100580","title":"Are there new states of matter at ultrahigh temperatures and densities?","text":"Under extreme energetic conditions, matter undergoes a series of transitions, and atoms break down into their smallest constituent parts. Those parts are elementary particles called quarks and leptons, which as far as we know cannot be subdivided into smaller parts.Is there exist new states of matter?"}
{"id":"39034","title":"Energy in electric field","text":"I'm having some trouble understanding a homework question and would appreciate some help. The question is as follows: > Jenny charges a capacitor with the help of a battery. She then removes the > battery and halves the distance between the two plates. How does the energy > conserved in the capacitor change? I solved the question like this: $E=\\frac{U}d$ and $U= \\frac{W}Q$ $\\implies W=EQd$ Hence, the energy doubles (as $E=\\frac{U\/d}2 \\implies 2E=\\frac{U}d$) where: $W$ = energy, $E$ = Electric field intensity, $Q$ = charge, $U$ = voltage, $d$ = distance This answer doesn't make any sense to me, considering the law of conservation of energy (So, Where did all that energy come from?) It also turns out that my answer is wrong, in fact the energy is halved. This also doesn't make any sense to me, what am I _missing_?"}
{"id":"16749","title":"How to set the electric potential on a certain boundary to a certain value experimentally?","text":"In electrostatic boundary value problems, usually the potential is set to some function or a constant on a certain boundary. It is never taught or told in textbooks how such assignment might be achieved experimentally. For example how to set experimentally the potential on the boundary of a ball of radius $R$ to some constant $V_0$, where $V_0$ could be $-100 V$ or $0 V$ or $5 V$ say?"}
{"id":"239","title":"What property of objects allow them to float?","text":"I used to think that the shape of an object determines its ability to float (boat-shaped objects are more likely to float, and spheres tend to sink). But my friend, who is fond of making me look stupid, took me to the local lake showed me a sphere that floated and a boat-shaped object made out of iron that sunk. Is it based on the mass? I'm not sure that is possible, because I've seen really heavy things (like airplane carriers) float, and really light things (like my friend's iron boat) sink. What property of certain objects allow them to float, if any?"}
{"id":"12058","title":"Photon statistics of an incandescent light source","text":"We usually calibrate the cameras on our microscopes by capturing 20 images of a blurry (not sharp) fluorescent particle. For each pixel in this stack of 20 images we calculate the intensity variance. By plotting the variance against the intensity we obtain the scale factor that allows us to convert the ADU (arbitrary data unit) into detected photoelectrons. However, this approach relies on a light source with Poisson statistics. This is applicable for fluorescence light. **But can we also use light of a halogen lamp instead of fluorescent light?** I heard that photons leaving a heated metal surface have a particular statistic. I think the argument was that they are coming out in bunches and it has something to do with the workfunction. It would be nice if someone could give a good reference on this. ![Calibration results of an EMCCD camera](http:\/\/i.stack.imgur.com\/9BX1D.jpg) The picture above shows the calibration curves for an EMCCD camera. The left graph corresponds to the readout mode you would use for bright images and has a readnoise of 15 electrons per pixel. The graph on the right shows the result for the low noise mode. In this mode the charge that has been accumulated under the pixels is transferred through an electron multiplication register. Impact ionization increases the number of electrons. The gain is usually up to 300x. The inset on the lower right is how a calibration image looks like. Note that this one is fake."}
{"id":"237","title":"Finding the volume of this irregular shape I have","text":"I have an approximately basketball-sized non-hollow piece of aluminum sitting in my house that is of irregular shape. I need to find the volume of it for a very legitimate yet irrelevant reason. What is the best way I can do this? In fact, what are **all** the ways I could feasibly do this without going to a lab? (I don't live near any labs)"}
{"id":"119827","title":"Charge inside a sphere","text":"Suppose I have a sphere of radius $r$ with all the charge residing on the surface, distributed uniformly i.e. charge density $\\sigma$ is constant. I want to find the electric field created by this sphere. All problems of this sort I have solved by enclosing the sphere in a larger imaginary sphere with radius $R$, and then using the assumption that the electric field lines are in a radial direction, in combination with the divergence theorem. If I use that method for this question, I find that the Electric field inside the sphere is zero. I can't understand this intuitively. I would have thought that if you are in the proximity of charge, there will be an electric field. Why would the electric field lines extend outwards radially from the sphere, and not inwards too?"}
{"id":"43787","title":"Generalized momentum conjugate and potential $U(q, \\dot q)$","text":"On Goldstein's \"Classical Mechanics\" (first ed.), I have read that if $q_j$ is a cyclic coordinate, its generalized momentum conjugate $p_j$ is costant. He obtained that starting from Lagrange's equation: $\\frac {d}{dt} \\frac {\\partial L}{\\partial \\dot q_j}- \\frac {\\partial L}{\\partial q_j}=0$ But this Lagrange's equation refers to a conservative system. What would happen if I considered a system in which the potential is $U(q, \\dot q)$?"}
{"id":"64630","title":"Derivation of the Poynting Vector without the Poynting Theorem","text":"My professor worked out the Poynting Vector without using the Poynting Theorem-I can't remember how. I can't seem to figure out where the cross product would come in."}
{"id":"63068","title":"Physical significance of effective wave function","text":"In Yanhua Shih's book on quantum optics, the coherence functions are expressed in terms of effective wave function. Here are the expressions for single photon wave packets. ![enter image description here](http:\/\/i.stack.imgur.com\/Cr13p.jpg) To derive the coherence functions we have used Heisenberg picture where the field operators, not the wave functions, are time dependent. We can also take a look at the effective wave function for thermal source. ![enter image description here](http:\/\/i.stack.imgur.com\/dSds4.jpg) So, it is confusing why we have a wave function with time as a parameter. What is the physical significance of effective wave function?"}
{"id":"21966","title":"How to find the value of the parameter a in this transfer function?","text":"> **Possible Duplicate:**   >  How to find the value of the parameter $a$ in this transfer function? I am given a transfer function of a second-order system as: $$G(s)=\\frac{a}{s^{2}+4s+a}$$ and I need to find the value of the parameter a that will make the damping coefficient $$\\zeta=.7$$ I am not sure how to do this but I might have found something that might have helped so I am going to take a stab at it. I found a transfer function in the book of a second order spring-mass-damper system with an external applied force in the book as: $$G(s)=\\frac{a}{m\\omega_{n}^{2}}(\\frac{\\omega_{n}^{2}}{s^{2}+2\\omega_{n}\\zeta+\\omega_{n}^{2}})$$ I was thinking that I could writeZx: $$2\\omega_{n}^{2}\\zeta=4$$ and $$\\omega_{n}^{2}=a$$ asd And then solve for a. Would this be possible?"}
{"id":"63063","title":"\"WLOG\" re Schwarzschild geodesics","text":"Why, when studying geodesics in the Schwarzschild metric, one can WLOG set $$\\theta=\\frac{\\pi}{2}$$ to be equatorial? I assume it is so because when digging around the internet, most references seem to consider this particular case... and some actually said \"wlog\". But why? I don't think the motion is necessarily confined to a plane? Correct me if I'm wrong, but isn't the Euler-Lagrange equations for the coordinate $\\theta$ $$\\ddot\\theta +\\frac{2\\dot r}{r}\\dot\\theta-\\dot\\phi\\sin\\theta\\cos\\theta=0?$$ I don't see why the motion can be \"wlog\" in $\\theta=\\frac{\\pi}{2}$."}
{"id":"90464","title":"What is the wave function outside the barrier region?","text":"I've been trying to learn how to apply WKB for several days now. I asked a similar question already about trying to find the wave function inside the barrier region. Now I would like to understand how to find the electron outside the barrier region. > _An electron, say, in the nuclear potential_ $$U(r)=\\begin{cases} & -U_{0} > \\;\\;\\;\\;\\;\\;\\text{ if } r < r_{0} \\\\\\ & k\/r \\;\\;\\;\\;\\;\\;\\;\\;\\text{ if } r > > r_{0} \\end{cases}$$ > > _What is the wave function outside the barrier?_ Here's my work so far: Outside the barrier region, $E>U(r)$ so $$\\psi(r)=\\frac{A}{\\sqrt{|p(r)|}}e^{\\frac{1}{\\hbar}\\int_{0}^{r}|p(r')|dr'}+\\frac{B}{\\sqrt{|p(r)|}}e^{-\\frac{1}{\\hbar}\\int_{0}^{r}|p(r')|dr'}$$ where notice that the exponents are real. My question is: are the bounds for the integral correct?"}
{"id":"38811","title":"Particles as a limit of classical field theory","text":"A common academic exercise has been to show that classical mechanics is a limit of quantum mechanics, usually by putting $\\hbar \\rightarrow 0$. Similarly is it possible to show that a limit to field theory will lead to a particle ?   May be by considering a initial field configuration such that the \"lumpiness\" of the particle is there, and then as it evolves it maintains that. Clearly such an initial field configuration will continue to evolve as it is in a free field theory as two different modes don't talk to each other, but what about an interacting model ? Are there any instances in the literature which attempts to establish this connection ?"}
{"id":"129459","title":"Why do some pressure cookers initially leak","text":"Why do some pressure cookers initially leak steam from the sides and after a while settle down and don't leak? Is it a positive feedback mechanism from the building steam which makes the sealing tighter and vice versa?"}
{"id":"38813","title":"Relationship between balloon size and pressure","text":"I see numerous websites talking about how a small balloon is at higher pressure than a big balloon. This is a fun counter-intuitive factoid... unless it isn't a fact at all. Young-Laplace gives $$T = \\Delta p \\frac R2 $$ Which means that for a given tension, increasing the radius does decrease the pressure. But why would the tension be constant? A latex balloon is like a 2D spring, right? So the tension in a inflated spherical balloon should be proportional to the surface area, correct? $$T \\propto R^2 $$ But this gives $$\\Delta p \\propto R$$ Which means the bigger the balloon, the higher the internal pressure. Which, again, would be the normal intuition - but it would burst the bubble of some simplified explanations. (har har)"}
{"id":"38819","title":"Calculating laser wavelength\/power to cause emission of light in a gas?","text":"I've become interested in volumetric displays. In particular I'm looking at a pair of lasers scanning a volume of gas such that the intersection of the lasers would supply enough energy to cause the gas to emit visible light. Since this is the start of my research I would like to investigate a number of gases in order to determine a number of gases which might be suitable for this application. I have, however, forgotten how to determine the amount of energy needed to cause an outer shell electron to emit a photon, and then trying to back-calculate the required wavelength(s) and power requirements of the laser emitters. I'm looking for some information on how to identify the required information for a given gas and how to calculate the energy required to cause visible light emission, and also information on how the wavelength of laser light impacts the ability of the gas to absorb the energy with the intended effect of emitting visible light. Links to introductory texts are always helpful, as are answers which help me get the terminology correct so that I can improve my searches for this information."}
{"id":"54204","title":"Quantum circuit, two control not gates","text":"Consider the quantum circuit in the picture below: ![Figure 1](http:\/\/i.stack.imgur.com\/s80aW.jpg) We have a Hadamard gate followed by a CNOT gate, this puts the 2nd & 3rd state in the bell state $\\beta_{00}=\\frac{1}{\\sqrt2}(\\mid 00\\rangle +\\mid 11\\rangle$ This occurs because the CNOT gate flips the target qubit iff the control qubit is $\\mid 1 \\rangle$. The two qubits are now entangled. When we proceed to the second CNOT gate we only care about what state the second wire is in? Would someone be able to help explain how we can compute how the top wire acts in this situation? Or if someone would be able to explain in general how we treat qubits entangled such as this if we want to say something about them separately? At the end of the circuit we have two detectors $x$ and $y$, these will collapse wave function, two two lower qubits will carry on unchanged in their bell state, how would we describe the middle wire being measured by $x$?"}
{"id":"48699","title":"Does quantum manybody theory with 2nd quantization completely equivalent to 1st quantization formulation?","text":"by equivalent I mean:   1. the Hilbert spaces are isomorphic;   2. the operators can therefore be mapped 1:1"}
{"id":"44756","title":"For a fixed power input, is it better to hit something with a light rod or a heavy rod?","text":"My friends and I got into an argument about whether it was more damaging to hit an object with a heavy rod or a light rod. The sides of the argument go something like: \"If you swing the light rod, you'll be able to swing it much faster and therefore do the most damage!\" \"Yes, but if it's too light there won't be enough mass behind it and it won't damage as much!\" I have been toying around with this question for a few months and I have yet to come up with a satisfactory answer. I'm setting up the problem like this: **Person A** is swinging a rigid rod of mass $m$ and fixed length $L$ at **Object B** located at a fixed distance $d$ away. Imagine **Person A** is swinging the rod much like a baseball player would swing to hit a ball. **Person A** has a fixed power $P$ to input to the rod over distance $d$ before making impact with **Object B**. Given input power $P$, rod length $L$, and swing distance $d$, what is the optimal mass $m$ of the rod to maximize the damage of **Object B**? For simplicity, I'm modeling the power application to be over a linear rather than angular distance, and I'm modeling **Object B** as a spring with spring constant $k$ attached to a fixed wall. Effects of gravity should be ignored. Also, for now ignore the constraints of human strength and assume we can swing anything as heavy as we like."}
{"id":"44753","title":"Do greenhouse gasses make the world habitable?","text":"An ageing gent says he does not accept this statement:               It's a fact that greenhouse gases, predominantly water vapour, keep the Earth      from being hostile and cold; that CO2 albeit in trace quantities is also a      greenhouse gas and therefore helps keep the planet warm...      I know no more about why he rejects this statement - maybe it is that water vapour alone would do the job. The gent in question has a first class degree and a doctorate in physics. Is there a debate within the physics community over this or am I misunderstanding?"}
{"id":"129143","title":"Why is Avogadro's law always true?","text":"Why is Avogadro's law always true? How and why do equal volumes of gases at equal pressure and temperature contain equal number of molecules? I know it is a fundamental principle in chemistry but I wonder how it works."}
{"id":"129142","title":"Increasing resistance, by heat?","text":"I was wondering the magnitude of effect to a conductor's resistance when heated by a few degrees. Can this be controlled way of changing conductor's resistance, or fixing in a system?"}
{"id":"40925","title":"Coverage of Quantum Electrodynamics (QED) in introductory Quantum Field Theory (QFT) books","text":"Which QFT books also cover QED? I am not very familiar with QED, so I am looking for QFT books which cover QED too (I know they cover Quantum Chromodynamics (QCD).)."}
{"id":"1135","title":"What is the difference between \"kinematics\" and \"dynamics\"?","text":"I have noticed that authors in the literature sometimes divide characteristics of some phenomenon into \"kinematics\" and \"dynamics\". I first encountered this in Jackson's E&M book, where, in section 7.3 of the third edition, he writes, on the reflection and refraction of waves at a plane interface: >   1. Kinematic properties: (a) Angle of reflection equals angle of incidence > (b) Snell's law >   2. Dynamic properties (a) Intensities of reflected and refracted radiation > (b) Phase changes and polarization > But this is by no means the only example. A quick Google search reveals \"dynamic and kinematic viscosity,\" \"kinematic and dynamic performance,\" \"fully dynamic and kinematic voronoi diagrams,\" \"kinematic and reduced-dynamic precise orbit determination,\" and many other occurrences of this distinction. **What is the real distinction between _kinematics_ and _dynamics_?**"}
{"id":"12876","title":"Heat & thermodynamics question based on heat loss","text":"![enter image description here](http:\/\/i.stack.imgur.com\/FkbmQ.jpg) A Sphere A is placed on a smooth table.Another sphere B is suspended as shown in the figure.Both the spheres are identical in all respects.Equal quantity of heat is supplied to both spheres.All kinds of heat loss are neglected.The final temperatures of A & B are T1 & T2 respectively,then   * T1>T2   * T1 = T2   * T1    * None of these Please give reasons for your answer."}
{"id":"12874","title":"Why is 55-60 MPH optimal for gas mileage of a passenger car?","text":"My driver's education teacher back in high school said 55 MPH is optimal for gas mileage of a passenger car. Just last week, I read an article in a magazine saying 60 MPH is optimal. These numbers are pretty close, so there's some validity in the statement. What's the physics explanation for this 55-60 MPH sweet spot?"}
{"id":"95234","title":"Toppling of a cylinder on a block","text":"A uniform cylinder rests on a cart.The height and diameter is given.coefficient of static friction is given.How can i find the minimum acceleration of block such that the block topples? Morever what is the reason of toppling? ![cylinder on block](http:\/\/i.stack.imgur.com\/eBbOx.png) Again, if i consider static friction to be the cause,then what is the point of rotation? If i consider the top to be te point of rotation,then why does the block topple with the lower end as point?I have conducted this experiment many times and if the block falls,it fell along the lower end."}
{"id":"119000","title":"Uncertainty principle - momentum so precise that uncertainty of position is outside light-cone?","text":"Thought experiment: what happens if we measure momentum of a particle so precisely, that the uncertainty of its position becomes absurd? For example, what if the uncertainty of the position exceeds 1 light year? We know for a fact that the particle wasn't a light year away from the measuring device, or else how could the momentum have been measured? What if the uncertainty extended beyond the bounds of the universe? Isn't there some point at which we know for certain the particle was closer than what the uncertainty allows for?"}
{"id":"95232","title":"Is Tube light polychromatic?","text":"Is tube light\/fluorescent lamp monochromatic or polychromatic? Please justify."}
{"id":"99888","title":"How do I calculate pressure drop across a reduction in pipe diameter?","text":"If reducing from one pipe size to another through a reducing fitting, say 6.25cm to 5cm, how do I calculate the pressure drop? Also, how does this impact a change in volumetric flow rate (or does it affect the volumetric flow rate)?"}
{"id":"14700","title":"The Time That 2 Masses Will Collide Due To Newtonian Gravity","text":"My friend and I have been wracking our heads with this one for the past 3 hours... We have 2 point masses, $m$ and $M$ in a perfect world separated by radius r. Starting from rest, they both begin to accelerate towards each other. So we have the gravitational force between them as: $$F_g ~=~ G\\frac{Mm}{r^2}$$ How do we find out at what time they will collide? What we're having trouble with is this function being a function of $r$, but I have suspected it as actually a function of $t$ due to the units of $G$ being $N(m\/kg)^2$. I've tried taking a number of integrals, which haven't really yielded anything useful. Any guidance? No, this is not an actual homework problem, we're just 2 math\/physics\/computer people who are very bored at work :)"}
{"id":"66545","title":"Solid in Liquid Heat Transfer","text":"If there is a solid immersed in a large (but finite) pool of water, where the solid has temperature $T_s$ and the water has temperature $T_w$, with $T_w>T_s$, how can I calculate $T_s(t)$ and $T_w(t)$? (Suppose that the masses and the material properties of the solid are known.)"}
{"id":"61012","title":"Graphene +1 extra carbon bond","text":"I'm not a physicist just a curious mind, so please go easy! I was just watching a BBC Horizon Documentary that featured a piece on the recently discovered material Graphene. One of the facts mentioned on the documentary was that a sheet of graphene, despite being a lattice of carbon atoms just one atom thick, could hold the weight of a household cat without breaking. They showed an illustration of graphene's hexagonal structure and how a crude form of graphene could be created using Sellotape and piece of graphite. OK so here's my question.. From what I can remember from Chemistry lessons in school, carbon has a valency of 4, so it can bind to 4 more carbon atoms. Surely (in theory at least) a sheet of carbon could exist that adopts a square grid lattice, as oppose to graphene's hexagonal structure. If so wouldn't such a material be stronger than graphene? Also would such a lattice naturally form under any circumstance or does carbon always assume a hexagonal structure when reduced to layers?"}
{"id":"26523","title":"Amateur Stargazing (30 degrees N)","text":"When I look at the sky I easily recognize Orion, The Big Dipper, and the North Star (in relation to the Big Dipper). Besides these (and the moon^^), what **_easily identifiable_** (unaided eye) astronomical objects should I look for? (Latitude = 30 degrees North)"}
{"id":"26521","title":"How did micrometeorite flux change with the age of the solar system?","text":"Do we also have functions for the micrometeorite flux on the Moon, Mars, and even any random body in space as a function of solar distance too?"}
{"id":"130466","title":"Minkowski to Euclidean","text":"When dealing with solutions to Einstein's equations given by a 4d metric with signature $(-,+,+,+)$, we're able to move to Euclidean space using some transformation so that our signature is now $(+,+,+,+)$ and our metric is changed slightly different. What I'm worried about is what this does to the properties of my metric. When I go to Euclidean space, I am able to deduce some nice relations, however, $\\textbf{do these relations hold for my original metric?}$ e.g. let's say I'm interested in the metric $$ds_1^2=-V_1dt^2+\\frac{dr^2}{V_1}+W_1d\\Omega^2,$$ where $d\\Omega^2$ is metric on $S^2$. Now my metric in Euclidean space is $$ds_2^2=+V_2dt^2+\\frac{dr^2}{V_2}+W_2d\\Omega^2.$$ where $V_1(a,b,c)=V_2(i a,b,c)$ and for $V_2$ I have some cool relation (e.g. $V_2(a=b)=0$, for $i$ the imaginary number. How can I relate this to the properties of $V_1$ if $a$, $b$ and $c$ are real parameters? The reverse transformation gives $V_1(-i a=b)=0$, but I don't know how to interrupt this $a$ and $b$ are real. Perhaps I haven't throughout deeply enough about it yet and it's a naive question, but I don't think I yet comprehensively understand what this transformation means for the properties of our space."}
{"id":"110187","title":"Why isn't angle of launch used in this kinetic energy equation?","text":"Question: A stick is thrown from a cliff 27 m high with an initial velocity of 18 m\/s at an angle of 37 degrees above the horizontal. a) use the law of conservation of energy to determine the speed of the stick just before it hits the ground. solution: ![a busy cat](http:\/\/i60.tinypic.com\/2qmdz74.png) Just wondering why the vertical initial speed vector isn't used? Should 18sin37 be used? I think I am confusing myself with projectile motion."}
{"id":"114401","title":"EM wave properties","text":"Here are two small questions about EM wave properties: **Question 1:** Why do E and B propogate simultaneously? See picture ![enter image description here](http:\/\/i.stack.imgur.com\/RcVwS.jpg) I mean why do not works a paradigm “changing E makes B and changing B makes E?” For such scenario we would have phase shift beatween E and B by $\\pi\/2$. I am asking because of f.e. maximum of B = dE\/dt it is when E=0. So why not E = E_max*sin(x) and B=B_max*cos(x)? **Question 2:** But for standing wave we have phase shift beatween E and B by $\\pi\/2$ (by theory). http:\/\/www.youtube.com\/watch?v=tbR93plA1WU ![enter image description here](http:\/\/i.stack.imgur.com\/FlZVj.png) So do you know some experiments to confirm that maximums of B are really shifted by $\\pi\/2$ for standing wave?"}
{"id":"110183","title":"Correlation between color of light visible to us and the intensity of transmitted light by the material","text":"I wanted to know if there is any relation between the color of light observed by the eye and the intensity of light transmitted by a material. Let us assume that I have a table of intensities of light transmitted at different wavelengths by a material for 2 materials. Is it possible to determine which material will appear more colored and which is that color? I know that if a material transmits with more intensity at the wavelength of green light, it would be more 'greener' but what will be the effect of a range of wavelengths?"}
{"id":"114403","title":"Hamiltonian Operator for Harmonic Oscillator","text":"I have been solving the harmonic oscillator problem in quantum mechanics using Algebraic Method and since then I am consulting the books of Tannoudji and Griffiths for that matter. While studying both of them I ham having a small confusion while defining the Hamiltonian operator Tannoudji used this expression $\\frac{H}{\\hbar w}$ for defining the Hamiltonian operator where $H$ is classical Hamiltonian while Griffiths have not used this expression at all. The confusion is how this expression from Tannoudji came across and why Griffiths did not use this expression at all."}
{"id":"26291","title":"Why did the ancients fail to discover that the Earth orbits the Sun?","text":"The ancients observed that the Sun and the 'fixed' stars rotated about the Earth. They were also aware that the Earth was spherical. They performed many astronomical measurements on the planets - which are far less 'obvious' than the Sun and the stars. Presumably they knew that the speed of rotation of the Sun and the 'fixed' stars differed and that difference happened to be one day a year. It would seem (in retrospect) obvious that this was no coincidence and no other heavenly motion was so clearly linked to any other. Could they not have deduced that the difference was due to the annual path of the Earth around the Sun?"}
{"id":"26295","title":"Spectral energy distribution fitting tools or routines","text":"I have observed magnitudes and fluxes for an object in different wavelengths from optical to mm. Now I need a tool, routine or something like that to fit a spectral energy distribution (SED) and derive some physical parameters from the fit. I'm looking for a ready tool which can be run remotely or installed locally. Mind you, I'm aware of this: http:\/\/caravan.astro.wisc.edu\/protostars\/"}
{"id":"93170","title":"How to get a $\\mathcal{N}=2$ SuperYang-Mills Lagrangian from a quiver","text":"How can one write down the $\\mathcal{N}=2$ SuperYang-Mills Lagrangian given a quiver graph? For concreteness consider the quiver $$(2)-(4)-[6]$$ where the node $(2)$ corresponds to a $U(2)$ factor of the gauge group, the $(4)$ node is a $U(4)$ factor and the $[6]$ node is a $SU(6)$ global symmetry (flavour). The lines correspond to two $\\mathcal{N}=2$ hypermultiplets charged in the fundamental representations of one node and the antifundamental representation of the other node into which the line ends. In particular, how can one read the superpotential from this graph? How can one read the Kahler potential? Is this latter one always assumed to be canonical? In case you would like to choose an easier quiver to make an easier example it is perfectly fine."}
{"id":"93176","title":"Lagrangian Mechanics - Commutativity Rule $\\frac{d}{dt}\\delta q=\\delta \\frac{dq}{dt} $","text":"I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are the same is presented in the book I am reading as a rule, commutativity, and possible non-commutative rules are mentioned too. I do not get it: given a path $q(t)$ and its variation $\\delta q(t)$, the equivalence between the variation of the derivative $\\delta \\dot{q}$ and derivative of the variation $\\dot{\\delta q}$ seems to me a fact descending straight from calculus, not an arbitrary choice."}
{"id":"118759","title":"Could this mean that Schrodinger's Cat thought experiment is false?","text":"Since it is uncertain on whether or not the cat could be dead or alive, it is concluded that it depends on the observer to make it either dead or alive. But lets look at a clock. It doesn't take observance of the gears to make it function. The clock ticks despite there being an observer. Could this mean that the famous thought experiment is false?"}
{"id":"32268","title":"Hamilton-Jacobi Equation","text":"In the Hamilton-Jacobi equation, we take the partial time derivative of the action. But the action comes from integrating the Lagrangian over time, so time seems to just be a dummy variable here and hence I do not understand how we can partial differentiate $S$ with respect to time? A simple example would also be helpful."}
{"id":"51324","title":"Can path integrals be used to understand entanglement?","text":"I like path integrals. I prefer to try to understand quantum phenomena in terms of path integrals rather than Hamiltonian mechanics. However, most of the standard texts on quantum mechanics start from Hamiltonian mechanics and mention the path integrals as a side note. This makes it difficult for me to gain insight into common phenomena (with a few exceptions like certain areas of optics) from the perspective of path integrals formulations. I am aware of some exceptions to this rule (books on path integrals in particular), but I have only seen entanglement explained from the perspective of eigenstates of a Hamiltonian. I was wondering how entanglement would be interpreted if you started from a path integral formulation. I would be more than happy to hear from a quantum field theoretic perspective as well, but the usual quantum mechanical path integral would certainly satisfy my curiosity."}
{"id":"79999","title":"My conundrum with Gauss’ law in electrostatics","text":"If I use Gauss’ law to calculate the electric field outside of a charged (conducting or insulating) sphere or a point charge, the fields are the same. However, as a test approaches a point charge, the field becomes infinite whereas the test charge approaching the surface of the sphere reaches a finite value of $Q\/4πϵ_0R^2$, where $R$ and $Q$ are the radius and charge of the sphere. Why wouldn’t the field also go to infinity as I get infinitesimally close to the surface of sphere? Those are just “point charges” on the surface. What ultimately led me to this question was trying to physical understand why Gauss’ law predicts the same field for a charge point and charged sphere (obviously $r > R$). I don’t understand it."}
{"id":"56315","title":"All mirrors always shrink to 50% scale?","text":"I have this geometric optics exercise here, in which a man is looking at himself in a mirror.   1. Determine the minimum height at which the bottom of the mirror must be placed so the man can see his feet (his eyes are at 1.7m).   2. How does this height vary depending on the distance from the mirror to the man's eyes?   3. What is the minimum length of the mirror so that the man can see his entire body (assume he is 1.8m tall). From the principle that the reflected angle is equal to the incident angle, we show that the minimum height of the bottom of the mirror is directly between the feet and the eyes (1), and that this does not depend on the distance to the mirror (2). The top of the mirror must be directly between the man's crown and eyes at the lowest, in order to reflect the top of his head, **therefore the mirror must be half the height of the man, at minimum**. Thus, the mirror reflects the man's image at a scale of 50%. But this doesn't depend on the distance from the mirror, and from what I can tell, doesn't even depend on the size of the mirror. These results, to me, indicate that no matter what kind of mirror you use or where you stand, your image should appear half way up the mirror, and at 50% scale."}
{"id":"79992","title":"Why is it difficult to differentiate between interference and diffraction?","text":"Why is it difficult to differentiate between interference and diffraction? Is it because we don't clearly understand how both of these phenomenon takes place? My thoughts: From an answer to one of my earlier question here I could understand that light waves behave differently when it strikes a slit which is of several times bigger than its wavelength and when it strikes a slit of size comparable to it its wavelength.In the former case the different points on the slit acts as wave-front sources producing circular wave fronts which interfere to form the diffraction pattern but in the later case plane wave-front striking a slit after passing through it forms circular wave-fronts and two such wave-fronts from two different slits interfere to form the interference pattern. But, I'm not satisfied by that because I don't get any intuition about points on the slit acting as wavefront sources. So, what is actually happening?"}
{"id":"116800","title":"Replacing diode in circuit analysis?","text":"While analyzing circuits containing diodes in reverse bias I replace it with a pure insulator for simplifying things. Similarly can I replace a forward biased diode with a conductor and a varying resistance in series?"}
{"id":"25360","title":"Is there a method to estimate the atomic yield of a supernova of a given size?","text":"Given a supernova with stellar mass $M$, is there a theoretical method to estimating the isotope yield? If so, what processes are taken into account, and how accurate can the estimate be? Would it be more accurate if more properties were known about the star (e.g. metallicity)?"}
{"id":"110231","title":"Is $\\frac{\\partial}{\\partial \\Phi(y)} \\Phi (x) = \\delta(x-y)$ correct?","text":"As stated in the heading: Is $\\frac{\\partial}{\\partial \\Phi(y)} \\Phi (x) = \\delta(x-y)$ correct? Here denotes $\\Phi(x)$ denotes a scalar field. And if yes, why? Any reference where I can read about this would be great."}
{"id":"104587","title":"anti-matter activation energy","text":"In chemical reactions, there are usually energy barriers that must be overcome before a reaction can take place. It seems that such a phenomena also exists at the atomic level as well. For instance, a certain amount of energy must be provided to two nuclei to bring them close enough together to fuse and release more energy than was input. In general, I read things like \"energetically favorable\" or \"a lower energy state is available\". However, in examining the behavior of anti-matter, it seems almost as if there is no such barrier between matter and anti-matter: they annihilate immediately on contact, converting most or all of their matter into energy. It seems like such a sharp divide between total annihilation with virtually no extra energy added to the equation, versus almost total stability for either matter or anti-matter by itself. Why such the big difference? Why does anti-matter seem to have no constraints to \"decaying\" into energy in the presence of matter (and vice versa) like other processes do? I know there are certain quantities like charge which must be conserved but it seems like a pretty abstract thing to say we don't all spontaneously decay into photons simply because charge must be conserved. Why isn't there some kind of energy barrier between matter and anti-matter which must be overcome before they can react? If anti-matter had negative mass, then it would have negative energy, in which the lack of such a barrier would make more sense. However, I don't know of any evidence that suggests anti-matter has negative mass. My naive guess would be that since a particle and anti-particle have opposite spin, they sort of \"slam into\" each other and cancel out, causing the particle to \"break apart\" into a free wave. However, since my understanding of spin is that it isn't really the same kind of spin we would observe at the macroscopic scale, i.e. actual physical rotation, this seems like an unreal explanation to me."}
{"id":"64575","title":"Many photons, one quantum field?","text":"If a photon can be described as an excitation in a quantum field, is this the same field for all photons, or does each photon exist in its own field?"}
{"id":"60057","title":"CP-symmetry and Ward identities and finite temperature","text":"I have a few questions about Ward-identities which I summarize here. For each I am very greateful for answers and references to literature. Wikipedia states about Ward-identities: > _The Ward-Takahashi identity is a quantum version of the classical Noether's > theorem, and any symmetries in a quantum field theory can lead to an > equation of motion for correlation functions._   1. Taken seriously this means that Ward-Takahashi identities also exist for discrete symmetries like CP (if present in the Lagrangian) in contrast to a conserved Noether current. If so, I wonder if CP relations between amplitudes, such as $|\\mathcal{M}|_{S_i\\rightarrow S_f} =|\\mathcal{M}|_{\\bar{S}_i\\rightarrow \\bar{S}_f}$ which I can get by computing the amplitudes in perturbation theory and comparing, can be considered Ward-identities. $\\bar{S}_{i,f}$ are the CP-conjugated states of ${S}_{i,f}$ respectively.   2. Since Ward-identities are considered to hold for the quantized theory, I wonder if they are invariant under the renormalization group running of parameters. If so, how does RGE running take these restrictions into account.   3. Finally I wonder if there is an established finite-temperature generalization."}
{"id":"53893","title":"NMR rotating frame","text":"I'm reading about a linearly polarized field (in the context of NMR). The field is given by $$ {\\bf H_{lin}}=2H_1({\\bf i}cos(\\omega_zt)).$$ This can be created by having a pulse field plus its mirror image; i.e. $${\\bf H_1}=H_1({\\bf i}cos(\\omega_zt)+{\\bf j}sin(\\omega_zt)).$$ The mirror image field in the lab frame is obviously the same as ${\\bf H_1}$ expect $\\omega_z \\rightarrow -\\omega_z.$ Now, my question is what is the expression for the counter-rotating \"mirror image\" field in the ROTATING frame? I've having some trouble working through this..."}
{"id":"53894","title":"Modeling non-quantum objects (in finance, sociology etc) using fermionic fields?","text":"Please provide (if any) applications of fermionic field theory in non-physics macro contexts (finance, sociology etc). I see only bosonic fields being used mostly. The only (minor) application of fermionic fields that I have come across is in section 7.9.1 in Baaquie's Quantum Finance book where nonlinear forward interest rates are modeled using fermionic fields. Is there an intuitive reason for such asymmetry between bosonic and fermionic field usage at such scales?"}
{"id":"117172","title":"How to calculate gravity path integrals about an AdS background?","text":"Suppose I have some Lagrangian of some higher derivative gravity coupled to a may be matter fields. Now I want to fluctuate it to quadratic order about an AdS background and calculate the 1-loop partition function. Can someone point to a reference where such a thing might be done? This sounds like something standard which someone would have done... I can't see how one can even write down the fluctuated Lagrangian to quadratic order...And even if I wrote that down, how does one impose that the background is AdS? Then the issues of gauge fixing and ghosts are only added complication.. * * * I can imagine calculating the first order variation in the Christoffel symbols and Riemann and Ricci tensors but that goes only so far."}
{"id":"98141","title":"Learning about super-symmetric quantum mechanics and integrable systems","text":"I'm an undergraduate interested in theory. I recently asked one of my professors, a physicist in the particle theory group at my school, if he'd be willing to take on an undergraduate for a senior project. He told me to check back in at the end of the quarter, and that he might have a project related to super-symmetric quantum mechanics and integrable systems that may be feasible for me to try out then. In the meantime, I want to familiarize myself as much as possible with the relevant math and physics so that I can impress him when I ask again. Do any of you have any books\/online sources that you recommend I take a look at?"}
{"id":"98146","title":"Matrix representation of a triplet state","text":"The $SU(2)$ triplet state is typically given in the fundamental representation as a column vector, e.g. \\begin{equation} \\vec{\\Delta} = \\left( \\begin{array}{c} \\delta^{++} \\\\\\ \\delta^+ \\\\\\ \\delta^0 \\end{array} \\right) \\end{equation} where I use this notation to be consistent with the references below. However, I'm reading through papers (in the context of the type II see-saw mechanism, but I don't think it has much relevance) and most references are using what they call the matrix representation of the $SU(2)$ triplet. My instinct was to assume that they were projecting the fundamental representation onto the Pauli matrices: \\begin{equation} \\Delta \\rightarrow \\vec{\\Delta } \\cdot \\vec{ \\sigma } = \\delta ^{++} \\sigma _1 + \\delta ^+ \\sigma _2 + \\delta ^0 \\sigma _3 \\end{equation} However, this doesn't quite give me the correct result, which is given by, \\begin{equation} \\Delta = \\left( \\begin{array}{cc} \\delta ^+ \/ \\sqrt{2} & \\delta ^{ + + } \\\\\\ \\delta ^0 & \\- \\delta ^+ \/ \\sqrt{2} \\end{array} \\right) \\end{equation} Is there some other way to get this \"matrix representation\"? Two sample references where they say they are using the matrix representation can be found here (pg. 124, equation 68) and here (pg. 4 equation 1)."}
{"id":"98148","title":"Uniform Circular Motion: v = v_0 + (a_T)*t as a vector VS as a scalar equation","text":"First, I'll explain my confusion, which may be enough for someone to clarify my misunderstanding. Second, to give context, I will give the homework problem (which I've solved successfully) which caused my confusion. Note: I'll use boldface to denote vector quantities. Also, I am in my third semester of Calculus. A particle accelerates around a track. Velocity is given by the below equation. $$ v = v_0 + a_Tt $$ When I substitute in values for this equation to solve homework problems, I use scalar values—in other words, I use magnitudes. I find $|\\mathbf{a_T}|$ to use in the above equation. However, if the equation is written as a sum of vectors: $$ \\mathbf{v} = \\mathbf{v_0} + \\mathbf{a_T}t $$ Then I take the magnitude of both sides . . . ($\\theta$ is the angle between the positive $x$-axis and $\\mathbf{a_C}$.) $$ \\begin{align} |\\mathbf{v}| &= |\\mathbf{v_0} + \\mathbf{a_T}t|\\\\\\ &\\text{. . . algebra . . .}\\\\\\ &= \\sqrt{|\\mathbf{v_0|}^2 + 2|\\mathbf{v_0}||\\mathbf{a_T}|t + |\\mathbf{a_t}|t^2}\\\\\\ &\\ne |\\mathbf{v_0}| + |\\mathbf{a_T}|t\\ \\ \\text{*} \\end{align} $$ But the starred equation is what I am using for homework problems and I get the answers with no problems. * * * http:\/\/i.imgur.com\/vUDZuKb.png To summarize my solution:   1. Use $\\mathbf{a}$ to find $|\\mathbf{a_C}|$ and $|\\mathbf{a_T}|$.   2. Substitute $|\\mathbf{a_C}|$ into $a_C = \\frac{{v_0}^2}{r}$ and solve for $v_0$.   3. Substitute $v_0$ and $\\mathbf{a_T}$ into $v_1 = v_0 + a_T t$ and solve for $t_1$ when $v_1 = 0$. I also run into sign issues because I'm treating the value I get for tangential acceleration as the magnitude of a vector. This means that the value for tangential acceleration is positive, but it needs to subtract from velocity. There is a similar issue with the centripetal acceleration. The only way to deal with it properly seems to be to treat vectors counter clock-wise around the circle or pointing away from the circle as positive, and other vectors as negative . . . which is akin to treating the circle as a straight line almost . . . It feels like a hell of a lot of hand-waving, though I am sure that's because I don't understand what is happening."}
{"id":"2702","title":"\"Magnetic mnemonics\"","text":"Over and over I'm getting into the same trouble, so I'd like to ask for some help.   1. I need to solve some basic electrodynamics problem, involving magnetic fields, moving charges or currents.   2. But I forgot all this rules about \"where the magnetic field should go if the current flows up\". I vaguely remember that it about hands or fingers or books or guns, and magnetic field should go somewhere along something, while current should flow along something else... But it doesn't help, because I don't remember the details.   3. I do find some very nice rule and use it. And I think \"that one I will never forget\".    4. ...time passes...   5. Go to step 1.  The problem is that you have to remember one choice from a dichotomy: \"this way or that way\". And all the mnemonics I know have the same problem -- I got to remember \"left hand or right hand\", \"this finger or that finger\", \"inside the book or outside of the book\". Maybe someone knows some mnemonics, that do not have such problem? ![By Bruno Touschek](http:\/\/i.stack.imgur.com\/SxrgK.gif)"}
{"id":"10552","title":"Is there literature on a continuous mass spectrum for the Higgs field?","text":"Various masses for the Higgs field are compatible with experiment, but is it possible that the Higgs field is not observable because it has a continuous mass spectrum? Work in the 60s and 70s on free fields that have a continuous mass spectrum, often called generalized free fields, established that there is no simple particle interpretation if the mass spectrum is not delta function valued, which I take it can be described as due to destructive interference between the different mass components. [I'm taking that largely from R. F. Streater, _Rep. Prog. Phys._ , **38** , 771-846 (1975).] Is the possibility that the Higgs has a continuous mass spectrum to be found in the literature, or is there some sufficiently obvious anomaly or other reason why it is not considered in any detail?"}
{"id":"61905","title":"Mie scattering intensity and spherical particles on","text":"I am running an experiment where I shine infrared light on (almost) spherical particles on the micron scale (PM2.5 - PM10). I then look at the (90 deg) scattering properties to try and size the particles. I am looking for the theory of Mie scattering and its intensity, specifically how the scattered intensity depends on the surface area of the particles. Looking for suggestions of good literature on this, or just simple equations."}
{"id":"61900","title":"Why do cold metal plate make less noise?","text":"I have a metal gate in my home so I have noticed that when it is a hot day and if I hit it it makes a lot of noise but when there is a rainy day and the weather and the gate re cold and if I hit the gate it makes very much less noise. I am assuming that when there is a cold day the metal is much dense so the vibration travel through it to the ground more easily as compared to that on a warm day, am I correct. ![enter image description here](http:\/\/i.stack.imgur.com\/8jh1H.png)"}
{"id":"119584","title":"Why the position of apparent horizon depend on choice of Sigma t of foliation?","text":"Whether a 2d surface is a trapped surface is independent of choice of Sigma t of foliation. Why its outer boundary (apparent horizon) depends on choice of Sigma t of foliation?"}
{"id":"18407","title":"I think there is something wrong with this problem. How do you know if the vertical acceleration is zero?","text":"Suppose a block with mass 3kg is on a frictionless table and a force of 15N pulls it from an angle of 38 degrees above the horizontal. 1) What is the apparent weight of the block? 2) What is the minimum force for the block to take off? **Solution Given** $\\sum F_y = n - mg - F\\sin\\theta = 0$ $\\sum F_x = F\\cos\\theta = ma_x$ Solve for n, the apparent weight or the normal force. n = 20.16N The minimum force to take off when n = 0 $mg = F\\sin\\theta$ $F = \\frac{mg}{\\sin\\theta}$ $F = 47.75N$ **Question** Why on earth is the force on the vertical direction 0? Shouldn't it be $\\sum F_y = n - mg - F\\sin\\theta = ma_y$ There is a horizontal acceleration component, but no vertical acceleration."}
{"id":"108288","title":"Perturbation of coupled spin","text":"I am given a system with Hamiltonian (all 1\/2 spins) $$H_0=\\alpha(S_1\\cdot S_2)$$ I broke it down and found that there were four eigenstates: $|1,[0,\\pm1]\\rangle$ and $|0,0\\rangle$. Each has an associated eigenvalues. I am then told that a weak magnetic field is applied giving a perturbation $$H_V=-\\mu B_0\\cdot(\\sigma_1+\\sigma_2)$$ I am supposed to use stationary state perturbation theory to find $E_n^{(1)}$. So I know that the perturbed energies are given as $$E_n^{(1)}=\\langle \\psi_n^{(0)}|H_V|\\psi_n^{(0)}\\rangle$$ I am a bit confused on what to use for the $\\psi_n$ and then the $\\sigma$ matrices. I know that the eigenstates are those given above, but I do not see how I can form a matrix from those and then I am unsure of what sigma would be in order to calculate the matrix element"}
{"id":"102324","title":"Do photons interact only with single particles or can a single photon be absorbed by 2 or more particles simultaneously?","text":"Say I have 100 particles in a vacuum that are spread out such that their movement caused by a time variant electromagnetic field does not have an effect on each other. That is the particles can not exchange any energy among themselves. The varying electromagnetic field must be experienced in quanta (a photon particle). Does this mean that it will take at least 100 photons worth of light to excite all the particles?"}
{"id":"63503","title":"Michaelis-Menten derivation for 2 enzyme substrates","text":"We know that the Michaelis-Menten derivation for the following reaction: $E + S \\rightleftharpoons ES \\rightarrow E + P$ However, what if the reaction took place in a different scenario whereby: $E + S \\rightleftharpoons ES_1 \\rightarrow ES_2 \\rightarrow E + P$ What would be the derived Michaelis-Menten equation now? [ES2] is assumed to be at steady state assumption. What would be the derived Michaelis Menten Equation?"}
{"id":"63505","title":"Dopant concentration and changes in band gap energy","text":"Thanks to this lovely website, I was able to pop out reasonable values for my band gap energies from a translucent material. As expected, I found a decrease in band gap energy due to my treatments. Now, I also need some numbers put to dopant concentration. My dopants, in this case, are reduced forms of the semi-conductor material I am treating, but I think the concept can be generalized to all dopants. So my query is as follows: If I know how the band gap changes between a doped and undoped material, is there a formula that will give me the amount or density of the dopants in the material? For example, say my band gap before doping was 3.0eV and after doping it was 2.5eV, can I pull out how many dopant atoms were added to my material?"}
{"id":"128218","title":"How is voltage generated due to difference in temperature in a thermocouple?","text":"How is voltage generated due to difference in temperature between the two junctions of a thermocouple? What happens to the electrons at the junctions of the two materials and how do they behave when they are immersed in hot and cold medium?"}
{"id":"112274","title":"Are intergalactic voids inertial frames?","text":"I know earth is a non-inertia frame, so is the solar system, the milky way, the local group and even our super cluster. But what about all the space, the void that is left outside of the superclusters. From wikipedia \"Interspersed among superclusters are large voids of space in which few galaxies exist. Superclusters are frequently subdivided into groups of clusters called galaxy clouds.\" When we talk of the voids the only movement we can talk about is the space expansion, not even a real movement as I have learnt after several debates and discussions. So are these voids actually inertial frames? If they are not, why are they not?"}
{"id":"112278","title":"Lattice pattern","text":"I'm trying to find and outline a non-primitive conventional unit mesh, I'd also like to find any mirrors planes and rotional symmetry axes. ![enter image description here](http:\/\/i.stack.imgur.com\/vNPjZ.png) I first thought it would be like this: , but I was told the answer was smaller in a different thread. ![enter image description here](http:\/\/i.stack.imgur.com\/dge86.png) This was the solution left by LDC3 ![enter image description here](http:\/\/i.stack.imgur.com\/NM7pS.png) Which one is correct? I don't know if his is correct because I thought the conventional unit mesh connected to points that looked the same as the original small pattern I picked(the small hexagon in the middle) Also tried the axis of symmetry, but am unsure if 8 lines from the topside or 6 for a hexagonal pattern should come out. https:\/\/imgur.com\/Bkjwhih I think for hexagonal structures there should be more lines like this https:\/\/i.imgur.com\/GopAzmM.png Any help is much appreciated."}
{"id":"24100","title":"mechanism of annihilation","text":"Can the annihilation of matter and antimatter be explained by the electro-weak interaction? Can pair-production be explained in the same way?"}
{"id":"82056","title":"Configuration space of particles in the box","text":"The notion of entropy says that we can count microstates that correspond to macrostate. But, I do not understand how this can be done. Does it imply that the state space is discrete (finite or countable)?"}
{"id":"15634","title":"Mathematical definition of Bogomol'nyi–Prasad–Sommerfield (BPS) states","text":"What is the mathematical definition of Bogomol'nyi–Prasad–Sommerfield (BPS) states, independent of any specific physical theory."}
{"id":"71966","title":"Leakage rate Calculation","text":"I have to calculate leak rate for helium\/air from an enclosure which is made of fabric. This is similar to helium balloons. There are small holes in the fabric. How should i proceed ?"}
{"id":"71967","title":"Proton Radius Puzzle, Is it possible that proton's radius differ depending on how you measure it? What does that even mean?","text":"I just read this NewScientist article, and I was stunned by its results. So I found the original paper here on arxiv. In the introduction of the paper it is stated: > The recent determination of the proton radius using the measurement of the > Lamb shift in the muonic hydrogen atom [1, 2] startled the physics world. > The obtained value of 0.84087(39) fm differs by about 4% or 7 standard > deviations from the CODATA [3] value of 0.8775(51) fm. The latter is > composed from the electronic hydrogenate atom value of 0.8758(77) fm and > from a similar value with larger uncertainties determined by electron > scattering [4] What Randolf Pohl _et al_ basically did, is that they measured the radius of the proton using more accurate techniques. While they were expecting to just get a more accurate result(few more decimal places), their proton radius is clearly smaller than all other experiments before. How they did the experiment is described in the paper I pointed to. The conclusion of the paper is: > The proton radius puzzle is real. A physical parameter should not depend on > the method of extraction. Yet here we have two highly precise methods based > on electronic and muonic hydrogen atoms that disagree significantly. The > possible origins of the disagreement are the subject of much of this text. > To summarize: the possibilities are > >   * The electronic hydrogen experiments are almost, but not quite, as > accurate as stated (Sect. 2.2). >   * The QED calculations are almost, but not quite, as accurate as stated > (Sect. 4.2). >   * The two photon exchange term that depends on proton polarizability has > not been correctly evaluated (Sect. 4.3). >   * The electron and muon really do have different interactions with the > proton (Sect. 4.4), so that there is physics beyond the Standard Model. > > > None of these possibilities seem very likely, but all must be pursued. My question is, what are the possible implications of this new result? Does anyone know how significant are the potential flaws in the experiments mentioned? It is stated in the paper that one possible solution might be Beyond the Standard Model theories; assuming the experiment is correct, do we really need BSM theories to explain the new results? If so, which specific branches of BSM will benefit from these results?"}
{"id":"68307","title":"Can one black hole suck in another black hole?","text":"In the recent news, scientists at NASA have found “unprecedented” black hole cluster near Andromeda’s central bulge. I wonder why doesn't all these black holes merge and such each other in until just 1 remains? Is that even possible? Can one black hole suck in another black hole?"}
{"id":"3004","title":"How to calculate the density of relic neutrinos?","text":"May be not neutrinos, but antineutrinos? Or both types? In the last case, why they didn't annihilate and what is the ratio of relic neutrinos to relic antineutrinos? Is that ratio somehow related to the barion asymmetry? For reference: Relic neutrinos or cosmic neutrino background > Like the cosmic microwave background radiation (CMB), the cosmic neutrino > background is a relic of the big bang"}
{"id":"71817","title":"General relativity theory","text":"As I understand general relativity theory (please correct me if I'm wrong), time becomes dilated and space becomes compressed around mass, and this is responsible for gravity. I'm struggling with precisely how that results in gravitational force between masses. Can someone explain that to me (please keep any required math understandable to a lay person)."}
{"id":"60726","title":"Einstein's theory tells us that gravity is a curve in space and time but how does that causes attraction in mass?","text":"The sun is incredibly massive object and it causes the space around it to bend. This causes the planets to pulled to the sun or the planets move in an elliptical path around the sun. But I don't understand how the curvature in space can cause the pulling or what cause the planets to move in elliptical path?"}
{"id":"27455","title":"Any use for $F_4$ in hep-th?","text":"In high energy physics, the use of the classical Lie groups are common place, and in the Grand Unification the use of $E_{6,7,8}$ is also common place. In string theory $G_2$ is sometimes utilized, e.g. the $G_2$-holonomy manifolds are used to get 4d $\\mathcal{N}=1$ susy from M-theory. That leaves $F_4$ from the list of simple Lie groups. Is there any place $F_4$ is used in any essential way? Of course there are papers where the dynamics of $d=4$ $\\mathcal{N}=1$ susy gauge theory with $F_4$ are studied, as part of the study of all possible gauge groups, but I'm not asking those."}
{"id":"10673","title":"What does plasmon look like in 3D band structure graph?","text":"Consider metal, and its reciprocal lattice representation with Fermi surface. What is the correct way to represent a plasmon in this system? M.b. rotating points on the surface? Or 3d membrane-like oscillation of surface? Or what?"}
{"id":"59999","title":"Work done by gravity on Water","text":"![enter image description here](http:\/\/i.stack.imgur.com\/qeiCo.png) Now according to me we would see change in potential energy of system and equate it to the work done by gravity. But when we see this the first column lowers by $H\/2$ and right one rises by $H\/2$ and thus there should be no net work done. But it's not so. Work done by gravity is +ve. Where is it wrong?"}
{"id":"7938","title":"Alternative career paths in physics?","text":"What do you guys think of alternative career paths in physics away from academia free from the usual academic shackles? Examples: Garrett Lisi who spends his time surfing and skiing while not working on E8. Or Daniel Bedingham who is able to support himself part-time as an investment banker when not working on quantum mechanics. Or Julian Barbour who supports himself by translating Russian journal articles into English when not working on quantum gravity. Is it feasible to count upon Foundational Questions Institute to support such unconventional career paths using Templeton money?"}
{"id":"88507","title":"Why put something in the middle of the pan would prevent soup coming out during cooking?","text":"I see a website which teach people to put something on top of the pot during cooking soup to prevent the soup coming out during cooking. What is the physical theory behind that? ![enter image description here](http:\/\/i.stack.imgur.com\/YAyKl.jpg)"}
{"id":"67330","title":"Art : A Particular Mirror","text":"There is a particular mirror used here (1 minute 26 seconds) in this video, which reflects light in a way I've never seen. I was hoping maybe one of you might know what kind of mirror this is, so that I may use it for my own personal use."}
{"id":"111911","title":"Real silicon atom structure?","text":"Text books say that a pure silicon atom has four electrons in its outermost orbit. How do the scientists know it has four electrons?"}
{"id":"111917","title":"Why doesn't rain fall down in streams (as opposed to drops)","text":"Why is it that raindrops don't collide and 'stick together' on their descent to Earth, arriving in streams rather than separate drops?"}
{"id":"30959","title":"Does water in the drain follow the distance-squared rule of force?","text":"When I shower I notice that the soapy water spinning around the drain often takes the form of a spinning galaxy, complete with spiral arms. Does this indicate that the water's speed around the drain is inversely proportional to the square of its distance from the drain? Actually, I have heard that spinning galaxies themselves don't follow this rule, hence the discovery of dark matter, so does the spinning water follow whatever rule that the spinning stars do? Thanks."}
{"id":"30952","title":"In hydrostatics, is it possible for the bouancy to be greater than the weight of a body?","text":"In the diagram we have two bodies, with density and volume $\\delta$ and $V$, and a fluid $\\ell$. $$\\delta_1=0,7 \\rm\\frac{gr}{cm^3}$$ $$\\delta_2=2 \\rm\\frac{gr}{cm^3}$$ $$V_1=100 \\rm {cm^3}$$ $$V_2=10 \\rm {cm^3}$$ $$\\delta_\\ell = 1,2 \\rm\\frac{gr}{cm^3}$$ The system is static. ![enter image description here](http:\/\/i.stack.imgur.com\/E0db1.jpg) I need to find $(1)$ The tension of the rope $(2)$ the buoyancy $1$ receives due to the liquid. I solved it, but I'm getting strange results: Let $W$ be the weighs, $B$ be the buoyancy and $T$ the tension. Then, since the system is static: $$B_1+B_2-W_1-W_2=0$$ (the tensions cancel out) But $W_1= \\delta_1 \\cdot g \\cdot V_1$, $W_2= \\delta_2 \\cdot g \\cdot V_2$ and $B_2 = \\delta_\\ell \\cdot g \\cdot V_2$ (since the body is fully submerged): $$\\eqalign{ 10\\frac{{\\text{m}}}{{{{\\text{s}}^{\\text{2}}}}}\\left( {70{\\text{gr}} + 20{\\text{gr}} - 12{\\text{gr}}} \\right) = B_1 \\cr 10\\frac{\\rm m}{{{\\rm s^2}}}\\left( {0,078\\rm{kg} } \\right) = B{_1} \\cr 0,78{\\text{N}} = B{_1} \\cr} $$ It seems strange that the buoyancy is greater than the weigh of the body ($0,7 \\rm N$), is that possible? Also, I can't figure out how to obtain the tension."}
{"id":"3917","title":"Resistance of a two-dimensional sample","text":"In this review of the QHE, Steve Girvin makes the following statement (bottom of pg. 6, beginning of Sec. 1.1.1): > As one learns in the study of scaling in the localization transition, > resistivity (which is what theorists calculate) and resistance (which is > what experimental- ists measure) for classical systems (in the shape of a > [d-dimensional] hypercube) of size L are related by [12,13] > > $$ R = \\rho L^{(2-d)} $$ where $R$ is the _resistance_ of the hypercube and $\\rho$ its _resistivity_. For $d=2$ it is clear that the resistance and the resistivity of the sample are the same, and as Girvin goes on to say, this fact is crucial for the universality of the quantization of the resistivity plateaus in the QHE. My question is: **Is there a simple physical explanation of the above relationship?** Girvin cites two papers. Both are RMP: \"Disordered Electronic Systems\" by Lee and Ramakrishnan and \"Continuous Quantum Phase Transitions\" by Sondhi et al. I am hoping that someone can spare me the effort of going through these two papers by providing an answer."}
{"id":"34706","title":"Chaos is predictable?","text":"I'm reading a book of computational physics [1] where the driven nonlinear pendulum is studied in deep. This is the equation used in the book: $$ \\frac{d^2\\theta}{dt^2} = -\\frac{g}{l}\\sin\\theta - q\\frac{d\\theta}{dt}+F_d\\sin(\\Omega_dt) $$ Obviously the authors know when (whit which conditions) the pendulum switch to a chaotic regime. My question is the following: looking at the ODE, we can see (predict) the existence of chaotic behavior? If yes, this is valid for all the ODEs or only for a strict family of ODEs? [1] _Computational physics_ , N. J. Giordano and H. Nakanishi, Second Edition, Pearson Prentice Hall, 2006"}
{"id":"72538","title":"For two planets in a 2:1 mean motion resonance , where will their periastron and apastron points be?","text":"I want to animate the changing orbits of planets when they enter a mean motion resonance. Using a 2:1 resonance, I want to show a low-mass inner planet and a low-mass outer planet being tugged by a Jupiter-mass planet in the middle. I know the orbits of both become increasingly elliptical but I don't know in which direction the ellipse becomes extended. I want to include images of the planets' in orbit, too, so I want my points of periastron and apastron to be realistic relative to the position of the planets in their orbits. > Would the periastron point of the inner planet be near the point of 2:1 > resonance where my Jupiter-mass planet and the inner planet are nearest each > other, or on the opposite side, as I believe the case is with Neptune and > Pluto?   >  The same question for the outer planet, would it's periastron be closest to > the point where it is closest to my Jupiter mass planet, or on the other > side? Illustrating my question, I have an jpg illustration and animation in Flash at this location: Mean motion resonance animation (Flash) ![enter image description here](http:\/\/i.stack.imgur.com\/8wpif.jpg)"}
{"id":"49960","title":"Do perfect spheres exist in nature?","text":"Often in physics, Objects are approximated as spherical. However do any perfectly spherical objects actually exist in nature?"}
{"id":"35251","title":"Are the limits to prediction by QM provable?","text":"Quantum mechanics states that only two aspects of a quantum system can be predicted with certainty: 1) the average and 2) the standard deviation of many measurements of identically prepared quantum systems. (At least, that's the message I took from my Quantum Chemistry textbook). I believe Einstein did not like the fact that you couldn't predict more than those two things -- that there's a fundamental limit to what science can help us predict about events that do occur. Any other measured information from the system is essentially \"random\". What I'm wondering is whether QM has definitively proven that we are unable to obtain any other information about the system? For example, assume that with each quantum system there is associated a pseudo-random number generator that we don't know about. The \"random\" information is actually the result of the next iteration of the generator. All statistical tests would say that the information is random, but if we knew what function the PRNG was using, we could predict this data! I remember reading that experiments have ruled out hidden variable theories. Is a PRNG function an example of a hidden variable? (By the way, I'm not trying to challenge QM; I learn best by asking contradictory questions like this)."}
{"id":"100216","title":"Is there current flowing though a power source?","text":"Would there be any current flowing though an ideal voltage source? I am asking this more specifically when you are analyzing a circuit and you are doing nodal analysis. I know i=v\/r, but since the source is ideal, resistance would be zero or infinity?"}
{"id":"55508","title":"Does Dirac's idea of filled negative energy states make sense?","text":"Please bear with me a bit on this. I know my title is controversial, but it's serious and detailed question about the _explanation_ Dirac attached to his amazing equations, not the equations themselves. Imagine for a moment that someone signed onto this group and proposed the following:   1. Empty space is a lot like a metal, or maybe a semiconductor, because like those materials its chock full of negatively charged electrons occupying different velocity states, only...   2. Unlike metals or semiconductors, the density of electrons in any one region of space is _infinite_ , because there is no limit to how fast the electrons can move. That is because these are negative energy states in which an electron can _always_ move faster simply by emitting a photon, so there's not \"bottom\" to how far they can drop and how dense they can become, and...   3. Unlike metals or semiconductors, there is no exactly balancing sea of positive atomic charges, well, unless maybe there are infinite numbers positively charged atoms too, and...   4. The resulting infinite negative charge density of real electrons not only doesn't matter but is in fact completely and totally invisible for some reason, and...   5. The resulting infinite mass density of electrons (recall that these are quite real electrons, only in odd negative-energy kinetic states) also doesn't matter, and...   6. Unlike the Fermi sea of a metal conduction band, removing an electron from this infinitely dense sea of electrons for some reason _doesn't_ cause other electrons to collapse into it and fill it, even though the negative kinetic velocity electrons are pushed by exactly the same Pauli exclusion forces as the ones in a Fermi band; in short, for reasons not clear, semiconductor-style hole stabilization applies while metal-style hole filling does not (is there a band gap going on here?), and...   7. Since the infinitely dense negative charges become invisible for no particular stated reason when the electrons fall into negative energy states, these unexpectedly stable open states in the negative energy sea have net positive charge, even though...   8. ... such missing states categorically should have _zero_ charge, since in sharp contrast to the positive ionic background of metallic and semiconductors, the vacuum has no background charge at all, which should leave holes in the mysteriously invisible negative kinetic energy just as uncharged and invisible as that sea for some reason is, and...   9. Even if you _do_ assume that the negative kinetic state electrons have visible charge, their infinite density would make the \"comparatively\" but infinitesimally smaller positive charge of such a hole invisible, and...   10. Repeat this process for every other kind of particle in existence, and...   11. If you have done all of this and done it correctly, congratulations: You now understand conceptually what anti-electrons (positrons) and other anti-particles are. First question: Have I misrepresented any of the implications of Dirac's explanation of positrons as holes in an infinite sea of negative-kinetic- energy electron states? What I have tried very hard to do is nothing more than make a list of the implications of a physics idea, just as people do all the time on this group. Who said it should not really be the issue, not if we are talking about an unelaborated _explanation_ rather than the math itself. Second question: If someone had proposed a theory in this forum like the one I just described, _and you had never heard of it before_ , what would you have thought of it? Please be honest. My point in all of this obviously is this: While Paul Dirac's amazing equations (they really are) managed to predict antimatter, his _explanation_ for why his equations require antimatter is... shall we say incompletely analyzed, to put a nice spin on it? A final thought: Has anyone ever seriously tried to make Dirac's _conceptual_ negative energy sea ideas, the ones that he espoused in his Nobel Lecture, into a real, working theory? And if so, how did they deal with the various issues I described above? (Me, I just think antiparticles are regular particles moving backward in time. Yeah, that's a pretty weird idea too, I know...)"}
{"id":"6084","title":"Dimensional reduction from $3+1$ to $2+1$ for $\\cal{N}=2$ vector superfield","text":"Let the supersymmetry transformations for the chiral multiplet $(z_k,\\psi_{kL},f_k)$ be, $\\delta z_k = 2i \\bar{\\alpha} \\psi_{kL}$ $\\delta \\psi_{kL} = D_\\mu z_k \\gamma ^\\mu \\alpha_R + f_k \\alpha_L$ $\\delta f_k = 2i\\bar{\\alpha} \\gamma^\\mu D_{\\mu} \\psi_{kL} - 2ie\\bar{\\alpha}\\lambda _R z^k$ Similarly let the gauge multiplet $(A_\\mu,\\lambda,D)$ transform as, $\\delta A_\\mu = i \\bar{\\alpha}\\gamma_\\mu \\lambda$ $\\delta \\lambda = -\\frac{1}{2}F^{4}_{\\mu \\nu}\\gamma_4 ^{\\mu \\nu} \\alpha + D \\gamma_5 \\alpha $ $\\delta D = i \\bar{\\alpha}\\gamma_5 \\gamma ^ \\mu \\partial _ \\mu \\lambda$ Here in $F^4$ the $\\mu$ and $\\nu$ go over indices $0,1,2,3$ and $\\gamma_4 ^{\\mu \\nu} = \\frac{1}{2} [\\gamma_4^\\mu, \\gamma_4^\\nu]$. (these gamma matrices are defined below) One does something called \"dimensional reduction\" of these to $2+1$ dimensional spacetime by assuming that the fields are independent of the third spatial coordinate. In $2+1$ dimensional spacetime the gamma matrices are defined as, $\\gamma ^0_3 = -i\\sigma^2$, $\\gamma^1_3 = \\sigma ^3$ and $\\gamma^2_3 = \\sigma ^1$ In the so called \"Majorana representation\" the $4-$dimensional Gamma matrices can be written such that $\\gamma_4 ^{0\\1\\2} = \\left [ \\begin{array}{cc} & \\gamma_3 ^{0\\1\\2} \\\\\\ \\gamma_3 ^{0\\1\\2} & \\end{array} \\right ] $ and $\\gamma_4 ^3 = \\left [ \\begin{array}{cc} I & \\\\\\ & -I \\end{array} \\right ]$ One renames the third component of the gauge field $A_3$ as $C$ and splits the $4-$component fermions into $2-$component spinors in $2+1$ dimensions as, $\\lambda = \\left [ \\begin{array}{c} \\lambda_1 \\\\\\ \\lambda_2 \\end{array} \\right ]$ $\\alpha = \\left [ \\begin{array}{c} \\alpha_1 \\\\\\ \\alpha_2 \\end{array} \\right ]$ One substitutes the above in the first set of supersymmetric transformations and sets to $0$ all derivatives with respect to the $3^{rd}$ spatial direction. Then one is supposed to get, $\\delta A_\\mu = i \\bar{\\alpha}_a \\gamma_\\mu \\lambda _a$ $\\delta \\lambda _a = - \\frac{\\epsilon ^{\\mu \\nu \\rho}F_{\\nu \\rho}}{2}\\gamma _ \\mu \\alpha_a + \\partial _\\mu C \\gamma ^\\mu \\alpha ^a + D \\alpha ^a$ $\\delta C = i\\bar{\\alpha}^a \\lambda _a$ $\\delta D = i \\bar{\\alpha}^a\\gamma ^\\mu \\partial_\\mu \\lambda_a$ where $a$ runs over $1,2$ and $\\alpha^1 = \\alpha_2$ and $\\alpha^2 = -\\alpha_1$ Doing the above I can get the above transformations for all the fields. (Just that I need to use a fact that is true only for these gamma matrices in the $3$-dimensions that $[\\gamma_3^\\mu,\\gamma_3^\\nu]=2\\epsilon ^{\\mu \\nu \\rho}\\gamma_{3\\rho}$)   * In this \"dimensional reduction\" the particular choice of the gamma matrices seemed crucial. Is this true?    * I don't know how but this is supposed to match the transformations of the $\\cal{N}=2$ vector multiplet in $2+1$ dimensions which has componets, $A_\\mu, \\lambda_a,C,D$ (why?)    * I don't know how one derives the field content (as above) of the $\\cal{N}=2$ vector multiplet in $2+1$ dimensions.    * From the above it is claimed that the following forms a lagrangian respecting the above symmetry and is what gets called the ``supersymmetric Chern-Simons theory\" $L = \\frac{\\kappa}{2} (\\epsilon ^{\\mu \\nu \\rho} A_\\mu \\partial _\\nu A_\\rho - i \\bar{\\lambda_a}\\lambda_a + 2CD)$ But to establish the above claim I need to assume two results,   * First is a property which again seems typical of $3$ dimensional gamma matrices that, $\\gamma_3^{\\mu \\dagger} \\gamma_3 ^0 = -\\gamma_3^0\\gamma_3^\\mu$    * Secondly I need to assume the following variation for the fermion self-coupling,  $\\delta (\\bar{\\lambda} \\lambda) = 2\\delta (\\bar{\\lambda}) \\lambda$ It is not clear to me as to why should the above hold.   * A crucial difference between the above and the usual Abelian supergauge theory is that the kinetic term for the fermion is missing. How does one understand this?"}
{"id":"97706","title":"Idea of integrable systems","text":"I do not quite understand the idea an integrable dynamical system. Does it mean that the EOMs are analytically and exactly solvable? What are the necessary and sufficient conditions such that a system is integrable?"}
{"id":"131351","title":"What is the motivation for the definition of a manifold?","text":"In Wald's _General Relativity,_ an $n$-dimensional $C^{\\infty}$ manifold $\\mathit{M}$ is defined as a set, with subsets $\\lbrace{O}_{\\alpha}\\rbrace$, which satisfies 3 properties. In particular, the third one (abbreviated) is: > _If any two sets $\\mathit{O}_{\\alpha}$ and $\\mathit{O}_{\\beta}$ overlap, we > can consider the map $\\psi_{\\beta}\\circ \\psi_{\\alpha}^{-1}$, which takes > points from subset $\\psi_{\\alpha}[\\mathit{O}_{\\alpha} \\cap > \\mathit{O}_{\\beta}]$ to points in subset $\\psi_{\\beta}[\\mathit{O}_{\\alpha} > \\cap \\mathit{O}_{\\beta}]$. We require these subsets of $\\mathbb{R}^{n}$ to > be open and this map to be $\\mathit{C}^{\\infty}$._ My question is: what is the purpose of this clause in the definition of a manifold? It appears to allow the existence of a transformation from different coordinate systems (from $\\psi_{\\alpha}$ to $\\psi_{\\beta}$). It requires the existence of an inverse. Other than that, is there any other motivation for defining the manifold in this way?"}
{"id":"43816","title":"The definition of entropy in quantum mechanics","text":"I have seen entropy with several different definitions. Like Von Neumann entropy and Rényi entropy, etc. So I am curious why there are so many different definitions in quantum mechanics while only one in classical mechanics named after Boltzmann?"}
{"id":"92089","title":"Forces in a balloon popping?","text":"How much difference between internal and external pressure can a balloon withstand? how would the value be calculated? I found an equation: http:\/\/library.thinkquest.org\/C003758\/Function\/laplacelaw.htm But, how does that relate to a material?"}
{"id":"100375","title":"Does our current notion of velocity\/acceleration (based on calculus) require absolute time?","text":"When people explain special relativity, they say that a velocity in one frame leads to time dilation in that frame relative to a stationary frame. When we say something like $v=\\mathrm{d}x\/\\mathrm{d}t$ or $a=\\mathrm{d}v\/\\mathrm{d}t$. Isn't dt something that cannot change? A smallest infinitesimal unchanging amount. Newton described it in terms of absolute space and time. How does it make sense to define a velocity of one frame as $\\mathrm{d}x\/\\mathrm{d}t$ yet say that the rate of time itself changes? Can $\\mathrm{d}t$ change?"}
{"id":"70171","title":"Position vector, length\/modulus function and differentials","text":"If $\\vec r$ and is the position vector of a point in motion and $r$ is its length\/modulus\/magnitude\/size, then: Can it be true that: $$\\|\\mbox{d} \\vec r\\| \\neq \\mbox{d}r? $$ I think that this is true in circular motion, but if this question is generic, i.e., for almost all cases, then do you think this is correct to say? **EDIT:** Thanks for the very prompt answers and comments, but please tell me, if this question is given in an exam, which option would you mark? (As a student, I am very interested in understanding physics, but at the end of the day, I also have to think of the exam point of view...)"}
{"id":"70179","title":"Time Reversal Operator","text":"I know that time reversal operator is an antiunitary operator. How does it work on wavefunctions? I believe in this way: $$T \\psi (k,+)=e^{i\\pi S_y\/\\hbar} K \\psi (k,+) = \\psi^*(-k,-),$$ but I am not sure. (\"+\" and \"-\" represent spin-up or spin-down states) Does anybody have a good explanation?"}
{"id":"4411","title":"Significances of energy conditions","text":"I know about these different energy conditions in GR, namely strong, weak and null, but never really understood the full physical significance of them or for example how to 'derive' them or how compelling is a certain condition (for that particular one to be applied). I know that they exist because you need some constraints for $T_{\\mu\\nu}$ to satisfy as otherwise its completely unconstrained. Can someone please elaborate?"}
{"id":"66765","title":"Diffraction-Long versus short waves","text":"Why is it that long waves diffract more than short waves? I know that refraction has a quantum mechanical explanation, does diffraction have a similarly complex explanation? My teacher in high school said that using ocean waves would help me understand-but it didn't. Is there something in the math, or some related physics concept that can help me better understand diffraction? Related Question: Bass and Treble-Car Steroes"}
{"id":"122052","title":"Does a hydrogen atom today have same mass as a hydrogen atom in the future?","text":"Does an atom of hydrogen today have the same rest mass energy as an atom of hydrogen a billion years in the future? Standard cosmology seems to tacitly make this assumption. But surely one can only make this assumption if the FRW metric has a time-like Killing vector? From V.Moretti's answer to my previous post the FRW metric does not have a time-like Killing vector so that we cannot assert that the energy of an atom of hydrogen is constant over cosmological time. Is this reasoning correct? I'm sorry to keep bringing this question up but I think it's an important point that is overlooked in current cosmology. **Answer to responses** I guess I am following the spirit of Sean Carroll's Energy is not conserved Blog post where he says: > When you thought energy was conserved, there was a reason why you thought > that, namely time-translation invariance. A fancy way of saying “the > background on which particles and forces evolve, as well as the dynamical > rules governing their motions, are fixed, not changing with time.” But in > general relativity that’s simply no longer true. Einstein tells us that > space and time are dynamical, and in particular that they can evolve with > time. When the space through which particles move is changing, the total > energy of those particles is not conserved. > > Energy and momentum evolve in a precisely specified way in response to the > behavior of spacetime around them. If that spacetime is standing completely > still, the total energy is constant; if it’s evolving, the energy changes in > a completely unambiguous way. According to the FRW metric spacetime is evolving so surely the energy of particles cannot be assumed to be constant? A hydrogen atom today does not necessarily have the same energy as it did a billion years ago or will have a billion years in the future. Taking the spatially flat FRW metric in Cartesian co-ordinates for simplicity we have: $$ds^2 = -dt^2 + a^2(t)(dx^2+dy^2+dz^2)$$ Now from this representation of the FRW metric one might conclude that it is only the spatial co-ordinates that are evolving. In fact we can transform to conformal co-ordinates: $$ds^2 = a^2(\\tau)(-d\\tau^2 + dx^2 + dy^2 + dz^2)$$ where $d\\tau = dt\/a(t)$. Now we can clearly see that spacetime is evolving so we cannot assume that the energy of massive particles is constant. However according to V.Moretti massless particles like photons that are described by conformally invariant fields do not have their energy changed by a conformally changing metric as described above. Thus we can rigorously argue that photon energy is constant if we are in fact moving forward in conformal time $\\tau$. So an increasing length scale is associated with a constant energy. Thus a massive particle with a _fixed_ length scale is associated with an energy that increases with the scaling factor $a(\\tau)$. In seems that the energies of all massive particles must scale with $a(\\tau)$ including the Planck mass. In fact I suppose as all particle masses are ultimately linked to the Planck mass then the the scaling particle masses is ultimately caused by the Planck mass increasing."}
{"id":"74952","title":"volume of phase space of composite microcanonical ensemble","text":"Let $N_1$ be number of particles in volume $V_1$ with momenta and coordinates $(p_1, q_1)$ and $N_2$ be particles with momenta and coordinates $(p_2, q_2)$ in $V_2$, if $E_1, E_2$ be the energies of subsystems satisfying $E< E_1 + E_2 < E + 2 \\Delta$ with $\\Delta << E$. My book says that > Obviously the volume of the region of $\\Gamma$ space that corresponds to > conditions with total energy lying between $E_1 + E_2$ and $E_1 + E_2 + 2 > \\Delta$ is $\\Gamma(E_1) \\times \\Gamma(E_2)$ Why is the volume of phase space multiplied? Shouldn't it be added?"}
{"id":"66768","title":"Equation for null geodesic around schwarzschild metric?","text":"I'm trying to find the path of a photon around the Schwarzschild black hole, given its initial conditions. After much tribulation, I've basically given up on solving the equations by myself. I just need equations which I can integrate for: $$ r(\\lambda) = \\text{function(initial direction, initial position)} $$ $$ \\theta(\\lambda) = \\text{function(initial direction, initial position)} $$ $$ \\phi(\\lambda) = \\text{function(initial direction, initial position)} $$ $$ t(\\lambda) = \\text{function(initial direction, initial position)} $$ given the initial conditions of the light ray. Surely this has been solved before?"}
{"id":"66769","title":"Origin of Charge Neutrality","text":"What is the origin of the \"charge neutrality\" requirement in solids? Why do we require the bulk to be charge neutral, yet the surface can have a net charge?"}
{"id":"87257","title":"Question about Majorana fermions","text":"I have a few questions about Majorana fermions.   1. What is Majorana mass? Does it have a different value compared to the mass in the Dirac equation for an arbitrary fermion? How exactly do they differ?   2. Can the Majorana equation be rewritten in form two-component spinor equation? Or it is two-component already?"}
{"id":"87250","title":"Single photon double slit experiment","text":"In the question Double Slit experiment with just one photon or electron, one of the answers says > There have been experiments recently where one can detect which the slit the > particle went through and still the interference pattern appears. This had 6 votes. Is it correct? If you put a detector to find out which way the particle went through won't that destroy the interference?"}
{"id":"53321","title":"Photons, no energy?","text":"I am only 14, so I don't know much about physics, and I would appreciate help on this topic. A photon, is said to have 0 mass. However, they carry energy, as electromagnetic radiation. Albert Einstein proved that $E = mc^2$. But if you plug the data in, $E = 0\\times (299791)^2$, you try to multiply the speed of light squared by $0$. Therefore, the energy of a photon would be $0$. How is that possible? Since all the photon does is carry energy?"}
{"id":"41502","title":"Compute closed line integral of electric field in circuit","text":"I have a circuit where resistor is parallel to capacitor, which is charged with voltage U. How to compute line integral around closed loop to get the result of Kirchhof second law - $U_{capacitor}+I\\cdot R=0$? For beginning I splited it in two parts, starting from the capacitor positive plate through resistor:   $$\\oint \\vec{E} \\cdot d \\vec{l} = \\int_{+R-}\\vec{E_{R}} \\cdot d \\vec{l} + \\int_{-C+} \\vec{E_C} \\cdot d\\vec{l}$$ Since intensity of resistor is with the same direction as current density $\\vec{j}$: $$\\int_{+R-}\\vec{E_{R}} \\cdot d \\vec{l} = \\int_{+R-} \\rho \\cdot \\vec{j} \\cdot d \\vec{l} = I \\cdot R$$ When I do the same for capacitor I gain negative line integral, because the $E_{c}$ is opposite to $d \\vec{l}$ when positive charge is moved from negative plate to the positive. So in which part I have made a mistake?"}
{"id":"132982","title":"Unequally charged hanging pith balls","text":"I have done a problem which asks us to find the charge on two equally 'massed and charged' pith balls which are left hanging on a string with a certain length that repels each other and attains an equilibrium point making an angle (the string) with the vertical. The problem was quite clear, what left me wondering was: Won't the angle be the same if the balls are unequally charged (but equally massed) ? Or does it deviate from the initial case?"}
{"id":"41509","title":"Divergence of non conservative electric field","text":"I'm looking for the proof that the 1st Maxwell equation is valid also on non conservative electric field. When we are talking about a electrostatic field, the equation is ok. We can apply the Gauss (or Flux) theorem and get Gauss' law: $$\\mathbf{\\nabla} \\cdot \\mathbf{E} ~=~ \\frac{1}{\\epsilon _0} \\rho (x,y,z).$$ The question is, why when there is a time dependent magnetic field, and then a time dependent (non conservative) induced electric field, the 1st Maxwell equation is the same? How we can prove that?"}
{"id":"39617","title":"What are the applications of Gauss's law in technology?","text":"Freshmen physics textbooks use Gauss's law plus symmetry to calculate the electric field. I was wondering if this method of finding the electric field using a symmetry is used in real applications in life, science, research, or technology. For example do researchers use symmetry to find the electric field due to a charged virus that is spherically symmetric, or a charged DNA that has some kind of symmetry or charged symmetrical objects in nanotechnology. I am looking for such specific applications."}
{"id":"39611","title":"What is the meaning of uncertainty in Heisenberg's uncertainty principle?","text":"The Heisenberg's uncertainty principle states the following: $$\\Delta p \\cdot \\Delta x \\ge \\frac{h}{4\\pi}.$$ While studying for my high school physics exams, I fooled myself into believing that I understood the uncertainty principle (at least the implications). But suddenly the question that's nagging me is the following. If the uncertainty $\\Delta x$ of an electron is 1.2 nm, does it imply that the probability that the x-coordinate lies within a 1.2 nm range, equal to 100%? Or does it mean that the probability is 95%? Or does it mean something totally different? I wonder why no author made it clear in the high school\/junior college level textbooks. I am uncertain about what uncertainty means."}
{"id":"37660","title":"Projectile Motion","text":"I know the angle at which a projectile is launched, how far it needs to go, and also the maximum height. How can I find the initial velocity needed (disregarding air resistance)? Currently, I am using: $Vf^2=Vy^2-2g\\Delta y$ From that, I can find the velocity in the y direction needed for the projectile to reach the known maximum height. Next, I am finding the time it takes to reach that maximum height by using the found y velocity."}
{"id":"108887","title":"How does a lightning striking stall a moving car?","text":"I am currently an undergraduate student leading a discussion section for an introductory electricity and magnetism course. I was asked this question that I was unable to answer and I am hoping for some help in understanding what is going on. I was told by this student that he and his girlfriend were driving along a highway when lightning struck the roof of their car. Although they were unharmed by the lightning (Faraday cage), my student claimed that his car stalled and was unable to turn it back on right away. However, after about 5 to 10 minutes, the car started up immediately. So my question is **how does lightning striking the roof of a car stall a moving car?**"}
{"id":"89903","title":"Thermographic camera vs infrared camera","text":"AFAIK, all objects above 0K emit infrared radiation. It is an electromagnetic radiation, like visible light. I guess thermographic and infrared cameras work on the same principles, but their images are very different, the former is false color and very different from the visible image, the latter is black and white and usually correlates to the actual visible image, at least that's what I've seen so far. So physically, how exactly are they different?"}
{"id":"108883","title":"A single valued function from a multi-valued function","text":"In Schrieffer's book \"Theory of Superconductivity\", there is said when he deals with multiple connected superconductors (and discuss London equations), that if one takes the line integral of the potential vector on the boundary of a hole where there is a magnetic flux inside we have : $$\\int{\\vec dl\\cdot \\vec A} = \\Phi$$ Then, if we go inside the superconductor ($\\vec B=0$ thus $\\vec A = \\nabla \\chi$ where $\\chi$ is a scalar function) and again take the line integral of $\\vec A$, we deduce that $\\chi$ must be multi-valued : $$\\int{\\vec dl\\cdot \\vec A} = \\Delta \\chi = \\Phi$$ But it is stated that now, if the magnetic flux is known in each hole, $\\vec A$ is determined and the physics is saved. But for me, it is not clear that if we know how much $\\chi$ raises with every $2 \\pi$ rotation around a hole, we don't know how it varies... Could someone help me on this ?"}
{"id":"30033","title":"Sound Produced due to motion of air inside a tube","text":"I'm trying to understand the sound produced by Air motion inside the respiratory system. If you have a long tube with a suction apparatus at one end , You turn on the apparatus and the air is sucked in. Now Sound waves are produced due to vibration of the molecules in the wall so that each molecule behaves like a harmonic oscillator and the whole is a system of coupled oscillators . I have some questions. If The air inside the tube is highly collaminated in its motion , That is it has zero component of velocity perpendicular to the axis , The there will not be any sound waves that are produced . If there is a very small inhomogeinity at the beginning then inside the tube will these inhomogenities be amplified ? How does The Frequency and intensity depends on the velocity of air , The diameter of the tube and the degree of disturbance in the motion of air ?(I expect that sound will be a linear combination of waves with different frequencies , the more the turbulence , the more terms will be in the series expansion , The less diameter the more velocity and subsequently the bigger values of frequencies that appear in the series) Will sound waves emitted have the same frequency along the tube, that is the velocity of and kinetic energy of air increases as they move along the axis and subsequently the energy of sound will increase or may be the higher velocity the more turbulence that occurs and subsequently sound waves will be a linear combinations of many waves with different frequencies ? Why is the sound produced from a cavity amplified and has low frequency?(I expect this is due to turbulence , but in this case why does turbulence amplify sound?, I think that the more turbulence , the more frequencies produced and when you add them together , the amplitudes will tend to vanish for some frequencies and increase for other frequencies , right? ) Lastly , What exactly is the pitch of the sound and how is it related to the frequency?"}
{"id":"8074","title":"Why is the Earth so fat?","text":"I made a naive calculation of the height of Earth's equatorial bulge and found that it should be about 10km. The true height is about 20km. My question is: why is there this discrepancy? The calculation I did was to imagine placing a ball down on the spinning Earth. Wherever I place it, it shouldn't move. The gravitational potential per unit mass of the ball is $hg$, with $h$ the height above the pole-to-center distance of the Earth (call that $R$) and $g$ gravitational acceleration. Gravity wants to pull the ball towards the poles, away from the bulge. It is balanced by the centrifugal force, which has a potential $-\\omega^2R^2\\sin^2\\theta\/2$ per unit mass, with $\\omega$ Earth's angular velocity and $\\theta$ the angle from the north pole. This comes taking what in an inertial frame would be the ball's kinetic energy and making it a potential in the accelerating frame. If the ball doesn't move, this potential must be constant, so $$U = hg - \\frac{(\\omega R \\sin\\theta)^2}{2} = \\textrm{const}$$ we might as well let the constant be zero and write $$h = \\frac{(\\omega R \\sin\\theta)^2}{2g}$$ For the Earth, $$R = 6.4*10^6m$$ $$\\omega = \\frac{2\\pi}{24\\ \\textrm{hours}}$$ $$g = 9.8\\ m\/s^2$$ This gives 10.8 km when $\\theta = \\pi\/2$, so the equatorial bulge should be roughly that large. According to Wikipedia, the Earth is 42.72 km wider in diameter at the equator than pole-to-pole, meaning the bulge is about twice as large as I expected. (Wikipedia cites diameter; I estimated radius.) Where is the extra bulge coming from? My simple calculation uses $g$ and $R$ as constants, but neither varies more than a percent or so. It's true the Earth does not have uniform density, but it's not clear to me how this should affect the calculation, so long as the density distribution is still spherically-symmetric (or nearly so). (Wikipedia also includes an expression, without derivation, that agrees with mine.)"}
{"id":"44953","title":"Why do the drift and diffusion components cancel for each type of carrier if EHP generation plays such big role in p-n-junctions?","text":"I have always argued to myself that drift and diffusion components of the current though a p-n-junction cancel for each type of carrier because any electron diffusing from n into p will sooner or later require the same electron to drift back. (And analogously for holes). However, it now seems to me that the generation of EHPs (electron-hole-pairs) profoundly contributes to the drift current: > \"The supply of minority carriers on each side of the junction required to > participate in the drift component of current is generated by thermal > excitation of electron-hole pairs.\" (In \"Solid-State-Electronic Devices\" , > Streetman and Banerjee) In thermal equilibrium, that implies an equal amount of recombination events. Anyway, one can imagine an EHP being generated on the p-side, with the electron drifting and the hole diffusing to the n-side, where thy recombine. In that scenario, there is no hole drift and no electron diffusion, but hole diffusion and electron drift. So, how in earth does one dare to argue that \"the drift and diffusion components must cancel for each type of carrier\"? I feel like I am missing the real argument."}
{"id":"47901","title":"Clarifying what metric counts as flat space","text":"In (2D) Cartesian coordinates, the Euclidean metric... $$\\begin{bmatrix} 1 & 0 \\\\\\ 0 & 1 \\end{bmatrix}$$ ...is flat space. If the diagonal elements are exchanged for other real numbers greater or less than zero, would this still count as flat space even though it is no longer Euclidean? I tbink the answer is yes but just want to make sure. Also, when a physicist says that the local tangent space on a manifold is flat, do they imply that the metric is locally Euclidean, or can the diagonal elements be any non-zero real number in that local tangent space?"}
{"id":"72947","title":"Field Strength Renormalisation in Peskin and Schroeder","text":"In chapter 7 of Peskin and Schroeder they define the field strength renormalisation $Z$ for a quantum field to be the residue of the Fourier transform of the correlation function $$\\langle \\Omega | \\phi(x)\\phi(0) | \\Omega\\rangle$$ They go on to refer to $Z$ variously as $Z_1$, $Z_2$ and $Z_3$ at different points in the text. Is the subscript meant to denote the **order to which we've computed $Z$**? In other words do we have $$Z = Z_1 + O(e^2) = Z_2 + O(e^3) = Z_3 + O(e^4)$$ and so on? This would seem to make no sense because then naturally $Z_1 = Z_2$ due to the Feynman diagram stucture. Alternatively are they just meant to be $Z$s for different systems (Lagrangians)? Or are they just genuinely different quantities depending on context? I'm getting confused because P&S generally use subscripts to denote the order to which a quantity is valid, but it doesn't seem to be the case here! If anyone has the book and wants page references, see p215, 221 and 243."}
{"id":"38225","title":"Statistical Weighting Factor on thermal neutron importance","text":"The problem is concerning the use of a thermal fluxed squared weighting factor in a thermal reactor. I have seen in sources the thermal flux in a reactor is squared as a statistical weighting factor, for example looking at temperature feedback. Say a number of thermocouples are measured in a reactor, the weighting of the importance of the thermocouples are to flux squared. Why is this when the reaction rate is equal to the flux? Is it simple statistics I'm not understanding."}
{"id":"72948","title":"A question about Lorentz invariance of the Polyakov action","text":"I have a super basic and stupid question about the Lorentz invariance of the Polyakov action (cannot skip the disclaimer..) $$S_p[X,\\gamma]=-\\frac{1}{4 \\pi \\alpha'} \\int_{-\\infty}^{\\infty} d \\tau \\int_0^l d \\sigma (-\\gamma)^{1\/2} \\gamma^{ab} \\partial_a X^{\\mu} \\partial_b X_{\\mu} $$ I may write the action as (using metric tensor $\\gamma^{ab}$) $$S_p[X,\\gamma]=-\\frac{1}{4 \\pi \\alpha'} \\int_{-\\infty}^{\\infty} d \\tau \\int_0^l d \\sigma (-\\gamma)^{1\/2} \\partial^a X^{\\mu} \\partial_a X_{\\mu} $$ The Lagrangian is obviously invariant under proper Lorentz transformation. $\\partial^a X^{\\mu} \\partial_a X_{\\mu}$ is a Lorentz scalar. $d\\tau d\\sigma$ transforms with a determinant equals 1 for proper Lorentz transformation. $(-\\gamma)^{1\/2} $ also transforms with a determinant equals 1. But for the range of integration $[0,l]$, there is length contraction under a boost. Why the action is still invariant under Lorentz transformation? (or I completely mistaken something....)"}
{"id":"18496","title":"Why did they omit the action-reaction pair force in this problem?","text":"**Problem** ![](http:\/\/img577.imageshack.us\/img577\/2205\/problemse.jpg)! **Solution** ![](http:\/\/img543.imageshack.us\/img543\/858\/solutionq.jpg)! **My solution** ![](http:\/\/img853.imageshack.us\/img853\/4305\/solution2.jpg)! Let me explain $n_{bg}$ means the force exerted by the block on the ground. This is because the normal force of the slab exerts (or touches) a force on the block and by Newton's Third Law, it repels with a force n. So $n_{ground}$ is actually a force exerted by the ground and the force exerted by the ground on the block Also I kinda intuitively knew to put a friction force on the slab on the right. Is this a consequence of Newton's third Law? As the block slides to the right, the friction force drags it from the left. On the slab, (imagining now) wouldn't the heat\/friction be sliding on top from the left?"}
{"id":"81506","title":"traditional transformer design improvement","text":"The traditional tranformers suffer a lot of losses in various forms, some of them are as follows : 1\\. Flux leakage 2\\. Eddy currents 3\\. Hysteresis Figure : ![Figure](http:\/\/lh3.ggpht.com\/-8_dBB49n_3Q\/UmPS6S2qMUI\/AAAAAAAAAd4\/FQy4MSOMjG8\/s1600\/2013-10-18%25252020.04.41.png) But as we are taught induction we come across the attached figure of two inductors placed coaxially various times. My point is that if we would actually place inductors in this way, we would have the following benefits : Note : taking inner coil as primary   1. NO NEED OF CORE : As the flux of inner coil passes from outer on its own, no core is needed for transferring flux from primary to secondary   2. NO FLUX LEAKAGE : As entire flux of inner inductor passes from outer, no flux is leaked.   3. NO EDDY CURRENTS: As no core is used no eddy currents are generated.    4. NO HYSTERESIS : Similarly due to absense of core, no loss due to hysteresis is encountered. So what really is the problem in creating transformers in such a way to reduce losses much more than the current status ? All relevant opinions and comments appreciated My original post : http:\/\/thinklo.blogspot.in\/2013\/10\/improving- transformers.html"}
{"id":"19014","title":"Why do physicists believe protons and electrons are present in equal numbers?","text":"I tended to consider that negative and positive charges are present in equal numbers in the universe to be a known, obvious fact. But is it so? How can we rule out the possibility that there is some kind of asymmetry in the numbers of protons and electrons? Of course, matter is neutral and even tiny deviations would give rise to enormous forces... But not all protons and electrons are in atoms in the universe. More importantly, is there some conceptual reason why we believe that this equality is exactly perfect or do modern theories allow tiny deviations?"}
{"id":"89277","title":"Energy loss of Earth","text":"How does the Earth lose the energy that it gains from the sun's radiation if it is surrounded by - and in contact with - nothing?"}
{"id":"96410","title":"Maxwell's equations and wave equations","text":"As can be proven, Maxwell's equations don't work for spherical or cylindrical waves, but these waves satisfy the wave equation. Are there any other examples of this type that satisfy the wave equation but not Maxwell's equations?"}
{"id":"69077","title":"Principle of Least Action","text":"Is the principle of least action actually a principle of least action or just one of stationary action? I think I read in Landau\/Lifschitz that there are some examples where the action of an actual physical example is not minimal but only stationary. Does anybody of know such a physical example?"}
{"id":"45663","title":"Is proper time renormalization gauge invariant?","text":"Proper Time Renormalization is achieved by putting: $$ \\int_0^\\infty e^{iat} dt = {1\\over ia} $$ Is it true that this is the only kind of normalization that is gauge invariant? If so, why do famous books ignore it even though it has such an important property (gauge invariance)? Does the reason behind the name have anything to do with this?"}
{"id":"52300","title":"Factors of $c$ in the Hamiltonian for a charged particle in electromagnetic field","text":"I've been looking for the Hamiltonian of a charged particle in an electromagnetic field, and I've found two slightly different expressions, which are as follows: $$H=\\frac{1}{2m}(\\vec{p}-q \\vec{A})^2 + q\\phi $$ and also $$H=\\frac{1}{2m}(\\vec{p}-\\frac{q}{c}\\vec{A})^2 + q\\phi $$ with $\\vec{p}$ the momentum, $q$ the electric charge, $\\vec{A}$ the vector potential, $\\phi$ the scalar potential and $c$ the speed of light. So basically the difference is in the term $1\/c$ multiplying $\\vec{A}$, present in the second form (which I use in my lectures) but not in the first one (used by Griffiths in _Introduction to quantum dynamics_ to treat the Aharonov-Bohm effect). Why does this difference exist and what does it mean? And how does the term $1\/c$ affect the dimensional analysis (the units) of the problem?"}
{"id":"17317","title":"Book against a wall and forces","text":"If you take a book with mass of 1kg and push it against the wall. With how much force do you have to push the book so it does not fall? The problem is I know how to calculate this problem, you say $F_{friction}$=$F_{gravitational}$ and $F_{wall \/ normal}$ = $F_{human \/ push}$. The problem is solvable if you say that $F_{gravitational}$ = $F_{normal \/ wall}$, but why is this true? How would you calculate this problem if you didn't know that $F_{normal}$ = $F_{gravitational}$, how would you prove this statement? For me the problem is that $F_{normal}$ = $F_{gravitational}$ * $\\cos(\\alpha)$, but cos(angle) is 0, I don't understand the relationship between $F_{gravitational}$ and $F_{normal}$ in this scenario."}
{"id":"105413","title":"Why is there no massless spin 1\/2 particles with charge? Is there any underlying theory?","text":"Whether any massless spin 1\/2 particles cannot have charge? If so, why?"}
{"id":"69502","title":"Are there (or could there be) electrically charged particles that move at the speed $c$?","text":"A photon, a neutrino (if it has zero rest mass) move at $c$ but what about charged particles? If the answer is no, is there a fundamental reason or just because of the radiation it emits?"}
{"id":"7906","title":"How might a resonant antenna and black body radiation interact?","text":"How does an antenna behave when it is cooled so that its black-body radiation is emitting energy at its resonant frequency? Edit: To clarify, its not how they're related in general, but how might thermal radiation and resonance interact with each other when their spectra are aligned well? Edit: Also, I'm sure that the thermal radiation spectra that have a significant peaks are associated with incredibly high temperatures, and peak at incredibly small wavelengths, rendering such an antenna completely impractical to build. Still, I'm still interested in the theoretical concept."}
{"id":"63299","title":"Air pressure relative to a force on a bag?","text":"Assume an airtight bag occupied by air such that the pressure inside the bag is equal to the atmospheric pressure. Assume the surface tension of the bag is negligible. What is the change in air pressure inside the bag relative to a weight placed on it? Is it possible to approximate the weight of the object on top of the bag by the air pressure inside the bag, without knowing the volume of the bag or the area of the object?"}
{"id":"46943","title":"How would natural (resonant) frequencies affect amplitudes?","text":"I read $y=A\\sin(2\\pi ft)$, where $A$=Amplitude, $f$=Frequency, $t$=Time and $y$=$Y$ position of the wave. Since natural frequencies only take the most effect when they are close to the frequency. How would one natural frequency and several natural frequencies affect the equation? Would I be correct in thinking it's something to effect of: `y=Y_Position*NaturalFrequency` where Y_Postion is the first equation and NaturalFrequency is similar to the first equation but with a low amplitude?"}
{"id":"46941","title":"Hamiltonian of the charged current in SM (related to the lorentz invariance)","text":"recently when I was studying the scatterings which involves a vector boson (like W boson) as an intermediate particle, I saw that the propagator is not Lorentz invariant, I read that there is another source of Lorentz non- invariance in the Hamiltonian of the charged current. I mean the Hamiltonian is not simply the minus of the Lagrangian in this case h=-l. the extra term is not Lorentz invariant and, these two source of the Lorentz non-invariance will cancel each other eventually in the S-Matrix. (you can see :The standard Model, A primer written by Burgess chapters 4,5 and problem 5.4). I don't understand this completely, I'll be grateful if anyone can help me and explain it or introducing a reference for it. I apologize for my bad English writing. Thanks"}
{"id":"21449","title":"Physics for taffy pulling?","text":"I am creating a simulation and am interested in pulling stretchy things and when they break, like taffy. I imagine this is a bit tougher then a simple equation like gravity, but I have no idea. Is there a general equation for an objects threshold and pulling it apart?"}
{"id":"63291","title":"Could entropy explain dark energy?","text":"This was 3rd beer idea, so please bear with me. What if the universe was not actually expanding but the speed of light was slowing? Wouldn't that be indistinguishable to our observations? Either way we should see an increase in the redshift the further a light source is from us? Suppose we're in a giant extremely advanced compute cluster, and the speed of light has to do with how long it takes data to travel between compute resources on the network\/bus. As more data travels across this bus (entropy increases in the universe) it takes longer to reach its destination (speed of light slows.) This seems vaguely reasonable considering fundamental limitations past and present on computers of varying types, there's always a trade off between size of data and speed of access that has it's roots in physics. So I know it's a bit wacky, but it sounds easily falsifiable to me, simply plot entropy over time and dark energy over time and see how well they correlate (or don't most likely.) Can anyone help me test my armchair theory? I think that would be a fun exercise. I know dark energy has been plotted over time, but what about entropy, is that possible?"}
{"id":"41990","title":"Basic question on experimental plots","text":"On the following Higgs $\\rightarrow$ Tau Tau plot, since we are plotting the ratio of $\\frac{\\sigma}{\\sigma_{SM}}$ on the y axis, shouldn't the expected for this be 1? i.e., shouldn't the expected 68% and 95% be centered at a dotted line at 1? Anything else seems to imply we are expecting something other than the Standard Model... ![Higgs ->Tau Tau](http:\/\/i.stack.imgur.com\/dffNF.png)"}
{"id":"13117","title":"Does tea stay hotter with the milk in it?","text":"A little thought experiment, similar to this one: Imagine you are making a cup of tea when the door bell rings. You've poured the boiling water into a cup with a teabag in it. As you're just about to pour milk in to the mix, the door bell rings. My question is: what will keep the resulting drink hottest after my chat with the person at the door: quickly pouring the milk in, or leaving it in the carton until after the interruption ends? Being a bit of a mathematician I'm happy to have the questions answered with a formula in terms of $t$ the length of the interruption, $m$, the temperature of the cold milk, $w$ the temperature of the hot water, $a$ the ambient temperature etc. I guess you'll have to use a model of how an insulated body loses\/gains heat... feel free to use simplifying assumptions like the environment stays the same temperature etc. My starting point from a physics point of view, is that heat is lost (or gained) proportional to the temperature difference between the object and the environment. So I imagine that would create some sort of exponential curve?"}
{"id":"81800","title":"How is the speed of light an absolute maximum speed in the universe?","text":"Under the heading _which came first, the chicken or the egg,_ which came first: the maximum velocity of electrons orbiting the atom, or the speed of light as the maximum velocity limit? Is it possible that when an object reaches the velocity at which its forward velocity equals its electron(s) orbital speed around its nucleus, then its advancing electrons would cease to spin around their orbit, lose their centrifugal force, and collapse into its nucleus, and the retreating electrons would lose their forward speed and the atom would run away from them, and shed them as radiation? Could this limit the maximum velocity at which a mass can travel, and this define the speed of light? If this is true, then at the speed of light its mass would not become infinite, in according to Einstein, but the mass would transmute into pure energy, in other words, become light or some other radiation frequency."}
{"id":"54078","title":"Ratio of distance between mirror and person","text":"In perspective of a given example, if a man was to stand $2\\ m$ away from a mirror which was $0.9\\ m$ in height and was able to see his full reflection, what would the height of the mirror have to be if the man was now $6\\ m$ away from the mirror and was to maintain a full reflection? Would the mirror be equal to or less than the original height and why? This scenario seems to have caught me out multiple times. So, some reasoning would be much appreciated."}
{"id":"94424","title":"General Thermodynamic equation of state","text":"I heard my professor saying that the equation $$ PV = \\frac{2}{3}U $$ is valid for any **non-relativistic** gas, be it Ideal or Real gas(includes quantum ideal gases). Is this true, If it is how can we prove with all generality ?"}
{"id":"19562","title":"Why (infra-)red light is not used in diagnostics?","text":"I noticed that human body conducts red visible light quite well. A hand placed over a powerful red lamp seems to be semi-transparent while over any other color it does not. Possibly the light conductivity in near infra-red range is even better. So my question is why (infra-)red light is not used for diagnostics instead of harmful X-rays?"}
{"id":"67532","title":"Dynamic equation","text":"I'm asked to give the dynamic equations of a charged particle moving on a potential, under relativistic considerations, basically $dp\/dt$, with $p$ the 4-momentum vector. The question for the three space coordinates is simply $d\\vec p =-q\\nabla\\Phi$, being $\\Phi$ the potential, that is: $$\\frac{d(\\gamma m\\vec v)}{dt}=-q\\nabla\\Phi$$ For the first term of the four vector, the answer given is this, and I have no idea where this comes from: $$\\frac{d(\\gamma mc^2)}{dt}=q(\\vec E\\cdot\\vec v)$$, being $\\vec E$ the electric field. Where does this come from? IS it even correct, isn't the first term of the four momentum vector just $\\gamma mc$ instead of $\\gamma mc^2 $"}
{"id":"128467","title":"Electromagnetic Inductance: Different voltmeter readings from different positions","text":"In the setup below, the voltmeter on the right would read differently than the voltmeter on the left even though they are both connected to point D and point A. ![enter image description here](http:\/\/i.stack.imgur.com\/3yrq7.png) This picture is taken from Lecture 16 of MIT course 8.02. It shows a circuit with a solenoid in the middle. The magnetic field of the solenoid is pointing out of the blackboard and at the moment it is turned on, there is an increase in magnetic field inducing a clockwise current in the wire. The EMF is given to be 1V. $ Given: V = 1 $   $ V=I(R_{1} + R_{2}) = I(100+900) = 1 $   $ \\Rightarrow I = 0.001 $ By Ohm's Law,   $ Clockwise: V_{D} - V_{A} = IR_{2} = 0.9 $   $ Counter clockwise: V_{D} - V_{A} = -IR_{1} = -0.1 $ So the Voltmeter on the right will read 0.9 while the left will read -0.1. This is shown in an experiment where two voltmeters connected to the same points A and D read different values. How is this possible? How does the voltmeter \"know\" whether it is on the left or right? It seems entirely symmetrical. How would the experiment need to be setup? If I were to just take a voltmeter and connect it to point A and D, how would I know what it would read?"}
{"id":"111285","title":"Finding current in a sphere while given a changing current density","text":"im trying to calculate the current running through a sphere while given a current density $$J=Jcos(2\\theta)e^{-t\/r} \\hat r $$ J is given in spherical coordinates. I know that: $$I=\\int J\\,da $$ But I can't seem to understand how to \"build\" my integral. I was thinking that i need to do something like $$ \\int r\\,dr\\,sin\\theta\\,d\\theta*J $$ but this gets very messy because of the exponent containing expression with \"r\". Any tips\/help please? Thanks"}
{"id":"101027","title":"Equilibrium of of water level in punctured bucket?","text":"Would a bucket with a hole in the bottom being filled at different rates with water come to to different equilibrium levels with in the bucket? Or would at any fill rate above initial flow rate of the hole cause the bucket to overflow? An answer with more math is preferred. **My thoughts:** If flow rates of water into bucket less than initial flow rate of hole then water level stays at or below height of hole in bucket. If water flow rates begin to increase, so does the depth of the water above the hole then since the product of the hole's size and pressure is positively proportional to flow rate, the water leaving and coming reach a equilibrium point with in bucket x height above hole. If water entering bucket becomes to fast then a equilibrium point in the size of the current bucket can not be reached, so the bucket over flows."}
{"id":"1307","title":"What happens if light\/particles exceeded the speed of light for a particular medium","text":"While the speed of light in vacuum is a universal constant (c), the speed at which light propagates in other materials\/mediums may be less than c. This is obviously suggested by the fact that different materials(especially in the case of transparent ones) have a particular Refractive Index. But surely, matter or even photons can be accelerated beyond this speed in a medium? If so, what are the effects?"}
{"id":"23758","title":"A dictionary of string - standard physics correspondences","text":"Motivated by the (for me very useful) remark ''Standard model generations in string theory are the Euler number of the Calabi Yau, and it is actually reasonably doable to get 4,6,8, or 3 generations'' in http:\/\/physics.stackexchange.com\/a\/22759\/7924 , I'd like to ask: Is there a source giving this sort of dictionary how to relate as much as possible from general relativity and the standard model to string theory, without being bogged down by formalism, speculation, or diluted explanations for laymen? Or, if there is no such dictionary, I'd like to get as answers contributions to such a concise dictionary. [Edit April 30, 2012:] I found some more pieces of the wanted dictionary in http:\/\/www.physicsforums.com\/showthread.php?p=3263502#post3263502 : ''branes and extra dimensions have proved to be implicit in standard quantum field theory, where they emerge from the existence of a continuous degeneracy of ground states. That multidimensional moduli space of ground states is where the extra dimensions come from, in this case! Branes are domain walls separating regions in different ground states, strings are lines of flux connecting these domain walls. Furthermore, in gauge theories with a small number of colors, it looks like the extra dimensions will be a noncommutative geometry, it's only in the \"large N\" limit of many colors that you get ordinary space.''"}
{"id":"126304","title":"About star groupings and there apparent positions in the sky","text":"Imagine a group of 3 stars that appear to be equally luminous ,star A, B, and C in the shape of an isosolese triangle. Imagine the 3 stars are in different $time-zones$. A,B and C look like they form an isosolese triangle right now on the Earth but when the light from star B reaches the Earth the light from star C still has say, 10 years of travelling to go. So when B is observed in its present position, the apparent position of C is from 10 years in the past. Where C is now relative to the present position of B might be different in such a way so as to not to appear to form an isosolese triangle. The 3 stars 'right now ' could form a zig-zag pattern as they appear in the sky but we might not 'see' that from the Earth for a 'while'. So should differences in many stars positions relative to the Earth be taken into account when the simultaneity or cause- and- effect of some stellar events are being analysed?"}
{"id":"16080","title":"Linearizing Quantum Operators","text":"I was reading an article on harmonic generation and came across the following way of decomposing the photon field operator. $$ \\hat{A}={\\langle}\\hat{A}{\\rangle}I+ \\Delta\\hat{a}$$ The right hand side is a sum of the \"mean\" value and the fluctuations about the mean. While I understand that the physical picture is reasonable, is this mathematically correct? If so what are the constraints this imposes? In literature this is designated as a \"linearization\" process. My understanding of a linear operator is that it is simply a homomorphism. I have never seen anything done like this and I'm having a hard time finding references which justify this process. I would be grateful if somebody can point me in the right direction!"}
{"id":"59244","title":"Lagrangian Coordinates in Fluid Flow","text":"I apologize if this is not the right place to ask this question: I am currently reading a paper by Y. Brenier, where for the fluid flow he introduces a Lagrangian label $a$ instead of the vertical coordinate $z$, and defines a \"Lagrangian foliation to be a family of sheets $z = Z(t,x,a)$, labelled by $a \\in [0,1]$, where $Z$ is a smooth function such that: $$ 0\\leq Z(t,x,a)\\leq 1, \\,\\,\\,Z(t,x,0) = 0, \\,\\,\\,Z(t,x,1) = 1\\\\\\ \\partial_a Z(t,x,a) > 0\\\\\\ \\partial_t Z(t,x,a) + u(x,t,Z(t,x,a))\\cdot \\nabla_x Z(t,x,a) = w(t,x,Z(t,x,a))\" $$ where $x \\in \\mathbb{R}^2$, $u = (u_1,u_2)$ is the horizontal flow velocity and $w$ is the vertical velocity and $\\nabla_x = (\\partial_{x_1},\\partial_{x_2})$. A possible initial choice is $Z(0,x,a) = a$. My question is: What is the physical interpretation for this? It seems like instead of labeling each individual fluid parcel in the classical Lagrangian coordinates, we are only labeling horizontal planes instead, and then seeing how these sheets move and deform with the flow?"}
{"id":"110884","title":"Calculation of error in focal length?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/P8Wsp.png) $u=-10cm$ $v=10cm$ ![enter image description here](http:\/\/i.stack.imgur.com\/wab5k.gif) Using the formula the focal length is 5 cm. But how do I get the fractional error in focal length when neither $\\Delta u$ nor $\\Delta v$ are specified? The options given are   A. (0.05±0.05) cm   B. (0.05±0.10) cm   C. (5.00±0.05) cm   D. (5.00±0.10) cm"}
{"id":"32543","title":"Emitting gravitational radiation","text":"Is the following true:   * Two massive bodies with variable distance between them do not emit GR in any direction   * Two bodies that revolve around common center will not emit in the plane of their orbits   * But they will emit in all other directions with the maximum along the axis of rotation perpendicular to the orbit's plane"}
{"id":"8331","title":"How is formula for converting pressure from mmHg to Pa derived?","text":"Today my younger brother asked me from where does the 1 Pa = 0.00750061683 mmHg formula for mercury barometer come. He needs a way to derive it, or an academic source which can be cited. After doing some calculations we got the formula for a standard U-tube manometer: $P=\\frac{h_2}{h_1}P_0$ where $P_0$ is atmospheric pressure, $P$ is pressure being measured, $h_1$ is height is mercury column exposed to atmospheric pressure and $h_2$ is height of the column exposed to pressure being measured. The problem is that in the case of a barometer, the $h_2$ is exposed to vacuum and I don't know how to use that. I've searched the Internet, and got countless sites which explain how a mercury column barometer works, but I was unable to find a site which explains which forces are acting there and how the number was derived. To make things even worse, none of the physics books I have access to have a detailed explanation."}
{"id":"111100","title":"A question on IR cancellation caculation in peskin schroeder","text":"In Peskin and Schroeder _Introduction to Quantum Field Theory_ book, above equation 6.64 on pg. 200, it was said that \"to gain better understanding, of the divergence, let us evaluate the coefficient of the log in the limit of $-q^2\\rightarrow\\infty$.\" Why does he want to take $-q^2$ to infinity rather than $0$ to extract the divergence? I thought the form factor $F_1(q^2)$ contains an infrared divergence at $q^2=0$."}
{"id":"56432","title":"Determine viscosity using falling sphere (Stokes Law, Ladenburg correction)","text":"## Introduction I am trying to determine the viscosity of a fluid. Therefore, I let a sphere of known mass `m` and radius `r` fall (sink) into the fluid. I then measure the time it takes to sink\/travel a given height. Then there are the following forces having a effect on the sphere: $F_g = mg$ and $F_a = \\frac{4 \\rho \\pi r^3 g}{3} $ (buoyant force) with $g = 9.81 \\ ms^{-2}$ and $\\rho$ being the density of the fluid. The result is a motion downwards (sinking), hence I get friction $F_r$. Assuming this friction gets that strong to have no resulting force on the sphere (i.e. its velocity remains constant), I can use the following equation: $F_g - F_a - F_r = 0$ According to Stokes Law, the friction of a moving sphere in a fluid with viscosity `n` is: $F_r = 6 \\pi r n v$ where `v` is the constant velocity. Using these equations, I can determine the viscosity: $n = \\frac{g}{6 \\pi r} (m - \\rho \\frac{4}{3} \\pi r^3)\\frac{1}{v}$ with $v = \\frac{s}{t}$. ## Question about Ladenburg correction The fluid is inside of a cylinder, so I cannot use Stokes Law \"as it is\" because the walls of the container add more friction. Hence I have to apply the so called Ladenburg correction with a correction factor $f = (1 + \\frac{(2.1 r)}{R}) (1 + \\frac{(3.3 r)}{H})$ When \/ where do I have to apply this factor? ## Question about extrapolation As I am doing this with spheres of different radiuses, I get multiple values for the viscosity:               radius\/mm | viscosity\/(kg\/(m*s))     --------------------------------     1.0       | 0.594     2.0       | 0.608     2.5       | 0.631      Extrapolation to $r = 0 \\ mm$ gives a value $n_0 = 0.575$. I do not understand: Why do I need this extrapolation?"}
{"id":"62155","title":"What Physical Quantity has SI Unit $kg \\cdot m$,?","text":"What is the physical quantity that has SI (System International) unit of $kg \\cdot m$ (kilogram meters)?"}
{"id":"12599","title":"Will a hole cut into a metal disk expand or shrink when the disc is heated?","text":"Suppose you take a metal disc and cut a small, circular hole in the center. When you heat the whole thing, will the hole's diameter increase or decrease? and why?"}
{"id":"26315","title":"Detecting dim objects","text":"What are the minimum conditions (switching times, sensitivity ...) for a camera to detect dim objects, e.g. Oort cloud or Kuiper belt objects, passing by in front of a star in the background? I've read the Hubble did this. Is it possible also from ground? How do these conditions change, if the camera would be in space? Or do I need Hubble?"}
{"id":"26317","title":"Is it possible to see satellites with the naked eye?","text":"Every now and then I notice some very bright \"stars\" in the sky. They tend to be very few (one or two, usually), and are quite much brighter than any other star out there. Often they're perfectly visible and bright even in dusk, when no other star is visible yet or will be for some time to come. Now, I'm writing the word \"star\" in quotes because I really don't think that they're stars. But what are they? Planets? Satellites? Latest Chuck Norris encounters leaving the atmosphere? My best guess would be that they are satellites, because planets should be about the same brightness as stars, but that sounds pretty implausible as well... And yes, I'm very certain that they are NOT airplanes."}
{"id":"25753","title":"How do we know the masses of single stars?","text":"I have recently read that we can only know the masses of stars in binary systems, because we use Kepler's third law to indirectly measure the mass. However, it is not hard to find measurements for the mass of stars not in binary systems. So how is the mass of these stars determined?"}
{"id":"56296","title":"Why does a photon colliding with an atomic nucleus cause pair production?","text":"I understand that the photon needs to have enough energy to produce a lepton and it's antimatter partner, and that all of the properties are conserved, but why does the photon do this in the first place? What's going on \"behind the scenes\" to transform a single, neutral particle (the photon) into two charged particles (i.e. electrons)? Why is the nucleus necessary?"}
{"id":"99914","title":"Five-component field","text":"Recently I was reading about 5-component field $(\\varphi , \\psi_{\\mu})$, for which $$ \\hat {p}^{\\mu} \\varphi = mc\\psi^{\\mu}, \\quad \\hat {p}_{\\mu}\\psi^{\\mu} = mc\\varphi . $$ This field refers to the spin-zero representation of the Lorentz group and obeys the Klein-Gordon massive equation (so it reproduces Poincare group irreducible representations). What physical system is described by it?"}
{"id":"64988","title":"Is every quantum measurement reducible to measurements of position and time?","text":"I am currently studying Path Integrals and was unable to resolve the following problem. In the famous book _Quantum Mechanics and Path Integrals,_ written by Feynman and Hibbs, it says (at the beginning of Chapter 5 _Measurements an Operators,_ on page 96): > _So far we have described quantum-mechanical systems as if we intended to > measure only the coordinates of position and time. Indeed, all measurements > of quantum mechanical systems could be made to reduce eventually to position > and time measurements (e.g., the position of a needle on a meter or time of > flight of a particle). Because of this possibility a theory formulated in > terms of position measurements is complete enough to describe all > phenomena._ To me this seems to be a highly non trivial statement (is it even true?) and I was unable to find any satisfying elaboration on this in the literature. I would be thankful for any answer to resolve this question and any reference to the literature!"}
{"id":"57747","title":"If light is linearly polarized, does it have some spatial extent?","text":"If light (a photon) is linearly polarized, say vertically, does it have some vertical _spatial_ extent (perhaps in amplitude)?"}
{"id":"57744","title":"How is celestial navigation done on a low-level?","text":"When we send a probe off to Jupiter or Saturn, or even Earth orbit, how are the rocket firings timed and coordinated? For instance, when I want to drive to another city I pull onto the highway and use the lines on the road to adjust my steering. I also use intuition to adjust my velocity, even though that won't make a difference _where_ I wind up (only _when_ ). However, spacecraft don't have lines on the road to line up with. Nor do they have a direct measurement to take for speed, which is critical in orbits and in planetary transfers. I suppose that some bright stars near opposition or off the ecliptic (Capella?) and GPS could be used for interplanetary navigation or LEO, but how was it done before GPS? Inertial navigation? How did the Soviets get the Vostok to enter a clean orbit (I understand that the Sputnik orbit was very elliptical)? How did early interplanetary probes orient themselves?"}
{"id":"119169","title":"How to distinguish between the spectrum of an atom in motion and the one of a scaled atom?","text":"**Galaxies are moving** dragged by the space expansion. When atoms are in motion the doppler effect will shift the spectra of the emitted photons. The proton-to-electron mass ratio, $\\frac{m_e}{m_p}$ has been measured constant along the history of the universe, but nothing was said about the constancy of the electron or proton's masses. The photon's energy obey the Sommerfeld relation, $E_{jn}=-m_e*f(j,n,\\alpha,c)$, as seen here, and **it is evident that a shifted (1) spectrum is obtained with a larger $m_e$.** The spectra lines are not only due to the Hydrogen atom; there are other spectral lines due to molecular interactions, due to electric\/magnetic dipoles, etc, and so the electromagnetic interaction,the Coulomb's law, $F_{}=\\frac{1}{4\\cdot \\pi\\cdot \\varepsilon}\\cdot \\frac{q1\\cdot q2}{d^2}$ must be analyzed. If we scale all masses by the relation $\\alpha(t)$ (not related with the above fine structure constant), where $t$ is time (past), and also scale the charges and the distances and time by the same factor, gives exactly the same value $F_{}=\\frac{1}{4\\cdot \\pi\\cdot \\varepsilon}\\cdot \\frac{q_1\\cdot q_2\\cdot \\alpha^ 2(t)}{d^2\\cdot \\alpha^2(t)}$. Thus the system with and without the transformation behaves in the same manner. The same procedure shows that the universal gravitational law is also insensitive to the **scaling of the atom** (2). This should not be a complete surprise because the scaling of masses, charges, time units and distances is routinely used on computer simulations that mimic the universe in a consistent way. **The conclusion is that there is no easy way to distinguish between the spectrum of an atom in motion and the one of a scaled atom.** The photons that were emitted by a larger atom in the past are received now without any change on its wavelength (here I'm assuming a static space). The mainstream viewpoint, not being aware that scaling the atom gave the same observational results, adopted the receding interpretation long time ago. As a consequence the models derived from that interpretation (BB, Inflation, DE, DM, ) do not obey the general laws of the universe, namely the energy conservation principle. My viewpoint offers **a cause for the space expansion**. Most physicists are comfortable with: 'space expands', period, without a known cause. Physics is about causes and whys, backed by proper references. **I used the most basic laws of mainstream physics to show that another viewpoint is inscribed in the laws of nature.** The question is strict, formulated in the title: How can someone make a distinction between those spectra? I've already opted for an answer, presented below to provide context, but that is not for discussion. When I graduated as electronic engineer, long time ago, I accepted naively that the fields (electrostatic and gravitational) are sourced by the particles, and expand at $c$ speed, without being drained. But now, older but not senile, I assume without exception, that in the universe there are no 'free lunches' and thus the energy must be transferred from the particles (shrinking) to the fields (growing). This new viewpoint is formalized and compared to the $\\Lambda CDM$ model in a rigourous document, with the derivation of the scale relation $\\alpha(t)$ that corresponds to the universe's evolution, at:   A self-similar model of the Universe unveils the nature of dark energy   preceded by older documents at arxiv:   Cosmological Principle and Relativity - Part I A relativistic time variation of matter\/space fits both local and cosmic data Again, my question is:   **Can someone provide a way to distinguish between the spectrum of an atom in motion and the one of a scaled atom ?** maybe by probing the atom's nucleus and find the isotope ratio's abundance, D\/H evolution and other isotopes as Mr Webb did with Mg (1998 paper) when in search of the $\\alpha$ variability. **PS** : To simplify the argument this question skips some details and to avoid misrepresentations a short resumé of my position is annexed. (1) - after the details are done it is a redshifted spectrum at the reception (2) - LMTQ units scale equally by $\\alpha(t)$, $c,G,\\varepsilon$ and the fine structure constant $\\alpha$ are invariant. * * * **A short introduction to the self-similar dilation model** The self-similar model arises from an analysis of a fundamental question: is it the space that expands or standard length unit that decreases? That analysis is not an alternative cosmological model derived from some new hypothesis; on the contrary, it does not depend on hypotheses, it has no parameters besides Hubble parameter; it is simply the identification of the phenomenon behind the data, obtained by deduction from consensual observational results. The phenomenon identified is the following: in invariant space, matter is transforming in field in a self-similar way, feeding field expansion. As a consequence of this phenomenon, matter and field evanesce while field expands since the moment when matter appeared. As we use units in which matter is invariant, i.e., units intrinsic to matter, we cannot detect locally the evanescence of matter; but, as a consequence of our decreasing units, we detect an expanding space. So, like the explanation for the apparent rotation of the cosmic bodies, also the explanation for another global cosmic phenomenon (the apparent receding of the cosmic bodies) lays in us. In units where space is invariant, named Space or S units, matter and field evanesce: bodies decrease in size, the velocity of atomic phenomena increases because light speed is invariant but distances within bounded systems of particles decrease. In standard units, intrinsic to matter, here called Atomic or A units, matter and all its phenomena have invariant properties; however, the distance between non-bounded bodies increases, and the wavelength of distant radiations is red-shifted (they were emitted when atoms were greater). The ratios between Atomic and Space units, represented by M for mass, Q for charge, L for length and T for Time, are the following: $$M=L=T=Q=\\alpha(t)$$ The scaling function $\\alpha(t)$ is exponential in S units, as is typical in self-similar phenomena: $$\\alpha(t_S)=e^{-H_0 \\cdot t_S}$$ Mass and charge decrease exponentially in S units, and the size of atoms decrease at the same ratio, implying that the phenomena runs faster in the inverse ratio; as A units are such that hold invariant the measures of mass, charge, length of bodies (Einstein concept of reference body) and light speed, they vary all with the same ratio in relation to S units. In A units, space appears to expand at the inverse ratio of the decrease of A length unit; the space scale factor in A units, a, is: $$a=1+H_0 \\cdot t_A$$ Therefore, space expands linearly in A units. In what concerns physical laws, those that do not depend on time or space (local laws), like Planck law, are not affected by the evolution of matter\/field and hold the same in both systems of units. The laws for static field relate field and its source in the same moment; as both vary at the same ratio, their relation holds invariant and so the laws – the classic laws are valid both in A and S units. The electromagnetic induction laws can be treated as if they were local, therefore holding valid in both systems, and then consider that, in S, the energy of waves decreases with the square of field evanescence (the energy of electromagnetic waves is proportional to the square of the field). In A, due to the relationship between units, the energy decreases at the inverse ratio of space expansion while the wavelength increases proportionally. This decrease of the energy of the waves (or of the photons) is a mystery for an A observer because in A it is supposed to exist energy conservation, which is violated by electromagnetic waves. The phenomenon is observed in the cosmic microwave background (CMB) because the temperature shift of a Planck radiation implies a decrease of the density of the energy of the radiation with the fourth power of the wavelength increase and space expansion only accounts for the third power (this is perhaps the biggest problem of Big Bang models, so big that only seldom is mentioned). Note that induction laws can be treated as time-dependent laws and the evanescence of the radiation be directly obtained from the laws; that introduces an unnecessary formal complication. Finally, there are the conservation laws of mechanics, which require a little more attention.   Although it is not usually mentioned, the independence of physical laws in relation to the inertial motion implies the conservation of the weighted mass summation of velocities and square velocities of the particles of a system of a particles. As the A measure of mass is proportional to the weighted mass, these two properties are understood as the conservation of linear momentum and of kinetic energy. Therefore, the correct physical formula of these conservation laws depends on the weighted mass and is valid in both systems of units; for simplicity, the A measure of mass can be used instead of the weighted mass; for instance, for the conservation of square velocity: $$\\sum_i{\\frac{m_i}{m_{total}}\\cdot v_i^2=\\text{const}}\\Leftrightarrow \\frac{1}{2}\\sum_i{(m_A)\\cdot v_i^2}$$ In the first expression the system of units is not indicated because the equation is valid in both systems; in the second equation, velocity can be measured in A or S (has the same value in both) but the measure of mass has to be the A measure. The second expression is the conservation of kinetic energy in A.   The conservation of the angular momentum is the only law modified in A units because the relevant measure of curvature radius is the S measure; the angular momentum L can be written as $$\\textrm{L}=\\mathbf{r}_s\\times m_\\text{A}\\mathbf{v}$$ This is the quantity that holds invariant in an isolated system; the usual A angular momentum, function of $r_A$ $$\\textrm{L}_\\text{A}=\\mathbf{r}_\\text{A}\\times m_\\text{A}{v}$$, of an isolated system increases with time: $$\\left ( \\frac{d\\textrm{L}_\\text{A}}{dt_\\text{A}} \\right )_0=H_0\\textrm{L}_0$$ This means that the rotation of an isolated rotating body increases with time. For an S observer, this is consequence of the decrease of the size of the body; an A observer can explain this considering that it is consequence of the local expansion of space, that tends to drag the matter.   Note that there is no conflict with the data that support standard physics because this effect is not directly measurable by now; yet, this insignificant alteration has an important consequence: the expansion of planetary orbits. A note on the value of constants in both systems of units. To change from one system to the other is like any change between two systems of units, but there is a difference: units are time changing one another. As a consequence, a constant in A may not be constant in S. For instance, the Planck constant change in S with the square of the scaling law (a simple image of the physical reason is the following: orbits radii are decreasing in S and also their associated energy; so, the wavelength of emitted radiation is decreasing with orbit radii and also the energy, which implies a Planck constant decreasing with the square of the scaling). Field constants are the exception: they hold constant in both systems of units. The Hubble constant is different: it is constant in S but not in A (in A, the Hubble constant is just the present value of the Hubble parameter). In short, the Hubble constant is the only time constant and is a S constant, field constants are constant is A and S and the other constants, including Planck constant, are relative to atomic phenomena and are constant in A; naturally, the dimensionless relations, like the fine structure constant, are independent of units.   Only the classical fundamental physical laws have been considered because that is what is required, the ground on which the analysis of all the rest of physics can be made, special and general relativity included."}
{"id":"60155","title":"Gravity mitigated by velocity regardless of mass?","text":"We know that a small object moving fast enough can pass by a planet and escape its gravity. Would this be (theoretically) true in reverse? Meaning a planet moving fast enough past a stationary smaller object would not assert its gravitational pull on the small object? Would this mean that gravity is mitigated by velocity regardless of whether the more massive object is moving as opposed to the small object?"}
{"id":"126961","title":"Does physics recognize the particle of light separately from the wave of light?","text":"To frame this question, we need to assume that time freezes when traveling at the speed of light. This is theoretically congruent with Einstein's theory of relativity and the theory of time dilation, as discussed here: Would time freeze if you can travel at the speed of light? If this is true, then the photon, which may only exist at the speed of light due to it's \"mass-less\" property, would not be observable or measurable by any experiment or method that required the passage of time. For the photon's entire existence, the rest of time and existence would not experience change. If the photon traveled 1,000 light years from it's origin in a star to where it was absorbed into some matter, the rest of time and existence would remain changeless for the duration of the photon's 1,000 light year journey. There is no opportunity for anything that experiences the passage of time to observe or measure a photon which never experiences the passage of time. In the fabric of space time, instead of traveling, the photon stretches. While any other particle not traveling at the speed of light would leave it's point of origin and travel, one point at a time along a path to its destination, it is different for a photon. Instead, a photon could be thought of as being created on all points along a path, including the origin and the destination, simultaneously, and for absolutely no amount of time. Can this be represented with a physics equation? So if the photon is the light particle and there is also an aspect of light that is known to be an electromagnetic wave form, then the relationship between the two is not that they are the same thing, it is that the photon particle exists before, and causes, the electromagnetic wave form, the way that a boat moving through water causes a wake or lightning through the atmosphere causes thunder. Is this currently recognized in physics?"}
{"id":"134874","title":"Why doesn't a resistor dissipate reactive power?","text":"Is there a (preferably microscopic) explanation for why a resistor does not dissipate reactive power?"}
{"id":"66779","title":"Is everything in the physical world composite?","text":"In philosophy there is a principle that anything composite cannot have existed eternally, since it is preceded by its parts and whatever forces assembled it. Is everything in the physical world composite in one sense or another or is there something not composite?"}
{"id":"41676","title":"Why do physicists believe that particles are pointlike?","text":"String theory gives physicists reason to believe that particles are 1-dimensional strings because the theory has a purpose - unifying gravity with the gauge theories. So why is it that it's popular belief that particles are 0-dimensional points? Was there ever a proposed theory of them being like this? And why? What reason do physicists have to believe that particles are 0-dimensional points as opposed to 1-dimensional strings?"}
{"id":"113394","title":"Rotational Potential Energy of a Hamster Wheel","text":"# Background (unimportant back story)A colleague of mine showed me what i considered flawed statistics, that Internet Explorer had faster Index and array functions than Chrome(we are software developers...FWIW I'm not proud that this is what we talk about either) the graphs showed was in operations per seconds. I replied by trolling him with an example of a Hamster in a Hamster Wheel, saying _Sometimes more work, isn't always faster_ . on the flawed premise that $S=d\/t$ for $0$ $d$, and $(PE_{initial} + KE_{initial}) + W = (PE_{final}+ KE_{final})$ where $KE = \\frac{1}{2}I\\omega^2$ are correlated. The basics of the equation make sense to me. However, apparently I've trolled myself, because i believe that the Potential Energy of the hamster wheel remains constant across the equation, but i don't believe it has 0 $PE$, I'm just not sure how to calculate the constant amount of $PE$ latent in a Hamster Wheel? # Research I like to show that i've done some work, I've done some searching on Rotational Potential Energy, i've found a few results such as $PE = T_{av} \\times θ$ but this seems specific to rotational $PE$ of a spring, which should not factor into a Hamster Wheel, I've found loads of simple examples of rotational $KE$ and Inertia like this site https:\/\/www.boundless.com\/physics\/rotational-kinematics-angular-momentum-and- energy\/conservation-of-energy\/conservation-of-energy-in-rotational-motion\/ # Summary and Question If a Hamster Wheel has inherent and constant Potent Energy as I believe, how would i calculate it, for a hamster wheel of a given size?"}
{"id":"113397","title":"Reason for establishing a magnetic hysteresis curve in Zeeman effect experiment","text":"I am new to StackExchange, and so apologize in advance for any issues with the formatting of the question\/appropriateness of the question to this forum. My lab partner and I were conducting a standard undergraduate laboratory experiment investigating the Zeeman splitting of the red spectral line from a cadmium lamp. The procedure is simple: one takes an electromagnet, driven by a DC power source, a cadmium-mercury lamp, which is placed in the region of a strong magnetic field generated by the electromagnet, and looks at the spectral lines through an interferometer of some kind (we used a Lummer- Gehrcke plate). In the process of looking up laboratory manuals from different universities to consider different techniques and methods of going about the measurements, we stumbled upon this one. In the very last paragraph of the document, the authors suggest establishing a magnetic hysteresis loop before conducting the measurements of the splitting. The apparatus in this manual differs from the one we use, but not significantly: both use a DC driven electromagnet with a cadmium lamp. The question is, what is the purpose of establishing this hysteresis loop? We have followed the instuctions in the last paragraph of the manual and concluded that doing so results in a degree of permanent magnetization in the electromagnet at zero current (as one would expect with a hysteresis loop) - but then why ask the students to first degauss the electromagnet, as in the second to last paragraph? Any comments\/ideas would be highly appreciated."}
{"id":"27860","title":"How much water is destroyed in photosynthesis, relative to the world's supply?","text":"Water is involved in the photosynthesis. How much water are we talking about compared with the total amount on water on Earth? Is it enough to have an effect on the average age of water molecules?"}
{"id":"99001","title":"If you suddenly move a piece of metal, will that disturb the free electron density?","text":"If we have a hollow pipe sitting at rest filled with gas and we moved the pipe suddenly along its length to the right, then the gas density will be momentarily higher near the rear of the pipe and lower near its front because gas molecules have inertia. If we start rotating the pipe then gas molecules near the pipe wall will have maximum speed and the speed will go down as we approach the axis of the pipe. What if this pipe was a long metallic cylinder. Metals have free electrons. Will the free electron density change in a similar way if we translate the cylinder suddenly? What if we rotate the cylinder? how the free electrons will respond to that motion? What if the metallic cylinder was charged (so now there is charge resides on its outer surface). How the charge on the surface of the cylinder respond to the sudden translation or rotation of the cylinder? (And what theoretical model(s) one should consider to answer the previous questions?)"}
{"id":"18526","title":"Thermal energy generated due to loss in kinetic energy when observed from two different frames of reference","text":"A body is moving with a velocity $v$ with respect to a frame of reference $S_1$. It bumps into a very heavy object and comes to rest instantaneously, its kinetic energy $$\\frac{1}{2}mv^2$$ as seen from the frame $S_1$ is completely converted to thermal energy. Now, a man moving with a uniform velocity $V$ (in the direction of the body) with respect to $S_1$ observes the body. He notes that its initial kinetic energy of the body is $$\\frac{1}{2}m(v+V)^2,$$ and that after it rams into the heavy body as $$\\frac{1}{2}mV^2,$$ and concludes that the thermal energy produced is the difference $$\\frac{1}{2}m(v+V)^2-\\frac{1}{2}mV^2~=~\\frac{1}{2} mv^2+mvV.$$ Which of the two answers is correct and why?"}
{"id":"131090","title":"Using Polyakov-Alvarez Anomaly Formula","text":"Take $\\Sigma=\\mathbb{D}$ to be the unit disk with metric $g=\\frac{4}{(1+|z|^2)^2}\\,|dz|^2$. If $\\phi$ is a nice enough function on $\\mathbb{D}$, then I want to compute $$\\int_{\\partial \\Sigma} k_g \\phi\\,ds_g$$ and $$\\int_{\\partial\\Sigma} \\partial_n \\phi\\,ds_g,$$ where $\\partial_n$ is the outer normal derivative, $k$ is the curvature of the boundary and $ds$ is an element of arclength. How exactly does one go about computing these quantities?"}
{"id":"107171","title":"Why can two (or more) electric field lines never cross?","text":"The the title is self explanatory, I guess. Why can two (or more) electric field lines never cross?"}
{"id":"113939","title":"Differentiating the gravitational redshift and the cosmological redshift?","text":"If general relativity accounts for a redshift, independent of inflation, how can we still know that inflation is viable? Moreover, how do we differentiate the the gravitational redshift and the cosmological redshift observationally?"}
{"id":"78151","title":"Naive quantum gravity","text":"My question involves an analogy I have to point out. Consider the Lagrangian density for the a complex scalar field: \\begin{equation} \\mathcal{L}=\\frac{1}{2}\\partial_{\\mu}\\phi^{\\dagger}\\partial^{\\mu}\\phi-\\frac{1}{2}m^{2}\\phi^{\\dagger}\\phi \\end{equation} which is invariant under the global gauge transformation $$\\phi \\rightarrow e^{i\\alpha}\\phi.$$ The divergence-less 4-current is $j=(\\rho,\\underline{j})$ and the associated conserved quantity is the electric charge. This illustrates a global $U(1)$ symmetry. To see the analogy we now look at spacetime translations $x \\rightarrow x + a$ which give $$\\phi(x) \\rightarrow \\phi(x+a) = e^{a^{\\mu}\\partial_{\\mu}}\\phi.$$ The Lagrangian is no longer invariant but the action: \\begin{equation} S = \\int \\mathcal{L} d^{4}x \\end{equation} is invariant due to the translation invariance of the Lebesgue measure. The Noether current is the stress-energy tensor $T^{\\mu\\nu}$ and the conserved quantity is the 4-momentum. Due to the role played by $T^{\\mu\\nu}$ in the Einstein field equations we can say that the 4-momentum plays the role of gravitational charge. Promoting the global U(1) symmetry to a local one couples $\\phi$ to the electromagnetic field. This involves modifying the Lagrangian by introducing a covariant derivative and adding the Lagrangian for the EM field. My question is: does promoting the global translation invariance of S to a local one give the correct coupling of $\\phi$ to the graviton field? An action invariant under general coordinate transformations is: \\begin{equation} S = \\int \\mathcal{L} \\sqrt{- g}d^{4}x \\end{equation} where $\\mathcal{L}$ should also be modified appropriately and $g_{\\mu\\nu}$ is the metric. I rephrase and extent my question: (1) Do general coordinate transformations include local translations $x \\rightarrow x + a(x)$ or are they only $SO(1,3)$ gauge-like transformations? (2) In view of the above analogy the gauge group for gravity is the translation group in Minkowski spacetime. The group is not compact. Shouldn't the group be compact to guarantee a positive-definite kinetic term for gravitons? (3) The translation group is also abelian. Isn't this inconsistent with the fact that gravitons self-interact? (4) In essence what I am saying is that the classical theory of gravity is likely incomplete. Are there any approaches to gravity that modify the underlying spacetime such that the isometry group of the new space admits a compact version of the translation group of Minkowski spacetime? Maybe the universe after formation was compact but due to expansion the compact nature is now hidden. (5) Related to (4). Shouldn't elementary particles correspond to irreducible representations of the isometry group of the universe? If the universe is Minkowski then the isometry group is the Poincare group. Have other possibilities in this regard been explored? (6) In c = h = 1 units Newton's constant G has mass dimension -2. The fine structure constant is given by: $$\\alpha = \\frac{e^{2}}{4 \\pi \\epsilon_{0}}$$ In gravitoelectromagnetism we have the correspondence $G \\leftrightarrow \\cfrac{1}{4\\pi \\epsilon_{0}}$. As the momentum $p$ plays the role of gravitational charge, shouldn't the coupling constant to gravity be given by $$g \\sim p^{2} G = m^{2} G$$ where $m$ is mass of the particle in question? $g$ is then dimensionless. Does this imply that the covariant derivative to couple matter to gravity is given by something like $D_{\\mu} = \\partial_{\\mu} - p^{\\nu} A_{\\mu\\nu}$? where $A_{\\mu\\nu}$ is some tensor potential related to $g_{\\mu\\nu}$. (7) The above ideas might imply something about spin 0 bosons and their mass renormalization. Any thoughts? Could the $\\Lambda^{2}$ dependence of $m^{2}$ be just an indication that gravity was ignored?"}
{"id":"78154","title":"Why does a street light cast a wavey shimmer across the water (at night)","text":"See image You will have to zoom in. Why is the reflection a **wavey** one? Could someone explain how this is occurring, to the layman, (with the style of feynman if possible -heh) :)"}
{"id":"31720","title":"Are there any objective wavefunction collapse theories which are local and forbid superluminal signalling?","text":"Are there any objective wavefunction collapse theories which are local and forbid superluminal signalling? GRW is nonrelativistic and nonlocal."}
{"id":"31726","title":"Why are rebar grooved?","text":"Just what the title states, please. Iron reinforcement bars used to provide the framework over which concrete solidifies always appear to bear grooves. Do the grooves behave as the thread of a screw and provide better grip? Are the grooves merely a manufacturing artifact? Are the grooves a historical artifact - the first bars had to be manufactured grooved for some reason, and hence they continue to be so produced without rhyme or reason. Why are rebar grooved?"}
{"id":"123188","title":"Principle Axes of inertia and moments of inertia","text":"1. Principal axes of inertia is defined as the eigenvectors of the inertia tensor matrices. I understand that a diagonalised tensor can yield these axes, but why are they necessarily form the axes of a rotating frame? Can anyone explain why?   2. Lastly, why is it true that any symmetric axis is always a principal axis?    3. If $I_{xx} = I_{yy}$, why is any linear combination of two orthogonal directions on the place spanned by the eigenvector $e_1$ and $e_2$ can always be chosen as principal axes? I'm really confused... Hoping that anyone could shed a light for me."}
{"id":"131966","title":"Looking for a simple proof of symmetry of linear susceptibility tensor","text":"**Getting a diagonal susceptibility tensor:** Define the linear susceptibility tensor as $\\chi_{ij}$: $P_i = \\epsilon_0\\chi_{ij}E_j$, using standard notation for the electric field and the polarization. If one assumes a toy model with atoms having a harmonic potential: $$\\ddot{x} + 2\\gamma\\dot{x} + \\omega_0^2x = -eE_x(t)\/m$$ Now if we put $E_x(t)$ in terms of exponentials\/sinusoid with a forcing frequency $\\omega$, we can find the steady-state solution in the same manner as a forced harmonic oscillator. Then the $x$ can be used to write $P = -Nex$, with $N$ denoting number density. Finally, we can end up with an expression where $P \\propto E$ and obtain the _scalar_ susceptibility. In principle, we can change the value of $\\omega_0$ for $y$ and $z$ and write three different equations for the Cartesian axes to generate a diagonal susceptibility tensor, where not all non-zero terms are equal. Obviously, this tensor is symmetric and if we do a similarity transformation, it will stay symmetric. **Proving symmetry?** I'm not sure how to extend this basic idea to actually prove that the susceptibility tensor ought to be symmetric in general, which is pointed out here on page 2. > The matrix $\\chi$ is known as the susceptibility tensor, and is a tensor of > rank 2. This is the most general representation of the susceptibility of a > linear and uniform dielectric medium. It may be shown that, for a lossless > and non-optically active material, $\\chi_{ij}=\\chi_{ji}$. While the optically-inactive part does seem intuitive, I'm not sure what is meant by lossless medium in this case. Does it mean $\\gamma = 0$ in the toy model? As far as I understand, the $\\gamma$ is introduced to account for retarding motions due to presence of other atoms in surroundings. Another question: how do you describe a material to be optically inactive, i.e., what equations can one write using it? Also, I did manage to find a proof for the symmetry of dielectric tensor. There are actually two proofs, one using Onsager's theorem and one using Fluctuation-Dissipation theorem. However, I'm looking for a much simpler proof, hopefully in the terms of energy and momentum conservation only, and not having any Thermodynamic machinery."}
{"id":"76003","title":"What is optical confinement?","text":"What is optical confinement? What is it intuitively? How does this 'optical confinement' affect the performance of a laser?"}
{"id":"4234","title":"Why is the relationship between atomic number and density not linear?","text":"What are the factors that affect the density of an atom?"}
{"id":"31183","title":"Where can one learn about dispersion relations for S-matrices?","text":"Most textbooks on quantum field theory never mention dispersion relations at all. Where can one learn about dispersion relations for S-matrices?"}
{"id":"79200","title":"Minus Sign in Feynman Diagram","text":"I've been reading these notes and I can't figure out the why on P.120, it is said that > The fermionic statistics mean that the first diagram has an extra minus sign > relative to the ψψ scattering of Figure 25. Would someone mind explaining it to me? I see that the calculations below, illustrating this, but is there an intuitive way of seeing that the signs of the respective first diagrams in Figures 25, 27 should be opposite? Thank you."}
{"id":"91354","title":"Moduli Stabilization in 6D Einstein-Maxwell theory - Fluxes and O3 planes","text":"I'd like to do the maths for the moduli stabilization of 6D Einstein-Maxwell Gravity $$ S= \\int d^6X \\sqrt{-G_6}(M_6^4R_6[G_6]-M_6^2|F_2|^2), $$ where the 6D metric is specified by $$ ds^2 = g_{\\mu\\nu}(x)dx^{\\mu}dx^{\\nu}+R^2(x)\\hat{g}_{mn}(y)dy^mdy^n, $$ and $\\hat{g}_{mn}$ is the metric of a compact 2D manifold with unit volume. The setup of this model can be found in Denef, Douglas, Kachru, starting on page 10. Now I want to perform the dimensional reduction of the theory and thereby obtain the correct effective potential $V(R) = \\frac{\\chi}{R^4}-\\frac{n^2}{R^6}$. I could manage to derive the first part of the potential by rewriting $$ ds^2= R^2(x)\\\\{\\frac{g_{\\mu\\nu}(x)}{R^2(x)}dx^{\\mu}dx^{\\nu}+\\hat{g}_{mn}(y)dy^mdy^n\\\\} $$ which allows me to rescale the 6D action by $G_{MN}=R^2\\tilde{G}_{MN}$. Then we are left with a product space, i.e. we can write $R_6[\\tilde{G}_6]=R_4[\\tilde{g}]+R_2[\\hat{g}]$, where $\\tilde{g}_{\\mu\\nu}=\\frac{1}{R^2}g_{\\mu\\nu}$. Further rescaling brings a factor $R^{-4}$ for the term $\\chi$, while a factor $R^2$ has been absorbed in order to write $M_4^2=M_6^4R^2$. However, I'm not sure how to obtain the term $\\frac{n^2}{R^6}$. I'd appreciate any help. My idea is as follows: $$ \\int d^6X \\sqrt{-G_6}M_6^2 |F_2|^2 = \\int d^6X R^6 \\sqrt{-\\tilde{G}_6}M_6^2 |F_2|^2 = \\\\\\ = \\int d^4x \\sqrt{-\\tilde{g}}\\int d^2y\\sqrt{\\hat{g}} R^6M_6^2 |F_2|^2 = \\int d^4x \\sqrt{-g}\\int d^2y\\sqrt{\\hat{g}} R^2M_6^2 |F_2|^2 \\sim \\\\\\ \\sim M_6^2 \\int d^4x \\sqrt{-g}R^2 (\\frac{1}{R^2})^2 \\cdot (\\frac{n}{R^2})^2 = M_6^2 \\int d^4x \\sqrt{-g} \\frac{n^2}{R^6} $$ In the last line, we obtain a factor $R^{-4}$ because of $\\gamma^{m\\hat{m}}\\gamma^{n\\hat{n}}F_{mn}F_{\\hat{m}\\hat{n}}$ and the second factor comes from $F_2 \\sim \\frac{n}{R^2}$, since $$ \\int_{M_2} F_2 =n. $$ However, I'm puzzled because the factor of $M_6^2$ remains. If I want to rewrite the action s.t. we have a factor $M_4^2$ in front, then the term I obtain reads $\\frac{1}{M_4R} \\frac{n^2}{R^6}$ (reminder: $M_4^2 =M_6^4R^2$). So, where is my mistake? Furthermore, it is claimed that O3 planes give rise to an additional term $\\frac{m}{R^4}$ (after Weyl rescaling), but I've no idea how it arises. I would start with a CS term $\\int C_4$ for the action... It would be great if anyone could help me with these questions. Thanks in advance!"}
{"id":"103466","title":"What would be the minimal size of an aerial vehicle capable of sustained suspersonic flight?","text":"I wonder, what's the smallest possible size for a UAV capable of sustained supersonic flight at the current technology level? Let's say 10 minutes of flight at 1.1 M."}
{"id":"79209","title":"A change of sign in the electron-hole second quantization form","text":"It is common to see people do a change of sign in the so called electron-hole representation, namely, $$ b^{\\dagger}_{-k}=a_{v,k} $$ similar argument also seen in 1992 mattuck's book \"guide to feynman's diagrams\" pp.138, eq.7.78b. Can somebody explain what is the physical meaning of changing the sign of momentum, since it make a difference in specific calculations, clearly it is not just a convention, meanwhile it seems to change momentum conservation law in a way that is unambiguous in the electron-electron representation.."}
{"id":"35351","title":"Is it easier to move or rotate an object?","text":"This is a problem I'm facing on designing a moving bookcase for my home, I just don't have enough physics background to tackle the problem. Which of the following requires more force\/effort?   * A bookcase with a pivot point in one of the back corners that is rotated 180° to its new position (therefore we now face the back of the bookcase).   * A bookcase that is pushed\/pulled to the same relative position but with a diagonal push (guided by one rail at the top). In both cases the bookcase will be supported by equivalent fixed caster wheels aligned with the direction of movement or rotation, with the pivot and the rail acting only as guide. I don't think it should matter, but the bookcase will be about 1 ft deep and 3 ft wide, I don't have any good source, but I think it would be around 200 kg if fully loaded."}
{"id":"35353","title":"Sum total distance of electrons on a spherical surface","text":"What is the sum total distance between every possible pair of point charges when there are n point charges on a spherical surface? All point charges can only and are located on the infinitesimal spherical surface. Basically, we're going to have a bunch of points staying as far apart as possible on the surface of a sphere. Is there a general equation for this? And whats the sum total distance between every pair of points? (Both the distances by travelling the shortest distance on the spherical surface and the shortest distance travelling through the interior volume of the sphere. I'm confident that the relationship between those two values require a relatively simple equation. Is there a similar mathematical model\/problem which can be used to solve this problem?"}
{"id":"73564","title":"Explaining to a five year old - why don't bubbles run like water","text":"My five-year old daughter was asking about astronauts the other day and why they float in space. After me showing her a few bits on the kids section on the NASA web site I started explaining about the planets and how the sun was a ball of gas and you couldn't stand on it. I then had to explain to her what a gas was, using ice, water and steam as examples for the three main states. A few hours later she asked me if the bubbles from the bubble-bath in the bath water were a liquid and if they were why didn't the gravity make them run off her hands like water. That had me stumped. I'm not a scientist but I've been doing my best to try and figure this out and explain it to her but I'm not fully sure, either. Firstly, are bubbles from bubble-bath a liquid or have I made a bad assumption? I thought it might be down to friction but I assume bubble-bath is a detergent and shouldn't encounter much friction. I then thought it might be because the bubbles had a large surface area relative to their weight and would suffer wind resistance as a result. My best guess now is that it's because the bubbles were more viscous than the water. Are any of these right, if not can someone explain it to me in simple terms so I can tell her, please? **UPDATE** Wow, thanks for all the responses! I'm usually on Stackoverflow and I don't think that in two and half years on there I've asked anything that's generated this level of response. Thank you to everyone who took the time to answer or comment - I've read all that's been written here. I explained this to my daughter last night saying that the bubble bath liquid was sticky and as a result would stick to things and slow down gravity's pull. I drew a bubble and explained about the liquid stopping the air in the middle getting out and by standing opposite and holding hands both I explained how we were like the liquid of the bubble squeezing the air in the middle which she understood. I expect that if she's asking questions like this now I'll be on here again soon asking something else. I've certainly learnt a lot! Thanks again."}
{"id":"73561","title":"What is the meaning of $G\/cm^2$?","text":"I'll prefix this question by stating that I am not a physicist, I am a programmer. I am working on updating a (very old) program that calculates the performance of a magnet. This whole program is in CGS units and it is required of me to change it to SI units. A particular equation in this program gives the Gauss\/square centimeter. I understand this as the gauss going through a 1 square centimeter area. Is there a proper name for the unit of $\\frac{G}{cm^2}$?, the closest I can find in CGS terms in a Maxwell unit which is $G \\cdot cm^2$ It's less about the name of the unit and more about being able to check whether this empirically derived equation is correct. Unfortunately I cannot post this equation as I do not have the necessary permissons. * * * In the comments it has now been shown that $$\\frac{G}{cm^2} = \\frac{T}{m^2}$$ So the question now becomes, what does this mean? The field strength in an area?"}
{"id":"128962","title":"What are \"finely subdivided\" substances?","text":"As in the title, what does it mean for a substance to be _finely subdivided_ , or _finely grained_? For example: > It is only in the cases for which the ratio area\/volume is very large (for > example, a _finely grained_ substance) that the surface must also be > considered. or: > In the case of a system composed by two homogeneous substances, it will be > possible to express the internal energy of the system as the sum of the > energies of the two substances only if we can neglect the surface energy of > the two substances where they are in contact. The surface energy can > generally be neglected only if the two substances are not very _finely > subdivided_ ; otherwise, it can play an important role."}
{"id":"87866","title":"Modular invariance of CFT","text":"I am looking at the Cardy formula for entropy in CFT, and in the article 'Kerr\/CFT correspondence and its Extensions' there is a sentence: > In any unitary and modular invariant CFT, the asymptotic growth of states... I've looked a bit, but couldn't find any simple explanation of the term modular invariant CFT. All I find is that modular invariance is the invariance under the group $SL(2,\\mathbb{Z})$. But what does that mean, what's the physical explanation of it and what's it's effect?"}
{"id":"23418","title":"Is there an upper frequency limit to ultrasound?","text":"Wikipedia has this diagram of the acoustic frequency spectrum: ![enter image description here](http:\/\/i.stack.imgur.com\/N8klp.png) Is there an upper limit to the frequencies you can transmit through the air? Are they absorbed more and more at higher frequencies, resulting in shorter travel distances? Are there \"windows\" of absorption and transmission like there are for radio waves? ![enter image description here](http:\/\/i.stack.imgur.com\/9x3VJ.jpg) **Update:** This says > An additional problem is created by the strong absorption of ultrasound in > air, at least for frequencies higher than 250 kHz. As a consequence, > ultrasound of e.g. 1 or 2 MHz can propagate in air over a distance of not > more than a few centimeter Also found some references, but I can't access them: http:\/\/scholar.google.com\/scholar?q=Atmospheric+absorption+of+sound"}
{"id":"109001","title":"Compatibility between solutions of explicit Maxwell equations vs. wave equation?","text":"When trying to solve for the allowed propagation frequencies in a cylindrical waveguide, I approached the problem by solving the wave equation for all three components of $\\bar{E}$, and subsequently apply boundary conditions - $E_z(R) = E_\\phi(R) = 0$ and $\\nabla\\cdot\\bar{E}=0$. This yields the modes: $$E_i=A_ij_m(k'\\rho)e^{\\pm im\\phi}e^{i(k_zz-\\omega t)},\\ k'^2\\equiv\\frac{\\omega^2n^2}{c^2}-k_z^2, \\ i=x,y,z$$ Where the boundary conditions set the degrees of freedom (the discrete values for $k'$, setting the $A_i$'s, etc.) On the other hand, reading through Jackson (3rd ed. chap. 8), solving directly Maxwell's equations, he states that only $E_z$ should satisfy a wave equation, and the other (traverse) components are given by $E_z$'s derivatives. This obviously leads to Bessel derivatives $j'_m(k'\\rho)$, as he demonstrates later in that chapter, which are absent from the modes I found above, if we ignore the relation between $j'_m$ and $j_{m\\pm1}$. Where does this discrepancy come from? I would imagine that the transverse elements can easily be found by   $$E_\\rho=\\cos(\\phi)E_x + \\sin(\\phi)E_y, \\ E_\\phi = -\\sin(\\phi)E_x+\\cos(\\phi)E_y$$ If one wishes to avoid solving the wave equation directly in polar coordinates, as then the components are mixed together."}
{"id":"128688","title":"Double rainbows","text":"In my garden, when I'm watering the plants I sometimes see a rainbow or two. How did two rainbows appear? Why can't I see three rainbows then, or how can I see three rainbows?"}
{"id":"97788","title":"Using $A = {\\lambda}N$ to find when a the amount of a radioactive source becomes constant","text":"The question and mark scheme I will write in bold and my own thoughts in normal sized text. I'm told that: **When a $\\bf{_{92}^{235}U} $ nucleus is exposed to free neutrons it can absorb a neutron. The resulting nucleus decays, first to $\\bf{_{93}^{239}Np}$ and then to $\\bf{_{94}^{239}Pu}$.** I'm then asked a few questions this, one of which I don't understand the answer to. They say that: **The number of $\\bf{_{93}^{239}Np}$ nuclei present eventually becomes constant, calculate this constant number of $\\bf{_{93}^{239}Np}$ nuclei, given that the half life of $\\bf{_{93}^{239}Np} = 2.04 \\times 10^{5} s$ and that the number of $\\bf{_{93}^{239}Np}$ nuclei produced is at a constant rate of $ \\bf{1.80 \\times 10^{7} s^{-1}} $** The mark scheme uses the equation: $ \\bf{A = \\lambda N}$ and rearranges this to say that: $\\bf{N=\\dfrac{A}{\\lambda}}$ **and we also know that** $\\bf{\\lambda = \\dfrac{0.693}{t_\\frac{1}{2}}}$ $\\bf{\\therefore \\lambda \\approx 3.397 \\times 10^{-6}}$ I understand all of this and I understand the maths of the next step, I just don't understand why it gives you they value of the number of nuclei of $_{93}^{239}Np$ when the rate of decay of $_{93}^{239}Np$ equals the rate of formation of $_{93}^{239}Np$. **Substituting this value of lambda into our equation for $\\bf{N}$ gives us:** $\\bf{N = \\dfrac{1.8 \\times 10^{7}}{3.397 \\times 10^{-6}} \\approx 5.3 \\times 10^{12}}$ **$\\bf{{\\therefore}}$ the number of $\\bf{_{93}^{239}Np}$ nuclei when this number becomes constant is $5.3 \\times 10^{12}$**"}
{"id":"33722","title":"Is Joule heating only between charged particles?","text":"The Wikipedia page for Joule heating explains _\"It is now known that **Joule heating is caused by** interactions between the moving particles that form the current (usually, but not always, electrons) and the atomic ions that make up the body of the conductor. **Charged particles** in an electric circuit are accelerated by an electric field but give up some of their kinetic energy **each time they collide with an ion**. The increase in the kinetic or vibrational energy of the ions manifests itself as heat and a rise in the temperature of the conductor. Hence energy is transferred from the electrical power supply to the conductor and any materials with which it is in thermal contact.\"_ However, I wonder why they only talk about ions. Of course, electrons can easier interact with particles, which also have a long range field, but what about the non-charged particles which are also all around? > Is Joule heating only between electrons and ions?"}
{"id":"106321","title":"Computer simulation of Schrödinger equation","text":"I am looking for a computer program which simulates the Schrödinger equation (say for a single particle) in two dimensions and for potentials and initial states specified by the user. Typical situations I have in mind are the double slit experiment, harmonic oscillator, Aharonov–Bohm, tunneling effect etc. Unfortunately I was only able to find the one dimensional case on the web. So are there some 2d programs available? PS: I don't think my question is a duplicate of this one. I only ask about a single particle, not about thousands. On the other hand I want freedom in my choice of potential. Also I want a numeric solution of a PDE, while the other question is about (as far as I understand, which is not ver far) a Monte Carlo simulation. Anyway these are completely different requirements."}
{"id":"121261","title":"Bragg diffraction and lattice planes","text":"Crystalline substances show, for certain sharply defined wavelength and incident directions, very sharp peaks of scattered X-ray radiation. From the illustration below we see that we get constructive interference when the path-length difference is a multiple of the wavelength $\\lambda$. ![Bragg diffraction](http:\/\/i.stack.imgur.com\/mxwmj.png) In real crystalline materials we have a large amount of closely packed lattice planes. This large amount accounts for the sharp peaks for certain $\\theta$. I do not understand how this follows from the Bragg reflection formula $$ n\\lambda = 2d \\sin \\theta , $$ since $d$ is not constant anymore. I understand the model for two lattice planes as in the illustration. Is it true that $d$ can only take on values of the seperation of lattice planes, so $d$ is defined to be the seperation of points in the reciprocal lattice, or in others words, is $d$ constrained to be the absolute values of some reciprocal lattice vector? How does the Bragg condition account for very sharp peaks when we let $d$ run through all such absolute values?"}
{"id":"95161","title":"Weighted average whenever a variance is 0?","text":"So I have data for the number of times a certain event happens. Each column is a different event and each row is one trial. A sample of data would look like \\begin{bmatrix} 1& 4& 7& 6& 5\\\\\\ 1& 3& 9& 4& 5\\\\\\ 1& 3& 7& 8& 8\\\\\\ \\end{bmatrix} I then collapse this data down into one row, and each number is just the average of the column and has its own variance. I want to take a weighted average of this row and obtain the associated variance as well, but in this example the first number has a variance of 0, and thus the weight would be 1\/0. What does one do in a situation like this?"}
{"id":"6292","title":"Gamma Ray Bubble at the center of our galaxy seen by Fermi Telescope","text":"How could we measure high energy photons, whithout measuring them ? I can't understand how we can \"see\" those Gamma Ray Bubbles if they are not reaching here In this graph from Nasa you can see those \"bubbles\" are not reaching solar system: ![Art about bubles](http:\/\/i.stack.imgur.com\/PEM2t.jpg) Then how could be measure that Gamma Ray without the Gamma Rays Thanks for any answer!"}
{"id":"95169","title":"Why does lambertian surface reflection result in uniform radiance measured from every direction?","text":"I'm reading through the light measurement handbook and page 28 is confusing me. I knew previously that lambertian diffuse reflections distribute the reflection of incoming intensity such that the surface looks equally bright from every direction from which it is measured (only dependent on incoming light direction and surface normal). Specifically, the measured radiance stays constant. However, the page shows that the actual reflected energy (the text says energy, I'm assuming it's talking either about radiant flux or intensity) isn't actually uniformly distributed, but forms essentially a cosine lobe centered about the surface normal: ![figure from page 28](http:\/\/i.stack.imgur.com\/AaJ0u.png) I'm assuming that the \"uniform radiance\" results somehow from the measured area in relation to Lambert's cosine law cancelling out this cosine distribution to somehow result in a uniform distribution. How can this be shown mathematically? Also, how does this influence the light received by surfaces receiving this reflected energy? Are they lit such that every surface placed at equal distances to the surface receives the same amount of light, regardless of its orientation w.r.t. the surface normal, or do they essentially receive this cosine distribution?"}
{"id":"38440","title":"Tried to do the double slit experiment, failed. Why?","text":"I tried, and failed, to do the double-slit experiment. I thought I had a good experiment setup, but obviously I was wrong. Can anyone offer some insight in what I might have done wrong? Most of this was done using some vague memory of experiments way back in High School, so there might be a better approach. I did search through past questions. This question was the closest to what I am shooting for, but there are significant differences and did not enlighten my particular situation. **Setup:** I used a variety of laser pointers as the light source, both red and green. One laser pointer was home built using a red laser diode (200-250 mW) from a DVD burner going through an adjustable lens. The slits were made by depositing soot from a candle onto a microscope slide, then \"cutting\" slits in the soot using razor blades. Two blades were held together and the slits \"cut\" at the same time with a single swipe of the razor blades. Initially I used old fashioned double-edged razor blades but the blades were too thin and only a single slit was made in the soot. I switched to X-Acto blades, which made two slits approximately 0.5 to 0.8 mm apart. I didn't measure this, I just eyeballed it. I did both a single slit and a double slit. I varied the distance from the laser to the slit, from inches to 20 feet. The distance from the slide to the wall was varied, inches to 8 feet. **Results:** I never clearly saw an interference pattern. I saw things that were probably some kind of interference pattern, but there was no consistency. One experiment showed a pattern of lines that was about 2 cm apart, while just changing the \"quality\" of the slit changed the pattern to about 0.25 cm apart (not changing the laser\/slit\/wall distances)-- this just seemed wrong to me and implied some fundamental experiment setup problem. I also expected a difference between the single and double slit, but that was not the case. The variation of patterns seen for a single slit was similar to what I saw with the double slit. When the laser-slit distance was 20 feet, and the slit-wall distance was inches then I could clearly see an impression of the slits on the wall with no hint of an interference pattern. **Other experiment notes:** The soot was surprisingly difficult to put slit into. If the soot was made thick enough to block all light then the edges of the slit were ragged and the width of each slit was not consistent. If the soot was thinner then the slit was smoother and consistent, but there was some light leakage in the areas that should have been solid black. **Questions:** Is the \"soot on a microscope slide\" a valid method for making the slits? Is the laser pointer with adjustable lens an appropriate light source? Should I expect to see a diffraction pattern with only the double slit, and not with a single slit? Did you see any other improvements I could make to this experiment?"}
{"id":"4591","title":"$N=4$ supersymmetric yang-mills theory and S-duality","text":"[Question suggested by @Lawrence B. Crowell's answer to the question on T-duality] There are three parts to my question: A. What is the action for $N=4$ SUSY Yang-Mills and what is the physics of the various terms in the action? B. Give a simple explanation for the origin of Montonen-Olive duality in this theory. This duality maps the physics at gauge coupling $g$ to the physics at $1\/g$, i.e. this is a duality between the strong and weak coupling regimes of the theory. C. **Auxiliary Question** : Explain how one can decompose a SUSY gauge field into its bosonic and fermionic parts. [ **Background** : I am familiar with non-abelian gauge theory and have had a class on SUSY some years back. So I pick up on the jargon but am missing a fuller picture.]"}
{"id":"4597","title":"Green functions in Quantum Mechanics","text":"How the Green's functions and the Quantum Mechanics are related? Do they can be used to solve the Schrödinger equation of an particle subjet to some potential that is not a Dirac's delta? And the proprieties of some Green's functions that are symmetrical, i.e. $ G(x|\\xi) = G(\\xi|x)^{\\ast} $, has some relation with the propriety of the inner product $ \\langle \\alpha \\vert \\beta \\rangle = \\langle \\beta \\vert \\alpha \\rangle^{\\ast} $?"}
{"id":"133428","title":"Weight held by jet stream of water","text":"> _Consider a bucket of weight $W$, held in air by a jet stream of water from > the ground (moving vertically upwards) at height $h$. The water is fired > from the ground at the rate of $\\mu$ (units kg\/s) with an initial velocity > of $v_0$. Find the maximum possible value of $h$. Find conditions to achieve > the same._ (To observers, this is a problem from Kleppner & Kollenkow: _An Introduction to Mechanics_ ). **What I did:** Consider a differential mass element $dm$ near the bucket before hitting it. Its velocity is $v_f = \\sqrt{v_0^2 - 2gh}$, so its momentum is $v_f dm$. Suppose after hitting the bucket it had a velocity of $v'$, and the time of contact was $\\Delta t$. Since we must exert a force of $W$ upwards on the bucket, the rate-of-change of momentum of $dm$ is $W$. Clearly $dm(v' + v_f) = W\\Delta t$, which in the limit $\\Delta t \\to 0$, becomes $W = \\mu (v' + v_f)$. Thus $2gh = v_0^2 - \\left(\\frac{W}{\\mu} - v'\\right)^2 $. This raises a question. Setting $v' = \\frac{W}{\\mu}$, we can attain $h = v_0^2 \/ 2g$, but at that height, the water would have no velocity, so no momentum change. What are my errors, and how to attack such problems (with continuous masses)? ( Please excuse me for any silly mistakes. I'm only a beginner :( ). A detailed solution which I can study would be extremely helpful."}
{"id":"80898","title":"How uncertainties affect values","text":"If I calculate the equivalent resistance of a circuit, for example: $$1\/R_{eq} = 1\/R_1 + 1\/R_2 + 1\/R_3 = 1\/1472 + 1\/3260 + 1\/5580 \\Rightarrow R_{eq} = 858.22\\,\\Omega$$ And then calculate its uncertainty: $$\\Delta R_{eq} = |\\delta 1\/\\delta R_1| \\times \\Delta R_1 + |\\delta 1\/\\delta R_2| \\times \\Delta R_2 + |\\delta 1\/\\delta R_3| \\times \\Delta R_3 = 0.001\\,\\Omega$$ Will the uncertainty affect the value $(858.22)$, given that it only has 2 decimal numbers?"}
{"id":"133422","title":"Can we make a camera that fits in one of the bands of the double slit experiment?","text":"So, in Young's double slit experiment, we see where the particle lands on the back wall. What if we were standing at the back wall, looking toward the slits? Could we gain more information not by simply observing the light bouncing at the back wall, but looking at it our selves? Of course, for electrons or lasers, that ain't a great idea. And the slits are too small for our eyes to really make out, I'd gather. Frogs can possibly detect a single photon. What if we trained a frog to lick its left eye if it detects a photon from the left slit, and its right eye if it detects a photon from the right slit. Or better yet, build a tiny robot with a camera. Would that robot see both slits illuminate as it detects the photon? Seems testable in theory."}
{"id":"80892","title":"Tool for Born-level Cross Section Generation with Color Decomposition","text":"I'm looking to find a tool, like CalcHEP, that calculates basic cross sections and allows for symbolic output which is decomposed by color. I understand that MADGRAPH 5 will decompose amplitudes by color in the calculation, but I'm not sure if I can extract those individual amplitudes in a reasonably easy way. Being unfamiliar with MADGRAPH, I'm also not sure how that output would look...or if it's symbolic in nature. I am also beginning to look at Comix* but I have not made much progress through the manual yet. I would like to be able to specify my basis, but I suppose that's a luxury. Any recommendations?"}
{"id":"49790","title":"Why does weak equivalence principle say gravity is equivalent to acceleration?","text":"I am told that the weak equivalent principle, that $m_i=m_g$ (inertial and gravitational masses are equivalent) is equivalent to the statement that in a small system you can't tell whether you are in a uniform gravitational field, or in an equivalent accelerating frame. My question is about light: Special relativity doesn't say anything about how light falls in a gravitational field, so if you are in a rocket ship accelerating at $g$ why should it be true that when you shine a light in the rocket ship, you see the same thing happening as if you shined a light on earth? (to clarify, if you shot a bullet in the rocket ship, I understand why the trajectory of the bullet would look the same as if the bullet were shot on Earth, but this is because special relativity (or classical mechanics) tells us what the trajectory of a bullet looks like on Earth)."}
{"id":"36075","title":"Is it really to solve problem below by using, in the main, Gauss law?","text":"There is an infinite cylinder surface which uniformly charged along and has a surface charge density, which can be represented as $$ \\sigma = \\sigma_{0}cos(\\varphi ), $$ where $\\varphi$ - polar angle in a cylindrical coordinate system (it means that z-axis is matches with cylinder axis). There need to find z-component $|\\mathbf E|$. I tried to learn use Gauss law, but tasks, where density isn't constant on the surface, are still hard for me. Can you help?"}
{"id":"14346","title":"Why Do Hurricane Balls Spin So Fast?","text":"I was wondering if anyone could offer an explanation as to why the balls described in this video spin so fast. Here's the setup: Two metal balls are wielded together. When spun with air, they acquire a massive amount of rpm."}
{"id":"18645","title":"Plausible explanations for 3 local space dimensions","text":"Every now and then I see accounts of models that claim to explain why we experience only 3 space dimensions (locally, i.e. within cosmic horizons and outside black hole horizons). One such of course is string theory; but I wondered if there might be others, and perhaps even a survey of this topic along the lines of those expository articles one often sees in the ArXiV."}
{"id":"33241","title":"Motion of a plank hinged about a point","text":"There is a plank hinged about a point O. At the free end of the plank a small impulse is given. Will the plank perform pure rotation or both rotation and translation?"}
{"id":"71070","title":"Measurement of event time by different observers","text":"Follow up question to Doppler redshift in special relativity A source of light pulses moves with speed v directly away from an observer at rest in an inertial frame. How will the time of the emission of a light pulse as measured by the emitter be different to the time as measured by the observer (how will it be different than the time of the pulse reaching the observer)? edit: to elaborate slightly, let us say $ t_{E_1} $ is the time of emission as measured by the emitter, $ t_{E_2} $ is the time of emission as measured by the observer, and $ t_O $ is the time of observation of the pulse by the observer. How are these three times different, how would they be measured in practice?"}
{"id":"45922","title":"Action for a point particle in a curved spacetime","text":"Is this action for a point particle in a curved spacetime correct? $$\\mathcal S =-Mc \\int ds = -Mc \\int_{\\xi_0}^{\\xi_1}\\sqrt{g_{\\mu\\nu}(x)\\frac{dx^\\mu(\\xi)}{d\\xi} \\frac{dx^\\nu(\\xi)}{d\\xi}} \\ \\ d\\xi$$"}
{"id":"45927","title":"How Scientist Came to Know the Shape of our own galaxy without going out of it?","text":"As far as I know, scientists have been able to see a lot of differently shaped galaxies in our visible universe through modern age telescopes. But I was wondering how it was possible to know how our own galaxy looked like without going out of it to see it's shape?"}
{"id":"44039","title":"Getting a given wavelength radio signal given an antenna with real-world constraints","text":"Supposing you are given a transmitting antenna of whatever type of metal is most commonly used these days, and supposing that you are applying an AC current with the intent of transmitting a 1 m wavelength radio signal, about how far is one electron likely to make it in one direction along the antenna conductor in one half cycle from a negative peak to a positive peak, factoring in the likelihood of the presence of antenna material atoms affecting the electron's path? Can individual charge carriers actually make it very far (on the order of meters) through metal? But if the answer is that individual charge carriers actually only make a microscopic or very small displacement in space, how could a 1 m electromagnetic wave be produced? I don't see how a large number of charge carriers each only making a very small displacement in space can add up to a 1 m wave. This site, especially Fig. 2 there, is where I'm getting my basic understanding of how alternating current gets an EM wave transmitted. From that is where I got the feeling that charge carriers would actually need to make a displacement in space on the same order as the intended EM wave."}
{"id":"126543","title":"Conservation of energy and momentum via the continuity equation in asymmetric time and space translation","text":"I am confused about energy and momentum conservation, time and space translation symmetry, and the continuity equation. Suppose we have a mass $m$ in inertial space far from any gravitational fields. Suppose the mass has an initial velocity $v$; thus, it has an initial momentum $p=mv$ and an initial kinetic energy of $T=\\frac12mv^2$. My understanding of the continuity equation is we imagine a closed surface this mass passes through and we say momentum is conserved if the momentum of $m$ exiting the imaginary surface is the same as the momentum of m entering the surface. This comes about by space translation symmetry. Now, suppose during the time $m$ is within the closed surface, an impulse is applied to the mass, so that when it exits the surface, it has a greater momentum. Do we say the momentum is still conserved because we include the impulse that acted on the mass during its transit within the closed surface? Also, the same question applies to the kinetic energy the mass gains by work being done on it by a force $f$ and displacement $d$ within the closed surface. We still say energy is conserved because work was done on the mass while it was in transit within the closed surface. Correct? Now, suppose the mass $m$ starts out in a gravitational field and free falls to the earth. During its fall, it also passes through an imaginary closed surface. The momentum and the kinetic energy of the mass will have increased from its initial values prior to entering the closed surface. But, relativity states gravity is not a force, but a curving of space-time. While the mass accelerates through the imaginary closed surface, it gains energy and momentum even though there is no force or \"weight\" acting on it. Is it correct to say that because there was no time or space translation symmetry within this closed surface-- spacetime is curved-- that energy and momentum is not conserved? Or, do we think of a real force acting on the mass due to the gravitational field so that we can assert energy and momentum is conserved locally?"}
{"id":"129643","title":"Entropy of a two-level system","text":"Consider a two-level system with energies and degeneracies $\\epsilon_0 = 0, g_0=1$ and $\\epsilon_1 = \\epsilon, g_1=4$. I can show that the temperature at which both levels are equally populated is given by $\\exp\\left(\\frac{\\epsilon}{kT_0}\\right) = 4$, at which the partition function, $q=2$ and the molar internal energy $U(T_0) = \\frac{1}{2}N_A\\epsilon$. This means that the entropy at $T_0$ is $S(T_0) = R\\ln 2 + \\frac{N_A\\epsilon}{2T_0}$, right? Is there any special interpretation of this value of the entropy? Can the two terms here be described separately by something physically meaningful?"}
{"id":"47711","title":"The Double Slit Experiment - delayed measurement","text":"I'm not a physicist; I'm just curious about what would happen. According to this video \"Explained ! The Double Slit Experiment\" (at 2:50): > \"Somebody decided to leave the detectors on but just not take any data. > [...] So if it was also going to a magnetic tape - there was no magnetic > tape loaded. [...] And what do you think happed? They got this. They got the > diffraction pattern.\" So the act of measuring (beeing concious of the data) destroys wave-form pattern? If so lets consider these cases:   1. What if I'd throw a dice after the experiment to decide if I should look at the data or to destroy it? (e.g. look at the data if the number on dice is even)   2. What if the information about measurement would need to travel longer period of time than the particle would need to hit the wall?  To above cases: Would it mean that information travelled instantly - faster than light or that \"the particle can look into the future\"?"}
{"id":"28935","title":"Amount\/flow rate of N2 needed to cool substrate","text":"I'm trying to cool a substrate inside a vacuum chamber with liquid nitrogen. The substrate holder is a block of copper suspended from the top of the chamber via nylon screws (for thermal insulation). A copper tube runs from the LN2 tank through the flange, through one of the substrate holder's through- hole, bend back through the other substrate holder through-hole, and exits the chamber. Upon exit, the LN2 is simply dumped into the atmosphere. The substrate holder starts at 288.2K, LN2 ~77K. The LN2 tank's line pressure is around 5 psi, and ends in standard pressure upon exit. I'm trying to calculate whether with my current setup I just described, I can actually cool the substrate (ITO coated with aluminum, with a foil of indium between it and the substrate holder surface) down to 77-90K. My problems right now are setting up the diff eqs describing the heat exchange inside the substrate holder. The flow rate of N2 inside the pipes depend on whether it would be heated to gas...I'm not exactly sure how to incorporate this into the diff eqs. I would really appreciate it if someone can walk me through how to approach this or point me to a sample calculation. Also, what would be the best way to achieve this kind of cooling? We've used a reservoir approach (having a vat of LN2 on top of the substrate holder) before with success, but it takes up a lot of space. If my current setup turns out not sufficient, can I just keep increasing the initial line pressure (to >5psi) to achieve the desired cooling? Thanks, Al"}
{"id":"6904","title":"Uses of the 'Golden Ratio' in Physics","text":"What are some physics applications of the golden ratio? $$\\varphi~=~ \\frac{1+\\sqrt{5}}{2}~\\approx~ 1.6180339887\\ldots$$ Does it ever function specifically as a constant in any formulas or theorems? EDIT: Original title said _Golden Radio... facepalm._ I originally asked this question at math.stackexchange but the answers there were all too abstract or useless for me."}
{"id":"105772","title":"Is the charge of a current carrying wire real or hypothetical?","text":"Realist753 Theory Relativistic Electrodynamics assert that a current carrying wire being neutral in a laboratory frame of reference appears electrostatically charged to an observer in relative motion to that wire (“Relativistic Wire”), like an ordinary pre-charged wire would appear at rest. The theory of this apparent charge of a Relativistic Wire is covered by most textbooks dealing with Relativistic Electrodynamics.Look here: {http:\/\/chip-architect.com\/physics\/Magnetism_from_ElectroStatics_and_SR.pdf} by Hans de Vries. Experimental evidence Experimentally such charge has indirectly been inferred from the electrostatic potential around a Relativistic Wire. My opinion however is that for a doubtless proof of such fundamental theory it would be justified to require more credible and immediate experimental evidence of the postulated nonzero charge of a “Relativistic Wire”. Proposed Experiment Missing evidence could preferentially be obtained by means of a sliding “galvanic” contact acting as a probe* to directly pick up the electrostatic potential of a “Relativistic Wire” and adequately store the collected charge as a measure of its potential. Subsequently, the sliding contact and the probe would have to be disconnected in motion, stopped and the conserved charge be measured. A nonzero collected charge could already be regarded as indisputable proof of a nonzero charge of the wire. Question: Does anybody know if the proposed experiment has ever been done? Your comments will be welcomed.   * To avoid disturbing homopolar induction in the vacuum field around the wire the probe should be magnetically isolated."}
{"id":"82136","title":"How does the curve ball drag air around it?","text":"In cricket or baseball there is a type of ball called the curve ball. ![enter image description here](http:\/\/i.stack.imgur.com\/HRcNA.jpg) This is the top spin of the ball.I read that due to spin the ball drags the air around it due to friction in the way shown above.Can you please explain why??"}
{"id":"123115","title":"who will win a race between man, horse and car of same power","text":"Other day I got into a chat with my friend on work power and energy where he asked an intelligence related question the question was like this > if a race was conducted between a man a horse and a car of SAME POWER then > who will most probably win the race and to his question i answered that obviously man could win it because the definition of power states that > power is the fraction of work done in a given time and as per the definition of work it is described as the dot product of force and displacement and from newton's second law we get F=ma the displacement of all the three is same as their end point. so the product of mass and acceleration must be same for all and in this situation a normal man will have less weight than horse and car so it clearly states that acceleration must be of more magnitude for man. but to my surprise he said my answer is wrong and he did'nt explained how it was wrong am i really wrong? please guide me if i am wrong in which aspect."}
{"id":"110718","title":"When to use which representation for an electric field","text":"In class we covered three types of possibilities to evaluate the electric field for static problems. Unfortunately, most physics textbooks cover these ways without addressing the question of applicability when they introduce these equations due to historical motivation. One possibility is Gauß's law: $$\\int_{\\partial V} \\langle E, n \\rangle dS = \\int_V \\frac{\\rho}{\\varepsilon_0} dx$$ This can only be used, when $E$ does by geometrical consideration not depend on the variables that the surface integral depends on(like in spherical symmetry, when it is not a function of the angles, but just of the radius), then it can be used to determine the electric field by a charged sphere. What we can get then, is an electric field on the surface of some volume that contains a particular charge. Then we have: $$ E(x) = \\frac{1}{4 \\pi \\varepsilon_0} \\int_{\\mathbb{R}^3} \\frac{\\rho(x')(x-x')}{||x-x'||^3} dx'.$$ This is more general in the sense that it does not require any symmetry or reduction to charges inside some volume. And finally we have $$\\Delta \\phi(x) = -\\frac{\\rho(x)}{\\varepsilon_0},$$ which is the most general way to think about electrostatic problems and contains (assuming that the solution fulfills all boundary conditions) all information. I hope my understanding was correct so far. In that case, my question is: How do I distinguish boundary value problems from problems where the first two equation apply? Many problems are like: Given a sphere with a particular charge distribution, calculate the field everywhere around! What does this mean, does this mean that I should think of this as a ball of charges assembling in vacuum and the material around being vacuum too? Cause if I assume that this is a metallic sphere or a sphere made out of some material which is different from the material (probably varcuum) around, then this would give me an interface which means that my first two equations don't apply anymore, as I would get boundary conditions. ( I am talking now about this in a strict theoretical sense, maybe they would give me the right result, but for the wrong reasons). Or do these two equations also apply, if the ball is an insulator?"}
{"id":"19193","title":"Gears and efficiency\/speed","text":"For Christmas me and my brother got an AR Drone, the propellers are on a large gear, which is rotated by a small gear that is connected to the motor. This means the small gear (and therefore the motor) must rotate many times in order to rotate the propeller just once. Surely swapping the gears, so that the propellers are on a small gear, which is rotated by the large gear that is connected to the motor, would mean the propeller would spin faster (given that the motor is operating at the same speed as before), or that the motor could operate at a slower speed (and therefore use less energy) and maintain the same speed of the propeller as before? Also what is the advantage of using any gears at all? Why not lose the gears, and get rid of the friction between the gears and between the gear and what holding them, and the added weight?"}
{"id":"19196","title":"How Relative Is Relativity With Respect To Visual Effects?","text":"This might seem naive, but here goes. Imagine Albert and Rick, of equal mass, accelerate together to a significant fraction of light-speed (call it v) relative to Earth, enough to clearly see the effects of light aberration when looking at the stars. They expend 2 units of energy to do this. The two are now in an inertial frame of reference. Can Albert now accelerate further so that his speed relative to Rick is v, expending 1 unit of energy? How can you be in an inertial frame of reference, and still witness light- aberration effects?"}
{"id":"117102","title":"Why do heavier objects fall faster in air?","text":"We all know that in an idealised world all objects accelerate at the same rate when dropped regardless of their mass. We also know that in reality (or more accurately, in air) a lead feather falls much faster than a duck's feather with exactly the same dimensions\/structure etc. A loose explanation is that the increased mass of the lead feather somehow defeats the air resistance more effectively than the duck's feather. Is there a more formal mathematical explanation for why one falls faster than the other?"}
{"id":"126891","title":"Mathematica package for supergravity and string theory","text":"I am looking for a Mathematica package that can manipulate tensors for supergravity, string theory or M-theory. I am particularly looking for a package that can do spinor and Clifford algebra computations. Also, I would like this package to be able to do wedge and hodge dual, and other computation relating to forms. Can anyone suggest a specific one? I looked for atlas2, but it seems I have to pay to use it without a trial version."}
{"id":"65616","title":"Magnitude of a photon?","text":"I encountered the following sentence in my textbook, which I don't quite understand, and after an unfruitful google search, I still can't figure out what they mean by **magnitude** in this context: > ...\"split a photon into N components with magnitude ϵ\"?"}
{"id":"65610","title":"Can a link between photons that don't exist at the same time provide communication with the past?","text":"They have published something about a link between photons that don't exist at the same time. Does this means that it is possible to build a device that will receive messages from itself but these messages will be received earlier than they will be sent? Can someone operating this device can win the lottery? How would you describe in numbers the bandwidth of such device? Larger than infinite? Negative?"}
{"id":"69919","title":"Will current pass without any resistance?","text":"I've learned that a resistor converts some electrical energy into heat energy while the current flows through it and thus causes a power loss, but what if there's not any resistor in a circuit. Will current still flow?"}
{"id":"81444","title":"Which is more characteristic of an element absorption or emission spectrum?","text":"Absorption and emission spectrum are used to reveal the identity of an element and even to find new elements. But, which of them is more characteristic? The mission spectrum would have more lines than the no. of lines missing in the absorption spectra simply because de-excitation can occur from any higher energy level but excitation always starts from the first energy level. So is absorption spectrum more characteristic than emission spectrum?"}
{"id":"81443","title":"Is projectile trajectory on Earth affected by the fact that it isn't a singular point?","text":"So in high school physics, we were told that projectiles traveled in a parabola, but Google searches have said that it's more like an ellipse, since the Earth isn't flat. However, I was thinking that since most projectiles stay so close to Earth, you can't really assume that gravity acts only towards the center, since gravity dictates that all particles are attracted to each other, so the core isn't the only thing on earth pulling the object \"down\". Is this logic valid? High School: $$d = v_{0}t + \\frac{1}{2}at^2$$"}
{"id":"5785","title":"Why are the Higgs phase and the confinement phase identical in Yang-Mills-Higgs systems?","text":"If we couple a Yang-Mills theory with a Higgs field and some quarks in the fundamental representation, we can have a Higgs phase and a confining phase. However, they are indistinguishable. The Wilson loops scale according to the perimeter law, not the area law in the confining phase because of hadronization. In the Higgs phase, there is no cluster decomposition for the Higgs field because distant points have to be connected by a Wilson line with exponential falloff as the length of the Wilson line. Why are both phases indistinguishable?"}
{"id":"60393","title":"Runge-Lenz vector and Keplerian Orbits","text":"Is the loss of _closed_ Keplerian orbits in relativistic mechanics directly tied to the absence of the Runge-Lenz vector?"}
{"id":"94247","title":"Does \"Crack The Whip\" actually speed you up?","text":"\"Crack The Whip\" is a game played on ice skating rinks where several individuals line up all facing the opposite end of the rink, and skate forward. When the group reaches the opposite end of the rink, the \"point man\" or the person on one end of the line stops, and everyone else pivots around him. Supposedly, the person on the opposite end swings around, and rockets forward at much greater speeds. However, I don't believe this is possible. Because the question remains, where would the extra speed come from? I was told that the momentum of all the people is transferred into the one person on the end. While there is a ponderous amount of outward force on the chain of people, I don't think it's possible to actually gain speed. And after watching my friends do it several times, I concluded the person on the end didn't seem to go any faster. Was it simply that the process was inefficient? Or is the theory even possible?"}
{"id":"19224","title":"Is it possible to create a hologram using X-Rays?","text":"Is it possible to create a hologram using X-Rays (or other penetrating EM- waves)?"}
{"id":"19222","title":"reflection, refraction and diffraction occur in radio waves, which one occurs the most?","text":"One of the things about Radio waves is that they undergo different behaviors such as reflection, refraction and diffraction when in contact with obstacles such as walls, buildings etc. what i want to understand is that, which of the behaviors would occur the most in let's say an environment filled with trees and the slope is rough?"}
{"id":"11431","title":"Finding Directions using moon","text":"Is it possible to find directions just by looking at moon(not full bright)? I was curious to use the clue that if moon is half lit then the direction in which it is lit will it be east?"}
{"id":"11436","title":"The acceleration of a particle in uniform electro-magnetic field","text":"A particle of charge $q$ and mass $m_0$ is subject to the action of a uniform electro-magnetic field $\\vec{E}=(0,E,0), \\vec{B}=(0,0,B)$ and at $t = 0$ moves with velocity $\\vec{v}=(v_1,v_2,v_3)$. What is the acceleration at $t = 0$ if $v_1 = v_3 = 0$?"}
{"id":"17102","title":"Does a Weak Energy Condition Violation Typically Lead to Causality Violation?","text":"In the answer to this question: ergosphere treadmills Lubos Motl suggested a straightforward argument, based on the special theory of relativity, to argue that light passing through a strong gravitational region which reaches infinity cannot reach infinity faster than a parallel light ray which does not pass through the strong gravitational region. The argument is just the standard special relativistic one, that if you can go faster than light, you can build a time-machine. To do this, you just have to boost the configuration that allows you to go faster than light, and traverse it in one direction, then boost it in the other direction, and go back, and you have a closed timelike curve. In the condition of the question, where the strong gravitational region is localized, you can do this using pre-boosted versions of the solution. Preboost two far away copies of the solution in opposite direction, one boosted to go very fast from point B to point A (at infinity) and the other boosted to go from point A to point B. Then if you traverse a path from A to B, crossing through the first boosted solution, then cross the second on the way back, you can make a CTC. I was skeptical of this argument, because the argument I gave required the Weak energy condition, not a no CTC condition. The argument from null energy just says that a lightfront focuses only, it doesn't defocus, the area doesn't ever spread out. Any violation of weak energy can be used to spread out a light front a little bit, and this corresponds to the geodesics passing through the violating point going a little bit faster than light, relative to the parallel geodesics nearby. So if both arguments are correct (and although I was skeptical of Lubos's argument at first, I now believe it is correct), this means that a generic violation of the weak energy condition which is not hidden behind a horizon could be turned into a time machine. Is this true? Question: Can you generically turn a naked violation of weak energy condition into a closed timelike curve? A perhaps more easily answerable version: If you have an asymptotically flat solution of GR where there is a special light ray going from point A to point B which altogether outruns the asymptotic neighbors, can you use this to build a time machine, using boosted versions of the solution?"}
{"id":"91563","title":"Finding out the minimum deviation angle","text":"![enter image description here](http:\/\/i.stack.imgur.com\/nsFgB.png) Let a light incident on a prisom at right angle. I want to determine the minimum deviation angle. I know a relation $$ \\frac{\\cos i_1}{\\cos r_1} = \\frac{\\sin i_1}{\\sin r_1}$$ and minimum angle of deviation $$\\delta_m = i_1 +i_2 -A$$ It can be seen that, $i_1$ is given but how to get $i_2$ and A. A is the prism angle. and $i_1 =90 $ and $i_2 $ is the angle of the outgoing ray."}
{"id":"80791","title":"Could the speed of light change outside our solar system","text":"Theory: The speed of light changes when it enters or exits the solar system due to a difference in medium (dark matter possibly). Potential problem 1: refraction If there was a speed change at the edge of the solar system, refraction would occur. This would manifest in ways similar to the sunset effect. Solution: the sunset refraction occurs because the angle against the edge of the atmosphere is extreme. No effect is noticed at high noon. Earth's proximity to the edge of the solar system, in contrast, makes it's angle nearly perpendicular in all directions. Wouldn't this negate any sunset effect? Sub-solution: My research on refraction seems to indicate that what I'll call the \"distinction level\" of the edge of the mediums has an effect on the level of refraction. So that, if the edge were not solid, but rather tapered, the refraction is reduced. SO, what would be effect on light entering our solar system if the light were severely slowed--let's say by a factor of 1,000,000 times, to what it is now. What, if any, refractive effects would be observed? How much would these effects change if the tapering of the edge were severe?--Could any refractive effect be overcome by tapering? Another question, what would be the effect of the light coming from Alpha Centauri if it's system had a similar medium transition at it's edge? How would it be noticable? This picture illustrates my thoughts: ![refraction](http:\/\/i.stack.imgur.com\/FSZzo.jpg)"}
{"id":"99249","title":"Understanding Normal Force Change","text":"_I know this has been asked around, my question is a bit more specialized towards my personal problem._ If I have an object the magnitude of the weight is usually equal to the normal force, right? I could be mistaken but from what I have read the normal force is the force the object is feeling opposite to the $(mass*gravity)$. If that is the case, and I put my hand on the object does my Force get added to the object's weight and thus the normal force of that object increases? Therefore if I keep adding hands to the object will the normal force keep compensating? Also is the normal force the negative charged electrons repelling each other (electromagnetism)? ![enter image description here](http:\/\/i.stack.imgur.com\/sPXjk.jpg)"}
{"id":"25509","title":"How far into space does one have to travel to see the entire face of earth?","text":"Virgin Galactic will take passengers aboard SpaceShipTwo as high as 65 miles above the surface of the earth. But from this altitude, passengers will only be able to see a certain segment of the curvature of the earth through windows as large as 17 inches in diameter. How much further into space would SpaceShipTwo have to travel to give passengers a view of the entire sphere of earth through one of these windows?"}
{"id":"64250","title":"Why is $r'\/r^2 = -1\/r$?","text":"If $r=r(t)$, why is $\\frac{r'(t)}{(r(t))^2}$ = $\\frac{1}{r(t)}$ where $'$ denotes the derivative? I saw it in a lecture. Can you please explain?"}
{"id":"64254","title":"Will the black hole evaporate in finite time from external observer's perspective?","text":"There is the problem that is bothering me with the black hole evaporation because of Hawking radiation. According to Hawking theory the black hole will evaporate in finite time because of quantum effects randomly producing particle-antiparticle pairs on the opposite sites of horizon. However, the gravity causes the time dilatation, which goes to infinity on the horizon itself, so the arbitrarily small amount of time from horizon's perspective is going to be infinitely long from external observer's perspective? So, from which perspective is the time the black hole will evaporate calculated? If it is calculated from horizon's perspective, it would mean, the black hole will never evaporate from the external observer's perspective?"}
{"id":"16855","title":"Is the equivalence principle strictly fulfilled by general relativity?","text":"The equivalence principle states _The outcome of any local experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime._ Any real local experiment needs some finite space, since nobody is a point particle himself. In principle I will always feel some deviation. Since in (pseudo-)Riemannian geometry I can only change to flat coordnates in a single point $p$ of the spacetime manifold (and not in a finite neighborhood), is the deduction right, that the equivalence princpil is only fulfilled by general relativity in an approximative sense?"}
{"id":"16851","title":"In layman's terms, what is a quantum fluctuation?","text":"What causes it and how does it occur? If you do post some mathematics, please explain what each term means too please."}
{"id":"16852","title":"Is 4-velocity normalized to -1 even for non-geodesic timelike curves?","text":"In Hartle's General Relativity book (\"Gravity\"), one of the problems (chapter 8 problem 6) is to prove that $g_{\\mu\\nu}u^\\mu u^\\nu$ is conserved along geodesics (really not hard to show), where $u^\\mu$ is the 4-velocity. My question is: Isn't it true that $g_{\\mu\\nu}u^\\mu u^\\nu$ is equal to $-1$ for _any_ timelike curve whether it is a geodesic or not? This follows (I think) from $$ g_{\\mu\\nu}u^\\mu u^\\nu = g_{\\mu\\nu}\\frac{dx^\\mu}{d\\tau} \\frac{dx^\\nu}{d\\tau} = \\frac{g_{\\mu\\nu}dx^\\mu dx^\\nu}{d\\tau^2}=\\frac{ds^2}{d\\tau^2} = \\frac{-d\\tau^2}{d\\tau^2} = -1. $$ Am I wrong about this? Why should we need the geodesic equation to prove this if it's true for any timelike curve?"}
{"id":"16859","title":"Where are the ice rings in X-ray crystallography located?","text":"I threw this image of one of my protein crystal's diffraction pattern (not the greatest) on a poster, and was wondering where (what reciprocal dimension) the ice rings are located as it might be a useful reference (or talking point). While I can find information about water ice's unit cell parameters (4.5 Å, 7.3 Å), I'm not sure if these are where the reflections would be located (the inner two rings seem far too close for that disparity, and I believe the outer two rings are at a much higher resolution (< 3 Å) ![enter image description here](http:\/\/i.stack.imgur.com\/FTPb4.png)"}
{"id":"95594","title":"Non-separable solution for the Schrödinger equation","text":"Schrödinger solutions are usually if not always of the type: $\\psi=\\operatorname{T}(t)*\\operatorname{X}(x)$ (we use the separation of variables method to arrive at the time independent Schroedinger equation). I was trying to find a non-separable solution. To this purpose I tried the following method: compose the product T(t)*X(x) into another function. For example: $\\sinh(\\operatorname{T}(t) \\cdot \\operatorname{X}(x))$ or $\\ln(\\operatorname{T}(t) \\cdot \\operatorname{X}(x))$ or $(\\operatorname{T}(t) \\cdot \\operatorname{X}(x))^2$ and so on. Then I try first time derivative of $\\psi = \\mathrm{second}$ position derivative of $\\psi$ (trying to find a solution for the null potential. In addition I know I need a constant but it's for simplification). I arrive at a differential equation. I try simple solutions for one of the functions like $\\operatorname{T}(t)=t$. I get to a very difficult differential equation that can't be solved even with a look at: http:\/\/www.amazon.com\/Handbook-Solutions-Ordinary-Differential- Equations\/dp\/1584882972 If there any obvious non separable solution that I am missing?"}
{"id":"46594","title":"paper about black branes and implications to 4d black holes","text":"This paper makes a case for piezoelectric response (electric dipole moment under mechanical oscillations) of black branes. This paper does not make an implication of their results for 4D black holes (do their results apply or not apply to them?). Can someone explain if this implies something for usual 4D black holes in De Sitter space? Do \"standard\" black holes have such a response as well?"}
{"id":"53643","title":"Any example of lower symmetry in high temperature phase than the low temperature phase?","text":"All the phase transition cases I came across so far have this property: the lower temperature phase has lower symmetry than the higher temperature one. But it is nowhere explicitly said that, lower temperature phase always has lower symmetry than the higher temperature phase. So, I was thinking, is there any counter-example to show that higher temperature phase in a phase transition can have lower symmetry too?"}
{"id":"21763","title":"Is a macroscopic pair where I observe one of them quantum entanglement?","text":"Of what little I know\/understand about quantum entanglement can somebody confirm if the following experiment is a good analogy to quantum entanglement of pair of particles? PS: please don't laugh as this could be very very lame! I take an orange and an apple (of similar shape & weight). I put each of them in a non transparent bag separately and seal it off. I put both the bags in a box. I close my eyes and shake it thoroughly such that I no longer know which bag contains what fruit. Then I randomly pick one bag and take a flight to other part of the country. Now given the above setup, I will not know what fruit is in my bag. In other words the fruit in my bag could be both apple\/orange at the same time until I open the bag and see what is inside. As soon as I open it it is determined (similar to wave function collapsing) that I have an orange (for example) and there by making the fruit in the bag that was left in the box an apple (or vice versa). Does this in anyway come close to what they are doing with quantum entanglement?"}
{"id":"8903","title":"What is the difference between Newtonian and Lagrangian mechanics in a nutshell?","text":"What is Lagrangian mechanics, and what's the difference compared to Newtonian mechanics? I'm a mathematician\/computer scientist, not a physicist, so I'm kind of looking for something like the explanation of the Lagrangian formulation of mechanics you'd give to someone who just finished a semester of college physics. Things I'm hoping to have explained to me:   * What's the overall difference in layman's terms? From what I've read so far, it sounds like Newtonian mechanics takes a more local \"cause-and-effect\"\/\"apply a force, get a reaction\" view, while Lagrangian mechanics takes a more global \"minimize this quantity\" view. Or, to put it more axiomatically, Newtonian mechanics starts with Newton's three laws of motion, while Lagrangian mechanics starts with the Principle of Least Action.   * How do the approaches differ mathematically\/when you're trying to solve a problem? Kind of similar to above, I'm guessing that Newtonian solutions start with drawing a bunch of force vectors, while Lagrangian solutions start with defining some function (calculating the Lagrangian...?) you want to minimize, but I really have no idea.   * What are the pros\/cons of each approach? What questions are more naturally solved in each? For example, I believe Fermat's Principle of Least Time is something that's very naturally explained in Lagrangian mechanics (\"minimize the time it takes to get between these two points\"), but more difficult to explain in Newtonian mechanics since it requires knowing your endpoint."}
{"id":"26939","title":"Models of higher Chern-Simons type","text":"It has long been clear that (the action functional of) Chern-Simons theory has various higher analogs and variations of interest. This includes of course traditional higher dimensional Chern-Simons theory (abelian and non-abelian) as well as the algebroid-version: the Courant-sigma model, but also seemingly more remote systems such as string field theory (and hence essentialy also its effective truncations), a fact that is already somewhat remarkable. In a recent article we claimed that there is a systematic sense in which also all AKSZ sigma-models are special cases of a general abstract notion of \"infinity-Chern-Simons theory\". These AKSZ models include, in turn, also the Poisson sigma-model (hence also the A-model and the B-model). Also BF-theory coupled to topological Yang-Mills theory fits in. Therefore in a precise sense all these systems are examples of a single underlying basic mechanism. My **question** is: can you point out other models of interest in the literature (or in your drawer) that look like they might be \"of generalized Chern-Simons type\", along these lines? (I am not just looking, say, for \"Chern-Simons term\"-summands in higher supergravity actions, even though these are related, but for example new variants of full higher dimensional Chern-Simons (super)gravity.) For instance: has there been a proposal for a nonabelian 7-dimensional Chern- Simons-type model that might be the holographic partner to the self-dual nonabelian 6d (2,0)-superconformal QFT (so that the state spaces of the former are the conformal blocks of the latter)? While we did come across a natural non-abelian 7-dimensional Chern-Simons type TQFT whose fields are string-2-connections (here), I am not sure how to see if this might be the relevant one. Do you?"}
{"id":"49905","title":"What causes an aircraft to roll when rudder is applied","text":"When continuous rudder is applied in a typical light aircraft during straight and level flight at \"normal\" flying speeds and altitudes, the primary effect is that the aircraft will yaw to the left - so it's flying \"sideways\" to some extent. It is commonly understood amongst pilots that the secondary effect in most aircraft, is that the aircraft will also start rolling in the direction of yaw (i.e. applying left rudder causes left yaw, and left roll). What causes this roll? And does the Dihedral angle have anything to do with it?"}
{"id":"67702","title":"Calculating rotor torque out of mass center","text":"Suppose I have an engine with propeller. I know there is aerodynamic force pushing the propeller opposite direction the engine is turning it. One component of this force is creating lift, but second is braking the prop. Now these forces are applied at some distance from engine shaft, thus they're creating torque, which is turning the whole engine if it wasn't fit somewhere. And now my problem: how to describe what happens, after fitting this engine out of mass center. For example take a ruler, that has pivot somewhere and at the end of it, there is this engine with propeller. How do I describe momentum at the center of mass - the pivot, instead at the engine (engine has no weight in this example)?"}
{"id":"24274","title":"String theory landscape and false vacua","text":"What is the multiverse of string theory? I always thought it would be from brane cosmology where there's chances for multiple collisions of the branes that can make a universe. So this \"multiverse\" would be in the \"bulk\". But when I read up on the string theory landscape it implies a relation to bubble cosmology and false vacua? How does that work? What's the connection? There's no need for strings in bubble cosmology and no need for bubble nucleation in brane cosmology so... what's going on?"}
{"id":"102730","title":"Electrons skip randomly around their orbits","text":"I read where the electron (as well as a few other particles) skips around in its orbit randomly rather than move around the orbit smoothly. This effect has been repeatedly observed in the laboratory over the years. I know if we treat this using quantum physics the electron exist in a fuzzy cloud of probability but in the old days before quantum physics what were they seeing in the laboratories when they found this effect?"}
{"id":"66255","title":"Electric field a distance $z$ above the center of a circular loop. The Hard way","text":"> _Problem 2.5: Find the electric field a distance $z$ above the center of a > circular loop of radius $r$ which carries a uniform line charge $\\lambda$._ This problem is in refereced here (with solution): http:\/\/teacher.pas.rochester.edu\/PHY217\/LectureNotes\/Chapter2\/LectureNotesChapter2.html But it is solved using a trick. The trick is: okay we know there is symmetry so let us just take the magnitude and not use the vectors. The problem I have is I am unable to solve it without using the trick and I would appreciate someones help in doing the problem without the shortcut, using the vectors. When I do it the r compontent doesn't cancel. Using polar coordinates here is the setup: $$dE = \\frac{1}{4 \\pi \\epsilon_0} \\frac{\\lambda}{\\rho^2} ~ \\hat{\\rho} ~ r d \\theta$$ $$\\hat{\\rho} = \\frac{z \\hat{k} + r \\hat{r}}{\\sqrt{z^2 + r^2}} $$ $$ E = \\frac{\\lambda}{4 \\pi \\epsilon_0} \\int_0^{2 \\pi} \\frac{r z \\hat{z} + r^2 \\hat{r} }{(z^2 + r^2)^{3\/2}} ~ d \\theta$$ $$ E = \\frac{\\lambda}{4 \\pi \\epsilon_0} \\int_0^{2 \\pi} \\frac{r z \\hat{z} }{(z^2 + r^2)^{3\/2}} ~ d \\theta + \\frac{\\lambda}{4 \\pi \\epsilon_0} \\int_0^{2 \\pi} \\frac{r^2 \\hat{r} }{(z^2 + r^2)^{3\/2}} ~ d \\theta $$ If I now evaluate the $z$ component I get the correct answer. When I evaluate the $r$ component, shown below the $r$ doesn't not cancel the way it should. why not? $$\\frac{\\lambda}{4 \\pi \\epsilon_0} \\frac{r^2 2 \\pi }{(z^2 + r^2)^{3\/2}} - \\frac{\\lambda}{4 \\pi \\epsilon_0} \\frac{r^2 0 }{(z^2 + r^2)^{3\/2}} $$ $$ = \\frac{\\lambda}{4 \\pi \\epsilon_0} \\frac{r^2 2 \\pi }{(z^2 + r^2)^{3\/2}} \\hat{r}$$"}
{"id":"79049","title":"Cross Section Peskin vs Srednicki","text":"in Peskin Schroeder after the derivation of the differential cross section there is a comment for the central mass system (CMS), which says: In the special case, where all four particles have identical masses (...), this [the general formula] reduces to the formula[...] (p.107): $$ \\left(\\frac{d\\sigma}{d\\Omega}\\right)_{CM}=\\frac{\\left|\\mathcal{M}\\right|^2}{64\\pi^2E_{CM}^2}$$ However in Srednicki the general formula for the differential cross section in the CMS is (p.97): $$\\left(\\frac{d\\sigma}{d\\Omega}\\right)_{CM}=\\frac{\\left|\\mathcal{M}\\right|^2}{64\\pi^2E_{CM}^2}\\frac{\\left|\\textbf{k}^\\prime\\right|}{\\left|\\textbf{k}\\right|} $$ where $\\left|\\textbf{k}^\\prime\\right|$ is the outgoing three-momentum in the CMS and $\\left|\\textbf{k}\\right|$ the incoming. Now to get from the formula from Srednicki to Peskin's formula I don't need the masses to be all the same but merely the incoming masses equal to the outgoing masses, so to say elastic scattering. I didn't see a further restriction in Srednicki's formula apart from being in the CMS. If I take e.g. Compton scattering I get with Srednicki's formula: $$\\left(\\frac{d\\sigma}{d\\Omega}\\right)_{CM}=\\frac{\\left|\\mathcal{M}\\right|^2}{64\\pi^2E_{CM}^2} $$ but Peskin gets on page 164 with his formula for the differential cross section: $$\\left(\\frac{d\\sigma}{d\\Omega}\\right)_{CM}=\\frac{1}{32} \\frac{1}{E_1E_2}\\frac{\\left|\\textbf{k}^\\prime\\right|}{E_{CM}}$$ where $E_1+E_2=E_{CM}$. I don't see, that this is equal? Is it actually equal? Where did I miss something?"}
{"id":"72019","title":"What is the relation between the representation the Higgs field transforms under, the types of couplings in the theory and Higgs\/Coulomb branches?","text":"When reading about Higgs and Coulomb 'phases' I came across two separate definitions: The first tells us that the Higgs\/Coulomb phases are determined by the representation that the Higgs field transforms under, as explained on Wikipedia. The Higgs field breaks the gauge symmetry; If the Higgs is in the adjoint then we're left with U(1)s and we have a Coulomb phase, if the Higgs transforms in any other representation then (typically) no U(1)s are left and we have a Higgs phase. However, the second definition tells us that the these phases are determined by the potential between two electric test charges (as read in Sannino's book). A potential proportional to 1\/r is the Coulomb phase and a constant potential gives the Higgs phase. Also, there are other phases: Confining, free-electric, free-magnetic and perhaps others. Question 1) Is there a link between these two definitions? I can see why the Coulomb phase as described in the Wikipedia article might lead to a 1\/r potential (due to the surviving U(1) groups). But why would the Wikipedia definition of a Higgs phase lead to a constant potential? Furthermore, if there is a link, what sort of phase (As in the Wikipedia article) would give us confining, free-electric and free-magnetic potentials? Then there are Higgs and Coulomb 'branches'. These are the moduli (scalar fields) of the N=2 hyermultiplet and vector multiplet respectively. Question 2) Do these link in with the phases described above or are the concepts of branches and phases very much distinct? Christian Samann's notes tell us (on page 13) that the Coulomb _branch_ is what we get when the gauge group breaks to U(1)'s. I think this is just an inconsistency in what different authors mean when they talk about branches and phases."}
{"id":"1402","title":"Is it possible to separate the poles of a magnet?","text":"It might seem common sense that when we split a magnet we get 2 magnets with their own N-S poles. But somehow, I find it hard to accept this fact.(Which I now know is stated by Gauss's Law) I have had this doubt ever since reading about the quantum-field-theory and I know I might sound crazy but is it really impossible to separate the poles of a magnet? **What about black holes**? They attract matter and everything else only inwards into them, right? So then, are they the only such case\/exception? Am I being naive? Is there some proof\/explanation that can give closure to this question of an independently existing pole?"}
{"id":"79042","title":"Current Carrying Wire Loop near Current Carrying Long Wire","text":"A rectangular wire loop of (width $a$ and height $b$) carrying clockwise current $I_1$ is a distance $d$ below a horizontal infinitely long wire carrying a current $I_2$ to the right. What is the force on the wire loop? I am not sure what equations I have to use here, it's been quite awhile. I know the magnetic field infinite wire is something like $\\frac{\\mu_0}{2\\pi} \\frac{I_2}{r}$ And F=v x B. For the horizontal segment of the loop nearest the wire, does $v=\\frac{I_1 a}{e}$? Is that right? does F = $\\frac{\\mu_0}{2\\pi} \\frac{I_2 I_1 a}{d e} - \\frac{\\mu_0}{2\\pi} \\frac{I_2 I_1 a}{(d+b)e} = \\frac{\\mu_0}{2\\pi e}I_2 I_1 a (\\frac{b}{d(d+b)})$ towards the wire. Is that the right answer?"}
{"id":"106969","title":"Wave or particle in the end","text":"In Young's double slit experiment I put a convex lens after double slit and direct interference pattern on to a fibre. Then I expose the other end of fibre to a screen. What will I observe - the interference pattern (slightly jumbled) or a dot? Will photons behave as wave or particle?"}
{"id":"107557","title":"Anti-symmetric forms on Dirac spinors","text":"In order to describe invariant forms on Dirac spinors $S$ one can find trivial subrepresentations in $S \\otimes S$. If we use $S \\cong (1\/2, 0) \\oplus (0, 1\/2)$ then \\begin{multline} [(1\/2, 0) \\oplus (0, 1\/2)] \\otimes [(1\/2, 0) \\oplus (0, 1\/2)] =\\\\\\ (0, 0) \\oplus (1, 0) \\oplus (1\/2, 1\/2) \\oplus (1\/2, 1\/2) \\oplus (0, 1) \\oplus (0, 0) \\end{multline} Therefore representation theory predicts existence of two invariant forms. It is usually claimed that this two forms are $$ D_1(\\chi, \\psi)=\\bar{\\chi}\\psi=\\chi^T\\gamma_0\\psi=\\chi_R^T\\psi_L+\\chi_L^T\\psi_R $$ and $$ D_2(\\chi, \\psi)=\\bar{\\chi}\\gamma_5\\psi=\\chi^T\\gamma_0 \\gamma_5\\psi=\\chi_R^T\\psi_L-\\chi_L^T\\psi_R $$ The form $D_1$ is symmetric and it's quadratic form (with complex conjugation on the first argument) is usually used to construct Dirac's Lagrangian. From the other hand, it is known that on Weyl spinors one can also find an antisymmetric invariant forms given as $$ \\chi^T_L\\sigma_2\\psi_L . $$ Let me use this to construct one more anti-symmetric invariant form on Dirac spinors as a sum of two forms on Weyl spinors $$ D_3(\\chi, \\psi)=\\chi^T_L\\sigma_2\\psi_L+\\chi^T_R\\sigma_2\\psi_R $$ Form $D_3$ is not a linear combination of $D_1$ and $D_2$ and thus I get a contradiction with representation theory prediction. Where I made a mistake?"}
{"id":"93665","title":"Ricci scalar higher dimensions","text":"I was wondering if there is a straightforward way to compute the Ricci curvature of a metric that has the form (à la Kaluza-Klein): $g_{MM}\\equiv\\begin{pmatrix}g_{\\mu\\nu}&g_{\\mu i}\\\\\\g_{i\\nu}&g_{ij}\\end{pmatrix}$ Ie, finding an expression of the form: $\\mathcal{R}(G_{MN})=R(g_{\\mu\\nu})+R(g_{ij})+(\\text{scalar depending on both metrics})$ I tried using the usual formula $R^M_{NPQ}=2\\partial_{[Q}\\Gamma^M_{P]N}+2\\Gamma^S_{M[P}\\Gamma^M_{Q]S}$ where $A_{[MN]}=\\frac{1}{2}(A_{MN}-A_{NM})$. But this becomes really quickly a mess due to the different indices. Does anyone has an idea how to find the result quickly? For a kaluza klein metric where $g_{ij}$ is a scalar and $g_{\\mu j}$ is a vector this is straightforward, but I can't find a nice solution."}
{"id":"93666","title":"What is going on with star's energy?","text":"The sun radiates $3.8 × 10^{26} J$ energy each second, this energy is absorbed by various celestial objects but most of it is reflected or emitted back in the space, similarly there must be energy pouring into empty space from various star systems, what is happening to all this energy ? In what form is it thriving if it is thriving ? Does it ultimately get absorbed by various objects ? Is it bringing some kind of change to the motion of galaxies etc ? Is there some sort of energy sink somewhere ( maybe getting sucked in by Black holes) ? Does it somehow combine and form new bodies via energy mass equivalence ? **what is happening ?** Addendum : This maybe a on the edge of site's mainstream physics discussion policy, but till now all I understood was that no definitive proof of where the energy is going is available, @ John rennie says that due to red shift the wave's are losing energy, he says that energy density is tending to zero, but there is no account for this energy. At the same time, @Thriveth mathematically asserts that the universe is losing energy, I find it a bit of a controversial topic regarding energy conservation and can't shake off the idea that maybe this energy could account for dark energy\/matter. I add, millions of stars have died\/ emptied their energy into the space since the beginning of time, by the logic of either answer, red shift or violation of energy conservation we can say that this energy is somewhere we can not account for, similarly we till now can't account reasons for dark matter\/energy can't we explain both of these with each other ?"}
{"id":"22300","title":"Black body balloon in vacuum","text":"**The problem statement, all variables and given\/known data** There is a perfectly spherical balloon with surface painted black. It is placed in a perfect vacuum. It is gently inflated with an ideal mono-atomic gas at Kelvin temperature $T_i$, slowly enough so that it reaches thermal equilibrium with the gas, and then it is sealed off. It has radius $r_i$ at this time and contains N atoms. The vacuum is large, so radiation from its walls can be ignored. a) Show that $\\frac{T}{T_i}= \\left(\\frac{r}{r_i}\\right)^3$ if the pressure inside the balloon is independent of its radius. b) How much energy does the balloon radiate per second when it is at radius $r$? Express your answer in terms of $r$ and constants. c) What is the rate of change of the internal energy of the gas? Express you answer in terms of $r$, $r˙$, and constants. **Relevant equations:** $$PV = nRT$$ $$J * A = σT^4 (4\\pi r^2)$$ **The attempt at a solution:** I already showed a), so I don't need help with that. In part b), I wrote pretty much $J * A = σT^4 (4\\pi r^2)$ but I am unsure how to get a speed of radiation. I also have no idea how to do part c), so any help on parts b) or c) is welcome. Thanks!"}
{"id":"61703","title":"Problem in geometrical optics","text":"I was trying to solve a problem taken from an Physics Olympiad when I came across a curious and complex mathematical expression. I can not prove with what I know so far about mathematics, does could anyone tell me if the answer is correct? My answer: $d\\tan(\\arcsin ((\\sin(180-t)*H)\/ \\sqrt(H² + 4d^2 - 4dH\\cos(180-t))))$ The correct answer:$(dH\\sin(t))\/(2d+H\\cos(t))$ The mathematical problem: $d\\tan(\\arcsin ((\\sin(180-t)*H)\/ \\sqrt(H² + 4d^2 - 4dH\\cos(180-t)))) = (dH\\sin(t))\/(2d+H\\cos(t))$ Is it true? My solution: Image 1: http:\/\/img825.imageshack.us\/img825\/4248\/46611781.png               Â = 180° - ϴ          (A’B)² = (AB)² + (AA’)² - 2. (AB). (AA’).cos(180° - ϴ)     A’B = v(H² + 4d² - 4.H.d.cos(180° - ϴ))          sen(Â’)\/AB = sen(Â)\/A’B     sen(Â’) = sen(180° - ϴ).H\/ v(H² + 4d² - 4.H.d.cos(180° - ϴ))     Â’ = arc sen (sen(180° - ϴ).H\/ v(H² + 4d² - 4.H.d.cos(180° - ϴ)))      Image 2: http:\/\/img23.imageshack.us\/img23\/7937\/466117811.png               tg(Â’) = CE\/A’C     tg(Â’) = L\/d     L = d.tg(Â’)     L = d.tg(arc sen (sen(180° - ϴ).H\/ v(H² + 4d² - 4.H.d.cos(180° - ϴ))))"}
{"id":"25650","title":"Why is the universe map shaped like an oval?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/W9EiD.jpg) I understand that the Mollweide projection is used to show the map of the universe. Although I understand how this projection can be interesting for Earth where most populated (and of interest) areas are not at polar latitudes, I imagine that in the sky the distribution of interesting places does not follow the same pattern of accentuating the equatorial plane. Do you think this the best projection to represent the Universe? Possible tags: map cosmos universe projection"}
{"id":"25652","title":"sun-moon-earth anomaly","text":"When one looks at the sun and the moon in the sky together, why is it that the illuminated crescent of the moon does NOT \"point\" at the sun? (More correctly the perpendicular bisector of the straight line joining the end points of the moon's terminator does not point at the sun. Intuitively one feels it should - as that is where the light is coming from. It is always around 20 - 30 degrees \"off target\".)"}
{"id":"74755","title":"What does \"quantum theory forbids promiscuous entanglements\" mean?","text":"The context is this article about black hole firewalls. The phrase appears on page 3. It appears to be saying that only pairs of particles can be entangled, never multiple particles, and that this is essential to the firewall paradox. Of course any number of particles can be entangled, at least in the usual sense of the word. Is Dr. Polchinski using the word in a specialized manner, or has the reporter just misunderstood or over-simplified? If the issue isn't about entanglement, what is it about - something to do with the no-cloning theorem, perhaps?"}
{"id":"74751","title":"Would an octahedral magnet work as a magnetic bottle?","text":"First, let me see if I understand all this: Charged particles are curved by magnetic fields, so if the magnetic field is strong enough, the particles \"follow\" the magnetic field lines, in N or S direction, by spiraling around them: ![particles spiraling around magnetic field lines to the left and to the right](http:\/\/i.stack.imgur.com\/vSXMg.jpg) (source) ![enter image description here](http:\/\/i.stack.imgur.com\/IDCml.gif) (source) If the magnetic field lines get closer together, it can create a magnetic mirror, which affects the rotation of the particles such that they turn back and go the opposite way: ![magnetic mirror](http:\/\/i.stack.imgur.com\/ODBgA.jpg) (source) If you combine 2 magnetic mirrors, you get a magnetic bottle, which can confine charged particles by both spiraling and reflection off the mirrors: ![magnetic mirror and magnetic bottle](http:\/\/i.stack.imgur.com\/XkOZ9.gif) (source) ![magnetic bottle](http:\/\/i.stack.imgur.com\/GlzGZ.jpg) (source) So my question is whether it's possible to combine _multiple_ magnetic bottles\/mirrors into other, more \"spherical\" configurations, and still have it confine particles in the center, such as an octahedral magnet (brown arrows are current, magnets show field direction): ![Octahedral magnet](http:\/\/i.stack.imgur.com\/lp4bE.gif) (source) or a cuboctahedral magnet, or other shapes that can be painted with 2 colors, so the field direction alternates from one face to the next (to avoid trying to create a magnetic monopole field, which obviously wouldn't work)."}
{"id":"78880","title":"Expectation value of total energy for the quantum harmonic oscillator","text":"A particles unnormalized wavefunction is given as $$\\psi(x)=2\\psi_1+\\psi_2+2\\psi_3.$$ How can I find $\\langle E\\rangle $ without calculating $\\langle T\\rangle$ or $\\langle V\\rangle $ first? I'm pretty sure I have $\\langle T\\rangle $ and $\\langle V \\rangle$ though, so is the only way to find $\\langle E\\rangle $ is to add these two values together? Any help would be appreciated."}
{"id":"78886","title":"Is there a known generalization of the Schmidt decomposition based on a maximal set of \"locally recorded branches\"?","text":"I came across an unusual multi-partite generalization of the Schmidt decomposition in my work, which I describe below. Usually, when people say \"a multi-partite Schmidt decomposition\", they mean a sum of product states which is in some sense _minimal_ , and there are different decomposition depending on what metric is used to measure minimality. In contrast, I'm looking at a decomposition of a state into a _maximal_ sum of locally-orthogonal (but not necessarily product) states. I would like to know if this structure has been studied before so I can read more about it. Suppose we are given a normalized quantum state $\\vert \\psi \\rangle$ from a Hilbert space $\\mathcal{H}$ composed of $N$ smaller subsystems $\\mathcal{H}^{(n)}$ that are tensored together: $\\vert \\psi \\rangle \\in \\mathcal{H} = \\mathcal{H}^{(1)} \\otimes \\cdots \\otimes \\mathcal{H}^{(N)}$. Now I am looking for a preferred decomposition $\\vert \\psi \\rangle = \\sum_i \\sqrt{p_i} \\vert \\psi_i \\rangle$ of $\\vert \\psi \\rangle$ expressed as a sum (weighted by the positive values $\\sqrt{p_i}$) of orthonormal vectors $\\vert \\psi_i \\rangle$ such that the $\\vert \\psi_i \\rangle$ live on orthogonal subspaces of _each_ subsystem. More precisely, we require $\\vert \\psi_i \\rangle \\in \\bigotimes_n \\mathcal{H}^{(n)}_i$ where $\\mathcal{H}^{(n)} = \\bigoplus_i \\mathcal{H}^{(n)}_i$ is a decomposition of $\\mathcal{H}^{(n)}$ into orthogonal subspaces. This is interesting because it means that the $i$-conditional reduced density matrices of any subsystem $\\mathcal{H}^{(n)}$, $\\rho^{(n)}_i = \\mathrm{Tr}_{\\overline{\\mathcal{H}^{(n)}}} \\left[ \\vert \\psi_i \\rangle \\langle \\psi_i \\vert \\right]$ are restricted to the same orthogonal subspaces. (Here, the trace is over all spaces except $\\mathcal{H}^{(n)}$.) Therefore, observers can make local measurements on _any_ of the systems $\\mathcal{H}^{(n)}$ and determine which \"branch\" $\\vert \\psi_i \\rangle$ they are on. This structure is special for the following reason. (a) There exists a unique decomposition which maximizes the number of vectors in the sum while still satisfying the requirement of orthogonal records in each of the subsystems. (b) This maximal decomposition can be obtained through a mechanical (i.e. algorithmic) fine-graining procedure; the same decomposition is obtained regardless of the order in which the fine-graining is done. (c) The quantity $E = - \\sum_i p_i \\ln p_i$ is maximized for the maximal decomposition and defines a measure of the _global_ entanglement (with respect to the particular choice of subsystems); if $v$ is such that any of the subsystems are unentangled with the rest, this quantity vanishes. (d) This structure reduces to the Schmidt decomposition (and $E$ reduces to the traditional entropy of entanglement) when $N=2$. (e) This structure, and the corresponding value of $E$, is insensitive to pairwise entangling interactions between subsystems (so long as they do not destroy the orthogonality of the conditional local states $\\rho_i^{(n)}$) even when the strength\/type of these entangling interactions are _conditional_ on the branch $i$. So: has this been explored before? * * * (I also submitted essentially the same question in math notation here.)"}
{"id":"32359","title":"About an electrostatics integral and a delta-function kernel","text":"I'm having trouble with an integral and I would like some pointers on how to \"take\" it: $$ \\int \\limits_{-\\infty}^{\\infty}\\frac{3\\gamma a^{2}d^{3}\\mathbf r}{4 \\pi \\left( r^{2} + \\frac{\\gamma^{2}}{c^{2}}(\\mathbf r \\cdot \\mathbf u)^{2} + a^{2}\\right)^{\\frac{5}{2}}} $$ Here $\\mathbf u$, $a$ and $\\gamma$ are constants, and the integrand converges to a Dirac delta $\\delta(\\mathbf r)$ as $a\\rightarrow 0$. The integral must be equal to 1."}
{"id":"104615","title":"Harmonic frequencies of electric guitar are detuned. What could be the cause?","text":"I have tried to record some music with my electric guitar and while playing around and looking the frequencies of single strings I noticed something strange. I took the spectra of the A string and I expected multiples of 440 Hz, but the Results were of by 6 Hz. I recorded again with a tuning fork and I got the 440 Hz as expected.   I used an electric tuning device to tune my guitar and it should have been perfectly tuned. What does this result mean? Is my tuning device broken or is there something I am missing? ![A String](http:\/\/i.imgur.com\/pAwAsxU.png) ![Tuning Fork](http:\/\/i.imgur.com\/cESDWvA.png) Here is the raw data from the 434 Hz peak:   Frequency[Hz] Dezibel[db]   ...   430,664063 -80,541397   431,336975 -77,470108   432,009888 -70,965828   432,682800 -62,022697   433,355713 -51,955708   434,028625 -38,634003   434,701538 -32,274822   435,374451 -34,905651   436,047363 -48,552521   436,720276 -55,133629   437,393188 -59,501816   438,066101 -72,979942   438,739014 -74,478226   439,411926 -83,606758   440,084839 -86,944931   440,757751 -85,912453   441,430664 -81,394493   442,103577 -93,493988   442,776489 -81,689384   443,449402 -83,698189   444,122314 -95,337143   444,795227 -106,526367   ..."}
{"id":"77565","title":"If matter and light have dual-nature, shouldn't we able to explain the observed phenomenons using either wave or particle?","text":"Historically wave and particle has been perceived as totally different phenomenons (before 20th century). Now is it widely accepted and there are experimental results to show that in fact both matter and light have a dual nature. Let's take light for example. Depending on the experiment it could behave as a wave(as in interference and diffraction) and it could be taken as a particle (as in explanation of photoelectric effect). If it has properties of both wave and particle at the same time, shouldn't we be able to explain the experiments with both wave and particle nature and not selecting either wave or particle? **Why particle or wave, why not particle and wave?** Please help me understand it and this question has been bugging me for quite some time now. I understand the standard explanation of both but I am having trouble putting them together. If there is anything I am missing please provide some link or reference to read."}
{"id":"26108","title":"Where to find Lunar Eclipse data","text":"I am wondering whether there's a good resource to find data about upcoming Lunar Eclipses. For example, showing the percentage of the eclipse over time. Such as: 17:23 GMT - 0%, 17:40 GMT - 10%, etc."}
{"id":"114641","title":"Frequency of cosmic microwave background","text":"Some people refer to cosmic microwave background's (CMB) frame of reference as an absolute one. If I understand correctly, we can measure 'absolute' velocity in this frame by using the Doppler effect. However, I don't know how we can know what the 'basis' frequency of CMB is (in its own frame)? Can it be obtained by some quantum-cosmological consideration?"}
{"id":"63834","title":"How to interpret the constraints on Poisson's ratio values, physically?","text":"If I'm not mistaken, a Young's modulus $E$ can theoretically take on any positive value without bound. Physically, I interpret this as though a solid can have any arbitrary \"stiffness\" (within the limits of its linear stress- strain region, of course). On the other hand, the range of possible Poisson's ratios is constrained to $[-1,0.5]$ for stable, isotropic, linearly elastic materials. I'm not quite sure how to interpret this physically. Why must the Poisson's ratio necessarily be constrained?"}
{"id":"75156","title":"Would this pump water up? and if so, how far?","text":"I had this idea of an osmotic pump way back in high school and I never got a satisfactory answer if it would work. If I had this configuration: ![osmotic pump](http:\/\/i.imgur.com\/xWsW29w.png) Would it continually pump water up given ambient heat so long as the bottom reservoir is full? **EDIT** To explain what is happening, there is pure water in each of the dark blue reservoirs, saline in the cyan containers (same concentration of saline in each container) and a semipermeable membrane at the bottom and near the top of each container. The pure water in each reservoir would be sucked up into each container directly above it due to osmotic pressure (high water concentration flows to lower water concentration), and then dumped out at the top also due to osmotic pressure (saline to air which is almost 0% water concentration). Since the membrane is not permeable to salt, only the water is released from the container into the next higher reservoir. _**NOTE_** that the membrane at the top of each saline container doesn't touch the pure water in the reservoir it empties into. I'm also thinking that the saline containers may have to be completely filled with extra osmotic pressure to spare to counteract the pure water that sticks to the outside of the upper membrane and cause a reverse osmosis effect. Other possible tricks relying on surface tension and gravity might also aid in pulling the water away from the membrane. **END EDIT** If enough were stacked, would this allow for transporting water higher than the maximum that trees can transport (about 138m according to this article)?"}
{"id":"79860","title":"Why is a laserpointer able to \"erase\" a glow-in-the-dark sticker?","text":"a while ago I tried to charge a glow-in-the-dark sticker using a simple red laser pointer. It was a large sticker, of the type used to mark emergency exits and fire extinguishers here in Germany. I thought I could draw a picture this way, with glowing strokes. To my surprise, the exact opposite happened - the sticker was already glowing a little bit and the areas I illuminated with my laser turned dark instantly, as if the charge of the sticker was removed by the laserlight. The effect was reversible, the darkened areas would charge and glow as usual if exposed to normal light. So I was able to draw a picture after all - however, it was a negative. Can anyone explain why the sticker was discharged instead of charged by the laserlight?"}
{"id":"2017","title":"How much does it cost every time your car stops for a traffic light?","text":"Every time we have to stop the car, it is costing us extra money, because we have to then accelerate to full speed again. I would like to know how much. In order to simplify the situation, we can make these assumptions.   * we disregard cost of car maintenance (brake pads wearing out etc.). This is focused entirely on fuel economy.   * A car is always traveling at 60km\/h when it needs to stop   * The car's fuel economy is 8L\/100kms when traveling at 60km   * The cost of fuel is $1.25 per litre.   * The car weighs 1000kg I will be happy with either a definitive answer, an explanation of how to arrive at the answer, or an explanation of why there is not enough information in the question to come up with the answer."}
{"id":"14103","title":"Can smoke stay still in the air?","text":"Can a small amount of smoke be dense enough to stay in the air keeping its shape for a minute or so?   Or does it always dissipate quickly?   If not smoke, can anything else stay in the air for a minute while keeping its shape?"}
{"id":"14102","title":"What happens to the absorbed light energy?","text":"When light comes across with a solid material, some of it is reflected, some of it passes through and some of it is absorbed. I understand the reflection and passing through, but I don't understand what happens when light is absorbed. Suppose that there is an enclosed box. Its material is thick enough so that no light can escape out. We put a Led diode in it whose internal resistance is zero (there is no loss as heat energy, all energy dissipated is given out as light). What are the possible steady state scenarios that can happen? Why and how some wavelengths of sun light heats up black objects while some other wave lengths do not? Why and how some wavelengths of light pulls the outer most electrons of atoms to higher energy levels while other wavelengths have no such effect? (I understand that energy of a photon must be enough to take electron to higher energy level, but why doesn't it take the electron temporarily a littler higher in the current energy level and then make it return back? I understand that an electron cannot stay between two energy levels, because those are unstable regions; but why doesn't electrons stay there even if it is as short as an instant?)"}
{"id":"14104","title":"Pollen Particle Attracted to TV due to Which Force?","text":"A pollen particle has no charge so I cannot understand how the Lorentz force $\\bar{F} = q \\bar{E} + q(\\bar{v} \\times \\bar{B} )$ could explain the event. I speculated that it is because of the electric field that is generated to the dipole with a charge in the TV and the distance, more here. I have not yet confirmed that it would attract the pollen particle. Any better ideas?"}
{"id":"71677","title":"quantization with constraints","text":"let be a Hamiltonian system $ H= H(x,p) $ for this system there is a conserved quantity namely $ C=xp $ so the total system is invariant under rotation if we 'quantizy' this function $ Cf(x)=\\lambda _{n}f(x) $ the result is known and $ f(x)= Ax^{{1\/2}+i\\lambda _{n}} $ for some constant 'A' then how is this function $ f(x) $ related to the quantization of the Hamiltonian $ H\\Psi (x,t)= E_{n}\\Psi (x,t) $ for example if the system has a SYMMETRY so there is a conserved quantity (there can be more constants of motion of course) how does this affect to the quantization of the TOTAL hamiltonian ??"}
{"id":"71674","title":"FFT distortion, harmonics (singing wine glass)","text":"I'm doing a school assignment on Singing Wine glasses (you rub the rim of the wine glass with a wet finger and it produces a pure tone). I have recorded $30\\,\\text{ms}$ of the \"singing\" at a sampling rate of $10\\,\\text{kHz}$ (total $300\\,\\text{samples}$) and have the following Sound pressure and FFT graphs: ![Sound pressure graph](http:\/\/i.stack.imgur.com\/OInFM.png) Sound pressure graph ![FFT graph](http:\/\/i.stack.imgur.com\/wvusd.png) FFT graph In the FFT graph, I'm assuming that the first peak is the fundamental frequency of the wine glass (natural frequency), but I'm not sure of the three others (they are integer multiples apart from the first peak). Are they due to the harmonics of the wine glass or the distortion? Also, if I were to have distortion in the graph, would it only occur at odd multiples of the fundamental frequency? **EDIT** : So the three peaks are harmonics. What are they caused by? Has it got something to do with modes of vibrations, distortion or something completely different? (this was my first post on SE, so please don't be harsh on me if I got it all wrong :) )"}
{"id":"108963","title":"Ising model. What is large fluctuations of magnetization?","text":"My background is in mathematics. I have studied the Ising model in $\\mathbb{Z}^2$. The main model of statistical mechanics. Yesterday, I was reading the preliminary notes of the book Statistical Mechanics by Sacha Friedli and Yvan Velenik (The Book Project can be seen here) that talked about **magnetization $M_N$** (chapter 1 pages 17 and 20). My question about the Ising model is the following. > What conceptually mean **fluctuation of the magnetization $M_N$**? And > **large fluctuations of $M_N$**? **Up date:** Surely this question is related to another concept: fluctuations of spins."}
{"id":"106103","title":"How stupid is this theory of gravity?","text":"As will be evident, I am _not_ a physicist. I've always been interested in physics but my education tapered out with general relativity and basic quantum mechanics, years ago. Several years ago a sort of thought experiment began to nag at me and I've wanted for those more knowledgeable to basically explain to me why this idea _must_ be wrong -- not simply that it is very silly or (obviously) at odds with existing models, but provably wrong. (It is those things.) The thought experiment: Imagine a piece of graph paper, with 2 circles of radius 1, centers 10 units apart. Now imagine that the circles are growing at a _steady rate_ of 1 unit\/sec. The edges of the circles approach each other and touch after 4 seconds. Now imagine the same sequence, but the graph paper is also growing at a steady rate, matching that of the circles. The circles are not \"tethered\" to the expanding graph paper; their centers remain the same distance, hence fewer graph paper units as the paper expands. Finally, imagine that you, the observer are growing at the same rate as the graph paper and the circles, so that the graph paper _appears fixed_. What you would perceive would be 2 circles of fixed size accelerating towards each other. This hypothetical scene is not describing physics, it's just a geometrical construct. It shows that what appears to be constant motion can be seen as acceleration (or vice versa) based on the perceived intertial frame of reference (the graph paper). We perceive gravity as a force that causes the distance between masses to decrease over time, accelerating just as the two circles described. This assumes the existence of a stable \"graph paper\" in space, where distance exists independent of the objects. What if we remove that assumption? What if distance in space cannot be separated from distance in time? Imagine that what appear to be subatomic particles are in fact standing waveforms, emergent from constantly expanding waves. Wave interference gives rise to particles at certain scales with consistent properties, just as fractals give rise to recurring forms at specific scales. These \"standing\" wave forms are themselves expanding. But, expanding relative to what? Depends on the scale of space\/time. At certain scales, wave interactions maintain fixed distances between \"particles\", e.g. electron valences, atomic distributions. This holds true for the scales of matter we directly experience, where Newtonian laws apply. Everything we perceive to be fixed in size is actually expanding at essentially constant rate, including every atom in our bodies, every electron, every photon. Every unit of measurement we think of as constant is in fact constantly expanding. Now imagine that this expansive property is not uniform (if it were uniform it would be meaningless), but diminishes with the distance in space\/time of the source of the waves. As masses get further apart, their expansion becomes less uniform, causing them to expand towards each other. As the wave sources move further apart, the resulting standing waves (particles) are not expanding in uniform directions (they don't share the same \"graph paper\"), and so appear to accelerate towards each other. These gravitational waves are not, therefore, caused by the presence of mass, but rather, are the substrate from which mass originates. The interference of waves at different scales not only causes particles with varying sizes and speeds, but causes complex motions like orbital ellipses, and non-uniform expansion with different arrangements of particles, as with the swelling of the oceans that causes the tides. Besides being silly, the radical idea here is that everything we assume about the shape of space, and the concept of distance, is distorted by the fact that the stuff we are made of is constantly expanding. Gravity as a force doesn't exist, it's an illusion arising from this distortion of perception. Please tell me this idea is ludicrous. How would one disprove it, other than citing centuries of accepted models of physical reality? Which parts of established physics is it incompatible with? As silly as it sounds, it has nagged at me for years because it does describe what we perceive as gravity, and explains things like how gravity's effect can be \"instantaneous\" at a distance (it isn't; it's just an illusion; there is no graph paper). It doesn't seem incompatible with Einstein's observation that mass bends space\/time, but rather a different conception of how mass and space\/time are related to each other. For a long time, I thought orbital motion was unexplainable with this model, until I realized that just as the fixed scale of imaginary graph paper can be seen to be an illusion, so can the fixed directionality of \"constant motion\". Motion cannot follow a fixed path relative to non-existent graph paper, it can only be relative to other objects, and wave interaction could conceivably result in orbital motion. I'm confident this question will either receive a great deal of flak or be closed as inappropriate or such. I am prepared for the flak but hope people will debunk it in earnest."}
{"id":"113215","title":"Physical interpretation of Green's theorem with Dirichlet boundary condition","text":"The potential is given by $$\\Phi(\\mathbf{x})=\\int_V d^3x' G_D(\\mathbf{x},\\mathbf{x'})\\rho(\\mathbf{x'})-\\frac{1}{4\\pi}\\oint_S d^2x'\\frac{\\partial G_D(\\mathbf{x},\\mathbf{x'})}{\\partial n'}\\Phi(\\mathbf{x'})$$ Is there a physical interpretation of the surface term? The only one I can think of is that it is the surface integral of $\\Phi(\\mathbf{x'})$ times $\\sigma(\\mathbf{x'})$, where $\\sigma(\\mathbf{x'})$ is the surface charge induced by the presence of a unit point charge at $x$, with the surface being grounded. But why it should be of this form?"}
{"id":"113217","title":"Proton-electron inability of annihilation","text":"Why would an electron annihilate with a positron if it sees one nearby but would prefer to orbit the proton rather than annihilate with it, despite that the force of attraction being the same?"}
{"id":"113213","title":"What are eigenfunctions in a box which includes two kinds of medium?","text":"I have to solve a problem of mathematical physics. There is a simple case which can inspire me to solving a more complex case. Considering a metal box in 2D $0<x<a$ and $0<y<b$. And there are two kinds of medium, $0<x<c$ and $c<x<a$ with different dielectic constants $\\epsilon_1$ and $\\epsilon_2$, respectively. I want to solve the equation $$\\nabla (\\epsilon \\nabla \\phi) = -Q\\delta (x-c)\\delta(y-\\frac{b}{2})$$ where $\\epsilon = \\epsilon_1$ if $0<x<c$ and $\\epsilon = \\epsilon_2$ if $c<x<a$ There is an electric charge $Q$ at $x=c$ and $y=\\frac{b}{2}$, then what the distribution of potential is in this case? How do I write the eigenfunctions in this condition? It is easy to write the eigenfunction in two different area, $\\phi_1$ and $\\phi_2$. $\\phi_1$ and $\\phi_2$ satisfy $\\nabla^2 \\phi_1 = - \\frac{Q}{\\epsilon_1} \\delta (x-c)\\delta(y-\\frac{b}{2})$ and $\\nabla^2 \\phi_2 = - \\frac{Q}{\\epsilon_2} \\delta (x-c)\\delta(y-\\frac{b}{2})$ respectively. Could I combine two parts directly with conditions $\\phi_1|_{x=c}=\\phi_2|_{x=c}$ and $\\epsilon_1 \\frac{\\partial \\phi_1}{\\partial x}|_{x=c} = \\epsilon_2 \\frac{\\partial \\phi_2}{\\partial x}|_{x=c}$ to get the eigenfunctions in the box? Are these functions complete?"}
{"id":"112165","title":"Would adding water after heating decrease the overall heat, when compared to adding the water before heating for the same period of time?","text":"I was rather cold last night, and warmed up a cup of water to drink in the microwave. I put it in for 60 seconds, but it came out boiling hot, so I put a bit of cold water in with the hot water, and it was warm for me to drink. So then it came to me - what if I had put that extra water in before I put it in the microwave? If I still heated it for 60 seconds, would the extra surface area of the extra water cause it to heat faster, or would the speed remain the same, due to the averaging of heat? Thanks."}
{"id":"38076","title":"Is Fractional quantum Hall effect proof that leptons are composite particles?","text":"The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantised plateaus at fractional values. Should this be considered evidence that electrons are composite?"}
{"id":"38070","title":"Is Heisenberg's matrix mechanics or Schrödinger's wave mechanics more preferred?","text":"Which quantum mechanics formulation is popular: Schrödinger's wave mechanics or Heisenberg's matrix mechanics? I find this extremely confusing: Some post- quantum mechanics textbooks seem to prefer wave mechanics version, while quantum mechanics textbooks themselves seem to prefer matrix mechanics more (as most formulations are given in matrix mechanics formulation.) So, which one is more preferred? Add: also, how is generalized matrix mechanics different from matrix mechanics?"}
{"id":"83224","title":"Spinor representation restricted under subgroup, a formula from Polchinski","text":"The question is about the spinor representation decomposed under subgroups. It's a common technique in string theory when parts of dimensions are compactified and ignored, and we are only interested in the remaining sub- symmetry. I'm learning it from the appendix B in Polchinski's big book volume II. For a particular decomposition, $SO(2k+1,1) \\rightarrow SO(2l+1,1) \\times SO(2k-2l)$ (B.1.43), the Weyl spinors decompose as the formula (B.1.44), $2^k \\rightarrow (2^l, 2^{k-l-1})+(2'^l,2'^{k-l-1})$ and $2'^k \\rightarrow (2'^l, 2^{k-l-1})+(2^l,2'^{k-l-1})$, where $2^k$ and $2'^k$ are the Weyl representations of Lorentz group $SO(2k+1,1)$ with chirality +1 and -1 respectively. Specifically on the case $SO(9,1)\\rightarrow SO(5,1)\\times SO(4)$ with decomposetions $16 \\rightarrow (4,2)+(4',2')$, which appears at (B.6.3). My question is **the contradiction with minimum representations**. By checking Majorana and Weyl conditions, the minimum spinors for d=6 and d=4 have 8 and 4 components respectively. (Ref. table B.1 Polchinski) So How can you find $(4,2)$ representation for $SO(5,1)\\times SO(4)$? Also, I'm very interested in the prove of (B.1.44) under (B.1.43)? How is it proved by comparing the eigenvalues of $\\Gamma^{+}\\Gamma^{-}-\\frac{1}{2}$ as claimed by Polchinski?"}
{"id":"83226","title":"Lagrangian and hamiltonian of interaction","text":"How to prove that lagrangian of interaction is equal to hamiltonian of interaction with minus sign? For example, I can't prove it for special case - quantum electrodynamics."}
{"id":"30687","title":"Why didn't Newton just propose the 2nd Law and leave it at that?","text":"**Why didn't Newton just propose the 2nd Law ($F=\\dot{p}$) and leave it at that?** The 2nd Law implicitly contains the first, doesn't it? If so, it seems he wasn't following his own Rule #1 of Book 3 of his _Principia_ : \"We are to admit no more causes natural things than such as are both true and sufficient to explain their appearances.\" There are intro textbooks (e.g., this one) that study momentum first and then force $F=\\dot{p}$ based on momentum. Cf. also this SE post and comments."}
{"id":"133151","title":"Scientists observe the laws of the physics but, Where do they come from?","text":"Has anyone ever considered how the laws of physics that we study came into being."}
{"id":"133158","title":"What is the meaning of this \"let there be light\" joke?","text":"A middle school teacher across the restaurant is wearing this shirt, and I certainly don't get it. ![\"Let There Be Light\" shirt with equations on it](http:\/\/i.stack.imgur.com\/YOlVZ.jpg)"}
{"id":"85974","title":"Electric field inside a conductor","text":"I would like to clear things up: How exactly the electric field inside a conductor _is_ zero? Let a really \"powerful\" electric field be outside of it, how can the \"few\" charges in a conductor balance the field? From my point of view, it's the conductor surface which stops their movement. I'm obviously missing something fundamental here. Thanks in advance."}
{"id":"85979","title":"How much extra distance to a CERN event horizon?","text":"How much extra distance would a scientist have to travel to get to the event horizon of a mini black hole if they ever make one?"}
{"id":"103058","title":"Fermi level and conductivity","text":"Can someone in a simple way explain me what the Fermi level is and what does it have to do with conductivity. My teacher said that Cu conducts electric current better than Al because of something in relation to Fermi level, I didn't get him very well, so please explain me what it is all about, but in a simple way."}
{"id":"119113","title":"What is 'heterotic string compactification'?","text":"I've read that some exceptional groups arises in the context of 'heterotic string compactification'. Could someone explain (to a person studying physics but who doesn't know string theory) what heterotic string compactification involves and what why exceptional groups have to do with it?"}
{"id":"102876","title":"Mutually unbiased bases","text":"This question can be formulated in two ways. Let there be two $d$-dimensional orthonormal bases $B_{1}$ and $B_{2}$. I refer to the elements of $B_{1}$ by $\\lvert\\nu_{i}\\rangle$ and to the elements of $B_{2}$ by $\\lvert\\omega_{j}\\rangle$. These two bases are mutually unbiased namely: $\\lvert\\langle\\omega_{j}\\lvert\\nu_{i}\\rangle\\lvert^{2}=1\/d$ for all $\\lvert\\omega_{j}\\rangle$ and $\\lvert\\nu_{i}\\rangle$. We define some vector $\\lvert\\tau\\rangle$ such that $\\lvert\\tau\\rangle$ is orthonormal to one element of $B_{2}$, say $\\lvert\\omega_{m}\\rangle$ and it is mutually unbiased with respect to all the elements in $B_{1}$: $\\lvert\\langle\\tau\\lvert\\nu_{i}\\rangle\\lvert^{2}=1\/d$ for all $\\lvert\\nu_{i}\\rangle$. Can we conclude that $\\lvert\\tau\\rangle$ is equal to a member of $B_{2}$ say $\\lvert\\omega_{p}\\rangle$ for some $p\\neq m$ up to a phase difference? Another way to make the same conclusion is to ask this: Let there be three orthonormal bases $B_{1}$, $B_{2}$ and $B_{3}$. $B_{1}$ and $B_{2}$ are mutually unbiased, so are $B_{2}$ and $B_{3}$. Can we conclude that $B_{1}$ and $B_{3}$ are either mutually unbiased or have equivalent Hadamard matrices? This question for me is particularly important since in the literature it is always talked about sets of pairwise mutually unbiased bases. But there's no proof given for the fact that three mutually unbiased bases should necessarily be pairwise MUBs."}
{"id":"128582","title":"Physical laws in all inertial frames","text":"Do there exist physical laws which do not take the same form in all inertial frames?"}
{"id":"114053","title":"Initial velocities of a collision","text":"This is the question: A car of mass 900 kg and a van of mass 1300 kg collide at a crossroads. Investigation into the collision discloses that the car was travelling south east and that the van was travelling due north. After the collision, the damaged vehicles remained locked together and skidded off in a direction of 30 degrees north of east at an initial speed of 10 m.s-1 before eventually coming to rest. Determine the speed of both vehicles before the collision. Im struggling to work out the y components of the initial velocities ive got the x components. Up to yet my workings out are; ![Physics](http:\/\/i59.tinypic.com\/ivc035.jpg) Thanks in advance.."}
{"id":"114051","title":"Can the thermal energy in an object be aligned, creating a thermal jerk?","text":"Thermal energy is related to the speed at which atoms vibrate, therefor thermal energy is a form of motion. In a stationary object, all of these vibrating atoms 'impact' each other in such a way that the net sum of their motion is zero. if all (or a significant number) of the atoms were aligned so that their thermal energy 'pointed' in the same direction, the net sum of thermal movement would no longer be zero, and the object should jerk. Does this ever happen? If so, wouldn't cooling one side of an object cause a difference in thermal motion vectors which would cause the object to move in the direction that the cooler side was facing? Why or why not?"}
{"id":"109707","title":"Interpretation of Conjugate Momentum in Field Theory","text":"The conjugate momentum density, following as a conserved quantity with Noethers Theorem, from invariance under displacement of the field itself, i.e. $\\Phi \\rightarrow \\Phi'=\\Phi + \\epsilon$, is given by $\\pi=\\frac{\\partial L}{\\partial ( \\dot{\\Phi})}$. On the other hand the physical momentum density, following as a conserved quantity with Noethers Theorem, from invariance under translations, i.e. $\\Phi(x) \\rightarrow \\Phi(x')=\\Phi(x+\\epsilon)$, is given by $\\Pi = \\frac{\\partial L}{\\partial ( \\dot{\\Phi})} \\frac{\\partial \\Phi}{\\partial x}$. Does anyone know a enlightening interpretation of the conjugate momentum? Furthermore why do we, in quantum field theory, impose commutation relations with the conjugate momentum instead of the physical momentum? (For brevity all possible indices are supressed)"}
{"id":"109706","title":"What provides the centripetal force for a glider?","text":"Whenever any object follows a circular path, there is always a centripetal force which is provided by something. However, in the case of a glider making a loop in air, what provides the centripetal force? This is the picture that made me think of this question: ![enter image description here](http:\/\/i.stack.imgur.com\/aXKqm.jpg)"}
{"id":"102879","title":"Can two media have same refractive indices?","text":"Can two mediums placed under same conditions* ever have same refractive indices? *Pressure, temperature,etc."}
{"id":"39306","title":"Mechanical effects of convective flow in steady-state stagnant film diffusion","text":"When analysing diffusion problems in the context of the simplified film model, two limiting cases are ussually introduced during the solution of the differential chemical species mass balance equation. Those cases are the so- called equimolecular counter-diffusion (which implies a purely diffusional movement of species) and the stagnant film diffusion (in which the net flux of every but one species is null). In the latter case, the solution of the equation gives rise to a convective term (i.e. bulk flow of the fluid) which contributes to the overall flux of the species being transferred, and opposes the diffusional flux of the stagnant species. Furthermore, this convective term implies that a net macroscopic (i.e. mass average) velocity field is established across the film. I wonder how is this field **mechanically established** on the film -given the widely recognized limitations of the model- because the mass average velocity field obtained from the solution of the mass-balance equation is **constant** through it (an example is derived in this paper by Brouwers and Chesters: http:\/\/doc.utwente.nl\/20661\/1\/Journal5New2.pdf). Solving the momentum balance equation with the characteristic boundary conditions of the model won't do the job, because the constant component of the velocity field in the direction perpendicular to the film boundary will lead to a non-informing pressure profile (constant too, assuming nearly-inviscid flow). What puzzles me is the transition between the velocity profile far away from the film (i.e. the turbulent, chaotic velocity of the fluid) which has a **null average velocity component** in the direction normal to the film, and the constant, net component of velocity in that direction inside the film which arises from the mass-balance resolution. I suppose the pressure profile should be distorted in some way, but I couldn't figure it out by myself."}
{"id":"39307","title":"Question on Sakurai's treatment of the Harmonic Oscillator:","text":"In Section 2.3 of the second edition of _Modern Quantum Mechanics_ (which discusses the harmonic oscillator), Sakurai derives the relation $$Na\\left|n\\right> = (n-1)a\\left|n\\right>,$$ and states that > _this implies that $a\\left|n\\right >$ and $\\left|n-1\\right>$ are the same up > to a multiplicative constant._ To my sensibilities, this is only implied if the $\\lambda$-eigenspace of the number operator $N:=a^{\\dagger}a$ corresponding to $\\lambda=n-1$ is one- dimensional. If it is multidimensional, then we cannot say that $a\\left|n\\right>$ and $\\left|n-1\\right>$ are proportional. So (unless I've made some fundamental error) how do we know that the $\\lambda$ eigenspaces of $N$ are one dimensional?"}
{"id":"88355","title":"One-loop beta functions of the Standard Model","text":"For my master's research on energy scale independent combinations of renormalization group equations in supersymmetric theories, I need an overview of all the one-loop beta functions of the Standard Model parameters. I just have not been able to find anything like this yet, which is weird (as they have been known for a long time). Does anyone in the field know of such an overview?"}
{"id":"88359","title":"How does the Einstein Equivalence Principle imply a spacetime with a metric (and a connection)?","text":"I have at hand the book by Clifford Will, \"Theory and Experiments in Gravitational Physics\", and the following Living Reviews in Relativity article. He quotes the Einstein Equivalence Principle (EEP) as follows [Will2006, 3.2]: >   1. \"mass\" is proportional to \"weight\", hence trajectories of freely > falling \"test\" bodies are independent of their masses, structures and > compositions; (Newton EP, or Week EP) >   2. local non-gravitational test experiments are independent of the > velocity of the freely falling reference frame in which they're performed. > That is, Special Relativity (as well as QED and so on) is locally valid; > (Local Lorentz Invariance, LLI) >   3. these experiments are also independent of where and when in the > Universe they're performed. (Local Position Invariance, LPI) > Then Will claims that > it's possible to argue convincingly that if EEP is valid, than gravity > **must** be a \"curved spacetime\" phenomenon. [...] The **only** theories > that can fully embody EEP are those that satisfy the postulates of \"metric > theories of gravity\", which are > >   1. The spacetime manifold has a symmetric metric $g_{\\mu\\nu}$; >   2. trajectories of freely falling test bodies are geodesics of > $g_{\\mu\\nu}$; >   3. locally, the non-gravitational laws of physics are those of Special > Relativity, QED and so on... > Ok, this claim is more than reasonable and experimental tests of EEP made with metric theories have yield optimal results and confirmations. But, is it rigorous or exhaustive? Is there the possibility of a technical fault with testing EEP with metric theories? I mean, can I test this claim at the same time as EEP, and vice versa? Is teleparallelism an alternative to the introduction of a metric? Though manifolds are a very general concept, and hence saying that the spacetime can be described by a manifold is too general to be wrong, embedding a metric is natural but more subtle and not so obvious to me asserting that gravity can be interpreted with its curvature, as a result of EEP. Can someone convince me that it's the only reasonable thing to do or, on the contrary, show me some alternatives?"}
{"id":"88358","title":"Can anyone explain the idea behind dS ∝ dV\/V?","text":"In a lecture on entropy, one of the equations $dS ∝ \\frac {dV}{V}$ was explained as \"a fractional change in volume as a measure of the increase in randomness\" (related to $\\frac{dQ}{T}$) How does this concept of $\\frac{dV}{V}$ work more precisely? What volume is $dV$ indicating exactly? I'd be interested to read a paper\/textbook notes on how this is conceptualised, since it seems relevant to molecular and structural biology (e.g. protein structural changes) I understand entropy and am not asking about the meaning of the term, rather I am interested in this aspect of its derivation"}
{"id":"100505","title":"Reaction force of the ground beyond the equator","text":"Let's imagine a person standing somewhere on Earth, but not on the equator, i.e. somewhere with a positive net value of latitude. Since the Earth spins around its axis and the person spins along, the sum of forces acting on them should give us the centripetal force: ![enter image description here](http:\/\/i.stack.imgur.com\/MAtGx.png) The only forces acting on the person are the gravitational force $F_g$ and the force exerted on the person by the ground, $N$. The gravitational force is directed into the centre of Earth: ![enter image description here](http:\/\/i.stack.imgur.com\/7uLZr.png) That means, that the other force $N$ should look like this: ![enter image description here](http:\/\/i.stack.imgur.com\/NCqfo.png) So now, when we \"zoom\" and look at our person, the situation looks like that: ![enter image description here](http:\/\/i.stack.imgur.com\/JjnSw.png) Which is rather peculiar, since the reaction force of the ground is usually a normal force, perpendicular to the ground. Is my deduction correct, and if yes, isn't the fact that $N$ is not perpendicular to the ground something we should consider while solving problems (for example problems related to friction, which is proportional to the normal force)?"}
{"id":"47082","title":"What is non-Abelian about non-Abelian Chern-Simons' theory?","text":"One is aware that in the axial gauge (say the light-cone gauge $A_{-}=0$) non- supersymmetric Chern-Simons' theory is a quadratic theory. Hence in this gauge there are no gauge-gauge interactions. Then what is non-Abelian about this theory?   * Are there effects which exist inspite of the above which make the theory different from the Abelian one?   * I am aware that there are calculations of exact partition functions for Chern-Simons' theory on compact 3-manifolds. Is that the subtle point that the non-Abelian-ness will somehow manifest itself when on compact space-times? (..or on manifolds with boundary?..) That there is some reason such convenient axial gauges may not exist for compact space-times? (..though i can't see why one can't always choose $A_{-}=0$..because 3-manifolds are parallelizable there aren't any topological restrictions but there could be an issue with Gribov ambiguities..I don't know and would like to know about)    * I guess that for supersymmetric Chern-Simons' theories this question itself doesn;t arise since I guess there are no gauge choices where the theory will become quadratic in the gauge fields.  (- related i would love to know of any proof\/argument as to why there can't be such a gauge choice for YM theory which will make it quadratic - axial gauge can kill the quartic term but I guess thats the best one can do..)"}
{"id":"47089","title":"Proper times of two observers in a three-torus","text":"Consider two observer in a tree-torus space of size $L$. Observer $A$ is at rest, while observer $B$ moves in the $x$-direction with constant velocity $v$. $A$ and $B$ began at the same event, and while $A$ remains still, $B$ moves once around the universe and comes back to intersect the worldline of $A$ without ever having to accelerate (since the universe is periodic). I've obtained proper time measured by $A$ (because observer $A$ is at rest, so the trajectory of $A$ is characterized by $v = 0$.): $$\\tau ~=~\\int d\\tau =\\int \\sqrt{1-v^{2}} dt~=~t.$$ For particle $B$, it moves at constant velocity $v$: $$\\tau ~=~\\int d\\tau ~=~\\int \\sqrt{1-v^{2}} dt~=~t\\sqrt{1-v^{2}}.$$ I think this result is inconsistent with my understanding of Lorentz invariance because the times don’t match and no one was accelerated and this is not the usual intuition for Lorentz invariance. Is this really consistent with Lorentz invariance?"}
{"id":"77144","title":"What is the speed of a neutral pion and how is it measured?","text":"Alvager et al 1964 purportedly disproved Ritz's emitter theory in an experiment that generated neutral pions with a vellocity of $v =B.c$ where B is slightly smaller than one. How is the velocity of a neutral pion determined? Does the determination invoke special relativity? Added: Alvager et al (http:\/\/mysite.verizon.net\/cephalobus_alienus\/papers\/Alvager_et_al_1964.pdf) beamed a burst of high energy protons at a beryllium crystal and measured arrival times of burst of emitted photons at downstream detectors. Their theory said that the photons were emitted from decay of neutral pions produced from collision of the protons with beryllium and that these pions had a speed of $v = B.c$. Time intervals between output burst being detected at different detectors indicated that photons were travelling at very close to $1.0c$ rather than at a range of values between $c -k.B.c$ and $c + k.B.c$ with $k>0$ which (they said) would have been the case if emission\/ballistic\/galilean theory\/model applied. ( Sorry I dont know where I got my earlier value of $k.B = 0.2$ from). I pose the question because it seems that Alvager et al were trying to prove special relativity (SR) by assuming pion velocities which were themselves determined assuming SR. Anna V has answered the question of pion velocity measurement for me (Thankyou!). Regarding the interpretation of the Alvager experiment the following note is interesting: (http:\/\/worldnpa.org\/pipermail\/memberschat_worldnpa.org\/attachments\/20090115\/db6f6bc5\/attachment.pdf)."}
{"id":"100879","title":"What does \"mathematically well defined\" quantum field theory mean?","text":"Reading Wald's book (page 380, end of the first paragraph of section 14.1) while the author is giving an overall discussion of quantum field theories you can read > However, for the more interesting case of a theory with interactions (i.e., > for a field or fields satisfying nonlinear equations) one is led unavoidably > to consider the products of field operators at the same spacetime point. > Such quantities have no natural mathematical meaning, and consequently, with > the exception of some simple models in lower spacetime dimensions, there are > at the present no known examples of quantum field theories of physically > reasonable interacting fields which are mathematically well defined. What does Wald exactly mean by _mathematically well defined_ , which are the problems, for example in QED that make it ill defined?"}
{"id":"9435","title":"What is the probability that a half dollar coin lands on heads?","text":"I recently found a half dollar coin in my basement. It's strange but I think when you flip it, that heads is a big favorite to land face up! I really could have sworn I heard that somewhere before; am I wrong or is the half dollar weighted for heads? It's funny, I actually said to my friend yesterday, \"Here look at this coin, I think it's weighted for heads, like 3 out of 4 times it lands heads.\" Well I flipped it 4 times and that's exactly what happened. In my opinion it's because the coin is large and the face seems to be pretty heavy; I'm curious to know if anyone can derive the exact chance of a half dollar landing heads, or to convince me it's not weighted."}
{"id":"21801","title":"Identification of the state of particle types with representations of Poincare group","text":"In the second chapter of the first volume of his books on QFT, Weinberg writes in the last paragraph of page 63: > In general, it may be possible by using suitable linear combinations of the > $\\Psi_{p,\\sigma}$ to choose the $\\sigma$ labels in such a way that the > matrix $C_{\\sigma'\\sigma}(\\Lambda,p)$ is block-diagonal; in other words, so > that the $\\Psi_{p,\\sigma}$ with $\\sigma$ within any one block by themselves > furnish a representation of the inhomogeneous Lorentz group. He continues: > It is natural to identify the states of a specific particle type with the > components of a representation of the inhomogeneous Lorentz group which is > irreducible, in the sense that it cannot be further decomposed in this way. My questions are:   1. How is the first blockquote true? Why is it possible? Please sketch an outline of proof or refer to some material that might be useful.   2. What does he even referring to in the second blockquote? I found some material on net and Physics.SE regarding this, but I din't find any treatment upto my satisfaction. Please be precise as to what the correspondence is and whether or not it is bijective (as some accounts seem to indicate).   3. What is the relation between Weinberg's \"specific particle type\" and \"elementary particle\" used in accounts of this correspondence?   4. What is the definition of \"one-particle state\"? Is this correspondence a way of defining it? If yes, what is its relation to how we think of such states intuitively? (Of course, the answer of this question depends largely on the answer of 2, but I just asked to emphasize what is my specific query.)"}
{"id":"43703","title":"Book reviewing current state of research on complex networks","text":"Can anybody recommend a book reviewing the current state of knowledge and active research on complex networks? Not primarily a textbook but a true review of the field - ideally with references to applications where relevant."}
{"id":"43700","title":"why dosen't a charged particle radiate energy in circular motion in a uniform magnetic field?","text":"I have studied in my Physics course that one of the drawbacks of Rutherford's atomic model was that when an electron will revolve around the nucleus, it is undergoing acceleration and so it should radiate energy and consequentially fall into the nucleus. Similarly when a charged particle is projected in the plane perpendicular to a uniform magnetic field it executes uniform circular motion withradius $r=mv\/qB$. My question is why isn't the charged particle radiating energy here? Even in this case the charged particle is accelerating, just as it was in Rutherford's model of the atom. So shouldn't the radius decrease in this case also?"}
{"id":"87673","title":"Did people realize that gravity accelerated things before Einstein's elevator thought experiment?","text":"I'm reading about the (very near) equivalence of gravitational mass and inertial mass in my undergrad GR course, and the text (Lambourne) describes this equivalence as the inspiration for Einstein's falling elevator thought experiment. (i.e. one cannot tell the difference between free-fall and lack of gravity\/upward acceleration and presence of gravity). What was the significance of this thought? Surely physicists understood that the gravitational force caused things to accelerate before this, and therefore were aware of this kind of equivalence of motion and gravitation. Why was understanding gravity as an acceleration so big for Einstein (and physics as a whole)?"}
{"id":"13369","title":"Trace of energy momentum tensor in $CFT_1$","text":"There is a standard procedure to show that for CFT's in dimension $d\\geq 2$, the trace of stress tensor vanishes. I think I can't apply those steps when I only have one dimension, say time, because if we could then we can show that stress tensor trace $T_{tt}$ (which is basically it) also vanishes which may not be true always. Things are definitely different in 1D (I think metric can just be $g_{tt}$ and conformal transformation just a $t\\rightarrow \\lambda t$ transformation. Please correct me if I am wrong.) So, where do basically things go wrong in such derivation?"}
{"id":"13367","title":"Theory of Complex Spectra, Applying Slater-Condon Rules","text":"C.W. Ufford and G.H. Shortley in Physical Review 1932, Vol. 42, pg. 167, separate the two $^2$D's of $d^3$. On page 173 (pg 7 of the PDF) of this article they determine the matrix of the electrostatic energy in eqn. 3.3 as: $$\\begin{array}{ccc} \\- &a & b \\\\\\ a & 3F_0+7F_2+63F_4 & 3(21)^{1\/2}(F_2-5F_4) \\\\\\ b & 3(21)^{1\/2}(F_2-5F_4) & 3F_0+3F_2-57F_4 \\\\\\ \\end{array}$$ I'm trying to figure out how they actually determine this matrix. How do they do it? Here is what I understand: This matrix comes from a description of the two terms, $$^2D^a_{2,1\/2} = 1\/2 [-A+B-C+E]$$ $$^2D^b_{2,1\/2} = (84)^{-1\/2}[-5A-3B-C+4D-3E-2(6)^{1\/2}F]$$ where $A,B,C,D,E,F$ are zero-order states of $d^3$ which are $$\\begin{array}{cccc} A &\\mathcal{A}u_1(2^+)u_2(2^-)u_3(-2^+) & B &\\mathcal{A}u_1(2^+)u_2(1^+)u_3(-1^-) \\\\\\ C &\\mathcal{A}u_1(2^+)u_2(1^-)u_3(-1^+) & D &\\mathcal{A}u_1(2^-)u_2(1^+)u_3(-1^+) \\\\\\ E &\\mathcal{A}u_1(2^+)u_2(0^+)u_3(0^-) & F &\\mathcal{A}u_1(1^+)u_2(1^-)u_3(0^+) \\\\\\ \\end{array}$$ In this notation $u_i(a^j)$ indicates the one-electron central field eigenfunction for electron $i$ with the $j^{th}$ set of numbers $n^il^im_l^im_s^i$. $\\mathcal{A}$ represents the antisymmetrizing operator $(N!)^{-1\/2}\\sum_P (-1)^p P$. We can say that $(ab|1\/r_{12}|cd)$ depends, in part, on $\\sum_{k=0}^\\infty c^k (l^a m_l^a, l^c m_l^c) c^k(l^dm_l^d,l^bm_l^b)$ There are tables of these values already determined, notably as $a^k$ and $b^k$, which can be used to evaluate the Coulomb and exchange integrals for $k=0,2,4$. E.U. Condon then extended Slater's method to determine the non-diagonal matrix elements in a paper (Physical Review, 1930, Vol. 36, pg. 1121) for separating multiplets of the same kind, as we have in our case. In case you are confused by the notation, $J(a,b)=\\sum a^k$ and $K(a,b)=\\sum b^k$ and $a^k(l^am_l^a,l^bm_l^b)=c^k(l^am_l^a,l^am_l^a)c^k(l^bm_l^b,l^bm_l^b)$ and $b^k(l^am_l^a,l^b m_l^b) = [c^k(l^am_l^a,l^bm_l^b)]^2$. When we get to dealing with the doublet D's, we see that they depend on the following wavefunctions: $(2^+,2^-,-2^+), (2^-,1^+,-1^+), (2^+,0^+,0^-), (2^+,1^+,-1^-), (1^+,1^-,0^+), (2^+,1^-,-1^+) $ -- or $A,B,C,D,E,F$. Using Slater's method we find that $E(^2H)+ E(^2G)+E(^4F)+ E(^2F)+E(^2D)+E(^2D)=18F_0-13F_2-152F_4$ and, $E(^2H)+ E(^2G)+E(^4F)+ E(^2F)=12F_0 -23F_2 - 158F_4$ Therefore, $E(^2D)+E(^2D)= 18F_0-13F_2-152F_4 - (12F_0 -23F_2 - 158F_4) = 6F_0+10F_2+6F_4$. Thus the mean energy is $3F_0 + 5F_2 + 3F_4$. But I don't understand how to get to the non-diagonal elements yet..."}
{"id":"80221","title":"A question abou $E=pc$ for massless particles","text":"Since photon has no (rest)mass and $$E^2=(pc)^2+(mc^2)^2$$ we derive that $E=pc$ for particle with no (rest)mass. However, if we transform the non-relativistic formula for kinetic energy $$E_k=\\frac{mv^2}{2}$$ $$E_k=\\frac{p^2}{2m}$$ $$E_k=\\lim_{m\\rightarrow0}\\frac{p^2}{2m}=\\lim_{m\\rightarrow0}\\frac{m^2v^2}{2m}=\\lim_{m\\rightarrow0}\\frac{mv^2}{2}=\\frac{pv}{2}$$ we derive that $E_k=\\frac{pc}{2}$. And this is bizarre because I'm expecting $E_k=E$. I've just started learning physics recently so I'm pretty sure I've got certain part wrong. And here are my hypotheses: 1.$ $ The net energy of a photon $E_{net}=pc$ is equivalent $E_{net}=E_k+E_0=\\frac{pc}{2}+E_0$ and thus $E_0=pc-E_k=\\frac{pc}{2}$. (If that's true, what exactly is $E_0$? The rest energy of a photon?) 2.$ $ It is incorrect to derive the kinetic energy formula from integrating the equation $$ \\frac{\\mathrm dE}{\\mathrm dv}=mv $$ I should integrate another equation instead (perhaps $\\frac{\\mathrm dE}{\\mathrm dv}=p$? If that's true, why do we break $p$ into $mv$ when deriving the formula $E_k=\\frac{pc}{2}$?) 3.$ $ I got the limit wrong: $$E_k=\\lim_{m\\rightarrow0}\\frac{p^2}{2m}\\ne \\frac{pc}{2}$$ $$E_k=\\lim_{m\\rightarrow0}\\frac{p^2}{2m}=pc$$ So is any of these hypothesis correct? And why?"}
{"id":"80220","title":"Superfluid Fountain","text":"I was wondering how a super fluid fountain can flow indefinitely without violating energy conservation. ![enter image description here](http:\/\/i.stack.imgur.com\/6Sd1R.jpg)"}
{"id":"135122","title":"Creation and Annihilation Operators","text":"Let $\\widehat{a}^{+}_{i}$ and $\\widehat{a}_{i}$ be the usual bosonic creation and annihilation operators. Consider $$\\widehat{q}_{i} = \\sqrt{\\frac{\\hbar}{2m_{i}w_{i}}}(\\widehat{a}_{i}+ \\widehat{a}^{+}_{i}).$$ We want to calculate time evolution of $\\widehat{q}_{i}(\\tau)$ in the interaction picture: $$\\widehat{q}_{i}(\\tau) = e^{i \\widehat{H}_{0} \\tau\/\\hbar} \\widehat{q}_{i} e^{-i \\widehat{H}_{0} \\tau\/\\hbar},$$ where $$\\widehat{H}_{0} = \\sum_{i} \\frac{1}{2m_{i}} \\widehat{p}_{i}^{2} + \\frac{1}{2} m_{i} w_{i}^{2} \\widehat{q}_{i}^{2},$$ where $m_{i}$ and $w_{i}$ is the mass and natural frequency of the $i$th harmonic oscillator, and $\\widehat{q}_{i}$ and $\\widehat{p}_{i}$ are the canonical position and momentum operator. I don't understand why the result is $$\\widehat{q}_{i}(\\tau) = \\sqrt{\\frac{\\hbar}{2m_{i}w_{i}}}(\\widehat{a}_{i} e^{-i w_{i} \\tau\/\\hbar} + \\widehat{a}_{i}^{+} e^{i w_{i} \\tau\/\\hbar}). $$"}
{"id":"134299","title":"A mirror in the sky to look at past Earth","text":"Earth reflects light from Sun into Space; are there any \"mirror like\" objects which reflect back to Earth this light? Has anyone been able to use this method to recover some information on how was the Earth in the past (how it looked, what was it temperature, etc)?"}
{"id":"129760","title":"About Paschos-Wolfenstein relation (Weinberg angle measuring)","text":"There is Paschos-Wolfenstein relation: $$ \\tag 1 \\frac{\\sigma^{NC}_{\\nu_{\\mu}} - \\sigma^{NC}_{\\bar{\\nu}_{\\mu}}}{\\sigma^{CC}_{\\nu_{\\mu}} - \\sigma^{CC}_{\\bar{\\nu}_{\\mu}}} = \\frac{1}{2} - \\sin^{2}(\\theta_{W}). $$ In the most simple case the neutral current (NC) processes from $(1)$ are caused by $\\nu q \\to \\nu q$, $\\bar{\\nu}q \\to \\bar{\\nu}q$ reactions, while the charged current (CC) processes are caused by $\\nu_{\\mu} X \\to \\mu^{-}Y$, $\\bar{\\nu}_{\\mu} X \\to \\mu^{+}Y$. Here $q, X, Y$ refer to $u$, $d$ quarks. It is not hard to derive $(1)$. Some historical experiments (like NuTeV) have used this relation for determination of $\\sin (\\theta_{W})$ value. Here is my question: I don't understand how to arrange the experiment which would give the data which correspond $(1)$ (and don't correspond usual $\\frac{\\sigma_{\\nu}^{NC}}{\\sigma_{\\nu}^{CC}}$). Maybe, I would understand why in the denominator of $(1)$ there is a minus sign: neutrinos make possible $\\nu d \\to \\mu^{-}u$ reactions, while antineutrinos make possible $\\bar{\\nu} u \\to \\mu^{+}d$ reactions. So we detect the number of neutrinos-caused events minus antineutrinos-caused events. But I don't understand why there is minus sign into the nominator of $(1)$ (from the viewpoint of the experimental data). I would be grateful for your help."}
{"id":"70263","title":"Deriving the canonical ensemble","text":"I'm confused by the derivation of the canonical ensemble, namely the origin of the probability density, that is the Boltzmann factor. Here's what I have: We have a system of particles with $(N_{tot},V_{tot},E_{tot})$ in thermodynamical equilibrium that we divide into two subsystems, $A$ and $B$. Let $(N,V,E)$ describe $A$, and let $(N',V',E')$ describe $B$. We have the relations $$ N+N' = N_{tot} $$ $$V+V' = V_{tot}$$ $$E+E' \\approx E_{tot}$$ where in the last one we've neglected any potential interaction between the particles, arguing that the number of degrees of freedom in any interaction is orders of magnitude smaller than the total amount of particles, because interactions are short ranged. Now, consider the probability that subsystem $A$ has energy $E$. This is the ratio of the number of microstates of the entire system wherein $A$ has energy $E$ to the number of microstates such that the entire system has $E_{tot}$: $$P(A \\mbox{ has }E)= \\frac{\\Omega_{tot}(A \\mbox{ has }E)}{\\Omega_{tot}(E_{tot})}$$ but since we've neglected the interactions, we can factor the numerator: $$P(A \\mbox{ has }E)= \\frac{\\Omega(E)\\Omega ' (E')}{\\Omega_{tot}(E_{tot})}$$ where the omegas follow the same convention with primes and subscripts as the other variables. But now I don't understand how to get the Boltzmann factor... this relies on having $$P(A \\mbox{ has }E) = C \\cdot \\Omega ' (E')$$ But I don't see why that's true... $\\Omega(E)$ isn't constant, is it?"}
{"id":"134743","title":"How to calculate gravitational force","text":"Suppose the centre of mass of 2 bodies coincide , then how will we calculate the gravitational force between the 2 bodies ??"}
{"id":"10126","title":"Mathematically rather than physically speaking, is there something \"special\" about 10 (or 11) dimensions?","text":"As I understand it, string theory (incorporating bosons and fermions) \"works\" in $9+1=10$ spacetime dimensions. In the context of dual resonance theory, I've read descriptions of why that is \"physically\", i.e., how in the process of formulating a quantum theory of relativistic interacting (closed and open) strings, the only way one can do things like get rid of ghosts, for example, is to have ten dimensions. I'm looking for a way to appreciate more intuitively this restriction on dimensionality. Is there something special about ten-dimensional spaces purely from the point of view of mathematics? Without reference to the particular detailed exigencies of building a physical theory based on strings, does ten- dimensional topology have some unique features that are lost when one dimension is added or subtracted? Another way to pose the question: For purposes of this discussion, let's stipulate by fiat that the universe has $8+1=9$ dimensions. Would we humans simply be back to the drawing board in terms of coming up with a unified theory of the fundamental forces? Would anything in string theory be salvagable? I realize that M-theory adds a dimension. So if someone wants to answer the question by telling me what's unique or special about eleven dimensions, that's fine."}
{"id":"135346","title":"Nature of Fields in QFT","text":"I'm not exactly an expert in quantum physics, but this seems to be a simple question, and I can't find an answer anywhere! There are specific types of fields used in physics: scalar fields (i.e. as in the case of the Higgs boson), vector fields (i.e. as in magnetic fields), tensor fields (i.e. as in general relativity), etc. But what types of fields are used in QFT to model elementary particles? Is my confusion simply a result of me thinking in purely classical terms?"}
{"id":"80919","title":"What causes atoms to have their specific colors?","text":"I understand that light (color) is part of the electromagnetic spectrum, and that it depends on what wavelengths are reflected\/absorbed. Though what property of an individual atom gives it its color? Do electrons, neutrons, and other subatomic particles have color? **Basically: What intrinsic property causes the differences between how the varying wavelengths of light are reflected at the atomic scale?** Also, how do photons factor into this?"}
{"id":"116267","title":"Is there a well known theory that states all particles are made up of smaller particles and so on and so forth?","text":"I've contemplated, for some time, the possibility that all things are made up of smaller particles, and all smaller particles are made up of even smaller ones. I'm wondering if there are any popular theories that specifically support or deny this possibility."}
{"id":"1646","title":"Turned to steel in the great magnetic field","text":"This is obviously a \"fun\" question, but I'm sure it still has valid physics in it, so bear with me. How great of a magnetic field would you need to transmute other elements into iron\/nickel, if that's in fact possible? The magnetic field near a neutron star, roughly of order 10^10 tesla, is so strong that it distorts atomic orbitals into thin \"cigar shapes\". (The cyclotron energy becomes greater than the Coulomb energy.) Certainly if a solid crystal were placed in such a field it would become very anisotropic, and at some field strength the lattice constant in the direction transverse to the field could become small enough for nuclear fusion rates between the nuclei to become non-negligible. How high do we need to crank up the field before the nuclei all equilibrate to the absolute energy minimum of iron and nickel in, say, a matter of hours or days? Update: From http:\/\/dx.doi.org\/10.1086\/152986 it appears that matter in strong magnetic fields forms into strongly-bound 1D chains along the field lines, which are only weakly bound to each other, and the parallel and transverse lattice constants are actually comparable."}
{"id":"41036","title":"Why are Euler's equations of motion coupled? Physical explanation","text":"I have a problem with one of my study questions for an oral exam: > _Euler’s equation of motion around the $z$ axis in two dimensions is > $I_z\\dot{\\omega}_z = M_z$, whereas it in three dimensions is > $I_z\\dot{\\omega}_z =-(I_y-I_x)\\omega_x\\omega_y+M_z$, assuming that the $xyz$ > coordinate systems is aligned with the principal axis. Why does Euler’s > equation of motion for axis $z$ contain the rotational velocities for axes > $x$ and $y$?_ How can one explain this physically? I mean I can derive Euler's equation of motion, but how can I illustrate that the angular velocities are changing in 3 dimensions?"}
{"id":"41031","title":"Adiabatic expansion","text":"I'll start off by saying this is homework, but I ask because I don't understand how the math should work (I don't just want an answer, I'd like an explanation if possible). I understand if this is still against the rules, and I apologize if it is. I have to calculate the work done by a sample of dry air on the environment in an adiabatic expansion to five times the original volume and an initial temperature of 20˚ C. The answer rounds to 100,000 (it's 99,???). So far, I have that $c_v dT = -p d\\alpha$. I tried integrating, which gives $c_v T = R_d T_0 \\ln{5}$ (because the limits of integration give 5 inside the natural log and $R * T_0$ is constant; the IGL gives $p\\alpha = RT$). Plugging numbers in gives $T = \\frac{(-287)(293)\\ln{5}}{717} = -188.76$. Since the change in internal energy is equal to the negative work done (because $\\delta q = 0$ in an adiabatic process), and $dT = 105$, then $\\delta w = 105 * 717 = 75,285$. This answer is incorrect, but I'm not sure where I went wrong."}
{"id":"41030","title":"Why is $\\int (dp\/2\\pi) |p \\rangle\\langle p| = 1 $?","text":"In quantum mechanics, why is $\\int (dp\/2\\pi) |p \\rangle\\langle p| = 1 $ where $|p \\rangle$ represents momentum eigenstate?"}
{"id":"29948","title":"Linear charge density, surface charge density and volume charge density","text":"What is the difference among:   1. linear charge density,    2. surface charge density, and    3. volume charge density?"}
{"id":"79791","title":"regarding nuclear magnetic resonance","text":"In NMR experiment we provide an external radio frequency signal which will induce transition of nuclear spins to higher states,and then it returns to ground state precessing around the magnetic field in larmor frequency,which is the same as that of absorbed radio frequency.One thing i don't understand is the underlying reason for resonance. Why is it that a system absorbs a certain frequency called the natural frequency? Another thing is both protons and neutrons have different magnetic moments so they should have different larmor frequencies(right?),but we obtain only one resonance frequency in the experiment so which one does it correspond to?Another thing is how do neutrons which are neutral particles have magnetic moment?"}
{"id":"114331","title":"What is chirality?","text":"I actually wanted to make the title as \"What is the difference between chirality and helicity\"? But I didn't do that because I don't understand properly what chirality is. I have gone through this Wikipedia article: chirality to get the meaning of chirality? and what I get from there is that something is said to have chirality if it is not identical to its mirror image. But I have often seen people saying a massless particle have same helicity (handedness, I think) and chirality? Now if the chirality has the above defination then how do people say it? Again for massive particle we can have changed helicity by changing our reference frame. Ok I understand that, but How does it have anything to compare with chirality? I have read these: Helicity and Chirality and many others in this site. But didn't find(or maybe didn't understand) the answer there which I am looking for."}
{"id":"114338","title":"Why could you not travel back in the time before a tipler cylinder is created?","text":"Is it because the lightcones are not tipped all the way over in which the time axis points downwards?"}
{"id":"111679","title":"Expansion of the Universe regarding Time and lightspeed","text":"According to our current understanding, the universe is expanding... it is often said that every galaxy is moving away from us (I'm assuming that isn't true for galaxies in our local group), and it's also often said that the _farther_ a galaxy is away from us -- the _faster_ it moves away from us. We also know that light has a finite speed, so when we look at those far away galaxies, we see them as they were a long, long time ago. But (1) if the _oldest_ galaxies are moving away _faster_ , and the _younger_ galaxies are moving away _slower_ , then wouldn't that mean that the expansion of the universe is slowing down? But (2) when I'm thinking of an expanding 2d sheet of paper with dots (scaled up in both directions), then it makes sense that the farther galaxies would move faster away from our perspective.. (although if the universe were expanding at a constant rate, we would still see the farther galaxies moving faster from our perspective, but I'm assuming that it has been accounted for, and that the galaxies move even faster than that). So... why is my first \"but\" incorrect? Something tells me that I'm looking at it at the wrong way, but I can't figure it out xD Thanks!"}
{"id":"111672","title":"What causes the 'tears' in yoghurt drinks?","text":"Fluid dynamical instabilities are present in many different everyday things. The famous tears in wine for example are a classical example of a Marangoni effect, where surface tension gradients due to evaporation cause an instability. I recently came across an instability that occurs when you have just emptied a glass with yoghurt drink and have only a film of the drink left on the walls of the glass. A tear-like pattern emerges over the time of about a minute that looks like shown below: ![enter image description here](http:\/\/i.stack.imgur.com\/KY0lG.jpg) I have been thinking what can cause this effect. It seems somewhat similar to a Marangoni effect, but given the ingredients of the yoghurt drink I think we can safely rule out evaporation on this timescale. Another potential effect explaining this would be viscous fingering, but for that to happen I believe the gas phase has to be driven, which it isn't here. Or perhaps it is a 'surface-version' of a Rayleigh-Taylor instability? Or maybe it is related to the non-newtonian nature of the yoghurt drink? To summarize: I don't know which instability I am looking at and I would love to know what causes this pattern on the glass with yoghurt drink! Does anyone know what is causing it?"}
{"id":"114487","title":"How does a unique electron probability distribution correspond to one wavefunction?","text":"I'm reading the Wikipedia article on DFT, and it says that there is a one-to- one correspondence between the ground state particle density $$n_0(\\vec{r}) = N \\int \\text{d}^3 r_2 \\int \\text{d}^3 r_3 \\cdots \\int \\text{d}^3 r_N \\Psi_0^{*}(\\vec{r}, \\vec{r}_2, \\ldots, \\vec{r}_N) \\Psi_0(\\vec{r}, \\vec{r}_2, \\ldots, \\vec{r}_N)$$ and the ground state wavefunction $\\Psi_0(\\vec{r}, \\vec{r}_2, \\ldots, \\vec{r}_N)$ such that $\\Psi_0$ is a functional of $n_0$: $$\\Psi_0 = \\Psi[n_0]$$ I don't understand how this is possible. It seems to me that you could have an infinite number of wavefunctions, with differing phases, as long as the squared amplitudes end up the same. Could someone explain to me why this isn't possible?"}
{"id":"89561","title":"What is the transmittance function of a photo filter?","text":"The effect of some photo filters can be summarized in transmission curves such as this one: ![http:\/\/micro.magnet.fsu.edu\/primer\/lightandcolor\/colortemperatureintro.html](http:\/\/i.stack.imgur.com\/xDw8d.jpg) I do not understand which physical value is modified by the value labeled as \"Diffuse Density\" in the picture above. It sometimes appear as a percentage, or a ratio included between 0 and 1. _Note: I originally posted a related question onphoto.stackexchange.com_"}
{"id":"89567","title":"Calculating angular velocity after collision","text":"Suppose I have a disc which doesn't move, just rotate around the axis going through its centre of mass perpendicular to its surface. The disk has a stick perpendicular to its surface at the edge. I also have a block. The block hits the stick tangentially with certain velocity and then stops. How can I calculate the angular velocity of the disc after collision."}
{"id":"89566","title":"Are high redshift masses corrected for relativistic mass dilation? They would appear more massive right?","text":"A distant quasar would be less massive in its frame of reference than our observations would suggest. Are such highly red-shifted objects corrected for relativistic mass dilation?"}
{"id":"56459","title":"Regarding real field Klein Gordon Equations","text":"Here are 2 doubts:   1. If we change the sign of the mass term in the free massive KG Lagrangian to get: $L = \\frac{1}{2}\\partial^\\mu\\phi\\partial_\\mu\\phi + \\frac{1}{2}m^2\\phi^2$, What would be the $physical$ implications of this change? (aside from on shell condition not being satisfied)?   2. Let $\\phi^{(1)}$ and $\\phi^{(2)}$ be 2 real scalar fields with the Lagrangian: $L = \\frac{1}{2}\\sum_{i}\\partial^\\mu\\phi^{(i)}\\partial_\\mu\\phi^{(i)} - \\frac{1}{2}m^2 \\sum_{i,j,k} \\phi^{(i)} M_{ij}M_{jk} \\phi^{(k)}$ . where $M_{11} = \\lambda$, $M_{12} = 1$, $M_{21} = 1$, $M_{22} = 0$ AND $\\lambda >> 1$. What is the mass ratio of the 2 particles in the theory? EDIT: As for the 2nd question, I found that $L$ should simplify to $\\frac{1}{2}\\sum_{i}\\partial^\\mu\\phi^{(i)}\\partial_\\mu\\phi^{(i)} - \\frac{1}{2} m^2(\\lambda \\phi^{(1)} + \\phi^{(2)})^2$. I tried continuing from the viewpoint that we can consider the superposition $\\phi^{(3)} = \\lambda \\phi^{(1)} + \\phi^{(2)}$ and try to eliminate cross-terms, thereby getting a simple sum of non-interacting Lagrangians, but that didn't work. (Now I'm out of my depths) Thanks in advance"}
{"id":"29673","title":"Modeling stochastic process with frequency-dependent power spectrum","text":"I'm trying to model of Johnson-Nyquist noise propagation in a nonlinear circuit. An ideal (linear) resistor can be modeled very nicely by the Fokker- Planck equation (equivalently, the drift-diffusion equation), where charge $V\/R$ flows through the resistor _on average_ , but there's also random flow of charge either way across the resistor characterized by \"diffusion coefficient\" $k_BT\/R$. I get a nice differential equation describing how charge probabilistically flows through my circuit. Everything is good. Then stage 2 is to have a resistor with frequency-dependent resistance (like all resistors in the real world). Here I get stuck... A time-domain-based analytical solution seems impossible because---with frequency-dependent resistance---it would seem that charge transport across the resistor right now depends on the entire history of charge transport in the past. A time-domain numerical (monte carlo) solution seems impossible because ~~the relevant frequencies vary over many orders of magnitude~~ I don't know how to construct a time-domain stochastic model with a predetermined power spectrum. Any kind of frequency-domain solution seems impossible because other parts of the circuit are extremely nonlinear and therefore mix different frequencies together. Any advice? Am I missing some trick?"}
{"id":"72272","title":"What's the simplest method for accurately measuring the speed of light?","text":"With common household objects in hand, and, say a budget of $20, what's the simplest experiment for accurately measuring the speed of light?"}
{"id":"7500","title":"Auto-ionization of water","text":"> $HCl(g) + H_2O(l) -> H_3O^{+}(aq) + OH^{-}(aq)$ Assuming $HCl$ completely ionizes in water, if we prepare a $HCl$ solution, will the $[HCl]=[H_3O]$ ($HCl$ concentration = $H_3O$ concentration)? That is, the $HCl$ before the reaction, and the $H_3O$ after reaction? If so, then if we observe the following equilibrium reaction for the auto- ionization of water: > $2H_2O \\rightleftharpoons H_3O^{+} + OH^{-}$ it can be concluded that $[H^{+}][OH^{-}]$ is a constant value, that is, the product of their equilibrium concentrations is a constant value. However, if the $H_3O$ concentration increases by the ionization of $HCl$, wouldn't the reaction shift to the left to counteract the change (by Le Chatelier's principle)? Consequently, this would decrease the concentration of $H_3O$ ions in solution, and contradict the equality of $HCl$ concentration = $H_3O$ concentration, so was the original proposition incorrect?"}
{"id":"96542","title":"How does $SU(2)$ group enters quantum mechanics?","text":"What is the reason that $SU(2)$ group enters quantum mechanics in the context of rotation but not $SO(3)$? What really rotates and which space it rotates? It cannot be the physical electron that rotating in real space. I think it is the state vector that rotates in spin space. Am I right? Is it the property of the \"weird\" spin space (Hilbert space) that rotation by $4\\pi$, brings it back to where it started?"}
{"id":"96547","title":"How to apply conservation of angular momentum with a shock?","text":"I got this tricky question, need help. A uniform rod of mass $M$ and length $L$ is attached to an axis at its top, a bullet with mass $m$ traveling at speed $U$ (horizontal) hits the rod at $2L\/3$ from the axis and then LEAVES the rod at speed $U\/2$. I need to find the angular speed of the rod right after collision assuming collision time is very short. If the bullet stayed in I would just do conservation of angular momentum, find the new moment of inertia with the bullet using Steiner's law and solve for $\\Omega$. But what do I do when the bullet leaves the rod?"}
{"id":"113696","title":"Current, Current density","text":"edit: Hi I'm trying to find the magnetic field generated by a time dependent oscillating current in the quasistatic case ($|z|,r <<c\\omega$) where r is the perpendicular distance from the z-axis. The current is flowing through a long, thing wire that is laying on the z-axis. If $z=0$, we can write the current as $$ I(t)=I_0\\sin \\omega t $$ and now I am trying to find the magnetic field $B(r,t)$ at $z=0$ with this current. How can we calculate $\\vec B$? Possibly, can we calculate $\\vec J$ to calculate $\\vec A$ to calculate the magnetic field? THanks a lot for your help on the last problem too."}
{"id":"95423","title":"What is the frictional force of this car?","text":"The breaking acceleration for a car is $-12.2\\frac{m}{s^2}$. The car has a mass of 925kg. I want to find the frictional force. Here is what I have so far: $$\\Sigma\\vec{F} = m\\vec{a}$$ $$\\Sigma\\vec{F} = (925kg)(-12.2m\/s^2)$$ $$\\Sigma\\vec{F} = -11285N$$ So now I have the net force of the car. Would this be the same as the frictional force (any action has an equal and opposite reaction)? Or do I need to use the Friction Law formula? ${}{}$ Thanks!"}
{"id":"78771","title":"Magnitude of magnetic field at the center of circular wire","text":"I'm preparing for an exam by solving the sample questions , here is the one I'm having difficulty with : Following is the given circuit.   Which contains two resistance $R_1$ and $R_2$ in form of circle of radius $r = 1 m$ with a battery having e.m.f. $V = 10π $ volt.   Upper resistance is having resistivity $p_1 = 4$ & lower resistance having resistivity $p_2 = 2$.   Angle between two points $A$ and $B$ is $60°$. (wires have same cross section $A_1 = A_2 = 2 cm^2$) Diagram : ![Circuit](http:\/\/i.stack.imgur.com\/5MiAz.png) Here is the question based on this passage that I'm having difficulty with :   Find magnitude of magnetic field at center. (A) 0T (B) 1T (C) 2T (D) πT The problem is I don't know any formula to calculate the magnetic field at a point nor am I expected to know (probably , at least it is not in syllabus) .   Also what is 'T' ? Is it unit of magnetic field?   The answer is 0 so I guess I should be able to reason why it should be 0 but I don't know how. Please help."}
{"id":"105103","title":"The temperature dependence of the electron-hole (or particle-hole) continuum","text":"In the theory of the electron gas, the particle-hole pairs are possible elementary excitations. I would like to know how the temperature T affects the particle-hole continuum which defines the domain where the imaginary part of the density-density response function (or Lindhard function) is different from zero. I need to understand it in order to calculate the Lindhard function at finite temperature."}
{"id":"10952","title":"Loss of Power at high frequencies","text":"One of my work colleagues told me that a cable he is sending a signal through is losing power at high frequencies. So he recommends the signal should be amplified before being sent. The explanation was given for the power loss is that higher frequency signals are lossier. As a newcomer to signal processing, I'd like to understand more about why or how this effect occurs. How is the frequency of the signal causing the power loss to occur? From Wikipedia's article on Coaxial cable I found this, which seems promising: > If an ordinary wire is used to carry high frequency currents, the wire acts > as an antenna, and the high frequency currents radiate off the wire as radio > waves, causing power losses. Is understanding how antennae work key to understanding why the high frequency results in power loss?"}
{"id":"69898","title":"Question about Matrix Integral","text":"I am stuck with the technique details of KKN's paper . How to get formula (2.11) $$Z= \\int \\frac{d \\phi }{ \\rm{ Vol(G)}} \\frac{1}{\\rm{Det}(\\rm{ad}(\\phi)+\\epsilon)} $$ from (2.10)? $$Z=\\int \\frac{d \\phi dX_a d\\psi_a}{ \\rm{ Vol(G)}} \\exp {\\kappa_1 \\rm{Tr}( i\\phi[X_1,X_2] -\\frac{1}{2}\\epsilon^2(X_1^2+X_2^2)+\\psi_1\\psi_2) } $$ I have no idea about this calculation and I even do not know the meaning of $\\rm{Det}(\\rm{ad}(\\phi)+\\epsilon)$. Any advices are welcoming."}
{"id":"10956","title":"Why does a thermometer in wind not show a lower temperature than one shielded from it?","text":"I'm a little familiar with the physics and thermodynamics of the wind chill effect, but this question seems to come up from time to time: Why, given two temperature sensors or thermometers in the same environment, do both report the same temperature if one is exposed to wind when the other is shielded from it? People often ask because the temperature reported by, for example, their vehicle, doesn't seem to change as they drive at different speeds, etc. (Other than of course the changes from one actual temperature to another as they change geography.) My understanding is that inert devices aren't endothermic like we are, so the effects of wind chill don't apply. Can you explain this in layman's terms?"}
{"id":"61989","title":"Why doesn't a stationary electron lose energy by radiating electric field (as per coulomb's law)?","text":"If an electron in a universe constantly generates an electric field why does it not get annihilated ? I am confused because I read that an accelerating charge radiates and loses energy. So, why won't a stationary electron radiate and lose energy?"}
{"id":"61983","title":"The electric field of a conductive sphere containing a charge - grounded vs not grounded","text":"Let's suppose we have a sphere but unlike theoretical ones it'll have some thickness say $\\Delta r$ and inner radius $R$. What I was wondering about is how will it behave if we place some charge $q$ in the center? What would the field look like and what will be the charge density on either side. When it's grounded and when it's not. My attempt: When it's not grounded I though that outside if we create a spherical surface Gaussian with $r>R+\\Delta r$ then there's only the charge $q$ to consider so it'll behave like a exactly like $q$ on it's own meaning $E=k \\frac{q}{r^2} \\hat{r}$ , and the same goes for $r<R$. inside the shell it'll be zero as it's a conductive surface so $\\phi = const$. On the inner surface and the outer surface the sum of all charge will have to be $-q , q $ accordingly as they have to cancel each other so the density will be $\\sigma = \\frac{-q}{4\\pi R^{2}} , \\frac{q}{4\\pi\\left(R+\\Delta r\\right)^{2}} $ . When it comes to the grounded version I'm a little confused as I don't really understand what's the difference apart from the fact that the initial $\\phi =0 $ but electrical charge will still be pulled to the sphere from $\\infty$ so it seems like there's no difference but I'm a beginner so I'm not sure whether my deduction are valid or not and it seems kind of fishy to me but I can't really point to what is essentially wrong."}
{"id":"81638","title":"Electron motion in a wire","text":"For my introductory course to electromagnetism (I'm an undergraduate student, so ELI5), I'm trying to get the right conceptual model of electron movement in a thin wire (with constant but non-zero cross section, like a cylinder) due to a constant current. It seems to me, that there are several contributing factors, and I wonder which dominate, and if some kind of order exists.   1. Electrostatic repulsion between electrons. If this was the only force at play, electrons should all travel on the surface of the wire.   2. Magnetostatic attraction caused by the magnetic field from charges in motion. If this was the only force then the electrons would all travel in the center of the cylinder.   3. Collisions with matter and thermal scattering, which would make the motion chaotic.   4. Material properties of the cylinder, and if in particular the cylinder is a metal, then I suspect that the lattice planes of the metal and orientation of these planes is somehow relevant.  Basically 1 and 2 create order, 3 and 4 seem to destroy it. I can't seem to find anything relevant in my courseware or online. Specifically I'm interested in common DC household wires."}
{"id":"3859","title":"Capillary action and conservation of energy","text":"When I dip a paper towel in a cup of water the water gets drawn up due to capillary action. How is this reconciled with conservation of energy, as it would seem on the surface that the potential energy of the system has gone up with no work being done?"}
{"id":"69765","title":"Is the neutral pi meson a relativistic electron positron pair?","text":"An electron position pair orbiting at relativistic speeds is proposed as a model of a neutral pi meson by Ernest Sternglass. However, his mathematics is semi-classical. Has anybody done a QED approximation?"}
{"id":"21351","title":"Neutrino Oscillations and Conservation of Momentum","text":"I would like to better understand how neutrino oscillations are consistent with conservation of momentum because I'm encountering some conceptual difficulties when thinking about it. I do have a background in standard QM but only rudimentary knowledge of particle physics. If the velocity expectation value of a neutrino in transit is constant, then it would appear to me that conservation of momentum could be violated when the flavor eigenstate at the location of the neutrino source is different from that at the location of the interaction, since they are associated with different masses. For this reason I would think that the velocity expectation value changes in transit (for instance, in such a way to keep the momentum expectation value constant as the neutrino oscillates), but then it seems to me that the neutrino is in effect \"accelerating\" without a \"force\" acting on it (of course, since the momentum expectation value is presumed constant, there may not be a real problem here, but it still seems strange). Any comments?"}
{"id":"52494","title":"Why aren't the graphs for black body radiation straight lines?","text":"We know that a wave which has greater frequency will have low wavelength and high energy. So, by decreasing the wavelength, the frequency and consequently energy (intensity) of that wave will increase or vice versa. Now, i want to ask a question in black body radiation by looking at the following graphs ![enter image description here](http:\/\/i.stack.imgur.com\/bod9S.jpg) My question is that if by decreasing the wavelength of a wave, the energy of that wave increases then why do these graphs fall down after reaching a maximum value. Shouldn't the graph just move straight on?"}
{"id":"22916","title":"Why can't a single photon produce an electron-positron pair?","text":"In reading through old course material, I found the assignment (my translation): > Show that a single photon cannot produce an electron-positron pair, but > needs additional matter or light quanta. My idea was to calculate the wavelength required to contain the required energy ($1.02$ MeV), which turned out to be $1.2\\times 10^{-3}$ nm, but I don't know about any minimum wavelength of electromagnetic waves. I can't motivate it with the conservation laws for momentum or energy either. How to solve this task?"}
{"id":"22912","title":"Does spin alone have any effect on the physical interactions of particles?","text":"In Hartree-Fock theory the time-independent electronic energy of a single (restricted) determinant electronic wavefunction consists of one electron terms, $h_{ii}$, Coulomb interaction energies, $J_{ij}$, and exchange interaction energies $K_{ij}$. Exchange interaction energy terms result from integration over the 2-electron repulsion operator $r_{12}^{-1}$ where the electrons in the 2-electron integral have parallel spin. I have read that the exchange interaction is not a true physical interaction, but arises because of the single determinant description of the system. It is a partial description of the correlated motion of electrons, which is better described in many-determinant methods. My question is this: does spin alone have any effect on the physical interactions of particles? For example, consider 2 uncharged particles with spin 1\/2. They may either have parallel or antiparallel spins. In physical reality, would these 2 particles repel one another if they have parallel spin? If I was to model this system with a Hartree-Fock calculation, I believe there would be no exchange interaction, because the Hamiltonian would feature no two particle operator, so no Coulomb or exchange integrals would arise. However, perhaps if I used a higher level of theory, I might see an interaction?"}
{"id":"56930","title":"Moose Models (Purpose, Examples)","text":"A problem set for my QFT class is titled \"Moose Models\" and deals with the moose model for a gauge symmetry of $U(1)\\times U(1)$. I was wondering if I could get an explanation of what a Moose Model is, and what it's useful for. Googling things such as \"Moose Model Particle Physics\" led me to a thesis from Harvard on Minimal Moose Models in $SU(5)$ but it's frankly too advanced for me to do much with. Any help would be appreciated."}
{"id":"11605","title":"When is the \"minus sign problem\" in quantum simulations an obstacle?","text":"The \"minus sign problem\" in quantum simulation refers to the fact that the probability amplitudes are not positive-definite, and it is my understanding that this leads to numerical instability when for example summing over paths, histories or configurations since large amplitudes of one phase can completely negate other large contributions but with the reverse phase. Without controlling this somehow, the sampling of alternative configurations has to be extremely dense, at least for situations where interference is expected to be important. Assuming I haven't misunderstood it so far, my main question is when this is a show-stopper and when it can be ignored, and what the best workarounds are. As an example, let's say that I want to simulate Compton scattering or something similar, numerically, to second order. I could evaluate the Feynman diagrams numerically to a certain resolution and sum them. I assume this won't work well. In Lattice QCD, complete field configurations are randomly generated and the action calculated and amplitude summed I guess (I have only superficial knowledge of Lattice QCD unfortunately)."}
{"id":"114166","title":"Road to String Theory","text":"I have a question for our theoretic SuperUsers. How much knowledge and which fields of physics you have to know to start studying string theory? I am now on QFT, after I think I'll start study Supersymmetry or maybe special cases of QED. How much more should I know to start Strings? :)"}
{"id":"62984","title":"Would the universe get consumed by blackholes because of entropy?","text":"Since the total entropy of the universe is increasing because of spontaneous processes, black holes form because of entropy (correct me if I'm wrong), and the universe is always expanding, would the universe be consumed by black holes or would its area increase too quickly for that to happen?"}
{"id":"9791","title":"Help getting the distance an object has moved","text":"_What is a distance an object went through at the speed of 14hm\/min for 6 hours?_ I am having problems with conversions. First, I tried passing 14hm\/min to m\/s. I did it alright. But now I am unsure. Do I multiply it by 6 hours and I'm done?"}
{"id":"9795","title":"What is the most natural definition of the weak hypercharge coupling constant if grand unification is wrong?","text":"A tricky question. Here is the famous graph of the running of the three coupling constants in the standard model: http:\/\/www-ekp.physik.uni- karlsruhe.de\/~deboer\/html\/Forschung\/unification_eng.eps . The graph shows, in its top curve, the running of the coupling constant alpha_1. This is the coupling of the weak hypercharge coupling constant for the weak hypercharge group U(1)_Y, which is one of the three gauge groups of the standard model of particle physics. But there is a tricky detail. In that curve, alpha_1 is multiplied with 5\/3. This factor 5\/3 comes from the assumption that GUTs are valid. The factor ensures that the various group traces of U(1)_Y, SU(2) and SU(3) are normalized in the correct way when they form the SU(5), SO(10) or any other grand unification gauge group. In the case that grand unification is wrong, the factor 5\/3 cannot be deduced. Which factor would be natural in this case? Clarification added, after remarks by Lubos Motl: it is assumed in the question that the usual definition of the weak hypercharge is used, Y_W=2(Q-T_3), in which left-handed quarks have hypercharge 1\/3."}
{"id":"9799","title":"Do and can phone signals come inside AC car which is glass-packed","text":"Can phone signals penetrate glass, so can I expect phone signals to come if I am sitting inside a closed AC car. Thanks,"}
{"id":"98598","title":"Basic Thermodynamics: Quasistatic Adiabatic Process","text":"I'm going through the exercises in a Thermodynamics book, just to revise and build my intuition. Right now, I'm working on: > Show that for a quasistatic adiabatic process in a perfect gas, with > constant specific heats: > > $PV^\\gamma = constant$ > > with $\\gamma = \\frac{C_p}{C_V}$ where P is pressure, V is Volume, and C is heat capacity. **I'm not looking for the answer, just for a hint (I'm stuck and want to find the solution myself).** So those are my thoughts:   * perfect gas means: $PV = RT$ , (R is the universal gas constant)   * adiabatic means: $dQ = 0$, (Q for heat)   * since there is no heat exchange, the process is reversible   * reversible means: $dW = -PdV$, (W is for work)   * heat capacity is defined as $C_V = \\left( \\frac{dQ}{dT} \\right)_V$, respectively $C_P = \\left( \\frac{dQ}{dT} \\right)_P$ If I draw a PV diagram for this situation, it looks like this: ![enter image description here](http:\/\/i.stack.imgur.com\/8LAQf.png) Now I want to show that $PV^\\gamma=const$ by going from state 1 to state 2 in the PV diagram. I've started like this: $W = -\\int_{V_1}^{V_2}PdV = -\\int_{V_1}^{V_2} \\frac{RT}{V}dV = RT ln (V_2\/V2)$ This leads me into the wrong direction though. I thought about using $R = C_p - C_V$ here, but it doesn't seem to work. Any suggestions? **Please just give me a hint, not the solution.**"}
{"id":"112413","title":"Model the gyroscopic effect for 3axes stabilisation","text":"**Context** I am trying to stabilise a platform on a guimbal using 3 axes gyroscopes. I have made tons of research (same problem for weeks even months now) without being able to properly model the 3 orthogonal gyroscope actuators (high inertia flywheels). **Question** What is the dynamic model of a 1-axis gyroscope actuator? Can it be used 3 times on different directions (I have a Simulink model) to get a 3-axes one or is it more complex? From what I have seen, $\\boldsymbol{w_{precession}} =\\boldsymbol{h_{gyro}}\\times \\boldsymbol{\\tau_{ext}}$ which may or may not be what I'm looking for (experts?) but I don't see the rotation being instantaneously triggered: what's the corresponding torque? **Additional info** My Simulink plant implements $\\boldsymbol{\\theta}=\\int_0^t\\int_0^t\\frac{\\boldsymbol{\\tau_{ext}}}{\\boldsymbol{J}}dt²$; and the gravity torque, to counter, is given by $\\boldsymbol{\\tau_{grav}}=\\boldsymbol{r}\\times\\boldsymbol{W}$ (weight with moment arm centre of guimbal-CG). If this is the right way to go, I could inject the $\\omega_{p}$ after the first integration, but what's the associated torque that counters the gravity's?"}
{"id":"20400","title":"What are the practical applications of decoherence?","text":"Let me clarify this question somewhat. I know decoherence is ubiquitous in nature and explains the emergence of a classical world from quantum physics. My question is really about how a knowledge of how decoherence actually works can be put to use in a practical application. An application we can't design in the absence of such a knowledge, even though decoherence is still happening all the time. Thanks"}
{"id":"99450","title":"Angular momentum in a rod rotating around one end?","text":"Sorry if I can't get straight to the point, I have to give a lot of details before I actually state the question. The formula for angular momentum is $L=I \\omega$. If we look up $I$ for a thin rod pivoted around one end we get $I=\\frac 13 ML^2$ so $L=\\frac 13 ML^2 \\omega$. However, $L$ is also equal to $p \\cdot d$, where $p$ is the linear momentum and $d$ is some distance, so $I \\omega=pd$ My question is as to why in this case $d$ comes out to be what it is $($it is $\\frac 23 L)$ We can work out the linear momentum of the rod: Assume the rod, which has a mass $M$, length $L$, and a constant angular speed of $\\omega$ is split into $n$ equal pieces. Then each piece has mass $\\frac Mn$ and a linear momentum of $\\frac Mn \\cdot \\omega r_i$, where $r_i$ is the distance from the pivot to the end of the $i^{th}$ segment. The linear momentum is approximately $\\sum_{i=1}^{n} \\frac Mn r_i \\omega$. We can make this exact by taking the limit as $n$ approaches infinity. We have $r_1=\\frac Ln$, $r_2=2\\frac Ln$, $\\cdot \\cdot \\cdot \\cdot \\cdot r_i=i \\cdot \\frac Ln$, so we can substitute this in the sum: $$ \\lim_{n \\to \\infty} \\sum_{i=1}^{n} \\left(\\frac Mn i \\cdot \\dfrac Rn \\omega\\right)$$ Since $M$, $\\omega$, $R$, and $n$ are not relevant to the summation, we can factor them out: $$ p=MR \\omega \\lim_{n \\to \\infty} \\dfrac {1}{n^2} \\sum_{i=1}^{n} i$$ $$p= MR \\omega \\lim_{n \\to \\infty} \\dfrac {1}{n^2} \\dfrac {n(n+1)}{2}$$ $$p=\\dfrac 12 ML \\omega$$ Again, we have $$L=\\dfrac 13 ML^2 \\omega=pd=\\dfrac 12 ML \\omega d$$ If we solve for $d$, we get $d=\\frac 23 L$. Why is this? I was expecting $d$ to be a more relevant point, such as $L$ or $\\frac 12 L$ (because that's where the center of mass is. But why $\\frac 23$? It seems pretty random."}
{"id":"99453","title":"How do speakers play sounds with multiple sources?","text":"I know that speakers produce sound by creating a pressure wave, but I don't understand how when they play music or games you can hear the different sources. I guess what I'm asking is how do they play (or seem to play) multiple frequencies and amplitudes simultaneously?"}
{"id":"131740","title":"Alcubierre drive and inertia.","text":"What is the inertia or velocity of a vehicle upon exiting or shutting down an Alcubierre bubble? Would the vehicle maintain the velocity it had in the bubble? I'm not sure I asked the question in a meaningful way so I'll try another question that targets what I'm after. If a vehicle was traveling in one of these bubbles how hard would it be to come to a full stop?"}
{"id":"111994","title":"Is it possible to find the hydrogen atom's radial wavefunctions?","text":"Is there a way to actually find the equation of $R(r)$ without looking at a table with these equations already given? I'm given $n$, $\\ell$, and $m$."}
{"id":"37920","title":"Do magnets work in outer space?","text":"Is there any media where magnet lose its property?"}
{"id":"95891","title":"A coincident stack of D3 branes vs a shell of them","text":"* I would in general like to understand how to derive the low energy metrics to describe D-brane configurations. Any pedagogic reference which explains the method? In particular I have in mind these two facts, (1) For N coincident D3 branes in IIB it seems that the metric is, $ds^2 = (1+ \\frac{L^4}{y^4})^{-0.5}\\eta_{ij}dx^idx^j + (1+ \\frac{L^4}{y^4} )^{0.5}(dy^2+y^2d\\Omega_5^2) $ where $L^4 = 4\\pi g_s N (\\alpha')^2$ ($L$ being called the radius of the Dbrane) (2) A \"shell\" of D3 branes is apparently described by, $ds^2 = h^{-1}(r)[-dt^2 + d\\chi^2 + dx_1^2 + dx_2^2 ] + h(r)[dr^2 + r^2d\\Omega_5^2 ]$ where $\\chi$ is a compactified dimension of length $L$ and $h(r) = \\frac{R^2}{r^2}$ for $r>r_0$ and $h(r) = \\frac{R^2}{r_0^2}$ for $r<= r_0$   * How are these two related? What exactly is a \"shell of Dbranes\"? How are these two derived?   * What exactly is meant by \"radius of the Dbrane\" (L) in the first one? Is that the coordinate in the spacetime which was compactified and T-dualized and is that the same as the $L$ in the second metric?"}
{"id":"37929","title":"What makes water-droplet\/dew stick to spider's web and what keeps them there?","text":"How water-droplet\/dew stick to spider's web? What keeps them there?"}
{"id":"79659","title":"How does volume of water in closed system relate to humidity at 25(C) degrees","text":"If I have a closed system with a volume of 100L, at ~24(C) or ~75(F) degrees, what volume must be filled with water to reach a humidity of 90%? I imagined if you fill it 100% with water, humidity is 100%? But I also imagine that with anything less than 100% filled with water, temperature becomes a limiting factor, and 90% humidity might not be achievable without raising the temperature. I checked out a max humidity ratio table to see that at 25(C) the saturation pressure is 3130 pa, with a maximum humidity ratio of 0.019826 -kg(w)\/kg(a) and that humidity ratio can be expressed with the partial pressure of water vapor:               x = 0.62198 pw \/ (pa - pw)            pw = partial pressure of water vapor in moist air (Pa, psi)          pa = atmospheric pressure of moist air (Pa, psi)          The maximum amount of water vapor in the air is achieved when pw = pws the saturation    pressure of water vapor at the actual temperature.      But this is as far as I got, as my understanding of humidity (relative, specific, etc).. is negligible... any suggestions or just a rough estimate?"}
{"id":"91781","title":"Physical meaning of the Rindler hyperbola vertex and the Rindler lines","text":"Two questions regarding the Rindler diagram: 1) Does the vertex of a given hyperbola in the diagram have physical meaning? I know it is the inverse of the constant proper acceleration ($\\alpha$) associated with that particular hyperbola, and I have heard mentioned several times (although never seen formally demonstrated) that acceleration is proportional to the gravitational potential. Can some sort of physical meaning be built out of these mathematical facts? (Even though the Rindler diagram describes flat space-time with no gravitational potential) 2) What sort of juicy information are we supposed to get out of the Rindler rays extending from the origin and intersecting all the hyperbolae? As far as I can tell these rays preserve the ratio of starting distances between the space ships taking off at the same time and traveling at different constant proper accelerations along different hyperbolae. But that is all. Do these rays tell us something about simultaneity? Bonus points for whoever can throw in a formal proof that vector potential is equal to acceleration. (Edit:) Oh, one more thing occurred to me--the thing people call the \"Rindler time\" ($\\omega$) should actually be called the proper velocity ($\\omega=\\alpha \\tau$). Why do people call it the Rindler time?"}
{"id":"72601","title":"Is there any physics behind covariance and contravariance of indices of tensors?","text":"Is there any physics behind covariance and contravariance (up and down) of indices of tensors?"}
{"id":"100115","title":"Can a wave exist on the “face” of a wave?","text":"It would seem that this would be possible with waves in water. What about other waves Clarification:   Given a wave starting from a point of impact, in water, at: Time 0:00 and xyz 0,0,0 and the impacting “object” came from 0,0,h. At some time it would reach its first maximum wave height, say z=10. If at that instant an impact coming from 0,0,5 traveling along the x axis impacted the “face” of the first wave, the second wave would initially propagate as though it had impacted a flat surface, but it is not a flat surface, nor is it a stationary nor a stable surface. It seems if would be a curved (initially around 0,0,0 and than over each peak and trough) “undulating” surface and additionally an EXPANDING surface. I suppose this is complicated enough although manageable to think about. And that brings me to the question: What about other waves (light . . ?) Added 2-22-2014: I really don’t think anyone has understood what I am getting at; undoubtedly my fault. The classic “Bulls Eye” water wave table pattern is relatively stable, long lived. Imagine an impact to the face (NOT another impact from above) of the first wave as it reached its first peak. . . Well it turns out what I was trying to describe with a wave table and then question if it would be applicable to light is well depicted; by\/as: Photon Electron Junctions in Feynman diagrams and the intricacies of particle interaction."}
{"id":"114814","title":"Alice sends random states in a channel, what Bob receives?","text":"Suppose Alice prepares $\\rho_x$ with probabilities $p_x$ and sends it to Bob. I would say this is the same thing as \"Alice prepares $\\rho = \\sum_x p_x \\rho_x$ and sends it to Bob\", but Preskill's lecture and Nielsen & Chuang seems to disagree. I will be more precise. If I want to compress the n-fold tensor product of $\\rho$ I can do it down to $n \\ S(\\rho)$ qubits ($S$ is the Von Neumann entropy), this is the Schumacher theorem as stated in Nielsen & Chuang, page 544. If I want to compress the ensemble $\\epsilon = \\\\{p_x,\\rho_x\\\\}$ Schumacher- like, I can do it down to $n \\ \\chi(\\epsilon)$ qubits, as stated in http:\/\/www.theory.caltech.edu\/people\/preskill\/ph229\/notes\/chap5.pdf, page 31. (Maybe if Preskill gave a definition for \"ensemble fidelity\" the issue would be easier to tackle...) Note that $n \\ \\chi(\\epsilon)$ is ensemble-dependent and in general different from the Von-Neumann entropy. But this seems to suggest that different ways of looking at a density matrix allows different ways of encoding it: If I think I am sending a $\\rho_x$ with probabilities $p_x$ I must use the $\\chi$ and hence a certain number of qubits, if I send $\\rho = \\sum_x p_x \\rho_x$ I must use the entropy, and hence another number of qubits... This bother me because I was intimately convinced that the ensemble description is not physical: Classical and quantum probabilities in density matrices,"}
{"id":"26768","title":"Does the sun cross over a given meridian line every day at the same time?","text":"Let's say I have an imaginary meridian line over new york going from north to south. I know that the sun will not always be exactly on top of my head here in New York, but every day it will have to cross somewhere on top of this imaginary line. Does it cross this at the exact same time every day?"}
{"id":"111338","title":"Interpreting the Partition Function and Free Energy Mathematically","text":"Given that > The partition function in statistical mechanics tells us the number of > quantum states of a system that are thermally accessible at a given > temperature > > http:\/\/vallance.chem.ox.ac.uk\/pdfs\/Equipartition.pdf How does one interpret the statistical mechanical partition function and free energy **mathematically** , in terms of the sample space, analogous to the way you can interpret microstates and macrostates in terms of a probability sample space? It seems very Shannon's entropy-like. If you take the example of flipping 2 coins:   * Random experiment: Toss two coins   * Example of an Outcome: $10 = (Heads, Tails)$   * Sample space: $S = {11,10,01,00}$, $|S| = 4$   * Examples of Events: 2 Heads $= 2H = \\\\{11\\\\}$, $|2H| = 1$, $1H = \\\\{10,01\\\\}$, $|1H| = 2$, $0H = \\\\{00\\\\}$, $|0H| = 1$  we can translate this into the language of statistical mechanics:   * A microstate is an element of the sample space, e.g. $10$ or $01$.   * A macrostate is an event (a subset of the sample space), e.g. $1H = \\\\{10,01\\\\}$.   * The statistical weight (statistical probability) of a macrostate is the cardinality of the event, e.g. $|1H| = 2$.   * The equilibrium distribution is the most likely macrostate which is the macrostate with the highest statistical weight which is the event with the highest cardinality, e.g. $1H = \\\\{10,01\\\\}$ since $|1H| = 2$. Finally, the Maxwell-Boltzmann distribution function $n_i$ for the coin toss is found by extremizing $$w(n) = \"number \\ of \\ heads \\ in \\ n\" = \\tfrac{2!}{n!(2-n)!}= \\tfrac{2!}{n_1!n_2!}$$ with respect to $n_i$ given the constraint equation $n_1 + n_2 = 2$, showing $n_1 = e^0 = 1$ maximizes $w$, i.e. $w(1) = |1H| = |\\\\{10,01\\\\}| = 2$ is the maximum. As fas as I can see, the MB distribution is a function of the total number of particles, $2$, but not the energy as there is no energy in this example. Here we see everything interpreted mathematically, however I don't see how one does this for the partition function or for the free energy. So to ask my question: **What exactly is the partition function in this example, and what is it's meaning in general?** _It sounds like it tells you how many elements are in an event for a given energy and given particle number, i.e. the cardinality of a subset of the sample space which varies as a function of the Lagrange multipliers._ **What is the free energy in this example, and in general?** _As a concept it seems very similar to the partition function, only it tells you how energies and particle numbers are distributed over all subsets of the sample space that we are considering, not just the most probable one, e.g. it says something about subset 0, subset 1 and subset 2. Though I'm not sure whether it works like this or whether it just relates to the most probable distribution the whole time, and says something about energies over all possible distributions (i.e. for 2, 3, 4, ... as particle numbers in the numerator of the MB distribution function given above)._"}
{"id":"107244","title":"Allowed interactions in bosonic string theory","text":"In a quantum field theory, only a finite set of interactions are allowed, determined by the Lagrangian of the theory which specifies the interaction vertex Feynman rules. In string theory, an $m$-point scattering amplitude is schematically given by, $$\\mathcal{A}^{(m)}(\\Lambda_i,p_i) = \\sum_{\\mathrm{topologies}} g_s^{-\\chi} \\frac{1}{\\mathrm{Vol}} \\int \\mathcal{D}X \\mathcal{D}g \\, \\, e^{-S} \\, \\prod_{i=1}^m V_{\\Lambda_i}(p_i)$$ where $S$ is the Polyakov action (Wick rotated), and $V_{\\Lambda_i}(p_i)$ is a vertex operator corresponding to a state $\\Lambda_i$ with momentum $p_i$. For example, for the tachyon, $$V_{\\mathrm{tachyon}} = g_s \\int \\mathrm{d}^2 z \\, e^{ip_i \\cdot X}$$ What I find troubling is that it seems the bosonic string theory does not impose any restrictions on which interactions may occur. For example, I could insert the vertex operators for a photon to compute photon scattering in the theory. But in the Standard Model direct interaction between photons is not permitted, but it could scatter via a fermion loop. So, how come any interaction is permitted in this theory? If this is an issue, how does the situation change in the case of the superstring?"}
{"id":"24401","title":"Confusion regarding geodesic of thrown ball - curved or Cartesian coordinates?","text":"I'm confused trying to understand what's happening in terms of spacetime geodesics when a ball is thrown and its trajectory plotted, height against time to give a parabola. I read (from more than one source) that although the curve is a parabola in three (spatial) dimensions “we must recognize that a massive body like the Earth actually curves the coordinate system itself, so that rather than following a curved path in a flat (Cartesian) coordinate system, the ball actually follows a minimum-distance path, or geodesic, in a curved coordinate system.” In other words, the problem here is “mapping curved space-time onto a flat piece of paper.” OK. But then I also read (again from more than one source) that, as an example of the weakness of the Earth's gravitational field, if the ball is in the air for, say, 1 second, using $ct$ units this is equivalent to $3\\times10^{8}\\,\\mathrm{m.}$ One author then talks about plotting this curve on an ordinary $ct\/x$ spacetime diagram saying “in this case the spacetime is practically flat, and thus the geodesic is very close to a straight line.” OK again, but aren't these two positions contradictory? If you aren't allowed to plot curved spacetime using Cartesian coordinates in the first example, how come you are in the second? And vice versa of course. Thanks"}
{"id":"478","title":"Is it possible to project a magnetic field at a location in space?","text":"A magnetic field strength drops-off quickly as the distance from a magnet increases. Is there any way to use electromagnetic fields to create a magnetic field at a location. For example, if there are strong electromagnetic fields intersecting at a location from many strong transmitters, could a constructive interference be created which creates a local magnetic field? The engineering idea is that if you create an array of radio transmitters at a strong strength working together, could there be a interference pattern at a point far from the transmitters (maybe 100 meters away) where the fields produce a magnetic field at that location. I think this could be done as a EM field is composed of a electric field and a magnetic field, shouldn't there be a way to make the magnetic field strong at a location in space. A positive consequence of this would be that a magnetic placed at that location 100 meters away could be moved by the field. Also, if the field was alternating, rather than steady, then a electromagnet at that location could be alternating, and the force on the electromagnet could provide motive force. I guess since I have never heard of this, it cannot be done. But then again, it was once said that levitating a normal magnet could not be done, then someone did it with a spinning magnet which levitates above another magnet, now it is a popular toy. The reason it works is that the magnet is spinning so that it cannot flip over. See spin stabilized magnetic levitation. \\--- Update Hi, I would like to clarify, I wish to move a physical object, a small magnet at a distance, not to transmit power to the object. Thanks! \\--- Update 2 My engineering goal for this is a small magnetic, spin stabilized, being pushed from a distant location. A small spin stabilized magnet pushed from below by some \"projected\" magnetic field could potentially overcome the force of gravity. Imagine a thin lightweight spinning disk with a magnet in the center. If a projected magnetic field is pushing it upwards then it would stay stable and could be projected from ground into space. This might be done with an array of thousands of transmitters on the ground, somewhere in the desert far away from people. All transmitters could focus a small magnetic field at this magnet. Alternatively, if the magnetic field alternates with a frequency, then the magnet in the levitated device could also be alternating via electromagnetism, it could be powered by an battery or microwave power transmitted to the object. This is more or less an engineering problem if there is some basic science which can allow this to be a potential reality."}
{"id":"35787","title":"Why do hydrogen atoms attract?","text":"That is, why is the potential energy with the orbitals overlapping less than with the Hydrogen atoms 'independent'. Similarly, why is a noble gas configuration stabler than if an electron were to be removed or added? Is this because pairs of electrons are more stable than single electrons? If so, why?"}
{"id":"116229","title":"How do we convert between the maximum value of the adhesive force (in kilograms or in Newton) of a magnet and its magnetic field strength (in tesla)?","text":"On the following website they indicate how powerful is a magnet by giving the theoretical maximum value of its adhesive force in kilograms or in Newton (its adhesive force at a distance of zero). http:\/\/www.supermagnete.fr\/eng\/faq\/How-strong-is-this-magnet I would like to know how to convert this in tesla or in gauss. More precisely, I would like to know how many tesla does a NdFeB magnet with a maximum adhesive force of 200 kg = 2000 N, a size of 10x10x1.5 cm and a mass of 1.3 kg generates at a distance of 50 cm ? (I don't need a very accurate result. So if you can give me a rough approximation\/estimation it will be good enough.)"}
{"id":"46015","title":"Why quantum mechanics?","text":"Imagine you're teaching a first course on quantum mechanics in which your students are well-versed in classical mechanics, but have never seen any quantum before. How would you motivate the subject and convince your students that in fact classical mechanics cannot explain the real world and that quantum mechanics, given your knowledge of classical mechanics, is the most obvious alternative to try? If you sit down and think about it, the idea that the state of a system, instead of being specified by the finitely many particles' position and momentum, is now described by an element of some abstract (rigged) Hilbert space and that the observables correspond to self-adjoint operators on the space of states is not at all obvious. Why should this be the case, or at least, why might we expect this to be the case? Then there is the issue of measurement which is even more difficult to motivate. In the usual formulation of quantum mechanics, we _assume_ that, given a state $|\\psi \\rangle$ and an observable $A$, the probability of measuring a value between $a$ and $a+da$ is given by $|\\langle a|\\psi \\rangle |^2da$ (and furthermore, if $a$ is not an eigenvalue of $A$, then the probability of measuring a value in this interval is $0$). How would you convince your students that this had to be the case? I have thought about this question of motivation for a couple of years now, and so far, the only answers I've come up with are incomplete, not entirely satisfactory, and seem to be much more non-trivial than I feel they should be. So, what do you guys think? Can you motivate the usual formulation of quantum mechanics using only classical mechanics and minimal appeal to experimental results? Note that, at some point, you will have to make reference to experiment. After all, this is the reason why we needed to develop quantum mechanics. In principle, we could just say \"The Born Rule is true because its experimentally verified.\", but I find this particularly unsatisfying. I think we can do better. Thus, I would ask that when you do invoke the results of an experiment, you do so to only justify fundamental truths, by which I mean something that can not itself just be explained in terms of more theory. You might say that my conjecture is that the Born Rule is not a fundamental truth in this sense, but can instead be explained by more fundamental theory, which itself is justified via experiment. **Edit** : To clarify, I will try to make use of a much simpler example. In an ideal gas, if you fix the volume, then the temperature is proportional to pressure. So we may ask \"Why?\". You could say \"Well, because experiment.\", or alternatively you could say \"It is a trivial corollary of the ideal gas law.\". If you choose the latter, you can then ask why that is true. Once again, you can just say \"Because experiment.\" or you could try to prove it using more fundamental physical truths (using the kinetic theory of gases, for example). The objective, then, is to come up with the most fundamental physical truths, prove everything else we know in terms of those, and then verify the fundamental physical truths via experiment. And in this particular case, the objective is to do this with quantum mechanics."}
{"id":"82241","title":"Why does the addition of paramagnetic ions decrease the spin lattice relaxation time of protons?","text":"My question pertains to NMR (nuclear magnetic resonance). $T_1$, the spin lattice relaxation time of protons in water is about 2.5 seconds. If you add some $CuSO_4$ (copper sulfate) to the water the spin lattice relaxation decreases dramatically. I.e. in a 1mM $CuSO_4$ solution, $T_1$ drops to about 0.46 seconds. Why is this? I've read that paramagnetic ions decrease spin lattice relaxation times, but how exactly does this work?"}
{"id":"3999","title":"Common approximations for fluid outflow beyond Torricelli's law","text":"Last month, a puzzle here in physics.stackexchange asked to calculate the dynamics of the outflow in a loose (over rails, no friction) tank using Torricelli's law as a simplification. The simplification turned out to be problematic; one would prefer some solution where the height $h(0)=h_0$ starts decreasing with the usual condition $h'(0)=0$. A first try to generalize is to use the potential energy stored in the tank to provide kinetic energy both to the CoM of the fluid and to the outflow. Consider for instance two horizontal equal nozzles, so that the CoM only goes down vertically. While this situation drives to a manageable second order differential equation, I am not sure if it is good enough. Particularly it seems that when we close the nozzle(s), so that the CM stops, this kinetic energy should change to be stored as internal energy, does it? We could imagine some configuration where we open the nozzle an interval $\\Delta t$, close it, open it at the next interval, &c, so that we force this conversion to internal, not CoM, energy. So, is there some generic condition where we can grant that all the energy goes either to the outflow or to the fluid CoM? If not, there is some usual parameter measuring how much of the energy goes to CoM and how much goes to internal (vorticity or so)?"}
{"id":"3998","title":"What is a maximal analytic extension?","text":"Can someone explain (as rigorously as possible) what is involved in analytically continuing, say, the Schwarzschild solution to the Kruskal manifold? I understand the two metrics separately but I'm not sure how analytic continuation is used, since I can't really see how the process of extending the domain of a complex function has anything to do with extending a manifold through a coordinate change."}
{"id":"3993","title":"Interpretation of QM as discrete derivative","text":"In a couple of letters dated 1923, Pauli writes to Sommerfeld and Lande about Zeeman effect, and he describes (a piece of) quantisation in a peculiar way: the substitution of the \" _Differentialquotient_ \" $$ {d \\over dj} {1 \\over j} $$ ($j$ is called a _impulsquantenzahlen_? is it an angular momentum, isn't it?) by the \" _Differenzen quotient_ \" $$\\frac 1j - \\frac 1{j-1}$$ (see page 3 of letter to Sommerfeld). The later being argued as plausible from the integration of $\\int {dj \\over j^2}$ (footnote in page 4 of letter to Landé). I wonder, did the concept of \" _Differenzen quotient_ \" survive in the literature, as a quantisation method or even as some fundational concept of quantum mechanics?"}
{"id":"56582","title":"Why does Quantum Electrodynamics Allow a Photon to Exist Temporarily as a Positron and an Electron?","text":"In this question... Why does a photon colliding with an atomic nucleus cause pair production? ...I asked why a photon colliding with a atomic nucleus can become an electron and a positron. The answer that I thought was most illuminating explained that a photon spends some of its travel time as a particle-antiparticle pair of an electron and a positron. If it strikes the nucleus at the right time, this pair will be separated. It was explained that this is because 'Quantum Electrodynamics allows it'. Why does Quantum Electrodynamics allow a photon to exist temporarily as an electron and a positron?"}
{"id":"73265","title":"Orthogonality of the wavefunctions on an subinterval?","text":"Lets say that functions (eigenfunctions) $\\psi_0$ and $\\psi_1$ are orthogonal on an interval $-d\/2 < x < d\/2$. Are they also orthogonal on any subinterval inside the interval $-d\/2 < x < d\/2$? For example if I take subinterval $$d\/4 < x < d\/2$$ Can I state that: $$ \\int\\limits_{d\/4}^{d\/2}\\overline{\\psi}_0\\psi_1\\, dx= 0 $$ This would easier a lot of integrations for me..."}
{"id":"56580","title":"Very large absorption lines in stellar spectrum","text":"I was puzzled by the wide absorption lines in a stellar spectrum I found. The following is what I expect absorption lines to look like - thin, crisp lines: ![enter image description here](http:\/\/i.stack.imgur.com\/RMHmB.gif) However, I found this stellar spectrum, which as large gaps, most notably from 6600 to 7100 angstroms: ![enter image description here](http:\/\/i.stack.imgur.com\/97J5O.gif) Why are the absorption lines so wide on the second spectrum? Are they even absorption lines at all, or another phenomenon altogether?"}
{"id":"56589","title":"Calculating the mass equivalency of a song?","text":"I've recently become fascinated with the idea of sound energy having a theoretical equivalent mass. I've read over this thread: Do light and sound waves have mass I understand this part: $m_{eq}=E\/c^2$ and $E=A\\rho \\xi^2\\omega^2$ Where I am getting tripped up is the way to measure $E$. Essentially, I want to measure $E$ and eventually $m$ of a song (for it's entire duration). The only equipment I have is an iPhone app that measures intensity (dB) (apparently it's fairly accurate for levels below 100 dB). I plan to play the \"music\" at about 80dB over speakers in a room that is 50'x30'x15' and I will assume the temperature is at room temperature. If your wondering, it's for a conceptual audio art piece."}
{"id":"56588","title":"Are two waves being in phase is the same as saying that the two waves are coherent?","text":"If two waves are coherent, is it the same as them being in phase? Please correct if I'm wrong."}
{"id":"35431","title":"Is the law of conservation of energy still valid?","text":"Is the law of conservation of energy still valid or have there been experiments showing that energy could be created or lost?"}
{"id":"32787","title":"motion in the body-fixed frame?","text":"This is really basic, I'm sure: For rigid body motion, Euler's equations refer to $L_i$ and $\\omega_i$ as measured in the fixed-body frame. But that frame is just that: fixed in the body. So how could such an observer ever measure non- zero $L$ or $\\omega$?"}
{"id":"27179","title":"Is ghost-number a physical reality\/observable?","text":"One perspective is to say that one introduced the ghost fields into the Lagrangian to be able to write the gauge transformation determinant as a path- integral. Hence I was tempted to think of them as just some auxiliary variables introduced into the theory to make things manageable. But then one observes that having introduced them there is now an extra global $U(1)$ symmetry - the \"ghost number\"   * Hence hasn't one now basically added a new factor of $U(1)$ to the symmetry group of the theory? How can the symmetry of the theory depend on introduction of some auxiliary fields?    * Now if one takes the point of view that the global symmetry has been enhanced then the particles should also lie in the irreducible representations of this new factor. Hence ghost number should be like a new quantum number for the particles and which has to be conserved!    * But one sees that ghost field excitations are BRST exact and hence unphysical since they are $0$ in the BRST cohomology.  I am unable to conceptually reconcile the above three ideas - the first two seem to tell me that the ghost-number is a very physical thing but the last one seems to tell me that it is unphysical.   * At the risk of sounding more naive - if the particles are now charged under the ghost number symmetry then shouldn't one be able to measure that in the laboratory?    * Lastly this ghost number symmetry is a global\/rigid $U(1)$ symmetry - can't there be a case where it is local and needs to be gauged?"}
{"id":"56059","title":"Is the crystallization of a heat pack the same as the crystallization of freezing water?","text":"Is the crystallization of a heat pack the same as the crystallization of freezing water? If so, why does one generate heat where the other does not?"}
{"id":"25447","title":"What does ionization of neutral Hydrogen have to do with \"transparency\"?","text":"Most accounts of the early history of the Universe make some reference to (re)ionization as being the _reason_ that the Universe becomes transparent after a period of opacity caused by the absence of ionized hydrogen. These accounts suggest that neutral H is opaque, wile ionized H is transparent. But I don't follow the reasoning. I understand that, while neutral H is primarily absorbing at a single relevant wavelength, those wavelengths are \"smeared out\" by expansion, as seen by an observer, so that a dense and nearly continuous neutral H at a large range of redshifts would result in absorption at a corresponding wide range of wavelengths (i.e. \"opacity\" at those wavelengths, as seen by an observer \"on the other side\" of the absorbing intervening H). But I'm not clear on why ionization of the same H, alone, would result in clearing up this opacity."}
{"id":"128277","title":"Molecular field meaning in Liquid Crystal Theory","text":"Given the Frank-De Gennes free energy $F = \\int f(\\boldsymbol{p},\\nabla\\boldsymbol{p}, ...)\\ d\\boldsymbol{x},$ for liquid crystals (see De Gennes-Prost, p. 107, formula 3.21), the vector $$h_{i}=-\\frac{\\delta F}{\\delta P_{i}}=-\\frac{\\partial f}{\\partial P_{i}}+\\partial_{j}\\frac{\\partial f}{\\partial\\partial_{j}P_{i}}$$ is usually called _molecular field_ , a \"notation derived from magnetism\". What is the physical meaning of this vector in this context? Does it have any sense to call it _molecular_ also here?"}
{"id":"75831","title":"A simple question on the projected wave function?","text":"For example, consider a spin-1\/2 AFM Heisenberg Hamiltonian $H=\\sum_{<ij>}\\mathbf{S}_i\\cdot\\mathbf{S}_j$, and we perform a Schwinger- fermion($\\mathbf{S}_i=\\frac{1}{2}f^\\dagger_i\\mathbf{\\sigma}f_i$) mean-field study. Let $H_{MF}=\\sum_{<ij>}(f^\\dagger_i\\chi_{ij}f_j+f^\\dagger_i\\eta_{ij}f_j^\\dagger+H.c.)$ be the resulting mean-field Hamiltonian, where $(\\chi_{ij},\\eta_{ij})$ is the mean-field ansatz. And let $\\psi_{1,2}$ represent two exact eigenstates of $H_{MF}$ with energies $E_{1,2}(E_1>E_2)$, say $H_{MF}\\psi_{1,2}=E_{1,2}\\psi_{1,2}$. Now we can construct the physical spin states $\\phi_{1,2}$ by applying the projective operator $P=\\prod_i(2\\hat{n}_i-\\hat{n}_i^2)$(Note that $P\\neq \\prod _i(1-\\hat{n}_{i\\uparrow}\\hat{n}_{i\\downarrow})$) to $\\psi_{1,2}$(where $\\hat{n}_i=f^\\dagger_{i\\uparrow}f_{i\\uparrow}+f^\\dagger_{i\\downarrow}f_{i\\downarrow}$), say $\\phi_{1,2}=P\\psi_{1,2}$, and _generally we don't expect that $\\phi_{1,2}$ are the exact eigenstates of the original spin Hamiltonian_ $H$. My question is: $\\frac{\\left \\langle \\phi_1 \\mid H \\mid \\phi_1 \\right \\rangle}{\\left \\langle \\phi_1 \\mid \\phi_1 \\right \\rangle}>\\frac{\\left \\langle \\phi_2 \\mid H \\mid \\phi_2 \\right \\rangle}{\\left \\langle \\phi_2 \\mid \\phi_2 \\right \\rangle} $ ? If it's true, then how to prove it rigorously ? Thanks a lot."}
{"id":"98269","title":"Free Will Theorem question","text":"The Kochen-Specker Theorem says, if I understand it correctly, that the results of spin measurements cannot be predetermined independent of measurement. They get to this conclusion by describing 33 possible measurements whose results cannot all be satisfied simultaneously. In the Free Will Theorem, this is developed further by taking two entangled particles sufficiently far away and measuring in 2 of these 33 directions. The proof seems to assume consistency of these results, so my question is this: If we had an entangled group of 33 particles rather than just 2, what would happen if all 33 directions were measured by space-like separated experimenters? The Kochen-Specker Theorem says that results can't all be consistent with one another. I have very little background in physics so I apologize in advance if I'm misunderstanding something fundamental. I would also request that answers assume little knowledge of physics beyond what I've displayed above, though knowledge of mathematics can be assumed."}
{"id":"101830","title":"Batteries and voltage?","text":"The _voltage_ of a battery gives you the difference in potential energy 1C of charge would have at the positive terminal vs the negative terminal. If I connect a wire to both terminals, the battery will generate a field within the wire. The field will cause surface charges to buildup on the wire. You essentially end up with surface charges distributed through the wire so that the field is always parallel to the surface of the wire. Now say I have an unconnected battery of 5V. Now hypothetically if I used \"tweezers\" to drag 1 coloumb of charge through a vacuum from the positive terminal to the negative terminal it would gain 5J of energy. Now consider a connected battery. If I used \"tweezers\" to drag 1 coloumb of charge through the _wire_ from the positive terminal to the negative terminal it would _still_ gain 5J of energy. My question is that since the surface charge in the wire sets up a field, how do we know that the field set up by the surface charge doesn't affect the potential energy difference between the positive and negative terminal? Voltage is defined as the line integral of the dot product of the electric field and distance, and since the electric field acting is not longer the same (since the surface charge also contributes a field) how do we know that 1C of charge would gain the same amount of energy if dragged from the positive terminal to the negative terminal? ![enter image description here](http:\/\/i.stack.imgur.com\/5nFRk.png)"}
{"id":"37550","title":"The \"official\" mass of the sun as unit?","text":"I'm searching for the \"official\" mass of the sun as a unit in astrophysics. The mass of the sun can be calculated by: $M_{\\odot}=\\frac{4\\pi^2\\times(1 \\ \\text{ua})^3}{G\\times(1\\ \\text{year})^2}$ * * * So in this formula: $\\pi = 3.1415926535898...$ $1 \\ \\text{ua} = 149597870700 \\ \\text{m}$ (it's an exact definition according to the IAU 2012 resolution http:\/\/www.iau.org\/static\/resolutions\/IAU2012_English.pdf) $G = 6.67384\\times10^{-11} \\ \\text{m}^3\\text{kg}^{-1}\\text{s}^{-2}$ according to CODATA (http:\/\/physics.nist.gov\/cgi-bin\/cuu\/Value?bg) but $G = 6.67428\\times10^{-11} \\ \\text{m}^3\\text{kg}^{-1}\\text{s}^{-2}$ according to IAU 2009 (http:\/\/maia.usno.navy.mil\/NSFA\/IAU2009_consts.html), so which one do we choose? Finally, for the year here, which year to choose? Is it the same year as for the light speed: $1 \\ \\text{year}=365.25\\times24\\times3600=31557600 \\ \\text{s}$ (Julian year)? Or the tropical year $1 \\ \\text{year}=365.2421897\\times24\\times3600=31556925.2 \\ \\text{s}$ (tropical year)? * * * For some of the preceding values we obtain: $M_{\\odot}=\\frac{4\\pi^2\\times(149597870700)^3}{6.67384\\times10^{-11}\\times(31557600)^2}=1.988622... \\ \\text{kg}$ which is not what gives wikipedia (http:\/\/en.wikipedia.org\/wiki\/Solar_mass) or google (https:\/\/www.google.fr\/search?q=solar+mass) So my question is: what are the correct preceding values, and consequently what is the current good value for the sun mass? (as some units may have been redefined since some measurements it's not that simple)"}
{"id":"134502","title":"Do small-scale physical effects lead to larger systems being required for long life (of dynamic systems, not static objects)","text":"I had a hard time finding a tag, but I think physicists can deal with this question best. I answered a question about \"immortality\" when some guy claimed I got it wrong that neurons die (I argued that even if you live a billion years you die slowly many times over because all your cells incl. nervous cells will have to be replaced, meaning you will regularly - but never suddenly - become a new person; also: continuous learning, and information storage is physical, and our brain has very limited capacity so memories will change), immortal means they are immortal. That got me thinking: Is it actually possible to have repair systems in each cell, or isn't it better to have a repair system on a much higher macro level and cell-level immortality is actually impossible? Background: The smaller the scale, the more events like quantum effects and Boltzman (energy) distribution (e.g. random atom movements, crystal structure defects, breaking bonds) dominate. Meaning small systems WILL break in unpredictable ways, so only large systems can live long and have reliable repair systems because the larger the system the less important those effects become. Summary: It is much better (or even possible) to have big long-living systems than small ones, because the inevitable repair system cannot be too small because it will be subject to random events of the micro-world. An \"immortal\" human would have a body-level repair system that repairs by letting broken cells die (we have that already) and build a new one, instead of having a repair system on the cell-level that would make cells immortal (too unreliable, won't work). Or in other words: The smaller the system the less likely a long life can be achieved. Note that I'm asking about complex dynamic systems, not static objects or information. Like living organisms or (complicated) machines, and about their chance to \"live long\"."}
{"id":"134508","title":"Is data which rides on the carrier frequency dangerous?","text":"My understanding of broadcasting data via electromagnetic radition is, that the data \"rides\" on a carrier frequency on which it is radiated. I am aware of the fact, that those carrier frequencies are below the frequencies of visible light, thus not dangerous like the higher ones (UV, Xray, Gamma). My question now is, if the data (be it an analog tune from a radio, or a digital WiFi signal) that \"rides\" on this carrier frequency is oscillating fast enough that it can be considered harmful, because it belongs to the high frequencies greater than visible light."}
{"id":"104981","title":"Deriving group velocity formula","text":"A formula for the group velocity of waves is: $u=k*dv\/dk + v$ But then, since $k=2π\/λ$, this equation can be rewritten as: $u=v-dv\/dλ*λ$ But how? My attempt: $k*dv\/dk$ = $((2π\/λ)*dv\/d(2π\/λ)$ Then what?"}
{"id":"83580","title":"What information is lost in the symmetrization necessary to derive the BBGKY hierarchy?","text":"The book on Kinetic theory I'm reading derives the BBGKY hierarchy after introducing the reduced distribution functions $f_s(q^1,p_1,q^2,p_2,\\dots,q^s,p_s):=\\int\\ \\rho\\ \\ \\mathrm d q^{s+1} \\mathrm d p_{s+1}\\cdots \\mathrm d q^N \\mathrm d p_N,$ where $\\rho$ is the phase space probability function. In fact, in the derivation, to get rid of most terms, he reasonably states that we must make the assumption that the $f_s$ are symmetric, i.e. $\\forall k.\\ f_s(\\dots,q^k,p_k,q^{k+1},p_{k+1},\\dots)=f_s(\\dots,q^{k+1},p_{k+1},q^k,p_k,\\dots).$ The step from going from the particle trajectory $\\pi$ (which solves the Hamiltonian equations of motions) to the distribution $\\rho$ (which solves the continouty equation with the vector field, generated by multiple $\\pi$-streamlines, as flow) can be a bit confusing. The fact that the author concerns himself two times with this and doesn't just say \"we can clearly assume that..\" suggests to me that it might not be so simple to put asside. He offers a symmetrization method $f_s\\mapsto \\frac{1}{s!}\\sum_{\\pi}f_s$, where the sum permutes the arguments. Now firstly, I wonder how this works with multi-particle distribution functions. What do the equations look like for an $f_{s,t}$ which describe $s$ particle of one sort and $t$ particles of another sort? Also, as this implies even the Boltzmann equation rests on this assumptions which therefore apparently isn't initially part of the Hamiltonian description - what information is lost in the process of symmetrization?"}
{"id":"113409","title":"Reflections in a double-glazed window","text":"Recently I observed reflections in a double-glazed window. I wondered that the double reflections were not the same if I changed my point of view. For example the two images of a candle would overlap at a certain angle but separated when I moved. The only idea I had to explain this behavior is that one (or both) of the glass panels was not totally plane but curved. Does that sound plausible or is there another explanation for this behavior?"}
{"id":"104985","title":"Sudden Approximation for Beta Decay of Tritium Atom","text":"I am working out this problem right now, and I'm confused by the answers I'm getting. Problem: A tritium nucleus (Z = 1) in a tritium atom undergoes beta decay, i.e., a neutron in the nucleus emits an electron and transforms into a proton (Z = 2). Assuming that the emitted electron acquires so much energy that becomes unbound and flies away almost instantly without interacting with the orbiting electron, (i) Compute the probability that the bound electron will be in a state with principal quantum number n in the Helium ion. Assume that initially the bound electron was in the ground state of the tritium atom. Attempt: The new wavefunctions differ by Z=2, and we want to calculate transition probability between the old ground state and the new wavefunctions, so we take the inner product between them. Looking at the hydrogen wave function, by orthogonality of the spherical harmonics the inner product will vanish if the l's aren't equal, so l'=l=0, which simplifies our inner product, which is now an integral over r. My final expression for the transition probablity is: (Integrate[(8*Sqrt[2]\/a^3)*1^(-5\/2)*LaguerreL[1 - 1, 1, 2*2*r\/(1*a)]* LaguerreL[0, 1, 2*r\/(a)]*Exp[-r\/a - 4*r\/(1*a)]*r^2, {r, 0, Infinity}])^2 or 128*(Integrate[ n^(-5\/2)*LaguerreL[n - 1, 1, 4*r\/n]*Exp[-r - 4*r\/(n)]*r^2, {r, 0, Infinity}])^2 So I get an expression of 512\/15625 for the probability it is in the new ground state. This is really low. I am supposed to calculate the infinite sum of these probabilities as well, with mathematica, but it can't figure it out. So either I'm doing something wrong in my thought process, or there is a way of writing of the laguerre polynomial that simplifies our problem considerably."}
{"id":"3206","title":"For which irreps can normalized spin vectors be chosen continuously, for all possible spins?","text":"Let $(u_x,u_y,u_z)$ be a unit vector. Then a normalized vector for spin-1\/2 in this direction is given by normalizing a column from the pure density matrix: $\\frac{1}{\\sqrt{2+2u_z}}\\left(\\begin{array}{c}1+u_z\\\\\\u_x+iu_y\\end{array}\\right)$. This solution works except when $u_z=-1$. So this gives a normalization choice of spin vector that is continuous over the possible eigenstates of spin with the exception of spin down = $\\left(\\begin{array}{c}0\\\\\\1\\end{array}\\right)$. It is impossible to choose normalized spin-1\/2 vectors for all possible eigenstates of spin-1\/2. This is a fact that is related to Berry-Pancharatnam phase. The best you can do is to leave a single point, as in the above, where there is an essential discontinuity. What I'd like to know is what other j have spin-j vectors unable to be normalized. That is, given all possible eigenstates of spin-j, is it possible to choose one from each ray, such that the choice is continuous? * * * Partial Solution: Per Dr. Motl's answer below, I'll down the situation for a natural representation of spin-1. Choose $S_x = i\\left(\\begin{array}{ccc}0&0&0\\\\\\0&0&+1\\\\\\0&-1&0\\end{array}\\right)$ $S_y = i\\left(\\begin{array}{ccc}0&0&-1\\\\\\0&0&0\\\\\\\\+1&0&0\\end{array}\\right)$ $S_z = i\\left(\\begin{array}{ccc}0&+1&0\\\\\\\\-1&0&0\\\\\\0&0&0\\end{array}\\right)$. It's easy to verify that these satisfy the $SU(2)$ rules $[S_x,S_y] = iS_z$, etc. Then the spin operator for spin in the $u = (u_x,u_y,u_z)$ direction is given by: $S_u = i\\left(\\begin{array}{ccc}0&u_z&-u_y\\\\\\\\-u_z&0&u_x\\\\\\u_y&-u_x&0\\end{array}\\right)$ and the solution with spin=0 in the $u$ direction is given by the vector $u$. And sure enough, it's impossible to choose the vectors for spin=1 as a continuous function of $u$."}
{"id":"28000","title":"Should the cold drink come out of a free falling bottle with an open lid?","text":"If I hold an open bottle and perform free fall, say sky diving, along with it from, say 10000ft, while I'm holding the bottle vertically upwards should the cold drink come out of the bottle? Kindly ask for any further details you need."}
{"id":"30521","title":"When does a function of an operator act in the same way as the operator?","text":"\"Consider an operator $A = r - a$, where $r$ is an operator and $a$ is a constant. Consider only those state kets $V_i$ in the Hilbert space such that $AV_i = 0$ ($A$ acting on $V_i$). Define a projection operator $P_A$ that projects an arbitrary ket on to the states $V_i$'s, i.e. the subspace of kets annihilated by $A$. Then $P_AAP_A = 0$. So $P_ArP_A = aP_A$.\" Now further in the literature which includes whatever I've quoted above, the following is stated: Suppose $H = (1\/r^2)p_\\theta^2 + \\lambda(r - a)$ (where $p_\\theta$ is an operator). It follows that $P_AHP_A = (1\/a^2)p_\\theta^2$. Could anyone please elaborate how the last implication comes about? Is it that since $rV_i = aV_i$, $(1\/r^2)V_i = (1\/a^2)V_i$? Thanks..."}
{"id":"5533","title":"Quantization of Gravitational Field: Quantization conditions","text":"I'm begining to study Quantization of field with the second quantization formalism. I've studied phononic field, electromagnetic field in the vacuum and a generic relativistical scalar field. I asked to me if is possible doing the same thing with the Gravitational Field Hamiltonian. I've heard that we can do it only in the condition of linearized Gravity and we obtain a field with spin 2, but we can't do it in the general condition because we don't have quantization conditions. What are these conditions? And how can we obtain only in linearized gravity the quantized field? And how it has got spin 2?"}
{"id":"20562","title":"As ISA Practical - Resistors in Parallel","text":"> **Possible Duplicate:**   >  Current against the inverse of resistance graph, I = V\/R +c How would you set up a circuit with a fixed resistor in parallel with a variable one. We are told to measure the current across the combination resistors and our Voltage is $1V$ and then construct an $I$ against $1\/R$ graph what sort of correlation would this be?"}
{"id":"89763","title":"Why does a window layer is on top of the absorber layer in solar cells and not below it?","text":"In solar cells there is a p-n junction. P-type semiconductor (for example CdTe) is often absorber layer because of its carrier lifetime and mobilities. In case of CdS\/CdTe, CdS is n-type window layer and everywhere it is said that it should be very thin and has large band gap – not to absorb any light and let it go through to the p-type absorber (that is why it is called a window layer). But why should it be on top of the absorber layer and not below it? If n-type layer is below, the light can hit the p-type absorber directly. I have some ideas that it is related to the distance between the place of absorption and p-n junction, but I am not sure. ![CdTe solar cell](http:\/\/i.stack.imgur.com\/8IajS.jpg) Image by Alfred Hicks\/NREL"}
{"id":"83055","title":"How can scintillation gamma-spectrometers work given that track length is different for different angles?","text":"As far as I understand, the basic principle of gamma spectrometer is simple - gamma ray hits scintillator, it generates number of photons which roughly proportional to gamma ray energy. Then we need to detect these secondary photons and plot a nice graph. But as far as I see, scintillator size is typically too small to completely absorb gamma photon. Which also means that gamma photons coming from different angles will have different track lengths inside scintillator - and should generate different amount of light for the same gamma ray energy. So, how can this work at all?"}
{"id":"47328","title":"Reference request for the Aharonov-Bohm effect","text":"I am looking for a good reference to an online source or book, on the magnetic Aharonov-Bohm effect. I have read the appropriate sections from the book by Griffiths and Ballentine, and still haven't understood it properly. The reference should also contain topological arguments. I have done QM on the order of 4 chapters of Griffiths, and the first 2.5 chapters of Sakurai."}
{"id":"132015","title":"Kinematic sign convention","text":"For example, if I drop a ball from a $50$ meters building, then I will consider   1. the ground is $0$ meter   2. downward is positive ( which makes gravity positive, downward velocity positive, etc) so with that if i use $X_f = X_i + V_it + \\frac{1}{2}at^2$ then I would get something like $0 = 50 + 4.9t^2$ which is not even possible. Instinctively I know what to do but when I think more about sign convention, it seems so confusing."}
{"id":"87774","title":"Deriving $p = mv$ from translational symmetry (momentum conservation law)?","text":"> \"In classical mechanics, momentum is defined as the quantity which is > conserved under global spatial translations or, alternatively, as the > generator of spatial translations.\" (G.Parisi, Quantum mechanics) From this definition (the generator of spatial translation), how can I obtain the analytical expression of the momentum (i.d. $p=mv$)?"}
{"id":"87775","title":"Is the covariance or contravariance of vectors\/tensors something that can be \"visualized\"?","text":"I'm taking an undergrad GR course, and our text (Lambourne) mentions covariant and contravariant vectors and tensors ad-nauseum, but never really gives a formal definition for what they are, and how they are unique from each other in any physical sense (other than their difference in transformations). Is there any physical intuition behind these two labels? There should be, right? If they differ in how they transform with transformation of coordinates, doesnt that indicate that there has to be some way of visualizing their difference, since coordinate transformations are easily visualized?"}
{"id":"88493","title":"Question about liquid \/ gaseous insulators","text":"I'm looking for information about liquid \/ gaseous insulators, but I can't find any, is there any website (because I can't find on Wikipedia) where I can get this information? Also, can I get a list of examples of liquid \/ gaseous insulators? Any kind of help is appreciated."}
{"id":"14179","title":"Where can I find simulation software for electricity and magnets?","text":"Is there easily-available* software to simulate coils, solenoids, and other magnetic and electromagnetic devices? I'd like to play around with some design ideas, such as Halbach arrays, but physics simulators like SketchyPhysics don't seem to have the laws of electromagnetism implemented. (*\"Easily-available\" means \"not cost prohibitive.\")"}
{"id":"39174","title":"If a magnetic monopole falls into a schwarzchild black hole, what happens to the magnetic field?","text":"By the no-hair theorem, black holes can only have mass, charge and angular momentum. Does \"charge\" include \"magnetic charge\" (such as from a magnetic monopole)? Can black holes have magnetic charge in addition to electric charge? [By the way, I believe theory says magnetic monopoles have to exist, the reason why we haven't discovered any is because inflation diluted them to such an extent that there are none or only a few in our Hubble volume.] * * * These three questions are similar but I think the answers will be different for each one: > What happens to an embedded magnetic field when a black hole is formed from > rotating charged dust? It seems to me a rotating charged black hole must > have a dipole magnetic field. But the strength of the dipole field seems > like an extra parameter that black holes are forbidden by the no-hair > theorem. > > If a magnetic monopole falls into a schwarzchild black hole, what happens to > the magnetic field? Here there would be only radial magnetic field lines > leaving from the event horizon to infinity. So if magnetic charge is counted > as charge this should be no problem. But if the black hole were rotating > wouldn't that produce an electric dipole field? > > When a neutral star with a magnetic field collapses to form a black hole, > what happens to the magnetic field? Here there is no charge so how can there > be a magnetic field associated with a black hole? That would definitely > violate the no-hair theorem."}
{"id":"74829","title":"Differences the nonlinerarties","text":"I want to comparison between oscillons based on non-linearities. Can someone elaborate it with the reason behind it : > When the sinusoidal vibrations are of the correct amplitude and frequency > and the layer of sufficient thickness, a localized wave, referred to as an > oscillon. > > Astonishingly, oscillons of opposite phase will attract over short distances > and form 'bonded' pairs. Oscillons of like phase repel. Oscillons have been > observed forming 'molecule' like structures and long chains. In comparison, > solitons do not form bound states. Reference: Wiki"}
{"id":"16669","title":"Hydrogen 2P Probability Density Question","text":"I'd like to calculate the probability density for Hydrogen in the $|2,1,1\\rangle$ and $|2,1,-1\\rangle$ states. There is an $\\exp(\\pm i\\phi)$ term attached to the wave function for these states. It seems to me that $|\\Psi|^2$ in this case would be the same for $|2,1,1\\rangle$ and $|2,1,-1\\rangle$ since $|\\exp(i\\phi)| = 1$ with the same being true for $|\\exp(-i\\phi)|$. Where am I going wrong here? Also, since I'm working in spherical coordinates the PD should be $r^2|\\Psi|^2$, right (this will result in those classic energy orbital diagram functions)?"}
{"id":"88143","title":"Effective mass of a black hole?","text":"Suppose a black hole forms from a given mass of particles such as the core of a star going supernova. The black hole formed will have an effective mass due to the curvature of space time induced. Such a mass is presumably the inferred mass deduced from the effect of the black hole on the motion of nearby bodies. How does this effective mass compare with the original mass of particles all of which ended up \"in the black hole\"? Is the effective mass less greater or the same as the formation mass?"}
{"id":"24461","title":"I want to charge 48V 108Ah battery from 48V 80Watt solar panel. Can anyone write that how much time will take this battery to fully charge?","text":"> **Possible Duplicate:**   >  Charging 12V 150Ah battery I want to charge 48V 108Ah battery from 48V 80 Watt solar panel. Can anyone write that how much time will take this battery to fully charge?"}
{"id":"88147","title":"Molecular formula of a compund","text":"I have no idea on how to get information out of this one. > A compound contains only hydrogen and nitrogen. Nitrogen makes up for 87.4% > of the mass. A sample of this have a density of 0.977 g\/L at 740 torr and > 100 C. What is the molecular formula of the compound? The best I could think of was to put it in $PV=nRT$, so $n=PV\/RT$. But since there is no defined volume I can't put in anything for $V$."}
{"id":"112934","title":"Can acceleration be achieved without reaction mass?","text":"In a previous question, I asked whether a flashlight can accelerate me. The answer is yes. This has led me to wonder, this is inferior to all other known propulsion methods in all aspects except one: Shooting light (photons) accelerates the shooter without ejecting any mass. This means we can space travel forever, assuming we have enough electricity. **Are there any other ways to accelerate without losing mass?** Note that there are lots of way to do it with external forces like laser propulsion, but I am only interested in doing it without those. e.g. consider a spacecraft swimming in the void trying to accelerate."}
{"id":"82040","title":"Why do excitonic absorptions have small bandwidth?","text":"Below is an image of the optical density (proportional to the absorption coefficient) of KBr crystal at low temperature. Indicated at 6.6 ev and 7.7 eV are the absorption by excitons. As you can see, it has a small width. Can you explain why? ![enter image description here](http:\/\/i.stack.imgur.com\/NQgRN.jpg) Source:Theory of Excitons, Robert S Knox, 1954"}
{"id":"16176","title":"How must you spin the ball to make it alternate between 2 positions?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/y6JZ2.jpg) Assume any parameters you may need. Thanks in advance."}
{"id":"69561","title":"What is the limit of vanishing density?","text":"I've been asked to examine the limit of vanishing density for the virial equation of state derivation of $\\left(\\frac{\\partial U}{\\partial V}\\right)_T$ and quite frankly, I have no idea what this is and a Google search doesn't seem to net anything extremely helpful. Assuming density can be isolated in my derivation above, am I just taking the limit as $\\rho \\rightarrow 0$ ?"}
{"id":"55222","title":"Can a black hole bounce?","text":"Is there a limit to the amount of matter that a black hole can accrete per second and if so could a certain sized black hole bounce off a dense enough surface?"}
{"id":"46495","title":"Questions about the speed of light and the extent of the universe","text":"What is the reason that speed of light is the maximum speed in the universe? Why there is no speed swifter than light? Or is it just we haven't found yet? A similar question is like this: why the light speed in vacuum has to be 299,792,458 m\/s? Why couldn't it be faster? Who or what is controlling this speed in vacuum? Who is stopping it from being faster? Another question confusing me is this, if light from a far away Galaxy travels to us and we see the stars tonight, couldn't it be fully possible that at this time, they actually are already dead? Which means by modern scientific methods, there is absolutely no way to know what is really happening out there, all we know is just what was there, right? So our knowledge is going to be forever limited unless we have something that can travel much faster than light?"}
{"id":"43513","title":"Relation between electric charge and gauge parameter of the moduli space of monopoles","text":"I am studying about the moduli space of a 2 monopole system from Harvey's notes, and Manton's paper. In both of these, (Harvey section 6.2), after constructing the Lagrangian for a two dyons system, the author replaces the electric charge $e$ with $\\dot{\\chi}$. $$e \\to \\dot{\\chi}$$ to obtain the the equation of motion is geodesic form, from which the Taub-NUT metric can be seen. Why is this identification made? Why does the rate of change of the gauge parameter, give the electric charge?"}
{"id":"135376","title":"A James Clerk Maxwell Disproof","text":"One of my favorite physicists to learn about was James Clerk Maxwell, for the fact that he unified the study of E&M in physics and he would often disprove theories that did not work as a Mathematician. I remember my physics professor mentioning a disproof Maxwell had done with what was then the current knowledge of how currents flowed or were held through capacitors... This is probably way off from what he actually did disprove since it has been over 5 years since I had been to this particular lecture but could anyone tell me or demonstrate the exact disproof by counterexample that Maxwell had done with capacitance?"}
{"id":"41919","title":"How do you super impose two or more signals to occupy a fix area of space with the resultant summed wave?","text":"Is it possible to super-impose two or more signals all sent from different directions as a standing wave with the resulting summed wave occupying a fix area of space that is also a complex area? Do you understand my question? It is not obvious probably what i am trying to say-feel free to ask for more of an explanation. These signals could be electro magnetic waves or micro waves designed to pass thru solid objects under normal circumstances but then excite the solid only when at a certain frequency - this excite frequency would come about when two or more signals interact within the solid."}
{"id":"13441","title":"In dimensional analysis, why the dimensionless constant is usually of order 1?","text":"Usually in all discussions and arguments of scaling or solving problems using dimensional analysis, the dimensionless constant is indeterminate but it is usually assumed that it is of order 1.   1. What does \"of order 1\" mean? 0.1-10?   2. Is there any way, qualitative or quantitative, to see why the dimensionless constant is of order 1?   3. Are there exceptions to that? I mean cases where the dimensionless constant is very far from 1? Could you give some examples? Can such exceptions be figured out from dimensional analysis alone?"}
{"id":"14082","title":"What is the physical significance of dot & cross product of vectors? Why is division not defined for vectors?","text":"I get the physical significance of vector addition & subtraction. But I don't understand what do dot & cross products mean? More specifically,   * why is it that dot product of vectors $\\vec{A}$ and $\\vec{B}$ is defined as $AB \\cos{\\theta}$   * why is it that cross product of vectors $\\vec{A}$ and $\\vec{B}$ is defined as $AB \\sin{\\theta}$ To me, both these formulae seem to be arbitrarily defined (Although, I know that it definitely wouldn't be the case). If Cross product could be defined arbitrarily, why can't we define division of vectors? What's wrong with that? Why can't vectors be devided?"}
{"id":"53033","title":"How can one say that if we subtract \"TS\" from \"U\" then what we get is free energy?","text":"> In thermodynamics, the Helmholtz free energy is deemed as a thermodynamic > potential which calculates the “useful” work retrievable from a closed > thermodynamic system at a constant temperature and volume. For such a > system, the negative of the difference in the Helmholtz energy is equal to > the maximum amount of work extractable from a thermodynamic process in which > both temperature and volume are kept constant. In these conditions, it is > minimized and held constant at equilibrium. (citation) My questions are   1. How can one say that if we subtract \"TS\" from \"U\" then what we get is free energy? Why do we have to subtract \"TS\"? What if we subtract pV instead of \"TS\".   2. If the system is in constant temperature and Volume , then how it is possible to do work?   3. Is it the statement true for Ideal gas also. If it is true then a thermodynamic system(with ideal gas) at a constant temperature and volume is also at constant pressure. so it is at equilibrium. Does that mean for a ideal gas system free energy is always minimum?"}
{"id":"16213","title":"Dependence of Friction on Area","text":"Is friction really independent of area? The friction force, $f_s = \\mu_s N$. The equation says that friction only depends on the normal force, which is $ N = W = mg$, and nature of sliding surface, due to $\\mu_S$. Now, less inflated tires experiences more friction compared to well inflated tire. Can someone give clear explanation, why friction does not depend on area, as the textbooks says?"}
{"id":"66159","title":"Friction force and contact area","text":"It's usually stated that the friction force is independent of the area of contact (Amontons' Second Law). I've always thought that this shouldn't be true, because the atraction between molecules would be higher and there would be more peaks. I've read that this is important for rubber surfaces. Is it important for nonelastic materials, like wood? Are there any quantitave studies of this phenomenon?"}
{"id":"111646","title":"De-excitation and vibrational frequency","text":"By utilising energetic beams of particles it is possible to set atomic nuclei into vibration. For a certain nucleus the subsequent de-excitation results in the emission of photons of energy $1$ $MeV$. Use this information to make an estimate of the vibrational frequency of the nucleus, stating any assumptions that you make. Using $E=hf$ I can calculate the frequency of the energetic beams. However I don't get the vibrational frequency bit..."}
{"id":"81961","title":"What is the difference between scale-free network and small-world network?","text":"What is the difference between scale-free network and small-world network? I can't understand from the definitions around the web if they are both the same name for one thing. Do both follow a power-law distribution of the nodes degree? In particular is it true that scale-free networks are a subset of small-world networks?"}
{"id":"20644","title":"What forms are theoretically predicted for orbitals or quarks in hadrons and of hadrons in tritium?","text":"We all know very illustrative spatial representations of predicted electron orbitals in atoms which are essentially spatial plots of the solutions of wave equations. In all atoms the electrons occupy the positions around the nucleus, sometimes, spatially symmetric. But I wonder which form could take orbitals of three particles of nearly- similar mass, such as quarks in a hadron or hadrons in tritium nucleus? Seems there is no place in the center for one of them..."}
{"id":"81968","title":"Objects falling on Slopes","text":"An object falls on a slope and then rebounces....and it is known it hits the slope again...how do I calculate it's second point of contact with the slope....how can a projectory equation be used in this case?"}
{"id":"118921","title":"Making elementary guess regarding current in AC circuits","text":"I was just studying the comparisons, between A.C, and D.C, and the effects they can produce in electrical circuits (viz, Joule-heating, Peltier-effect and Seebeck-effect in thermocouples, Electrolysis, Magnetic effect). I found from what is written in my textbook, that all of these effects could be produced by a purely D.C circuit, but a purely A.C circuit can produce only Joule-heating, and none of the others. So, I made some elementary guess, which, if correct, I think I could use to save time, in competitive-exams, based on multiple-choice questions: _Guess:_   * _Effects, that are directly, or inversely proportional to odd-powers of current through a circuit, are not exhibited in a purely AC circuit, for a complete cycle, and hence, forever._   * _Effects, that are directly, or inversely proportional to even-powers of current through a circuit, are not exhibited in a purely AC circuit, for a complete cycle, and hence, forever._ But I just found out, that in purely capacitive, and inductive AC circuits, no power is dissipated through the respective element, for a complete cycle, and thus, forever,and as well in A.C circuits containing all 3 elements in series- resistor, inductor, and capacitor, power-loss in circuit is zero for a complete cycle, if the current and voltage-drop throgh combination, differ by a phase angle of 90 degrees. The current through these circuits is called **wattless current**. Does the last paragraph oppose my guess?"}
{"id":"13191","title":"Could dark matter be expanded atoms, and dark energy expanded radiation?","text":"After watching Lawrence Krauss' lecture 'life, universe, and nothing', an interesting idea lighted up in my mind. Is it possible that dark matter is just spread out matter waves and dark energy extremely long wavelength light? matter -> expansion -> dark matter (delocalized matter waves\/huge atoms) visible and xray radiation -> expansion -> invisible light with wavelength greater than the diameter of universe (perhaps the source of energy of the empty space?!!!!) so dark matter today may be what we will become in the future. and we are what dark matter was in the past. eternal expansion just spreads out everything but new stuff is created all the time. anyone living in such a universe at any point of time will see the universe almost as we see it today, expanding with a beginning."}
{"id":"106641","title":"Can somebody explain the graphical integration method in a better way?","text":"After hours of research all what I can find is this: http:\/\/www.public.iastate.edu\/~fanous\/ce332\/virtualwork\/homepage.html can somebody explain this method in a better way. > If the moment diagram due the applied loads or the moment diagram due to the > virtual load is not continuous, one MUST divide the integration into > segments, each of which is continuous over the integration length.see this > example: The example is somehow hard to understand, Does anybody has a simpler example. I'v been trying to do this method in the conjugate beam method and then trying it with the virtual work method but the results are different. See this example: ![enter image description here](http:\/\/i.stack.imgur.com\/ot0jD.png) Now assume that $w=50$ and $L=8$. According to the conjugate beam method, the deflection at the center will be -2666.66 but using the virtual concept it is: $\\frac{2}{3} \\times 400(8)(1.25) = 2666.66$ which is positive, why is this? Also even if I use this method to another load it still give me a different result."}
{"id":"122557","title":"Formation of atoms","text":"If a Proton goes toward an Electron with a trajectory that forms a circular motion, these particles will form an atom ?"}
{"id":"134894","title":"Silicon\/silicon dioxide interface","text":"Suppose we have an interface of silicon\/silicondioxide. If we shine light on the top surface electron-hole pairs are generated in silicon. since there is always a field directing from surface to bulk in silicon (taking n type silicon), electron moves to the bulk and hole moves to the interface states. Is there any change of holes being reached in silicon dioxide surface?"}
{"id":"20394","title":"Was the universe a black hole at the beginning?","text":"Big bang cosmology, as far as I understand it, says that the universe was super hot and super dense and super small. It looks like that all the current matter, seen and unseen, were compressed to infinitesimal distance, which means it was a black hole. 1- Is the big bang, and our universe expansion in this case, hence is nothing but a black evaporation via Hawking radiation? 2- Are we living inside that primordial black hole explosion?"}
{"id":"10088","title":"Why does gravity need to be quantised?","text":"The electroweak and strong forces seem to be completely different types of forces to gravity. The latter is geometric while the former are not (as far as I'm aware!). So why should they all be described in the same way? - Why are gravitons needed? Alternatively, I suppose my question is why a theory of everything has to be some form of quantum gravity. Is there a reason why a theory of everything requires gravity to be quantised? Is it possible for the quantum world and gravity to be described by some overarching theory which does not require gravity to be quantised? Or is there some reason why this is not the case?"}
{"id":"7322","title":"References on the non-compositeness of the known elementary particles","text":"What paper(s) or theory(s) describe or prove that the elementary particles that we have determined today cannot be made up of smaller more fundamental particles?"}
{"id":"10081","title":"Has Cosmological Natural Selection been disproved?","text":"I've been reading Lee Smolin's Life of the Cosmos. Great book and it makes a lot of sense that the conditions in black holes are the same as conditions at the big bang. Question is, has his theory about Cosmological Natural Selection been disproved as the wikipedia article states?"}
{"id":"29578","title":"What is a good reference for the mechanical properties of materials at cryogenic temperatures?","text":"I want to research various the properties of various materials, namely tensile strength at cryogenic temperatures (~20K). I was wondering if a freely available database of material properties exists."}
{"id":"8620","title":"Newton's color Disk","text":"How does Newton's color disk work? Newton's disk - Take a circular white color disk, make 7 equal intersections and paint section with respective VIBGYOR colors, now when you spin the disk in certain speed, it appears white."}
{"id":"108722","title":"In what sense do Goldstone bosons live in the coset?","text":"Goldstone's theorem says that if a group, $G$, is broken into its subgroup, $H$, then massless particles will appear. The number of massless particles are given by the dimension of the coset, $G\/H$. It is then often said that the Goldstone boson's live in the coset. In what sense is this statement true? The Lagrangian is not invariant under transformations of the coset so what does this \"living\" explicitly mean? To be explicit we can consider the linear sigma model: \\begin{equation} {\\cal L} = \\frac{1}{2} \\partial _\\mu \\phi ^i \\partial^\\mu \\phi ^i - \\frac{m ^2 }{2} \\phi ^i \\phi ^i - \\frac{ \\lambda }{ 4} ( \\phi ^i \\phi ^i ) ^2 \\end{equation} We define, \\begin{align} & \\phi _i \\equiv \\pi _i \\quad \\forall i \\neq N\\\\\\ & \\phi _N \\equiv \\sigma \\end{align} and give $\\sigma$ a VEV. The spontaneously broken Lagrangian is, \\begin{equation} {\\cal L} = \\frac{1}{2} \\partial _\\mu \\pi _i \\partial ^\\mu \\pi _i + \\frac{1}{2} ( \\partial _\\mu \\sigma ) ^2 - \\frac{1}{2} ( 2 \\mu ^2 ) \\sigma ^2 - \\lambda v \\sigma ^3 - \\frac{ \\lambda }{ 4} \\sigma ^4 - \\frac{ \\lambda }{ 2} \\pi _i \\pi _i \\sigma ^2 - \\lambda v \\pi _i \\pi _i \\sigma - \\frac{ \\lambda }{ 4} ( \\pi _i \\pi _i ) ^2 \\end{equation} The Goldstone bosons, $\\pi_i$, exibit a $O(N-1)$ symmetry, but this is not the coset group symmetry. So where in the Lagrangian do we see this symmetry?"}
{"id":"128578","title":"Peak at zero in one device and not the other","text":"I was wondering if anyone could shed some light on this problem. I have placed two accelerometers on an animal one sampling at 50 Hz the other at 100 Hz. They were placed in the same position. I then computed the FFT on my results and found that the spectra from the device sampling at 100 Hz had a large peak at zero whilst the other device didn't display this large zero peak. Any ideas why this might be? I thought that it may be because of the increased sampling rate thus displaying more points that were the same or 'stationary'."}
{"id":"112069","title":"Balloon, lighter than air and vacuum?","text":"As I understand, in a balloon\/air balloon, a gas burner is used to heat air or using some lighter atom like helium. Since helium or hot air is lighter and less dense than the cool air around the balloon, the heated air or helium causes the whole balloon to rise. And as I understand, vacuum is a space where there are no atoms inside. So no atom inside means it's lighter than to have atom an inside? Isn't it? So why we can't make something fly by giving it a vacuum container?"}
{"id":"9982","title":"Brans-Dicke gravity and Black-Hole \"vacuum\"","text":"The field equations of the Brans-Dicke gravity are $$\\Box\\phi = \\frac{8\\pi}{3+2\\omega}T$$ $$G_{ab} = \\frac{8\\pi}{\\phi}T_{ab}+\\frac{\\omega}{\\phi^2} (\\partial_a\\phi\\partial_b\\phi-\\frac{1}{2}g_{ab}\\partial_c\\phi\\partial^c\\phi) +\\frac{1}{\\phi}(\\nabla_a\\nabla_b\\phi-g_{ab}\\Box\\phi)$$ In general relativity, the singularity in the Schwarzchild black hole seems to be considered either a 'delta function' like mass source term, or the \"edge\"\/boundary condition of a manifold. Either way, the equations can't tell us how the \"infinite curvature\"\/singularity evolves and therefore we are forced in some sense to \"impose it\" as a boundary condition either way. However, in Brans-Dicke gravity the effective gravitational coupling constant appears to be sourced by $T$. So it appears in a true vacuum $T \\rightarrow 0$, $\\phi \\rightarrow 0$, and therefore no Schwarzchild solution is possible _regardless_ of the value of the parameter $\\omega$. But I've heard that Brans-Dicke is equivalent to General Relativity at least in some limit, so what is going on here? Brans-Dicke seems to really care precisely how we deal with singularity, while General Relativity doesn't care what is behind the event horizon. **1) Can Brans-Dicke gravity support black hole vacuum + curvature singularity solutions?**   **2) Does it force us to interpret the singularity a certain way, and if so does the Brans-Dicke $\\longrightarrow$ General Relativity limit tell us the singularity should be interpretted as non-vacuum to make sense?**"}
{"id":"1380","title":"Teleportation by quantum isolation","text":"This is a thought experiment, so please don't treat it too harsh :-) Short: If we could isolate two places A and B in the universe from all and any interaction with the surroundings, is there a physical law which states \"if something is dropped in place A, it has to stay there\"? Long version: Let's assume that the energy of the whole universe is fixed. Let's further assume that it is (by some trick) possible to completely isolate a box of 1m^3, say in the center of a planet (all gravitational and centrifugal forces cancel themselves out), the mass of the planet shields against radiation and we use a trick to shield against neutrinos or we ignore them since the rarely interact with matter). How does an object behave when it has no interaction with the rest of the universe whatsoever? If I put an object in a box described above and I have several such boxes, would it matter in which box the object is? Is there a law which says \"even if no one knows, the object **has** to stay where it is\"? Or is that just our expectation based on everyday experience?"}
{"id":"95998","title":"What is average life in radioactivity and what is its significance?","text":"By definition, average life of radioactive sample is the amount of time required for it to get decayed to 36.8% of its original amount. But what is the significance of 36.8% and why has that value been chosen?"}
{"id":"55740","title":"Evaluate Commutator with Partial Derivatives","text":"I need to evaluate the following commutator... $[x(\\frac{\\partial}{\\partial y})-y(\\frac{\\partial}{\\partial x}),y(\\frac{\\partial}{\\partial z})-z(\\frac{\\partial}{\\partial y})]$ i tried applying an arbitrary function $\\psi(x)$ but I am unsure of how to evaluate the whole thing. My first step was: $=(x(\\frac{\\partial}{\\partial y})-y(\\frac{\\partial}{\\partial x}))*(y(\\frac{\\partial}{\\partial z})-z(\\frac{\\partial}{\\partial y}))\\psi(x)-(y(\\frac{\\partial}{\\partial z})-z(\\frac{\\partial}{\\partial y}))*(x(\\frac{\\partial}{\\partial y})-y(\\frac{\\partial}{\\partial x}))\\psi(x)$ This is where i am lost, I am unsure of where to go from here. Maybe there is something else I can do as my first step to simplify the question? please let me know what I can do. Thanks."}
{"id":"55745","title":"How can the nucleus of an atom be in an excited state?","text":"An example of the nucleus of an atom being in an excited state is the Hoyle State, which was a theory devised by the Astronomer Fred Hoyle to help describe the vast quantities of carbon-12 present in the universe, which wouldn't be possible without said Hoyle State. It's easy to visualise and comprehend the excited states of electrons, because they exist on discrete energy levels that orbit the nucleus, so one can easily see how an electron can excite from one energy level into a higher one, hence making it excited. However, I can't see how the nucleus can get into an excited state, because clearly, they don't exist on energy levels that they can transfer between, but instead it's just a 'ball' of protons and neutrons. So how can the nucleus of an atom be excited? What makes it excited?"}
{"id":"56213","title":"Why the CFL light bulb flashes even if there is a power cut?","text":"Today when there was a power cut I saw that the CFL light bulbs were flashing time to time at regular interval. I was amazed as there was no source of power to the CFL bulbs even they were flashing as if they were sucking energy from surrounding, collecting it and then releasing them with a flash. Why was it happening?"}
{"id":"119282","title":"What would be physics of a star totally consisting of electrons?","text":"What is the minimum radius and mass (if any) for such a structure to be stable? What is the ultimate future of such structure? Will the electron repellation always be stronger than any gravitational attraction? Even at any density?"}
{"id":"2167","title":"Can magnets rotate infinitely?","text":"There are many videos on youtube in which people arranged magnets in circle and rotated one placing in middle of that circle on a shaft, and the magnet (magnet motor) starts madly and continues its movement.   Do they really rotate infinitely?   If so, do they from where they get that extra energy to move so fast and infinitely? EDIT: This question is about real world scenario UPDATE: I am unable to find that particular video about which i talked about -- I added them to my favourites, but they are all now deleted :( -- there are hundreds of that that kind of videos. Some example videos links   http:\/\/www.youtube.com\/watch?v=dT9s33X9D4I   http:\/\/www.youtube.com\/watch?v=yei0NMqUaZ8   http:\/\/www.youtube.com\/watch?v=5DvPLFX5gwA   http:\/\/www.youtube.com\/watch?v=i0oUaPZ_wF8   http:\/\/peswiki.com\/index.php\/OS:Screw-Magnet_Motor   http:\/\/www.youtube.com\/watch?v=aHVBu77jz4w"}
{"id":"24566","title":"Are perpetual magnetic motors a scam?","text":"I read articles and watched videos and more videos where people say they've built perpetual magnetic motors. I do not see any proof that this can work and also I do not see proof that it is wrong. I'm not really into physics. People tend to say this is scum and cannot happen. Can somebody explain what is wrong in the first video?"}
{"id":"104460","title":"Lenz's law and bar magnet falling from some height above","text":"I am reviewing my electromagnetism studies and I became confused with where energy of induced emf due to Faraday\/Lenz' law come from. Suppose that there is a curved track that starts with some height above loop of wires. The path passes through the center of the loop at the lowest point of the track. The track is symmetrical and it also ends at the same height as the starting height. Now the bar magnet is placed at the starting point. As there is gravitation potential energy converted to kinetic energy, magnetic flux through the loop will change causing an induced emf. But this suggests that by conservation of energy, this electrical energy must have come from somewhere. Otherwise, we would be able to create electrical energy forever using a bar magnet. This is how my confusion arises. Where does the energy come from?"}
{"id":"127033","title":"Physics of Snow Globe","text":"What is the physics behind snow flakes inside a 3D snow globe? If I were to implement a snow globe in computer graphics what kind of model do I need to the flakes motion like a real snow globe? What field of physics does this problem belong to?"}
{"id":"127034","title":"How to find the force to cause certain trajectory?","text":"So I got a little board today and decided to do some physics. What I came up with is confusing me alot. Say someone throws a rock and when this happens you get the known's of. From here is there a way to find the force that took it to have that trajectory. (kind of like http:\/\/imgur.com\/2ozNMgH) Angle it was thrown at. (45 degrees) The weight of the rock. (.1 kilo) The distance that the rock was thrown. (82.9056m) Missing Time in Air. Force it was thrown at. Trying to find Speed it was thrown at\/force. So with this information we can't really get alot at first but I can almost feel a solution at the top of my tongue. I think that the most critical piece of information in this situation is the angle at which it was thrown (being 45 degrees). From this you can get that the force was split equally between the Y and X component of the situation. I have racked my head for a possible equation (not very advanced in physics of math only high school freshman). All I can think of is that there is some sort of calculus wizardry behind finding this (possibly integral?). I really really have a feeling this can work. Sense I don't know how to write the fancy equations on this and I don't know how to explain what I am thinking enough to explain well. Maybe this will help some more http:\/\/www.imgur.com\/LHfQJhZ) I understand that because the normal D=1\/2gt^2 that there would be two different answers due to it being a quadratic but with it being at 45 degrees they would both be the same. Any sort of help towards finding the answer would be greatly appreciated. Like the old saying,\"the fun is getting there, not being there\"."}
{"id":"82089","title":"Loop-the-Loop with Friction","text":"Let's consider a track that begins vertically becomes a 450 degree loop, and level off. (See diagram) ![enter image description here](http:\/\/i.stack.imgur.com\/Kuy52.png) We drop a block from height $H$ that falls and goes around the loop. Ignoring air resistance, but taking into account a friction coefficient of $\\mu$ against the track, find the smallest $H$ for which the block will complete the loop. By conservation of energy I have obtained the following equation: $$ mgH = \\frac{m v(\\theta)^2}{2}+mgR(1+\\mu-\\sin(\\theta)) + m\\mu R \\int_0^\\theta\\left( g \\sin(\\theta) + \\frac{v(\\theta)^2}{R}\\right) d\\theta $$ Where $v(\\theta)$ the the block's speed at $\\theta$, and $\\theta$ is an angle measured between the block's position (centred at loop center) and a horizontal line going left-wards. The left hand side is gravitational potential energy, the first term on the right is kinetic energy, the next one is also gravitational potential energy, and the last one - the work done by friction. Dividing by $m$, differentiating with respect to $\\theta$, and using the chain rule on the first term on the right hand side we obtain: $$ 0=v(\\theta)\\frac{dv(\\theta)}{d\\theta} +gR(\\mu\\sin(\\theta) -cos(\\theta)) + \\mu v(\\theta)^2$$ Assuming the block doesn't make it, because it lost's its speed to friction, let's find $\\theta$ for which $V(\\theta) = 0$. Substituting we get: $$ 0 = \\mu \\sin(\\theta) - \\cos(\\theta) $$ But it is totally counter intuitive (and incorrect by experiment), that this point depends only on $\\mu$ and not on the initial height $H$. So where did I make the mistake? Could somebody please clarify."}
{"id":"25135","title":"How many earth-sized planets have been discovered outside the solar system?","text":"How many earth-sized planets have been discovered outside our solar system? Is there a combined registry of them anywhere? Where might I look for more information?"}
{"id":"25137","title":"After the Space Shuttle is retired, how will the new sections of ISS be launched?","text":"The International Space Station (ISS) had many of its components launched by the Space Shuttle. After the Space Shuttle's retirement, what is the proposed launch vehicle for new ISS sections?"}
{"id":"53991","title":"The theory of strings stretching between intersecting D-branes","text":"I am trying to understand various aspects of intersecting D-branes in terms of the gauge theories on the worldvolume of the D-branes. One thing I'd like to understand is the worldvolume action for strings stretching between the D-branes. One thing I have considered is $M+N$ D-branes initially coincident but then an angle $\\theta$ developing between $M$ and $N$ of them. The gauge symmetry is broken from $U(M\\times N)$ to $U(M) \\times U(N)$ with the off- block-diagonal terms of the gauge field becoming massive. I anticipate that $\\theta$ is the vev of some Higgs field that mediates this transition. What is the action of this Higgs field and where in the string spectrum does it come from?"}
{"id":"16349","title":"Why does an octave on a piano have the divisions of 8 white keys and 5 black keys?","text":"An octave corresponds to a doubling or halving of the frequency. Each octave on a piano and in classical music score is broken downs into 8 white keys and 5 black keys. Is there a physics explanation for that division?"}
{"id":"105738","title":"The shape of the graph of the equation: $V= -r I + E$","text":"I have recently collected data (for a school experiment) in order to measure the EMF and the internal resistance of a solar cell. The data complied with the equation: $V = -rI + E$, i.e. the voltage decreased as the current increased and I have a negative gradient. However, I am having some trouble as to explaining why the graph is the way it is, that is, why is there a negative gradient? Here's my rationale: Using Ohm's law, we can see that decreasing the (external) resistance increases the current flowing throughout the whole circuit (it is a simple series circuit with just a solar cell, ammeter, variable resistor and a voltmeter (in parallel, obviously)). Now, the voltage drop across the solar cell should increase as well (assuming its internal resistance in constant) because $v = V_r = Ir$, where $v$ is the 'lost volts'. For this reason the voltage drop across the load must decrease(which is what we see from the graph) because $E = V + v$. Thus, the voltage drop across the resistor decreases as current increases. I don't know to what extent this is correct or if it is correct at all but I would appreciate any and all help given. Also, as you can probably tell, the level of physics used here is very basic, so I would also appreciate it if the answers explained it in this way as well, although, where possible, do not sacrifice accuracy for the sake of simplicity."}
{"id":"65174","title":"Is there any case where one would use, snap, crackle or pop?","text":"As we all know, if you differentiate distance with reference to time, you get speed, and likewise, differentiating speed you get acceleration. However, if you keep differentiating, to the rate of change of acceleration and so forth (more emphasis on the so forth): is there any point to this? Does it have any useful application in any industrial or practical industry, as opposed to theoretical physics? Also, what could we possibly learn about a situation by understanding the rate of the rate of the rate of change of the objects distance?"}
{"id":"116692","title":"transforming experimental rate constant to monte-carlo rate constants?","text":"I am trying to write a kinetic monte carlo code for polymerization process, but i am confused about how to compare my monte carlo equilibrium with analytical calculations. as an e.g. $X + X <=> X_2$ with a forward rate $k_a^+$ and backward rate $k_a^-$. $X_2 + X <=> X_3$ with a forward rate $k_e^+$ and backward rate $k_e^-$. $X_3 + X <=> X_4$ with a forward rate $k_e^+$ and backward rate $k_e^-$. ............ etc. Now i want to specify concentration of monomer X. volume of the container V and avogadro's number. But what i am confused in is that how to transform experimental rate constants to monte carlo rate constants. real or experimental units for $k_a^+$ are $mol^{-1}s^{-1}$ and $k_a^-$ are $s^{-1}$ and same for $k_e^+, k_e^-$ the equilibrium constant $K_e = \\frac{k_e^+}{k_e^-}$ and $K_a = \\frac{k_a^+}{k_a^-}$ is usually given in (\/mol) units in literature. Nitin"}
{"id":"105732","title":"Total Electrical potential energy of two particle system","text":"I recently have been studying Electro-statics and I couldn't understand properly how the potential energy of two particle system is found. > Suppose you have two particles with charges $Q_1$ and $Q_2$ respectively. > The distance between them is $r$. What is the total electrical potential > energy of the system comprising of the two particles? Well, I know the > answer. I want to know the reasons behind the answer. My book says \"Fix one of the charges and then bring the other from infinity. Hence we get the answer. I am not satisfied with the answer, as I saw in Wikipedia they find the potential of each other with respect to the other and then add and divide by $2$. Why is this taking potential with respect to each other valid? Why do they divide by two? What does taking potential with respect to other mean? Do they fix one charge and bring another one? what about three charges $Q_1, Q_2$ and $Q_3$ which are situated in the vertices of an equilateral triangle with side length $r$?"}
{"id":"105761","title":"Alpha scattering, observations and conclusions","text":"For Rutherford's alpha scattering experiment where he fired alpha particles at a thin sheet of metal to determine the size of the nucleus, please can someone give me the direct link between his observations and his conclusions e.g. this observation lead to this conclusions."}
{"id":"53371","title":"Calculating displacement amplitude of ultrasonic power transducer","text":"I am not a physicist, but my current project drives me to some physics-related computations, hence seeking help. I have some ultrasonic transducers, 5938D-25LBPZT, for which very limited information is available. ![Transducer Diagram](http:\/\/i.stack.imgur.com\/PCWcM.jpg) ![Transducer Photo](http:\/\/i.stack.imgur.com\/j5dKc.jpg) In order to define the parameters for an ultrasonic booster and amplification horn for this device, I need to compute the displacement amplitude of its radiating surface. The procedure I am following, and the **base data** ( _or where data is unavailable, my assumptions_ ) are thus:   * Operating power: **50 Watts** RMS (the transducer is rated for 60 Watts)   * Emitting horn material: Stainless Steel **SS316** (best-guess, unfortunately, as SS316 is typical in other such transducers)   * Emitting face diameter: **5.865 cm** (measured)   * Frequency at resonance: **_f_** = **24989 Hz** (measured)   * Best-case power transmission: **90%** (in forward direction, according to various sources) **Constants** :   * Density of SS316: **ρ** = **7.8 gm\/cm^3**   * Velocity of longitudinal sound propagation: **C** = **5.8 mm\/μSec** (from this reference, approx)   **_Should the axial velocity of sound be considered instead for this calculation?_** **Calculations so far** :   * Face area: **27.016 sq cm** (ignoring area lost to bolt-hole)   * Sound intensity at face: **I** = 90% x Power \/ Area = **1.666 Watts \/ sq cm**   * Displacement Amplitude: **A**   ![Amplitude Formula](http:\/\/i.stack.imgur.com\/dfeib.jpg)   * ... where **ω** = 2 x π x _f_   * Thus, **A = 172.8296514 nanometers** * * * **My questions are:**   1. I am not sure I have managed to convert everything to consistent SI units, so I need help confirming that I've got that part right   2. If anyone has experience in such power ultrasonic piezo transducers, a confirmation that my results are more or less within the realms of plausibility would really help.   3. Is there some significantly incorrect assumption or constant value in the foregoing?   4. Should I be using some online resource that trivially calculates all this from the rated power and other inputs, instead of struggling with it manually? * * * **Update** : _Adding diagrams to show the excitation direction and materials involved._ ![Ultrasonic Stack](http:\/\/i.stack.imgur.com\/wknTr.gif)   (please ignore _Diamond insert_ ) This is the final arrangement of the Ultrasonic Stack. The _booster_ and _horn_ in the diagram above will both be SS316. In order to compute dimensions for these two parts for resonance, I need the displacement amplitude of the front face of the steel \"radiation head\" in the transducer ( _first diagram here_ ), which is to be bolted to the booster. The bolt-hole in the radiation head can be seen in the photograph above. Another diagram, for a similar \"Ultrasonic Stack\", with the Langevin transducer itself split up. This diagram has a cylindrical radiation head, as opposed to the cone section shape in the transducers I need to compute for. ![Ultrasonic Stack](http:\/\/i.stack.imgur.com\/5zuUG.jpg)   (please ignore _Wedge_ ) (Last two diagrams are from articles on ScienceDirect.com)"}
{"id":"21257","title":"How do we see different colours?","text":"* Why do different wave lengths cause electrons to behave(?) differently, causing us to see different colors?   * What is happening at the quantum level which causes the colour black to absorb all of the wavelengths of white light?   * Why is it that a black can of soft drink will heat up more than a white one?    * How does the energy, while passing through different colours, increase the temperature differently?"}
{"id":"63467","title":"Wavefunction in first Brillouin zone","text":"We know that with symmetry, Brillouin zone is nothing but copies of its irreducible zone, so can we conclude that we can find all possible wavefunctions in its irreducible zone? What about wavefunctions in high- symmetric points?"}
{"id":"1819","title":"What does an atom radiate: a wave packet or a single photon?","text":"What does an atom radiate: a wave packet or a single photon?"}
{"id":"68020","title":"Why two connected 1.5 volt battery become 3 volt","text":"Why two connected 1.5 volt battery become 3 volt? If one battery plus side is connect to minus side of another battery, there should be a current flow between them until no free charge move anymore, then the final state of these two side is 0 volt. So the total two connected battery should be still 1.5 volt. This is wrong actually, but I can't think out why."}
{"id":"78487","title":"Motion in insulating fluid under high voltage","text":"I observed the following phenomenon in an experiment (I'm not a student of physics, just an amateur) and was hoping for an explanation. A metal pan is electrically grounded and a layer of insulating fluid (vegetable oil) is placed in the pan. If you like, assume the pan is infinite in diameter. An electrode (a needle) is placed above the pan, pointing downward at the oil. A high voltage (-12kV) is applied to the electrode. The electrode is close enough to the pan to generate an ion wind, but not close enough to overcome the breakdown voltage of the air and oil, so there is no arc. The oil is repelled from the electrode (perhaps because it acquires a negative charge from the ion wind) and also, more remarkably, the surface of the oil begins to undulate. My theory to explain the undulation is that the negatively charged oil on the surface is repelled from the electrode until it touches the pan, where it loses its charge and therefore is permitted to rise to the surface again, and the process repeats. So the undulations are something like convection cells, only with electric charge instead of heat. Is this explanation correct? Is there a name for this phenomenon? Would a similar phenomenon create leakage current in capacitors with liquid dielectrics?"}
{"id":"112598","title":"How does one produce entanglement?","text":"In practice, when entangling two particles (say two electrons), is creating an entangled state purely a matter of bringing them close enough together? What I mean is, to make their wave functions correlated, is it just a matter of bringing them close enough together (regardless of what is required to accomplish this), or are there other processes necessary to cause the entanglement?"}
{"id":"92671","title":"What are the practical applications of quantum foundations?","text":"Many quantum foundation researchers keep emphasizing that For All Practical Purposes (FAPP), quantum foundations are irrelevant. They even invented an acronym for it! Does that mean that quantum foundations have no practical applications? If so, why bother working on quantum foundations?"}
{"id":"78730","title":"How is relativity equation approximated","text":"What is the technique with which I can approximate the equation $$\\frac{mc^2}{\\sqrt{1-(\\frac{v}{c})^2}}-mc^2$$ when $v\\ll c$? Any hint would be much appreciated"}
{"id":"132258","title":"Is \"microbunching\" in a free electron laser limited by the Pauli exclusion principle?","text":"As I understand it, a free electron laser can basically be pictured as a synchrotron light source with an undulator which by the particular setup causes the electrons to self-attune so that they produce light which is both coherent and monochromatic (as opposed to the usual synchrotron which produces monochromatic but incoherent light). The process of self-attuning can be also replaced by attuning the electrons by an external laser. Skipping the details, the resonance caused by the right set of parameters causes something which is called \"microbunching\" in the electron beam, i.e., an effect which can be classically pictured as all the electrons clumping around points in the beam with a distance of one optical wavelength in between. Same spatial points means same undulation, so they oscillate throughout the beam with the same phase causing coherence of emitted light. (I have just read about it, this might be a slightly naive description.) All the electrons at one place and with one velocity? Pauli exclusion principle comes quickly to mind. It seems to me that at least for small wavelengths (say ångströms $\\to$ X-rays) the bunching has to be done at the cost of a noticeable velocity dispersion of the electrons due to the Pauli exclusion principle. This would mean a dispersion of the produced wavelengths and possibly a tendency to further dilute the bunched electron packets. Also, a larger required brightness of the output means more electrons and once again a wider peak in the phase space to fit the electrons into. This would in a very hand-waivish manner suggest that there is a certain constraining relation between the wavelength\/monochromaticity\/coherence\/brightness of the outcome. So the question is: **_Does such a fundamental constraint of a free electron laser exist? And is it relevant for the current state of art or the near future?_**"}
{"id":"63149","title":"What is the lowest possible theoretical temperature that nuclear fusion can occur at?","text":"I am not talking about the pseudo-science of so called cold fusion I am interested in what temperature you can get away with to produce fusion reaction. I was thinking in terms of micro-fusion or at least at levels below the huge tokamaks being produced because obviously small scale devices couldn't handle millions of degrees of temperature. Or correct me if I am wrong please as I am no expert just a curious student!!"}
{"id":"111811","title":"Physical question on an RLC circuit","text":"I do not know if this is an appropriate place to ask this question here but this is the only website that I am a member of so I hope that it is okay. If not, I can delete it. This circuit question was brought to me while tutoring yesterday that I was not able to answer physically. Initially the capacitor had been charged to 2V so $v_C(0^−) = 2V$. Before the switch is thrown there is no current through either the resistor or inductor, so $i_L(0^−) = 0A$. ![enter image description here](http:\/\/i.stack.imgur.com\/jlIuR.jpg) Since the inductor and capacitor cannot have an instantaneous change in their values just before and after the switch is thrown, these values must remain the same: $i_L(0^−) = i_L(0^+−) = 0A$, $v_C(0^−) = v_C(0^+) = 2V$ This implies that at the moment the switch is closed, the inductor shorts out the capacitor and resistor, and no current occurs along any branches of the circuit. That is, the voltage drop across all three elements is zero. What is happening to the capacitor’s and inductor’s current and voltage in the circuit? How can the capacitor have a voltage drop of 2V and yet the inductor appears to short it out simultaneously?"}
{"id":"106410","title":"When polarized light hits a polarizer, what happens to the polarization which is not transmitted?","text":"It is typically said that in polarization of light only those waves can pass through a polaroid which has a plane of vibration parallel to the axis of polaroid. My question is: what happens to the other waves (the other polarization)?"}
{"id":"98035","title":"Why does the windshield of my car freeze even if the outside temperature is above freezing?","text":"Under what conditions does the windshield of a car freeze even if the outside temperature is above freezing? It is not clear whether this is related to the question why bridges freeze with non-freezing outside temperatures (see related topic), as I did not see an explanation in terms of radiation there. It indeed seems that electromagnetic radiation is the culprit for windshields, which would predict that this phenomenon can only occur with an open sky. Does anybody know whether this is true? And why does ice only form on top of the windshield and not on the metal surface of the car? Is this because heat is conducted much faster on the metal surface?"}
{"id":"132784","title":"Why does frequency increase as the length of an open air column shortens?","text":"I am curious as to why the frequency of a wavelength increases as an open air column becomes smaller in length. Is it because an open air column will always contain half a wavelength, therefore if the tube is smaller, it means a shorter wavelength, which in itself means a higher frequency?"}
{"id":"95708","title":"Why does a flat clover-like shape fall slower when it is rotating?","text":"The plastic flat clover-like shape pictured below falls noticeably slower when it is rotating fast. I wonder why. Its three edges are flat, so I don't think that the rotation makes it act like a propeller. An ideal explanation would include an analogy that a young child can understand (e.g. rain bouncing on a car's wind-shield). Here is a photo of the object. ![object that falls slower when rotating](http:\/\/i.stack.imgur.com\/i9pNt.jpg)"}
{"id":"108596","title":"Why do we see laser beams?","text":"A laser produces a coherent beam of photons with particular $\\mathbf{k}$ and $\\mathbf{\\omega}$. So, if there is no particular $\\mathbf{k}_0$ directed toward our eyes, why do we see laser beams? I think this is related to the Huygens–Fresnel principle, but I wasn't sure."}
{"id":"73879","title":"Magnetic moment precession around magnetic field","text":"I have a question regarding the magnetic moment of an atom. If you just have one atom, and you apply an external magnetic field in lets say the z-direction. Then the magnetic moment will precess around the direction of the magnetic field applied, and depending on the quantum numbers it can precess in different quantized directions. The question is rather simple: WHY does it precess around the magnetic field ? My book only states that it do indeed precess if a magnetic field is applied, but as I'm asking, I'm not sure WHY it does ? Why can't it just point in one direction all the time ?"}
{"id":"73877","title":"Do we need infinite energy to make 2 similar charges touch only in theory?","text":"By Coulomb's law, say if we have 2 point particles each having a charge of +1C then by the formula, F = k\/(d)^2 if we need to make the distance between them zero, clearly y the formula, we need to have an infinite amount of force. However, in a real life scenario, we see that if we rub two balloons with our head, they acquire the same charge. However, it is still possible to make them touch. Is this because the two balloons are not point objects? OR is the definition d=0 actually mean \"to occupy the same space\". for d to be 0, do the objects only need to touch or do their centers need to be together?"}
{"id":"3810","title":"Examples of circuitry using proton currents","text":"Proton cuircuits and proton motive force are part of standard discussion in biology and processes involving photosynthesis. The sort of proton currents discussed in biology are obviously slightly different than the magnetically confined proton currents in an acclerator like LHC, however, although we are very familar with electronic circuitry, I was wondering if there are example of protonic circuitry outside of biological processess."}
{"id":"34626","title":"Is it Possible to have Adiabatic Processes other than $PV^\\gamma$ for the ideal Gas?","text":"Is it possible to represent an adiabatic process for an ideal gas by a formula other than $PV^\\gamma=Const$?: _Relevant Considerations_ : We always need to connect a pair of _arbitrary_ points\/states $A:(P_1,V_1)$ and$B:(P_2,V_2)$ on the P-V indicator diagram to define the Internal Energy function(U) If the two points do not lie on some $PV^\\gamma=Const$, we may think of some abrupt process connecting the two states. Such a process cannot be represented by a continuous curve on the diagram since the intermediate states are non- equilibrium states. Pressure will not have a unique value wrt the system for such processes.It will be different from point to point.We have a pair of points A and B without any curve connecting them Now we connect the points A and B by an arbitrary \"continuous\" path on our diagram(on paper) and select close discrete points on it. The consecutive \"close\" discrete points may be connected by abrupt adiabatic \"processes\" as described previously. Therefore in the pragmatic sense we can have an arbitrary quasi-static \"adiabatic\" path connecting the points A and B : The \"close\" points ie physical; states,will lie on a curve different from $PV^\\gamma=Const$ [In the physical\/practical implementation of the $PV^\\gamma=Const$ curve we do consider\/implement a chain of discrete equilibrium states]"}
{"id":"76994","title":"Polchinsky's Evaluation of the One Loop String Path integral","text":"I try to evaluet the matrix M in the Polchinsky's article(Communications in Mathematical Physics,1986, Volume 104, Issue 1, pp 37-47,\"Evaluation of the one loop string path integral\",Joseph Polchinski) (1)$$\\det\\begin{bmatrix}1 & 0 & 0\\\\\\ -D_{a} & \\delta_{a}^{c} & 0\\\\\\ \\frac{1}{2} g^{ef}g_{ef,i} &0 & \\delta_{ik}\\end{bmatrix} \\cdot \\det\\begin{bmatrix}2+4C & 0 & 0\\\\\\ 0 & 2 \\Delta_{c}^{d} & -2D_{e} {\\chi_{l}^{e}}_{c}\\\\\\ 0 &2 \\chi_{k}^{ed} D_e & \\chi_{kef} \\chi_{l}^{ef}\\end{bmatrix} \\cdot\\det\\begin{bmatrix}1 & D^{b} & \\frac{1}{2} g^{ef}g_{ef,i}\\\\\\ 0 & \\delta_{d}^{b} & 0\\\\\\ 0 & 0 & \\delta_{ij}\\end{bmatrix}=J N J^{T}$$ where $\\Delta_{c}^{d}= -\\delta_{c}^{d} D^2-D^{d}D_{c}+D_{c}D^{d} and \\chi_{iab}=g_{ab,i}-\\frac{1}{2} g_{ab}g^{cd}g_{cd,i}$. I'm considering a torus, so the covariant derivatives become $\\partial$. Also the determinant of $J$ and $J^T$ are $1$. So the determinant of the matrix N become $$\\det(N)=\\det(2+4C)\\det(-2\\delta_{c}^{d}\\partial^2) \\det(\\chi_{kef} \\chi_{l}^{ef}).$$ This is right? (2) $\\det(\\chi_{kef} \\chi_{l}^{ef})=\\chi_{kef}(g^{e \\alpha}g^{f \\beta} \\chi_{l\\alpha \\beta})=\\partial_k g^{\\alpha\\beta}\\partial_l g_{\\alpha \\beta}-\\frac{1}{2} (g_{\\alpha \\beta}D_kg^{\\alpha \\beta})g^{\\gamma \\delta} \\partial_l g_{\\gamma \\delta}-\\frac{1}{2} g^{\\alpha \\beta} \\partial_l g_{\\alpha \\beta} g^{cd} \\partial_k g_{cd}+\\frac{1}{2}(g^{cd}\\partial_k g^{cd})(g^{\\gamma \\delta} \\partial_k g_{\\gamma \\delta})$ this is also right? (3) If steps 1 and 2 are correct, how can I get to the form (given by Polchinski in the article)? $$\\det'(N)=\\det(2+4C)\\det'(-2\\delta_{c}^{d} g^{ab} \\partial_a \\partial_b) \\frac{4}{{\\tau_2}^4}$$ where the prime denotes the restriction to the zero mode."}
{"id":"76993","title":"Is the Copenhagen interpretation still valid?","text":"\"The secret lives of photons revealed\" - physicsworld.com. It seems that this experiment violates the Copenhagen interpretation. Is it still valid? Can you please give me more insight on this experiment?"}
{"id":"100150","title":"Is it possible to have a negative attraction to gravity just like how magnets with same pole does?","text":"Layman question here, is it possible for an object to be negatively attracted to a gravitational pull of another object? Like how same pole magnets repels each other, can it happen on gravity too?"}
{"id":"92621","title":"Is it possible to Vectorialize Quantum Field Theories?","text":"If I take the rules for classical electrodynamics in the covariant formulation (the closest to QFT), I have a tensor that describes the field, $F_{\\mu\\nu}$. Now we know that we can take some of the components of that tensor and find two vectors, $\\mathbf{E}$ and $\\mathbf{B}$, that obey Maxwell equations and have an identity on their own. In my understanding the \"photon field\" has been reduced to something much more material, that I can visualize. I can see easily, for example, a wave propagating in a particular direction. I am aware that I tacitly assumed that the \"photon field\" is described by the classical theory, but I think that the concept is clear nonetheless. Now, is it possible to do something like that to the \"electron field\", the \"positron field\", the \"gluon field\" and so on? If you take all the forces together and you have a giant field with many components (as in my understanding is the Standard Model), is it possible to find some combinations of those components that behave like vectors?"}
{"id":"109640","title":"Hydrogen Transition Rate from n',l to n',l'","text":"I am trying to solve Problem 9.22 in Griffiths' Intro to Quantum Mechanics (2nd Ed.) that asks to show the spontaneous emission rate for a transition from n,l to n',l' in hydrogen is given by $\\frac{e^2\\omega^3I^2}{3\\pi\\epsilon_0\\hbar c^3} \\begin{cases}\\frac{l+1}{2l+1} & if l'=l+1 \\\\\\ \\frac{l}{2l-1} & if l'=l-1 \\end{cases}$ where $I=\\int_0^\\infty r^3R_{nl}(r)R_{n'l'}(r)dr$ I found this question rather interesting because it generalizes transitions for the hydrogen atom and that the result is independent of m. Why should this be the case? Attempt at solution. As shown below, I am having difficulty adding everything up to prove the desired result above. The spontaneous emission rate for a transition from state a to state b is given by $\\frac{e^2\\omega^3|p|^2}{3\\pi\\epsilon_0\\hbar c^3}$ where $p=q\\langle b|r|a\\rangle $ (the text notes that $p$ in general may be complex but is treated as real). The hint in the text suggests that you integrate over the x,y and z coordinates between $|nlm\\rangle$ and $|n'l'm'\\rangle$, and then calculate $|\\langle n',l+1,m|\\boldsymbol{r}|n,l,m\\rangle|^2+|\\langle n',l+1,m-1|\\boldsymbol{r}|n,l,m\\rangle|^2+|\\langle n',l+1,m+1|\\boldsymbol{r}|n,l,m\\rangle|^2$ This hint is based on section 24 of Fermi's Notes on Quantum Mechanics (The spherical harmonic relations shown below including conversion to Cartesian coordinates are obtained from this section. The solution to the integrals shown below for the case $l'=l+1$ perhaps unsurprisingly agree with those in this text. The integrals are not provided for the $l'=l-1$ case). In order to make this hint meaningful, we first need to remember (or in my case look up) some relations for spherical harmonics: $\\sqrt{\\frac{8\\pi}{3}}Y_{1}^{1}Y_{l}^{m-1}=\\sqrt{\\frac{(l+m)(l+1+m)}{(2l+1)(2l+3)}}Y_{l+1}^{m}-\\sqrt{\\frac{(l-m)(l+1-m)}{(2l+1)(2l-1)}}Y_{l-1}^{m}$ $\\sqrt{\\frac{4\\pi}{3}}Y_{1}^{0}Y_{l}^{m}=\\sqrt{\\frac{(l+1)^{2}-m^{2}}{(2l+1)(2l+3)}}Y_{l+1}^{m}-\\sqrt{\\frac{l^{2}-m^{2}}{(2l+1)(2l-1)}}Y_{l-1}^{m}$ $\\sqrt{\\frac{8\\pi}{3}}Y_{1}^{-1}Y_{l}^{m+1}=\\sqrt{\\frac{(l-m)(l+1-m)}{(2l+1)(2l+3)}}Y_{l+1}^{m}-\\sqrt{\\frac{(l+m)(l+m+1)}{(2l+1)(2l-1)}}Y_{l-1}^{m}$ We will also need some rather useful relations between spherical harmonics and Cartesian coordinates: $\\sqrt{\\frac{8\\pi}{3}}Y_{1}^{1}(\\theta,\\phi)=-\\frac{x+iy}{\\sqrt{x^{2}+y^{2}+z^{2}}}$ $\\sqrt{\\frac{8\\pi}{3}}Y_{1}^{-1}(\\theta,\\phi)=-\\frac{x-iy}{\\sqrt{x^{2}+y^{2}+z^{2}}}$ $\\sqrt{\\frac{4\\pi}{3}}Y_{1}^{0}(\\theta,\\phi)=-\\frac{z}{\\sqrt{x^{2}+y^{2}+z^{2}}}$ We can see from looking at the above equations that the integrals with z will be nonzero when the m values are equal while the integrals containing $x\\pm iy$ will be nonzero when the m values are different (i.ei $m'\\neq m$). First let us focus on the $l'=l+1$ case. From the above relations we can see that $\\langle n,l+1,m+1|x+iy|n,l,m\\rangle=-I\\sqrt{\\frac{(l+m+2)(l+1+m)}{(2l+1)(2l+3)}}$ $\\langle n,l+1,m+1|x-iy|n,l,m\\rangle=0$ $\\langle n,l+1,m|z|n,l,m\\rangle=I\\sqrt{\\frac{(l+1)^{2}-m^{2}}{(2l+1)(2l+3)}}$ $\\langle n,l+1,m-1|x+iy|n,l,m\\rangle=0$ $\\langle n,l+1,m-1|x-iy|n,l,m\\rangle=I\\sqrt{\\frac{(l+1-m)(l+2-m)}{(2l+1)(2l+3)}}$ Now let us add up the above elements. This is where I am stuck. I obtain $\\langle n',l',m'|r|n,l,m\\rangle=|\\langle n,l+1,m+1|x+iy|n,l,m\\rangle|^{2}+|\\langle n,l+1,m|z|n,l,m\\rangle|^{2}+|\\langle n,l+1,m-1|x-iy|n,l,m\\rangle|^{2}$ $\\langle n',l',m'|r|n,l,m\\rangle=I^{2}\\left\\\\{ \\frac{(l+m+2)(l+1+m)}{(2l-1)(2l+3)}+\\frac{(l+1)^{2}-m^{2}}{(2l-1)(2l+3)}+\\frac{(l+1-m)(l+2-m)}{(2l+1)(2l+3)}\\right\\\\} $ $\\langle n',l',m'|r|n,l,m\\rangle=I^{2}\\left\\\\{ \\frac{l^{2}+lm+2l+l+m+2+lm+m^{2}+2m}{(2l-+1)(2l+3)}+\\frac{(l^{2}+2l+1)-m^{2}}{(2l-+1)(2l+3)}+\\frac{l^{2}+l-lm+2l+2-2m-lm-m+m^{2}}{(2l+1)(2l+3)}\\right\\\\} =I^{2}\\left\\\\{\\frac{(3l^{2}+8l+5+m^{2})}{(2l+1)(2l+3)}\\right\\\\} $ which is clearly not equal to what we want. I apologize if I simply do not know how to add, but the extra $m^2$ makes me believe there is some trick that I am missing. In any case, we may continue on for the $l'=l-1$ case. Similar to the above, we obtain $\\langle n,l+1,m+1|x+iy|n,l,m\\rangle=I\\sqrt{\\frac{(l-m-1)(l-m)}{(2l+1)(2l-1)}}$ $\\langle n,l+1,m+1|x-iy|n,l,m\\rangle=0$ $\\langle n,l+1,m|z|n,l,m\\rangle=-I\\sqrt{\\frac{l^{2}-m^{2}}{(2l+1)(2l-1)}}$ $\\langle n,l+1,m-1|x+iy|n,l,m\\rangle=0$ $\\langle n,l+1,m-1|x-iy|n,l,m\\rangle=-I\\sqrt{\\frac{(l-1+m)(l-2+m)}{(2l+1)(2l-1)}}$ Now let us add the above together like we did for the previous case. With any luck, we will get stuck on this part again. $I^2\\left\\\\{ \\frac{(l-m-1)(l-m)}{(2l+1)(2l-1)}+\\frac{l^{2}-m^{2}}{(2l+1)(2l-1)}+\\frac{(l-1+m)(l-2+m)}{(2l+1)(2l-1)} \\right\\\\}$ $=I^2\\left\\\\{ \\frac{l^2-lm-l- lm+m^2+m}{(2l+1)(2l-1)}+\\frac{l^{2}-m^{2}}{(2l+1)(2l-1)}+\\frac{l^2-l+lm-2l+2-2m+lm-m+m^2}{(2l+1)(2l-1)} \\right\\\\}$ $=I^2\\left\\\\{ \\frac{3l^2-4l+2-2m+m^2}{(2l+1)(2l-1)}\\right\\\\}$ Again, we appear to get stuck. Thank you for reading this post to the end. Any help would be greatly appreciated in eliminating the m variable from the above sum and proving the answer."}
{"id":"109648","title":"Need help solving a task - laws of conservation (leptons, baryons, charge)","text":"We got a task from our teacher, a multiple choice task; and here it is: Which of the following reactions is correct, considering the conservation laws? Note: **We have only learned to use the conservation laws for Electric charge, Baryon number and Leptonic number.** ![The Tasks:](http:\/\/i.stack.imgur.com\/BQRl2.png) However, i found out that **B** and **C** must be wrong regarding to those laws, and that **A** and **D** is both correct. But the task ask specificly about the correct answer, and i am stuck here. And please also, explain why one of them should be wrong, regarding to those 3 laws, only. My teacher had problems solving it aswell, so it might be an error in the task itself. Thankyou!"}
{"id":"38996","title":"Are the electrons in a quantum hall edge state entangled?","text":"I am reading the paper on Quantum Energy Teleportation by Yusa, Izumida and Hotta(This article), and it seems that they are assuming that the quantum hall edge state is a quantum correlated state, which I understand as it being an entangled state. How can all the electrons in the edge state be entangled? Or have I misunderstood the assumptions of the article?"}
{"id":"31817","title":"Angular momentum conservation in a central field through the Hamiltonian","text":"In my teacher's notes there is a discussion of the Hamiltonian for a central force field with potential $V(r)$. The Hamiltonian is formulated in spherical polar coordinates: $$H=\\frac{p_r^2}{2m}+\\frac{p_\\phi^2}{2mr^2\\sin^2\\theta}+\\frac{p_\\theta^2}{2mr^2}+V(r)$$ Then the conservation of $L_z$ is trivial, because $\\phi$ is cyclic. However it is asserted, as if it were evident, that $L^2$ is also conserved. One consequence is that the Hamiltonian can be written in the form $H=\\frac{p_r^2}{2m}+\\frac{L^2}{2mr^2}+V(r)$. Now, I know that for any radial force it is trivial to prove that $\\vec L$ is conserved, but how can you prove that $L^2$ is conserved just from looking at the previous Hamiltonian? Moreover, does it suffices to prove that $L^2$ and $L_z$ are conserved in order to find that all the three components of $\\vec L$ are conserved? I'm a bit confused, so any help is appreciated"}
{"id":"31811","title":"Calculate quadrotor propeller torque due to aerodynamic drag","text":"I'd like to know the right formula to calculate torque of a quadrotor propeller (propeller pitch is constant) due to aerodynamic drag in terms of angular velocity $ \\omega $. For the sake of my simulation I'm assuming `APC 10x4,7` propellers. I'd like to refer to UIUC Propeller Data Site, so please assume only the set of data given there is available while answering. I know really little of aerodynamics, I'm planning to design a control system in Simulink later on, but would like to derive a non-linear mathematical model first. Please note I need to use data from UIUC Propeller Data Site for APC 10x4,7, which means the propeller below. I'm guessing all is there to derive a formula $\\tau_d(\\omega)$, I just don't understand how to do it. ![Top view of APC 10x4,7 propeller, taken from \\[UIUC Propeller Data Site\\]\\(http:\/\/www.ae.illinois.edu\/m-selig\/props\/propDB.html#APC\\)](http:\/\/i.stack.imgur.com\/hAWT6.jpg) ![Side view of APC 10x4,7 propeller, taken from \\[UIUC Propeller Data Site\\]\\(http:\/\/www.ae.illinois.edu\/m-selig\/props\/propDB.html#APC\\)](http:\/\/i.stack.imgur.com\/02YnN.jpg) **EDIT:** By now I figured out, lift force of a wing can be calculated as: $ F_T = C_T \\rho A \\frac{V^2}{2} $, where $C_T$ is lift coefficient, $\\rho$ means air density, $A$ is wing total area and $V$ is relative air speed. I also asked a friend who told me, for a helicopter propeller this transforms into: $F_T(\\omega) = C_T \\rho A r^2 \\omega^2 = C_T \\rho \\pi r^4 \\omega^2 $, with $A = \\pi r^2$ being the area of the rotor, $r$ being its radius and $\\omega$ being the propeller's rotational velocity. I guess, assuming the latter formula is true, one could derive one for drag force from it substituting $C_D$ for $C_T$ and rotor blade's frontal area (its projection, rather) for $A$. Could this be right? Thrust and drag coefficients from UIUC:               RPM    CT       CP     2377   0.1039   0.0431     2676   0.1058   0.0437     2947   0.1059   0.0437     3234   0.1083   0.0444     3494   0.1096   0.0450     3762   0.1121   0.0460     4029   0.1136   0.0466     4319   0.1155   0.0474     4590   0.1177   0.0484     4880   0.1199   0.0494     5147   0.1213   0.0500     5417   0.1228   0.0508     5715   0.1239   0.0513     5960   0.1253   0.0520     6226   0.1261   0.0524     6528   0.1274   0.0531      Geometry data from UIUC below. I believe, $c$ is aerodynamic chord length, $R = 0.127m$ (propeller radius), $\\beta$ is propeller's pitch at the respective point. I believe I can calculate propeller's side area using simple quadratic integration rule. But first I need to know my reasoning is right. ![taken from \\[UIUC Propeller Data Site\\]\\(http:\/\/www.ae.illinois.edu\/m-selig\/props\/propDB.html#APC\\)](http:\/\/i.stack.imgur.com\/nQUsv.png)               r\/R    c\/R     beta     0.15   0.109   21.11     0.20   0.132   23.90     0.25   0.156   24.65     0.30   0.176   24.11     0.35   0.193   22.78     0.40   0.206   21.01     0.45   0.216   19.00     0.50   0.223   17.06     0.55   0.226   15.33     0.60   0.225   13.82     0.65   0.219   12.51     0.70   0.210   11.36     0.75   0.197   10.27     0.80   0.179   9.32     0.85   0.157   8.36     0.90   0.130   7.27     0.95   0.087   6.15     1.00   0.042   5.04"}
{"id":"77641","title":"Is the number 1 a unit?","text":"In dimensionless analysis, coefficients of quantities which have the same unit for numerator and denominator are said to be _dimensionless_. I feel the word _dimensionless_ is actually wrong and should be replaced by \"of dimension number\". For example, the Mach number is of dimension one. Many people write, for this case: > Mach-Number | Dimension: \"-\" | Unit: \"1\" As mentioned before, I would say 'Dimension: \"1\"' in this place. But what about the unit? $\\text m\/\\text s$ divided by $\\text m\/\\text s$ is equal to one. But is the number one a unit by definition? Or should one say that the Mach number has no unit and therefore 'Unit: \"-\"'?"}
{"id":"77642","title":"Are the symmetry operators well defined in the context of Projective Symmetry Group(PSG)?","text":"Consider the Schwinger-fermion approach $\\mathbf{S}_i=\\frac{1}{2}f_i^\\dagger\\mathbf{\\sigma}f_i$ to spin-$\\frac{1}{2}$ system on 2D lattices. Just as Prof.Wen said in his seminal paper on PSG, the enlarged Hilbert space and gauge redundancy complicate our symmetry analyses. Now let's take the translation-symmetry as an example. The unitary translation-symmetry operator $D$ is defined as $D\\psi_iD^{-1}=\\psi_{i+a}$, where $\\psi_i=(f_{i\\uparrow},f_{i\\downarrow}^\\dagger)^T$ and $a$ is the lattice vector. As we know, the transformation $\\psi_i\\rightarrow \\widetilde{\\psi_i}=G_i\\psi_i(G_i\\in SU(2))$ doesn't change the spin operators and the projective opearator $P=\\prod_{i}(2\\hat{n}_i-\\hat{n}_i^2)$(Note here $P\\neq \\prod _i(1-\\hat{n}_{i\\uparrow}\\hat{n}_{i\\downarrow})$). Similarly, in the new basis $\\widetilde{\\psi_i}$, we can define another translation-symmetry operator $\\widetilde{D}$ as: $\\widetilde{D}\\widetilde\\psi_i\\widetilde{D}^{-1}=\\widetilde\\psi_{i+a}$. But $D\\widetilde\\psi_iD^{-1}=G_i\\psi_{i+a}\\neq \\widetilde\\psi_{i+a}$, which means that **$\\widetilde{D}\\neq D$, the translation operators depend on the 'fermion basis' we choose**. Does this imply the translation operators unphysical? But the translation operators should be physical, so are they equivalent in the physical subspace, say for any physical spin-state $\\phi=P\\phi$, does $\\widetilde{D}\\phi=D\\phi$? If this is true, then how to prove it? Thanks in advance."}
{"id":"86882","title":"A commutation problem in Hubbard model","text":"Does the Hubbard Hamiltonian $$H=-t\\sum_{\\langle ij\\rangle \\sigma}c_{i\\sigma}^{\\dagger}c_{j\\sigma}+h.c.+U\\sum_{i}n_{i\\uparrow}n_{i\\downarrow}$$ commute with $\\sum_{i}\\mathbf{S}_i^2$? where $\\mathbf{S}$ is the spin angular momentum."}
{"id":"31818","title":"How the car keys prevent the shock caused by Static Electricity","text":"I was reading around about the Radio Frequency and its effect on the GAS\/Petrol forecourts, suddenly i came across something posted by someone, \"I personally went on a trip with some friends, one who received a nasty shock every time he tried to enter the car. Since he was the only one receiving these shocks I theorize that the source of his shocks were the shoes he was wearing. I told him to hold his key out toward the door and a spark jumped between the key and the body of the car. This prevented him from receiving the painful shock. But I could see how this could also ignite a gas fire.\" I couldn't understand how the static electricity could cause the shock from the car ? and how could be the source of this is Shoes ? and how holding the key could prevent someone from this shock ?"}
{"id":"105238","title":"How do I recover the 1D wave equation from the Lagrangian?","text":"Consider small displacements, $y(x,t)$, of an element of a string (circled in red and shown below) from equilibrium. ![enter image description here](http:\/\/i.stack.imgur.com\/T8tnU.png) The force balance in the vertical direction yields: $$+\\uparrow \\Sigma F: -T\\sin(\\theta_1)+T\\sin(\\theta_2)=(mdx)\\frac{\\partial^2 y}{\\partial t ^2}$$ Where we (well.. my prof) have equated the tension on either side of the element because we are considering the case in which there is no horizontal acceleration and because we are making a small angle approximation $$T_{left}\\cos(\\theta_1)=T_{right}\\cos(\\theta_2).$$ After linearizing and simplifying, we end up with the 1D wave equation: $$c^2 \\frac{\\partial^2 y}{\\partial x ^2} = \\frac{\\partial^2 y}{\\partial t ^2}$$ where $c^2 = \\frac{T}{m}$. For the Lagrangian approach, I am able to write down the kinetic energy of the element and of the entire string (I think): For the string: $$T=\\frac{1}{2}\\int_{_0}^{^L} \\rho \\dot y(x)^2 dx.$$ Where $y$ is a function of $x$ because we can have a different velocity in the $y$ direction at every point $x$. For an element of the string: $$T=\\frac{1}{2}m(dx)\\dot y^2.$$ I am stumped at the potential energy for both the element and the string as a whole and I can't seem to figure out how to recover this wave equation from first principles. The potential energy should be proportional to the displacement $y$ from equilibrium but from what I'm seeing online it doesn't look anything like the potential energy due to a spring. What I've seen here is that \"for a taut string, the stored energy of system described by the height function $u(x)$ is assumed to be proportional to the change in length\": $$\\int_a^b (\\sqrt{1+u'^2}-1)dx$$ Which is a complete mystery to me. So my question is how do I recover the EOMs for a taught string using the Lagrangian? I am also interested in how this particular case can be recovered from a more general solution but I may not have the background to understand that."}
{"id":"93571","title":"How can you have positive electricity","text":"In a circuit, you have two opposite charges. One would be negative, which would consist of electrons. The other would be positive charge, and protons carry this. But protons aren't mobile, so how does this work? This might be tagged as a duplicate, as it is similar to a question posted here, but that post didn't have a very clear answer."}
{"id":"98546","title":"Looking For The Derivation Of Gellmann-Nishijima Relation","text":"How is the Gellmann-Nishijima relation $Q=I_3+\\frac{Y}{2}$ derived? Either a derivation or a link will be helpful."}
{"id":"98541","title":"Why can you touch two positively charged rods?","text":"For a physics lab on the Triboelectric effect, we rubbed two rods with fur which gave both of them a positive charge. We then brought them close together, and they obviously repelled. We then held one rod down firmly and touched the rods together. We were able to do so. Why? Coloumb's Law says that as the distance between two like charges become zero, the force becomes infinite. Using superposition on all the charges in the rod, shouldn't the infinite force have prevented us from touching the rods together?"}
{"id":"75642","title":"Has a system of natural units been designed for human-like scales?","text":"A long time ago I was thinking about how the Imperial system of measurements is arbitrary and annoying, and I decided to design the best system of units ever (I wasn't very old then). I worked on this idea occasionally for years without making any progress. When I finally got serious about it, I discovered Planck Units and that seemed to settle the issue. Now my problem is that Planck Units are so small that they can't be used for \"normal\" things without huge exponents, most of these being very different depending on the quantity being measured. Solving human-scale equations with these units by computer would thus either be very inaccurate due to floating- point errors (for numbers that are even within range) or very slow due to the need for extended precision. They also go \"up\" but not \"down\", which renders almost half of the possibile signed floating-point values as useless. I considered the idea of creating a new system by raising each Planck Unit with a standard exponent or multiplier, but I think this would bring some into an acceptable range but not others. So what I want to know is whether a system of natural units exists that uses units that are appropriate for work with technology that humans can interact with, or can be made appropriate without introducing too many arbitrary elements."}
{"id":"98542","title":"Why the speed of light is represented by $c$?","text":"In almost every textbook, I've found that the speed of light is $c \\approx 3 \\times 10^8\\: \\mathrm{m\/s}$. I wonder why it's just $c$ ?"}
{"id":"130124","title":"Is it possible that matter\/antimatter collisions emit resonance frequencies of H\/He?","text":"Is it possible that matter\/antimatter collisions emit resonance frequencies of H\/He? I am researching a high energy phenomenon that occurs between the frequencies of 1200-1580 MHz. This come from a very very long way away though, so the signal does change a little bit. I'm trying to find out if matter\/antimatter collisions could be the cause, and my mass estimates are asteroid sized masses of matter and anti-matter colliding, and it all seems very promising. Other than the fact that such collisions probably emit high energy Gamma Ray Bursts, not Radio Bursts. However, I found this to be interestingly near the resonance frequency of hydrogen (1420MHz) and I saw a figure that said Helium was 1400, but I'm not sure if that is correct. Is it possible for Matter-Antimatter collisions to emit the resonance frequency of there respected elements? Such as Hydrogen-antiHydrogen collision, emitting a radio signal at 1420 MHz?"}
{"id":"70034","title":"Baryogenesis only at the Planck scale, or none at all?","text":"I can think of three general ways of explaining why the universe contains more matter than antimatter: (1) Near the Planck time, the universe had zero baryon asymmetry, but at some later time, determined by some GUT energy scale, the Sakharov conditions were satisfied, and the baryon asymmetry became nonzero. (2) Nonconservation of baryon number occurs only at Planckian energies. Near the Planck time, the baryon asymmetry evolved from zero to some nonzero value. (3) The Sakharov conditions have never been satisfied. The baryon asymmetry has always been nonzero, and has simply scaled as expected. (Apparently one expects $\\Delta n=n_B-n_\\bar{B}\\propto s$, where $n$ is number density and $s$ is entropy density). It seems like most theorists are interested in #1, but is there any reason why 2 and 3 aren't possible? 2 seems pretty reasonable, since for the reasons given in this answer, we have good reasons to think that baryon number is not conserved under Planckian conditions. 3 also seems reasonable to me, since even if baryon number is nonconserved at Planckian energies, that's only one of the three Sakharov conditions. I don't see any obvious fine-tuning objections to #3, since the scaling of baryon asymmetry with cosmological expansion isn't particularly drastic (not an exponential decay or anything). Is there something unphysical about maintaining $\\Delta n\\propto s$ all the way back to the Planckian era? Some people might object to #3 on aesthetic grounds, since we \"expect\" the initial conditions of the universe to be symmetric, but that seems weak to me. After all, we don't object aesthetically to the fact that homogeneity was an imperfect symmetry of the early universe, and we even accept that the early universe was in a thermodynamically unlikely state."}
{"id":"104101","title":"How do we know that $F = ma$, not $F = k \\cdot ma$","text":"It seems intuitive that $a\\; \\propto \\frac{F}{m}$, as the greater the force that is applied on an object, the greater its acceleration will be. Inversely, the greater the mass of the object, the slower the acceleration will be. However, when rewriting proportions as equations, you must introduce a constant of proportionality, and in this case of a direct proportion, if $a \\propto \\frac{F}{m}$ then when rewriting as an equation you will have $$a = k\\cdot\\frac{F}{m}$$ In order to get the standard formula $F = ma$ this constant must be $1$. However, how do we know that this is the case? How do we know that the constant isn't $2$ and the formula $F = \\frac{1}{2}ma$, for instance?"}
{"id":"70696","title":"Transformer: primary side & secondary side current 180 degree out of phase","text":"I am a novice in electrical engineering. I notice that in transformer the secondary side current & current referred to as primary are 180 degree out of phase from each other. But why it is so, I don't know.I did some theoretical analysis on this topic but no phasor analysis(why 180 degree). So, please give me suggestions. I also thought a phasor analysis but I need 10 reputations."}
{"id":"12931","title":"Finding the position of a planet between two other planets of known mass and distance","text":"Here is the question: > A planet with mass $m$ and a second with mass $M$ are separated by a > distance $d$. A third planet with mass $m_3$ happens to be midway between > $M$ and $m$. Where could the third planet be positioned (distance from the > larger planet $M$ in meters) so that the net gravitational force is zero? My confusion lies with how to solve for the position of the third planet. I am given this equation to find the force between two planets at a given distance $r$: $$F = {GMm}\/{r^2}$$ With this I can then set the sum of the two forces to zero: $$0 = {GMm_3}\/r_1^2 + {Gmm_3}\/r_2^2$$ My confusion lies in the fact that both $r_1$ and $r_2$ are unknown. However, we do know that $r_1 + r_2 = d$. But I am confused with how to solve for either $r_1$ or $r_2$."}
{"id":"12936","title":"decay with initial velocity:","text":"decay with initial velocity: All static particle decay to particles with momentum distribution in the same shape like ![enter image description here](http:\/\/i.stack.imgur.com\/i8gD5.jpg) that's implied by structure function and calculated by MG. but the decay of W emits e like (similar to) ![enter image description here](http:\/\/i.stack.imgur.com\/mXogV.jpg) Why that? The only explanation is the W generated by p--p has initial velocity. The superficial explaination of resonance are contradictory to the logical deduce. This will modified the mass of it to 15 Gev. Is it reasonable?"}
{"id":"76691","title":"Which pressure to use in the $T~ds$ equation?","text":"Let's say I have an adiabatic, rigid, open container that has an amount of air at some pressure, $P_\\text{cv}$, and some temperature, $T_\\text{cv}$. I have heated pressurized air coming into the container. I am trying to calculate the change in specific entropy of the air in the container. I use the $T~ds$ equation, but I am not sure which pressure to use. The equation used is $$T\\frac{ds_\\text{cv}}{dt}=\\frac{du_\\text{cv}}{dt}+P\\frac{dv}{dt}.$$ Would the pressure term in the equation be the pressure of the incoming air, or would it be the pressure of the air in the control volume?"}
{"id":"37746","title":"Math or Physics degree?","text":"I am hoping to become a physicist focusing mainly on the theoretical side in the future. I am trying to decide whether to go for a physics or math undergrad course. Assuming that I am capable of doing either, what are the pros and cons of either route? I know that mathematics is essential to doing physics, and in most math courses, there are applied math modules that are very much related to physics. Also that many research physicists have math degrees. But surely there is a reason why people choose the physics course over the math course and vice versa? Thank you."}
{"id":"37743","title":"273 + degree Celsius = Kelvin. Why 273?","text":"Temperature conversion: 273 + degree Celsius = Kelvin Actually why is that 273? How does one come up with this? My teacher mentioned Gann's law (not sure if this is the one) but I couldn't find anything related to this, which law should it be?"}
{"id":"127280","title":"Invariance of canonical Hamiltonian equation when adding the total time derivative of a function of $q_i$ and $t$ to the Lagrangian","text":"The following is exercise 8.2 in 3rd edition (and exercise 8.19 in 2nd edition) of Goldstein's _Classical Mechanics._ Adding the total time derivative of a function of $q_i$ and t to the Lagrangian will not change the the Euler-Lagrangian equation. So if we make the following change to Lagrangian, $$L'(q,\\dot{q},t) = L(q,\\dot{q},t) + \\frac{dF(q_1,q_2,...,q_n,t)}{dt}$$ we can get $$ \\frac{d}{dt}\\frac{\\partial{L'}}{\\partial{\\dot{q_i}}} - \\frac{\\partial{L'}}{\\partial{q_i}} = 0 $$ from $$ \\frac{d}{dt}\\frac{\\partial{L}}{\\partial{\\dot{q_i}}} - \\frac{\\partial{L}}{\\partial{q_i}} = 0 $$ How can we get the corresponding Hamiltonian equation part? This is to prove $$ \\dot{p'_i} = \\frac{\\partial{H'}}{\\partial q_i} $$ $$ -\\dot{q_i} = \\frac{\\partial{H'}}{\\partial p'_i} $$ from $$ \\dot{p_i} = \\frac{\\partial{H}}{\\partial q_i} $$ $$ -\\dot{q_i} = \\frac{\\partial{H}}{\\partial p_i} $$ where $p'_i = \\frac{\\partial L'}{\\partial \\dot q_i}$. **Edit** The corresponding $H'$ is $$ H' = \\sum_k{p'_k \\dot{q_k}} - L' $$ where $p'_k = \\frac{\\partial L'}{\\partial \\dot q_k}$."}
{"id":"6987","title":"running of coupling constant as a function of distance?","text":"There are many papers about the running of coupling strength as a function of momentum\/energy scale, but are there any experimental papers about coupling strength as function of distance? Also, are there any good books about this topic? oops ,really sorry i forgot to specify \"experimental\"paper"}
{"id":"86717","title":"QCD mass gap finite temperature","text":"QCD has a mass gap. Does heating up QCD to a finite temperature change this fact, or is it a property which is independent of temperature?"}
{"id":"86715","title":"Entanglement entropy at finite temperature","text":"The entanglement entropy is computed on a constant time slice. At finite temperature the time is periodic. How then can the entanglement entropy be calculated on a constant time slice? For example this is done in Eq. (3.34) of the paper 0605073."}
{"id":"123192","title":"Why do batteries become weaker","text":"I was thinking about my phone (Sony Xperia Z2) and when I put it to charge and remove it, the phone says \"Please remove the charger, to save battery life,\" which is very intuitive question for a middle school student because of my lack of scientific knowledge Why do batteries become weaker as I use it more and charge it more?"}
{"id":"14699","title":"Interference of EM Waves with Orbital Angular Momentum","text":"If you have two coherent collinear e-m beams of same frequency and polarization, but 180 degrees out of phase, they will destructively interfere. If you introduce orbital angular momentum of L=3 (three revolutions per wavelength) to one beam and L=1 to the other beam, ceteris paribus, will they still destructively interfere in the exact same manner?"}
{"id":"122860","title":"Energy definition in special relativity","text":"I'm going through the early homework assignments for my special relativity course and I've got myself a little confused about energy. I've got a basic understanding of what the 4-momentum is, having defined it as $m\\dfrac{dx^{\\mu}}{d\\tau}$, and shown that this is equal to $m\\gamma (|\\vec{v} |)(c,\\vec{v})$ where $\\vec{v}$ is the classical velocity in the inertial reference frame associated to the above Cartesian coordinates. Now one of my assignments has began by saying > Denote the components of $p^{\\mu}$ as $(E\/c, \\vec{p})$... I don't understand why the time component of the 4-vector is being _denoted_ as $E\/c$. You can't just go arbitrarily denoting quantities by other quantities! So I'm left wondering whether this is a definition of relativistic energy, or if it follows from some other result in special relativity? Thanks for any clarification."}
{"id":"47828","title":"Fighting a black hole: Could a strong spherical shell inside an event horizon resist falling in to the singularity?","text":"As a thought experiment imagine an incredibly strong spherical shell with a diameter a bit smaller than the event horizon of a particular large black hole. The shell is split into two hemispheres, and the two hemispheres are released from either side of the black hole such that they will meet and join inside the event horizon, forming a stationary uniform shell around the singularity. Of course, we consider that the shell is flawless, and exactly centred around a stationary singularity - clearly this is an unstable equilibrium but we imagine it is possible. Is is possible that the structural forces provided by such a shell could be strong enough to resist the extreme gravity conditions inside the event horizon, and hence avoid being sucked into the singularity? (For the purposes of this question we can consider taking the shell to extreme levels of strength, e.g. consider that the shell is made from a theoretical maximally densely packed neutronium-like material that is strongly resistant to any further compression due to the Pauli exclusion principle)."}
{"id":"47829","title":"The Moon is slowly moving away from the earth. Does this mean that a total solar eclipse wasn't possible at some point in earth's history?","text":"When the moon was closer to earth, was it still possible to witness a total solar eclipse millions of years ago? Or was the view-able space so small that it was impractical to even witness it?"}
{"id":"47791","title":"What do massive photons have to do with superconductivity?","text":"I keep reading that the idea of massive photons leads to an explanation of the Meissner effect but I fail to see how photons are involved with the repulsion of fields inside a superconductor. How can a photon effect the electromagnetic fields inside a superconductor?"}
{"id":"47798","title":"Parallel universe and Infinite monkey theorem","text":"Is the Infinite monkey theorem helpful for determining the existence of the very same our universe somewhere else?"}
{"id":"38141","title":"Complete list of nuclear fusion reactions","text":"Could anyone suggest good source of all fusion reactions and their cross section vs energy graphs, including ones which were ruled out for nuclear fusion reactors (i.e. including endothermic and reactions with only 1 product)?"}
{"id":"43238","title":"Are Newton's three laws of motion correct?","text":"New technology brings new ideas with these new ideas we have to look at the old ones. Where else is a better place to start then Newton's three laws of motion! With our common age of technology do we apply these \"old laws of physics\" into our new age?"}
{"id":"43231","title":"Does the length of the sidereal day vary systematically?","text":"I'm confused about some properties of the sidereal day, in particular whether its duration varies systematically over the course of the year.1 It seems to me that that must be the case, but the details are a bit confusing to me. I understand that length of a sidereal day is the time interval between successive transits of the line of equatorial longitude corresponding to a specified right ascension, $R$ (usually $0^\\text{h}$). Each sidereal day, this line shifts against the earth's direction of rotation by some amount, $\\Delta\\alpha_{SID}$, to a new position, so that the sidereal day is always shorter than the earth's rotational period on its axis (the \"stellar day\"). What's confusing me is that the length of the sidereal day is often described as if it were constant; but it can't be, nor can the corresponding changes in the equatorial longitude of $R$, $\\Delta\\lambda_{SID}$, even assuming a constant rate of precession:2 If the sidereal day has a constant length, $\\Delta\\alpha_{SID}$ must be constant, but then $\\Delta\\lambda_{SID}$ will vary (between $\\Delta\\alpha_{SID}\/\\cos\\varepsilon$ and $\\Delta\\alpha_{SID}\\cdot\\cos\\varepsilon$) as a consequence of the varying equatorial latitude at which the the line of equatorial longitude corresponding to $R$ intersects the ecliptic. But this is impossible, since equatorial longitude should change uniformly, so that $\\Delta\\lambda_{SID}$ should be the same for each (same-length) sidereal day. The reverse can't be the case either: that each sidereal day, the position of the line corresponding to $R$ shifts by the same fixed amount, $\\Delta\\lambda_{SID}$, along the ecliptic. If that were the case, $\\Delta\\alpha_{SID}$ and thus the length of the sidereal day would vary (between $\\Delta\\lambda_{SID}\\cdot\\cos\\varepsilon$ and $\\Delta\\lambda_{SID}\/\\cos\\varepsilon$) resulting in a sidereal day of varying length; but if the length of the sidereal day varies $\\Delta\\lambda_{SID}$ can't be constant. So the only conclusion I can come to is that _both_ $\\Delta\\lambda_{SID}$ and the duration of the sidereal day, and thus $\\Delta\\alpha_{SID}$, vary, subject to the constraints imposed by spherical trigonometry: $$\\Delta\\lambda_{SID}\\cdot\\cos\\varepsilon<\\Delta\\alpha_{SID}<\\Delta\\lambda_{SID}\/\\cos\\varepsilon$$ and $$\\langle\\Delta\\lambda_{SID}\\rangle=\\langle\\Delta\\alpha_{SID}\\rangle$$ Is this correct?3 Do both of these quantities vary systematically in this way, with these properties.4 Is \"the\" sidereal day really a _mean_ sidereal day?5 * * * (1) Though a distinction between a \"mean sidereal day\" and an \"apparent sidereal day\" is sometimes mentioned, the terminology I've come across is confusing, and descriptions that do discuss variations seem to be referring to different, unspecified, phenomena. Here, I am asking about the properties of an idealized \"mean sidereal day\", specifically whether even this \"mean\" day has systematic variation in duration. (2) My thinking went like this. Take the plane of the ecliptic as reference. Then precession is just the a rotation of the equatorial coordinate system around an axis perpendicular to that plain. The earth's rotational axis is not collinear with that axis, so that the great circle of the equator intersects the ecliptic plane at two points, one of which can be used as a reference to \"track the progress\" of precession. Clearly, the progress of that point along the ecliptic — i.e., the change in the intersection's ecliptic longitude, $\\Delta\\lambda$ — has a direct one-to-one relationship with the rate of precession. What's not clear to me is how this relates to the corresponding change in right ascension, $\\Delta\\alpha$. But what's being measured in that case is the distance between the reference point and some initial point on the ecliptic, within the equatorial system, an arc that passes through different declinations and thus $\\Delta\\alpha$ values that vary from $\\Delta\\lambda$ as described above. (Moreover, this seems to me the only way that $\\langle\\Delta\\lambda\\rangle$ and $\\langle\\Delta\\lambda\\alpha\\rangle$ to be the same, which must be the case for the method described in (3) to work.) (3) I'm reasonably confident that the first is true, but I'm not sure how to arrive at the second; though it must also be true, since otherwise it would not be possible to determine the precessional period (in sidereal days) from $24^\\text{h}\/\\langle\\Delta\\alpha_{SID}\\rangle$, as is commonly done. (4) The corresponding (by definition, assuming constant rotational rate for the earth) _constant_ quantities for the stellar day must thus be $$\\Delta\\lambda_{ST}=\\langle\\Delta\\lambda_{SID}\\rangle\\cdot\\frac{d_{ST}}{d_{SID}}=\\Delta\\alpha_{ST}=\\langle\\Delta\\alpha_{SID}\\rangle\\cdot\\frac{d_{ST}}{d_{SID}}$$ where ${d_{SID}}$ is the length of a mean sidereal day, and ${d_{ST}}$ the length of a stellar day. (5) Or perhaps a _mean_ mean sidereal day. (A very mean sidereal day?)"}
{"id":"5702","title":"Can we transport energy over infinite distances through vacuum using light?","text":"I know that light (or electromagnetic radiation in general) attenuates in intensity as the square of the distance it travels.   * Why does it attenuate?    * Are the photons being scattered by the medium they pass through? I also know that the energy carried by light _quanta_ is a function of their frequency only.   * If the answer to B is yes, then can we possibly transmit light energy (and information) over infinite distances through a perfect vacuum? I may be wrong on some assumptions, so please enlighten me."}
{"id":"88841","title":"Deriving expression for motional emf","text":"I'm having a trouble deriving the expression for emf caused by the Lorentz forces in a moving conductor. Consider the following classical example of a slidewire generator which consists of U-shaped conductor and a rod which is able to slide on the conductor. The system is placed in a uniform non-varying magnetic field. ![Illustration](http:\/\/i.stack.imgur.com\/FifpA.jpg) I know that because the rod is moving with a certain speed, the charge carriers (i.e. electrons) are also moving, and because of that they experience Lorentz force, which eventually causes them to concentrate on one end of the rod and leaving the positive ones at the other. The force itself is equal: $$|\\mathbf{F}|=q |\\mathbf{v} \\times \\mathbf{B}|= q v B \\sin \\theta$$ (I include the sine on purpose, for the general case) We can describe a field: $$|\\mathbf{M}|=\\frac{|\\mathbf{F}|}{q}=|\\mathbf{v} \\times \\mathbf{B}|= v B \\sin \\theta$$ The work done by the non-electrostatic Lorentz force on unit of charge is: $$\\mathcal{E}=|\\mathbf{M}| L=L |\\mathbf{v} \\times \\mathbf{B}|= L v B \\sin \\theta$$ (where $L$ is the length of the rod) We can multiply this expression by $\\frac{\\mathrm{d}t }{\\mathrm{d} t}$: $$\\mathcal{E}=|\\mathbf{M}| L \\frac{\\mathrm{d}t }{\\mathrm{d} t}=L |\\mathbf{v} \\times \\mathbf{B}|\\frac{\\mathrm{d}t }{\\mathrm{d} t}= L v B \\sin \\theta\\frac{\\mathrm{d}t }{\\mathrm{d} t}=B \\sin \\theta \\frac{L v~\\mathrm{d}t }{\\mathrm{d} t}$$ Because $L v ~ \\mathrm{d} t = \\mathrm{d}S$ we can rewrite it as: $$\\mathcal{E}=B \\sin \\theta \\frac{\\mathrm{d}S }{\\mathrm{d} t}$$ So because of the Lorentz force we have a sine instead of a cosine like in Faraday's law $$\\frac{\\mathrm{d} \\Phi}{\\mathrm{d} t}=\\frac{\\mathrm{d} (B S \\cos \\theta)}{\\mathrm{d} t}$$ as we would expect. Where is my mistake here?"}
{"id":"88844","title":"A Global Measure on a Gravitational System","text":"I am a computer science engineer and I am trying to simulate the gravity on my computer; several objects with different masses randomly distributed in a Three-dimensional space. I would like to compute a global measure (scalar) that describes the dynamic of that system at each instant T. My question is what kind of measure can I use?"}
{"id":"103347","title":"Does quantum mechanics require classical measurement apparatus?","text":"I am trying to learn quantum mechanics and I have a question. Landau, in his quantum mechanics book says that it is in principle impossible to formulate basic concepts of quantum mechanics without using classical mechanics. But I learned a bit about consistent histories and I think it makes sense to talk about closed quantum mechanical systems using this language if I didn't understand something wrong. Then I have listened to a lecture video of a very famous physicist (Milner prize winner) who says that quantum mechanics does not make sense if you can't go out of the system and measure it using an infinite apparatus (he considers decoherence and when you calculate expectation value of an observable you see that you can talk about classical probabilities, that is non-diagonal terms disappear). Hence wavefunction of universe does not make sense. I believe what he says is actually same as Landau. Is it meaningful to talk about closed systems which does not interact with outside such as universe? What does framework of consistent histories says about this?"}
{"id":"103340","title":"AdS3 soliton of Witten - for Hawking-Page transition","text":"> Are there explicit AdS$_3$ soliton solution? in the sense of Witten's Anti De Sitter Space And Holography and Hawking-page transition paper, by doing a $$\\tau_E, y ,r \\to y, \\tau_E ,r$$ from a geometry with a Euclidean time $\\tau_E$ cigar space and a compactified $y$ circle ![enter image description here](http:\/\/i.stack.imgur.com\/UshaH.png) to a geometry with a $y$ cigar space and a compactified Euclidean time $\\tau_E$ circle ![enter image description here](http:\/\/i.stack.imgur.com\/JmbBC.png) > If yes, with a AdS3 soliton solution, what are the form of solutions? > > If not, without a AdS3 soliton solution, what are the obstructions?"}
{"id":"43492","title":"Does high entropy means low symmetry?","text":"According to Bogolubov postulate (various texts name it differently) in Non- equilibrium thermodynamics, the number of needed parameters to describe our system is decreasing with time, and finally at infinity, once our system reaches its equilibrium state, we will need only the usual three thermodynamic parameters $T,V,N$ to define the state. Ok, this seems to be very logical, however, how does entropy fit in there? Generally speaking we can say that entropy describes the disorder in our system, and the higher the entropy, the less order we have, and more disorder means less symmetry in the system. (Or is this statement wrong?)   And when a system has high symmetry, we need less parameters to describe it, which uncovers a contradiction than I can't understand: A system has reached equilibrium state, when the system's entropy reaches its maximum (we are talking here about isolated systems only), but maximum entropy means maximum disorder, maximum disorder means the need for a lot of parameters to describe it due to luck of order. But that is the opposite of what the postulate above states. So where is the mistake in this statement? (I asked my professor and he answered that defining entropy in a non-equilibrium state is, for now, an open question but I think that the answer should lie somewhere else...) (in case there is really some relation between Entropy and Symmetry, please provide some math how this is accomplished)"}
{"id":"43497","title":"Understanding Work and the conservation of energy","text":"_We have a car with a mass of $780 kg$ with travels with a speed of $50 km\/h$. The car brakes and after $4,2m$ is stops completely. Warmth is created. Calculate the friction._ I solved this easily, by simply filling in the data like a headless chicken (the solution I'm showing is from the correction model, I got the same answers but I did it without writing anything down, so): $E_{total 1} = E_{total 2} $ $0.5mv_1^2 = 0.5mv_2^2 + Q = 0+ F_w . s $ $0.5 . 780 . (50\/3.6)^2 = F_w . 4.2 $ $F_w = 1,8.10^4 N$ I didn't have any trouble with this, I got the same answer, but than I started thinking about it, and I am increasingly finding the solution illogical. The LHS is completely logical to me, but the right hand side. Normally, the formula of work is the resultant force times the distance d. But, after you've travelled the 4,2 meters, your $F_w$ is $0$. So how can you say that $F_w \\times d$ = LHS, because by the time the total $d$ (4.2) is reached, the resistance has already turned into 0. What is the logic behind this? I know this is high school level so it is simplified, but even then, knowing it's simplified a lot, I don't understand the logic. Can someone explain?"}
{"id":"21346","title":"Equipartition theorem for flowing gas","text":"If an ideal gas is flowing with a velocity $v$, how is the equipartition theorem applied. Normally, we can say that $\\frac{1}{2}mv_{x,rms}^2=\\frac{1}{2}k_BT$. We can do the same thing for $v_y$ &c. But, when a gas is flowing in the x direction, I don't think that $v_{x,rms}=v_{y,rms}$. I'm not too sure of this, the distribution may be such that the rms velocities are preserved. If the rms velocities aren't preserved, obviously we cannot use $\\frac{1}{2}k_BT$ as the temperature is the same (or is it?). So how does one analyse such a situation with the equipartition theorem? I'm not very good at Hamiltonian mechanics, so Wikipedia isn't helping. Question sparked off by Air velocity in a double-skin facade"}
{"id":"80718","title":"Derivation of the Hartree-Fock equations. Functional varitation","text":"I asked this question at chemistry.stackexchange.com, but the attendance of that source is a little bit lower than here. I would like to ask a question about mathematical derivation of the HF equations. Some moments related to the functional variation are not clear for me. I read a book of Szabo and Ostlund \"Modern quantum chemistry\". There the procedure of the Single determinant energy minimization is presented. Omitting the whole procedure i present the most obscure moment. Given the single determinant $| \\Psi_0 \\rangle = | \\chi_1 \\chi_2 \\ldots \\chi_a \\chi_b \\ldots \\chi_N \\rangle$, the energy $E_0 = \\langle \\Psi_0 | \\mathscr{H} | \\Psi_0 \\rangle$ is a functional of the spin orbitals $\\\\{ \\chi_a \\\\}$. $E_0$ is the expectation value of the single determinant $\\vert \\Psi_0 \\rangle$, $E_0[\\\\{ \\chi_a \\\\} ] = \\sum\\limits_{a=1}^N [a\\vert h \\vert a] + \\frac{1}{2} \\sum\\limits_{a=1}^N \\sum\\limits_{b=1}^N [aa \\vert bb] - [ab \\vert ba]$, Here $[ij|kl]$ is the chemist's notation for a two-electron integral involving spin orbitals: $[ij|kl] = [\\chi_i \\chi_j | \\chi_k \\chi_l] = \\int d\\mathbf{x_1} d\\mathbf{x_2} \\chi_i^{*}(\\mathbf{x_1}) \\chi_j(\\mathbf{x_1}) r_{12}^{-1} \\chi_k^{*}(\\mathbf{x_2}) \\chi_l(\\mathbf{x_2}) $ And one-electron integral is $[i|h|j] = \\int d \\mathbf{x_1} \\chi_i^{*}(\\mathbf{x_1}) h(\\mathbf{r_1}) \\chi_j(\\mathbf{x_1})$ This equation can be varied (1) $\\delta E_0 = \\sum \\limits_{a=1}^N [\\delta \\chi_a | h | \\chi_a] + [\\chi_a | h | \\delta \\chi_a] $ $+ \\frac{1}{2} \\sum \\limits_{a=1}^N \\sum \\limits_{b=1}^N [\\delta \\chi_a \\chi_a | \\chi_b \\chi_b] + [ \\chi_a \\delta \\chi_a | \\chi_b \\chi_b] + [ \\chi_a \\chi_a | \\delta \\chi_b \\chi_b] + [\\chi_a \\chi_a | \\chi_b \\delta \\chi_b]$ $-\\frac{1}{2} \\sum \\limits_{a=1}^N \\sum \\limits_{b=1}^N [\\delta \\chi_a \\chi_b | \\chi_b \\chi_a] + [ \\chi_a \\delta \\chi_b | \\chi_b \\chi_a] + [ \\chi_a \\chi_b | \\delta \\chi_b \\chi_a] + [\\chi_a \\chi_b | \\chi_b \\delta \\chi_a]$ Authors suggest a reader as an exercise to manipulate this equation for $\\delta E_0$ to show that (2) $\\delta E_0 = \\sum \\limits_{a=1}^N [\\delta \\chi_a | h |\\chi_a] + \\sum \\limits_{a=1}^N \\sum \\limits_{b=1}^N [\\delta \\chi_a \\chi_a | \\chi_b \\chi_b] - [\\delta \\chi_a \\chi_b | \\chi_b \\chi_a] + \\text{c.c.}$ It is clear that the first sum in (1) can be easily converted to that in (2) because $[\\delta \\chi_a | h | \\chi_a]^* = [\\chi_a | h | \\delta \\chi_a]$ Analogously for the second sum in (1) one can show that $[\\delta \\chi_a \\chi_a | \\chi_b \\chi_b]^* = [\\chi_a \\delta \\chi_a | \\chi_b \\chi_b]$ and $[\\chi_a \\chi_a | \\delta \\chi_b \\chi_b]^* = [\\chi_a \\chi_a | \\chi_b \\delta\\chi_b]$ I did the same manipulations with the third sum in (1) and obtained (3) $\\delta E_0 = \\sum \\limits_{a=1}^N [\\delta \\chi_a | h | \\chi_a] + $ $+ \\frac{1}{2} \\sum \\limits_{a=1}^N \\sum \\limits_{b=1}^N [\\delta \\chi_a \\chi_a | \\chi_b \\chi_b] + [ \\chi_a \\chi_a | \\delta \\chi_b \\chi_b] $ $-\\frac{1}{2} \\sum \\limits_{a=1}^N \\sum \\limits_{b=1}^N [\\delta \\chi_a \\chi_b | \\chi_b \\chi_a] + [ \\chi_a \\delta \\chi_b | \\chi_b \\chi_a] + \\text{c.c.}$ So, comparing (3) and (2) i suppose that one should show that both terms in each sum are equal, i.e. $[\\delta \\chi_a \\chi_a | \\chi_b \\chi_b] = [ \\chi_a \\chi_a | \\delta \\chi_b \\chi_b]$ and $[\\delta \\chi_a \\chi_b | \\chi_b \\chi_a] = [ \\chi_a \\delta \\chi_b | \\chi_b \\chi_a]$ So, how to do that?"}
{"id":"52223","title":"Finding optimal angle for projectile, taking into account linear (Stokes) drag","text":"If you throw a projectile from the ground at a certain angle, it's not hard to see that, assuming we're in a vacuum, throwing it at 45º from the ground will always make it go farthest before it hits the ground again. I wanted to find out what's the optimal angle if we take into account air resistance. Before I go on, let me say that there's really no point to this question. I'm not trying to solve a practical problem. I just thought it would be fun to try to solve the equations. This is the model I'll use: We'll assume that air resistance is roughly proportional to velocity (aka. Stokes drag; this fails for high speeds\/Reynolds numbers, so let's not go there); in symbols, $$\\mathbf{F} = -k\\mathbf{v}.$$ We'll throw the projectile with mass $m$ from $(0,0)$ with initial speed $v_0$, forming an angle $\\alpha$ with the ground, and let's say $\\gamma = k\/m$ because it will turn up frequently. The differential equations are: $$\\begin{align} \\ddot{x} + \\gamma \\dot{x} &= 0 \\\\\\ \\ddot{y} + \\gamma \\dot{y} &= -g \\end{align}$$ with initial conditions $(x(0), y(0)) = (0,0)$ and $(\\dot{x}(0), \\dot{y}(0) )= v_0(\\cos \\alpha, \\sin \\alpha)$. Solving (assuming I haven't made any mistakes), we get: $$ \\begin{align} x &= \\frac{v_0}{\\gamma}\\cos \\alpha (1- e^{-\\gamma t}) \\\\\\ y &= (\\frac{v_0}{\\gamma} \\sin \\alpha + \\frac{g}{\\gamma^2})(1-e^{-\\gamma t}) - \\frac{g}{\\gamma}t. \\end{align} $$ Now the thing to do would be to solve $y(t) = 0$, substitute that into $x$, differentiate with respect to $\\alpha$ and set it equal to $0$. However, that gets messy quickly, because the solution involves the Lambert-W function and it's all a mess. I haven't even tried to substitute into $x(t)$. So finally, my question is: is there a simpler or numerical way to solve this? Is there even a single angle which will always work, or does it depend on the conditions? Can we find that out without actually solving?"}
{"id":"90597","title":"What will happen to a compressible fluid filled inside a sealed container when the container begins to accelerate?","text":"Assume the fluid to be air and the container to be a car, will there be a pressure gradient which will result the fluid to stack at one end of the car or there is something else that I'm missing?"}
{"id":"88653","title":"General wavefunction and Schrödinger Equation","text":"I'm starting with quantum mechanics and the book I follow (Griffiths) first introduces the wavefunction as the probability density of the position of a 0-spin single particle. Later on I've realized that there is a bigger picture: the state of a system can be expressed as a superposition, which is a linear combination of the vectors of a particular basis, (depending on the observable we choose), with a (maybe continuous) set of coefficients that are probability amplitudes of the eigenvalues. If the chosen observable happens to be the position we obtain the wavefunction. So now I'm wondering the following: is this the definition of a wavefunction or is there a wavefunction for every observable, being this wavefuntion just a map from the possible values of the observable to their probability amplitudes? If so: is the time-dependence of this wavefunction always restricted by the Schrödinger Equation? For example, suppose that in the case of a 2-dimensional system with eigenstates |0> and |1> for a certain observable, with respective eigencalues 0 and 1. We have the superposed state f(0,t)|0> + f(1,t)|1>, where f(state, time) is the \"wavefunction\" that assigns probability amplitude to each eigenstates. Does this f function have to satisfy the Schrödinger Equation or other constraints?"}
{"id":"17188","title":"What is the status of gravitational wave searches?","text":"What is the status of gravitational wave searches such as LIGO?"}
{"id":"75635","title":"link between real particles, representation of algebra and Young tableau","text":"I know that different representations of this algebra correspond to different spin. One can sort the representation according to the casimir. For any simple Lie algebra, the operator $$ T^2 = T^{\\alpha}T^{\\alpha} $$ comutes with all the group generators such that $$ [T^2,T^{\\alpha}]=0 $$ So $T^2$ takes a constant value on each irreductible representation. Thanks to Shur's lemma, one knows that $T^2$ is proportional to the identity for all representation. If one considers the rotation algebra $o(3)$, the Casimir takes the form $$ C=\\delta^{ij}J^iJ^j=s(s+1) $$ where $J^i$ are the generators of $o(3)$ and $s$ is a half integer that labels each representation of dimension $$ d = 2s + 1 $$ For instance :   * $s=0 \\Rightarrow d=1$ and $S_i=0$   * $s=1 \\Rightarrow d=2$ and $S_i=\\frac{1}{2}\\sigma_i$   * $s=2 \\Rightarrow d=3$ and $S_i=\\Sigma_i$   * ... where the $S_i$ are matrix representations of $J^i$ and $\\sigma_i$ are the Pauli matrix. Moreover there is an isomorphism between $su(2)$ and $o(3)$. $\\\\\\$ Now if one looks at the Lorentz algebra, $o(1,3)$, one can do the same work but this time, there are two Casimirs that can be written $$ C_{\\pm}=s_\\pm(s_\\pm+1) $$ Thus one has :   * $(s_+,s_-)=(0,0)\\Rightarrow d=1$ scalar   * $(s_+,s_-)=(1\/2,0)\\Rightarrow d=2$ spinorial chiral (not sure of the english terminology)   * $(s_+,s_-)=(0,1\/2)\\Rightarrow d=2$ spinorial antichiral    * $(s_+,s_-)=(1\/2,1\/2)\\Rightarrow d=2$ vectorial    * ... And again there is an isomorphism between $o(1,3)$ and $sl(2,\\mathbb{C})$. $\\\\\\$ **So my question is :** what is the correct algebra and representation to describe a spin one-half fermion (an electron for instance) ? From what I understand I could choose between $su(2)$, $o(3)$, $o(1,3)$ or $sl(2,\\mathbb{C})$ but with a given representation. Same question for a spin one boson ? higher spins... ? $\\\\\\$ My guess would be that :   * spin 1\/2 fermion is described by the $su(2)$ in the fundamental representation (labeled by a single box within the Young tableau formalism)   * spin 1 boson is described by the $su(2)$ with the symmetrical combination of two fundamental representation (two box in a single row)   * spin 3\/2 is described by the $su(2)$ with the symmetrical combination of two fundamental representation and one antisymmetrical one(two boxes in the first row and one box in the second) $\\\\\\$ I know that this question is confused... and so am I... My goal is to undersand the link between the physical particles, the $su(N)$ algebra (for any $N$), the representation of the algebra and the Young tableaus. So instead of answering the question above, it might be easier to explain my those links... $\\\\\\$ Thanks !"}
{"id":"64122","title":"Coherent $U(N)$ intertwiners in Loop Quantum Gravity (LQG) and a measure on the Grassmannian","text":"This is a detailed question about $U(N)$ intertwiners in LQG, and it comes from the the paper by Freidel and Livine (2011 - archive). It is very specific but related to finding a measure on a quotient space so I hope it will get more general mathematical physics interest. We start by casting LQG into the spinor language. Most of the details are not necessary, but we have a set of $N$ spinors $$|z_i\\rangle =\\left(\\begin{array}{c} z^0_i\\\\\\ z^1_i\\end{array}\\right).$$ In addition to the usual spinor relations, they satisfy _the closure relation_ , which is $$\\tag{23} \\sum_i |z_i\\rangle\\langle z_i| =A(z)\\mathbb{1}, \\qquad A(z):=\\frac{1}{2}\\sum_i\\langle z_i|z_i\\rangle.$$ Now the states in the coherent intertwiner framework are holomorphic functions of $\\mathbb{C}^{2N}$ that are homogenous functions under $GL(2,\\mathbb{C})$ transformations. Thus, they are elements of a Grassmanian $$\\tag{81} Gr_{2,N}=\\mathbb{C}^{2N}\/GL(2,\\mathbb{C}).$$ Now, we want a measure on this Hilbert space (and a resolution of unity) so we start by relating the measures on the three spaces above (Eq (93) in that paper): $$\\int_{\\mathbb{C}^{2N}}\\left(\\prod_jd^4z_j \\right)e^{-\\sum_k\\langle z_k|z_k\\rangle}F(z)=\\int_{GL(2,\\mathbb{C})}\\\\!\\\\!\\\\!\\\\!\\\\!\\\\! d\\lambda~ e^{-Tr(\\lambda^\\dagger\\lambda)}|\\det(\\lambda)|^{2(N-2)}\\int_{Gr_{2,N}}\\\\!\\\\!\\Omega(z)F(z) .$$ 1) Here is my first question; **Why does the measure on $\\mathbb{C}^{2N}$ have the exponential?** Is it there because we are looking at coherent states? I am also a bit confused about the measure on $GL(2,\\mathbb{C})$, but for the moment I'm ok just thinking it probably comes from the Haar measure and not the main point of my question. The next thing they do in that paper is to say the measure on $Gr_{2,N}$can be represented as $$\\tag{94} \\Omega(z)=\\left(\\prod_j d^4z_j\\right) \\delta^{(4)}\\left(\\sum_i|z_i\\rangle \\langle z_i|\\right).$$ Then they go on to find the volume of $Gr_{2,N}$ as $\\frac{\\pi^{2(N-2)}}{(N-1)!(N-2)!}$ which gives a normalized measure on the Grassmanian. 2) So the next question is **How does this measure jive with the closure constraints?** If we perform the integration over the Grassmanian we enforce the constraint $$\\sum_{i}|z_i\\rangle\\langle z_i|=0\\rightarrow A(z)=0.$$ While that doesn't directly contradict the closure constraints, it does seem like it is adding an extra condition which is not implied by anything else. Can someone help me understand this measure? (and extra points for anything about that $GL(2,\\mathbb{C})$ part.)"}
{"id":"52591","title":"why do lightbulbs sometimes unscrew by themselves?","text":"I've seen some light fixtures which exhibit the behavior of lightbulbs gradually unscrewing by themselves."}
{"id":"119950","title":"Why do we need high energy to explore small dimensions?","text":"I am taking a quantum physics class, and for the life of me, I can not remember why we would need a vast amount of energy to understand the microscopic universe."}
{"id":"9364","title":"Does opening or closing the window in a non-AC car in the summer affect how much heat is felt inside?","text":"This will perhaps look like a very basic and trivial question. But I find it confusing. As an experience, when you are travelling in non-AC car in summer, have people felt if putting on or putting off the windowpane affect how much heat it feels inside to passengers. On one hand, I feel that heat can still come inside thru glass particles (This makes me ask if heat needs media to travel?) and so putting on windowpane should not reduce heat felt inside. But psychologically it feels like by closing the window, you are blocking heat. Also the windowpane will at least block the heated air from coming inside. But does this heated air really brings much heat apart from the heat coming through anyway or it actually in practical acts more like a ventillator thus reducing heat felt inside. I realize that this question may appear strange, but it pops up to my head everytime I travel in non-AC car in summer. I have tried to articulate it as well as I could. Do insulator windowpane glasses reduce heat felt inside? because they will transmit lesser heat inside. How do black shade glasses work?"}
{"id":"8981","title":"Group Cohomology and Topological Field Theories","text":"I have a two-part question:   1. First and foremost: I have been going through the paper by Dijkgraaf and Witten \"Group Cohomology and Topological Field Theories\". Here they give a general definition for the Chern-Simons action for a general $3$-manifold $M$. My question is if anyone knows of any follow-up to this, or notes about their paper?   2. To those who know the paper: They say that they have no problem defining the action modulo $1\/n$ (for a bundle of order $n$) as $n\\cdot S = \\int_B Tr(F\\wedge F)$ $(mod 1)$, but that this has an $n$-fold ambiguity consisting of the ability to add a multiple of $1\/n$ to the action - What do they mean here? Also, later on they re-define the action as $S = 1\/n\\left(\\int_B Tr(F\\wedge F) - \\langle \\gamma^\\ast(\\omega),B\\rangle\\right)$ $(mod 1)$ - How does this get rid of the so-called ambiguity?  Basically my question is if anyone can further explain the info between equations 3.4 and 3.5 in their paper. Thanks."}
{"id":"8980","title":"Half-massive sphere in space","text":"A sphere contains all the mass on one side while the other side is empty of matter. Both sides are separated by a strong material, you name it, that will keep the sphere together. Given that the mass side is more massive than the sun or at least very massive, how would it behave in respect to gravity? Would it spin?"}
{"id":"45537","title":"Example of two linearly independent, nowhere vanishing vector fields in $\\mathbb{R}^{2}$","text":"I knew that two linearly independent and nowhere-vanishing vector fields provide a basis for the tangent space at each point in $\\mathbb{R}^{2}$. Is it necessary that these two vector fields commute? Would you give me an example for these two vector fields?"}
{"id":"109120","title":"Electric potential energy and signs","text":"I know that electric potential is negative near a negative charge and positive near a positive charge. But does this mean a small positive 'test' charge has a negative electric potential energy near a negative charge and a small negative charge would have a positive?"}
{"id":"40738","title":"Reflection of light","text":"> **Possible Duplicate:**   >  What is the difference between a white object and a mirror? When I look at a red object under white light, I see it as red because it absorbs the other colours and reflects the red light. When I wear black clothes in summer, I get hotter than if I wear white clothes because the black absorbs all colours of light and the energy associated with it. So my question is, what makes the difference between a white piece of paper and a mirror? The white paper doesn't absorb any colours, that's why it looks white. But what makes that different from a mirror, which also reflects the light that hits it? Or is my basic understanding of light and colour wrong? I'm not exactly experienced in Physics, I just like knowing how some things work."}
{"id":"1952","title":"Why is the ground state of the ferromagnetic tetrahedron threefold degenerate?","text":"I'm preparing a presentation on Spin-Ice, but something's been bugging me for a while. On the Wikipedia page for Geometrical Frustration, it says the following about easy spins on a tetrahedron with ferromagnetic interactions: > There are three different equivalent arrangements with two spins out and two > in, so the ground state is three-fold degenerate. ![](http:\/\/i.stack.imgur.com\/7CRYL.png) I just don't understand why.   * We're considering easy spins with ferromagnetic interactions, so any ground state needs 2 spins pointing towards the center (in) and 2 spins pointing away from the center (out).    * Given a tetrahedron, I can think of 6 different ways to distribute 2 spins in and 2 spins out on the vertexes. $(iioo,ioio,oiio,iooi,oioi,ooii)$    * You can rotate any of the above states to reproduce another, which might mean they are not different.  **But then, why does wiki state there are three _different_ arrengements?** **Note** Each of the above configurations has a total magnetic moment in a different direction. We can place our $(x,y,z)$ axis so that each one of the first three cases has a total moment in the positive direction of one of the axis, and each of the last three has a total moment in the negative direction of one of the axis."}
{"id":"49985","title":"How a spacecraft travelling near light speed avoid asteroids?","text":"How would a spacecraft traveling near light speed avoid a (relatively dense group of) asteroids? Or suppose such spacecraft is designed, how would the physics work for steering it inside such a \"cloud\" of asteroids? If it simply by designing the spacecraft to have huge acceleration, wouldn't that mean lots of constraints on the material the spacecraft can be made of?"}
{"id":"59819","title":"Vacuum to vacuum transition amplitude","text":"I have two questions about Vacuum to vacuum transition amplitude.   1. Can any particle stay in $|0\\rangle$?   2. I was studying this topic from Srednicki's QFT book. He writes in eq.$(6.22)$ $$\\langle0|0 \\rangle_{f,h} = \\int \\mathcal{D}p \\mathcal{D}q \\, \\exp \\Big[i\\int_{-\\infty}^\\infty dt (p \\dot{q} - H_0(p,q) - H_1(p,q) + fq + hp \\Big]$$ $$ =\\exp \\Big[-i \\int_{-\\infty}^\\infty dt H_1 \\Big(\\frac{1}{i} \\frac{\\delta}{\\delta h(t)}, \\frac{1}{i} \\frac{\\delta}{\\delta f(t)} \\Big) \\Big] \\times \\int \\mathcal{D}p \\mathcal{D}q \\,\\exp \\Big[i\\int_{-\\infty}^\\infty dt (p \\dot{q} - H_0(p,q) + fq + hp \\Big] $$ Here in the second line the arguments of $H_1$ suddenly changes from $p$,$q$ to $\\frac{1}{i} \\frac{\\delta}{\\delta h(t)}, \\frac{1}{i} \\frac{\\delta}{\\delta f(t)}$. How can I derive this line from the first line?"}
{"id":"47611","title":"Level of calculus required for physics","text":"First time for me here so kindly let me know if I violate the rules - especially if this is a duplicate. After reading the page how to become a good theoretical phycist, I started a serious revision of calculus. For exercises, Math.SE is a good place. I was attempting exercises from this user (question s\/he asked and answered) and realized I couldn't solve the majority of them. So the question is what is the level of mastery of calculus required for physics? What is(are) the best book(s) for that? I'm interested in limits, integration and infinite series for now. Background: standard mathematics with limits, integration, infinite series, differential equations (ODEs and PDEs) and numerical analysis and optimization. Everything was fast paced because professors wanted to finish the program as fast as possible ( during 3 semesters ) so I really never got good at any of the listed above.   Note: self-study kind of books will be most helpful."}
{"id":"90047","title":"Physical Interpretation of Poynting Vector","text":"I'm looking for a physical interpretation of the Poynting Vector. I understand that it should be thought of as an energy flow due to the electromagnetic field, but would I be correct in saying that in some sense it's equivalent to a photon flow\/packet? After all, for an electromagnetic wave, the poyntig vector yields the direction of propogation, or the direction in which the photon is moving. Does this analogy apply in general?"}
{"id":"59813","title":"Real world usage; pump flow vs pressure vs pump head, what is more useful in this situation?","text":"I am currently making a water cooling system for my computer and decided to see if I could get some better help here than what people think they know on the computer forums. So for my cooling system I have three, what are called, water blocks. They have a inlet and an outlet and are the biggest source of restriction. I specifically have three, and they each have 1\/4in input and output barbs, I then connect them all with two manifolds. The manifold is made of 3 PVC T fittings and an end cap. So all the T's are setup such to look like this TTT . One of the ends of the tops on the T's is capped and the other is the input\/output, while the bottom of the T's goes to and from the water blocks. (I hope this is very easy to visualize, if you cannot ask me and I will try to explain more) The input for each manifold is 1\/2in tubing and the T's run to x3 1\/4in tubing lines. So the water enters in a 1\/2in tube, is split to x3 1\/4in tubes, then combines back into a 1\/2in output tube. Each waterblock is responsible for, at max, 190watts of heat to dissipate. So the entire loop must remove ~600watts of heat + the pumps heat = ~625 watts. This is where the radiator comes in, I found a nice chart that tells me the size I need to remove X watts from a system so I know I have the right size radiator. The radiator has 1\/2in tubing inputs and outputs (but it is restrictive cause of the inside geometry. So my question is, what should I consider in a pump when trying to find one for this system? People talk of flow rate, vs system pressure, vs pump head. I know the head is how hard the pump can push with no restriction in its flow. So I figure pressure is combined in the pump head figure. What no one talks about though is how to find out how much flow and pressure is needed for the water cooling loop to work as it should. Even if I have the right size radiator to remove the amount of heat I have it is worthless if the water is not moving in the right manner. So can I possibly buy a pump that has too much flow; how can I figure the minimum amount of flow needed for my loop to work? EDIT: Ok I have found out a little bit more.. So I figure part of what I need to know is the bernoulli equation, which will describe the flow and pressure of the water at a certain point in the system. What I guess I need to do is equate the amount of heat water can remove at a specific velocity. So each water block gives off 190watts, and water has a thermal conductivity of .6Watts\/m*K at 0C. How do I equate these two figures to figure out how much flow and head my pump should have?"}
{"id":"22080","title":"Does a ski racer with a larger mass have an advantage?","text":"Does a ski racer with a greater mass have an advantage over a racer with a lesser mass? If mass of one racer is 54 kg and the mass of a more slender racer is 44 kg I know the speed at which they descend should be equal if they were to fall in a vacuum. How does friction force, air resistance and momentum play a part in determining the advantage or disadvantage of larger mass in ski racing?"}
{"id":"56276","title":"Skiing downhill","text":"The other day on skiing holiday we've been arguing about whether an adult has weight advantage over a child when skiing downhill. I was claiming that gravity is a constant regardless of object's weight. I was claiming that weight doesn't affect skier's speed. I can still remember primary school experiment where we dropped same sized objects of different matter in vacuum. Styrofoam and iron ball of the same size and they both fell equally. It is true that skier's maximum speed is determined by his air drag (and snow\/ski resistance) and heavier objects have higher persistence making them move through resistive medium easier. But compared to a child an adult would have more drag due to larger _surface_ so I suppose this somehow evens out. I was basically claiming that weight doesn't matter at all but that other factors are much more important when it comes to skier's speed like: ski stiffness (weight force much better distributed over the whole ski surface than with softer skis) and ski sliding surface smoothness and especially skier's position (to reduce drag as much as possible). I also got a bit confused when I started to think about force vectors where greater weight results in longer vectors resulting in stronger movement pull. ## Question If we approximate is it safe to say that a child can ski just as fast as an adult? Or that a thinner adult would ski faster than his same high companion that is overweight due to larger drag surface? Or does his weight benefits his speed? **What is the correct answer here?**"}
{"id":"89936","title":"How do I find angular and linear velocity after normal force and \"infinite\" friction force?","text":"Look at this picture: http:\/\/i.imgur.com\/mY8ShkV.png Here we have a ball resting on top of a platform (the contact point is marked red). I know the normal vector (marked green). I also know the mass, radius, and current angular and linear velocity of the ball. Let's say I set the new angular velocity after friction which is always big enough (\"infinite\") has been applied to: $V_a = V_l \\times rN$ where $V_l$ is the linear velocity, $r$ is the radius of the ball and $N$ is the normal vector. Does this math check out, is this correct? Because this model doesn't work if I leave the ball leaning on the side of the platform where it should obviously roll of the platform and fall. But it doesn't. What is the correct mathematical model for simulating \"infinite\" friction? I suppose you will also have this problem when you want to simulate finite friction and the friction doesn't give it's max force."}
{"id":"45514","title":"About free quarks and confinement","text":"I simply know that a single free quark does not exist. What is the reason that we can not get a free quark? If we can't get a free quark then what is single-top-quark?"}
{"id":"28529","title":"How \"to take\" this integral?","text":"When I learned anharmonic model of crystal, I read that considering anharmonic oscillations and Boltzmann distribution for the \"atoms\" of crystal we can get the dependence of distance between the \"atoms\" from a temperature as $$ \\langle r \\rangle = r_{0} + \\alpha T. $$ As I understood the words below, it's like $$ \\langle r \\rangle = \\frac{\\int \\limits_{0}^{\\infty}re^{-\\frac{U}{kT}}dr}{\\int \\limits_{0}^{\\infty}e^{-\\frac{U}{kT}}dr} \\approx \\frac{\\int \\limits_{0}^{\\infty}re^{-\\frac{U_{0} + a(r - r_{0})^{2} + b(r - r_{0})^{3} }{kT}}dr}{\\int \\limits_{0}^{\\infty}e^{-\\frac{U_{0} + a(r - r_{0})^{2} + b(r - r_{0})^{3} }{kT}}dr} = |x = r - r_{0}| = \\frac{\\int \\limits_{0}^{\\infty}(x + r_{0})e^{-\\frac{ax^{2} + bx^{3}}{kT}}dr}{\\int \\limits_{0}^{\\infty}e^{-\\frac{ax^{2} + bx^{3}}{kT}}dr}, $$ and then - $$ \\langle r \\rangle \\approx r_{0} + \\frac{\\int \\limits_{0}^{\\infty}xe^{-\\frac{ax^{2} + bx^{3}}{kT}}dr}{\\int \\limits_{0}^{\\infty}e^{-\\frac{ax^{2} + bx^{3}}{kT}}dr}. $$ What can I do on the next step?"}
{"id":"54529","title":"Is the speed of light the ultimate speed limit?","text":"As we all know nothing can go faster than the speed of light as mentioned by most of our pioneer's in physics. But as I was listening to one of the statements of Sir. Stephen Hawkins he stated that at the time of big bang the universe expanded to about a thousand light years in just few trillionth of a second, how it's possible. Are our laws of physics wrong as if they are correct they should be applicable everywhere in every conditions. Please answer thihs question as answer in above link was not about it."}
{"id":"108501","title":"'Warp'- type travel; is it possible?","text":"I just re-read an article about space travel, and the same article claimed that Star Trek-type 'warp' nuclear engines ARE theoretically possible, but that it will take around 500 years for humans to develop and adapt them for use in the engines of space craft. How true is this statement?"}
{"id":"29438","title":"Why do rockets accelerate fastest horizontally?","text":"I've heard that rockets accelerate fastest when travelling horizontally to the ground, not downwards or upwards. Is that true, and why?"}
{"id":"40850","title":"Wavelength and resolution","text":"I'm reading some texts that seam to assume knowledge of light that I'm not too familiar with. How does wavelength of light relate to the minimum distance span that can be observed (i.e. you cannot make a lens big enough to resolve individual atoms), and is this a light phenomena or an intrinsic wave phenomena?"}
{"id":"29432","title":"How do we describe the relationship of balloon size and related lift?","text":"We know that hydrogen and helium are lighter that air. Therefore it is certain that they would create a lifting force when enclosed in a balloon. How do we describe the relationship of balloon size and related lift?"}
{"id":"1227","title":"Calculation of the G-force","text":"I have a formula which is $\\text{G-force} = \\frac{v\\omega}{9.8}$, where $v$ is speed and $\\omega$ is the angular velocity. I've seen on the internet that G-force is actually $\\text{acceleration}\/9.8$. I'm confused as to which formula is correct. For simulating the motion of particle taking a turn, would omega simply be velocity divided by radius of turn? Assuming Cartesian coordinates.   Another funny thing I noticed is that while simulating particle motion, a 7G turn showed up as an almost straight line (while using a constant turn motion model) with a velocity of 900m\/s and time interval of 1second. Am I simulating wrong or is my use of the first equation wrong?"}
{"id":"28385","title":"Why isn't the force modelled which confines excess charge to remain inside a conductor?","text":"Excess charge placed on a conductor distributes itself on the surface of a conductor. But why isn't the force which confines the charge to remain there, modelled in some way?"}
{"id":"126226","title":"Problem on speed of light","text":"As I was reading special relativity, my book says the speed of light is $c$ with respect to any other thing. Does that mean the speed of an individual photon is $c$ even with respect to another photon? I mean, shouldn't the relative velocity be zero?"}
{"id":"129561","title":"How does quantization solve the ultraviolet catastrophe?","text":"I understand how classical physics leads to the UV catastrophe. But I cannot understand how quantization solves it.   * How can quantization prevent the body from radiating a lot of energy? I know this question is similar to many other questions but I need a layman's explanation (without too much mathematical details).   * Finally why does the peak reach zero intensity after a certain frequency? ![enter image description here](http:\/\/i.stack.imgur.com\/2vACu.png) Image source:- Wikipedia"}
{"id":"59362","title":"Are there 2 kinds of photons, one that mediate the electromagnetic interaction and the other the quanta of light?","text":"It is usually said that photons are the force carriers or the mediators of the electromagnetic forces between electric charges. At the same time we know also that electromagnetic waves on the quantum level are described in terms of photons. It seems to me that we have 2 types of photons:   * the photons which mediate the electromagnetic interactions which are offshell hence their masses do not necessarily have to be zero, I suppose    * and the photons which are the quanta of electromagnetic waves which are massless. Am I correct, or am I confused and missing something?"}
{"id":"97802","title":"Force when acceleration is zero","text":"i tried to find it in books but i was unable to do so and i am not able to understand this and I couldn't think of a better example. i repeat. its just an example. if i punch someone's face with my arm having constant velocity, it will mean that the acceleration of my arm is zero. F=ma, therefore force exerted by my arm is zero. Now if the force applied is zero the why does the person whom i punched experiences pain. please help. thanks."}
{"id":"35114","title":"How can a non-conservative field be a scalar multiple of a conservative field?","text":"Okay so I was reading this from University Physics by Freeman and Young and on the topic of inductors as circuit element, they wrote that $\\mathbf{E_c} + \\mathbf{E_n} = 0$ which makes no sense to me Here is an excerpt of the text and underlined the confusion with green ![enter image description here](http:\/\/i.stack.imgur.com\/Ae1lO.jpg)"}
{"id":"35112","title":"Proper notation when working with three Euclidean spatial coordinates in a setting with a time parameter","text":"The How does the Euclidean metric is the symmetry group of Euclidean space. It includes rotations and translations. Say I consider an Euclidean space and a time parameter. How does the Euclidean metric (which in canoncial coordinates is $\\delta_{ij}$, with $i,j\\in \\\\{1,2,3\\\\}$), transform under a Galilean transformation, more specifically a velocity boost $t\\mapsto t,x\\mapsto x-vt$? In a way, from the three dimensional pov, which doesn't see $t$, this is just a translation. So here is what I'd like a derivation for: > $$(t\\mapsto t'=t,\\ x\\mapsto x'=x-vt)\\ \\Longrightarrow\\ > h_{ij}\\equiv\\delta_{ij}\\mapsto\\ ?$$"}
{"id":"78804","title":"Surface Area Relative to Element Distribution","text":"We tend to find heavier elements like iron and others toward the center of celestial bodies. Lighter elements like carbon, hydrogen, and oxygen are almost always towards the surface. That makes sense because heavier elements have a greater gravitational effect and would naturally want to move toward the center. Now think about the chaos that ensued during initial formation of the celestial body. Heavier elements could get jumbled in with lighter elements throughout the depths. Even to this day, we have lighter elements being pushed to the surface through volcanic gases and ocean fissures. **Question** : If elements were not always arranged from heaviest to lightest during creation, would this in anyway introduce volatility in the overall surface area of the celestial body? Imagine the worst case scenario that would be utterly unimaginable: gold and iron wrapped around the surface with oxygen and hydrogen trapped deep within the depths. Would the overall surface area be any different over time as heavier elements pushed their way in and lighter elements rise to the surface? Intuitively, I would say it would change over time, but to what degree and at what rate of change depends on how elements were initially distributed."}
{"id":"32469","title":"Increase the signal-to-noise ratio of photomultiplier","text":"It is a GRE physics problem. A technique would most likely increase the signal- to-noise ratio of a photomultiplier tube is to   1. operate the tube at a lower temperature   2. operate the tube at higher voltage per dynode than usually specified    3. use a radioactive source to saturate the noise level   4. use a retarding potential on the first dynode   5. use a photocathode of lower work function I tried searching for information regarding photomultiplier, but none seems to fit any of the above options."}
{"id":"111746","title":"Why doesn't this model plane fly?","text":"I have been designing a model plane for Design Technology for the past month or so, and today I laser cut my final design and assembled, it then tested it. Upon testing the plane does not get any lift, whereas the previous testing model which was virtually the same did. The plane is built using Balsa Wood, and Assembled with hot glue (I used as little glue as possible to reduce weight :) ) Any Ideas? Images: http:\/\/imgur.com\/a\/fdKxZ"}
{"id":"119179","title":"How to transform material permittivity tensor from Cartesian coordinates to another orthogonal coordinate system?","text":"I have a material specified by a permittivity tensor written in Cartesian coordiantes: $$\\begin{pmatrix} \\epsilon_{xx} & \\epsilon_{xy} & \\epsilon_{xz}\\\\\\ \\epsilon_{yx} &\\epsilon_{yy} & \\epsilon_{yz}\\\\\\ \\epsilon_{zx} & \\epsilon_{zy} &\\epsilon_{zz} \\end{pmatrix}$$ I want to write all my equations in cylindrical coordinates and so I have to write the permittivity tensor in cylindrical coordinates too. How can I do that?"}
{"id":"68902","title":"Why is there an emission of gamma rays?","text":"When a spontaneous radioactive reaction happens, there is an emission of gamma rays (in most cases) What causes this emission?"}
{"id":"66436","title":"Covariant derivative-Differential","text":"I was trying to prove that the derivative-four vector are covariant. This can be proved only if you consider the time and space derivatives to be $\\dfrac{\\partial}{\\partial t^\\prime}=\\dfrac{\\partial}{\\partial t}\\dfrac{\\partial t}{\\partial t^\\prime} + \\dfrac{\\partial}{\\partial x_1} \\dfrac{\\partial x_1}{\\partial t^\\prime}$ $\\dfrac{\\partial}{\\partial x_1^\\prime}=\\dfrac{\\partial}{\\partial t}\\dfrac{\\partial t}{\\partial x_1^\\prime} + \\dfrac{\\partial}{\\partial x_1} \\dfrac{\\partial x_1}{\\partial x_1^\\prime}$ I was trying to understand but I'm afraid I can't.. if oyou for instance add the terms in the second part of the equations you get $\\dfrac{\\partial}{\\partial t}\\dfrac{\\partial t}{\\partial t^\\prime} + \\dfrac{\\partial}{\\partial x_1} \\dfrac{\\partial x_1}{\\partial t^\\prime}=\\dfrac{\\partial}{\\partial t^\\prime} + \\dfrac{\\partial}{\\partial t^\\prime} =2 \\dfrac{\\partial}{\\partial t^\\prime}\\neq \\dfrac{\\partial}{\\partial t^\\prime}$ How is that possible?"}
{"id":"127176","title":"Classification of small extra-solar system bodies","text":"A Small Solar System Body (SSSB) is an object in the Solar System that is neither a planet, nor a dwarf planet, nor a satellite. This encompasses all comets and all minor planets, as well as all asteroids (that are classed as minor planets). I want to know if there is a term for classifying Small Extra Solar System Bodies. What about comets and minor planets in systems other than our own. Do they have a classification? Is there also a recognised categorisation for Small Bodies regardless of whether they are in the solar system or not? In the same way the 'solar system' pertains to our system but a 'planetary system' means any system including but not limited to our own."}
{"id":"60873","title":"A question on the doped Kitaev-Heisenberg model?","text":"Recently, some groups have studied the effects of doping the Kitaev model on honeycomb lattice(e.g.,http:\/\/arxiv.org\/abs\/1109.6681 and http:\/\/arxiv.org\/abs\/1109.4155) and their calculations show the existence of a topological p-wave superconducting phase. My question is as follows: As we know,one of the amazing properties of the Kitaev model is that the low energy physics(ground states properties and excitations above it) of the model does not depend on the signs of the coupling constants in the Hamiltonian. So for the (hole)doped case, is the existence of topological p-wave superconducting ground states still insensitive to the signs of the couplings we choose? Thanks in advance."}
{"id":"75403","title":"What is the the Ehrenfest-Oppenheimer rule on the statistics of composite systems?","text":"Ehrenfest 1931 gives an argument to the effect that the application of the spin-statistics theorem to composite systems is valid, but only as an approximation and under certain conditions. Unfortunately this article is behind a paywall. The abstract reads as follows: > From Pauli's exclusion principle we derive the rule for the symmetry of the > wave functions in the coordinates of the center of gravity of two similar > stable clusters of electrons and protons, and justify the assumption that > the clusters satisfy the Einstein-Bose or Fermi-Dirac statistics according > to whether the number of particles in each cluster is even or odd. The rule > is shown to become invalid only when the interaction between the clusters is > large enough to disturb their internal motion. The quaintness of the abstract is enhanced by the mysterious failure to mention neutrons -- this being explained by the fact that the neutron was not discovered experimentally until the following year. Pretty amusing that the American Physical Society still owns the copyright on a paper this old, one of whose authors was probably paid through my grandparents' and great- grandparents' income taxes. Googling quickly turns up various loose verbal characterizations of this result, without proof. Can anyone supply an outline of how the argument works and possibly a more precise statement of what the result really is? Some sources seem to refer to this as the Wigner-Ehrenfest-Oppenheimer (WEO) result or Ehrenfest-Oppenheimer-Bethe (EOB) rule. Historically, Feynman seems to have used this result to show that Cooper pairs were bosonic. Although I've tagged this question as quantum field theory, and the spin- statistics theorem is relativistic, the 1931 date seems to indicate that this would have been a result in nonrelativistic quantum mechanics. This answer by akhmeteli sketches what seems to be a similar relativistic result from a book by Lipkin. In the limit of low-energy experiments, the EO rule says the usual spin- statistics connection applies, and this seems extremely plausible. If not, then it would be too good to be true: we would be able to probe the composite structure of a system to arbitrarily high energies without having to build particle accelerators. Bethe 1997 has the following argument: > Consider now tightly bound composite objects like nuclei. It then makes > sense to ask for the symmetry of the wavefunction of a system containing > many identical objects of the same type, e.g., many He4 nuclei. This > symmetry can be deduced by imagining that the interchange of two composites > is carried out particle by particle. Each interchange of fermions changes > the sign of the wavefunction. Hence the composite will be a fermion if and > only if it contains an odd number of fermions[...] Note the qualifier \"tightly bound,\" which implies a restriction to low-energy experiments. The abstract of Ehrenfest 1931 seems to say that this is a necessary assumption, but it is never used in Bethe and Jackiw's argument, and this suggests that their argument is an oversimplification. Fujita 2009 recaps the Bethe-Jackiw argument, but then says: > We shall see later that these arguments are incomplete. We note that Feynman > used these arguments to deduce that Cooper pairs [5] (pairons) are bosonic > [6]. The symmetry of the many-particle wavefunction and the quantum > statistics for elementary particles are one-to-one [1]. [...] But no one-to- > one correspondence exists for composites since composites, by construction, > have extra degrees of freedom. Wavefunctions and second-quantized operators > are important auxiliary quantum variables, but they are not observables in > Dirac's sense [1]. We must examine the observable occupation numbers for the > study of the quantum statistics of composites. In the present chapter we > shall show that EOB's rule applies to the Center-of-Mass (CM) motion of > composites. Unfortunately I only have access to Fujita through a keyhole, so I can't see what the references are or any of the details that they promise in this beginning-of-chapter preview. Related: Huge confusion with Fermions and Bosons and how they relate to total spin of atom Bethe and Jackiw, Intermediate Quantum Mechanics P. Ehrenfest and J. R. Oppenheimer, \"Note on the Statistics of Nuclei,\" Phys. Rev. 37 (1931) 333, http:\/\/link.aps.org\/doi\/10.1103\/PhysRev.37.333 , DOI: 10.1103\/PhysRev.37.333 Fujita, Ito, and Godoy, Quantum Theory of Conducting Matter: Superconductivity, 2009"}
{"id":"75404","title":"Nuclear Weapon Effects: Cloud Rise","text":"I am interested in modeling the rise of a nuclear cloud to stabilization height post detonation. I am interested in any ideas you may have on the approach. I have collected some research on the topic but I am curious as to how someone might think through the problem without being exposed to one possible solution."}
{"id":"75409","title":"Do bifacial solar panels operate at lower temperature than normal solar panels?","text":"Bifacial solar panels have a glass backplane instead of the usual aluminum and are designed to absorb and convert to electricity light incident on either face. Many manufacturers claim that their bifacial panels operate at lower temperature than normal (monofacial) panels because bifacial panels transmit the 20% of the solar spectrum that is too low-energy to be absorbed by silicon. See for example this manufacturer's datasheet. Meanwhile, the only reputable study I could find (apologies for the paywall) was of a lab-scale cell and not a commercial module. That study argues that convection is the dominant heat transfer mechanism and so the differences in optical absorption and emission aren't relevant. However, temperature differences of a few degrees centigrade can have a noticeable impact on solar cell efficiency and useful life. Anybody have a compelling argument or know of other relevant experimental work on this topic?"}
{"id":"65316","title":"Kinetic energy when $F(t)$ graph is given","text":"![enter image description here](http:\/\/i.stack.imgur.com\/jBPTL.png) The very basic problem here is that the point where this graph ends is ambiguous. I mean, what is the corresponding $y$-value for the point whose $x$-valu is $4.5s$? Then I think we should find the area, then that will be the momentum, so use $\\frac{p^2}{2m}$, as mass is given. So if theres any error in this, pls let me know..."}
{"id":"26185","title":"What will the universe look like for anyone falling into a black hole?","text":"I've heard that, from the perspective of an external observer, something falling into a black hole will eventually look \"frozen\": light waves will move to the infrared and further into lower frequencies. But what about that unfortunate individual falling into such a black hole? What will he\/she see when looking back at our universe? For instance, I wonder, will it be the ultimate end of the Universe? Can someone who just crossed events horizon of the black hole still receive messages from our side?"}
{"id":"93001","title":"Smoothed Particle Hydrodynamics, Chemical Reactions and Stiffness","text":"Dear people from stackexchange physics. I have been using SPH for quite a while to simulate free boundary flow, and just recently we tried to include some kind of (simple) chemical reaction in our code. We are still doing some tests, but one curious thing that we found was that we needed much smaller time steps than we usually used to be necessary. I have never dealed with chemical reactions, but I don know that some of them fall on the category of 'Siff Equations', and as far as the wikipedia description goes, it seems to be my case, but I would like to ask if anyone tried anything similar and also found that their equations were stiff. I never studied stiff equations, I just knew that they existed and recently read a bit more about them, but that's it. Last but not least, do anyone know how to check if a set of SPH equations is stiff or not? The wikipedia recomendation seemed at least complicated to calculate for SPH."}
{"id":"106273","title":"Work done in non-uniform circular motion","text":"In non-uniform circular motion, the work done by force should not be zero because tangential force acts on body. Is this correct? If I am wrong, please give a proper explanation."}
{"id":"116717","title":"Thomas-Fermin approximation of cold atoms in a harmonic trap","text":"In Bose-Hubbard model of cold atoms, one can use Thomas-Fermi approximation to get a rough number of total particle number. In Thomas-Fermi approximation, the site density is $n(R_i)=（\\mu-\\kappa R_i^2）\/U$,where $\\mu$ is the chemical potential and $\\kappa$ is the trapping potential. The atomic cloud extends to the Thomas-Fermi radius $r_{TF}=\\sqrt{\\mu\/\\kappa}$. Why in Thomas-Fermi approximation, one can neglect the kinetic energy. Does it mean it only work for strong interaction regime? When I use Gutzwiller mean field method to calculate the total particle number in a trap, I find the total particle number varies a lot as the hopping amplitude varies."}
{"id":"104691","title":"Magnetic field of uniformly charged rotating hollow sphere","text":"I want to compute the magnetic field due to a homogeneously charged, rotating sphere with radius $R$, angular velocity $\\vec{\\omega} = \\omega\\hat{z}$ and total charge $Q$. I want to use Biot-Savart law, so I first computed the current density $\\vec{j}(r,\\theta,\\phi) = \\frac{\\omega Q}{4 \\pi R} \\delta(R-r) \\cdot (-\\sin\\theta \\sin\\phi,\\sin\\theta\\cos\\phi,0)^T$. After taking out all the constants with respect to integration and removing the integral over the radial component with the $\\delta$ function, what stays in the Biot-Savart integral is $$\\vec{B}(\\vec{r}) \\propto \\iint_{\\theta,\\phi}^{\\pi,2\\pi}\\frac{\\vec{j}(\\vec{y}) \\times (\\vec{r} - \\vec{y})}{|\\vec{r}-\\vec{y}|^3}\\sin{\\theta}\\;\\text{d}\\phi\\;\\text{d}\\theta$$ where $\\vec{r} = |\\vec{r}|\\cdot(\\sin\\theta_r\\cos\\phi_r, \\sin\\theta_r\\sin\\phi_r,\\cos\\theta_r)^T$ is the point where the $B$-field is evaluated, $\\vec{y} = R\\cdot(\\sin\\theta\\cos\\phi, \\sin\\theta\\sin\\phi,\\cos\\theta)$ is a point on the sphere. and the $\\vec{j}$ is now missing the $\\delta$ part, and some constants. The problem I have is, that the vector inside the integral is $$\\begin{pmatrix} \\sin\\theta\\cos\\phi(|\\vec{r}|\\cos{\\theta_r} - R\\cos\\theta )\\\\\\\\\\sin\\theta\\sin\\phi(|\\vec{r}|\\cos\\theta_r-R\\cos\\theta)\\\\\\\\-|\\vec{r}|\\sin\\theta\\sin\\phi\\sin\\theta_r\\sin\\phi_r-|\\vec{r}|\\sin\\theta\\cos\\phi\\sin\\theta_r\\cos\\phi_r+R\\sin^2\\theta\\end{pmatrix}.$$ Since all terms involving $\\sin\\phi$ or $\\cos\\phi$ will integrate to zero, this implies that I will get a $B$-field that is along the $z$ axis, no matter where it's evaluated. This can't be, because in the next part I'll show that far away the field looks like that of a dipole. I have been sitting over this for hours, can someone help me clear this up? Where am I going wrong?"}
{"id":"106184","title":"Streamlines for steady flow problem","text":"I need to calculate the streamlines for the steady flow given by the velocity field $$\\mathbf{u}=(u,v,w) = \\left( 0,\\frac{y}{y^2 +z^2} , \\frac{z}{y^2 +z^2}\\right)$$ Now I've been able to notice that $\\dfrac{ \\partial y}{ \\partial z} = \\dfrac{y}{z}$ therefore $y = ze^{c}$ but I'm not sure how I should continue. I've tried to solve $\\dfrac{ \\partial y}{ \\partial s} = \\dfrac{y}{y^2+z^2}$ but I get a $y^{z^2}$ term in my answer which makes me doubt whether I'm doing it correctly."}
{"id":"105560","title":"Nonequilibrium thermodynamics in a Boltzmann picture","text":"The Boltzman approach to statistical mechanics explains the fact that systems equilibriate by the idea that the equillibrium macrostate is associated with an overwhelming number of microstates, so that, given sufficiently ergotic dynamics, the system is overwhelmingly likely to move into a microstate associated with equilibrium. To what extent is it possible to extend the story to nonequilibrium dynamics? Can I make concrete predictions about the approach to equilibrium as passing from less likely macrostates to more likely macrostates? (Wouldn't this require us to say something about the geometric positioning of macrostate regions in phase space, rather then just measuring their area? Otherwise you'd think the system would immediately equilibriate rather than passing through intermediate states.) Can the fluctuation-dissipation theorem be explained in this way? **Edit:** Upon a bit more poking around, it looks like the fluctuation- dissipation theorem _cannot_ be explained in this way. The reason is that this theorem discusses the _time-independent distribution_ of fluctuations in some macroscopic parameter (e.g. energy of a subsystem) but, as far as I understand, it does not describe the _time dependence_ of such a parameter. In particular, I'd really like to understand is if it's possible to explain things like Fourier's Law of thermal conduction (that the rate of heat transfer through a material is proportional to the negative temperature gradient and to the cross-sectional area) with a Boltzman story. According to these slides, it's surprisingly hard."}
{"id":"105563","title":"Non-Hermitian operator with real eigenvalues?","text":"So we know that in Quantum Mechanics we require the operators to be Hermitian, so that their eigenvalues are real ($\\in \\mathbb{R}$) because they correspond to observables. What about a non-Hermitian operator which, among the others, also has real ($\\mathbb{R}$) eigenvalues? Would they correspond to observables? If no, why not?"}
{"id":"105565","title":"Magnetism due to relativity?","text":"So I have been reading in some books that magnetism does not have to be assumed a priori, but can be obtained from the electric field + special relativity. And I have seen how this leads to the common formula for the magnetic field of a current carrying wire. Fine. What about materials that are inherently magnetic? Such as iron, or magnetite? Surely their magnetic field is not a consequence of relativity? (if yes, who's moving and with respect to whom?)"}
{"id":"107582","title":"Time dilation in special relativity","text":"Suppose a star ship is moving with some velocity. Two light pulses one in direction similar to star ship another opposite to it is shot towards the space ship. Then how time inside space ship adjust to make velocity of light constant?"}
{"id":"107585","title":"Spin with Stern gerlach Experiment","text":"Within a week, I am trying to conduct a seminar on the topic \" Spin with Stern gerlach Experiment\" at my University. It would be great if you guys could provide me some inputs regarding the things I should say about the experiment during the seminar. I am still unsure about the outlines I should draw in my mind about what I should present serially.......Since it would be the first seminar of mine infront of more than 100 people, I am really confused as well as afraid."}
{"id":"18409","title":"How does the focal length of the lens change the distance from the camera to an object?","text":"I am reviewing some old image processing notes and have the following problem: > I have a camera with a projection plane of $2cm\\ by\\ 2cm$ and a $15mm$ lens. > If you want an $1.5m$ object to take up half of the resulting image, how far > away should it be? It seems like the standard $x = \\frac{fX}{Z}$ equation should be useful here, but I am not sure if this information gives focal length at all. How should this be solved?"}
{"id":"33401","title":"Relativity - time dilation","text":"I'm learning about relativity and I'm having some issues with it and the twin paradox. I found many questions and answers on this subject but they did not answer my specific problem. In my thought experiment I added a third twin - triplet. So here it goes: Let's have triplets $A$,$B$ and $C$. sitting still at same point in space. They want to test relativity so they do this experiment.. They all reset their clocks, and $B$ and $C$ start moving together away from $A$ at some ridiculous speed $v_{365}$ which coincidentaly has lorentz factor $365$. they move at uniform motion in only one direction (let's say acceleration time nears 0). State 1: $$\\begin{align}t(A)=t(B)=t(C)&=0 \\\\\\ v(A,B) = v(A,C) &= v_{365}; \\\\\\ v(B,C) &= 0\\end{align}$$ $t(A)$ being A's clock, $v(A,B)$ being velocity at which $A$ and $B$ are moving away from each other, etc.. Now after some time, let's say when $t(A) = 1\\text{ year}$, brother $B$ starts to decellerate to full stop. It also shows up that brother $C$ was sleeping through whole first part of the experiment. But now he wakes up and sees that his brother $B$ is accelerating away from him in the opposite direction than was planned but that doesn't bother him. State 2: $$\\begin{align}t(A)&=1\\text{ year} \\\\\\ t(B)=t(C) &\\stackrel{?}{=} 1\\text{ day} \\\\\\ v(A,B) &= 0 \\\\\\ v(A,C)=v(B,C) &= v_{365}\\end{align}$$ Now after another while lets say when $t(A)=2\\text{ years}$, brother $C$ sees that brother $B$ isnt stopping any time soon so he decides to match his speed.. (brother $C$ stops now from point of $A$) State 3: $$\\begin{align}t(A)&=2\\text{ years} \\\\\\ t(B)&= 1\\text{ year }1\\text{ day} \\\\\\ t(C)&= 2\\text{ days} \\\\\\ v(A,B)=v(A,C)=v(B,C)&=0\\end{align}$$ That's it.. now they stand still some light year away from each other, but they can communicate and compare their clocks.. here are my questions: 1] Is my assesment of time dilation effects correct in this scenario? if not which state contains first error and why? 2] If my assesment is correct, then from point of brother $C$ it was brother $B$ who was accelerating away, yet it was also brother $B$ who aged when they got in same speed - therefore it seems to me its always the one who is matching speed who doesnt age - is this a general rule? 3] If answer to questions 1 and 2 is yes, then can this effect be used to gather infomation that would take much longer to extract in normal speeds? Regards"}
{"id":"54717","title":"What is the highest speed time dilation has been tested?","text":"What is the highest speed time dilation has been tested? How close to the Special Relativity prediction did it get?"}
{"id":"102031","title":"Kirchoff Loop and current","text":"The pic of the circuit\/problem: ![enter image description here](http:\/\/i.stack.imgur.com\/kS5Pm.jpg) I got the problem right, but i want to make sure my reasoning is correct for the current left of the 2.1 V battery. Its the same current as $I_1$ because the charge carriers do not split in anyway, and the battery only does work on them moving the carriers from a low potential to a high potential. However, if $I= \\frac{dQ}{dt}$, wouldn't supplying energy via moving the charges from low potential to high potential cause the charges to move more quickly, i.e. increase the current? A little fuzzy on why it is the same current, if someone could enlighten me I would very much so appreciate it."}
{"id":"18616","title":"Projective Transformations in GR","text":"A Thought Experiment: We are in flat spaceime provided with a reference frame—a rectangular Cartesian frame. The coordinate labels[the spatial labels] are visible to us. Each spatial point is provided with a clock—and the different clocks are synchronized wrt to each other. Gravity is now turned on and made to vary upto some final state. During this process of experimentation the physical separations change while the coordinate labels remain fixed to their own positions. [Coordinate separations remain unchanged]. The length and the orientation of a vector changes in this process both in the 3D and in the 4D sense. We are passing through different\/distinct manifolds in our thought experiment and if the 4D arc length does not change we are simply having a transition between manifolds for which ds^2 is not changing but the metric coefficients are changing.We consider the option of $ds^2$ changing in this posting. Query: Our experiment indicates at projective transformations operating in the physical sense[considering changes in the metric and in the value $ds^2$].A time dependent field is being observed where the metric coefficients are not being preserved. Is it important to include projective transformations[concerned with the non-preservation of the metric] in the mathematical framework of GR?"}
{"id":"24034","title":"Is the Avogadro's constant equal to one?","text":"**Question: Is the Avogadro's constant equal to one?** I was tasked with creating a presentation on Avogadro's work, and this is the first time I actually got introduced to the `mole` and to `Avogadro's constant`. And, to be honest, it doesn't make any mathematical sense to me.               1 mole = 6.022 * 10^23     Avogadro's constant = 6.022 * 10^23 * mole^(-1)      What? This hole field seems very redundant. There are four names for the same thing! Since when is a number considered to be a measurement unit anyway?!"}
{"id":"20437","title":"Are matrices and second rank tensors the same thing?","text":"Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified: 1-Are matrices and second rank tensors the same thing? 2-If the answer to 1 is yes, then can we think of a 3rd rank tensor as an ordered set of numbers in 3D lattice (just in the same way as we can think of a matrix as an ordered set of numbers in 2D lattice)?"}
{"id":"71898","title":"String total cross sections at asymptotically high energy","text":"I only have a vague understanding of string theory, but a solid understanding of particle physics. At asymptotically high energy (Regge limit), the string cross section is dominated by the exchange of the leading trajectory -- the graviton trajectory, corresponding to an intercept $\\alpha(0)=2$. Regge phenomenology dictates that the string-string total cross should rise linearly with energy: $$\\sigma_\\text{tot} \\sim s^{\\alpha(0)-1} = s$$ which is startling to me. Given that gravitons are massless, they are not subject to the Froissart-Martin bound $\\sigma_\\text{tot}<\\log(s)^2$. So, is it really true that string-string total cross sections exhibit this growth?"}
{"id":"71899","title":"Two charges in a medium where different permittivity is adjacent","text":"I was thinking about the following: Assuming that you have NON SPHERICALLY SYMMETRIC charges and they are both in medium 1. Does it make a difference(whereby I am referring to the total force) if there is a second medium 2 in the local area of both, such that both charges are NOT in the second medium, but very close to it to the case where only medium 1 is around them?"}
{"id":"71891","title":"Parseval's Theorem on a Random Signal","text":"NB - I'm re-posting this question in physics because I haven't had any luck getting a response from the maths StackExchange site - it's a rather applied problem so is probably better suited here anyway. ** I'm struggling with Parseval's Theorem. I'm trying to relate variation in the time domain to the average value in the frequency domain. To do this, I'm performing the Fourier Transform on an arbitary random signal that I've generated with 2048 points (although the graph below only shows 100 of them) and a standard distribution (in this particular case) of 0.58: ![Time domain trace of random signal - NB only showing 100 of 2048 points!](http:\/\/i.stack.imgur.com\/O0zlF.png) The frequency domain, after an FFT, then looks like this: ![Frequency domain trace of random signal](http:\/\/i.stack.imgur.com\/coWig.png) with an average value of 0.022 - although this depends on the number of samples used. Now when I try to apply Parseval's Theorem, where: $\\sum_{n=0}^{N-1} |x[n]|^2 = \\frac{1}{N}\\sum_{n=0}^{N-1} |X[k]|^2 $ I run into a problem. When I do the summations, I get 678 for the time domain and 0.662 for the frequency domain. When I apply the factor of $1\/N$ I end up with 0.0003 for the frequency side - clearly a long way off! Obviously I'm going wrong somewhere, but can't see where. This many orders of magnitudes off isn't very encouraging... Thanks for your help!"}
{"id":"76325","title":"(Exercise review) Find the moment of the electric dipole","text":"I have solved an exercise, but the result I have obtined is wrong and I can't understand why. If you can help me, I'll be so grateful. Let's consider a charged cylinder, its radius is equal to R and its height=4R. Its center of is in the origin of the cartesian axes and its axis is parallel to z axis; we know the volumic density of charge: $\\rho= 2az \\epsilon$ where $a, \\epsilon$ constant $>0$. There is an electric field inner to the cylinder, described by: $E_{ox}=0, E_{0y}=0, E_{0z}=az^2$ We want to find the electric potential to a great distance from the origin of the axes. I have thought as follow. Using Maxwell's first equation, I have found the volumic density of charge: $\\rho=2\\epsilon_0 a z$. At a very big distance to the origin, I can consider the cylinder like an electric dipole, with the positive charge concetrated in the center uf the upper part of the cylinder and the negative one in the center of the lower part: So $Q_+$ will be in (0,0,R) and $Q_-$ in (0,0,-R). Now, I need the moment of dipole, ${\\bf p}=q{\\bf a}$ where ${\\bf a}$ is the vector from the negative charge to the positive charge. I have integrate $\\rho$ and I have obtained $q=\\int_0^{2R}\\int_0^R\\int_0^{2\\pi}2az\\epsilon r d\\theta dr dz$ and I have obtained $q=4\\pi a \\epsilon_0 R^5$ Then I have posed a=2R, so I have obtained $p=8\\pi a \\epsilon_0 R^4$.The correct result is $p=32\/3 \\pi a \\epsilon_0 R^5$ I can't understand where I was wrong and I wouldn't like to have made an error of proceeding."}
{"id":"76326","title":"Black bodies, energy per unit volume","text":"While reading about black bodies, I found that the average energy per mode per unit volume is $KT$ classically, but I don't understand why this is per unit volume. The linear oscillators are considering to be in the walls of a cavity modeling a blackbody, so shouldn't it be modes per area?"}
{"id":"52165","title":"Newtonian gravity vs. general relativity: exactly how wrong is Newton?","text":"Is there a simple function I can use to describe the difference between simple Newtonian dynamics and the actual observed motion? Or maybe some ratios for common examples of, say, the motion of stars and planets? I know for example that Newtonian physics is sufficient to fire a rocket at the moon, so the error there must be miniscule. What exactly is that ratio, and could a layperson calculate it generally for other bodies? The context is this: in a debate regarding how drastically wrong science can be, I want to make the case that even though Newton was wrong regarding gravity in the bigger picture, he was only ___ % wrong regarding what he'd observed. I know science can often be wrong, but I want to emphasize the fact that our system of observation isn't fundamentally dysfunctional."}
{"id":"103635","title":"Max. speed that a space probe can communite with the Earth?","text":"**Question** : _Which is the average or maximum speed of a telematically sent message that a space probe can send_? **Is as simple as radio waves travel at the speed of light in a vacuum?** **Context** : I have been asked to make a first estimation about how much time is needed to send a space probe and then receive the signal. I am supposing that the space probe is sent at $15 \\ km\/s$ but I have not idea of the telematic signal speed. Thank you in advance!"}
{"id":"132423","title":"Quadratic versus other types of degrees of freedom","text":"In my text book I have the following question and after thinking about this one for hours, I can't figure out what they're getting at: In Section 2.5* I quoted a theorem on the multiplicity of any system with only quadratic degrees of freedom: In the high-temperature limit where the number of units of energy is much larger than the number of degrees of freedom, the multiplicity of such a system is proportional to $U^{Nf\/2}$, where $Nf$ is the total number of degrees of freedom. Find an expression for the energy of such a system in terms of its temperature, and comment on the result. How can you tell that this formula $\\Omega$ cannot be valid when the total energy is very small? * * * (*) = In section 2.5 of the text book, the formula for multiplicity $\\Omega$ is given to be $$\\Omega\\left(U,V,N\\right)=f\\left(N\\right)V^NU^{3N\/2}.$$ with $f\\left(N\\right)$ some function of $N$. (This formula is for three degrees of freedom.) * * * What I got so far is: The initial expression (for $f$ degrees of freedom) is $$\\Omega_N\\approx\\frac{1}{N!}\\frac{V^N}{h^{3N}}\\frac{\\pi^{Nf\/2}}{\\frac{Nf}{2}!}\\left(\\sqrt{2mU}\\right)^{Nf}.$$ Using $S=k\\ln\\Omega$ and solving $T=\\left(\\partial S\/\\partial U\\right)^{-1}$ for $U$. the final answer for $U$ becomes $$U=\\frac{1}{2}NfkT.$$ (Which is really just the equipartition theorem all over again???? Or did I make a mistake?) Now, I don't get why this formula can somehow tell me that my initial formula for multiplicity is not sufficient when the energy is small:   * Both $N$ and $k$ are constants.   * Although $f$ sort of \"jumps\" per the rule of Dulong and Petit, assuming that more degrees of freedom freeze out would seem baseless to me. Do I have to treat this as a constant as well?   * From this I conclude that $U$ is proportional to $T$, which really doesn't seem so problematic to me. How do I have to interpret the role of the quadratic vs. the non-quadratic degree of freedom here? What is the consequence?"}
{"id":"130374","title":"Changes in planetary orbit?","text":"Forgive a novice and her naive curiosity, but I was wondering about planetary orbit and the aging of planets. As a **planet ages** , does its transit around its sun\/star **decrease** or **increase** in **speed**? ## Does the speed and proximity of a planet's orbit around the star, have more to do with the age and type of the star, and less to do with the age of the planet and its distance from the star? Also, is it possible for a planet to be **pushed** or **pulled** closer to its star, into a closer orbit, for example by increased gravitational pull of the star, or the planet being knocked off orbit by a comet or asteroid, or even a shockwave from an implosion -- one far enough away so that no debris comes near, but near enough that the shockwave is powerful enough to knock the planet. I realise such an occurance would probably cause catastrophic upset on the planet, but if it remained relatively intact, could it find itself on a new path around its star??   Or could the comet\/asteroid\/shockwave hit the planet from the opposite direction of its transit, perhaps glancing it at such an angle as to cause it to slow in pace around the star, or to turn faster or slower on its own axis? And would its poles be reversed? Living in a world with sci-fi like **Doctor Who** , **Eureka** , **Stargate** , **Star Trek** , **X-files** , **Fringe**.. it makes you wonder what is **theoretically possible**. Often the explanations they give sound **far- fetched, yet plausible**. Just as every myth evolves from a truth, you tend to wonder how much truth exists within these stories. Hope one of you can shine a light on my wonderings and find a kernel of truth among them."}
{"id":"91541","title":"Are \"phase-specific\" masks ever used in the context of double exposure lithography? Can these sorts of things exist?","text":"I was recently reading an 08' paper on double exposure lithography: and I was wondering if there existed _some_ material that could be used to create a mask for e.g. a 193 nm light source that was _only opaque to light with a specific phase_ $\\theta$. The idea, in the context of double exposure lithography, would be to overlay two masks, one opaque for light with some phase $\\theta_a$ and another opaque for light with some phase $\\theta_b$, where (in radians) $$\\theta_a = \\theta_b + \\pi\\cdot n$$ where $n$ is any non-zero integer value. Thus, one could just phase-shift the source of coherent light used for lithography by $\\pi$ radians to pattern with one mask and then the other. This would save an alignment step for the substrate being patterned. Is there any precedent for this sort of thing? Or is this question nonsense?"}
{"id":"101257","title":"Doing a Gaussian Integral","text":"When you integrate over p you get: ![enter image description here](http:\/\/i.stack.imgur.com\/yEUbh.png) by using ![enter image description here](http:\/\/i.stack.imgur.com\/9gXXP.png) What are the steps to this? Do you integrate by parts?"}
{"id":"128627","title":"Why aren't units with powers, like cm³, surrounded by parentheses?","text":"Since $\\renewcommand{\\unit}[1]{\\,\\mathrm{#1}} 1\\unit{dm} = 10^{-1}\\unit{m}$, it follows that $1\\unit{dm^3} = 10^{-1} \\times 10^{-1} \\times 10^{-1} \\unit{m^3} = 10^{-3} \\unit{m^3}$. However, in regular mathematics the following equation holds true: $$a\\,b^{3} = a\\,b\\,b\\,b$$ By the above, the cube unit should expand as follows: $$\\mathrm{dm^3} = \\mathrm{dmmm}$$ While in actual usage (as seen in the second equation) the expansion is $\\mathrm{dddmmm}$, which would arise from using $\\mathrm{(dm)^3}$ instead. $$\\mathrm{(dm)^3} = \\mathrm{dddmmm}$$ So shortly: why aren't parentheses (commonly?) used in units?"}
{"id":"82048","title":"Is the 74.2km\/sec per megaparsec accidental?","text":"The rate of expansion of the universe is 74.2km\/sec per megaparsec The age of the universe is 13.798 bln years That places the furthest theoretically observable edge of universe at 13.798 bln light years away - this is how long light would take to reach us from there, and there were no light sources before that date. Now, how fast does the observable part of the universe expand? A bit of unit conversion and multiplication: 13798000000 light years in megaparsecs = 4 230.39567 megaParsecs [1] 74.2km\/sec in light years\/year = 0.000247504559 light years \/ year [2] Multiply the two, and I get the expansion rate: 1.04704221 light years per year per distance to the observable edge of the universe. Speed of 1 light year per year is the speed of light. The 0.047 seems awfully small. Probably within error margin of the universe expansion speed and age of the universe estimates. Is this a coincidence, or is this somehow bound - and if so, what are the consequences?"}
{"id":"109127","title":"Integrating the divergence of a velocity field (Fluid Mechanics)","text":"I'm looking more for verification of my answer as I'm not 100% sure how valid my integration is here: If I have a 2-D incompressible fluid velocity field given by $\\vec{U}(x,y,t)=u(x,t)\\vec{x}+v(y,t)\\vec{y}$, I know that $\\nabla \\cdot \\vec{U} = 0$, call this eqn (1), by the definition of incompressibility. In order to find out more about the velocity components $u(x,t)$ and $v(y,t)$, i have been told to integrate eqn (1). Is the following integration w.r.t. space valid?: \\begin{align} &\\int \\nabla \\cdot \\vec{U} \\mathbb{d} r = \\int 0 \\\\\\ &\\int \\left(\\frac{\\partial u}{\\partial x}+\\frac{\\partial v}{\\partial y}\\right) \\mathbb{d}r = C = \\text{const.} \\\\\\ &\\int \\frac{\\partial u}{\\partial x}dx+\\int \\frac{\\partial v}{\\partial y}\\mathbb{d}y = C \\\\\\ &u(x,t) + v(y,t) = C(t) \\end{align} In other words, the magnitude of the sum of the fluid velocity in the $x$ and $y$ directions are constant everywhere in the fluid at any instant in time? Is there a better way of obtaining more information?"}
{"id":"109122","title":"Tuned Mass Damper","text":"I was learning about the different applications of resonance and one of them is the tuned mass damper used in buildings. One thing I am confused about is as to why the mass attached to the building oscillates $180^\\circ $ out of phase when in resonance. Is there any mathematical reason behind this or an intuitive way of understanding the concept?"}
{"id":"128969","title":"Determine the acceleration in a double Atwood machine","text":"Trying to solve the following double Atwood machine: > _Suppose there is a mass of 12kg hanging on an ideal rope that wraps around > an ideal pulley, and that the other end of the rope is attached to another > ideal pulley's axis. This second pulley has a rope around it which on one > side holds a mass of 4kg and on the other a mass of 8kg. What is the > acceleration of the first mass?_ So here's what I've done so far: First I've determined that the tension in the first rope is twice the tension in the rope for the \"subsystem\", and that the acceleration of the first mass is the average of the accelerations of the two other masses but in the opposite direction. That is to say $$T' = T\/2$$ $$a_{1}=-\\frac{a_{2}+a_{3}}{2}$$ From free-body diagrams we can determine that $$-12a = T-12g$$ $$4a_{2} = T\/2 - 4g$$ $$-8a_{3} = T\/2 - 8g$$ However, when I solve this system I obtain $T=32a$ which seems impossible that the tension would be greater than the greatest force of gravity of any mass. Part of me wonders if this has anything to do with it: How do I know which direction the third mass is traveling? The first mass is causing the subsystem to accelerate upwards but the third mass is accelerating downwards relative to the subsystem. So how do I know the direction of acceleration relative to the ground?"}
{"id":"74417","title":"Electromagnetic radiation and quanta","text":"since electromagnetic radiation possess the property of both wave and particle(photon). and both theory are applicable but how we have to find out that which theory is suitable or applicable in particular explanation. for example in laser traping of atom we use photon concept rather than wave why?"}
{"id":"128963","title":"CFT Entanglement Entropy - relation between translations and the stress-energy tensor","text":"In a recent paper on CFT entanglement entropy, I want to understand the defintion of a certain partition function. They consider a metric space $S^1 \\times \\mathbb{H}^{d-1}_q$ with metric: $$ ds^2_{H_q^{d-1}} = d\\tau^2 + du^2 + \\sinh^2 u \\; d\\Omega_{d-2}^2 $$ Here $d\\tau$ is probably a Wick-rotated time, $u$ is a radial variable and $d\\Omega$ is the spherical area measure. Then they define a partition function $ Z_q = \\mathrm{tr}(e^{-2\\pi q H_\\tau}) $ where $H_\\tau$ \"generates translations along the $S^1$\". What does that mean? Could it mean this? $$ H_\\tau = \\frac{d}{d\\tau}$$ This is the generator for translations along the $\\tau$-axis. However, they also say this is related to the stress-energy tensor: $H_\\tau = \\int_{\\mathbb{H}^{d-1}} dx^{d-1} \\sqrt{g} T_{\\tau\\tau}$ This seems like a very complicated way of describing translations. Could there be another meaning for the phrase \"generates translations along $S^1$? * * *   1. Jeongseog Lee, Aitor Lewkowycz, Eric Perlmutter, Benjamin R. Safdi   Renyi entropy, stationarity, and entanglement of the conformal scalar"}
{"id":"6649","title":"Can the Kramers–Kronig relation be used to correct transfer function measurements?","text":"In experimental physics, we often make measurements of linear transfer functions; these are complex-valued functions of frequency. If the underlying system is causal, then the transfer function must be analytic, satisfying the Kramers-Kronig relations. Our measurements, however, are corrupted by random (and perhaps systematic) errors. **Is it possible to improve a measurement of a linear transfer function of a causal system in the presence of noise by applying some kind of constraints derived from the Kramers-Kronig relations?**"}
{"id":"132931","title":"Which of these two textbook equations of geodesic deviation is correct?","text":"My previous question Geodesic deviation on a 2-sphere - is this the right track? got shot down as “off topic”, so I'm having a second stab at it. Misner _et al_ 's _Gravitation_ (p34) gives the geodesic deviation equation as$$\\frac{D^{2}\\xi^{\\alpha}}{D\\tau^{2}}+R_{\\phantom{\\mu}\\beta\\gamma\\delta}^{\\alpha}\\frac{dx^{\\beta}}{d\\tau}\\xi^{\\gamma}\\frac{dx^{\\delta}}{d\\tau}=0,$$ with the right-hand side $\\xi$ index $\\gamma$ equal to the _second_ lower index on the Riemann tensor. Lambourne's _Relativity, Gravitation and Cosmology_ (p185), on the other hand, gives $$\\frac{D^{2}\\xi^{\\mu}}{D\\lambda^{2}}+R_{\\phantom{\\mu}\\alpha\\beta\\gamma}^{\\mu}\\xi^{\\alpha}\\frac{dx^{\\beta}}{d\\lambda}\\frac{dx^{\\gamma}}{d\\lambda}=0,$$ with the right-hand side $\\xi$ index $\\alpha$ equal to the _first_ lower index on the Riemann tensor. My question is, which of these two equations is correct? I tried to answer this question myself by using the two equations to calculate the geodesic deviation on the surface of a unit 2-sphere. With Misner's equation (substituting $\\lambda$ for $\\tau$) I got $$\\frac{D^{2}\\xi^{\\theta}}{D\\lambda^{2}}=\\left(\\sin^{2}\\theta\\right)\\left(u^{\\phi}u^{\\theta}\\right)\\xi^{\\phi}-\\left(\\sin^{2}\\theta\\right)\\left(u^{\\phi}u^{\\phi}\\right)\\xi^{\\theta}$$ and $$\\frac{D^{2}\\xi^{\\phi}}{D\\lambda^{2}}=\\xi^{\\theta}\\left(u^{\\theta}u^{\\phi}\\right)-\\xi^{\\phi}\\left(u^{\\theta}u^{\\theta}\\right).$$ You can see my calculation on my previous question Geodesic deviation on a 2-sphere - is this the right track? With Lambourne's equation I got $$\\frac{D^{2}\\xi^{\\theta}}{D\\lambda^{2}}=0$$ and $$\\frac{D^{2}\\xi^{\\phi}}{D\\lambda^{2}}=0.$$ This didn't seem right to me so I concluded that Lambourne's equation is incorrect."}
{"id":"92014","title":"Polarization vectors of massive and massless particles","text":"I read from Mandl & Shaw that when quantizing massless vector particles such as photons in Lorentz gauge, there are 4 linearly independent polarization vectors (2 of them being able to \"gauged away\"), while when quantizing massive vector particles such as W bosons, there are only 3 linearly independent space-like polarization vectors. It also says that when W and Z gauge bosons \"eat\" the Goldstone bosons from Higgs field, they become massive and acquire longitudinal polarization. So apparently there is some rule that says massive vector particles have longitudinal polarization, while massless particles don't. Why is that? From what principle is it derived? Their Euler-Lagrangian equations are essentially the same (both look like Klein-Gordon equation), just one of them has $m=0$. Is it the case that when $m=0$, it has four linearly-independent solutions while only three when $m\\neq0$ ?"}
{"id":"92013","title":"Initial pressure of explosion","text":"I would like to know, if given a TNT equivalent of explosive (or its Energy) is it possible to determine what would be the initial pressure that will be the result of that explosion ? To be more precise let's say I have a vessel and given the vessel volume can I determine pressure inside that vessel (assuming vessel won't break)."}
{"id":"121341","title":"CPT symmetries for a free Klein-Gordon equation and in minimal coupling","text":"I'm studying for an exam on relativistic quantum mechanics and one of the issues to prepare is about symmetries of Klein-Gordon equation concerning $C$, $P$, $T$ transformations for a free field, and with an electromagnetic interaction. So could you help me out a bit? As I understand the general free solution can be written as: $$ \\Psi(\\mathbf{x},t) = \\int\\limits_{ \\mathbb{R}^3 }\\frac{d^3p}{(2 \\pi \\hbar)^3} \\frac{1}{2 E_p} \\left( A_\\mathbf{p}^{(+)}e^{-\\frac{i}{\\hbar}E_p t} e^{\\frac{i}{\\hbar}\\mathbf{p} \\cdot \\mathbf{x}} \\+ A_\\mathbf{p}^{(-)}e^{\\frac{i}{\\hbar}E_p t} e^{\\frac{i}{\\hbar}\\mathbf{p} \\cdot \\mathbf{x}} \\right) $$ where: $E_p := \\sqrt{\\mathbf{p}^2c^2-m^2c^4}$ For a special solution (in a box): $$ \\psi^{\\pm}_{\\mathbf{p}}(\\mathbf{x},t) = \\frac{1}{\\sqrt{V}} e^{\\mp\\frac{i}{\\hbar}E_p t} e^{\\frac{i}{\\hbar} \\mathbf{p} \\cdot \\mathbf{x}} $$   * $P$ transformation gives: $P\\psi_{\\mathbf{p}}^{\\pm}(\\mathbf{x},t) = \\psi_{-\\mathbf{p}}^{\\pm}(\\mathbf{x},t)$,   * $T$ transformations gives: $T\\psi_{\\mathbf{p}}^{\\pm}(\\mathbf{x},t) = \\psi_{\\mathbf{p}}^{\\mp}(\\mathbf{x},t)$ and   * $C$ transformations gives: $C\\psi_{\\mathbf{p}}^{\\pm}(\\mathbf{x},t) = \\psi_{-\\mathbf{p}}^{\\mp}(\\mathbf{x},t)$ \/\/Corrected a mistake thanks to a comment by TwoBs So what conclusion (if any) I can draw from this about the general solution? And do the conclusions stay valid for a KG equation with minimal coupling ($\\partial_{\\mu} \\rightarrow D_{\\mu} := \\partial_{\\mu} + \\frac{i e}{\\hbar c}A_{\\mu}$)?"}
{"id":"6397","title":"Low frequency electromagnetic waves","text":"Some frequencies of electromagnetic waves are used for transmission of information, like radiowaves, microwaves, light, but some not.. What about Low Frequencies?: Perhaps low frequencies aren't used for transmision simply because the waves haven't got enough energy to reach destiny, but is well known that if you go near a high voltage transmission tower, you can turn on a fluorescent lamp http:\/\/www.electric-fields.bris.ac.uk\/ Perhaps the intensity of the high voltage gives more photons to interact there, than in a home installation, but the fact is waves are traveling anyway at 50\/60 Hz without need any modulation. There is a limit on how low a frequency can be to put photons into travel?"}
{"id":"134217","title":"Is the charge of an ion evenly distributed?","text":"This question relates to: Gauss' law and ions? Is the charge distribution in an ion spherically symmetric due to quantum mechanical effects or do we assume it when using Gauss's law, as in the linked question, to make the calculation easier? I think it is the former but I am not sure."}
{"id":"134214","title":"Can we multiply an electromagnetic field?","text":"![photomultiplier](http:\/\/i.stack.imgur.com\/vNvVw.jpg)   * can we multiply light by a sequence of reflections in a closed vessel?   * also, is light a part of EM wave?   * can we multiply the radio wave or EM wave from a transformer?"}
{"id":"80998","title":"For semi-insulating GaAs is the bandgap bigger or smaller than the pure GaAs?","text":"The carrier concentrations are lower in semi insulating semiconductors, therefore, can we say that their bang gap is wider in terms of energy (eV)? If they have wider band gap how much bigger is it approximately and what parameters affect it? i.e. is temperature one of them?"}
{"id":"80995","title":"Reason for strange \"canyons\" of dust on the cooling platform","text":"I took apart my cooling platform Something like this :![Cooling Platform](http:\/\/i.stack.imgur.com\/kIPFr.jpg) And I found something like this inside : ![Canyon of dust 1 ](http:\/\/i.stack.imgur.com\/IyZes.jpg) ![Canyon of dust 2 ](http:\/\/i.stack.imgur.com\/4yX7K.jpg) ![Canyon of dust 3 ](http:\/\/i.stack.imgur.com\/udFJ5.jpg) I wonder why around the hollows are the \"canyon like\" structures. PS: I also drop this platform on the floor, maybe this could be a reason ? EDIT : Everybody seems (except Brandon Enright answer) to ignore the word \"dust\" in this question. Those structures are not in plastic itself. It is dust on the ON the plastic. Proof : ![Dust canyon](http:\/\/i.stack.imgur.com\/SCmXW.jpg) Completely wiped out with my finder : ![Dust canyon wiped](http:\/\/i.stack.imgur.com\/my18U.jpg)"}
{"id":"14223","title":"Representations of gamma matrices","text":"I have to do this exercise for homework. Find a representation of the gamma matrices unitarily connected to the standard representation for wich the spinors $u(p)$ that satisfy the equation $(p_\\mu \\gamma^\\mu - m)u(p)=0$ of motion for a generic momentum $p^2=m^2$ have only the first two components different from zero. I tried to find the transformation $V$ such that $Vu(p)$ meet the demand but was not able to complete my idea. Can anyone help me?"}
{"id":"71135","title":"Coordinate system","text":"**Quoting from 'Nuclear Physics - Theory and Experiment' by RR Roy, BP Nigam 2005 edition** Link to text * * * > How did the author arrive at equations (23a, 23b,23c)? * * * _Chapter 8 Nuclear model II, 8.7 **Quadrupole Deformation** , page 286_ In the body-fixed reference frame in which the coordinate axes coincide with the principal axes, we denote the deformation parameters $\\alpha_{2\\mu}$ by $a_{2\\mu}$. The relationship between deformation in the two coordinate systems is $$\\begin{align} \\sum_\\mu a^*_{2\\mu}Y_{2\\mu}(\\theta,\\phi)&=\\sum_\\nu a^*_{2\\nu}Y_{2\\nu}(\\theta,\\phi)\\\\\\ &=\\sum_\\nu a^*_{2\\nu}\\sum_\\mu D^2_{\\mu\\nu}(\\theta,\\phi,\\psi)Y_{2\\mu}(\\theta',\\phi') &(22c) \\end{align}$$ so that $$\\begin{align} \\alpha_{2\\mu}&=\\sum_\\nu a_{2\\nu}D^{2*}_{\\mu\\nu}(\\theta,\\phi,\\psi) \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad&(22d) \\end{align}$$ Since, in terms of the principal axes, the product of inertia is zero, we define the following $$\\begin{align} a_{20}&=\\beta\\cos\\gamma \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad&(23a)\\\\\\ a_{21}&=a_{2-1}=0&(23b)\\\\\\ a_{22}&=a_{2,-2}=\\tfrac1{\\sqrt 2}\\beta\\sin\\gamma&(23c) \\end{align}$$ where $\\beta$ and $\\gamma$ are new parameters whereby the $a$'s are defined. The deformations $\\delta R_j$ along the principal axes $j=1,2,3$ (nuclear body-fixed axes) are obtained from $$\\delta R(\\theta,\\phi)=R_0 \\sum^2_{\\mu=-2}a^*_{2\\mu}Y_{2\\mu}(\\theta,\\phi)$$ * * * > and how can we write the equations $\\delta R_1$? * * * _page 287_ and are given by $$\\begin{align} \\delta R_1\\left(\\frac\\pi2,0\\right)&=\\sqrt{\\frac5{4\\pi}}\\beta R_0\\cos\\left(\\gamma-\\frac{2\\pi}3\\right)\\\\\\% i am so depressed \\delta R_2\\left(\\frac\\pi2,\\frac\\pi2\\right)&=\\sqrt{\\frac5{4\\pi}}\\beta R_0\\cos\\left(\\gamma-\\frac{4\\pi}3\\right)\\\\\\ \\delta R_3\\left(0,\\phi\\right)&=\\sqrt{\\frac5{4\\pi}}\\beta R_0\\cos\\left(\\gamma\\right)\\\\\\ \\end{align}$$ * * *"}
{"id":"71137","title":"Nillson Potential","text":"We define a Hamiltonian to derive Nilsson potential, $$\\hat{H}=-\\frac{\\mathbf{\\hat{p}}^2}{2m}+ \\frac{1}{2}m\\left[\\omega_\\bot^2(\\hat{x}^2+\\hat{y}^2)+\\omega_z^2\\hat{z}^2\\right]$$ Nilsson model is a famous model, but I didn't find the potential pertaining to it, so I wrote the Hamiltonian and Asked that , is there any way to get the Nilsson potential from the Hamiltonian. **EDIT: according to Thomas** So are you trying to say that, when symmetric potential is broken, that is the Nilsson potential? if someone ask me to write down the potential, then I can not write that down, because it is very subtle amount? **What are the failures of collective model that Nilsson potential introduced? I mean what are the difference between these model?**"}
{"id":"71136","title":"The boundary between polycrystalline and crystalline","text":"My current understanding of solid crystalline-like materials (please correct me if I'm wrong!) is that it is a continuum in terms of crystallinity, from amorphous (basically no periodicity) to single crystals (complete periodicity), with \"polycrystalline\" pretty much covering everything in the middle, like the wiki article shows. So my question is, what should you expect from a polycrystal in terms of properties you know about a single crystal of the same material, such as dispersion relation, band structure, etc? Obviously, it's a continuum, so you should expect a polycrystal that's _almost_ a single crystal (maybe like, a single crystal split in two, and put back together the tiniest bit off) to behave pretty damn similar to a single crystal. It seems like each single crystallite in a polycrystal should act as a tiny crystal with the band structure of a perfect crystal. But what do you do as the grains get smaller? Is there some way to predict how close to a single crystal you should expect the polycrystal to behave, given its average and variance of grain sizes? As the grains get smaller, it seems like you should expect the thin layers of amorphous solid holding the crystallites together to be more and more important. In fact, is there any sort of formalism for working with amorphous solids? Thanks!"}
{"id":"44170","title":"Does the equivalence principle mean that the Earth is expanding?","text":"Popular books suggest that for an observer in the Einstein lift the following situations are equivalent: 1, the lift hangs motionless (relative to the Earth) on a cable in the gravitational field of the Earth 2, there is no gravity but the lift is accelerated up The natural question then is: \"If I stand on the Earth and feel its gravity does it mean that there is no gravity in fact and the Earth is expanding with acceleration?\" I know the short answer ( **no** ) but as a devil's advocate I would appreciate a longer version."}
{"id":"44178","title":"Are there any electro-optic crystals that are also pyroelectric but not birefringent?","text":"As the title says, a crystal that is electr-optic and pyroelectric can it be non-birefrigent?"}
{"id":"123870","title":"Sign of Feynman rules with derivative couplings","text":"Feynman rules for derivative couplings always make me confused. For example, the derivative in $gV^\\mu\\phi^+\\partial_\\mu\\phi^-$ will give you $\\pm ip_{-\\mu}$, where $\\pm$ depends on whether the particle is outgoing or incoming. I found \"$-ip_{-\\mu}$ for incoming\" in Peskin & Schroeder's QFT book, BUT I also found \"$-ip_{-\\mu}$ for outgoing\" in some other literature. So I'm wondering if the latter one is wrong. Or are they just in different conventions so that we can choose either of them?"}
{"id":"123685","title":"Effect of small variable transverse force on satellite orbital precession","text":"Consider a small satellite which moves in a 2D elliptical orbit around a much larger body (e.g. the Sun) under the influence of Newtonian gravitational acceleration $$Ar = G.M\/d^2 $$ Next imagine that a small, time-varying force acts upon the satellite which produces acceleration of the satellite in the direction transverse to the instantaneous radial direction. The general pattern of variable transverse acceleration (At) which I am interested in is given by the following equation:- $$At = (K\/c^2)*Vr*Vt*Ar $$ I have modelled such effects in a reiterative computer model and would like to validate those measurements. The principal effect is the introduction of a rotation of the aspides axis of the orbit (\"precession of the line of aspides\"). QUESTION Is there an analytical technique I can use to produce a formula for the rate of orbital rotation? TERMS & ASSUMPTIONS   * K is a constant (e.g. +\/- 3.0).    * Ar is the radial acceleration of the satellite towards the Sun due to Newtonian gravitational influence of the Sun.   * At is the time-variable acceleration of the satellite in the direction transverse to the Sun:satellite radial direction (positive when directed forward along the future orbital path of the satellite, negative when directed backwards)   * c is speed of light.    * Vr is satellite radial velocity relative to the Sun (positive when satellite is moving further away from the Sun)    * Vt is satellite transverse velocity relative to the Sun (always forward)    * G is Newton's Universal Gravitational Constant.    * M is mass of the Sun.    * d is distance of satellite from centre of the Sun (assume at least 10 million km).   * Assume satellite mass and diameter are extremely small relative to the Sun (e.g. 20kg and 2m respectively).   * Satellite is spherical with uniform density.    * No other bodies are in the system.   * All motions and accelerations are confined to a two-dimensional plane.    * Transverse acceleration is produced by a very small nuclear-powered mass-reaction rocket attached at the centre of the satellite.    * Consider the mass ejected by the rocket to be insignificant. Thus kinetic energy is added to and taken away from the system at different times, but there is no significant loss in mass.    * Assume a purely Newtonian System (no General Relativity please). I have seen various treatments of small _radial_ accelerations which use Lagrangian and Hamiltonian principles but I do not think these are extendable to _transverse_ accelerations (e.g. http:\/\/www.mathpages.com\/home\/kmath527\/kmath527.htm). UPDATE I have accepted Walter's excellent answer at http:\/\/astronomy.stackexchange.com\/questions\/632\/determining-effect-of-small- variable-force-on-planetary-perihelion-precession. (He used peturbation analysis to produce a precise formula for the rotation) But second opinions are welcome."}
{"id":"41610","title":"The difference between heat and temperature","text":"So as I understand it, heat energy of an object is the SUM of all the kinetic energies of the molecules of the object (upto constant factor). The temperature on the other hand is the AVERAGE of the kinetic energies of all the molecules of the object. Now when ice is melting at 0 degrees Celsius, the temperature as measured on a thermometer does not go up. The common explanation is that any heat being absorbed by the ice is being used to break the somewhat strong solid solid bonds between the molecules of the ice. Here is my question. if heat of an object is what I defined above, then since all the molecules are increasing in kinetic energy, the average of the kinetic energy should also increase, meaning the temperature should increase. But at 0 degrees celsius for ice that does not seem to be the case. Where am I going wrong in my understanding?"}
{"id":"129764","title":"The Physics Behind Chemical Bonding","text":"Ok so here's the problem: say I'm a sodium atom. It is 'charged' at +1e. A partner Chlorine atom is flying about, also 'charged' at -1e. According to chemistry (or rather the measurement of the reaction) energy is released if the two bond together. The problem is, no electrons changed energy states. Both electrons in the shared band have the same state as before. I've heard a number of conflicting answers. One is that since the extra space is given to the shared electrons, there is a relaxing of the shared electrons which gives energy. That makes sense, but I have a few more questions: Why is a valence band such a limiting factor? Not all atoms bond together to share electrons, mainly just the ones closest to a full shell, yet why does the atom care? Shouldn't the extra space for the electrons always be a good thing? Why do covalent reactions occur solely for these atoms with near- perfect valence shells? Also, sometimes atoms just swap electon partners, in redox reactions specifically. Why do opposite spins attract heavily enough to do work against the coulomb potential? Also, even if electrons with opposite spin attract, since when do we forget the coulomb attraction? That is, electrons repel each other on the average. Too many questions? That's fine. Let's also have too many answers. I want a clear and well-explained response, and would not encourage a rushed reply. Thank you."}
{"id":"15571","title":"Does the dimensionality of phase space go up as the universe expands?","text":"Ever since Hubble, it is well known that the universe is expanding from a Big Bang. The size of the universe had gone up by many many orders of magnitude as space expanded. If the dimensionality of the quantum phase space is finite because of spatial cutoffs at the Planck scale, does it go up as space expanded? If yes, how can this be squared with unitarity? If no, would this lead to what Tegmark called the Big Snap where something has got to give sometime. What is that something which gives?"}
{"id":"8274","title":"Can you give me some examples of recreational physics?","text":"Can you give me some examples of recreational physics? Thx"}
{"id":"8272","title":"net displacement and path dependence","text":"reading the paper about spacetime swimming by Wisdom   (something related to this has been previously asked here) can't help but think that there is more to this than what is on the paper. Basically the paper just proves that inertially, an object can get a net displacement by using the fact that phases are path-dependent due to non- commutativity in space paths, and he uses the rotational argument to properly ambient his point. The punchline here is that this same argument seems to be applicable when the masses have also for instance, a magnetic dipole, and there is a geometric phase in the magnetic field due to curvature Do you think that the same argument are applicable to obtain a similar effect due electric charges or magnetic dipoles in a curved manifold? i'm aware the question is vague but i hope this can entice (or stir) at least some deep discussion on the subject As a side (but very important) question; is not clear from the paper if the displacement is constrained to go only down in the gravitational potential well, but most discussions and papers referencing it says it should be so (check the conclusions section and comments)"}
{"id":"123255","title":"Question about an integration by parts in Feynman's Quantum Mechanics","text":"I have begun reading Feynman & Hibbs _Quantum Mechanics and Path Integrals_. Knowing little about variational calculus or Lagrangians I found the following integration by parts opaque. I think if I saw it done once methodically it would clarify a lot. I have no problem with integration by parts in general. On p. 27 he says that upon integration by parts the variation in $S$ becomes $$ \\delta S = \\left[\\delta x \\frac{\\partial L}{\\partial \\dot{x}} \\right]_{t_a}^{t_b} - \\int_{t_a}^{t_b}\\delta x\\left[\\frac{d}{dt}\\left( \\frac{\\partial L}{\\partial \\dot{x}}\\right)-\\frac{\\partial L}{\\partial x}\\right] dt. \\tag{2-6} $$ Now $$ S = \\int_{t_a}^{t_b}L(\\dot{x},x,t) dt \\tag{2-1}$$ in which $L$ is the Lagrangian $$ L = \\frac{m}{2}\\dot{x}^2 - V(x,t)\\tag{2-2}$$ and he says that to a first order $$\\delta S = S[\\bar{x}+ \\delta x] - S[x] = 0. \\tag{2-4}$$ He shows $S[x+\\delta x]$ explicitly in (2-5) and from this derives (2-6)."}
{"id":"44820","title":"Could dark energy be heat? Could dark matter be stuff that is not seen because of its refraction limits?","text":"Heat has to go somewhere. Would the light and heat of the stars in the universe amount up to being dark matter ...and dark energy ... Dark matter would just be small pieces of stuff. Dark energy would just be heat that has gone into space. If dark energy was just heat then a warp-drive could be as simple as a sphere with one very hot side."}
{"id":"126755","title":"Position Representation in Quantum Mechanics","text":"How does the 3d position operator look like in position representation? I know that in 1d the position operator $\\hat{x}$ is just multiplication by $x$."}
{"id":"65156","title":"Refraction of light and frequency dependence","text":"Why do higher frequency waves refract more, both ocean waves and light waves? Also why is energy stored in the frequency as opposed to the wavelength."}
{"id":"65151","title":"Pulley system with rope","text":"> Two masses $m_1$ and $m_2$ are connected through a weighty, uniform and an > unstretchable rope of length $L$. The surface is frictionless. Masses of the > objects are $m_1=m$, $m_2=\\frac{2}{3}m$, mass of the rope is > $m_r=\\frac{1}{3}m$. At which length $x_1$ of the vertical part of the rope, > the tension forces exerted by the rope on the bodies would be equal? What > acceleration does the system have? What is the type of the acceleration > (increasing\/decreasing). ![http:\/\/oi42.tinypic.com\/e9956e.jpg](http:\/\/i.stack.imgur.com\/nMkXx.png) I had no particular problems with this question, however, the book wrote the first second law applied to the rope as: $m_r a = N_1-N_2+\\frac{x_1}{l}m_rg$ What I got is a bit different: $m_r a = N_2 - N_1 + \\frac{x_1}{l}m_rg$ In this particular question, the signs of the tension forces aren't important, as we have to calculate the acceleration when both of them are of the same magnitude, and thus cancel each other out ($N_1=N_2=N$), but I want to know, how it should really look like. Also, I have a conceptual question - if we have a similar question, where the rope has some mass, and there's no information regarding the uniformity of the rope, can I assume that the rope is actually uniform (assuming all the other details are the same as in this question)? And what if the rope is non- uniform? What happens to the acceleration if the rope isn't uniform? Is it possible that the acceleration of such a system wouldn't be increasing nor decreasing, but rather be some arbitrary function (e.g. the acceleration may increase and decrease randomly)? And what if I have only a rope (no masses, no pulley), lying on such a frictionless surface, where part of it is hanging (vertical part) - will the hanging part always pull the horizontal one, and the whole rope will slide down? Will it just be the same equation as above, just without $N_1$ and $N_2$?"}
{"id":"96271","title":"Inner products in relativity","text":"In physics, the definition of a dot (inner) product is often between a vector (“contravariant vector”) and a covector (“covariant vector”). However, in mathematics, a dot product is always defined between two vectors (“contravariant vectors”). What is the mathematical justification for redefining the dot product? I understand why we do it and I know that it works, but I'd like to understand how to justify it from the point of view of mathematics. I'm asking because I've been told that it is not mathematically _a priori_ correct to do it that way."}
{"id":"96272","title":"Interesting identity on $SU(3)$","text":"In arXiv:hep-ph\/1307.5414 Grabovsky use an interesting identity which is not derived in the paper: $$\\tag{2.30}\\varepsilon^{ikm}\\varepsilon^{jln}\\left((U_{2}U_{4}^{\\dagger}U_{1})_{ij}+(U_{1}U_{4}^{\\dagger}U_{2})_{ij}\\right)(U_{4})_{kl}(U_{3})_{mn}=\\\\\\ -B_{123}+\\frac{1}{2}(B_{144}B_{324}+B_{244}B_{314}-B_{344}B_{214}).$$ Where: $$\\tag{2.6}B_{123}:=\\varepsilon^{ikm}\\varepsilon^{jln}(U_{1})_{ij}(U_{2})_{kl}(U_{3})_{mn},$$ which is known as \"Baryon\". I'm looking forward for some help to derive this identity and generalize it."}
{"id":"82908","title":"Impulse, relative velocties","text":"![enter image description here](http:\/\/i.stack.imgur.com\/TbAeS.jpg) Why is (m)(v)=Impulse or as they put it here, Vs = I\/Ms? Shouldn't (m)(v) be equal to momentum, not I? I don't understand why that is the solution. I was trying to solve for relative velocities, V2f - V1f, using momentum conservation but I failed hopelessly. I was thinking perhaps I should set momentum equal to zero so I have a center of mass frame, but I do not know how I should approach that. How and Why is this solution true?"}
{"id":"116551","title":"Effective medium approximation for calculating relative permeability of cylindrical shells of alternating magnetic\/nonmagnetic layers","text":"I'm trying to apply the effective medium approximation to my own research. http:\/\/ieeexplore.ieee.org\/ielx5\/20\/30863\/01430899.pdf?tp=&arnumber=1430899&isnumber=30863 As far as I understand, these approximations are for elliptical magnetic materials in a non-magnetic medium. I would like to have an approximation for a cylinder consisting of alternating layers of ferromagnetic and non-magnetic material. I have yet to find one. Can someone help me? Many thanks in advance. -Raphael"}
{"id":"116553","title":"Dirac sea entanglement","text":"The theory of the Dirac sea is in disrepute, but I still wonder about it sometimes. In this theory, the particles are all electrons, and obviously an electron with positive energy can be entangled with an electron of negative energy. But how, on this theory, can an electron with positive energy be entangled with a hole?"}
{"id":"89306","title":"Gravitational Stark Effect","text":"Could gravity induce line splittings in the optical spectrum of a molecule similar to the Stark or Zeeman Effects? Naively, a gravitational potential would be a simple addition to the Hamiltonian that should behave similarly to a constant electric field (i.e. the simplest condition for the Stark Effect). Is there a reason that things should not be so simple? If such splittings do occur, what would the typical energy difference be between the states? Would they be observable at tenable energies?"}
{"id":"19302","title":"In a universe where the speed of light is infinite, are relativistic models and Newtonian models equivalent?","text":"Take our universe. Observations are consistent with relativity, but not consistent with Newtonian mechanics. Assume that our current (relativistic) model of gravitation is correct. Now increase $c$ to infinity. Experimental observations in this altered universe are still consistent with relativity (by the assumption that relativity is correct). Are they now consistent with Newtonian mechanics as well?"}
{"id":"51948","title":"Does hidden variable quantum mechanics imply the collapse of atoms(electrons falling into the nucleus)?","text":"why do atoms not collapse on themselves. Doesnt this problem rule hidden variables as invalid as the heisenburg uncertainty is the solution to the problem because it says electrons exist in a probability cloud and is in many places at once, not at a single location. If hidden variables are right the electron has a well defined position orbiting the nucleus and where does the energy come from? Doesnt this mean if HV are right it would collapse in again. whats the answer?"}
{"id":"16179","title":"Poles, wavefunctions, transmission","text":"Why is it said that $\\operatorname{sech}x$ (a transmission amplitude) has a simple pole on the imaginary axis?"}
{"id":"5911","title":"Calculating the deflection of an arbitrary beam of light from a thin lens","text":"Suppose we have a lens with known focal length and are using the thin lens approximation. Is it possible to determine how an arbitrary beam of light will be deflected by it? Or, is it the case that there are multiple kinds of thin lenses, and so more information is required. I'm asking this question just so I can solve first year university optics problems."}
{"id":"17227","title":"How do we know Dark Matter isn't simply Neutrinos?","text":"What evidence is there that dark matter isn't one of the known types of neutrinos? If it were, how would this be measurable?"}
{"id":"97871","title":"Angular Velocity and Translation velocity after ball bounce","text":"A ball with radius $r$, angular velocity $\\omega$ and linear velocity $v$ hits a surface with coefficient of friction $\\mu$ at an angle $\\theta$ with the surface. I want to calculate the final angular velocity and translation velocity after the ball leaves the surface. Let $dt$ be the impact time."}
{"id":"121809","title":"Why $e$ in the formula for air density?","text":"I am reading a book that says that the density of air is approximately $D = 1.25 e^{(-0.0001h)}$, where h is the height in meters. Why is Euler's number $e$ used here? Was a differential equation used in deriving this formula?"}
{"id":"121800","title":"-Thought Experiment- At superluminal speeds, to what degree would vision be impaired?","text":"I was thinking about superluminal travel when thinking about advanced civilisations in the future. I was also thinking about the Doppler effect; so I decided to think about how light may be affected in a similar way (besides redshift). I was thinking about thing appearing to go back in time (when looking from behind a superluminal object), but then I found a fundamental flaw with that thought - any light that had previously hit the object before it travelled at such superluminal speeds would not somehow radiate out of the object again, therefore rendering everything behind invisible (shown in this diagram): ![v=c](http:\/\/i61.tinypic.com\/2957lp4.jpg) (The circles represent light waves that would radiate from the source if it were luminous.) But then I thought: \"If I went past light speed, and onto higher speeds altogether, would I have further limitations of vision?\" Here is another diagram of that: ![v=xc](http:\/\/i58.tinypic.com\/334l0f7.jpg) **My question:** If superluminal travel were possible, then would the \"funnel of vision\" be possible, and if so, how would we see it? Would we see it as a small, bluish (due to the Doppler effect) circle with darkness surrounding it or would light seem to bend, blueshift and contort itself?"}
{"id":"27908","title":"Pseudo-Superluminal Motion and the Synchronization of Clocks","text":"Let's consider two points A an B separated by a finite distance in curved space time. A light ray flashes across an infinitesimally small spatial interval at B. Metric: $$ds^2=g_{00}dt^2-g_{11}dx^2-g_{22}dy^2-g_{33}dz^2$$ ----------- (1) We may write, $$ds^2=dT^2-dL^2$$ Where, $dT= \\sqrt{g_{00}}dt$ and $dL=\\sqrt{ g_{11}dx^2+g_{22}dy^2+g_{33}dz^2}$ dL is the physical infinitesimal(spatial) at B. Observer at A records the same value of dL But for dT the observers at A and B record different values: $$dT_{A}=\\sqrt{g_{A(00)}}dt$$ $$dT_{B}=\\sqrt{g_{B(00)}}dt$$ Local Speed of light[observation from B]$$=c=\\frac{dL}{dT_B}=\\frac{dL}{\\sqrt{g_{B(00)}dt}}$$ Non-Local Speed of Light:[speed measured from A] $$C_{non-local}=\\frac{dL}{dT_A}=\\frac{dL}{\\sqrt{g_{A(00)}}dt}$$ Therefore : $$c_{non-local}=c \\times \\sqrt{\\frac{g_{B(00)}}{g_{A(00)}}}$$ --------- (2) For a particle moving across an infinitesimally small spatial interval at B wehave in a similar fashion: $$V_{non-local}=V_{local} \\times \\sqrt{\\frac{g_{B(00)}}{g_{A(00)}}}$$ -----(3) Since $V_{local}<c$ $$V_{non-local}<C_{non-local}$$ ----------- (4) From (2) we find that the non local speed of light may exceed the local barrier. The non_local speed of a particle may also exceed the local speed barrier. _But the light ray is always ahead of the particle [in purely local or non- ocal observations]according to relation (4)_. There is no violation of relativity as such. _Average speed of light for a specified path_ : Let's consider a light ray is passing from P to Q along some path.Observation is being made from the point P.Time interval observed from P is given by: $$dT=\\frac{dL}{C_{non-local}}$$ ------(5) $$=\\frac{dL}{c\\times \\sqrt{\\frac{g_{00}}{g_{P(00)}}}}$$ $g_{00}$ is the value of the metric coefficient at some arbitrary point of the path and $g_{P(00)}$ is the value of the metric coeff. at P. The average speed of light for the path between P and Q obtained by integrating relation (5),to obtain the time of travelof the light ray, may be different from the constant standard value of “c” _Query_ : Is it important to take care of such differences[differences in the local and non-local speed of light] in the synchronization of clocks in sensitive experiments?"}
{"id":"27906","title":"Angular momentum operator in terms of ladder operators","text":"I wanted to show that the angular momentum of the particle state with zero momentum $| \\vec{0} \\rangle$ is $0$, that is to say the intrinsic spin of a scalar field is $0$ using a mode expansion. There is a clever argument which is essentially that $e^{i \\vec{\\theta} \\cdot \\hat{\\vec{J}}} \\, | \\vec{p} \\rangle = | R(\\vec{\\theta}) \\vec{p} \\rangle$ and thus we have that for any $\\vec{\\theta}$ $e^{i \\vec{\\theta} \\cdot \\hat{\\vec{J}}} \\, | \\vec{0} \\rangle = |\\vec{0} \\rangle + i \\vec{\\theta} \\cdot \\hat{\\vec{J}} |\\vec{0} \\, \\rangle + \\cdots = |\\vec{0} \\rangle$ and hence it is clear that $\\hat{\\vec{J}} |\\vec{0} \\, \\rangle = 0$ which shows the desired result. I would like to show the same thing by using $\\hat{\\vec{J}} = -\\int d^3x \\, \\vec{x} \\times ( \\hat{\\pi} \\, \\nabla \\hat{\\phi})$ and use the mode expansions for $\\hat{\\phi}$ and $\\hat{\\pi}$: $\\hat{\\phi} = \\int \\frac{d^3p}{(2 \\pi)^3} \\frac{1}{\\sqrt{2 E_{\\vec{p}}}} (\\hat{a}_{\\vec{p}} + \\hat{a}^{\\dagger}_{-\\vec{p}})e^{i \\hat{p} \\cdot \\hat{x}}$ $\\hat{\\pi} = \\int \\frac{d^3p}{(2 \\pi)^3} \\sqrt{\\frac{E_{\\vec{p}}}{2}} (\\hat{a}_{\\vec{p}} - \\hat{a}^{\\dagger}_{-\\vec{p}})e^{i \\hat{p} \\cdot \\hat{x}}$ Doing this (and normal ordering) I get the following $\\hat{\\vec{J}} = - i \\int \\frac{d^3p}{(2 \\pi)^3} \\hat{a}^{\\dagger}_{\\vec{p}} ( \\vec{p} \\times \\nabla_{\\vec{p}}) \\hat{a}_{\\vec{p}}$ I am unsure of how to proceed with derivatives of the ladder operators and how to apply them to states in the Fock space. Could someone argue as to how one can apply this mode expansion to the state $| \\vec{0} \\rangle$ and show that it annihilates it, as it must."}
{"id":"69109","title":"3D coherence of light speckles- difference between Airy and gaussian speckles","text":"In this article, http:\/\/www.dfm.uninsubria.it\/laboferri\/web_page\/articoli_pdf\/PRA_2009-Magatti- Three_dimensional_coherence_of_light_speckles-experiment.pdf, the 3D-coherence properties of light speckles are studied at different distances from the source. What I don't understand is the distinction drawn between what the authors call Airy speckles and gaussian speckles. In particular, it's not clear under which experimental conditions Airy or gaussian speckles are produced. How should I choose my setup so that I can know a priori that the speckles will be Airy or gaussian?"}
{"id":"27900","title":"Dynamics - Beginner question","text":"I am trying to understand this question, but I keep coming to a dead end: Two identical rectangular masses rest on top of one another, the top is tied to a wall on the left, while the one on the bottom has a roped attached to it on the right side. What is the max force that can be applied to the rope on the bottom before the block will begin to slide out from underneath the top one. (the coefficient of friction is equal between the block and the ground, and the bottom block with the top block). Ok, so I am pretty sure this question is just asking me to find the net force on the bottom block, as applying a force larger then this will result in an acceleration of the lower block: $$ F_{net}=ma$$ $$ 2F_f=ma$$ $$2F_n\\mu=ma$$ $$2mg\\times2\\mu=ma$$ Ok, I am totally lost as to how solve this question. My intention is to find the net force acting on the lower block. I am pretty sure the the acceleration will be zero, resulting in the right side of the equation equaling zero. But I am not sure how to handle the coefficient of friction."}
{"id":"99364","title":"Continuum limit for solid mechanics","text":"Is there a rigorous derivation of the limits for continuum properties in solid mechanics? For instance, the stress-strain relationship may be linear for large samples (the slope being the Young's Modulus) but at what limit does that break down? Even if a linear relationship exists for the stress-strain curve between two atoms, or two crystals, the slope likely isn't the same as for the large sample. What spatial scale separates the two regimes and is it similar to fluid dynamics where there is the Boltzmann equation at the non-continuum regime, the Navier-Stokes equations in the continuum regime, and a bridge between the two regimes (Chapman-Enskog equations)?"}
{"id":"99363","title":"Why meet at the center of mass?","text":"If two objects of different masses are held at a distance $d$ and then I let them go, they will meet at the center of mass of the particle system due to mutual gravitational attraction My question (which may be rather silly) is that why do they meet a the center of mass, why not anywhere else? Is there some sort of mathematical proof behind this?"}
{"id":"90366","title":"Friction acting on mass on string","text":"> _Assume we have some stationary point mass that would slide, if not for some > coefficient of friction, along a light, inextensible string, attached to two > points at different heights. Considering forces on the mass whilst it is > stationary, which direction does the frictional force act?_ My first thought is that it will act along the steeper of the two strings, but the concept of a reaction force (which surely friction must still be acting perpendicular to) seems less clear for a string? ![enter image description here](http:\/\/i.stack.imgur.com\/293gN.gif) This diagram gives an idea of the kind of set up i'm thinking about (from a random webpage with a sufficiently useful diagram, not specific to this problem), although the string can be of any length and thus don't assume that either string is horizontal."}
{"id":"10799","title":"Quantum circuit decomposition","text":"I need to construct a universal quantum circuit decomposition for a three- qubit operation where one qubit is the control bit, and a two-qubit unitary operator acts on the other two depending on the control bit. I can do the decomposition when there's a one-qubit unitary operation with as many control bits as needed; this two-qubit unitary with one control bit I am not clear about. Anyone has any clue what the recipe for this construction is?"}
{"id":"95709","title":"Can I feel (the results of) electromagnetic induction from overhead rail lines in the human body?","text":"A few weeks back, I was standing at a Hornsey rail station (in the UK) which uses overhead lines, and particularly has a number of parallel rail lines, all close to the platform. While I was standing there, I went to hold hands with my girlfriend, and experienced a peculiar sensation in my fingertips, almost as if there were very large ridges on her hands. I've felt this before, touching a metal bodied laptop computer that was seemingly not earthed - dragging my finger across the surface my finger felt as if it was jumping along the surface, despite it being a smooth aluminium surface. Is this a result of the overhead lines inducing a reciprocating current through my arm and\/or fingertips? Is there some other feature of the railway that could cause this effect? And were we in any danger of any sort?"}
{"id":"10793","title":"Why are smaller animals stronger than larger ones, when considered relative to their body weight?","text":"I am interested in why many small animals such as ants can lift many times their own weight, yet we don't see any large animals capable of such a feat. It has been suggested to me that this is due to physics, but I am not even sure what to search for. Could someone explain why indeed it is easy for smaller objects\/lifeforms to support several times their own weight, but this is harder as objects\/animals become larger?"}
{"id":"21683","title":"Correlation\/relationship between human magnetic field and body temperature","text":"Is there any relationship between human magnetic field and body temperature? I am looking for studies on this topic."}
{"id":"78214","title":"Collision of 2 neutrons","text":"If two neutrons collide in 3D space and we want to determine the final velocities of both nuetrons (3 components for each neutrons), we can use the conservation of momentum equations and the conservation of energy equation. Those equations are 4 while the number of unknowns we have is 6. There are 2 free variables. For classical perfectly spherical objects, the solution is strait forward as explained here. However, neutrons because of their quantum nature can't be treated as classical perfectly spherical objects. We can't define a precise point of collision, so the trick we use for classical objects won't work for neutrons or any quantum level particle, or will it? If it doesn't work, does that mean we can't avoid having free two degrees of freedom in this system?? NOTE: What is meant by a collision is a momentum-exchange event"}
{"id":"10248","title":"Why trimming the mainsail gives a larger torque to head up","text":"I am learning sailing on a 5m catamaran (Nacra 5). I am familiar with basic aerodynamics and the physics of the sail and keel. We learned that when sailing closed hauled, too tight a mainsail tends to bring the boat up to the wind. And that the opposite is true for the jib. For example, one may steer up to the wind to come about, using the mainsail alone, by trimming it tight. My question is why the trimmed mainsail in the above setup gives a larger torque. This also seems opposed to the rule: \"The more the mainsail is sheeted out the more the boat tends to come up.\", as explained in http:\/\/www.sailtheory.com\/mandf.html#sailsteering Edit: To my understanding, there are several possible competing effects involved: (1) Effects that tend to INCREASE the mainsail torque to head up: (1A) The direction of the sail force becomes more perpendicular to the boat. This increases the heeling torque. Since heeling motion happens faster than turning, the boat will heel more. This moves the sail force out, which increases the lever arm and the torque to head up. see diagram in: http:\/\/www.sailtheory.com\/mandf.html#hellingstuur (1B) The sail force moves backwards since the sail is stretched backwards. This increases the lever arm and the torque. (2) Effects that tend to DECREASE the mainsail torque to head up: (2A) The direction of the sail force becomes more perpendicular to the boat. Assuming that the force is perpendicular to the sail, and that the center of rotation is between the mast and the center of force, one sees that pulling the sail in, reduces the torque to head up, as can be seen in the following diagram: ![diagram of the sail force torque](http:\/\/i.stack.imgur.com\/9I2TX.jpg) As Theta gets smaller, the torque Tau is reduced. This is opposed to what is stated in some of the answers below. (2B) The sail force is reduced since the sail was pulled beyond its optimal angle of attack, thus losing lift and reducing torque. (2C) Due to (2B) heeling torque is also decreased. With a similar reasoning to (1A) this decreases the lever arm and torque. (2D) The sail force moves forward since the aft part of the sail is too flat and stalls. This shortens the lever arm and reduces torque. This is described in http:\/\/www.sailtheory.com\/mandf.html#sailsteering We know from reality that 1A+1B > 2A+2B+2C+2D. Considering heeling alone, we also know that 1A > 2C. It remains to be explained why in general this is the case."}
{"id":"63360","title":"Stress energy tensor of a perfect fluid and four-velocity","text":"In the following demonstration, there is an error, but I cannot find where. (I explicitely put the $c^2$ to keep track of units). We consider a metric $g_{\\mu\\nu}$ with a signature $(-, +, +, +)$ : $\\boxed{g_{\\mu\\nu} = \\begin{pmatrix} -c^2 & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 1 \\end{pmatrix}} \\qquad\\qquad \\boxed{ g^{\\mu\\nu} = \\begin{pmatrix} \\frac{-1}{c^2} & 0 & 0 & 0 \\\\\\ 0 & 1 & 0 & 0 \\\\\\ 0 & 0 & 1 & 0 \\\\\\ 0 & 0 & 0 & 1 \\end{pmatrix}} $ The tensor of a perfect fluid is: $ \\boxed { \\begin{aligned} T^{\\mu\\nu} &= \\left(\\rho c^2+P\\right)u^{\\mu}u^{\\nu}+Pg^{\\mu\\nu}\\\\\\ T_{\\nu}^{\\mu} &= \\left(\\rho c^2+P\\right)u^{\\mu}u_{\\nu}+P\\delta_{\\nu}^{\\mu}\\\\\\ T_{\\mu\\nu} &= \\left(\\rho c^2+P\\right)u_{\\mu}u_{\\nu}+Pg_{\\mu\\nu} \\end{aligned} }$ where $\\rho$ is the mass density. We can easily check that $\\rho c^2$ and $P$ have the same unit. If we consider only the first component $00$, there is no pressure $P$ involved (I consider a cosmological case where the fluid is at rest in comoving coordinates). The only solution to have that is that $u^0u^0=-g^{00}$, so that the contravariant perfect fluid tensor become : $T^{00} = \\left(\\rho c^2+P\\right)u^{0}u^{0}+Pg^{00}=\\left(\\rho c^2+P\\right)\\times{-g^{00}}+Pg^{00}=-\\rho c^2 g^{00} = -\\rho c^2 \\times -\\frac{1}{c^2}=\\rho$ which is the expected result. So, to have the expected result, one should have $\\boxed{\\left(u^0\\right)^2=+\\frac{1}{c^2}}$. Now come the weird part. The metric can be written: $ds^2=g_{\\mu\\nu}dx^{\\mu}dx^{\\nu}=-c^2dt^2+dx^2+dy^2+dz^2=dt^2\\left(-c^2+\\frac{dx^2}{dt^2}+\\frac{dy^2}{dt^2}+\\frac{dz^2}{dt^2}\\right)\\approx-c^2dt^2$ which is equivalent to : $\\frac{dt^2}{ds^2}\\approx-\\frac{1}{c^2}$ which means : $\\boxed{\\left(u^0\\right)^2=-\\frac{1}{c^2}}$ So we clearly have a problem of sign here between the two expressions of $u^0$. **QUESTION: where is the mistake ?**"}
{"id":"122861","title":"A question on an exercise from Gravitation by Misner, Thorne and Wheeler","text":"My question is on problem 4.1 of Gravitation. In a generic case of electric field and magnetic field(i.e not $E=0$ or $B=0$ or $E$ and $B$ perpendicular), define the direction $\\hat{n}$ unit vector , $$\\hat{n}\\tanh (2\\alpha)=\\frac{2\\vec{E}\\times\\vec{B}}{\\vec{E}^{2}+\\vec{B}^2}$$ and $\\vec{\\beta}=\\tanh(\\alpha)\\hat{n}$ is the velocity vector. Show in the frame of the rocket with velocity $\\vec{\\beta}$, the Poynting vector vanishes. I tried the following but I am stuck at the cancellation. Let the $\\bar{\\vec{E}}$ and $\\bar{\\vec{B}}$ be the field in the rocket frame and the field without bars be the field of the rest frame. direction parallel along the velocity of rocket is denoted as subscript $\\parallel$ and direction perpendicular to velocity of rocket direction is denoted by $\\perp$ as the subscript. By lorentz transformation. \\begin{align} \\bar{\\vec{E_{\\parallel}}}&=\\vec{E_{\\parallel}}, \\\\\\ \\bar{\\vec{E_{\\perp}}}&=\\frac{\\vec{E_{\\perp}}+\\vec{\\beta}\\times\\vec{B_\\perp}}{\\sqrt{1-\\beta^{2}}},\\\\\\ \\bar{\\vec{B_{\\parallel}}}&=\\vec{B_{\\parallel}}, \\\\\\ \\bar{\\vec{B_{\\perp}}}&=\\frac{\\vec{B_{\\perp}}-\\vec{\\beta}\\times\\vec{E_\\perp}}{\\sqrt{1-\\beta^{2}}}. \\end{align} Note $\\beta\\times X_{\\perp}$=$\\beta\\times X$ In barred frame, \\begin{align} \\bar{\\vec{E}}\\times\\bar{\\vec{B}} &=(\\bar{\\vec{E_{\\perp}}}+\\bar{\\vec{E_{\\parallel}}})\\times(\\bar{\\vec{B_{\\perp}}}+\\bar{\\vec{B_{\\parallel}}})\\\\\\ &=\\bar{\\vec{E_{\\perp}}}\\times\\bar{\\vec{B_{\\parallel}}}+\\bar{\\vec{E_{\\parallel}}}\\times\\bar{\\vec{B_{\\perp}}}. \\end{align} Plug in the lorentz transformation. One gets $$\\frac{\\vec{E}\\times\\vec{B}-\\vec{E}_{\\parallel}\\times(\\vec{\\beta}\\times\\vec{E})-\\vec{B}_{\\parallel}\\times(\\vec{\\beta}\\times\\vec{B})}{\\sqrt{1-\\beta^{2}}}.$$ Now the last two terms looks generically like $\\vec{X_{\\parallel}}\\times(\\vec{\\beta}\\times\\vec{X}$) simplifies to $\\vec{X_{\\perp}}(\\vec{X}\\cdot\\vec{\\beta})$. However, I could not see the cancellation at this stage. Did I do something wrong?"}
{"id":"107234","title":"Wave function not normalizable","text":"Does the solution of the Schrodinger equation always have to be normalizable? By normalizable I mean, given a wavefunction $\\psi(x)$ $$\\int_{-\\infty}^{\\infty}|\\psi(x)|^2 dx<\\infty \\qquad \\text{or}\\qquad \\int_{0}^{\\infty}|\\psi(x)|^2 dx <\\infty$$ What would be the physical implications if one (or both) of those integrals diverges. From the viewpoint that the Copenhagen interpretation is one of the most popular and the wave function is interpretated as a probability distribution in this case; will the wavefunction diverging be valid for any other interpretation of quantum mechanics. Does anyone know any wavefunctions which are not normalizable? What if there was a singularity at 0 that made it diverge at all times. For instance $$\\int_{0}^{\\infty}|\\psi(x)|^2 dx \\to\\infty\\quad \\text{but}\\quad \\int_{a}^{\\infty}|\\psi(x)|^2 dx <\\infty $$ where $a>0.$ Would the second integral count as a valid pdf?"}
{"id":"68476","title":"What if the Moon was 100x more massive: Binary planets?","text":"I was reading a couple of Earth-Moon related questions (Mars just collided with Earth! A question of eccentricity, Could the earth have another moon?) and they got me thinking about planet-moon systems in general. Binary star systems are pretty common. The types of the two stars in the binary can vary pretty widely (main sequence, puslar\/neutron star, black hole, white dwarf, giant phases, etc, etc), but some are formed of a pair of roughly equal mass (within a factor of <10). I can't say I've ever heard of a binary planet system, though. Of course a planet with a moon is sort of a binary, but I've never heard of an equal mass binary planet. I think the closest thing in the solar system would be the Pluto-Charon system, with a mass ratio of about 10:1. Is there any reason a binary planet would be unstable? Obviously this is a three-body system, which has some inherent instability, but Earth-Moon-Sun seems pretty stable. Would increasing the mass of the Moon to match that of Earth make the system unstable? How about gas-giants? I think a Jupiter-Jupiter binary close to a star would be short lived because of three-body interactions, but what about further out? Would, for instance, a double-Jupiter or double-Saturn be stable in our solar system? Or is there some tidal effect that would cause the orbit to decay and the binary planet to merge? As an aside, it seems that binary asteroids aren't terribly difficult to find... perhaps we just haven't seen any binary planets yet because they're only stable relatively far away from their star, making them difficult to detect outside our own solar system?"}
{"id":"77393","title":"Mass of particle w.r.t. dimension","text":"I heard in a lecture recently - just as a comment - that a particle which is massive in say $D=4$ can be seen as a massless particle in higher dimensions and vice versa. Our prof didn't give any equations to explain this and I don't really see how this comes about. What's the trick?"}
{"id":"122868","title":"Determinant of ray transfer matrix","text":"Quick question on the wikipedia entry for the _ray transfer matrix_ , which is defined : ![enter image description here](http:\/\/i.stack.imgur.com\/Jt51r.png) where $x$ is the distance of a point from the optical axis and $\\theta$ the angle that the beam profile at that point makes with the optical axis. Now, wikipedia says:![enter image description here](http:\/\/i.stack.imgur.com\/2eMFv.png) but does NOT provide a reference. Does anyone have any idea on how to get that results? Do you know a website\/paper where this is discussed?"}
{"id":"68470","title":"Under what conditions is a pure fusion nuclear bomb possible?","text":"Is it possible to build a pure fusion and powerful nuclear bomb? (because due to the harsh conditions of enriched uranium, I would guess most of today's world countries will want to build pure fusion bombs.) A pure fusion bomb is a nuclear bomb that does not need fission so does not need enriched uranium."}
{"id":"67628","title":"Is it theoretically possible for a person to pass through a solid wall\/object?","text":"I understand that matter cannot pass through other solid matter because of the electrons that orbit an atom prevents this but I was curious to know if it is theoretically possible to somehow get passed this problem caused by the electron shell and by doing so would this allow th effective passage of matter through matter. My thinking was that on the scale of the atom the amount of space taken up by actual particle mass is small compared to the space between the nucleus and the electron cloud so if the electron problem can be circumvented then all that space could possibly be used to pass through. It would probably take some fanciful piece of tech based on unknown physics but i though i would ask!! Thanks."}
{"id":"67627","title":"Is it possible see an invisible light with something similar to a normal lens of an eyeglass?","text":"Let's say that I am a spy in a room with a lot of people and I want to communicate with my friend using Morse code via a light signal. Does there exist a type of light (laser or otherwise) that is invisible to the human eye, but that I can see using a modified transparent lens? In other words, can I make an infrared or ultraviolet light visible using something similar to a normal lens of an eyeglass? If yes, is it easy to buy or make it?"}
{"id":"67621","title":"Why are wires in simple circuits approximated as equipotentials?","text":"I just answered this question: Voltage in a circuit by asserting that any two points joined only by a wire, but no other circuit elements, are at the same value of the electric potential. It is, of course, true that in the context of electrostatics, the values of the electric potential at any two points inside of a perfect conductor are the same, but the electrostatics argument seems irrelevant for circuits. **Why are we justified in using this same fact for circuits in which we are clearly no longer faced with electrostatics?** **Edit.** As prompted by user twistor59 in the comments below, let me include a bit more detail to clarify the question and hopefully indicate why I don't think the answer is obvious. If we consider any segment of the wire as a small ohmic resistor, then as the resistance of any segment drops to zero, the voltage drop across that segment will be correspondingly small for a given current. In particular, zero resistance implies zero voltage drop. However, if I just have a piece of a perfect conductor sitting somewhere, then it is totally unclear to me why, or that it is even true, that the electric field will vanish in its interior if the situation is non-electrostatic. Perhaps, therefore, the magic lies in the microscopic physics that (in particular) lead to Ohm's Law?"}
{"id":"24338","title":"How fictitious are fictitious forces?","text":"How fictitious are fictitious forces? More specifically, in a rotating reference frame i.e. on the surface of the earth does an object that is 'stationary' and in contract with the ground feel centrifugal and Coriolis forces? Or are these forces purely fictional and used to account for differences in observed behaviour relative to an inertial frame? To give a practical example a turreted armoured vehicle is sitting stationary and horizontally somewhere in the UK. The turret is continually rotating in an anti-clockwise direction. Do the motors that drive the turret's rotation require more power as the turret rotates from east to west and less power as the turret rotates from west to east? i.e. are the turret motors cyclically assisted and hindered by the earths rotation?"}
{"id":"24489","title":"What exactly is the difference between advection and convection?","text":"After reading Wikipedia articles on advection and convection, I still cannot determine whether there is a consensus on a difference between these two terms. Sometimes, the term convection seems to include advection and diffusion, sometimes not. After all, equations exhibit two different terms for advection and diffusion, and we don't call the sum of them a convective term. When we talk about a convective term, it is indeed the advective term! It seems:   * transport of pollutants in a river by bulk water flow downstream relates to **advection**   * boiling water for pasta relates to **convection** Maybe, a conserved property such as energy can be advected, but we only use convection for transport of \"ensembles of molecules\". So, is there a difference? Could we figure it out with examples?"}
{"id":"54102","title":"Was the 2013 meteor over Russia stronger than an atomic bomb?","text":"Reports of the Russian meteor event (2013) say that it released more energy than 20 atomic bombs of the size dropped on Hiroshima, Japan: > Scientists estimated the meteor unleashed a force 20 times more powerful > than the Hiroshima bomb, although the space rock exploded at a much higher > altitude. Amy Mainzer, a scientist at NASA's Jet Propulsion Laboratory, said > the atmosphere acted as a shield.   >  The shock wave may have shattered windows, but \"the atmosphere absorbed the > vast majority of that energy,\" she said. > > http:\/\/abclocal.go.com\/kfsn\/story?section=news\/national_world&id=8994311 Really? Wouldn't that have done more damage than was seen? How does the damage depend on how quickly the energy is released?"}
{"id":"62014","title":"Torque, lever and mass","text":"The Force used in a catapult is exerted near its axis. If we double the length of the arm of the catapult, but still use the same Force at the same point as before near the same axis, does the distance in the Torque equation change?Edit (In an experiment we have found that the longer arm\/lever makes the ball fly a longer distance, but how do we compute this into the equation?) Also, how can one combine the mass of the ball used by the catapult with the calculation of the torque?"}
{"id":"106840","title":"A bomb can be made to explode or implode?","text":"A bomb can be made to implode or explode. Can one be made to explode while creating a vortex at the same time?"}
{"id":"115141","title":"Question: if a nuclear reactor of a submarine explodes will it contaminate the surrounding waters?","text":"After watching a Naval War movie I began to ask myself this question and I would like to know."}
{"id":"34479","title":"Is there really time reversibility in physics?","text":"First, how i got to the question. I was randomly looking at this page: http:\/\/en.wikipedia.org\/wiki\/Langton%27s_ant describing some sort of cellular automaton. Here is the path of this automaton after more then 10000 steps: ![Langton's ant trajectory](http:\/\/upload.wikimedia.org\/wikipedia\/commons\/d\/d3\/LangtonsAnt.png) You can clearly see that after some time ant gets stuck in the loop (actually its a highway) and will be going on it forever. Here comes the problem with many faces: If we turn back time and start ant from one of the loop(highway) points we will never get to the original messy part, so obviously this computation is not time reversible since there is no way out of the loop! :) (But there should be ! :D) Another thing is that information is also not time reversible - since if at some point in computation operation DELETE is used to erase all the previous data there is no way to get that data back. Ok. So far so good. Lets take this question one step further. How is that so? They always told us laws of physics are time reversible and so on. My intuitive answer is following. We can say that in the universe there are so called LAWS (physics and so on) and OPERATIONS. What is the difference? LAWS do not change the objects on which they are operating and hence probably are time reversible. OPERATIONS on the other hand can change objects, their number in the system and hence are not time reversible (if I delete some files there is no algorithm how to restore them again) I understand its very close to the idea and interpretation of information. So when they show us movie backwards and we think time got reversed its actually false! Yes you can reverse the motion of the people and objects, but you can not reverse speach for example since there is no logical connection between phrases being said. Here we can see how the content of the movie (life) is actually twofold - time revesible part (geometry in the poorest sense) and informational part (speech, computations and so on). From all above it seems that the universe overall is NOT time reversible, and there is no way to go back in time! :) Or is there?"}
{"id":"115145","title":"Do photons in interferometer violate the law of conservation of mass?","text":"I mean Mach-Zehnder Interferometer, where light split into two shortest paths. Is light after splitting wave or particle? Is it a particle? How could one photon change to two? If the wave, does the sum of the energies of both waves divided by the square of the speed of light equal to the weight of the original photon? ![Mach-Zender interferometer](https:\/\/upload.wikimedia.org\/wikipedia\/commons\/d\/d8\/Mach_Zehnder_interferometer.svg) Thank you."}
{"id":"34473","title":"maximum positive electric charge of solid body","text":"What are the limiting factors on the positive charge of a solid body? If I assume a 'perfect insulator' environment that would not exchange charge with my solid body, I would guess that I can remove electrons from the body until it's internal chemical bond structure made by them breaks down and the body disintegrates. Are there any other other limits that would be hit first? How to calculate the charge at which the body breaks away?"}
{"id":"23596","title":"How do we measure $i[\\hat\\phi(x),\\hat\\phi(y)]$ in QFT?","text":"What operational procedure is required to measure $i[\\hat\\phi(x),\\hat\\phi(y)]$ in an interacting (or non-interacting) QFT? [assume smearing by test- functions, or give an answer in Fourier space, for $i\\bigl[\\widetilde{\\hat\\phi}(k),\\widetilde{\\hat\\phi}(k')\\bigr]$, to your taste; give electromagnetic field indices to $\\hat\\phi(x)$ and transpose the question to QED if the scalar field case seems unreal in an essential way]. In a non-interacting QFT, $i[\\hat\\phi(x),\\hat\\phi(y)]$ is a c-number, but in an interacting QFT it may presumably be an observable independent of $\\hat\\phi(x)$ and $\\hat\\phi(y)$. How do we measure it, however? If we cannot measure it, is it effectively chosen by convention? An answer that describes the (operational) difference between $A=\\hat\\phi(x_1)...\\hat\\phi(x_i)\\hat\\phi(x_{i+1})...\\hat\\phi(x_n)$ and $B=\\hat\\phi(x_1)...\\hat\\phi(x_{i+1})\\hat\\phi(x_i)...\\hat\\phi(x_n)$ might be equally good (though $A^\\dagger\\not=A$ and $B^\\dagger\\not=B$). As always, a reference would be good. EDIT: Following Luboš' answer, and more particularly his comments, for which again thanks, $i[\\hat\\phi(x),\\hat\\phi(y)]$ is zero if $x-y$ is space-like, both in non-interacting and interacting quantum fields. If a quantum field $\\hat\\phi(x)$ is the basis for an empirically adequate quantum theory, measurement operators at time-like separation from a given space-like hyperplane at time $t$ can in principal be expressed as a nontrivial function of the field and field momentum at time $t$, $\\hat\\phi(y)=\\int[\\hat\\phi(t,\\mathbf{x}')f_t(\\mathbf{x}')+\\hat\\pi(t,\\mathbf{x}')g_t(\\mathbf{x}')]\\mathrm{d}^3\\mathbf{x}'$, for two functions $f_t$ and $g_t$, which for a given theory are determined by the Lagrangian. Given many implicit assumptions, measuring the commutator $i[\\hat\\phi(t,\\mathbf{x}),\\hat\\phi(y)]=-g_t(\\mathbf{x})$ (because $[\\hat\\phi(t,\\mathbf{x}),\\hat\\pi(t,\\mathbf{y})]=i\\delta^3(\\mathbf{x}-\\mathbf{y})$) is equivalent to measuring $g_t(\\mathbf{x})$. If we also measure $i[\\hat\\pi(t,\\mathbf{x}),\\hat\\phi(y)]$, we can determine (or at least significantly constrain) the Lagrangian. _If we do not know the Lagrangian_ , therefore, it would seem to be desirable to have an operational way to measure $i[\\hat\\phi(x),\\hat\\phi(y)]$ in order to _measure_ the Lagrangian (instead of the usual approach of guessing the Lagrangian and seeing how close the consequent predictions are to the measurements; the haphazardness of this process is something of a crisis in HEP). I would welcome comments on this or an answer that takes this EDIT into account. Alternatively, an operational approach to _measuring_ the Lagrangian would implicitly be a measurement, in principle, of $i[\\hat\\phi(x),\\hat\\phi(y)]$; is there such a thing?"}
{"id":"27689","title":"Does Kaluza-Klein theory successfully unify GR and EM? Why can't it be extended to the Standard Model gauge group?","text":"As a quick disclaimer, I thought this might be a better place to ask than Physics.SE. I already searched there with \"kaluza\" and \"klein\" keywords to find an answer, but without luck. As background, I've been reading Walter Isaacson's biography of Einstein, and I reached the part where he briefly mentions the work of Kaluza and Klein. Also, I did my undergraduate degree in theoretical physics, but that was quite a few years ago now... The way I understand the original work of Kaluza and Klein is that you can construct a theory that looks like a 5-D version of GR that reproduces the Field Equations and the Maxwell Equations. An important bit is that the 4th spatial dimension is the circle group $U(1)$, which we now know as the gauge group of EM. I guess the first part of the question should be whether I've understood that correctly. Then, if you build a Kaluza-Klein type theory, but use the SM gauge group $U(1)\\times SU(2)\\times SU(3)$ instead of $U(1)$, what do you get? Is it gravity and the standard model together? If not, then what? The Wiki article on Kaluza-Klein theory says that this logic \"flounders on a number of issues\". The only issue is explicitly states is that fermions have to be included by hand. But even if this (Kaluza-Klein)+(SM gauge) theory only describes interactions, isn't that okay, or at least a big help?"}
{"id":"51576","title":"Does electric potential have a temperature?","text":"When I took my first thermo class a tucked away chapter introduced Exergy in terms of electrical energy, meaning that the amount of electrical energy you could get from something is functionally its exergy. Wikipedia doesn't go so far, but this has bothered me for years. > Forms of energy such as macroscopic kinetic energy, **electrical energy** , > and chemical Gibbs free energy **are 100% recoverable** as work, and > therefore have an exergy equal to their energy. However, forms of energy > such as radiation and thermal energy can not be converted completely to > work, and have exergy content less than their energy content. Let's say that we have 1 Volt of potential. The smallest unit of energy we could isolate from this would be a single electron accelerated over the potential. Numerically and naively I could write: $$ 2\/3 k T = V e$$ Where $e$ is the electron charge. Doing this I would get something like $17,000 K$ for the temperature. This is high enough to approximate to infinity for conversion processes where your sink is room temperature. I can imagine accelerating something with a greater charge than electron over that potential. This method would find a higher temperature, which presents a counter-argument. So what is it, **is there a theoretical limit to the energy you can get from electricity, giving it a functional temperature?** That would imply an unavoidable loss from DC-DC voltage converters. Another thing that bothers me is saying that radiation can't be converted completely to work. Thermal radiation, yes, but most nuclear radiation is at $keV$ or $MeV$ energies, which would give a dramatically higher temperature than electrical energy (if I've understood correctly)."}
{"id":"113587","title":"Gravitational force from spherical shell","text":"Say we have a point mass $\\mu$ located at $(0,0,R)$ and a spherical shell ( _not_ considering it's volume!) of radius $R$ located in the origin. So we have a particle standing right on top of the sphere and we want to determine the total gravitational force exerted on the particle by the spherical shell. Since the force is directed along the z-axis, we only have to consider the z-component of the force. Moreover, suppose the shell has homogeneous density $\\sigma$. We can then write $dM = \\sigma dA = \\sigma R^2 \\sin \\theta d\\theta d\\phi$. So we have $$ dF_z = \\frac {(z-R)G\\mu dM}{\\sqrt{x^2 + y^2 + (z-R)^2}^{\\ 3}} = \\sigma \\mu G R^2 \\frac { (z-R)\\sin \\theta d\\theta d\\phi }{ \\sqrt{x^2 + y^2 + (z-R)^2}^{\\ 3} } $$ We then integrate over the entire sphere: $$ F_z = \\iint \\sigma \\mu G R^2 \\frac { (z-R)\\sin \\theta d\\theta d\\phi }{ \\sqrt{x^2 + y^2 + (z-R)^2}^{\\ 3} } $$ This turns out to equal $-2\\pi G \\sigma \\mu$ and $\\sigma = \\frac M{4\\pi R^2}$, where $M$ is the total mass of the spherical shell, which yields $-\\frac {GM\\mu}{2R^2}$. What I do not understand why there is a factor 2 in the denominator, should it really be there? Just for the protocol, we get the classical formula if the particle is outside the shell and zero if it is inside the shell."}
{"id":"126413","title":"Difference between weight of water and pressure of water","text":"_(I didn't even have a basic formal education in physics. I'm learning through the internet out of my own interest, so if there are any silly mistakes, kindly bear with and guide me through.)_ Everywhere, everyone is saying that pressure will same be for a given height. How is it possible? When volume changes, doesn't pressure change in calculations? [Pressure will be 2.5 bars approximately (including atmospheric pressure) for a height of 15 metres from ground level. I checked on many sites that 2.5 bars equals 2.5 kg\/cm².] Now let's take **base area as** **2000 cm² and height of 15 m as constant** in three scenarios: EACH SCENARIO IS INDEPENDENT AND TUBES ARE SEPARATE, I.E NOT CONNECTED WITH EACH OTHER In **scenario 1** a straight vertical tube from ground level will have capacity of 3000 litres with a weight of 2.5 kg per cm² [3000 litres \/ 2000 cm² = 1.5 kg\/cm² + 1 kg\/cm² atmospheric pressure]. In **scenario 2** the tube becomes narrow from the base area, resulting in a volume of 2000 litres. In **scenario 3** the tube becomes wide from the base area, resulting in a volume of 4000 litres Question: When they say pressure remains the same at a given height, does it mean that weight will be 2.5 kg\/cm² in all 3 scenarios, where volume is 3000, 2000, or 4000 litres (base area 2000 cm², height 15 m constant)? How is it possible? Where am I wrong?"}
{"id":"126410","title":"Noncanonical commutation relation and noncanonical wave mechanics","text":"Consider noncanonical operators $\\hat{x}_1,\\hat{x}_2,\\hat{p}_1,\\hat{p}_2$ satisying the following condition in the $q_1,q_2$ - basis ($\\psi=\\langle q_1,q_2|\\Psi\\rangle$)(similar to wave mechanics): $$\\hat{x}_1\\psi=q_1\\psi-ia\\hbar\\frac{\\partial}{\\partial q_2}\\psi$$ $$\\hat{x}_2\\psi=q_2\\psi-ib\\hbar\\frac{\\partial}{\\partial q_1}\\psi$$ $$\\hat{p}_1\\psi=-i\\hbar\\frac{\\partial}{\\partial q_1}\\psi+cq_2\\psi$$ $$\\hat{p}_2\\psi=-i\\hbar\\frac{\\partial}{\\partial q_2}\\psi+dq_1\\psi+i\\hbar\\gamma\\frac{\\partial}{\\partial q_1}\\psi$$ The commutation relations matrices are (similar to Hilbert-space quantum mechanics): $[x_i,x_j]=i\\hbar\\left(\\begin{array}{cc} 0 & (b-a) \\\\\\ (a-b) & 0 \\end{array}\\right)$   $[p_i,p_j]=i\\hbar\\left(\\begin{array}{cc} 0 & (c-d) \\\\\\ (d-c) & 0 \\end{array}\\right)$ $[x_i,p_j]=i\\hbar\\left(\\begin{array}{cc} 1 -ac & -\\gamma \\\\\\ 0 & 1-bd \\end{array}\\right)$ Because $\\hat{x}_1,\\hat{x}_2$ are not commute so I have to introduce 2 other parameter $q_1,q_2$. My question is: is it true or is it not true that all wave mechanics theories with the same commutation relations will predict the same results?"}
{"id":"110463","title":"Is frequency quantized?","text":"I'm aware that there're some questions posted here with respect to this subject on this site, but I still want to make sure, is frequency quantized? Do very fine discontinuities exist in a continuous spectrum like the black body spectrum? The quantization of photon energies"}
{"id":"110460","title":"Why does gravitational collapse occur suddenly in a supernova progenitor?","text":"I was reading the Wikipedia article on Supernovae, and it says that one of the reasons why a supernova occurs is due to sudden gravitational collapse when the core of the star has little fusable material. But why is this collapse sudden? Shouldnt the collapse be gradual as the fusable material gradually decreases? Why is the sudden collapse like driving off a cliff? (if I've interpreted the article rightly.)"}
{"id":"28945","title":"Variational wavefunctions and \"spread\" of potential in quantum mechanics","text":"A particle in a box has an energy that decreases with the size of the box. In the general case, it is often said that a variational solution for a \"narrow and deep\" potential is higher in energy than a variational solution for a \"wider and shallower\" potential, by appealing to the particle in a box as a special case. The former wavefunction is said to be \"more confined\" in the former than the latter, and the act of being given more space to spread out is said to lower the energy variationally. Question: are there _rigorous mathematical statements_ of this argument?"}
{"id":"51683","title":"Is enthalpy defined under non-isobaric conditions?","text":"My question is this, Does Enthalpy have a meaning under non-isobaric conditions? Is its existence as a property of a system independent of whether the system is under isobaric condition or not? Edit:- I wanted to know if enthalpy as a property of a system is valid only if from its creation to current state, it undergoes only isobaric processes."}
{"id":"60869","title":"What is the definition of a timelike and spacelike singularity?","text":"What is the definition of a timelike and spacelike singularity? Trying to find, but haven't yet, what the definitions are."}
{"id":"61593","title":"How do we know that there isn't a classical solution to the measurement problem\/Quantum Mechanical uncertainty?","text":"It was mentioned to me that it can be shown that there is no classical explanation for the uncertainty in Quantum Mechanics -- i.e. that there are no hidden workings that we have just not yet seen, which could be explained classically and would explain the probabilistic nature of Quantum events in a 'deterministic' fashion. Can someone explain how this is known please?"}
{"id":"73151","title":"Can we live forever by taking advantage of the time dilation effect?","text":"I have been reading about Time dilation and twin paradox in Relativity. So,I thought that if It can be the difference of 10 yrs between the Girl in space and on Earth, then why can't anyone not age rather. Is it possible? [I'm 14 and new in the field of modern physics, so pls correct if I'm wrong. I regard All of You as teachers.]"}
{"id":"73158","title":"Symmetry of Euler-Lagrange equations and conservation laws","text":"1. Continuous symmetry of the action implies a conservation law, but what if equations of motion have a continuous symmetry? Does it imply a conservation law?    2. Also is symmetry of equations of motion also a symmetry of the action?  I assume that the equations of motion I am talking about follow from the action, i.e. the equations of motion are Euler-Lagrange equations. Related examples could help and references also."}
{"id":"95288","title":"Are there any materials that change the electromagnetic wavelength of visible light?","text":"What i mean is for example exclusively red light comes in but gets reflected or refracted as blue or xray wavelength or whatever. Does this happen at all?"}
{"id":"98774","title":"Are magnetic hysteresis losses relevant to alternating currents flowing in a wire?","text":"Say we have an AC in a magnetically lossy material, like iron. Because of iron's relatively high permeability, skin effect will be more pronounced than it is in say, copper, so this iron wire isn't so great a conductor, practically speaking. However, skin effect arises due to eddy currents, which are themselves due to the time varying magnetic field due to the time varying current. So, this suggests there is a magnetic field at work, and there's the potential for magnetic hysteresis to be an additional source of loss in the material. Yet, I'm not sure if this is true or not, if somehow the geometry of the conductor and the fields around it make this a non-issue. I would think that if a time-varying magnetic field exists inside the iron, then there would be hysteresis losses. Is this true? Does such a field exist? To be clear: the issue is with iron _as an electric conductor_ , not as part of a magnetic circuit such as a transformer core as is the more common application."}
{"id":"56502","title":"Probability current density : Isn't there something wrong with this proof?","text":"Is there something wrong with the following proof (see below)? To me, it seems like the third line should show $$\\frac{dP_{ab}}{dt}=-\\int_a^b \\frac{\\partial}{\\partial t}J(x,t)dx$$ Am I missing something obvious? * * * ![enter image description here](http:\/\/i.stack.imgur.com\/9L5fT.jpg)"}
{"id":"32167","title":"Hermitian operator ?? is this possible","text":"we know that the operator $ H= - \\hbar ^{2} \\frac{d^{2}}{dx^{2}}+ V(x) $ is hermitian isn't it ?? however what would happen if the potential were still real but it depends on the Wave function, for example huypothetically $ V(x) = |\\Psi (x)|^{2} $ or $ V(x) = arg \\Psi (x) $ since the functions $ |x| $ and $ arg (x+iy) $ are ALWAYS real, then the hamiltonian with potentials (1) and (2) should be Hermitian but they depend on the solution so i am not sure about the Hermiticity of a Hamiltonian in the form $ H =p^{2}+ |\\Psi (x) |^{2} $ i believe that a real potential makes the Hamiltonian hermitian even in the case that we do not exactly know what potential is"}
{"id":"98199","title":"Adjoint of Gamma Matrices - Dirac","text":"I just started to learn how to quantise Dirac field. Meanwhile, as we can write the Dirac equation in terms of gamma matrices : $$ (i\\hbar\\gamma^\\mu\\partial_\\mu - m)\\psi = 0 $$ where $\\gamma_\\mu$ matrices obey Clifford algebra $$ \\\\{\\gamma^\\mu,\\gamma^\\nu\\\\} = \\eta^{\\mu\\nu} $$ Now I just came across that the adjoint of the gamma matrices can be written as : $$ \\gamma^{\\mu\\dagger} = \\gamma^0\\gamma^\\mu\\gamma^0 $$ I already checked this question, but it doesn't suggest a way to prove it in a representation independent manner. Also the Wikipedia page suggests that gamma matrices are chosen such that they satisfy the above relations, since they are arbitrary upto a similarity transformation. So is that the way this identity comes, or is there something else ?"}
{"id":"79945","title":"Can entanglement with an inaccessible system be useful?","text":"Quantum phenomena in bipartite pure state systems like teleportation are pretty well understood. What I'm interested in is the following situation: Alice, Bob and Charlie hold some general tripartite pure state $\\rho_{ABC}$. As usual, Alice and Bob are trying to perform some task, and are allowed only LOCC. Charlie's subsystem, however, is completely \"inaccessible\", by which I mean that Charlie can't do anything at all- no local operations and no classical communication with either of the other two. His role is solely to hold onto his part of the system. So my question is, is there any situation where Alice and Bob, with full knowledge of the tripartite state, would be able to perform some task 'better' (e.g. with a greater probability of success) with this setup than they would if they just traced out Charlie's system and were left with $\\rho_{AB}$? If the answer is yes, then I'd be interested to know of any general results in this area."}
{"id":"79946","title":"Does Saturation velocity in semiconductors have a relation with the wavelength in which the peak in the absorption spectrum occurs?","text":"> Saturation velocity is the maximum velocity a charge carrier in a > semiconductor, generally an electron, attains in the presence of very high > electric fields. (source) I want to know if the absorption peak can be calculated with this parameter when a semiconductor like GaAs is under illumination."}
{"id":"2111","title":"How long a straw could Superman use?","text":"To suck water through a straw, you create a partial vacuum in your lungs. Water rises through the straw until the pressure in the straw at the water level equals atmospheric pressure. This corresponds to drinking water through a straw about ten meters long at maximum. By taping several straws together, a friend and I drank through a $3.07m$ straw. I think we may have had some leaking preventing us going higher. Also, we were about to empty the red cup into the straw completely. ![alt text](http:\/\/i.stack.imgur.com\/VCLN5.jpg) My question is about what would happen if Superman were to drink through a straw by creating a complete vacuum in the straw. The water would rise to ten meters in the steady state, but if he created the vacuum suddenly, would the water's inertia carry it higher? What would the motion of water up the straw be? What is the highest height he could drink from? Ignore thermodynamic effects like evaporation and assume the straw is stationary relative to the water and that there is no friction."}
{"id":"90557","title":"Temperature rate of change inside a cylinder","text":"I was wondering about this today, mainly for baking a cylindrical cake. Say you have a solid cylinder which is at temperature `T0` inside. Say you then move the cylinder to a warm location (like an oven) which is at temperature `T1`. ![Solid Cylinder](http:\/\/i.stack.imgur.com\/4hE0W.png) **Is it possible to achieve a rate equation for`dT\/dt=f(t)` where `T` is the temperature exactly in the centre of the cylinder and `t` is time?** And therefore work out the time it would take this location to reach a temperature `T2` somewhere between `T0` and `T1`. To keep it as general as possible I would prefer to assume the cylinder is height `h` and radius `r`. I've been trying to use the thermal diffusivity equation: ![Diffusivity Eqn](http:\/\/i.stack.imgur.com\/XZa4O.png) But my working gets messy quickly, I figured as this is quite a simple shape it should be less complicated than some similar examples I've found. * * * EDIT: So using my notes from a differential equations course a few years back I managed to do the following: ![Working](http:\/\/i.stack.imgur.com\/0YeHf.png) But I'm stuck here.."}
{"id":"14325","title":"Electrostatics:Basic question on electric current","text":"I've got two questions for you.   1. Electric current is the flow of electrons across a conductor.Why has it always got to do with electrons and not with protons? (I know neutrons are not eligible for electricity as they carry no charge.)   2. Sometimes my computer starts properly after I disconnect all the cables and press and hold the power button for a few seconds and then re-connect all the cables (same thing with my modem). What's going on? I've heard that some amount of left over current get's drained. But where do they go? How did that excess current get accumulated in the first place?"}
{"id":"106021","title":"Expectation of dipole is vector in x direction","text":"So I am asked to find a state $|\\Psi(t)\\rangle$ in terms of the hydrogen wave functions, such that the expectation of the dipole operator -$e\\hat r$ is a vector in the x direction. I am not completely sure how to approach this. I am having a hard time visualing what this is trying to say in a sense of wavefunctions. I know what a dipole is, but I would expect the dipole moment for hydrogen states to be a vector in two directions. Any ideas on where to start?"}
{"id":"106020","title":"How exactly do protons and electrons interact with each other?","text":"How do these particles exchange information about charge and position between themselves, even though there's mostly empty space between them? Also what happens if a free electron passes closer to a nucleus of a hydrogen atom than its own electron? My random guesses so far:   1. Proton won't react because it's somehow locked with its existing electron   2. Proton will catch it and release the previous one    3. Proton will catch it and keep both of them with half the force   4. Existing electron will simply repel incoming one   5. Something else?"}
{"id":"106023","title":"Transformer and magnetic field\/flux through the iron core?","text":"I am familiar with how a transformer works. This is the diagram that I found on the wikipedia page:![Mangetic flux in a transformer](http:\/\/i.stack.imgur.com\/9ccpO.png) The current through the red solenoid varies with time, which creates a time- dependent magnetic flux (green) that also goes through the blue solenoid therefore inducing an electromotive force. Both solenoids are wrapped around a magnetic object so that this \"carries\" the magnetic flux created _inside_ the solenoid (which is much bigger than the one outside of it) around. I am not quite sure why the magnetic flux bends in the way displayed in the diagram. The magnetic material is ferromagnetic right? So shouldn't the magnetic field point produced inside the red solenoid force all the other domains to align with it? In other words shouldn't the magnetic field point in the same direction everywhere in the magnetic material (as I have tried to illustrate below) ?![enter image description here](http:\/\/i.stack.imgur.com\/vNeEM.png)"}
{"id":"106025","title":"Are there any musical instruments which use resonance tubes with two closed ends?","text":"Many musical instruments use resonance tubes with one closed end - all brass instruments (I think), clarinet, etc. There are also instruments where both ends are open (flute, pipes) Are there any which use two closed ends, that is, a resonance tube with nodes at both ends? I am aware of the problems which would exist for such an instrument - essentially not be very loud - but I am just curious if anyone knows of any."}
{"id":"106027","title":"Excited Energy levels of Hydrogen vs Solids","text":"My question has to do with Excited Energy Levels. I keep reading and learning that all objects\/most solids emit infrared radiation. But to emit radiation, an object's particles must first enter a higher energy orbital (electrons) which will then immediately lead to the atom returning to its ground state, releasing the energy of the level jump from ground state to level 2 in eV in the form of a photon wave. BUT! When I looked for information about energy levels of elements, I could only find hydrogen, and when I plugged in the energy for: l=hc\/E where l is wavelength, h is Plancks constant, c is the speed of light, and E is energy of the photon. The wavelength came up to be 1.121x10^-7! That is in the ultraviolet range! That means that it takes a ton of EM Energy to emit a photon wave! This doesn't make sense unless solids require much less energy to move up orbitals. Is this true? What happened to Infrared or Radio? And how do I find out about energy levels of solid elements and understand the data?"}
{"id":"57360","title":"2-body interaction energy of 3 particles","text":"If only two-body interaction is considered then what's the energy if I put three particles on one site? Assume delta-interaction and interaction strength is proportional to scattering length $a_F$ which depends on the total spin of the two-particle state. For two particle case, the interaction is simply $\\hat U |F,M\\rangle = U_F |F,M\\rangle $ Suppose the three particles are fermions, then I will take a completely anti- symmetrized spin wave function. But now, how to express the interaction energy in terms of two-body scattering lengths?"}
{"id":"53866","title":"Young experiment: square of classical real wave function","text":"I can't understand why the sum of two real waves result in a time dependent wave, but not so for the complex waves. In details, I can't get this passage on p.38-39 in A.C. Phillips, _Introduction to Quantum Mechanics:_ $$\\tag{3.9}\\Psi ~=~ A\\cos (kR_1 - \\omega t) + A\\cos (kR_2 - \\omega t)$$ $$\\Downarrow$$ $$\\tag{3.10}\\Psi^2 ~=~ 2A^2\\cos^2\\left(\\frac{k(R_1-R_2)}{2}\\right)\\cos^2\\omega t.$$ This is driving me crazy!"}
{"id":"36428","title":"In what way do Cooper pairs of electrons bond and stay bonded in superconductors?","text":"I understand how electrons initially move into another's vicinity, but nowhere can I find a fathomable answer to this. Also, does the pairs forming 'a condensate' mean a Bose-Einstein condensate?"}
{"id":"18238","title":"What electric potential is found over individual resistors connected in series to an AC power supply?","text":"Given a set of resistors connected in series to an AC power supply:   1. What formula governs the peak-to-peak voltage which will be measured when voltage is probed over individual resistors and sets of resistors in that circuit?   2. What is a good way to think about this construction so the answer to question #1 is obvious and intuitive? Intuitively, I would think that the measured voltages would be the same as in a DC circuit such that:               V_out = (R_partial\/R_total) V_in      Where V_out is the voltage which is probed, R_partial is the total resistance of the sub-set of resistors which are being probed and R_total is the total resistance in the circuit. This formula makes sense intuitively because the current throughout a circuit of series components is everywhere constant and according to Ohm's law for a constant current, the higher a given resistance over an area the higher the voltage potential measured there. Is this correct for AC circuits as well?"}
{"id":"27775","title":"Why Are Even and Odd Regge Trajectories Degenerate?","text":"The Gribov-Froissart projection treats even angular momentum differently from odd angular momentum. But in QCD, I believe that the odd trajectories interpolate the even trajectories--- the two trajectories are degenerate. Is there a way of understanding why the even-odd trajectories are degenerate? Is is a symmetry argument? Can you find a natural system where they are not degenerate? A quick google search revealed this reference, which doesn't answer the question, but gives an experimental signature for the degeneracy: http:\/\/www.slac.stanford.edu\/cgi-wrap\/getdoc\/slac-pub-0576.pdf , There is probably excellent data by now on this."}
{"id":"34867","title":"Parity and Helicity of the Higgs Boson","text":"I have been studying how the spin and parity of the new boson discovered at the LHC will be studied and have run into some confusion. The Standard Model Higgs is expected to be a scalar (i.e. have even parity with spin 0). My question is how is the intrinsic parity of a particle experimentally determined? Is it related to the helicity amplitudes? Why is parity even an important quantum number to determine?"}
{"id":"83502","title":"Relationship between boiling point and pressure","text":"In the bubble chamber, liquid hydrogen is kept at high pressure. When a stream of charged particles are injected into the bubble chamber, pressure on the liquid hydrogen is released and bubbles of gas are produced around the charged particles. However my question is that:                  why would the boiling point of hydrogen decrease when pressure decreases?"}
{"id":"133058","title":"Torque due to a magnetic field","text":"We have two infinite perpendicular wires, each carrying a current \"i\" and the distance between them is l. If both the wires are free to move, how will they arrange themselves because of the torque exerted by the magnetic field produced by the two wires?"}
{"id":"133056","title":"If a photon has no mass, how can it be attracted by the Sun?","text":"I read that the photon doesn't have mass, but my teacher says that the photon has mass because the sun can attract it (like in the experiments to prove the theory of relativity). I think that there is another reason to explain that. How can I explain that the photon doesn't have mass and the sun attracts photons?"}
{"id":"113482","title":"How thick (in molecules) would a piece of paper be if it was folded to the moon?","text":"Apparently you can fold a piece of paper 42 times and reach the moon! 42 is the answer the everything after all! Oh, Douglas Adams, you clever son of a bitch. The reasoning behind this is that as you fold a paper, it becomes twice as thick. It's exponential and can be modeled by the following formula for a paper that is say- 0.001 cm thick. 0.001(2)^42 = space travel I want to test this. You have to figure that as you fold a paper and it becomes twice as thick, the area is also halfed. So, for a standard piece of A4 paper, being 8.5x11, would have an area of 92 in^2 or about 603 cm^2. The equivalent decrease in area can be modeled by this formula: 603(0.5)^42 Now, for the number of molecules in a sheet of paper. The number of molecules for a piece of paper consisting solely of cellulose (C6H10O5) strains can be found by using the molar mass of cellulose and the mass of the paper in question. So the calculations for a standard piece of A4 paper with a mass of 5 grams should look something like this: 5g × (1 mol C_6 H_10 O_5)\/162g =3.09 × 〖10〗^(-2) mol damn stackexchange formatting And since a mole of cellulose contains 6.022 x 10^23 molecules, you would have about: 1.8586 x 10^22 molecules in that piece of paper This is where that math breaks down, I've tried to divide the number of molecules by the resulting area, but the answer doesn't seems quite right. :\/"}
{"id":"94974","title":"electron flow in capacitor with dielectric","text":"if I make a circuit with a battery and a capacitor (with a dielectric inside), how it is possible to get a current in the circuit? If electrons go from one pole of the battery and they arrive to one plate of the capacitor they cannot flow to the other plate of the capacitor because the plates are separated by a dielectric and this dielectric doesn't have free charges, they are bound charges. How can I explain a net flow of electrons in this circuit? Thanks."}
{"id":"6860","title":"estimating ion collisions from Vlasov equations","text":"Suppose you have a distribution function $ f_{\\alpha}( \\vec{r} , \\vec{p} , t)$ obtained from Vlasov equations for a certain $\\alpha$ species, say some ions. Is there a rigorous way (in the domain of validity of the model, that is) to estimate a collision density function $ p^{collisions}_{\\alpha}( \\vec{r} , E_{CM} , t)$ , where $E_{CM}$ is a variable describing collision energy parameters? A link or reference would be great"}
{"id":"6768","title":"Why didn't the control rods in Fukushima shut down the reactor?","text":"With all the hubbub over the reactors in Japan and nuclear safety, I was wondering why the insertion of the control rods in Fukushima did not shut down the reactor? Shouldn't they slow the fission reaction so far that cooling is not needed? Why do they need to continue cooling the core when I would expect no more exothermic reaction to take place?"}
{"id":"36414","title":"What mechanism produced the baryon excess in the universe?","text":"A baryon is a composite subatomic particle made up of three quarks (as distinct from mesons, which comprise one quark and one antiquark). Baryons and mesons belong to the hadron family, which are the quark-based particles"}
{"id":"71444","title":"decay time for a blast wave","text":"The blast wave that is formed as a result of an explosion has some import parameters. One of them is the peak pressure which is basically the pressure measured when the blast wave arrives at a given location. It is the case that this peak pressure decreases as one moves further and further from the center of the explosion. Ignoring the actual mathematical form, this is plausible. Now the other parameter of interest is the decay time. It is the time it takes for the peak overpressure to decay to the atmospheric pressure at a fixed point in space. One could imagine an experimentalist putting a pressure gauge at some location. When the blast wave arrives, the pressure rises to the peak pressure and after some time $t_{d}$, it decays to atmospheric pressure. This $t_{d}$ is the decay time. From my readings so far, I don't seem to have an intuitive grasp of the decay time. The books I read suggest that the decay time increases as one moves away from the source. Could any body provide me with some simple explanation as to why this might happen?"}
{"id":"39467","title":"Einstein's special relativity beyond the speed of light","text":"Could someone with access to this paper which claims to have new transformations between frames with relative motion faster than light which are supposedly consistent with special-relativity, say what the new transformations are and why they might make sense ?"}
{"id":"77841","title":"Mass-Energy Equivalence","text":"I've asked before, but I'm still confused as to what the mass energy equivalence implies. I've taken an introductory course in relativity, so I only covered special relativity. From what I gather, all things have a rest mass. As you increase your energy, your mass increases as well, and the rest mass behaves as some potential. But then when we talk about fusion and fission, we are splitting the atoms to generate energy-breaking apart the potential stored in forces between atomic components. This kind of makes sense to me. But then are _all_ energies and masses equivalent? An electron has a rest mass, is this considered energy as well? Is the mass stored in objects I hold due to the gravitational field simply too small for me to notice? Furthermore, if mass is energy, then why do nuclei weigh less? Is some of their mass going into the bonds? The fission and fusion also confuse me, both are building towards a more stable atom (32?) but somehow both paths generate energy."}
{"id":"72368","title":"Why are most metals gray\/silver?","text":"Why do most metals (iron, tin, aluminum, lead, zinc, tungsten, nickel, etc.) appear silver or gray in color? (What atomic characteristics determine the color?) What makes copper and gold have different colors?"}
{"id":"88416","title":"How to show that value is conserved along geodesics?","text":"Let's have the motion of charged particle in a field of Reissner-type black hole. The equation of motion looks like $$ \\frac{d^{2}x^{\\mu}}{d \\tau^{2}} + \\Gamma^{\\mu}_{\\nu \\lambda}\\frac{dx^{\\nu}}{d \\tau}\\frac{dx^{\\lambda}}{d \\tau} = \\frac{q}{m}F^{\\mu}_{\\quad \\nu} \\frac{dx^{\\nu}}{d\\tau}. $$ How to show that value $$ E = m\\left( 1 - \\frac{2M}{r} + \\frac{Q^{2}}{r^{2}}\\right)\\frac{dt}{d\\tau} + \\frac{qQ}{r} $$ is conserved along geodesics, $u^{\\mu}D_{\\mu} E = 0$? The first step, maybe, must be the attention to find the $\\frac{dt}{d\\tau}$. But what to do after that? It doesn't seems like convolution $\\varepsilon_{\\mu}u^{\\mu}$ or to $u_{\\mu}u^{\\mu}$, where $u^{\\mu}$ is the 4-speed. So what to do next?"}
{"id":"43048","title":"Can we create a magnet with only one Pole?","text":"> **Possible Duplicate:**   >  How to make a monopole magnet? No matter how many times you cut a magnet, we always end up with 2 poles. Is there any possibility of creating a monopole magnet?"}
{"id":"90085","title":"Observables still commute even if fields only anti-commute","text":"In Peskin & Schroeder page 56, after introducing _anti_ commutation relations for the fields instead of commutation relations (in order to fix the negative energy problem as well as to have proper causality conditions stemming from particles propagating from y->x and anti-particles propagating from x->y), it is stated that even though now we have anti-commutation relations, any two observables evaluated outside of the lightcone should still commute (and not _anti_ commute) and thus our theory still preserves causality, in the sense that if two observables commute then measuring one does not affect the measurement of the other. The reason they give for this is that any observable is ultimately going to be composed out of an even number of fields. I have tried to put this claim into mathematical sense, by doing the following:   1. Assume $g(x-y,x-y)<0$ where $g$ is our metric.   2. Thus we know that $<0|\\\\{\\Psi(x),\\,\\bar{\\Psi}(y)\\\\}|0>=0$ from our previous calculations.   3. Define $O_1(x):=<0|\\bar{\\Psi}(x)A_1(x)\\Psi(x)|0>$ and $O_2(x):=<0|\\bar{\\Psi}(x)A_2(x)\\Psi(x)|0>$. $A_1$ and $A_2$ are two differential \/ spinor operators which are thus far arbitrary.   4. Try to show that $[O_1(x),\\,O_2(y)]=0$, because $g(x-y,x-y)<0$. I have failed at showing 4. assuming 2. (I have tried to somehow make linear combinations of the commutator as anti-commutators of the fields and it didn't work out). My question is:   * Is this the most general formulation of the claim that one could make? If not, how would you better formulate it?   * Once we have the formulation we want, how to prove it assuming 1. thus 2.?"}
{"id":"100464","title":"Otto cycle efficiency described in pressure ratio (instead of compression ratio)","text":"The Otto cycle's thermodynamic efficiency is often described using the compression ratio: $$\\eta_{th} = 1 - \\left( \\tfrac{1}{CR} \\right ) ^ {k-1},$$ (where CR is the compression ratio, and k the ratios of specific heat). Now I tried describing this using a pressure ratio (So instead of $\\tfrac{V_1}{V_2}$ using $\\tfrac{p_1}{p_2}$. Taking into account the compression in a cycle is (in the ideal case) an isentropic process, one could use $$p_1V_1^k = p_2V_2^k$$ $$\\tfrac{P_2}{P_1} = \\left ( \\tfrac{V_1}{V_2} \\right ) ^ {k} = \\Pi$$ Combining these I would get: $$\\eta_{th} = 1 - \\left( \\tfrac{1}{\\Pi ^ {\\tfrac{k-1}{k}}} \\right ) $$ However this is **exactly** the same as the brayton efficiency.. While the cycles are different (otto uses isochoric combustion, while brayton uses an isobahric expansion during combustion). I wanted to compare those two, what did go wrong? EDIT: PS why is in the mathjax \"\\frac\" not working, yet \\tfrac does?"}
{"id":"133948","title":"Physics problems requiring optimization?","text":"I would like to know if there are simple benchmark physics problems to test a Genetic Algorithm C++ library I developed. The library supports single and multi-objective constrained optimization and I would like to publish the code showing its performance in the optimization of some physics examples."}
{"id":"132613","title":"A curious case of Relativistic Velocity Addition","text":"The relativistic velocity addition formula is $$u = \\frac{v+u'}{1+ \\frac{vu'}{c^2}}$$ Where $u$ = velocity of projectile seen by rest observer \"A\" $v$ = velocity of moving observer \"B\" as seen by rest observer \"A\" $u'$ = velocity of projectile seen by B Now the question is this: If $v=c,$ and $u'=-c $ I get an undefined answer. i.e. the relavistic velocity addition formula is undefined. Whats wrong with setting $v=c,$ and $u'=-c $?"}
{"id":"109877","title":"Is there no radioactive decay between nuclear fusion and solid material formation?","text":"I'm aware my question might be considered a duplicate of this one: Radio- dating and the age of the earth I read that one and I looked everywhere and I still can't find my piece of mind. I would really like to understand the following points:   * does one uranium nucleus start the radioactive decaying process the instant it is born inside a supernova ? (I understand that for a single nucleus the chance of decaying which is an instantaneous event, grows from 0% to 100% in the time from _the beginning_ to the half-life)   * while talking about a collection of atoms, Wikipedia states here that \" _This predictability allows the relative abundances of related nuclides to be used as a clock to measure the time from the incorporation of the original nuclides into a material to the present._ \"   * @JohnRennie helped me understand that chemically speaking zircon and lead do not mix whereas zircon and uranium do. So basically macroscopic contamination via accretion with flying lead is an event too rare to take into consideration right ? Also... I understand that the uranium-lead decay series has very short half-lives for every nuclide except for the beginning and the end (uranium and lead) and so, from a cosmological perspective it's quite improbable that zircon would mix with anything other than uranium, should a chemical compatibility exist with another nuclide. Please bare with me and excuse my \"highly scientific\" vocabulary. I have an experiment I would like to imagine in order to discover what I'm doing wrong:   1. A star goes supernova   2. We concentrate our attention on one uranium nucleus which is fused together during the violent event   3. The uranium nucleus travels for millenia towards the an accretion cloud surrounding a newborn star   4. The nucleus collides with _brand new molten zircon crystal_ and becomes part of it. Let's say that event 4. was happening after 0.99 uranium half-lives and the uranium nucleus still hadn't decayed. Isn't it imminent that the nucleus will decay pretty soon after event 4. ? I mean, I understand that if the nucleus would've become **thorium** or **protactirium** or **lead** it wouldn't've gotten stuck in the crystal for chemical reasons, but if it was still uranium what does that say about the internal clock of the nucleus which **had** been ticking for 99% of the half- life ? Is that clock reset because of external electromagnetic reasons the moment the nucleus becomes part of the crystal ? Thank you for your indulgence"}
{"id":"133941","title":"A Variation on Laplace's equation (context: Yang-Mills N-Instantons, Rajaraman's book)","text":"**Statement of the problem** I need to solve the equation \\begin{align} 0 = \\frac{1}{\\phi} \\partial_{\\sigma}\\partial_{\\sigma} \\phi \\hspace{20mm} (1) \\end{align} where $\\phi$ is a scalar field and we're working in Euclidean 4-space. More specific questions are asked below. Basic advice on this would also be appreciated. **Context** I'm trying to work through 't Hooft's Euclidean Yang-Mills N-instanton solution. The idea is that we make the ansatz for the gauge potentials \\begin{align} A_{\\mu} = i \\overline{\\sigma}_{\\mu \\nu} \\partial^{\\nu}(\\ln \\phi(x)) \\end{align} where $\\overline{\\sigma}_{\\mu \\nu}$ can be thought of as a constant 4x4 matrix that is unimportant to my question. We use the self-duality of the field strength tensor that arises from this ansatz to obtain the equation (1) for $\\phi$ **The Difficulties** One generally good source has been Rajaraman's \"Solitons and Instantons.\" The relevant section is displayed ![Relevant section in Rajaraman](http:\/\/i.stack.imgur.com\/mdLWX.png) My difficulties are:   1. I am assuming by \"singular\" Rajaraman means that the function contains singularities. Why, explicitly, does $\\phi$ being non-singular imply that we we need only solve $\\Box \\phi = 0$? Surely if $\\phi$ contains any zeros then $1\/\\phi$ raises issues?   2. Why does $\\Box \\phi = 0$ only permit the constant solution and not say $\\phi(x) = a_{\\mu} x^{\\mu} + b$ where $a_{\\mu}$ and $b$ are constants? Is this because of the form of the ansatz, or is it something I'm missing?   3. In the case where $\\phi$ is allowed to be singular, why does $\\phi = 1\/|x|^2$ solve the equation (1) for $x \\neq 0$? Surely we have \\begin{align} \\frac{1}{\\phi} \\partial_{\\sigma}\\partial_{\\sigma} \\phi &= |x|^2 \\partial_{\\sigma}\\partial_{\\sigma} \\frac{1}{|x|^2}\\\\\\ &= |x|^2 \\partial_{\\sigma}\\bigg(-\\frac{2 x_{\\sigma}}{|x|^4}\\bigg)\\\\\\ &=|x|^2\\partial_{\\sigma} (\\frac{-2 x_{\\sigma}}{|x|^4})\\\\\\ &= |x|^2\\partial_{\\sigma} (\\frac{-2 x_{\\sigma}}{(\\sum_{i} x_i^2)^2})\\\\\\ &= |x|^2\\frac{-2}{(\\sum_i x_i^2)^2} + |x|^2\\frac{8 x_{\\sigma}^2}{(\\sum_i x_i^2)^3}\\\\\\ &=\\frac{-2}{|x|^2} + \\frac{8 x_{\\sigma}^2}{|x|^4}\\\\\\ &\\neq 0 \\end{align} ( _edit_ ) I see where I went wrong above, the last three lines should be \\begin{align} &= \\sum_{\\sigma}(|x|^2\\frac{-2}{(\\sum_i x_i^2)^2}) + |x|^2\\frac{8 x_{\\sigma}^2}{(\\sum_i x_i^2)^3}\\\\\\ &=\\frac{-8}{|x|^2} + \\frac{8 |x|^2}{|x|^4}\\\\\\ &= 0 \\end{align}"}
{"id":"114210","title":"Linear independence of the Covariant Derivative","text":"What's the easiest way to show that the covariant derivative $\\nabla U^{\\mu}$ is linearly independent to $U^{\\mu}$, which is a vector? I mean I'm assuming they are since I'm proving the second Bianchi identity and my proof requires this to be the case; and I'm quite confident that the Bianchi identity is true! So I start with the Jacobi identity $$[\\nabla_a,[\\nabla_b,\\nabla_c]]+\\text{cyclic}=0$$ I then substitute in the definition of the Riemann tensor, giving me $$\\nabla_a(R^d_{\\space\\space\\space ebc}U^e)+\\text{cyclic}=0$$ which gives $$\\nabla_a(R^d_{\\space\\space\\space ebc})U^e+R^d_{\\space\\space\\space ebc} \\nabla_aU^e+(\\text{cyclic in }a,b,c)\\space\\space=0.$$ So we have 6 expressions, from which if the second Bianchi identity is true, the vector and the covariant derivate of the vector must be linearly independent. Or am I just OK to say that as the covariant derivative is (1,1) tensor and a vector is a (1,0) tensor, the result must follow straight away?"}
{"id":"87996","title":"normal ordering from anti normal ordering of creation and anhilation operators","text":"I am working on entanglement of systems in beam splitters. I constantly come across equations of creation and annihilation operator (ladder operators) in anti normal ordering. I want those equations in normal ordered form. Does anyone know the the relation between normal ordered and anti normal ordered. ( a^m * a' ^n)= ?"}
{"id":"86023","title":"derivation of noise spectral density for Poisson distribution","text":"I am trying to follow this article online to derive the noise spectral density of a Poisson distribution. I was able to follow it until equation (18). I cannot understand how he applied Wiener-Khintchine theorem to get that. It doesn't even look like the WK theorem that I know of (link to wiki). Does anyone know how he got that? The full article is from here. ![enter image description here](http:\/\/i.stack.imgur.com\/YlVF5.png)"}
{"id":"100792","title":"Quantum frequency vs classical frequency and Energy dependence","text":"I recently realized what seems to be a contradiction in what is called \"frequency\" for light, and would enjoy some clarification of some more educated mind on the matter, thanking you in advance for your time. At the classical level, light is, in an ideal case, a plane EM wave. The electric field $\\mathbf E$ for some light frequency $\\nu$ is $$ \\mathbf E(t,\\mathbf x) = \\mathbf E_\\nu \\sin(2 \\pi\\nu t-\\mathbf k \\mathbf x ) $$ with $\\nu=c|\\mathbf k|$ and $c$ is the speed of light. The energy density of the said wave is readily found to be proportional to $|E_\\nu|^2$ (magnetic field is omitted for clarity). Light is described as a wave packet of such waves. The light spectrum : Radio, visible light, UV , $\\gamma$ and all others in between is classified by the value of $\\nu$. In quantum mechanics (or rather QED) one readily defines the energy of a light quantum, a photon, as (having usually put all constants to 1) a frequency $\\nu'$. $$ E= \\nu'. $$ Now to my question. For sure jumping from photons to classical wave is no light matter (!), but at a first glance, one can reasonably assume that $\\nu'$ is linked to $|\\mathbf E_\\nu|$ due to the energy relations. One would get something like $$ |\\mathbf E_\\nu|^2 = \\mbox{density of photons} \\times \\nu'. $$ However, while discussing radioactive decay for example, photons of fixed energy i.e. $\\nu'$ are often emitted. At the same time the same terminology is used. That is we say some $\\gamma$ ray is emitted. In what sense is that to be understood, as it refers to $\\nu$ and not $\\nu'$ ? It seems to me that by definition there are NO links between $\\nu$ and $\\nu'$, even if one introduces wave-packets to do things more rigorously. Wishing you a good day, Frédéric Edit: Thank you for your comments. The reasoning from QED, i.e. on some state clarifies the identification of $\\nu$ and $\\nu'$. I stupidly only recalled the classical average energy over time (which is $\\nu$ independent and thus mislead me), where one should really consider the Poynting vector. Thanks again for your quick comments."}
{"id":"4129","title":"Could the measurement problem be solved by string theory \/ other ToE?","text":"A quick wikipedia search will reveal the many different approaches to the measurement problem in quantum theory. What I find strange is that they even make up all these weird hypothesis like true randomness, many worlds, bohm and other interpretations when there isn't any ToE they're working it out from. I'm just a layman so maybe my assumption is wrong, but wont a fundamental theory inevitably change quantum theory? Is really a realistic model where superpositions are nothing but lack of information on our part not likely to come from the ToE? To my layman mind it seems premature to even begin to try to answer these questions if a theory of everything isn't in place. Or is there actually a reason that people are trying to solve this now? Such as: no matter what the ToE is, quantum theory wont change, superpositon will always exist etc.?"}
{"id":"4123","title":"Given temperature, composition, column density, and radial velocity, can I find the bulk flow of a gas cloud?","text":"We've got a hot star in the middle of a gas cloud. We point a spectrometer at the star, calculating the following attributes of our line of sight at the star through the cloud:   * Total number of various elements and molecules (H, H2, C, O, N, CO, lots of ions of all of the above) along the line of sight (units are number\/cm^2)   * Average temperature of the atoms in our line of sight   * Average radial velocity of the atoms in our line of sight All the radial velocities are roughly the same, so we infer that the line of sight is more or less a single coherent blob of gas. Good! Now, the star itself which illuminated the line of sight is moving about 40 km\/s towards us, while the gas cloud chunk we observed is moving at around 20 km\/s away. Also, from comparing the luminosity of the star to the temperature of the gas, we conclude that the gas cloud is a light year or two away from the star. Given all this information, here is your task: If we take another spectrum of the _same line of sight_ ten years from now, will it be the same blob of gas? I will tell you right away that the radial velocity will only make the gas 0.002LY closer to the star, which is way less than what we need. So, in other words, given all the above, **can you conclude anything about the transverse velocity of the cloud?** Can you tell anything about the bulk flow and determine whether it is probable that the gas in our line of sight has moved with respect to the star? My physical intuition says that you'd expect yes, since the radial velocities are so different, but I'm not totally sure - the radial velocity doesn't necessarily imply anything about the transverse velocity. Or does it?"}
{"id":"4121","title":"How to compute the speed of sound in relativistic hydrodynamic?","text":"In Weinberg: Gravitation and Cosmology chapter 2.10 (Relativistic Hydrodynamics) the speed of sound is derived as $v_s^2 = \\left(\\frac{\\partial p}{\\partial \\rho}\\right)$ and the equation of state is given as $p=(\\gamma - 1) (\\rho - nm)$ with $\\gamma = 4\/3$ for the case of a relativistic gas and $\\gamma=5\/3$ for the case of a nonrelativistic monoatomic gas ($nm$ is the rest mass density and $\\rho$ the energy density). Then Weinberg computes the sound speed for the relativistic case as $v_s= \\sqrt{\\frac{1}{3}}$ in natural units and states that this is still safely less than unity (the speed of light). However what he doesn't show is the same calculation for the nonrelativistic case. This would yield $v_s^2 = \\frac{\\partial ((5\/3 - 1) (\\rho - nm))}{\\partial \\rho} = 2\/3$ and so the speed of sound in the non-relativistic case would actually be $v_s =\\sqrt{\\frac{2}{3}}$. So the speed of sound for a non-relativistic gas is actually higher than the speed of sound for an relativistic gas!? However one usually computes the non-relativistic speed of sound with another equation of state $p=K\\rho^{\\gamma}$ which yields $v_s^2=\\frac{p}{\\rho} = K \\gamma \\rho^{\\gamma-1} = \\gamma \\frac{p}{\\rho}$ This is a completely different result compared to the one we get from the equation of state given by Weinberg. So is the equation of state given by Weinberg wrong? If so, what is the correct equation of state for a relativistic gas and what is the real maximum speed of sound for a relativistic gas? If not, what is wrong with my calculation of the non- relativistic sound speed based on the equation of state given by Weinberg?"}
{"id":"110771","title":"Electrons in Rutherford's gold foil experiment","text":"I have been reading a chapter in my physics text. In it there is Rutherford's gold foil experiment. There it is written that doubly charged helium particles (alpha particles) were projected onto very thin gold foil. Most of the alpha particles passed straight through, but some of the alpha particles bounced back because positive particles (protons) in the nucleus repelled them. Positive and positive always repel each other. If charge was the reason some of the alpha particles bounced back, then why didn't the electrons (negative) of the gold atoms which are much smaller then protons get attracted to the alpha particles? Positive and negative attract. I would have thought the doubly charged alpha particles would be attracted to the negative electrons."}
{"id":"53682","title":"What causes ballpoint pens to write intermittently?","text":"After a while, a ball point pen doesn't write very well anymore. It will write for a little distance, then leave a gap, then maybe write in little streaks, then maybe write properly again. It seems to be worse with older pens, but I have observed this with new pens right out of the box too. Experiments I have done:   * Take the cartridge out and look at the amount of ink. There is still plenty.   * Inspect the ball with a jewler's loupe, no obvious damage, everything looks smooth and clean.   * Stored new pens unused tip-down to eliminate gravity slowly pulling the ink from the ball and leaving a air pocket. Some of the pens exhibit the symptom even when used the first time with the cap never removed before and stored this way for a year.   * Stuck a wire in the open end of the ink reservoir to see if maybe the end dried to a hard plug so that new ink couldn't move down as it was removed from the reservoir by writing. I have never found anything hard, and observed the same symptoms even after \"stirring\" the top of the reservoir with a wire a little.   * When a pen stops writing, shake it hard, like resetting a fever thermometer. That seems to help for a brief while, but so does just waiting a few seconds, so I'm not sure the shaking is relevant.   * Stored a pen ball-down in a glass of water overnight. The thought was if the ink just above the ball had dried, maybe this would re-constitute it. Some ink clearly dissolved in the water since it was colored, but once the pen was started up again there was no apparent change to the symptoms.   * Cold seems to exaggerate the symptoms, but warming to body temperature doesn't fix them. This is not just a single pen or a single model. I have bunch of different pens of different models that do this. I'm curious, what causes this? ## Added: I have done some more experimenting, and it seems Emile Jetzer was right. The cause seems to be that the ink is so viscous that new ink doesn't flow down to replace what is removed via the ball fast enough. Two experiments support this hypothesis:   1. A pen will write again after a while by just letting it sit ball-down, but the time is significantly decreased when you shake it, like you would resetting a fever thermometer.   2. Some stick pens are sealed except for a small air hole at the top. Putting lips around the top of the pen and applying pressure as if you were trying to blow into it resets the writing action quickly. Even better, I can write with such pens much longer than they would normally go by holding my mouth over the top and applying constant air pressure. So, I think the mystery is solved. Probably ink in the reservoir dries out slowly over time by losing water vapor from the top. That makes the entire ink more visous, which explains why old but unused pens also exhibit this symptom. The next experiment is to take such a pen and add a little water at the top of the ink reservoir, then let it stand for a week and see what difference that makes."}
{"id":"103844","title":"Why does material resistance increase with temperature but electrolyte internal resistance decrease with increase in temperature?","text":"The question is pretty much in the title itself, why does the internal resistance decrease with the increase in the temperature of electrolyte? Is it related to the electrolyte bring a fluid or something?"}
{"id":"135049","title":"Find acceleration along normal and tangent","text":"A small block can move in a straight horizontal line a along AB. Flash lights from one side projects its shadow on a vertical wall which has horizontal cross section as a circle. Find tangential & normal acceleration of shadow of the block on the wall as a function of time if the velocity of the block is constant (v). ![enter image description here](http:\/\/i.stack.imgur.com\/qaMRn.png)"}
{"id":"135048","title":"Is there a simpler way to calculate the diminishing fringe pattern of a single edge?","text":"A single edge diffraction pattern is different than a slit or multiple slit diffraction pattern. The pattern is not equally spaced and the fringe spacing's get smaller and smaller diminishing infinitely. The equations I have found for calculating these patterns are very complex and complicated. \"Is there a simpler way to calculate the single edge diffraction pattern?\" That is the question but what I'm really wondering is why the equations I use below are not used to work this out? I have found a way to calculate the diminishing single edge fringe pattern but explaining it can be quite involved especially when introducing a new or different way so I posted a link to my paper https:\/\/www.dropbox.com\/s\/lu5irtlxxe4hpis\/Single%20Edge%20Certainty.docx?dl=0 If there is a better place to post and discuss this please let me know because I would love to talk about it. I can show that all slit diffraction patterns come from single edge diffraction patterns and calculating these unique diminishing fringe patterns is much simpler than the equations I have found anywhere else. Depending on the wavelength and distance from edge to screen one simple equation based on The Pythagorean Theorem will calculate the single edge fringe pattern infinitely and accurately. From there it can be shown how slit patterns can be constructed from overlapping single edge fringe patterns. Below is a few pages taken from my paper. If you could please take a look at the whole paper I would appreciate it and as I said would love to discuss it. Thanks Bill Alsept The following is based on 500nm light and a distance of one meter (1,000,000,000nm) edge to screen. The first time I calculated the distance (Y) (measured from the shadows edge) to the 1st bright fringe (B1). I tried adding ½ wavelength (250nm) to L to create the hypotenuse and then I calculated: Y=√((L+250nm)²-(L)²) or Y=√((1,000,000,000nm+250nm)²-(1,000,000,000nm)²) or Y=707,106.82nm But this was not the observed Y distance for the first bright fringe B1. I had nothing to compare my findings with because the equations I did find for single edge fringe patterns where extremely complicated. One day I came across this site about lunar occultation’s http:\/\/spiff.rit.edu\/richmond\/occult\/bessel\/bessel.html and discovered another way to look at single edge fringe patterns. This was the first time I had read about occultation. By chance I happened to recognize a sequence of numbers that were similar to the calculations I was doing for single edge fringe patterns and realized I was on the right track. Based On what I had learned about lunar occultation I could now calculate the distance of the fringes the moon would make on the earth. The creator of this web site had already calculated these distances so I finally had something to compare to. For my experiment all I had to do was convert (L) back to 1 meter instead of the 184,000 kilometers their experiment had used for the distance to the moon’s edge. Using Pythagorean Theorem and calculating backwards I converted the occultation’s measurements for Y and L and applied to them to my experiment and then calculated for the hypotenuse. What I discovered was that no matter what wavelength of light or distance you input (when calculating for the 1st bright fringe B1) the hypotenuse will always be L+3\/4*1\/2λ or L+3\/8λ. Not quite 1\/2λ This told me that these photons (maybe all photons) were emitted at the same phase of their amplitude cycle. This also told me that at the point of emission the photons were already part way into their cycles or more precisely 1\/8 beyond the negative amplitude. That's why 3\/8λ is added to (L) so that the first hypotenuse is the exact length needed to complete an oscillation just as it’s impacting the screen in a positive amplitude. The distance Y from the shadows edge to the first bright fringe is calculated as: Y=√((L+.75*.5*λ)²-(L)²) or Y=√((L+.75*.5*500)²-(L)²) or Y=√((1,000,000,000nm+187.5nm)²-(1,000,000,000nm)²) or Y=612372.46nm See Fig 12 below Figure 12"}
{"id":"46412","title":"Is the liquid\/solid line infinite?","text":"Starting from the triple point, is the melting line between solid-phase and liquid-phase infinite? If not, why does it end? Because pressures are so high that classical inter-molecular interactions break down? Is this related in any way to the concept of degenerate matter? Can such pressure (at which the line ends) be easily computed?"}
{"id":"46413","title":"What happens at the end of Coriolis Deflection","text":"Consider we launch a cannonball due south from a point at 45 degrees latitude in the Northern Hemisphere (e.g the point defined with the co-ordinate system on this diagram). The cannonball travels for 5 minutes (through vacuum - ignore air resistance), experiences Coriolis deflection and drifts 'right\/west' ending up at 40 degrees latitude. A few nanoseconds before it hits the ground, the rotational\/tangential speed of the cannonball is slower than the tangential speed of the ground below. In order for the cannonball to stick\/stay at the point of impact, its tangential speed must match the speed of the ground. What effect does this difference in speed have on the cannonball on landing\/impact? Does it experience a sudden west-east acceleration due to friction? If yes, would this effect be even more pronounced in this case where the project is fired from North Pole?"}
{"id":"134398","title":"Understanding the effective low-energy Lagrangian for hadrons","text":"My course in Higgs Physics is discussing a two-nucleon low-energy effective theory of hadron interaction. With $\\psi=(p,n)$, the pion is defined as $\\vec{\\pi}= i \\bar{\\psi}\\vec{\\tau} \\gamma_5 \\bar{\\psi} $ and the sigma as $\\sigma = \\bar{\\psi}\\psi$ Then the Lagrangian is: $L=$ $\\bar{\\psi} i \\partial_\\mu \\psi + \\frac{1}{2}(\\partial_\\mu \\vec{\\pi} \\partial^\\mu \\vec{\\pi} + \\partial_\\mu \\sigma \\partial^\\mu \\sigma$) (kinetic) $+ g_\\pi (i \\bar{\\psi} \\gamma_5 \\vec{\\tau} \\bar{\\psi} ) \\vec{\\pi} + g_\\pi \\bar{\\psi}\\psi\\sigma$ (nucleon-pion-interaction) $+ \\frac{\\lambda}{4}((\\pi^2+\\sigma^2)^2-f_\\pi^2)^2$ (potential) First, I am surprised that the nucleon current in the nucleon-pion-interaction $ (i \\bar{\\psi} \\gamma_5 \\vec{\\tau} \\bar{\\psi} )$ is almost the same as the definition of the pion $(i \\bar{\\psi}\\vec{\\tau} \\gamma_5 \\bar{\\psi})$. In fact they seem to be equal since $[\\tau_i,\\gamma_5]=0$. ( $\\tau_i$ acts on the SU(2)-space while $\\gamma_5$ acts on the left-right space.) _Why are we reading the same term once as a pion and once as a nucleon current?_ Secondly, $f_\\pi$ is referred to as the pion decay constant. It is supposed to be proportional to the coupling of a pion to the W-Boson. _Why is $\\pi$ included into the potential in the way it is? How do I see that it is related to the W-Boson coupling?_"}
{"id":"80070","title":"Reynolds number and inertial force","text":"The Reynolds number is defined as the ratio of ´inertial´ forces to viscous forces. $$ Re = \\frac{\\text{Inertial Forces}}{\\text{Viscous Forces}}$$ Now, viscous forces make sense to me. They are frictional shear forces that come about due to the relative motion of the different layers in a flowing fluid, resulting in different amount of friction, hence, different viscosity values. However, I am not really sure how to think about the 'inertial force'. This, to me, is somewhat of a dynamic effect since large Re numbers indicate turbulence in most cases, where there is a lot of motion, vortices and eddies. But what exactly is the inertial force and how can it be explained physically?"}
{"id":"27168","title":"How can one build a multi-scale physics model of fluid flow phenomena?","text":"I am working on a problem in Computational Fluid Dynamics, modeling multi- phase fluid flow through porous media. Though there are continuum equations to describe macroscopic flow (darcy's law, buckley leverett equations, etc), these models do not apply for heterogenous media (with transport properties). We could, however, try to use the microscopic model (lattice boltzmann, or pore network models) which would be more faithful to the dynamics of the macroscopic heterogeoneous media. But any computational simulation of this model would run too slowly to be worth it. The principals of conservation laws apply at both scales (conservation of mass, momentum, energy), but the equations that describe these laws differ at each scale. How then, can we upscale microscopic physics in a computationally efficient manner? Are there any techniques for describing microscopic phenomena at the macroscopic level without such a heavy computational cost? Is there any technique to build a continuum description at all scales of the problem?"}
{"id":"6052","title":"Starting a physics Ph.D. at 60?","text":"Decades ago I was in a physics Ph.D. program, made a hash of things, and quite rightly was dropped from the program. Since then I've had a very satisfying career as a computer programmer, and picked up an M.S. in Applied Math. I'm currently working in a bioinformatics research lab, and really enjoying myself. I think I might want to retire in five or six years, and lately I've been daydreaming about returning to my first love, and entering a physics Ph.D. program. This would be entirely for my personal satisfaction. I have no intention of seeking a further career in physics, I'd just like to spend a few years working in a physics research group. I've been out of physics too long to really know what area I'd like to work in, but I'm thinking quantum computing or quantum optics. While I wouldn't require any financial support, I recognize that taking on a Ph.D. student is still a major investment of resources for a researcher. My hope is that my years of experience in software development would be enough of an asset to make it worth their while. I'm just curious if there is anybody in the audience who's done something similar, or any faculty who can give an opinion about whether this is a realistic notion. Update: Thanks to all for your answers and comments. I think nibot's comment that the thing to do is to establish relationships with some faculty I might like to work with is the most \"actionable\" answer. I really appreciated all the other anecdotes and advice though."}
{"id":"2952","title":"Does the equivalence between inertial and gravitational mass imply anything about the Higgs mechanism?","text":"For example: the role it might play in a theory of quantum gravity (ie causing space-time curvature)? I realize that inertial mass can result from binding energy alone. Has the equivalence principle been tested on elementary particles (like the electron) whose mass would be entirely due to the Higgs coupling? It seems like an obvious question, but I have never heard it discussed before."}
{"id":"77459","title":"Is there a definition of force?","text":"Well, Newton's three laws talks about forces, but no definition is given. In truth, Newton's second law gives an idea of what **total** force is: the time change rate of momentum. But, if we have a force amongst others, this force alone isn't the time rate change of momentum, so that we cannot use Newton's second law to define force, but just to try defining total force. Knowing all of this, is there a definition of force or we take force as a primitive concept like that of time? Thanks very much in advance!"}
{"id":"93692","title":"Isn't Newton's Second \"Law\" merely a definition of force?","text":"Is it just me, or is Newton's Second \"Law\" in fact no law at all, but a mere definition of force? The concept of force does not exist prior to the statement of N2, and thus it would seem that the law is entirely analytic and does not in itself improve our knowledge about nature. Things were different if the concept of force had been introduced separatedly, and if N2 then postulated a _proportionality_ between $\\mathbf F$ and $d\\mathbf p\/dt$. The fact that N2 is a statement of _equality_ now has the consequence that no better unit can be given for force than the traditional Newton = metre kilograms per second squared. As I see it, the only synthetic statements that could be considered part of N2 are: 1) all behaviours of and interactions between bodies may be described in terms of forces, which are applied _to_ bodies _from_ other bodies (note that this fact is not at all evident in itself); and 2) the values of the individual forces are independent of any other simultaneous forces acting from other objects. 1) now allows us to state _real_ laws about concrete interactions, like gravitation. 2) gives rise to the _superposition principle_ , while 1) and 2) together allow us to extend the concept of force from N2, which only defines the _net force_ acting on a body. The _force_ applied from a body on another body is then defined as what _would_ be the net force on the body if all other objects did not exist. In fact, it is not even evident that the concept of force is necessary for the physical science and its ability to describe the nature around us. The use of force is probably a consequence of its great practical applicability, which is in turn the result of a desire to describe nature in terms of linear differential equations. For entirely mathematical reasons, the study field of linear differential equations is rich, and thus its use in physics gives us invaluable tools that allow us to easily analyse physical systems. Work and energy are examples of concepts with properties that are again direct consequences of the mathematical framework (at least I think most of them are). However, were it not for those rich mathematical systems, it is not obvious that the mathematical framework for classical physics is even the most natural one. For example, a concept like _velocity_ (central to the force- oriented mathematical framework) seems less natural in a relativistic context, where _rapidity_ in some ways seems more in accordance with the physical system itself. So any thoughts about whether Newton's Second Law actually contains synthetic information about the physical world?"}
{"id":"70185","title":"Operator norm of creation and annihilation operators","text":"Are the creation and the annihilation operators $a(f)$ and $a^{\\dagger}(f)$ for the bosonic Fock space bounded? What is their norm? So far I did not have found any note about this in the linked Wikipedia article."}
{"id":"94708","title":"Spinor inner products","text":"The spinor inner product in particle physics is given by $\\overline{\\psi} \\psi = \\psi^{\\dagger} \\gamma_0 \\psi $, where I take the convention that the zeroth gamma matrix is hermitian while the rest are anti-hermitian. This is invariant under spin group transformations, $\\psi \\rightarrow e^{\\omega_{ab} S^{ab} }\\psi$, with $\\omega_{ab}$ real parameters and $S^{ab} = \\frac{1}{4}[\\gamma^a, \\gamma^b]$. However, there is a second invariant inner product given by $\\psi^{\\dagger} \\gamma_0 \\gamma_5 \\psi$, with $\\gamma_5 = i\\gamma_0 \\gamma_1 \\gamma_2 \\gamma_3$. My question is, why not the other one? Is it down to the fact that there are observable differences between the two, and one is favoured by experiment?"}
{"id":"119989","title":"What is the appropriate scaling for a Fermi-liquid fixed point?","text":"I am puzzled with the RG description of Fermi liquids. Consider non- interacting spinless fermions in 2D that form a circular Fermi surface, with an energy cutoff $\\Lambda\\ll E_F$ ($E_F\\equiv |\\mathbf{k}_F|^2$ is the Fermi energy). Now, what is the appropriate transformation under which the observables (e.g. $\\mathcal{G}(\\mathbf{k},\\omega)\\,=\\,\\langle \\psi_{\\mathbf{k},\\omega}\\bar{\\psi}_{\\mathbf{k},\\omega} \\rangle$) remain invariant? Following Polchinski's and Shankar's RG approach, one is tempted to say, scaling $\\mathbf{k}-\\mathbf{k}_F$, i.e., scaling momenta measured from the Fermi surface. In the present case of a circular Fermi surface, this means scaling $r-r_F$, where $r$ is the radial coordinate in the momentum space ($(k_x,k_y)\\to (r,\\theta)$) and $r_F=|\\mathbf{k}_F|$. I seem to recall that Shankar mentions in his paper that the fact that there is a scale, namely $k_F$, does not mean that Fermi-liquid fixed-points are not scale invariant. Strictly speaking, the scaling that maintains the Fermi surface is a non-linear scaling. Does this mean that for systems that possess a Fermi surface (not necessarily Fermi liquids) we should always cook up a non-linear scaling transformation? Is there a general recipe to deal with cases where the \"zero-energy manifold\" is not an isolated point?"}
{"id":"105382","title":"Question about equation 2.27 from Pachos's Introduction to topological quantum computing","text":"http:\/\/quince.leeds.ac.uk\/~phyjkp\/Files\/IntroTQC.pdf above is the PDF that is hosted on his website. The equation is on page 22 (pg 30 in the pdf). In chapter 2. It is the second equation of the subsection \"derivation of the holonomy\". On equation 2.27 he splits up the integral from the exponential of a time evolution operator into finite steps (that are infinitely small). Because they are within an exponential the infinite series becomes an infinite product. I get all of that. WHat I don't get is how the two Unitary matrices (the $U$ and the $U^{\\dagger}$) come out of the exponential. This has been vexing me for a while. This is as he goes from the center series to the right-hand side product. I'm wondering if this is a mistake or not. I'm not asking for a complete solution since I want to work it out but a hint would be nice."}
{"id":"121685","title":"Naive unification of scalar QFT and GR is possible?","text":"I am thinking on the Klein-Gordon equation with curved (non-diagonal) metrics. Is it possible? Doesn't have it some inherent contradiction? If yes, what? If no, what is this combined formula?"}
{"id":"111973","title":"Lorentz force, multiple conductors?","text":"3 conductors are placed in a magnetic field $B$, that have equal current $I$ flowing, is the Lorentz force totaled as the sum of all those conductors? Assuming that all of them have the same direction of current."}
{"id":"134024","title":"Prove scalar product is distributive","text":"The scalar product is defined as r*s = the sum of all r*s. Using this definition, prove that r*(u+v) = r*u + r*v. Also, if r and s are vectors that depend on time, prove that the product rule for differentiation applies to r*s. Ok, so I'm new to proofs and I literally do not know where to even start. Any suggestions?"}
{"id":"134026","title":"4-vector velocity to Newtonian?","text":"The four-vector condition for a particle free of forces is: $\\frac{du}{dτ} = 0$ and the equivalence of this to the statement of newton's first law follows from the expression for four-velocity: ![enter image description here](http:\/\/i.stack.imgur.com\/DtUxu.png) Then I came across this part in my lecture notes: ![enter image description here](http:\/\/i.stack.imgur.com\/01vnu.png) Now this got me curious. How exactly does one get to the Newtonian statement i.e. what algebra steps are involved? Thank you to anyone who can enlighten me~"}
{"id":"111971","title":"Gauge symmetry for p-forms","text":"It is well known that the Lorentz invariance of the S-matrix implies Gauge redundancy for 1-forms,'photons'. Does this argument go through to p-forms? That is does lorentz invariance of s-matrix of these fields imply that the following is a symmetry of the action? $A_{a_1 \\dots a_p} \\to A_{a_1 \\dots a_p} + \\partial_{[a_1}\\lambda_{a_2 \\dots a_p]} $ where $\\lambda$ is a (p-1) form."}
{"id":"18453","title":"Is there a magnetic field that satisfies $\\nabla \\times B = 0$?","text":"In electrostatics Maxwell's equations for the magnetic field are $\\nabla \\cdot B = 0$ and $\\nabla \\times B = \\mu_0 J$ Now, take $B = xi-yj$, where $i$ and $j$ are the usual unit vectors, then one can show that $\\nabla \\times B = 0$ which consequently means that $J=0$. But in Maxwell's equations, isn't that $J$ supposed to be the source of the magnetic field B? Then how come $J$, the source of the magnetic field, is zero yet $B$ is not zero? What am I missing?"}
{"id":"123682","title":"Experimentally measuring coherence length of laser","text":"So I am trying to figure out the coherence length of my laser source. I am using a mach zehnder interferomter (image attached). I have theoretically calculated $L_c$ to be somewhere between 800-1200 $\\mu m$ (equation on wiki page). So anyways. When I scan the movable mirror through a large range (50 mm), I observe interference fringes everywhere. I thought that outside of the coherence length, there should be no interference. Am I doing something incorrect?![Interferometer](http:\/\/i.stack.imgur.com\/ULNEk.jpg) NOte: The top part of the MZI interferomter is an adjustable mirror. NOT TO SCALE."}
{"id":"5373","title":"Quantum death like heat death possible?","text":"Quantum decoherence is an irreversible process which is the result of interaction of the system with its environment. It prevents interference due to lack of coherence. Environment acts just like a heat bath. Now my question is, is the different branches of the wave function of the universe becoming gradually more and more decoherent so that in a far future no trace of interference can occur? In other words I would like to know whether there will be any quantum death (like heat death of thermodynamics) of the universe if one waits for a sufficiently long time?"}
{"id":"128240","title":"How to feel or sense magnetic fields from earth?","text":"I know that earth has magnetic fields\/forces, but why we don't feel them? So if I hold a magnet and earth's magnetic field is positive and my magnet is from the positive side, then why the magnet does still fall down? Shouldn't be there a pullsive force since they will be of the same polarity?"}
{"id":"122339","title":"How stellar aberration is measured?","text":"A simple calculation shows that stellar aberration due orbital motion of earth is roughly 20 arcseconds. My questions are:   1. Practically how this small value is measured?    2. Does this value is in the range of accuracy of a 11 inch reflective telescope with a camera?    3. And how this measurement originally was done by Bradley in the 18th century?"}
{"id":"129487","title":"Regarding Non-renormalizatibility of GR","text":"I've been doing some reading trying to get to a better understanding of some renormalization issues with the Einstein-Hilbert action. But, something odd came into mind that I'm hoping some users may have the background to comment on it. Specifically, most of the sources I have seen are based on expanding the Einstein Hilbert action about a flat background to achieve a Lagrangian of the form $(\\partial h)^2+{1\\over{k}}(\\partial h)^2h+{1\\over k^2}(\\partial h)^2h^2+...$with terms that become more and more divergent the further out one goes. But, for some reason I find I'm a little pre-occupied with the notion that perhaps the particular expansion itself has issues with convergence that may or may not fundamentally have to do with the E-H action in its non expanded form. We commonly entertain the notion of a cut-off energy range to deal with the renormalization issues in the 'effective field theory' speak. How do we know that the expansion of the E-H itself isn't only valid as a low energy approximation though? Do we know that there isn't an expansion of an alternate form which has convergent terms for a range of energies where the first runs into problems?"}
{"id":"15981","title":"How can airplanes fly upside down?","text":"I've read many times, including here on this very site that the commonly known explanation of flight is wrong, and that airplanes can fly because the shape of their wings deflects air down. This makes sense, but as far as I can tell it doesn't explain upside down flight or symmetric wings. The images I've seen show an inclined wing, which forces the air to go downwards. But how can planes fly upside down then?"}
{"id":"110660","title":"Static Electricity and Grounding Confusion","text":"1. In an electroscope, you take metallic balls, and rub them with either a comb or a glass. Touching the balls with your finger is said to undo the effect of rubbing them with the comb or glass, since your finger is connected to ground, and ground is so enormous that it absorbs or releases the electrons to neutralize the metallic ball.   2. When you rub your shoes against a carpet and then touch a metallic doorknob, a shock results, since the build-up of electrons that was built up by rubbing against the carpet is released via the doorknob to ground. What I don't understand in #2 is how the buildup of electrons occurs in the first place. We've already seen in #1 that you are connected to ground, so in #2, why don't any excess electrons released from the carpet at the time of the rubbing immediately go back to ground? (I guess I'm asking a larger question: how can any rubbing of anything create a surplus or deficit of electrons if the thing being rubbed is connected to ground -- why doesn't that connection immediately discharge any imbalance of electrons created by the rubbing?)"}
{"id":"110662","title":"Can the Lorentz force stabilize the Hydrogen atom?","text":"I've recently been working on relative equilibria for some systems of particles. (ie. studying equilibrium solutions in a rotating frame. Saturn's rings for example.) This has evolved into some classical notions of atomic physics and questions of stability for Coulomb interactions there. Entertain for a few moments a classical(planetary) model for the atom. Take an isolated hydrogen atom for example. It seems to me that the $q\\left(v\\times B\\right)$ term of the Lorentz force provides an intrinsic stabilization for the two particle system. Naively play the right hand rule game with the system and you'll see. If we assume that the electron is in a classical orbit and not radiating, then could we establish the stability of atoms with the Lorentz force? The non-radiating electron is a topic of its own and appears to be far from solved. For example, if you would like to make the statement that accelerating charges radiate, therefore...... Try to go one step further and show why this must be true in general and you imediately run into difficulties. Gyro-stabilization might be a relevant topic here. Order of Magnitude investigation: \\begin{align} &\\text{Proton Charge:}\\qquad&q &\\approx 1.602\\times 10^{-19}\\: C\\\\\\\\[2mm] &\\text{Permeability of free space:}\\qquad &\\mu_0 &\\approx 4\\pi\\times10^{-7}\\frac{volt\\cdot s}{amp\\cdot m}\\\\\\\\[2mm] &\\text{Permitivity of free space:}\\qquad &\\epsilon_0 &\\approx 8.854\\times10^{-12}\\frac{farad}{m}\\\\\\\\[2mm] &\\text{Electron Speed: }\\qquad &v &= \\alpha c\\approx c\/137 \\approx 2.19\\times10^6\\frac{m}{s}\\\\\\\\[2mm] &\\text{Electron Radius:(Bohr) }\\qquad &r & = \\frac{\\hbar}{m_ec\\alpha}\\approx 5.29\\times10^{-11}m\\\\\\\\[2mm] \\end{align} Maximum Fields(Im thinking of the fields generated by the electron in a linear approximation for an infinitesimal section of the circular orbit: \\begin{align} \\text{Biot - Savart}\\qquad{\\bf B}_{\\text{max}} &= \\frac{\\mu_0 q v}{4\\pi r^2}&\\approx& 12.537\\:\\text{T}\\\\\\\\[2mm] \\text{Coloumb}\\qquad{\\bf E}_{\\text{max}} &= \\frac{q}{4\\pi\\epsilon_0 r^2}&\\approx& 5.145\\times 10^{11}\\frac{N}{C} \\end{align} With Force ratio: \\begin{align} \\frac{q(v\\times{\\bf B})_{max}}{q{\\bf E}_{max}} =\\frac{\\frac{\\mu_0 q^2 v^2}{4\\pi r^2}}{\\frac{q^2}{4\\pi\\epsilon_0 r^2}} = v^2\\mu_0\\epsilon_0 = \\frac{v^2}{c^2}\\approx\\frac{(2.19\\times10^6)^2}{(3\\times10^8)^2}\\approx 5.329\\times10^{-5} \\end{align} This shows the magnetic force is 5 orders of magnitude smaller than the electrostatic force for this crude model of the ground state hydrogen. Not sure what this conclusively says about the internal dynamics but at least I think I can foresee some problems with this force stabilizing the system. A colleague of mine who does work with NMR(nuclear magnetic resonance) informed me that the magnetic fields they work with in the lab are on the order of 3$T$ and that they do not observe the large fields I am envisoning in practice. The concept he quoted as the reasoning behind this was \"orbital quenching\". This appears to be a sort of averaging to zero of this field. However, it was also stated that the magnetic field that I am interested in can be measured in beam type experiments."}
{"id":"118418","title":"Why does the coriolis effect dissapear at the equator?","text":"I'm studying from the book \"Classical Mechanics\" by Goldstein and from a coursebook my Professor provided me. In the coursebook, it says that \"the Coriolis effect disappears at the equator (Where the rotation vector $\\vec\\omega$ of the Earth is horizontal)\" Now here's my reasoning: Take the Coriolis term: > $-2m(\\vec\\omega\\times\\vec v_r)$ Nearby the north pole, this would cause a force pointing towards the equator if you were moving from west to east. The closer you get to the equator, the more this force starts pointing \"upwards\" if you were to describe the vector from the surface of the Earth.   At the equator, this vector is perpendicular to the tangent of the equator and pointing outwards.   Intuitively, this would mean that if you move from west to east on the equator you would be accelerating away from Earth. Now if my reasoning is correct, this doesn't mean the Coriolis effect disappears, only it turns into some sort of centrifugal force, but this seems weird because it's described by another term. Where did I go wrong?"}
{"id":"89405","title":"Bullet: Piercing vs. Deflection","text":"I am trying to create a simulation where a projectile (imagine a bullet) is fired at a surface of some kind. When it collides with the surface, I need to figure out what happens between the bullet and surface. > Example: > > Bullet fired at extremely sturdy object, bounces off the object. > > Bullet fired at very soft object, pierces into object. > > Bullet fired at thin sturdy object, dents the object but does not pierce it. 1) Does what I'm looking for have a name? I am having problems finding any type of equations that may relate to my examples. 2) Can somebody please explain to me what the forces are in this example and how they interact? 2b) What are the variables that I need to consider?"}
{"id":"127301","title":"Coleman-Weinberg potential: resum at 2 loops?","text":"Say we want to compute the Coleman-Weinberg potential at 2 loops. The general strategy as we know is to expand the field $\\phi$ around some background classical field $\\phi \\rightarrow \\phi_b + \\phi$, and do a path integral over the quantum part of the field, $\\phi$. We can retrieve the effective action by doing a path integral, something like eq.42 in this reference. There are 2 ways to do this at 1 loop, we can either evaluate a functional determinant or do the classic Coleman-Weinberg thing where we sum up all diagrams we get by inserting any number of background fields $\\phi_b^2$ into the loop integral. This is eq. (56) of that same reference again. My question is, why do we not need to do this resummation over background field insertions at 2 loops? For example, in this (quite standard) reference, as well as in chapter 11 in Peskin and Schroeder, the authors seem to claim that the 2 loop contribution to the path integral are simply the \"rising sun\" and \"figure 8\" vacuum diagrams, and no summing over classical field insertions is even mentioned. What am I missing? EDIT: To give some more details, in perturbation theory, each diagram contributing to the path integral is spacial integral of some functional derivative acting on the free field path integral with a source: the loop diagram with n insertions of external field $\\phi_b$ is the term: $$\\left( \\phi_b^2 \\int dx \\left( \\frac {\\delta}{\\delta J(x)}\\right)^2 \\right)^n Z_0[J]$$ The 2 loop figure 8 is $$\\int dx \\left( \\frac {\\delta}{\\delta J(x)}\\right)^4 Z_0[J]$$ The 2 loop diagrams that it seems like the papers cited above are excluding are contributions like $$\\left( \\phi_b^2 \\int dx \\left( \\frac {\\delta}{\\delta J(x)}\\right)^2 \\right)^n\\int dx \\left( \\frac {\\delta}{\\delta J(x)}\\right)^4 Z_0[J]$$ It seems to me that these terms will indeed arise in the exponential expansion of the interacting lagrangian, so it seems that a resummation over $n$, as in the 1 loop case, is still necessary. Where is my error?"}
{"id":"93992","title":"Capacitance and energy","text":"So I've been reading about capacitance (Specifically in capacitors) and I'm wondering if this reasoning is sound, and why\/why not: C = Q\/U U = Q\/C We also know that: U = E\/Q So: E\/Q = Q\/C Q^2 = EC E = Q^2\/C However, if I try to use this it conflicts with some of the answers in my textbook, but I don't see where I made any mistake."}
{"id":"28804","title":"Does using batteries in a series raise the overall Wh's of the batteries used? Why\/Why Not?","text":"If one has 2 batteries of 6 Volts each with 500 mAh each, then separately the batteries should provide 6 Volts * 500 mAh = 3 Wh each or 6 Wh total. However would it be the case that if someone where to connect the two batteries in a series that they would be able to double the Wh, as now the equation would be 12 Volts * 1000 mAh = 12 Wh. Why\/Why not?"}
{"id":"82009","title":"Can a very thin sheet of any material float inside a liquid?","text":"When a body is immersed in a liquid,buoyancy is the net force of all the forces acting on it. Now the forces are equivalent to those which will act on the same volume of liquid.Rightly so,but considering the chaotic motion of the molecules of the liquid,for every force acting on that particular volume of liquid,there will be another force in the opposite direction but of different magnitude.The magnitude differs because of the difference in height. Now let's consider a very very thin sheet immersed in the liquid.The opposite forces now will be of the same magnitude as difference in height is negligible and hence will cancel each other!so the only force acting will be its weight. So the thin sheet will never be balanced!So does that imply that no thin sheet will ever float inside a liquid??"}
{"id":"12510","title":"Is it possible to mechanically isomerize an sp3 hybridized carbon center?","text":"Imagine I have an sp3 hybridized carbon attached to four separate polyethylene chains. By pulling on the polyethylene chains in some manner, is it possible for me to mechanically isomerize the chiral center prior to breaking any carbon-carbon bonds? Perhaps a more realistic scenario would involve shining a laser on a compound similar to 2,2-dimethylpropane (i.e. a fully saturated carbon compound where four carbons are attached to an sp3 hybridized carbon), but with asymmetric functionalizations on the bonded carbons to allow for detection of isomerization. Just as in the previous scenario, can mechanical isomerization occur prior to breaking any carbon-carbon bonds? * * * Quoting from Georg's answer: \"In \"practice\" there is a big problem, You would need two Laplacian demons (with two hands each) to do the experiment...\" It's not clear to me that this is an impossible experiment. For example, you could attach one polyethylene chain to a surface and then try to make a covalent bond between the other relevant chain and, say, an AFM tip. If this is successful, one would then proceed in a manner similar to that reported in \"How Strong is a Covalent Bond?\" by Michel Grandbois et. al. (http:\/\/www.sciencemag.org\/content\/283\/5408\/1727.short), which estimates a ~4.0 nanonewton rupture force for a C-C bond (See Fig. 4) and directly measures the rupture of a single Si-C bond to be ~2.0 +\/- 0.3 nanonewtons. In the case where one wants to look for mechanical isomerization of the sp3-hybridized carbon, one would look for an earlier sub-4.0 nN peak in the force vs. extension curve (or whatever critical force is experimentally measured for breaking a C-C bond), perhaps allow relaxation at some critical force to complete the isomerization process, and then compare the observed displacement with a geometrical model. Sounds ridiculously hard, but I see no fundamental reason it couldn't be done."}
{"id":"12511","title":"Simplest interferometer","text":"I want to build simplest interferometer which should be able to measure movements down to fraction of wavelength. What is the simplest scheme for that, and what are the requirements for a laser? I have a bunch of laser diode-based ones, and I guess they might be not coherent enough... Are green DPSS ones any better?"}
{"id":"95919","title":"What is this form of notation called?","text":"$$^{14}_6C \\rightarrow ^{14}_7N + E^{-} + \\bar{\\nu}_e$$ Just curious!"}
{"id":"63302","title":"Optimal tunnel shape for travelling inside the earth","text":"Say you were to travel from Paris to Tokyo by digging a tunnel between both cities. If the tunnel is straight, one can easily compute that the time for travelling from one city to the other (independently from where both cities are actually) is a constant (about 42 minutes). It goes without saying : only gravitationnal potential and no friction. But let's upgrade on the idea. Fix the two cities : **what is the optimal shape of the tunnel** , so that the time it takes for going from one city to the other is shortest ? I assume a planar movement, where the plane contains both cities and the center of the earth. With Gauss' law, it is easy to compute that the gravitationnal potential on a mass m inside the earth at any distance $r$ from the center : $V(r) = m\\alpha r^2$ where $\\alpha$ is a constant. I parameterize the tunnel by a polar function $r(\\theta)$ where the origin is taken at Earth's center. The problem reduces to minimizing the following functional $\\mathcal{T}(\\dot{r},r,\\theta) = \\int \\limits_{\\theta_1}^{\\theta_2} \\frac{ds}{|v|} = \\int \\limits_{\\theta_1}^{\\theta_2} d\\theta \\frac{\\sqrt{\\dot{r}^2 + r^2}}{|v|} = \\sqrt{\\frac{m}{2}}\\int \\limits_{\\theta_1}^{\\theta_2} d\\theta \\frac{\\sqrt{\\dot{r}^2 + r^2}}{\\sqrt{E - m\\alpha r^2}} \\equiv \\sqrt{\\frac{m}{2}}\\int \\limits_{\\theta_1}^{\\theta_2} d\\theta G(\\dot{r},r,\\theta)$ $\\theta_1$ and $\\theta_2$ are the angular coordinates of both cities. $\\dot{r}$ is the derivative of $r$ with respect to $\\theta$. With this parametrisation, the 'Hamiltonian' is conserved : $\\frac{\\partial G}{\\partial \\dot{r}}\\dot{r} - G = c$ Which leads to $\\frac{-r^2}{\\sqrt{E-2\\alpha r^2}\\sqrt{\\dot{r}^2+r^2}} = c$ And here I get stuck. Even if I try to play with special initial conditions, I cannot solve this equation. The easiest way is to perform separation of variables, but the integralcannot be carried out. Any ideas on how to solve this problem ? I am sure the solution is a nice well known symmetric function. I just don't know how to tackle this problem. Would you tink it'd be easier to solve this in cartesian coordinates ?"}
{"id":"95917","title":"How do I write the Hamiltonian for a 3-level system?","text":"I came across following types of three-level systems like V-system, Λ-system and 2-photon absorption ![enter image description here](http:\/\/i.stack.imgur.com\/Xq1Lo.png) It seems that their Hamiltonians can be written intuitively by checking out the coupled levels and the gap between levels both in general and in the dressed picture (rotating frame approximation). Do V and Λ systems have the same Hamiltonian due to the symmetry in the level structure ? By following the analogy, I designed the Hamiltonian for 2-photon absorption process to be same as the other 2 cases but it is not so :- What are the coupled levels in the 2-photon absorption process and how can the Hamiltonian for this process be written ? How do I extend the design to N-level system -lets say for N-photon absorption in N-level system ?"}
{"id":"117038","title":"Can you get a playground to swing from stationary","text":"I feel this might be a FAQ but I would love a definitive answer. Imagine a frictionless stationary idealised child's playground swing. If you are sitting on the seat of the swing, is it possible in principle to set it into motion by simply moving your body and pulling on the chains of the swing? If so, is there a simple explanation for how you can change your center of mass without an external force acting on you?"}
{"id":"30154","title":"extracting energy from cosmological expansion","text":"This question is a more concrete reincarnation of an old question about energy conservation in GR. Are there mechanisms to extract energy from the cosmic rate of expansion? putting some extremely long ropes across the universe and extract work from their increase of separation? assuming this is possible ( i don't see why it would not ), what form would take the back-reaction force? in all physics, every time you extract work from some system by some force, this system will exert a force back. Can we slow down cosmic expansion by variations of these mechanisms?"}
{"id":"117030","title":"Quick question on Perturbation theory - How do I evaluate this probability?","text":"We know that hamiltonian for interaction between an electron and external field along $z$ is: $$\\hat H = \\frac{e\\hbar B}{2m}\\hat \\sigma_z = \\frac{\\hbar \\omega}{2} \\hat \\sigma_z $$ This has energy eigenvalues of $\\pm \\frac{\\hbar \\omega}{2} $ Time evolution operator is given by: $$ exp(i\\frac{\\omega}{2}\\sigma_z) = \\left( \\begin{array}{ccc} e^{-\\frac{i\\omega t}{2}} & 0 \\\\\\ 0 & e^{\\frac{i\\omega t}{2}} \\end{array} \\right) $$ At time $t=0$, the electron is in state $ \\left( \\begin{array}{ccc} a_0 \\\\\\ b_0 \\end{array} \\right) $ and at $t>0$, the state of the electron is: $$ |\\psi_{(t)}\\rangle = \\left( \\begin{array}{ccc} a_0 e^{-\\frac{i\\omega t}{2}} \\\\\\ b_0 e^{\\frac{i\\omega t}{2}} \\end{array} \\right) $$ Suppose now we have a small perturbation field in the x-y plane: $$B_1= B'\\left( \\begin{array}{ccc} cos (\\omega t) \\\\\\ sin (\\omega t) \\\\\\ 0 \\end{array} \\right) $$ Thus the perturbation is: $$V_1 = \\frac{e\\hbar}{m}\\vec B_1 \\cdot \\vec S = \\frac{e\\hbar}{2m}B' \\left( \\begin{array}{ccc} 0 & e^{-i\\omega t} \\\\\\ e^{i\\omega t} & 0 \\end{array} \\right) $$ I now want to find the probability the electron transits from the lower energy state to the higher energy state. By first order perturbation theory, $$\\frac{\\partial a_m^{(1)}(t)}{\\partial t} = e^{\\frac{i\\Delta E}{\\hbar}t}\\frac{1}{i\\hbar} \\sum_n a_n^{(0)} \\tilde V_{mn} \\space dt $$ . Thus for this case, it is represed by this matrix: $$i\\hbar \\frac{\\partial}{\\partial t} \\left( \\begin{array}{ccc} a^{(1)} \\\\\\ b^{(1)} \\end{array} \\right) = \\left( \\begin{array}{ccc} \\langle-|V_1|-\\rangle & \\langle -|V_1|+\\rangle \\\\\\ \\langle+|V_1|-\\rangle & \\langle+|V_1|+\\rangle \\end{array} \\right) \\left( \\begin{array}{ccc} a_0 \\\\\\ b_0 \\end{array} \\right) $$ Since the particle is originally in the lower state, and we want to find the chance electron jumps to higher (+) state, we want to find first order perurbation $|a_{+}^{(1)}|^2 = |b^{(1)}|^2 $ with $b_{(0)} = 0 $ $$ i\\hbar \\frac{\\partial}{\\partial t} \\left( \\begin{array}{ccc} a^{(1)} \\\\\\ b^{(1)} \\end{array} \\right) = \\frac{e\\hbar B'}{2m} \\left( \\begin{array}{ccc} 0 & e^{-i\\omega t} \\\\\\ e^{i\\omega t} & 0 \\end{array} \\right) \\left( \\begin{array}{ccc} a_0 \\\\\\ 0 \\end{array} \\right) $$ Thus $b^{(1)} = \\frac{e\\hbar B'}{2mi\\omega}a_0e^{i\\omega t} $ and thus probability of jumping from lower to higher state is: $$ (b^{(1)})^2 = \\left(\\frac{e\\hbar B'}{2m\\omega}a_0\\right)^2 cos^2 (\\omega t) $$ Is it okay to simply take the real value? Because if I don't, it will be $|bb^*| = const.$ which is wrong, as it should oscillate with time."}
{"id":"88758","title":"A change in the mass of the moon","text":"Would a change in the mass or radius of the moon have any effect on its speed? When asked this question in an assignment, I stated that it did not. My reasoning being that a change in the mass or radius of the moon would in no way change the acceleration or velocity of the moon. The moon would travel along the same path, the only difference being that the gravitational force being exerted by\/on the Earth and Moon would be of a lesser magnitude. As the mass of the Moon decreases by ½, does the gravitational force. F=ma ½F=½m*a The original gravitational force being exerted is double that of the new gravitational force 2(G MiMii\/r2)=G Mi(1\/2Mii)\/r2 I got this answer wrong and I am unsure why. Was I wrong in my approach? Are there other forces that I should have taken into account when coming to my conclusion? Edit:The mass of the moon simply vanishes."}
{"id":"19537","title":"How can Kaluza-Klein particles be told apart from winding modes at the LHC?","text":"I`ve already asked this in the comments below this article http:\/\/profmattstrassler.com\/articles-and-posts\/some-speculative-theoretical- ideas-for-the-lhc\/extra-dimensions\/how-to-look-for-signs-of-extra-dimensions\/ from Prof. Strassler but did not get a satisfactory answer... In the article Prof. Strassler explains that a possible sign of extra dimensions at the LHC could be the detection of Kaluza-Klein particles. The energy (or mass squared) spectrum of the corresponding particle tower would have a step size proportional to 1\/r (where r is the radius of a compact extra dimension). Now I`m wondering how such a (hypothetical) KK particle candidate can be told apart from a winding mode particle (with the step size of the energy spectrum proportional to r) which is probably more easily produced depending on the size r of the extra dimension ? By the way what's the actual state of these searches for extra dimensions considering the data already available ? **Clarification** Considering an extra dimension that is much larger than the particle, the KK modes with the 1\/r spectrum arise from the quantized angular momentum a particle can have due to it' movement in the extra dimension. If the extra dimension is smaller the particle can wind around the extra dimension leading to the winding mode particles. For these the energy spectrum is determined by the potential energy of stretching. The spectra of the KK and winding modes are T-dual to each other."}
{"id":"34147","title":"Is there a non-perturbative remormalization? If so, how does it work?","text":"Is there a method to renormalize a theory without using perturbative expansions for the divergences? For example, is there a method to get masses and other renormalized quantities without using expansions and counterterms? O have heard about Lattice Gauge theory but what other solutions of examples of non perturbative renormalization (numerical or analytical ) are there?"}
{"id":"107775","title":"Acceleration and Circular Motion","text":"Lets assume that there is a force that makes our body moves in circular motion. We know that the acceleration of a body that moves in circular motion is Velocity ^ 2 \/ Radius . How is it possible that if we grant velocity to a body with mass M , Its acceleration is not dependent on M ?"}
{"id":"65725","title":"How sharp is the direction of magnetic field of a cone magnet?","text":"I heard recently that cone magnets can be used to focus and direct the field sharply at one particular point in a particular direction. I jus want to kno how accurate and sharp this field is? And if its so sharp that it does not influence the area surrounding that point but only straight above it, how small can the cones be fabricated and whaether we can fabricated cone electromagnets?"}
{"id":"43382","title":"By how much will water resistance and upthrust slows down a fully submerged object?","text":"Consider the following example: While running, I do work and I have a certain velocity and acceleration. When I'm in a pool (my feet are touching the bottom), I do the same work but, I have less acceleration. As more of my body is submerged I begin to be slower (doing the same work) . So I require an explanation, by how much water slows me."}
{"id":"43385","title":"Scales and Hooke's Law","text":"Bob weighs $B$ kg and when he steps on a scale, the spring inside gets compressed by $c$ meters. Assuming the scale obeys Hooke's Law, if Alice weights $A$ kg, how much work is done compressing the spring when Alice stands on the scale? There is obviously a flat surface serving as the interface between the spring of the scale and the part of the scale Bob and Alice step on. For simplicity, I will assume that this interface has no mass, so that the only forces acting on it are the weight of the person and the spring force. When Bob is on the scale, the interface moves an amount $c$ at which point the spring force equals Bob's weight, i.e. $B = kc$, where $k$ is the spring constant. Solving for $k$ yields $k = B\/c$. When Alice is on the scale, the interface moves until the spring force equals Alice's weight. If we call the displacement the interface undergoes $d$, then $A = kd = (B\/c)d$, hence $d = (A\/B)c$. The work done by Alice's weight to move the interface $d$ meters is $Ad$. The work done by the spring force to move the interface $d$ meters is $\\int_0^d kx \\cdot x \\, dx=kd^3\/3$. The net work on the interface is therefore $Ad - kd^3\/3$. Apparently I am wrong because the answer is $Ad\/2$. Where did I go wrong here?"}
{"id":"69601","title":"Suggestions of a non-linear example for a small research project on numerical solution of ODEs?","text":"I'm a first year undergrad and I'm doing a small research extension on numerically solving ODEs. I have done the main ODE course at my university, as well as physics. The second part of the project constitutes using a method (linear multistep in my case) to solve a system of our choice and writing a report about it. I spoke to my supervisor and she said that a nonlinear example would be instructive since I am interested in plasma physics in the future. Also I'd like to do something involving fluids. Problem is I have no idea where\/how to find a specific example of a system modelled by a nonlinear ODE (I have found plenty of nonlinear PDE examples). So I was wondering if someone could tell me or guide me to an example, or how to find an example. I am aware that this style of question may be frowned upon, but I've been looking for a while without success (and posted on other forums) so I thought I'd give it a shot."}
{"id":"69606","title":"SUSY and the Large hadron collider","text":"Did the LHC rule out SUSY as a solution to the hierarchy problem ? What's the common belief among theorists regarding the possibility of discovering SUSY in the 2015 run of the LHC ? Did the absence of a SUSY signal so far depress the enthusiasm that SUSY is an important element of our universe ?"}
{"id":"46353","title":"Is observable universe an explanation against Olbers' paradox?","text":"First of all, let me tell you that I'm not a physicist but rather a computer scientist with a mere interest in physics at nowhere near a professional level so feel free to close this question if it doesn't make any sense. I remember a physicist friend mentioning me about an argument about the finiteness of the universe. I have looked it up and it turned out to be Olbers' Paradox. We computer scientist like to use astronomical numbers to help us imagine the complexity of an algorithm. One of the most common one is the number of atoms in the Observable Universe (which we take as $10^{80}$) so I have a crude understanding about the observable universe concept. I had known these two for some time hence I woke up with the dilemma today. So my question is, how come it can be argued that universe is finite just because it is dark if we know that we can only observe a finite portion of it? Can't it be the case that the universe is infinite even if the sky is dark because not all the light from all the stars reach the earth? I have searched this a little bit but I think I need an explanation in simpler terms (like popular physics). A historical perspective would also be welcomed."}
{"id":"1894","title":"Path integral vs. measure on infinite dimensional space","text":"Coming from a mathematical background, I'm trying to get a handle on the path integral formulation of quantum mechanics. According to Feynman, if you want to figure out the probability amplitude for a particle moving from one point to another, you 1) figure out the contribution from every possible path it could take, then 2) \"sum up\" all the contributions. Usually when you want to \"sum up\" an infinite number of things, you do so by putting a measure on the space of things, from which a notion of integration arises. However the function space of paths is not just infinite, it's _extremely infinite_. If the path-space has a notion of dimension, it would be infinite-dimensional (eg, viewed as a submanifold of $C([0,t] , {\\mathbb R}^n))$. For any reasonable notion of distance, every ball will fail to be compact. It's hard to see how one could reasonably define a measure over a space like this - Lebesgue-like measures are certainly out. The books I've seen basically forgo defining what anything is, and instead present a method to do calculations involving \"zig-zag\" paths and renormalization. Apparently this gives the right answer experimentally, but it seem extremely contrived (what if you approximate the paths a different way, how do you know you will get the same answer?). Is there a more rigorous way to define Feynman path integrals in terms of function spaces and measures?"}
{"id":"81064","title":"Drag\/air-resistance impulse","text":"I'm simulating snowfall in a discrete-time physics engine, using this formula for the drag force on a snowflake: $F_d=CAv^2$ (where $A$ is the flake's cross-sectional area and $C$ is a catch-all constant). I need to calculate the drag impulse over a given $dt$, and that's where I'm stuck. I've read this similar question, but it's been so long since I've dealt with differential equations that I can't follow the accepted answer well enough to adapt it for my purposes."}
{"id":"81066","title":"Not satisfied with \"trick\" in zeta function regularization","text":"I am not satisfied with the explanations of the $\\sum_n \\log \\lambda_n = - \\frac{d}{ds} \\sum_n \\lambda_n^{-s}|_{s=0}$ \"trick\" used in zeta function regularization, discussed here and here, or the references therein. If someone could give a step-by-step explanation or a reference if the explananation is more complicated than I'm guessing, that would be greatly appreciated."}
{"id":"11726","title":"Can a photon exiting from a gravity well ever reach a frequency of zero \/ wavelength of $\\infty$?","text":"In reading another question about gravity's effects on a photon, I wondered if it were possible for a photon to ever be redshifted to zero wavelength. I know that black holes have a gravity field strong enough to keep light from escaping but what about the affect on the photon that is escaping. Say we have a single photon escaping on a path directly away from a non-spinning singularity. Also conjecture that we can release a single photon at various distances from the singularity (still on a trajectory directly away) so as to select the strength of the gravity field it must traverse. I know the distance has already been calculated (the event horizon I believe) but what of the photon itself? Is there a point where the amount of redshifting (e.g. gravitational) causes the wavelegth of the photon to become infinitely large? This answer mentions the change in potential and kinetic energy. Is it possible for all of a photon's kinetic energy be transformed to potential energy? What are the implications, both if it were possible and\/or impossible? More importantly if impossible, what makes it impossible? This would beg the question, what is the smallest amount of energy possible (or quanta) for a photon? If that minima-photon attempted to escape from within a gravity field, it can't slow down so what does it do? As the wavelength increases approaching infinity, so must it's frequency decrease to approaching zero."}
{"id":"11724","title":"Stress vs Strain for mild steel","text":"![two muppets](http:\/\/i.stack.imgur.com\/mELg2.jpg) After Yield point on stress strain diagram the under curve occurs what does it mean what will happen for the mild steel at that particular time and again why the curve goes to up and reaches ultimate point i am not getting that WHY that under curve comes"}
{"id":"11722","title":"Argument for quantum theoretic conformality of $\\cal{N}=2$ super-Chern-Simon's theory in $2+1$ dimensions -Part 2","text":"This is in continuation to what I was asking here earlier - Argument for quantum theoretic conformality of $\\cal{N}=2$ super-Chern-Simon's theory in $2+1$ dimensions Or one can look at this paper - http:\/\/arxiv.org\/pdf\/0704.3740 I am mainly having in mind the $\\cal{N}=2$ lagrangian that would be obtained by combining the equations 2.4, 2.5 and 2.7 of that paper.   * In that light I would like to understand the meaning of the comment made in that paper just below equation 2.9    * I have heard that one way of arguing the existence of the $\\cal{N}=3$ fixed point in the flow of the matter couplings of the $\\cal{N}=2$ theory is to see that for large values of the coupling the theory is effectively behaving like the Wess-Zumino model and for small values it is like pure-super-Chern-Simons theory.  What is known about the signs and the derivatives of the beat functions of these two limiting theories? Like if it known that the sign of the beta function is different for these two cases then it might suffice to argue by continuity that the beta function does go to zero somewhere in between and then that is a strong candidate to be the $\\cal{N}=3$ theory."}
{"id":"93989","title":"What happens to a conducting ring when exposed to an electric field?","text":"It might be a silly question, but one of my friends just got asked this question at an oral exam, and he could not answer it, and didn't receive the answer either (Or at least he forgot). And I've been thinking a while, and I'm not sure what would really happen. If I take, lets say a wedding ring, made from a conducting material, placed it on a table, and turned on an external electric field, what would happen ? The inside of the ring is field free, right ? So does anything even happen, and if so, is it only on the surface of the ring ?"}
{"id":"94660","title":"General motion of a cone on an inclined surface","text":"Suppose that a solid cone is placed horizontally on an inclined surface and is initially at rest. How will the cone move when it starts motion due to its weight? I know that its motion depends on the incline angle and also on the friction coefficient of the surface (as I observed by doing some experiments), but I can't establish the relation between them. Can anyone help me?"}
{"id":"17072","title":"Are there irreducible tensors of half integral degree in quantum mechanics?","text":"According to Ballentine, an irreducible tensor of degree k can be defined as a set of $2k + 1$ operators $\\\\{T_q^{\\;\\;(k)}:(-k \\le q \\le k)\\\\}$ satisfying the following commutation relations: $$ [J_\\pm, T_q^{\\;\\;(k)}] = \\hbar \\left( (k \\mp q)(k \\pm q + 1) \\right)^\\frac{1}{2} T_{q \\pm 1}^{\\;\\;(k)} $$ $$ [J_z, T_q^{\\;\\;(k)}] = \\hbar q T_{q}^{\\;\\;(k)} $$ The usual examples employ scalars, vector operators and the irreducible subspaces of cartesian tensor operators. The irreducible tensors that arise from these constructs all have integral degree. Is it possible to have irreducible tensors of half integral degree, for example $k=\\frac{1}{2}$? If so is there a simple example?"}
{"id":"18301","title":"Value of the Hubble constant over time","text":"There is something I don't understand about the Hubble \"constant\" H, as it seems to clump two concepts together that I can't quite unify in my head. On the one side, we have $$V = D H$$ which means that for a given distance D, there is a certain amount of new space created over time - and H is simply the factor that makes this relationship work. So, for example, say we have two points 1 Mpc apart, this would mean they recede at about 70 km\/s away from each other (given our current approximation of H). Now the thing I can't wrap my head around is that $$T =\\frac{1}{ H}$$ is also the age of the universe. Contrary to claims made on, say, Wikipedia this means that H cannot possibly have been a constant throughout the past 13 billion years, because mathematically 1\/H means that H _must be_ continually shrinking as the universe ages. So if H _did_ start out as some huge value and is now shrinking over time, doesn't this mean the expansion of the universe is slowing down? Because if H is shrinking, I'll get a lower value of V today than I'll get tomorrow. Shouldn't the notation then be more like $$V = DH(t)$$ So which one is it? If 1\/H is simply the solution for D=0, how can we use it as the expansion-velocity-per-unit-of-distance at the same time? What's worse, how can literature say H has probably been more or less constant forever and simultaneously assert that 1\/H is the current age of the universe? What am I missing?"}
{"id":"9690","title":"resistance of a flywheel to change of angular momentum","text":"I've got a flywheel with angular momentum $\\vec L$. Now I'd like to change the orientation of $\\vec L$ by an angle $φ$ (in degrees). How do I calculate the angular momentum $\\vec M$ for that change?"}
{"id":"104958","title":"Differentiate wave speed, don't understand","text":"The speed $v$ of some wave is $ω\/k$ and I want to differentiate this with respect to $k$. Apparently this equals: $dv\/dk = d(ω\/k)\/dk-ω\/k^2$ But I don't understand why. Isn't this just saying \"the derivative minus the derivative\"? Should it not simply be: $-ω\/k^2$?"}
{"id":"62731","title":"Can a photon exhibit multiple frequencies?","text":"Can a photon be a superposition of multiple frequency states? Kind of similar to how an electron can be a superposition of multiple spin states."}
{"id":"112514","title":"Semiconductor doping and conductivity","text":"Why is the conductivity of Si dopped with P and As much bigger than the conductivity of Si dopped with In and P?"}
{"id":"48295","title":"Do we exert more force when we walk?","text":"When we walk, is the magnitude of the earth's force on us greater than if we were standing still?"}
{"id":"20567","title":"Could a Dyson sphere destroy a star?","text":"Freeman Dyson proposed that the power needs of an advanced civilisation would eventually require the entire energy output of a star to be collected, so that the star would end up surrounded by a dense network of satellites extracting power from the radiation. In science fiction this idea has often mutated into a solid shell that completely surrounds the star. My question is, if it were possible to build such a shall, what effect would it have on the dynamics of the star inside? I suspect the effect would be catastrophic, but I'd like more details. Here's my reasoning: Wikipedia implies that the temperature gradient between the core and outer layers of a star plays an important role in its stability. The shell around the star would reflect or re-emit a lot of the star's radiation. (I'm assuming the shell isn't composed of perfect solar collectors that would simply absorb all the radiation.) This reflected radiation would reduce the loss of heat from the outer layers of the star. This would reduce or even eliminate the temperature gradient, which I guess would cause the star to expand. My question is whether this is correct, and if so whether it would be enough to disrupt the process of fusion in the star's core. Or would there be some other, less obvious effect on the star's dynamics? A closely related question is, would stars be stable in a static or contracting universe? In this case all of space would be filled with the radiation emitted from other stars, and I'd be interested to know what effect this would have on stellar dynamics."}
{"id":"80338","title":"Ball on a slope","text":"I have a lab report, but I can not write it in a correct way. The lab experiment was about a ball rolling on a slope, I have a height of the slope, the distance, and the time in which the ball spent to reach the end of the slope. I have data from 4 trials. My question is: How I can know the gravity is constant in all these trials, and which equation do I need to use?"}
{"id":"78405","title":"A thermodynamic transformation that can be represented by a continuous quasistatic path in its state space may still be irreversible. Why?","text":"A thermodynamic transformation that has a path (in its state space) that lies on the surface of its equation of state (e.g., $PV=NkT$) is always reversible (right?). However, if the path is a continuous quasistatic path in state space but not on the surface of equation of state, it is considered irreversible. Why is this so? In this case the gas still has a uniform thermodynamic variables (because all the intermediate states are points in the state space). Why can it not be reversed exactly along the same path?"}
{"id":"111272","title":"Trajectory of a photon around a Schwarzschild black hole?","text":"Consider a photon coming from the infinity in a unbounded orbit to a Schwarzschild black hole (Schwarzschild radius $r_{s}$) (see this for illustration). Its impact parameter is $b$ and its distance of closest approach is $r_{0}$ with $$b^2=\\frac{r_{0}^{3}}{r_{0}-r_{s}}$$. Then its trajectory in polar coordinates is defined by : $$\\frac{d\\varphi}{dr} = \\frac{1}{r^2\\sqrt{\\frac{1}{b^2}-\\left(1-\\frac{r_s}{r}\\right)\\frac{1}{r^2}}}$$ Consequently : $$\\varphi\\left(r\\right) = \\int_{r_{0}}^{r} \\frac{dp}{p^2\\sqrt{\\frac{1}{b^2}-\\left(1-\\frac{r_s}{p}\\right)\\frac{1}{p^2}}}$$ and one can compute the total deviation using : $\\Delta\\varphi = 2\\times\\left(\\lim_{r\\to+\\infty}\\varphi\\left(r\\right)-\\frac{\\pi}{2}\\right)$ But my question is : how can I plot\/draw the trajectory using the integral expression of $\\varphi\\left(r\\right)$ ? * * * Because if I compute : $$f\\left(r\\right) = 2\\times\\left(\\int_{r_{0}}^{r} \\frac{dp}{p^2\\sqrt{\\frac{1}{b^2}-\\left(1-\\frac{r_s}{p}\\right)\\frac{1}{p^2}}}-\\frac{\\pi}{2}\\right)$$ I obtain $f\\left(r_{0}\\right) = -\\pi$, and then $f$ increases up to zero, crosses zero, and tends to its positive value at infinity $\\Delta\\varphi$. It does not make sense for me and I do not understand how to compute the trajectory from that. * * * If I compute : $$g\\left(r\\right) = \\int_{r_{0}}^{r} \\frac{dp}{p^2\\sqrt{\\frac{1}{b^2}-\\left(1-\\frac{r_s}{p}\\right)\\frac{1}{p^2}}}$$ it starts from $0$, and increase up to $\\frac{\\pi}{2}+\\frac{\\Delta\\varphi}{2}$. * * * I would like to compute the trajectory in the $\\left(x, y\\right)$ plane, so how to use the values of $f\\left(r\\right)$ or $g\\left(r\\right)$ to compute the function $y\\left(x\\right)$ ?"}
{"id":"67812","title":"Dirac magnetic monopoles and quark fractional electric charge quantization","text":"When applying the Dirac quantization rule for electric and magnetic charge, I assume one is considering unit electric charges such as electrons. How does the Dirac quantization rule apply for the fractional electric charges of quarks?"}
{"id":"9319","title":"How many $fb^{-1}$ for the most likely $5\\sigma$ 115 Gev Higgs at the 7 Tev LHC?","text":"How many $fb^{-1}$ of integrated luminosity at the 7 Tev LHC do physicists expect are needed, to make a $5\\sigma$ discovery of the most likey 115 Gev Higgs, if it exists?"}
{"id":"22234","title":"Can a system that holds information about it's past ever be Markovian?","text":"To my (basic) understanding a Markov process is a process wherein the future state of a system only depends on the current state, and not on the past states of the system. I was wondering on what the standard approach is on systems that inherently hold information about their past in them. One could image, in the discrete-time case, system carrying in it a representation of it's three previous states. There are two cases to distinguish here:   * The future state of the system is not a function of this memory and only of factors that don't have anything to do with it. This is perhaps a boring case (although there might be interesting examples I don't know of).   * The future state of the system is a function of this memory information, yet in a \"Markovian\" way. This is the case I am most interested in. Are these things that would be considered as Markovian systems (perhaps on a different scale)? Are there examples of these kinds of systems?"}
{"id":"22231","title":"Does a cycle (in Simple Harmonic Motion) have to equal 2π?","text":"So, I search for the definition of cycle and I get this in Wikipedia: > A turn is a unit of angle measurement equal to 360° or **_2π radians_** (or > ...). A turn is also referred to as a revolution or complete rotation or > full circle or **_cycle_** or rev or rot. When I look for motion in pendulum, I get:- > The time for **_one complete cycle_** , **_a left swing and a right swing_** > , is called the period. So, in the first quote, it says that one cycle has to be 2π radians, **_but_** in the second quote it says \"one complete cycle\" is \"a left swing and a right swing\" (i.e the full cycle doesn't have to be equal to \"2π radians\"). Can you please clarify what I see as contradicting statements?"}
{"id":"24698","title":"How could $\\textbf{S}^2$ not be a multiple of the identity?","text":"I'm self-studying quantum mechanics with Sakurai's book ( _Modern Quantum Mechanics_ , 2nd edition) and came across the following in reference to the operator $\\textbf{S}^2$: > As will be shown in Chapter 3, for spins higher than $\\frac{1}{2}$, > $\\textbf{S}^2$ is no longer a multiple of the identity operator; however, > $[\\textbf{S}^2, S_i] = 0$ still holds (for $i = x, y, z$). (page 28) The square of the total spin commuting with the components, I'm comfortable with. But the first part just confuses me: for a system with spin $s$, is it not true that $$\\textbf{S}^2|\\cdot\\rangle = \\hbar^2s(s+1)|\\cdot\\rangle$$ whether or not $s = \\frac{1}{2}$? Or do I have a fundamental misunderstanding of the situation? (I have read through Chapter 3, but apparently I kept missing the part where the book addresses this.)"}
{"id":"135151","title":"How to calculate the buoyant force on a balloon at different altitudes","text":"For a project into balloon simulation I'd like to know how the force on a balloon changes with altitude: I know that the Buoyant force on a balloon is: $F = (\\rho_{air} - \\rho_{helium})gV$ Using changing air density from the international standard atmosphere and changes in gravitational acceleration I can calculate the buoyant force and add this to the (negative) force of gravity _mg_. However I know that it isn't this simple, the balloon changes volume with altitude as the air pressure outside the balloon changes. I'd like to know how the balloon inflates - this would be a great help! I think it may be something to do with ideal gas laws - I can calculate air temperature and pressure from the international standard atmosphere but do I use pressure as the air pressure? - or air pressure minus balloon pressure?"}
{"id":"107988","title":"Visual Fields testing: Decibels of sensitivity - meaning","text":"Automated Visual Field test measures the patient light sensitivity in decibels. Questions: If one point has sensitivity of 30 decibels and another point has sensitivity of 27 decibels does it mean that point 1 is 2 times more sensitive than point 2? If point 3 has sensitivity of 31dB and point 4 has 21dB does it mean that point 3 sensitivity is 10 times greater than of point4? Third and most important question - How these decibels calculated mathematically? Is there way to reconstruct this (theoretical) calculation?"}
{"id":"114007","title":"Ladder operator on momentum basis","text":"Since in Quantum mechanics momentum operator can be written in terms of ladder operators $$\\widehat{p}=-i\\sqrt\\frac{{\\hbar m \\omega}}{2}(\\widehat{a}-\\widehat{a}^\\dagger)$$ these operators operate on energy eigen-state $|n\\rangle$ of particle and we get $$\\widehat{a}|n\\rangle=\\sqrt{n}|n-1\\rangle$$ $$\\widehat{a}^\\dagger|n\\rangle=\\sqrt{n+1}|n+1\\rangle$$ But what if these ladder operator or momentum operator operate on discrete momentum basis, for example $|k_x\\rangle$, $|k_x-k_o\\rangle$, $|k_x-2k_o\\rangle$, $|k_x-3k_o\\rangle$........etc. Will it looks like $$\\widehat{a}|k_x-k_o\\rangle=\\sqrt{k_x- k_o}|k_x-2k_o\\rangle$$ and $$\\widehat{a}^\\dagger|k_x- k_o\\rangle=\\sqrt{k_x}|k_x\\rangle$$"}
{"id":"101201","title":"Optical Bloch Oscillation","text":"I have a doubt about how the optical Bloch oscillations happen in a 1D photonic crystal. I try to explain: in a photonic crystal with discrete translational symmetry in one direction I superimpose a second bigger periodicity by mean of microcavities. These reduce the amplitude of the first Brillouin zone. Now it is found that minibands are formed, that correspond to delocalized states. If now i introduce a gradient in the refractive index in the longitudinal direction, the minibands are tilted and a Wannier-Stark ladder of states is created. How can I explain the oscillation of an optical pulse from the Wannier-Stark states in the real space? (I ask this beacuse in many article the Bloch oscillations in real time are explain as the fourier transform of the Wannier-Stark ladder but I'm not able to appreciate this point of view). Thanks."}
{"id":"61417","title":"How fast did hydrogen atoms travel when they were first formed in the early universe?","text":"I can't seem to find any data on this, is it a known value?"}
{"id":"51728","title":"Non-destructive measurement of qbits","text":"Yale news \"New qubit control bodes well for future of quantum computing\" (Original paper) says: > \"The Yale physicists successfully devised a new, non-destructive measurement > system for observing, tracking and documenting all changes in a qubit’s > state, thus preserving the qubit’s informational value.\" Which would be the major applications of quantum theory that are impacted by this new technique? Wouldn't e.g. the security of quantum key distribution in cryptography be rendered questionable?"}
{"id":"48048","title":"How can crystal structures be determined using X-ray diffraction?","text":"You have the intensity peaks and the diffraction angles. Let's say you suspect the material is cubic, how do I find if it's simple cubic or BCC or FCC? I've googled and all my textbooks just state that xrd is really nifty for determining crystal structure but no further information is presented."}
{"id":"56711","title":"Is it possible to record the 'blue air\" effect when a core goes critical?","text":"I've been studying about radiation, and I saw something called the Demon Core. Apparently, it was a core that was under experimentation by the government in Los Alamos to see the exact point at which a mass of plutonium would go critical. It was composed of two hemispheres of beryllium partially enclosing a mass of plutonium and made to reflect the radiation back to the center. But the top dome slipped and shut off the opening and the whole structure briefly went critical. During this brief time, the surrounding are turned blue because the air was being ionized. I'd like to know if this effect can be recorded, or if perhaps the radiation would corrupt the recording system(s) at work. If anyone can give me an explanation, I would greatly appreciate it."}
{"id":"56717","title":"Why is there a linear relationship between charge and isospin?","text":"So the title basically says it all. Something that's really bothering me is the fact that the Gell-Mann Nishijima relationship can be extended to provide a linear relationship between charge and all the flavor quantum numbers for baryons as follows: $Q = I_{3} + (A + S + C + B + T)\/2$ Where Q is the charge, A is the baryon number, S is the strangeness number, B is the bottomness number and T is the topness number. What is the physical interpretation behind this relationship? Is charge really isospin or vice-versa? Are they manifestations of each other in the same sense as electricity and magnetism? Is it just a coincidence?"}
{"id":"111707","title":"Why do excited states in $^4$He not decay by photon emission?","text":"Here's a level scheme for the $^4$He nucleus (source; click image to see full size): ![Level scheme for he-4](http:\/\/i.stack.imgur.com\/d6h4F.png) Notice that all of the confirmed decay modes are by disintegration — emission of a neutron, proton, or deuteron. There are a few states with the right spin- parity to decay to the ground state by an E1 (from $1^-$) or E2 (from $2^+$) electromagnetic transmission, but there seems to be no evidence that those internal transitions have been observed. Is there a simple reason why electromagnetic transitions are so strongly disfavored? If I were giving a talk and this question came up I would mumble something about the strong force winning out over electromagnetism in few-nucleon systems and get back on topic, but I'd like a slightly more sophisticated answer than that."}
{"id":"8959","title":"Uncertainly Principle in orthogonal directions","text":"The Heisenberg Principle states that for each direction, $\\Delta x\\cdot \\Delta p_x \\ge \\hbar , \\Delta y\\cdot \\Delta p_y \\ge \\hbar$ and $\\Delta z\\cdot \\Delta p_z \\ge \\hbar$. But, can anything be said about $\\Delta x\\cdot \\Delta p_y$? Can I measure the position in one direction and the momentum in an orthogonal direction in any required precision?"}
{"id":"75970","title":"Entanglement entropy for finite interval using heat kernel method","text":"I am looking for some reference for calculation of entanglement entropy(EE) for a finite interval in 4 spacetime dimensions using heat kernel method. Till now I have only found references where the calculation has been done for a halfspace. The paper by Wilczek http:\/\/arxiv.org\/abs\/1007.0993v2 says that the heat kernel method works well only when the region of interest is a half space. So can anyone please let me know whether the heat kernel method is suited for finite size interval calculations of EE or suggest a reference for the same? Thanks in advance."}
{"id":"70051","title":"Why can the Euler beta function be interpreted as a scattering amplitude?","text":"The Wikipedia article on the Veneziano Amplitude claims that the Euler beta function can be interpretted as a scattering amplitude. Why is this? In another word, when the Euler beta function is interpreted as a scattering amplitude, what features does it have that make it able to explain strong force of mesons? What properties of (or why) the Euler beta function (when interpreted as scattering amplitude) has string behavior?"}
{"id":"27345","title":"Random bond Ising model and computational efficiency","text":"If you want to find the ground state of the 2d random bond Ising model (no field), a computationally efficient algorithm exists to do it for you (based on minimum weight perfect matching). What about the thermal state for some finite temperature, T? Is there an efficient (i.e. time scales polynomially with system size) way to sample states of different spin configurations from this distribution?"}
{"id":"25275","title":"Why is this radio telescope's reflector spherical and not parabolic?","text":"This is the Arecibo Observatory in Arecibo, Puerto Rico. Its reflector is spherical, measuring 1,001 ft. in diameter. It is considered the most sensitive radio telescope on Earth, but the fact that its reflector is spherical and not parabolic makes me wonder how much more sensitive it could be if the reflector were parabolic. What are the pros and cons of a sphere vs. parabolic? ![Picture of the telescope](http:\/\/i.stack.imgur.com\/0CmCC.jpg) ![Plaque at Arecibo](http:\/\/i.stack.imgur.com\/9i9q1.jpg)"}
{"id":"25278","title":"Video of Earth spinning?","text":"If the Earth is spinning or rotating at a really fast speed, why haven't we seen any videos from space of it spinning when we get a lot of photos of it?"}
{"id":"70613","title":"current operator in Hubbard model","text":"How to derive the particle current operators for the non-interacting and interacting Hubbard model ? Hubbard Hamiltonian is given here with the interaction term: http:\/\/en.wikipedia.org\/wiki\/Hubbard_model Here's an introduction : http:\/\/quest.ucdavis.edu\/tutorial\/hubbard7.pdf"}
{"id":"101979","title":"Time Reversal in a Black Hole","text":"I had a lively discussion with a person about black holes recently, and was making the point about gravitational acceleration in GR being paralleled by speed in SR. One thing that I know people talk about with special relativity, is that if you could reach a speed faster than the speed of light then time would reverse (I don't get this as the Lorentz factor gives an imaginary number, and who knows what that means). This is impractical for things with mass due to mass increasing asymptotically with velocity. However, since the event horizon for black hole behaves exactly like asymptotic barrier to reaching the speed of light (we'll say for a stationary non-charged black hole to keep things simple), shouldn't the effects translate to a black hole? Ie once you're in a black hole (ie the mass that was there when the black hole formed), wouldn't the mass be essentially traveling backwards in time, fastest the further it was from the event horizon? For clarity sake, for matter, approaching the speed of light and approaching the event horizon of black hole are essentially the same with respect to redshift\/blueshift, space and time dilation. There for approaching an event horizon from the inside of a black hole would be like de-accelerating toward the speed of light (v>c). So matter in a black hole should be falling outward toward the event horizon. All of this I am of course imagining from the reference frame of a distant observer. Just curious if I'm way off base on this."}
{"id":"101972","title":"Valley meaning explanation for foreigner","text":"English is not my native language and I have some hard time translating this word. I was searching in couple dictionaries(both paper and online) and could not find it. Could anyone provide me definition of word \"Valley\". example usage: > Suggest using only the valley degree of freedom   > > > valley and spin degeneracy > > Valley relaxation    Thanks in advance!"}
{"id":"101976","title":"Is TRUE invisibility possible?","text":"First off, this question may seem highly stupid at first, but it arose from past cartoon memories. In the cartoon Tom & Jerry from the 1940s, there was already hypothesizing over cream that, when applied, makes you invisible. I have looked into invisibility, and I see that nothing surprising has been done. For example, only success in certain magnitudes, presets, lights, etc., but not enough for this to be any breakthrough yet. So my question is: Is an invisibility cloak that can make a completely visible object in visible light (me, for example) completely invisible (after I put it over me, my torso\/body appears to magically vanish) to someone whilst wearing it possible, with just a cloak itself and nothing else? Truly invisible means: 1.It can't be camouflaged; you must literally appear invisible with no distortion, traces, or signaling that someone or something is\/was there. 2.It must be only accomplished by a cloak itself, and no add-ons or special presets must be met; no light tricks, external items needed, etc. How about \"vanishing cream\"? Is It somewhat possible?"}
{"id":"33139","title":"What is the speed for an object that travel close to the speed of light?","text":"I have some questions regarding Einsteins theory of Relativity that should be fairly easy to answer. Lets say we make an experiment where we have a rocket (with an astronaut inside) that travels very close to the speed of light and it is heading away from Earth. Also back on earth I have an observer that makes measurements regarding the rocket that travels very close to the speed of light. **Question 1:** What is the speed of the rocket that the earth observer would measure? Lets change the above experiment and place the rocket very close to the event horizon of a black hole. According to what I have heard and read the rocket would seem to stand still as our observer see it back on earth. **Question 2:** What is the speed that the earth observer would make about the rocket in this situation? **Question 3:** Are the two experiments the same or are they somehow fundamentally different? Would an object seem standing still while traveling very close to the speed of light?"}
{"id":"33133","title":"Gauge fixing choice for the gauge field $A_0$","text":"In many situations, I have seen that the the author makes a gauge choice $A_0=0$, e.g. Manton in his paper on the force between the 't Hooft Polyakov monopole. Please can you provide me a mathematical justification of this? How can I always make a gauge transformation such that $A_0=0$? Under a gauge transformation $A_i$ transforms as $$A_i \\to g A_i g^{-1} - \\partial_i g g^{-1},$$ where $g$ is in the gauge group."}
{"id":"105278","title":"Why do we must initially assume that the wavefunction is complex?","text":"The sound waves are real, and they can interfere, so corresponding apparat may be used in quantum mechanics. We also may use the time dependence in a form of orthogonal matrix multiplying the initial constant vector. So why do we must postulate that the wavefunction is initially complex? This question is very similar to the corresponding questions at the site, but there I didn't find the answer."}
{"id":"113360","title":"Why is Kirchoff's Current Rule valid?","text":"The fact that Kirchoff's Current Rule is valid means that whatever current flows in flows out. But this will only be valid for a steady current circuit, that is, when there is no accumulation of charge. But there is no reason why there shouldn't be even small accumalation of charge and therefore inconsistencies in current, is there? Somehow it all just doesn't seem very intuitive. Am I looking at something the wrong way? Is there a proper explanation?"}
{"id":"113368","title":"Why is there an $E^2$-term in the expression for the polarization density in nonlinear optics","text":"In nonlinear optics the polarization density can be expressed as $$P=\\epsilon_0(X(1)(E)+X(2)(E,E)+\\dots)$$ As far as I know, $P$ arises from the dipole moment in materials, and $E$ is an applied electric field in this equation. Terms involving $E^2$ and $E^3$ cause the phenomenon of nonlinear optics. However, I don't understand why these $E^2, E^3 \\dots$ terms appear in this equation? What's the physical meaning?"}
{"id":"41469","title":"Effect of charges near a parallel plate capacitor","text":"If I charged a parallel plate capacitor. And then, I insert a charged body near one of the plates. Will there be any interactions like attraction or repulsion?   What if I disconnected the battery?"}
{"id":"41463","title":"Parabolic Motion","text":"I was wondering if I was right about this simple problem: A projectile is shot at an angle of 37.0° with the horizontal. The velocity of the projectile at its peak height is 16 m\/s. Then I was told to find the velocity in x direction. So I wrote: Given that at its peak, the velocity of the projectile in y direction is equal to 0, but stated that at its peak, the velocity of the projectile is 16 m\/s, this must be the velocity in x direction. So my final answer was: 16 m\/s. But then, my teacher just said it was wrong. That I had to find Vx using the formula Vx= Vcos(37°). I told her that 16 m\/s could not be the initial velocity, since the problem says that 16 m\/s is the velocity at its peak, therefore it must be Vx. Am I right? If so, can you tell me a good explanation besides the one I said to prove my teacher wrong. If not, what am I missing?"}
{"id":"41464","title":"Why is the gas halo of the Milky Way so hot?","text":"I have read on the webpage of NASA that there is a massive hot gas halo around our galaxy. Its temperature is between 100,000 and 1 million Kelvins or more. I do not understand why is it so hot. The temperature of the cosmic background radiation is only 2.7 Kelvins. The stars of our galaxy warm it up or one (or more) process in the gas? UPDATE: The question is about circumgalactic medium (CGM) not about the gases within our galaxy."}
{"id":"704","title":"Are the physical laws scale-dependent?","text":"If you read the article \"More Is Different\", by P.W. Anderson (Science, 4 August 1972), you will find a deep question: are the physical laws dependent of the size of the system under study? As an example, we can ask ourselves, are the description of a hundred of atoms more than simply one hundred times the description of one atom alone? Of course we have interactions, but, are these interactions dependent of the number of particles implied?"}
{"id":"86794","title":"Principle of relativity - a second, equivalent form, using invariants","text":"Most people state the principle of relativity like this: > \"The rules of physics must take the same form in all inertial frames.\" Question: is this an equivalent way of saying the same thing: > \"The rules of physics must be _expressible_ using only (Lorentz) > invariants.\" Note the word _expressible_. A given rule could be expressed in two different ways. A statement using action integrals with a Lagrangian would be an example of using only Lorentz invariants. A statement using covariant differential equation would not."}
{"id":"703","title":"Is gravity a vector or tensor function and does gravity have velocity or is it instantaneous?","text":"If gravity exists quantumly it must take up space, have frequency, velocity, etc. Perhaps gravity is too fine a frquency for detection."}
{"id":"106615","title":"Solving the 1D Schrodinger Equation for a Free Particle - Two Different Methods?","text":"So starting from the time dependent schrodinger equation I perform separation of variables and obtain a time and spatial part. The spatial part is in effect the time independent schrodinger equation. Since we are dealing with a free particle I can take the time independent equation, set V = 0 and solve. I can do this successfully to obtain : $Ae^{+i\\sqrt{{2mE}\/{\\hbar^{2}}}x}+Be^{-i\\sqrt{{2mE}\/{\\hbar^{2}}}x}$ My lecturer has a small section titled : > Solving for the Free Schrodinger Equation > > $$V=0$$ > > $$\\frac{\\hbar^{2}}{2m}\\frac{\\partial^2\\psi}{\\partial x^2}+E\\psi=0$$ > > $$E=\\frac{p^2}{2m}$$ > > $$\\psi=Ce^{-{iEt}\/{\\hbar}+{ipr}\/{\\hbar}}$$ > > This is the solution to the free TISE and TDSE. He seems to be doing the same thing as me initially but he's obtained a different result ? Also, the first section of his answer :$e^{-{iEt}\/{\\hbar}}$ is the solution to the time part of the equation (described above)."}
{"id":"102724","title":"Comprehensive list of $\\mathcal{N}=2$ supersymmetric topological quantum field theories in $d=4$","text":"Could anyone provide a comprehensive list of $\\mathcal{N}=2$ supersymmetric topological quantum field theories in $d=4$? I know of one - Kapustin-Witten TQFT, but I do not know of any more. Thanks!"}
{"id":"102722","title":"Computing Curvature via Cartan Formalism","text":"Given a metric $g_{\\mu \\nu}$, one can select an orthonormal basis $\\omega^{\\hat{a}}$ such that, $$ds^2= \\omega^{\\hat{t}}\\otimes\\omega^{\\hat{t}} - \\omega^{\\hat{x}} \\otimes \\omega^{\\hat{x}} - ...$$ By employing the Cartan structural equations, one can 'read off' the components of the Ricci tensor and Riemann curvature tensor **in the orthonormal basis**. My question is: **how do I take my curvature tensors in the orthonormal basis back to the coordinate basis after?**"}
{"id":"30446","title":"Explaining Walter Lewin's \"Complete Breakdown of Intuitiion\"","text":"This Walter Lewin lecture hinges on the solenoid producing a non-conservative field. I read in the Feynman Lectures that \"all the fundamental forces in nature appear to be conservative\" (Vol 1, 14.5) which comes decades after Faraday's Law of Induction so surely he would have known about it. 1) Who is right? Or, what did Feynman _really_ mean? 2) In Lewin's final diagram there are two voltmeters which, apart from a section of wire, attach in the same place. What happens as you bring the two attachment points together to meet? I can't see, in a physical on-the-table experiment, how one voltmeter is \"measuring in the other direction\"."}
{"id":"89641","title":"Proof of the inverse square law for a uniformly bright sphere","text":"On page 8 of \"Radiative processes in Astrophysics\" by Rybicki and Lightman they have a proof that the flux of a uniform sphere is inversely proportional to the distance of the sphere from the observer. I don't understand why they claim $\\theta_c=\\sin^{-1}R\/r$, where the geometry is shown on pg 8, figure 1.6. It seems to me that its should rather be $\\theta_c=\\tan^{-1}R\/r$. But then I don't think the proof would work. Any help in clarifying this for me would be greatly appreciated."}
{"id":"31790","title":"Euclidean quantum gravity and gravitational instantons","text":"I have some questions concerning the calculations made by Hawking in the 80-90's using Euclidean (canonical) quantum gravity on gravitational instantons. Were those tunnelings only between identical closed FRW metrics (in the minisuperspace approximation where the only unfrozen mode is a(t) ) with different a (tunneling from $a_1$ to $a_2$) but with identical topology, or could those worm-holes represent topological changes or something more exotic? I tried to read this paper: http:\/\/siba.unipv.it\/fisica\/articoli\/P\/Physical%20review%20D_vol.28_no.12_1983_pp.2060-2975.pdf (Hawking Hartle 1983) but couldn't clearly find the answer to my question. Said in other words, what kind of transitions can be described by these instantons processes? and, if Euclidean gravity is a dead-end as it seems widely considered today, are those results totally irrelevant nowadays? Thank you."}
{"id":"33867","title":"How much force is required to propel a bicycle?","text":"A friend of mine had an idea to attach a motor to a bicycle with the idea to be able to turn it on and have it maintain average bike speed, apparently 15-20 mph. Neither of us have taken physics classes before, though, so we aren't too sure how to calculate how much force (torque?) the motor has to put out. Any help would be appreciated, and I apologize for my general ignorance (and Imperial units)."}
{"id":"134623","title":"Square of the Pauli matrices and the identity matrix","text":"The square of any of the three Pauli Spin matrices is equal to the identity. Is there any physical meaning to this? Would you expect it? Maybe in the context of the $SU(2)$ group?"}
{"id":"47174","title":"What conductance is measured for the quantum spin Hall state when the Hall conductance vanishes?","text":"It's probably just a definition, but what did König et al. actually measure when he confirmed the existence of surface states in CdTe\/HgTe\/CdTe quantum wells (see http:\/\/arxiv.org\/abs\/0710.0582)? According to most reviews (e.g. http:\/\/www.annualreviews.org\/doi\/pdf\/10.1146\/annurev- conmatphys-062910-140538): > \"However, because the Hall conductance of the QSH state vanishes, it is > clear that the TKNN or Chern number discussed above, which corresponds to > the the value of the Hall conductance in units of e2\/h, cannot provide a > useful classification of the QSH state. This issue has been addressed within > both the topological band theory (23) and the topological field theory (23). > Accordingly, the proper topological invariant is valued in the Z2 group > containing only two elements, 0 or 1, with 1 corresponding to the > topologically nontrivial QSH insulator and 0 corresponding to a > topologically trivial insulator with no robust gapless edge states.\" and > \"Because spin-orbit coupling destroys spin conservation, there is no such > thing as a quantized SH conductance in the QSH effect. This is another way > to understand why the correct topological invariant for the QSH effect is Z2 > and not Z. Finally, the BHZ Hamiltonian predicts a single helical edge state > per edge.\" I don't understand this. So there is no charge conductance, but we measure charge conductance? What is the difference between spin and charge conductance? I thought König did measure a charge conductance which was exactly twice the Hall conductance (e^2\/h) (For me that's quantized...). Does this mean there actually are only two states that lead to twice the Hall conductance? Also: Why is there only a single helical edge state per edge? Why must we have at least one and why can't we have, let's say, two states per edge?"}
{"id":"127137","title":"Ziolkowsky Rocket Equation","text":"I've looked at multiple sources on the internet and just can't wrap my mind around it. If I reference the wikipedia page on it; http:\/\/en.wikipedia.org\/wiki\/Tsiolkovsky_rocket_equation I am getting stuck on certain places. $V_e=V-v_e $ Why does this occur? What integration limits do we use to derive this and why? $\\Delta V\\ = v_e \\ln \\frac {m_0} {m_1} $ (I am only concerned with this equation up to this part) If someone could explain it from the beginning in an easier fashion that would be great but solving these doubts for me would also help."}
{"id":"87543","title":"How does the proof of operator commutativity work with non-continuous operators?","text":"In some books, a proof that if two self-adjoint operators $A$ and $B$ share a common eigenbasis $\\\\{\\phi_n\\\\}$, then they commute is given as follows : > For any $\\phi_n$, $$AB\\ \\phi_n = a_n\\ b_n\\ \\phi_n = BA\\ \\phi_n,$$ and so $AB > - BA = 0$. > > Then, for any vector $\\psi = \\sum\\limits_{n=1}^\\infty c_n\\ \\phi_n$, > > $$(AB-BA)\\ \\psi = (AB-BA)\\sum\\limits_{n=1}^\\infty c_n\\ \\phi_n= > \\sum\\limits_{n=1}^\\infty \\ (AB-BA)c_n \\phi_n = 0$$ But how is the third step on the to last line valid, where the operator $C = (AB-BA)$ is moved across the summation sign? $C (\\lim f_n) = \\lim (C f_n)$ is only valid if $C$ is continuous. But here, while $C$ is linear and self- adjoint, it may not be continuous!"}
{"id":"87549","title":"Is it possible for a material to have a polarization field greater than the applied field?","text":"In the case of a dielectric (LIH at least, since that is all I've studied), the polarization field is always less than the applied field. In the case of a conductor, the polarization field is equal to the applied field, which is why there is no electric field inside conductors. Is there any sort of material which has a polarization field greater than the field applied to it? My initial thought is no, since you would be creating a more powerful electric field than you started out with, and that would be a way of creating energy, right?"}
{"id":"15251","title":"Entropy of radiation emitted into space","text":"In several papers I see something equivalent to the following expression for the entropy of radiation given by an astronomical object such as the Sun (assuming the object can be approximated as a black body): $$ s = \\frac{4}{3}\\frac{u}{T}, $$ where $u$ is the total flux of radiation energy through a spherical shell surrounding the object, $s$ is the entropy flux through the same imaginary surface, and $T$ is the black-body temperature of the object (and hence also of the radiation). It is also generally stated that no entropy is produced during the process of emitting radiation. In a much smaller number of papers I see a formula that corresponds to just $s=u\/T$, which is what I would expect. So the short version of my question is, which of these is correct for an astronomical body? But please read the rest, so you can understand why I'm so confused about it. I understand the derivation of the 4\/3 formula by considering a photon gas enclosed in a piston (see below), but in the context of a continuously emitting body like the Sun it doesn't seem to make sense. The issue is that if, over a given time period, the body loses an amount of heat $Q$ then this must balance the increase in energy of the radiation field $U$, i.e. $U=Q$. The body loses entropy at a rate $Q\/T$, and if radiation is really a reversible process then this should equal the gain in entropy of the radiation, which should therefore be $U\/T$. But according to the above formula it's actually $4U\/3T$, meaning that the total entropy increases by $Q\/3T$. The 4\/3 formula above was derived by Planck (in his book \"the Theory of Heat Radiation\", of which I've read the relevant chapter), who considered a photon gas in a sealed cylinder of finite volume. At one end of the cylinder is a black body and at the other is a piston. The radiation comes into equilibrium with the black body and exerts a pressure on the piston. If one reversibly (i.e. slowly) allows the piston to move then this causes some heat to be lost from the black body. In this case it turns out that $U=3Q\/4$, with the discrepancy being due to the fact that the radiation field loses energy when it does work on the piston. The entropy balance then requires that the entropy of the photon gas increases by $4U\/3T$, as above. The point is, I can't see how the radiation being emitted by the Sun can be seen as doing work on anything. At first I thought it might be doing work on the outgoing radiation field. So let's draw an imaginary shell around the Sun again, but this time let the shell be expanding at the speed of light. Perhaps the radiation inside the shell is doing work on the radiation outside it, and that's what's \"pushing\" it away from the Sun? But it seems to me that for anything inside the shell to have an effect on anything outside it, some kind of influence would have to travel faster than light, so I don't think that can be right. In any case it's well known that for a normal gas (made of matter), expanding against a piston is quite different from just expanding into a vacuum. In the former case the temperature and internal energy decrease, because the molecules lose energy in pushing the piston, whereas in the latter case they both remain constant. I haven't found any source that addresses the question of why this would be different for a photon gas. So it seems like the emission of radiation from a body like the Sun into space is quite different from the emission of radiation into a sealed piston, and I'm puzzled as to how the same formula can apply. Below I've listed some possible resolutions to this which seem plausible to me. I haven't been able to find any sources that directly address this issue, or state any of these positions.   1. The emission of radiation into space is an irreversible process after all. $U=Q$, and the total entropy increases by $Q\/3T$. (But then, what happens when radiation is absorbed by a body at a similar temperature? Surely the entropy doesn't decrease.)   2. There is some weird sense in which the outgoing radiation can be thought of as doing work on something, so $U\\ne Q$. (If so, why does nobody ever explain this subtle and important point?)   3. $Q=U$, but the entropy of radiation emitted into space is actually different from the entropy of a photon gas in a sealed cylinder, such that its entropy is given by $U\/T$, not $4U\/3T$. (This one actually seems the most reasonable to me. The radiation in the closed cylinder has rays travelling in all directions, whereas the radiation coming off the Sun only has rays travelling in directions away from its surface, so it seems reasonable that they would have different entropies. This would mean that the 4\/3 formula doesn't apply to the radiation emitted by astronomical bodies after all - but if that's the case then it's an extremely widespread mistake.) Any insights into which, if any, of these is correct would be greatly appreciated - as would any references to sources that directly address the relationship between Planck's photon-gas piston and the emission of radiation into empty space."}
{"id":"35967","title":"How can I determine density of a gas only given temperature?","text":"I have a homework problem where exhaust is traveling through an exhaust system (assumed to be air for simplicity) from and engine and then released into the atmosphere. The exhaust is at a temperature of 250F. I need to find the pressure at the beginning of the exhaust system. I've been able to set this up and created an equation to solve for the pressure. I assumed that the pressure of the atmosphere where it it exits is 0 atm. But the problem is that the pressure depends on the density of the exhaust (air). I was only given temperature of the exhaust in the problem and I need a second property to determine density. Is there another way I can determine density? I am also given the volumetric flow rate and the area of the piping system"}
{"id":"16507","title":"Gravitational waves as dark energy?","text":"Is the energy carried by gravitational radiation a viable candidate for $\\Lambda$ \/ dark energy?"}
{"id":"19731","title":"What is a dynamical variable","text":"I see the phrase 'dynamical variable' in several of my texts and they don't really define it. A google search doesn't yield much either. I have gathered that a dynamical variable can be represented in terms of position and momentum, and examples are velocity and kinetic energy, but what is the definition?"}
{"id":"127855","title":"Is it possible to produce images of pair production in home-made cloud chamber?","text":"There are some nice pictures on the web showing the counter-spiralling paths of an electron positron pair produced in a bubble chamber with a uniform magnetic field, for example:- ![e+ e- pair production](http:\/\/www.bigganblog.com\/wp- content\/uploads\/2012\/03\/I15-70-positron.jpg) Would it be practical to produce a cloud chamber and Helmholz Coils capable of imaging such events in an ordinary domestic garage? What sort of source would I need? How often might such events be detected?"}
{"id":"127857","title":"Reluctance of torus shaped iron core with embedded wire loop","text":"**Imagine a circular wire loop** (r = 50mm), the wire has an assumed diameter of zero, **which is embedded in a torus shaped iron core with a circular cross-section** of R = 10mm. A current in that loop would cause a circular magnetic field around the wire. **Is there any possibility to calculate the reluctance of that core?** I'm looking for a solution for weeks now, without any success. A solution for harmonic currents is desired, but I would even be happy for a DC solution. ![enter image description here](http:\/\/i.stack.imgur.com\/y65Cw.png) * * * # CONTEXT **to explain what this is all about.** My real geometry looks as following: ![enter image description here](http:\/\/i.stack.imgur.com\/gVvQ5.png) A torodial coil surrounded by a core with a cross section of a rounded rectangle. So I'm interested in the reluctance of the greyish part (and the other corners). If you put all corners together you'd get the mentioned torus. The green lines are the magnetic flux, the rectangle in the middle the torodial coil. For high frequencies and\/or high conductive and\/or high permeable materials the influence of the corners is negligible, for my case unfortunately not. I'd guess there is no analytic solution, but any idea which could get me close to it, would help. Thank you! * * * * * * # Attempted solution # Preface If one wants to calculate the permeance $P$ of a rectangular bar: ![enter image description here](http:\/\/i.stack.imgur.com\/YWrk8.png) it is an easy task: $$P = \\frac{\\mu a b}{L} ~~~~ \\rightarrow ~~~~ P\\propto ab ~~~~and~~~~ P\\propto\\frac{1}{L}$$ where $\\mu$ is the material constant. (Permeability) But my geometry is a **torus with just a quarter of its circular cross section** and the field $V$ **passes through it parallel to the circumference** of the (full) cross section: ![enter image description here](http:\/\/i.stack.imgur.com\/Uq1n1.png) How can I calculate the permeance of this geometry, when there are the same proportional relations as above? * * * # Attempted solution I divide my geometry in $N$ hollow toruses with constant wall thickness $\\Delta R$ and medium length element $\\Delta L$, so the field passes an area of $\\Delta A$: ![enter image description here](http:\/\/i.stack.imgur.com\/nXbbl.png) A little piece of the radius $R$ is $\\Delta R = \\frac{R}{N}$. Now one can calculate: $$\\Delta P_{n} = \\frac{\\mu \\Delta A_n}{\\Delta L_n} $$ with $$ \\Delta A_n = \\pi \\bigg( (r+(n+1) \\Delta R)^2-(r+n \\Delta R)^2\\bigg) $$ (Consider the full torus circumference, not just a quarter as displayed) and $$ \\Delta L_n = \\frac{\\pi}{2} (2n+1) \\frac{\\Delta R}{2} $$ (but quarter cross section!) follows: $$P = \\sum^{N-1}_{n=0} \\Delta P_{n} = \\mu\\sum^{N-1}_{n=0} \\frac{\\pi(2r\\Delta R+(2n+1)(\\Delta R)^2)}{\\frac{\\pi}{2}(2n+1)(\\frac{\\Delta R}{2})}~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$= 4\\mu\\sum^{N-1}_{n=0} \\frac{2r\\Delta R+(2n+1)(\\Delta R)^2}{(2n+1)(\\Delta R)} $$ $$= 4\\mu\\sum^{N-1}_{n=0} \\Bigg( \\frac{2r}{(2n+1)} + \\Delta R \\Bigg)~~~~~~~~~~ $$ $$= 4\\mu \\Bigg( R + 2r \\sum^{N-1}_{n=0} \\frac{1}{(2n+1)} \\Bigg)~~~~~~~~~~ $$ **And this series does not converge for $N\\rightarrow\\infty$.** Which is physically seen not possible, so there must be a problem with the math. Do you see what I'm missing?"}
{"id":"12614","title":"On numerically solving the Schrödinger equation","text":"I just read a paper 'A pocket calculator determination of energy eigenvalues' by J Killingbeck (1979). Link: http:\/\/iopscience.iop.org\/0305-4470\/10\/6\/001 I have some questions about section 2.   1. Why $R<0$ implies the $E$ estimate was too high and $R>1$ implies too low? And is it possible that $0<R<1$ remains?   2. Why \"if the starting point is chosen at $r=h(l+1)$, then $R(hl)$ can be assigned any finite value without disturbing the calculation at larger $r$ values\" ?   3. How to guess a trial energy $E$ (especially the ground state energy)? (Actually I'm interested in a later paper by Killingbeck (1982) 'Finite- difference methods for eigenvalues'(Link: http:\/\/iopscience.iop.org\/0022-3700\/15\/6\/009), in which the author applies the method in 'A pocket calculator determination of energy eigenvalues' to some potentials with highly singularity at origin) * * * _mod note: copying the text of whoplisp's post here since it isn't actually an answer to the question_ They convert the Schroedinger equation into a finite difference equation (instead of a variational approach): $-\\frac{1}{2}\\Delta\\psi +V\\psi=E\\psi$ $V$ is a central potential Ansatz with solid harmonic $\\xi_l$ of degree $l$ (e.g. $x$ or $xy$) $\\psi=\\xi_l\\phi(r)$ leads to $2r(V-E)\\phi=rD^2\\phi+(2l+2)D\\phi$ Notation: $D=\\frac{d}{dr}$ finite difference approximation $h^2D^2\\phi \\rightarrow \\phi(r+h)+\\phi(r-h)-2\\phi(r)$ $2hD\\phi \\rightarrow \\phi(r+h)-\\phi(r-h)$ crucial step: **introduce ratio variable** $R(r)$: $\\phi(r+h)=R(r)\\phi(r)$ convert Schroedinger equation in recursive equation to calculate $R(r)$: $2r[1+h^2(V-E)]=R(r)[r+h(l+1)]+[r-h(l+1)](R(r-h))^{-1}$ > the corresponding 1D Schroedinger equation is: > > $2[1+h^2(V-E)]=R(x)+(R(x-h))^{-1}$ if $r=h(l+1)$ is chosen then $R(hl)$ can be assigned any finite value without disturbing the calculation at large $r$ values to find the ground states with angular momentum $l$ start with an estimate for $E$, some non-zero $R(hl)$ and some small $h$. $R(r)$ as $r$ increases will either become negative or pass through unity from below. first thing happens when guessed $E$ was too high second thing indicates $E$ was too low (because $\\psi$ increases with $r$) you just try various values and 'sandwhich' the solution example for $V=\\lambda r-1\/r$: $\\lambda=1$, $h=.1$, state=2p: 1.9759"}
{"id":"69443","title":"Fundamental Representation of $SU(3)$ is a complex representation","text":"Let in a $D(R)$ dimensional representation of $SU(N)$ the generators, $T^a$s follow the following commutation rule:   $\\qquad \\qquad \\qquad [T^a_R, T^b_R]=if^{abc}T^c_R$. Now if $-(T^a_R)^* = T^a_R $, the representation $R$ is real. Again if we can find a unitary matrix, $V(\\neq I)$ such that $ \\qquad \\qquad \\qquad -(T^a_R)^*=V^{-1} T^a_R V \\quad \\forall a $ then the representation $R$ is pseudoreal. If a representation is neither real nor pseudoreal, the representation $R$ is complex. **Claim: One way to show that a representation is complex is to show that at least one generator matrix $T^a_R$ has eigenvalues that do not come in plus- minus pairs.** Now let us consider $SU(3)$ group. The generators in the fundamental representation are given by $T^a =\\lambda^a\/2; \\quad a=1,...8$,   where $\\lambda^a$s are the Gell-Mann matrices. We see that $T^8$ has eigenvalues $(1\/\\sqrt{12}, 1\/\\sqrt{12}, -1\/\\sqrt{3} )$. My doubt is: **According to the claim, is the fundamental representation of $SU(3)$ a complex representation?**"}
{"id":"69448","title":"Are there old aged particles?","text":"To measure the lifetime of a specific particle one needs to look at very many such particles in order to calculate the average. It cannot matter when the experimentalist actually starts his stopwatch to measure the time it takes for the particles to decay. If he measures them now or in 5 minutes makes no difference, since he still needs to take an average. If he measures later there will be particles out of the picture already (those who have decayed in the last 5 min), which won't contribute and the ones his measuring now behave (statistically) the very same, of course. I have just read the following in _Introduction to Elementary Particles_ by Griffiths: > Now, elementary particles have no memories, so the probability of a given > muon decaying in the next microsecond is independent of how long ago that > muon was created. (It's quite different in biological systems: an 80-year- > old man is much more likely to die in the next year than is a 20-year-old, > and his body shows the signs of eight decades of wear and tear, But all > muons are identical, regardless of when they were produced; from an > actuarial point of view they’re all on an equal footing.) But this is not really the view I had. I was imagining, that a particle that has existed for a while is analogous to the 80 year old man, since it will probably die (decay) soon. It just doesn't matter because we are looking at a myriad of particles, so statistically there will be about as many old ones as babies. On the other hand it is true that I cannot see if a specific particle has already lived long or not; they are all indistinguishable. Still I am imagining particles as if they had an _inner age_ , but one just can't tell by their looks. So is the view presented in Griffiths truer than mine or are maybe both valid? How can one argue why my view is wrong?"}
{"id":"21015","title":"Finding distance when the force is a function of time","text":"I'm having trouble with this homework question > A mysterious rocket-propelled object of mass 49.0 kg is initially at rest in > the middle of the horizontal, frictionless surface of an ice-covered lake. > Then a force directed east and with magnitude $F(t) = (16.3\\text{ N\/s})t$ is > applied. > > How far does the object travel in the first 5.50s after the force is > applied? For some reason I'm not getting the correct answer. I think maybe I'm not understanding how to use the magnitude of the force they are giving me. I know how to use a constant force, but is this different because the force is a function of time? I tried starting it like this: $$\\begin{align}F &= ma\\\\\\ 16.3(t) &= (49)(a) \\\\\\ 16.3(5.5) &= (49)(a) \\\\\\ 89.65 &= (49)(a) \\\\\\ a &= 1.82959\\ \\mathrm{m\/s^2}\\end{align}$$ So now we know that: $$\\begin{align}t &= 5.5\\text{ s} \\\\\\ a &= 1.82959 m\/s^2 \\\\\\ V_o &= 0\\end{align}$$ So I plug it into my equation: $$\\begin{align}\\Delta X &= V_o t + 1\/2 a t^2\\\\\\ \\Delta X &= (0)(5.5) + (1\/2)(1.82959)(5.5)^2\\\\\\ \\Delta X &= 27.7\\text{ m}\\end{align}$$ But that's not the right answer."}
{"id":"57253","title":"\"Redshifting\" of forces in stationary space - times","text":"Here's the problem statement: Let $(M,g_{ab})$ be a stationary spacetime with timelike killing field $\\xi ^{a}$. Let $V^{2} = -\\xi _{a}\\xi ^{a}$ ($V$ is called the redshift factor). (a) Show that the acceleration $a^{b} = u^{a}\\triangledown _{a}u^{b}$ of a stationary observer is given by $a^{b} = \\triangledown^{b}\\ln V$. (b) Suppose in addition that $(M,g_{ab})$ is asymptotically flat. Then, the \"energy as measured at infinity\" of a particle of mass $m$ and 4 - velocity $u^{a}$ is $E = - m\\xi _{a}u^{a}$. Suppose a particle of mass $m$ is held stationary by a (massless) string, with the other end of the string being held by a stationary observer at infinity. Let $F$ denote the magnitude of the local force exerted by the string on on the particle. According to part (a), we have $F = mV^{-1}(\\triangledown ^{a}V\\triangledown _{a}V)^{1\/2}$. Use **conservation of energy** arguments to show that the magnitude of the force exerted on the **other end** of the string by the observer at infinity, is $F_{\\infty } = VF$. My attempt: (a) A stationary observer's 4 - velocity must be proportional to the time - like killing vector and it must be normalized to -1 so we find that for a stationary observer, $$u^{a} = \\frac{\\xi ^{a}}{(-\\xi ^{c}\\xi _{c})^{1\/2}}$$. Now we compute, $$\\triangledown _{b}u^{a} = \\frac{\\triangledown _{b}\\xi ^{a}}{(-\\xi ^{c}\\xi _{c})^{1\/2}} + \\frac{\\xi ^{a}\\xi ^{c}\\triangledown _{b}\\xi _{c}}{(-\\xi ^{c}\\xi _{c})^{3\/2}}$$ so $$u^{b}\\triangledown _{b}u^{a} = \\frac{\\xi^{b}\\triangledown _{b}\\xi ^{a}}{(-\\xi ^{c}\\xi _{c})} + \\frac{\\xi ^{a}\\xi^{b}\\xi ^{c}\\triangledown _{b}\\xi _{c}}{(-\\xi ^{c}\\xi _{c})^{5\/2}} = -\\frac{\\xi^{b}\\triangledown ^{a}\\xi _{b}}{(-\\xi ^{c}\\xi _{c})}$$ where the second term vanishes because it is a contraction of a symmetric tensor with an anti - symmetric one and I have swapped the indices in the first expression using killing's equation. Therefore, $$a^{a} = u^{b}\\triangledown _{b}u^{a} = \\frac{1}{2}\\frac{\\triangledown ^{a}(-\\xi^{b}\\xi _{b})}{(-\\xi ^{c}\\xi _{c})} = \\frac{1}{2}\\triangledown ^{a}\\ln V^{2} = \\triangledown ^{a}\\ln V$$ as desired. (b) This is where I'm totally stuck. As far as conservation of energy goes, we know that $E$, as defined above, is a conserved quantity **along** the worldline of the stationary particle; physically $E$ is the energy needed to bring in the particle from infinity to its orbit. Here we have a stationary observer at infinity holding this particle stationary by a long thread. There is a tension force at the end the observer holds and at the end the particle hangs by. Let's say the observer exerts the force $F_{\\infty }$ at event $P_{1}$ and the particle feels the local force $F$ at event $P_{2}$. As far as I can tell, all we know is that in between these events, $E$ is constant. But how do I relate $E$ to $F_{\\infty}$ and how do I do this using the conservation of energy explained above? I should note that I tried something on a whim and looked at $\\triangledown _{b}E$. We know that for the stationary particle hanging from the string, to which this total energy is attributed, $$E = -m\\xi _{a}u^{a} = -\\frac{m\\xi_{a}\\xi^{a}}{(-\\xi _{c}\\xi^{c})^{1\/2}} = m(-\\xi^{c}\\xi_{c})^{1\/2}$$ so if we compute the derivative we get $\\triangledown _{b}E = mV\\triangledown _{b}\\ln V$ thus $(\\triangledown ^{b}E\\triangledown _{b}E)^{1\/2} = mV(\\triangledown ^{b}\\ln V\\triangledown _{b}\\ln V)^{1\/2} = VF$ but I have NO idea how this quantity is related to $F_{\\infty }$, if at all. If it is related somehow I have no idea how one uses conservation of energy to arrive at the relation. Thanks in advance!"}
{"id":"81710","title":"How can enzyme\/substrate reactions that adhere (largely) to quantum theory also require 'Newtonian' consideration of gravity?","text":"I'd just like to ask for a little clarification here due to confusion from interdisciplinary studies. I'm currently reading the 1976 paper related to the recent 2013 Nobel Prize for Chemistry, by 2 of the laureates Warshel and Levitt — _Theoretical Studies of Enzymic Reactions: Dielectric, Electrostatic and Steric Stabilisation of the Carbonium Ion in the Reaction of Lysosyme_ — and have recently been reading into quantum chemistry, as well as studying biochemistry and have had further lectures in Physics to boot. I'm wondering if the points made here are comparable to points I've heard in mechanics lectures, about a cannonball not being modelled as a quantum object, since every subatomic particle would be too computationally intensive, so a coarse-grain model is used (i.e. Newton's \"classical\" laws). I wouldn't be so quick to draw parallels between mechanics lectures and enzyme biochemistry but it seems to be in the language they are using here (see highlights below). From what I understand (in mechanics) only 3 of the 4 'fundamental forces' (strong, weak, and EM) only strictly apply at the quantum\/microscopic scale, and scientists are as it stands struggling to work gravity into the setup. Is this worth considering in this paper - if an enzyme is considered all of a sudden due to its size as a quantum object does this entail ignoring gravity QM now in favour of 'Newtonian', inertial frame-of-reference type mechanics, or considering it in some other 'Newtonian' manner? The introductory section is as follows (full paper linked above): > The conformation of the enzyme-substrate complex can be adequately studied > using empirical energy functions based on the “classical” contributions of > bond stretching, bond angle bending, bond twisting, non-bonded interactions, > etc. The mechanism and energetics of the enzymic reaction can only be > studied using **a quantum mechanical approach**. Previous quantum mechanical > calculations on enzymic reactions have been limited in several respects. In > the first place, they deal with an over-simplified model system consisting > of only small fractions of the atoms involved in the real enzyme-substrate > interaction. Second, all available quantum mechanical methods treat the > reaction as if in a vacuum and are, therefore, unable to include the > dielectric, which has a very important effect on the energy and charge > distribution of the system. This limitation does not depend on the actual > quantum mechanical scheme, being equally valid for the simplest Hiickel > treatment and most extensive _ab initio_ calculation. > > Because the **treatment of the whole enzyme-substrate complex quantum > mechanically is computationally impossible** , it is necessary to simplify > the problem and use a **hybrid classical\/quantum mechanical approach**. The > method described here considers the whole enzyme-substrate complex: the > energy and charge distribution of those atoms that are directly involved in > the reaction are evaluated quantum mechanically, while the potential energy > surfac e of the rest of the system, including the urrounding solvent, is > evaluated classically. An important feature of the method is the treatment > of the “dielectric” effect due to both the protein atoms and the surrounding > water molecules. Our dielectric model is based on a direct calculation of > the electrostatic field due to the dipoles induced by polarizing the protein > atoms, and the dipoles induced by orienting the surrounding water molecules. > With the inclusion of the microscopic dielectric effect we feel that our > model for enzymic reactions includes all the important factors that may > contribute to the potential surface of the enzymic reaction. Because we use > analytical expressions to evaluate the energy and first derivative with > respect to the atomic Cartesian co-ordinates, it is possible to allow the > system to relax using a convergent energy minimization method. Here we use > the method to study the mechanism of the lysozyme reaction, with particular > emphasis on factors contributing to the stabilization of the carbonium ion > intermediate. We first outline our theoretical scheme, paying attention to > the **partition of the potential surface into classical and quantum > mechanical parts and to details of the microscopic dielectric model**. Then > we use the method to investigate factors that contribute to the > stabilization of the carbonium ion intermediate and deal with the relative > importance of each factor. The effect of steric strain is considered for > both the ground and the carbonium states and found to be of minor > importance. **Electrostatic stabilization** is treated at some length with > special emphasis on the balance between stabilization by induced dipoles > (polarizability) and stabilization by charge- charge interactions. The > results demonstrate the importance of electrostatic stabilization for this > reaction."}
{"id":"33687","title":"What is the Principle of Maximum Conformality?","text":"I'm trying to understand this article about an advance in the theoretical understanding of QCD which centers on the Principal of Maximum Conformality. What is this Principle? In other words, what is being maximized and what does this tell us about the structure of QCD? Also, is this a new principle or a new application of an old principal? Here's the full paper on the principle's application to top physics."}
{"id":"135231","title":"How many atoms in the particle (AuNP)","text":"Let's say I have to determine number of atoms of gold in ball-shaped nanoparticle with diameter of 75 nm. I need general method of solving such problems more than specific answer for only this question."}
{"id":"41816","title":"Why does a tire need to be filled with air?","text":"The **Mini 4WD** 's tires aren't full of air, and it can run. Also, the tank doesn't have tires with air. So, the question is: why do real cars on the road need to be filled with air? What is the idea behind it? ![](http:\/\/i.stack.imgur.com\/0ciAs.jpg)"}
{"id":"17676","title":"Why is storage of spent nuclear fuel dangerous?","text":"Just what the title states; there's a good deal of noise made about transport, and storage of spent nuclear fuel. Why all the hullabaloo when the fuel is all spent?"}
{"id":"95504","title":"Bending a flat sheet","text":"I am interested in the following problem: starting from a flat sheet, once can bend it into a (sector) of a cylinder isometrically. If further (orthogonal to the first plane) curvature is induced, the sheet will necessarily be stretched. The question is, how much stretch is locally caused by the increase in curvature? I could attempt to answer the point by classical continuum mechanics tecniques, but I am told differential geometry allows an answer to be given immediately. Could somebody please point me towards the answer, or suitable material to study? Many thanks"}
{"id":"101565","title":"Cheap launching objects to orbit","text":"Why slingatron hasn't been successful ? Same thing with gigantic nail clipper. Since we dont need great force when arm length increases, why couldn't we build gigantic nail clipper orbital launcher :) ? Provided of course that we had material able to sustain great compression stress. Disclaimer, no im not affiliated in any way with creators, dont know them, not even same country. Just thought that idea was cool."}
{"id":"101560","title":"Schrödinger's cat and the difficulty of macroscopic superposition state","text":"The Schrödinger's cat was regarded as peculiar since we seldom encounter a superposition state in macroscopic scale: $$ | \\mathrm{dead \\,\\,cat} \\rangle + | \\mathrm{alive \\,\\, cat}\\rangle $$ We more often describe an unknown cat as $$ | \\mathrm{dead \\,\\,cat} \\rangle \\langle \\mathrm{dead \\,\\,cat} |+ | \\mathrm{alive \\,\\, cat}\\rangle \\langle \\mathrm{alive \\,\\, cat}| $$ without superposition. I often heard that it is difficult to prepare and maintain a large-scale superposition state. Similar difficulty also occurs in quantum computing. My question is, actually what is the reason for the difficulty to prepare and maintain a large-scale superposition state? If it is decoherence, why decoherence happens? Is that because of entropy?"}
{"id":"122984","title":"Why is the $SO(4)$ symmetry of the Hydrogen atom called dynamical?","text":"Why _dynamical?_ My previous quantum mechanics teacher could not answer it."}
{"id":"62534","title":"What is to be considered a \"body\" in physics?","text":"Well, the question says it all; is there a definition of body in physics? What is to be considered a physical object and what it cannot?"}
{"id":"122988","title":"Why a transistor is activated when I touch its base with my finger?","text":"Is my body charged eletrically? If so, how much? Will it work with all types of transistors?"}
{"id":"20768","title":"How to find maximum velocity","text":"I stack a question about projectile question. The question was > _A projectile is being launched from ground level with no air resistance. > You want to avoid having it enter a temperature inversion layer in the > atmosphere a height $h$ above the ground (a) what us the maximum launch > speed you could give this projectile if you shot it straight up ? Express > your answer in terms of $h$ and $g$.(b)Suppose the launcher available shoots > projectiles at twice the maximum launch speed you found in part (a). At what > maximum angle above the horizontal should you launch the projectile ?_ I could solve the (a) part. How did was following (a) using the following formula to drive $V$ $\\delta x$ = $\\frac{V^{2}-Vi^{2}}{2g}$$ also we have $Vi = 0$, $\\delta x = h$ I got $V = \\sqrt{2gh}$ after that I think I need to use some kind of angle relative formula to create $arccosx$ or $arcsinx$ will be equal to some number then find the angle but I still don't have idea which formula do I need to use and find max angle. also Do I need to split $Vx$ and $Vy$ from $V$ ? One more question, I have seen some minimum spe"}
{"id":"20763","title":"Force and Laws of Motion","text":"While catching a fast moving cricket ball, why do fielders gradually pull back their hands backwards with the moving ball? Please advice which among the following is the correct answer. If both are incorrect kindly give the correct answer:      **A** : By pulling the hands backward the fielder increases the time during which the high velocity of the moving ball decreases to 0. Thus, the acceleration of the ball is decreased and therefore the impact of catching the fast moving ball is also reduced.      **B** : By pulling the hand backwards, the fielder gradually and continuously decreases the momentum of the moving ball and thereby reducing the velocity to zero."}
{"id":"7246","title":"Light bending surfaces and energy funneling properties","text":"In the last year there has been fuzz regarding metamaterials studied for the purpose of cloaking, and it seem to me they are somewhat glorified 2D waveguides, but in any case it seems a reasonable application of this to funnel radiation captured over a wide area of incidence into smaller area, and i thought that this could be a cheap way to produce ignition radiation for a fusion reactor. Reasons why this wouldn't work, or not produce enough energy density? All stuff i've read about fusion ignition with lasers doesn't seem to depend too heavily on the coherence of the radiation"}
{"id":"44664","title":"Possible colors of fire?","text":"I have learnt that depending on the various gases those are involved in the reaction that produces fire, different colors (yellow, red or blue) of flames become visible. I have a question .. what are the possible colors that fire can possess? and do they (how, if yes) relate to the intensity? like I know blue is more intense than red, etc.."}
{"id":"44668","title":"How does Delta baryon decay conserve angular momentum?","text":"I'm a chemist so bear with me: I understand the Delta baryons $\\Delta^{+}$ and $\\Delta^{0}$ to be in some sense spin (and isospin) quartet states of the proton and neutron. These can decay straight to a proton or neutron through emission of a pion, which is spin-0. I've been puzzling about the mechanism by which this decay conserves spin. I note via Wikipedia that pions decay into either a lepton pair or a photon pair, for which it's easy to show spin is conserved, however pion production seems to induce a loss of $J=1$. This seems to suggest momenta of: $\\Delta^{+} (J=\\frac{3}{2}) \\rightarrow{} p^{+} (J=\\frac{1}{2}) + \\pi^{0} (J=0)$ Am I missing a boson? Is my understanding of spin flawed? I assume that if this decay exists, similar things turn up everywhere else."}
{"id":"54442","title":"Electric field singularity","text":"What's a general proof for the log-singularity of electric field near the edge of a uniformly charged surface?"}
{"id":"54443","title":"How much better is salt water at letting through microwaves than pure water?","text":"> **Radio Wave Attenuation** > > There are two general types of matter (substances) in the universe that > affect electromagnetic waves, conductors and insulators which are called > dielectrics by scientists. Most, but not all, conductors are metals, such as > copper, aluminum, silver and gold. However, salt water is also a rather poor > conductor! Most, but not all, dielectrics are non metals. Examples of > dielectrics are paper, plastic, Teflon, glass, ceramic and dry wood. Pure > water is a good dielectric substance! Source: http:\/\/www.madsci.org\/posts\/archives\/2002-03\/1015162213.Eg.r.html I read this paragraph, and was truly intrigued. Is it really true that salt water (NaCl) passes radio waves (specifically 3cm) through better than pure water? Why is this, and how large is the difference? Would it be noticeable in an experiment?"}
{"id":"59673","title":"Why doesn't fusion contradict the 1st law of thermodynamics?","text":"I was reading up on the 1st law of thermodynamics for my Chemistry exam and I was wondering why doesn't fusion contradict the 1st law of thermodynamics? The 1st law states that > The energy of an isolated system is constant or that whatever is put into the system, you get out of it, but in fusion you get more out of the initial reactions than you put in Hydrogen + Hydrogen = Helium etc. I am still a bit confused... Thanks for any help!"}
{"id":"59678","title":"What is the definition of correlation length for the Ising model?","text":"The correlation length $\\xi$ is related to critical temperature $T_c$ as $$ \\xi\\sim|T-T_{c}|{}^{-\\nu}, $$ where $\\nu$ is the critical exponent.   1. Is this the **formal definition** of correlation length? If not, what is the formal definition of correlation length (for phase transition in the Ising model)?   2. Can you give a physical understanding of correlation length?"}
{"id":"102575","title":"Lie algebra of axial charges","text":"Starting from the lagrangian (linear sigma model without symmetry breaking, here $N$ is the nucleon doublet and $\\tau_a$ are pauli matrices) $L=\\bar Ni\\gamma^\\mu \\partial_\\mu N+ \\frac{1}{2} \\partial_\\mu\\sigma\\partial^\\mu\\sigma+\\frac{1}{2}\\partial_\\mu\\pi_a\\partial^\\mu\\pi_a+g\\bar N(\\sigma+i\\gamma_5\\pi_a \\tau_a)N$ we can construct conserved currents using Noether's Theorem applied to $SU(2)_L\\otimes SU(2)_R$ symmetry: we get three currents for every $SU(2)$. By adding and subtracting them, we obtain vector and axial currents.   We could have obtained vector charges quickly by observing that they are just isospin charges, so nucleons behave as an $SU(2)$ doublet (fundamental representation), pions as a triplet (adjoint representation) and sigma as a singlet (so basically it does not transform): $V_a=-i\\int d^3x \\,\\,[iN^\\dagger\\frac{\\tau_a}{2}N+\\dot\\pi_b(-i\\epsilon_{abc})\\pi_c]$ But if I wanted to do the same with axial charges, what Lie algebra\/representation must I use for pions and sigma?   I mean, axial charges are $A_a=-i\\int d^3x \\,\\,[iN^\\dagger\\frac{\\tau_a}{2}\\gamma_5N+i(\\sigma\\dot\\pi_a-\\dot\\sigma\\pi_a)]$ and I would like to reproduce the second term using a representation of Lie algebra generators of axial symmetry which act on $\\sigma$ and $\\pi$, but I don't know the algebra (I think it is $SU(2)$), neither the representation to use.   I tried to reproduce that form using the three matrices $T^1=\\begin{bmatrix} 0&-i&0&0\\\\\\i&0&0&0\\\\\\0&0&0&0\\\\\\0&0&0&0 \\end{bmatrix}\\quad T^2=\\begin{bmatrix} 0&0&-i&0\\\\\\0&0&0&0\\\\\\i&0&0&0\\\\\\0&0&0&0 \\end{bmatrix}\\quad T^3=\\begin{bmatrix} 0&0&0&-i\\\\\\0&0&0&0\\\\\\0&0&0&0\\\\\\i&0&0&0 \\end{bmatrix}$ which should act on the vector $(\\sigma,\\pi_1,\\pi_2,\\pi_3)$, but I calculated their commutator and they don't form an algebra, so I think I'm getting wrong somewhere in my reasoning."}
{"id":"88892","title":"Why can movies get away with 24 frames per second, but games can't?","text":"Movies look reasonably smooth at 24 frames per second; unless the screen is huge and you use your peripheral vision, you usually can't see each frame individually. This has lead to a large amount of textbooks claiming that the visual response time of the retina is slower than 1\/24 seconds. However, games at 24 frames per second absolutely suck. The motion is so choppy it is unplayable. For a game, you seem to need at least 40 frames per second, and even at that rate you still see individual frames if the objects are moving quickly. Why is this so? Is this related to the fact that movies have motion blur due to a non-zero shutter speed? Is \"jerkiness\" mostly perceived in terms of \"gap between object in frame 1 and frame 2\" rather than visual response time (the gap is smaller in films due to blurred objects \"connecting\" across frames)? If so, doesn't this prove that our visual response time is much faster than 1\/24 seconds? To make the question more precise: is this response determined by physical mechanisms, such as the relaxation times of pigments in the eye or the firing times of neurons? If so, which ones? Or is it mostly due some sort of postprocessing done by the brain, i.e. more to do with the cognitive than the physical side of perception?"}
{"id":"14939","title":"Does Gödel preclude a workable ToE?","text":"Gödel's incompleteness theorem prevents a universal axiomatic system for math. Is there any reason to believe that it also prevents a theory of everything for physics? * * * _Edit:_ I haven't before seen a formulation of Gödel that included time. The formulation I've seen is that any axiomatic systems capable of doing arithmetic can express statements that will be either 1) impossible to prove true or false or 2) possible to prove both true and false. This leads to the question: Are theories of (nearly) everything, axiomatic systems capable of doing arithmetic? (Given they are able to describe a digital computer, I think it's safe to say they are.) If so, it follows that such a theory will be able to describe something that the theory will be either unable to analyse or will result in an ambiguous result. (Might this be what forces things like the Heisenberg uncertainty principle?)"}
{"id":"95033","title":"Covariance of rayleigh channel","text":"**f** =[ _f1_ _f2_ _f3_ ]; **g** =[ _g1_ _g2_ _g3_ ]; _f1_ ,_f2_, _f3_ ,_g1_, _g2_ ,_g3_ are all independent identically distributed zero mean complex random variable. **h** = elementwise wise Schur-Hadamard product of **f** and **g**. What is the covariance matrix of **h**? Is that diagonal matrix?"}
{"id":"44086","title":"The synthesis of $^{254}\\text{No}$","text":"How is $^{254}\\text{No}$ synthesised? Could you explain the reaction where it is preceded by $^{208}\\text{Pb}(^{48}\\text{Ca}, 2\\text{n})$? References to articles are well enough—I was somehow unable to find anything sufficiently detailed and informative."}
{"id":"119244","title":"electric-circuits","text":"Why can't you connect the + of a device to one battery and the - to another battery (and have it work)?![enter image description here](http:\/\/i.stack.imgur.com\/W9x5Q.png)"}
{"id":"119247","title":"How can I find the motion equations of the 2-dim harmonic oscillator?","text":"First of all: I am no physicist, so I am rather helpless. I need to find the moving equations of the 2-dim. harmonic oscillator. If it is possible it should be rather elementary, because, as I said, I am no physicist. I've heard that ONE possibility is Hamilton mechanics, but to be honest, I hardly know what that is! Maybe you can give me help? I would be very thankful. **Update** : For the general spherical pendulum I get $$ \\ddot{\\theta}=\\sin\\theta\\cos\\theta\\dot{\\varphi}^2+\\frac{g}{l}\\sin\\theta,~~\\ddot{\\varphi}=\\frac{-2\\cos\\theta\\dot{\\theta}\\dot{\\varphi}}{\\sin\\theta}. $$ Isn't the 2-dim. harmonic oscillator a special spherical pendulum so that I maybe can use this results?"}
{"id":"123762","title":"Is it possible to prove that projectiles with drag will not travel in a parabola under the theory of classical mechanics?","text":"Can you prove that projectiles will travel in a parabola, provided the acceleration due to gravity $g$ does not vary significantly along the trajectory? This should take air resistance into account. I know how to prove a parabolic trajectory without air resistance (see my user page on Wikipedia, for instance, where I have successfully proven this), but with air resistance, the equations get more complicated and difficult to solve analytically."}
{"id":"123761","title":"A question on the Bousso-Polchinski paper","text":"In this famous paper by Bousso and Polchinski, Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant an example in M-theory compactification is given in section 2.2, page 7. Since I am new to string theory, I wonder how exactly (2.16) $$ S= V_72\\pi M_{11}^9\\int d^4x \\sqrt{-g_4}\\left(R - {1 \\over 2 \\cdot 4!} F_4^2\\right) + S_{branes}$$ is obtained from (2.12) $$ S= 2\\pi M_{11}^9\\int d^{11}x\\sqrt{-g_{11}}\\left(R-{1 \\over 2 \\cdot 4!} F_4^2 \\right) +S_{{branes}}$$ Is there any assumption made which is not stated in the paper? Especially, why \"eleven dimensional $\\bf F_4$ is directly reduced to a four dimensional $\\bf F_4$\" Since I am new to string theory, I really want to have a thorough understanding of this work. Detailed explanation is very much appreciated!! Thank you very much!"}
{"id":"129697","title":"What's the point of NASA's impossible space engine? Wouldn't a mirror work better?","text":"My understanding (which is very little) is that the point of NASA's recent virtual-particle engine is to convert solar energy into momentum. That's fine, but what's the point? If the spacecraft is already receiving solar energy, can't it just carry around a mirror to reflect light and accelerate via radiation pressure? Or is the quantum version somehow injecting \"more\" momentum than is actually being delivered via radiation pressure?"}
{"id":"129696","title":"Leonard Susskind said that the small value of the cosmological constant could mean that the universe is much bigger than we thought. Why?","text":"He said that near the beginning of this Youtube video that it could mean that the universe is much bigger than he had thought that it was, and that the cosmological constant wasn't even a constant. I don't have a clue what his reasoning was though."}
{"id":"113538","title":"Jet rapidity vs. Pseudorapidity","text":"thanks to this post Why is pseudorapidity defined as $-\\log \\tan \\theta\/2$ I got a better understanding of what makes (pseudo) rapidity an interesting an often-used concept. Tangentially working with jet physics (the spray of hadrons kind of jet :), I am wondering why people work with rapidity (no pseudo here) when it comes to jets. What is the reason for this (doesn't pseudo rapidity offer the same convenient properties of Lorentz invariance wrt z boosts)? More importantly, how does one determine the rapidity (or rapidity difference) in a measurement at a hadron collider, where one does not know the the full four vector of any physics object? I suppose that the above question might be related to the topic of jet mass, which I have read in papers before. Any hints on the definition\/usefullness of jet mass would also be appreciated. Thanks"}
{"id":"61689","title":"Conservation of angular momentum tensor $L^{\\mu\\nu}$ in special relativity","text":"I have edited this question because I don't think that the related post answers my question fully. It refers to Noether's theorem but I would like an explicit illustration in an easier fashion: The angular momentum tensor is defined: $$L^{\\mu\\nu}~=~x^\\mu p^\\nu-x^\\nu p^\\mu$$ I would like to show that if a particle is not acted on by an external force that the angular momentum is conserved. Despite it being a direct consequence of the theorem I want to compute it in an inferior way, would this be sufficient? Without loss of generality take the rest frame of the particle $$\\frac{dx^\\mu}{d\\tau}=(c,\\underline{0})$$ Since the particle is not acted on by an external force, we have $$\\frac{dp^\\mu}{d\\tau}=\\left(\\frac{1}{c}\\frac{dE}{ d\\tau},\\underline{0}\\right)$$ $$L^{\\mu\\nu}L_{\\mu\\nu}=\\left( 2 (x\\cdot x) (p\\cdot p) - 2 (x \\cdot p)^2 \\right)$$ Can I show the conservation by $$\\frac{d(L^2)}{d\\tau}=0$$"}
{"id":"89754","title":"Do electrons collapse into nucleus, if electrons in the atom are constantly excited?","text":"From the Bohr's atomic model, it is clear that electron can have only certain definite energy levels. When the electron is present as close to the nucleus as possible, the atom has the minimum possible energy and is said to be in the _ground state_. When energy from some outside source is supplied to it, it can absorb a definite amount of energy and jumps to higher energy state. Such a state of an atom in which the atom possesses more energy than possessed in the ground state is called _excited state_. These excited states are unstable and the electron tends to come back to lower energy level. This transition (change) from upper to lower energy level occurs with a jump and energy is emitted in the form of a quantum equal to difference in energies between the two levels. I have a doubt here, if electron absorbs energy from the outside source and jumps to the higher energy state, it will be now storing absorbed energy, as potential energy at the excited state. If electron emits all the absorbed energy (quanta-difference in energies between the two levels). By virtue of what energy does electron come down? I mean, in common situations, we say that an object at any height comes down converting it's potential energy into kinetic energy. Here in case of electron, it has already emitted absorbed energy as quanta. So, is it that electron losses some energy other than the energy absorbed from the source, to come down to ground state. I thought, if it was a possibility, then electron would constantly need to lose energy, whenever excited, at last, it would collapse into the nucleus. But, this not we really observe. I think there might be some misunderstanding by me or there might be any of the existing model like quantum mechanical model, which could account for this. If any were the case, please explain."}
{"id":"118591","title":"Irreversible heat engines strictly less efficient than reversible ones","text":"I undertand how Canot's theorem implies that irreversible heat engines must be no more efficient than reversible one's, but it isless clear why they need to be less efficient, as I have seen stated in some places. If they could be equally efficient then an irreversible engine engine could be used to drive a reversible engine operating between the same heat reservoirs, without any net energy transfer between the reservoirs. It would then be unclear what is irreversible about the irreversible engine. Does that constitute an actual contradiction though? If so can the argument be stated more tightly? It feels a litle sloppy as is. It could also be a quation of how an irreversible engine is defined in Carnot's theorem. I understood it to mean one that cannot be run in reverse as a heat pump, which could presumably include a Carnot engine with a one way ratchet attached. If it really means an engine whose thermodynamic effects can not be undone, the implication would be trivial."}
{"id":"110038","title":"Spring Constant - Waves","text":"I tried to prove that for the same length of spring , a one dimensional wave that moves faster in spring 2 than spring 1, its spring constant will be lower than the spring constant of spring 1. That's the begining of the proof I tried to make: $$w = \\sqrt{\\frac km} = 2π \\cdot \\nu$$ $$\\nu = \\frac{1}{2π} \\cdot \\sqrt{\\frac km}$$ $$k = m \\cdot \\nu^ 2 \\cdot 4π ^ 2$$ What am I missing?"}
{"id":"118623","title":"Why don't you always use a high sampling rate?","text":"I was just curious, as I was reading about aliasing. As far as I understand, aliasing comes from the fact, that you use a bad sampling rate, resulting in getting a wrong waveform compared to the one you had from the beginning. So why is it you just don't use a fast sampling rate all the time ? Or maybe you do that, and I just don't know ?"}
{"id":"110033","title":"Does a black hole have enough time to actually form a singularity?","text":"I am trying to wrap my head around black holes, singularties and hawking radiation. Physics.se contains many intresting questions and answers, but from none I could so far read about the interaction between formation of a singularity and hawking radiation (most seem to talk about the point at which matter plunges into a black hole that already has a singularity). From what I understand, from the perspective of a star collapsing into a singularity, the perceived time is just like in classical mechanics: the mass is accelerated towards the center of gravity, and when \"everything\" is there, it is a singularity. Now this takes a finite time, and the matter itself perceives just that. Part of the question is: how long is this time? I am assuming in the magnitude of milliseconds. Now as I understand hawking radiation, as soon as there is an event horizon, it will start emitting negative mass\/energy into the direction of the singularity. Time dilation gets infinitely stronger as we approach the singularity, so I would assume that from the perspective of those negative particles, they would all the time reduce their perceived distance to the particles of positive energy\/mass that are already on their way to the singularity. Since approaching the singularity, time dilation becomes kind of infinite, even the $10^{100}$ years or so it is supposed to take for hawking radiation to evaporate a black hole seem to be enough to send the necessary amount of negative energy\/mass particles down to the singularity so that they can \"chase\" the positive energy\/mass paticles very closely. But will they reach them before a singularity forms, so that they cancel out each other to the point there is not enough gravitational force? If this is just a matter of calculating \"$n$s until matter reaches singularity\" and \"$m$s until negative mass\/energy reaches singularity\" and calculation shows that always $n<m$ then I would be intrested in the calculation (if possible presented in a way that doesn't require too exotic concepts). If this is a qualitative matter, in that some of my ideas are fundamentally wrong, please point those out (again if possible in a way that doesn't require too exotic concepts)."}
{"id":"118629","title":"Do negative numbers have any physical meaning?","text":"So, mindlessly wandering off into space, thinking about quantum and how cool physics is, I came to a realization that... well.. negative numbers to me make 0 sense. You have either something, or not something. It's always a yes or no answer, with everything And everything always has an opposite; it's nonexistence. I either have a red scarf, or not. If there are 3 doors, labeled 1 2 3. I have to choose one of them. The choice isn't me choosing which door, the choice is what door i'll pick. I'll either pick door 1, or not pick door 1. The opposite of door 1 is no door 1. Lets go really small. I have 2 grains of salt. I can't take 3 grains away. I believe negative numbers were invented by man to create debt. Because it doesn't make any sense in physics. Can someone please explain this to me!!"}
{"id":"110036","title":"Balloon tethering cable tension model (catenary tether)","text":"If a moving balloon (let's say it's a blimp system with some rotors for movement) is connected to the ground by a cable that describes a catenary curve, how can we model the tension being applied on the blimp by the cable? ![MSpaint draft](http:\/\/i.stack.imgur.com\/mmAmG.png) This force has a vertical component that will be related to the mass of the cable and a horizontal component with the opposite direction of the movement of the blimp. Is there any model for this kind of behavior?"}
{"id":"82219","title":"Cauchy's Second Law of Motion understanding","text":"I'm told that Cauchy's Second law of motion is $$\\tau_{ik} ~=~ \\tau_{ki}.$$ Now my notes don't go into great details for simplicity sake, but they mention that the order of magnitude of body force $\\rho F \\delta V$ and inertia $\\frac{D}{Dt}(\\rho \\delta V v)$ on a volume is smaller than that of stress $t(n) \\delta S$ on a surface. What I'm not understanding is the order of magnitude stuff? I'm not sure why stress has a larger order of magnitude."}
{"id":"82214","title":"Force on impact in case of a free fall","text":"In a free fall, on impact with the ground, the body experiences a massive force because of a huge acceleration since the velocity goes to 0 rather quickly. Since this force has to come as a result of reaction, can it be assumed that the body applied a massive force on impact far greater than its weight?"}
{"id":"57285","title":"Are oscillations of electron chirality experimentally observable?","text":"Is there any plausible experiment by which chirality oscillations in electrons could be observed experimentally, such as through some analogy to neutrino oscillation experiments?"}
{"id":"105070","title":"Electromagnetism duality theorem","text":"Concerning Electromagnetism, textbooks often refer to the Duality Theorem. Sometimes it is presented like this: «Consider the Maxwell's Equations (with phasors) and a known field $\\mathbf{E}_1$, $\\mathbf{H}_1$: $\\nabla \\times \\mathbf{E}_1 = - j \\omega \\mu \\mathbf{H}_1$ $\\nabla \\times \\mathbf{H}_1 = j \\omega \\epsilon \\mathbf{E}_1$ If $\\mathbf{E}_1$ is replaced with $\\mathbf{H}_2$ (the magnetic field of another electromagnetic field: $\\mathbf{E}_2$, $\\mathbf{H}_2$) and $\\mathbf{H}_1$ is replaced with $-\\mathbf{E}_2$, $\\mu$ is replaced with $\\epsilon$ and $\\epsilon$ with $\\mu$, then the above equations become respectively $\\nabla \\times \\mathbf{H}_2 = j \\omega \\epsilon \\mathbf{E}_2$ $\\nabla \\times \\mathbf{E}_2 = - j \\omega \\mu \\mathbf{H}_2$» They are valid Maxwell's Equations too. But now what does follow from this substitution? 1) Should be $\\mathbf{E}_1 = \\mathbf{H}_2$ and $\\mathbf{H}_1 = -\\mathbf{E}_2$? It is dimensionally incorrect. 2) I read also $\\mathbf{E}_1 = \\eta \\mathbf{H}_2$ and $\\eta \\mathbf{H}_1 = -\\mathbf{E}_2$. My question is double: what is the advantage of Duality Theorem and which is the correct form between the two just written? Thank you anyway!"}
{"id":"117263","title":"Coulomb potential in atoms other than hydrogen","text":"The energy of an electron on $H$ atom is given by the formula: $-13.6 \\; \\text{eV}\/n^2$. The constant value is born from $H$ dielectric constant and efective mass of the electron. My question is: this formula can be used to estimate the energy of a electron on a $Si$ Coulomb potential? Is it sufficient to user the proper values of $\\epsilon$ and electron effective mass?"}
{"id":"105078","title":"Direction of future time cone inside Schwarzschild horizon","text":"Following is the picture copied from enter link description here, illustrating the future time cone inside the Schwarzschild horizon. ![enter image description here](http:\/\/i.stack.imgur.com\/ZP1w4.png) My question is, on the left side of $r=2M$, why must the time cone be left- oriented, but not right-oriented?"}
{"id":"116793","title":"Does the Anti-Up, Anti-Strange, Anti-Strange Quark combination form a Baryon?","text":"I was using this applet: http:\/\/www.lon-capa.org\/~mmp\/applist\/q\/q.htm And I saw that the up, strange, strange quark combination formed this Baryon: ![enter image description here](http:\/\/i.stack.imgur.com\/1Oa2a.png) But the anti-up, anti-strange, anti-strange combination didn't form anything. So is that a fault of the applet or is there an actual reason? What would the reason be?"}
{"id":"983","title":"Why do those dolphin bubble rings I saw in a video behave the way they do?","text":"This (see http:\/\/www.youtube.com\/watch?v=TMCf7SNUb-Q&feature=related) is so cool, I wanted to show it. As a pro forma question for this forum, any idea what is going on with the breaking of larger, presumably vortex related bubble rings and reconnection making the smaller rings? The dolphins appear to be making smaller vortices by suddenly moving their snouts through a ring. On a simpler level, why do the rings get larger? Any other good questions come to mind?"}
{"id":"64447","title":"Thermodynamics, PV diagrams?","text":"My teacher told me that the total amount of work done on or by a gas can be represented by the area enclosed in the process in a PV diagram. This is only valid for non isothermic processes, right?"}
{"id":"64442","title":"First law of thermodynamics?","text":"The first law says that the change in internal energy is equal to the work done on the system (W) minus the work done by the system (Q). However, can $Q$ be any kind of work, such as mechanical work? For example, does a string heat up if it is attached to a block and a tension is applied to move the block? And what about the work done on the system by gravity when an object falls? Is there any internal energy change when that happens?"}
{"id":"22868","title":"How do I find the Gain of this Transfer Function","text":"I found the transfer function for the spring mass damper system to be $$G(s)=\\frac{1}{ms^{2}+bs+k},$$ and now I need to find the gain of this transfer function. I know that the gain is $G=|G(j\\omega)|$, but I'm not really sure how to go about finding the gain of a transfer function with a quadratic term in the denominator."}
{"id":"22867","title":"Abstract Algebra in Relativity and Cosmology?","text":"Is Abstract Algebra useful in theoretical Relativity and\/or Cosmology? If so can anyone give me some examples or point me towards a good book with that emphasis if it is one? Thanks in advance."}
{"id":"2672","title":"Calculation of the cross section","text":"Why, when we calculate the total cross section, we make the average other initial states and the sum over final states?"}
{"id":"133650","title":"Magnet spinning between two other magnets","text":"Suppose, we have two magnets, MA, MB, and we have a third magnet MC in between the two magnets. Each magnets' north pole faces the other magnets south pole, and the magnets are placed horizontally side by side. We spin the magnet MC at a speed between super fast and slow. What will happen then?   I mean, once MC's north pole faces MA's south pole and MC's south pole faces MB's north pole, then the next moment MC's south pole faces MA's south pole and MC's north pole faces MB's north pole. So what will happen then?![the image](http:\/\/i.stack.imgur.com\/E0yGZ.jpg)"}
{"id":"71812","title":"Atom emitts a photon - what is the frequency of the energy of the photon?","text":"Lets say I have an atom of known mass who goes from excited state to the ground state whose energy is $14.4 keV$ lower. I know that the emitted energy $14.4keV$ got converted into an energy of a photon and kinetic energy of the Atom - this means energy of the photon $E_\\gamma < 14.4keV$. I tried to calculate it like this and got a nonsense: \\begin{align} E_{1} &= E_{2}\\\\\\ \\sqrt{ {E_{0~Fe}}^2 + {p_1}^2c^2 } &=\\sqrt{ \\left(E_{0~Fe} - E_\\gamma\\right)^2 + {p_2}^2 c^2 } \\longleftarrow \\substack{\\scriptsize \\boxed{p_1 = 0}~\\boxed{p_2 = E_\\gamma \/c}}\\\\\\ \\sqrt{ {E_{0~Fe}}^2 + 0 } &= \\sqrt{ \\left(E_{0~Fe} - E_\\gamma\\right)^2 + \\frac{{E_\\gamma}^2}{c^2} c^2 } \\\\\\ E_{0~Fe} &= \\sqrt{ \\left(E_{0~Fe} - E_\\gamma \\right)^2 + {E_\\gamma}^2 }\\\\\\ {E_{0~Fe}}^2 &= {E_{0~Fe}}^2 - 2E_{0~Fe}E_\\gamma +{E_\\gamma}^2 + {E_\\gamma}^2 \\\\\\ {E_\\gamma}^2 + (-E_{0~Fe})E_\\gamma + 0 &= 0\\\\\\ &\\Downarrow\\\\\\ E_\\gamma &= 0\\\\\\ E_\\gamma &= E_{0~Fe} \\end{align} None of the solutions to the quadratic equation make sense. Can anyone give me a hint where did i go wrong? * * * **EDIT:** After i read some of your comen I did rethink the situation and wrote equation like this. But still energy of a photon $E_\\gamma$ is too large. Shouldnt it be smaller than $14.4keV$? \\begin{align} E_1&=E_2\\\\\\ E_{0~Fe} + \\Delta E &= \\sqrt{{E_{0~Fe}}^2 + {p_2}^2c^2}\\\\\\ E_{0~Fe} + \\Delta E &= \\sqrt{{E_{0~Fe}}^2 + \\frac{{E_\\gamma}^2}{c^2}c^2}\\\\\\ E_\\gamma &= \\sqrt{\\left(E_{0~Fe} + \\Delta E\\right)^2-{E_{0~Fe}}^2}\\longleftarrow{\\substack{\\text{Lets assume that we have a \"Fe\" atom}\\\\\\\\\\text{whose rest energy is $E_{0~Fe}=53GeV$}}}\\\\\\ E_\\gamma &= \\sqrt{\\left(53\\times10^9eV + 14.4\\times10^3 eV\\right)^2 - \\left(53\\times10^9 eV\\right)^2}\\\\\\ E_\\gamma &\\approx 3.91\\times10^7 eV \\end{align} * * * **EDIT:** By even improving the equation I got better result which I think must be the right one. \\begin{align} E_1&=E_2\\\\\\ E_{1~atom}&=E_{2~atom} + E_\\gamma\\\\\\ E_{0~Fe} + \\Delta E &= \\sqrt{{E_{0~Fe}}^2 + {p_2}^2c^2} + E_\\gamma\\\\\\ E_{0~Fe} + \\Delta E &= \\sqrt{{E_{0~Fe}}^2 + \\frac{{E_\\gamma}^2}{c^2}c^2} + E_\\gamma\\\\\\ E_{0~Fe} + \\Delta E - E_\\gamma &= \\sqrt{{E_{0~Fe}}^2 + {E_\\gamma}^2}\\\\\\ \\substack{\\text{square of}\\\\\\\\\\text{a trinomial}}\\longrightarrow(E_{0~Fe} + \\Delta E - E_\\gamma)^2 &= {E_{0~Fe}}^2 + {E_\\gamma}^2\\\\\\ \\big((E_{0~Fe} + \\Delta E) - E_\\gamma\\big)^2 &= {E_{0~Fe}}^2 + {E_\\gamma}^2\\\\\\ (E_{0~Fe} + \\Delta E)^2 - 2(E_{0~Fe} + \\Delta E)E_\\gamma + {E_\\gamma}^2 &= {E_{0~Fe}}^2 + {E_\\gamma}^2\\\\\\ (E_{0~Fe} + \\Delta E)^2 - 2(E_{0~Fe} + \\Delta E)E_\\gamma &= {E_{0~Fe}}^2\\\\\\ E_\\gamma &= \\frac{(E_{0~Fe} + \\Delta E)^2 - {E_{0~Fe}}^2}{2(E_{0~Fe} + \\Delta E)}\\\\\\ E_\\gamma &= \\frac{\\left(53\\times10^9eV + 14.4\\times10^3eV\\right)^2 - \\left(53\\times10^9eV\\right)^2}{2\\left(53\\times10^9eV + 14.4\\times10^3eV\\right)}\\\\\\ E_\\gamma &= 14.399 keV \\end{align} Is it possible that atom takes so little energy?"}
{"id":"104854","title":"How clean will first generation fusion reactors be compared to fission reactors?","text":"Googling the topic seems to indicate that fusion reactors will produce less waste and less toxic radioactivity, but this fact never seems to be mentioned during the current debate over nuclear power. Will fusion reactors be cleaner than fission reactors? If so, how much cleaner?"}
{"id":"93657","title":"In which field theories with fermions do string- and fivebrane structures not come up?","text":"A year ago, username @Greg Graviton asked in a thread here about the Spin group as covering of the spatial rotations. A subquestion was: _What other groups, even larger than SU(2) are there that (could) act on \"physics\" and are an extension of SO(3) Is it possible to classify all possibilities, in particular the ones that are not direct products?_ My answer to this was: _Of course, there are many more groups relevant for Quantum Mechanics and other physical theories (like gauge groups, as you mentioned). But concerning your second question, as far as \"rotations only\" go, SU(2) is already theuniversal covering group. These are the simple connected Lie-Groups, which lie on top of these upward chains you're searching for._ Now recently I came across a construction for structures lying above the spin structure, namely string stucture and then fivebrane structure. There are mainly the nLab references, and there is a thread on MathOverflow, relating the Whitehead tower to physics. Without knowing too much homotopy, I can still see the definitions seem geometrical and hence natural to me. The definitions seem to be quite general constructions over the manifold and are the discussed in the context of QFTs. I make some observations which surprise me a little and I want to ask these specific questions:   * The title of the thread tries to address this question. Their names kind of suggest it, but do these mathematical objects, the objects above the spin structure, necessarily have to do with string theory? Now this question is problematic in as far as string theory people will say QFT is generally the necessary conclusion when studying QFTs. I want to filter this thought out for a moment and ask: If you want to do generally non-stringly quantum field theory, do you for practical reasons want to spend your time learning about these objects? Do you come across them just by learning about Dirac operators?    * The presentation for field theories discussing this focus on a map $\\Sigma\\to X$, were $\\Sigma$ gives the field configuration and $X$ is a manifold. I figure I have to view this just as generalized element of the manifold. Now I find it curious that the Spin, String and fivebrane structure seems to be something build over $X$ first, prior to considering physically propagating things in that world. This reverses my thinking of the spin being something associated with the particle, not with the space it lives in. When following this approach, to what extend to I think of the spin (bundle, etc.) to be a property of the world the object lives in, and then what is the object itself, if we say it only gets these feature when it's put into $X$?   * Why *five*brane? Why not some other number?"}
{"id":"91017","title":"Why are the laws of thermodynamics \"supreme among the laws of Nature\"?","text":"Eddington wrote > The law that entropy always increases holds, I think, the supreme position > among the laws of Nature. If someone points out to you that your pet theory > of the universe is in disagreement with Maxwell's equations — then so much > the worse for Maxwell's equations. If it is found to be contradicted by > observation — well, these experimentalists do bungle things sometimes. But > if your theory is found to be against the second law of thermodynamics I can > give you no hope; there is nothing for it but to collapse in deepest > humiliation. and Einstein wrote > [classical thermodynamics] is the only physical theory of universal content > which I am convinced will never be overthrown, within the framework of > applicability of its basic concepts. Why did they say that? Is it a very deep insight they had, or is it something one can be convinced of quite easily? Or even, is it trivial?"}
{"id":"3170","title":"Detecting coherence","text":"Is there a way to know if a particle is acting as a wave or a particle? Alternatively, if an entangled particle was already measured? A - Yes So any experiment over an entangled particle that let you know if they are coherent or decoherent, so would let we know that the distant particle is same \"state\" (coherence or decoherence), so independiently of any quantum state measured, having detected that \"the wavicle has been measured\", let you send information coded as decoherence time intervals. B - No So, does still have sense to talk about a coherence\/decoherence \"state\" if there is no way to detect it? **EDIT** The wave has his meta-existence as a probability of a detectable particle interaction, but.. if there is no way to know about the system coherence or decoherence -before- it have \"expressed\" his behaviour (it's done always as a particle) then we could say that coherent state is never detectable?, I mean if you have measured a photon by a detector, you already know coherence is lost, but with engangled particles A, B, if you decoherence A, then what happend to B? only we could know that there is no \"random\"(umpredictable) value anymore.. then could we know if a particle is entangled with other? or explicitly could we know if a particle was measured or decoherenced from another entangled partner ? Fantastic way to say it: \"Suposse you are employed to discover if our entire world is being spied from other distant world via entangled particles and the random we see (and its subsecuent evolution) was already known via an engangled model of our world, so.. there is a way to know if someone had got access from a distant engangled particle?, or there is no way to know if any measured value was already determined or is the current experiment where it's first known and fixed (and last years as a lab note in a yellow paper..) or was the experiment just redundant, because the value was already known in an entangled distant experiment (and by a distant world, with beings laughing at our surprise) it doesn't matter if the value is random, anyway is a known value, and they saw it first\""}
{"id":"135047","title":"Moment of inertia as a tensor","text":"A professor at my university briefly stated that moment of inertia is a tensor and can be represented by a $3×3$ matrix. I don't have a good idea of what a tensor is, so I would be grateful if someone could explain how to **intuitively** think of moment of inertia as a tensor."}
{"id":"135041","title":"H-theorem and Boltzmann equation applied to Boltzmann distribution","text":"Using the Boltzmann equation: $$ \\frac{dH}{dt} = \\int_0^{\\infty} dr \\int_0^{\\infty} ds W(r,s)[p_r - p_s][\\ln{p_r} - \\ln{p_s}]$$ and assuming $p_r = e^{-\\beta r}$, the equation looks like $$ \\frac{dH}{dt} = \\beta \\int_0^{\\infty} dr \\int_0^{\\infty} ds W(r,s)[e^{-\\beta r} - e^{-\\beta s}][s - r]$$ I would like to prove that as long as the transition rate satisfies detailed balance ($W(r,s) = W(s,r)$) the system will be in equilibrium (meaning that the whole expression is equal to zero) However, if I assume that $W(r,s)=1$, the integral doesn't seem to converge Just for giggles I attempted $W(r,s)=e^{-(r-s)^2}$, which makes the integral converge, but it is clearly non-zero Any idea what step is required here?"}
{"id":"27562","title":"status of +4\/3 scalar as explanation of $t\\bar t$ asymmetry","text":"One of the early proposals for the Tevatron asymmetry on $t \\bar t$ was a \"fundamental diquark\" with a charge (and hypercharge) +4\/3, either in a triplet or a sextet colour. I am interested on the current status of this proposal: Generically, has some particular model of been favoured in ulterior research? For starters, you can see the short report in JHEP 1109:097,2011, at the end of section 2, points 5 and 6. I only become aware of the survival of this hypothesis after seeing http:\/\/arxiv.org\/abs\/1111.0477 last week."}
{"id":"67134","title":"Imaginary masses","text":"While watching this video, at around 5:00, the man mentions a certain type of particle having imaginary mass. He also says that these kind of particles can go faster then light. But how it is possible that a mass is imaginary? For me, this make no sense. And also what does this even mean? For example, how would you calculate a gravitational force involving particles having imaginary masses? The result of the force will also be imaginary. Or will it interact gravitationally with other objects?"}
{"id":"67133","title":"How can one get the eccentricity of the orbit of the Sun around center of the Milky Way?","text":"How can one get the eccentricity of the orbit of the Sun around center of the Milky Way? Can it be measured?"}
{"id":"64406","title":"Light-Particle Wave Duality","text":"There is a lot of reading to do on this to fully understand it, but without doing that reading is there a short explanation as to why and how light behaves as a wave and a particle?"}
{"id":"25841","title":"How long would it take to scan the visible universe for unique signals?","text":"The article _Amazing rays as star succumbs to dark side_ talks about a very large black hole swallowing up a star. The report goes on to say that the only reason it was discovered was because it shot out a signal directly at our Solar System, by chance. A discovery that was 3.8 billion years in the making, as our Solar System spins around and happened to collide with that signal at the right moment in time. Given that the universe is around 14 billion years old and this was only 3.8 billion years ago (the Earth may not have existed when this event occurred). But I digress. This discovery implies to me that it's very hard to even detect significant events. That is, we have to be pointing our listening equipment at very minute areas of the sky to detect an astronomical event. So even with our available array of detection hardware here on Earth as well as orbital telescopes, how much of the visible sky have we already scanned? And how long will it be until we've \"scanned\" the universe? To help clarify the question, I know that there may be areas of the sky that may be completely void, so perhaps scanning \"the relevant universe\" is a more appropriate scope of this question, but if you can guestimate the whole visible sky, it would be interesting to know."}
{"id":"29177","title":"Using CO2 to air condition a room","text":"I'm trying to determine how much dry ice or liquid nitrogen I would need to cool 3300 cubic feet, about 90,000 liters of air, from about 100F (37.78C or 310K) to about 90F (26.67C or 299.81K). I'm assuming that the air is comprised of about 21% oxygen. How would I go about doing this?"}
{"id":"76898","title":"Wavefront RMS errors, mirror surface roughness and Gaussian beam scattering","text":"What I'm interested is in the scattered power of Gaussian beams reflected from mirrors with a given surface RMS. Usually the surface RMS $\\sigma_{s}$ of a mirror translates in an error for the wavefront $ \\sigma_{w} \\sin{\\theta} = 2 \\sigma_{s}$, where the beam has incidence angle $\\theta$. Now, What I' concerned is specifically, what fraction of power is lost from a Gaussian beam that reflects in a mirror with a given $\\sigma_{s}$ **(Ignoring the portion of beam that is absorbed or transmitted, considering only the light that is scattered)**. It sounds like _maybe_ , this is somewhat related to the Strehl ratio $1 - (\\frac{2 \\pi \\sigma_{w}}{\\lambda})^2$. What I know is that the Strehl ratio gives a qualitative broadening of the diffraction- limited ideal beam. But it is usually applied to incoherent sources, like point sources generating spherical wavefronts. The interesting part here is to ask: if a perfect Gaussian beam, with far- field divergence of $\\frac{\\lambda}{\\pi W_0}$ is incident on a mirror with surface RMS $\\sigma_s$, what portion of the reflected beam will have the same divergence in the far-field? The reflected beam divergence is usually characterized by $\\frac{M^2 \\lambda}{\\pi W_0}$, where $M^2$ is a factor describing the beam quality: **What is the relationship between $M^2$ and the surface errors?** **Important insight** one of the reasons that I'm not quite convinced that Strehl ratio is the only figure of merit for analize the power loss is that a lower-quality beam that is not diffraction limited will have a beam divergence larger than the Gaussian beam. But if the mirror reflects the beam near its Rayleigh range, and tries to focus it at focal length $D$, the Strehl ratio only says that the beam waist will be thicker by an amount $W_0'= MW_0$. **But there is no inmediate reason to conclude that the beam will be any wider at the Rayleigh range at twice the focal length $2D$.** , since the Strehl ratio only concerns itself with the focal plane, which concides with the waist in the case of a Gaussian beam: a lower-quality beam will have a higher divergence, but a thicker waist also means a lower divergence, so both effects seem to cancel each other, leaving the beam width at the Rayleigh range unchanged. If this conclusion is wrong, please explain"}
{"id":"107084","title":"Do Pulsar Stars produce sound?","text":"Can I hear electromagnetic radiation coming from a pulsar star? Or can I hear it if I stand outside it?"}
{"id":"107083","title":"Orbital motion with varying radius","text":"The model of a orbital motion in many Physics textbooks and online resources (e.g., Wikipedia on circular motion) assumes that the orbit is a circle, i.e. the radius is constant and the speed is constant, when deriving the many relationships such as $v = R \\omega$ and $a_{\\text{centripetal}} = R\\omega^2 = \\frac{v^2}{R}$. Would the many relationships hold when the orbit's radius is no longer constant but varies with time, for example? I argue that the many relationships would not hold because the model from which the relationships are derived does not take into account the change in the orbit's radius. If my argument is wrong, what is the logical basis to say that the relationships still hold even when the model does not take into account the change in the orbit's radius? If my argument is right, any pointer to a better model that takes into account the change in the orbit's radius?"}
{"id":"55347","title":"Is the curvature of spacetime invariant? Could it be characterized as the ether?","text":"I'm writing a paper for a Philosophy of Science course about GR\/SR and I'm wondering if I can (1) characterize the curvature of spacetime as invariant and (2) argue that this is what Einstein referred to in 1920 when he said \"space without ether is unthinkable.\" I take (1) from Gauss' proof that the curvature of 2-surfaces has one invariant which seems to be an intrinsic quality of the space (i.e. non- reference-frame-dependent). Shown by: $K=\\frac{(\\nabla_{2}\\nabla_{1}-\\nabla_{1}\\nabla_{2})e_{1},e_{2}}{det(g)}$ where $\\nabla_{i}=\\nabla_{e}$ is the covariant derivative and $g$ is the metric tensor. And (2) from a rather terse paper (which I don't fully understand) found here. I would just like to know if I'm completely off-base as even though I have the math background, my physics knowledge is spotty at best."}
{"id":"74704","title":"Thought experiment-shell theorem","text":"![](http:\/\/i.stack.imgur.com\/p2Al9.png) Consider a uniformly dense perfectly spherical planet. At any point inside the planet, the gravitational attraction is only due to the mass below the point. (red circle) according to shell theorem. * * * Now onto the thought experiment, as per Newtons law of gravitation, _every particle_ attracts _every other particle_ with $\\frac{G m_1 m_2}{r^2}$ By symmetry, the particles in the blue region should nullify their(net) attraction at the point. Similarly, all particles in the green region would have a component of the force along the -ve X-axis. _Even the particles in the green region **but not inside the red circle** would have a small component of attraction along -ve X-axis at the point._( **contradicting shell theorem** ) Again, their Y-axis components would nullify each other by symmetry. The same would hold true by spinning this cross-section along the X-axis to a 3d planet. * * * Now what's wrong with this explanation?"}
{"id":"24719","title":"Compatibility of economics and physics?","text":"This is an open-ended and broad and multi-part question; but I would be willing to accept an answer that simply corrects my misconceptions. I am sorry if it is inappropriate or too soft for this site. Essentially I am wondering how can a financial system generally operate in the presence of the two actual pinnacles of physical theory, General Relativity and Quantum Mechanics? Please forgive or else enjoy the following science-fiction... In the former case, suppose in the future we make contact with an alien civilization on Alpha Centauri which is about four light-years away. Assume even that it is possible to travel at close to the speed of light, so ambassadors from each of our stars can travel back and forth within a lifetime. And suppose that subsequently many economic benefits of the interaction become apparent: perhaps they wish to buy our computers because they are faster than theirs, and we wish to buy their rocket ships because they are more efficient than ours. Soon an interstellar trade is established at a certain exchange rate. Alien corporations are established on both worlds. What are some general strategies to minimize risk in these situations? Given that a single currency is superior to a multi-good barter system here on Earth, is the same true in such an interstellar distributed system with high communication latency? Does the nature of stocks and bonds change in the presence of this latency? For example, does the Black-Scholes formula apply just as well as it does locally? This aspect is quite analogous to foreign trade prior to radio. How did radio influence the theoretical development and applications of economics? I recall reading somewhere that HFT benefits from the placement of data centers at intermediate points (e.g. the mid-Atlantic or mid-Pacific). Is this true, and is there a reference for the existence or effectiveness of any such practice? One tool that comes to my mind is the CAP theorem which seems to place limits on the effectiveness. In a context of distributed systems, one might imagine bartering persistent storage space for serial computation speed (e.g., I have a 386 with a 1TB hard-drive; you have a 3GHz processor with only a floppy disk; and we are connected with a certain latency). On the other hand, Quantum Mechanics suggests the possibility of win-win situations that would not be possible in a classical or purely relativistic universe. Can this somehow mitigate the confusion caused by considering General Relativity? If we place a computer in interstellar space midway between here and Alpha Centauri, that helps, but what if we place a quantum computer there? Would that help even more? Ultimately I am wondering how can financial risk be calculated in the future when our current physical knowledge is exploited in the form of technology?"}
{"id":"130867","title":"Why does reparameterisation invariance lead to gauge-fixing?","text":"In Becker, Becker and Schwarz, the point particle action is given in terms of an auxiliary field $e(\\tau)$ as: \\begin{align} \\tilde{S}_0 = \\frac{1}{2}\\int \\,d\\tau \\left(e^{-1}\\dot{X}^2 - m^2e\\right) \\end{align} It is then shown that under infinitesimal reparametrizations of $\\tau$, the action is unchanged. This allows us to pick a gauge, in particular $e(\\tau) = 1$. I'm not sure I'm understanding this right, but I have a few issues with this.   1. Doesn't this assume that $e$ takes the value $1$ somewhere?   2. Although $\\tilde{S}_0$ (sorry, not $e(\\tau)$) may be reparametrization invariant, I don't see how you could pick a reparametrization that can leave $e$ constant. Such a reparametrization $\\tau'(\\tau)$ would need to map all $\\tau$ to a constant, but then $\\frac{d\\tau'}{d\\tau} = 0$, which can't be the case.   3. Invariance is only under infinitesimal transformations. This is related to 2: how do we know that an infinitesimal reparametrisation could make $e$ constant?"}
{"id":"110957","title":"Hamiltonian flow is volume preserving","text":"I was reading about advantage of Hamiltonian over Lagrangian. One of the advantage is \"Hamiltonian flow is volume preserving\". Can anyone help me to understand this? Means what is advantage of preservation of volume vs volume is not preserve?"}
{"id":"73066","title":"Caimir effect regularization for every divergent sum or series","text":"can we use the tools of renormalization of casimir effect to get finite results for any divergent series in QFT ?? for example let be the divergent series $ \\sum_{n=1}^{\\infty}n^{l} $ for positive 'l' then instead of the sum we interpretate this as the difference $$ \\sum_{n=1}^{\\infty}n^{l}e^{-n\\epsilon}-\\int_{0}^{\\infty}dtt^{l}e^{-t\\epsilon}=finite$$ this is made to evaluate the infinite sums of the vaccuum energy in casimir effect but could it be done for any divergent series??"}
{"id":"22001","title":"At what angle does a single atom \"reflect\" a single photon?","text":"Does this question make sense in the quantum world? Imagining a single photon (wave packet?) interacting with a single atom (its electrons etc) how do we currently describe\/define the emitted photon in terms of its direction in relation to the incoming photon? Now \"scaling up\" to a surface of atoms actually reflecting \"light\" according to the simple reflection rules like angle-in equals angle-out how do we manage to explain this effect in terms of the quantum world? How comes the probabilities work out for the out-going angle depending on the incoming- angle?"}
{"id":"77769","title":"How can you prove generalised Jacobi identity?","text":"This $$[A, [B,C]] + [C, [A,B]] + [B, [C,A]] = 0$$ This proof is not wanted, since there is an attachment of some variables. **How can you prove the equation?**"}
{"id":"77763","title":"What is the experimental evidence for quantization of lattice vibrations?","text":"Is there any direct experiment to show the quantization of lattice vibrations?"}
{"id":"74490","title":"Can water exist as liquid in a vacuum chamber at room temperature?","text":"I know boiling point at vacuum is far below the room temperature. But I still wondering if the humidity reaches the saturated humidity (100% relative humidity), will the vapor condense to liquid, although it is probably going to evaporate again soon?"}
{"id":"74494","title":"Difference between parallel and series capacitors","text":"Basically, I can't figure out that whether a capacitor is connected in series or parallel. The same problem occurs while solving for resistors. Can someone help me understand this difference conceptually?"}
{"id":"74499","title":"What does one need to learn to become a experimental particles physicist?","text":"I know one needs E&M, classical, quantum, statmech at the graduate level. But what more does one need? Are quantum field theory, relativistic QM, advanced math needed? or just an advanced course in elementary particles on top of basics suffice? Please bear in mind that the intended profession is an experimenter and not a theorist."}
{"id":"98466","title":"Radians to Rotate Earth to Match ECI Lat\/Lon with ECEF Lat\/Lon","text":"I am attempting to model GPS Satellite positions on the globe for a set of ephemerides. I have a verified set of ECI ( http:\/\/en.wikipedia.org\/wiki\/Earth-centered_inertial ) XYZ Coordinates and a verified set of ECEF ( http:\/\/en.wikipedia.org\/wiki\/ECEF ) XYZ coordinates.               For example ( m ):     ECEF X 4076514.50     ECEF Y 14673598.00     ECEF Z -21793674.00          ECI X 8004604.50     ECI Y 12956032.00     ECI Z -21793674.00      Sometimes the signs of the ECI X and Y coordinates are opposite of the signs for the ECEF X and Y coordinates. When I plot these locations on a globe and, obviously they do not match unless you rotate the earth by some number of radians. I can see that if you rotate the earth, the ECI coordinates follow a groundtrack that is consistent with the ECEF groundtrack, but with my current code, it is offset by different varying amounts, depending on the time of day I plot it. Here is my code:               \/\/ The radians earth rotates in 1 second     double earthRotation = 0.000072921151467;          \/\/ Do something different depending on the switch position     if ( self.shouldAnimate == true ) {              \/\/ The seconds since we started rotating         double newRotationMultiplier = [self.eciDate timeIntervalSinceDate:self.lastEciDate];              \/\/ Seconds           \/\/ Rad\/Sec        \/\/ Multiplier         self.timeSinceOpenGlStarted = newRotationMultiplier * earthRotation;         matrixToRotate = GLKMatrix4Rotate(matrixToRotate, self.timeSinceOpenGlStarted, 0.0, 0.0, 1.0);              \/\/ Latch it so that when we stop rotating, it doesn't revert         self.effect.transform.modelviewMatrix = matrixToRotate;          }      How do I determine the radians to rotate the earth to show an ECI positions for a time to be the same location on the rotated globe ( in terms of latitude and longitude )? That is, how do I calculate for any UTC time in a day, how much to rotate the earth about the Z ( \"Up\" ) axis so that the ECI coordinates agree with the ECEF coordinates and the satellites appear to be in the same position over my 3D globe? UPDATE: I got it to work - thanks to the answer below! Here is the code:                   \/\/ Get the NSDate components         NSCalendar *currentCalendar = [NSCalendar currentCalendar];         [currentCalendar setTimeZone:[NSTimeZone timeZoneWithAbbreviation:@\"UTC\"]];         NSDateComponents *components = [currentCalendar components:NSCalendarUnitDay | NSCalendarUnitMonth | NSCalendarUnitYear | NSHourCalendarUnit | NSMinuteCalendarUnit | NSSecondCalendarUnit fromDate:self.eciDate];         [components setTimeZone:[NSTimeZone timeZoneWithAbbreviation:@\"UTC\"]];         NSInteger hour = [components hour];         NSInteger minute = [components minute];         NSInteger second = [components second];              \/\/ Get UTC in terms of decimal hours         float utc = hour + (float)minute\/60.0f + (float)second\/3600.0f; \/\/ UTC is expressed in hours as a decimal number              \/\/NSLog(@\"UTC: %f H %f M %f S %f\",utc,(float)hour,(float)minute\/60.0f ,(float)second\/3600.0f);              \/\/ Get the seconds since the last vernal equinox         double secondsSinceVernalEquinox = [self.eciDate timeIntervalSinceDate:appDelegate.lastVernalEquinox];              \/\/ Do the math         \/\/ http:\/\/physics.stackexchange.com\/questions\/98466\/radians-to-rotate-earth-to-match-eci-lat-lon-with-ecef-lat-lon         double d = secondsSinceVernalEquinox  \/ ( 60.0 * 60.0 * 24.0 ); \/\/ Give decimal days         double p = 365.242187; \/\/ Length of tropical year         double result = ( M_PI * 2 \/ 24 ) * ( utc - 12.0 ) + ( 2 * M_PI * d  \/ p );              \/\/NSLog(@\"Radians = %f\",result);              \/\/ Seconds           \/\/ Rad\/Sec        \/\/ Multiplier         self.timeSinceOpenGlStarted = result;         matrixToRotate = GLKMatrix4Rotate(matrixToRotate, self.timeSinceOpenGlStarted, 0.0, 0.0, 1.0);              \/\/ Latch it so that when we stop rotating, it doesn't revert         self.effect.transform.modelviewMatrix = matrixToRotate;"}
{"id":"75723","title":"What happens (in terms of pressure) when there is a leakage in a horizontal pipe?","text":"Consider a horizontal pipe that has water being pumped continuously through it, with a leak at a given point. What will the leak's effect be on the pressure in the pipe, both above and below it in the flow?"}
{"id":"98463","title":"induced emf when a wire or coil travel through a magnetic field","text":"I've recently learned that If you move a conductor through a magnetic field, an emf is induced across the ends of the conductor: $E = BLv $ I've also been told that the same is true for a coil but that the equation is: $E = BLvN$ And that for a coil, there is only an emf induced when the current is entering or leaving the magnetic field as once the coil is inside, there is no change in magnetic flux linkage. I've been told that this is not the case with just a wire moving through a magnetic field, but I haven't been told why and I can't seem to figure it out. I've thought about it, and I think it perhaps has something to do with the area covered by the wire moving at speed $v$ per unit time compared to the area covered by a coil moving at the same speed $v$ in the same unit time. Unfortunately, I can't quite work it out. Thank you."}
{"id":"80077","title":"Can a particle with non-zero angular momentum pass through the center of a spherical potential?","text":"Suppose you have a particle of mass $m$ moving in a potential $V(r) = -\\frac{k}{r^2}$, with $r^2 = x^2+y^2+z^2$ and $k > 0$. Since the angular momentum $l$ is conserved, the particle will move in a plane: Let's use polar coordinates $(r, \\theta)$. The expression for the energy is then: $$ E = \\frac12m\\dot{r}^2+\\left(\\frac{l^2}{2m}-k\\right)\\frac1{r^2} $$ Suppose $l < \\sqrt{2mk}$. Then the effective potential is of the form $V_{eff} = -\\frac{a}{r^2}$ with $a > 0$. This implies that no matter what the initial energy $E$ of the particle is, it can and will at some point pass through $r=0$. Indeed, if you find the trajectory $r = r(\\theta)$, you get things that look like either $r = \\cos \\theta$ or like $r = \\sinh \\theta$, which are $0$ at some point. I don't understand how this can be. When the particle is very close to $r=0$, its momentum has to pointed towards $r=0$, but this would imply that $\\vec{r}$ and $\\vec{p}$ are parallel, and so $l$ would be $0$! How can this be? Doesn't conservation of angular momentum prevent the particle from reaching the center, no matter how strong the potential is?"}
{"id":"122612","title":"Electric field represented by parallel curves","text":"In a region of space, I have an electric field represented by parallel curves as shown in the diagram. ![enter image description here](http:\/\/i.stack.imgur.com\/GnB8T.jpg) The curves are equally spaced. We know that a uniform electric field can be represented by lines that are parallel and evenly spaced. Can this definition of uniform electric field be extended to parallel and equally spaced curves?"}
{"id":"100","title":"How is it possible to calculate the energy liberated by a given fission process?","text":"How is it possible to calculate the energy liberated by a given fission process? For example, in the fission of a $^{235}$U induced by capturing a neutron?"}
{"id":"101","title":"Production of antiproton at the Tevatron","text":"The Tevatron is a proton-antiproton collider; it collides a beam of protons against a beam of antiproton. I can understand how we obtain the protons, but for the antiprotons ? How are they produced ?"}
{"id":"36336","title":"Physics and what it means","text":"I got into a debate with a friend about the meaning of physics and its purpose, he is the sort who will test you and if you get it wrong it somehow gratifies his own self-reflection and self-worth. That aside, I have loved physics since I got into it when I was at school, I had a great teacher who fostered that love and helped grow my appreciation for physics and ever since I have said to myself that I would one day go do my degree in physics. At the moment I have just finished my degree in computer security and forensics, I plan to study physics part-time hopefully in the next 5 years after I settle into permanent work in the software engineering field. Anyway the conversation started with a question from him asking what temperature water boils at and what temperature does water freeze at. For the life of me I couldn’t remember, even as a general question most would know the answer, well that’s what we thought. For him this was a big disgrace and he continued to laugh and \"then\" call a friend to verify that I am being moronic. Of course he phoned our friend and pitched the same question, of which he neither knew the boiling point nor the freezing point of water. Regardless, for him he classed this as \"true physics\", but to me it could have been any random number for all I cared. What he was trying to prelude to and was expecting as an answer was \"it depends\" and the dependency was altitude (from a documentary he watched). Of course I never got past that stage and all it became was a ridicule session. For me it didn’t matter I thought the true physics was understanding why water boiled in the first place. How exciting molecules or increasing the energy within water gradually increases its temperature and when the molecules become extremely excited water gets to its boiling temperature. Understanding this allows us to understand all the states of water (phases), from a solid (ice) to a liquid or to a gas (vapour, altho vapour is essentially two phases) in one, the phase change occurs via nucleation and growth - small liquid drops spontaneously form and evaporate, but if a drop is larger than some critical radius, it will continue to grow ('nucleates'). This is the most common phenomenon of a gas-liquid (or liquid-solid) phase transition. Of course I discussed this with him but he argued blind that \"knowing\" the boiling point of water was \"physics\" but to me that was essentially just a factoid. That the boiling point of water did not matter but understanding the process behind it did. So I ask those of you in the community who was wrong or what physics trully is. If we take the standard definition of Physics it is a natural science that involves the study of matter and its motion through space and time, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves. For me knowing the boiling temperature of water does not equate to understanding how water as a liquid, boils. Or the transfer of energy to change a liquids state occurs. It is simply just a consequence of the action or a \"factoid\" of the means in which we can reliably get the same testable results."}
{"id":"36332","title":"Power, Force Velocity","text":"In any case, if the force and power is constant , does velocity also need to be constant? If no, can you tell me a case. Further can this be extended for non-rigid bodies."}
{"id":"87361","title":"Allowed interactions for the neutral weak vector boson","text":"Just wanted to double check a couple of things for my own sanity! I am looking at scattering amplitudes for 2 partons going to two partons with the emission of a Z boson (eventually decaying to e+ e-). I have used madgraph to plot the possible tree level Feynman diagrams for a couple of processes and I'm a bit surprised by the output...See link below for the output diagrams. Initially I thought that diagram 2 cause of the virtual anti-down quark (displayed as d~). It looks like a d quark has radiated a Z boson which changed the d quark to an anti-down quark which in turn radiated a gluon which changed it back to a down quark. I am 99% sure this is not allowed as the Z can't change q's to qbar's. An alternative way of thinking about it is that the u quark radiates a gluon which then _pair produces_ a d-dbar pair, one of which is final state particle but the other one goes and annihilates the incoming d quark to give a Z...But I am not certain that this is allowed either! ![Feynmann diagrams](http:\/\/i.stack.imgur.com\/x5aYE.gif)"}
{"id":"87367","title":"Shining object in the sky resembling a star","text":"From last 2 days I am observing a shiny star like shining object in the sky in almost same place at same time. Is there any possibility that it could be a satellite? Will satellite shine like a star in the sky due to any kind of reflection from the sun. I see it in the evening and in white light. I don't know how to verify but I am very curious to know what it is."}
{"id":"46097","title":"What is the origin of flavor?","text":"> **Possible Duplicate:**   >  Origin of lepton\/quark generations? In the standard model (and in nature), Fermions appear in different generations, or flavors. Besides up and down quarks and electrons, there are strange, charm, muon, tau, etc.. I am wondering if there is any theoretical explanation for the existence of heavier generations. What I'd like to know is especially: _Why_ are there different, heavier copies of $e$, $u$, $d$? What makes these particles different from their lighter relatives _besides_ their mass? What is flavour, where does it come from, and why is it approximately conserved (at least for charged leptons)? * * * Note: I'm posting this question as a follow up on my previous one _\"Origin of lepton\/quark generations?\"_, which didn't reallty get the type of answers I was expecting. Sorry if the old question caused some confusion. To be clear, I'm _not_ asking:   * \"Why must there be _three_ generations (and not 2 or 4)?\" -> e.g. CP Violation   * \" _How do we know_ that there are 3 generations?\" -> Experiment   * \"Why do we _organize_ particles in this manner?\" -> Quantum numbers But rather: What possible theoretical explanation is there that our universe exibits flavor quantum numbers, and maybe how are they are related to particle masses? Conceivable (although probably wrong) answers would be along the line of:   * \"Because electrons and up quarks have substructure, and muons and charm quarks are the first excited states.\"   * \"In XY theory, $e$, $\\mu$ and $\\tau$ are actually the same at first order, but the thingamajig symmetry is broken at low energies.\"   * \"It is explained by string theory, if your vacuum has such-and-such properties it comes out naturally.\"   * \"It's accidential in the SM and just experimentally observed, but in SU(...) GUTs it is a prediction.\""}
{"id":"41343","title":"What's the difference between background field and dynamical gauge field?","text":"Dynamical gauge fields are assumed to be able to respond to sources. What's the difference in the _Lagrangians_ between a background field and a dynamical field?"}
{"id":"83877","title":"Why does $\\langle a_-\\alpha|\\alpha\\rangle = \\alpha $for harmonic oscillator","text":"Why does $\\langle a_-\\alpha|\\alpha\\rangle = \\alpha$ doesn't the ladder operator lower the $\\alpha$ so that it became $\\sqrt{\\alpha}*\\delta_{\\alpha, \\alpha_{-1}}$ Maybe its because that \\alpha can be imaginary? In that case how would you prove the above relation?"}
{"id":"133747","title":"Degenerate vibrating modes of a metal plate","text":"My question is based around this video: https:\/\/www.youtube.com\/watch?v=1yaqUI4b974 The video shows sand settling in vibrational nodes of a plate at certain resonance frequencies. It appears that all the patterns are radially symmetric however at 5:15 I see a pattern that is bilaterally symmetric. My question is why this pattern occurs horizontally instead of vertically? For a rectangular plate I understand that the two situations would occur at different frequencies, but what about a square plate? Is this a random occurrence?"}
{"id":"123564","title":"What is the error on measuring the phase of a sine wave?","text":"Let's say I have a wave, with frequency $\\omega$ and phase $\\phi$, of the form: $$y(t) = 1 + A \\sin(\\omega t+\\phi)$$ where $A < 1$. I have measured this wave N times, and we can assume these measurements are evenly spaced over one period. Each of those measurements has an uncertainty in the y-direction of $\\sigma_y$, and no uncertainty in the time of measurement (for practical purposes). How well can I measure the phase of this sine wave? I've tried constructing a Fisher matrix for the problem, but Mathematica thinks it's singular, and I can't see how to invert it either."}
{"id":"18920","title":"Confused on newton's second law being invariant under relaitivity","text":"I am a math student with some interests in physics. I picked up a book called \"A First Course in General Relativity\", and I am confused on the second page. I am assuming by notation or convention. The chapter is on special relativity, and at this point they are just talking about how measurements of velocity are invariant by a constant. That is $v'(t) = v(t) - V$, where $v(t)$ is a measurement by one observer and $v'(t)$ is a measurement by another observer whose relative velocity to the original is $V$. Then it says Newton's second law is unaffected by this replacement. It offers as an explanation, $$a' \\- dv'\/dt = d(v - V)\/dt = dv\/dt = a.$$ I am confused in how this explains anything. Also as I read the notation, shouldn't $a' = dv'\/dt$? So $a' \\- dv'\/dt = 0$. Also the first equality is confusing, since $v' = v - V$, I thought, so using that replacement I get $$a' \\- dv'\/dt = a' \\- d(v - V)\/dt.$$ And I don't see where they go from there. I assume I am just misunderstanding what is meant by a particular variable. Can anyone help shed some light on my confusion on this point? Thanks. **EDIT** I think I am being confused by a typo, replacing the $=$ with $-$ seems to make things make sense. I probably should have seen that. $$a' = dv'\/dt = d(v - V)\/dt = dv\/dt = a.$$"}
{"id":"7674","title":"How to simulate a crashing wave?","text":"I'd like to create a very rough animation of a wave crashing on a beach. I'm guessing it would have to be a particle simulator, where you code in the forces between the particles and then integrate forward in time. I've done similar things, like simulations of charged particles, but there the forces are pretty straightforward, whereas here I guess I'd have to account for 1) tides 2) gravity 3) water surface tension. These seem like widely different forces acting on different scales. I don't even know where to begin. Any hints or links to papers related to this topic?"}
{"id":"132833","title":"How do charged particles interact?","text":"You'll have to forgive me if this question is too wrapped up in \"classical\" thinking. I've read that electrons and protons interact by trading photons, but this only raises more questions. What determines the energy and direction of the emitted photon? How often can a particle emit a photon? How often can a particle absorb a photon? Can one particle emit and absorb multiple photons at once? Where does the energy to emit a photon come from? Is the destination of the photon somehow pre-determined or is the photon simply emitted in the hopes of being absorbed?"}
{"id":"80597","title":"Quantum Thrusters and Warp Drives","text":"Any reservations about the potential of this (given in the link below\/title above)? Does it seem like a helpless attempt or something which might have the potential of developing into something real? http:\/\/news.discovery.com\/space\/quantum-thruster-warp-drive-physics-130823.htm"}
{"id":"47693","title":"Would an object hit the ground faster when it gets launched vertically upwards or downwards?","text":"A while ago I had a discussion with someone about the speed of an object when it hits the ground. When does an object hit the ground faster:   * When you launch the object to the sky so it can fall down;   * Or when you launch the object to the ground so it doesn't have to deal with much air resistance. The attraction “The Booster” has an arm that has a constant speed (about 120 km\/h) is about 80M high and undergoes full revolutions. ![enter image description here](http:\/\/i.stack.imgur.com\/iAbtQ.png) In this situation the cart gets disattached at 90 degrees. ![enter image description here](http:\/\/i.stack.imgur.com\/gLB1j.png) I think that the cart hits the ground with a higher speed when it's launched to the ground because of:   * The air resistance, when you launch it to the sky it has a longer path to travel, so it needs to travel through much more air.    * The cart has the same initial speed, when you travel a shorter path there will be less to slow the cart down;   * Nothing provides the cart with additional energy;   * I guess that the \"final gravity\" is the same."}
{"id":"13832","title":"Integrals of Motion","text":"Landau writes in chapter 2 of his Mechanics book \"The number of independent integrals of motion for a closed mechanical system with s degrees of freedom is 2s-1\". Then he goes on to show how -- \"Since the equations of motion for a closed system do not involve time explicitly, the choice of the origin of time is entirely arbitrary, and one of the arbitrary constants in the solution of the equations can always be taken as an additive constant t0 in time. Eliminating t + t0 from the 2s functions, we can express the 2s-1 arbitrary constants as functions of q and q’ (generalized co-ordinates and velocities) and these functions will be the integrals of the motions.\" Could someone elaborate?"}
{"id":"85980","title":"tracking a satellite","text":"The following article says that because the GOCE satellite has a polar orbit, it is impossible to track where it will land. > \"The satellite is one of the few satellites in a Polar Orbit. Consequently, > it could land almost anywhere,\" Mark Hopkins, chair of the National Space > Society's executive committee told FoxNews.com. I was looking for an explanation based on the physics of the orbit, as I am not familiar with orbital mechanics. http:\/\/www.foxnews.com\/science\/2013\/11\/08\/track-europe-falling-2000-pound- satellite-in-real-time\/"}
{"id":"5627","title":"Light bending by black holes","text":"In the center of our milky way, it is assumed that a black hole exists with a mass of $\\approx 4\\times 10^6$ times our sun's mass. How much light bending (in degrees) would arise for stars that are in perspective near the position of this black hole in the sky? Could the mass of a black holes become so great that it actually distorts the area in the sky we see around it (through telescopes)?"}
{"id":"38352","title":"Am I making the right assumption about a jump discontinuity in the acceleration?","text":"> A train heads from Station A to Station B, 4 km away. If the train begins at > rest and ends at rest, and its maximum acceleration is 1.5 m\/s^2 and maximum > deceleration is -6 m\/s^2, what's the least time required to complete the > journey? **My attempt at a solution** : I assumed the acceleration function, $a(t)$, has a jump discontinuity at some time, say $t_1$, where it jumps from 1.5 to -6. Under this assumption, it's easy to use the velocity boundary conditions to get the total time $t_f$ is $\\frac{5}{4}t_1$, because the velocity $v(t)$ for $t>t_1$ is just $1.5t_1-6(t-t_1)=7.5t_1-6t$. Setting this equal to 0 at $t_f$ gives this result. The position $x(t)$ for $t\\leq t_1$ is just $\\frac{1}{2}(1.5)t^2$ and for $t>t_1$ it's just $\\frac{1}{2}(1.5)t_1^2+(1.5t_1)(t-t_1)-\\frac{1}{2}(6)(t-t_1)^2$. Using the fact that $x(t_f)=4000$, and substituting $t_f=\\frac{5}{4}t_1$, I find $t_f\\approx 86.06$ seconds. But my teacher's solution says the minimum time is $81.65$ seconds. Who's wrong? Is my assumption about the jump discontinuity of $a(t)$ what's wrong?"}
{"id":"82456","title":"Why removing one slab in Michelson-Morley experiment causes an elliptical fringe pattern?","text":"I performed Morley experiment using He-Ne laser. Two glass slabs were there and circular fringes were formed but when I removed one slab elliptical fringes were there (and less intense fringes). Why?   In general is there any benefit of using two slabs?"}
{"id":"19383","title":"Would the north poles of two magnets repel each other if a weaker south pole was inserted between them?","text":"My son asked me this question and I was stumped - my intuition says that is the south pole was strong enough the attraction between the north and south poles would outweigh the repulsion between the two north poles - but how strong would the south pole need to be in this case?"}
{"id":"131872","title":"Ideas for high school physics paper, including simulations","text":"I have to write, with two friends, a simulation based physics paper at school. The problem is, that will I am quite familiar with the concepts of calculus and algebra, my friends aren't. So, we could find a good topic. The task is, to write a paper about a physical topic, with the use of a selfmade program. The development of the program shouldn't be a problem at all. we have a good programmer. Any suggestions, for a topic that involves only pre-calculus math or only just some calculus?"}
{"id":"131654","title":"How to calculate the electrostatic force applied on an object that is along the axis of a charged tube's surface?","text":"Let's say there is a charged tube(a cylinder with no top and bottom) with radius r and length l, charge q1 which also made out of insulating material. And also if there is an object with charge q2 along the axis of the surface of the tube, how can I calculate the force applied on this object?(the distance between tube and object is d) Here is my calculations: From the Gauss' law: $$ \\phi_E= \\frac Q{\\epsilon_0} $$ I took cylinder as Gaussian surface. $$ E \\times 2\\pi dl= \\frac {\\lambda \\bullet l}{\\epsilon_0} $$ $$ E = \\frac {\\lambda}{\\epsilon_0\\times 2\\pi d} $$ or $$ E = \\frac {q_1}{\\epsilon_0\\times 2\\pi dl} $$ Thus we found the electric field applied to a point d away from the tube.To calculate the force we use $F=qE$. And finally the electrostatic force is: $$ \\frac {q_1q_2}{\\epsilon_0\\times 2\\pi dl} $$ However what I found is the force applied to an object d away from the tube but perpendicular to the surface of the tube. Would this calculation be valid for an object which is along the axis of the surface of the tube? P.S: I know I asked that question but I couldn't get any answers so I hope this question won't be closed. Thanks in advance."}
{"id":"39001","title":"How to obtain Dirac equation from Schrodinger equation and special relativity?","text":"I'm reading the Wikipedia page for the Dirac equation: > _The Dirac equation is superficially similar to the Schrödinger equation for > a free massive particle:_ > > A) $-\\frac{\\hbar^2}{2m}\\nabla^2\\phi = i\\hbar\\frac{\\partial}{\\partial > t}\\phi.$ > > _The left side represents the square of the momentum operator divided by > twice the mass, which is the non-relativistic kinetic energy. Because > relativity treats space and time as a whole, a relativistic generalization > of this equation requires that space and time derivatives must enter > symmetrically, as they do in the Maxwell equations that govern the behavior > of light — the equations must be differentially of the same order in space > and time. In relativity, the momentum and the energy are the space and time > parts of a space-time vector, the 4-momentum, and they are related by the > relativistically invariant relation_ > > B) $\\frac{E^2}{c^2} - p^2 = m^2c^2$ > > _which says that the length of this vector is proportional to the rest mass > m. Substituting the operator equivalents of the energy and momentum from the > Schrödinger theory, we get an equation describing the propagation of waves, > constructed from relativistically invariant objects,_ > > C) $\\left(\\nabla^2 - \\frac{1}{c^2}\\frac{\\partial^2}{\\partial t^2}\\right)\\phi > = \\frac{m^2c^2}{\\hbar^2}\\phi$ I am not sure how the equation A and B lead to equation C. It seems that it is related to substituting special relativity value into quantum mechanics operators, but I just keep failing to get a result..."}
{"id":"80582","title":"Physical meaning of Impedance","text":"So I have been thinking about the way impedance is defined for electrical systems, and the way it is derived. Even after looking through some websites, I cannot seem to grasp something, which every website I visited seemed to skip on. In an ideal resistor, if we take the ratio of the voltage to the current at ANY time, the ratio is a constant. This ratio is then the resistance (also the impedance) of the resistor. I have been interpreting this as the opposition the resistor bears to the current flow through it upon application of a voltage. I thought of it like a constant that relates the otherwise-would-be- infinite current to the applied voltage. Coming to a capacitor, there seems to be some sort of change in the definition of impedance. The impedance for a capacitor is no longer the ratio of voltage to the current, but the ratio of the complex voltage to the complex current. In other words, mathematically it is equivalent to $$\\dfrac{1}{i\\omega C},$$ and not $$\\dfrac{\\tan(\\omega t)}{\\omega C},$$ for a capacitor with capacitance C at a frequency w. I am finding this difficult to interpret because the ratio of the voltage to current is definitely varying in time. How can the analogy of a resistor's impedance be applied to the impedance of a capacitor when the ratio of the voltage across it to the current through it is changing (it's even 0 or undefined sometimes). Why is the magnitude of the quantity $$\\dfrac{1}{i\\omega C}$$ now the resistance of the capacitor circuit? What happened to the plain old voltage to current ratio? Have books been skipping this subtlety, or I am just not understanding something very simple? Clarification: $$ if \\hspace{.5 pc} v(t) = V_m \\sin (\\omega t), \\hspace{0.5 pc} then \\dfrac{d V(t)}{dt} = \\omega V_m \\cos (\\omega t) $$ $$ so \\hspace{0.5 pc}the\\hspace{0.5 pc}ratio\\hspace{0.5 pc} \\dfrac{v(t)}{i(t)} = \\dfrac{V_m \\sin (\\omega t)}{C\\omega V_m \\cos (\\omega t)} = \\dfrac{\\tan (\\omega t)}{C\\omega} $$ $$ In \\hspace{0.5 pc}phasor \\hspace{0.5 pc}notation, \\hspace{0.5 pc}this \\hspace{0.5 pc} ratio\\hspace{0.5 pc} seems\\hspace{0.5 pc} to\\hspace{0.5 pc} be\\hspace{0.5 pc}different \\hspace{0.5 pc}in \\hspace{0.5 pc}that: \\\\\\ \\dfrac{\\mathbf{V_c}}{\\mathbf{I_c}} = \\dfrac{1}{Cj\\omega} = -\\dfrac{1}{C \\omega }j $$ $$ and \\hspace{0.5 pc} \\Re\\\\{-\\dfrac{1}{C \\omega }j \\\\} = 0$$ $$but \\\\\\ |\\dfrac{\\mathbf{V_c}}{\\mathbf{I_c}}| = \\dfrac{1}{C \\omega } $$ So my question is why does the magnitude of the ratio of the complex voltage to the complex current now suddenly carries a physical meaning (if my understanding is correct, it is the resistance which can be measured in ohms, just like in the resistor). Also, why is going from the complex domain to real domain not possible for this ratio, because clearly the real part of the complex impedance is 0 (@Alfred Centauri mentions that the impedance is not itself a phasor). I understand that the math works, but the thing that is not making sense to me is this ratio of two complex quantities and the apparent emergence in its physical meaning."}
{"id":"19963","title":"How can we represent the motion of a particle in 2D space using Lagrange's equations?","text":"Can we represent the motion of a particle in 2D space using Lagrange's equations? This is what I tried. Please tell me what is wrong? Consider a particle on a plane have the co-ordinates $(x,y)$ with a velocity $v$ and mass $m$. Now $v=\\dot{x}+\\dot{y}$ where $\\dot{x}$ and $\\dot{y}$ are the derivatives with respect to time. According to Lagrange's equation, $L=\\frac{1}{2}m(\\dot{x}+\\dot{y})^2-mgy$ $\\frac{{\\partial}{L}}{\\partial{y}}=-mg$ ---(1) $\\frac{{\\partial}{L}}{\\partial{\\dot{y}}}=m(\\dot{x}+\\dot{y})$ Therefore $\\frac {\\mathrm d t}{\\mathrm d t}(\\frac{{\\partial}{L}}{\\partial{\\dot{y}}})=\\frac{{\\partial}{L}}{\\partial{\\dot{y}}}=m(\\ddot{x}+\\ddot{y})$ Now $\\ddot{x}$ and $\\ddot{y}$ are the vertical & horizontal components of the acceleration $a$ of the particle. So let $\\ddot{x}=a_x$ and $\\ddot{y}=a_y$ Hence $\\frac {\\mathrm d t}{\\mathrm d t}(\\frac{{\\partial}{L}}{\\partial{\\dot{y}}})=m(a_x+a_y)$ ---(2) From Lagrange's equation, Eq (1)= Eq(2) => $-mg=m(a_x+a_y)$ $-g=a_x+a_y$ Does the above equation make any sense? I tried to apply it to a real problem and it gave me a wrong result. What is wrong with my working? Thanks in advance. ## **Update:** Thanks a lots guys. Yes, the expression for $v$ was wrong. When I calculated with the correct expression I got the result $a_y=-g$ What does it mean? Does it mean that the Y component of the acceleration is always equal to acceleration due to gravity? And can someone explain how Lagrangian mechanics can be used to solve simple mechanical problems? When I tried applying it to problems all I got were few known results & other nonsensical ones (like the one in this question. Probably due to my erroneous application of the Lagrange.) PS: Will someone explain why this question was downvoted?"}
{"id":"100005","title":"Vertex for quartic interaction of complex scalar multiplet","text":"Since I'm new to QFT and I tend to do a lot of errors during calculations, I would like you to tell me if I got the four-point vertex of the quartic interaction with a multiplet of complex scalar fields right: $$ L = (\\partial_\\mu \\Phi)^\\dagger (\\partial^\\mu\\Phi) - m^2\\Phi^\\dagger\\Phi-\\lambda(\\Phi^\\dagger\\Phi)(\\Phi^\\dagger\\Phi),$$ where $\\Phi= (\\phi_1,\\ldots,\\phi_N)$ and $\\phi_i$ are complex scalar fields. I have $$-2i\\lambda ( \\delta^{ik}\\delta^{jh} + \\delta^{ih}\\delta^{jk}),$$ where $i,j$ are the indices of the two $\\Phi^\\dagger$, while $k,h$ are the indices of the two $\\Phi$. Is it correct?"}
{"id":"64706","title":"Air pressure in balloon","text":"I have to calculate the air pressure inside of an hot air balloon. After some searching I found out that I can use the ideal gas law: PV = nRT (from Wikipedia) So to get the pressure in the balloon I would have to know n, which is the amount of gas in moles. For that amount I'm currently using the pressure (kg\/m3) * the volume. So for the pressure I need the pressure which of course doesn't work. Is there a way to calculate this? Right now I do know the normal air pressure (outside of the balloon) volume of the balloon etc. etc., but I can't know the weight of the amount of air in the balloon, because I use the pressure to calculate it. One thing to note is that it isn't an actual hot air balloon. It's just one made up from physics formulas in a program that calculates them every 0.1 seconds (for example) for me. Is there some thing I'm missing?"}
{"id":"69551","title":"Viscous force in a conductor","text":"I was told that in a conductor there is a dissipative force and if Ohm's Law is valid for a component, the component has to be dissipative. I'm trying to prove it, but I have some problems. I know that for a viscous force: $$\\tag{1}\\mathbf {F}=m\\mathbf {a}=-\\gamma \\mathbf {v}$$ so $$\\tag{2}\\mathbf {a}=\\frac{-\\gamma \\mathbf {v}}{m}$$ and in a conductor (considering the motion of electrons): $$\\tag{3}-e\\mathbf {E}=m\\mathbf {a}$$ so $$\\tag{4}\\mathbf {a}=\\frac{-e \\mathbf {E}}{m} $$ Considering the drift velociy given by $$\\tag{5}\\mathbf {v_d}=\\frac {-e \\tau}{m}\\mathbf {E}$$ I could find $\\mathbf {E}$, put it in (4) and obtain an acceleration in function of $\\mathbf {v}$. So I could say that I'm treating a viscous force. But I'm not sure that this is a correct way... How could I verify that if Ohm's Law is valid for a component, the component has to be dissipative?"}
{"id":"95788","title":"$\\Delta T$, temperature units. How to use fahrenheit in this formula","text":"To calculate the power in a fluid with a flow, the following formula is applied: $$P = \\Delta T \\rho c_p \\times flow$$ $\\Delta T$ can be entered as both Kelvin and Celsius. But what if I want to use Fahrenheit? Is there any formula for this? Reason is I'm making a web application that calculate some stuff. I have full unit support. I live in Europe so we always use Celsius or Kelvin. But, are americans using fahrenheit for calculations in general? If not, then maybe I can skip it all together."}
{"id":"9242","title":"Definition and difference between the R-symmetry and the $U(1)_R$ internal symmetry","text":"* For a general ${\\cal N}$ the R-symmetry group is $U({\\cal N})$ but for the ${\\cal N}=2$ case why is it $SU(2)$ ? I guess it is again different for ${\\cal N}=4$. How does one understand this? One denotes the generators of the R-symmetry group by $B^i$ and let $\\alpha$ be the spinor index and let $a,b$ be the R-charge indices in a ${\\cal N}=2$ theory. Then one writes the defining equations as, $[Q_{\\alpha a}, B^i] = -\\frac{1}{2} (\\tau ^i)_a ^b Q_{\\alpha b}$ and $[\\bar{Q}_{\\dot{\\alpha}}^a, B^i] = \\frac{1}{2} \\bar{Q}_{\\dot{\\alpha}}^b (\\tau ^i)_b ^a$   * But it is not clear to me as to why over and above the aforementioned equations there should be yet another charge $R$ (called the $U(1)_R$ charge) with the defining equations, $[Q_{\\alpha a},R] = Q_{\\alpha a}$ and $[\\bar{Q}_{\\dot{\\alpha} a}, R] = - \\bar{Q}_{\\dot{\\alpha} a}$ I would like to know why the above charge should exist in ${\\cal N}=2$ supersymetry separate from the R-symmetry. Is there an analogue for this $R$ for other values of ${\\cal N}$ ?   * Typically this R-charge as defined above is anomalous. How does one see that? Also what is the analogous statement for R-symmetry?"}
{"id":"46788","title":"How efficient is the human body?","text":"This question sort of comes to mind when hearing how efficient an internal combustion engine is turning chemical energy in mechanical energy (something like 20-40%) with lots of excess heat. As an analog, how efficient is (or potentially) the human body at turning food into energy? Please bare with me, I realise there LOTS of different variables (how much the person weighs vs mass, metabolism, diet, etc). But I would imagine that there shouldn't be much margin of error given that most people maintain the same constant temperature (98 F +\/- 1 degree)."}
{"id":"21607","title":"Can an Electromagnetic Gauge Transformation be Imaginary?","text":"The Hamiltonian of a non-relativistic charged particle in a magnetic field is $$\\hat{H}~=~\\frac{1}{2m} \\left[\\frac{\\hbar}{i}\\vec\\nabla - \\frac{q}{c}\\vec A\\right]^2$$. Under a gauge transformation of the magnetic potential: $$\\vec A ~\\rightarrow~ \\vec A + \\vec\\nabla \\chi,$$ the wavefunction of the particle transforms as $$\\Psi~\\rightarrow~ \\Psi\\exp(\\frac{iq\\chi}{\\hbar c}).$$ When $\\chi$ is real, the wavefunction simply gains an extra phase factor. However, when $\\chi$ is imaginary, there is a measurable change to the wavefunction. This seems to contradict the fact that the magnetic field is invariant under the gauge transformation. How do I resolve this?"}
{"id":"90494","title":"How to be sure that a He-Ne laser light is monochromatic","text":"How can I be sure that the emission of a He-Ne laser contains only one single mode of laser cavity? The only thing that I know is that if I use a diffraction grating and the light isn't monochromatic, I'll see maximums of the same order $m$ at different angles, but I also know that if wavelengths are very close I may not see them. I have to mind the resolutive power of the grating ($R=mN$). If N1=1000 lines\/mm and N2=500 lines\/mm and the grating paces are D1=10^-6 m and D2=2*10^-6 m, will I see different maximums if the light isn'tmonochromatic? Do you know other ways to know if the light of a He-Ne is monochromatic?"}
{"id":"63731","title":"I reached a result concerning displacement with quantized time intervals. Am I on to something?","text":"A few days ago, I realized a similarity between distance with constant acceleration, $d = v_i t + 1\/2 a t^2$, and the sum of integers up to n, $(n^2 + n)\/2$. This came up again today when I decided to work out some formulas for distance and velocity with constant acceleration updated at discrete intervals, as happens in physics simulations I've programmed. > Discretely incremented $v$ and $x$ with constant acceleration, $v_i = 0$ > >   * Add $a\\mathrm{d}t$ to $v$ every tick >   * Add $v\\mathrm{d}t$ to $x$ every tick > > > $$\\begin{align} v_f &= a\\mathrm{d}t + a\\mathrm{d}t + a\\mathrm{d}t + \\cdots > \\\\\\ x_f &= v_1\\mathrm{d}t + v_2\\mathrm{d}t + v_3\\mathrm{d}t + \\cdots \\\\\\ v_2 > &= (a\\mathrm{d}t) + (a\\mathrm{d}t) = 2(a\\mathrm{d}t) \\\\\\ v_n &= > n(a\\mathrm{d}t) \\\\\\ x_f &= (a\\mathrm{d}t)\\mathrm{d}t + > 2(a\\mathrm{d}t)\\mathrm{d}t + 3(a\\mathrm{d}t)\\mathrm{d}t + \\cdots \\\\\\ x_n &= > \\sum_{i=0}^{n}i(a\\mathrm{d}t)\\mathrm{d}t = \\frac{n^2 + > n}{2}(a\\mathrm{d}t)\\mathrm{d}t \\\\\\ n &= > \\left\\lfloor\\frac{t_\\text{total}}{\\mathrm{d}t}\\right\\rfloor \\\\\\ x_f &= > \\frac{1}{2}\\Biggl[\\biggl(\\frac{t_\\text{total}}{\\mathrm{d}t}\\biggr)^2 + > \\frac{t_\\text{total}}{\\mathrm{d}t}\\Biggr](a\\mathrm{d}t^2) = > \\biggl(\\frac{t_\\text{total}^2}{2\\mathrm{d}t^2} + > \\frac{t_\\text{total}}{2\\mathrm{d}t}\\biggr)(a\\mathrm{d}t^2) \\\\\\ x_f &= > a\\mathrm{d}t^2\\times\\frac{1}{2}\\times\\frac{t_\\text{total}^2}{\\mathrm{d}t^2} > + \\frac{1}{2}a\\mathrm{d}t\\frac{t_\\text{total}}{\\mathrm{d}t} \\\\\\ x_f &= > \\frac{1}{2}at_\\text{total}^2 + a\\mathrm{d}t\\frac{t_\\text{total}}{2} \\\\\\ > \\lim_{\\mathrm{d}t\\to 0} &= \\frac{1}{2}at^2 = \\text{normal $v_f$ equation > where $v_i = 0$ and $\\Delta a = 0$} \\end{align}$$ I'd like to know what relations to theory this or an idea like it has if any, and if it touches on anything else."}
{"id":"45471","title":"How do I know the position of some mass in space (starting at rest) over time when applying earth's gravitational force to it?","text":"I know how to use Newton's law of universal gravitation. But I'm not sure where to go from there. This is for my understanding while working on a game. I'm not trying to do anything fancy here."}
{"id":"46879","title":"The Role of Gravity among the Fundamental Forces of Nature","text":"If we look at the standard model, we have 4 fundamental forces which include   1. Gravity,   2. Electromagnetism,   3. Nuclear weak force,   4. Nuclear strong force. I would like to look at Gravity for a minute. Scientists are still searching for what causes gravity. But is it possible that gravity is only a property of the space-time fabric? For example take a blanket that is stretched tight and put different weight balls on the blanket. As one would see, the balls sink into the blanket differently. If you look at the space-time fabric as the blanket and the different balls as objects in the space-time fabric when it comes to planets and more on the macro scale, thinking gravity as a property of space-time fabric seems to work. Since the standard model deals more at the quantum level though and we are looking at it here on earth where gravity is $9.8\\,m\/s^2$, it seems to me that the standard model would also need to stand up to say if we had gravity that was $11.4\\, m\/s^2$ or something more absurd like $20.6\\,m\/s^2$ . Also could we not say that gravity is really just a property of space-time fabric?"}
{"id":"58069","title":"Why does a superconductor obey particle-hole symmetry?","text":"We normally solve the Bogoliubov-de Gennes (BdG) equations in order to compute the energy spectrum of a superconductor. The Nambu spinor is a common object that is used in formulating these equations. In other words, the problem is formulated with the a priori assumption that the superconductors obeys particle-hole symmetry. But how do we know that it will obey particle-hole symmetry? I'm sure that by assuming particle-hole symmetry, computing the spectrum by solving the BdG equations, and comparing with experiments we get a good agreement. So we know it's true. I just don't understand why that is the case. Is there any physical reason behind it?"}
{"id":"46871","title":"Why is my approach to the equation of time off by a constant?","text":"I'm trying to better understand the causes for the equation of time by deriving an approximation from first principles. My naive approach, $EOT_{NAIVE}$, is to take the difference between the right ascension of the mean sun, $\\alpha_M(t)$, and the right ascension of the \"real\" sun $$EOT_{NAIVE}(t)=\\langle\\dot{\\alpha }\\rangle\\cdot(t-t_0)-\\alpha(t)$$ where $\\alpha(t)$ is simply the \"actual\" position of the sun at time $t$ from ephemeris data (e.g. either of the RA values reported by JPL HORIZONS, or a similar source) and where $\\langle\\dot{\\alpha }\\rangle = 24\/365.242$ and $t_0=t:\\alpha_M(t_0)=0$. To confirm that this is about right, I compare my result with what I get using the USNO definition of GMST $$EOT_{GMST}(t)=GMST(t)-(t-t_{noon})-\\alpha(t)$$ and with $EOT_{POLY}$, a standard polynomial expression from Dershowitz & Reingold as described in The Clock of the Long Now documentation. But the values I get for $EOT_{NAIVE}$ (blue dashed line) are consistently about $7.4$ minutes greater than these reference methods (exactly $7.4537$ for $EOT_{GMST}$, red line; and within a couple seconds of that for $EOT_{POLY}$, gray outline) give: ![enter image description here](http:\/\/i.stack.imgur.com\/cqWAE.png) Surprisingly, I can fix this by simply changing $t_0$ from the date, $t_E$, on which the most recent vernal equinox occurred (2012-03-20T05:14:33Z) — which I had assumed would give a \"starting\" RA of 0 — to 2012-03-22T02:20:32.41Z, which brings my approximation (blue line) exactly in register with the GMST approach (gray band), and within a couple seconds of the polynomial approach (red line), as can be seen by taking the difference between the GMST approach (gray band) and each of these approaches (note the change of scale, this figure essentially takes the difference between $EOT_{GMST}$ and each of the plots in the first figure above, and \"zooms in\" on the gray band): ![enter image description here](http:\/\/i.stack.imgur.com\/njj15.png) Why should changing the date in this way \"fix\" my approximation? Why doesn't my approximation work with $t_0=t_E$? Why does it work perfectly with a different date? Are the approaches I'm using as references ($EOT_{GMST}$ and $EOT_{POLY}$) the right ones for this kind of exploration? * * * Note that the reported RA of the sun at 2012-03-22T02:20:32.41Z is $0.1146^h$, and that the fit between $EOT_{GMST}$ and $EOT_{NAIVE}$ using that date is quite good (this figure zooms in further on the gray band in the second figure above): ![enter image description here](http:\/\/i.stack.imgur.com\/1Mtl5.png) Note also that despite the documentation for the ephemeris I use claiming that \"light time delay is not included\", the values I get match those from JPL's \"airless apparent right ascension and declination of the target center with respect to the Earth's true-equator and the meridian containing the Earth's true equinox-of-date. Corrected for light-time, the gravitational deflection of light, stellar aberration, precession and nutation.\""}
{"id":"46877","title":"Does a particle in a spherically symmetric infinite square well potential exert a force on the inner and outer shell barrier?","text":"For a particle in the potential: $$V(r) = \\begin{cases} 0 & \\text{a < r < b}\\\\\\ \\infty & \\text{otherwise.} \\end{cases}$$ Does this guy in the ground-state exert a force on the shells a and b? I don't think so because at ground-state both the centrifugal term of the radial equation and the angular momentum of the particle is zero. What say you brethren?"}
{"id":"45785","title":"Why isn't temperature measured in units of energy?","text":"Temperature is the average of the kinetic energies of all molecules of a body. Then, why do we consider it a different fundamental physical quantity altogether [K], and not an alternate form of energy, with a dimensional formula derived from three initial fundamental quantities, length [L], mass [M] and time [T]?"}
{"id":"13062","title":"Simple friction formula for a car","text":"I am making a 2D driving video game, and I would like to know the \"simple\" formula for calculating the friction force between the car and the road. I have read lots of friction diagrams involving balls rolling down inclined planes, but I'm having trouble applying it to this problem. Consider a car driving on a horizontal road, with some coefficient of friction, $\\mu$. Its velocity is $v$, mass is $m$, and a constant driving force is accelerating the car at $a\\ \\mathrm{m\/sec}^2$ in the forwards direction. What is the friction force, $F$, that acts on the car in the backwards direction? I _guess_ that the friction force is proportional to the car's speed, but I can't explain this guess. The reason I think that is because of a handful of case studies. Say that the driver's foot is on the pedal such that the car will get up to $16\\ \\mathrm{m\/sec}$:   * If the car's velocity is $0\\ \\mathrm{m\/sec}$, there is no force accelerating it backwards (it is stationary).   * If the car's velocity is $8\\ \\mathrm{m\/sec}$, the friction force accelerating it backwards must be less (in magnitude) than the driving force, and hence the car keeps accelerating.   * If the car's velocity is $16\\ \\mathrm{m\/sec}$, the friction force accelerating it backwards is equal (in magnitude) and opposite to the driving force, and hence the car maintains a constant velocity.   * When the driver releases the pedal, the same friction force accelerates the car backwards, but now there is no driving force, so the car drifts slowly to a halt. It would therefore make sense that $F \\propto -\\mu v$, such that in the first case, $F = 0$; in the second case $F = -\\frac{1}{2}ma$; in the third and fourth cases, $F = -ma$, where $a$ is whatever acceleration is required to maintain a constant velocity of $16\\ \\mathrm{m\/sec}$. But I can't figure out the exact relationship between $F$ and $v$, or explain why they are proportional."}
{"id":"118819","title":"Change of variables in calculating the integral of multivariable differential entropy","text":"I have already asked this question in math.SX but here might be more proper. So I decided to put a copy here and delete the one which is not the one that got an answer: I know that for one dimensional differential entropy of a density function $p(x)$, one has the following formula by change of variables: $$H(p)=\\int p(x)\\log(p(x))dx=\\int\\limits_{0}^{1}\\frac{d}{dp}[\\log F^{-1}(p)]dp$$. Where $F$ is the cumulative distribution function of $p$. To extend the idea for multivariate density functions my idea was to try to integrate on level sets of $p(x)$ to simulate somehow what happens in one dimension. As a results I took a look at this question and topics relevant to pullback method in differential geometry and co-area formulation but couldn't come up with anything by myself. Does anybody know of any method to get a similar formula for differential entropy when $p$ is a multivariable density function? **EDIT** : As it has been clarified by \"NowIGetToLearnWhatAHeadIs\" CDF does not have a unique generalisation. What I thought to be the natural one is: $$F(x_1,...,x_n)=\\int\\limits_{-\\infty}^{x_1}\\int\\limits_{-\\infty}^{x_2}...\\int\\limits_{-\\infty}^{x_n}p(x_1,...,x_n)dx_ndx_{n-1}...d_2d_1$$"}
{"id":"62111","title":"Functions of time","text":"Following are the functions of time:   $s(t)$ = distance a particle travels from time $0$ to $t$   $v(t)$ = velocity of a particle at time $t$   $a(t)$ = acceleration of a particle at time $t$. If we want to see how the position of a particle changes with respect to time only, then its velocity must remain constant with time. Likewise, if we want to see how velocity varies with time, then the distance between the former position of the particle and the current position should remain constant with time. Similarly, if we want to see how acceleration varies with time, then the difference between the initial velocity $U$ and final velocity $V$ should remain constant with time. Is this what the above functions of time tell us?"}
{"id":"44430","title":"Momentum And Energy Problem","text":"The information to the question is, \"A $59.0~kg$ boy and his $38.0~kg$ sister, both wearing roller blades, face each other at rest. The girl pushes the boy hard, sending him backward with a velocity $3.00~m\/s$ toward the west. Ignore friction.\" The specific question I am working on is, \"How much potential energy in the girl's body is converted into mechanical energy of the boy–girl system?\" Well, I said that $E_{mech,~i}=0$, because no one has kinetic energy (no one is moving), and neither of them have the potential to move, that is, until the girl commences with the push; but, $E_{mech,~f}=PE=KE$, when she uses her potential energy to apply a force over a distance, on her brother, and by doing so she changes the energy of the system, by changing the speed of both of them, meaning the potential energy will convert to kinetic energy. I've tried to plug every piece of data given to me, but I still couldn't find how much $PE $ was converted to mechanical energy. How do I find it? Is my analysis correct; is there anyway to improve it? Edit: I have another question: why can't internal forces of a system cause momentum to not be conserved?"}
{"id":"44433","title":"Solution of a partial differential heat equation with derivative and boundary conditions","text":"I want to solve the following partial different equation. Find $u(x, t)$, satisfying $u_t = u_{xx}$ , $u(x, 0) = x − x^2$ , $u(0, t) = T_0$ , $u_x (1, t) = 0$ and $|u|$ is bounded. Using separation of variables, and eliminating solutions that diverge with time, one gets $$u(x,t)~=~e^{ -\\lambda^2t}(A \\cos(\\lambda x)+B\\sin(\\lambda x)).$$ How do I proceed after this? The condition $u_x (1, t) = 0$ gives $ A=-B \\tan(\\lambda )$ which is giving me a contradiction for the condition $u(0, t) = T_0$ , as you get $$-e^{ \\lambda^2}B\\tan(\\lambda)~=~T_0.$$"}
{"id":"40957","title":"Number of Grassmann generators for Dirac field?","text":"How many Grassmann generators are sufficient for the description of a Dirac spinor in 4 dimensions? _i.e._ The Dirac field is a map to $\\Lambda_N$, the space of supernumbers with $N$ real Grassmann generators. What is $N$? This is a follow up question to my previous question Grassmann Paradox Weirdness. I am following Prakash's book Mathemaical Perspectives on Theoretical Physics, where they say a supernumber $z\\in\\Lambda_N$ can be thought of the extension of complex numbers by the addition of $N$ Grassmann generators $\\zeta^1,\\, \\zeta^2,\\,\\ldots \\zeta^N$. The most general supernumber is written $$z = z_0+z_i\\zeta^i+\\textstyle\\frac{1}{2!}z_{ij}\\zeta^i\\zeta^j+\\ldots,$$ where $z_i$, $z_{ij}$, $\\ldots$, are complex-valued and antisymmetric. The odd part of this is anticommuting and is used to describe Fermion fields. The book says that for finite $N$, it takes $2^{N-1}$ complex numbers to specify an anticommuting number. How do I figure out how many Grassmann generators $\\zeta^i$ I need to specify a Dirac spinor in 4 dimensions."}
{"id":"44438","title":"Finding an efficient strategy for walking","text":"Let's say you are already walking at a maximally efficient combination of pace and stride (or $\\omega$ and $X_0$ I guess) but you need to reach your destination faster. Should you increase\/decrease your pace and\/or stride to achieve this goal (assuming that any adjustment will result in a net increase in speed)? How can I interpret this question as a question about kinetic energy of oscillators? I have tried solving the equation $KE'=0$ to find a minimum $KE$ but this seems like the wrong approach, for all that seems to tell me is the if you do not walk (ie $w$ or $X_0$ is $0$) then yes, of course you have kept your kinetic energy at a minimum. Where does this idea about \"efficiency\" come into play? Edit: Pace is the frequency of steps, stride is the length of steps, and average speed is the product of pace and stride."}
{"id":"23517","title":"How to find that a molecule has zero spin?","text":"I read that Ne has $S=0$. How can this be found, knowing the electron configuration? Electrons, protons and neutrons all have 1\/2 spin. The Pauli exclusion principle implies that the even number of electrons are half up-spin and half down-spin, so that they sum to 0. Is that argument valid, and if so, can the same be applied to show that the total spin of neutrons and protons, which occupy the same space, is 0? I suppose that would imply that $S=0$ for all molecules with even numbers of electrons and protons+ńeutrons."}
{"id":"27609","title":"precise definition of \"moduli space\"","text":"I'm curious what the precise definition of the moduli space of a QFT is. One often talks about the classical moduli space, which then can get quantum corrections. Does this mean the quantum moduli space is something like the set of minima of the effective potential in the 1PI action (or does the 1PI action already presume a choice of vacuum?)? Is there a non-perturbative definition, without referring to field configurations or effective actions (eg, how do we define it in string theory, where there are no fields)? Maybe it can be identified with some submanifold of the (projective) Hilbert space? Can the metric or other structure be defined in such an invariant way?"}
{"id":"97929","title":"How to calculate the total angular momentum (J) values","text":"How can I calculate the values of $J$ (total angular momentum) for a particular term, for instance, ${}^3P$?"}
{"id":"95209","title":"Migration of an ion in a non-conductive medium","text":"Someone asked me why, when performing DNA electrophoresis we need to put the gel inside a conducting buffer. Couldn't we just run it in distilled water? My answer was that if we had distilled water we would not be able to close the circuit between the anode and the cathode of the electrophoresis apparatus, no current (or very little) would flow and no movement would be possible. The next question then was: why do we need a current to pass through the system in order to have movement of the DNA (or, in general, any charged particle) towards the anode? What physical forces enter into play? In other words: if we had a bath of distilled water, we put two electrodes connected to a generator at each end and put a charged molecule in the middle of the bath, would it be able to migrate? Why? Which forces would be involved? I found this document which essentially confirms my first answer: > **Electrophoresis Buffer:** Several different buffers have been recommended > for electrophoresis of DNA. The most commonly used for duplex DNA are TAE > (Tris-acetate-EDTA) and TBE (Tris-borate-EDTA). DNA fragments will migrate > at somewhat different rates in these two buffers due to differences in ionic > strength. Buffers not only establish a pH, but provide ions to support > conductivity. If you mistakenly use water instead of buffer, there will be > essentially no migration of DNA in the gel! Conversely, if you use > concentrated buffer (e.g. a 10X stock solution), enough heat may be > generated in the gel to melt it. However, I have not been able to find a proper explanation of the physical forces involved in the process. PS: of course in a real DNA electrophoresis the charged molecule is trapped in a gel, which itself has been made using a conducting buffer, so the question becomes probably a tad more tricky..."}
{"id":"95201","title":"what is the 'resolution' of the reality in pixel terms","text":"1.What resolution should a TV screen have so that its image were so faithful as reality as if the TV were a window ? 2.Also what would happen if Physics could reproduce a 'pixel' of the size $ l_{p}^{2} $ the square of the Planck length ? Would be then this the resolution of our reality like watching through a real window ?"}
{"id":"91262","title":"Rigorous definition of frame of reference","text":"I'm looking for a mathematical definition of frame of reference. Most of the textbooks I have seen take it for granted and they just refer to some set of spacetime coordinates. A more mathematical definition I found defines spacetime $E$ as an affine space over $\\mathbb{R}^4$ and then defines a frame of reference as a 5-tuple $(P,v_1,v_2,v_3,v_4)$ where $P\\in E$ and $(v_i)$ is a basis of $\\mathbb{R}^4$. But with this definition I cannot make sense of the phrase \"a frame moving with respect to another\"."}
{"id":"111558","title":"What is the cause of Special Relativity","text":"Why does special relativity exist? Via studying the outcome(s) of such relativity, one can end up with all the equations involved in special relativity. Thus one runs into things such as contraction, dilation, transformations, velocity addition, etc. But these are the outcomes. My question is what is the cause of special relativity, and is the knowledge of this cause currently being shared in the world of physics? Or, to put it another way, is it currently accepted despite there being no cause that is known of, thus an acceptance of an effect without a cause? If you give this question a negative vote, please present your reasoning and or evidence to support such a vote."}
{"id":"52809","title":"Thermodynamics of a bar, specific heat","text":"I have a problem and I don't understand how the book solves it. It says I have a bar such that $$F=aT^2(L-L_0)$$ where $F$ is the force or tension of the bar, $T$ is the temperature, and $L$ the longitude. $L_0$ is the longitude when $F=0$. It says that $C_{L_0}=bT$ and it asks me about the general function $C_L(L,T)$, being $C_L$ the specific heat at constant $L$. The book starts solving it with this expression: $$\\left(\\frac{\\partial C_L}{\\partial L} \\right)_T=-T\\left(\\frac{\\partial^2 F}{\\partial T^2}\\right)_L$$ From there, I know how to go on, but I have no idea where that comes from. I have tried to get that out of the generalised Mayer relation but if that's the way, I can't do it."}
{"id":"60932","title":"Where are exactly the charges of charged capacitor?","text":"If a capacitor is connected to a battery and is charged, are there charges inside the wires or do they just accumulate on the surfaces of capacitor?"}
{"id":"65210","title":"The units of gain and number of atoms in population inversion in a laser","text":"I am following my university course notes on amplification in laser media, and have come across expressions for the **gain** of a medium, but the notes are not exactly rigorous... The expression given for the gain $G$ in units of dB m$^{-1}$ is: $$G = 10 \\log_{10}(e^\\gamma) \\mathrm{dB} \\mathrm{m}^{-1}$$ where $\\gamma$ is _also_ the gain coefficient for the media with units m$^{-1}$. However later in the notes it gives the expression $$\\gamma = \\Delta N \\sigma$$ where $\\Delta N$ is the \"number of atoms in population inversion\" and $\\sigma = [\\mathrm{m}]^{-2}$ is the \"cross-section for stimulated emission\". This clearly shows that $\\Delta N = [\\mathrm{m}]$, which doesn't make much sense to me. Also the first expression implies that $\\gamma$ is unitless, so I am mighty confused."}
{"id":"383","title":"Derivation of the centrifugal and coriolis force","text":"I was wondering how easily these two pseudo-forces can be derived mathematically in order to exhibit a clear physical meaning. How would you proceed?"}
{"id":"114455","title":"Displacement of a body in free fall","text":"Suppose a ball is thrown up from the ground with some velocity. Let vertically up be the positive direction and vertically down be the negative direction and let the thrower be at the origin. What will be the sign of displacement after it has reached it's maximum height and is still descending? I think it will be negative since it's displacement will be in the negative direction, although it's position will always be positive. Am I correct?"}
{"id":"114454","title":"Where does the idea gravity=curvature of spacetime really come from?","text":"I have been searching for quite a while but mostly found the answer: Einstein's genius. Quite unsatisfactory. I know and understand that the idea gravity=curvature of spacetime works. Furthermore I know that the starting point for Einstein's considerations was the equivalence principle. Nevertheless I can't make the connection from there, to why gravity is the curvature of spacetime. Any help or reference to where this is explained would be much appreciated."}
{"id":"114452","title":"Definition of force, kinetic energy and momentum","text":"**I've edited the post. Q1 and Q4 are the important ones but I didn't delete Q2 and Q3 since some older answers would not make sense anymore.** To begin with, the formula of the kinetic energy $T$ is $\\frac{mv^2}{2}$. Furthermore momentum is conserved $\\Sigma m_{i}\\vec{v_{i}}=const.$ Then you have the definition that force is the change of momentum with respect to time $\\vec{F}=\\dfrac{d(m\\vec{v})}{dt}$. I've read the chapters concerning mechanics of Physics for scientists and engineers by Giancoli and the Feynman Lectures. Giancoli introduces the arbitrarily work as $W=\\int\\vec{F} \\cdot d \\vec{s}$. From this definition of work he derives the kinetic energy to be $\\frac{mv^2}{2}$. In contrast to that, in the Feynman lectures you never get a derivation of $\\frac{mv^2}{2}$, but it is shown that $\\dfrac{dT}{dt}=\\vec{F}\\cdot\\vec{v} = \\vec{F}\\cdot\\dfrac{d\\vec{s}}{dt}$. Then it is shown that $dT=\\vec{F}\\cdot d \\vec{s}$ and as a consequence $\\Delta T = \\vec{F} \\cdot \\vec{s}$ which is called work. http:\/\/www.feynmanlectures.caltech.edu\/I_13.html#Ch13-S1 **Now I've got some questions:** Q1 _Is $F=\\dfrac{d(mv)}{dt}$ **just an arbitrary definition** or is there something \"more\" behind the formula for force?_ Q2 _Is $W=\\int\\vec{F} \\cdot d \\vec{s}$ **just a definition** or is there something more behind? I mean, can you derive the formula for work not by taking the formula for the kinetic energy as given._ Q3 **How to derive the formula for the kinetic energy and work only form the conservation of momentum $\\Sigma m_{i}\\vec{v_{i}}=const.$?** Q4 **How are work and kinetic energy defined?** I have found both: A) Kinetic energy is doable work and work is $F=\\dfrac{d(mv)}{dt}$. B) The formula for kinetic energy is: $\\frac{mv^2}{2}$. Then, after some maths it follows that $\\Delta T = \\int \\vec{F} \\cdot d\\vec{s}$. **It doesn't make sense to define 2 things in that way. This would be circular logic.** P.S.: I am not a native English Speaker so feel free to edit."}
{"id":"32256","title":"The possibility of free electrical energy?","text":"Please excuse my lack of knowledge\/understanding. **Question:** Why Nikola Tesla's Free Energy concept was never worked upon? Even today. **Context:** Now that we know Nikola Tesla was a genius and did exceptionally well with his works and the **Free Energy** concept. Why his idea was never adopted or was worked upon, even now? I know, back then, Edison didn't let him do much. But now, in our times, why GE or any other company or well the physics `PhD`s aren't following up? Why CERN is not working over it? By the articles that I have read over the internet I am under an impression that everyone (now) feel that we have \"missed\" the free-energy _chance_ that we had as a planet, that Nikola Tesla was trying to bring in. I see a lot of _brag_ (to an extent, as well) which leaves no doubt that his Free-Energy concept would have been **must-have**. But if I look at the actual followups, no one _it seems_ got **actually** interested to pursue his theories; or is it that no one is capable enough to do it? Which would be strange, given that we are at the peaks of technological advancements. **_How hard is it to build a prototype? Really?_** Or people might have worked over it, but Tesla's blue-prints were not correct, and that is why no one came up with the actual implementation of it? Or is it that he was _really_ a crackpot as politically-publicized by Edison? I would really appreciate an answer, because this is beyond me, to understand why no one followed up on this concept? Thank you, in advance."}
{"id":"104263","title":"A small object on the opposite side of the earth's orbit with 0 velocity,\"in line\" with the core - would it collide with the earth?","text":"If a small object (say mass 1 kg), at the opposite side of the earth's orbit (six months away), was placed in the same orbital position as the earth's core, at zero velocity in all directions, would the object collide with the earth or would it have been drawn by the sun's gravity away from a collision with earth? A naive answer would simply calculate how far \"inward\" the object would travel in six months, when accelerated by the sun's gravity, and if the distance covered is greater than the earth's radius, a collision would be averted. But I started thinking about the earth's own gravitational influence as it orbited closer to the object. How would we get an answer to this question?"}
{"id":"75298","title":"How did Jackson get the exact solution to potential in polar coordinates","text":"On page 45 of Jackson's Classical Electrodynamics (3ed), He solved a polar potential problem with known source distribution. $$ \\frac{1}{\\rho}\\frac{\\partial}{\\partial \\rho}\\left( \\rho\\frac{\\partial \\psi}{\\partial \\rho} \\right)=-g(\\rho)\\\\\\ \\psi(\\rho)\\bigg|_{\\rho=1}=0\\\\\\ $$ with the given source $$ g(\\rho)=-5(1-\\rho)+10^4\\rho^5(1-\\rho)^5 $$ To me, this second order Sturm-Liouville ODE can only be solved if we have two boundary conditions, but in this case, we don't know the other BC."}
{"id":"98119","title":"Must every isometry have an associated Killing vector?","text":"I understand that the flows of Killing vector fields are isometries, and that one-parameter groups of isometries have an associated Killing vector which generates them, but are your Killing vectors guaranteed to give you all possible isometries of the manifold? I guess what I'm trying to ask is, do isometries necessarily come in families, or can you have \"isolated\" isometries which can't have an associated Killing vector field?"}
{"id":"98708","title":"Twin Paradox, calculating spacetime intervals from both perspectives","text":"I've very recently started to try to understand special relativity. I'm want to get a decent understanding of the twin paradox. I'll post what I've done so far and highlight what's gone wrong for me. The situation is that Alice and Bob both begin at point $x_1$ in Alice's coordinate system $(x,t)$ (we have orientated the axis so that y and z will not matter). Bob then instantaneously moves off with velocity $v$ in the positive $x$ direction. Bob's coordinate system is now $(x',t')=(\\gamma (x-vt), \\gamma (t-v\\frac{x}{c^2}))$. In Alice's coordinates, Bob reaches point $x_2$ after having moved distance $d$, then instantly turns around and travels back to Alice at $-v$. I want to show that Alice's space time interval $\\Delta s_A^2$ is greater than Bob's $\\Delta s_B^2$, as each path's space time interval is proportional to the proper time passed along that path. * * * **According to Alice:** $\\Delta s_A^2=-c^2\\Delta t^2 + \\Delta x_A^2$ where $\\Delta t=\\dfrac{2d}{v}$ and $\\Delta x_A=0$ as she didn't move, but the time elapsed is the time it took bob to travel the distance and then come back. Thus $\\Delta s_A^2=-\\dfrac{4c^2d^2}{v^2}$ Still according to Alice, Bob's $\\Delta s_B^2=-c^2\\Delta t^2+\\Delta x_B^2$, where the change in time is the same, but now bob has moved distance $d$ twice, thus $\\Delta x_B=2d$ ($\\Delta x$ refers to total distance traveled rather than displacement which is 0 in this case). Now we have $\\Delta s_B^2=-\\dfrac{4c^2d^2}{v^2}+4d^2$. * * * The |size| of Bob's spacetime interval now definitely smaller than Alice's, and this would be all OK, except that when I do the calculations in Bob's frame, they don't agree. This is contradictory to the fact that $\\Delta s^2$ is conserved under Lorentz transformations. * * * **According to Bob:** $\\Delta {s'}_A^2=-c^2\\Delta t'^2 + \\Delta {x'}_A^2$ where $\\Delta t'=\\dfrac{2d'}{v}$. I'm not sure if I'm right in saying that $v$, the relative velocity of the two frames, is the only velocity upon which both Alice and Bob will agree on, other than the speed of light. Anyway, $d=\\gamma d'$ as lengths contract by $\\gamma$ i.e. $d'$ is smaller than $d$ by factor $\\gamma$. Also, $\\Delta {x'}_A=2d'$, so we have $\\Delta {s'}_A^2=-\\dfrac{4c^2d'^2}{v^2}+4d'^2=-\\dfrac{4c^2d^2}{\\gamma ^2 v^2}+\\dfrac{4d^2}{\\gamma ^2}=4d^2\\dfrac{v^2-c^2}{\\gamma^2 v^2}\\neq \\Delta s_A^2$ Although I haven't included them here, my calculations for $\\Delta {s'}_B^2$ and $\\Delta s_B^2$ agree. Sorry for the ultra-long post, but any help would we well appreciated!"}
{"id":"57679","title":"Why is carbon dating limit only 40,000 years?","text":"For an example, when they tried to get the carbon dating for presence of Aboriginal people in Australia they get to the number 40,000. But it could be much earlier. Why is that 40,000 years limit for carbon dating methods?"}
{"id":"106288","title":"Boltzmann distribution: derivation from canonical distribution","text":"I'm trying to understand the Maxwell-Boltzman Distribution, and in particular the derivation from the boltzman distribution for energy. I have successfully created an incorrect derivation, but I'm not sure what's wrong with it :). Any guidance would be much appreciated! I believe that the probability density function for the energy, $E$, of a random particle in a system at equilibrium is meant to look like this: $$f_E(x) = Ae^{-\\frac{x}{kT}}$$ To simplify things I'll just consider the particular case: $$f_E(x) = e^{-x}$$ The cumulative distribution function is then: $$Pr(E\\leq x) = F_E(x) = 1-e^{-x}$$ Now suppose all the particles in our system are of mass $1$, so, ignoring potential energy, the random variable governing the speed of a given particle is $V = \\sqrt{2E}$. Now we want to find the distribution of the speeds. We can do this as so: $$F_V(v)=Pr(V\\leq v) = Pr(\\sqrt{2E}\\leq v) = Pr\\left(E\\leq \\frac{v^2}{2}\\right) = F_E\\left(\\frac{v^2}{2}\\right)$$ and so $$F_V(v)=1-e^{-\\frac{v^2}{2}}$$ and finally $$f_V(v)= ve^{-\\frac{v^2}{2}}$$. This is _almost_ right, however, the Maxwell-Boltzman distribution would predict a pdf with $v^2$ rather than $v$, something like: $$f_V(v)= \\sqrt{\\frac{2}{\\pi}}v^2e^{-\\frac{v^2}{2}}.$$ Can anyone point out the mistake in my logic? Thanks!"}
{"id":"78739","title":"Commutator problem","text":"I am trying to calculate the following commutator $$[\\mathcal{H}_0(r',t'),\\psi(r,t)]_-$$ where $\\mathcal{H}_0 = (\\frac{1}{2m}\\nabla^2 + e\\mathbf{A}(r',t'))^2 + e\\phi(r',t') - \\mu$, and $\\mu$ is the chemical potential. It is supposed to be $0$, but I don't understand why it is not a a delta- function $\\delta(r-r')[\\mathcal{H}_0(r',t'),\\psi(r,t)]_-$ which is not necessarily $0$. Could someone please help me understand why the anticommutator is 0?"}
{"id":"102957","title":"Why is there a Majorana zero mode in the $\\pi$ flux core of the p+ip superconductor?","text":"In this review paper (http:\/\/arxiv.org\/pdf\/1202.1293.pdf), the author shows that threading a $\\pi$ flux through a 2D $p_x+ip_y$ superconductor will trap a Majorana zero mode at the flux center. The deduction involves solving differential equations, and is hard to generalize to arbitrary dimensions or arbitrary gauge defects. Since Majorana zero mode is topologically robust, I would expect a simpler and less detail-dependent way to show that there is indeed a Majorana zero mode in the $\\pi$ flux of a $p_x+ip_y$ superconductor. Does anyone know such a simpler approach? PS: The answer I expect should be general enough that can be applied to $\\pi$-flux tubes in 3D topological insulator or topological superconductors as well."}
{"id":"60060","title":"Why does the index of refraction change the direction of light","text":"I've been studying in optics the macroscopic maxwell's equations, and how electromagnetic fields propagate through different mediums. Over there, the index of refraction appears, as a complex function that generally depends on $\\omega$, with both refraction and absorbtion terms: $n_c=n+i\\kappa$. I understand how this affects the speed at which light propagates, both macro and microscopically, and how this affects to how light is absorbed when it transmits through the medium, but I don't get yet how this index changes the direction of light, producing dispersion. I mean, all time I'm seeing those effects as something that separates dispersive from non dispersive mediums and all that stuff, and I don't know how light is actually dispersed. As it's something that happens when changing medium, I guess it's an interface effect, and we haven't seen those effects yet, but I would like to know an explanation."}
{"id":"60065","title":"Why is the energy spectrum continuous for a plane wave when it has energy less than the potential barrier?","text":"Please explain it in the context of this task: we have a potential barrier that looks like $\\prod$, with $E<U$. There are 3 regions: 1) no field   2) barrier   3) no field Solution could be found as * * * 1) $\\psi_{1} = A\\cdot e^{-ikx}+B\\cdot e^{ikx} $   2) $\\psi_{2} = C\\cdot e^{-ik_{0}x}+D\\cdot e^{ik_{0}x}$   3) $\\psi_{3} = 0 +F\\cdot e^{ikx}$ (cause we haven't got a reflected wave here) where $k_{0}^{2}=2m_{0}(U_{0}-E)\/h^{2}$ and $k^{2}=2m_{0}E\/h^{2}$ * * * So why in this case spectrum is continuous?"}
{"id":"2043","title":"Why are von Neumann Algebras important in quantum physics?","text":"At the moment I am studying operator algebras from a mathematical point of view. Up to now I have read and heard of many remarks and side notes that von Neumann algebras ($W^*$ algebras) are important in quantum physics. However I didn't see where they actually occur and why they are important. So my question is, where they occur and what's exactly the point why they are important."}
{"id":"113152","title":"Question about the parallel axis theorem","text":"I've a question about the parallel axis theorem. So I'm perfectly OK with the derivation of the proof in 2D. However in the typical derivation they just say this in my textbook \"Because the Zi coordinates are not taken into account into this calculation, we can generalise the summations to include all particles for all slices\" How is this possible? I don't get it how you can generalise for all slices! Please help me, and thank you very much."}
{"id":"113150","title":"Is there a mistake in this example in my mechanics textbook?","text":"This is a worked example in my mechanics book: ![enter image description here](http:\/\/i.stack.imgur.com\/1FKOF.png) ![enter image description here](http:\/\/i.stack.imgur.com\/rxymK.png) But I think this worked example is wrong. In the 6th line. shouldn't $X=4R$ as the expression for distance above the surface is $(X-R)$ and when the particle P is a distance of 3R ABOVE the surface, $v=0.5(gR)^{0.5}$ could someone please tell me if I am wrong or the book is wrong... I mean books a re rarely wrong and so I am not sure what to follow"}
{"id":"2731","title":"How to count quarks using Deep Inelastic Scattering?","text":"The Wikipedia article on deep inelastic scattering suggests that the experiment shows baryons have three point of deflections (corresponding to three quarks) and mesons have two points of deflection. How are the electrons fired in this experiment being detected, and how exactly do the two or three points of deflection appear in the data? Are they fired at a target consisting entirely of baryons, or are collisions with non-baryons somehow filtered from the data?"}
{"id":"105605","title":"Electric arc due to static discharge in a T-Shirt possible?","text":"Yesterday, when I came home, I went to the bathroom (lights off) and i took off my T-Shirt (100% cotton) which I wore under a Shirt (50% cotton, 50% polyester). I _believe_ to have seen a small but well visible light arc. **Question:** Is the amount of static discharge in T-Shirt big enough to produce a visible light arc?"}
{"id":"105609","title":"Can it be considered as time travel?","text":"India is about 8.5 hours ahead of US. If some person starts from India at 9 pm(according to his wrist watch) and reaches US in 2 hours(just assume),at that time his wrist watch will b showing 11 pm but as he his in US so he will have to match his watch according to US time so he will adjust his watch to 4:30 pm.This means that he is goin to live those 6.5 hours again in his life,but this time in US.So can this be considered as time travel or i am wrong.Please explain."}
{"id":"1836","title":"Why is air invisible?","text":"I think that something is invisible if it's isolated particles are smaller than the wavelength of visible light. Is this correct? Why is air invisible? What about other gases and fumes which are visible?"}
{"id":"7437","title":"Why glass is transparent?","text":"Once I asked this question from my teacher and he replied \"because it passes light\", \"and why it passes light\" I asked and he said \"because it is transparent\". Same question again, Why glass is transparent. Why it passes light while opaque objects do not?"}
{"id":"24179","title":"Find vector position in time give this graph?","text":"Graph is position x position. There are 3 points, $A$, $B$ and $C$.   * $A(0,2)$    * $B(4,2)$    * $C(6,0)$  Particle travels from $A$ to $B$ and from $B$ to $C$ at constant $v = 2 ~m\/s$. Find vector position in time for the $BC$ part of the motion? For the AB part I did this: $v$ is constant, so position in time is given by $2\\hat{i}\\cdot \\mathrm{t} + 2\\hat{j}$. Velocity in $i$ direction x time, plus the constant $2\\hat{j}$ vector component. For BC the \"angled\" displacement vector is making things a bit harder. Displacement vector has two components, $2\\hat{i} + (-2\\hat{j})$. Absolute value of it is $2\\sqrt{2}$. Tried to decompose the velocity vector in two: $\\hat{i}\\sqrt{2}$ $-\\hat{j}\\sqrt{2}$. Then integrate each component and add up the two. But the result is not quite right... Integrated each component independently: $\\hat{i}\\mathrm{t}\\cdot\\sqrt{2} + c$ and $-\\hat{j}\\mathrm{t}\\cdot\\sqrt{2} + c$"}
{"id":"10631","title":"Why do Engineers manipulate coordinate systems?","text":"When tackling a physics problem, An Engineer will manipulate the axes\/coordinate system where a Mathematicians and\/or Physicists will use the original coordinate system and math. Why do Engineers think differently? I know its likely because that is how they are taught, but why are they taught that way?"}
{"id":"105536","title":"How does someone move space?","text":"I've heard that we've known components of the universe appear to be flying apart faster than $c$, and that this can happen because the space containing these materials are also moving apart. How does someone move space? How efficiently can it be done? I would like to get on a rocket and then move the space very close to Sedna and see what it looks like. What are the physical barriers to doing that?"}
{"id":"68333","title":"Orthochronous Lorentz transformations are time-preserving and $SL(2,\\mathbb{R})$","text":"Let's consider the psuedosphere\/hyperboloid in $\\mathbb{R}^{1,2}$ given by $$x^2+y^2-z^2=-R^2.$$ We know that the Lorentz group $$O(1,2)=\\\\{ A \\in Mat(3,\\mathbb{R}): A^tGA=G \\\\},$$ where $G=diag(-1,-1,1)$ leaves the pseudosphere invariant. Now we are interested in the following facts:   1. How can we show that the orthochronous Lorentz group $O_+(1,2)=\\\\{ A: a_{33}>0 \\\\}$ is subgroup and, more important, maps upper cone to upper cone?   2. What is the relation between groups $O_+(1,2)$ and $SL(2,\\mathbb{R})$?"}
{"id":"106791","title":"Whether the holographic universe of string theory prove the hypothesis of the holographic universe created by David Bohm or not?","text":"Bohm, David (1980), Wholeness and the Implicate Order, London: Routledge, ISBN 0-7100-0971-2 As we can see from the book above, David created the hypothesis of the holopraphic universe in 1980. And we can see from the words below, the holographic universe of string theory is created in 1995. Using this approach he has argued that near a black hole, quantum fields could be described by a theory in a lower dimension.[Stephens, C. R.; 't Hooft, G.; Whiting, B. F. (1994). \"Black hole evaporation without information loss\". Classical and Quantum Gravity 11 (3): 621] This led to introduction of the holographic principle by him and Leonard Susskind.[Susskind, L. (1995). \"The world as a hologram\". Journal of Mathematical Physics 36 (11): 6377–6371.] So Whether the holographic universe of string theory prove the hypothesis of the holographic universe created by David Bohm or not? Or what is the link or differences between them? I have read the book of \"The Elegant Universe\" and the book of \"The Fabric of the Cosmos:Spaca,Time, and the Texture of Reality\", but Brian Greene doesn't mention about David Bohm. Thank you in advance!"}
{"id":"108111","title":"Latent heat of vaporization","text":"The molecular weight of water is 18.015 gram. The number of moles of water in one liter (1000 gram) will be: $3.34\\times 10^{25}$ molecules (in 1kg). We know that latent heat of vaporization of water is $L_v= 2.26\\times 10^6$ then the amount of heat required to rise one molecule of water will be $1.47\\times 10^{-19}$ joule: Does this calculation is correct according to macroscopic and microscopic physics?"}
{"id":"33470","title":"Understanding Tensors","text":"I don't seem to be able to visualize tensors. I am reading The Morgan Kauffman Game Physics Engine Development and he uses tensors to represent aerodynamics but he doesn't explain them so I am not really able to visualize them. Please explain in very simple ways. I just want to understand the basics."}
{"id":"32917","title":"Size of universe after inflation?","text":"Wikipedia states the period of inflation was from $10^{-36}$sec to around $10^{-33}$sec or $10^{-32}$sec after Big Bang, but it doesn't say what the size of the universe was when inflation ended. Just saw a Brian Greene show on the Multiverse and I thought I heard him say size was galactic scales when inflation ended. However I've also read size was about a basketball. Are there multiple theories with different resulting sizes? Does 'size' even mean anything in this context?"}
{"id":"30344","title":"Straightforward questions about calculating SUSY F-terms","text":"So in the Lagrangian for a SUSY theory we have the F-terms, which I have seen written (e.g., in Stephen Martin's SUSY primer) as $F^*_i F^i$ where $F^i = \\frac{\\partial W}{\\partial \\phi^i}$. I have a few questions about this. 1) What do we do in the case of a gauge multiplet? For instance, in the MSSM we have $Hu = (Hu^+,Hu^0)$. Let's look at the term $y_{jk} Q_j \\bar{U_k} Hu$ in the superpotential, where $Q$ is the left-handed SU(2) squark doublet. Should we take $F^{Hu} = \\frac{\\partial W}{\\partial Hu} = y_{jk} Q_j \\bar{U_k}$? Or should it rather be two terms, like $F^{Hu^+} = \\frac{\\partial W}{\\partial Hu^+}$ $F^{Hu^0} = \\frac{\\partial W}{\\partial Hu^0}$? 2) If it is $F^{Hu} = \\frac{\\partial W}{\\partial Hu}$, then where Martin has $F^*_i$, should it really be $F^\\dagger_i$? Is that implied by the raised and lowered indices? 3) In either case, why is it not a problem that we will get terms like ($y_{33}Q_3\\bar{U_3})^*y_{22}Q_2\\bar{D_2}$? I'm worried about whether such terms will form gauge invariants and whether they imply vertices that should not be there. I know quarks can mix, but this looks like it describes something like a vertex where a 3rd-gen up-type quark-antiquark pair directly annihilate to a 2nd-gen down-type quark-antiquark pair. Am I misreading this? Thanks so much for any help! This seems like a straightforward thing to answer but I haven't seen it addressed in the primer, etc."}
{"id":"76172","title":"Symmetry Breaking and Vacuum Expectation Values","text":"Consider a real scalar field $\\phi$ in a theory with a Lagrangian $$ \\mathcal{L}:=-\\frac{1}{2}\\partial _\\mu \\phi \\partial ^\\mu \\phi -V(\\phi ), $$ where $$ V(\\phi ):= -\\mu ^2\\phi ^2+\\frac{\\lambda}{4!}\\phi ^4, $$ where both $\\mu$ and $\\lambda$ are positive real numbers. We see that the potential has a couple of non-zero minima: $$ V'(\\phi )=-2\\mu ^2\\phi +\\frac{\\lambda}{3!}\\phi ^3=0\\Rightarrow \\phi =0,\\pm 2\\mu \\sqrt{\\frac{3}{\\lambda}}=:\\pm V_0 $$ (It turns out that $\\phi =0$ is a local max, and $\\phi =\\pm V_0$ are local mins; check the second derivative.) As the usual story goes, we must define a new field $\\psi :=\\phi -V_0$ and re- write the theory in terms of this $\\psi$ to get the appropriate Feynman rules of the quantum theory. If I did my algebra correctly (the details aren't exactly relevant here anyways), this substitution gives us $$ \\mathcal{L}=-\\frac{1}{2}\\partial _\\mu \\psi \\partial ^\\mu \\psi -2\\mu ^2\\psi ^2+\\mu \\sqrt{\\frac{\\lambda}{3}}\\psi ^3+\\frac{\\lambda}{4!}\\psi ^4-6\\frac{\\mu ^4}{\\lambda}. $$ (Our Lagrangian no longer admits the symmetry $\\psi \\mapsto -\\psi$, hence the term \"symmetry breaking\".) Th question arises: Why is this substitution special? This form of the Lagrangian has some nice properties (namely that the potential has a local min at $0$), but surely there are some other substitutions that could given us some other nice properties as well. What about those? My understanding of this was the following: The LSZ Reduction Formula, among other things, requires a priori that the fields one is working with _have vanishing vacuum expectation value_. Thus, when applying the LSZ formula, we must be working with $\\psi$, not $\\phi$, and so the appropriate Feynman rules can be read off only when the Lagrangian is written in terms of $\\psi$. I have just recently discovered a problem with this explanation, however. Before, I was under the impression that $\\langle 0_\\pm |\\phi |0_\\pm \\rangle =\\pm V_0$ (this theory evidently has two physical vacuums, whatever that precisely means), so that the definition of $\\psi$ forces $\\psi$ to have vanishing expectation value, so that the LSZ formula can be applied. However, I recently learned that $\\pm V_0$ is _only an approximation_ to $\\langle 0_\\pm |\\phi |0_\\pm \\rangle$, which implies that $\\psi$ _only approximately has vanishing vacuum expectation value_ , which means that LSZ doesn't technically apply. It seems that the proper substitution is in fact $\\psi :=\\phi -\\langle 0_+|\\phi|0_+\\rangle$. There are several problems I see with this:   1. The Lagrangian re-written in terms of $\\psi$ should have a small, but non-zero, _linear_ term in $\\psi$.   2. The Feynman rules I've been using all along that arise from the substitution $\\psi :=\\phi -V_0$ are only approximation.   3. The coefficients that arise from the 'proper' substitution $\\psi :=\\phi -\\langle 0_+|\\phi |0_+\\rangle$ are going to be written in terms or something that can (to the best of my knowledge) only be calculated perturbatively (namely $\\langle 0_+|\\phi |0_+\\rangle$), but we need to know these coefficients to obtain the Feynman rules to begin with (resulting in a 'circularity' problem). How does one go about resolving all these issues? (Disclaimer: I asked a very similar question here not quite a year ago, but my understanding of the situation has improved since then, and as is usual, my improvement of understanding has only brought forth many more questions regarding this, so I felt it was appropriate to address the issue once again.)"}
{"id":"109861","title":"Why can't I use Bell's Theorem for faster than light communication?","text":"I read this description of Bell's theorem. I understand he's restating it slightly, so there may be incorrect assumptions there, or I may have some. I think Bell's theorem _should_ lead to FTL communication, and I'll try to lay out my assumptions as to how, and propose an experiment. So:   1. A and B are entangled photons.   2. With 100% probability, if I measure A at X°, and you measure B at X°, our measurements will be opposite (mine will go through iff yours is absorbed). 3. So this is the experiment:   1. Generate A and B, and send B lightyears away.   2. Whoever's manning the experiment at B does the following:  **1.** If they want to send binary 1, measure B at 20°. **2.** If they want to send binary 0, don't do anything.   3. Now whoever is at A measures A first at 0°, then (if it goes through), at 40°.   4. The results should be (it seems to me):     1. If **1** above, then B was absorbed with 50% probability and was transmitted with 50% probability. In either case, the probability of seeing A get transmitted at 0° and get absorbed at 40° is less than or equal to 11.6% (because the probability of case 1 or 2 in the article is 50% for the measurement at 20° times either the probability of A transmitted at 0°, or B transmitted 40° (which under assumption 2, is the same as A **not** transmitted 40°) so the probability of both must be less than or equal to 11.6%).     2. If **2** above, then the probability of seeing A get transmitted at 0° and get absorbed at 40° is 20.7%. So A can determine a binary message from B, violating the no-communication theorem. If done with enough photons, you can send much longer messages. I know it is far likelier that I've made a mistake here than that I've disproved the no-communication theorem, so what's wrong? If anyone doesn't understand the experiment, ask and I'll try to clarify."}
{"id":"91220","title":"Probability of having energy $E$ when $E$ is bounded","text":"For a canonical ensemble the probability of a system to have energy $E$ is $P(E)=e^{-\\beta E}$. For that we consider the that the system can have any energy between $0$ to $\\infty$. What will be the probability if the system can have energy only within certain range say, $E_{min}\\leq E \\leq E_{max}$ ($E_{min},E_{max}>0$)? **Derivation for canonical ensemble** For a canonical ensemble I follow this derivation. Consider the total energy of system and reservoir ($E_1+E_2$) is constant and then optimize the total number of possible configuration of system and reservoir, i.e. $\\frac{\\partial}{\\partial E_1} (\\Omega_1 \\Omega_1) =0$ (Statistical Mechanics, R K Pathria). This along with the relation $\\frac{\\partial E_2}{\\partial E_1}=-1$ gives $\\frac{\\partial}{\\partial E_1} \\ln \\Omega_1 = \\frac{\\partial}{\\partial E_2} \\ln \\Omega_2 = \\beta$. Assuming all the configurations are equally probable, the probability is given by $P(E_1)=\\Omega_1^{-1} = e^{-\\beta E_1}$. But this doesn't imply $P(E)=0$ for $E>E_{max}~or~E<E_{min}$. The reason is probably that the boundedness condition ($E_{min}\\leq E \\leq E_{max}$) is not used in the derivation. How to obtain the correct probability factor if the energy is bounded?"}
{"id":"35260","title":"Difference between deltaV and specific impulse","text":"When reading any literature regarding space propulsion, I keep getting about these terms deltaV and ISP or specific impulse. I know specific impulse is supposed to be the velocity the propellant mass leaves the exhaust, but I'm not sure what deltaV means. For instance, I've read that from Earth low-orbit to Mars orbit we require a deltaV of 6.1, what units are those? Also, I would expect ISP to be in units of $\\frac{meter}{second}$, but most texts use units of second."}
{"id":"101738","title":"the action of $\\bar \\psi \\psi$ on a one-particle state","text":"Suppose, in the quantized theory of Dirac field, we start with some one- particle state $|\\vec p\\rangle\\equiv a^\\dagger_{\\vec p}|0\\rangle$ in the Fock space. What is the action of the operator $\\bar\\psi\\psi$ on this state? What is the intermediate physics? First I want to understand how $\\psi$ changes one-particle state $|\\vec p\\rangle\\equiv a^\\dagger_{\\vec p}|0\\rangle$ and then how does the term $\\bar \\psi$ changes the resulting state after that?"}
{"id":"101731","title":"Inequality of Clausius","text":"I'm having some trouble understanding the inequality of Clausius. My treatment of the inequality applies to an engine operating in a cycle between two temperatures (i.e. like a Carnot engine). The inequality states that for an irreversible process: $$ (1)\\ \\int _{closed\\ path} \\frac {dq_{irr}}{T}<0 $$ From what I've been reading, $ {dq_{irr}} $ is the heat added to the system (the engine) and $ T $ is the temperature of the reservoirs, $ T_{res} $. If this is the case, then $-{dq_{irr}}$ is the heat added to the reservoirs, $ dq_{irr,res} $, and $$ (2)\\ \\int _{closed\\ path} \\frac {-dq_{irr}}{T_{res}}=\\int _{closed\\ path} \\frac {dq_{irr,res}}{T_{res}}>0 $$ I read somewhere that if the reservoir is infinite then any heat added is actually occurring infinitely slowly. Is this true? If this is the case, then $$ (3)\\ dq_{irr,res} = dq_{rev, res} $$ $$ (4)\\ \\int _{closed\\ path} \\frac {dq_{rev, res}}{T_{res}}=∆S_{res}>0 $$ This seems reasonable. Since the engine itself is operating in a cycle, $ ∆S_{sys} = 0 $. This therefore satisfies the second law of thermodynamics: $$ (5)\\ ∆S_{univ}=∆S_{sys}+∆S_{res}=0+∆S_{rev}>0 $$ Does this all look right? $ (5) $ seems to say that the entropy change for an irreversible process is 0 for the case of the system itself, but is > 0 for the case of the surroundings. Is this only the case for the situation I have described? I am also unsure about $ (3) $. If the heat is being added to the reservoirs infinitely slowly, then wouldn't the same be true of the system? This would make $ ∆S_{sys} = -∆S_{res} ≠ 0 $ and is obviously not correct."}
{"id":"109242","title":"Why maximum energy transfer at natural frequency even if max amplitude occurs below $f_0$","text":"This is a paragraph from my book: \"For a damped system, the resonant frequency at which the amplitude is a maximum is lower than the natural frequency.However, maximum transfer of energy, or energy resonance always occurs when applied frequency is equal to natural frequency\" This doesn't make intuitive sense to me. I understand that if there is damping, maximum amplitude occurs below $f_0$... so, shouldn't maximum energy be transferred at this driving frequency as amplitude maximum instead of at $f_o$?"}
{"id":"130212","title":"Can \"vacuum be brought\" from outer space?","text":"Instead of creating vacuum in chambers on earth, why can't vacuum be brought from outer space in chambers? Outer space pressure ranges from $10^{-6}$torr to $10^{-17}$torr very very low. Is it possible at all?"}
{"id":"30968","title":"Second Law of Thermodynamics, heat and quantum-level explanation","text":"1. I heard somewhere that the second law of thermodynamics is stating that some heat (energy) becomes unusable for work (at the scale of the whole universe). Why is it like that? Can't heat be converted into some energy forms? I do understand that entropy is the measure of spontaneity, but I am having hard time understanding this.   2. How can we depict heat in relativistic (classical) level and quantum level? Is there heat energy that is not mediated by any form of particle?"}
{"id":"112068","title":"Confused about the concept of time and time dilation","text":"I am having a hard time understanding what is time. If scientists define time as a multiple of caesium frequency, then time itself is dependent on motion, so what if I have a number of particles that are at 0 Kelvin? Do these particle experience time by the fact it exist in the fabric of spacetime? And if so, is this the fundamental reason why time dilation exists? Because you cannot exceed the speed of light therefore your movement is restricted, thus the time dilates as your movement is restricted?"}
{"id":"128848","title":"Why would classical correlation in Bell's experiment be a linear function of angle?","text":"Sorry if it's a newbie question, but I have trouble understanding the following part in the Wikipedia's explanation for the Bell's theorem: > With the measurements oriented at intermediate angles between these basic > cases, the existence of local hidden variables could agree with a _linear > dependence_ of the correlation in the angle but, according to Bell's > inequality (see below), could not agree with the dependence predicted by > quantum mechanical theory, namely, that the correlation is the _negative > cosine_ of the angle. Experimental results match the curve predicted by > quantum mechanics. Accompanied with the following plot showing corellation as a function of angle: ![quantum mechanics predicts correlation to be a cosine function](http:\/\/i.stack.imgur.com\/zCAMO.png) But, the thing that troubles me is that I can't find the proof to that reasoning. A projection of a line onto an axis is the cosine of its angle, right? To clarify, if I slightly modify the image from that article, let's say you measure particles A and B using parallel detectors: ![Parallel case](http:\/\/i.stack.imgur.com\/O616g.png) The results are perfectly correlated (well, anti-corellated, but that's due to spins being different), as expected. Now, if you deflect the B measurement device slightly ($\\pi\/8$, since that's where the \"large classical error\" happens), we can represent it like this: ![pi\/8 angle](http:\/\/i.stack.imgur.com\/XeSgj.png) As a complete newb, my knee-jerk reaction would be that detector B would detect a projection of $cos(\\pi\/8)$ length, compared to the detector B, as shown in the detail: ![Detail \\(projection at the detector B\\)](http:\/\/i.stack.imgur.com\/PY8nL.png) Why does the classical theory predict linear correlation, then?"}
{"id":"131097","title":"Gauge Field Tensor from Wilson Loop","text":"It is possible to introduce the gauge field in a QFT purely on geometric arguments. For simplicity, consider QED, only starting with fermions, and seeing how the gauge field naturally emerges. The observation is that the derivative of the Dirac field doesn't have a well-defined transformation, because: $$n^\\mu \\partial_\\mu \\,\\psi = \\lim\\limits_{\\epsilon\\rightarrow 0}\\big[\\psi(x+\\epsilon n)-\\psi(x)\\big],$$ i.e. the derivative combines two fields at different spacetime points (having different transformation rules). We need to introduce a parallel transporter $U(y,x)$ that transforms as $$U(y,x) \\rightarrow e^{ig\\alpha(y)}U(y,x) e^{-ig \\alpha(x)},$$ such that we can adapt the defintion of the derivative into a covariant derivative, that transforms in a well-defined way: $$n^\\mu D_\\mu \\,\\psi = \\lim\\limits_{\\epsilon\\rightarrow 0}\\big[\\psi(x+\\epsilon n)- U(x+\\epsilon n,x)\\psi(x)\\big].$$ From geometric arguments, it is straightforward to show that the parallel transporter is a Wilson line: $$U(y,x) = \\mathcal{P}\\,e^{ig\\int\\limits_x^y dz^\\mu\\, A_\\mu(z)},$$ which introduces a new field, namely the gauge field $A_\\mu$. See e.g. Peskin & Schroeder Chapter 15 for more detail. However.. Where the interaction term $\\bar{\\psi} A_\\mu \\psi$ emerged in a natural way, I totally don't see how the kinetic terms emerge. The standard way to proceed, is to consider a Wilson loop (a Wilson line on a closed path), and use Stokes' theorem: $$\\text{exp}\\left\\\\{ig\\oint_\\mathcal{C}dx^\\mu\\, A_\\mu \\right\\\\} = \\text{exp}\\left\\\\{ig\\int_\\Sigma dx^\\mu \\wedge dx^\\nu\\,\\left(\\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu \\right)\\right\\\\},$$ where of course $\\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu \\equiv F_{\\mu\\nu}$. In Peskin & Schroeder, they then consider a small rectangular loop, and see that in the limit $\\epsilon\\rightarrow 0$, $F_{\\mu\\nu}$ is invariant. But what's the point? I mean, the transformation law for $A_\\mu$ is easily calculated from the definition of the Wilson loop: $$A_\\mu\\rightarrow A_\\mu+\\partial_\\mu \\alpha,$$ making $F_{\\mu\\nu}$ invariant by definition: $$F_{\\mu\\nu}\\rightarrow \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu +\\square \\alpha-\\square \\alpha. $$ I would have liked to see a calculation, starting from a particular loop parameterisation, that naturally leads to the correct kinetic terms in the Lagrangian, as was the case for the interaction term. In other words $$\\text{exp}\\left\\\\{ig\\oint_\\mathcal{C}dx^\\mu\\, A_\\mu \\right\\\\} \\leadsto -\\frac{1}{4}\\left(F_{\\mu\\nu}\\right)^2,$$ but I have no idea how to do it. Or is the idea simply 'look I have found some quadratic derivative terms that are invariant, now let me fiddle a bit and put its square in $\\mathcal{L}$'? If yes, then why do Peskin & Schroeder bother calculating a loop parameterisation (p484), if using Stokes' theorem would have been enough to find $F_{\\mu\\nu}$ somewhere? **EDIT** Based on the comments, a few clarifications. This is not a question about why we need kinetic terms or how they should like. I perfectly know how to construct the SM Lagrangian the standard way. You have a vector field $A_\\mu$, you need kinetic terms so you use some Klein-Gordon like structure, adapt it a bit (because of gauge behaviour of $A_\\mu$) and so on. Standard QFT. But this is all _manual_ , almost a bit trial and error, putting in terms because you know they work the way they should. It is like making the Lagrangian as a puzzle: just put in those pieces that fit. No problem with that, it works and is a widely accepted way of doing physics, but from a theoretical point of view, it is not so elegant. But we can do it in a more elegant way. If we start only from a Dirac field and the Dirac equation, then purely by mathematical and geometric arguments, the gauge field appears, as discussed above. The question is if we can also make the kinetic terms for the gauge field appear purely based on mathematical and geometric arguments. If you believe the QFT book by Peskin & Schroeder, you can, starting from a Wilson loop (as discussed above see p484-494). But then you end up with a factor $\\epsilon^4$, the loop area squared, in front of the field tensor. You could restrict to a class of loops with area $S=1$, but there are two problems with that:   * In the calculation, the exponent and integral where expanded, using the fact that $\\epsilon <\\\\!\\\\!< 1$. This contradicts our restriction $S=1$.   * This restriction invalidates the elegance somewhat, as now the loop area has to be fine-tuned. I expected to find something like 'in the limit $\\epsilon\\rightarrow 0$, the field tensor is what remains'. So, is it not possible then to get the kinetic terms for the gauge field purely from geometric arguments? If the answer is no, it doesn't make sense to brag about the natural emergence of the interaction term. It is not elegant when the interaction term emerges naturally, but you have to hand-pick the kinetic terms. For one, how do you prove that the naturally emerging field (interaction terms) and the one you put in (kinetic terms) are the same field?"}
{"id":"131099","title":"Hooke's Law: why is only one force considered?","text":"I have a question about Hooke's Law springs. What I learned is that a Hooke's Law spring gives a force F pushing left and right and each force equals $k \\cdot |l-l_o|$ Let's say there is a mass block resting on a spring on a table. The amount of force on the mass block by the spring is thought to be $F$. But I am thinking that that is only considering the force of the spring pushing against the block. Shouldn't we consider the $F$ the spring is pushing against the table as well?"}
{"id":"92754","title":"Which of these two different forms of spin-orbit interaction is correct?","text":"I am seeing the spin-orbit interaction in two different ways:   1. $\\lambda [\\mathbf{p} \\times \\nabla V]\\cdot \\sigma$   2. $\\lambda [\\nabla V \\times \\mathbf{p}]\\cdot \\sigma$ I don't see how these two expressions can be equivalent though; in (1) there will be derivative operators from momentum acting on the gradient whereas in (2) this will not be the case. Which one of these is correct?"}
{"id":"73459","title":"Moving electron - finding the wavefunction","text":"On our modern physics class my professor did a problem: > Write down a wavefunction of an electron which is moving from left to right > and has an energy $100\\text{ eV}$. At first i said: \"Oh i know this!\" and solved the case like this. **My solution:** The energy $100eV$ must be the kinetic energy of the electron. So i said ok this kinetic energy is very small compared to the rest energy and i can say that $pc \\ll E_0$ which means i have a classical limit where: \\begin{align} E=\\sqrt{{E_0}^2 + p^2c^2}\\\\\\ E=\\sqrt{{E_0}^2 + 0}\\\\\\ \\boxed{E=E_0} \\end{align} So now i can write the general wavefunction for a free right -mooving particle like this: $$\\psi=Ae^{iLx}\\quad L=\\sqrt{\\tfrac{2mE}{\\hbar^2}}$$ So if i want to get the speciffic solution i need to calculate the constant $L$ and then normalise the $\\psi$. Because $E=E_0$ i calculated the constant $L$ like this: $$L=\\sqrt{\\frac{2mE_0}{\\hbar^2}}$$ while my professor states that I should do it like this: $$L=\\sqrt{\\frac{2mE_k}{\\hbar^2}},$$ where $E_k$ is the kinetic energy of the electron. Who is wrong? I mean whaaaaat? The constant $L$ is after all defined using the full energy and not kinetic energy..."}
{"id":"101065","title":"Why are reflected objects blurry?","text":"While this may seems like a strange question, I'll try to explain it the best I can. Like 75% of Americans, I wear glasses and contacts to correct my worsening vision. I'm not far sighted so objects in the distance appear blurry. I took my glasses off to look at my iPad. After doing so I noticed in the reflection of the glass, my lamp. The lamp is in the distance and looks blurry to me. But on my iPad's glass it was blurry unlike the rest of the objects on screen, including the glass which I could see clearly because I'm near-sighted. It is strange to me why the reflection was blurry even though it was close up. How can this be?"}
{"id":"43849","title":"Is it possible to recover the old Bohr-Sommerfeld model from the QM description of the atom by turning off some parameters?","text":"Is it possible to recover the old Bohr-Sommerfeld model from the QM description of the atom by turning off some parameters? Can we use Ehrenfest's theorem (or some other scheme) to reduce the QM model to the Bohr-Sommerfeld model? If not, why not? The issue seems to be significant because it might shed some deep conceptual issues."}
{"id":"121579","title":"Differential or integral form of the conservation equations?","text":"Is there a 'rule' for when it is best to use either the differential or integral form of the continuity and momentum equations in calculations?"}
{"id":"75884","title":"Why planes have propellers in front but watercraft have them behind?","text":"Why do propeller airplanes mostly have their propellers in front (of the fuselage or wings) while ships and boats mostly have them at the back? I realize that there are aircraft with pusher configurations but they are relatively rare."}
{"id":"75882","title":"What is the difference between phase difference and path difference?","text":"I have learnt that path difference is the difference between the distance travelled by two waves meeting at a point. If that is path difference,then how will one know what is phase difference and how can one calculate or determine it?"}
{"id":"93392","title":"Biconvex vs planoconvex lens","text":"As I understand, less spherical aberration is obtained when a collimated beam is focussed with a planoconvex lens as opposed to a biconvex lens. What would be a situation when a biconvex lens should be used? When should one use a biconvex lens vs a planoconvex lens?"}
{"id":"134310","title":"Special Relativity Question: Doppler shift","text":"Imagine an observer watching a moving rocket carrying on it both, a light source and a clock. If on the rocket the clock is synchronized with the frequency of the light being emitted then will not a stationary observer see not only the clock on the ship slow but then necessarily the wave length of the light increase. Again, imagine those on the rocket are measuring the distance of one wavelength comparing it with some high precision ruler, then will not the observer watch the ruler contract and then necessarily the wavelength of the light along with it, a contradiction to the original conclusion?"}
{"id":"128151","title":"On the coordinate independence of general relativity","text":"I've been having a bit of trouble with the idea of coordinate independence in general relativity. Let me start with a simple example that I think illustrates my question conceptually: Say you have two objects, A and B. A is at rest and B accelerates directly away from A. From a mathematical point of view, there is absolutely nothing which distinguishes the two objects - A is accelerating away from B as much as B is from A. To distinguish the two, we need something extra - something empirical which tells us that B feels a force acting on it and A does not. In this sense, there are infinitely many coordinate motions which might be the truly unaccelerated ones, and they can only be distinguished by the _physical_ fact that they feel no forces. So, at some point, doesn't any physical law, even if it is written in such a way that it is satisfied in all coordinate systems, have to make reference to truly unaccelerated frames, or something similar? In other words, in order to connect an abstract mathematical description of motion, which in some sense can be transformed in any way, to physical observations, doesn't one have to anchor the mathematical reference somehow to physically meaningful concepts like unaccelerated frames? Now, as I've been studying the Einstein field equations, I've wondered about the following: if you compare the equations for empty, flat space, in inertial coordinates versus some weird accelerating coordinates they'd be exactly the same - the space must be Ricci flat, or $R^{\\alpha \\beta} = 0$. Of course, there are many solutions of $R^{\\alpha \\beta} = 0$ (including non-flat ones, but ignore those for a second). Without some physical knowledge, wouldn't it be impossible to say which coordinates lead to $g_{\\alpha \\beta} = \\eta_{\\alpha \\beta}$, and which coordinates lead to $g_{\\alpha \\beta}$ being some complicated (still Ricci flat) function of the coordinates? I imagine that this means that while the field equations must be satisfied, they alone are not enough to say what someone will observe - one must also know how his coordinates relate to locally inertial ones. So I now ask: by the \"coordinate independence of General Relativity\", does one really just mean that the expression for curvature makes no reference to a coordinate system - that the expression for curvature is coordinate independent, and so the law relating curvature and mass-energy, the Einstein field equation, is valid in all coordinate systems? And yet, even though the law must be satisfied, in order to know what one will observe, it's necessary to also know which coordinate systems are (locally) inertial? Finally, if you're feeling up to it, is this a real point I'm making in general? Don't the mathematical laws of physics all have to \"give up\" and some point and make reference to the solely empirically-defined concept of unaccelerated motion? **EDIT** : I don't think I made myself totally clear - let's just say this. If I asked you to solve Einstein's field equations in flat space in the coordinates $x^\\mu$, wouldn't you, in general, have to say that there's not enough information? How could you possibly know whether $x^\\mu$ were inertial coordinates and $g_{\\mu \\nu} = \\eta_{\\mu \\nu}$, or if $x^\\mu$ were some weird accelerated coordinates, and $g_{\\mu \\nu}$ were the same flat metric but written in these accelerated coordinates? Both are solutions of the Einstein field equations. In this way, aren't the field equations underdetermined? Don't they have to be supplemented with information about how the coordinates relate to locally inertial coordinates?"}
{"id":"33082","title":"Is the Chern-Simons integral of gauge fields over black hole singularities zero?","text":"Suppose we have an evaporating black hole and a nonabelian Yang-Mills theory with a $\\theta$ topological term. This counts the total number of instantons minus antiinstantons. Consider the total number of instantons minus antiinstantons inside the black hole, i.e. between the event horizon and the singularity. This is equal to the difference between Chern-Simons integral between the singularity and event horizon. According to black hole complementarity, we replace the interior with a stretched horizon. Do we have to add an effective Chern-Simons term to the stretched horizon action? If so, is this equivalent to saying the CS integral over the singularity has to be zero? Otherwise, we get decoherence by CS singularity value, converting pure states to mixed states?"}
{"id":"33084","title":"How is this a condition for a Majorana zero mode?","text":"EDIT: Nevermind, I figured it out. When you apply the expression on a state, you can clearly see what it does: The new state created by applying the Majorana operators must have the same energy as the original one. So when applied on a zero-energy state, this creates a degeneracy and the so-called Majorana zero mode. * * * I am currently trying to grasp this lecture by Frank Wilczek about Majorana zero modes on an 1-dimensional wire. In equation (31), he states that a general condition for such a zero mode is $$ \\left[ H, \\sum_{j=1}^{2L} c_j \\gamma_j \\right] = 0, $$ where the $c_j$ are coefficients and the $\\gamma_j$ are the Majorana operators for the sites of the chain. I'm not sure I understand why this is a general condition for a zero mode if the two operators commute. Can anyone shed some light on this? Thanks in advance!"}
{"id":"122009","title":"Help in Basics for Capacitance","text":"How does capacitor store a charge? I'm a little bit confused, as I know that a capacitor stores charge but how does it do so?"}
{"id":"44231","title":"Positivity of Total Gravitational Energy in GR","text":"I read the following statement in the introduction to an article: > _Over the last 30 years, one of the greatest achievements in classical > general relativity has certainly been the proof of the positivity of the > total gravitational energy, both at spatial and null infinity. It is > precisely its positivity that makes this notion not only important (because > of its theoretical significance), but also a useful tool in the everyday > practice of working relativists._ Since I haven't been involved in GR for about 30yrs, I've missed this. Could someone briefly explain how these results are stated and give some references if possible."}
{"id":"44238","title":"Fock picture of bosonification in condensates","text":"I want to understand how bosonification in a condensate must be interpreted in the Fock states picture Say i have uncoupled fermions in a set of states $E_1$, $E_2$ ... over the vacuum $E_0$. They occupy all the levels up to the Fermi energy. Now, i introduce some coupling between them so, they become effectively boson pairs. Are the $E_i$ energy states out of the picture for these bosons? My random guess is that fermions would couple in states like $$ | E_1 \\times E_2 \\rangle - | E_2 \\times E_1 \\rangle $$ $$ | E_1 \\times E_2 \\rangle + | E_2 \\times E_1 \\rangle $$ $$ | E_3 \\times E_4 \\rangle - | E_4 \\times E_3 \\rangle $$ $$ | E_3 \\times E_4 \\rangle + | E_4 \\times E_3 \\rangle $$ And so on, but this is the aspect i'm not sure and i want clarification What are the states where the boson pairs are condensing into??"}
{"id":"33558","title":"Could you make a bar of metal glow by forcing the charge density on the surface to oscillate?","text":"Since light is just a changing electromagnetic wave through space, could you create light by changing a charge density at a certain frequency?"}
{"id":"33557","title":"Why equivalence principle is principle and not law?","text":"We can prove that the inertial mass and the gravitational mass should be the same (equivalence principle) from the $f=mg=ma$ then $g=a$, so we have equivalence _law_! But why we said equivalence _principle_?"}
{"id":"114271","title":"What makes laminar cascade break?","text":"Near my house there is a mall that have a **cascade, which has a pratically constant flow, and doesn't seem to have perturbations** (at least near the edge where water falls), between its two levels. What botters me is thar that laminar flow that seem unperturbed, break in the middle after falling for a height of approximately 1.1m (it falls another 0.5m, before being colected by a pool, and, I think, repumped up). I don't know exactly how to explain that behavior but I think it must have to do with waves propagating on that filament of water and that make it **break after falling a height $h$**. Part of that flow is directed forward, and part is directed backwards (see the front view of the sketch below). From this view I can conclude also that there is a perfect left-right simetry (in relation to the vertical red line). ### Update1: I incluided also what I expect from the velocity vectors due to viscosity (with purple arrows on the sketch), similar of what we can see on Flow velocities due to viscosity of the Laminar flow text. Unfortunately I don't have a real picture, or vídeo, of the local. * * * ## What is causing this effect? How can it be explained? * * * ![Cascade Views - Sketch](http:\/\/i.stack.imgur.com\/2YgkR.png)"}
{"id":"127760","title":"Calculation of pressure from flow rate of water","text":"Anybody kindly help me to find how to calculate pressure in bar from flow rate. I have a pipe and from that I am transferring water at a constant flow rate of 5ml\/min. At this flow rate, with a 0.5 cm diameter pipe, what will be the pressure generated by water?"}
{"id":"71761","title":"Independancy of angular momentum with respect to origin for rotation about center of mass","text":"Why is angular momentum for rotation about the center of mass independent of origin of the coordinate system?"}
{"id":"89959","title":"Find the force used to press a book against a leaned desk so it doesn't slip","text":"I know there are several threads concerning this issue, but none of it helped me, so I decided to open a new one. So, the task is to find a force $F_{\\mathrm{push}}$ on a desk which is leaned for an angle of $\\varphi=45^{\\circ}$ I know that for a vertical system you calculate that $F_{\\mathrm{push}}=F_{\\mathrm{friction}}=k F_{\\mathrm{\\mathrm{normal}}}$ and that's how it is calculated, but for this particular case I do not know... I also know that $F_{\\mathrm{static}} = mg\\cos(\\varphi)$ and $F_{\\mathrm{dynamic}} = mg\\sin(\\varphi)$, so I guess that the $F_{\\mathrm{friction}} = -kmg\\cos(\\varphi)$ Please help me, are any of these formulas correct?"}
{"id":"8045","title":"I lost a factor of two in the electromagnetic field tensor","text":"I apologize for this simple question, but I lost a factor of 2 and can't find it anymore, so now I'm looking on the internet, perhaps one of you has some information about its whereabouts. :-) Consider the electromagnetic field tensor $F_{\\mu\\nu}$ which corresponds to a differential form $F =\\frac12 F_{\\mu\\nu} dx^\\mu \\wedge dx^\\nu$. The wedge product of $F$ with itself can be expressed as $$ F \\wedge F = \\frac14 F_{\\mu\\nu}\\mathcal F^{\\mu\\nu} $$ where $\\mathcal F^{\\mu\\nu} = \\frac12 \\epsilon^{\\mu\\nu\\sigma\\tau}F_{\\sigma\\tau}$ is the dual field tensor. Writing $A = A_\\mu dx^\\mu$ for the vector potential, we have $$ \\int_V F \\wedge F = \\int_V (d A) \\wedge F = \\int_V d(A \\wedge F) = \\int_{\\partial V} A \\wedge F$$ But it appears to me that the latter integral is equal to $$ \\dots = \\int_{\\partial V} dx^\\mu\\ A_\\nu \\mathcal F^{\\mu\\nu} .$$ Yet, at the same time, we can write $$ \\int_V dx^\\alpha \\frac12 F_{\\mu\\nu}\\mathcal F^{\\mu\\nu} = \\int_V dx^\\alpha (\\partial_\\mu A_\\nu)\\mathcal F^{\\mu\\nu} = \\int_V dx^\\alpha \\partial_\\mu (A_\\nu\\mathcal F^{\\mu\\nu}) = \\int_{\\partial V} dx^\\mu\\ A_\\nu \\mathcal F^{\\mu\\nu} .$$ Clearly, that cannot be. > I have lost a factor of two. Where is it?"}
{"id":"8049","title":"Are these two quantum systems distinguishable?","text":"Suppose Stanford Research Systems starts selling a two-level atom factory. Your grad student pushes a button, and bang, he gets a two level atom. Half the time the atom is produced in the ground state, and half the time the atom is produced in the excited state, but other than that you get the exact same atom every time. National Instruments sells a cheap knockoff two-level atom factory that looks the same, but doesn't have the same output. In the NI machine, if your grad student pushes a button, he gets the same two-level atom the SRS machine makes, but the atom is always in a 50\/50 superposition of ground and excited states with a random relative phase between the two states. The \"random relative phase between the two states\" of the NI knockoff varies from atom to atom, and is unknown to the device's user. Are these two machines distinguishable? What experiment would you do to distinguish their outputs?"}
{"id":"89484","title":"Doubt regarding inclined plane in feynamn lectures","text":"I recently read something in the feynman lectures which I find difficult to understand. Imagine an inclined plane on the top of which is a smooth pulley. There is a weight hanging from the pulley and another weight at the base of the inclined plane on the other side. Feynman says since the weights are just balanced, the system is reversible and the weights can move up and down? What does feynamn mean here and why is the word reversible significant here? Also if the weights are balanced how can they move up and down?"}
{"id":"28887","title":"Rayleigh diffraction by circular aperture","text":"I am a beginner to physics and would need an explanation on a statement in a book \"Karttunen, _Fundamental astronomy_ \". In a section named _\"Rayleigh diffraction by circular aperture\"_ , author states: > The sum of the amplitudes of the waves from a small surface element is > **proportional** to the area of the element $\\operatorname{d}x$ > $\\operatorname{d}y$ And then author writes down the equation for amplitude vector $\\boldsymbol A$ $$ \\boldsymbol A = \\operatorname{d}x \\operatorname{d}y (\\cos{\\delta} \\boldsymbol{\\hat{i}} + \\sin{\\delta} \\boldsymbol{\\hat{j}}), $$ where $\\cos{\\delta} \\boldsymbol{\\hat{i}} + \\sin{\\delta} \\boldsymbol{\\hat{j}}$ is a direction of an amplitude vector $\\boldsymbol A$ and vectors $\\boldsymbol{\\hat{i}}$ and $\\boldsymbol{\\hat{j}}$ are unit vectors. **And my question is, how does the author know that amplitude is proportional to element $\\operatorname{d}x$ $\\operatorname{d}y$ ???**"}
{"id":"28880","title":"Supersymmetry in Quantum Mechanics","text":"I was reading _Supersymmetry in Quantum Mechanics_ and got stuck in the various mathematical terminology like \"Graded-Lie Algebra\", \"Super Algebra\". Is there any good lecture notes concerning these topics, explaining the basic mathematics needed to understand supersymmetry. I know basic analysis, topology, and first course in group theory, but never encountered ring theory, modules etc."}
{"id":"69686","title":"Definition of Kinetic energy","text":"In class we had that $ T= \\frac{1}{2}T_{ij}v_iv_j$ where we used the Einstein summation convention. Hitherto we only discussed examples where the kinetic energy was dependent of the square of one coordinate and our teacher assured us that this is the case in most examples. Now I was wondering whether anybody if you knows an example where the kinetic energy depends in ONE term on different $ v_i , v_j$?"}
{"id":"19028","title":"Conjugate Variables and Fourier Transforms in Classical Physics","text":"Let q be a generalized coordinate with a conjugate momentum p and a potential resulting in a periodic motion of q. What is the meaning of the Fourier transform of q(t) over its period? Can this be interpreted as the distribution of momenta? I know this is the case in QM but what happens in the Classical case? For instance for a particle in a box (classical) with an initial velocity of p\/m, the motion is a triangle wave. The Fourier transform is a squared sinc function. As the triangle wave gets smaller, the F transform spreads out and vice versa. Very nice but the only two momenta for the particle are p and -p as the particle either moves up with constant speed or moved down at constant speed."}
{"id":"116340","title":"Quick question on sketching wavefunction in well","text":"![enter image description here](http:\/\/i.stack.imgur.com\/kjjrv.png) Usually for an infinite well, the sketch for n=3 level is this: ![enter image description here](http:\/\/i.stack.imgur.com\/I6Nqf.png) Now I think if one side of the potential barrier is higher, the particle will be more likely to spend time on the left side than the right side, so the wavefunction should have higher amplitudes on the left (skewed to the left): ![enter image description here](http:\/\/i.stack.imgur.com\/iepnV.png)"}
{"id":"82005","title":"Why are all basic physical measurements ratios?","text":"Why is it that \"all basic physical measurements [...] are ratios\"? This question comes from the book _Mathematical Physics_ by Donald H. Menzel. Here is the explanation given: > **1\\. The significance of an observation.** Any direct observation of a > physical nature ordinarily results in a number expressing the magnitude of > the measured quantity. The simpler measures are those of lengths, of masses, > or of times. More complicated ones may be of velocities, of energies, or of > angular momenta. The number, by itself, does not indicate what is being > observed; its magnitude depends on the upon the type of measuring scale > employed. We may express lengths in centimeters, miles, or light years; we > may define masses in grams, tons, or in units of solar mass. _All basic > measurements are, therefore, ratios._"}
{"id":"60266","title":"Higgs VEV in terms of measurements on an ensemble?","text":"Let $A$ be a Hermitian operator corresponding to some observable. If we prepare $N$ identical systems in the state $\\psi$ and measure this observable in each system, the average of the measurements (for large $N$) will be $\\langle \\psi | A | \\psi \\rangle$. In the standard model, $\\langle 0 | \\phi^0(x) | 0 \\rangle = v\/ \\sqrt{2} = 174\\ GeV\\ \\ \\forall x$. Questions:   1. What observable, if any, corresponds to $\\phi^0(x)\\ $?   2. Is $v\/ \\sqrt{2}$ really the average of a large number of measurements of identical systems in the vacuum state? (For precision, you could modify question 1 to \"what observable corresponds to $\\int d^4 x\\ f(x) \\phi^0(x)$,\" where $f:\\mathbb{R}^4 \\to \\mathbb{R}$ is a smooth function localized about some $x_0$.)"}
{"id":"64011","title":"Christoffel symbol for Schwarzschild metric","text":"I know that the christoffel (second kind) can be defined like this: $$\\Gamma^m_{ij} = \\frac{1}{2} g^{mk}(\\frac{\\partial g_{ki}}{\\partial U^j}+\\frac{\\partial g_{jk}}{\\partial U^i}-\\frac{\\partial g_{ij}}{\\partial U^k})$$ but I don't know how $U^i$ is defined (specifically for the Schwarzschild metric."}
{"id":"20875","title":"Does the existence of dualities imply a more fundamental structure?","text":"I was wondering if the existence of some kind of duality in physics always implies the existence of some underlying more fundamental structure\/concept? Let me give a few example from history:   1. Wave-particle duality$~\\Rightarrow~$ Existence of quantum particle.    2. Heisenberg's Matrix Mechanics $\\&$ Schrodinger's wave formulation of QM $~\\Rightarrow~$ Existence of Dirac formulation of QM.   3. Magnetic field $\\&$ electric field $~\\Rightarrow~$ Existence of an electromagnetic theory. Similarly, can one conclude that   1. for example, from AdS\/CFT correspondence,   2. or more generally, because there is an holographic equivalence between quantum gravity in $D+1$ dimensions, and QFT in $D$ dimensions,  then there must be a more fundamental underlying structure that incorporate both sides of the correspondence?"}
{"id":"20871","title":"Blandford-Znajek process: Why\/how does the current flow along the magnetic field lines","text":"Related: How would a black hole power plant work? _I have put a bit of commentary enumerating my confusions in parentheses_ I read in _Black Holes and Time Warps_ (Kip Thorne), that quasars can generate their jets from four different processes. These all involved the accretion disk, but there was one which doesn't make quite as much sense. It was called the Blandford-Znajek process, and it involved magnetic field lines carrying current. The process was visualized in two ways. A black hole, with magnetic field lines, is spinning. In the first visualisation (viewpoint actually), the magnetic field lines 'spin' along with the black hole, and nearby plasma is anchored onto the field lines by electrical forces (where did the electrical fields come from?). The plasma can slide along the field lines but not across them (why?). Since the field lines are spinning, centrifugal forces will fling them up and down the field lines, forming jets. The other viewpoint is this, and it makes even less sense (to me that is, I haven't had a formal education in GR): The magnetic fields and the swirl of space generate a voltage difference across the field lines (Why? How?). The voltage carries current across the magnetic field lines (why are the _field lines_ behaving like wires?). This current travels across plasma, which accelerates it, creating the jets. Now the main thing that doesn't make sense, is that magnetic field lines are behaving like wires. Why would they? I suspect the answer lies hidden somewhere in the equivalence of EM waves in different frames, but I can't think up any convincing argument from that side. If the answer involves GR equations, you don't need to solve it here (wouldn't make sense to me), but if you have to, just refer to the equation and what you did to it, along with the final result. Thanks!"}
{"id":"90287","title":"Classical electrodynamics formulated in terms of forces","text":"The Newton's law of universal gravitation is described in terms of a force, which is produced by an action at a distance. It also can be described using the concept of a field, and that would be an equivalent formulation. Now, classical electrodynamics is formulated in terms of fields in a more complicated way that Newton's gravitation. My question is: Is there an equivalent formulation of classical electrodynamics in terms of action at a distance that is completely equivalent to the formulation in terms of fields (Maxwell's equations)? I would tempted to think that it is not possible, since in Maxwell's equations you may have electromagnetic waves without the presence of charges, and I don't know how to do it only in terms of forces. However, I'm not entirely sure, that's why I'm asking."}
{"id":"99554","title":"Point group symmetries and unit cell","text":"I was wondering if the unit cell (of a given lattice) had to have every point group symmetries of the lattice it defines ? I guess there is no unique way to define a unit cell and that it may not have all point group symmetries. However, is it possible to define a unit cell that has all point group symmetries ?"}
{"id":"45622","title":"Does the Chandrasekhar Limit scale for a Black Hole?","text":"No physicist\/astrophysicist I; All I know about the Chandrasekhar limit is that it apparently limits the mass a star may survive, beyond which it degenerates to a neutron star, or a black-hole. Does something similar apply to a black-hole? Is there an upper limit to black-hole mass? Can it acquire infinite mass? Is there a limit beyond which it may no longer acquire mass? Does a black-hole die? Is there a corresponding time-frame applicable?"}
{"id":"45623","title":"What is the meaning of the word \"particle\" in particle physics?","text":"I want to use Matt Strassler's definition of the word \"particle\" as a specific example: Matt Strassler writes: > (1) \"...all the elementary “particles” (i.e. quanta) of nature are quanta of > waves in fields...\" My first question is, in this sentence, is the word \"quanta\" short for \"quanta of waves in fields?\" Sentence (1) simply defines the word \"particle\" to be another name for \"quanta\". I looked up quanta and quanta means > a wave that for a given frequency, have minimum energy and amplitude. So, a quantum is a wave. Is this correct? From this analysis, I conclude that, in particle physics, a particle is a wave, because a particle is defined as a quantum. In mathematical terms: $quantum \\equiv wave$ or $particle \\equiv wave \\equiv quantum$ So, in particle physics, the word particle, has the properties of a wave, and it has no other properties. A particle, in particle physics, for instance, is not a spherical object with finite radius. This is confirmed by Matt Strassler, who writes (2) \"...the word “particle” in particle physics has the same meaning as the word “quantum”...\" Sentence (2) is incomplete, because the word \"particle\" has several other meanings besides being another name for quanta. This is very confusing to non- physicists because the word \"particle\" has been defined many times and physicists pick and choose a meaning that saves the context. My questions are: 1) The word \"quantum\" is a perfectly good word meaning a wave with minimum amplitude; why are physicists insisting on calling a quantum a particle? 2) Is there a unique definition of the word particle in physics not dependent on context (not casuistic) and that all physicists can agree upon? If you cannot answer any of these, can you clarify if \"particle\" is another name for \"quantum\"."}
{"id":"90288","title":"Does zero change in magnetic flux always imply zero emf induced?","text":"If you have a uniform B field, with a finite piece of wire inside it. Assuming the B field spans all space and the wire cannot leave the field. Are you able to create an emf by moving the wire ? I say there is only one possible way, and that is if you rotate it so from its starting perpendicular to the field position it does parallel to the field. That would create a change in flux. But are you able to create emf if the wire stays perpendicular to the wire?"}
{"id":"9169","title":"Correlators at large N and large N factorization","text":"I am having this very basic problem. In e.g Maldacena's AdS\/CFT review (0309246) (page 5), he has defined operators as $\\mathcal{O}=N\\,{\\rm tr}[f(M)]$ for some matrices $M$ and got the connected correlators as $\\langle\\mathcal{O}\\mathcal{O}\\rangle_c\\sim N^2$ and $\\langle\\mathcal{O}\\mathcal{O}\\mathcal{O}\\rangle_c\\sim N^2$. Whereas in the MAGOO review (9905111) (page 14-15 e.g) we see some operators defined as $G$ which is added to the action as $S\\rightarrow S+N\\sum g_jG_j$ which gives the correlators as $\\langle\\prod_j G_j\\rangle\\sim N^{2-n}$. So that three points and higher vanish at large N. I thought this one is generic. If so I wanted to understand how is the first conventions\/case are different and their difference (has it something to do with connected\/disconnected diagram? etc.). Related to that I wanted to know about large N factorization. For a particular model if I calculate disconnected diagrams, I see that at large N I indeed get this factorization. I learned that its like WKB in $\\hbar\\sim 1\/N$. But I wanted to feel it more generally in possibly a mathematical+insightful way if possible. Thanks for the help. Edit: If it has a long answer please at least let me know which way to think or any references where I can get some ideas. Thanks again."}
{"id":"53082","title":"Does Mansuripur's Paradox involve fictitious charges?","text":"Mansuripur's Paradox involves a magnet moving at relativistic speeds in an external electric field. _Additional:_ thanks to Retarded Potential, who found the original paper. If I understand correctly, the fictitious charges on either side of the magnet should cause a torque, which of course does not actually exist. Apparently this is conventionally resolved by adding an (equally fictitious) internal angular momentum. Mansuripur appears to be arguing that a better resolution would be to add a (fictitious) additional term to the Lorentz force law. Am I right in thinking fictitious charges are involved? If so, why are they introduced, i.e., what is the advantage in doing so? If not, where does the torque really come from?"}
{"id":"53087","title":"What's the difference between these two formulas and how are these called?","text":"I just want to know the differences between these two formulas: $h = h_0 + v_0 t ± \\frac{1}{2} g t^2$ and $y = y_0 + v_{0y} t + \\frac{1}{2} g t^2$ Also, how are these called in English?"}
{"id":"63790","title":"Radioactive Decay","text":"> **Problem** :Nuclei of a radioactive element $\\Bbb X$ having decay constant > $\\lambda$ , **(** decays into another stable nuclei $\\Bbb Y$ **)** is being > produced by some external process at a constant rate $\\Lambda$.Calculate the > number of nuclei of $\\Bbb X$ and $\\Bbb Y$ at $t_{1\/2}$ I tried to create an equation for rate of change of the number of nuclei a: $$\\dfrac{dN_{X}}{dt}=\\Lambda-N_X\\lambda $$ I did that because in simple decay $\\dfrac{dN}{dt}=-\\lambda N$ holds and here it's also being produced by rate. But after integration should we write $$ln\\Bigg(\\dfrac{\\lambda N_X-\\Lambda}{\\lambda N_0-\\Lambda}\\Bigg)=-\\lambda t$$ or $$ln\\Bigg(\\dfrac{\\lambda N_X-\\Lambda}{N_0}\\Bigg)=-\\lambda t$$ First one because limit was on $N: (N_0\\to N)$ And next what to substitute for $t$ (ie. what is $t_{1\/2}$? $ln2\/\\lambda$ or something else?) Also how to do it for $\\Bbb Y$? Just write $$\\dfrac{dN_Y}{dt}=\\lambda N_x $$?"}
{"id":"52680","title":"How is North Korea planning to test a nuclear device underground?","text":"I am reading the news about North Korea planning to test a nuclear device **underground**, and while fascinating, I have no idea what this phrase actually means. Could somebody explain, in absolute layman's terms, how an underground nuclear test actually works, what the dangers are, and how North Korea will use the results?"}
{"id":"12274","title":"Resistance from voltage and charge","text":"I'm sure this is pretty basic, but I'm not very good at electricity. > At both ends of a conductor we apply 12 volts. For one minute an electric > charge of 72 C passes through the conductor. What is the resistance of the > conductor?"}
{"id":"118995","title":"Why is number of collisions equal to number of scattered particles times number of scatterers?","text":"In Landau, Lifshitz \"The Classical Theory of Fields\" $\\S12$ \"Invariant cross- section\" the following is said1: > ... according to the usual definition of the cross-section $\\sigma$, the > number of collisions occuring in volume $dV$ in time $dt$ is $$d\\nu=\\sigma > v_{\\text{rel}}n_1n_2dVdt,$$ where $v_{\\text{rel}}$ is the velocity of > particle 1 in the rest system of particle 2 ... I'm trying to make sense of this formula, and here's what I got so far: $\\sigma v_{\\text{rel}}n_1dt$ is total number of scattered particles in time $dt$, and $n_2dV$ is total number of scattering centers. Why isn't total number of scattered particles equal to the number of collisions? Is it considered that there's more than one collision per scattered particle? If yes, then why should it appear that each scattered particle collides with _each_ of scattering centers (only in this case would we just multiply by number of scattering centers)? 1\\. In English version which I cite the formula lacks a $dV$ factor. I put it here according to original Russian book."}
{"id":"90135","title":"Do algorithms have an intrinsic time direction?","text":"This article says > There is no intrinsic time direction in Newton's mechanics nor in the > differential equations of the new physics. My question is, do other types of mathematics, say a cellular automata as an example, have the intrinsic time direction that the equations lack? With a CA, all steps have to be completed sequentially. **UPDATE** : Here's what I mean. On this page, it describes projectile motion. It uses one approach with equations, and another approach with an algorithm. Very clearly both approaches yield the same answers. The biggest difference between them is how time works. Question: Is having \"an intrinsic time direction\" the same thing as \"time having no choice but to march forward in the algorithm\"?"}
{"id":"90131","title":"Slowing of photons in a EB Condensate","text":"Help me understand this. I have heard tell that they are now able to slow the forward progress of a photon traveling through an EBC to somewhere near 35mph. Is this actually the physical slowing down of the photon, or is it the slowing down of the absorption and emission process? If the photon is actually slowed down, does it gain mass? Or am I miss interpreting this all together. Thanks"}
{"id":"8870","title":"Heat reflection on distant planets","text":"I was watching a documentary last night on the first planet discovered outside our solar system. The first one apparently is a giant gas planet that orbits very close to its sun over a very fast period (~1 week per orbit), and it's so hot all metal elements vaporise and form parts of it's atmosphere (so the current theory states anyway). Out of curiosity, when they estimate the surface heat of the planet (they give big numbers), did they take into account reflection of heat from it's metallic atmosphere, and internal reflection from heat bouncing off the surface and it's atmosphere (like global warming) when estimating surface temperature of the planet?"}
{"id":"81286","title":"Homework Question Transformation Energy","text":"A $1400kg$ car is approaching the hill shown in the figure at $14.0m\/s$ when it suddenly runs out of gas. ![enter image description here](http:\/\/i.stack.imgur.com\/RDU61.png) What is the car's speed after coasting down the other side? I think I have to use this equation $mg(\\Delta h)=\\frac 12 m(\\Delta v)^2 $ $(1400)(9.81)(\\Delta h)= \\frac12 (\\Delta v)^2$ I am not sure if this is correct. If it is, what do I plug in for $\\Delta h$?"}
{"id":"21474","title":"Can a wave propagate in an elastic fluid in the absence of volume forces?","text":"A motion (wave) $\\mathbf{x}: \\mathcal{B}_0 \\times [t_0,t_1] \\to \\mathcal{E}:$ such that   $q-o = \\mathbf{x}(p,t)=(p-o)+\\mathbf{a}_0 cos(\\mathbf{k}_0\\cdot(p-o) - \\omega_0 t)$ can propagate in an elastic fluid with constitutive equation:   $\\mathbf{T}=-\\pi(\\rho)\\mathbf{I}$ not subject to any volume force? What restrictions on polarization vector $\\mathbf{a}_0$ and the vector $\\mathbf{k}_0$ are needed? $\\mathbf{a}_0$ and $\\mathbf{k}_0$ should be parallel? What restrictions on $\\pi(\\rho)$ are needed? I can write the balance equation of linear momentum, in an eulerian description I suppose: $\\rho \\mathbf{a} = \\mathrm{div}_q \\mathbf{T} = -\\nabla_q \\pi = - \\pi' \\nabla_q \\rho$ But it is easier to compute the acceleration in a lagrangian description:   $\\ddot{\\mathbf{x}}(p,t) = -\\omega_0^2 \\mathbf{a}_0cos(\\mathbf{k}_0\\cdot(p-o) - \\omega_0 t)$ At this point computing $\\nabla_q\\rho$ is not easy I think. There is some mistake here? There is a better and simpler way to prove the results? _Thanks and Please excuse my bad english_"}
{"id":"44570","title":"Reflectance and phase shift vs incident angle and wavelength","text":"Plot the refectance and phase shift on reflection $\\varphi$ for p and s polarized light versus incident angle $\\theta$ for various wavelength $\\lambda_1=.., \\lambda_2=..,..$ in the simple case of reflection from a bare substrate given the complex refractive index $N=..$ of the substrate. Since the reflectance is $$R=(\\frac{1-\\eta_{s,p}}{1+\\eta_{s,p}})(\\frac{1-\\eta_{s,p}}{1+\\eta_{s,p}})^* $$ where $\\eta_{s,p}$ is $Ncos\\theta$ and$\\frac{N}{cos\\theta}$ for s and p polarized light. Also, the phase shift on reflection is:$$\\varphi=\\arg(\\frac{1-\\eta_{s,p}}{1+\\eta_{s,p}})$$ I can see the dependence of $R$ and $\\varphi$ on the incident angle, but how can i plot it at different wavelength ? i don't see the dependence of $R$ and $\\varphi$ on wavelength at all."}
{"id":"44573","title":"Böotes Void and Dark Galxies","text":"Wouldn't it be possible that Böotes Void, a space we generally consider to be \"empty\" be filled with a number of undetectable dark galaxies?"}
{"id":"44579","title":"What is (or where can I discover) the Burke Potential?","text":"I have very much enjoyed William L. Burke's _Applied Differential Geometry_. Reading around on the web it seems that he discovered something which is called the (retarded) Burke Potential, but I have not been able to find any references to it. Can anyone point me at a reference or explain succinctly what it is?"}
{"id":"68647","title":"Is there a way to define a function of a meniscus curvate?","text":"If you cut a thin slit in thin opaque material and then put it into water and pull it out, the meniscus will be formed in the slit. For my research I need to know if it is analytically possible to define the curvature of meniscus( is it hyperbola, parabola, circle or something else). I need the function of curvature so I will be able to make calculations of how the light passes through the meniscus. I know there are numerical methods to make an approximate function, but I want to know if there is any other way."}
{"id":"68648","title":"Time taken for collision","text":"We have three particles at the vertices of equilateral triangle of side $d$. At $t=0$ they start moving in such away that at all instant of time each of them has a speed $v$ towards adjacent one. We have to find the time after which they will collide. I know that you can solve this by calculating from reference frame of any one particle, but I want a way to solve this by calculating the total distance traveled first in ground frame and then dividing by speed."}
{"id":"117035","title":"tritium beta decay - probability of being in 1s state","text":"Hydrogen-like wavefunctions have the form: $$R_{10} = \\left(\\frac{Z}{a_0}\\right)^{\\frac{3}{2}} 2\\space e^{-\\frac{Zr}{a_0}}$$ $$Y_{00} = \\frac{1}{\\sqrt {4\\pi}} $$ where $a_0 = \\frac{4\\pi \\epsilon_0 \\hbar^2}{\\mu e^2}$ Suppose Tritium $^3$H that's in its ground state beta decays to $^3$He suddenly. I want to find the probability the helium atom is in the 1s state. Firstly, is the reduced mass $\\mu$ the same for both helium and $^3$He? I assumed it is $m_e$. Wavefunction of helium is $R_{10}' = \\left(\\frac{2}{a_0}\\right)^{\\frac{3}{2}}2\\space e^{-\\frac{2r}{a_0}} $. Wavefunction of tritium is $R_{10} = \\left(\\frac{1}{a_0}\\right)^{\\frac{3}{2}}2\\space e^{-\\frac{r}{a_0}} $. To find the probability, do I simply overlap the two and square them? $$\\langle (1,0,0)_H| (1,0,0)_T \\rangle = \\left(\\frac{1}{a_0}\\right)^3 (2)^{\\frac{3}{2}} (2)^2\\left(\\frac{1}{4\\pi}\\right) \\int e^{-\\frac{3r}{a_0}} r^2 sin\\theta \\space d\\theta d\\phi $$ $$ = \\frac{2^{\\frac{9}{2}}}{27}$$ Thus probabiliy is $\\frac{2^9}{27^2} = \\frac{512}{729} $?"}
{"id":"91421","title":"Deriving the Electromagnetic Tensor","text":"The electromagnetic tensor is given as: ![enter image description here](http:\/\/i.stack.imgur.com\/b7xa5.png) How do you derive this? And how come there is a partial derivative in front of $A_\\mu$? Do you multiply the derivatives or what?"}
{"id":"34339","title":"How would a physicist move?","text":"Let's assume that Johnny Physicist has decided to move from his poor dingy second story apartment, into his much deserved home. Without making modifications to the existing structures? What would be the most work efficient way of moving all of his possessions? [Lets assume that the apartment is outside facing, and has stairs leading up to it] [Also lets assume that his possessions are a mix of carryable boxes, and large items]"}
{"id":"115062","title":"What is the leading order Feynman diagram for nucleon-anti-nucleon annihilation into two mesons ($\\psi^{\\dagger} \\psi \\to \\phi\\phi$)?","text":"I am working with a standard basic scalar Yukawa theory. I.e. the only interaction term is $-g\\psi^\\dagger\\psi\\phi$, where the $\\phi$ field quanta are the mesons, the $\\psi$ field quanta are the nucleons and the $\\psi^\\dagger$ field quanta are the anti-nucleons. I have tried fiddling around with various combinations and have looked at some relevant papers, but I have not found a solution which does not invoke other interaction terms. Is this process possible?"}
{"id":"34330","title":"Two-loop regularization","text":"Working out some quantum field theory computations, I have to find out the value of the two-loop Feynman integral $$ I(p)=\\int\\frac{d^4p_1}{(2\\pi)^4}\\frac{d^4p_2}{(2\\pi)^4}\\frac{1}{(p_1^2+m_1^2)(p_2+m_2^2)[(p-p_1-p_2)^2+m_3^2]}. $$ This integral is rather common and so, its value at small $p$ should be already well-knwon. But I was not able to find out it in literature. I would also appreaciate to see all the procedure to get the right value with whatever regularization procedure one likes. Thanks beforehand."}
{"id":"110916","title":"Particle Physics : Conservation laws","text":"As a whole, is conservation of a Lepton number valid, or is it basically the generalization of conservation of electron and muon numbers?"}
{"id":"100329","title":"How is the direction of the instantaneous acceleration determined?","text":"I know from the text book that the direction of velocity at any point on the 2D path of an object is tangential to the path at that point and is in the direction of motion. But how would one determine the direction of instantaneous acceleration? In case, the only force influencing is gravity, direction of instantaneous acceleration is downward towards the earth's center. but what in the other cases?"}
{"id":"67891","title":"Parity of spin states","text":"Since orbital angular momentum commutes with the parity operator and since both are hermitian it is possible to build a common basis. These are the spherical harmonics, whose parity is known. Now, we also know that since the spin operator commutes with the parity operator we can build common eigenkets, but I have never been told which is the parity of a spin state. Moreover, when calculating the parity of, for example, a particle I always multiply the contribution of the orbital angular momentum by the intrinsic parity of the particle, and the spin part is never mentioned. I guess that the spin contribution is in that intrinsic parity. So i would like someone to tell me about the parity of the spin states and how they are related to the intrinsic parity of particles(if in fact my guess is right)"}
{"id":"129361","title":"What is anti-screening?","text":"In his book \"The Lightness of Being\" Frank Wilczek (page 48) writes about screening and anti-screening. In screening, a bare charge attracts virtual particles of opposite charge which lessen the effect of the bare charge. He writes that anti-screening is the \"opposite of screening\" with no further physical explanation. This is an attempt to explain why two quarks attract weakly at close range and more strongly with distance. Is there any physical explanation of this anti-screening?"}
{"id":"129363","title":"How can heat turn into light","text":"I am confused about how hot surfaces can radiate light to their surroundings. When I shine a light on a surface the light turns to heat spontaneously, and when I leave that hot surface it radiates light spontaneously. To me this suggests that the process isn't driven by entropy increasing because it is two directional. Could anyone explain how this process complies with the second law of thermodynamics?"}
{"id":"110509","title":"If I were standing on the surface of the sun, how far could I see?","text":"Assuming it would be possible to handle the intense heat... Is the surface even? If not, would average surface elevation be an appropriate definition of \"surface\"?Does the sun have an atmosphere that would degrade visibility? How far apart are small solar flares that would block the horizon? What affect would the heat have on the light rays? Assume I am 2 meters tall."}
{"id":"114086","title":"Will we see interference in a double slit experiment in these three cases?","text":"First case:   1. A detector registeres particles going through the slits, but doesn’t reveal the information to conscious observes   2. A detector irreversibly erases the information about particles before they hit the film   3. The particles hit the film   4. A person observes the film. Second Case:   1. A detector registeres particles going through the slits, but doesn’t reveal the information to conscious observes   2. The particles hit the film   3. A detector irreversibly erases the information about particles before they hit the film   4. A person observes the film. Third case:   1. A detector registeres particles going through the slits, but doesn’t reveal the information to conscious observes   2. The particles hit the film   3. A person observes the film   4. A detector irreversibly erases the information about particles without anyone ever knowing what it was"}
{"id":"13","title":"What is sound and how is it produced?","text":"I've been using the term \"sound\" all my life, but I really have no clue as to what sound exactly is or how it is created. What is sound? How is it produced? Can it be measured?"}
{"id":"27038","title":"What Hermitian operators can be observables?","text":"We can construct a Hermitian operator $O$ in the following general way:   1. find a complete set of projectors $P_\\lambda$ which commute,   2. assign to each projector a unique real number $\\lambda\\in\\mathbb R$. By this, each projector defines an eigenspace of the operator $O$, and the corresponding eigenvalues are the real numbers $\\lambda$. In the particular case in which the eigenvalues are non-degenerate, the operator $O$ has the form $$O=\\sum_\\lambda\\lambda|\\lambda\\rangle\\langle\\lambda|$$ **Question:** what restrictions which prevent $O$ from being an observable are known? For example, we can't admit as observables the Hermitian operators having as eigenstates superpositions forbidden by the superselection rules. **a)** Where can I find an exhaustive list of the superselection rules? **b)** Are there other rules? **Update:** **c)** Is the particular case when the Hilbert space is the tensor product of two Hilbert spaces (representing two quantum systems), special from this viewpoint?"}
{"id":"59211","title":"Faraday's law in a ring","text":"What role does the induced emf in Faraday's law play in generating current in a ring in which the magnetic flux is changing ?"}
{"id":"73613","title":"Torque of a winch with pulleys","text":"I'm working on a project that involves using a remote-controlled electric winch to raise and lower my bicycle using a pulley system. The problem is that I don't know how much torque will be required of the electric motor in this system. A somewhat rudimentary diagram is as follows: ![enter image description here](http:\/\/i.stack.imgur.com\/dR3xC.png) I haven't bought the winch yet, but it'll probably be more like 2-3 cm radius rather than 5 cm as labelled, so I'll assume 3 cm. My calculation is as follows: The pulley system has a 4:1 mechanical advantage, so the weight on the rope is ~50 N, and then $$T = rmg = (3\\,\\mathrm{cm})(50\\,\\mathrm{N}) = 1.5\\,\\mathrm{N\\,m}$$ Which means that this motor should do the job. Is my reasoning correct?"}
{"id":"74047","title":"Any projects to build a telescope in space?","text":"Like in the title. With E-ELT we are pushing the limit of what we can build on the ground. With JWST we are pushing the limit of what can we launch into space. So instead, why don't we launch tools and materials to build a telescope and polish its mirror in orbit? This has several benefits. The environment is vacuum and microgravity, so very large components can be moved with small and light tools. The structure can be very light and fragile, bacause it doesn't have to survive the launch. It can be huge, because it doesn't have to fit into a rocket. Errors can corrected before it is commisioned. I am sure, I'm not the first one to have this idea, but I failed to find any existing projects (upcoming or future) on the web. Do you know any such projects being considered?"}
{"id":"32597","title":"Superconductor: What form of paramters (like London penetration depth) to use?","text":"I am dealing with thin film and LN2 temperature, and am trying to calculate values such as the Pearl Length, but am trying to find clarification as to which value of $\\lambda_L$ to use in this case. e.g. $\\lambda_L(t,T)$ with T = 0 or T = 77K? I have other formula, such as the kinetic inductance for a CPW, but the articles I am pulling them from are somewhat unclear which forms of the parameters they are using. Any help is appreciated."}
{"id":"9954","title":"How to get Intensity over a range of wavelengths from a Total Intensity","text":"This is a theory vs experiment problem for me I think in that - as usual - I think I know the theory, but when it comes time to run an experiment, I can't actually extract squat. So, here is my light source: I am using the Cold White LED (MCWHL2). If you scroll down a bit, you'll see the LED normalized intensity vs. wavelength profiles for all the LEDs. The black line on this graph is my LED. As you can see, it meanders instead of being a simple Guassian-like thing. I measure the intensity of this light by using this power meter: Scientech Astral AD30 meter - which displays power and energy - with an AC2501 detector. Specs are here . So I'm calculating total intensity by dividing by the area of the beam. Now, here's my question: say I get a reading of $ 10\\rm\\frac{mW}{cm^2} $. If I want to know what is the intensity of the output only over the range 400nm to 455 nm, based on the Intensity vs Wavelength graphs and the characteristics of the meter and detector, what do I do? Now, Planck's equation for a black body was where I first looked. My procedure was to find the total intensity for the light temperature (6500 K) and integrate from 0 to infinity. I then numerically integrated from 440 nm to 455 nm and divided the two. Multiplying this by the output on the detector gave a ridiculous solution. Given the intensity profile - not really black bodyish - the ridiculous results make sense. It was at this point that I started searching and found nothing but Planck stuff. Thus, I humbly come before all of you. Please help! There must be some sort of standard procedure for those without specialized light analyzing equipment! Thanks! Sam"}
{"id":"67197","title":"Cauchy Problem for Boltzmann Equations","text":"One of the first profound analysis about the solutions of the Boltzmann Equation was given by DiPerna and Lions in the late 1980s. You can find one of their main papers here: http:\/\/www.jstor.org\/discover\/10.2307\/1971423?uid=3737528&uid=2&uid=4&sid=21102076873573 If you do not have access to the paper then use this link: http:\/\/www.4shared.com\/office\/eeIqgvxB\/MathematicsOn_the_Cauchy_Probl.html I was reading through this and similar papers and one of the things I do not understand is the _interpretation_ of the solution. The paper delivers more or less a proof for the existence of a global solution of the Boltzmann Equation, but what can we do with it? If you have an idea how to interpret it I would be very thankful. P.S: I put a question about the proof of one of the Lemmas on http:\/\/mathoverflow.net\/questions\/132871\/cauchy-problem-for-boltzmann- equations"}
{"id":"126047","title":"What is the deepest cause of the such high specific heat capacity of water?","text":"Yes, I know about the hydrogen bridges. But I think, it isn't the deepest cause. Anyway, they are only second-order bindings, although quite strong. I think, somehow should have the water a significant number of \"hidden degrees of freedom\", or some like internal structure with a lot of degrees of freedom. But it is only my naive intuition. So, what is the deepest cause?"}
{"id":"75318","title":"How exactly do I calculate this correlation function?","text":"I found a research paper (from 1977) that has a particular equation I need to reproduce. The paper essential calculates dynamic light scattering correlation functions. The full equations I need to work out is this: $$ f(q,\\tau) = \\frac{\\int_{-1}^{+1}(1-2w^2\\nu^2) \\exp(-w^2\\nu^2)\\ d\\nu \\times \\int_{0}^{2\\pi} A[q,\\mu(t)]A^*[q,\\mu(0)]\\ d\\psi}{\\int_{0}^{2\\pi} |A[q,\\mu(0)]|^2\\ d\\psi} $$ Essentially I need to calculate this for values of the delay time $t$. The bit I am not sure how to do is the $\\int_{0}^{2\\pi} A[q,\\mu(t)]A^*[q,\\mu(0)]\\ d\\psi$. The actual functional form of A isn't that important, but it consists of spherical Bessel functions and the term $\\psi$ which is to be numerically integrated over. Now, where I am stuck is what do I actually do to calculate that function for the delay times. I have tried a lot of things but I just cant get it looking like they have in the paper. Here is what I have tried:   * Calculate $\\int_{0}^{2\\pi} A[q,\\mu(t)]$ for all delay times, and then use a built in autocorrelation function to obtain the results.   * Take my first $t$ value (say 0.01 sec), and my second $t$ value (say 0.02 sec) and then calculate $\\int_{0}^{2\\pi} A[q,\\mu(0.02)]A^*[q,\\mu(0.01)]\\ d\\psi$... etc for all t values (so $t0$ lags 1 behind t.   * A few other things. As you can see I am a bit confused, if anyone has any insight as to how I would calculate this by hand that would be extremely helpful. * * * ### Edit: The full functional form of $A(q,\\mu(t))$ is $$ A(q,\\mu(t))\\propto \\frac{3j_1(u)}{u} + F^3 \\frac{m_2-m_1}{m_1-1}\\frac{3j_1(f u)}{fu} %Old version: \\frac{F^3 (m_2-m_1) (3 j_1(F \\mu(t)))}{(m_1-1) (F \\mu(t))}+\\frac{3 j_1(\\mu(t))}{\\mu(t)} $$ where $j_1$ is a spherical Bessel function of the first kind, $u=q\\sqrt{a^2\\mu^2(t)+b^2(1-\\mu^2(t))}$, $F=1-\\frac ta$, and $\\mu(t)=\\cos(\\psi)\\sin(\\theta)\\sin(\\alpha(t))+\\cos(\\theta)\\cos(\\alpha(t))$, for $\\nu=\\cos(\\theta)$ and $\\alpha(t)=\\omega t+\\phi$. Yes, I am after $f$ vs $t$ for a $q$ of my choice. I am just not sure I am computing the integral properly for the values of $t$. For example, lets say my $t$ of interest was $0.1, 0.2,0.4, 0.8, 1.6$ (just an example). For the very first $t$, would I simply do, $$ \\int_{0}^{2\\pi}A[q,\\mu(0.2)]\\times A[q,\\mu(0.1)] $$ Then for the next $t$ $$ \\int_{0}^{2\\pi}A[q,\\mu(0.4)]\\times A[q,\\mu(0.1)] $$ or $$ \\int_{0}^{2\\pi}A[q,\\mu(0.4)]\\times A[q,\\mu(0.2)] $$ And dont I need to average over the same delay time $\\tau$? So for $\\tau 1$ its $$ \\int_{0}^{2\\pi}A[q,\\mu(0.2)]\\times A[q,\\mu(0.1)]\\\\\\ \\int_{0}^{2\\pi}A[q,\\mu(0.4)]\\times A[q,\\mu(0.2)]\\\\\\ \\int_{0}^{2\\pi}A[q,\\mu(0.8)]\\times A[q,\\mu(0.4)]\\\\\\ \\int_{0}^{2\\pi}A[q,\\mu(1.6)]\\times A[q,\\mu(0.8)] $$. I am using Mathematica to evaluate these functions. As you can see I am rather confused. Thanks so much for your help. The paper I am referencing is: > Rotational—translational models for interpretation of quasi-elastic light > scattering spectra of motile bacteria. Michael Holz and Sow-Hsin Chen. > _Appl. Optics_ **17** no. 20, pp. 3197-3204 (1978)."}
{"id":"29167","title":"Does General Relativity require that Spacetime must be a orientable?","text":"> **Possible Duplicate:**   >  Can spacetime be non-orientable? Apart from the constraints put on the topology of spacetime by QFT (Parity For example), if the global topology of a universe is that of a Mobius band and at one point there is a gravitational field pointing towards a star and a passenger travels along the band and returns to the same point. Then he will find the gravitational field pointing away from the star causing an anti- gravity effect. Does this mean that space-time must be orientable?"}
{"id":"86924","title":"Is thermal\/heat energy the same thing as internal energy?","text":"I learned that internal energy of an object is the sum of the kinetic energy and potential energy of the molecules of the object. Is this energy the same thing as the heat energy the object contains?"}
{"id":"61340","title":"One-loop $\\phi^4$ theory in $d = 3$","text":"I'm trying to calculate the 1 loop correction to the propagator in massless $\\phi^4$ theory, in $d = 3$, just for fun. The diagram just looks like a straight line with a circle touching tangently to it. I'm not sure whether to take $ m = 0$ straightaway or to compute it for $m > 0$ before taking $m \\to 0$, because the two methods give different answers. The diagram is $\\sim \\int d^3k \\frac{1}{k^2 - m^2}$. If I set $m = 0$ straightaway, then I get $\\sim \\int k^2 dk \\frac{1}{k^2} \\sim \\Lambda$, where $\\Lambda$ is the cutoff. On the other hand, if I do the integral, using the results in the appendix from Peskin and Schroeder \\begin{align} \\int d^dk \\frac{1}{k^2 - m^2} \\sim \\frac{\\Gamma(1-d\/2)}{(m^2)^{1-d\/2}} =\\Gamma(-1\/2)(m^2)^{1\/2}. \\end{align} $\\Gamma(-1\/2)$ is finite, so taking $m \\to 0$ this diagram is $0$. So one one hand it's UV divergent, on the other hand it's 0. Which method should I be using? Are the theories where $m=0$ and $m \\to 0$ not the same? I also seem to have missed out the $i \\varepsilon$ prescription in the propagator in the first case - if I include that in and evaluate the diagram before sending $\\varepsilon \\to 0$, then the diagram is $0$ too. Or perhaps it does not matter too much because the mass renormalization term can take care of either divergence and such discrepancies will not show up in calculations of any observables anyway? Thanks."}
{"id":"69583","title":"What's an \"Action\" and what does the Lagrangian equation mean exactly?","text":"How and why would a particle take the shortest path? $L=KE-PE$? What's the $KE-PE$ mean in English? I understand the 'mechanics' but not the idea itself. Please explain simply, I do know Calculus but the simpler the better. Please don't say because that's how nature is, because I get that a lot."}
{"id":"25188","title":"Ways of verifying the origin of a meteorite?","text":"Assume you have a small sliver of a lunar or martian meteorite (or an object asserted to be so). Without using any special scientific equipment, is it possible to verify (or give a high probability) that it is indeed of lunar or martian origin? Are there any visual or other tests one can do to confirm the source? _(If this question is off-topic or not quite right, feel free to close.)_"}
{"id":"104764","title":"Alternative symmetries for the Maxwell Lagrangian?","text":"I'm wondering about how to show that $A_a\\rightarrow A_a+\\alpha\\partial_0A_a$, with $\\alpha$ infinitesimal, is an infinitesimal symmetry of $\\mathcal L=-\\frac14F_{ab}F^{ab}$. \\begin{equation} F_{ab}\\rightarrow F_{ab}+\\partial_a(\\alpha\\partial_0A_b)-\\partial_b(\\alpha\\partial_0A_a)=F_{ab}+\\partial_0(\\alpha F_{ab}). \\end{equation} \\begin{equation} \\Rightarrow \\mathcal L\\rightarrow\\mathcal L-\\frac12F_{ab}\\partial_0(\\alpha F^{ab})=\\mathcal L-\\frac14\\partial_0(\\alpha F_{ab}F^{ab}). \\end{equation} We require $\\delta\\mathcal L=0$ for it to be called \"a symmetry of $\\mathcal L$\", right? Or do we only require $\\delta\\mathcal L=$ a total derivative? Either way, I don't seem to get what we want."}
{"id":"104765","title":"Phase factor for nearest neighbor hopping in the Haldane Model","text":"In Haldane's model, he imagines a staggered magnetic field in graphene where the net flux through a unit cell is zero. To model this, he has a phase factor in next nearest neighbor hopping on the honeycomb lattice. Why does he not consider a phase factor for hopping between nearest neighbors as well?"}
{"id":"113051","title":"What happens to electrons when they are detected?","text":"I know photons are absorbed and annihilated when they are detected but a) when electrons are detected, are they absorbed and annihilated? and b) if electrons are ever absorbed, what happens when they are? Are photons omitted?"}
{"id":"33713","title":"Condition for circular orbit","text":"I am a little confused about the condition for circular orbit. Goldstein's Classical Mechanics has the condition for circular orbit as $$f'=0\\tag1$$ where $f'$ is the effective force. I understand that the reason for this requirement is that the corresponding $V'$is at an extremum. However, setting $f'=0$ yields $$f(r)=-\\frac{l^2}{mr^3}\\tag2$$ and the one of the equations of motion becomes $$m\\ddot r=0\\tag3$$ This can't be right as it is saying that the net force on the object is zero, and yet the object is supposedly in a circular orbit. Am I interpreting equation (3) wrong?"}
{"id":"134843","title":"Frequency modulation using visible light","text":"Currently I am doing research into visible light communication from an Embedded Systems background (MSc), but I am struggling to relate the concepts of modulation of radio to visible light. I already apply modulation to a working VLC system, but the underlying system still puzzles me. My goal of this post is to create an understanding of the underlying fundamental concepts, so if there are flaws in my assumptions below please point me to it. **Radio** In RF communication, two modulation techniques are used: amplitude modulation and frequency modulation. All other modulation techniques are variations on these types. These modulation techniques change the electromagnetic field in terms of coherent photons with a specific frequency and amplitude (source). **Visible light communication** In VLC, amplitude modulation is the most common technique (changing the intensity), but frequency modulation is also used. As frequency relates to color, some research uses RGB LEDs to do this in combination with color filtered photodiodes. **(1)** However, and this is where the problem is, some research applies frequency and phase modulation using a single white LED. They use a photodiode as a receiver which is sensitive to intensity but not to color variations, so how is frequency modulation then possible? **(2)** It seems that VLC has a third kind of main modulation technique in comparison with RF, is this correct? **(3)** If so, what makes light so different from RF then, as it also is electromagnetic radiation? And can this third type be applied to RF?"}
{"id":"102097","title":"Why not drop $\\hbar\\omega\/2$ from the quantum harmonic oscillator energy?","text":"Since energy can always be shifted by a constant value without changing anything, why do books on quantum mechanics bother carrying the term $\\hbar\\omega\/2$ around? To be precise, why do we write $H = \\hbar\\omega(n + \\frac{1}{2})$ instead of simply $H = \\hbar\\omega n$. Is there any motivation for not immediately dropping the term?"}
{"id":"27858","title":"Fermi statistics and Berry phase","text":"When the positions of two fermions are exchanged adiabatically in three- dimensional space, we know that the wave function gains a factor of $-1$. Is this related to Berry's phase?"}
{"id":"27851","title":"Producing photons with same frequency, different amplitude wave","text":"I don't understand how two photons of the same _frequency_ can have different _amplitudes_ , neither how to produce them. I know that classically the square of the amplitude is proportional to the energy, but photons aren't classical particles. My understanding is that a photon's energy is $h\\nu$ - what does the square of its amplitude represent, then? Are there bounds to the amplitude of an EM wave? Take two waves of amplitudes $A_1$ and $A_2$ and frequency $f_0$. If $A_2 = 2A_1$, can the wave with amplitude $A_2$ be said to carry\/be two $A_1$ photons of frequency $f_0$?"}
{"id":"83393","title":"Explanation of homogeneity of space and time by giving examples?","text":"while reading landau lifshitz i came across these three terms:-   1. homogeneity of space.   2. homogeneity of time.   3. isotropy of time. it will be a great help for me if someone can explain it to me by giving examples ."}
{"id":"789","title":"Energy of the electron-muon reaction","text":"Lets see the reaction: $e^- \\mu^- \\to e^- \\pi^- \\nu_\\mu \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; {(1)}$ I suppose, that this reaction occurs as follows $e^- \\mu^- \\to e^- \\mu^- \\pi^+ \\pi^- \\to e^- \\pi^- \\nu_\\mu$ Is it possible at energy less than 2*140 MeV? The same is for the analogous proton-muon reaction $p^+ \\mu^+ \\to p^+ \\pi^+ \\bar{\\nu}_\\mu \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;{(2)}$ Once more reaction: $p^+ p^+ \\to p^+ p^+ \\pi^- \\pi^+ \\to p^+ n \\pi^+ \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;{(3)}$ What are the experimental data? * * * **P.S.** This question is important enough. The main solar reaction is $p^+ p^+ \\to d^+ e^+ \\nu_e $ If this reaction occurs as follows $p^+ p^+ \\to p^+ p^+ e^- e^+ \\to p^+ n \\nu_e e^+ \\to d^+ e^+ \\nu_e$ then it would require the energy of more than 2*0.511 MeV to take place. Cross section of this reaction will be much less, so the main solar reaction should be $p^+ p^+ e^- \\to d^+ \\nu_e $ * * * * * * **Update 20.02.11** Why I think so? I suppose that new particles are created in pairs particle- antiparticle. So the reaction $e^- e^- \\to e^- e^- \\pi^- \\pi^+ $ requires the energy of more than 2*140 MeV to take place As well as the reaction $e^- e^- \\to e^- e^- \\pi^- e^+ \\nu_e $ from symmetry considerations, since there is a decay $\\pi^+ \\to e^+ \\nu_e$ And the result is $e^- e^- \\to e^- e^- \\pi^- e^+ \\nu_e \\to e^- \\pi^- \\nu_e $ contrary to $e^- e^- \\to e^- \\nu_e W^- \\to e^- \\pi^- \\nu_e$ with the minimum reaction energy of 140 MeV The same is valid for the reactions (1) (2) (3) So what is the **experimental data** on minimum energy of these reactions?"}
{"id":"18576","title":"How to derive Fermi-Dirac and Bose-Einstein distribution using canonical ensemble?","text":"My textbook says that microcanonical ensemble, canonical ensemble and grand canonical ensemble are essentially equivalent under thermodynamic limit. It also derives Fermi-Dirac and Bose-Einstein distribution from grand canonical ensemble. My question is then: How to derive Fermi-Dirac and Bose-Einstein distribution using canonical ensemble. The expression of canonical ensemble $$\\rho\\propto e^{-\\beta E}$$ seems to imply Boltzmann distribution only."}
{"id":"39540","title":"Something I don't understand in Quantum Mechanics","text":"I've just started on QM and I'm puzzled with a lot of new ideas in it. 1.On a recent lecture I've attended, there is an equation says: $\\langle q'|\\sum q|q\\rangle \\langle q|q' \\rangle =\\sum q \\delta(q,q')$ I don't understand why $\\langle q'|q\\rangle \\langle q|q' \\rangle =\\delta (q,q')$ Can you explain this equation for me? 2.Actually, I'm still not clear about the bra-ket notation. I've learnt the bra and the ket could be considered as vectors. Then what are the elements of the vectors? Thank you very much!"}
{"id":"48151","title":"Question on inflation as a phase transition","text":"I have just finished watching the following video http:\/\/www.youtube.com\/watch?v=beQ9fZ0jVdE where Laughlin, Gross and some students discuss e.g. about inflation. The following question is risen: Is inflation a phase transition (e.g. of the geometry of space time)?"}
{"id":"72209","title":"Does 'maximum principal stress fracture criteria' only apply in tensile loading?","text":"I'm working on 3d simulation of solid fracture, primarily for graphics applications. I have one question about the Rankine fracture criteria, which is widely used in the computer graphics community. The maximum principal stress fracture criteria, aka the Rankine criteria, states that the fracture is initiated when the greatest tensile principal stress in the solid reaches a critical magnitude: $$\\sigma{}_{1 max} = \\sigma{}_{TS}$$ where the subscript TS stands for tensile strength. My question is: Does this criteria only apply in tensile loading? Is it that this criteria can only model the 'Mode I' fracture? Note: the 'Mode I' fracture means the 'Open' fracture, as can be seen here: Fracture wikipedia."}
{"id":"34074","title":"Can superdeterminism resolve contextuality, entanglement and Shor's algorithm in quantum mechanics?","text":"Superdeterminism is the idea that the apparent freedom for the choice of experimental apparatuses and their settings are nothing but an illusion. Contextuality is the dependence of the properties of a system upon the choice of measuring device. However, what if there is no \"free\" choice of measurement settings, i.e. the measurement settings aren't a freely variable independent degree of freedom? Bell inequalities require counterfactual polarization settings, but what if the polarization setting could only have been one way and not any other? As for the ability of Shor's algorithm to factorize large integers and compute discrete algorithms, maybe the answers gotten at the end retrocausally determine the numbers fed into the quantum computer in the first place so that it couldn't have been any other number? Are there any absolutely fatal objections?"}
{"id":"88510","title":"Fluid Mechanics Piston Problem","text":"![enter image description here](http:\/\/i.stack.imgur.com\/OEbe7.png) **The actual problem:** A hydraulic car lift has a reservoir of fluid connected to two cylindrical fluid filled pipes. The pipe directly below the car has a diameter of 1.8 m. the pipe on which the plunger acts has a diameter of 0.045m. the plunger is depressed a distance of 1.5m. How much does the car rise? **My attempt:** Since the piston movess downaward thorugh a displacement Δx1 equals the volume of liquid pushed up on the right as the right piston moves upward through a displacement $\\ π (\\frac{0.0225m^2}{4}) (1.5m) = π (\\frac{0.9m^2}{4})(x)$ x=0.00094m **My thoughts:** My attempt is wrong but I am confused as to why. Since $(A1)(Δx1)=(A2)(Δx2)$ , I thought that if I used the depressed distance of 1.5m for $Δx1$, then I would be able to find the hight of the car. I understand the diameter of the pipe on the left is very small and unrealistic if you ask me (typical text book problem). **My question** Any hints to how I can look at this problem differently? Maybe I am looking at the problem wrong. Any hints would be appreciated!"}
{"id":"109419","title":"Visualizing Tension in FBD -- Rope With Mass","text":"Asked to find the tension of a point that is halfway up a rope (where the rope has mass), given the system: ![enter image description here](http:\/\/i.stack.imgur.com\/OQWpl.png) The solution of the problem (FBD included) is as follows. In this diagram (Tm = tension at mid-point, and Tt = tension at top). Question(s): (1) Why is Tt incorporated into the FBD? (2) Why does Tm point downward? ![enter image description here](http:\/\/i.stack.imgur.com\/4OG2i.png) Also if it helps the FBD's for the individual components are given here: ![enter image description here](http:\/\/i.stack.imgur.com\/ZvJeq.png) Thanks!"}
{"id":"133865","title":"Gravity and electromagnetism","text":"If light bends due to the curved spacetime,would not the act of bending light warp space? How does one describe curved light?"}
{"id":"108099","title":"Radiation heat transfer","text":"I'm looking at the radiosity equations for heat transfer http:\/\/en.wikipedia.org\/wiki\/Radiosity_(heat_transfer)#Radiosity_method Specifically, I'm hesitant to accept the equation: $$ \\dot Q_i = \\frac{ A_i \\epsilon_i }{ 1 - \\epsilon_i } ( \\sigma T_i^4 - J_i )$$ My analysis is as follows: power goes out via Stefan-Boltzmann term, which is given by a term like $$ - A_i \\epsilon_i \\sigma T_i^4 $$ and radiation goes in via absorption of all emissions from nearby surfaces, that is: $$ A_i \\epsilon_i H_i = A_i \\epsilon_i \\sum_j F_{ij} J_j $$ where the radiosity $J_j$ expression is correct on the article, and we are assuming Kirchoff's law of thermal radiation to hold. But the above implies that the actual expression for the net heat transfer is: $$ \\dot Q_i = - A_i \\epsilon_i ( \\sigma T_i^4 - H_i ) = - A_i \\epsilon_i ( \\sigma T_i^4 - \\sum_j F_{ij} J_j ) $$ Thoughts? Is my derivation correct, or is the one on the article?"}
{"id":"132195","title":"A Sphere of Black Holes","text":"Imagine a sphere of black holes surrounding a piece of space. Will this piece be separated from the rest of normal spacetime (at least for some time, till these black holes finally attracted themselves). So, seen from the outside, we have a black hole, but with a non-singular interior. Is this possible?"}
{"id":"81477","title":"Why is classical physics not valid for a harmonic oscillator in its lowest energy state?","text":"I am reading Born's interpretation of wave function in quantum physics by Eisberg & Resnick and I am not able to understand this description about comparison between the classical and quantum validity for an oscillator at lowest energy.![enter image description here](http:\/\/i.stack.imgur.com\/aDFYf.png) ![enter image description here](http:\/\/i.stack.imgur.com\/5Hjwt.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/z64Pc.jpg)"}
{"id":"73555","title":"Fractionalization and the structure of spin rotation group?","text":"As we know, the phenomena of fractionalizations in condensed matter physics is fantastic, like fractional spin, fractional charge , fractional statistics, .... And one key point is that the quasiparticals must be created or annihilated by _pair_. On the other hand, consider the groups $SU(2)$ and $SO(3)$, they are the rotation groups for _half-integer_ and _integer_ spins, respectively. And we know that $SU(2)\/\\mathbb{Z}_2=SO(3)$, which means that each element in $SO(3)$ can be viewed as **one pair** $(U,-U)$, where $U\\in SU(2)$ (otherwise put: the coset $\\left\\\\{U, -U\\right\\\\} \\subset SU(2)$ in the quotient group $SU(2)\/\\mathbb{Z}_2$ is our element in $SO(3)$). So I wonder that whether is there any underlying connection between the _pair_ nature of quasiparticals in topological phase in physics side and the _pair_ structure relating $SU(2)$ and $SO(3)$ in mathematics side? Thank you very much."}
{"id":"73553","title":"Applicability of the concept of voltage in electrodynamic circuits","text":"In electrostatics, we have $$\\nabla \\times \\vec{E} = 0$$. Hence, we can define a scalar potential $V$, where $$\\vec{E} = -\\nabla V$$. We know from Faraday's law that $$\\nabla \\times \\vec{E} = -\\frac{\\partial \\vec{B}}{\\partial t}$$. In situations involving constant currents(D.C.) the rate of change of the magnetic field is zero, so we can continue to use the concept of scalar potential, also known as voltage in circuit theory, for analysis of DC circuits. However,in practice, in the literature, we find the concept of voltage being used to study alternating current circuits as well. Further, it is also used in the study of high frequency alternating current circuits used for generation of electromagnetic waves. In such situations $$ \\oint \\vec{E}.\\vec{dl} \\neq 0$$. So strictly speaking, we cannot use the concept of voltage to study alternating current circuits, especially in the high frequency applications. Yet, in practice we find that it continues to be used in the literature. So is it really justified to use this concept outside of its realm of validity, and why or in what way?"}
{"id":"77079","title":"Is continuous evolution from one eigenstate of operator $O$ to another $O$-eigenstate possible?","text":"Eigenvectors associated with distinct values of an observable are orthogonal, according to quantum mechanics. Does this entail that a quantum system cannot continuously evolve from one eigenstate into another, for ANY observable? At first, that seems strange: it seems like a particle should be able to \"travel\" in the sense of continuously moving from one eigenstate of position to the next. Another example: can't a particle \"speed up\" (i.e. go continuously go from one velocity eigenstate to the next)?"}
{"id":"3309","title":"What is the world's biggest Schrodinger cat?","text":"How big is it by a truly quantum measurement? I am thinking of comparing Science magazines \"Breakthrough of the Year\" (BYOT) with the Zeilinger buckyball. The BYOT is a piezoelectric mechanical oscillator (PO) the size of the cross section of a human hair. It is placed in a superposition of its ground state and its first excited state. The well known buckyball experiment is a two slit experiment using buckyballs. (A third candidate might be a macroscopic Josephson junction oscillator conducting both ways at once.) I have made some basic calculations. For instance, the BYOT contains about 10^14 atoms compared to 60 or 72 atoms in the buckyball. By this measure the BYOT is bigger by a factor of about 10^12. On the other hand, the two slits are separated by 50 to 100 nanometers, or 10^-7 meters. In its first excited state, the top of the BYOT PO moves about 10^-15 meters per cycle, according to my calculations. By this measure the buckyball wins by a factor of about 10^8. Calculating energy of the moving parts, I find a much closer horserace, but the buckyball is about 100 times bigger. However, none of these calculations is at all quantum mechanical (QM). ArXiv lists at least five papers proposing truly quantum mechanical measures of the size of a macroscopic Schrodinger cat. The most recent is Lee and Jeong http:\/\/arxiv.org\/abs\/1101.1209 which references the other four. Can someone competent (or expert) in QM apply one or more of these quantum measures to the BYOT and the buckyball and tell me which is larger? TIA. Jim Graber"}
{"id":"77070","title":"Vertical circular motion followed by projectile motion","text":"If you have a stone tied to a string, and it is given a certain velocity at the lower most point such that it completes a quarter circle but then the string becomes slack, and it undergoes projectile motion. The problem is, is it necessary that this parabola will be symmetric about the verticle? If so, is it impossible that the stone passes through the center of this circle during downward motion? Please give a detailed solution Thanks in advance"}
{"id":"103923","title":"Spring Damper System : Recoil Reduction","text":"I am designing a man-\"portable\" 30mm sniper rifle. Although it may be classified as a anti-materiel cannon. I have the mass of the round in total, and the mass of the projectile leaving the barrel, as well as the pressure inside the barrel. I have the weapon's weight as well. Here is my source : http:\/\/www.pmp.co.za\/index.php?page=mediumcalibre6 Now, my problem.   1. How do I calculate the force that this bullet imparts upon the rifle?   2. If I use a spring-damper system to reduce the force on the shooter, how do I actually determine the force on the shooter? If you do not find the need to supply me with a direct answer, which is understandable, could you please redirect me to a website\/book that might help me?"}
{"id":"105256","title":"Why does soda fizz when it meets ice?","text":"I read that the soda's temperature allows it to contain its $\\text{CO}_2$: colder means it can retain its bubbles better. So, why does soda fizz when it meets ice? If anything, because the soda is made colder, it should retain its gas better, as per above. The ice is taking away energy from the soda, thus freezing. Won't heating, not freezing, encourage the gas to escape, or is it not an energy matter?"}
{"id":"130775","title":"Non Zero correlation function (for large separations) in one particle state?","text":"So i computed the following equal time correlation function for a one particle state. The vacuum correlations give the function $$\\langle \\phi(\\vec x)\\phi(\\vec y)\\rangle_0=D(\\vec x-\\vec y)\\\\\\ =\\frac{e^{-m|\\vec x-\\vec y|}}{4\\pi |\\vec x-\\vec y|}$$ while the one particle correlation function looks like: $$ \\langle\\vec q| \\phi(\\vec x)\\phi(\\vec y)|\\vec q\\rangle=2\\omega_q\\langle0|a_q \\phi(\\vec x)\\phi(\\vec y)a^{\\dagger}_q|0\\rangle\\\\\\\\\\propto 2cos[\\vec q\\cdot(\\vec x-\\vec y)]+D(\\vec x-\\vec y)\\\\\\=2cos[\\vec q\\cdot(\\vec x-\\vec y)]+\\frac{e^{-m|\\vec x-\\vec y|}}{4\\pi |\\vec x-\\vec y|}$$ I recognize that since this is a spacelike separation that it can't be observed, the commutator of this type vanishes since the terms are invariant under $\\vec x\\rightarrow \\vec y$. Since there is no elapsed time there is also no valid interpretation as particle propagation. it just seems odd to me that the field has non vanishing correlations for large spatial separations. I was wondering if there is any consequence to this that people know about. Thanks, Selman"}
{"id":"130779","title":"What is the status of massless photons traveling through a medium?","text":"Photons in vacuum have no proper time, and they are not considered as observers and not as reference frame. But what about photons travelling through matter? Their velocity is lower than light speed, thus the proper time formula (multiplication with reciprocal gamma) would yield a proper time, but do we consider that they have a proper time\/ that they are observers\/ that they may be reference frames?"}
{"id":"81803","title":"Tensor contra- and covariance concept","text":"Imagine we have $p$-contravariant and $q$-covariant tensor, such that $$ t~=~t^{i_1, \\dots, i_p} _{j_1, \\dots, j_q} e_{i_1}\\otimes \\dots \\otimes e_{i_p} \\otimes e^{j_1} \\otimes \\dots \\otimes e^{j_q}~\\in ~T \\left( p,q, L\\right),$$ where $e_{i_l}$ are basis vectors of $L$ (arbitrary linear space), $e^{j_k}$ its dual basis in $L^{*}$. My question is following: what is the physical meaning of vector and its dual basises? I mean, for example, how do I interpret 2 times contravariant and 1 time covariant tensor in physics."}
{"id":"74524","title":"Quantum Cascade Laser","text":"I want to learn about quantum cascade laser in a six month time. Can anyone please suggest me any reading materials or video lectures. I have background in calculus and linear algebra. I am learning quantum mechanics now."}
{"id":"118483","title":"Infinite Round Square-Well zeroes of Spherical Bessel function","text":"Consider a potential well where the potential is zero within a spherical well of radius a centered on the origin. Taking $a$ as our length scale and $\\hslash^2\/(2ma^2)$ as our energy scale, we gain the dimensionless form $$ -\\biggl(\\frac{\\partial^2}{\\partial r'^2}+\\frac{2}{r'}\\frac{\\partial}{\\partial r'}+\\frac{1}{r'^2}[\\frac{1}{sin\\theta}\\frac{\\partial}{\\partial\\theta}(sin\\theta\\frac{\\partial}{\\partial\\theta})\\frac{\\partial^2}{\\partial\\phi^2}]\\biggr)\\psi(r',\\theta,\\phi)=(E'-U')\\psi(r',\\theta,\\phi) $$ Seeking a solution where the dependence on $r'$ factors from the dependence . on angles $$ -\\biggl(\\frac{\\partial^2}{\\partial r'^2}+\\frac{2}{r'}\\frac{\\partial}{\\partial r'}-\\frac{l(l+1)}{r'^2}\\biggr)R(r')=(E'-U')R(r') $$ We substitute: $ \\rho=\\sqrt{E'-U'}r'=k'r' $ where $E'>U'$ Dividing the above equation by $E'-U'$: $$ \\biggl(-\\frac{\\partial^2}{\\partial \\rho^2}-\\frac{2}{\\rho}\\frac{\\partial}{\\partial \\rho}+\\frac{l(l+1)}{\\rho^2}\\biggr)R(\\rho)=R(\\rho) $$ The case where U is infinite and so the wavefunction must be zero for $r>a$, i.e., $r'>1$. Thus we must have: $j_l(k'r')=0$ for $r'=1$ That is $k'$ must be a zero of $j_l$. The energy of the wavefunction can then be calculated from: $E'=k'^2$. Here is the overall solution: $$ \\psi_{n_rlm}(r',\\theta ,\\phi)=\\frac{\\sqrt{2}}{j_{l+1}(k'_{n_r,l})}j_l(k'_{n_r,l}r')Y_{lm}(\\theta , \\phi) $$ where $j_{l}(k'_{n_r,l})=0$ If i want to plot the radial part of the function I need $k'_{n_r,l}$. should I use `fzero` in $Matlab$ for example to find theme first? Thanks for any help to draw it.               jnu=sqrt(pi.\/(2*x)).*besselj(nu+1\/2,x);"}
{"id":"110906","title":"Why do electrons move towards regions of lower potential?","text":"Can someone explain why electrons move from high to low potential regions in quantum mechanics? By the way, what is the very definition of this potential? Moreover, how does this generalize to other particles?"}
{"id":"107460","title":"Force on a point charge q inside a cavity in an uncharged conductor","text":"This is problem 2.40 from Introduction to Electrodynamics by D. J. Griffiths: > A point charge $q$ is inside a cavity (not necessarily spherical or anything > similarly regular) in an uncharged conductor. Is the force on $q$ > necessarily zero ? What my understanding tells me is that if the charge on the inner surface of the conductor is just enough to cancel the field from the point charge from every direction, then the force from any two opposite pieces on the inner surface will produce equal but opposite forces. Thus, the net force is zero. Is my reasoning correct?"}
{"id":"33979","title":"What is better than time-dependent perturbation theory if the pointer states aren't energy eigenstates?","text":"Time-dependent perturbation theory works excellently if the interaction is weak and the pointer states are approximately energy eigenstates. However, what if the pointer states are not remotely energy eigenstates? Then, the pointer states themselves evolve in time nontrivially by more than just a phase rotation."}
{"id":"6263","title":"III-V semiconductor compounds - band gap approximations","text":"I'm looking for the most common and accurate approximation functions for III-V semiconductor compounds. So far, I found two (InGaAs as the sample compound):   1. The simplest one: $E(Ga_{1-x}In_xAs)=(1-x)E_{GaAs}+x \\cdot E_{InAs}$   2. Binomial: $E(Ga_{1-x}In_xAs)=(1-x)E_{GaAs}+x \\cdot E_{InAs} - b \\cdot x \\cdot (1-x)$ For InGaAs, b=1.44 Are there more? I mean the methods for which the parameters can be found in articles, not measured by myself. When using the first one is sufficient? Do the scientists bother to calculate next corrections? Are they useful?"}
{"id":"6269","title":"Projectile motion without air resistance","text":"Does the mass of projectile affect its trajectory when it is projected in no air resistance environment and why?"}
{"id":"122909","title":"Is there a general H-theorem?","text":"In statistical mechanics, Boltzmann showed that for dilute gases the H-function increases. I remember from a lecture that there is no general H-theorem, e.g. for non-dilute gases or in the quantum case but there seem to be also claims to the contrary. So what is the status of the (generalised) H-theorem?"}
{"id":"70315","title":"Costant volume gas thermometer","text":"I have two constant volume gas thermometers, one with an ideal gas $pv=RT$ and the other following the Van der Waals equation $(p+\\frac{a}{v^2})(v-b)=RT$. They are calibrated using the freezing point and the boiling point of water. Will they show the same temperatures?"}
{"id":"32532","title":"Can ship\/boat propellers be placed, with adequate protection, alongside the fuselage instead of at the back?","text":"On small private aircraft the engine is placed at the front of the fuselage, or on the wings E.g. Stationair, Otter On combat aircraft the engine\/s is placed at the end of the fuselage E.g. Raptor, Typhoon On larger aircraft constructed to carry cargo\/passengers the engines are placed on the wings E.g. DC3, A380 The positioning probably has to do with the amount of load to be lifted into the air, and manouverability desired... which seems sensible. Yet most water-going craft (boats, power-boats, merchantmen, naval craft) have their propellers\/push-pull placed at the back of the fuselage. What is the advantage in placing the driving component for boats in this position in addition to protecting them against damage? Can the propellers on a boat, with adequate protection say- a tubular enclosure, placed alongside the fuselage not drive the watercraft with equal efficiency and power?"}
{"id":"134101","title":"Suppose that the position of some object is goiven by the expression: x = 6.8t + 9.0t^2. Determine the object's acceleration","text":"x is in meters t is in seconds Answer = 18 m\/s\/s How do I solve this problem? Thank you for your time."}
{"id":"80864","title":"Average of $\\sin^2 \\omega t$, $\\cos^2 \\omega t$ for long times","text":"I don't understand how the averages mentioned in excerpt below work out. The question is regarding calculating the power spectrum of an oscillating classical dipole. IS averaging over long periods, for example, For $\\sin^2 \\omega_o t$ given by $$<\\sin^2 \\omega_o t> = \\frac{\\int t \\sin^2 \\omega_o t}{\\int \\sin^2 \\omega_o t} =?? \\frac{1}{2}$$ Also, how is the time important in doing this average. What I mean is, how is it justified to replace the function with its average value for times longer than its time period, how does this make sense physically? Thanks ![enter image description here](http:\/\/i.stack.imgur.com\/kbBqN.jpg)"}
{"id":"44003","title":"Schrödinger function: Separable wave function with even potential function of x","text":"I have done the Problem 2.1 in Griffiths' quantum mechanics, and it seems not making sense to me. What if the wave function isn't symmetric at all? Then obviously the proof doesn't work. The solution confuses me. > If $V(x)=V(-x)$ then the \"position\" wave function can always be taken to be > either even or odd."}
{"id":"41187","title":"The Heisenberg uncertainty principle: Interpreting $\\Delta p$, $\\Delta t$, etc","text":"(1) I have a textbook question that states the following: An electron has a speed of 500 m\/s with an accuracy of 0.004%. Calculate the certainty with which we can locate the position of the electron. The solution starts with $$\\frac{\\Delta p}{p}=\\frac{0.004}{100}$$ I don't understand how the question is being interpreted to give the above expression. (2) The very next question says: The average lifetime of an excited atomic state is $10^{-9}$s. If the spectral line associated with the decay of this state is 6000 Angstrom, estimate the width of the line. The solution begins with $\\Delta t = 10^{-9}$s. Again, I don't understand how this is arrived at. I'd expect it to be $t=10^{-9}$s. * * * Can somebody give me an intuitive understanding of uncertainty (perhaps in words) that I can apply to these two problems? I know the statistical definition, but I haven't found it very helpful in this context."}
{"id":"45972","title":"Measuring nucleons using electron beams","text":"sorry if the question is too elementary. From: The Britannica Guide to Particle Physics: > The sizes of atoms, nuclei, and nucleons are measured by firing a beam of > electrons at an appropriate target. The higher the energy of the electrons, > the farther they penetrate before being deflected by the electric charges > within the atom. For example, a beam with an energy of a few hundred > electron volts (eV) scatters from the electrons in a target atom. The way in > which the beam is scattered (electron scattering) can then be studied to > determine the general distribution of the atomic electrons. > At energies of a few hundred megaelectron volts (MeV; 106 eV), electrons in > the beam are little affected by atomic electrons. Instead, they penetrate > the atom and are scattered by the positive nucleus. Therefore, if such a > beam is fired at liquid hydrogen, whose atoms contain only single protons in > their nuclei, the pattern of scattered electrons reveals the size of the > proton. At energies greater than a gigaelectron volt (GeV; 109 eV), the > electrons penetrate within the protons and neutrons, and their scattering > patterns reveal an inner structure. Thus, protons and neutrons are no more > indivisible than atoms are. Indeed, they contain still smaller particles, > which are called quarks. I'm confused by this. I understand that the increasing velocity of the electron particles in the beam can overcome repulsive force of the electron cloud in an atom. Thus detecting size of the atom and even particular electrons. But what happens to beam particles in MeV range? At higher velocities, when they do reach nucleus, doesn't the positive charge then attract beam particles? How does it repel them? Is it that they have so much speed that they on near miss of a proton deflect or is something else happening here? What happens on GeV range? Do electron beam particles pass through proton to allow detection of quarks? Or is there happening something completely differentTM, like total annihilation of collided particles and recreation of particles from energy. Thanks for your time..."}
{"id":"87073","title":"Will a relativistic rocket be destroyed by incident photons?","text":"I have read, that as a rocket approaches light speed, objects in front of it become blue shifted and also intensity of light increases. Does this mean that at some point any rocket will be destroyed, just because the blue-shift will cause impacting photons to become gamma rays of arbitrary high energy - so high in fact that no material could withstand it?"}
{"id":"17957","title":"Would a high-explosive in a vacuum be less harmful?","text":"Putting aside shrapnel effects, I believe that high-explosives cause damage by producing a shockwave. How do shockwaves work in space? I've managed to convince myself that a high-explosive shockwave would work \"about the same\" by producing a shell of high-speed gas that would arrive at the surface with pretty much the same amount of energy that you'd get in air. Is that right?"}
{"id":"66923","title":"How charge distribution takes place when a battery is connected to a conductor?","text":"When one terminal of a battery say of 1.5 volt connected to a short length wire, few electrons get transferred from battery terminal to the wire raising the potential of the wire also to 1.5 volt. We know that raising the potential of a wire requires transferring certain numbers of electrons. Now consider the same battery is connected to very long wire of some thousand kilometer length. Will the battery be able to raise the potential of the wire?"}
{"id":"66926","title":"Does it matter in which direction I travel in relativity theory?","text":"I'm not a physicist, but I'm still very interested in the relativity theory, especially in how the twin paradox is explained. Actually, it does not make sense to me and I hope you can answer my following question to help me understand how it works: Does it matter in which direction i travel in the relativity theory? When I first read about the theory, I thought, that it does not matter if A moves away from B, or B moves away from A, because every party will see the effect of \"time going slower\" for the other pary, meaning A sees B aging slower and B sees A aging slower. But regarding the twin paradox, when A leaves the earth with insane speed, and returns years later, A has aged less than the people on earth. If this is actually true, then my previous assumption is wrong and it **does** matter if A moves away from B or the other way round, because the effect could be turned around, so that the earth ages less than A, if the earth moved away from A instead. If this is true, then this brings up a next question: How can we know if B moves away from A or A moves away from B? Since the earth, our solar system, and even our galaxy is already traveling in the universe at some speed in a specific direction, we could actually stop moving by flying in the exact opposite direction at the same speed. **but** would this be considered as the earth moving away or as we moving away? I really hope you can help me with that :) **Update for @Alfred Centauri's answer** `the stay-at-home twin does not change direction while the travelling twin does.` When I look at it from only the travelers perspective, then yes. But when I look at it from the earth's perspective, then the earth indeed changes its direction, while the traveler does not. Why is that changing direction thing that important? As I could see in Will's answer, the change of speed in universe seems to be the critical difference. In the time where the traveler is still on the earth, both the traveler and the earth move at the same speed in the universe, which means they age exactly the same. As soon as the traveler starts to accelerate or decelerate (move faster or slower than the earth) he **slows** his _own_ time, no matter if he accelerates or decelerates. If this is true, this means that not only the relative speed of the objects influences the time. Also the _change of speed_ in the universe does. But this opens another question: What if 2 travelers move to the exact same position in the universe, but traveler A goes there at twice the speed? Are the two travelers the same age when they meet at the destination? Or is one traveler older because he moved faster?"}
{"id":"66927","title":"Application of Calculus in Physics","text":"Why do we apply Calculus in Physics when most of the quantities are not continuous and are not symmetrical at all levels of magnification? Aren't most, if not all, forms of Matter and Energy discrete? We talk about differential elements of energy, charge, fields, current, liquids and so many other quantities which don't or can't exist and we derive results from such assumptions. How is this right?"}
{"id":"118518","title":"Shell model of an odd-odd nucleus: $^6$Li","text":"Lithium-6 isotope has an approximate magnetic momentum of $0.88\\ \\mu_N$ in its fundamental nuclear state. I'm trying to find its angular momentum and parity. I found in a standard table: $I=1^+$ and while I get the parity part, I don't understand the rest. What I have done is filling the shells separately for protons and neutrons and finding that both have their outer particle in the $1 P_{3\/2}$ state. Sum of these gives possible total angular momenta to be either $0$, $1$, $2$ or $3$. So my question is: how do you choose between these to find the real value $I=1$?"}
{"id":"45751","title":"Quantum cryptography: encryptions","text":"I am studying quantum cryptography and I have a very basic question. Suppose A and B share a secret key k, where k=0 or 1. A wants to send one qubit to B. What A does is, if k=1, she 'flips' the qubit (i.e. applies the Pauli X matrix), otherwise she leaves the qubit as it is. She sends this to B. Now B, knowing the value of k, either flips it or leaves it as it is. I am wondering why this protocol is insecure. How can an eavesdropper E cause problems?"}
{"id":"129619","title":"Time reversal operator symmetry of dirac lagrangian","text":"I want to prove time reversal symmetry of Dirac Lagrangian, I have some problems with calculations. I start with \\begin{eqnarray} T\\psi T = U \\psi \\end{eqnarray} \\begin{eqnarray} T\\bar{\\psi } T = \\psi^\\dagger U^\\dagger\\gamma_0 \\end{eqnarray} and I take time reversal transformation of the first part of Dirac Lagrangian \\begin{eqnarray} T\\bar{\\psi } \\gamma_aD^a\\psi T = {\\psi }^\\dagger U^\\dagger\\gamma_0 \\gamma_aUD_a\\psi \\end{eqnarray} I find, in order to parity invariant of the last part of Dirac Lagrangian \\begin{eqnarray} U^\\dagger\\gamma_a U = \\gamma^a \\end{eqnarray} this relation must satisfy. But it is not true for the part \\begin{eqnarray} T\\bar{\\psi } \\psi T = {\\psi }^\\dagger U^\\dagger\\gamma_0 U \\psi \\end{eqnarray} from here I find \\begin{eqnarray} U^\\dagger\\gamma_0 U = \\gamma_0 \\end{eqnarray} This is not consistent with $U^\\dagger\\gamma_a U = \\gamma^a $ for my convention. Here I used (-+++) metric and $TV^aT =V_a$, $\\bar{\\psi}= \\psi\\gamma_0$. Can you show me my fault."}
{"id":"47723","title":"Lennard Jones liquid vapour phase transition","text":"I am trying to generate a phase diagram of pressure versus temperature in a NPT ensemble using molecular dynamics method. I am using the Lennard Jones potential with 108 particles and Berendsen thermostat and barostat to fix the parameter. In order to observe phase transition I plot the radial distribution function (RDF) and also density versus temperature graph for various fixed temperature. However it is hard to differentiate liquid and gaseous phase from RDF. Another peculiar thing is that there should be a discontinuity in density as the transition temperature is reached, which I observe only for solid liquid transition but not for the liquid vapor transition. The change in density should be huge for liquid vapor transition. Any idea on what possibly goes wrong? Periodic boundary   cut off radius = 2.5   Truncated and shifted lennard jones potential   number of particle = 108   number of steps = 2000 timestep = 0.004   Berendsen barostat of scale coeefficient of 0.1 is used to scale the size of simulation box The density versus temperature graph for pressure of 1.0MPa:   http:\/\/i.stack.imgur.com\/x0d0p.jpg The solid liquid transition can be seen but not the liquid vapor transition."}
{"id":"89029","title":"How to interpret band structures","text":"I'm currently taking a Solid State Physics class, and is currently reading about the quantum mechanical description of solids. I then came across the following figure: ![Band structure for Al](http:\/\/i.stack.imgur.com\/9ydGp.png) It's supposed to be the band structure for aluminium. My question is basically: How do I interpret these band structures ? I can't even see why they cut of the region from $X$ to $\\Gamma$ at the dotted line in the first figure, and not in the second :\/ Why is the last bit of $X$ to $\\Gamma$ that you see in figure 2, not important in figure 1 ? I understand that, the $X$, $\\Gamma$, $K$, $L$ and $W$ are points of symmetry on the brillioun zone. But what does it mean for the electron(s) ? Only that they have higher energies at some points ? I'm just having some trouble understanding what this band structure tells me, and what I'm able to see\/do with it ? So can anyone maybe explain it, maybe even easily, what I actually have to understand when I see this ? Thanks in advance."}
{"id":"82160","title":"what happens during collision at molecular level?","text":"When two massive bodies collide we treat them simply by their mass interactions with momentum and energy conservations. Down at the molecular level it must be more like a field particle collision for the particles coming in close contact, why dont we consider all these effects ? Are they included in our equations already ?"}
{"id":"110747","title":"How come squeezing a water bottle makes the water come out?","text":"This seems natural, but I can't wrap my head around it when I think about it. When I squeeze an open bottle filled with water, the water will spill out. When I squeeze a bottle, the material collapses where I squeeze it, but expands in other areas, resulting in a constant volume. If the volume is constant, then I would think that the water shouldn't spill out. If I were to guess, there is something related to the pressure my hand is creating inside the bottle, but I'm not entirely sure."}
{"id":"110740","title":"Rate of radiation of infra-red ray from a hot body?","text":"I have the equation $\\frac{dE}{dt}=a(\\tau^4-\\tau_s^4)$ however I can't find anywhere what E is supposed to be. It is not pointed out anywhere in the text and what is confusing is that in engineering E was always Young's modulus. Though the pages heading is Radiation of infra-red rays. Is E the radiation of infra-red rays? Because I have another equation down the bottom saying that the equation $\\frac{dT}{dt}=-k(T-T_s)$ was derived from it. I can't see the relation. It just looks like they have changed the letters.. without any meaning."}
{"id":"96384","title":"Charge or Charges on the oil droplets?","text":"Due to my poor understanding of various paper and other material provided by google on Millikan's oil drop experiment I ask the following question here.   1. As the drops were charged by either friction or x-ray radiation or both, how did they ensure that they found out the charge of a **single** electron? For all we can say it just as well may have been 2 or more electrons!    2. Also even after the drops were charged by lets say a single electron (in excess) how could one be sure that it did not form a dipole? Why assume spherical symmetry?"}
{"id":"88897","title":"What is the motivation behind $\\tau=\\mu \\frac{\\mathrm{d}u}{\\mathrm{d}y}$ for Newtonian fluid?","text":"Where does the motivation come from? $\\tau$ is the shear stress, $u$ is the velocity and $\\mu$ is the shear viscosity. EDIT: Since I wrote the question on phone I wasn't clear enough about what I was actually wondering about (see comments below); thus let me clarify. I was simply wondering where did the derivative term come from and not why the expression is linear in the derivative term or why it is the first derivative and not e.g. the second and so on. The diagram in the answer I accepted shows exactly how the gradient comes into the picture."}
{"id":"119747","title":"Potential of the pseudo Nambu-Goldstone boson","text":"Pseudo Nambu-Goldstone bosons appear in breaking approximate symmetries, for instance, the QCD axions. Many literatures just write the action of those pseudoscalars without any explanation. In general, how is the effective action, especially the potential, of these Pseudo Nambu-Goldstone bosons obtained?"}
{"id":"48299","title":"How can gas from compressed air can \"take\" heat from surrounding environment?","text":"I have recently been reading about why a can of compressed air gets cold when the air it contains is discharged. From what I understand the change from a liquid to a gas requires energy and therefore heat is drawn from the surrounding environment. What confuses me is how the phase change sucks up heat? If energy is required to complete the phase change wouldn't that mean that the phase change would not take place unless heat was added? I guess I am confused as to how the phase change extracts the heat from the surrounding environment. If I wanted to compress a gas into a liquid I would have to use force or my own energy to accomplish this, yet when going from liquid to gas the energy is just taken from the surrounding environment. This is confusing to me."}
{"id":"82834","title":"Non-zero charge density due to Lorentz contraction in current carrying wires","text":"In trying to answer this question I came across the following problem. The original question relates to the idea that what looks like a magnetic field in one reference frame, ends up as an electrostatic field in another frame through special relativity. I'm going to imagine that the wire is just two blobs, one of positive charge, the other of negative charge (same magnitude of charge density), and I can choose to move them relative to one another along the length of the wire (Basically I'm ignoring all of the real physics of currents in wires to get a pristine situation). We have a test charge; we're going to describe the world in it's rest frame. Case 1 Both the positive and negative charge densities are at rest, in the test particle's frame. Obviously there is no electric force on the test charge, since the two charge distributions cancel each other out. There is no magnetic force either Case 2 Positive charge density is moving with velocity +v, negative charge density is moving with velocity −v. By symmetry, any relativistic effects on the the charge density must affect both densities in the same way; therefore no electric force. But we do have moving charges, so we'd see a magnetic force. Case 3 Positive charge is stationary relative to our test charge; negative charges are moving relative the positive charge and our test charge; length contraction yields higher negative charge density, and an electrostatic force on the test particle. Case (2) would seem to imply that we only see pure magnetic fields in a special symmetric inertial frame -- that frame where the effects of length contraction serve to make the two charge densities exactly the same. However, Case (3) is the normal lab measurement situation. I'm not sure if anyone has tried to put an upper bound on the magnitudes of any electrostatic forces related to running a real current through a real wire. If I run a computations based on a numerical example for drift velocity I get a result that the \"extra\" linear charge density for a 1mm diameter copper wire is <<1 one electron per meter; a deviation that would be very hard to measure experimentally. If this logic holds we'd have:   1. Theoretically, lab measurements of real wires do have some electrostatic forces involved, although these forces are way too small to have significant effects (and are probably smaller than thermal effects),   2. And we have to go to a reference frame where both charge densities are moving in order to see pure magnetic effects; I guess that this could be true, but I've never heard anyone refer to it in the application of special relativity to classical E&M; is there a flaw in my logic?"}
{"id":"46099","title":"Materials with different gravitomagnetic permeability?","text":"If you start with general relativity, and assume small perturbations around a nearly flat metric, it is possible to obtain linearized equations of gravity that look a lot like Maxwell's equations, also called GEM equations. One of them reads: $$\\nabla \\times \\mathbf{B}_\\text{g} = 4 \\left( -\\frac{4 \\pi G}{c^2} \\mathbf{J}_\\text{g} + \\frac{1}{c^2} \\frac{\\partial \\mathbf{E}_\\text{g}} {\\partial t} \\right)$$ Where $\\mathbf{E}_g$ is the (static) gravitational field, $\\mathbf{B}_g$ is a gravitational analogon to the magnetic field that encodes some general- relativistic effects. You see that it is coupled to the mass current density $\\mathbf{J}_g$, which has the effect that rotating or moving masses have a distinct (attractive or repulsive) gravitational effect. For example, two massive rings rotating in the same direction along a common axis will experience a small additional repulsion (or a slightly reduced net gravitational attraction). These effects are typically very small, but have been observed, e.g. by Gravity Probe B. Now compare with the corresponding electromagnetic equation: $$ \\nabla \\times \\mathbf{B} = \\frac{1}{\\epsilon_0 c^2} \\mathbf{J} + \\frac{1}{c^2} \\frac{\\partial \\mathbf{E}} {\\partial t}$$ The factor in front of the electric current $\\mathbf{J}$ is the magnetic permeability $\\mu_0 = 1\/\\epsilon_0 c^2$. In a medium, it becomes $\\mu_r \\mu_0$ instead. This relative permeability $\\mu_r$ is material dependent and source of a lot of interesting phenomena and technical applications, including transformators. You can simularly identify the gravitomagnetic permeability $\\mu_0^g = 4\\pi G\/c^2$. (Don't know if I should put the minus sign in the constant or not.) **Now I'm wondering if there is a corresponding relative $\\mu_r^g$. Are there materials that show a different GM permeability? Maybe event time-dependent, or manipulable through external fields?** From a microscopic persepective, it seems this would be somehow related to the spin or angular momentum of particles in the material. The effect is probably tiny, but if one would be able to vary it on very short time scales, it could maybe produce measurable effects. If found a mention of this idea in a 1963 paper (there's a PDF on the net if you Google the title):   * Robert L. Forward: _Guidelines to Antigravity_ \\- American Journal of Physics, March 1963, Volume 31, Issue 3, pp. 166 Unfortunately, more recent publications on this topic seem to be of dubious veracity (going in the direction of Podkletnov et al.)."}
{"id":"129351","title":"Superconductors and magnetic field","text":"I am working on a project, and I have a question: Concerning superconductors considered as diamagnetic material, what is the equation that describes the repulsive force to a diamagnetic material in a magnetic field?"}
{"id":"11442","title":"Does magnetised material lose weight?","text":"I've had argument on a forum about this. People think, that: if the same object is magnetised, it has less energy. so by $E = mc^2$ it weighs less than an object that is not magnetised (hence spin of its particles is chaotic). I've thought about experiment :) Let's say we have square of 4 protons and another square of 4 antiprotons. If we magnetise each square and annihilate them, do we get the same energy that we would get if only the antiprotons were magnetised?"}
{"id":"11449","title":"Conversion of mass to energy in chemical\/nuclear reactions","text":"Is mass converted into energy in exothermic chemical \/ nuclear reactions? My (A Level) knowledge of chemistry suggests that this isn't the case. In a simple burning reaction, e.g. $C+O_2\\to \\text{CO}_2$, energy is released by the CO bonds forming; the atoms lose potential energy when they pull themselves towards each other, in the same way that a falling object converts GPE to KE. There are the same number of protons, electrons etc. in both the reactants and products. I would have assumed that this reasoning extends to nuclear fission\/fusion as well, but one physics textbook repeatedly references very small amounts of mass being converted into energy in nuclear reactions. So I just wanted to know if I was wrong about either of these types of reactions, and if so, what mass is lost exactly?"}
{"id":"43116","title":"Time dilation $t'=(1-\\frac{v}{c})t$ instead of $t'=\\frac{1}{\\gamma} t$","text":"Suppose we have a train moving. When the origin of train's frame coincides with the origin of observers frame; the the time is set to zero. At that very instant, a photon is emitted from train towards the direction train is moving. After time $t$ measured by observer at rest the photon will be at a distance $ct$ and the train at a distance $vt$; but the photon is at a distance $ ct- vt$ from the train, now the time that the passenger on train measures for the photon to reach that point is the distance divided by the velocity of light ( which of course is $c$) so time measured by observer on train is $t'=\\frac{ct-vt}{c}= (1-\\frac{v}{c})t$ . where $t$ is time measured by observer at rest. Where is the mistake in this reasoning?"}
{"id":"46549","title":"How do \"holographic plates\" work?","text":"I asked a question about laser stage lighting over at Audio Video Production, and received an excellent answer that explained that laser clusters are generated from a single beam via something called a \"holographic plate\". I'd imagine they're made up of a crystal-like structure of materials with different refractive indices, but I'm not exactly sure. If that's the case, surely such tricks with refraction would cause component colours to be refracted at different angles, but this doesn't seem to be the case. How do these really work? **Update:** There seems to be some confusion. Cluster laser lighting units seem to be able to produce arbitrary numbers of output beams, and have some rudimentary control their direction, all from a single laser diode. I'm not asking about the principles of holograms, but rather about the mechanism by which a single laser beam passing through the plate results in multiple distinct beams coming out of the other side."}
{"id":"46544","title":"Force through quantum mechanics","text":"In classical physics force is: $$F=\\frac {dp}{dt}$$ How about quantum mechanics? In Old Quantum Mechanics momentum is: $p=\\hbar \\cdot k$ so force will be: $$F=\\hbar \\frac {dk}{dt}$$ what does $\\frac {dk}{dt}$ mean?"}
{"id":"64197","title":"Is there anything physically infinite?","text":"I can't think of a single thing that could be infinite. Because the universe is expanding, isn't it? But there is an ever-changing barrier, so why could there be anything infinite, both hypothetically and realistically?"}
{"id":"52520","title":"Why quantum hall effect has chiral edge state?","text":"The most popular explaination may be the following: in magnetic field, electrons move in cycolotron orbits, such cycolotron orbits ensure electrons to move in one direction at the edge. That is why the edge state is chiral. But I think this picture is too classical. Could anyone provide me with other explaination about this question? Does it has relation with time-reversal broken, chern number, or something else with topology? Thanks in advance~"}
{"id":"64198","title":"Why elastic materials are discribed by tensors?","text":"I am starting to read about elasticity of thin surfaces and I don't understand why tensors play such a major part? What are the tensors describing about the material? And just to clarify - Is there some calculation often done with 2nd order tensors (describing 3d materials) that is common for elastic materials?"}
{"id":"52527","title":"Can low-gravity planets sustain a breathable atmosphere?","text":"If astronauts could deliver a large quantity of breathable air to somewhere with lower gravity, such as Earth's moon, would the air form an atmosphere, or would it float away and disappear? Is there a minimum amount of gravity necessary to trap a breathable atmosphere on a planet?"}
{"id":"102528","title":"saturation property of nuclear forces ? and its relation binding energy per nucleon constantcy?","text":"Please explain why binding energy per nucleon is constant(pratically) for atomic number, A, larger than 30 and less than 170, and explain the saturation property of nuclear forces with analogy that is easy to understand?"}
{"id":"60431","title":"Given expectation values for E and B, can you find an associated state?","text":"When we quantize the electromagnetic field, we develop the concept of the field operator $A(\\vec{r},t)$ and the simultaneous eigenstates of momentum and the free field Hamiltonian (i.e., each eigenstate is given by specifying the number of photons with momentum $k$ and polarization $\\mu$). We can then construct the operators for the electric and magnetic fields, and we can calculate their expectation values for an arbitrary state. Now, suppose the expectation value of the electric field is $E(\\vec{r},t)$ and the magnetic field is $B(\\vec{r},t)$. Assuming $E$ and $B$ obey Maxwell's Equations, can we construct a state that has these expectation values? Is it unique, or could there be multiple states with the same expectation value for $E(\\vec{r},t)$ and $B(\\vec{r},t)$? What if the expectation values are time independent (i.e., static fields $E(\\vec{r},t)=E(\\vec{r},0)$ and $B(\\vec{r},t)=B(\\vec{r},0)$ for all $t$)?"}
{"id":"60437","title":"How do you tell if a metric is curved?","text":"I was reading up on the Kerr metric (from Sean Carroll's book) and something that he said confused me. To start with, the Kerr metric is pretty messy, but importantly, it contains two constants - $M$ and $a$. $M$ is identified as some mass, and $a$ is identified as angular momentum per unit mass. He says that this metric reduces to flat space in the limit $M \\rightarrow 0$, and is given by $$ds^2 = -dt^2 + \\frac{r^2 + a^2 \\cos^2\\theta}{r^2 + a^2}dr^2 + \\left(r^2 + a^2 \\cos^2\\theta \\right)d\\theta^2 + \\left(r^2 + a^2 \\right)\\sin^2\\theta d\\phi^2 $$ and $r$, $\\theta$ and $\\phi$ are regular spherical polar co-ordinates. But I don't understand why this space is obviously flat. The Schwarzschild metric also contains terms involving $dt^2$, $dr^2$, $d\\theta^2$ and $d\\phi^2$ but is curved. I always thought that a metric with off diagonal elements implied a curved space, but clearly I was very wrong. **Question:** How do you tell if a metric is curved or not, from it's components?"}
{"id":"80475","title":"How long will it take to temperatures of two solids to become equal?","text":"If I have two solids with commmon area where they are touching each other, is there any way for calculating the time it will take to them to become equal? For example, I have two blocks of different materials with different temperatures. They touch each other on one side. Can I calculate the time until their temperatures become equal? ![enter image description here](http:\/\/i.stack.imgur.com\/dxNsp.png) EDIT: According to your answer, I found this formula: $$ \\Delta Q=\\frac{\\lambda S(T_2-T_1)}{L}\\,\\Delta t $$ Can I calculate $\\lambda$ for my example as proportion of weights? $$ \\lambda_{r} = {m_{1}\\lambda_{1}\\over m_{1} + m_{2}} + {m_{2}\\lambda_{2}\\over m_{1} + m_{2}} $$"}
{"id":"48025","title":"How is quantum mechanics compatible with the speed of light limit?","text":"Consider a free electron in space. Let us suppose we measure its position to be at point A with a high degree of accuracy at time 0. If I recall my QM correctly, as time passes the wave function spreads out, and there is a small but finite chance of finding it pretty much anywhere in the universe. Suppose it's measured one second later by a different observer more than one light second away and, although extremely unlikely, this observer discovers that electron. I.e. the electron appears to have traversed the intervening distance faster than light speed. What's going on here? I can think of several, not necessarily contradictory, possibilities:   1. I'm misremembering how wave functions work, and in particular the wave function has zero (not just very small) amplitude beyond the light speed cone.   2. Since we can't control this travel, no information is transmitted and therefore special relativity is preserved (similar to how non-local correlations from EPR type experiments don't transmit information)   3. Although the difference between positions is greater than could have been traversed by the electron traveling at c, had we measured the momentum instead, we would have always found it to be less than $m_e c$ and it's really the instantaneous momentum that special relativity restricts; not the distance divided by time.   4. My question is ill-posed, and somehow meaningless. Would anyone care to explain how this issue is resolved?"}
{"id":"45018","title":"Irreversibility and the Fermi golden rule","text":"When a quantum system is perturbatively coupled to a continuum of states, one uses the Fermi's golden rule to compute the rate of transition form an initial state to a set of states contained in an infinitesimal volume of phase space (the energy density factor). However, I am still puzzled by the fact that the continuum feature of the spectrum implies this non reversible evolution whereas a discrete one would imply Rabi oscillations and finite time reversibility. How does one makes the transition from one to the other?"}
{"id":"99744","title":"Two masses attached by a spring","text":"> Suppose there is a system of two masses $M_1$ and $M_2$ connected by a > spring with constant $K$ on a frictionless surface. What is the frequency of > this system's oscillation? So I have fixed a coordinate system with the first mass at position $x_1$ and the second at $x_2$ so that if the spring has length $l$ at equilibrium, we have: $$m_1 x_1'' = k \\left([x_2-x_1]-l\\right)$$ $$m_2 x_2'' = -k \\left([x_2-x_1]-l\\right)$$ but I don't know how to solve this to find frequency. Any hints? I suppose I could try to solve this system of differential equations, but I don't really know how."}
{"id":"63167","title":"Is my textbook in error about this geometric optics relation?","text":"Given a centered optical system (having an axis of rotational symmetry), let $H_1$ be the intersection of the optical axis and the \"principal object plane\" (I'm studying in French and have no idea how to translate that... it's the plane whose image is a plane with a scaling factor of 1). Let $H_2$ be the intersection of the optical axis and the \"principal focal plane\", in other words the image of $H_1$. ![](http:\/\/i.stack.imgur.com\/j9bQC.gif) My book states that the distances between those points and the focal points, $f_1=H_1F_1$ and $f_2=H_2F_2$, verify the following relations: $$n_1f_1=-n_2f_2$$ $$\\frac{n_2}{f_2}=-\\frac{n_1}{f_1}=V$$ Where V is the vergence. But these seem to contradict eachother. From relation 1, we have $$n_1=-\\frac{n_2f_2}{f_1}$$ Substituting into the second relation: $$\\frac{n_2}{f_2}=\\frac{n_2f_2}{f_1^2}$$ $$1=\\frac{f_2^2}{f_1^2}$$ $$f_1=\\pm f_2$$ But the book then goes on to say that that's only true when $n_1=n_2$."}
{"id":"95677","title":"Breaking ocean waves on other planets possible?","text":"I have always wondered if there were an ocean on another planet with stronger or weaker gravity, would it be possible to have breaking waves?"}
{"id":"59533","title":"Free body diagram and Newton's second law","text":"> Two bodies of mass $m_1=50 \\text{kg}$ and $m_2 = 10 \\text{kg}$ are connected > with a light rope through a pulley (no friction between the rope and the > pulley). $m_2$ moves on surface with angle $37^{\\circ}$ with a friction > coefficient $\\mu_k=0.5$. Force $\\vec F$ acts on $m_1$. The system was moving > with a constant velocity for some time until the force stopped and then the > system started to accelerate. What's the acceleration of the system? ![Free body diagram](http:\/\/i.stack.imgur.com\/cp4JM.png) I assumed that the system accelerates to the left, therefore the equation for the $m_1$ would be: $ \\begin{cases} N_1=m_1 g\\\\\\ T-N_1 \\mu=m_1a \\end{cases} $ Therefore $T=m_1(a+g \\mu)$ For $m_2$ the equation would be: $ \\begin{cases} N_2=m_2 g \\cos 37^{\\circ}\\\\\\ m_2 g \\sin 37^{\\circ} - T - N_2 \\mu=m_2a \\end{cases} $ Therefore $m_2 g \\sin 37^{\\circ} -T- m_2 g \\cos 37^{\\circ} \\mu = m_2a $ The solution would be: $100 \\cdot 0.2024 - 50a -250=10a \\\\\\ a \\approx -3.829 \\frac{\\text{m}}{\\text{s}^2}$ Which is wrong according to the answers. The answer should be $\\approx -5.83\\frac{\\text{m}}{\\text{s}^2}$ which you can get pretty close if instead of $10a$ in the last step it was $-10a$. So I'm pretty sure that my mistake here is with the signs (of the acceleration), but I can't figure out why I'm wrong."}
{"id":"79097","title":"Non-conservative electric fields due to changing magnetic flux?","text":"What I read in several places, tells me that, the fact Coulomb's Law follows _inverse-square law and gives a force which is radial_ , implies that a static electric field must be _conservative_.(In short, Coulomb's law _is_ conservative!) But the electric field in presence of a time-changing magnetic flux (either time changing fields or a change of area of the loop, Motional EMF) is non-conservative, since closed-loop line integral of electric field is no longer zero but is $-\\frac{d\\phi}{dt}$ where $\\phi$ is the magnetic flux through the surface bound by the loop along which the line integral is carried out. I have the following question:- Consider a situation for the sake of presenting my argument, where there are only steady currents and no accelerated charges, and the changing flux is due to a wire sliding over another U-shaped wire and thus changing the area bound by the loop formed by the U-shaped end and the moving wire.![enter image description here](http:\/\/i.stack.imgur.com\/B4b5X.png) In this case, all the components and the charges (metallic free electrons and the positive kernels) are unaccelerated (i.e. on an average) and hence, there should be no radiation of electromagnetic waves. Therefore, whatever the electric field produced, which is responsible for the existence of current in the loop, must be produced by a particular non-accelerated charge distribution in the apparatus. But, whatever the field produced by any complicated arrangement of charges, through superposition must follow the inverse-square and the radial dependence of Coulomb's law and must be conservative, since all the superposed fields are conservative in themselves as the individual charges follow the conservative coulomb's law. > **So, how can a non-conservative configuration of the field, result from the > superposition of fields produced by an arrangement of individual charges > which follow a law (Coulomb's) which is conservative , assuming the absence > of electromagnetic waves?**"}
{"id":"34272","title":"Gauss's Law in action","text":"![enter image description here](http:\/\/i.stack.imgur.com\/gJq0p.jpg) Need someone to tell me if I got this done correctly (a) Draw Gaussuian cylinder inside the black cylinder to find charge enclosed $Q_{en} = Q(\\frac{r}{a})^2$ Apply Gauss's Law $E2\\pi r \\ell = \\dfrac{Q\\left ( \\frac{r}{a} \\right )^2}{\\varepsilon_0}$ $E = \\dfrac{Q\\left ( \\frac{r}{a} \\right )^2}{2 \\pi r \\ell\\varepsilon_0} = \\dfrac{Qr}{2\\pi \\varepsilon_0 a^2\\ell}$ $E = \\dfrac{\\rho r}{2 \\varepsilon_0}$ (b) Basically same principle, but the total charge of the black cylinder $E = \\dfrac{\\rho a^2}{2r \\varepsilon_0}$ (c) 0 under static equilibrium (d) The enclosed charge should be a sum of cylinders' charges. $Q_{en} = \\rho V + \\lambda \\ell = \\rho \\pi a^2 \\ell + \\lambda \\ell = \\ell ( \\rho \\pi a^2 + \\lambda)$ Hence $E2\\pi r \\ell = \\dfrac{\\ell ( \\rho \\pi a^2 + \\lambda)}{\\varepsilon_0} \\implies E =\\dfrac{ \\rho \\pi a^2 + \\lambda}{2 \\pi r \\varepsilon_0} $ The term looks very awkward, so it's probably wrong. Thank you for reading"}
{"id":"135119","title":"Determine radius, speed, radial and tangential component of velocity","text":"![enter image description here](http:\/\/i.stack.imgur.com\/Qdhv5.png) I need help mainly with the first question. If you can solve also the second problem,I'd greatly appreciate it."}
{"id":"37893","title":"gravitational convergence of light","text":"light has a non-zero energy-stress tensor, so a flux of radiation will slightly affect curvature of spacetime > **Question:** assume a flux of radiation in the $z$ direction, in flat > Minkowski space it will propagate and spread slightly in the $x$ and $y$ > directions. But what happens if we take into account self-gravitational > effects of light onto itself? will the radiation flux self-gravitate and > eventually reconverge?"}
{"id":"105891","title":"An EM problem of polarized sphere","text":"Today, I'm doing an EM problem and my question is exactly the same as this site shows: http:\/\/www.physicsforums.com\/showthread.php?t=133501 To reiterate my confusion, why we can't use that integral formula at the end V(r)=1\/4 pi epislon...+1\/4 pi epislon... This formula appears right in the same chapter of the problem. I know we can solve it by applying the Gauss Law, but why this formula won't work. I calculated it by using this formula, and I get a totally different answer."}
{"id":"105894","title":"Is changing angle of grooves in a Fresnel lens important for the lens to work?","text":"I'm thinking of making a huge Fresnel lens out of a plastic sheet I found in my storage. As it's for fun, and I'm not very concerned about precision, I thought I could make it using a nail, rope and two blades positioned so as to form a sharp angle between their cutting edges (or simply form a \"V\" shape). From looking at a flat Fresnel lens condensation diagram and sketching my own \"expanded\" lens, I figured I would need to account for groove angle changing when cutting grooves to allow the resulting lens to focus light. But as I'm not very knowledgeable in optics, I don't know if it's important. ![enter image description here](http:\/\/i.stack.imgur.com\/PwxAW.png) Will the lens still focus light in one spot if the grooves are cut at the same angle? If so, which angle should I choose then?"}
{"id":"105897","title":"What causes drag in a fluid?","text":"What causes resistance of an object to motion within a fluid like water? Please explain to me the molecular dynamics of the situation."}
{"id":"26657","title":"What happens when astronomical bodies made up of Dark Matter collapse or collide?","text":"Does dark matter not produce heat or radiation after a collision etc?   Or, we detect radiations having no generators?"}
{"id":"87412","title":"Bloch wave function orthonormality?","text":"there is this text book that is giving me a hard time for a while now: It shows that Bloch wave functions can be written as $$\\Psi_{n\\vec{k}}\\left(\\vec{r}\\right) = \\frac{1}{\\sqrt{V}}e^{i\\vec k \\vec r}u_{n\\vec k}\\left(\\vec r\\right),$$ which is fine to me. It also states that the Bloch factors $u_{n\\vec k}\\left(\\vec r\\right)$ may be orthonormalized on the (primitive) unit cell volume $V_{UC}$: $$\\frac{1}{V_{UC}}\\int_{V_{UC}}d^3r u^*_{n\\vec k}\\left(\\vec r\\right) u_{n'\\vec k}\\left(\\vec r\\right) = \\delta_{n n'}$$. However, and here starts my problem, it then concludes that therefore, the Bloch functions $\\Psi_{n\\vec{k}}\\left(\\vec{r}\\right)$ fulfill $$\\int_V d^3r \\Psi^*_{n\\vec{k}}\\left(\\vec{r}\\right) \\Psi_{n'\\vec{k'}}\\left(\\vec{r}\\right)=$$ $$=\\frac 1 V \\sum_\\vec{R} \\int_{V_{UC}\\left(\\vec R\\right)}d^3r e^{-i\\vec k \\left(\\vec R + \\vec r\\right)} u^*_{n\\vec k}\\left(\\vec R + \\vec r\\right) e^{i\\vec {k'} \\left(\\vec R + \\vec r\\right)} u_{n'\\vec{k'}}\\left(\\vec R + \\vec r\\right)=$$ $$=\\frac 1 N \\sum_{\\vec R} e^{i\\left(\\vec{k'}-\\vec{k}\\right)\\vec R} \\frac{1}{V_{UC}}\\int_{V_{UC}} d^3r u^*_{n\\vec{k}}\\left(\\vec r\\right) u_{n'\\vec{k'}}\\left(\\vec r\\right)=$$ $$=\\delta_{\\vec{k'}\\vec{k}}\\delta_{n'n}$$ with lattice vectors $\\vec R$ and crystal volume $V = N V_{UC}$. **But I just don't get the last two lines.** I mean, the integral in the second last line actually should read $\\int_{V_{UC}} d^3r e^{i\\left(\\vec{k'}-\\vec{k}\\right)\\vec r}u^*_{n\\vec{k}}\\left(\\vec r\\right) u_{n'\\vec{k'}}\\left(\\vec r\\right)$, shouldn't it? And, if that is true and I didn't miss something important already, I can't understand how that would yield these two $\\delta$s ... I'm almost sure I missed something, but I just desperately keep fail getting it, so any help would be greatly appreciated! **EDIT** Thank you very much for your reactions! However, it seems I failed to state my problem clear enough, so I figured it might be best to tell you what my approach so far was step by step so may be someone can see where I actually go wrong: Starting with $$\\int_V d^3r \\Psi^*_{n\\vec{k}}\\left(\\vec{r}\\right) \\Psi_{n'\\vec{k'}}\\left(\\vec{r}\\right),$$ I partitioned the integration domain $V=NV_{CU}$, thus getting a sum of integrations over the unit cell volume, yielding $$\\sum_{\\vec R}\\int_{V_{UC}} d^3r \\Psi^*_{n\\vec{k}}\\left(\\vec{r} + \\vec R \\right) \\Psi_{n'\\vec{k'}}\\left(\\vec{r} + \\vec R\\right),$$ which happens to be exactly the second line when exploiting $\\Psi_{n\\vec{k}}\\left(\\vec{r}\\right) = \\frac{1}{\\sqrt{V}}e^{i\\vec k \\vec r}u_{n\\vec k}\\left(\\vec r\\right)$. I then proceeded using $\\Psi\\left( \\vec r + \\vec R\\right)=e^{i\\vec k \\vec R}\\Psi\\left(\\vec r\\right)$. But this yields $$ \\sum_{\\vec R} e^{i\\left(\\vec{k'}-\\vec{k}\\right)\\vec R} \\int_{V_{UC}} d^3r \\Psi^*_{n\\vec{k}}\\left(\\vec r\\right) \\Psi_{n'\\vec{k'}}\\left(\\vec r\\right)$$ which disagrees with the third line in the book where the integral is over the Bloch factors $u_{n\\vec k}\\left(r\\right)$ only. However even assuming this is just a typo (which I'm not so sure of ...), I would be confronted with the integral $\\int_{V_{UC}} d^3r e^{i\\left(\\vec {k'} - \\vec k\\right)\\vec r} u^*_{n\\vec{k}}\\left(\\vec r\\right) u_{n'\\vec{k'}}\\left(\\vec r\\right)$ and I can't see how those two $\\delta$s would arise from that either. Thank you all again for your reactions and I hope I know actually stated my problem clearly."}
{"id":"77271","title":"Explanation for the minus sign in $\\Omega_3$ in the Kappa symmetry of the Green - Schwarz formalism for F1 strings","text":"> _Just so that there can be more higher - level physics questions here, let > me post this question + answer. Also because I'm a bit sad that there are > almost no questions on the Green-Schwarz formalism._ In the Green-Schwarz formalism, for F1 strings, we have the action $$S=S_1+S_2$$ Where $${S_1} = - T\\int_{}^{} {\\sqrt { - \\det \\left( {{\\Pi _{\\alpha \\mu }}\\Pi _\\beta ^\\mu } \\right)} {{\\text{d}}^2}\\sigma } $$ and $S_2$ is the additional action term that arises through Kappa Symmetry. The transformations of kappa symmetry for F1 strings are intuitively related to that for D0 branes, etc. They are given by: $$\\begin{gathered} \\delta {X^\\mu } = {{\\bar \\Theta }^A}{\\gamma ^\\mu } \\\\\\ \\delta {\\Theta ^A} = - \\delta {{\\bar \\Theta }^A}{\\gamma ^\\mu }{\\Theta ^A} \\\\\\ \\end{gathered} $$ So that: $$\\delta \\Pi _\\alpha ^\\mu = - 2\\delta {\\bar \\Theta ^A}{\\gamma ^\\mu }{\\partial _\\alpha }{\\Theta ^A}$$ From this transformation, the variation in $S_1$ by kappa symmetry, is almost trivially (not that trivial, thiough...): $$\\delta {S_1} = \\frac{2}{\\pi }\\int_{}^{} {\\sqrt { - \\lambda } {\\lambda ^{\\alpha \\beta }}\\Pi _\\alpha ^\\mu \\delta {{\\bar \\Theta }^A}{\\gamma _\\mu }{\\partial _\\beta }{\\Theta ^A}{{\\text{d}}^2}\\sigma } $$ Here, $$\\begin{gathered} \\lambda = \\det {\\lambda ^{\\alpha \\beta }} \\\\\\ {\\lambda ^{\\alpha \\beta }} = {\\Pi _{\\alpha \\mu }}\\Pi _\\beta ^\\mu \\\\\\ \\end{gathered} $$ To determine $S_2$, howefver, is not all that trivial since the simple procedure I've mentioned here, for example, is just not practical by any means. We there fore use a 2-form, form $\\Omega_2$, such that: $${S_2} = \\int_{}^{} {{\\Omega _2}} = \\int {{\\epsilon ^{\\alpha \\beta }}{\\Omega _{\\alpha \\beta }}{{\\text{d}}^2}\\sigma } $$ We may also (formally) define a 3-form $\\Omega_3=\\mbox{d}\\Omega_2,$ so that by Stokes' theorem, we have it that: $$ \\int_M {{\\Omega _2}} = \\int_D^{} {{\\Omega _3}} $$ $M$ would be the worldsheet whereas $D$ would be it's interior, ; $M=\\partial D$. There are 3 supersymmetric 1-forms prominent to us namely, ${\\text{d}}{\\Theta ^1},{\\text{d}}{\\Theta ^2},{\\Pi ^\\mu }$. If ${\\Omega _3}$ is to be supersymmetryic, which it better be, then, we shoul'd have it that ${\\Omega _3}$ involves just these 3 1-forms. A very sensible choice of 3-form for $\\Omega_3$ is: $${\\Omega _3} = A\\left( {{\\text{d}}{{\\bar \\Theta }^1}{\\gamma _\\mu }{\\text{d}}{\\Theta ^1} -{\\kern 1pt} {\\kern 1pt} {\\kern 1pt} {\\text{d}}{{\\bar \\Theta }^2}{\\gamma _\\mu }{\\text{d}}{\\Theta ^2}} \\right){\\Pi ^\\mu }$$ Now, when I first learnt this, I was pretty confused at the \"$-$\" si1gn, (probably because of the fact that I was too confused to bother reading the next line of bbs : ) ...). As I guess many others may also be confused at this, let me ask and answer the question: > Is this minus sign, by any chance, related to the fact that Type IIA String > Theory is chiral? Does this mean that the above expression only holds for > the Type IIA, and not for the Type IIB ? ."}
{"id":"22377","title":"Why does change in speed of a wave make it refract?","text":"When a light wave enters a medium with a higher refractive index (e.g. from air to standard glass) and its speed decreases, why does that make it refract\/bend? I understand that wavelength decreases and frequency stays the same and therefore its speed decreases, but I can't find anywhere whatsoever why the speed decrease cause the wave to refract. So could someone please explain this?"}
{"id":"24793","title":"Possibility for radiation in dark matter that is not interactive with regular matter?","text":"**Definition:** Radiation in this case does not refer to electromagnetic radiation. It refers to any kind of emission of energy, even energy that does not interact with regular matter. Just like dark matter does not interact with electromagnetic radiation, could regular matter not interact with \"dark matter radiation\" (I'm not talking about the usual \"really high wavelength radiation\" kind)?"}
{"id":"24794","title":"Will a stone thrown in space move forever?","text":"If I throw a stone on space, in a place where gravity is equal zero, will the stone move forward forever, because no air, so no friction?"}
{"id":"79549","title":"Working with a Routhian for a specific system","text":"I asked a more general question earlier about the Routhian, but I'm still having trouble working with it. Here's my specific case. Given the following Lagrangian: $$L=(1\/2)m(\\dot{r}^{2}+r^{2}\\dot{\\phi}^{2}sin^{2}\\theta_{0})-mgrcos(\\theta_{0}),$$ I obtained the corresponding Routhian: $$R=(1\/2)p_{\\phi}\\dot{\\phi}-(1\/2)m\\dot{r}^{2}+mgrcos\\theta_{0}.$$ I'm being asked to do the following: obtain the Hamilton equation of motion for $p_{\\phi}$, show that it's constant in time and find its initial value, obtain the Lagrange equation of motion for $r$ from the Routhian, and obtain the Hamiltonian When I try to do the first part (obtain the Hamilton e.o.m.) I get $∂R\/∂p_{\\phi}=(1\/2)\\dot{\\phi}$ which doesn't show that it's constant in time or that angular momentum is conserved, which I think is what we're trying to show here. What am I doing wrong? Also, how can I obtain the Lagrange e.o.m. for $r$ using the Routhian? Do I just take $$\\frac{d}{dt}\\frac{∂R}{∂\\dot{r}}-\\frac{∂R}{∂r}=0,$$ since $r$ appears in the Routhian and plug in $r=r_{0}$ to find the initial value? Similarly, how do I go about obtaining the Hamiltonian from the Routhian?"}
{"id":"35633","title":"Buoyancy in gases","text":"If buoyancy is caused by pressure gradient and if the pressure of a gas is same everywhere in its container, which means there's no difference of pressures on surfaces of the lighter-than-air object, how come we observe such an effect?"}
{"id":"32943","title":"Shouldn't LHC have used $p\\bar{p}$ collisions, instead of $pp$ collisions, to study baryogenesis?","text":"Baryogenesis is the physical process(es) that produced baryon antibaryon asymmetry in the early universe. That means, the laws that governed the bigbang was baryon-antibaryon symmetric. On the other hand it is usually said that LHC is probing the early universe by creating similar conditions of the early universe. My question is, by using $p\\bar{p}$ collisions, instead of $pp$ collisions, we are not exactly reproducing the early conditions of the universe because the conditions we created are not baryon antibaryon symmetric, are we?"}
{"id":"95398","title":"Calculating angle of projectile motion","text":"I realize that similar questions have been asked but I'd like to put my problem here and what I have to make sure everything is right. So my problem is: I've got a rock $2.5$ meters above the ground and I shoot it out of a slingshot. It hits another rock which is $3.6$ meters above the ground, $14$ meters away from it's initial position $x$. The initial velocity of the rock is $15.9$ m\/s. So the formula I know is: $ \\frac{1}{2} \\cdot g \\cdot t^2 - (v_o \\cdot \\sin ( \\theta ) ) + y $ I also know: $x = (v_o \\cdot \\cos ( \\theta ) ) \\cdot t$ But these don't seem to help me solve my problem as I don't have $t$ nor $ \\theta$."}
{"id":"32947","title":"R charge of the chiral multiplet in $2+1$ dimensions","text":"These are two examples that I am puzzled by,   * One can see in this paper on page 16 that for ${\\cal N} =2$ theory on $2+1$ the R-charge of the $\\phi$ and the $\\psi$ is determined to be $\\frac{1}{2}$ and $-\\frac{1}{2}$. How?  Further in this case since the supercharges are in a vectorial (self- conjugate!) representation of the R-symmetry group why is the R-charge of $\\psi$ and $\\psi^*$ different? Since post radial quantization $\\bar{Q}=S$ it follows that $Q$ and $\\bar{Q}$ transform in exactly the same way under $SO(2)$ and hence shouldn't the R-charges $\\psi$ and $\\bar{\\psi}$ been the same?   * One can see in this paper at the bottom of page 7 a statement about how the scalar component of the ${\\cal N}=3$ chiral multiplet in $2+1$ dimensions is in a $SU(2)_R$ triplet. So does this (what?) determine the R-charge of the scalar field and other components in the multiplet?  What is puzzling is why did the authors in the second paper think of the R-symmetry group as $SU(2)$ for ${\\cal N}=3$ rather than the $SO(3)$ that would be gotten by definition? I would have thought that the R-charges of a field are the same as the eigenvalue under the Cartan subalgebra of the R-symmetry group of the supercharge(s) which acted on the lowest helicity state state of the multiplet. But isn't the R-charge of the lowest helicity state of a multiplet merely a convention? Can't the scalar field be always chosen to be uncharged (R=0) under the $SO({\\cal N})$ R-symmetry of $2+1$?"}
{"id":"93632","title":"Is boson sampling a problem in 'continuous variable' quantum information?","text":"When people generally speak of quantum information in the context of continuous variables, what is generally meant is that observables, like position\/momentum or the field quadratures of quantum optics, can take a continuous range of values and that information processing tasks are performed using operators on these variables. However, most work in this area restricts attention to a countable number of modes of the radiation field (in quantum optics). In this case, the state of the system can be described using a countably infinite number of variables, like the number eigenstates in the field. But this is still called 'continuous variable' quantum information because the relevant observables still take continuous values. Is that so? If so, is the boson sampling problem in the domain of continuous variables? I don't know what exactly the relevant variables are in this case. A preliminary reading seems to suggest to me that the observable is the photon number, since what is observed is the probability after measuring in some basis. This would now mean that in the sense I think of continuous variables, boson sampling is not a problem in this domain. Am I right\/wrong? Which of my statements here are false?"}
{"id":"23682","title":"Free particle propagation amplitude calculation","text":"I have a quick calculational question. In Peskin and Schroeder, Chapter 2, they want to look at the amplitude for a particle to propagate between two arbitrary points, $x$ and $x_0$, in an arbitrary amount of time t given the Hamiltonian $\\sqrt{p^2c^2+m^2c^4}$. To do this, we look at the inner product of the time-evolved particle that was at $x_0$ with the $x$ eigenket: $\\langle \\vec{x}|\\exp(-i\\hat{H}t\/\\hbar)|\\vec{x_0}\\rangle$, correct? If we insert two complete sets of momenta, the integral becomes (give or take factors of $\\hbar$): $\\frac 1 {(2\\pi)^3}\\int d^3p\\; \\exp(-it\\sqrt{p^2c^2+m^2c^4}\/\\hbar)\\exp(i\\vec{p}\\cdot(\\vec{x}-\\vec{x_0}))$ I'm not quite sure where to proceed from here... Peskin and Schroeder somehow reach: $\\frac 1 {2\\pi^2 |x-x_0|}\\int_0^\\infty dp\\;p\\sin(p|x-x_0|)\\exp((-it\\sqrt{p^2c^2+m^2c^4}\/\\hbar)$. To get here, however, it seems that you would have to assume that $p$ points in the same direction as $x-x_0$ to get spherical symmetry or something. Why can we do this? Aren't we summing over all possible momenta? Thanks!"}
{"id":"25681","title":"Is the Apophis asteroid a concern?","text":"I was watching the TED lecture _How to defend Earth from asteroids_ , and the speaker Phil Plait spoke of the possibility of this asteroid hitting Earth. He also said something about a Mayan legend. He says it's not a concern, is there a consensus on that? And what is the source for this Mayan legend?"}
{"id":"23970","title":"So where is mistake in the formula of wave number (magnitude of wave vector)?","text":"I have following form for wave vector $k_2=n_2 \\omega\/c_0$. Now because $\\omega=2 \\pi c\/ \\lambda$, then $k_2=n_2 \\omega\/c_0=\\frac{n_2 2 \\pi c_0}{c_0 \\lambda}=\\frac{n_2 2 \\pi}{\\lambda}$. But problem is that $k$ (is magnitude of wave vector~ wave number) and according to wikipedia $k=\\frac{2 \\pi}{\\lambda}$. So where is mistake?"}
{"id":"23971","title":"Building some measurement appratus that distinguish between two mixtures","text":"We have a measurement $M$ that distinguishs between $\\rho_1$ and $\\rho_0$, if it has three possible answers 1,2,3 and whenever it answers something different than 3 it's correct. $M$ succeeds with probability $p$ if for every $\\rho s$ above it answer the correct answer with probability at least $p$. Now I found the most efficient value of $p$ that $M$ succeeds to disntiguish between $|0\\rangle\\langle0|$ and $|+\\rangle\\langle+|$, but I am not sure how to contruct a three dimensional measurement that achieves this value of $p$. Any hints, or references? Thanks."}
{"id":"65908","title":"The quickest 6-3 play in baseball: to bounce or not to bounce?","text":"What's the quickest way the get the ball (say from shortstop) into the first- baseman's glove, given some fixed initial (throwing) speed? Directly or with one bounce? I'm fairly sure that the answer depends on the initial throwing speed, but where's the cut-off point (roughly)? * * * PS 1: Numbers, _if needed_ : > Under the current rules, a major league baseball weighs between $5$ and > $5\\frac1 4$ ounces ($142$ and $149\\,\\mathrm g$), and is $9$ to $9\\frac1 4$ > inches ($229–235\\,\\mathrm{mm}$) in circumference ($2\\frac7 8–3$ in or > $73–76\\,\\mathrm{mm}$ in diameter). The horizontal distance to be travelled is $36\\,\\mathrm m$. PS 2: The surface is \"clay\" (around first base). The ball is covered in leather. PS 3: Let's assume that throwing accuracy and catching abilities are not issues."}
{"id":"27268","title":"Visualization of 1D spin chain wave fuction","text":"What are the known methods for visualizing quantum states of one-dimensional spin chains? They can be based either on their wave functions or density matrices. My particular interest is in plotting eigenstates of 1D spin chains (e.g. Heisenberg model) in a way in which correlations between particles are related to the self-similarity of the image. **As a side note** , I found a Wikipedia Commons image with its description on the topic (a thumbnail below). However, it is not supported neither by a peer reviewed publication nor by a decent preprint. The only text related is extremely drafty. However, I'm the most interested in published results, and even more when they relate properties of the image to the properties of the state. ![enter image description here](http:\/\/i.stack.imgur.com\/MM0hN.png)"}
{"id":"22084","title":"How is information related to energy in physics?","text":"I recently attended a talk by Dr. Ravi Gomatam on 'quantum reality', where the speaker suggested, that conservation of energy is not a fundamental law, and is conditional, but the conservation of information is fundamental. What exactly is the meaning of information? Can it be quantified? How is it related to energy?"}
{"id":"22087","title":"Neutron electric dipole moment and $T$ symmetry violation","text":"Our textbook (and other sources I have found) says that non-zero electric dipole moment of neutron would violate $T$ symmetry. They prove this statement by first assuming $\\boldsymbol{D}=\\beta\\boldsymbol{J}$, where $\\boldsymbol{D}$ is the dipole moment, $\\boldsymbol{J}$ is the angular momentum, and $\\beta$ is a constant. But why? Why is $\\boldsymbol{D}$ proportional to $\\boldsymbol{J}$? Why is $\\boldsymbol{D}$ related to $\\boldsymbol{J}$ at all? And how can't this argument be applied to other composite particles such as atoms and molecules, thereby breaking T symmetry for most of the world?"}
{"id":"27260","title":"Are lens spaces classified via a Weinberg angle?","text":"I am thinking about Kaluza Klein theory in the 3 dimensional lens spaces. These have an isometry group SU(2)xU(1), generically, and in some way interpolate between the extreme cases of manifolds $S^2 \\times S^1$ and $S^3$ (or, arguably depending of your parametrisation, across $S^2 \\times S^1 ... S^3 ... S^1 \\times S^2$). My question is, simply, if the application of Kaluza Klein construction forces different coupling constants to the two groups $SU(2)$ and $U(1)$, and then a kind of discrete Weinberg angle, or if they have always the same coupling. Note that when you build a lens space bundle over CP2 (which is equivalent to a $S^1$ bundle over $CP^2$ x $CP^1$, if I read correctly a remark of Atiyah reported by Kreck and Stolz), you get an isometry group of SU(3) times SU(2) times (1), so the same question could be asked here, in this more widely known setup. Of course that odd dimensional spaces do not carry chirality, but my question is just about having different couplings, that is all. It could be interesting, as a plus, to compare with coupling relationships coming from Non Commutative Geometry..."}
{"id":"65906","title":"What happens when a ball stops bouncing?","text":"If I were to drop a bouncy ball onto a surface, each successive bounce will be lower in height as energy is dissipated. Eventually, however, the ball will cease to bounce and will remain in contact with the ground. _What happens during that small moment of time that is the transition period?_ Here's my line of thought so far: Let's say, for example, that the ball could be modeled as an \"ideal\" bouncy ball. Each time it hits the floor, it rebounds with one-half of the impact velocity, the other half of which is absorbed and dissipated by various things. Will this ideal bouncy ball ever stop bouncing? The answer, as far as I can tell, is yes (well, _maybe_ ). Since each bounce has half of the initial velocity of the previous bounce, the time between each bounce also halves. The first bounce might take $1$ second, the second takes $0.5$ seconds, the third $0.25$ seconds, etc. This infinite series converges to a finite amount of time, but with an infinite number of bounces. Since it's obviously not possible for an actual ball to bounce an infinite number of times in finite time, it is clear that the above model doesn't work. Eventually, I decided that the problem must be in the fact that the model doesn't take into account the fact that ball spends time compressing and decompressing while in contact with the floor. Even once the ball touches the ground, the center of gravity continues to move downward as the ball compresses. Soon, the ball expands, and the center of gravity moves upward. Lets $h_1$ be the height of the center of gravity when the ball is uncompressed, and $h_2$ the height of the center of gravity of the ball when compressed, and $D = h_1 - h_2$. Eventually, the height of the bounce will be smaller than $D$, so that center of gravity simply vibrates between those two locations, and the ball never leaves the ground."}
{"id":"25531","title":"Maximum depth for bacteria on rogue planets","text":"If microscopic life evolves on a planet which is subsequently ejected from orbiting its star, then the planet would cool down. How deep into the interior of the planet could organisms reach, assuming the mantle had cooled down enough? For example, on Earth there are rock-boring bacteria that have tunneled their way 3 km into the crust. The limiting factor for depth of bacteria on Earth seems to be temperature, but presumably if the mantle had cooled down then at some depth other factors would come into play such as pressure or rock hardness or composition."}
{"id":"30671","title":"Time and distance where velocity is a function of time","text":"I'm creating a video game set in space in which spaceships travelling between places accelerate until they hit a maximum velocity. They then travel at that velocity until a time when they need to start decelerating so as to arrive at their destination and not overshoot. There is no relativity or friction or anything else 'fancy' involved For the algorithm I'm writing to do this, I need to know a) the time it will take a ship to reach the half way point in its journey (assuming it continues to accelerate forever), and b) the time it will take to reach maximum velocity, and c) the distance travelled at time t (assuming continuous acceleration). I can then use that to tell the ship at what time to start decelerating, depending on how far the whole journey is. However, to accelerate the ships, I multiply their current velocity by 1.5 every second (to decelerate, I divide by 1.5). i.e. v(t) = v(t-1) * 1.5 I can also assume Initial velocity (velocity at time 0) is a constant and always positive If anyone can point me in the right direction as to what formula to use (or even better, the pseudo-code to do it), I'd be most appreciative"}
{"id":"26139","title":"Why did the asteroid belt between Mars and Jupiter form as it did?","text":"I'm curious about why the asteroid belt wasn't pulled by Mars's or Jupiter's gravity or formed into either moons or planets. Why did it form into an asteroid belt instead?"}
{"id":"104454","title":"List of experimental band gap","text":"does anyone know where one can find a list, database, book of experimental bandgap values of semiconductors? Is there such a collection? Or do I need to scour papers one at a time to get the values?"}
{"id":"70555","title":"LHC vacuum tube diameter","text":"What's the diameter of the tube where the protons fly around? I dont mean the overall diameter (~1-10 m), instead the inner most ultra-high vacuum tube where the protons are. The inner pipe, the capillary, the channel.. I would guess maybe 1 mm?"}
{"id":"2029","title":"Can I levitate an object without using an electromagnet?","text":"I know that it's possible to make an object levitate using an electromagnet to hold it up. But is it also possible to do this with regular magnets? Is there a special kind of magnet I need in order to have one powerful enough to hold an object up? I'm asking because I have this idea in mind where I want to make a decorative item that is levitating an inch or so above its container. ![alt text](http:\/\/i.stack.imgur.com\/30uUp.png) **A** is the object I want to levitate.   **B** is the container.   **C** shows the magnetic field that is repelling **A** and **B** from each other achieving the levitation. The size of this would be as small as a food plate, maybe even smaller. **B** is barely a pound or two in weight. Is that possible?"}
{"id":"71683","title":"Dipole moment induced in a spherical particle","text":"Consider a spherical metal particle made out of gold. If there is an external charge somewhere near the gold particle, is there a way to calculate the resulting dipole moment that is induced by the external charge? Or is there a way to calculate the induced dipole moment that results from an external dipole moment instead of an actual charge? Would be glad if you could give me some ideas or equations to tackle these two problems."}
{"id":"71688","title":"How to write the single electron spin-orbit coupling under an external magnetic field?","text":"As we know, without the external magnetic field, the single electron spin- orbit coupling(SOC) has the form $\\boldsymbol{\\sigma}\\cdot(\\boldsymbol{\\nabla} V\\times \\mathbf{p})$ up to a coefficient, where $V$ is the static electric potential. Now if we add an static external magnetic field $\\mathbf{B}=\\boldsymbol{\\nabla}\\times \\mathbf{A}$, does the SOC preserve the above form $\\boldsymbol{\\sigma}\\cdot(\\boldsymbol{\\nabla} V\\times \\mathbf{p})$ or change into $\\boldsymbol{\\sigma}\\cdot(\\boldsymbol{\\nabla} V\\times (\\mathbf{p}-e\\mathbf{A}\/c))$? The notation $\\mathbf{p}$ always represents $-i\\hbar(\\partial_x,\\partial_y,\\partial_z)$ in all of our formulas. This reminds me another question, may be worthless: Does this expression $\\mathbf{r}\\times(\\mathbf{p}-e\\mathbf{A}\/c)$ make sense? How to write the orbital angular momentum under an external magnetic field? Thank you very much."}
{"id":"106158","title":"State of constant motion","text":"Why does an object remains in its state of constant motion if there are no forces acting on that object? My understanding is that all the energy of the motion will be kept inside and a change in the speed needs a change in the amount of the energy stored by that object. But how is that energy stored in a form of speed? Does anything changes inside the object's structure if it is moving?"}
{"id":"70883","title":"How much iron ware to make a Faraday cage","text":"In a thunderstorm I was thinking the following: suppose I am rowing in a lake during a thunderstorm. How big a Faraday cage do I need to make to protect myself? If lightning strikes the cage, will it dissipate away through the water?"}
{"id":"70887","title":"Where does the light go when I turn off the light switch?","text":"As above... Surely it must go somewhere? My child asked me this question and it was a difficult one to answer."}
{"id":"14484","title":"Why is the string theory graviton spin-2?","text":"In string theory, the first excited level of the bosonic string can be decomposed into irreducible representations of the transverse rotation group, $SO(D-2)$. We then claim that the symmetric traceless part (i.e. the **35** rep) is the spin-2 graviton - but isn't the label \"spin-2\" intrinsically 3+1 dimensional? I.e. it labels the representation under the little group $SU(2)$?"}
{"id":"14489","title":"Acceleration vector - deceleration vs direction","text":"If acceleration of something $= - 10 \\text{ m s}^{-2}$ And forwards is define as north. Does that mean the object is getting slower (decelerating) or accelerating in the reverse direction (south) How can you tell the difference?"}
{"id":"38021","title":"What happens when we cut objects?","text":"What is the role of the molecular bonds in the process of cutting something? What is the role of the Pauli exclusion principle, responsible for the \"hardness\" of matter? Moreover, is all the energy produced by the break of bonds transformed into heat?"}
{"id":"76484","title":"Book on optics in curved space-time","text":"As evidenced from my earlier questions on vision and curved space, I am struggling a little bit with visual perception in curved space-time. I would like a book recommendation on optics and vision in curved space-time.   1. I find books on Gravitational lensing to be inadequate as Gravitational lensing itself deals mostly with specific cases in astronomy. It does not have the breadth and scope I require.   2. I want the book to discuss in detail how space (of low curvature) could affect visual perception in everyday life scenarios. Very high mathematical rigor is desired.   3. I would also really like the book to have pretty colors and pictures for aesthetic value. If you know any book that come close to satisfying my conditions, please do tell me. I will be very grateful if you can. Also, as per the books policy, it is also OK to provide an explanation for any subdiscipline of this topic instead, especially if that subdiscipline is not covered by most resources. But please do not make such an answer Community Wiki."}
{"id":"73956","title":"Why does a stack of transparencies appear white when a single sheet is clear?","text":"My daughter and I found a box of old transparencies (clear plastic sheets) and something struck me as odd: even though each transparency is clear, the whole stack of transparencies looks whitish - why is that?"}
{"id":"103066","title":"Chirality and helicity","text":"I have massless Dirac equation and chirality and helicity operators which are given as $$ \\hat {P}_{ch}\\Psi = \\gamma_{5}\\Psi, \\quad \\hat {P}_{h}\\Psi = \\frac{(\\hat {\\mathbf S} \\cdot \\mathbf p)}{|\\mathbf p|}\\Psi . $$ I showed that both of them are lorentz invariant operators, they commutes with dirac free hamiltonian and their eigenvalues are $\\hat {P}_{ch}\\Psi (x) = \\pm \\Psi (x), \\quad \\hat {P}_{h}\\Psi (x) = \\pm \\Psi (x)$. Is it enough for showing their identity? Or I also need to connect the given eigenvalue of $\\hat {P}_{h}$ with corresponding eigenvalue of $\\hat {P}_{ch}$?"}
{"id":"103069","title":"Is there a wave function for anyons?","text":"People talk about anyons a lot. But i have never seen an anyon wave function. I suspect that there is no such thing as a wave function for anyons. I mean, anyons are not generalizations of bosons or fermions. For bosons and fermions, one can have many-body hilbert spaces, but for anyons, there is no such thing."}
{"id":"88649","title":"What really happens with Time Dilation?","text":"I know if you move your time moves slower than someone who is stationary, by Lorentz's transformation. However, I don't get how this happens. What does it mean when time moves slower? How does it affect my watch for example when I move close to the speed of light? Does an electron for example feel time-dilation?"}
{"id":"15687","title":"decomposing forces","text":"I have a question about something I'm not sure about. I will first explain the story. Image If you calculate the force of a truck going downhill and applying its brakes:   * The mass is $50000\\text{ kg}$   * The deceleration is $7\\ \\mathrm{m \/ s^2}$   * And the angle is 7 degrees. $F_1$ is the componenent of the gravity that is parallel to the to surface of the ground ( the F\/\/ or FII). $$F_1 = 50000 *9.81* \\sin(7^\\circ) = 59777\\text{ N (force downwards)}$$ The resulting force is $50000 \\times 7 = 350000\\text{ N}$ (using the deceleration value) (force downwards). $$F_\\text{brake} = 59777 + 350000 = 409777\\text{ N (force upwards)}$$ If your truck goes up instead of down this time and then applies its brakes, do you need to do this? $$F_\\text{resulting} = F_\\text{brake} + F_1 = 409777+59777 = 469554\\text{ N}$$ Is it right to count $F_1$ twice, the first time in the F included in the brake and the second time separately?"}
{"id":"87468","title":"Why does sweetness of coke change after freezing completely","text":"I freeze my coke in a freezer completely to solid and then keep it out to melt and as it melts portion by portion I starts to drink, initially It will be very sweet and later it wont be sweet at all. why does this happen? We know that all sugar in coke will be dissolved (Solid solution) but how a major portion of this melts faster than water in the juice their by getting more sweet? How these sucrose can escape through the crystals formed by the ice and join in the water. I measured the sweetness (Brix) and found it varies."}
{"id":"31504","title":"Why are the physical sciences described perfectly by mathematics?","text":"Why are the physical sciences described perfectly by mathematics?"}
{"id":"127952","title":"Could two identical stars revolve around each other in a common orbit if we only account for Newtonian physics?","text":"Both a parent star and its planet revolve around the center of mass of the system, the reason we see stellar wobble. But if we take this to be true, which it is, there can be a configuration in which two identical stars revolve around their center of mass, in a common orbit. What I find astonishing in this case is that they will be revolving around something having no mass at all, in a shared orbit, like two runners trying to catch each other but never quite being able to do so. In that case, in a purely Newtonian system, the centripetal force must be provided by the gravitational attraction between the stars. Now if I assume both the stars to have a mass $m$, at a separation of $d$ from each other, revolving in the common orbit diametrically opposite to each other, $$\\frac{mv^2}{r} = \\frac{Gm^2}{d^2}\\; {\\rm where}\\; r = \\frac{d}{2}$$ solving for $v$, we get a velocity where a stable orbit is formed, $v = \\sqrt{\\frac{Gm}{2d}}$ Note: I have failed to find such an expression, I am not certain about the math. I did find a system described in such a way that two stars orbit a common point in separate ellipses though. Is my conventional wisdom correct, or is my derivation of this expression somehow intrinsically flawed? Does this expression already exist? And I suppose the odds of observing such a system are 'astronomical', but has something like it ever been observed?"}
{"id":"131707","title":"What physical forces allows for electromagnetic induction","text":"In electromagnetic induction, what force is actually doing the work? what physical force actually drives the electrons around the circuit? Let's say we have a coil and an increasing magnetic field through the loop. The electrons start to flow by faraday's law. Is there a very real electric field that is induced around the wire which does the work? because magnetic fields can most certainly not do work. Also, why do the charges even start moving in the first place? At each instant in time, there is some instantaneous value for the magnetic field. The magnetic field cannot cause the electrons to start moving because it doesn't affect stationary charges"}
{"id":"73964","title":"Accretion disk physics - Stellar formation","text":"I was going through the Wikipedia page for Accretion disks, and I couldn't comprehend what the meaning of this is: > \"If matter is to fall inwards it must lose not only gravitational energy but > also lose angular momentum.\" What does that mean? Also, why are these accretion disks planar? I learned that the reason is conservation of angular momentum, but what is the physics behind it?"}
{"id":"43750","title":"vacuum level alignment","text":"When two materials are connected like the case in heterojunctions, we always firstly align the so called vacuum level[1], and then decide the relative position of other energy levels like conductance and valence bands. But in the wikipedia page it says \" The vacuum level alignment approach may or may not hold due to details of the interface\", so my question today is, how to judge whether the vacuum levels align or not? [1]http:\/\/en.wikipedia.org\/wiki\/Vacuum_level"}
{"id":"43751","title":"How to relate photon's higher frequency to time dilation?","text":"The usual explanation for photon's higher frequency in lower altitudes (higher gravity), when the photon is going downward towards a massive body, is that gravitational potential energy is converted into photon's energy which, by its turn, through $E=hf$ implies higher frequency. Because, since $c$ is constant, it doesn't make sense to turn that potential energy into kinetic energy. That, I understand. My question is, why is it affirmed that this implies that time runs slower in the lower altitude? I mean why is it such that this higher frequency is equivalent to slower running clocks?"}
{"id":"100084","title":"Identifying an optics equation in a code refactoring project","text":"I apologise if this a somewhat naive post. My knowledge of optics is limited, and I am learning on the job, so to speak. I'm a computational biologist, dabbling in a bit of physics. I'm currently reading Sonke Johnsen's \"The Optics of Life\" to try and get a better picture. I am involved in an academic project to refactor some obsolete code from a program created over 15 years ago, which models the resolution and sensitivity in reflective superposition compound eyes. The problem is that a lot of the code uses non-descriptive variables that are difficult for me to interpret. I am stuck with working out what this is doing:               DA = sqrt((ER * ER) - (AR * AR))    \/* DA??? *\/     AC = atan(AR \/ DA) \/ CONV           \/* AC??? *\/      This is then used to adjust the aperture diameter:               AD = (AC \/ 360.0) * EC              \/* adjust aperture diameter *\/      In this code the variables definitions are as follows: > ER = eye radius > > AR = aperture radius > > CONV = pi \/ 180.0 (converts radians to degrees) > > AD = aperture diameter > > EC = eye circumference What might `DA` and `AC` be? Any help should be greatly appreciated :)"}
{"id":"4058","title":"Variational method applied to brownian motion","text":"It's possible apply the variational method to the brownian motion ? I mean, one of requisites on $y(t)$ is that it must be continuous and $\\partial_t{y(t)}$ too, and in this case, $\\partial_t{y(t)}$ is not."}
{"id":"109092","title":"Charge distribution inside a parallel plate capacitor","text":"I'm having difficulty deriving a known expression for the magnitude of the Electric Field within a parallel plate capacitor with applied potential difference $V$ and an immobile charge distribution within the capacitor with uniform density $\\rho$. The expected result is: > $$\\frac{\\rho(d-2x)}{2\\varepsilon_{0}}-\\frac{V}{d},$$ Where $d$ is the > distance between the plates and $x$ is the distance from the positively > charged plate. My initial attempt involved using Gauss' law in a Gaussian surface $\\Sigma$ projecting out of the plate a distance $x$: $$\\iint_{\\partial \\Sigma}\\vec{E}\\cdot\\mathrm{d}\\vec{A}=\\frac{\\rho}{\\varepsilon_{0}}\\iiint_{\\Sigma}\\:\\mathrm{d}V+\\frac{Q}{\\varepsilon_{0}},$$ Where $Q$ is the magnitude of the charge on each plate. We therefore get: $$EA=\\frac{Ax\\rho}{\\varepsilon_{0}}+\\frac{Q}{\\varepsilon_{0}}\\implies E=\\frac{\\rho x}{\\varepsilon_{0}}+\\frac{Q}{\\varepsilon_{0}A}=\\frac{\\rho x}{\\varepsilon_{0}}+\\frac{V}{d}$$ As $V=\\frac{Q}{\\varepsilon_{0}A}d$. However, this is not what we should get, so I'd appreciate any pointers in the right direction."}
{"id":"101686","title":"What is the fringe separation in Young's double slit experiment?","text":"In the double slit experiment, a pattern of dark and bright bands appears on the screen. Is the fringe separation the distance between two bright bands? Question: > In a double slit experiment using light from a helium-neon laser, a student > obtained the following results: > > width of 10 fringes 10x = 1.5 cm > > separation of of slits a = 1.0 mm > > slit-to-screen distance D = 2.40 m > > Determine the wavelength of light. How would I find the fringe separation here? If I divide 1.5 by 10, I would get the width of 1 fringe. Is that the same thing as the fringe separation? A diagram showing what the fringe separation is would be really helpful. The textbook that I'm currently reading defines the fringe separation as the distance between the centres of adjacent bright fringes. How would I find out the fringe separation here in this question? Thank you in advance."}
{"id":"17445","title":"Number of conditions for a two-particle state to be decomposable","text":"Suppose we have a general two-particle state $ \\Phi (x_1, x_2 ) = \\sum_{n_1,n_2} \\phi_{n_1,n_2}(x_1,x_2)|n_1,n_2> $, where $n_1$ can be any of $n$ possible states, and $n_2$ can be any of $m$ states. If the state is decomposable then the coefficients $\\phi_{1,2}$ can be decomposed into $\\phi_{n_1} * \\phi_{n_2} $. It seems to me that for the decomposable state there are $n + m$ independent coefficients ($n$ coefficients describing the state of partice 1, and $m$ coefficients describing the state of particle 2), and if the system is not decomposable, i.e., if there is entanglement between the two particles, then the number of independent coefficients is $n * m$ ($n$ possibilities for $n_1$ times $ m$ possibilities for $n_2$ in $\\phi_{n_1,n_2}$). If this logic is correct, then the number of conditions to be fulfilled by a decomposable state is $nm - n - m$. However, according to the book I am studying the number of conditions is $nm - n - m +1 $. I wonder why there is an extra condition. I am not considering normalization, because the states are seen as rays in projective space, and furthermore both the decomposable and entangled states would have to fulfill the same normalization requirements so I guess there would be no difference."}
{"id":"46700","title":"When is the right ascension of the mean sun 0?","text":"I understand that the right ascension of the mean sun changes (at least over a specified period) by a constant rate, but where is it zero? I had naively assumed that it would be zero at the most recent vernal equinox, but when I try to calculate the equation of time using this assumption and true sun positions, all my values are about 7.5 minutes larger than they should be. When (at what date and UT time) is the right ascension of the mean sun 0? And why?"}
{"id":"101350","title":"Adding stuff to the path integral (Faddeev-Popov method)","text":"I'm wondering about the Faddeev-Popov method described in Peskin Schroeder and also on page 7 in this link. What gives them the right to simply add the Gaussian $\\omega$ and thus introduce the $\\xi$ parameter? It seems so arbitrary to me. Are there any rigorous derivation of the Faddeev-Popov method?"}
{"id":"46709","title":"Why is information in a volume proportional to it's surface area?","text":"I read the wiki article on the holographic principle, but it never answered this question. Can anyone explain the math that leads to this conclusion?"}
{"id":"101354","title":"How many celestial bodies could be in stable orbit at roughly the same distance from a star?","text":"**How many planet-like celestial bodies** of roughly the same mass (say within 50%) could orbit a star at roughly (say within 10%) the **same distance from the star** and be in stable orbits. By stable I mean they would not require tethers or significant energy to stay there. I feel sure the answer is at least 2, Earth plus another body at Lagrangian Point L3. I have heard it's not 3 -- that putting Earth-size bodies at Earth's L4 and L5 would destabilize. Is that true? Please explain concisely and cite reputably. I'm asking in the interests of Earthling living space within the solar system, in the far future of course. (Or other stellar systems in the uber far future.) Thanks to @dgh for steering the terminology away from \"planet\" which is defined by the IAU as having cleared its neighborhood (i.e. orbital radius) of other bodies. ## Research Possibilities   * Klemerer Rosette (theoretically unlimited, but practically no more stable than a ball on a hill)   * Iridium Satellites (66 are active in stable, Low Earth Orbit, in six planes)"}
{"id":"20157","title":"Why does electron-positron annihilation prefer to emit photons?","text":"If gravitons are massless, and neutrinos nearly so, why aren't pairs of either of them normally expected products of electron-positron annihilations? Are they possible but simply unlikely, or is there actually some conserved quantity prohibiting their creation? Edit: I'm talking about the low energy limit, not in accelerator beam collisions."}
{"id":"44703","title":"Using the monte carlo method to compute the magnetic field of a curent carrying loop","text":"I have written a program in cpp that computes the magnetic field at a point from a current carrying loop. It uses the biot savart law and the monte carlo technique to carry out the integral. The program functions as expected for points that are not too close to the loop but from looking at a plot of the vectors you can see strange errors close to the loop. I checked my results against http:\/\/hyperphysics.phy- astr.gsu.edu\/hbase\/magnetic\/curloo.html#c3 for a point on the axis and http:\/\/www.netdenizen.com\/emagnet\/offaxis\/iloopcalculator.htm for a point off the axis. My program agrees with these sources for points not too close to the loop. This is my code. http:\/\/pastebin.com\/4BH9qEZz My main question is if there is anything inherent in the monte carlo technique that would cause this error?"}
{"id":"118893","title":"Proving that Measurement increases von Neumann entropy","text":"Let $V$ be a finite dimensional complex inner product space. Let $\\mathcal{M}$ be the classical sample space of measurement outcomes that may occur in a given experiment, and $M_\\mu$, $\\mu \\in \\mathcal{M}$ are a family of measurement operators acting on the pure states of $V$ that satisfy $\\sum\\limits_{\\mu}M_\\mu^\\dagger M_\\mu = Id_V$. The state of the system before the measurement is given by the density operator $\\rho$. The probability that the outcome of the experiment is $\\mu$ is $P_\\rho (\\mu) = \\operatorname{Tr} ( M_\\mu^\\dagger M_\\mu \\rho)$. The system is now in the mixed state \\begin{equation} \\rho' = \\sum\\limits_{\\mu} P_\\rho(\\mu) \\frac{M_\\mu \\rho M_\\mu^\\dagger}{P_\\rho(\\mu)} = \\sum\\limits_{\\mu}M_\\mu \\rho M_\\mu^\\dagger \\end{equation} Now I am asked to prove that $$ S(\\rho') \\geq S(\\rho)$$ where $S$ denotes the _von Neumann_ entropy. The hint says, that i should use that $$ - {\\rm Tr} \\left[ \\rho \\log \\tau \\right] \\geq - {\\rm Tr} \\left[ \\rho \\log \\rho \\right] $$ which I have proven already in a previous exercise. Unfortunately I do not have any idea how to start. I'm not even sure that the definitions I took from the lecture notes are correct. All I know is that I have to get to $$ - {\\rm Tr} \\left[ \\sum \\limits_{\\mu}M_\\mu \\rho M_\\mu^\\dagger \\log \\rho' \\right] = - {\\rm Tr} \\left[ \\rho \\log \\rho' \\right]$$ in order to apply the hint. I tried to write it in a basis where either $\\rho$ or $\\rho'$ are diagonal but I found no way of achieving the above, I'm still feeling uncomfortable with density and measurement operators and appreciate any hint. PS: I'm not sure whether this is more of a math.SE question, but given the physical meaning of the calculation I decided to post it here. I'm happy to move it if it is considered unsuitable for physics.SE."}
{"id":"59730","title":"Electromagnetic inertia due to advanced radiation?","text":"The scalar potential $\\phi$ and vector potential $A$ at a distance $r$ from a charge $q$ are given approximately by $$\\phi = \\frac{q}{r}$$ $$\\mathbf{A} = \\frac{q\\mathbf v}{r}$$ where the constants have been suppressed. The corresponding electric and magnetic fields are given by $$\\mathbf{E} = -\\nabla \\phi - \\frac{\\partial \\mathbf A}{\\partial t}$$ $$\\mathbf{B} = \\nabla \\times \\mathbf A$$ Now the gradient and curl terms fall off as $1\/r^2$ so that at large distances from the charge $q$ we only have an electric field $E$ given approximately by $$\\mathbf{E} = - \\frac{\\partial \\mathbf A}{\\partial t}\\ \\ \\ \\ \\ \\ \\ \\ \\ (1)$$ where $$\\mathbf{A} = \\frac{q\\mathbf v}{r}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ $$ Thus at large distances an accelerating charge produces an electric field that is proportional to its acceleration. This is the standard (retarded) radiation field of an accelerated charge. But perhaps this is not the only solution to equation (1). Imagine that the charge is fixed with respect to an inertial frame and that the observer is moving with velocity $-v$. In the rest frame of the observer the charge is moving with velocity $v$. This implies that in the observer's rest frame there is an $A$-field. Now if the observer changes his velocity then in his comoving frame he experiences a changing $A$-field. My proposal is that Equation (1) still applies in this situation so that this \"apparently\" changing $A$-field induces an electric field in the observer's accelerating frame. This field causes any charges that are co-moving with the accelerating observer to feel a kind of electromagnetic inertial force. I realise that this is non-standard as Maxwell's equations are usually only applied in an inertial frame. My hypothesis is that the retarded solutions apply to an inertial frame whereas the previously unused advanced solutions apply to accelerated frames."}
{"id":"134734","title":"How far will the stone travel in 2.0 s?","text":"A stone is thrown vertically down from the top of a 40 story building with an initial speed of 1 m\/s. How far will the stone travel in 2.0 s? the answer is 22 m and my answer is 17.6 m I used this equation to solve the problem $x=v_0t-(1\/2)gt^2$ = (1)(2)-(0.5)(9.8)(4)=-17.6 = 17.6m So, where is the mistake?"}
{"id":"129134","title":"Electron shells in atoms: What causes them to exist as they do?","text":"I have seen similar posts, but I haven't seen what seems to be a clear and direct answer. Why do only a certain number of electrons occupy each shell? Why are the shells arranged in certain distances from the nucleus? Why don't electrons just collapse into the nucleus or fly away? It seems there are lots of equations and theories that describe HOW electrons behave (pauli exclusion principle), predictions about WHERE they may be located (Schrödinger equation, uncertainty principle), etc. But hard to find the WHY and\/or causality behind these descriptive properties. What is it about the nucleus and the electrons that causes them to attract\/repel in the form of these shells at regular intervals and numbers of electrons per shell? Thank you! Please be patient with me, new to this forum and just an amateur fan of physics."}
{"id":"100441","title":"Particle physics conservation law checking tool","text":"I'm just starting out with simple particle physics, and I'm doing a ton of exercises where I have to check if a certain reaction is allowed, from the point of lepton\/baryon\/energy conservation and such. Now in principle I am able to figure most of it out of course, as it's simply checking a few numbers and whatnot, but from time to time I'm asked to 'add' the missing particles to the equation, and I'm not always sure if my answer is sensible. Now, as I'm doing approximately 40 of these exercises posting them here and asking if they are sensible is neither time-efficient nor very useful for anyone other than myself, so I was wondering if there's something out there such as a simple tool that checks if the conservation laws are obeyed. Something where you'd say I begin with an antitau and I end up with a tau neutrino, an antimuon and a mu neutrino, and it'll tell you yes this is possible, or no this or that conservation law is not satisfied. I'm assuming something like this should be relatively simple to make (as it simply has to know a few particles and count some numbers), but I haven't been able to find one as of yet. Does anyone know if one exists?"}
{"id":"126482","title":"BBC radiation: What is it?","text":"I'm reading articles about blazars, and in a discussion about models concerning Blazar SEDs (Spectral Energy Densities) there are mentions of different processes: Sychrotron scattering, inverse Compton scattering, etc. Now concerning models for blazar emission, there are a lot of acronyms. I managed to figure out that SSC stands for \"Self Synchrotron-Compton,\" but I have no idea what the acronyms \"XC\" and \"BBC\" stand for. And it isn't easy to google out \"bbc + anything\" without finding just references to... you know: http:\/\/en.wikipedia.org\/wiki\/BBC Does anyone know or have a good sense of what those acronyms might stand for?"}
{"id":"75305","title":"Motivating the Legendre Transform Mathematically","text":"If I begin with a functional of the form $$J[y] = \\int_a^b f(x,y,y')dx$$ and find its Euler-Lagrange equations $$\\frac{\\partial f}{\\partial y} - \\frac{d}{dx}\\frac{\\partial f}{\\partial y'} = 0 = \\frac{d}{dx}\\frac{\\partial f}{\\partial y'} - \\frac{\\partial f}{\\partial y}.$$ I end up with a second order ODE $$\\frac{d}{dx}\\frac{\\partial f}{\\partial y'} - \\frac{\\partial f}{\\partial y} = \\left( \\frac{\\partial ^2 f}{\\partial y' \\partial y'} \\right) \\frac{d^2 y}{dx^2} \\+ \\left(\\frac{\\partial ^2 f}{\\partial y \\partial y'}\\right) \\frac{dy}{dx} \\+ \\frac{\\partial ^2 f}{\\partial x \\partial y'} \\- \\frac{\\partial f}{\\partial y} = 0$$ Now every higher order ODE can be broken up into a system of first order ODEs in $y$ and the derivative $M = \\frac{dy}{dx}$, giving $$ \\frac{d}{dx}\\frac{\\partial f}{\\partial y'} - \\frac{\\partial f}{\\partial y} = \\left( \\frac{\\partial ^2 f}{\\partial y' \\partial y'} \\right) \\frac{d M}{dx} \\+ \\left(\\frac{\\partial ^2 f}{\\partial y \\partial y'}\\right)M \\+ \\frac{\\partial ^2 f}{\\partial x \\partial y'} \\- \\frac{\\partial f}{\\partial y} = 0. $$ From this perspective, Hamilton's equations $$\\left\\\\{ \\begin{array} & \\frac{dy}{dx} = \\phantom{-}\\frac{\\partial \\mathcal{H}}{\\partial p}\\\\\\ \\frac{d p}{dx} = - \\frac{\\partial \\mathcal{H}}{\\partial y} \\end{array}\\right.$$ are merely a system of first order equations making my above system of first order ODEs look more symmetric, after a suitable change of variables. My question is, looking at $$\\frac{d}{dx}\\frac{\\partial f}{\\partial y'} \\- \\frac{\\partial f}{\\partial y} = \\left( \\frac{\\partial ^2 f}{\\partial y' \\partial y'} \\right) \\frac{d^2 y}{dx^2} \\+ \\left(\\frac{\\partial ^2 f}{\\partial y \\partial y'}\\right) \\frac{dy}{dx} \\+ \\frac{\\partial ^2 f}{\\partial x \\partial y'} \\- \\frac{\\partial f}{\\partial y} = 0$$ it should be possible to see why the Legendre transformation arises, first because its a transformation using derivatives to change variables but also because it should just make some terms go to zero in this second order ode so that everything looks nicer, but how do you see this explicitly? It'd be great if you could use my notation, ie. $J[y]$ etc... as you see I snuck in $\\mathcal{H}$ in above which shouldn't really be there, I'd love to see how that comes about in my notation - thanks!"}
{"id":"129793","title":"A confusion about notation in Goldstein","text":"On treating systems of particles, Goldstein starts with the consideration that whenever there are $k$ particles on a system, the $i$-th one obeys the relation $$\\dfrac{d}{dt}{\\bf p}_i = {\\bf F}_i^{(e)}+\\sum_{j=1}^k {\\bf F}_{ji}$$ Where ${\\bf F}_{i}^{(e)}$ is the external force on the $i$-th particle and ${\\bf F}_{ji}$ is the force that the $j$-th particle applies on the $i$-th one. Now if we compute the work on moving the system along a curve $\\gamma$ we have: $$W=\\sum_{i=1}^k\\int_\\gamma {\\bf F}_i = \\sum_{i=1}^k \\int_{\\gamma}{\\bf F}_i^{(e)}+\\sum_{i,j=1}^k\\int_{\\gamma}{\\bf F}_{ji}$$ Now he says the following (on p. 10) > In the special case that the external forces are derivable in terms of the > gradient of a potential, the first term can be written as $$\\sum_{i=1}^k > \\int_{\\gamma} {\\bf F}_i^{(e)} = -\\sum_{i=1}^k \\int_{\\gamma} {\\bf > \\nabla}_iV_i=-\\sum_{i=1}^k V_i\\bigg|_{a}^b$$ where the $i$ subscript on the > del symbol indicates that the derivatives are with respect to the > coordinates of $\\mathbf{r}_i$. Now I can't understand why this $i$ on the del symbol. Basically we are considering the configuration manifold to be $\\mathbb{R}^3$ so the natural coordinate functions are $x,y,z: \\mathbb{R}^3\\to \\mathbb{R}$ which are just projections onto the axis and with those coordinates, there's just _one_ del operator, namely: $${\\bf \\nabla} = \\dfrac{\\partial}{\\partial x} e_1 + \\dfrac{\\partial}{\\partial y}e_2+\\dfrac{\\partial}{\\partial z}e_3$$ So what really means \"differentiate with respect to the coordinates of $\\mathbf{r}_i$? And why do we need it? Does it have any relationship with the idea of taking the configuration space for a system of $k$ particles as $\\mathbb{R}^{3k}$?"}
{"id":"64964","title":"Force Between Two Conducting Spheres","text":"**Aakash PHYSICS JEE (Main & Advanced) Study Package - 5 & 6 (Class XII)** **Chapter -** Electric Charges and Field               Assignment (page 12)      _SECTION - A; Q.no - 1_ > The force of repulsion between two point charges is F, when they are d > distance apart. If the point charges are replaced by conducting spheres each > of radius r and the charge remains same. The separation between the center > of sphere is d, then force of repulsion between them is > > (1) Equal to F > > (2) Less than F > > (3) Greater than F > > (4) Cannot be said               Answer - (2) Less than F      What I expected the answer to be was (1). The electric field due to a conducting sphere of charge Q is equivalent to the same due to a point charge at the center of the sphere as total charge on it appears as concentrated at the center for the points outside the charged sphere. So, the force should have remained the same. Any suggestions?"}
{"id":"29990","title":"Hairy ball theorem: references to applications","text":"I'm looking for references to applications of the Hairy ball theorem. This is a result of mathematics (topology), but I am interested in applications. I already visited wikipedia and cited references, but I need a little more explanation in magnetic fields, meteorology and applications in computer graphics. If any other application would be excellent. I've also read some questions on this site and some articles uploaded (an article of John Milnor for example, in math.stackexchange), but none deals specifically with applications. Maybe it's a bit difficult because I seek not know much about magnetic fields or computer graphics, but I'm just interested I would like to know if this result is really effective to model these situations, it is mentioned briefly in some texts, but not how they actually work. Thanks for any help."}
{"id":"94882","title":"flavor splitting of quark masses","text":"It seems pretty clear that flavor symmetry splits the quark masses. It seems extremely odd that a global gauge symmetry of the Standard Model fields could or would produce such extreme splitting as is calculated in the typical models and confirmed in experiment. The range seems more suited to the magnitude of the color force itself, and yet the color force is flavor agnostic. Does this bother anyone else? Perhaps it's neatly explained somewhere that I've missed?"}
{"id":"129862","title":"Bose Einstein condensation and macroscopic occupation","text":"If have been thought, that Bose Einstein condensation occurs of the ground- state is occupied macroscopically, so $n_0\\in \\mathcal{O}(N)$ when performing the thermodynamic limit. So naively, this seems odd as of course, if I put in more particles without increasing system size this will always be fulfilled (when keeping the temperature constant). The problem with my imagination is now, that of course when performing the thermodynamic limit, I have to change the systems volume while I put in new particles and thus \"create\" more low energy states. How do I see, that above the critical temperature these effects cancel, so that due to the additional states $n_0$ does not scale linearly with $N$?"}
{"id":"90448","title":"Do you need energy to exert a force?","text":"Is force dependent or independent of energy?"}
{"id":"95826","title":"Electric current in quantum mechanics?","text":"Quick question: I want to find an expression for the (electric) current density of an electron, in quantum mechanics. Either a single electron or a general charge distribution $\\rho$. Classically **j** =$\\rho $ **v**. What should I use here? Maybe the electric charge multiplied by the probability current? Thanks."}
{"id":"89532","title":"Feynman propagators for scalar fields","text":"If there are few massless scalar field, are the propagators of those different massless scalar fields indistinguishable?"}
{"id":"51353","title":"Crystal momentum and the vector potential","text":"I noticed that the Aharonov–Bohm effect describes a phase factor given by $e^{\\frac{i}{\\hbar}\\int_{\\partial\\gamma}q A_\\mu dx^\\mu}$. I also recognize that electrons in a periodic potential gain a phase factor given by $e^{\\frac{i}{\\hbar}k_ix^i}=e^{\\frac{i}{\\hbar}\\int k_idx^i}$. I recall that $k_i$ plays a role analogous to momentum in solid state physics. I also recall that the canonical momentum operator is $P_\\mu=-i\\hbar\\partial_\\mu-qA_\\mu$. Notice that when you operate with the momentum operator on a Bloch electron, $\\psi(x)=u(x)e^{\\frac{i}{\\hbar}k_ix^i}$, you get $e^{\\frac{i}{\\hbar}k_ix^i}(-i\\hbar\\partial_i+k_i)u(x)$. My question is whether a parallel can be drawn between the crystal momentum, $k$, and the vector potential $A$. It seems they play a similar role quantum mechanically, but I have never seen Bloch's theorem described in terms of vector potentials. I suppose one does not even need a nontrivial vector potential for Bloch's theorem to hold. Still, crystal momentum and the vector potential play very similar roles in phase factors and I wonder whether there is any deeper meaning to that."}
{"id":"82315","title":"Deriving an equation involving Killing vectors","text":"I'm currently studying Carroll's GR book _Spacetime & Geometry_, and ran into some trouble understanding the text. When discussing Killing vectors, Carroll mentions that one can derive $$K^{\\lambda}\\nabla_{\\lambda}R=0$$ That is, the directional derivative of the Ricci scalar along a Killing vector field vanishes (here, $K^{\\lambda}$ is a Killing vector). He remarks that the only necessary ingredients to derive this equation are: $$\\cdot\\;\\;\\text{The Killing equation:}\\ \\nabla_{(\\mu}K_{\\nu )}=0$$ $$\\cdot\\;\\;\\text{The Bianchi identity:}\\ \\nabla_{[\\mu}R_{\\nu\\rho ]\\sigma\\lambda}=0$$ $$\\cdot\\;\\;\\ \\nabla_{\\mu}\\nabla_{\\nu}K^{\\rho}=R^{\\rho}_{\\lambda\\mu\\nu}K^{\\lambda}\\longrightarrow \\nabla_{\\mu}\\nabla_{\\nu}K^{\\mu}=R^{\\mu}_{\\lambda\\mu\\nu}K^{\\lambda}=R_{\\lambda\\nu}K^{\\lambda} $$ With $R_{\\mu\\nu\\rho\\sigma}$ the Riemann curvature tensor and $R_{\\mu\\nu}$ the Ricci tensor. Although I was able to derive all of the equations involved, I don't see how to put them together to get the sought-after result. The only way I saw to get started is the following: $$K^{\\lambda}\\nabla_{\\lambda}R=\\nabla_{\\lambda}(K^{\\lambda}R)-R\\nabla_{\\lambda}K^{\\lambda}=g^{\\mu\\sigma}(\\nabla_{\\lambda}(R_{\\sigma\\mu}K^{\\lambda})-R_{\\sigma\\mu}\\nabla_{\\lambda}K^{\\lambda}) $$ which immediately runs me into trouble because I don't see how to simplify\/manipulate either of the terms to even make use of any of the three 'ingredients'"}
{"id":"16180","title":"Is there a contradiction of the theory of relativity here? -- Length contraction and EMR amplitude","text":"Suppose there is a laser beam powerful enough to burn through iron aimed at a piece of iron. You observe this event while you are in the same frame as the piece of iron and the laser-beam generator. In this frame, there is a certain part of space that you know that the light is traveling through. Now say that you get in a rocket that travels a few meters away from the laser beam in a direction perpendicular to eventually at a constant speed arbitrarily close to the speed of light. As you do so, the area that you knew the light was traveling through contracts in the direction that you are traveling in. Say that the amplitude that you think that the light has decreases to the point where the light would no longer be carrying enough energy to burn through the piece of iron. If you see the iron stop being burned by the laser, our universe is seriously weird. I don't think this would happen. Would you see the laser continue to burn through the iron even though it does not seem to you to have the energy necessary to do so? Would this mean that, instead of having the length contract, that there is such a thing as absolute distance? (If we suppose that the amplitude you perceive the laser to have remains constant regardless of what inertial frame you are in, then the laser would appear to have a constant amplitude even as objects around it continued to contract in the direction you are traveling through. This would mean that the laser would have to seem to affect more and more of space as you traveled faster and faster, so it seems it would have to seem to burn through more and more objects as you went faster, which doesn't seem right to me.) Does this contradict the theory of relativity? Is there an error in here somewhere? Thanks."}
{"id":"16188","title":"Optical element is heated by laser: is it possible to get oscillating heat distribution?","text":"Imagine that we have thin optical element, which is irradiated by laser. Laser heats element, so there is some heat distribution in element. There is a heat sink through upper and lower element faces proportional to temperature. There is also a heat conduction in element. The most important characteristic of the model is that heat consumption depends not only on power of laser, but on heat of the element too and relation between current heat and heat change may be very complex. With given parameters after some time heat distribution in element usually sets to some stable state (with equal heat income and outcome in all element points). Stable heat distribution can vary depending on initial heat distribution. So, the question is: if laser radiation power distribution depends on element coordinates and heat consumption is related to temperature in current part of element, but all the parameters do not change in time, is it possible that heat distribution will not come to some stable state but instead will oscillate, or behave in more complex way?"}
{"id":"105138","title":"Materials about S-matrix and S-matrix theory","text":"What is the best book or paper to learn about analytical structures of S-matrix and S-matrix theory? I already know books as _The Analytic S-matrix_ by RJ Eden, PV Landshoff, DI Olive, JC P and _Quantum Field Theory_ by S. Weinberg."}
{"id":"10923","title":"Can the electroweak\/strong forces, and\/or quantum mechanics be thought of as geometric?","text":"Can the electroweak and strong forces be written as geometric theories? - Why and why not? Can quantum mechanics in general? For example, the Kaluza-Klein theory explains the electromagnetic field as \"twists\" that include an extra dimension of space. (As written by Lubos Motl in a previous question)."}
{"id":"28145","title":"Thermodynamics, bullet fired into ice","text":"I'm beginning to study basic thermodynamics and I want to solve this exercise: It fires a lead bullet from $3g$ to $30^oC$ with a speed of $240\\frac{m}{s}$ to a block of ice at $0^0C$. How much ice melts? (the bullet remains embedded in the ice). I think that the energy becomes kinetic friction: $$Q=K=\\frac{1}{2}mv^2=86,4J$$ Then, I can find the mass of ice from the ecuation of latent heat: $$Q=mL_{f}$$ $$\\implies m= 0,26 g$$ Is this correct? Do not influence the temperature of the bullet? Thanks for your help."}
{"id":"56363","title":"$\\nabla({\\bf u}^2)=2({\\bf u}\\cdot \\nabla){\\bf u} - 2(\\nabla \\times {\\bf u}) \\times {\\bf u}$","text":"Please see the next link: http:\/\/www3.kis.uni- freiburg.de\/~peter\/teach\/hydro\/hydro02.pdf In (2.13), he used: $$\\nabla({\\bf u}^2)=2({\\bf u}\\cdot \\nabla){\\bf u} - 2(\\nabla \\times {\\bf u}) \\times {\\bf u}.$$ Is this is a formula? can someone let me some details\/link about this formula? Any help appreciated!"}
{"id":"93870","title":"What is the \"inner\" force behind attracion\/repulsion?","text":"First of all, I'm sorry for any grammatical big error - English is not my native language. I have a question that maybe does not have an answer beyond those which we already have, but maybe there is someone who can help me understand deeper than high-school level: what is exactly an electrical charge? I already know something is charged (negative or positive) when it attracts or repulses another object or particle. But, stoping at that point, the electrical force coming from a charged body seems like magic. I know it's analog to gravity: given at least 2 bodies with really big masses, they will one attract each other. So to friction\/rubbing (similar forces to opposite sides)... But, why do all these happen? Sintetizing: what exactly produces electrical attraction or repulsion? How can a particle move itself in one or another direction depending its \"charge\"? Sorry for confusion... Thanks"}
{"id":"65438","title":"Doubt on the analysis of $RC$ circuit","text":"I'm studying $RC$ circuits and I'm in doubt on how to deal with one kind of situation. Well, first when we have simply a circuit with just one resistor and one capacitor in series, or a circuit that is equivalent to one of those simplest ones I know how to deal with it: we just use Kircchoff's loop rule and then it leads to a first order linear differential equation which can be easily solved: from here I obtain many results, like the final charge when charging is $Q\\varepsilon$ and so on. However, there are some cases in which it's not possible to reduce to this simple case, and so I'm unsure on how to proceed. The example I've trying to tackle for instance is the same I've asked here: ![enter image description here](http:\/\/i.stack.imgur.com\/cwNW7.jpg) Now, as I've been answered on that question I made it's not possible to reduce what's on the right to an equivalent resistor. I cannot work with impedance, just resistance and capacitance really. I've tried using Kircchoff's loop rule again, however the currents are different and I don't know how to tackle this. Can someone give a little help on how do we tackle this kind of problem? Thanks very much in advance."}
{"id":"133030","title":"How far is the range of an ultrasonic sensor as far as it can transmit?","text":"How far can ultrasonic waves propagate in the air? (if the energy is enough), how far can ultrasonic waves propagate in the ground (concrete floor)? Are there any articles or experiments about attenuation of ultrasonic waves in different mediums?"}
{"id":"61958","title":"What is the difference between Radiation and Electromagnetic Radiation","text":"Are the two equivalent or is Electromagnetic Radiation a subset of Radiation. I am further confused by the fact that electromagnetic radiation includes both ionizing and non ionizing types of radiation. The closest answer I can think of is that Electromagnetic radiation is the set of all forms of radiation except alhpa and beta particles, because they are **not** massless photons. The only site which I found which supports this is this one. Could someone please give a definitive distinction of the two. I would also appreciate an exhaustive list of everything that is, and is not Electromagnetic Radiation."}
{"id":"63557","title":"Why does the comb attract the pieces of papers if they're neutral?","text":"When we rub our hairs with a comb, and then try to attract small pieces of paper, they're attracted by the comb. The pieces of the paper were not electrified before they were attracted. Then they might be neutral. Why does the comb attract the pieces of the paper if they're neutral, and have no influence of a specific charge on them (Pieces of paper)."}
{"id":"56944","title":"Inertial Frames of Reference - Description of an Inertial Frame of Reference","text":"An inertial frame of reference is described as being a frame of reference in which the first law of Newton (the law of inertia) holds. This means that all events as described with respect to this frame of reference must have a zero net force acting on it and therefore traces a straight line with a uniform non-translatory motion. But, I have read in some books, especially \"Introduction to Special Relativity\" by the well-known Robert Resnick, wherein his definition of an inertial frame of reference also refers to such a frame of reference as being an unaccelerated system. This is where I am confused.      How can we describe a frame of reference as being unaccelerated if we occupy the frame of reference itself? No mechanical experiment conducted solely confined to a single frame of reference can determine the absolute motion of the frame of reference relative to another frame of reference. All that can be understood is that there is a certain uniform relative motion between frames of reference and no more. Is Robert Resnick saying that the inertial frame of reference is unaccelerated with reference to another frame of reference?"}
{"id":"90991","title":"PhD or master in physics","text":"I am 35 years old and I've just got a bachelor in Science. I will pursue higher education in Physics but I have my doubts if either a PhD or a master is more convenient for me. My areas of interest are Plasma-Fusion Physics, Condensed Physics, and Applied Physics. If I enter a PhD program, will take me around 6 years to complete it and probably I would be kind of old to take advantage of it...in the other hand If I choose the master (which is shorter in time) probably won't help 'cause Universities and Companies in these areas mostly look for doctors. These are my doubts and concerns. Please help with your expertise."}
{"id":"90999","title":"Are communications between computers faster by electrical signals via copper cables or electromagnetic signals?","text":"Assume that there are two computers which are connected with a copper cable, e.g. Ethernet. Also, there is a radio connection between them, e.g. AM radio, in order to exchange data. When we try to send data from one to the other, which one would be faster? Assume they are the same distance away. As far as I know, the propagation speed of electromagnetic signals in the air is better than the copper. On the other hand, frequency of the AM signal is smaller. Is there any other factors which affects the speed?"}
{"id":"108620","title":"Protons (as opposed to neutrons) to mediate nuclear fission?","text":"I am just wondering why are protons (as opposed to neutrons) not used to mediate nuclear fission? Is it because it is charged, so we will have to input more unnecessary energy to overcome the Coulomb force?"}
{"id":"71911","title":"What is heat and how does it effect an atom?","text":"What happens in the atom when it gets heated or cooled and turns into gas, liquid or solid? I just want to know how does heat affect an atom."}
{"id":"113735","title":"Are there any causeless phenomena from the mainstream physical viewpoint?","text":"**EDIT:** The orginal version did not produce any answers about physics. I know what life is, I have studied that for decades. I wanted to hear how the border between matter and spirit looks from the physical point of view, and I would prefer references to published works and widely accepted terminology rather than some hand-waving. If we conclude by symptoms that what everyone else calls \"life\" is called \"chaos\" in the physics parlance, then the answer is simply \"no\", physics does not deny life, it calls it **chaos**. Probably, what I called freedom is called \"lack of causality\" in physics parlance and it corresponds well with my definition of life in the sense that there cannot be something half-spirit-half-matter. Either there is a **causal relationship** between two phenomena, or there isn't, no third option. (I will leave for future the related question whether there is an original cause of all causes.) * * * Life means **freedom** and freedom means laws can be broken. If the laws can never be broken (including physical laws), there is no freedom and no life. Everything is completely **determined** and the time dimension is a mere decoration. **EDIT:** I missed a big option here, that physics might not describe the entire world. Besides the physical world of causal relationships there might be other parts of existence also, which are free from causal relationships. The might not break the laws of physics, but also not be subjected to them. There might even be some interaction between the physical and non-physical worlds. In that context my question becomes very precise: **is it possible to describe all phenomena with mathematical-physical-logical laws** or (as I expect it) there will always be some phenomena that in principle cannot be described with such laws (chaos, spirit, life etc.). * * * From Michio Kaku's Physics of the Impossible I gather that the single biggest problem in physics is radioactive decay, there is only **half-life** , but no causality whatsoever for a particular particle to decay at a particular moment. Is it the loophole through which life enters physics? In related question the accepted upvoted answer starts with **\"physics deals with the explanation of observable phenomena.\"** That subjects physics to the ill-defined concepts of _explanation_ and _observation_ but I want a strict definition (probably it is not possible to define in mathematical-physical- logical terms what physics is, that would be a perfectly sensible conclusion and might even be provable). In that post the proponent presumes that all consciousness is **limited** (and probably the entire physical existence is limited). Is it really a necessary conclusion of the mainstream physics? The life-accepting eastern philosophies unequivocally declare both the physical nature and the consciousness to be **infinite**. That brings about the Goedel's incompleteness theorem, Banach-Tarski paradox and other conclusions common to both eastern and western sciences. I am myself mathematician and consider the **Goedel's incompleteness theorem** one of the greatest achievements in mathematics. Am I correct to interpret that in context of physics it means: \"not all phenomena can be described by physics\"?"}
{"id":"54736","title":"about the 1D singularity of black hole","text":"I saw some responses here saying that the singularity into the black hole is one dimension object so my question is : is it possible that the singularity is simply a merger of the 4 dimensions of the spacetime ? Is it possible physically and mathematically that many dimensions can merge into one ?"}
{"id":"54735","title":"Why can we use Gauss' law to compute electric field?","text":"For simplicity I'm considering only the sphere case. In the Gauss' Law formulation we have some field $E$ introduced by charges $Q$ inside some sphere, then we compute flux and integrate, and we get result $Q\/e_0$. Right. But this formulation doesn't take into account any possible outer charges, because $E$ used in this law comes only from $Q$. **I suppose that any outer field effects will cancel out, because flux coming from it must go through surfaces with opposite normals. But we didn't make any proof of that, as far as I have seen in Gauss' Law proofs.** So, when we measure electric field inside the sphere with uniformly distributed $Q$, we make some assumptions on uniformity of $E$ inside this sphere and we consider hollow Gauss' sphere with radius smaller than radius of the outer sphere. And then we use $E$ to compute total flux going through inner sphere and conclude that it must be zero. Right. But why can we use this $E$? We don't know that there aren't any effects from outer charges fields. So why can we use Gauss' Law?"}
{"id":"78531","title":"What is known on violations of unitarity or locality?","text":"Recently the amplituhedron become a hot topic. I realized that two of the central pillars that QFT is based on, unitarity and locality, are no longer playing an important part (due to gravitational interactions). I would like to get more details on why we should give up these seemingly innocent assumptions. In particular: **are black holes the only source of concern here?**"}
{"id":"37911","title":"Are probabilities really tangible physical real numbers?","text":"Probabilities are usually considered to be a real number between 0 and 1. A real number has an infinite decimal expansion. Are probabilities really real numbers? Is the infinite decimal expansion really physical? Ordinarily, when we deal with probabilities in practice, we only deal with the first few significant digits. Are the later digits physical? If one wants to measure the 100th decimal digit of a probability, one has to sample an ensemble of size slightly over $10^{200}$ to measure the right answer with probability near 1. Does the necessity for such a large ensemble mean thinking of probability as a real tangible physical quantity is incorrect? There are two cases to consider here. If the probability is \"actually\" around say $4.83 \\times 10^{-100}$, then one would expect the 100th decimal digit to be a lot more physical compared to the case where the \"actual\" probability is say $.804\\cdots 3\\cdots $, if such a precise probability even makes any sense. But even in the former case, isn't the probability effectively zero for all practical purposes? Effectively, isn't $3\\times 10^{-145}$ indistinguishable from $7\\times 10^{-82}$? How can one distinguish between both cases _in practice_? If probabilities are even more tangible than that, shouldn't we be able to set up a gadget which behaves in one way if the 100th digit is even, but in a different manner if it's odd? That's just not how probabilities behave in practice. This question leads up to the nature of the complex coefficients of the wave function in quantum mechanics. Are they really physical tangible complex numbers? What about their absolute square (a real number), or relative phases? What if we set up a case where we have a nearly exact destructive interference, with the coefficients of two basis terms nearly cancelling up to $10^{-50}$? In a Bayesian sense, it is ridiculous to suppose our knowledge or ignorance of a system can be quantified so exactly. If considered as betting strategies, a typical \"rational\" agent might as well flip a coin to decide between bets when their expectation values match up to the 100th digit. In a frequentist sense, an ensemble of size at least $10^{200}$ is needed. Only propensitist interpretations can make the real value of a probability physical. _Theoretically_ , for _idealized_ systems, one can have actual real numbers with an infinite decimal expansion for probabilities, but do such concepts apply to the real world? If not, then what are probabilities really?"}
{"id":"37912","title":"What is the easiest way to stop a star?","text":"On long enough cosmological time scales, hydrogen and helium nucleii will become scarce in the Universe. It seems to me that any advanced civilisations that might exist in that epoch would have the motivation to try and prevent the stars from using them up, in order to burn the fuel more slowly and extract a greater proportion of the energy as usable work. One might say that, on time scales measured in trillions of years, the stars are an unsustainable use of the universe's fuel. This question is about whether such civilisations would have the means to do something about it. My questions are:   1. What would be the most energy-efficient way (using known physics) to blow apart a star or otherwise prevent or greatly slow the rate at which it performs fusion? We're assuming this civilisation has access to vast amounts of energy but doesn't want to waste it unnecessarily, since the aim is to access energy from the hydrogen the star would have burned. In order for this to be worthwhile, the energy gained from doing this would have to be substantially more than the energy the process takes.   2. What would be the astronomical signature of such an activity? If it was happening in a distant galaxy, would we be able to detect it from Earth? **Update** I'm still interested in this question. There are some great answers below, but currently I don't feel that any of them approach the most efficient way to solve the problem, _i.e._ a way likely to be attempted by a civilisation with the resources and motivation to do so. Luboš' answer makes it clear that the energy requirements are not too stringent in themselves (you just need to skim off a small fraction of a nearby gas giant and fuse the hydrogen into heavier elements; this question already presupposes the ability to extract power from fusion on a literally astronomical scale) but points out that no \"bomb\" made of atoms can enter a star and blow it up. However, this does not rule out other methods, such as increasing the star's angular momentum through some means (as in AlanSE's answer) or somehow (perhaps electromagnetically) removing plasma from its surface rather than blowing it up from the inside. If removing plasma from the surface is possible then ideas based on this have a certain appeal, because the resulting hydrogen could be fused, releasing energy that could be used to skim off more, and so on exponentially. (Kind of like the opposite of that bit in the film _2010_ when a self-replicating monolith turns Jupiter into a star.) Some of the resulting energy could be put into increasing the star's angular momentum as well. However, I don't know enough about plasma physics to know whether there's a way to suck the plasma up from the star's surface. (It can be done gravitationally, of course, but then the hydrogen just ends up being depleted by runaway fusion on the surface of another star, so this isn't very useful.) Given this, my current specific questions are:   * Is there a plausible way to remove material from the surface of a star at an appreciable rate, either electromagnetically or through some other means? If so, is there anything other than the (virtually unlimited) availability of energy that limits the rate at which this could be done?   * Given a sufficiently huge supply of energy, is there an effective way to rapidly increase a star's angular momentum? It seems (per AlanSE's answer) that redirecting the star's radiation would take too long, but perhaps firing massive objects or particle beams at it would accomplish the same effect in a shorter time.   * Aside from these ideas, would adding heavier elements than hydrogen to the star suppress fusion by absorbing neutrons? What quantities would be required in order to affect the star's evolution by this method? (A civilisation that eats stars will generate a huge supply of stable nucleii as a waste product, so maybe it makes sense to simply dump some of them into the next star.)"}
{"id":"27408","title":"Where do theta functions and canonical Green functions appear in physics","text":"In the beginning of Section 5 in his article, Wentworth mentions a result of Bost and proves it using the spin-1 bosonization formula. This result provides a link between theta functions, canonical Green functions, and Faltings' delta invariant on a compact Riemann surface of positive genus. Are there any other situations in physics which involve canonical Green functions on compact Riemann surfaces?"}
{"id":"104778","title":"How can a wave interfere with itself?","text":"We know that waves can interfere with themselves to form interference patterns for example photons and electrons. How is this possible? I know it does happen but is there a proper representation or a proof of this? It would be good if someone could represent it with a diagram. I just need to understand the reason for the concept and not the concept. **EDIT** I strongly mean a single photon interfering with itself. By a single wave I meant a single photon. I just do not understand the whole concept of a wave packet and a wave interfering with itself."}
{"id":"106954","title":"Tensile strength of a braided steel wire vs. a solid steel bar?","text":"I am wondering which one is the strongest between a steel rope and a steel bar with same diameter and made of the same steel? **EDIT :** The test could be: attach rope\/bar to ceiling and suspend a mass until it break **EDIT2 :** Can I have also some order of magnitude? For example, is the rope 2 times stronger than the bar?"}
{"id":"134103","title":"How is dark matter meant to explain the faster than expected rotation of galaxies?","text":"The stars on the outer edges of galaxies go around faster than they should be. How is dark matter meant to account for this? if you just add more mass wont that just give it a greater gravitational force? So we would still expect the outer stars of galaxies to be going slower than the ones near the center. Whats the flaw in my reasoning?"}
{"id":"68957","title":"Can we disprove Artificial Gravity (like in the movies) with a thought experiment?","text":"Suppose you have a Device that creates an 'artificial gravity' (AG) inside a Box, with these properties:   * the system does not use inertial forces (like centrifugal force) or a huge mass to create AG   * the AG 'feels' just like gravity\/inertia over most of the volume of the box   * the Device, the Box and its content are still subject to any gravitational forces that may be present, but the AG force adds to other gravity\/inertia forces, net force may be zero   * Matter, Energy and Information can leave and enter the box   * there is a smooth but strong gradient in the AG at the walls of the box -    * conservation of energy still holds: if the AG does work, this energy has to be supplied to the device in some way, vice versa for work done against the AG   * conservation of momentum still holds: for every force due to the AG on an object, there is a corresponding counteracting force on the device There's two ways this could be ruled impossible:   * Something in our current understanding of gravity rules out such a system - though our current understanding    * AG like this could be used as a perpetuum mobile, a time machine that can create paradoxes or another device that can be shown to be impossible.  Even if AG can be easily ruled out by the first way (our understanding of Gravity), I'd be interested in the kinds of shenanigans one could pull of with such a system! Edit: I'll rephrase what I want to achieve with this question. I'm trying to sketch an AG system that 'looks' like the plot device known from many SF- Movies, and I try to define it in a way that does not obviously violate basic laws of conservation. Now, I ask you, esteeemd SE-Physicists to take this device and find ways to break the laws of physics (or maybe even logic) with it. In other words, I'm looking for a thought experiment to test the idea of AG to destruction."}
{"id":"67380","title":"How to calculate incident solar energy on earth in a given day with a spectral filter","text":"The daily solar radiation irradiation is often quoted as being between $3.2 \\text{ kW hours}\/\\text{m}^2$ and $7 \\text{kW hours}\/\\text{m}^2$ per day . If you filter out a range of wavelengths, how can you calculate the energy from the remaining wavelengths? To make it concrete, say we filter everything below 450nm or alternatively above 700nm."}
{"id":"442","title":"How much force would be necessary to hold a 6-foot steel staff weighing 20 lbs from one end of it, parallel to the floor?","text":"This isn't a homework problem. I'm asking cause I'm wondering how strong a person would have to be to pull it off. Their arms would be extended, both hands gripping the staff at one end, and the staff is facing directly away from them, parallel to the floor, going through their 'z-axis' as it were (with x being left-right and y being top-bottom)."}
{"id":"115197","title":"Polar coordinates explanation needed on calculation","text":"This is the question. ![enter image description here](http:\/\/i.stack.imgur.com\/feXTM.png) ![enter image description here](http:\/\/i.stack.imgur.com\/PyOt3.png) Here is the answer. But honestly I cant figure it out. Maybe my lecturer's handwriting is quite illegible too (just kidding). Sorry if I ask too simple question but seriously I want a clarification on this. Thanks :) ![enter image description here](http:\/\/i.stack.imgur.com\/H907B.jpg)"}
{"id":"26755","title":"Does Celestia take the lunar orbit precession into account?","text":"Does the lunar orbit plane rotate (8 year period precession)? Does Celestia take this into account while drawing the moon orbit?"}
{"id":"37","title":"How to check Einstein-like equations on their correspondence with the real world?","text":"Physicists studying the grounds of physics and some mathematicians often come to a theories which are like a general-relativity, but do not coincide with it. Often these theories contradict the real world. What should one do in order to check one of such theories? Which papers\/books should one read in order to be able to understand about some mathematical theories, that \"the mathematical theory is nice, but this is not our world\"?"}
{"id":"35","title":"Where do magnets get the energy to repel?","text":"If I separate two magnets whose opposite poles are facing, I am adding energy. If I let go of the magnets, then presumably the energy that I added is used to move the magnets together again. However, if I start with two separated magnets (with like poles facing), as I move them together, they repel each other. They must be using energy to counteract the force that I'm applying. Where does this energy come from?"}
{"id":"70054","title":"Why do electrons emit radiation?","text":"I know how you can emit light with an alternating current, running back and forth, creating an electric field in addition to the magnetic field. But why does an electron emit light when it gets accelerated, there is no positive end to create an electric field. Further more, as opposed to talking about currents, talking about acceleration seems to be a different beast entirely."}
{"id":"87569","title":"Critical radius in heterotic string theory","text":"If I have understood the basics of heterotic strings correctly, one requires a critical radius of compactified dimension to reconcile the apparent difference in spacetime dimension of left and right movers, $$ R=\\sqrt{\\alpha} $$ Why would the radius of the compactified 16 dimensional torus be related to the string tension\/length etc in such a way? They seem completely unrelated to me."}
{"id":"50122","title":"If there was no Earth, what would be the acceleration of gravity from here from the Sun?","text":"Suppose there were no Earth to pull us back and we're hanging in the solar system. Assuming no other force of gravity from other bodies, what would be the $g$ from the Sun?"}
{"id":"107213","title":"$L_+$, $L_-$ Old terminology and Quantum mechanics","text":"Two Quantum mechanics Questions one simple one, and one stupid one: For angular momentum: Considering the appropriate                   i) Mathematical motivation      How do we derive $L_+$ and $L_-$   I think I am looking for an explanation that would mention something about roots and weights. May be mention something about restricted simultaneous diagonalization.                  ii) Physical motivation      Defining some nice properties with associated upper and lower bounds. Just to mention I just want to know where $L_+ $ and $L_-$ come from. I know its almost trivial but I'm looking to see something. . . I know its not defined, its derived in two ways. For some general stuff about the importance of preserving the old language. In Weinberg's text on QM(It's ok to claim I don't understand it), the questions in chapter 2 seem to employ a lot of old, and largely still used terminology. Question 2p and 1s type language, and Question 6 has something along the lines of \"Lyman $\\alpha$\" transitions, which is fairly understood language among experts. I have heard these things before when taking QM, but I just feel so put off by it that I have to keep reminding myself what these things mean when I see them. I know \"par excellence\" Weinberg is correct to promote such language, but his reputation aside, is this the best possible language* to describe these things? I recall feeling very stupid until I reminded myself of what this meant. I want to know if this sort of language* is used by modern researchers working in AMO, or if perhaps the language has evolved. Should I just stop complaining and memorize these balmer, Lyman . . . . and other historically relevant words, that are still everywhere including Record exams and other places I don't want to find myself in. The reason this bothers me is I don't want memorize values of l m n ; s p d f and other such things for specific cases and make associations along these lines. Is there some modern cure to this problem I know its a really stupid question. In the event the question is too childish just leave some references"}
{"id":"100244","title":"If Black Hole never forms, how important will be to study Black Hole paradoxes?","text":"I recently came across a paper Black Hole - Never Forms, or Never Evaporates. It is claimed that under general evaporation conditions, before particles come into the Black Hole, the Black Hole itself will completely evaporate. In other words, Black Hole cannot form from gravitational collapse. If this paper was correct, (I haven't went through all the details, but did not find a technical error yet. And I would like to know your opinions about this paper, though i am not sure if it will be an opinion-based question), do we still need to study many paradoxes of Black Holes?"}
{"id":"66036","title":"Can random constructive interference with sound waves cause damage?","text":"Any two sound waves have a random chance to constructively interfere with each other at a given spot and this will cause an increase or decrease in pressure. So is it possible that a large number of local noises in a loud environment could interfere perfectly at a single point, such as on a person or building, and cause damage? I'm not worried, just curious if this is a daily occurrence or if the probability of it is extremely low or if it can even happen."}
{"id":"129738","title":"Which is the most accurate way to model the fundamental atmospheric parameters of the Earth?","text":"I need to calculate the fundamental atmospheric parameters, I mean the actual ones, this is not an exercise but a model for a real world application so I need to take account of the humidity of air. I need to calculate the density of the air, and I've solved this by implementing this calculator in MATLAB. But now I need to find a way to calculate the input parameters which are:   * Air Temperature   * Altimeter Settings (Which is the barometric pressure right?)   * Relative Humidity given a certain altitude in meters above the sea level. If it's not possible to calculate all the three parameters, given an altitude, please show me a way to calculate the possible ones. For example I know that for the barometric pressure the best equation is the Laplace equation which is an improved version of the barometric formula right?"}
{"id":"22124","title":"How the nucleon structure has been identified experimentally?","text":"It is known that nucleons (proton, neutron) are composed of partons (quarks, etc.). How was this identified experimentally? In particular, how it has been identified that nucleons comprise of more than one constituent?"}
{"id":"35404","title":"Would it be possible to develop special relativity without knowing about light?","text":"My understanding of special relativity is that it is fundamentally based on the constancy of the speed of electromagnetic radiation - that this speed is a physical law (or derivable from physical laws and facts about the vacuum) and not a contingent fact. Does that mean that, if humanity did not know about light (or electromagnetic radiation, generally), would we then be unable to come up with special relativity? To put it another way, imagine a species of intelligent life insensitive to electromagnetic radiation - no eyes etc. Would they still have a reason to produce the theory of relativity, or would they likely have created a physics framework which doesn't involve the speed of light so centrally?"}
{"id":"35400","title":"The origin of femto, atto and zepto SI prefixes","text":"Do you know why the SI prefixes: femto, atto, zepto have been accepted by Scientific Community, if this triad of metric units, is neither greek nor latin?"}
{"id":"73235","title":"Imaginary time?","text":"If we begin with the equation for time dilation: $$ \\Delta t^\\prime = \\frac{\\Delta t}{\\sqrt{1-\\frac{v^2}{c^2}}} $$ Now, for a particle moving in space with no potential energy, $$ E = \\frac{1}{2} mv^2 $$ So $$ v^2 = \\frac{2E}{m} $$ Then $$ \\Delta t^\\prime = \\frac{\\Delta t}{\\sqrt{1-\\frac{2E}{mc^2}}} $$ $$ = \\frac{\\Delta t}{\\sqrt{1-2}} $$ $$ = \\frac{\\Delta t}{\\sqrt{-1}} $$ Explain please?"}
{"id":"73237","title":"Pauli matrices as measurement operators","text":"I am trying to understand a paper on the Bell test experiments. I understand that if we wanted to measure the spin of a spin-1\/2 particle in state $\\psi$ along the z-axis we would apply the operator $$\\sigma_z= \\begin{pmatrix} 1&0\\\\\\0&-1 \\end{pmatrix}$$ and to measure along the x-axis we would use $$ \\sigma_x = \\begin{pmatrix} 0&1 \\\\\\ 1&0 \\end{pmatrix}.$$ But if, given an ensemble of spin-1\/2 particles we wanted to measure spin along either the x-axis or z-axis in random alteration, how would we construct the relevant measurement operator? What about if we wanted to measure, again choosing randomly, spin either along the line 45 degrees between the positive x and z axes, or along the line 45 degrees between the negative x and positive z axes? What is the operator in this case? The paper I am reading assumes the reader is already familiar with such details."}
{"id":"73230","title":"Why is the Higgs boson created so infrequently at the CERN collider?","text":"Although each collision at CERN has the available energy to create a Higgs boson each time, the vast majority of the time it does not create one. I suspect that part of the reason for this is that most of the energy goes into creating mostly lighter, secondary particles which carry a lot of kinetic energy away. As a result, this would leave very little energy left to create more massive Higgs bosons. So I started thinking and realized that there is more that is going on (e.g. cross sections sizes). I have purposely left this question open-ended to just see what people are thinking since I know very little about this. Therefore, as the title of this question puts it: why is the Higgs boson not created more frequently at the CERN collider? Aside: I guess I could of asked this question about some X-particle instead of the Higgs boson. Either way, my question would still be the same."}
{"id":"51653","title":"Looking backwards in time at yourself","text":"If a person on Earth today is looking at a star, say, 10 billion light years away, is it possible that some of the atoms he is looking at will eventually go on to make him?"}
{"id":"101114","title":"Neutrinos and anti-neutrinos in the Standard Model","text":"In standard model neutrinos and the left handed electron forms SU(2) doublet.   1. What about the anti-neutrinos in the standard model? Do they also form some doublet?    2. If neutrinos have tiny masses will it not imply indirectly and conclusively that right-handed neutrinos must exist in nature? **EDIT :** _Neutrinos will have Majorana mass term if they are Majorana fermion. Is that right? Now, if neutrinos are Majorana fermions, will they have definite handedness? For example, does $\\nu_M=\\begin{pmatrix}\\nu_L\\\\\\ i\\sigma^2\\nu_L^*\\end{pmatrix}$ have definite handedness? Therefore, doesn't it imply that if neutrinos are massive then a right-handed component of it $\\begin{pmatrix} 0\\\\\\ i\\sigma^2\\nu_L^*\\end{pmatrix}$ must exist? Although we are not using $\\nu_R$ to construct this column, does it imply $\\nu_M$ do not have a right handed component? It is the column $\\nu_M$ which we should call a neutrino. Then it has both the components. However, one can say that a purely right-handed neutrino need not exist if the neutrino is a Majorana fermion. Therefore, it seems that if neutrinos are massive a right handed component of it must exist (be it a Dirac particle as well as a Majorana particle). Correct me if I am wrong._"}
{"id":"56005","title":"Do the proton PDFs change much with Q?","text":"Specifically, the second moments, do they change much from say 100 to 1000 GeV? Why or why not?"}
{"id":"25475","title":"What effects will the heliosheath have on the Voyager spacecraft?","text":"I wanted to know what effects the heliosheath will have on the Voyager 1 & 2 craft. I know they used the slingshot method to travel through our solar system; so, with being way out in the heliosheath, how fast are the Voyagers going considering:   * the amount of fuel still available,   * and having no objects to slingshot around anymore?"}
{"id":"25477","title":"Telescope for 6 year old and dad new to astronomy","text":"What's a good telescope for my 6 year old? We're both brand new to telescopes in general, but love space, stars, planets, cosmology, etc. Happy to spend up to $200 to start. Maybe more for a good reason... **_UPDATE_** : Thank you guys for all the really great answers. I'm about to purchase the Celestron AstroMaster 114 EQ. Seems like the best value for the money based on the feedback below."}
{"id":"4460","title":"Using energy from sun magnetic field","text":"Knowing               * earth spin and translation movement around sun,     * sun have a big magnetic field,     * a variable flux can generate a induced voltage (and current) into the circuit      Would it be posible to receive in a coil that energy? Perhaps it's just enough to detecting the earth movement relative to magnetic field, or if it were strong enough even for use the energy. It doesn't seem to be so strong because if it where perhaps every coil in earth without a current generated by this flux variation."}
{"id":"128221","title":"The formation of shockwave inside pellets in inertial confinement fusion","text":"In the explanations i have read on the internet about ICF they all explain that the top layer of the pellet or the ablator is heated till it explodes and this produces a reaction force which compress the D+T fuel. They then say shock waves are formed due to the collapsing fuel which compress it even further and eventually ignite the fuel. My question is why and when are the shock waves formed?"}
{"id":"36273","title":"How to find focal length of a single convex lens?","text":"I need help finding the focal length of a single convex lens. The radius of curvature is 200mm. left side is air and the glass has a index of 1.5. I search on google but there was only formulas for think lens, convex and converge lends. I need the formula for a single convex lens. Any idea? I know the ray transfer matrix. $$\\left(\\begin{matrix} 1 & 0 \\\\\\ \\frac{n-n'}{nR} & \\frac{n'}{n} \\end{matrix}\\right)$$"}
{"id":"87205","title":"Electric heating rod","text":"I usually heat my bathing water with electric heating rod, I always thought that the base of rod is an insulator so that it can develop high heat. But when I tried touching water while rod was dipped in it, I got current, damn! Why that happened? I am talking about this type of rod :![enter image description here](http:\/\/i.stack.imgur.com\/nS3jk.png)"}
{"id":"41261","title":"Rays in Symmetric Resonator","text":"I'm having some trouble figuring out how to get started on this question: If I have a symmetric resonator with two concave mirrors of radii $R$ separated by a certain distance, after how many round trips through the resonator could I expect the ray to retrace its path? It seems to me that this implies some sort of periodic function should exist that models the behavior of a ray in such circumstances."}
{"id":"36182","title":"How does Sol's magnetic field continue to exist at such high temperatures?","text":"The temperature at the surface of Sol is apparently well above 5000C; I'm assuming the layers beneath the surface may be even hotter. At school we learnt heating a metal beyond a certain temperature specific to each metal would demagnetize the magnet. How does Sol's magnetic field continue to exist at such high temperatures?"}
{"id":"119647","title":"Derivation of the full generator of the lorentz transformations","text":"Let us study the subgroup of the Poincare group that leaves the point $x=0$ invariant, that is the Lorentz group. The action of an infinitesimal Lorentz transformation on a field $\\Phi(0)$ is $L_{\\mu \\nu}\\Phi(0) = S_{\\mu \\nu}\\Phi(0)$. By use of the commutation relations of the Poincare group, we translate the generator $L_{\\mu \\nu}$ to a nonzero value of $x$: $$e^{ix^{\\rho}P_{\\rho}} L_{\\mu \\nu} e^{-ix^{\\sigma}P_{\\sigma}} = S_{\\mu \\nu} - x_{\\mu}P_{\\nu} + x_{\\nu}P_{\\mu}\\,\\,\\,\\,\\,\\,\\,\\,\\,(1),$$ where the RHS is computed using the Baker-Campbell-Hausdorff formula to the second term. Then we can write the action of the generators $$P_{\\mu} \\Phi(x) = -i\\partial_{\\mu}\\Phi(x) \\,\\,\\,\\,\\text{and}\\,\\,\\,\\,L_{\\mu \\nu}\\Phi(x) = i(x_{\\mu}\\partial_{\\nu} - x_{\\nu}\\partial_{\\mu})\\Phi(x) + S_{\\mu \\nu}\\Phi(x)\\,\\,\\,\\,\\,(2)$$ Could someone explain the contents of Eqn (1) and how (2) is deduced from it? I think there is an unitary transformation on the LHS of Eqn (1), but I don't understand what the equation is doing and why the contents are what they are in the exponents of the exp terms. Many thanks for any clarity."}
{"id":"101802","title":"Did a racer on a \"restless\" course race at some particular average velocity? (1: Starting blocks, finish line moving uniformly wrt. each o.)","text":"Consider two participants, $A$ and $B$ moving uniformly wrt. each other; first having approached each other, then having met and passed each other, and eventually having separated from each other such that their mutual speed wrt. each other is given quantitatively as $\\beta ~ c$ with some particular positive real number value $\\beta < 1$. (Evidently, therefore, $A$ and $B$ had not been at rest to each other.) After having observed $B$'s passage, $A$ has been preparing starting blocks;   and at duration $\\mathop{\\Delta}\\limits_{\\stackrel{\\normalsize{\\text{until}}}{\\small{\\text{start}}}} \\\\! \\tau_A$ after having observed $B$'s passage, $A$ issued a starting signal to a racer ($R$) upon which the racer took off from $A$'s starting blocks. After having observed $A$'s passage (i.e. soon enough after that), $B$ has been preparing a finish line;   and at duration $\\mathop{\\Delta}\\limits_{\\stackrel{\\normalsize{\\text{until}}}{\\small{\\text{finish}}}} \\\\! \\tau_B$ after having observed $A$'s passage, $B$'s finish line ribbon was passed by $R$. The ratio of those durations is given as some particular positive real number value: $r := \\mathop{\\Delta}\\limits_{\\stackrel{\\normalsize{\\text{until}}}{\\small{\\text{finish}}}} \\\\! \\tau_B ~ \/ ~ \\mathop{\\Delta}\\limits_{\\stackrel{\\normalsize{\\text{until}}}{\\small{\\text{start}}}} \\\\! \\tau_A$ which (of course) satisfies $r > \\sqrt{\\frac{1 + \\beta}{1 - \\beta}}$. **Question:** Can an \"average velocity\" of racer $R$ with respect to the race course, as far as it is specified by $A$'s starting blocks and $B$'s finish line be defined,   such that its magnitude could be expressed in terms of the given quantities $\\beta ~ c$, $\\beta$ and $r$   (and where its direction is obviously \"from $A$ to $B$\") ?"}
{"id":"99026","title":"branes-collision -> big-bang -> 2nd Universe?","text":"If the cosmological model of the Ekpyrotic universe is correct should there not always a 2nd universe be created on the 2nd brane? Pictures and articles I found speak only of one universe. ![branes-collision](http:\/\/i.stack.imgur.com\/Pzm82.jpg) Why is that? Did I miss something?"}
{"id":"86557","title":"How does the light waves make up to my eye?","text":"I was thinking about a situation were there is a group of people in a room. While all those people are observing something(none of them are blind) the light ways need to reach their respective eyes. But, how does the light ways make to one's eyes without being disturbed by the other light waves moving towards the eyes of others? Even there are a million people in a room one can see everything clear. How is it possible?"}
{"id":"132974","title":"Axioms behind entropy!","text":"The concept of entropy is very ubiquitous, we learn about its uses starting from Information Theory (Shannon entropy) up to its basic definition in statistical mechanics in terms of number of micro-states. Limiting the discussion to physics, when studying a physical system, can be a box filled with an ideal gas, a melt of polymers or the state of rods\/molecules in a liquid crystalline system, in all such scenarios, there are specific entropic terms that we define in describing the evolution of the system (by including it in the free energy expression). From a statistical mechanics point of view, we use Boltzmann's definition of: $$S=k_B \\ln\\Omega$$ where $\\Omega$, can be the partition function in an ensemble, or simply the number of microstates within a given macrostate. But of course we almost never use this exact form of entropy in studying real systems, as it is impossible to count the microstates by any means. Instead we define entropic terms based on macroscopic variables of a system, like the followings (examples among the usual ones):   * For a perfect gas one can write the entropy per atom of N atoms in volume V as: $$S_{\\rm ideal}=k_B \\ln\\left(a\\frac{V}{N}\\right)$$ with $a$ a constant.    * In soft condensed matter, studying liquid crystallinity, often we define an orientational entropy, describing the entropy lost when molecules are oriented. In its most general form defined as: $$S_{\\rm orient}=-k_B \\int f(\\theta)\\ln f(\\theta)d\\Omega$$ where $f(\\theta)$ is an orientation distribution function and $d\\Omega$ a small solid angle.   * In polymers physics, often there are entropic terms attributed to homogeneity in number density distribution of monomers $n(i)$ along the chains (often called Lifshitz entropy), described as $$S_{\\rm homogeneity}\\propto -\\left(\\nabla \\sqrt{n(i)}\\right)^2$$ which is just the gradient of a density distribution. In all such cases, it is relatively straightforward to see as to why we refer to such state functions as \"entropic\", as they all boil down to describing a certain kind of disorder in the system, and how their inclusion would effect the equilibrium state of a system. But the underlying question is about, how we give physical and mathematical bounds to these entropic terms must be consistent, hence there must be a set of axioms that a function must fulfill in order to qualify as an entropic term. **To elaborate on the question:** On the one hand, from a physical point of view, the entropy is always set to reach its maximum at the most disordered state, and a minimum for the most ordered case (i.e. 1 possible micro-state, complete certainty over its state). For example in liquid crystals again, we want $S_{\\rm orient}$ to be at maximum in the orientationally disordered state, the isotropic, and for it to be at minimum in the completely ordered states, like the smectic. On the other hand, mathematically we require that:   1. $S$ to be continuous at every $\\Omega$   2. To be extensive with system size   3. Differentiable (does it have to be always?)   4. Path independent: state function   5. ...what else? Clearly if we don't know how to bound such functions based on a set of axioms, we cannot know if they will make sense physically and mathematically. (Helpful thought scenario: Imagine studying a system of particles in suspension, where their positional distribution comes with a certain periodicity, if we are to attribute an entropic term to the state of periodicity of the system, what conditions should such state function satisfy.)   * What are the axioms to be satisfied by a state function for it to qualify as entropy?"}
{"id":"99882","title":"How to prove that a spacetime is maximally symmetric?","text":"In Carroll's book on general relativity, I found the following remark: > In two dimensions, finding that $R$ is a constant suffices to prove that the > space is maximally symmetric [...] In higher dimensions you have to work > harder Here, $R$ is the Ricci scalar. This begs the following questions:   * How _does_ ones prove that a $d$-dimensional spacetime is maximally symmetric?    * If the general case is very complicated, then is there a simpler way to obtain the result in four dimensions?"}
{"id":"71493","title":"Identify the weight of operator under conformal transformation","text":"I have a stupid (homework tag may be suitable =_=) question about the problem 2.7 in Polchinski's string theory volume 1. Why the weight of operator $:e^{ik\\cdot X}:$ is $(\\frac{\\alpha'k^2}{4}, \\frac{\\alpha' k^2}{4})$? From the solution manual http:\/\/arxiv.org\/abs\/0812.4408 and a similar procedure in the answers of this post Identity of Operator Product Expansion (OPE), $$ T(z):e^{ik\\cdot X}: \\sim \\frac{ \\alpha' k^2}{4z^2} :e^{ik\\cdot X(0,0)}: +\\frac{1}{z} i k_{\\mu} : \\partial X^{\\mu} (0) e^{ik\\cdot X(0,0)}: (1)$$ (skip the antiholomorphic part.) How Eq.(1) is related to the weight $(\\frac{\\alpha'k^2}{4}, \\frac{\\alpha' k^2}{4})$ without the contribution of $1\/z^2$? For my understanding, consider $T(z) :\\mathrm{operator}:$ is to use the Ward identity (2.3.11) in Polchinski's string theory $$\\mathrm{Res}_{z \\rightarrow z_0} j(z)A(z_0, \\bar z_0) + \\overline{\\mathrm{Res}}_{\\bar z \\rightarrow \\bar z_0} \\tilde{j}(z)A(z_0, \\bar z_0) = \\frac{1}{i \\epsilon} \\delta A(z_0, \\bar z_0)$$ with $j(z)=iv(z)T(z)$ and $\\tilde{j}(z)=iv(z)\\tilde{T}(z)$ thus we relate $T(z) :\\mathrm{operator}:$ to $\\delta :\\mathrm{operator}:$ For example, the operator $X^{\\mu}$, we got $\\sim \\frac{1}{z} \\partial X^{\\mu}(0)$. For conformal transformation $z \\rightarrow \\zeta z$, $\\partial$ increase weight 1, $1\/z$ decrease 1, thus the weight is (0,0). However, how Eq. (1) is related to $(\\frac{\\alpha'k^2}{4}, \\frac{\\alpha' k^2}{4})$ without the contribution of $1\/z^2$?"}
{"id":"113104","title":"Why do solar panels not have focusing mirrors?","text":"Most of the solar panels that I have seen do not have any mirrors, etc., but usually solar cookers have mirrors. What is the reason for solar panels not having focusing mirrors?"}
{"id":"14725","title":"Formula for polarized \"light\" transmission through close filters?","text":"I'm still trying to understand photons (or polarized electromagnetic radiation).   This question is similar to one of my previous questions, but different:   Consider the famous demonstration of crossed polarizers.   Two polarizers at right angles block all light. Insert a polarizer at 45 degree angle between the two crossed polarizers, and 25% of the light is transmitted (neglecting absorption). Now perform the experiment with \"light\" of wavelength on the order of centimeters to meters, ie radar or radio band.   (Let's just choose 10 centimeters to be specific.)   Use parallel wire filters as the polarizing filters.   Now place the two crossed filters a meter apart, say at postions 0 and 100 centimeters.   Consider inserting the 45 degree filter in the middle at position 50 centimeters.   We should get 25% transmission. What happens if we move the 45 degree filter from position 0 to position 100?   I expect zero percent transmission at positions 0 and 100.   I expect 25% transmission at all positions between 10 and 90, but I can't prove it.   I expect transmission intermediate betwween 0% and 25% as the filter moves from position 0 to position 10, and the mirror image as the filter moves from position 90 to position 100.   What formula describes this transition?   More generally what formula describes the transmission as a function of position all the way from 0 to 100?"}
{"id":"14726","title":"right hand rule using left hand?","text":"im kind of confused, the right hand rule is used to know where the direction the magnetic field created by the current is going . is there any particular reason why the right hand is used to determine the direction of the Magnetic field?"}
{"id":"38789","title":"Why do we treat time as parameter in non relativistic QM instead of treating position as parameter?","text":"1. Why do we treat time as parameter in non relativistic QM instead of treating position as parameter?   2. What is actually a parameter?    3. I have read this Wikipedia page, but could not understand why should we take only time as parameter?"}
{"id":"38780","title":"Odd number of second class constraints (!)","text":"For my thesis, I have calculated the constraints for a system using Dirac method of constraint analysis. The problem is I got odd number of second class constraints (!), which gives me unusual numbers of degrees of freedom in phase space. I might have made some mistakes. Is there any other method beside Dirac, to analyze the constraints of the system?"}
{"id":"38783","title":"What would a length contracted object look like?","text":"I have multiple sub-questions but they are related.   * What would the object look like if in were passing by?    * What would a star look like if we were traveling near $c$? Would the perspective be a a large blue-shifted disk??"}
{"id":"38782","title":"How does a supersymmetry transformatio relates bosons to fermions?","text":"A superfield is defined as a function of of a spacetime coordinate and two spinor coordinates, $\\Phi (x, \\theta, \\theta^*)$. A supersymmetric transformation is then generated by the operator $\\partial_{a}+i\\sigma^{\\mu}_{a\\dot{c}}\\theta^{\\dot{c}}\\partial_{\\mu}$. What does it mean when we say this transformation relates fermionic to bosonic fields? Does the parameter $\\theta$ measure the 'bosonic' content of a superfield?"}
{"id":"77897","title":"Integrals of Power","text":"This question has two parts: a question about a specific expression and a question about a family of expressions. _In what follows I will omit all constants of integration_. If we integrate force with respect to time we get momentum: $\\int F \\, \\operatorname{d}\\\\!t = mv$. If an object accelerates along a line then we can find its acceleration at any given point and write force as a function of distance. Doing so, and integrating gives kinetic energy (or work done): $\\int F \\, \\operatorname{d}\\\\!x = \\frac{1}{2}mv^2$. I decided to try the same thing with power using the definition $P = Fv$. When I integrated with respect to time I got, as expected, $\\frac{1}{2}mv^2$ which is work done. If an object accelerates along a line then we can measure the power at regular distances and write power as a function of distance. Then we may integrate power with respect to distance: $$\\int P \\, \\operatorname{d}\\\\!x = \\int mav \\, \\operatorname{d}\\\\!x = m\\int\\frac{\\operatorname{d}\\\\!v}{\\operatorname{d}\\\\!t}v\\, \\operatorname{d}\\\\!x = m\\int\\frac{\\operatorname{d}\\\\!x}{\\operatorname{d}\\\\!t}\\frac{\\operatorname{d}\\\\!v}{\\operatorname{d}\\\\!x}v\\,\\operatorname{d}\\\\!x = m\\int v^2\\operatorname{d}\\\\!v = \\tfrac{1}{3}mv^3$$ **Here is the first question** : _What is $\\frac{1}{3}mv^3$? Does it have a name? How can I understand it conceptually?_ **Here is the second question** : Momentum is $mv$, kinetic energy is $\\frac{1}{2}mv^2$, this new quantity $\\frac{1}{3}mv^3$ has turned up. It seems that $\\frac{1}{k}mv^k$ for positive whole numbers $k$ plays a role. These can all be combined into one power series. Provided $v$ is very small, we have $$mv + \\tfrac{1}{2}mv^2 + \\tfrac{1}{3}mv^3 + \\cdots + \\tfrac{1}{k}mv^k + \\cdots \\equiv m\\ln\\left(\\frac{1}{1-v}\\right)$$ _Do the expressions $\\frac{1}{k}mv^k$ have any general meaning; and does this logarithmic expression have any use?_ (I realise that the units in the sum are all different.)"}
{"id":"18111","title":"Knowing the mass and force acting on a particle, how do we derive the relativistic function for velocity with respect to time?","text":"Use this scenario: An electron gains speed in the Stanford Linear Accelerator (SLA) across 3000 meters, reaching a final velocity of 0.95c due to a constant force pushing on the electron. Given the mass of the electron and the constant force pushing on the electron, what is the function for velocity in terms of time? How do we derive this?"}
{"id":"38253","title":"How is charge measured and mass of the electron at the same time?","text":"1. There are few constants that usually come together, $e\/m$ also $h\/e$. How are they decoupled?    2. If the speed of light is \"derived\" as Wikipedia states how meter defined and time?"}
{"id":"38252","title":"How can the properties of the Earth be determined?","text":"How can the properties of the Earth be determined? And how is the gravitational constant then determined from them? A satellite can determine the G.M.. But, how they are individually determined nowadays ?"}
{"id":"38259","title":"Extracting acceleration vector from rotated aircraft","text":"Suppose we have an aircraft with accelerometer measuring accelerations along each axis. It is mounted in a way so it is perpendicular to the plane in all axes (that should be obvious). We also have gyrometer sensor mounted on the plane - it can tell us how is the plane currently aligned (rotated) in air. **1st case:** Suppose we have an aircraft on the ground. Then we can get our desired acceleration component just by reading the accelerometer's Z axis reading (Z axis is pointing from sky to the center of the earth). **2nd case:** The plane is in the air, doing some loops for example - now to get the desired reading as previously (acceleration towards earth) we would need to compensate the Z axis acceleration with current rotation and\/or accelerations. So, I have all the data but I don't know what physical\/mathematical formulas I should use to extract the acceleration component pointing towards the earth. I would also be glad to see some tips about what literature I should read to be able to do this."}
{"id":"103188","title":"What is the place of an electromagnetic field in the electromagnetic spectrum?","text":"What place should I give an electromagnetic field (produced by a current conducting coil) in the electromagnetic spectrum? What will its wavelength and frequency be?"}
{"id":"72918","title":"After normalizing a wavefunction I don't know how to calculate probability on an interval (-0.1 + 0.1)","text":"This is quite large homework where I 1st had to normalize the wavefunction $\\psi = Axe^{-x^2\/2a}$ and I got a constant $A=\\sqrt{2\/(a\\sqrt{\\pi a})}$. How do I calculate the probability now for the interval (-0.1nm , + 0.1nm) if $a=1nm^2$? * * * I first tried to calculate the normalisation factor but the units are weird... is this possible? Is most likely my normalisation factor wrong? $$A=\\sqrt{\\frac{2}{1nm^2 \\cdot \\sqrt{\\pi\\cdot1nm^2}}}=1.06\\frac{1}{nm\\sqrt{nm}}$$ * * * **EDIT:** After thinking again I decided to include the derivation of the normalisation factor $A$: \\begin{align} &\\phantom{=}\\int\\limits_{-\\infty}^\\infty A^2x^2 \\exp\\left[\\tfrac{-x^2}{a}\\right]dx = 1 = A^2\\int\\limits_{-\\infty}^\\infty \\underbrace{x}_f\\cdot \\underbrace{x\\exp\\left[\\tfrac{-x^2}{a}\\right]}_{dg\/dx} dx= \\\\\\ &=A^2\\left[\\smash{\\underbrace{\\left.x\\left(-\\tfrac{a}{2}\\right)\\exp\\left[\\tfrac{-x^2}{a}\\right]\\right|_{-\\infty}^{\\infty}}_0}-\\int\\limits_{-\\infty}^\\infty\\left(-\\tfrac{a}{2}\\right)\\exp\\left[\\tfrac{-x^2}{a}\\right]dx\\right]=\\\\\\ &=\\tfrac{A^2a}{2}\\int\\limits_{-\\infty}^{\\infty} \\exp\\left[\\tfrac{-x^2}{a}\\right]dx = \\tfrac{A^2a}{2}\\int\\limits_{-\\infty}^{\\infty} e^{-t^2}\\sqrt{a}\\,dt = \\tfrac{A^2a\\sqrt{a}}{2}\\int\\limits_{-\\infty}^{\\infty} e^{-t^2}dt = \\frac{A^2a\\sqrt{a}}{2}\\sqrt{\\pi} \\end{align} Which means that: \\begin{align} \\frac{A^2a\\sqrt{a}}{2}\\sqrt{\\pi} &= 1\\\\\\ A&=\\sqrt{\\frac{2}{a\\sqrt{a\\pi}}} \\end{align}"}
{"id":"47375","title":"separation between slip rings of DC generator","text":"![enter image description here](http:\/\/i.stack.imgur.com\/9lj41.gif) this is a DC generator which converts mechanical energy to Direct Current. The commutators in a DC generator are separated (as you can see in the image). It is explained in our book that it prevents the change in direction of current. But i don't understand, even if we don't keep them separated, how will current change direction? In short, what is the function of the gap between the two split rings?"}
{"id":"90581","title":"Calculating the gravity gradient of a black hole","text":"Given a black hole of, say, $10^8 \\odot$ (solar masses), how can I calculate the gravity gradient ($\\frac{\\Delta g}{m}$?) at a given distance from it?"}
{"id":"103233","title":"Gauss's Law problem","text":"I've been thinking about this for a while, but I'm not sure how to proceed. I understand uniform charge-density problems, but the added non-uniform deal makes me uneasy:   * find the e. field inside a sphere (rad. = R) with nonuniform charge density $\\rho = {\\rho _0}\\frac{r}{R}$ **Edit** I did this: ![](http:\/\/s8.postimg.org\/qpv82s4xx\/img.jpg). Can someone confirm if this is correct?"}
{"id":"87789","title":"Hermitian conjugation in Radial Quantization","text":"I'm a little confused about Hermitian conjugation in a radially quantized CFT. Now, in the Minkowski theory, Hermitian conjugation leaves the coordinates invariant, i.e. $t^\\dagger = t$ and $x^\\dagger = x$. We then Wick rotate the theory to Euclidean time $t \\to - i \\tau$ and therefore $\\tau^\\dagger = - \\tau$. Next, we define the complex coordinates on the Euclidean plane $w = x + i \\tau$ and ${\\bar w} = x - i \\tau$. The Hermitian conjugate of acts on this coordinate as $w^\\dagger = w$ and ${\\bar w}^\\dagger = {\\bar w}$. On the radial plane with $z = e^{- i w}$ we get $$ z^\\dagger = e^{i w^\\dagger} = e^{i w} = \\frac{1}{z} $$ However, Francesco explicitly says (pg. 152, above equation (6.4)), that $z^\\dagger = \\frac{1}{z^*}$. Where am I going wrong? Can anyone explain this?"}
{"id":"82661","title":"Where are the L3, L4, and L5 of a hyperbolic orbit?","text":"Do the L3, L4, L5 points exist in hyperbolic orbits? If yes, then where do they lie?"}
{"id":"109936","title":"Functional field integral in condensed matter field theory (Altland)","text":"This is the action for the 1+1 dimensional interacting electron system; $$S_{cl}[\\theta , \\phi]= \\frac{1}{2\\pi} \\int dxd\\tau \\left(g^{-1}v(\\partial_x \\theta)^2 + gv(\\partial_x \\phi)^2 + 2i\\partial_{\\tau} \\theta \\partial_x \\phi \\right).$$ I want to integrate out the Gaussian field $\\phi$. This book says that it is just an \"elementary\" Gaussian integration. So, I tried some modification to this action; $$S_{cl}[\\theta , \\phi]= \\frac{1}{2\\pi} \\int dxd\\tau \\left(g^{-1}v(\\partial_x \\theta)^2 + (\\sqrt{gv}\\partial_x \\phi + \\frac{i}{\\sqrt{gv}}\\partial_{\\tau} \\theta)^2 + \\frac{1}{gv}(\\partial_{\\tau} \\theta)^2 \\right).$$ For this action, partition function is given by $$\\int D\\theta D\\phi \\exp[-S_{cl}].$$ Maybe, the second term in the action is related to Gaussian integral. But, I don't know how to calculate it. How can I calculate this?"}
{"id":"39122","title":"Dimensional reduction and Schwarzschild solution","text":"It is well-known that, given a Killing vector $\\xi_\\mu$ for a metric, this can be dimensionally reduced. This is exactly the case of the Schwarzschild solution due to the rotational symmetry. This procedure is quite general and one can always reformulate a general 2d (or 3d) model, as the Jackiw- Teitelboim model, of general relativity and obtain solutions for the 4d case with the given symmetry properties.A reference about this is this. The effect, as always happens in dimensional reduction, is to introduce another field into the gravity action. So far, I was not able to see the derivation of the reduced Schwarzschild metric from the dimensionally reduced Einstein equations. This is generally given for granted. So, I would appreciate to see such a derivation or to get some tutorial about this. Thanks."}
{"id":"74811","title":"Can a Celestial body have a mountain rise all the way to Synchronous Orbit?","text":"On Earth, the tallest mountain is Mauna Loa. In the solar system, to date, the tallest mountain is Mons Olympus. Both fall far short of GSO. Mars gravity is 0.376 of Earth gravity. If Mauna Loa were on Mars it would have an altitude of ~80000 ft. If Mons Olympus were on Earth it would similarly be no more than ~27000 ft. Yet these are almost comparable altitudes. Is it possible for a mountain to rise as high as GSO? Are all known tallest peaks on planets in the Solar System similarly limited?"}
{"id":"74812","title":"What are the forces used in this weightlifting example?","text":"You have a person doing a bench press. Their hands are equidistant apart and they are centered on the bar - each hand about 1.5 feet from center of the bar. The bar weighs 45 pounds. In example 1 there are two 45 pound weights on each side at 2.5-3 feet each from the center (lets say each weight is 3 inches wide). In example 2 there are nine 10 pound weights on each side starting at 2.5 feet again and then ending 27 inches later. What are the forces that cause example to to feel heavier and is there any way to calculate it."}
{"id":"16616","title":"returning to physics research?","text":"What the the most realistic and practical ways for a PhD in physics who has left the fold to return to a career of research in physics? The traditional employer of choice, universities won't hire such people. What alternative career choices are there for someone without an impressive track record so far? Are there alternative forms of support, or part time careers earning just enough money to get by while not interfering with research?"}
{"id":"64720","title":"Some sort of conservation equation","text":"As far as I know, in General Relativity, an expression of the kind $\\nabla_{\\mu} X = 0$ states that, associated to $X$, there exist a charge which is conserved. The first example that comes to mind is the conservation of energy-momentum $\\nabla_{\\mu} T^{\\mu\\nu} = 0$, as a consequence of the contracted Bianchi identities and assuming Einstein's field equations. My question regards the meaning of the following expression: $\\nabla_{\\mu} \\left( R^{\\mu\\nu} K_{\\nu} \\right) = 0$ where $R^{\\mu\\nu}$ is the covariant Ricci tensor and $K_{\\nu}$ is a **generic** vector field. Does this equation mean something as it stands? Or do I have to impose a metric (and hence a form of the Ricci tensor) in order to find more informations? The fact is that I don't understand what kind of conserved quantity could be related to the vector $R^{\\mu\\nu} K_{\\nu}$. The equation comes from a rather peculiar way to obtain a modified theory of gravity. The reference is here, but the original source is there I admit that the procedure is not entirely clear to me, but basically it seems to appeal to a \"gauge theory\"-like approach in order to obtain modified gravitational field equations. It's kind of lengthy, but ultimately they consider commutators of covariant derivatives of successive orders to obtain relevant quantities, such as the Riemann tensor (first-order commutator) and matter current (second-order commutator). For the latter, the analogy with gauge theories is necessary. Finally they get the modified gravitational field equations, eq.(16) in the first reference. The very nature of the approach makes it necessary to use a vector field $K_{\\nu}$ in order to carry on the calculations, but the character of such a vector field is never specified (there's no need to, in fact). Only after eq.(16) they comment that Einstein's case is recovered if $\\nabla_{\\mu}K_{\\nu}=0$. Going back to the equation that I posted... in order to have some hints about the meaning of it all, starting from eq.(16) in the paper, I just impose the vacuum condition $T^{\\mu\\nu}=0$ and see what happens: $\\left(\\nabla_{\\mu}R^{\\mu\\nu}\\right) K_{\\nu} + R^{\\mu\\nu} \\left( \\nabla_{\\mu} K_{\\nu} \\right) = 0$ and that's it. I'm not at all convinced about the comments they make about the vector field (they call it \"substratum\") but the overall procedure doesn't seem entirely bad."}
{"id":"64721","title":"Diagonal matrix in k-space","text":"I'm having some trouble with an integration I hope you guys can help me with. I have that: ${{\\mathbf{v}}_{i}}\\left( \\mathbf{k} \\right)=\\frac{\\hbar {{\\mathbf{k}}_{i}}}{m}$ and ${{\\mathbf{v}}_{j}}\\left( \\mathbf{k} \\right)=\\frac{\\hbar {{\\mathbf{k}}_{j}}}{m}$ Now according to my teacher when I multiply these two, and integrate the angular part ($\\,d\\theta \\,d\\varphi$) over $d{\\mathbf{k}}$, with $d\\mathbf{k}={{k}^{2}}\\,dk\\sin \\left( \\theta \\right)\\,d\\theta \\,d\\varphi$, I should end up with: ${{\\mathbf{v}}_{ij}}{{\\left( \\mathbf{k} \\right)}^{2}}=\\frac{{4\\pi{\\hbar }^{2}}}{3{{m}^{2}}}{{k}^{2}}$ My question is why?I see that when there is no angle dependency I get the $4\\pi$, and nothing touches $\\hbar$, $m$ and $k$. But where to the $1\/3$ come from? That's probably my main concern. So anyone with a hint or something ? Thanks in advance."}
{"id":"120076","title":"Why are black mosquito nets so much less visible than white ones?","text":"I have just been installing a couple of mosquito nets for windows in our apartment. The material is polyester, fibers are approximately 0.2mm thick, and the hexagonal \"holes\" apprixmately 1.5mm in diameter. Just like the packaging says (\"buy black ones if you want to have a better view\"), the black ones are almost invisible from a little distance (~2 meters, its a little bit darker, almost in effect like an ND photo filter). From the same distance, the white ones are clearly visible, with good eyes you can also spot the \"holes\" (the effect looks almost like if you put the black point in photoshop into a much whiter area, making the contrast much less). Why is that so? Is there a physical effect that objectively reduces the \"image information\" that reaches the human eye? I would appreciate an answer that -- if applicable -- contains a short average human explanation, and additional one that goes a bit more in-depth into the \"hobbyist physics\" level"}
{"id":"63719","title":"Is this 2D structure triclinic?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/v8Uhd.jpg) The only rotation axis obvious to me is rotation by 360 degrees, the identity. Vertical mirror planes I've been dicing and cutting it through several planes and I still see none. Yet, the structure looks fairly symmetric and I feel like I'm missing something. No center of inversion either. Not too terribly sure about horizontal mirror planes. I suppose that one bisecting the plane horizontally would be there but I am not entirely sure if this is appropriate. Right now I'm saying this structure belongs to the C1 point group."}
{"id":"21161","title":"True or false? Particle physics","text":"1. It is not possible to prove the point of origin of a photon   2. It is not possible to prove the point of origin of a free electron   3. It is not possible to prove that protons or neutrons exist inside a nucleus Can anyone confirm?"}
{"id":"64098","title":"Gas Circulation Using Pressure Difference","text":"Dear all, see attached picture Please, is it possible to have the gas recirculated from the gas phase to the liquid as described in the diagram assuming the gas is not soluble in the water. These are the conditions. 1\\. The valve remained closed, 2\\. The liquid level didn’t change 3\\. P1 increases. Could P1 get to a level where it would force itself through the pipe and go through the liquid. If P1 increases, what happens to P2. ![enter image description here](http:\/\/i.stack.imgur.com\/CHf0S.jpg)"}
{"id":"46939","title":"Electric field inside and outside a metallic hollow sphere","text":"1) It is known that inside a metallic hollow sphere it will not experience outside electric field because of the charge separation of electrons and holes at the surface of sphere and creating an equal and opposite field. So it is better to stay inside the car during lightning. Why stay in the car during thunderstorms? But isn't there a maximum value of electric field the metallic surface can resist because at some point (higher value) of electric field the movement of electrons will get stopped because it can't generate more electron holes pairs and which will limit the further change in increment of electric field to zero. Since lighting produce such a high value of electric field is it safe to stay inside a car during lightning? _assuming no heat generated during lightning_ 2) Similar case about charge inside a earthed metallic hollow sphere. If I keep some charge inside the metallic hollow sphere, outside field is zero and inside field is dependent upon the charge. If I increase the charge inside the metallic hollow sphere to such a level which is more than the total electron charge on the surface of sphere then the earthing has no effect because it has already neutralized the maximum amount of holes (+ve) by accepting the electrons from earth. So the field will escape outside."}
{"id":"53003","title":"Trace of stress tensor vanishes ==> Weyl invariant","text":"You often see in textbooks the statement that ${T^\\mu}_\\mu = 0$ implies Weyl invariance or conformal invariance. The proof goes like $\\delta S \\sim \\int \\sqrt{g} T^{\\mu\\nu} \\delta g_{\\mu\\nu} \\sim \\int \\sqrt{g} {T^\\mu}_\\mu$ where I have varied the action with respect to the metric and assumed $\\delta g_{\\mu\\nu} \\propto g_{\\mu\\nu}$ (i.e. a Weyl transformation). This does not seem to be completely general because I can imagine a Lagrangian containing matter fields with non-trivial conformal weights. Then the full variation under Weyl tranformation contains a term proprotional to the matter equation of motion. So I would conclude that the correct statement is more like ${T^\\mu}_\\mu = 0, \\quad \\& \\quad \\frac{\\delta S}{\\delta \\phi} = 0\\implies \\textrm{Weyl invariant}$ Is it true that Weyl invariance only holds when the matter fields are on-shell or am I missing something? Thanks."}
{"id":"90204","title":"Optimal velocity in rain","text":"**_Problem statement:_**   > A sphere is rolling in rain from point A to point B. The vertical speed of > rain is V and horizontal speed of rain is v, as shown in the picture. Angle > between horizontal component of rain's velocity and sphere's velocity is > $\\varphi$. What is the optimal speed of sphere so that it would be as dry as > possible?   >   > ![Problem picture](http:\/\/i.stack.imgur.com\/sljvY.png) So here's what I tried to do:   The velocity of rain in respect to the sphere is $\\vec{v} = \\vec{v_s} - \\vec{v_r}; v=\\sqrt{v_s^2 + v_r^2} = \\sqrt{v_s^2+(\\sqrt{V^2+v^2})^2}=\\sqrt{v_s^2+V^2+v^2}$. I thought that maybe if I found the derivative of this function in respect to $v_s$ and equated it to 0, that could be the answer. But I don't know how to find the derivative of such function and the solution doesn't seem right to me, though I couldn't come up with anything better.. Any help appreciated!"}
{"id":"13495","title":"How do I solve for the phase constant given the amplitude and the angular frequency?","text":"> A piston (with mass M) in a car engine is in vertical simple harmonic motion > with amplitude A. The engine is running at a period T. Suppose a small piece > of metal with mass m were to break loose from the surface of the piston when > it (the piston) is at the lowest point z and velocity is v. At what position > in meters does the piece of metal lose contact with the piston? Also the maximum velocity and acceleration are given. The text book gives these equations: $x(t) = Acos(wt+φ)$ $v(t) = -wAsin(wt+φ)$ $a(t) = -w^2Acos(wt+φ)$ So I propose that the best way to figure this out would be to solve for the time at which acceleration is at its max, then plug that time into $x(t)$. But I must first solve for the phase constant. Is this correct? If so, how do I find the phase constant?"}
{"id":"62448","title":"Voltage of open circuit","text":"> A battery with emf $\\varepsilon$ and internal resistance $r$ is connected > with a resistor $R$ in the following open circuit. What is the voltage > $V_{ab}=V_a-V_b$? > > ![enter image description here](http:\/\/i.stack.imgur.com\/8RVNX.png) The answer is $- \\varepsilon$. \" _No current. There is no voltage change across R and r._ \". But I don't really understand why ... I was thinking intuitively it should be $0$? Then thinking of how to get 0, I was thinking ... $V_a = - \\varepsilon$ since its on the negative terminal, the $V_b = + \\varepsilon$ since its on the positive terminal. But $V_a - V_b = -2 \\varepsilon$ ... how do I make sense of this?"}
{"id":"62441","title":"Inserting metal into parallel plate capacitor","text":"> The plates of an isolated parallel plate capacitor with a capacitance C > carry a charge Q. The plate separation is d. Initially, the space between > the plates contains only air. Then, an isolated metal sheet of thickness > 0.5d is inserted between, but not touching, the plates. How does the > potential difference between the plates change as a result of inserting the > metal sheet? The answer is: The potential difference will decrease. How do I get this result, it seems right but how do I reason it out? $$V = Ed$$ There was mention about \" _E is the same outside the conductor. But the field inside is 0._ \". But in the formula, E is the field outside conductor right? If so it remains the same? Then $d = 0.5d$?"}
{"id":"13140","title":"Where the angular momentum has gone?","text":"I came across an interesting problem when I prepared for the preliminary exam on electromagnetism. Below is the problem in its original words: > A metallic sphere of mass, $m$, and radius, $a$, carries a net charge, $Q$, > and is magnetized with uniform magnetization, $M$, in the $z$-direction. > > (a) Determine the total angular momentum associated with the electromagnetic > field. > > (b) The magnetization of the sphere is now reduced to zero. Assuming that no > external mechanical forces act on it (PS: this is possible. For instance, > just heat up the sphere and the magnetization will be eliminated ), > determine the angular velocity and the sense of rotation of the sphere when > $M=0$. > > (c) Describe the origin of the torque that causes the sphere to rotate. For part (a), it is straight forward to calculate electric and magnetic field E, H inside and outside of the sphere. Generated by the net charge $Q$ uniformly distributed on the surface, the electric field $E$ is isotropic outside the sphere and vanishes inside. The magnetic field H is generated by magnetization $M$ and can be calculated using the \"magnetic scalar potential\" formalism since there is no free current. The result is that H is also uniform inside the sphere and takes the dipole form outside the sphere. The total angular momentum $J$ is obtained by integrating over the region where a local angular momentum density of the EM field exists, i.e. outside of the sphere. It is in the $z$-direction and I find it to be $$J= \\frac{2 Q M a^{2}}{9 \\epsilon_0 c^{2}}$$ For part (b) and (c), it is interesting to ask why the sphere would rotate after magnetization $M$ is erased. What I think up is that as $M$ decreases to zero, a electric field with non-zero curl is generated, as implied by Faraday's law. This electric field is in the azimuthal direction, and will act on the surface charge $Q$ to provide a torque. However, a straightforward calculation can show that the angular momentum this torque can transfer is $$J'= \\frac{Q M a^{2}}{9 \\epsilon_0 c^{2}}$$ Obviously a factor of 2 is missing, so this only accounts for half of the angular momentum stored originally in the EM field. So I am perplexed by where the other half has gone? I come up with 2 possibilities:   1. since the charged sphere begins to rotate, there is also non-zero angular momentum stored in EM field after magnetization is removed. But it could hardly account for the missing amount, since in that case the electric field $E$ is proportion to $Q$, the magnetic field $H$ to $Q^2$ ( one $Q$ from the charge and a second $Q$ from the angular velocity). Hence one expects the angular momentum stored is proportional to $Q^3$, which is very different from $J$'s linear dependence on $Q$.   2. maybe in the process of magnetization removal, EM radiation carries away some angular momentum. But this possibility has to be validated quantitatively before it can convince people. What I feel is that if the process of removal is infinitely slow, then the effect of radiation can be neglected. Then what is the correct answer? I provide quantitative result for $J$ and $J'$ in the hope that someone can check them and point out that I have made a wrong calculation lol. But I am really eager to hear from any explanation that sounds real ... Electromagnetism is as interesting as it was 3 years ago, but I just don't remember as much as I did then ..."}
{"id":"13143","title":"What did Marie Curie do for atomic theory?","text":"There appears to be a distinct lack of agreement in the physics community on what exactly Marie Curie did for atomic theory. Many journals state that Curie was responsible for shifting scientific opinion from the idea that the atom was solid and indivisible to an understanding of subatomic particles. However, JJ Thompson had already discovered the electron prior to Curie's work, and Rutherford proved the nature of atomic structure - so what did Marie Curie really do for atomic theory?"}
{"id":"118915","title":"How can $J_1^2, J_2^2, J_{1z}, J_{2z}$ commute mutually?","text":"I'm reading through J. J. Sakurai's Modern Quantum Mechanics book and currently looking at the \"Angular-momentum addition\" part. Here, it says you have two options and that one option is to construct simultaneous eigenket $\\vert j_1j_2;m_1m_2\\rangle$ of $J_1^2, J_2^2, J_{1z}, J_{2z}$ since the four operators commute with each other. I understand that $J_1^2$ and $J_{1z}$ commute, but I'm not sure how $J_1^2$ and $J_{2z}$ can commute intuitively. \"commute\" means that one can measure both at once right? But total angular momentum of spin 1 and angular momentum of spin 2 are independent. Where am I wrong here?"}
{"id":"118910","title":"Does mass equal angular momentum?","text":"At the wikipedia pages for angular momentum ($L$) and moment of inertia ($I$) we find the equations: $$L=I \\omega$$ $$I=m r^2$$ where $m$ is mass and $r$ is the distance between said mass and barycenter and $\\omega$ is angular velocity. Now rename the angular velocity as follows: $\\omega=\\frac{d\\theta}{dt}$. Note that the area of an ellipse sector swept out from a focus is given by: $$A(\\theta)=\\frac{1}{2}\\int_0^{\\theta} r^2 \\:\\: ds$$ And so: $$\\frac{dA}{d\\theta}=\\frac{r^2}{2}$$ Note that for a mass in an elliptical orbit this area is proportional to time. Therefore, we can write: $$\\frac{d\\theta}{dt}\\sim\\frac{d\\theta}{dA}\\sim\\frac{1}{r^2}$$ Substituting this back into the first two equations at the top of this post gives: $$L\\sim m$$"}
{"id":"20345","title":"Wave function of hydrogen atom including spin of nucleus","text":"How do I write the wave function of hydrogen atom taking into consideration of nucleus spin? For example consider $2S_{\\frac{1}{2}}$ state with nucleus spin $I$, then wave function $\\psi=\\langle2S_{\\frac{1}{2}},F,F_{3}|$ where $F$ is the total angular momentum of hydrogen atom $F=J+I$ and $F_{3}$ is the projection of it along z axis. Now what will be the explicit form of the wave function? Thanks in advance."}
{"id":"20347","title":"What did Hawking mean? 'Time started at the big bang'. Book suggestions please","text":"**After writing down this question, I have come to realize that. What I really want is _reading materials_ on the questions below.** Before the big bang there was no such thing as 'time' (Steven Hawking on why the pope was wrong when he stated that physics and cristian beleves could go together as someone 'must have started the big bang'). Hawking stated that according to his theories time started at the big bang. Before the big bang, no time, hence no creator. Hawking actually said this in a speach, in memorial to Gallileo when the pope made official appologices on behalf of the church for banning Gallileo. I read this years ago so I might be wrong in some detail. **I would like something to read to understand what Hawking meant when he said that there could be no creator since time 'started' at big bang and did not exist before that**. I found this paper but I am not sure I understand what it means. Concentric circles in WMAP data may provide evidence of violent pre- Big-Bang activity. I think I have to read other things before actually digging in to that paper since I find it pretty hard. **What can I read as an introduction before reading this paper. Frankly, I don't understand it.** From a physics phd student I was informed that many physics researchers actually are unsure that the big bang really existed. Since they don't believe that everything could have existed in a singular point. Do they? (Might be a tricky question to answer). When you spread out time it turns out that time can not be spread out. I.e. it does not work like a rubberband. Or perhaps, When you compare time (maybe as a result of jouries at extreme speed where the two objects then have different ideas of how long time it has taken) does not follow form a bell curve statistically. Why not? The uncertainty evaluated at t = 0 also gives strange results but I am not sure it is related. _Perhaps it is perhaps not_. **I know a lot of calculus, a lot of linear algebra and basic quantum mechanics.**"}
{"id":"20349","title":"Synchronising the Earth's rotation via mass redistribution","text":"How much material would have to be moved per year from mountain-tops to valleys in order to keep the Earth's rotation synchronised with UTC, thus removing the need for leap seconds to be periodically added? Would it be a feasible project to undertake in order to resolve the current disagreement about the future of the leap second?"}
{"id":"62191","title":"A sphere rolling down a rough wedge which lying on a smooth surface","text":"A sphere of mass $m$ and radius $r$ rolls down from rest on an inclined (making an angle $\\phi$ with the horizontal ) and rough surface of a wedge of mass $M$ which stays on a smooth horizontal floor. Find the acceleration of the wedge and the acceleration of the sphere relative to the wedge. I have written out the equations of motion for the sphere and the wedge and also the equation of motion for the rotational motion of the sphere. But then I get stuck. I don't know how I should relate the angular accleration $\\alpha$ of the sphere with the acclerations $a_{\\perp},a_{\\parallel}$ appearing in its equations of motion. Can anyone please help me out? Thanks!"}
{"id":"94451","title":"How does a liquid crystal on Silicon affect the phase of incident light?","text":"When a polarized light beam is directed to a phase-only LCoS system how is the phase modulated ? What is the physical effect behind ? Does voltage modulation has an impact on birefringence ?"}
{"id":"29543","title":"Does the vacuum energy problem of quantum field theory only occur in the Hamiltonian approach, or also in the path integral approach and in AQFT?","text":"In a standard QFT class, you're being indoctrinated that there is the \"infinite vacuum energy density problem\". (This is sometimes paraphrased as the \"cosmological constant problem\", which is in my opinion a misnomer since the calculation that calculates a finite value for the vacuum density and relates it to the cosmological constant is ill-founded and dubious.) The argument goes as follows: When defining the Hamiltonian $H$ from the Lagrangian density $\\mathcal{L}$, we first define the Lagrangian function $L := \\int \\operatorname{d}^3 x \\mathcal{L}$ and the momenta $\\pi(x) := \\frac{\\partial L}{\\partial \\dot \\phi(x)}$ and finally the Hamiltonian $H := \\int \\operatorname{d}^3 x \\pi \\dot \\phi - L$. We then promote all the quantities that occured to operators and calculate the decomposition of $H$ into creation and annihilation operators $a^\\dagger$, $a$. Then, the lecturer makes a big fuss (prototypical example) about the Hamiltonian being of the form $\\int \\operatorname{d}^3 p E_p \\frac{1}{2} \\left( a^\\dagger a + a a^\\dagger \\right)$ = $\\int \\operatorname{d}^3 p E_p \\left( a^\\dagger a + \\delta(0) \\right)$ and the $\\delta(0)$ being an infinite contribution to the vacuum energy density. (I always found the argument fishy because we can't know beforehand if some quantisation strategy will succeed or not, so we should rather start with a quantised Hamiltonian and ask whether its classical limit is the theory we started with, which can be answered with yes.) My question now is: Does a similar problem also occur in the Feynman path integral approach, or in any approach from algebraic quantum field theory?"}
{"id":"29542","title":"interpretation of Green function","text":"Is there a physical interpretation of the existence of poles for a Green function? In particular how can we interpret the fact that a pole is purely real or purely imaginary? It's a general question but I would be interested in the interpretation in quantum systems."}
{"id":"131269","title":"Is there a method which quantizes non-abelian gauge theories without path integrals formalism?","text":"In the most QFT books there is a method of quantization of non-abelian theories through path integral methods. But I want to learn also the other methods without using of this formalism. Does anyone know these methods?"}
{"id":"110852","title":"Unit Conversion","text":"I need to convert this expression $ 81752 \\left( \\frac{BTU}{in^2 h ºR} \\right )$ to $ \\left( \\frac{Kg}{s^3 ºC} \\right )$ Is my draft correct? $ 81752 \\left( \\frac{BTU}{in^2 h ºR} \\right ) \\rightarrow \\frac{1 BTU}{in^2 h} = \\frac{454,3 W}{m^2} = \\frac{1 ºR}{0,5555 K} = \\frac{66,846533 W}{m^2 k} = \\frac{66,896533 kg}{s^3 K} $ $ = \\frac{1 K}{-272,2 ºC} = \\frac{-2456 Kg}{s^3ºC} $"}
{"id":"110850","title":"Interference - the shortest way from the point of constructive one to the point of destructive one","text":"So this is a problem from Polish maturity exam. ![2 speakers](http:\/\/i.imgur.com\/TtzkFnS.png) The image shows 2 speakers (G1, G2) and point B. The wavelength of sound coming from both speakers is 0.155 m, and the wave coming from both speakers is in phase. So point B is where the constuctive interference appears (you can calculate this quite easily). And the problem - \"Draw an arrow in the direction of the shortest way possible from point B, where the sound intensity is high, to point A, where the sound intensity is low\". Essentialy what they're asking is in which direction should we turn from point of constructive interference in order to, by going in a straight line, achieve point of destructive interference in the shortest way possible. So any thoughts on how to calculate, solve this?"}
{"id":"92117","title":"Does light have mass\/weight?","text":"If I plug in my memory stick\/USB into my laptop and load it up with documents etc. will its mass\/weight increase?"}
{"id":"61364","title":"General physics question involving Heisenberg Uncertainty Principle","text":"Question: > An unstable particle produced in a high-energy collision is measured to have > an energy of $483\\ \\mathrm{MeV}$ and an uncertainty in energy of $84\\ > \\mathrm{keV}$. Use the Heisenberg uncertainty principle to estimate the > lifetime of this particle. Attempt at answer: I thought $$ (\\Delta E)(\\Delta t) = \\hbar = 1.055 \\times 10^{-34}\\ \\mathrm{J}\\cdot\\mathrm{s} = 6.591 \\times 10^{-16} \\mathrm{eV}. $$ So I tried solving for $\\Delta t$, my time, with $\\Delta E = 84\\ \\mathrm{keV}$. However, this answer is wrong (I get something like $7.85 \\times 10^{-21}\\ \\mathrm{s}$, which is incorrect). I assume I'm supposed to incorporate $483\\ \\mathrm{MeV}$ somehow, but am not sure how. Any suggestions?"}
{"id":"111458","title":"Comparing Graphene and Ni3 HITP2","text":"A recently announced advance in Metallized Organic Films at MIT: http:\/\/pubs.acs.org\/doi\/abs\/10.1021\/ja502765n discusses a self-organizing material: Ni3HITP2 = Ni3(2,3,6,7,10,11-hexaiminotriphenylene)2 with a film conductivity of 40 Scm^-1 The wikipedia article talks about resistance of graphene of 10^-6 ohm-cm but mentions a sheet resistance of 1ohm\/square under certain circumstances at the edge. I'm confused by the different metrics. How do the two materials compare?"}
{"id":"93408","title":"Equation for a galaxy's overall gravitational pull?","text":"Can the Newton's formula be used to express $g$ between two Galaxies in proximity? Gravity on Earth uses two concepts, one is the radius (R) of earth and the other is the distance $h$ from the surface of the Earth. Can you use similar concepts for galaxies? If that is the case, how do you measure $R$ and $h$, considering that: 1 - a galaxy does not have a spherical shape. 2 - a galaxy does not have much density. 3 - a galaxy does not appear to have a surface as would a star or a planet. What then would be the equation that I would use to calculate the pull from the Andromeda galaxy?"}
{"id":"26257","title":"How fast is the Cosmic Microwave Background Radiation (CMBR) changing?","text":"I know that the Cosmic Microwave Background Radiation (CMBR) is the leftover radiation from the \"surface of last scattering\". However, at every instant the surface is changing (at the rate of flow of time). So how constant is the CMBR?"}
{"id":"110589","title":"Relativistic Time dilation and travel to exoplanets","text":"NASA recently discovered the first habitable exoplanet with the Kepler space telescope. Recently I was on this site: http:\/\/www.cthreepo.com\/lab\/math1\/ With the Long Relativistic Journeys calculator on that page, I calculated the amount of time it would take according to the traveler to travel 500 light years with half of the journey at a constant acceleration of `2g` and the second half with equal deceleration. The result was under 7 years. Does this require going near the speed of light? Or does it only require acceleration? If, say, we were to accelerate the space ship up to a certain speed and then decelerate periodically, could the same result be achieved? Or would no significant time dilation occur?"}
{"id":"130795","title":"What exactly is Electromagnetic Radiation?","text":"I don't understand how and why the electric and magnetic fields oscillate in the electromagnetic radiation wave, and any way where do these fields originate from, for there are no charged particles in the wave (there are no particles in it at all). Till now I've taken this for granted that the electric and magnetic fields oscillate at 90° to each other but recently I'm trying to study quantum physics but due to this confusion I couldn't understand it."}
{"id":"53493","title":"computer vs heater coil comparison","text":"If I have a computer and a heater coil that consume exactly the same amount of energy, which would be more efficient at heating my room? This is assuming that they both have the same fans and heat dispersion characteristics. And if the answer is that they are the same, then why don't we change all of our personal household heaters into computers?"}
{"id":"55778","title":"Renormalization Group and Ising with d=1 and D=1","text":"I have a question about the results of RG on Ising model. I know it's possible to obtain two couple of relations   1. $K'(K)$, $q(K')$   2. $K(K')$, $q(K)$ between the coupling costants. My problem arise when i try to draw the flow diagram of the coupling constants. I know that this model dont allow phase transition except the trivial case $T=0$, but if we reiterate the relation (1) or (2) we increase or decrease the coupling costant. In one case i obtain $$K=0,T=\\infty>>>>\\cdots>>>>K=\\infty,T=0$$ but in the other? **Boolean Ising Model** with $d=1$ dimension of lattice, $D=1$ dimension of vector space of the spins on lattice. The energy with zero external field is $$H=-J\\sum_{<ij>}S_iS_j$$ note that there are overcounting. Then the partition function can be put in the following form $$Z=\\sum_{\\\\{S\\\\}}\\prod_ie^{KS_iS_{i+1}}$$ With a partial summation on even spins it became $$Z'=\\sum_{S}\\prod_ie^{K(S_i+S_{i+1})}+e^{-K(S_i+S_{i+1})}=\\sum_{S}\\prod_if(K)e^{K'S_iS_{i+1}},$$ where in the last i used the scaling proprieties   $$Z(N,K)=f(K)^{N\/2}Z(N\/2,K').$$ The relations for $f(K)$ and $K'(K)$ are: $$f(K)=2\\cosh^{1\/2}2K,$$ $$K'=\\frac{1}{2}\\log\\cosh{2K}.$$ The extensivity of free energy states $-\\beta F=\\log Z=Nq(K)=\\frac{N}{2}\\log f(K)+\\frac{N}{2}q(K')$."}
{"id":"98019","title":"Triple axis accelerometer direction-finding","text":"Can I detect the direction of the disturbance source which propagated in the medium by waves, using only triple axis accelerometer? I guess yes, because I already have projections of all three axes (i.e. three accel detectors) on first arrived wavefront. Or maybe I'm wrong? What mathematics could be involved?"}
{"id":"98011","title":"Binary Star system with one star stationary?","text":"Can a Binary Star system be possible where in one star is stationary and the other star revolves around it? (Just like a planet revolving a star. i.e planets in the system and the star revolving around their own center of masses to balance the system)."}
{"id":"110322","title":"Number of zero-modes on the sphere","text":"Is it true that a field of conformal dimension $h$ (integer or half integer) has $1-2h$ zero-modes on the sphere, if $1-2h \\geq0$. This seems to be right for different ghost fields :   * $c$ has weight $-1$ and 3 zero-modes   * $\\gamma$ has weight $-1\/2$ and 2 zero-modes   * $\\xi$ has weight $0$ and 1 zero-modes   * $\\eta$ has weight $1$ and 0 zero-modes I'm not sure how to compute the number of zero-modes on the sphere... Is it the argument given at p.167 in volume 1 of Polchinski's book ?"}
{"id":"114669","title":"Is Newton's third law always correct?","text":"Newton's third law states that every force has an equal and opposite reaction. But this doesn't seem like the case in the following scenario: For example, a person punches a wall and the wall breaks. The wall wasn't able to withstand the force, nor provide equal force in opposite direction to stop the punch. If the force was indeed equal, wouldn't the punch not break the wall? I.e., like punching concrete, you'll just hurt your hand. Doesn't this mean Newton's third law is wrong in these cases?"}
{"id":"52011","title":"Sound level of organ pipe driven by helium","text":"The sound pressure level of an organ pipe is a function of the gas flow rate delivered to the pipe. Source Would the sound level of an organ pipe driven by helium be lower than that of a pipe driven by air at the same blowing pressure (same volume flow rate) due to the lesser mass flow rate of the helium (helium being of lesser density than air)?"}
{"id":"118992","title":"Trace operation on dynamic equation: physical meaning?","text":"Suppose we have Heisenberg equation of motion for some observable $A$, $$ i\\hbar\\frac{dA}{dt}= -[H,A] $$ since the trace of any finite dimensional commutator structure vanish(not something like $[x,p]=i\\hbar$ ), $$ Tr\\left(\\frac{dA}{dt}\\right)=\\frac{i}\\hbar~Tr([H,A])=0 $$ so my question is, what does $Tr(\\frac{dA}{dt})=0$ indicate?"}
{"id":"60105","title":"Is lattice enthalpy positive or negative?","text":"I've learnt that the lattice enthalpy (defined as the energy change from a solid ionic lattice to separate gaseous ions) is always positive, obviously. However, I've seen it explained as the opposite other places, so it's negative. What is correct?"}
{"id":"82502","title":"Some questions about calculation central charge in a CFT in $d$ spacetime dimensions","text":"This is based on this paper, http:\/\/arxiv.org\/abs\/hep-th\/0212138   * For a CFT on a $S^d$ spacetime (of radius R) it seems to be claimed that the central charge is given by, $ c = \\langle \\int_{S^d_R} d^dx \\sqrt{g} T_\\mu ^\\mu \\rangle $    * Their equation 23 (on page 6) seems to indicate that if $W = -log Z$ is the free energy of the theory then it further follows that, $c = \\frac{1}{d}R \\frac{\\partial }{\\partial R} W$ (...I believe that the derivative is being evaluated at the value of the radius of the sphere..)    * Just below equation 26 it is claimed that, \"...the central charge can be read off from the coefficient of log R in an expansion of W[R]...\"  I would like to know the proof\/derivation of three methods that have been spelt out to calculate the central charge of a CFT."}
{"id":"93930","title":"Work is force multiplied by distance What is this distance?","text":"![Work](http:\/\/i.stack.imgur.com\/8XUua.gif) What is the distance which the force is multiplied by it to get the work done? a) The distance of the body motion while the force was touching it? b) The total distance of the body motion?"}
{"id":"117315","title":"Why my simple electromagnetic experiment does not work?","text":"I make a setup (see attached picture).   * ONE coil (1\" diameter x ~120 turns), which is separated with a gap, is mount on two sides. Each side has an iron rod (1\" diameter x 6\" long) to increase the focus of the magnetic wave.The wire for the core is 14 gauge motor wire (5.9 amp power rating max).   * Two super strong magnets (1\" diameter x 0.5\" thick each) are mount at the middle block (plastic) under Duck tape. Each of them is rated ~40 lbs holding force. Each side of the block has one magnet.   * I use 6 power resistors in series to create 50 ohm resistance. I connect the circuit to wall power 120VAC. Since the impedance of the coil is small, I think I give 2.4 amp @ 60 HZ to the circuit.    * Everything for holding is by very smooth plastic (Acetal Resin).    * Before I connect the power, I put the system at rest. The magnet is so strong that, they make the middle piece to go to either one of the iron.  I thought the MIDDLE PIECE would swing to the left and right with 60Hz. However, the system does NOT move at all. Any idea? ![Experiment setup](http:\/\/i.stack.imgur.com\/PoK8O.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/qBCen.jpg)"}
{"id":"117316","title":"Which charge value to use and Why","text":"$1\\mu C=10^{-6}C$ vs $e=1.60\\times10^{-19}C$. I am having trouble understanding when to use for $q$, $10^{-6}C$ or $1.60\\times10^{-19}C$. I had a problem where $a=Eq\/m$. I don't know which value of a charge go into the formula. I need to understand why."}
{"id":"99055","title":"Double slit experiment and representation of light waves","text":"Consider the following image from Wikipedia and based on it I have a doubt. ![Image Credits: Wikipedia](http:\/\/i.stack.imgur.com\/X0QtM.gif) I do not understand why are the light waves represented like the waves in water. Shouldn't the waves be like sine waves. Why is the slit part the starting of a new wave. Secondly, Why does this lead to to destructive interference rather than constructive because the waves are emitted at the same time. And if anyone can tell me how is double slit possible even when there is a single photon?"}
{"id":"56833","title":"Slinky base does not immediately fall due to gravity","text":"Why does the base of this slinky not fall immediately to gravity. My guess is tension in the springs is a force > mass*gravity but even then it is dumbfounding. ![](http:\/\/i.imgur.com\/qr3dIsC.gif)"}
{"id":"69754","title":"Potential and Kinetic Energy","text":"In engineering school you learn the basic swing problem. Essentially that there is a transfer of kinetic energy (as seen in the velocity at the bottom of as swing) to potential energy at the top of the swing's arc. My question: What is the nature of KE and PE and how do they seamlessly \"transfer\" between the two? How is this explained by classical physics?"}
{"id":"9647","title":"Space-time hospitality","text":"Are there any reasons to assume that organic processes cannot arise in relatively flat spacetime environments?.. An argument for the existence of a critial value or window .."}
{"id":"112367","title":"Noether's theorem and gauge symmetry","text":"I'm confused about Noether's theorem applied to gauge symmetry. Say we have $$\\mathcal L=-\\frac14F_{ab}F^{ab}.$$ Then it's invariant under $A_a\\rightarrow A_a+\\partial_a\\Lambda.$ But can I say that the conserved current here is $$J^a=\\frac{\\partial\\mathcal L}{\\partial(\\partial_aA_b)}\\delta A_b=-\\frac12 F^{ab}\\partial_b\\Lambda~?$$ Why do I never see such a current written? If Noether's theorem doesn't apply here, then is space-time translation symmetry the only candidate to produce Noether currents for this Lagrangian?"}
{"id":"112363","title":"Taylor expansion of the metric","text":"Consider a coordinate change $$ x^a\\mapsto \\tilde x^a=x^a+\\epsilon y^a $$ In the note I am reading, the author calculate the change of metric by $$ g_{ab}(x) = \\tilde g_{ab}(\\tilde x)=\\tilde g_{ab}(x^a+\\epsilon y^a)=\\tilde g_{ab}(x^a)+\\epsilon\\mathcal{L}_Y\\tilde g_{ab}(x^a)+\\cdots $$ My question is, why do we use $\\mathcal{L}_Y\\tilde g$ rather than $\\nabla_Y\\tilde g$ in the first order term? * * * Update Following is my newly try. As Prahar pointed out, I should write $$ g_{ab}(x)\\mapsto\\tilde g_{ab}(\\tilde x)=\\frac{\\partial x^c}{\\partial {\\tilde x}^a}\\frac{\\partial x^d}{\\partial {\\tilde x}^b} g_{cd}(x) $$ Then we have $$ g_{ab}(x)=\\tilde g_{ab}(x^a+\\epsilon y^a)=\\tilde g_{ab}(x^a)+\\epsilon\\nabla_Y\\tilde g_{ab}(x^a)+\\cdots $$ And I have to show $$ \\nabla_Y\\tilde g_{ab}(x^a)=\\mathcal{L}_Yg_{ab}(x^a) $$ We have $$ \\begin{align} \\tilde g_{ab}(x^a)&=\\frac{\\partial x^c}{\\partial {\\tilde x}^a}\\frac{\\partial x^d}{\\partial {\\tilde x}^b} g_{cd}(x^a)\\\\\\ &=\\left(\\delta_a^c-\\epsilon\\frac{y^c}{\\tilde x^a}\\right)\\left(\\delta_b^d-\\epsilon\\frac{y^d}{\\tilde x^b}\\right)\\left.\\right|_{x^a-\\epsilon y^a}g_{cd}(x^a-\\epsilon y^a)\\\\\\ &=g_{ab}(x^a-\\epsilon y^a)-\\epsilon\\left[\\frac{y^d}{\\tilde x^b}g_{ad}+\\frac{y^c}{\\tilde x^a}g_{cd}\\right]\\left.\\right|_{x^a-\\epsilon y^a}+O(\\epsilon^2)\\\\\\ &=g_{ab}(x^a) - \\epsilon\\nabla_Yg_{ab}(x^a)-\\epsilon\\left[\\frac{y^d}{\\tilde x^b}g_{ad}+\\frac{y^c}{\\tilde x^a}g_{cd}\\right]\\left.\\right|_{x^a} +O(\\epsilon^2) \\end{align} $$ So we get $$ \\nabla_Y\\tilde g_{ab}(x^a)=\\nabla_Yg_{ab}(x^a)-\\epsilon\\nabla_Y\\left(\\nabla_Yg_{ab}+\\frac{y^d}{\\tilde x^b}g_{ad}+\\frac{y^c}{\\tilde x^a}g_{cd}\\right)\\left.\\right|_{x^a} + O(\\epsilon^2) $$ On the other hand, we have $$ \\mathcal{L}_Yg_{ab}(x^a)=\\nabla_Yg_ab{x^a}+g_{ac}\\nabla_bY^c+g_{cb}\\nabla_aY^c $$ I cannot see these two are identical. Is there anything wrong in my deduction?"}
{"id":"81606","title":"Stability theory","text":"I'm studying stability theory recently and met a lot of phrases like linear stability and nonlinear instability. After searching on Google, I became more confused. Thus I wonder if there is any approperiate or famous books discussing these issues of dynamical systems."}
{"id":"105495","title":"How fast would a truck have to go to pull a pedestrian onto the road?","text":"Let's suppose a pedestrian **P** is walking or standing next to a highway.   Suppose a truck **T** will drive by the said pedestrian at speed **V** leaving distance **L** between the two. Assuming **L** is a reasonably small finite size how would one calculate the force acting upon the pedestrian from both the general differences in pressure and turbulence? Shortly put, how fast would the truck have to go _in practice_ so that the air flow left in its wake would pull the pedestrian onto the road? ![Sample image](http:\/\/i.stack.imgur.com\/Q8R1P.png)"}
{"id":"113940","title":"Is Initial Coherence a Problem in Experiments?","text":"In experiments, or optical setups, is the initial coherence of lasers something to be concerned about? You can calculate interference patterns, but those generally rely on the phase shift between the two sources being zero, right? In a laboratory, how do you know two lasers will be in phase?"}
{"id":"125965","title":"Tricks at manipulating creation\/annihilation operators","text":"Manipulation of terms in algebras different from the standard one (e.g. boolean algebra) can be a bit unnatural but there are always shortcuts that can help you. I was wondering if there is a list with the standard tricks for manipulating creation and annihilation operators of bosons and fermions, instead of using intensively their commutation relations and having to rediscover the wheel every time. This would prevent me (and many others) from losing a big amount of time at blind guess-checking... Even a good set of exercises can count as an answer."}
{"id":"98901","title":"POVM positive density matrix decomposition","text":"I have to prove that given a density matrix $\\rho$ in a finite d-dimensional Hilbert space $\\mathcal{H}_d$, it always exists at least one informationally complete POVM measure $\\\\{E_i\\\\}_{i=1}^{d^2}$ such that $$ \\rho=\\sum_{i=1}^{d^2} \\alpha_i E_i, $$ with $\\alpha_i\\geq 0$. Without the last constraint on the $\\alpha_i$ it is obvious, how can I prove it imposing the constraint $\\alpha_i\\geq 0$? Any idea?"}
{"id":"112094","title":"Having trouble understanding the concept of current and Ohms law","text":"Circuit in series: $10\\,\\text{V}$ power supply, 2 resistors connected. Each resistor is $10\\,\\Omega$, total of $20\\,\\Omega$ resistance. If I put an ammeter at the end of the circuit, will the current be calculated using 20 Ohms or 10? Electrons flow through the first 10 Ohm resistor and are slowed down, yes? Then they flow through the second 10 Ohm resistor. They have passed through a total of 20 Ohms BUT at 10 Ohms per resistor. Will the current be the same as if it had passed through 1 20 Ohm resistor? That violates everything I understand."}
{"id":"127285","title":"Are there any known (closed form even if approximate) solutions to problems in relativistic elasticity?","text":"There are several useful known solutions to the EFE with relatively simple \/ trivial stress-energy-momentum tensor, such as the Schwarzschild solution. Despite the idealizations made therein they are enormously important and insightful. I am curious to know 1) Do any such solutions exist for the case of relativistic elastic solids? 2) If not, what are the principal obstacles in constructing solutions, even approximate (say linearized) ones? 3) If I understand correctly, various competing versions of a \"theory\" of relativistic elasticity have been around for several decades. Is there a reason why there are multiple theories? 4) Also, IIUC, matters of existence and uniqueness for the Cauchy problems arising from such theories have also been dealt with. I came across this paper by Beig and Schmidt, in which the authors deal with questions of local existence and uniqueness for a theory they describe as a \"relativistic version of classical elasticity\". Is this a generally accepted state of affairs or are there caveats? 5) There seem to be several disparate sources of information about this topic, is there a nice \"unified\" resource to get started? Thanks."}
{"id":"108018","title":"One problem on Classical Mechanics","text":"![enter image description here](http:\/\/i.stack.imgur.com\/L1h39.jpg)![I'm wondering why in this problem I can't apply newton second law? The mass m of the moving part will be the mass solved in the problem.](http:\/\/i.stack.imgur.com\/HXxjS.jpg) I'm wondering why in this problem I can't apply newton second law? The mass $m$ of the moving part will be the mass solved in the problem."}
{"id":"108014","title":"About photons and mirrors","text":"If a photon hits a 'perfect' mirror (with no environment interference) would the mirror move a bit?"}
{"id":"3372","title":"Non Classical mechanic answer to : length of time a thrown object spends in rest before falling down?","text":"When an object is thrown upwards, when it eventually comes to rest and starts falling, for how long is it stationary? What about an particle in electric field having an initial velocity towards it's same charge? That too would come to rest and reverse velocity, the question is for how long is it at full stop with rest? Classical physics gives a time of 0, but is that correct? Is it really at rest for 0 seconds? That answer seems a bit counter intuitive."}
{"id":"30260","title":"How to liquefy Hydrogen?","text":"I have got a science project and my teacher has recommended me to do \"Liquefying Hydrogen\". I have been continuously thinking about that but I have not come to a solution. Can anyone please tell me how to liquefy and store hydrogen??"}
{"id":"30261","title":"Analogy between magnetic bottle and Van Allen's radiation belt","text":"A magnetic bottle is an arrangement that permits to confine charged particles. Here you can find a review for charged particle rotating in a magnetic field and at the bottom of the page a description of the magnetic bottle: Motion of a charged particle in magnetic field Let's use this image since it's more clear ![enter image description here](http:\/\/i.stack.imgur.com\/y45wc.jpg) What results from this picture is that a particle inside a magnetic bottle makes a circular motion following magnetic field lines which have a sort of cylindrical symmetry as it's expressed by this simulation: Magnetic Bottle 1 So you can imagine to put the earth in the middle of the picture: in this way the two rings are to the two magnetic poles. In the case of a Van Allen's radiation belt motion simulation (Radiation Belt as a Magnetic Bottle), particles move around a magnetic field lines instead of moving around the earth. But from this simulation: Magnetic Bottle 2 you can see that a motion around a magnetic line is possible but i can't understand heuristically why. So the questions are:   1. how a particle can move around a magnetic line if there is closed to it another one? so how can a particle choose around which line move (losing in this way the cylindrical symmetry)?   2. why the two type of motions are in disagreement? Is the motion in the first picture just an approximation perhaps?"}
{"id":"30269","title":"Wind Turbine Impact","text":"OK, I'm not sure this is not a silly question, but here goes. I've often wondered what the impact is of wind turbines on the weather. My question is, since wind turbines transform wind energy into electricity, this means that the winds lose in power. I'm well aware that for the effect to be noticeable, you'd need a pretty large wind turbine park. But I wondered how large? How would the effect scale with the size of the park? Bonus question: what about tidal turbines? Most closely related questions: Is it possible that windmills can consume the wind? What does the quantification of causes and effect look like for clouds in offshore wind turbine wakes?"}
{"id":"4247","title":"Solving a Young Laplace equation for a meniscus against a flat plate","text":"This is more of a math question and one, furthermore, that I know the final answer to. What I am asking is more of a \"how do I get there\" question as this question was generated during a self study situation. So, for a flat plate with a meniscus on it at some contact angle the form is: a y''$\/(1+(y')^2)^{3\/2}$ + b g y = 0 where a is surface tension, b is density, and g is the usual 9.8 m$\/s^2$. Now, I know that when you \"A first integration, together with the boundary condition dy\/dx=y=0 as x goes to infinity yields: 1$\/(1+y'^2)^{1\/2}$=1-(bg\/2a)$y^2$ \" \\--From de Gennes on Menisci My problem is with the integration. How do I integrate something like the 1st equation? I find myself running in circles. Worse, I know that this type of integration is something I've run across numerous times, but I can't seem to find the methodology in the current stack of books. So I beg, can someone either tell me where to look to find the appropriate methodology to attack this problem or take me through the process? I would be grateful for as long as it would be useful for you. Thanks, Sam"}
{"id":"31131","title":"Why does Joule-Thomson expansion of a gas do no (external) work?","text":"I understand one of the benefits of the Hampson-Linde cycle is that there are no cold-side moving parts, but isn't one losing an awful lot of energy in the throttling process? There must be something basic I'm missing here, because it looks almost like this cycle destroys energy: pressure drops while volume increases, and temperature drops at the same time. The expansion is isentropic (How does that differ from an adiabatic expansion? I thought the condition that dS = 0 constrained a gas to a unique trajectory - an adiabat - along which it does work. But unless I'm mistaken, a Joule-Thomson expansion does no external work, despite not being an expansion into an evacuated space. Where did the work \"go\"? Maybe an example might illustrate. If I let air out of my car's tyres, is that a Joule-Thomson expansion? If so, doesn't the expansion do work in lifting the atmosphere by 20 attometers? If not, what _is_ an everydayish example?"}
{"id":"130276","title":"How fast can fire spread?","text":"You can build a home-made flamethrower by using a spray and a lighter: ![enter image description here](http:\/\/i.stack.imgur.com\/uxv6S.png) (from this YouTube video) But this can be dangerous if the fire gets in the can and the can explodes. So I guess the speed at which the particles come out of the box has to be high enough to prevent this. > **NOTE** : I don't want to do this at home, because I think it's too > dangerous. This is just a thought experiment! I guess the question \" **how fast can fire spread?** \" highly depends on the details (like: What kind of exothermic reaction is going on; how big is the pressure at which the spray exits the can; temperature around the can). I would like one theoretical answer (choose everything to make it as fast as possible \/ slow as possible) and a practical answer (make a reasonable assumption of what could go on with the lighter \/ spray stuff)."}
{"id":"110694","title":"Trying to solve 2D Toda Lattice Equation with Lax Pair Approach","text":"I am working on this Hamiltonian: $$ H = \\frac{p_1^2 + p_2^2}{2m} + e^{q_2-q_1} + e^{q_2} + e^{-q_1} -3 $$ Thank you for the hint that it is a modification of the Toda Lattice Equation. Let me sketch what I tried until now and why it is not working: Analogous to the mentioned publications I introduced $$b_n:= \\frac{1}{2}Exp{(\\frac{q_n-q_{n+1}}{2})}\\\\\\ a_n:= -\\frac{p_n}{2} $$ where it follows directly with $\\frac{\\partial H}{\\partial q_i}=-p_i$ and $\\frac{\\partial H}{\\partial p_i}=q_i$: $$\\dot{b_n} = (a_{n+1} - a_n)b_n \\\\\\ \\dot{a_n} = 2 (b_{n}^2 - b_{n-1}^2)$$ When now using the Lax Pair $L$,$B$: $$ L f_n = b_n f_{n+1} +b_{n-1} f_{n-1} + a_n f_{n}$$ $$ B f_n = b_n f_{n+1} - b_{n-1} f_{n-1} $$ it can be shown that $\\partial_t L=[B,L]$. My problem arises in defining the border conditions of my couple $q_1$ and $q_2$ in the 2d lattice above, since one needs to shift to the 3d representation $\\\\{b_0,b_1,b_2\\\\}$ in order to satisfy the periodic conditions (One mutual coordinate $q_3 = 0$ coupled to the others). Since it can be shown easily that $\\dot{\\lambda} = 0$ (where $\\lambda$ is an eigenvalue $Lv=\\lambda v$) the constants of motion reduce to the calculation of the eigenvalues. But in this case the eigenvalues of $L$ dont seem to simplify, in fact it doesnt seem to be a solution, which was my inital goal. In general this approach seems to be at overkill for the 2d problem since it solves the n-dimensional Toda lattice.   1. Anyone knows of an easier approach to the 2d problem?   2. The Matrix $L$ seems to yield the wrong solution: $$ L = \\begin{pmatrix} a_0 & b_0 & 0 \\\\\\ b_0 & a_1 & b_1\\\\\\ 0 & b_1 & a_2 \\end{pmatrix}$$ The Matrix does neigher solve $\\partial_t L = [B,L]$ (with $B=L_+ - L_-$) nor are eigenvalues constants of motion. Was has gone wrong?   3. Since the inverse scattering method can be applied here, I tried to get the scattering data, but actually I was not able to do the task. Any literature?"}
{"id":"72281","title":"Why do we only have complete particle generations?","text":"There are 3 generations of fermions in the standard model. I know that there is a theorem that states, that only complete generations are allowed. This means that there have to be quarks with three colors. With this, if we sum over all leptons and quarks in a generation the overall charge is zero. Why is it assumed that this must hold?"}
{"id":"91308","title":"Two spinor tensors and Maxwell's equations","text":"Let's have two symmetric (by the indices) spinor tensors $F_{ab}, F_{\\dot {a}\\dot {b}}$ and conditions $$ F_{ab}, \\partial^{\\dot {a} a}F_{ab} = 0, \\quad F_{\\dot {a}\\dot {b}}, \\partial^{\\dot {a}a}F_{\\dot {a}\\dot {b}} = 0, \\qquad (1) $$ $$ \\partial^{a \\dot {a}} = (\\tilde {\\sigma}_{\\mu})^{\\dot {a} a}\\partial^{\\mu}, \\quad (\\tilde {\\sigma}_{\\mu})^{\\dot {a} a} = \\varepsilon^{\\dot {a}\\dot {b}}\\varepsilon^{a b}(\\sigma_{\\mu})_{b \\dot {b}} = (\\sigma_{0}, -\\sigma_{i}). $$ by which these tensors may represent massless field with helicity $1, -1$ respectively. If our field is real, we must take the direct sum of these representations. It may be described through antisymmetrical 4-tensor rank 2. I know that there is the correspondence between an arbitrary antisymmetric tensor $F_{\\mu \\nu}$ and two symmetric spinor tensors $F_{ab} , F_{\\dot {a} \\dot {b}}$: $$ F_{\\mu \\nu} = \\frac{1}{2}(\\sigma_{\\mu \\nu})^{a b}F_{ab} - \\frac{1}{2}(\\tilde \\sigma_{\\mu \\nu})^{\\dot a \\dot b}F_{\\dot {a} \\dot {b}}, $$ $$ F_{ab} = (\\sigma^{\\mu \\nu})_{ab}F_{\\mu \\nu}, \\quad F_{\\dot {a} \\dot {b}} = (\\tilde {\\sigma}^{\\mu \\nu})_{\\dot {a}\\dot {b}}F_{\\mu \\nu}. \\qquad (2) $$ Then it obvious that I must substitute $(2)$ into $(1)$ and get Maxwell's equations. But I can't do it, because I have the problem with convolution of sigma-matrices. Can you help me?"}
{"id":"111847","title":"Noether's Theorem For Functionals of Several Variables","text":"My question is on using a form of the single variable Noether's theorem to remember the multiple variable version. Does Noether's theorem, for functionals of a single independent variable, just say that, because $\\mathcal{L}$ is invariant, we have $$\\mathcal{L}(x,y_i,y_i')dx = \\sum_{j=1}^n p_i d y_j - \\mathcal{H}dx = \\mathcal{L}(x^*,y_i^*,y_i'^*)dx^* = \\sum_{i=1}^n p_i d y_i^* - \\mathcal{H}dx^*?$$ It is usually stated by saying that $$\\sum_{i=1}^n \\frac{\\partial \\mathcal{L}}{\\partial ( \\tfrac{d y_i}{dx})} \\frac{\\partial y_i^*}{\\partial \\varepsilon} - \\left[\\sum_{j=1}^n \\frac{\\partial \\mathcal{L}}{\\partial ( \\tfrac{d y_j}{dx})} \\tfrac{\\partial y_j }{\\partial x} - \\mathcal{L}\\right]\\frac{\\partial x^*}{\\partial \\varepsilon}$$ is conserved, so is this equivalent to what I've written above (I've hopefully justified all of this below)?. (If there is any ambiguity in this question, I've offered a hopefully unnecessary explanation of the details below) The only problem with it I see is that it works fine for conservation of linear momentum, but it seems to show that conservation of energy and angular momentum are always zero, since $dx^* = 0$, unless I add a constant to the above, i.e. $$\\mathcal{L}(x,y_i,y_i')dx = \\mathcal{L}(x^*,y_i^*,y_i'^*)dx^* + C$$ which seems fine as adding constants to Lagrangians do not affect the EOM, so that's the only problem I see with it. I like the above expression, it's great for remembering Noether's theorem, it's more general than what almost every physics book presents as Noether's theorem (it's not just conservation of energy!), explains what Landau's additivity of integrals of motion comment means mathematically etc... **Can we generalize it to Lagrangian densities?** The statement of the multivariable Noether I know is that, for $$\\mathcal{L} = \\mathcal{L}(x_i,u_j,\\frac{\\partial u_j}{\\partial x_i})$$ we have that $$\\sum_{i=1}^n\\frac{\\partial}{\\partial x_i}\\left[\\sum_{j=1}^m \\frac{\\partial \\mathcal{L}}{\\partial (\\tfrac{\\partial u_j}{\\partial x_i})}\\left(\\frac{\\partial u_j^*}{\\partial \\varepsilon_k} - \\sum_{i=1}^n\\frac{\\partial u_j}{\\partial x_i}\\frac{\\partial x_i^*}{\\partial \\varepsilon _k}\\right) + \\mathcal{L}\\frac{\\partial x_i^*}{\\partial \\varepsilon _k}\\right] = 0$$ I can hardly remember this, and as I've indexed it I can't turn it into anything involving what I _think_ is the Hamiltonian for a functional of several independent variables $$\\mathcal{H} = \\sum_{j=1}^np_{ij}\\frac{\\partial u_j}{\\partial x_i} - \\mathcal{L} = \\sum_{j=1}^n \\frac{\\partial \\mathcal{L}}{\\partial (\\tfrac{\\partial u_j}{\\partial x_i})}\\frac{\\partial u_j}{\\partial x_i} - \\mathcal{L}$$ (I have no idea why Jackson adds a $g^{\\alpha \\beta}$) Can this be turned into something similar to my main equation, perhaps using $\\delta_{ij}$s or $g_{\\mu \\nu}$s or something? An attempt: $$\\sum_{i=1}^n\\frac{\\partial}{\\partial x_i}\\left[\\sum_{j=1}^m \\frac{\\partial \\mathcal{L}}{\\partial (\\tfrac{\\partial u_j}{\\partial x_i})}\\left(\\frac{\\partial u_j^*}{\\partial \\varepsilon_k} - \\sum_{i=1}^n\\frac{\\partial u_j}{\\partial x_i}\\frac{\\partial x_i^*}{\\partial \\varepsilon _k}\\right) + \\mathcal{L}\\frac{\\partial x_i^*}{\\partial \\varepsilon _k}\\right] = 0$$ $$\\sum_{i=1}^n\\frac{\\partial}{\\partial x_i}\\left[\\sum_{j=1}^m \\frac{\\partial \\mathcal{L}}{\\partial (\\tfrac{\\partial u_j}{\\partial x_i})}\\left(d u_j^* - \\sum_{i=1}^n\\frac{\\partial u_j}{\\partial x_i}d x_i^* \\right) + \\mathcal{L}d x_i^* \\right] = 0$$ $$\\sum_{i=1}^n\\frac{\\partial}{\\partial x_i}\\left[\\sum_{j=1}^m \\frac{\\partial \\mathcal{L}}{\\partial (\\tfrac{\\partial u_j}{\\partial x_i})}d u_j^* - \\sum_{k=1}^m \\frac{\\partial \\mathcal{L}}{\\partial (\\tfrac{\\partial u_k}{\\partial x_i})}\\sum_{l=1}^n\\frac{\\partial u_k}{\\partial x_l}d x_l^* + \\mathcal{L}d x_i^* \\right] = 0$$ $$\\sum_{i=1}^n\\frac{\\partial}{\\partial x_i}\\left[\\sum_{j=1}^m \\frac{\\partial \\mathcal{L}}{\\partial (\\tfrac{\\partial u_j}{\\partial x_i})}d u_j^* - \\sum_{k=1}^m \\frac{\\partial \\mathcal{L}}{\\partial (\\tfrac{\\partial u_k}{\\partial x_i})}\\sum_{l=1}^n\\frac{\\partial u_k}{\\partial x_l}d x_l^* + \\sum_{l=1}^n\\delta^l_i \\mathcal{L}d x_l^* \\right] = 0$$ $$\\sum_{i=1}^n\\frac{\\partial}{\\partial x_i}\\left[\\sum_{j=1}^m \\frac{\\partial \\mathcal{L}}{\\partial (\\tfrac{\\partial u_j}{\\partial x_i})}d u_j^* - \\sum_{l=1}^n (\\sum_{k=1}^m \\frac{\\partial \\mathcal{L}}{\\partial (\\tfrac{\\partial u_k}{\\partial x_i})}\\frac{\\partial u_k}{\\partial x_l} - \\delta^l_i \\mathcal{L})d x_l^* \\right] = 0$$ $$\\sum_{i=1}^n\\frac{\\partial}{\\partial x_i}\\left[\\sum_{j=1}^m \\frac{\\partial \\mathcal{L}}{\\partial (\\tfrac{\\partial u_j}{\\partial x_i})}d u_j^* - \\sum_{l=1}^n \\mathcal{H}d x_l^* \\right] = 0$$ $$\\sum_{i=1}^n\\frac{\\partial}{\\partial x_i}\\left[\\sum_{j=1}^m p_{ij}d u_j^* - \\sum_{l=1}^n \\mathcal{H}d x_l^* \\right] = 0.$$ I don't know if this is right. References:   1. Gelfand, Calculus of Variations - Ch. 3, 4 & 7   2. Jackson, Classical Electrodynamics - Ch. 12 Sec 10 * * * Noether's theorem for functionals of a single independent variables states that, given a continuous one-parameter family of infinitesimal transformations of the form $$T(x,y_i,\\varepsilon) = (x^*(x,y_i,\\varepsilon),y_i^* (x,y_i,\\varepsilon))= (x^*,y_i^*)= (x+\\frac{\\partial x^*}{\\partial \\varepsilon}\\varepsilon,y_i+\\frac{\\partial y_i^*}{\\partial \\varepsilon}\\varepsilon),$$ if a Lagrangian $$\\mathcal{L} = \\mathcal{L}(x,y_i,\\frac{dy_i}{dx})$$ is invariant $$\\mathcal{L}(x,y_i,\\frac{dy_i}{dx}) = \\mathcal{L}(x^*,y_i^*,\\frac{dy_i^*}{dx})$$ under $T$ then the scary-looking quantity $$\\sum_{i=1}^n \\frac{\\partial \\mathcal{L}}{\\partial ( \\tfrac{d y_i}{dx})} \\frac{\\partial y_i^*}{\\partial \\varepsilon} - [\\sum_{j=1}^n \\frac{\\partial \\mathcal{L}}{\\partial ( \\tfrac{d y_j}{dx})} \\tfrac{\\partial y_j }{\\partial x} - \\mathcal{L}]\\frac{\\partial x^*}{\\partial \\varepsilon}$$ is conserved (more precisely, you do it in terms of the functional but that is irrelevant for this discussion). Examining this quantity for two seconds let's us notice we can write this conserved quantity as $$\\sum_{i=1}^n \\frac{\\partial \\mathcal{L}}{\\partial ( \\tfrac{d y_i}{dx})} \\frac{\\partial y_i^*}{\\partial x} - \\mathcal{H}\\frac{\\partial x^*}{\\partial \\varepsilon}$$ in terms of the Hamiltonian $$\\mathcal{H} = \\sum_{j=1}^n \\frac{\\partial \\mathcal{L}}{\\partial ( \\tfrac{d y_j}{dx})} \\tfrac{\\partial y_j }{\\partial x} - \\mathcal{L} = \\sum_{j=1}^n p_i \\tfrac{\\partial y_j }{\\partial x} - \\mathcal{L} $$ or, taking away the parameter, as $$\\sum_{i=1}^n p_i d y_i^* - \\mathcal{H}dx^*$$ Notice the similarity between the quantity in Noether's theorem and this: $$\\mathcal{H} = \\sum_{j=1}^n p_i \\tfrac{\\partial y_j }{\\partial x} - \\mathcal{L} \\rightarrow \\mathcal{L} dx = \\sum_{j=1}^n p_i d y_j - \\mathcal{H}dx$$ Does Noether's theorem just say that, because $\\mathcal{L}$ is invariant we have, solving for $\\mathcal{L}$, that $$\\mathcal{L}(x,y_i,y_i')dx = \\sum_{j=1}^n p_i d y_j - \\mathcal{H}dx = \\mathcal{L}(x^*,y_i^*,y_i'^*)dx^* = \\sum_{i=1}^n p_i d y_i^* - \\mathcal{H}dx^*?$$"}
{"id":"111841","title":"A rigid rotating rod that breaks in two pieces","text":"Suppose we have a rigid rod of lenght $L$ and homegenous mass density. One of its extreme points, say $P$, is fixed so that the rod can rotate around the axis passing in it. Initially the rod is held horizontally, then it is left and when it reaches the vertical position it breaks, without generating any impulsive force, in two equal pieces. I'm wondering about how to find the angular velocity of the two fragments immediately after the rod breaks. I tried but without succes, here is what I did. I suppose that energy is conserved, that is: $$ E_{rod}= E_{1} + E_{2} $$ where $1$ labels the upper fragment and $2$ the lower fragment of the rod. We have: $$ E_{rod}=\\frac{1}{2}I_{rod}\\omega_{rod}^{2} $$ $$ E_{1}= \\frac{1}{2}I_{1}\\omega_{1}^{2} $$ $$ E_{2}=\\frac{1}{2}I_{2}\\omega_{2}^{2} + \\frac{1}{2}\\frac{m}{2}V^{2}_{2} $$ where $V_{2}$ is the velocity of the center of mass of the lower fragment and the moment of inertia with respect to P are: $$ I_{rod}=\\frac{mL^{2}}{3} $$ $$ I_{1}=\\frac{mL^{2}}{24} $$ $$ I_{2}=\\frac{7mL^{2}}{24} $$ **EDIT** According to V.Moretti's comments we know that energy and angular momentum are conserved, thus we have the system: $$ \\left\\\\{\\begin{array}{c} E_{rod}= E_{1} + E_{2} \\\\\\ \\vec{L}_{rod}=\\vec{L}_{1} + \\vec{L}_{2}\\end{array}\\right. $$ I think it is: $$ \\vec{L}_{tot}=I_{rod}\\vec{\\omega}_{rod} $$ $$ \\vec{L}_{1}=I_{1}\\vec{\\omega}_{1} $$ $$ \\vec{L}_{2}=I_{2}\\vec{\\omega}_{2} + \\frac{m}{2}\\vec{r}\\wedge\\vec{V}_{2} $$ Moreover I think that $\\vec{V}_{2}$ can be simply calculated considering the center of mass of the lower fragment as a point of the rod and then use the well-known relation between angular velocity and velocity: $$ \\vec{V}_{2}=\\vec{\\omega}_{rod}\\wedge \\vec{r} $$ and thus, in a right-handed coordinate system with the z-axis exiting the screen, it is: $$ \\vec{V}_{2}=(-\\omega_{rod}\\hat{z})\\wedge(-\\frac{3}{4}L\\hat{y})=-\\frac{3}{4}L\\omega_{rod}\\hat{x} $$ and: $$ \\vec{r}\\wedge\\vec{V}_{2}=(-\\frac{3}{4}L\\hat{y})\\wedge(-\\frac{3}{4}L\\omega_{rod}\\hat{x})=-\\frac{9}{16}L^{2}\\omega_{rod}\\hat{z} $$ Now there is nothing that indicates motion in the x-z and y-z planes, thus the conservation of the angular momentum should be a scalar equation, namely: $$ L_{tot}=I_{rod}\\omega_{rod}=I_{1}\\omega_{1} + I_{2}\\omega_{2} + \\frac{m}{2}\\vec{r}\\wedge\\vec{V}_{2}= $$ $$ =I_{1}\\omega_{1} + I_{2}\\omega_{2} -\\frac{9}{32}mL^{2}\\omega_{rod}=L_{1} + L_{2} $$ Now I am trying to solve the system: $$ \\left\\\\{\\begin{array}{c} \\frac{1}{2}I_{rod}\\omega_{rod}^{2}= \\frac{1}{2}I_{1}\\omega_{1}^{2} + \\frac{1}{2}I_{2}\\omega_{2}^{2} + \\frac{9mL^{2}\\omega_{rod}^{2}}{64}\\\\\\ I_{rod}\\omega_{rod}=I_{1}\\omega_{1} + I_{2}\\omega_{2} -\\frac{9}{32}mL^{2}\\omega_{rod}\\end{array}\\right. $$ but, quite surprisingly, I am not having good results. What am I doing wrong?"}
{"id":"35304","title":"Tensor product decomposition of SU(2)","text":"I have a rather trivial question. I am looking for the decomposition of $1\/2\\otimes 1\/2\\otimes 1\/2$. It should give, $0,1\/2$ and $3\/2$. I thought one must get as the overall dimension of this space 8, but counting, I just get 7. Does one have 2 singlets?"}
{"id":"23190","title":"Where is the flaw in deriving Gauss's law in its differential form?","text":"From the divergence theorem for any vector field E, $\\displaystyle\\oint E\\cdot da=\\int (\\nabla\\cdot E) ~d\\tau$ and from Gauss's law $\\displaystyle\\oint E\\cdot da=\\frac{Q_{enclosed}}{\\epsilon_0}=\\int \\frac{\\rho}{\\epsilon_0}~d\\tau$ Hence, $\\displaystyle\\int\\frac{\\rho}{\\epsilon_0}d\\tau=\\int (\\nabla\\cdot E)~d\\tau$ Textbooks conclude from the last equation that $\\displaystyle \\nabla\\cdot E=\\frac{\\rho}{\\epsilon_0}$ My question is how can we conclude that the integrands are the same? Because I can think of the following counter example, assume $\\displaystyle \\int_{-a}^a f(x)~dx=\\displaystyle \\int_{-a}^a [f(x)+g(x)]~dx$ where $g(x)$ is an odd function. Obviously the 2 integrals are equal but we cannot conclude that $f(x)$ is equal to $f(x)+g(x)$ so where is the flaw?"}
{"id":"23191","title":"The Double Slit Experiment and the changing of electron behaviour","text":"As you will all know, when one tries to detect which slit an electron has gone through with close up observation, it changes from behaving like a wave and producing an interference pattern to behaving like a particle and producing two lines on the screen. Is there a full explanation as to why the electron changes its behaviour when observed?"}
{"id":"79127","title":"Coordinate transformations of the metric tensor","text":"Let's have metric (it describes the space-time of uniformly accelerating observer in Minkowski space-time) $$ ds^2 = v^2du^2 - dv^2. \\qquad (.0) $$ I need to find expressions for $u = f(x, t), v = g(x, t)$, which leads to $$ ds^2 = dt^2 - dx^2, \\quad u = f(x, t), \\quad v = g(x, t). $$ How to get $f, g$? My attempt. I substituted $u = f(x, t), v = g(x, t)$ in $ds^2 = u^2dv^2 - du^2$ and then, by equating it to $dt^2 - dx^2$, got system of PDE: $$ g^2 (\\partial_{x}f)(\\partial_{t}f) - (\\partial_{x}g) (\\partial_{t}g) = 0, \\quad g^{2}(\\partial_{x}f)^{2} - (\\partial_{x}g)^{2} = -1 , \\quad g^{2}(\\partial_{t}f)^{2} - (\\partial_{t}g)^{2} = 1 . $$ It may be \"simplified\" to $$ (\\partial_{x}g)^{2} - (\\partial_{t}g)^{2} = 1 , \\quad g^{2}\\left((\\partial_{t}f)^{2} - (\\partial_{x}f)^{2}\\right) = 1. \\qquad (.1) $$ How to solve it? Or maybe there is more simple method to get an expressions for $f, g$? **Edit**. I got an equation $$ \\partial_{tt}f - \\partial_{xx}f = 0 . \\qquad (.2) $$ But I can't get some conditions, which can help me to choose some partial solution. By the other words, I can't use $(.1)$ in the right way. So, what conditions may I use to solve $(.2)$? **Edit**. I got the solution by getting a solution for transition from Minkowski space-time to original space-time which metric is given by $(.0)$. I used method of separation of variables."}
{"id":"87817","title":"Fundamental invariants of the electromagnetic field","text":"It is a standard exercise in relativistic electrodynamics to show that the electromagnetic field tensor $F_{\\mu\\nu}$, whose components equal the electric $E^i=cF^{i0}$ and magnetic $B_i=-\\frac12\\epsilon_{ijk}F^{jk}$ fields in the taken frame of reference, has two Lorentz invariant quantities, $$\\frac12F^{\\mu\\nu}F_{\\mu\\nu}=\\mathbf{B}^2-\\mathbf{E}^2$$ and $$\\frac14F_{\\mu\\nu}\\ {}^\\ast F^{\\mu\\nu}=\\frac14\\epsilon^{\\mu\\nu\\alpha\\beta }F_{\\mu\\nu}F_{\\alpha\\beta}=\\mathbf{B}\\cdot\\mathbf{E}.$$ There is, however, a further Wikipedia article which states that these two quantities are _fundamental_ , in the sense that any other invariant of this tensor must be a function of these two. While I find this plausible, I have never seen a proof of this fact, and it is absent from e.g. Jackson. **Is there a simple proof of this fact?** I'm particularly interested in higher- order invariants, but I would also like answers to include a proof that these are the only two bilinears. To be more precise, I would like to see a proof that > Any function $I:F_{\\mu\\nu}\\mapsto I(F)\\in\\mathbb{R}$ that takes > electromagnetic field tensors to real scalars and which is Lorentz invariant > (i.e. $I(\\Lambda_{\\mu}^\\alpha \\Lambda_{\\nu}^\\beta F_{\\alpha\\beta})= > I(F_{\\mu\\nu})$ for all Lorentz transformations) must be a function > $I(F)=I'(F^{\\mu\\nu}F_{\\mu\\nu},F_{\\mu\\nu}\\ {}^\\ast F^{\\mu\\nu})$ of the two > fundamental invariants described above. If there are multiple ways to arrive at this result, I would also appreciate comments on how they relate to each other."}
{"id":"73885","title":"Eigenvalues of the square of Pauli-Lubanski operator","text":"Let's have Pauli-Lunanski 4-operator: $$ \\hat {W}^{\\nu} = \\frac{1}{2}\\varepsilon^{\\nu \\alpha \\beta \\gamma}\\hat {J}_{\\alpha \\beta}\\hat {P}_{\\gamma}, $$ which easy transforms to $$ \\hat {W}^{\\nu} = \\frac{1}{2}\\varepsilon^{\\nu \\alpha \\beta \\gamma}\\hat {S}_{\\alpha \\beta}\\hat {P}_{\\gamma}, $$ where $\\hat {S}_{\\alpha \\beta }$ is the spin-tensor. For the square of this operator it's easy to get $$ \\hat {W}_{\\nu}\\hat {W}^{\\nu} = \\hat {S}_{\\alpha \\beta}\\hat {P}^{\\beta}\\hat {S}^{\\alpha \\gamma}\\hat {P}_{\\gamma} - \\frac{1}{2}\\hat {P}_{\\delta}\\hat {P}^{\\delta}\\hat {S}_{\\rho \\varepsilon}\\hat {S}^{\\rho \\varepsilon}. \\qquad (.1) $$ In the case of the rest frame $\\hat {P}_{\\gamma}\\psi = 0$, so $$ \\hat {W}_{\\nu}\\hat {W}^{\\nu} \\psi = -\\frac{1}{2}\\hat {P}_{\\delta}\\hat {P}^{\\delta}\\hat {S}_{\\rho \\varepsilon}\\hat {S}^{\\rho \\varepsilon}\\psi = -\\frac{m^{2}}{2}\\hat {S}_{\\rho \\varepsilon}\\hat {S}^{\\rho \\varepsilon}\\psi . $$ In the special case of 4-tensor I know, that $$ \\hat {S}_{\\alpha \\beta}\\hat {S}^{\\alpha \\beta} = 2(\\hat {S}_{1}^{2} - \\hat {S}_{2}^{2}), $$ where $\\hat {S}_{1}$ is the spin-vector 3-operator. But what is $\\hat {S}_{2}$? And what it do with eigenvalue? Without it $\\hat {W}_{\\nu}\\hat {W}^{\\nu} \\psi = -m^{2}s(s + 1) \\psi $, as it must be. **Addition**. Maybe, I understand my mistake. I ignored the first term in $(.1)$. At the rest frame in a case of diagonal operators of $\\hat {P}_{i}$ it has structure $\\hat {P}_{i} = (\\hat {P}_{0}, 0, 0, 0) = (m, 0, 0, 0)$, so $$ \\hat {W}_{\\nu}\\hat {W}^{\\nu}\\psi_{\\mathbf p = 0} = \\left(\\hat {S}_{\\alpha \\beta}\\hat {P}^{\\beta}\\hat {S}^{\\alpha \\gamma}\\hat {P}_{\\gamma} - \\frac{1}{2}\\hat {P}_{\\delta}\\hat {P}^{\\delta}\\hat {S}_{\\rho \\varepsilon}\\hat {S}^{\\rho \\varepsilon}\\right)\\psi = \\left(\\hat {S}_{\\alpha 0}\\hat {P}^{0}\\hat {S}^{\\alpha 0}\\hat {P}_{0} - \\frac{1}{2}\\hat {P}_{0}\\hat {P}^{0}\\hat {S}_{\\rho \\varepsilon}\\hat {S}^{\\rho \\varepsilon}\\right) \\psi_{\\mathbf p = 0} = $$ $$ =m^{2}\\left( - \\hat {\\mathbf S}_{2}^{2} - \\frac{1}{2}2 \\left( \\hat {\\mathbf S}_{1}^{2} - \\hat {\\mathbf S}_{2}^{2}\\right)\\right) \\psi_{\\mathbf p = 0} = -m^{2}s(s + 1)\\psi_{\\mathbf p = 0 }. $$ But, anyway, what is $\\hat {\\mathbf S}_{2}$?"}
{"id":"76965","title":"Fermion vs. Bosons and particle vs. wave: is there a link?","text":"I'm puzzled since several years on this basic aspect of quantum mechanics. Quantum theory is supposed to describe particle-wave symmetry of our world. It also describes our universe in term of bosons and fermions. Now the experimental facts:   * It seems rather easy to observe wave particle of bosons (think about photon of course) whereas they particle behaviour is pretty complicated to observe. You need stringent experimental condition to observe the particle behaviour of photons for instance.   * It seems rather easy to observe particle behaviour of fermions (think about electron of course) whereas they wave behaviour is pretty complicated to observe. You need stringent experimental condition to observe the wave behaviour of electrons for instance. I was just wondering if some people figure out the wave-particle behaviour in terms of fermion vs. boson classification, or if the above experimental facts are just pure coincidence. I admit it would require a less stringent dichotomy boson\/fermion, perhaps like in supersymmetry theories, that I do not know enough. Any comment are welcome."}
{"id":"72421","title":"Connection between $\\Delta x \\Delta p \\geq \\frac{\\hbar}{2}$ and $\\Delta E \\Delta t \\geq \\frac{\\hbar}{2}$","text":"Is there a way to derive second equation from the first one? I mean is there a connection between those two uncertainty relations? \\begin{align} \\Delta x \\Delta p &\\geq \\frac{\\hbar}{2}\\\\\\ \\Delta E \\Delta t &\\geq \\frac{\\hbar}{2} \\end{align}"}
{"id":"23445","title":"How did the inflaton field \"add\" energy to the universe?","text":"How did inflation add energy to the universe? What mechanism did this occur by? In other words, where did that energy come from? Was it due to the quantum fluctuation (or that scalar field rolling down a potential energy hill) to a lower, more stable energy state? This energy then manifested as exponential expansion, and then switched from expansion into heating (by filling up the universe with hot quark-gluon plasma)? Do we know the mechanism for this expansion->reheating switch?"}
{"id":"109055","title":"Kronecker delta in inertial tensor","text":"I feel confused in (11.9) how does the book prove the following identity: $$\\sum\\limits_{i} w_{i}x_{\\alpha,i} \\sum\\limits_{j} w_{j}x_{\\alpha,j} = \\sum\\limits_{i,j}w_{i}w_{j}x_{\\alpha,i}x_{\\alpha,j}$$ ![enter image description here](http:\/\/i.stack.imgur.com\/w8u1I.jpg) I tried to derive it myself: $$\\sum\\limits_{i} w_{i}x_{\\alpha,i} \\sum\\limits_{j} w_{j}x_{\\alpha,j} = \\sum\\limits_{i}\\sum\\limits_{k}w_{k}\\delta_{ik}x_{\\alpha,i} \\sum\\limits_{j}\\sum\\limits_{m}w_{m}\\delta_{jm}x_{\\alpha,j} $$ Exactly from here, I got stuck, but I'm still not very much convincing about the result that the book got, becasuse it does not contain kroncker delta function for the part I tried to derive. Can anyone give me a mathematical proof?"}
{"id":"92656","title":"Does the Earth rotate vertically too?","text":"We know that the earth has seen several Ice Ages. And we know the Earth's magnetic poles seem to flip once in a while. Could all of this be explained by a once every 250 Million Years or so the Earth actually rotates head over heels, so to speak?"}
{"id":"92659","title":"How can the reduction postulate be removed with the other postulates of QM still leading to correct predictions?","text":"In the axiomatic presentation of QM, I've seen it stated many times that the reduction postulate is not needed and\/or incorrect, and could be gotten rid of. However, without the reduction postulate, wouldn't tests of QM simply yield wrong results? If I measure a system that is in some linear superposition $\\psi=\\sum c_i |e_i\\rangle$ of eigenstates of my observable $O$, the measurement postulate says I will get one of the eigenvalues as a result, with probability $|c_i|^2$. But now, since I no longer have the reduction postulate, I will calculate that as a result of the interaction with my measurement apparatus $\\psi$ evolved linearly and unitarily into $\\psi'$, and therefore still consists of (possibly a different) linear superposition $\\sum c_i' |e_i\\rangle$ of eigenstates of $O$. Therefore, again applying the measurement postulate, I calculate that I will now get $|e_i\\rangle$'s eigenvalue with probability $|c_i'|^2$. After a sequence of $n$ such measurements, close together in time, the probability that I will get some particular $|e_j\\rangle$'s eigenvalue in all the $n$ measurements is $(|c_j|^2 * |c_j'|^2 * ... * |c_j^{(n)}|^2) << 1$. However, we know from experience that making $n$ close in time measurements of the same observable will yield the same value with a probability of $1$ (or very close to $1$). So I don't see how the reduction postulate can just be dropped without either QM giving incorrect predictions, or without modifying some other postulates somehow?"}
{"id":"113184","title":"Higgs Boson and its relation to the vacuum","text":"The Higgs boson and the electroweak theory used symmetry breaking from condensed matter physics as its inspiration. The BCS theory of superconductivity is one such condensed matter symmetry breaking theory. I believe this theory uses some kind of model for the underlying atomic structure of the metal. In a similar manner the vacuum works in the same way with particles popping in and out with charges all the time. Is there a relation between the vacuum and the atomic structure of condensed matter. If so, could one define the vacuum with particular properties to match the Higgs potential? I know the Higgs potential is a vaccuum, but can properties be assigned to space itself and the fluctuating charged particles that pop in and out that are analogous to the charged atoms in a condensed matter metal and produce the Higgs fields this way? Maybe the vacuum is discrete like a metal lattice instead of continuous."}
{"id":"6153","title":"Quantum Hall effect for dummies","text":"In the past few days I've become increasingly intrigued by the QHE, mainly thanks to very interesting questions and answers that have appeared here. Unfortunately, I am as of yet very confused by all the (seemingly disparate) stuff I learned. First, here are some random points that I've been able to gather   1. I(nteger)QHE occurs due to the presence of Landau levels   2. IQHE is an embodiment of topological order and the states are characterized by the Chern number that tells us about topologically inequivalent Hamiltonians defined on the Brillouin zone   3. IQHE requires negligible electron-electron interations and so is dependent on the presence of impurities that shield from Coulomb force   4. F(ractional)QHE occurs because of formation of anyons. In this case Coulomb interaction can't be neglected but it turns out an effective non-interacting description emerges with particles obeying parastatistics and having fractional charge   5. FQHE has again something to do with topology, TQFT, Chern-Simons theory, braiding groups and lots of other stuff   6. FQHE has something to do with hierarchy states So, here are the questions >   * Most importantly, do these points make sense? Please correct any > mistakes I made and\/or fill in other important observations >   * How do explanations 1. and 2. of IQHE come together? Landau quantization > only talks about electron states while topological picture doesn't mention > them at all (they should be replaced by global topological states that are > stable w.r.t. perturbations) >   * How do explanations 4., 5. and 6. relate together >   * Is there any accessible introductory literature into these matters? >   * Do IQHE and FQHE have anything (besides last three letters) in common so > that e.g. IQHE can be treated as a special case? My understanding (based on > 3.) is that this is not the case but several points hint into opposite > direction. That's also why I ask about both QHE in a single question >"}
{"id":"128589","title":"Is Newtonian physics invariant under fast-forward?","text":"I am aware of how the laws of physics are invariant with a time reversal transformation, so that (ignoring entropy) a film of billiard balls running backwards cannot be differentiated from one running forwards. What about a film of billiard balls running in fast-forward motion? Ignoring relativistic effects, I think that there would be no way to tell the playback speed. OK, then imagine that one of the billiards were made out of delicate, thin glass, and that in the regular play-back film the motions are slow enough that it does not break. Playing the film in fast-forward motion, the delicate ball still would not break despite being hit by very fast-moving ones. Does this mean that Newtonian physics is not invariant under fast-forwarding? Is there a way of expressing this mathematically (analogous to saying t goes to -t for time reversal)? And finally, since the glass ball does not break, is this violation of intuition an illusion having to do with entropy, like with running a film in reverse, or is there some kind of modification that needs be made to the forces of the molecular bonds under fast-forward motion to prevent the glass from shattering?"}
{"id":"128586","title":"From where comes the raindrop","text":"First off all, I've no knowledge of how all this works. I've only got some common sense of \"how things work\"... But I've been wondering this for quite a long time now, so maybe you guys can help me. The question is rather simple: Where does the raindrop come from? And with that I mean: The raindrop that just hit the ground, where did that come from? From the clouds that are right above me, in front of me, or back of me? So let's say that we've got these 3 possibilities, which one is the most likely one? The first one, where the rain is traveling with the same speed horizontal as the cloud it came from: (so it came from the cloud directly above me) ![Raindrop 1](http:\/\/i.stack.imgur.com\/FLkOU.jpg) The second one, where the rain is traveling faster than the clouds (due to wind?). So that the rain came from the cloud that yet has to pass me. ![Raindrop 2](http:\/\/i.stack.imgur.com\/oHvYk.jpg) The third one, (And i do think that this is the right one) is that the rain is falling slower (horizontally seen) than the cloud (because the cloud is lighter). So that the rain that hits the ground came from the cloud that already past me. ![Raindrop 3](http:\/\/i.stack.imgur.com\/0U8uE.jpg) An another possibility is that the rain is going left and right, (keeping it 2D for simplicity) because of the different wind regions (in the high clouds, the wind goes from left to right, but lower it goes from right to left). Is there even an \"right\" answer for this? Or are there too many variables that could change everything? And could you please explain it as simple as possible? I've got zero knowledge in this area... Thank you in advantage! -Also, i don't know if this is the right stackexchange site to ask..."}
{"id":"89580","title":"How long does it take to scan a typical scanning electron microscope image?","text":"I suppose the answer depends on resolution and imaging area, but can you provide some ball park measures of imaging times with an SEM?"}
{"id":"38435","title":"Is $w=mg$ the right way to calculate mass?","text":"How can you find the mass of an object by using a weight and a gravitational acceleration from another planet then earth? I was given $5.8\\text{ N}$ and an acceleration of $4.0\\ \\mathrm{m\/s^2}$. The answer I keep getting is $1.45\\text{ kg}$, but the real answer is supposed to be $.58\\text{ kg}$. I'm using the formula $w=mg$. Am I doing something wrong?"}
{"id":"75614","title":"Factor of two differences for free field Green's functions in conformal field theory","text":"I have a question about the expressions for free field Green's functions in conformal field theory. It comes from three origins 1) In Polchinski's string theory volume I p36, it is given > $$ \\frac{1}{\\pi \\alpha'} \\partial_z \\partial_{\\bar{z}} X^{\\mu}(z,\\bar{z}) > X^{\\nu} (z',\\bar{z}') = -\\eta^{\\mu \\nu} \\delta^2 (z-\\bar{z}', > \\bar{z}-\\bar{z}') \\tag{2.1.20} $$ 2) In David Tong's string theory lecture note, p 186 (in the number at the bottom) or p 193 (counting by acrobat reader), > $$\\langle Y^a(z,\\bar{z}) Y^b(\\omega,\\bar{\\omega}) \\rangle = G^{ab} > (z,\\bar{z}; \\omega,\\bar{\\omega} )$$ $$ \\partial \\bar{\\partial } G^{ab} > (z,\\bar{z}) = - 2 \\pi \\delta^{ab} \\delta (z,\\bar{z}) \\tag{7.33} $$ here $\\alpha'$ has been scaled by $X^a = \\bar{x}^a + \\sqrt{\\alpha'} Y^a$ in p 185\/192. Why there is a factor of 2 difference in the RHS of (2.1.20) (Polchinski) and (7.33) (Tong)? In page 77, it is given > $$ \\langle \\partial^2 X(\\sigma) X(\\sigma') \\rangle = - 2\\pi \\alpha' > \\delta(\\sigma-\\sigma') \\tag{4.20}$$ I am not sure what is the definition of $\\partial^2$. Since $\\partial \\bar{\\partial}=\\frac{1}{4} (\\partial_1^2 + \\partial_2^2) $ and $\\delta^2(z,\\bar{z})=\\frac{1}{2} \\delta(\\sigma^1) \\delta(\\sigma^2)$ (Polchinski p33),the factor at RHS of Eq. (4.20) in David Tong's lecture note seems to agree with Polchinski (2.1.20), but not Tong's (7.33) In addition, David Tong's lecture is related to > We'll be fairly explicit here, but if you want to see more details then the > best place to look is the original paper by Abouelsaood, Callan, Nappi and > Yost, \\Open Strings in Background Gauge Fields\", Nucl. Phys. B280 (1987) 599 3) In the original paper, it is given > $$\\frac{1}{2\\pi\\alpha'} \\square G(z,z')= - \\delta(z-z') \\tag{2.7}$$ where $\\square=\\partial_{\\tau}^2 + \\partial_{\\sigma}^2$. Again, a factor of two difference between (2.7) and (2.1.20) of Polchinski. In Eq. (2.1) of that paper states > $$ S= \\frac{1}{2\\pi \\alpha'} \\left[ \\frac{1}{2} \\int_{M^2} d^2 z \\partial^a > X_{\\mu} \\partial_a X^{\\mu} + i \\int_{\\partial M} d \\tau A_{\\mu} > \\partial_{\\tau} X^{\\nu} \\right] $$ In Polchinski's string theory volume I, p32, it is given > $$ S = \\frac{1}{4\\pi \\alpha'} \\int d^2 \\sigma \\left( \\partial_1 X^{\\mu} > \\partial_1 X_{\\mu} + \\partial_2 X^{\\mu} \\partial_2 X_{\\mu} \\right) > \\tag{2.1.1}$$ Since $d^2 z = 2 d \\sigma^1 d \\sigma^2 \\tag{2.1.7}$ , is that the origin of factor of 2 difference? Because combininig Eqs.(2.1.1) and (2.1.7) leads to $$ S = \\frac{1}{2\\pi \\alpha'} \\int d^2 z \\left( \\partial_1 X^{\\mu} \\partial_1 X_{\\mu} + \\partial_2 X^{\\mu} \\partial_2 X_{\\mu} \\right) $$"}
{"id":"75613","title":"Space translation of operators, states, and particle densities","text":"In Sidney Coleman's Lectures he talked about space translations such that $$\\tag{1} e^{ia P}\\rho(x) e^{-ia P} ~=~ \\rho(x-a),$$ but when I expanded the exponentials and used the commutation relation of $P=-i\\frac{d}{dx}$ and $x$, I got $$\\tag{2} e^{ia P}\\rho(x) e^{-ia P} ~=~ e^{ia [P,~\\cdot~]}\\rho(x) ~=~e^{a\\frac{d}{dx}}\\rho(x)~=~\\rho(x+a)$$ with a plus instead of $\\rho(x-a)$ with a minus as Coleman (1). This was on Coleman's lectures on QFT (1975-76), Lecture 3, about 12:20 minutes into the lecture. Does anyone see where I may have gone wrong? References:   1. Coleman's lectures on QFT (1975-76), Lecture 3, about 12:20 minutes into the lecture.   2. S. Coleman, _Notes from Sidney Coleman's Physics 253a,_ arXiv:1110.5013, p. 23."}
{"id":"134189","title":"Why does the amount of dark matter increase the further away from the galaxy's center?","text":"Why isn't the dark matter just randomly distributed? How does it know where to go?"}
{"id":"134181","title":"Derivative chain rule in a triangle, confusing but interesting problem","text":"I asked the question in math.stackexchange. But I think it is better to ask here again. I am new to these sites. Please forgive me if it is not polite. http:\/\/math.stackexchange.com\/q\/921001 You can reply any poster. I paste it again here. ![variables in triangle](http:\/\/i.stack.imgur.com\/RmuHH.png) Refer to the above figure. Assuming the length of the 3 edges of triangle are $r_0,z_0,\\xi_0$. And we have $\\xi=\\sqrt{r^2+z^2}$ (Eqn.1)and $\\xi_0=\\sqrt{r_0^2+z_0^2}$. The normal vector on the hypotenuse is $\\hat{n}$, and the components are $n_r=\\frac{z_0}{\\xi_0}$ and $n_z=-\\frac{r_0}{\\xi_0}$ Meanwile$\\frac{\\xi}{r}=\\frac{\\xi_0}{r_0}=\\frac{1}{\\cos{\\theta}}$(Eqn.2) and $\\frac{\\xi}{z}=\\frac{\\xi_0}{z_0}=\\frac{1}{\\sin{\\theta}}$(Eqn.3). Now I have a funtion $T=T(r,z)$ on the hypotenuse, and I need to calculate $f=\\frac{\\partial{T}}{\\partial{z}}n_r-\\frac{\\partial{T}}{\\partial{r}}n_z=\\frac{\\partial{T}}{\\partial{\\xi}}\\frac{\\partial{\\xi}}{\\partial{z}}n_r-\\frac{\\partial{T}}{\\partial{\\xi}}\\frac{\\partial{\\xi}}{\\partial{r}}n_z$ (Eqn.4) If using Eqn.1, then $\\frac{\\partial{\\xi}}{\\partial{z}}=\\frac{z}{\\sqrt{r^2+z^2}}$ (Eqn.1*a)and $\\frac{\\partial{\\xi}}{\\partial{r}}=\\frac{r}{\\sqrt{r^2+z^2}}$(Eqn.1*b) and $f=\\frac{\\partial{T}}{\\partial{\\xi}}(\\frac{z_0z}{\\xi_0\\sqrt{r^2+z^2}}+\\frac{r_0r}{\\xi_0\\sqrt{r^2+z^2}})$. (Eqn.5) If using Eqn.2 and Eqn.3, $\\frac{\\partial{\\xi}}{\\partial{z}}=\\frac{\\xi_0}{z_0}$ (Eqn.6)and$\\frac{\\partial{\\xi}}{\\partial{r}}=\\frac{\\xi_0}{r_0}$(Eqn.7) and $f=\\frac{\\partial{T}}{\\partial{\\xi}}(\\frac{\\xi_0z_0}{z_0\\xi_0}+\\frac{\\xi_0r_0}{r_0\\xi_0})=2\\frac{\\partial{T}}{\\partial{\\xi}}$(Eqn.8) It looks to me that Eqn.6 and Eqn.7 are right. but Eqn.8 is wrong. because when $\\theta$ goes to 90degree,it should be $f=\\frac{\\partial{T}}{\\partial{z}}=\\frac{\\partial{T}}{\\partial{\\xi}}$ So what is the right expression for $f$, and why? ($f$ is azimuthal component of the $\\nabla\\times({T\\hat{n}})$) as somebody asked."}
{"id":"54141","title":"Stresses in asteroid during close flyby","text":"The acceleration of an asteroid (such as 2012DA14) as it approaches earth is proportional to the reciprocal of distance $r$ from earth center, squared. the derivative of the acceleration, or jerk, is then proportional to the cube of the reciprocal of distance $r$, and the speed $v$. Thus the jerk is rapidly increasing as the asteroid approaches. Will the asteroid experience stresses from the jerk that could possibly cause it to break up?"}
{"id":"118734","title":"Why does a large train cause the ground to shake?","text":"I work in a 4 story building that is approx. 150 feet away from a set of train tracks. When a large (40+ car) freight train goes by, the shaking in the building is perceptible. As I've watched the train go by, there does not appear to be any side to side movement and the speed is constant. What exactly is causing the vibrations in the ground? Is it simply the train's traversal over each segment of track? Surely the train itself is not \"bouncing\" along the track with enough energy transfer to shake the ground, right?"}
{"id":"128057","title":"Can stimulated emitted photons be absorbed?","text":"Typically a stimulated photon will be one of a pair with its stimulating photon. If the leading photon is absorbed by a particle in the ground state, will it then be re-emited by the stimulated emission caused by the second photon? If it causes a stimulated emission from an excited particle, will there then be a train of three photons, or will the new pair leave the excited particle excited? I am interested in the case where there is not a population inversion, but presumably the same rules would apply to lasers."}
{"id":"36026","title":"Is the fuel burn for a satellite launch affected by the position of the moon relative to the launch site?","text":"The gross mass of a satellite rocket is tiny compared to that of Earth, and Luna. Between them, however, the two bodies set up tides in bodies of water which itself is again considerable mass. At the point of launch a satellite rocket might not be affected by the Moon's gravitation attraction. As it goes higher\/ more distant from earth\/closer to escape\/ target orbit do it's fuel burn statistics change depending upon the position of the moon vis-a-vis launch location\/payload release location?"}
{"id":"79699","title":"Dynamics of counter-rotating flywheels","text":"I've wondered about this for ages. If we create a pair of flywheels that rotate in the opposite direction with the same angular momentum, but are co- located and have the same mass and inertial moment (one can imagine various ways to accomplish this, at least approximately) -- it is clear to me that there cannot be any precession forces, but, if we try to rotate the entire assembly around an axis perpendicular to the axis of flywheel rotation, will the force needed to produce this secondary rotation be the same as if the flywheels were stationary, or will it require a proportionally greater force to rotate in this way, as it does with a single flywheel?"}
{"id":"33212","title":"Reflectance of Titanium as Function of Thin Film Thickness","text":"As far as I know, transmittance equals $e^{-\\alpha x}$, where $\\alpha$ is absorption coefficient and $x$ is thin film thickness($100-300\\,nm$). My team and I have engineered a way to find absorption. Transmittance, T= (Output intensity)\/(initial intensity). And, absorption A=(initial intensity-output intensity)\/(initial intensity). After simplifying the equation, one should get $$A= 1-T$$ The problem I am facing is I do not know any equation, which will give me reflectance of Ti as a function of thin film thickness at $808\\,nm$ wavelength. If someone gives me an equation of transmittance, please include the effects of absorption so that I can calculate the reflectance afterwards."}
{"id":"83901","title":"Harmonic Oscillator Expectation Value","text":"In Calculating the expectation value of the quantum harmonic oscillator, I've come across a problem for finding $\\left \\langle x \\right \\rangle$ for the coherent state $\\left| \\alpha \\right \\rangle$ $$x = \\sqrt{\\frac{\\hbar}{2m\\omega}}(a_+ + a_- )$$ $$\\sqrt{\\frac{\\hbar}{2m\\omega}}\\left \\langle \\alpha \\right| (a_+ + a_- ) \\left| \\alpha \\right \\rangle =\\sqrt{\\frac{\\hbar}{2m\\omega}} \\alpha ^* (a_+ \\alpha + a_- \\alpha )$$ This doesn't agree with the result my book has given, which is $\\sqrt{\\frac{\\hbar}{2m\\omega}} (\\alpha + \\alpha ^* )$. I'm not used to working in bra-ket notation, should it's extremely likely that I'm missing something small here. The general principle is that $\\left \\langle \\beta \\right| A \\left| \\beta \\right \\rangle = \\beta ^* (A\\beta)$, right?"}
{"id":"41767","title":"Is the Schwarzschild black hole unphysical?","text":"To obtain the Schwarzschild metric from Einstein equations of general relativity, we suppose that the energy density is a distribution : $$ \\rho (\\vec{r}) = M \\delta(\\vec{r})$$ The Schwarzschild radius, corresponding to the horizon, is $R_s = 2 M$, in units $G=c=1$. A physical interpretation of this, is that you cannot put more energy than $M$ in a sphere of radius $R_s$. The way I understand this, is because the total energy of the black hole, sum of its mass energy (positive) and its auto-gravitational energy (negative), should remain non-negative. However, for the Schwarzschild black hole, for each radius $r$ between $0$ and $R_s$, the total mass in the sphere of radius r, is : $$ m(r) = \\int_0^r \\rho(\\vec{u}) d^3u =\\int_0^r \\delta(\\vec{u}) d^3u = M$$ So it seems to violate the principles cited above, and so the Schwarzschild black hole should be unphysical. If it makes sense, a physical black hole (static, with spherical symmetry) should have a mass\/energy density $\\rho (r)$, such that, whatever the value of $r$ is, we have the inequality (in units $G=c=1$): $$m(r) = \\int_0^r \\rho (u) 4\\pi u^2du \\le \\frac{r}{2}$$"}
{"id":"41765","title":"Is there a symmetry associated to the conservation of information?","text":"Conservation of information seems to be a deep physical principle. For instance, Unitarity is a key concept in Quantum Mechanics and Quantum Field Theory. We may wonder if there is an underlying symmetry, in some space, which may explain this conservation of information."}
{"id":"41763","title":"two electron spins line up in opposite direction and the Ising model","text":"What cofuses me with the Ising Model is the claim that these nearby little magnets want to face the same direction. facing the same direction, they have lower energy. This seems to conflict with the fact that north poles of magnets repel each other. So little magnets act different from big everyday magnets? But then, it's said that if you bring two electrons near enough that they can feel each other's magnetic field, then if you wait enough, a photon will be emitted and they will get into an entangled state such that their spins are facing opposite directions. This seems to conflict with the claim that little magnets facing the same direction have lower energy. I am confused. It just looks like there are two kinds of little magnets and one kind's north poles repel each other, and another kind's north poles attract each other. What is really going on?"}
{"id":"91747","title":"Why does a \"speed of sound\" exist?","text":"I've recently read that wind cannot be faster than the speed of sound (german source). But why is there a speed of sound? I understand (well, mostly accept to be honest) that the speed of light in vacuum is a maximal speed for all matter. And I understand that you need more energy the more you accelerate particles. But why can't you make wind faster?"}
{"id":"7077","title":"Ratio of radiation sources in the uranium cycle","text":"Sorry, I know y'all are probably getting a lot of questions re: Fukushima, but I had a very specific one and no-one has been able to answer it. I am specialised in medical radiation, and have been explaining the risks to workers in this accident, but I need to know what are the rough proportions of alpha, beta (+\/-) and gamma radiation flying around. If I have this right, most of the decays in the uranium cycle are alpha or beta+ and beta-. Some isomeric transitions (and obviously the beta+ decays) will release gammas. Obviously there will be a small amount of xrays too, from Bremsstrahlungs and such. But what are the percentages of this in general uranium fuel material (I guess 'spent' fuel is the real problem here, not fuel undergoing chain reaction)? Is 50% of the radiation alphas? Is 80% gammas? It makes a huge difference from a biophysics perspective as alphas are easy to protect against (for the workers) but gammas are unshieldable. Beta- decays are in between, pretty easy to deal with. Anyone know the uranium fuel cycle well enough to give a rough estimate here? EDIT: Added some thoughts in my response to Bars, who helped a lot. Any other thoughts that follow on from those insights?"}
{"id":"52757","title":"Depends river stream on river level?","text":"If we have referral particle in the middle of the river, will it's velocity increase with higher river level?"}
{"id":"18631","title":"Effect of introducing magnetic charge on use of vector potential","text":"It is well known that Maxwell equations can be made symmetric w.r.t. $E$ and $B$ by introducing non-zero magnetic charge density\/flux. In this case we have $div B = \\rho_m$, where $\\rho_m$ is a magnetic charge density. But this means that $B$ cannot be expressed as the curl of vector potential anymore. Does it mean that it is impossible to develop electrodynamics of non-zero electric and magnetic charges in terms of vector and scalar potentials? What happens to $U(1)$ gauge invariance then? P.S.: I know that for the point-like magnetic charge vector potential can still be introduced, but not in the entire space. My question is related to non point-like magnetic charge densities."}
{"id":"126510","title":"how does a lizard walk on vertical walls?","text":"One day I was observing a lizard walking on walls as easily as we walk on horizontal surfaces. I got curious to know the physics behind this but I couldn't get to any useful conclusion. I tried to Google it but all the hypothesis like lizards create vacuum while they walk or they secrete some sort of chemical (and few others) were rejected due to some or the other reason, so I thought maybe here I could get some help on the subject."}
{"id":"30740","title":"What's the most efficient strategy to shake a bottle to create a global flow?","text":"Consider a bottle of orange juice with pieces of the fruit gathered at the bottom of the bottle. You need to shake the bottle to evenly distribute the pieces in the bottle. The first part of the process seems straight forward, just giving the bottle a few sharp shakes to create turbulence and destroy the conglomerate of fruity pieces. However, what's the most energy-efficient strategy to create a global flow to distribute the stuff? Since at the start of the process, the interesting parts are gathered at the bottom of the bottle, a strong axial component would be neccesary, for energy reasons a laminar flow would be interesting (assuming the dispersal has allready happened). Edit to add: Efficient means the least energy input from me. If your mixing method involves throwing the bottle off a cliff, the energy for cyrrying it up there counts.   Where I live, juice comes in round bottles or in boxy containers."}
{"id":"76574","title":"Rate of evaporation of water","text":"In our everyday life, does the temperature or humidity have a greater effect on how fast things dry? Are there any theories\/models that explain this quantitatively?"}
{"id":"30748","title":"Exchange operator in terms of rotation operator","text":"I have studied about exchange operators and rotation operators and I know that an exchange between 2 particles in a combined state is the same as rotating each particle 180 degrees (according to http:\/\/en.wikipedia.org\/wiki\/Exchange_symmetry). Could someone explain the mathematical equation behind this? I have tried to search but to no avail."}
{"id":"15449","title":"Dark-energy sources and respective implications","text":"Firstly, I want this to be a very layman\/simple level. According to Sean Carroll, we have certain possibilities for where dark-energy comes from: http:\/\/www.youtube.com\/watch?v=v6o2bUPdxV0 Including   * A modification of general relativity   * Vacuum energy   * A modification of vacuum energy And he goes on to say that **if vacuum energy is the right answer the universe will continue to expand**. What would be the implications if the right answer is a modification of general relativity? Are there any other theories that explain dark energy that make predictions about the future like this?"}
{"id":"126512","title":"Why doesn't matter pass through other matter if atoms are 99.999% empty space?","text":"The ghostly passage of one body through another is obviously out of the question if the continuum assumption were valid, but we know that at the micro, nano, pico levels (and beyond) this is not even remotely the case. My understanding is that the volume of the average atom actually occupied by matter is a vanishingly small fraction of the atom's volume as a whole. If this is the case, why can't matter simply pass through other matter? Are the atom's electrons so nearly omnipresent that they can simultaneously prevent collisions\/intersections from all possible directions?"}
{"id":"127648","title":"How many state configurations are possible for $N$ particles in completely different states?","text":"How many state configurations are possible for $N$ particles in completely different states? I cannot remember if the total number of state configurations for $N$ particles in completely different states is $n^n$ or $n!$. Also, if $n$ particles are in completely different states, does this represent the maximum possible entropy of this system?"}
{"id":"108470","title":"Was Titanic's captain's decision correct?","text":"I just saw a documentary on Titanic. Someone in that documentary told Captain Smith > \"Shall we close the water tight doors, sir?\" and the Captain replied > \"No, let it be open. Let the water be distributed uniformly throughout ship, > so that we could lower the life boats.\" Note: The quotes are not exact but very near to exact. * * * ## My Question If and if, the watertight doors were shut, was there a chance of Titanic to escape from its tragic fate? And what exactly is a water tight container? I presume that this is related to physics :)"}
{"id":"28963","title":"Carrying water on person, or on the frame when bicycling","text":"So, the question is as follows: What is the difference in work exerted by the rider in the two following scenarios? a) Rider + bike. Water carried in a holder on the frame   b) Rider + bike. Water carried in a backpack (for simplicity, assumption that weight of holder = weight of backpack and backpack is perfectly aerodynamic, not increasing drag) I would assume that in case a), as the weight is closer to the bike's centre of mass, it would require the least amount of extra work, compared to b), where the weight is more distant from the centre of mass. Thanks in advance."}
{"id":"43266","title":"What's the difference between exchange spin wave and magnetostatic spin wave?","text":"So far I've heard of three kinds of spin waves   * Magnetostatic spin waves (MSW)   * Dipole-exchange spin waves (DESW)   * Exchange spin waves (ESW) What's the difference?"}
{"id":"69966","title":"Electron revolving in an atom","text":"When an electron revolves around the nucleus in P or d-orbitals why does not it collide with the nucleus.   I mean to say that the shape of the orbital narrows near the nucleus , so shouldn't it collide with the nucleus?"}
{"id":"96308","title":"Twisted supermultiplets","text":"What is a twisted supermultiplet, in a generic supersymmetric theory? Which ordinary fields belong to one of such twisted supermultiplets? I am confident with the idea of a supermultiplet or a superfield, in $\\mathcal{N}=1,2,4$. However, I don't know what the term _twisted_ stands for. Also, why is it useful to consider such things?"}
{"id":"11376","title":"Complementarity between the laws of physics?","text":"Is this following proposal plausible, worth considering, or dismissable as lunatic fringe science? What if the universe isn't really what we think it is but some universal quantum computer where we read off one output at a time? This universal quantum computer is initialized to a superposition of program instructions -- which are identified with the laws of physics -- and after reading off one output after another, one finds there are in general two different kinds of questions we can ask; general relativity type questions, where we get GR type answers, and quantum mechanical type questions, where we get QM type answers. But we have to choose between one or the other, and whichever we choose, we get the answer appropriate for that domain. The place where both are united is the Planck scale, but unfortunately, the Planck scale as way too inaccessible to us, and is hence unobservable. Can this explain the confusion about UV\/IR mixings? String compactifications are computer programs determining the parameters of the effective field theory describing the domain of answerable experiments. A complementarity between the laws of physics themselves. GR\/QM complementarity. Is there any flaw with my proposal?"}
{"id":"19141","title":"Is it a total or an explicite time derivative in the Schrödinger equation?","text":"I am always dubious when I need write Schrödinger equation: do I write $\\partial \/ \\partial t$ or $d\/dt$ ? I suppose it depends on the space in which it is considered. How?"}
{"id":"60344","title":"Perturbed stress-energy tensor in a cosmological context?","text":"In the theory of cosmological pertubations, we can write the metric of a null- curvature expanding Universe as : $ds^2 = -c^2\\left(1+2\\frac{\\psi}{c^2}\\right)dt^2 + a^2 \\left(1-2\\frac{\\phi}{c^2}\\right)\\left(dx^2+dy^2+dz^2\\right)$ From that metric we can compute the perturbed Einstein tensor $G_{\\mu\\nu}$. Now, I search what is the form of the stress energy tensor associated to this in order to write the Einstein equation : $G_{\\mu\\nu}=\\frac{8\\pi G}{c^4}T_{\\mu\\nu}$. So, starting from the fact that we consider a perturbed fluid : $\\rho=\\bar{\\rho}+\\delta\\rho$ $P=\\bar{P}+\\delta P$ what is the $T_{\\mu\\nu}$ associated to that fluid ? * * * Remark : Here are some indications (see 4.4.1) : http:\/\/www.damtp.cam.ac.uk\/user\/db275\/Cosmology\/Chapter4.pdf but I do not understand very well why the author write the $T_{\\nu}^{\\mu}$ instead of the $T_{\\mu\\nu}$ (so if someone knows why, please explain it to me)."}
{"id":"60347","title":"Switching from an accelerated frame of reference to a locally inertial reference system","text":"_Using the equivalence principle, show that the interval for an accelerated observer ($\\textbf{g}$ uniform and constant) has the form_ $$ ds^2|_{\\text{first order in }\\textbf{g}}=(1+2\\varphi\/c^2)c^2dt^2-d\\textbf{x}^2+o(1\/c^2), $$ where $\\varphi=-\\textbf{x}\\cdot\\textbf{g}$."}
{"id":"60343","title":"Why does the wave description say that probability oscillates, while the phase interpretation says constant amplitude?","text":"The wave description of a particle illustrates an oscillating probability of the particle being found in any point in space. When a particle travels, it carries along with it a phase that oscillates in the complex plane. When the particle travels, the amplitude never changes although the phase does, which doesn't matter. The only property that matters is the amplitude. Why does the wave description say that probability oscillates and the phase interpretation says constant amplitude? What am I missing?"}
{"id":"52251","title":"Kinetic energy of a cylinder","text":"It is a long cylinder (you can approx $R=0$), and it has a fixed point in one os its ending points, it's rotating on a plane and I have to calculate the kinetic energy from reference systems situated in the center of mass, and both extremes. If I calculate it in the fixed point, it has only rotational energy: $$\\frac{1}{6}ML^2\\dot{\\varphi}^2$$ If I choose the center of mass, the moment of inertia changes but I have to add to that the energy of that moving point $1\/2Mv_{CM}^2$, and the result is the same. My problem is that when I choose the moving extreme, the result changes: The moment of inertia is the same as with the fixed point: $1\/3ML^2$, and the speed at that point is $1\/2ML^2\\dot\\varphi ^2$, so adding them gives you something different from the $1\/6$ result, and they should be the same, ¿What am I doing wrong?"}
{"id":"95035","title":"Higgs maths video Brian Greene","text":"What do you think of the video below? ==> http:\/\/www.youtube.com\/watch?v=KWj00MCqSxs Is the explanation in this video largely correct, or missing something? Is the mathematical description in the video of the higgs mechanism right? He said that phi is re-written as a m + h. What does it mean? Is the m in this video the vacuum expectation value? Thanks for all answers!"}
{"id":"87597","title":"How close are humans to creating gold from other elements?","text":"Gold is made when dying stars collide. How close are humans to doing that? Or is it even possible to do that unless gravity is absurdly high?"}
{"id":"65647","title":"Where does the extra equation come from to determine the forces from an object on a table?","text":"I have a question about basic statics but somehow I cannot manage to find the answer on my own (btw, this is not a homework. It's been so many years since school for me...). The problem is very simple: we have an object with weight $D$ at a given location on table wit with four legs ($F_1$ to $F_4$). What is the force applied on each leg? (for simplicity, I'm just using the same labels $F$ and $D$ for both the location and the force) ![object on a table](http:\/\/i.stack.imgur.com\/ZXhRB.png) $W$, $H$, $x$, $y$ and $D$ are given. To find the forces on each leg, as far as I remember, I have to consider two general equations: $\\sum F=0$ and $\\sum M=0$. So I have: $$ F_1 + F_2 + F_3 + F_4 - D = 0 $$ Also, considering the moments round the point $F_1$: $$ W(F_2+F_3) - xD = 0 $$ $$ H(F_3+F_4) - yD = 0 $$ But this just give me 3 equations! I missing one more equation and cannot figure it out."}
{"id":"81292","title":"Harmonics in an open organ pipe","text":"How to solve the following question? > _An open organ pipe has two adjacent natural frequencies of 500 and 600 Hz. > Assume the speed of sound in the air 340m\/s. The length of the organ pipe > is?_ What is **open organ pipe**? It means two open ends? Like this? Or one closed and one open? ![enter image description here](http:\/\/i.stack.imgur.com\/DmRsb.jpg) Should I use this formula? $L=\\frac{(2n+1)v}{4f}$"}
{"id":"64111","title":"Convolution kernel of poisson equation by FFT","text":"I'm trying to solve poisson equation using FFT. In genral it is a convolution of the charge density with potential well of point charge ( Green's function of laplace equation ) which is $1\/r$ I'm not quite sure about fourier transform of this convolution kernel. the 2D fourier transform of $1\/r$ is $ \\frac{1}{\\sqrt{k_x^2+k_y^2}}= 1\/k_r$ acording to this: http:\/\/sepwww.stanford.edu\/public\/docs\/sep103\/jon3\/paper_html\/node3.html while in many papers about FFT solution of poisson equation is used kernel like this: $ \\frac{1}{(4-2cos(2*\\pi*k_x)-2cos(2*\\pi*k_y))} $ For example here: http:\/\/www.physics.buffalo.edu\/phy410-505-2004\/Chapter6\/ch6-lec2.pdf In one dimension the kernel is $1\/k^2$ **Basicaly, I would like to make clear what is the kernel in 1D, 2D and 3D case both in real space and Fourier space**"}
{"id":"20478","title":"Solutions of damped oscillator differential equation","text":"I am reading about damped harmonic motion in my Physics book (Gerthsen Physik) and there are two things that irritate me: # Stokes friction It says that Stokes friction would be $$F = -m \\gamma \\dot{x}.$$ To me, it does not make any sense why it depends on the mass of the object. It means that the deceleration does not depend on the mass. Or is it just since it is proportional to the volume and the volume usually depends on the mass. # linear combination of solutions Then the differential equation is set up, I fully understand it, except the confusion about the friction. $$ m \\ddot{x} + \\gamma \\dot{x} + \\omega_0^2 = 0 $$ With $$\\delta = \\frac{\\gamma}{2}$$ and $$\\omega = \\sqrt{\\omega_0^2 - \\delta^2}$$ the solution to the $$x(t) = \\exp(\\lambda t)$$ approach are $$\\lambda = -\\delta \\pm i \\omega t$$. One of the plus or minus variants are a valid solution, but the linear combination of both are all solutions. I understand that. The book then says that the general solution is the combination of real and imaginary parts. Alternatively, one can add a phase angle and have one \"compact\" solution like this: $$ x(t) = x_0 \\exp(-\\delta t) \\exp(i ( \\omega t + \\phi ) ) $$ The $\\exp(i \\phi)$ certainly makes up a linear combination of real and imaginary parts, but how is it a combination like $$c_1 \\exp(i\\omega t) + c_2 \\exp(-i\\omega t) ?$$"}
{"id":"80741","title":"Fluid properties","text":"I cannot understand the following sentence, > Without flow , however, shearing stresses cannot exist, and compression > stress or pressure is the only stress to be considered. (from _A Physical Introduction to Fluid Mechanics_ , by A. J. Smits, John Wiley & Sons, 2000) Could someone explain this to me?"}
{"id":"99211","title":"about satellite is it possible to get signal from two satellite at a time?","text":"1. Is it possible to get signal from two satellites at the same time?   2. If we go far away from a certain country to another, and be able to continue to recieve the same signal?"}
{"id":"99214","title":"Are Black Holes set to take over the Harmonic Oscillator in the 21st century?","text":"A few years ago I attended a talk given by Andy Strominger entitled Black Holes- The Harmonic Oscillators of the 21st Century. This talk, http:\/\/media.physics.harvard.edu\/video\/?id=COLLOQ_STROMINGER_091310 was quite formidable and has had a lasting impression on my physics studies since. I'd consider the talk transformative in my approach. Now, imagine hearing the power of the harmonic oscillator model for the first time before taking an EM or QM class and simply having a spring mass system in mind... Fast forward a few years and consider learning the eigenfunctions of the harmonic oscillator Hamiltonian ${\\Omega_n}$ are an orthonormal basis set for $\\mathcal{H}=L_2(\\mathbb{R^d},d\\mu)$. A statement equivalent to: 1) Schwartz Space $\\mathscr{S}$ being Fourier invariant 2) The Fourier inversion theorem holding Now, the power of the harmonic oscillator is coming to fruition in what I can solve in physics. Admittedly, there is still much in the field that is beyond me at this point that is well represented by the harmonic oscillator model. This leads me to first ask what are the best known fits for the harmonic oscillator model? However, this question is not the main focus of this post as I am also deeply interested in models extending beyond the harmonic oscillator paradigm. Namely, what are examples of objects currently being modeled as Black holes? It's interesting (for me, hopefully more) to ask this question in the \"firewall\" era, given that I attended this talk before the now famous AMPS paper. So, in summary I am asking for examples of physical objects modeled by harmonic oscillators, or black holes, relevant in the field today. It would be interesting to compare the usage of the models in light of the theme of Andy Strominger's talk."}
{"id":"63692","title":"Modelling the movement and jumps of a chalk while drawing a dashed line on a blackboard","text":"You probably know that if you try to draw a line using a piece of chalk on a blackboard , under some conditions (for example, $\\alpha<\\frac{\\pi}{2}$ in the picture below) you will have a dashed line pattern instead of a continuous line. My question is : (how) Can you model this special movement of the chalk ,and specially , **find the length of line segments and the distance between them** (which are the _visible_ characteristics of the motion)? Note:gravity is present. (I think it affects the solution, at least in some models) ![the chalk](http:\/\/i.stack.imgur.com\/jy0C5.png) The pattern: ![enter image description here](http:\/\/i.stack.imgur.com\/8awdr.png)"}
{"id":"16800","title":"Mathematical Modelling for Efficiency of PV thermal technology","text":"As the efficiency of PV thermal or wind energy decreases with time(from the installment). Is there any mathematical model that gives its relation with time. i.e., whether it is a linear decrease or some other function of time. I need to use the result of this in my research on energy markets."}
{"id":"16806","title":"WKB approximation to loop diagrams","text":"I'm a bit confused with the terminology here. This paper claimed to use WKB method to calculate the usual loop diagrams. Notice that the vertex is approximated by expanding around the classical (on- shell) solution of the massive particle propagator. However this method doesn't look like the usual WKB I know. AFAIK in WKB (eikonal) approximation we are supposed to expand the propagator around the particle limit of the solution, but NOT the classical field limit. So, how do you describe the WKB method in QFT? Is it practical enough to give us quick intuitive calculation on our system? (probably in this sense, the WKB+1loop is the crudest semiclassical approximation we can do -> first order in $\\hbar$ and coupling constant)"}
{"id":"122239","title":"Conservation of energy in transformers","text":"How does energy remain conserved in a transformer if emf is increasing, or decreasing? Does the current decreases to accomodate? Does Ohm's law still hold here? Although we know, Ohm's law is not universal."}
{"id":"9337","title":"What is the smoking gun signature of string theory?","text":"What is the smoking gun signature of string theory? Suppose we have a complete and consistent model of quantum gravity with a zero or negative cosmological constant, but all we are given is its complete S-matrix. Using only this information, how do we figure out whether or not this model corresponds to a string theory compactification? If the cosmological constant is positive, what can we use in place of the S-matrix?"}
{"id":"90070","title":"Rayleigh-Taylor Instability dependence on acceleration direction","text":"I'm trying to bolster my understanding of the Rayleigh-Taylor instability, and I've gotten stuck on the point of which fluid (more or less dense) is being accelerated into the other. Cases of uniform acceleration make sense (e.g. plane-parallel fluids under gravity; or being accelerated by a piston), but my understanding falters in more exotic configurations. Consider the following: A wind tunnel(-like) setup---without gravity---in which one fluid (density $\\rho_1$) is at rest, while another ($\\rho_2$) is blown into the first with a constant velocity. Because the two substances are colliding --- they'll both feel an acceleration. But because there is no acceleration between the **reference frames** of $\\rho_1$ and $\\rho_2$, how do you tell which configuration is stable, and which is not?"}
{"id":"90076","title":"Transferring CFT correlations from $\\mathbb{R}^3$ to $S^3$","text":"There seems to be a simple method to transfer a CFT's correlations from $\\mathbb{R}^3$ to $S^3$ but I am not understanding why it is supposed to work. The idea is that somehow because, $ds^2_{S^3} = \\frac{4}{(1 + \\vert \\vec{r} \\vert^2 )^2}ds^2_{\\mathbb{R}^3 }$ it follows that all that one needs to do is that replace all occurences of $\\vec{r} - \\vec{r}'$ by $\\frac{2(\\vec{r} - \\vec{r}') }{\\sqrt{ (1 + \\vert \\vec{r} \\vert ^2 )(1 + \\vert \\vec{r}' \\vert ^2 ) } } $   * Why is this supposed to work?    * What is the necessary relationship between the two CFTs for this to work? (like if one uses this on the free scalar field theory then it will turn out that w.r.t the Laplacians on $\\mathbb{R}^3$ or $S^3$ the two 2-point functions satisfy _different_ P.D.Es respectively and hence they come from two different Lagrangians - since we anyway knew that the Lagrangian for the conformally coupled scalar on the flat space is not the same as the Lagrangian for the conformally coupled scalar on $S^3$)   * Is it part of some general idea that would work between other pairs of manifolds?"}
{"id":"54961","title":"Quantum computing problem","text":"Suppose that a qubit is in the state $|\\varphi\\rangle=a|0\\rangle+\\sqrt{1-a^2}|1\\rangle$, where $a\\in[-1,1]$. If we first perform a standard basis measurement on this qubit and then perform a $\\\\{|u\\rangle,|u^\\perp\\rangle\\\\}$-basis measurement, where $$|u\\rangle=b|0\\rangle+\\sqrt{1-b^2}|1\\rangle \\text{ for some }b\\in[-1,1],$$ what is the probability that the outcome of the second measurement is $|u\\rangle$, in terms of $a$ and $b$? Edit: I've found a solution, it should be $ab+\\sqrt{(1−a^2)(1-b^2)}$."}
{"id":"62876","title":"Translator Operator","text":"In _Modern Quantum Mechanics_ by Sakurai, at page 46 while deriving commutator of translator operator with position operator, he uses $$\\left| x+dx\\right\\rangle \\simeq \\left| x \\right\\rangle.$$ But for every $\\epsilon > 0$ $$\\langle x+ \\epsilon \\left| x \\right\\rangle = 0.$$ Therefore this limiting process $$\\lim_{\\epsilon \\rightarrow 0} \\left| x+ \\epsilon \\right\\rangle = \\left| x \\right\\rangle$$ does not make sense for me. I couldn't derive commutator relation without using these."}
{"id":"30550","title":"Recommendations for Statistical Mechanics book","text":"I saw Book recommendations   No reference to Statistical Mechanics there. I learned thermodynamics and the basics of statistical mechanics but I'd like to sit through a good advanced book\/books. Mainly I just want it to be thorough and to include all the math.   And of course it's always good to give as much intuition about the material. Some things I'd be happy if it includes (but again it mostly just needs to be a good clear book even if it doesn't contain these) are:   1. As much justifications for the postulates if possible, I'm very interested in reading more about how Liouville's theorem connects to the postulates   2. Have examples of calculating partition functions, hopefully not just the partition function for the ideal gas. Thank you"}
{"id":"89833","title":"Statistical mechanics prerequisite","text":"I`ll be taking senior level and graduate course in statistical mechanics in a month. I was wondering what would be the best statistics and probability textbook to prepare for it? I`m currently reviewing classical mechanics and thermodynamics. I forgot quite a bit. Should I focus on a book that teaches mainly probability or both statistics and probability? I'm getting flags for this being a possible duplicate. I already have the statistical mechanics book I'll be studying from. But it doesn't have a decent enough review of statistics and probability which is what I'm trying to prepare for."}
{"id":"68765","title":"Blocking electro-magnetic waves between two near field antennas","text":"Its been a while since I studied physics, so pardon my ignorance. I have two near field antennas. The range of the antennas is about 2-3 cm. I am looking for a way where I can block and unblock the communication between them. One of the way that I tried is to place a flat tightly woven wire and if I short the two ends of the wire, I was able to block the communication. My assumption is that it is effectively changing the frequency of the signal and thus blocking the signal. Is that what is happening? However, if the radio signal has good strength, the coil is not doing a good job. Is there any other better solution that I can achieve blocking\/unblocking with electrical\/electronics (without mechanical parts)?"}
{"id":"13263","title":"Can the distance of a quasar be determined accurately?","text":"As noted in A doubt about the age of the universe, the wiki about quasars still contains the following misleading sentence: \"The highest redshift quasar known (as of June 2011) is ULAS_J1120+0641, with a redshift of 7.085, which corresponds to a proper distance of approximately 29 billion light-years from Earth.\" But even if the \"strange 29 billion\" is replaced by the \"correct 12.9 billion\", the fact remains that the actual measurement is \"a redshift of 7.085\". The \"proper distance\" is only a different way to express that measurement. It's not clear to me how \"accurately\" this describes the distance of the quasar, because the quasar surrounds a black hole and rotates quite fast. So there are at least two additional sources for the redshift, but how significant is their contribution?"}
{"id":"59829","title":"Relativistic Lorentz force law","text":"If we consider the the relativistic Lorentz force law: $$\\frac{d}{dt} (m\\gamma \\vec{u})=e(\\vec{E}+\\vec{u} \\times \\vec{B})$$ How can we deduce: $$\\frac{d}{dt} (m\\gamma c^2)=e \\vec{E} \\cdot \\vec{u}$$ Clearly dotting with $\\vec{u}$ will give us the RHS. Which leaves us: $$\\vec{u} \\cdot \\frac{d}{dt} (m\\gamma \\vec{u})=e \\vec{u} \\cdot \\vec{E}$$ Could anyone help explain how to proceed and if this is the correct method? EDIT: If it helps: with reference to these notes i'm working through: http:\/\/www.maths.ox.ac.uk\/system\/files\/coursematerial\/2012\/2393\/8\/WoodhouseLectures.pdf Page 86, eq (178), the paragraph underneath states 'The first equation (which follows from the second)', this is what i'm trying to prove (a warning, the notes are riddled with errors..)."}
{"id":"13269","title":"Can temperature be defined as propensity to transmit thermal energy?","text":"I was recently surprised to learn that defining temperature isn't easy. For a long time, it was defined operationally: how much does a thermometer expand. Also surprising, temperature isn't a measure of thermal energy: 0 C water has more thermal energy than 0 C ice, but the same temperature. I came up with this definition, and I'd like to know if it's correct: Temperature is the propensity of something to give off heat, that is transmit thermal energy. Equivalently, it is the resistance of something to receive heat. Is that correct? Please justify your answer."}
{"id":"28514","title":"Trouble with the Lorentz law of force: Incompatibility with special relativity and momentum conservation?","text":"In Physical Review Letters, there was a paper recently published: > Masud Mansuripur, _Trouble with the Lorentz Law of Force: Incompatibility > with Special Relativity and Momentum Conservation,_ Phys. Rev. Lett. 108 > (2012) 193901, arXiv:1205.0096. The paper considers the case of a stationary point charge at a fixed distance from a stationary magnetic dipole in some frame, stating that the magnetic dipole neither experiences a torque nor force from the stationary charge. Upon transforming to another frame, they argue that the magnetic dipole is transformed into a magnetic dipole and electrical dipole. This combination will therefore experience both an electrical force, and magnetic torque from the field of the moving charge. The authors therefore claim that there is a contradiction and therefore a problem with the consistency of the Lorentz force and special relativity. Is the paper correct, or has the author overlooked something?"}
{"id":"54344","title":"Finding the Electric Field (and other information, besides)","text":"The problem I am working on is: > Two parallel plates having charges of equal magnitude but opposite sign are > separated by 29.0 cm. Each plate has a surface charge density of 33.0 nC\/m2. > A proton is released from rest at the positive plate. > > (a) Determine the magnitude of the electric field between the plates from > the charge density. > > (b) Determine the potential difference between the plates. > > (c) Determine the kinetic energy of the proton when it reaches the negative > plate. > > (d) Determine the speed of the proton just before it strikes the negative > plate. > > (e) Determine the acceleration of the proton. > > (f) Determine the force on the proton. > > (g) From the force, find the magnitude of the electric field. So far, I've only attempted to solve part (a). For part (a), seeing as they didn't provide the dimensions of the plates, I assumed them to be infinite-sized; furthermore, they both have to have the same area (although, I can't justify why this is so). So, for plate 1, which we'll consider positive: $\\sigma = 33.0 \\cdot 10^-9 C\/m^2$ Since they don't say otherwise, that the plates do not have uniform charge distribution, we'll assume that the charge is in fact uniformly distributed; this is reasonable, seeing as they give one surface charge density for the entire plate. $\\sigma = \\frac{dq}{dA} \\rightarrow dq = \\sigma \\cdot dA \\rightarrow \\int dq = \\sigma \\int dA \\rightarrow q_1 = \\sigma \\cdot a^2$, $q_1$ being the charge of the entire plate. A similar process can be done for the second plate, which we'll regard as negative: $q_2 = \\sigma \\cdot a^2$ Hence, $q_1 = - q_2$ This is the only way of starting that I could think of; and as you may see, it wasn't very helpful. How should I approach this problem? * * * As per Michael Brown's suggestion, should I generate a Gaussian plate, infinite in size, and place it between the two sheets? Then, find the magnitude of the electric field from sheet, and multiply it by two?"}
{"id":"20065","title":"Energy formula for separating $O_2$ from mixture of $O_2$, $NH_3$ and $H_2O$","text":"I have a physics problem I'd like to make sure I get correct. The practical aspect of this problem is that the photosynthetic efficiency of algae is inhibited with dissolved O2 in the growth medium, and the economics of any algae cultivation system are very sensitive to photosynthetic efficiency. The problem I'm trying to solve is determine the theoretic energy to separate a gas mixture of, O2, NH3 and H2O (in boiling point order) into two gas streams, one of which is 99.99% pure O2 but the other stream can have as much as 50% of the original O2 along with NH3 and H2O. The input and output temperatures and pressures are STP and the input, at least, is at saturated humidity. Presumably, the energy is derived from the equations in this paper. Any approximate ideas of what the formula would look like? PS: The detailed model of the algae system is here."}
{"id":"20063","title":"A terminological question about work and energy","text":"Work is force applied over distance. Is it also reasonable to say that work is (the same thing as) the transfer of energy? When work is done, the equivalent energy is transferred. But if energy is transferred, such as by heating something, is that (or could that be) called work? Also, if a black body is subjected to a stream of energy, which it absorbs and radiates away, is any work done? The black body, let's assume, remains at a constant temperature, but energy is transferred to it and then radiated away. Is any work done? So what I'm really asking about is how the term \"work\" should be used."}
{"id":"67751","title":"How to use Ising Spin model for prediction of time series \"Phase\"","text":"I am investigating a 2d Ising Spin Lattice. I have been able to generate a Monte Carlo app that gives me the changing spin matrix through my iterations - like the many examples on the web. However, I am trying to predict an independent variable (atmospheric pressure). I believe it is possible to measure the pseudo \"Magnetism\" and \"Energy\" of the timeseries using the Ising approach - but I am not sure how to do that. I assume it entails flipping the \"spin\" based on the timeseries changes, but I am not really sure weather that is the right approach. If anyone could give me some advice that would be great. EDIT 1 I have a series of pressure readings:                Xt :1..N      Instead of using Metropolis flip where               deltaE = 2 * J * Si * sumOf(Sneighbours)     if (deltaE < 0 || exp(-deltaE\/TEMP) > rand())            Si+1 = -Si      I need to set               Si+1 = someFunctionOf(Xt)      But I don't know what \"someFunctionOf\" should be. If I did I could update the Lattice from Xt, and workout a \"live\" Magnetism and I am hoping that might tell something about the current phase of pressure changes."}
{"id":"67756","title":"Quantum Collapse","text":"When we observe a quantum object does it collapse into a point? Or does it collapse into a smaller wave of area that is out of our range of accuracy? My gut tells me the latter."}
{"id":"67755","title":"(Canonical) Partition function - what assumption is at work here?","text":"The canonical partition function is defined as $$Z=\\sum_{s}e^{-\\beta E_s}$$ with the sum being over all states of the system. The way I saw this derived was by assuming that for each state, the probability of the system occupying that state is proportional to the Boltzmann factor: $P(E=E_s) = c \\cdot e^{-\\beta E_s}$. Summing the probabilities to one gives $Z=\\frac{1}{c}$. My question is : What principles were used in order to get this probability distribution for the energy?"}
{"id":"129089","title":"How does angular momentum transfer between a planet and its moon?","text":"Could you explain how a moon draws angular momentum from a planet? I know that the gravitational force transfers momentum, but I don't understand the mechanics behind it."}
{"id":"89804","title":"Is forward scattering = no scattering?","text":"What is forward scattering? If it is equivalent to no scattering, then why not call it \"no scattering\"?"}
{"id":"126256","title":"The mathematical physics of saltatory neuron action potential transmission","text":"Some time ago, I tried to understand saltatory electrical potential transmission on a myelinated axon but have not found a clear mathematical model of this mechanism. I understand the cable theory, but that seems only explains the normal continuous transmission within the cable either with or without insulation, instead of the saltatory one between two un-myelinated synapsis of an axon. What is the model that describes the interior and exterior interaction that results in the SALTATORY transmission mechanism? It would be even better if the answer to my question could be accompanied with reference to pertinent papers on the mathematical model on the saltatory transmission. Here is a reference on neuronal transmission modeled as electrical potential transmitted in a cable and as an application of the cable theory. My problem with this theory is that it does not account for the saltatory transmission mode of the action potential in a myelinated neuron."}
{"id":"66407","title":"What are the physics behind the Coriolis effect?","text":"What causes the Coriolis effect?"}
{"id":"27776","title":"Do Gauge Theories (CFTs) Have Phase Transitions as the 't Hooft Coupling is Varied?","text":"With an eye toward AdS\/CFT, I'm wondering if large $N$ CFTs have a (quantum) phase transition as the 't Hooft coupling is varied. To be more specific -- if I look at correlation functions of low-dimension, single trace operators as a function of $\\lambda$, are they analytic, continuous, discontinuous in $\\lambda$? How much do we know \/ any references? I'm already aware of 0811.3001."}
{"id":"27778","title":"Do Killing spinors know global information?","text":"The conformal Killing spinor equations on $R\\times S^3$ in Minkowski signature are \\begin{equation} \\nabla_\\mu \\epsilon=\\pm \\frac{i}{2}\\gamma_\\mu\\gamma^0\\gamma^5\\epsilon \\end{equation} whose solution is \\begin{equation} \\epsilon=e^{ix^0\\gamma^5\/2}\\epsilon_0 \\end{equation} where $\\epsilon_0$ is a constant spinor. Note that we identify $S^3$ with $SU(2)$ and use the left-invariant vector fields of $SU(2)$ as an orthonormal frame. For more detail, please see around Eq.2.20 in the following paper. http:\/\/arxiv.org\/abs\/hep-th\/0605163v3 When the superconformal index is interpreted as a partition function on $S^1\\times S^3$ in the Euclidean signature, the conformal Killing spinors are modified by the Wick rotation $\\epsilon =e^{-x^0\\gamma^5\/2}\\epsilon_0$ as in the following papers. http:\/\/arxiv.org\/pdf\/1104.4482v3 http:\/\/arxiv.org\/abs\/1104.4470 However, $\\epsilon =e^{-x^0\\gamma^5\/2}\\epsilon_0$ is not well-defined on the temporal circle $S^1$. This issue also happens on the Killing spinor on $AdS_3$. The metric of $AdS_3$ in the Minkowski signature can be written as \\begin{equation} ds^2=-\\cosh^2 \\rho dt^2 + \\sinh^2\\rho d\\phi^2+d\\rho^2~. \\end{equation} The Killing spinors on this coordinate takes the form \\begin{equation} e^{\\frac 12\\rho\\gamma_3}e^{-\\frac {i }2 (\\phi+ t)\\gamma_1}\\epsilon_0 \\end{equation} as in http:\/\/arxiv.org\/abs\/hep-th\/9310194 . However, when you consider the elliptic genus $Tr(-1)^Fq^{L_{0}-c\/24} \\overline{q}^{\\overline L_{0} -\\overline c\/24} y^{J}$, the bulk duals which satisfy the E.O.M of the supergravity theory are the Euclidean thermal $AdS_3$ and the family of its $SL(2,Z)$ transformations such as the BTZ BH. Then, again the Killing spinors on the Euclidean thermal $AdS_3$ take the form \\begin{equation} e^{\\frac 12\\rho\\gamma_3}e^{(-\\frac {i\\phi }2 + \\frac{t}{2})\\gamma_1}\\epsilon_0 \\end{equation} which are also not well-defnined on the temporal circle. The Killing spinor equations themselves are local, so does the Killing spinor know only local information? If so, how you distinguish the Killing spinors in the NS from in the R boundary condition on 2-torus?"}
{"id":"51441","title":"Pure mathematical exposition vs A \"for physicists\" approach: Which is better?","text":"I'm an undergrad interested in theoretical physics. Don't know enough to decide which specific subfield I may go in, but particle theory (strings, QFT, GR) and mathematically heavy condensed matter seem appealing. I have taken quite a few pure math courses in my undergrad career so far (algebra, analysis, topology etc), but now that I've exhausted most of the undergraduate curriculum, its time to start some advanced topics such as Algebraic Topology and Lie Algebras etc. My concern however is that these courses are offered by the mathematics department and thus the exposition and focus is aimed towards training mathematicians rather than someone who may find them useful in another field. So as expected in such a class, the main goal is to understand and prove the main theorems of these fields. However, as a physicist I'm not sure to what extent I can benefit from such an exposition. It would be highly time consuming to take such an approach to mathematics for someone to which mathematics is not the ends itself, but rather the means. On the other hand, there is the option of using \"math for physics\" type books such as Nakahara or Frankel to learn these topics, which would be a lot quicker and would have a physicist as a reader in mind, but no doubt, depth of understanding would be lost with such an approach. So my question is: Is it worth it for an aspiring theoretical physicist to learn advanced mathematical topics from graduate math classes (and as a result sacrificing the time spent physics), or would it be a better use of his time to focus on the physics and pick up the math from a text or a class of the \"for physicists\" type? How would the answer change if applied to a mathematical as opposed to theoretical physicist? Note that over here I'm talking about topics such as Algebraic Topology, Differential Geometry, Representation Theory etc."}
{"id":"61758","title":"Ground state energy of hydrogen molecule ion","text":"In this paper, it is mentioned: > Furthermore, since the energy of $H_2^+$in the ground state must be lower > than that of an H atom in the ground state,the negative (attractive) forces > in the $H^+ – H$ interaction must play the dominant role How do we intuitively expect the ground energy of hydrogen molecule ion $H_2^+$ should be less than that of hydrogen atom without doing exact quantum mechanical calculation."}
{"id":"61757","title":"How (or why) equivalence principle led to Einstein field equations?","text":"If equivalence principle was origin of general relativity what was the process that this principle led Einstein to developed his theory of general relativity?"}
{"id":"129558","title":"Can current and resistance result in establishing voltage?","text":"I want to know whether current & resistance can cause a voltage across the circuit, according to Ohm's Law $V=IR$. Is it possible to use a low current to produce a high voltage?! How can this be, as current is directly proportional to voltage?"}
{"id":"10211","title":"a question about singularities in gravity and Physics in general","text":"I had this doubt bugging my mind for a long time about singularities in Physics. I heard that R.Penrose and S.Hawking have proposed that there could be singularities at Blackholes and at the time of the origin of universe (at the time of Bigbang). The concept of singularity is essentially a mathematical one. (unless you call anything that you cannot explain physically as a singularity). My question is (which i want to get it resolved once for all so that i stop bugging people with it :-))....We know that the gravitational field has a singularity at points where there are point masses, so is the case with point charges and electric field. So what is it they have proved about existence of singularities. I assume that they most probably have started with a universe which doesn't have point masses. If they did'nt start with point masses, then what did they actually started with. I want to know what they have started with in mathematical terms and physical terms, once for all. I am open to read some references as well to resolve this doubt. **EDIT :** I am more puzzled as to why I am not getting an answer to the question I've asked which doesn't involve any analysis but a plain request for information. Did they (R.P and S.H) assume existence of point masses in their model of universe (or objects or whatever it is) or they have assumed smooth matter distributions ? This is a fairly straight forward request for information....I do not understand why I don't get the answer for the question I've asked...rather I get something else."}
{"id":"55922","title":"A man running on the treadmill","text":"Imagine a man is running on a treadmill. His inertia with respect to floor will be zero because he is not moving with respect to floor. If both he and the tread mill suddenly stops he will not fall, according to Newton's first law. Imagine an ant on the conveyor belt. He doesn't worry about our reasoning. He just believes in 1st law, but can't explain it using the 1st law. (Note: the ant will measure two momentum of the man. same but opposite in direction) Now imagine we are a 2D being, living in a closed world like the surface of the Earth. Is Newton's first law not applicable here? Then what will be the definition of inertia in this world? I think I should mention my current level of understanding. I read general relativity only in popular books."}
{"id":"104391","title":"Deriving diffusion coefficients from velocity field?","text":"If I know the velocity, $\\mathbf{v}(\\mathbf{r},t)$, everywhere, is it possible to determine the diffusion coefficient, $D(\\mathbf{r},t)$ everywhere as well? Would it be possible from using Fick's Law $$j = -D\\cdot\\nabla L$$ Or can it be found from the mean-squared displacement with $$\\frac{\\partial \\langle r^2\\rangle }{\\partial t} = 4D$$ ($6D$ in 3D) What would be the best way to go about this?"}
{"id":"104393","title":"Does a high energy photon experience deceleration or direction-change when it impinge into water or something due to the change of refraction index?","text":"Wave experiences refraction when it propagates into another medium which has different refraction index. Lights surely does experience refraction at the border of mediums which have different refraction index because of its wave- like property. But I am wonder if high energy photons, whose energy exceeds MeV or GeV, also experiences refraction."}
{"id":"24958","title":"How can a black hole emit X-rays?","text":"Considering that a black hole's gravity prevents light from escaping, how can a black hole emit X-rays? Visible light and X-rays are both electromagnetic radiation, shouldn't the black hole's gravity prevent X-rays from escaping?"}
{"id":"24955","title":"What happen when two galaxies collide?","text":"As the title suggests, I know collision of galaxies happens. Questions are   1. Why do they collide (obviously because of gravity, but in what conditions)?   2. Aftermath of the collision?   3. If there is a super massive blackhole in the center of the galaxies, what happen to them?"}
{"id":"54758","title":"Why do we need different ensembles in statistical mechanics?","text":"Why do we study these different ensembles, microcanonical, canonical, grand canonical ensemble ? Are they used for studying different physical system or scenarios?(e.g. in some system you can only treat it as mirocanonical, and in other cases you can only apply canonical ensemble) Do they have the same result at thermodynamic limit?"}
{"id":"79836","title":"Transformation rule of a partial derivative","text":"We know the following transformation rule: $$ \\partial'_b = \\frac{\\partial}{\\partial x'^b} = \\frac{\\partial x^c}{\\partial x'^b} \\, \\frac{\\partial}{\\partial x^c} = \\frac{\\partial x^c}{\\partial x'^b} \\, \\partial_c. $$ Here $\\partial_c$ is covariant tensor of rank $1$. But what is the corresponding transformation rule for $$ \\partial^m :=g^{mn}\\partial_n?$$ How can we derive the transformation rule?"}
{"id":"54759","title":"References on wave solutions in continuum mechanics","text":"I am interested in literature on known wave solutions in continnum mechanics, precisely the following mechanical equation: $$\\rho\\partial_t^2u_i = C_{ijkl}\\nabla_j\\nabla_ku_{l}$$ My interest is spread around its study for cubic lattices, which doesn't seem straightforward to solve analytically to me. But I couldn't find a reference book or articles illustrating a peculiar solution of it."}
{"id":"116749","title":"Derivation of neutrino oscillation phase factor","text":"As we know, the neutrino $\\nu_{\\alpha}$ with flavor $\\alpha=e,\\mu,\\tau$ is a linear combination of mass eigenstates: $$ |\\nu_{\\alpha}\\rangle=\\sum_iU_{\\alpha i}|\\nu_i\\rangle,\\quad i=1,2,3 $$ where the propagation of the mass eigenstates $|\\nu_i\\rangle$ can be described by plane wave solutions: $$ |\\nu_{i}(t)\\rangle = e^{ -i ( E_{i} t - \\vec{p}_{i} \\cdot \\vec{x}) } |\\nu_{i}(0)\\rangle. $$ Suppose that $t=T$ and $|\\vec{x}|=L$. Then the phase factor can be given by $$ \\begin{split} E_iT-p_iL &= E_i\\frac{L}{\\sqrt{1-m_i^2\/E_i^2}}-\\sqrt{E_i^2-m_i^2}L \\\\\\ &=E_iL\\left(1+\\frac{m_i^2}{2E_i^2}\\right)-E_iL\\left(1-\\frac{m_i^2}{2E_i^2}\\right) +\\mathcal{O}\\left(\\frac{m_i^4}{E_i^3}\\right) \\\\\\ &=\\frac{m_i^2L}{E_i}+\\mathcal{O}\\left(\\frac{m_i^4}{E_i^3}\\right) \\end{split} $$ However, this result is not correct. The artilce Neutrino oscillation on Wikipedia says that $$ |\\nu_{i}(L)\\rangle = e^{ -i m_{i}^2 L\/2E }|\\nu_{i}(0)\\rangle $$ which differs in a factor of $1\/2$. I can't figure out what is wrong with my derivation above (Maybe the formula for time interval $T$ is not correct, but I am not sure). Any help is greatly appreciated."}
{"id":"104649","title":"Information carried by single photon","text":"In Quantum Information we can use photons for quantum bits (qubits). What i often read is that each photon can carry one unit of information, i.e. using the polarization state of a single photon.   I have two questions:   1) i read on this article that it is possible to send 1.63 bits of information per photon, what does that mean?   http:\/\/www.newscientist.com\/article\/dn13522-twisting-light-packs-more- information-into-one-photon.html#.Uy1cKBx22QE   2)If i can take a single photon state as a tensor product of its polarization state and its orbital angular momentum state (we could also add a frequency state), can i say that i am sending 2 (or 3) qubits of information in one single photon?   Thank you"}
{"id":"105511","title":"Equivalent Chern Simons Theories","text":"This is a follow-up question to FQH Edge Theory as decoupled chiral bosons . The document that I will be refering to is http:\/\/dao.mit.edu\/~wen\/pub\/toprev.pdf . On page 14 in Eq.(2.33) the author argues that two Chern-Simons theories describe the same \"Fractional quantum hall physics\" when the K-matrices are the same up to conjugation with a SL(n,Z)-matrix. Also the charge vector of the two theories has to transform in a specific way. At the bottom of page 31 the author outlines a way to simultaneously diagonalizes the K-Matrix and the V-Matrix in the edge theory. Here however there doesn't seem to be a restriction that the conjugation matrices need to be in SL(n,Z). So I am wondering whether one may \"accidentaly\" perform a transformation on the fields that may leed to a theory describing a different fractional quantum hall state? For example we choose $K=\\left( \\begin{matrix} 1 & 2 \\\\\\ 2 & 1 \\end{matrix} \\right)$ to describe a $\\nu=2\/3$ state. We diagonalize the matrix by conjugating it first with $\\Lambda_{1}=\\frac{1}{\\sqrt{2}}\\left( \\begin{matrix} 1 & 1 \\\\\\ 1 & -1 \\end{matrix} \\right)$ and then with $\\Lambda_{2}=\\left( \\begin{matrix} \\frac{1}{\\sqrt{3}} & 0 \\\\\\ 0 & 1 \\end{matrix} \\right)$. This yields a new K-Matrix K'=diag(1,-1). Then I am performing a Lorentz transformation that leaves K invariant and diagonalizes V. However none of these matrices is in SL(2,Z). (Also generically $U^{T}=U^{-1}$ is false in SL(2,Z).) I would be very happy if someone could clear up this confusion for me:) I am looking forward to your responses."}
{"id":"105518","title":"Why is the water on one side of a bridge choppy?","text":"My daily commute take me across the Evergreen Point Floating Bridge on Lake Washington. I have noticed on several occasions that the water on the southern side of the bridge will be quite choppy while the water on the northern side is relatively still (not completely still but there is a noticeable difference from the other side). What causes this to happen?"}
{"id":"134967","title":"What happens to objects pulled in by black hole?","text":"What happens to the particles\/elements\/objects that sink into the gravitational pull and ultimately go to the interior part of a black hole? If, according to popular notion, it is crushed by the excessive gravitational high density thrust, what is that object reduced to? Which theory provides a basis for calculation of this effect?"}
{"id":"134960","title":"How is Meissner effect explained by BCS theory?","text":"Someone says we can derive the GL equations from BCS theory, which can explain Meissner effect, but I want a more clear physical picture of this phenomena."}
{"id":"127851","title":"Moon's orbit period as seen from a spaceship traveling at 0.8c","text":"I am studying special relativity and I am trying to figure out the following small problem which occurred to me: An observer, the pilot of a spaceship flying to or from earth at v = 0.8c, is observing earth from his spaceship. He observes the moon orbiting the earth once every 45 days, 1.667 times slower than the period seen from earth. He also observes the earth is 1.667 times more massive than it seems to a person on earth. However, doesn't that mean the moon should orbit earth faster than normal, not slower? What is the solution to this problem?"}
{"id":"8143","title":"Fields and Newton's Third Law","text":"I'm studying basic physics. I'm using the text available at http:\/\/www.anselm.edu\/internet\/physics\/cbphysics\/downloadsI.html. It develops the universal law of gravitation by postulating the existence of a vector at each point of the form $$ g_P = \\sum G\\frac{m_i}{|r|_i^3}r_i, $$ Where $m_i$ and $r_i$ are the mass of and separation vector from $P$ for all particles that aren't at the point $P$. It examines the effect of one particle on another. If the separation vector is $r$ then from the above equation, we see that $$ g = G\\frac{m_1}{|r|^3}r $$ and that when a particle of mass $m_2$ is placed at the given point, the force will be $$ F = G\\frac{m_1m_2}{|r|^3}r $$ The authors then go on to claim that we can either repeat the development to see the effect of the second particle on the first one or apply Newton's third law. How does Newton's third law apply through a field? If it's the field exerting the force, then Newton's third law would require a force on the field and not the object \"generating\" the field, correct?"}
{"id":"99154","title":"What does k denote in this equation?","text":"Recently, I was attempting to compare the rate of flow of different liquids using the formulas given below: $$P = \\rho g h$$ $$ \\frac{dh}{dt}\\varpropto h$$ $$ \\frac{dh}{dt} = -k h $$ $$ \\frac{dh}{h} = -k dt $$ $$ \\int \\frac{dh}{h} = -k \\int dt$$ $$\\ln h = -k t + c$$ My question here is: What does $k$ denote? Is it a physical quantity like density?"}
{"id":"99158","title":"Finding electric field between overlapping surfaces","text":"The problem is: > A sphere with radius R is centered at the origin, an infinite cylinder with > radius R has its axis along the z axis, and an infinite slab with thickness > 2R lies between the planes z=−R and z=R. The uniform volume densities of > these objects are ρ1, ρ2, and ρ3, respectively. The objects are superposed > on top of each other; the densities add where the objects overlap. > > How should the three densities be related so that the electric field is zero > everywhere throughout the volume of the sphere? What do you think the ratios > $\\frac{\\rho_{1}}{\\rho_{2}}$, $\\frac{\\rho_{1}}{\\rho_{2}}$ and > $\\frac{\\rho_{1}}{\\rho_{2}}$ are equal to? > > Hint: Find a vector expression for the field inside each object, and then > use superposition. ![enter image description here](http:\/\/i.stack.imgur.com\/wmKpo.jpg) The way I approached the problem: Find individual Electric fields and add them up to 0. Find the relation between the charge densities. To do that, I used Gauss' law. I bounded the region (the sphere) and that gives me $E_{sphere} = \\frac{\\rho_{1}R}{3\\epsilon_{o}}$. However, because my region of interest is the sphere, I used the same bounding surface for the cylinder and the slab as well thereby getting the equations $E_{cylinder} = \\frac{\\rho_{2}R}{3\\epsilon_{o}}$ and $E_{slab} = \\frac{\\rho_{3}R}{3\\epsilon_{o}}$ Then, since the $E_{net} = 0$, $$E_{sphere} + E_{cylinder} + E_{slab} = 0$$ which, when simplified, gives $$\\rho_{1} + \\rho_{2} + \\rho_{3} = 0$$ However, I'm not sure if a) what I've done so far is correct, b) how to proceed from here to calculate the ratios of the charge densities. Any help would be really appreciated. Thank you."}
{"id":"52848","title":"Falling through the rotating Earth","text":"Suppose you were standing on the rotating Earth (not necessarily Equator or the poles) and suddenly your body lost the ability to avoid effortlessly passing through solid rock. Because the earth's rotation at the surface is considerably below escape velocity, you would slip below the earth's surface. If the earth's gravity were a consequence of a central point mass, you'd have an elliptical orbit (mostly) within the earth. With a planet of constant density, the gravity you feel underground is equivalent to standing on the surface of an identically dense planet with a radius equal to your current distance from the centre. So effectively, as you fall the gravity you experience lessens.   1. What would be the shape of the trajectory?    2. How close would you get to the centre?    3. How long would it take before your orbit brought you back to the surface (assuming no losses and a stationary planet)?   4. What complications would arise from the planet being in orbit around a star?   5. Bonus points for finding two well-known places on the Earth that you could travel between in one \"orbit\" using this method."}
{"id":"18430","title":"2d simulation shooting projectiles that inherit the gun's veocity at a moving target while also moving","text":"Overview: I'm working on programming a simulation that requires 'shooting' projectile-type objects at other moving objects. How can I calculate the angle at which to shoot the object to hit? Details: Imagine you're holding some gun-type object that shoots projectiles at velocity $v_d$ in straight lines. You are moving with velocity $v_p$ at angle $\\theta_p$, in a straight line. Another object, which we will name the target, is located a distance $R$ away from you at an angle $\\theta_{tp}$, and is moving in a straight ine with velocity $v_t$ at angle $\\theta_t$. I need to calculate the angle $\\theta_a$ I should aim the gun in order to hit the target. Note that there is no decision on the TIME to fire. I have to fire immediately, at $t=0$, so this adds a helpful constraint but also makes it so that some targets are not able to be hit based on the numerical values of the parameters mentioned, which is fine. The target is moving so you need to lead it with your shot to compensate. Also, the projectile will inherit your velocity on top of its own since you are also moving, which will further modify the angle you need to shoot at. It seems to me that the strategy should be that I find a point in space which will be occupied by both the projectile and the target. I know all the points in space that the target will take along its linear path, and the times at which it will take all points. However, because the total velocity of the projectile varies based on the angle you shoot it (e.g. the magnitude of the velocity vector will be different depending on what Theta_a you decide to shoot at), I'm not sure how to figure out where to shoot it. Any help would be appreciated! In addition, if it is easy, a solution to a target moving in a circular path with radius $R$ (same $R$ as the distance between you and it, so it's making a perfect circle around you) would be appreciated!"}
{"id":"33451","title":"direct conversion of heat to electric potential \/ current - are there theories that contradict the 'Kelvin Statement'?","text":"In particular, William Thomson (Kelvin) appeared to be wrong about key things in physics (initially X-rays, aether, even aviation feasibility).   en.wikipedia.org\/wiki\/William_Thomson,_1st_Baron_Kelvin#Pronouncements_later_proven_to_be_false Are there any theories that contradict the idea that a heat differential must exist in order to convert heat energy to another form (causative) vs a heat differential being a secondary effect? I have seen some discussion about the potential for room-temperature thermionics (and whether or not that by definition relies on a heat differential to function?) jap.aip.org\/resource\/1\/japiau\/v94\/i7\/p4690_s1 If such a device existed to convert heat without a 'classical' heat differential being a primary causative of the conversion, what implications would it have for irreversibility? How does a purely homogeneous heat \/ kinetic energy pool in the classical sense behave when considering different atoms \/ elements with the same kinetic energy at the atomic level? Is it possible that different elements can never truly have the same kinetic energy in such a case, causing a situation where an exploitation can be found to do the conversion of energy? One hypothesis could be that heterogeneous atomic mixtures is the true cause of brownian motion (unless this has been proven to not be the case). In other words, is it not the case that homogeneous bulk matters behave predictably at any given kinetic energy level (heat)? This would have implications for the brownian ratchet being based on randomness, when in fact physical matter could potentially be biased to a predictable movement (even if only in a very confined system, i.e. nano-scale), where possibly extremely small kinetic differences can be engineered and exploited to generate electricity (or mechanical movement useful for work) from a single reservoir of heat, disproving the Kelvin statement."}
{"id":"71603","title":"What is the probability density function over time for a 1-D random walk on a line with boundaries?","text":"If a single particle sits on an infinite line and undergoes a 1-D random walk, the probability density of its spatio-temporal evolution is captured by a 1-D gaussian distribution. \\begin{align} P(x,t)&=\\frac{1}{\\sqrt{4 \\pi D t}}e^{-\\frac{(x-x_0)^2}{4Dt}} \\end{align} However, suppose there are impassable boundaries on the line; on one side, or on both sides. Are there any boundary conditions for which there exists a closed form probability density function for how this particle will behave over time? Any references to such solutions would be extraordinarily helpful. EDIT. Attempting to generalize Emilio's result below for an arbitrary initial particle position $-L\/2 < x_0 < L\/2$. I had to work it out by example. I found the following \"images\" were required to account for reflections of an off-center particle at position $x_0$: for the first and second reflections on both sides the new gaussians had to be centered on ($-2L+x_0$, $-L-x_0$, $x_0$, $L-x_0$, $2L+x_0$). From the pattern I think the full solution can be expressed, for all integers $n$, as: \\begin{align} P(x,t)&=\\frac{1}{\\sqrt{4 \\pi D t}}\\sum_{n=-\\infty}^\\infty e^{-\\frac{(x-nL-(-1)^nx_0)^2}{4Dt}} \\end{align} where the old $x_0$ is now defined as $nL+(-1)^{n}x_0$"}
{"id":"30185","title":"Why is radioactive decay dependent on amount of substance available?","text":"Radioactive decay is an attribute of unstable nucleus. When we represent it in equation, we don't involve any macroscopic attribute of substance. But still, rate of radioactive decay is proportional to amount of substance available (a macroscopic attribute). Why? What's really the relation with others when it comes to bleeding out its unstability?"}
{"id":"76377","title":"Is it possible to float, starting at an elevated point, with a board and enough strength?","text":"Premise: I have a structure like this.                  O      _____     |     |     |     |      I'm the `O`, and I'm sitting on a flat surface or board while being supported by two poles. (it looks like I'm floating, but just imagine I'm sitting. it's hard to draw stuff with only characters! :P) Now let's say one of the poles is removed.                  O      _____           |           |      Of course, if nothing else happened then the structure would fall over. But what if I pulled up on the board?                            O\/ < left arm     right arm > \/______                 \\\/ ^for|ce                        |      It's not a very good drawing, but my right arm is below the board, pulling up on it. Assuming that I was limitlessly strong, is it possible to not fall over? If so, is this possible?                             O     right arm > \/___\\ < left arm                 \\\/ \\\/      In other words, **is it possible to float, assuming you already start out elevated, with enough strength?** If not, why? My research has been quite useless, although I think I have found the name for this kind of thing: Bootstrapping. That article is not particularly helpful though. This question looks related, although I want to know whether this is theoretically possible with the aid of a board and when you are already elevated."}
{"id":"30189","title":"Euclidean Geometry in Classical Thought - Realization or Representation?","text":"First post! :] This has been bothering me for a while now: [Taken from John J. Roche's \"The Mathematics of Measurement: A Critical History\"] > When physico-mathematicians in the seventeenth century needed to represent > their physical quantities mathematically they almost always turned to > geometry rather than to numbers. The author goes on to explain this further, saying how the Ancient Greeks had used geometrical figures to represent physical quantities rather than numbers (influencing the physicists later on) and how they were most appropriate for representation since they denoted \"continuous physical quantities\". I don't understand this; what makes physical quantities continuous? Wouldn't you figure them to be discrete considering that they are quantified by units? More importantly, is the author saying that physicists and mathematicians realized physical relationships geometrically? Or is he just saying that they merely chose to _represent_ physical quantities and their relationships to others with geometrical figures? It seems extremely non-intuitive to view physical quantities as being segments\/planes\/solids instead of as quantifiable values with defined dimensions. Of course it's easier to view\/realize proportionalities between physical quantities through geometrical\/graphical means, but there must have been a numerically-based intuition even _before_ this. In other words in finding a relationship between 2 quantities, there must have been a numerically-based line of thought that preceded a geometrical _demonstration_ of it, simply through my reasoning that it is more intuitive. I wanted to discuss this though, to see if anyone had any input\/corrections to make about my thoughts here; I'd love to hear what you guys have to say :] Here's a direct link to the book I'm referring to (Google Books). The quote stated above is on page 87, and it continues on from there. Thank you!"}
{"id":"107323","title":"Angular momentum of two rotating spheres","text":"I am trying to calculate an instantaneous merger of two rotating spheres into one. Two spheres each rotating around their own axis of rotation (which are generally not aligned) and moving relative to each other. Imagine two entirely inelastically colliding planets. While I can calculate their individual spin angular momentum and the orbital angular momentum, it is unclear to me how to combine this into one (spin) angular momentum for the resulting sphere. It seems to me I cannot simply add the four vectors together... or can I? After merging the two into a single body maintaining the initial momentum, can I calculate the released energy (in the inelastic collision) simply as the difference in kinetic energy, using the same reference frame? I realize this should be pretty trivial, but I have thus far not been able to neither derive nor locate and answer."}
{"id":"107601","title":"Question regarding 2 conducting spheres connected by a long fine line (electric potentional)","text":"I'm having a bit of trouble understanding a pretty simple issue. Assuming I have 2 conducting spheres uniformly charged connected by a long fine line (as shows in the added photo) and im being asked about what will happened, how much charged will move from on spheres to the other. I was told that the electric potential of these 2 spheres are: $k\\frac{Q_1}{R_1}$ and $k\\frac{Q_1}{R_1}$ The thing is, that I don't get why, I mean, $R_1$ and $R_2$ are the radiuses of the 2 spheres, these are not vectors or anything and they do not meet the same coordinates and this is what I should take when calculating the potential using integral the good way... So, how can $k\\frac{Q}{R}$ be the potential of the spheres? ![enter image description here](http:\/\/i.stack.imgur.com\/aDzN4.gif)"}
{"id":"18387","title":"Physics Paradox about Newtons Second Law $F=ma$","text":"I randomly thought of this- I'm surprised it took me so long to think of it, and I'm SURE I'm making a simple oversight of something completely trivial, but here it goes. (Im only in high school taking normal physics, so don't laugh if this is stupid.)   1. If force equals mass times acceleration, wouldn't a basketball dropped from the top of the Eiffel tower exert the same force on the ground as a basketball dropped a foot off the ground? They both have the same mass, and they both are accelerating towards the ground at a rate of g=9.81 m\/s^2. (I don't know what terminal velocity is that well as im only in physics 1 in highschool, but just assume that air drag is not important and the ball doesn't reach terminal velocity.)   2. Also, if a ball IS dropped high enough to reach terminal velocity, then it accelerates at 0 m\/s^2, so it has a force of ZERO when it hits the ground?"}
{"id":"33989","title":"what is the relationship between these two sorts of anomalies?","text":"Recently there has been a few questions about anomalies in QFTs: Why do some anomalies (only) lead to inconsistent quantum field theories Classical and quantum anomalies In these, people have been discussing anomalies that (let me know if I am wrong in my understanding here) come from essential symmetries which cannot be realised by finite energy unitary representations on a suitable Hilbert space. Meanwhile, my understanding of \"anomaly cancellation formulas\" that appear for instance in 11d supergravity is that, a priori the action functional is not an honest function on the configuration space of the theory, but a section of a line bundle over the config space. Nontriviality of the line bundle prevents integration of the action functional. This is my definition of an anomaly. Anomaly cancellation happens for example when the bosonic and fermionic sectors have action functionals in dual line bundles, so that in the tensor product theory there is no anomaly, though either sector is by itself anomalous. My question is : what is the relationship (if any) between these two perspectives? I suspect there is some on the basis of the Borel-Weil-Bott theorem, but this is idle speculation. Thanks."}
{"id":"63568","title":"Finding Surface Tension of water at certain Temperature and Pressure","text":"So the question is: > Using the Young-Laplace Equation (if applicable), find the surface tension > (dynes\/cm) for water at 20 degrees Celsius with 2.5 psi. Round to the > nearest tenth. Well, I didn't use the Young-Laplace equation, not sure if needed though. What I did was use the Eötvös rule and it's special case for water to solve the question. The equation is: $$\\gamma = 0.07275\\;\\frac{N}{m}\\;\\times\\;(1-0.002\\times(T-291K))$$ What I did was convert 20 Celsius to Kelvin (293K) and then put in the equation to get: $$\\gamma = 0.07275\\;\\frac{N}{m}\\;\\times\\;(1-0.002\\times(293K-291K))= 0.072459\\frac{N}{m}$$ However, I think I may be wrong as this does not account for pressure at all. Which ends up becoming about $72.46\\frac{dynes}{cm}$ Am I right or wrong? And is there a better\/correct way of doing this?"}
{"id":"44453","title":"Is there a \"present state\" of distant stars if simultaneity is relative?","text":"Special relativity theory says simultaneity is relative, meaning that different observers will not agree on what happened first and what second. Does it then make sense to say that looking at distant stars, we see them how they looked \"billions of years ago\" and not how they look now? Does it make sense to talk about what these stars look like _now_? How do we define this \"now\" if simultaneity is relative?"}
{"id":"52","title":"How do 2 magnets spin by themselves if positioned precisely?","text":"A few years ago I went to a museum, where there was a board with 2 bar magnets, on a pole each so they could rotate. If you rotated them so the lined up with the same poles (N) facing each other, they'd repel spinning until the south poles lined up, then they'd repel and spin so the north poles lined up and so. This was without any extra help, once you lined them up they would just keep spinning by their selves. How exactly is this possible that the 2 magnets kept producing kinetic energy?"}
{"id":"68241","title":"Are higher order mixed partial derivatives of wave function with different ordination equal?","text":"For example, given two operators:   $$A = \\frac{\\partial}{\\partial x}+\\frac{\\partial}{\\partial y},$$   $$B =\\frac{\\partial^2}{\\partial x^2}+\\frac{\\partial^2}{\\partial y^2} + 1.$$ Deriving commutator $[A,B]=0$ involves mixed partial derivative of higher order with opposite signs, and it is very temptative to zero them to make commutator equals zero. But is this allowed, i.e., does any valid quantum wave function must have continues (mixed) partial derivatives of higher order so one can use Schwartz theorem?"}
{"id":"79354","title":"Coulomb force on the center of a hexagon?","text":"Six point charges $q$ are at the corners of a regular hexagon that has sides of length $a$. What is the force on another charge $Q$ which is located in the center of the hexagon? What is the force on $Q$ if we remove one charge $q$ from one corner? * * * I know I'm supposed to use Coulombs law and that the force on $Q$ is the sum of all other forces from the six point charges $q$. But how do I compute the forces on each of the six points? Thanks much."}
{"id":"77139","title":"Is there valid physics behind the bodybugg?","text":"The bodybugg is a faddish gadget whose marketers claim it can measure your body's daily energy expenditure. Their sales literature says: > As reported in the British Journal of Sports Medicine, 2008 edition, the > bodybugg® armband achieved average accuracy rates of 90% in studies > comparing calorie expenditure measurements from the bodybugg device to > calorie expenditure measurements in a metabolic chamber (the \"gold standard\" > of calorie expenditure measurement). \"Accuracy rates of 90%\" is obviously not how anyone with any level of science literacy would describe error bars. Their FAQ says: > In addition to measuring motion and steps, the bodybugg is able to see how > much you are sweating, your skin temperature and the rate at which heat is > being dissipated from your body. They also say it doesn't measure your heart rate. Is there valid physics behind this device, and if so, what principles would it be based on? Measuring \"motion and steps\" would seem to involve an accelerometer, but it seems unlikely that that would be able to tell the difference between, say, walking on a horizontal sidewalk and walking uphill. How can they separately measure skin temp and rate of heat dissipation? Maybe there's a thermometer in contact with your skin and one sensing the air temperature? It's not obvious to me how one could combine the mechanical data with the thermodynamic data."}
{"id":"22974","title":"How can a Higgs decay to heavier products than its mass?","text":"How is it possible that a higgs at ~125 GeV can decay into 2 W bosons @ ~ 80 GeV a piece (for example)? Shouldn't a particle only be allowed to decay to lighter particles + energy? Diagram copied from this question ![enter link description here](http:\/\/i.stack.imgur.com\/eljaD.jpg)"}
{"id":"103601","title":"Can the two electromagnetic field tensors be combined into a more general tensor?","text":"Given the electromagnetic field tensor $$\\begin{align} F_{\\mu\\nu} = \\begin{pmatrix} 0 & -E_{x} & -E_{y} & -E_{z} \\\\\\ E_{x} & 0 & B_{z} & -B_{y} \\\\\\ E_{y} & -B_{z} & 0 & B_{x} \\\\\\ E_{z} & B_{y} & -B_{x} & 0 \\end{pmatrix} \\end{align}$$ and its dual $$\\begin{align} G_{\\mu\\nu} = \\begin{pmatrix} 0 & B_{x} & B_{y} & B_{z} \\\\\\ -B_{x} & 0 & E_{z} & -E_{y} \\\\\\ -B_{y} & -E_{z} & 0 & E_{x} \\\\\\ -B_{z} & E_{y} & -E_{x} & 0 \\end{pmatrix} \\,, \\end{align}$$ does there exist a general tensor $H_{\\mu\\nu}$ that combines $F_{\\mu\\nu}$ and $G_{\\mu\\nu}$ into a more general tensor in Minikowski space?"}
{"id":"128930","title":"Quantum Mechanical States","text":"What can be the precise answer to the question that > Quantum states are complex and infinite dimensional. Why is this so? Is it because they belong to the complex Hilbert space? Even if they belong to the Hilbert space, why they are infinite dimensional, is this because of the projection operator?"}
{"id":"111365","title":"What should I look for at SAR levels for cell phones?","text":"I've been trying to wrap my head around SAR levels when comparing cell phones. Considering that I spent most of my time not talking on the phone but carrying it around near my body, would a phone **lower body** value but **higher head** value be preferable one with the reverse combination? Or, suppose one phone has a value that is 2x of another. How should I think of it? Is it significantly worse or slightly worse?"}
{"id":"135175","title":"Work done against gravity","text":"The work done against gravity is $mgh$, well at least that's what my textbook says. I have a question: I can apply a force say 50N, so total work done = $mgh + mah$. Where $ma$ = Force. But the truth is irrespective of the force applied, the work done against gravity is always $mgh$. Why? For example, when I move an object with a force, the work done is more, so work depends on the Force. But in case of gravity it is always depends upon the weight"}
{"id":"6345","title":"Do photons interact with each other, or with themselves only?","text":"Even if the double slit experiment gives interesting (weird) results, it only concludes that each photon interacts with itself after passing the two slits. I have been thinking about a different experimental setup, where you have two well defined light sources (with specific wave lengths and phase) but no slits. And now to my questions: Has anyone ever done such an experiment, and will there be an interference pattern on the wall? If the answer to the second question is \"no\", light can not be a true wave - it only has some wavelike properties. But if it is \"yes\", things become much more interesting. If there is an interference pattern on the wall, there has to be an interference pattern even if both light sources are emitting single photons at random, but as seldom as, say, once per minute. That in turn would mean that the photons know about each other, even if they are separated in time with several seconds, and the light sources are independent (not entangled)."}
{"id":"107275","title":"Thomson's jumping ring experiment","text":"For the jumping ring experiment (I think it is also called Thomson ring experiment) why does the ring float and not move up and down the ring as would be expected, I think by Lenz's law, since would the ring when the current thru the coil is decreased (due to it been ac) not fall back down to increase the magnetic flux thru it that has been lowered due to the smaller current??"}
{"id":"134226","title":"Why does Snell's law not factor in wavelength?","text":"If dispersion is caused due to different wavelengths bending with different angles, how is Snell's law right in generalizing it as $\\sin(i) \/ \\sin(r)$? Or am I missing something?"}
{"id":"44125","title":"The Deflecting System in a Hot Cathode Ray Tube","text":"In an HCR-Tube, the deflecting system used to deviate the _electron_ beam is made of _positively_ charged plates. How is this justified? If, due to some malfunction, the electron beam deflects from its original path before it is actually supposed to, won't it stick to the plates? And why is it not wiser to use _negatively_ charged plates? If this is done, then the electrons of the beam will return to their stream even if they are accidentally deflected, and when the actual deflection is required positive plates can be introduced.   So why only positive plates?"}
{"id":"94585","title":"Dealing with dirac notation with regards to different basis'","text":"So this should be a pretty simple question. So we say that $\\langle x | \\psi \\rangle = \\psi(x)$. In other words $\\psi(x)$ is the ket $|\\psi\\rangle$ expressed in terms of the $x$ basis. Now suppose I wanted to get $\\psi(x+a)$. How would I act on $|\\psi\\rangle$ to get it? I think I could just do $$ \\langle x' | \\psi \\rangle = \\psi(x+a)$$ Where $x' = x+a$. But I don't really know what this means in terms of how $\\langle x' |$ and $\\langle x |$ are related, or the formal justification for this. Perhaps my trouble is I don't thoroughly understand what the $x$-basis is or what $\\langle x | \\psi \\rangle$ is really doing."}
{"id":"38177","title":"Lorentz transformations for sine wave in random direction","text":"Hello i have been struggling to derive Lorentz transformations for a sine wave, which is traveling at random direction. I started by prooving that phase $\\phi$ is invariant for relativity and that equation $\\phi = \\phi'$ holds. By using the above equation i am now trying to derive Lorentz transformations for angular frequency $\\omega$, and all three components of the wave vector $k$, which are $k_x$, $k_y$ and $k_z$. This is my attempt: \\begin{align}\\phi' &= \\phi\\\\\\ \\omega'\\Delta t' + k' \\Delta r' &= \\omega \\Delta t + k \\Delta r\\\\\\ \\omega'\\Delta t' + [{k_x}' , {k_y}' , {k_z}'] [\\Delta x' , \\Delta y' , \\Delta z'] &= \\omega \\Delta t + [k_x , k_y , k_z][\\Delta x , \\Delta y , \\Delta z]\\\\\\ \\omega' \\Delta t' + {k_x}'\\Delta x' + {k_y}' \\Delta y' + {k_z}' \\Delta z' &= \\omega \\Delta t + k_x \\Delta x + k_y \\Delta y + k_z \\Delta z \\end{align} Now with just the left side: \\begin{gather} \\omega' \\gamma \\left(\\Delta t - \\Delta x \\frac{u}{c^2}\\right) + {k_x}' \\gamma \\Bigl(\\Delta x - u\\Delta t \\Bigl) + {k_y}' \\Delta y + {k_z}' \\Delta z\\\\\\ \\gamma \\left(\\omega' \\Delta t - \\omega' \\Delta x \\frac{u}{c^2}\\right) + \\gamma\\Bigl({k_x}' \\Delta x - {k_x}' u \\Delta t \\Bigl) + {k_y}' \\Delta y + {k_z}' \\Delta z\\\\\\ \\gamma \\left(\\omega' \\Delta t - {k_x}'c\\, \\, c \\Delta t \\, \\frac{u}{c^ 2}\\right) + \\gamma \\Bigl({k_x}' \\Delta x - \\frac{\\omega'}{c} u\\frac{\\Delta x}{c} \\Bigl) + {k_y}' \\Delta y + {k_z}' \\Delta z\\\\\\ \\Delta t \\, \\gamma \\Bigl(\\omega' - {k_x}' u \\Bigl) + \\Delta x \\, \\gamma \\Bigl({k_x}' - \\omega' \\frac{u}{c^2} \\Bigl) + {k_y}' \\Delta y + {k_z}' \\Delta z \\end{gather} From this I can write down the Lorentz transformations. \\begin{equation} \\begin{split} \\gamma\\Bigl(\\omega' - {k_x}' u \\Bigl) &= \\omega\\\\\\ \\gamma \\Bigl({k_x}' - \\omega' \\frac{u}{c^2} \\Bigl) &= k_x\\\\\\ {k_y}' &= k_y\\\\\\ {k_z}' &= k_z\\\\\\ \\end{split} \\end{equation} My professor said that my signs are wrong, but what am i doing wrong?"}
{"id":"106896","title":"Entanglement entropy and area law","text":"I am currently reading a review \"Area law for the entanglement entropy\" by Eisert, Cramer and Plenio (2010). From what I understand:   1. In one dimension, for local gapped models, we have an area law for entanglement entropy.   2. In one dimension, some models with long range interactions + all critical systems obey a log(N) law, where N is the size of the subregion of the 1D space, the pre-factor depends for CFT models on the central charge.   3. For higher dimensions, local gapped (equivalently quasi-free) models obey an area law.   4. For higher dimensions, critical bosonic models obey an area law, fermionic ones obey a divergent log law with pre-factor dependent on the topology of the Fermi surface (which is also probably related to the central charge of the CFT, but no exact results are known). However I am a bit confused about the treatement of the Klein-Gordon field. The massive Klein-Gordon field is a local gapped model, hence not critical. Papers by Bombelli and Srednicki show that entanglement entropy obeys an area law for dimension greater than 2, in agreement with point 3. However in the review they seem to say that this is a critical system. Thanks to point 4 there is still no contradiction with the area law result. However, this becomes problematic for the 1D case: a massive or massless KG system obeys a divergent log law, as shown here: http:\/\/arxiv.org\/pdf\/hep- th\/9401072.pdf and in the review. To me this is only true if the field is massless. A massive KG field is a local gapped system, so not critical and hence it should follow an area law according to point 1, not a divergent log law. So it seems to me that the problem is that these papers consider the KG field (massive or massless) as critical, whereas to me only a massless KG field is critical. What did I misunderstand?"}
{"id":"70216","title":"What is the geometry of our universe?","text":"I have some questions about the notion of space in current cosmological theories. I have been trying to decipher a few papers written on this topic such as this one, but I keep running into some trouble. Most of the papers I have been skimming talk about anti-de Sitter space (AdS) and about how the best definition of nothing is > _the limit of anti-de Sitter space in which the curvature length goes to > zero._ This is great and all, but I have read on Wikipedia that AdS implies a negative cosmological constant, which would imply that dark energy should be attractive. Is my reasoning here incorrect? If not, could someone explain why cosmologists spend so much time devoted to studying AdS when the universe is clearly expanding? I have read somewhere about local geometry vs. global geometry, but I wasn't sure if that is what is going on here. The papers are so complex."}
{"id":"71168","title":"Divergence theorem in complex coordinates","text":"This question is related to Stokes' theorem in complex coordinates (CFT) but, I still don't understand :( Namely how to prove the divergence theorem in complex coordinate in Eq (2.1.9) in Polchinski's string theory $$\\int_R d^2 z (\\partial_z v^z + \\partial_{\\bar{z}} v^{\\bar{z}})= i \\oint_{\\partial R} (v^z d \\bar{z} - v^{\\bar{z}} dz ) (1) $$ I may try $$ \\int_R dx dy ( \\partial_x F_y - \\partial_y F_x) = \\oint_{\\partial R} (F_x dx + F_y dy)(2) $$, but what kind of substitution I should use to get Eq. (1)?"}
{"id":"5462","title":"Does the potential energy for a given photon increase or decrease in quanta?","text":"As a photon leaves a strong gravitational field, it loses energy and redshifts. Is the exchange in potential energy of a photon characterized by energy quanta?"}
{"id":"123843","title":"Existence of negative temperatures and the definition of entropy","text":"How negative temperatures can be possible has been treated on StackExchange before (several times in fact), but in light of some recent academic discussion, most of these answers seem to be possibly wrong or incomplete. The literature I am referring to is Dunkel & Hilbert, _Nature Physics_ **10** , 67 (2014) arXiv.1304.2066, where as I understand it, it is shown that negative temperatures are an artefact of choosing an incorrect definition of entropy. The Wikipedia article on the matter has also been amended to reflect this. This was later challenged by similarly well-known scientists, in arXiv:1403.4299 where it was, among other things, pointed out that this argument is actually decades old (citing Berdichevsky _et al._ , _Phys. Rev. A_ **43** , 2050 (1991)). The original authors quickly countered the arguments made in the comment, by what seems to be a rigorous treatment of the matter, arXiv:1403.6058. The first arXiv comment (arXiv:1403.4299) has been updated since and it still reads that \"Obviously severe points of disagreement remain\". What I am asking, then, is whether someone on StackExchange might be able to shed some light on the matter as to how there can be a disagreement about something that seems should be a mathematical fact. I would also be interested in hearing whether changing the definition of entropy from that of Boltzmann to that due to Gibbs might potentially change any other results. Might for example the Wang-Landau algorithm be affected seeing that it does use the density of states and that you can never simulate infinite systems (although as I understand it, even in the present context with finite scaling you should be able to get consistent results)?"}
{"id":"123846","title":"Intermolecular Hydrogen Bonding","text":"I learnt about intermolecular hydrogen bonding today, which occurs between molecules such as ortho-nitro-phenol. What I was told is that in case of intermolecular bonding, the molecules separate from each other, opposite of what happens during intramolecular hydrogen bonding. I don't understand why is this. Why would attraction within a molecule cause separation of the molecules? There is obviously some intramolecular hydrogen bonding present in this case, but I understand that this might be negligible. Still, why should the molecules spread away from each other?"}
{"id":"87628","title":"Does a four-divergence extra term in a Lagrangian density matter to the field equations?","text":"Greiner in his book \"Field Quantization\" page 173, eq.(7.11) did this calculation: > ${\\mathcal L}^\\prime=-\\frac{1}{2}\\partial_\\mu A_\\nu\\partial^\\mu > A^\\nu+\\frac{1}{2}\\partial_\\mu A_\\nu\\partial^\\nu > A^\\mu-\\frac{1}{2}\\partial_\\mu A^\\mu\\partial_\\nu A^\\nu $   >  $\\space\\space\\space\\space=-\\frac{1}{2}\\partial_\\mu A_\\nu\\partial^\\mu > A^\\nu+\\frac{1}{2}\\partial_\\mu[A_\\nu(\\partial^\\nu A^\\mu)-(\\partial_\\nu A^\\nu) > A^\\mu]$ > > The last term is a four-divergence which has no influence on the field > equations. Thus the dynamics of the electromagnetic field (in the Lorentz > gauge) can be described by the simple Lagrangian > > ${\\mathcal L}^{\\prime\\prime}=-\\frac{1}{2}\\partial_\\mu A_\\nu\\partial^\\mu > A^\\nu$ Yes, **if it is a four-divergence of a vector whose 0-component doesn't contain time derivatives of the field** , indeed according to the variational principle this four-divergence will not influence the field equation. And actually I calculated the time-derivative dependence of 0-component of $[A_\\nu(\\partial^\\nu A^\\mu)-(\\partial_\\nu A^\\nu) A^\\mu]$, in which only $[A_0(\\partial^0 A^0)-(\\partial_0 A^0) A^0]$ could possibly contain time- derivative, which vanishes fortunately, so whatever the general case it doesn't matter in this present case. But how can he seem to claim that **it holds for a general four-divergence term** ,`The last term is a four-divergence which has no influence on the field equations`? EDIT:   I only assumed the boundary condition to be $A^\\mu=0$ at spatial infinity, not at time infinity. And the variation of the action $S = \\int_{t_1}^{t_2}L \\, dt$ is due to the variation of fields which vanish at time, $\\delta A^\\mu(\\mathbf x,t_1)=\\delta A^\\mu(\\mathbf x,t_2)=0$, not having the knowledge of $\\delta \\dot A^\\mu(\\mathbf x,t_1)$ and $\\delta \\dot A^\\mu(\\mathbf x,t_2)$, which don't vanish generally, so the four-divergence term will in general contribute to the action, $$\\delta S_j=\\delta \\int_{t_1}^{t_2}dt\\int d^3\\mathbf x \\partial_\\mu j(A(x),\\nabla A(x),\\dot A(x))^\\mu =\\delta \\int_{t_1}^{t_2}dt\\int d^3\\mathbf x \\dot j^0 =\\int d^3\\mathbf x [\\delta j(\\mathbf x, t_2)^0-\\delta j(\\mathbf x, t_1)^0]$$ which does not vanish in general!"}
{"id":"29914","title":"Is Schrodinger's Cat a real conceptual problem or just a problem with approximations?","text":"In this thought-experiment a cat is placed in a box set with a bottle poison that will release and kill it depending on whether or not a certain radioactive particle decays. The box is kept closed and it is asked \"is the cat alive or dead?\" Since the decay of the particle is a quantum mechanical process- it is represented by a state function. This function remains in a \"superposition state\" until it is observed whether the particle has decayed or not. The cat is presumably and unfortunately entangled with this function- it's own life\/death is in a superposition state until it is indeed observed to have dodged the bullet or not. Schrodinger used this thought experiment in an attempt to show the ridiculousness of certain aspects of quantum theory. His motives were clear in devising it and are well documented in his correspondences with Einstein. My question is not about the cat, the particle, or Schrodinger. I want to know about the box. The box that contains this scenario must effectively isolate the outside observer from any information about what is (or could be is perhaps more correct) going inside of it. So what sort of box would qualify? It seems that such a box could not have any interaction at all with the outside universe- no gravitational or thermal effects- anything. Otherwise would not some kind of information about what was going on inside escape without it being opened? And this being said- were such a box constructed would it not be entirely cut off from reality and how could we be sure it even contained a cat still or was even there at all? Is this experiment possible or are the problems I outline above a deterrent to it's physical possibility?"}
{"id":"118780","title":"Why an unstable particle is not the exact eigenstate of the exact hamiltonian?","text":"According to Srednicki's QFT textbook, page151 \"According to our development of LSZ formula in section5, each incoming and outgoing particle should correspond to a single-particle state that is an exact eigenstate of the exact hamiltonian. This is clearly not the case for a particle that can decay.\" Why an unstable particle is not the exact eigenstate of the exact hamiltonian? Because its eigenvalue of the full hamiltonian is the $m_\\phi$. Where is my fault?"}
{"id":"8282","title":"Why does microwaved food get cold faster","text":"My evidence is entirely anectodal and non-scientific, but I've noticed food gets cold faster when it's been heated in a microwave instead of a stove. Is this true? And if it is, why does it happen?"}
{"id":"109016","title":"What is the electric field between and outside infinite parallel plates?","text":"I know that Gauss's law says $$\\oint_S {\\vec{E} \\cdot d\\vec{A} = \\frac{q_{enc}}{{\\epsilon _0 }}}$$ and that because $\\vec{E}$ is always parallel to $d\\vec{A}$ in this case, and $\\vec{E}$ is a constant, it can be rewritten as $$\\left | \\vec{E} \\right |\\oint_S {\\left | d\\vec{A} \\right | = \\frac{q_{enc}}{{\\epsilon _0 }}}$$ which is also equal to the electric flux through a Gaussian surface. I've included a picture to make it easier to ask my question. ![Parallel plates of uniform, opposite charge densities with Gaussian cylinders I - IV](http:\/\/i.stack.imgur.com\/Mloif.png) What I don't get is how, mathematically, there is no electric field \"outside\" of the plates and how the electric field between them is determined. I - IV are Gaussian cylinders with one face on a plate. $$\\vec{E} = \\vec{E_+} + \\vec{E_-}$$ Where $\\vec{E_+}$ is the electric field from the positive plate and $\\vec{E_-}$ is the electric field from the negative plate. For I: $$\\left | \\vec{E_+} \\right | \\pi r^2 = \\frac{\\sigma \\pi r^2}{{\\epsilon _0 }}$$ $$\\left | \\vec{E_+} \\right | = \\frac{\\sigma}{\\epsilon_0}$$ $$\\left | \\vec{E_-} \\right | \\pi r^2 = \\frac{0}{{\\epsilon _0 }}$$ $$\\left | \\vec{E_-} \\right | = 0$$ This doesn't make sense to me because it's saying that the magnitude of the electric field due to the negative plate is 0, but even if I just assume that's because Gauss's law only works for surfaces that enclose some charge and disregard the 0 I got for the negative electric field, I'm still confused for the following reason: For III: $$\\left | \\vec{E_+} \\right | \\pi r^2 = \\frac{0}{{\\epsilon _0 }}$$ $$\\left | \\vec{E_+} \\right | = 0$$ $$\\left | \\vec{E_-} \\right | \\pi r^2 = \\frac{-\\sigma \\pi r^2}{{\\epsilon _0 }}$$ $$\\left | \\vec{E_-} \\right | = \\frac{-\\sigma}{\\epsilon_0}$$ Now I have values for $\\left | \\vec{E_+} \\right |$ and $\\left | \\vec{E_-} \\right |$, but when they're going in the same direction (as they are between the plates), they sum to 0, which isn't right. To the left, when you add them going in opposite directions, you get $\\frac{2\\sigma}{\\epsilon_0}$ and to the right you get the same thing. What am I doing wrong?"}
{"id":"60195","title":"How much of current flows through a bird sitting on a power line?","text":"I've been googling for hours and went through over a hundred answers. Now, some say the bird doesn't form a closed loop, some say the current is so small that it doesn't kill the bird. From as much as I understand electricity, I'm sure the bird does make a parallel connection to the wire, so there must be some electrons moving through it's body. Could someone please explain which is the right answer and why? And if it's the case that a parallel circuit is formed, how can I calculate the current going through the birds body?"}
{"id":"109799","title":"How does light get from one bulb to another in a sequence of light bulb?","text":"How does light get from one bulb to another in a sequence of light? Like in the advertisements or a smaller group? Is it the same light?"}
{"id":"82397","title":"Calculate the acceleration of a moving elevator","text":"An elevator is moving down with uniform acceleration. A man inside the elevator who wants to measure the acceleration drops a coin as soon as the elevator starts. The coin is 6ft above the floor of the elevator at the moment it is dropped. It takes 1 second for the coin to reach the floor of the elevator. Calculate the acceleration from all this data. How do I go about solving it ? I have absolutely no idea. Can anyone give at least some hints ?"}
{"id":"19332","title":"Is there any literature that compares the candle-power to the candela?","text":"I am unable to find anything in the literature to back up the claim that appears all over the internet that:               The modern candlepower now equates directly to the number of candelas.      Can anyone please suggest something in the scientific literature to back up this statement? If this is not the correct stack exchange for this question, please suggest a better one."}
{"id":"88669","title":"Physics of the inverted bottle dispenser","text":"![Look at the red arrow](http:\/\/i.stack.imgur.com\/aBnic.jpg) When you invert a water-bottle in a container, the water rises and then stops at a particular level --- as soon as it touches the hole of the inverted bottle. This will happen no matter how long your water-bottle is. I understand this happens, because once the water level touches the hole, air from outside cant go inside and therefore there is nothing to displace the water that falls out of the container. Now according to the laws of pressure ---- the pressure at the water level must be same everywhere --- whether it's inside the water bottle or outside. And that must be equal to the atmospheric pressure. Therefore the pressure of the water column + air column inside the inverted bottle must be equal to the atmospheric pressure. What I dont understand is, no matter how long a bottle you take, the water level will always stop at the hole. So that means that no matter how long a bottle you take, the pressure of the water column + air column inside the water bottle will be equal to the atmospheric pressure. How could this be possible? Also I'd like to let you know that, if you pierce the upper part of the bottle with a small pin, then the water level rises and overflows out of the container. I'm assuming air from outside rushes in and pushes the water out."}
{"id":"64832","title":"NP-completeness of non-planar Ising model versus polynomial time eigenvalue algorithms","text":"From the papers by Barahona and Istrail I understand that a combinatorial approach is followed to prove the NP-completeness of non-planar Ising models. Basic idea is non-planarity here. On the other hand, we have polynomial time algorithms for calculating eigenvalues for matrices. This confuses me. Should not be solving an Ising model be equivalent to calculating the eigenvalue of the Hamiltonian matrix of that model? In that case why is it NP-complete? Is coming up with such a Hamiltonian matrix at the first hand is NP-complete? If it is so, it makes the whole project (coming up with such Hamiltonian and solving it) NP-complete."}
{"id":"2227","title":"Paradox?: What is the form of radiation experienced by a harmonically accelerated observer?","text":"Theory predicts that uniform acceleration leads to experiencing thermal radiation (so called Fulling Davies Unruh radiation), associated with the appearance of an event horizon. For non uniform but unidirectional acceleration the shape of the experienced radiation changes from thermal to other spectral densities, but also is predicted to exist. But suppose the acceleration is periodic and oscillatory, i.e. no permanent horizon persists? In particular, what about the case of harmonic motion, for a full cycle, half a cycle, etc.? Here is an even simpler related problem that makes the apparent paradox easier to see. Suppose at proper time t=0, I accelerate at constant acceleration k in the x direction for t0 seconds, presumably experiencing Unruh radiation. Then I accelerate with acceleration -k, (in the -x direction,) for 2*t0 seconds, seeing more Unruh radiation coming from the opposite direction, and then I finish with with acceleration +k for the final t0 seconds. At the end of the 4*t0 proper seconds, I'm back where I started, at rest, without any event horizon. Was the Unruh radiation I felt when reversing acceleration secretly correlated or entangled with the radiation I initially and finally saw? Otherwise, from a more macro scale, I didn't actually necessarily move much, and the acceleration event horizon was instantaneous, evanescent and fleeting, so whence arose the Unruh radiation?"}
{"id":"96519","title":"Collision of multiple fluid flows at a point","text":"I would like to do an analysis when two or more fluid streams flowing from different direction with different velocity, meet at a point. What happen after the collision? Are there any related studies or papers?"}
{"id":"103355","title":"Is there a standard reference where entropy is set equal to zero in property tables?","text":"For practical considerations, it seems that entropy is only meaningful as a difference between states, like $\\Delta s$ going from state A to state B. For an ideal gas, for instance, standard formulas give some infinities for entropy at absolute zero. But we still tabulate values. Looking at these tables, I see the s=0 point to be around -10 to -5 degrees C for air. This doesn't seem to be consistent with other substances (notable steam tables). Is there a commonly accepted rule about where entropy to zero. While that point is arbitrary physically, different steam tables would need to use the same offset to be compatible among each other. I'm interested to know if there's a standard rule for this. Particularly for ideal gases."}
{"id":"120183","title":"Why does my gravity simulation do this?","text":"For a school project i created a simple 2D gravity sim in Matlab using the simplest possible method. There are 2 nested loops so that the total force and acceleration of every object can be calculated. I found that conservation of momentum in x and y works perfectly. I also use a damping so if two object are very close, their acceleration doesn't explode. Then i place two objects of equal mass on coördinates (-100,0) and (100,0) with initial speeds 0. If i start the sim, the two objects will move towards eachother, pass though eachother and move away again. However, they move away much farther than where they originally started. Would that mean that kynetic energy + pot energy is violated, or is this normal? In the real world this obviously never happens because objects can't pass through eachother. It is not very important to solve the problem, the project is from a few months ago. I just wonder what went wrong. Here is my code. The circle.m must be in the same folder as the gravity.m Some names of variables are in Dutch, but i translated the comments. http:\/\/speedy.sh\/4wVqm\/Matlab-gravity2.rar"}
{"id":"62935","title":"Relativistic Computation?","text":"Is it possible to employ relativity to develop computational technology? Here is a really basic example: Build a Computer and Feed it the Problem (say the problem is projected to take 10 years to solve) Start the solution of the problem using algorithm. Now go on to your convenient relativistic jet... accelerate very near the speed of light (utilizing time dilation to your advantage) and make a round trip back to the planet, which takes a reasonable amount of time for the observer within the relativistic jet to complete (say a couple weeks) and allows them to arrive back to their computer with the problem solved. This sort of system could solve very difficult problems \"relatively\" quickly in the sense that the user would be able to find the solution to their problem in much less time ,relative to themselves, then normal. A more user convenient system can be found if research is being conducted in a heavy gravitational field (time speeds up). For example: If you have a lab close to the surface of a black hole, and you send your computer far away (much less gravitational attraction) a mere couple of minutes could pass in your lab's frame of reference while the computer had thousands of years to work, in it's frame of reference, so then the computer can be fetched back to the lab with the solution found. Unfortunately Earth's gravitational time dilation is not nearly enough to take advantage of this sort of system (which is convenient in that it allows the user to remain stationary). Are there any better ideas? Just for the sake of though experiment and logic."}
{"id":"60681","title":"What causes a soccer ball to follow a curved path?","text":"Soccer players kick the ball in a linear kick, though you find it to turn sideways, not even in one direction. Just mid air it changes that curve's direction. Any physical explanation? Maybe this video will better emphasize this phenomenon: http:\/\/www.youtube.com\/watch?feature=player_detailpage&v=ZEv7QEFNVq0#t=265s"}
{"id":"107859","title":"Heating a black body","text":"By definition, Black body has absotivity=emmisstivity=1. This means the black body radiates all energy it accepts. Does this mean the black body cant be heated?"}
{"id":"35460","title":"The physics behind The Great Flood","text":"The book of Genesis floats (pardon the pun) some interesting numbers when discussing the Great Flood. For example, it rained for 40 days and 40 nights, and at the end of that time, the entire planet was covered in water. I think we can deduce how much water that would have had to be, estimating that the highest peaks in the Himalayas were covered with water. (8,848 meters above sea level) My questions are, how fast would the rain have had to come to raise the ocean level that high in 40 days and nights, how much would the mass of the earth have changed for this event, and would that significantly alter the strength of gravity on earth?"}
{"id":"80535","title":"Beam splitters and Mach-Zender interferometers","text":"I have a question (my very first here) related to 50\/50 beam splitters as used in the Mach-Zehnder interferometers (see for example http:\/\/en.wikipedia.org\/wiki\/Mach-Zehnder_interferometer). Let's concentrate on the input beam splitter: A continuous light beam (the input) is split 50\/50, one in the 90deg direction and one in the forward direction of the input beam. Momentum\/energy conservation teaches us that deflected part of the beam must exert a (tiny) force on the beam splitter. The forward part of the beam will not. In the single photon (quantum) case, the beam-splitter should get a slight \"kick\" if the photon is deflected, while the beam splitter is left in its original state if the photon goes straight through. If I got this right so far, we can now increase the energy (and momentum) of the photon and decrease the mass of the beam splitter (now coupled to a sensitive piezo transducer or something) so that a photon \"kick\" can be recorded if deflected. Now \"which way\" information is available and interference should be destroyed as I understand this. Would interference be restored if the transducer is disconnected? If yes, how do the photon \"know\" whether the transducer is connected or not? Maybe this is related to how much the photon\/beam splitter system can become entangled to the environment? If the answer is \"No\", how weak must the photon kick be in order for the interferometer to work? We already know that it works for ordinary light (very tiny photon kicks). Update: Maybe I can reformulate a bit: Why does not the momentum exchange (or lack thereof) between the photon and the beam-splitter (and the trace this leaves in the environment) destroy the interference? Anyway, I see that no-one is interested in this question. Is the question very poor? How can I improve it? If someone just could point me to relevant some literature were transfer of momentum between between photons and optical elements are discussed.. Regards, Carl"}
{"id":"27482","title":"Gauge invariance and the form of the Rarita-Schwinger action","text":"in Weinberg Vol. I section 5.9 (in particular p. 251 and surrounding discussion), it is explained that the smallest-dimension field operator for a massless particle of spin-1 takes the form of a field strength, $F_{\\mu\\nu}$. This is because a massless vector field does not transform properly under Lorentz transformations (Weinberg, Vol. I eq. 5.9.22) and requires additional gauge symmetry\/redundancy to 'mod out' unphysical degrees of freedom. One of the questions in Banks' Modern QFT textbook (problem 2.10) asks to repeat this general analysis for spin-3\/2 and spin-2 fields, for which we know the main examples are the Rarita-Schwinger field and the graviton. For spin-3\/2, however, I am confused as to why the Rarita-Schwinger field is described by an action which looks very similar to the Dirac action. How is it that the massless Rarita-Schwinger field, which carries a vector index as well as an additional spin-1\/2 index, is able to avoid the ill-behavior under certain Lorentz transformations (which Weinberg calls $S(\\alpha,\\beta)$) that is exhibited in the massless vector? Naively I would have wanted to write a spin-3\/2 field as a $(1,1\/2) = (1\/2,1\/2)\\oplus(1\/2,0)$ representation of the Lorentz group so that the vector index can be treated independently of the spinor index. Shouldn't the argument for massless photons carry over and imply that I should have some kind of `field strength' for the spin-3\/2 field? I suspect that what I'm missing is something trivial and silly. Thanks!"}
{"id":"20454","title":"Orientation of Magnetic Dipoles","text":"Does a magnetic dipole (in a permanent magnet) tend to align with the B-field or with the H-field? The current loop (Ampère) model of the magnetic dipole suggests the former, while the magnetic-monopole (Gilbert) model suggests the latter."}
{"id":"45753","title":"How to express answers where inequalities are involved?","text":"Part of the floor of a workshop is inclined at 10 degrees to the horizontal. This is to allow the safe storage of steel cylinders. A cylinder of mass 7000 kg is stored as shown in the diagram (c the link below). The magnitudes of the forces exerted on the cylinder by the wall and the floor are Rw and Rf newtons respectively. Find the values of Rw and Rf. If the sloping part of the floor was inclined at alpha degrees (0 < alpha < 45) instead of at 10 degrees, show that a) Rw < 70000< Rf, whatever the value of alpha. b) Rw and Rf both increase with alpha. http:\/\/www.flickr.com\/photos\/8625954...in\/photostream i did this: R (II to slope) : Rwcos10 = 70000sin10 ---- Rw = 12342.9 N R (perpendicular to slope) : Rf = 70000cos10 + 12342.9sin10 ----- Rf = 71079.9 N I am able to get Rw and Rf on the cylinder. But I don't know how to approach mathematically expressing the second part of the question (the proof part). Parts (a) and (b) of the second part. Could you please help?"}
{"id":"52379","title":"Specific electron energy gap values $E_{i+1}-E_i$ vs. photons with arbitrary energy $\\hbar \\omega$","text":"The energy levels of electrons in an atom are quantized $E_i$. A photon of a specific momentum $\\vec p$ and energy $$\\omega=(E_{i+1}-E_i)\/\\hbar$$ hits an atom and gets absorbed. Okay now say the energy of the photon is $$\\omega'=(E_{i+1}-E_i)\/\\hbar+\\Delta E\/\\hbar,$$ where the value $\\Delta E$ is some energy, which is not even near the value of any electron state gap energy. hence the quantity $\\Delta E$ cannot be absorbed by means of excitation of the other particle. What happens in this case? Does this energy- wise non-fitting photon just go through the material? Does the photon \"spit\" in a way that the part $E_{i+1}-E_i$ of the photon energy gets absorbed and a new photon with energy $\\Delta E$ will move away? Microscopically, has this anything to do with soft radiation? Okay well, if we model the atom as only being able to suck up energy by changing it's energy levels, this would violate momentum conservation. I guess the atom must start to move too, or if we view it as part of a bigger solid state system, the photon will make the thing wobble a little. From that perspective, we'd have to consider the kinetic energy distribution of the particles composing the material and this would be able to suck up any real energy or momentum value. If we leave the microscopic description (in terms of which I'd like to see an answer to the first question - after all, I'm speaking of the behaviour of a single photon here) and go on and view this as a \"gray body\" problem, and in case the answer really is that after the photoelectric effect the rest energy gets termalized, then I'd like to know the relation (for a class of materials) between the absorption per frequency and material characterizing parameters like (electron\/atom density? emissivity? ...)."}
{"id":"63083","title":"Hamiltonian of Harmonic Oscillator with Spin Term","text":"We have the usual Hamiltonian for the 1D Harmonic Oscillator: $\\hat{H_{0}}=\\frac{\\hat{P^2}}{2m} + \\frac{1}{2}m \\omega \\hat{X^2}$ Now a new term has been added to the Hamiltonian, $\\hat{H} = \\hat{H_0} + \\mu B\\hat{S_z}$ The system has a spin degree of freedom with $s=\\frac{1}{2}$ What are the new eigenstates of the Hamiltonian and what are the energy eigenvalues? We have been denoting the stationary states as $|n,s_z\\rangle$ and the non-spin eigenvalues as usual $E = h \\omega (n+1\/2)$ Can anyone give me some help with this question? I'm guessing that the eigenstates don't change and then it's straightforward to find the eigenvalues but I'm not sure if this is correct (or why it would be correct)."}
{"id":"10233","title":"Proving that the free energy is extensive","text":"If I have two system of an Ideal gas $A$ and $B$ each of these system has a partition function: $Z_{A,B} = \\left ( \\frac{V_{A,B}}{\\lambda_T} \\right )^{N_{A,B}}$ Where: $\\lambda_T = \\left ( \\frac{m}{2\\pi\\beta \\hbar } \\right )^{\\frac{1}{2}} $ The free energy is: $F_{A,B} = -kT \\ln \\left ( Z_{A,B} \\right ) = -kT N_{A,B}\\ln \\left (\\frac{V_{A,B}}{\\lambda_T}\\right)$ For the free energy to be extensive the following must be true: $F_{A} + F_B = F_{A+B} \\Rightarrow Z_A \\cdot Z_B = Z_{A+B}$ However: $Z_A \\cdot Z_B = \\left (\\frac{V_A}{\\lambda_T} \\right )^{N_A} \\left ( \\frac{V_B}{\\lambda_T} \\right)^{N_B}$ and: $Z_{A+B} = \\left ( \\frac{V_{A+B}}{\\lambda_T} \\right )^{N_A + N_B}$ So, for $Z_A + Z_B = Z_{A+B}$ to be true $V_A^{N_A} V_{B}^{N_B} = \\left(V_A + V_B \\right)^{N_A + N_B}$ must be true as well but this isn't true for any system. Since we cannot create energy by mixing two containers of Argon in the same pressure and temperature, something in my understanding is wrong. Where is my fault?"}
{"id":"53422","title":"How do particles, such as electrons become visible?","text":"Quantum mechanics says that atoms are invisible - they do not have some specified location, only a probability distribution. So, how can we see them? If there is to be particle-antiparticle annihilation (or other interactions), the particles must have a fixed location, right? So, is this process just random? Is it impossible to know whether a pair will annihilate, but only know the probability?"}
{"id":"112466","title":"How does a Trumpet loud speaker work?","text":"I pierced a hole in a cone shaped cardboard's tip and attached it to my phone speaker. Surprisingly the sound produced when attached is three times louder than the sound when it is removed. I do know that it is something relevant to trumpet loud speaker.What makes it to produce louder sounds? ![enter image description here](http:\/\/i.stack.imgur.com\/VWBpM.png)"}
{"id":"126565","title":"Relativity of Simultaniety","text":"![enter image description here](http:\/\/i.stack.imgur.com\/73vNK.jpg) In the first figure, A and B are two equidistant points from the observer O in S. In the second figure (reference frame S') the corresponding points are A'and B' such that A'O'=O'B', where O' is the observer in S'. The first figure represents the state of S and S' at time t=0 when two events occur at A and B in S and the corresponding points In S' are A' and B'. Now S' moves with a velocity v w.rt. S. The second figure represents the state of S and S' at time t=t1 when the rays from A and B reaches O simultaneously (if possible). Clearly, since at that time O and O' don't coincide, to O' the events should not be simultaneous. Conversely, if at time t=t1' the rays from A' and B' reaches O' simultaneously (if possible), for the same reason O will not find the events to be simultaneous. Since the light rays are unique, they cannot meet more than once. So only one observer should find the the events to be simultaneous. In this case which observer will find the events to be simultaneous and why?"}
{"id":"115119","title":"Rotation of object on another object under rotation","text":"First, I would like to know the rotational velocity of disk2 if disk1 turns at $\\omega1$. The axis \"x\" is fixed to the ground and disk1 is allowed to turn around it. Axis \"y\" is fixed to disk1 and disk2 is allowed to turn around it. For me, disk2 turns around axis \"x\" and around axis \"y\" but not in the same manner. I can understand its rotational velocity is $\\omega2$ and for me even disk2 turns around axis \"x\" and because disk1 is turning, this doesn't change its rotational velocity $\\omega2$. I'm not sure. Can you explain please ? In fact, what's a rotational velocity in this case for disk2 ? ![enter image description here](http:\/\/i.stack.imgur.com\/Uo4sT.png) Second, with $\\omega1=-\\omega2$. A motor is fixed between disk1 and disk2 (in axis \"y\"), stator on disk1 and rotor on disk2 (idea of Floris, thanks). If a torque from rotor increase $\\omega2$ this would say another torque from stator increase $\\omega1$. The motor need energy for give a torque to disk2, I'm agree with that. But I don't understand why the motor need energy for apply its torque from stator to disk1. For me, like stator and rotor turn at the same rotational velocity around axis \"x\" it's not difficult to add torque to disk2 even the stator+rotor turn around \"x\" in the same time. The motor turn around the axis \"x\", when stator turn $d\\theta$ degree in one direction, rotor turn $d\\theta$ degree in the same time. So, it's not \"difficult\" (need energy) to apply torque on disk1 for me. If rotor moves far away from stator, in this case I'm agree, I will be more difficult to accelerate it, but here stator and rotor never move from each other. If I need to give forces that stator gives to disk1, I'm agree I need energy, but these forces are only a reaction from the stator because rotor apply a torque. I'm not sure I explain very good what I think, I hope someone can clear this idea. I would like to understand why the motor need energy for apply its torque to disk1. \\------------------------Added from comment of Floris: I'm agree, if I'm on object that move back and I try to move an object in front of me, I need more energy. I drawn disks and 2 points A and B: ![enter image description here](http:\/\/i.stack.imgur.com\/OJYLN.png) Imagine, disk2 don't turn around its center of gravity (axis y). Motor and disk2 turn only around axis \"x\". You could look at distance AB, it is always the same even the stator rotates of 45°, because the rotor rotates of 45° too. It's not like stator move in one direction and rotor in other. Now, the rotor turn around axis \"y\" but why it's not the same with a linear trajectory ? Now, distance AB change but only because stator \"attack\" rotor, in the same time stator rotates around axis \"x\" but rotor rotates around axis \"x\" too. Maybe my confusion come from I dissociate 2 movements: first the rotation around axis \"x\" and second the rotation around axis \"y\". For me, inertia and centrifugal forces rotate motor and disk2 around axis \"x\" and torque from motor add rotational velocity of disk2 around axis \"y\", I don't see where the stator move \"back\" like your example, could you explain more please ? Maybe I don't have the right to think like that. Sorry if my level of physics is low. I added forces, maybe like that it's easier to explain the problem: ## ![enter image description here](http:\/\/i.stack.imgur.com\/qEhuC.png)"}
{"id":"105993","title":"Water falling, why does it spin?","text":"I have noticed that water, when falling will rotate. Looking closely at a thin stream from a faucet and placing a flat object mid stream you will see the water is rotating. The further down the stream you go you will notice the rotation is much quicker. Am I confused at this occurrence or is there a reason? I can only come up with the rotation of the earth taking into affect."}
{"id":"92420","title":"Is it possible to calculate the half-life of a nucleus?","text":"Is it possible to calculate theoretically the half life of a nucleus (how?) or they are all known through measurements? As an example how do you calculate the half life of Carbon-14?"}
{"id":"66305","title":"Molecular Hamiltonian","text":"I was reading some material on the Molecular Hamiltonian on Wiki. It said that, > _Almost all calculations of molecular wavefunctions are based on the > separation of the Coulomb Hamiltonian first devised by Born and Oppenheimer. > The nuclear kinetic energy terms are omitted from the Coulomb Hamiltonian > and one considers the remaining Hamiltonian as a Hamiltonian of electrons > only._ I am not sure why this is done. This won't be the \"full\" Hamiltonian if we omit the KE terms of the nuclei. Can somebody please explain to me why the nuclear kinetic energy terms can be ignored from the Coulomb Hamiltonian."}
{"id":"24364","title":"Relationship between gauss and decibels","text":"In my ongoing effort to understand the world around me, I want to wrap my head around the relationships between two units of measure. Specifically gauss and decibels. The quandary comes from my understand of waves. Hertz is the measure between wave peaks and describes the frequency of the wave. And from what I know about sound, decibels is the measure of intensity. I've played around with a spectrum analyser designed for sound and really got an understanding of how it all works. Then I turned my attention to electromagnetic waves. I was looking for a tool that I could use to measure electromagnetism at any given location. The first tool I came across was an electromagnetic field meter (EMF meter) which is measured in gauss. Wouldn't it be equally effective to measure in decibels? I also could not find a conversion from one to another, so I would assume they measure different things. Can you clear this up for me?"}
{"id":"24360","title":"Thermal Energy from Friction on a Rope","text":"Question: A capstan is a rotating drum or cylinder over which a rope or cord slides to provide a great amplification of the rope's tension while keeping both ends free. Since the added tension in the rope is due to friction, the capstan generates thermal energy. If the difference in tension ($T_0-T'$) between the two ends of the rope is $520N$ and the capstan has a diameter of $10.0 cm$ and turns once in $0.90s$, find the rate $P_{thermal}$ at which thermal energy is being generated. My Work: $F = 520N$ $d = 0.1m$ $\\Delta T = 0.90s$ $\\Delta U = Q - W \\implies Q = \\Delta U + W$ $W = \\tau \\Delta \\theta = Fd(2\\pi) \\implies W = 326.7J$ $P = \\frac{\\Delta W}{\\Delta T} = 363W$ However, this is the incorrect answer. (I don't know the correct one.) Can anybody give me any guidance here?"}
{"id":"79191","title":"Testing the speed of gravitational waves in superfluids","text":"Ok. Light can be slowed down in superfluids. I would like to know whether if a Cavendish like experiment could be possible within inside superfluids i.e, to rotate two balls such that to observe their gravitational attraction just like the big G is determined and test whether gravitational waves(might take debroglie wavelength) are slowed down or not? Or whether it could be tested for celestial bodies' gravitational wavelength (or debroglie wavelength) redshifted well enough to an observable limit. Would deduce something about gravitons?"}
{"id":"79199","title":"Quantum tunneling effect in a potential of the kind $V(x)=A\\frac{x^2}{1+x^4}$","text":"Given a potential: $$V(x)=A\\frac{x^2}{1+x^4}$$ with $A\\gt 1$ and a quantum particle inside the well around the point $x=0$. I'm stuck on the calculation of the transmission and reflection coefficients for this particle vs. its energy."}
{"id":"77335","title":"Is inertia present in a body while in state of motion?","text":"According to Newton's law of inertia (written in book: **The Mathematical Principles of Natural Philosophy**** ) it is said that in all objects having mass, have always inertia included in their property but if a body is in motion well then, would there be inertia in that body. For instance: A ball is hit by a bat and is in state of motion in air so would there be inertia present in that current state of ball or not? According to friction law of physics friction is any thing which exists when two or more bodies come in contact with one-another and it is against the direction of force (pushing a body) which is against inertia so would friction be helpful too in supporting inertia of a body consequently? If yes then could it be said that, inertia is directly proportional to the friction?"}
{"id":"77485","title":"Are principle of Conservation of energy and principle of conservation of momentum consequences of Newton's laws?","text":"It is known that principle of Conservation of momentum and principle of conservation of energy are two fundamental principles of physics.But in RP Feynman's _Lectures of physics,_ in the chapter of conservation of momentum, it is said that conservation of energy and conservation of momentum are consequences of Newtons laws of dynamics.Is it correct?"}
{"id":"73105","title":"Photon temperature above Electroweak Symmetry Breaking (EWSB) transition","text":"In discussions of the history of our Universe, photon temperature is substituted for time. As the Universe cools, phase transitions break symmetries, including electroweak symmetry. Why does it make sense to speak of photon temperature prior to Electroweak Symmetry Breaking (EWSB)? Is this a misnomer? Is it shorthand for \"boson for unbroken $U(1)$\" temperature? Or did I misunderstand something."}
{"id":"127456","title":"What will be the effect of placing a light source very close to a photodiode?","text":"What will be the effect of placing a photodiode really close to a laser source and what should be the appropriate distance between a light source and photodiode to get maximum output current?"}
{"id":"55060","title":"How do we show that no hidden variable theories can replace QM?","text":"I've always hit two big stumbling blocks in conceiving of the proof or disproof of hidden variable theories as being even valid idea, let alone an answerable question... I feel I must be misunderstanding some very fundamental things.   1. Quantum mechanics is deterministic, ignoring the bit where we take our complex number and fudge it into a probability.   2. We use computers to make calculations to provide the predictions of quantum mechanics - how is this not itself a hidden variable theory? From what I understand the famous Bell theorem and work around it doesn't disprove hidden variable theories, just one specific flavour of them, and even then I am not convinced. I feel like I must be missing something again, because this result seems to me in no way profound... although it certainly fits into the picture of the EPR 'paradox' and is a good demonstration of the validity of QM. I guess the crux of my problem is that this 'area' really addresses wave function collapse and the problems it causes, but I don't see the need for some collapse mechanism to begin with... again I feel like I am missing something really fundamental. Apologies if this is a bad question. I have had a look around and there are similar questions, but I don't feel that either of my points are addressed..."}
{"id":"93770","title":"Energy required to remove both electrons","text":"The question is: **Q.** An energy of $24.6$eV is required to remove one of the electrons from a neutral Helium atom. The energy(in eV) required to remove both the electrons from a neutral Helium atom is: $a.$ 38.2 $b.$ 49.2 $c.$ 51.8 $d.$ 79.0 **My try:** Since after removing an electron the Helium atom becomes a single electron atom, Bohr's Model can be applied. The energy of the single electron is hence given by $-13.6\\frac {z^{2}}{n^{2}}$ (where z is atomic no. and n is the no. of the shell in which the electron resides). Thus the energy comes out to be $-13.6\\frac {4}{1}$, which is $-54.4$. Thus the total energy required to remove both the electrons is $24.6+54.4=79.0eV$, which should be option $d.$. But the answer is $a.$. Did I go wrong somewhere? Please correct.."}
{"id":"91889","title":"Crash course how cosmologists develop mathematical model for the Big Bang theory","text":"First of all, probably it might be useful to introduce me. I have sufficiently good enough mathematics and physics knowledge but only in bachelor level in engineering course. I am extremely interested in cosmology and right now I cannot wait to know how the cosmologists develop mathematical model to formulate the Big Bang theory chronologically from the earliest stage up to the conclusion that the space- time is expanding. The explanation should not be too detailed but please keep the mathematical aspects intact by providing me with the main, basic formulae. Ten paragraphs with 5 sentences each might be more than enough. Hopefuly I can have a good guidance to start learning cosmology from the given answer."}
{"id":"111343","title":"Given partial derivatives find the state function","text":"I have to find the state function of a gas (which under very low pressures behaves as an ideal gas). So, first I tried writing it as part of an equation $ dV = \\dfrac{\\partial V}{\\partial T}\\ dT + \\dfrac{\\partial V}{\\partial p}\\ dp $ and then checking if the \"switched\" derivatives will be equal. And then I tried to find an integrating factor as a function of $p$ and then $T$. The result in both cases was overly complicated which suggests that it is not the correct answer. Could you please explain to me how to solve it? And when is it alright to change $V, p, T$ using the ideal gas equation to make calculations easier? Moreover, I have a conceptual difficulty - what would the integrating factor times $dV$ represent? I only used integrating factor to prove that $1\/T$ was the integrating factor of heat (which gives entropy). ![the partial derivatives](http:\/\/i.stack.imgur.com\/YWeR4.jpg)"}
{"id":"61652","title":"Rotational velocity of face-on spiral galaxies using spectroscopy","text":"I am doing my first steps in spectroscopy (IFS actually) and how we can learn more about galaxies by using it. I came up with a simple question which, unfortunately, I can not answer: How can we measure the rotational velocity of a spiral galaxy given that the galaxy is perfectly face-on? I understand how to do so if the galaxy was tilted or edge-on: By measuring the Doppler shift of the spectral lines for each spatial position. One side will be red, the other blue, and the rest somewhere in between. But what about if the galaxy is perfectly face-on? The Doppler shift will be zero right? Most of the resources, papers etc I find online are about tilted galaxies."}
{"id":"35486","title":"Is there a thought experiment which brings to light the contradiction between General Relativity and Quantum Mechanics?","text":"I've been told that GR and QM are not compatible, is there an intuitive reason\/thought experiment which demonstrates the issue? (Or one of the issues?)"}
{"id":"35483","title":"Is there a strong force analog to magnetic fields?","text":"In special relativity, magnetism can be re-interpreted as an aspect of how electric charges interact when viewed from different inertial frames. Color charge is more complex than electric charge, but also has deep similarities to it. This leads to three questions:   1. _Is it mathematically possible to re-interpret standard chromodynamics in terms of frame-dependent combinations of “chromostatic” and “chromomagnetic” interactions?_ The obvious analogy would be that moving color charges would create chromomagnetic fields that could in turn alter the paths of other nearby moving color charges. (Note: My just-created word “chromomagnetic” is analogical only, since the component of chromodynamics that it describes would be no more related to ordinary magnetism than color charge is to electric charge.)   2. _If such re-interpretation is possible, has there been prior work on this idea?_ That is, has the idea of a chromomagnetic analog to the magnetic fields proposed or assessed? If so, does anyone know the correct keywords for searching for such works, or have any specific references?   3. _If such a re-interpretation is possible, is there any real advantage to it?_ That is, even if chromodynamics _can_ be split into two frame-dependent components, that does mean it should be. It could be an idea that is plausible in terms of the some mathematical framework, but has no real conceptual or modeling merit. Conversely, if meaningful cases exist in which a sustained chromomagnetic field could be generated, the concept of chromomagnetics might prove useful for the same reason that we still have a concept of independent magnetic fields, more than a century after such fields were shown fundamentally to be only another aspect of moving electrical charges. **2012-09-09 Comments on Answer by David Bar Moshe** I am awarding the answer to David Bar Moshe specifically for the excellent reference he provided to a popular-style 1997 Edward Witten paper on the issue of quark confinement. In particular, on page 3 of that paper in the bottom left corner Witten says this: > \"As a non-Abelian gauge theory, QCD has fields rather similar to ordinary > electric and magnetic fields but obeying a nonlinear version of Maxwell’s > equations. Quarks are particles that carry the QCD analog of electric > charge, and are confined in vacuum just as ordinary magnetic charges would > be in a superconductor. This analogy led in the 1970s to the idea that the > QCD vacuum is to a superconductor as electricity is to magnetism. This is an > important idea, but developing it concretely was out of reach in that > period.\" This answers all three of my questions: 1. = Yes, QCD analogies to EM were explored decades ago (!), back in the 1970s, albeit incompletely due to lack of sufficient mathematical support at that time (and possibly still today, according to David Bar Moshe's comments); 2. = Yes, there was prior work and continuing work, to which David Bar Moshe has provided some references; and 3. = Yes, the interpretation does appear to have have value, since it enables a useful analogy between monopole magnetic fields in a superconductor and QCD \"magnetic-like\" color force components that in a vacuum produce a similar effect. So, David, thanks! For anyone else reading this, I would note that I'm pretty sure we are still talking here about an _analogy_ of magnetic-like components within the color forces and the separate but mathematically similar description of true magnetic fields between (hypothetical) monopoles. Dave, if I am mistaken about that -- if you are suggesting that there is more than just an analogy going on between the component of QCD that produces confinement and the monopoles with conventional magnetic fields -- then you may want to add a note to that effect, since I'm still assuming (as Witten seemed to do) that this is an analogy or mathematical similarity only, versus an actual link between QCD and magnetic fields. I may add more comments after reading more from your other references. I appreciate your observations about unknown areas, also. So again: Interesting answer, and one that for me had a very unexpected outcome. Since I was only basing my question on an analogy to relativistic electromagnetics, I never would have guessed that it was used decades ago for something as important as quark confinement."}
{"id":"130328","title":"Electromagnetic fields and human behaviour","text":"To what extent does electromagnetic fields and\/or radiation from space have influence on human behaviour and\/or physiology?"}
{"id":"74682","title":"Introduction to spinors in physics, and their relation to representations","text":"First, I shall say that I am familiar with the intuitive idea that a spinor is like a vector (or tensor) that only transforms \"up to a sign\" when acted on by the rotation group. I have even rotated a plate on my palm to explain this to my fiancee! I have also looked at spinors as mathematical objects, such as the 2d subspace of complex 3-space such that $X·X = 0$, and feel that I understand this well also. I am confused by spinors in physics. Are they still vectors that are isotropic (inner product with themselves is 0)? In what vector space? Normally states are vectors in infinite dimensional spaces! Every attempt to find literature which pins down specifically what a spinor is (in physical terms) seems to assume that one is already well acquainted with the idea. Take, for example, the Dirac equation. I can see that the solutions are four- component wavefunctions, which then splits into two parts. Is this a spinor? Why? What vector space do these solutions live in? I believe I've heard that the answer has something to do with representation theory, perhaps of the Poincare group? I am also familiar with the basics there, so don't hesitate to explain in terms of representations."}
{"id":"65829","title":"Is there a term for the argument of the sine function outside of geometry?","text":"Are there similar terms in other areas for the idea the \"angle\" conveys in geometry? I find that functions for abstract things such as pressure, electrical currents (nothing geometric there) on AC circuits and are described by the sine function (argument is an angle, therefore a geometric entity). Do we really have that abstract things are necessarily described by geometric entities? Maybe the concept of angle we know is a simply an example of a greater idea (like a a measure of difference between instances of the same thing) applied to geometry or am I tripping?"}
{"id":"65825","title":"Does a lightning rod prevent lightning strikes?","text":"I was always under the impression that a Lightning rod worked to protect a building by being the tallest part of the building and having a low path of resistance to the ground, thus being a good target for lightning to hit, reducing the chances of lightning strikes elsewhere on the building. However when reading an Ars technica article, one comment (from user \"malor\") said that theory is incorrect, and explained that a lightning rod actually _prevents_ lightning strikes in the first place: > Lightning happens when a really huge negative charge builds up in the > ground, corresponding to a positive charge overhead, and the differential > becomes sufficient to jump the gap. (and, as others are pointing out > repeatedly, air is an excellent insulator, so it takes a whacking huge > voltage differential to make the arc.) > > A lightning rod doesn't exist to provide a spot for the lightning to strike. > Rather, it exists to dissipate the charge so that the strike never happens > at all. This comes from early experiments with the Leyden jar; if a pointed > metal rod was attached to the jar, it wouldn't charge. The electrons are > able to leap off a pointed tip, and into the air, dissipating charge as fast > as it accumulates. The lightning rod does the same thing, on a larger scale; > it spits out electrons into the air like crazy, so that the charge won't > build up sufficiently, and the lightning never hits at all. > > Now, if the charge accumulates faster than the rod can dissipate it, there > can still be a lightning strike, and the rod typically IS the best route to > ground. But everything around the rod is going to take a hell of a jolt > anyway, including, most likely, your electronics. A lightning rod getting > hit means it failed to work adequately; ideally, it should never be hit at > all. How does a lightning rod actually work? Does a lightning rod actually prevent lightning strikes?"}
{"id":"55687","title":"Causality and Quantum Field Theory","text":"I have a problem with proof of causality in Peskin & Schroeder, _An Introduction to QFT,_ page 28. To avoid confusion I use three vectors notation, rewriting the Eq. (2.53) for $y=0$ as follows: $[\\phi(x,t),\\phi(0,0)]=\\int \\frac{d^3p}{(2\\pi)^3}\\frac{1}{2\\sqrt{p^2+m^2}}\\left(e^{-i\\mathrm{p}.\\mathrm{x}-it\\sqrt{p^2+m^2}}-e^{i\\mathrm{p}.\\mathrm{x}+it\\sqrt{p^2+m^2}}\\right)$ The book goes on about how the integrand being Lorentz invariant makes this integral zero for the x out of the light cone. But I (not being a special relativity expert) want to see it more rigorously: after changing variables $p\\to-p$ in the first term, the equation simplifies to: $[\\phi(x,t),\\phi(0,0)]=\\int \\frac{d^3p}{(2\\pi)^3}\\frac{-2i}{2\\sqrt{p^2+m^2}}e^{i\\mathrm{p}.\\mathrm{x}}\\sin\\left(t\\sqrt{p^2+m^2}\\right)$ using spherical coordinates: $[\\phi(x,t),\\phi(0,0)]=\\int \\frac{dpd\\phi d\\theta p^2\\sin\\theta}{(2\\pi)^3}\\frac{-i}{\\sqrt{p^2+m^2}}e^{ipx\\cos\\theta}\\sin\\left(t\\sqrt{p^2+m^2}\\right)\\\\\\ [\\phi(x,t),\\phi(0,0)]=\\int_0^{\\infty}\\frac{dpp}{(2\\pi)^2}\\frac{-2i}{x\\sqrt{p^2+m^2}}\\sin (px)\\sin\\left(t\\sqrt{p^2+m^2}\\right)$ again after another change of variables $u=\\sqrt{p^2+m^2}$, $[\\phi(x,t),\\phi(0,0)]=\\frac{-2i}{x}\\int_m^{\\infty}\\frac{du}{(2\\pi)^2}\\sin (x\\sqrt{u^2-m^2})\\sin\\left(tu\\right)$ I cannot see how this integral should be zero for $x>t$ !!! Can somebody please explain this to me?"}
{"id":"55682","title":"Violation of Lorentz invariance (Lagrangian for particle)","text":"I'm trying to get the relativistic action (or Lagrangian) for a free particle in the case of violation of Lorenz invariance. Suppose we have the modified dispersion relation: $$ E^{2}=\\Omega^{2}(p^{2}) $$ Here $E$ and $p$- energy and momentum of the particle, $\\Omega^{2}$ is a function that takes the form $\\Omega^{2}(p^{2})=m^{2}c^{4}+p^{2}c^{2}$ if we have Lorentz invariance. In the other word, we have equation $$ \\left(v\\frac{\\partial L}{\\partial v}-L\\right)^{2}=\\Omega^{2}\\left(\\left[\\frac{\\partial L}{\\partial v}\\right]^{2}\\right) $$ **We want to define the Lagrangian**. For example, if $E^{2}=m^{2}+(1+\\xi)p^2$, we can obtain(overriding $\\tilde{v}=v(1+\\xi)^{-1\/2}$, we present the equation written above to the standard relativistic form, above which we know everything): $$ L=-m\\sqrt{1-\\frac{v^{2}}{1+\\xi}} $$ **I want to find lagrangian and action for the following case** : $$ E^{2}=p^{2}c^{2}+m^{2}c^{4}+\\frac{p^{4}}{M^{2}} $$ Here $M\\gg m$. _So, we have equation_ : $$ \\left(v\\frac{\\partial L}{\\partial v}-L\\right)^{2}=\\left(\\frac{\\partial L}{\\partial v}\\right)^{2}c^{2}+\\frac{1}{M^{2}}\\left(\\frac{\\partial L}{\\partial v}\\right)^{4}+m^{2}c^{4} $$ -I tried to find the solution in the following way: $L(v)=L_{0}+L_{1}v$, where ${{ L_0}}^{2}={{ L_1}}^{2}{c}^{2}+{\\dfrac {{{ L_1}}^{4}}{{M} ^{2}}}+{m}^{2}{c}^{4}$, and $|L_0| \\ge mc^2$. I set it in the original equation, but it didn't give significant results.   * _Then I've found the following article_ : http:\/\/arxiv.org\/abs\/1209.0464 (arxiv: 1209.0464) Here (pages $6$-$7$) they obtaining the Lagrangian for the case I described above, and they say that in the general case Lagrangian can be defined as (equation $18$) $$ L=-m\\sqrt{1-v^{2}}F\\left(\\frac{1}{\\sqrt{1-v^{2}}}\\right) $$ I tried to find a solution in this form (by substituting in the equation) but I didn't succeed. **Can you help me to obtain Lagrangian (action) for the relation I wrote? Also it would be great if you gave the usefull references devoted to the topic** (for example, where they obtain Lagrangians, etc)"}
{"id":"32288","title":"Stokes' theorem and alike for fractal surfaces","text":"Has anyone ever read an article dealing with the extension of integral theorems to fractal surfaces ?"}
{"id":"104568","title":"Detecting absolute motion with Doppler (reconciling receiver frequencies)","text":"Suppose a transmitter emits a pulse of light every $T$ seconds. The spacial distance between pulses is $cT$. Suppose the receiver is approaching it with speed $v$. ![enter image description here](http:\/\/i.stack.imgur.com\/M2EGq.png) Because, after receiving the first pulse, the next pulse is at distance $cT$ and it will be covered by both light and receiver approaching each other at speed $(v+c)$ so that the next pulse will be detected after time $$t_1 = {cT \\over v+c}$$. Particularly, when receiver is moving at v=c, we have $t_1 = T\/2$ -- the receiver detects twice the frequency emitted. Suppose that light source is moving instead ![enter image description here](http:\/\/i.stack.imgur.com\/a7bof.png) Now, at a time the first pulse reaches the receiver, the next pulse is closer, by distance $cT-vT$ because the transmitter has traveled $vT$ meters when light traveled $cT$. This distance, $cT-vT$, will be closed by light in $$t_2 = {(c-v)\\times T \\over c}$$ seconds. Particularly, when transmitter speed is equal to the speed of light, v=c, the observer detects infinite frequency ($t_2 = 0$). By measuring the frequency we can detect who is moving, receiver or transmitter. Do you know that this contradicts to the principle of relativity and Michelson-Morley experiment which both say that we must not be able to detect who is moving? Einstein has corrected this by time dilation. Can you show how Lorentz arises here naturally for the Doppler correction? I can guess that time dilation can be modeled by scaling the time period of moving body. That is, when receiver is moving, we can have the observed frequency of $t_1 = {cT \\over (c+v) } \\times {1 \\over f_1(v)} $ and when transmitter is in motion then true observed period $t_2 = {(c-v) T\\over c} \\times f_2(v)$. I have placed $f_1(v)$ into denomitator because receiver slowdown can be considered as transmitter's speedup. Now, receiver must observe the same frequency regardless of who is moving, $${c \\over (c+v) } \\times {1 \\over f_1(v)} = {(c-v) T\\over c} \\times f_2(v)$$ or $$f_1(v) \\times f_2(v) = {c^2 \\over c^2-v^2}$$ which looks suspiciously identical to the square of the Lorentz factor squared. Should I guess that $f(v) = f_1(v) = f_2(v) = \\sqrt{1 \\over 1 - v^2\/c^2}$?"}
{"id":"79970","title":"Gravitational binding energy and integrated potential energy not the same?","text":"Before looking up the formula for the gravitational binding energy of a uniform sphere, I simply figured that the general formula for binding energy of an arbitrarily-shaped mass distribution would be $\\left\\langle V,\\rho\\right\\rangle $, where $V$ is the potential as a function of space due to the distribution, $\\rho$ is the density distribution as a function of space, and $\\left\\langle ,\\right\\rangle $ is the inner product (ie, integral over all space). Going ahead for the special case of a uniform sphere of density $\\rho$ and radius $R$, I used the well-known result for the potential inside of a sphere of uniform density, $$V(r)=\\frac{2}{3}\\pi G\\rho(r^{2}-3R^{2})\\mbox{ for }r\\leq R. $$ I then computed the inner product, $$\\left\\langle V,\\rho\\right\\rangle =\\int_{0}^{R}r^{2}dr\\int_{0}^{\\pi}\\mbox{sin}(\\theta)d\\theta\\int_{0}^{2\\pi}d\\phi V(r)\\rho=-\\frac{32}{15}G\\pi^{2}R^{5}\\rho^{2}=-\\frac{6GM^{2}}{5R}, $$ which is twice the correct result, $\\frac{-3GM^{2}}{5R}$. (The last equality follows from $\\rho=\\frac{M}{4\/3\\pi R^{3}}$). I fully understand the geometric proof that the binding energy of a sphere is $\\frac{-3GM^{2}}{5R}$ that proceeds by successively moving shells in from infinity, but was a bit confused when the inner product approach gave an extra factor of 2. My question is as follows:   * Is the extra factor of two in $\\left\\langle V,\\rho\\right\\rangle$ due to double-counting the interaction energies?   * If so, is the correct formula for the binding energy of an arbitrary mass distribution $U=\\frac{1}{2}\\left\\langle V,\\rho\\right\\rangle$? I am trying to estimate the total energy released when two planets which are just barely touching each other collapse to form a single large sphere, and in the course of the derivation this issue sprung up. Any help explaining the extra factor of 2 would be greatly appreciated!"}
{"id":"75225","title":"What causes electron to orbit the nucleus in an atom?","text":"What causes the electron to orbit the nucleus? Which is the force that causes it to do so? Is it related to the Electro - Magnetic force? ."}
{"id":"2144","title":"Potential functions","text":"Can someone please explain what a potential is? Example. velocity potential in ideal flows, acoustic potential (gradient of which gives the particle velocity in a sound wave). Whenever I see potential functions complex analysis is applied to compute integrals, complex functions for conformal mapping. I vaguely understand that potential functions are independent of path and in complex plane the integration around a contour is independent of path as well (the integration depends only on the end points, hence in a closed curve say region $Z$, if function, $f$ is analytic then integration of $f$ over $Z$ is 0). Can you please explain on this?"}
{"id":"106058","title":"Water and electricity? Hot or Cold?","text":"Is hot water or cold water more conductive for electricity or is it the same regardless?"}
{"id":"70762","title":"What can be done to increase the strength of a given electromagnet?","text":"There's a lot of information online about increasing the strength of an electromagnet with more turns, different cores, etc, but not much about factors in the power supply that affect the strength of the magnet. I've got a commercial electromagnet rated for 12VDC that's supposed to produce 200 lbs of pull. What factors in the power supply will maximize this magnet's strength? Thanks!"}
{"id":"30573","title":"Why is plasma the highest \"state\" of matter?","text":"As far as solid, liquid, gas, plasma go, why is plasma the highest state? Are there any other states of matter?"}
{"id":"34858","title":"If gauge symmetries are fake, then why do we care if they are anomalous?","text":"My understanding is that gauge symmetries are fake in that they are only redundancies of our description of the system that we put in (either knowingly or unknowingly) see Gauge symmetry is not a symmetry?. Moreover it seems like we could in principal work without the gauge redundancy, and things would just be a bit nastier at intermediate stages of a calculation. At the same time, people put a great deal of emphasis that gauge symmetries cannot be anomalous. I understand that if we work with the gauge redundancy then we do indeed need it to be anomaly free so that the unphysical polarizations will cancel. However, what if we work without the gauge redundancy, with just the physical polarizations, what goes wrong then if we don't make sure the would-be gauge symmetry is not anomalous?"}
{"id":"30574","title":"What is a mass gap?","text":"I read that Quantum Chromodynamics is a theory with a mass gap. What is a mass gap in laymen terms? Why some theories have it? Which theories does not have it? **Note: I searched for mass gap before asking, all topics about it assume that the reader knows what it is, or define it in an advanced and technical way. So I'm looking for a simple answer if possible.**"}
{"id":"34854","title":"Filming light in motion?","text":"Regarding this video http:\/\/www.youtube.com\/watch?v=Y_9vd4HWlVA, which claims to \"film\" light in its motion. Is it not an absolute nonsense? Even if photons could even be \"seen\" (meaning, returning light them self), isn't this video look unrealistic? You can see that the area around the \"light\" is getting brighter, but that absurd.. light is not a flash light... light does not illuminate the area around it, because it does not \"emit\" more light to all the directions.. Am I missing something?"}
{"id":"133006","title":"Is the second law of Thermodynamics an immense tautology?","text":"Let's imagine for a second a system that is changing constantly from one microstate to another one. It could be a given volume of a gas with its atoms moving and bouncing around, or a deck of cards being constantly shuffled by a monkey. If the starting microstate belongs to a macrostate with very few microstates, chances are that in the next step the system will be in a macrostate with more microstates in it. There is nothing mysterious about this, it is simply a matter of probabilities and how we define them. Now we have the second law of thermodynamics, that says that entropy always increases. It could have been reformulated like: a system that is permanently visiting different states, will spend more time in those which have a higher probability of being visited. Things more probable occur more times. And, since we define probabilities in terms of frequency: Are not we simply saying that things more likely to occur, occur more times? **Isn't it true then, that the second law is simply an immense tautology?**"}
{"id":"83556","title":"Relativistic Time Difference","text":"I thought that it might be interesting to calculate the time difference between a clock placed on Earth when I was born and a hypothetical clock placed at the centre of the sun at the same time. I thought that I might use special relativity and the Lorentz factor. However, even if the speed of the Earth were constant (which it is not), the Earth still experiences acceleration as it traverses its (almost) elliptical orbit around the sun. Given there is acceleration, I suppose that I must use general relativity. How might one calculate the time difference experienced between me and my barbecued brother?"}
{"id":"83550","title":"Mean field theory = large-N approximation?","text":"Wikipedia entry of 1\/N expansion (or 't Hooft large-N expansion) mentions that > _It (large-N) is also extensively used in condensed matter physics where it > can be used to provide a rigorous basis for mean field theory._ I would like reference(s) that reviews this connection between MFT and large-N."}
{"id":"47482","title":"In a double slit experiment are any particles lost because they hit the space between the two slits?","text":"In its wave-form a particle should pass through every time because it propagates in all directions. So there shouldn't be any losses of particles landing in between the slits, right?"}
{"id":"112700","title":"Is having full information about the resonances of a rigid body equivalent to having full information about its material parameters?","text":"Lets say I have a mechanical system whose mechanical resonances (mode shape and frequency) I can measure with perfect accuracy. Is this theoretically equivalent to knowing the materials parameters, including spatial variation, with perfect accuracy? My intuition says that the answer is yes, but I am not sure how to prove it. For concreteness, here is a specific example. Given a right circular cylinder such as the one shown below; will measurements of its resonant modes allow me to reconstruct its spatially non-uniform Young modulus and Poisson ratio? ![enter image description here](http:\/\/i.stack.imgur.com\/fFUyi.jpg)"}
{"id":"501","title":"Simple three-body-problem?","text":"Consider the problem of three bodies two of which having mass _M_ , one of them having mass _m_. Body _m_ is in the middle between the other two, coupled to them by two equal linear springs in rest. Now fix the two bodies _M_ and move body _m_ for a small amount perpendicular to the connection line. Now let loose the three of them. The solution is trivial when we assume _m_ to be finite and _M_ to be infinite (or vice versa), or when the spring constant is 0. But the problem is so symmetric, that there might be hope to get a closed or approximate solution even for finite _m_ >> _M_ , _M_ >> _m_ , or even _M_ = _m_?"}
{"id":"15019","title":"The Opera Neutrino Experiment and the Supernova 1987","text":"So this probably stems from my massive ignorance about post-Newtonian physics but the supernova results of 1987 which measured neutrinos arriving 3 hours before the light from the supernova have been presented as a non-trivial objection to the Opera result. However if the Opera result is confirmed, then doesn't that weaken the supernova objection since star mechanics relies heavily on relativity and such a result might imply that relativity is not the complete picture which in turn could imply different star mechanics than the one we understand? Thanks Edit: To clarify, I'm asking whether the supernova objection may not apply given that this experiment seems to challenge relativity which is what has been used to understand the behavior of stars, supernovae and such."}
{"id":"80492","title":"Newton's laws of motion","text":"According to Newton's laws, if net force is zero on a particle then the particle is at rest or in uniform motion, and if it is not zero then it is accelerating. So when a car moves on the road, friction helps it to accelerate. If the acceleration is zero, (i.e) it is moving with constant velocity, the net external force must be zero, so friction is zero. Then with friction equal to zero how is the car moving on the road?"}
{"id":"31626","title":"Best Materials Science Introduction Book?","text":"Why is this here? > Sorry, I know that this isn't the best place to put this however there does > not seem to be any other stack exchange specific to this so I thought this > would be the best place since it seemed the most related. Question: > What would you suggest as a good introduction to materials science? (Prior > usage and examples would be wonderful, my particular focus is neutron > scattering) What I've Found So Far: >   * Materials Science and Engineering: An Introduction by Callister (Seems > to have good reviews) > Thank you for the help. Note: I didn't see anything related on the book request wiki."}
{"id":"63674","title":"Is there a closed form solution to the Esdale river problem?","text":"This is probably not well known problem but it looks like open problem. What kind of methods there are to find a closed form solution to the physical situation? * * * Can you solve this problem? You're passively flowing downstream in the middle of a river that has its fastest flow in the middle. Most places along the riverbank are too steep or too vegetated, but suddenly you spot a perfect location for coming ashore a distance y away (measured along the river bank). Note that y can be zero so that you only spot it when you're already drifting past it; or if you're slower still, y can be negative. Now assume your swimming speed is a constant v, while the river's flow is V in the middle and linearly decreasing towards the shore. What angle should you take, as a function of your distance to the shore (x), to reach the desired location using minimum effort (i.e. shortest journey or, equivalently, shortest time)? ![enter image description here](http:\/\/i.stack.imgur.com\/ha4VF.jpg) Source and image credit: Hanna Kokko."}
{"id":"88463","title":"How to make a black hole?","text":"Many Physics discussions I have often conclude with: > Well you will then form a black hole... My questions are:   * Is there a general recipe for making a black hole?    * If not, then can you list the families of black hole causes? This means I can put a more rapid halt to many future Physics discussions."}
{"id":"88462","title":"When experimental search for exotic dark matter-like particles began exactly?","text":"I am looking for information about :   * **what was the first experiment that claimed to look specifically for non baryonic dark matter particles ;**   * **when occurred the first serious(*) reporting of an experimental result regarding a non baryonic dark matter candidate.** (*: published in a peer reviewed journal)"}
{"id":"88461","title":"Want to achieve maximum magnetic force & find out the max point","text":"![setting](http:\/\/i.stack.imgur.com\/rcorp.jpg) Hi Sir\/ Madam Want to achieve maximum magnetic force & find out the max point for this setting, want to know   1. is the strongest magnetic force must in the 'dot' area ( middle poin of setting )   2. would the length of D1, D2, D3 affect the 'dot' location?   3. if add a metal sheet ( material AISI1010 ) between the magnet, would the magnetic force increase? would the thicknes of the sheet D4 affect the force? Magnet material: NdFeB 52 5x10x18mm D1 would be <4mm"}
{"id":"35812","title":"Relating theta_QCD to neutron EDM","text":"How do I relate the topological $\\theta_\\text{QCD}$ parameter to the electric dipole moment (EDM) of the neutron? I am very familiar with chiral perturbation theory. I just need to know how to take $\\theta_\\text{QCD}$ into account. Any good sources?"}
{"id":"103814","title":"Action and reaction pair problem","text":"If I sit on a chair, the first action and reaction pair force is my weight and the force acting on the earth by me. And the 2nd pair is the force acting on the chair by me and the opposite force. According to newton third law, the action and reaction force must be equalI. 'm confused because how come the force acting on the chair by me is equal to normal force? F=gmm\/r2"}
{"id":"103812","title":"Construction of Angular Momentum eigenvectors","text":"I have a problem that asks (verbatim) > Carryout the construction of the eigenvectors of total angular momentum and > its z component for $j_1$=3\/2 and $j_2$=1\/2 I am not completely sure where to start out with this. Our professor has not given the best explanation. The equations I have been trying to use are the following: $$\\textbf{L}^2|l,m\\rangle =\\hbar^2l(l+1)|l,m\\rangle$$ $$\\textbf{L}_z|l,m\\rangle=\\hbar m|l,m\\rangle$$ I figure I probably want to use the same equations, but for $\\bf J$ instead of $\\bf L$. I am also confused what the differences between $j_1$ and $j_2$ would represent. I would greatly appreciate any help!"}
{"id":"106454","title":"Number of wave modes in a cavity","text":"I'm trying to calculate the number of acoustic modes that can exist in a room in a certain range of frequencies. I thought of using the Rayleigh-Jeans formula for the electromagnetic standing wave modes possible in a cavity ($\\frac{dN}{d\\nu}=\\frac{8\\pi V}{c^3}\\nu^2$), dividing by two to account for the fact that longitudinal waves such as sound do not have two different polarizations. I'm not sure that this formula is valid for a non-cubical cavity, though, because the development represents the modes as points in 3d \"mode space\", and counts them by looking how many of the points are included within a sphere. If the cavity is not a cube, wouldn't there be more modes in a direction than in another, stretching the sphere into an ellipsoid? My book (Eisberg-Resnick, \"Quantum Physics\") says that the formula is valid for an arbitrary cavity, but doesn't explain that statement. Can someone explain to me why I'm wrong?"}
{"id":"103656","title":"What is the perturbative vacuum?","text":"Will someone please supply me with a good definition of the perturbative vacuum vis a vis quantum cosmology?"}
{"id":"74894","title":"computing wavefront deviations for off-axis mirrors","text":"Suppose you have two off-axis aligned mirrors which are nearly planar, with small deformations such that a wavefront at plane $P_I$ at distance $-D$ from the mirrors, is refocused at distance $D$ at plane $P_O$. I only care about monochromatic waves of wavelength $\\lambda$ I want to compute how much orders of Zernike mode expansion are required to account in the deformations of the mirrors, such that an input gaussian beam with intensity $I_I$ is projected to a gaussian beam with intensity $I_O$, such that $\\frac{I_O}{I_I} > 1 - f$. That is, only a fraction $f$ of the beam is lost at most from the $T_{00}$ mode I've been looking into the literature but most of it does the wavefront calculations for on-axis elements. I wanted to find some proper literature for the off-axis case that would suit the particulars of my problem"}
{"id":"74896","title":"experimental bounds on microcausality violation","text":"In \"The Great Soviet Encyclopedia\", 3rd Edition from 1970-1979, (evidently an old book), some V. I. Grigor’ev has a well-written little note on microcausality. Towards the end he states an experimental bound on the vialotation of microcausality. According to that ancient encyclopedia article, if such violation occurs at all then it must be on scales smaller than $10^{-15} cm$. I don't see a citation for that number in his text (I don't have the original book in my hand though, to be frank) and I am wondering which experiment Grigor’ev might have been thinking of. And: if there are decent such experiments indeed, then probably there are more recent numbers on whatever that experiment actually measured. My question is if anyone might be able to help me with finding whatever useful there might be to say about experimental bounds of the kind Grigor’ev seems to be referring to, or anything similar. Maybe I should say that I am not trying to find such violations. Rather, I am in some discussion with some mathematicians interested in axiomatic QFT, and I'd find it entertaining to be able to accompany the statement of the microcausality axiom (or a similar axiom) with something substantial that has the word \"experiment\" in it."}
{"id":"109825","title":"Uncoupling a coupled oscillator Hamiltonian by change of variables","text":"I'm working on the problem of two entangled harmonic oscillators with Hamiltonian: $$H = \\frac{1}{2} [p_1^2 + p_2^2 + k_0(x_1^2 + x_2^2) + k_1(x_1 - x_2)^2].$$ Introducing the variables $x_± = \\frac{1}{\\sqrt{2}} (x_1 ± x_2)$, and frequencies $\\omega_+ = k_0^{1\/2}$, $\\omega_- = (k_0 + 2k_1)^{1\/2}$ I've tried to write this Hamiltonian in an uncoupled way. I have found that $(x_+^2 + x_-^2) = (x_1^2 + x_2^2)$ and $2x_-^2 = (x_1 - x_2)^2$. Furthermore, since $ p_{1,2}^2 = \\frac{\\partial^2}{\\partial x_{1,2}^2}$ when ignoring constants, I find $$p_±^2 = \\frac{\\partial^2}{\\partial x_{1}^2} + \\frac{\\partial^2}{\\partial x_{2}^2} ± 2 \\frac{\\partial}{\\partial x_{1}} \\frac{\\partial}{\\partial x_{2}}.$$ Then $\\frac{1}{2} (p_+^2 + p_-^2) = p_1^2 + p_2^2$. And: $$H = \\frac{1}{2} [ \\frac{1}{2} (p_+^2 + p_-^2) + k_0 x_+^2 + (k_0 + 2k_1)x_-^2] = \\frac{1}{2} [ \\frac{1}{2} (p_+^2 + p_-^2) + \\omega_+^2 x_+^2 + \\omega_-^2 x_-^2]. $$ The paper I'm working from gives the normalised ground state as $$\\psi_0(x_1, x_2) = \\pi^{-1\/2} (\\omega_+ \\omega_-)^{1\/4} e^{-(\\omega_+ x_+^2 + \\omega_- x_-^2) \/ 2}.$$ However, when I try to check that $H \\psi_0(x_1, x_2) = E \\psi_0(x_1, x_2)$ I don't find anything sensible ($E$ is negative in the constant terms, and still dependent on $x_±$). I suppose then that my Hamiltonian is wrong, but I can't find my error."}
{"id":"46420","title":"What are anomalous threshold singularities","text":"In the papers of the 1950s and 1960s, I see reference to anomalous threshold singularities. What are these? Is there a good reference that discusses this subject?"}
{"id":"45199","title":"Thermodynamic relations from Gibbs-Duhem","text":"Given the Gibbs-Duhem relation $V dp = S dT - N d \\mu$, I am having trouble deriving the following identity: $\\ (\\frac{\\partial N}{\\partial \\mu})_{V,T} = N (\\frac{\\partial \\rho}{\\partial p})_T$ The problem is that the variables $N$ and $\\rho$ don't appear as infinitisemals in the equation. How can I proceed?"}
{"id":"123767","title":"How can we describe the trajectory of an object projected into a viscous fluid?","text":"Can we quantitatively describe the motion of a particle inside a fluid? Let us consider the case when an object (say a sphere, so that stoke's law holds true) is projected into a fluid of viscosity $\\mu$, with a velocity $v$, at an angle $\\theta$ with the horizontal. I myself approached it through Newtonian mechanics. First off the problem is when the object enters the fluid. The buoyant force as well as the viscous force are varying. There will be some threshold velocity for which it enters the fluid. Let's assume that $v$ given to the ball is enough to take it into the fluid. But how do we describe the motion of the ball while it is midway into the fluid? I tried writing the forces (Buoyant, Viscous and Gravitational) with respect to displacement and form some sort of a differential equation but it din't work for me. So i skipped that part and considered that the ball has entered the fluid and has a velocity $v'$ and makes an angle $\\phi$ with the horizontal. This part i think is simpler. At an arbitrary time, i assumed the $x$ component of velocity to be $v_x$ and the $y$ component to be $v_y$. $$\\vec{F_x}=-6 \\pi \\eta r \\vec{v_x}$$ &$$\\vec{F_y}=-6 \\pi \\eta r \\vec{v_y}+m\\vec{g}+\\vec{F_{B}}$$ I do not know what to do next. Help will be appreciated, in both the cases- first, when the ball is entering the fluid, and the second after entering the fluid."}
{"id":"80047","title":"Is it correct to assume that a stretched rubber-band has negative entropy change?","text":"If so, how could we express it in equations connecting S,T,Q? I was wondering if the net change is heat transfer was positive; Since we could feel the heat when it is stretched."}
{"id":"48479","title":"Quantum superposition and fate","text":"First of all, sorry for my knowledge of physics. Maybe my question is too obvious but I want to ask it. I am thinking about fate and if it exist or no. According to my assumption if I take any system, for example cube, throw into it some atoms and I will know all the laws which exist at this cube, I can predict position of this atoms at any moment of time. On the other hand I know about such experiment called \"Schrödinger's cat\" and quantum superposition. So, it brings uncertainty to my system. If this uncertainty will exist at cube than I can't predict anything. Taking into account this assumptions I can say that at our universe fate doesn't exist. But I know that during this choices parallel universe can be created. It's horrible for me to imagine numerous number of parallel universes. What you can say about it? It's naive or it makes sense?"}
{"id":"11928","title":"Relationship of the abundance of an isotope and the vapor pressure","text":"Say I have a sample of some alkali metal - Rubidium for example. A sample of Rubidium consists of approximately 27% $^{87}$Rb and 73% $^{85}$Rb. Does the vapor pressure of each isotope individual _necessarily_ have to be in the same proportion? More specifically, assuming I have enough Rubidium around that I have enough atoms of each isotope to reach the saturated vapor pressure, is there any physical reason why the ratio of the vapor pressures would match their relative abundances? I guess this is coming from this perspective - the linear absorption spectrum for a sample of natural rubidium at a specific fine transition (D2 line - 5S1\/2 ->5P3\/2 if you are interested) looks like image below, the $^{87}$Rb peaks (outer) are clearly smaller than the $^{85}$Rb peaks (inner), and this amplitude ratio is directly related to their natural abundances. If the atoms are in a gas, what forces their vapor pressure to have the same ratio as their relative abundance? ![enter image description here](http:\/\/i.stack.imgur.com\/0WgIG.jpg) Along a similar line of questioning. A sample of Rb will naturally have some trace amount of Cs in it. The saturated number density of Cs at room temperature is ~$10^9$ atoms per cm$^3$, and I'm looking at a volume of say 1cm$^3$. If I have a droplet of 1 gram of Rb in the volume, which is .001 moles, even at .001% impurity I have $>10^{17}$ atoms of cesium in the sample, clearly enough to reach saturated density. -Does this imply that you can do spectroscopy on a volume with Rb and obtain the same spectrum as a volume with a droplet of Cs in it? If not, what prevents the Cs impurity in Rb to reach full saturation. *Note: Perhaps I should make it clear : The linear absorption spectrum is proportional to the theoretical absorption (for a single atom) times the number density of atoms in the sample. The relative amplitudes of the peaks are therefore a proportional to the relative abundances of the two isotopes."}
{"id":"70409","title":"Rate of twinkling of stars","text":"According to \"Why do stars flicker?\", the twinkling of stars (stellar scintillation) is caused by the thick layers of turbulent air in the Earth's atmosphere. While the explanation is convincing, it would be better to have some quantitative justification. Since the twinkling of stars is easily observed, it is reasonable to estimate the twinkling rate as ~ $1\\textrm{s}^{-1}$. How can one obtain a similar estimate from the underlying theory?"}
{"id":"121658","title":"How do I use the direction of a cross product?","text":"The question is: > An electron is projected at a speed of $3.70×10^6\\text{ m\/s}$ in the > $u=\\frac{(i+j+k)}{\\sqrt3}$ direction into a uniform magnetic field $\\vec {B} > = 6.43 i + 3.43 j −8.29 k$. Calculate the $x$ component of the vector force > on the electron. So I know that I'll need to use $F = q\\vec{v}\\times\\vec{ B}$ and do the cross product between the velocity and the magnetic field but I'm still getting the hold of cross products and vectors and the direction of the velocity is really throwing me off so I don't know how to write it down and solve. If anyone can put me in the right direction that would be awesome. Thanks."}
{"id":"29897","title":"Change in the Higgs mechanism","text":"The most important thing about the Higgs mechanism is that the higgs mechanism is one of the simplest mechanisms ( _certainly the simplest way to break the electroweak symmetry_ ) of elementary particles. There is an unwritten law in particle physics: When the mechanism is simple, just light Weight particles arise. What should be change to Higgs mechanism for the production of heavier particle?"}
{"id":"123659","title":"Thorium based nuclear reactor","text":"My question is how thorium works as a radioactive nuclear fuel and what are the special properties of thorium based nuclear reactor."}
{"id":"29898","title":"spectral eigenvalue staircase and quantum system","text":"in a d-dimensional system of Quantum physics , does the Eigenvalue staircase $ N(E)= \\sum_{E_{n}\\le E} 1 $ determine ALL the properties of Quantum System ?? for example, let us assume that the System is Chaotic and that Spectral eigenvalue staircase is given by $ N(E)= <N (E)>\\+ \\frac{1}{\\pi}ArgZ(1\/2+i \\sqrt E )$ with $Z(s)= \\prod_{n=0}^{\\infty}\\prod_{l_{p}}(1-l_{p}^{-s-n})$ the Gutzwiller zeta function"}
{"id":"113593","title":"An error in Gravitation by Misner Thorne and Wheeler?","text":"I was studying on Gravitation the PPN formalism. Since in equation (39.41) pag. 1087, the term $1 + \\dfrac{v^2}{2}+(2+\\gamma)U = 1 + \\dfrac{v^2}{2}+3U$ (the second in GR) looked odd, I tried (several times) to derive the formula, but I always find: $1 + \\dfrac{v^2}{2}+U$ Is this one of the two famous errors metioned by Wheeler? I am really going crazy over this..."}
{"id":"129908","title":"Are the cylindrical and spherical form of Jeans' equations equivalent?","text":"The question kind of says it all, what I really want to know is are the differences in their forms only due to the co-ordinate transform? And as such should a suitable spherical system satisfy axisymmetric Jeans equations? (numerically rather than analytically, otherwise it would all get very messy...) More background: Providing them in sufficient detail to be helpful would be very longwinded (as presumably you'd need to compare all three of the first moment equations in both co-ordinate systems) but for a bit more background people can always look here http:\/\/en.wikipedia.org\/wiki\/Jeans_equations, or in for an in depth discussion look in Galactic Dynamics by James Binney and Scott Tremaine."}
{"id":"113592","title":"plane wave move out","text":"I have a plane wave that is recorded by a set of receivers with $x$ spacing between them (see the sketch). in the sketch (plane wave in black slant line, receivers are the little circles in black and the line in blue is the surface). The place wave arrives with an angle to the normal (alpha) at the receivers. The wave was generated from 2000 cells to the left of my receivers line and 5000 cell deep below my line. Given I have the recording time of the arrival at each receiver (black circle) I was trying to correct for it and shift it into a normal incident rather than slanted as of the plane wave originated from deep below the receivers (5000 below). I do have the velocity of the medium below, arrival time, and angel of incident and I want to accomplish that. My final gaol is not to linear move out the arrive but rather shift it is if it was originated below the receivers. one approach I am thinking of is by multiplying the recording time at the first station by the cosine of the angel and then subtracting that from the first recording time $t_1-\\cos(\\alpha)\\times t_1$.![enter image description here](http:\/\/i.stack.imgur.com\/zoqTu.png)"}
{"id":"129907","title":"WHY is a Refracting telescope image distance $S'=-\\infty$?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/vUi8g.png)![enter image description here](http:\/\/i.stack.imgur.com\/kRqij.png)![enter image description here](http:\/\/i.stack.imgur.com\/lOO0v.png) In the above slide, it says that \"Eyepiece '2' makes the image distance $S'$ approach $-\\infty$.\" However, What I don't understand is that if I place my eye right at the Eyepiece 2, then I would see the image as if the object (for example, moon) is right in front of me as if I'm observing a rock in my hand... Therefore, I feel like $S'$ has to be closer to zero or some finite number! not negative infinity... Also, eventhough $S'$ is actually negative infinity, I quite don't understand how the moon can be observed as if I am observing a rock right in my hand... If $S'=-\\infty$, then my brain says that we can't really observe the cracks, rocks, or even sand in the moon since the object is so much farther away from me... Could somebody explain why I am confused???"}
{"id":"28279","title":"Are right handed neutrinos actually antineutrinos and vice versa?","text":"Is it experimentally ruled out that right-handed neutrinos are actually antineutrinos, and left-handed antineutrinos are neutrinos ?"}
{"id":"11078","title":"Stronger than Newton's laws?","text":"According to the Newton mechanics, the force is responsible for changing the body velocity, and the body mass is the body inertia - a property to resist to the applied force. These two things make a clear sense in the Newton equation (Second Law) $ma=F$. Both \"parameters\" (the body mass and the force strength) are observable and measurable. Now I am reading an **article** of G. 't Hooft where he states that \"The interactions among particles have the effect of modifying masses and coupling strengths\". So not only velocities are modified with forces but the particles masses and the forces themselves. I see here a contradiction with the Newton's definition of force and mass. As soon as the QFT is more fundamental than Classical Mechanics, does it mean that the 't Hooft's law is stronger than the Newton's ones? What is the impact (effect) of this stronger law on Classical Mechanics? How to define now the masses and forces if they are self-modifiable? **EDIT:** OK, is there a classical physics example to demonstrate that a potential interaction can modify mass and charge?"}
{"id":"28276","title":"Cloud chamber video showing large particles - What are they?","text":"Watching a video of a cloud chamber on wikipedia (http:\/\/en.wikipedia.org\/wiki\/File:Cloud_chamber.ogg), I cannot help noticing the large collisions that take place at 00:12 and 00:24. What are they? Alpha particles? They are huge compared to the small ones (i guess they are electrons)"}
{"id":"54915","title":"kirchhoff's law.Simple question","text":"So far I calculated total resistance and it is 4.66 ohms and there is 12 volt voltage across them, what is the best way or HOW to calculate current for each resistor ? ![enter image description here](http:\/\/i.stack.imgur.com\/DZWKG.jpg)"}
{"id":"8675","title":"Voltage drop along an idealized resistance-free wire in a circuit?","text":"If you connected the positive terminal of a battery to the negative terminal to a battery with a wire with (hypothetically) no resistance, and are asked to give the voltage drop of a segment of wire in the circuit, how would you determine this?"}
{"id":"89476","title":"How to solve Schrödinger's equation for this potential algebraically?","text":"I want to solve Schrödinger's equation with the potential $$V(x)=\\frac{1}{2}mx^2+\\lambda x$$ algebraically? Is there any way to construct ladder operators that are similar to the one for the harmonic oscillator in order to get the full Hamiltonian written as $H=k_1(b^*b+k_2)$? A good reference may be suitable as well."}
{"id":"48336","title":"Lightest flying solid material","text":"I need to use a number of lanterns, to write something with it! I would like to use a light material which will bind the lanterns together (separated by space) so it looks like dotted letter. I know that lanterns will lift up like helium balloon, but I don't know if I can think of a material which will be strong enough to not ruin\/mis-shape the letter (one letter at a time), but not heavy enough to stop the lanterns from raising up or causing harm to life on earth (people mainly) when they fall back down to earth (gently). I hope I have given all the variables for the above."}
{"id":"127373","title":"Does \"dark matter\" explain how I can have -1 apples?","text":"If I have 3 apples and you take 4 of them, that means I have -1 apples... is that apple made of dark matter?"}
{"id":"28856","title":"What are the topics of string theory that are comprehensible with only a mathematical background on Manifolds and Algebraic Topology?","text":"What are the topics of string theory that are comprehensible with only a mathematical background on manifolds and algebraic topology? Also, I have read only the first four chapters in Peskin & Schroder. And I have studied QM."}
{"id":"126974","title":"How to properly construct the electromagnetic tensor in curved space-time? (Part II)","text":"In this question, I am testing what was previously discussed. I can't seem to get my results to match D'Inverno's electromagnetic tensor for a charged point (page 239 of his book - _Introducing Einstein's Relativity_ ). **Here are D'Inverno's steps:**   * The line element in spherical coordinates is ($\\eta$ and $\\lambda$ are functions of $r$ only) $$ \\mathrm{d}s^2 = \\mathrm{e}^\\eta \\mathrm{d}t^2 - \\mathrm{e}^\\lambda \\mathrm{d}r^2 - r^2 (\\mathrm{d}\\theta^2 + \\sin^2\\\\!\\theta\\ \\mathrm{d}\\phi^2). $$   * He defines this covariant electromagnetic field tensor: $$ F_{\\mu\\nu} = E(r) \\begin{pmatrix} 0 & -1 & 0 & 0 \\\\\\ 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 \\end{pmatrix}. $$   * He then proceeds to find the electric field, and consequently the electromagnetic field tensor by using the source-free Maxwell equations: \\begin{align} \\nabla_\\nu F^{\\mu\\nu} & = 0 \\\\\\ \\partial_{[\\lambda} F_{\\mu\\nu]} & = 0. \\end{align}   * Solving the differential equation that appears from the equations above, he finds the electric field: $$ E(r) = \\mathrm{e}^{(\\eta+\\lambda)\/2} \\varepsilon\/r^2. $$   * He then notes that this field is that of a point charge at infinity ($\\eta$ and $\\lambda$ go to zero at infinity) where $\\varepsilon$ is the electric charge. I managed to reproduce these steps. **Now, here are my steps, using the four-potential procedure (the line element is the same):**   * I define my contravariant four-potential (there is just the first element which is the electric potential of a point charge, just as D'Inverno found): $$A^\\mu = (\\varepsilon\/r, 0, 0, 0). $$   * Then I lower the index of this four-potential to find the covariant one: $$A_\\mu = (\\mathrm{e}^\\eta \\varepsilon\/r, 0, 0, 0). $$   * Finally I apply this equation to build the covariant electromagnetic tensor: $$ F_{\\mu\\nu} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu. $$   * The result is $$ F_{\\mu\\nu} = \\frac{\\mathrm{e}^\\eta \\varepsilon}{r^2}\\\\! \\left(r\\frac{\\partial\\eta}{\\partial r}-1\\right) \\begin{pmatrix} 0 & -1 & 0 & 0 \\\\\\ 1 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 \\\\\\ 0 & 0 & 0 & 0 \\end{pmatrix}. $$   * Where: $$\\frac{\\mathrm{e}^\\eta \\varepsilon}{r^2}\\\\! \\left(r\\frac{\\partial\\eta}{\\partial r}-1\\right) = E(r)$$ And this is different from D'Inverno's electric field. I don't know what I am doing wrong. The calculations are not difficult for this simple case. The question is, due to these calculations: Do my contravariant four-potential needs to contain my metric funcions in some way? I was assuming it is just the four-potential for a electric charge in the flat space: $$A^\\mu = (\\varepsilon\/r, 0, 0, 0). $$ If everything is right, the wrong assumption must be here."}
{"id":"133981","title":"Physical interpretation of the relation $\\dot{x}(t) \\rightarrow i \\omega \\tilde{x}(\\omega)$","text":"If $x(t)$ is some time-dependent real quantity i can interpret its Fourier Transform $\\tilde{x}(t)$ as representing, in a generic sense, the frequency components of $x(t)$. What about the FT of $\\dot{x}(t)$? Can this quantity be correlated in some intuitive way to the \"motion\" $x(t)$ itself? In other words, is there some intuitive picture explaining the fact that $\\dot{x}(t) \\longrightarrow i \\omega \\tilde{x}(\\omega)$? This discussion should, of course, not be limited to functions that depend on time (only)."}
{"id":"11142","title":"Properties of liquid and air bubbles","text":"What is\/are the properties of a liquid (e.g. viscosity, surface tension) which is conducive to formation of stable bubbles floating in air (not the bubble inside the liquid)? E.g., if soap dissolved water is bubbled through a straw, big-small air bubbles are formed which comes out of the liquid and keep floating in the air. Edit:   This Wikipedia page (http:\/\/en.wikipedia.org\/wiki\/Surface_tension#Surfactants) attributes this to low surface tension of the liquid, but I can enumerate several liquids like spirit, petrol which have quite low surface tension, still if you blow through a straw into it, you won't get bubbles."}
{"id":"116959","title":"Electric & Magnetic fields - clarification","text":"If (changing electric field)=(magnetic field) and (changing magnet field)=(electric field), then why is there a constant magnetic field around a current carrying wire when the electric field does not change?"}
{"id":"10800","title":"How can the Hall effect ever show positive charge carriers?","text":"The Hall effect can be used to determine the sign of the charge carriers, as a positive particle drifting along the wire and a negative particle drifting the other direction get deflected the same (as $F = q \\vec{v}\\times\\vec{B} = (-q) (-\\vec{v})\\times\\vec{B}$). But I don't understand how positive charge carriers are ever possible. As I've understood, a positive hole is nothing more than the absence of an electron. As all the electrons in the valence band are still negatively charged, why would this hole behave in a magnetic field as if it were positive? Also, a hole is created if an electron is excited into the conductance band. If there is always the same numbers of holes as electrons, how can any Hall effect ever occur? Thank you very much"}
{"id":"41770","title":"Why do birds, sitting on electric wires, not get shocked?","text":"If we would touch electric wires, we would get a shock, even if we are not touching ground (so that no connection is complete form wire to ground through us). I always see that birds sit on electric wires. Why don't they get shocked?"}
{"id":"62996","title":"Electric shock from a power line","text":"A bird perches on a bare high power line and nothing happens to it. A man standing on ground touches the same line and gets a fatal shock. why?"}
{"id":"80679","title":"How does a crow sitting on one of the electric lines attain the same potential as that of the line so as to prevent a charge flow through its body?","text":"The crow doesn't get an electric shock while sitting on only one electric line because it has the same potential as that of the line. Since there is no potential difference the charge does not flow through the crow's body unless the crow touches the neutral line and creates a potential difference between one end of its body to the other. How could the crow attain this much electrical potential? The crow was at a zero potential when it came and sat on the line. Was there any charge flow to the crow's body from the line? If there was, how could the crow survive such a potential hike?"}
{"id":"63980","title":"Experiment to find Ammeter Resistance","text":"I've been studying experimental electrodynamics and I needed to describe an experiment to find the resistance of an ammeter if I just have the ammeter together with one voltmeter and a protection resistance. My idea was the following: since the ammeter has much less resistance than the voltmeter we connect them in paralel and connect this in series with a protection resistance connected to the battery. The idea is that the current will be reduced by the protection resistance avoiding damages to the ammeter. Then the current will divide and a much greater part will flow through the ammeter. Since the ammeter and the voltmeter are in parallel the voltage seen in the voltmeter is the same on both devices, and the ammeter will show the current that flows through it so that we can use Ohm's Law to find it's resistance: $R=V\/I$. If we knew then the errors of the ammeter and of the voltmeter we could propagate to find the error in the resistance. Is this correct? Is this a valid way to measure the resistance of the ammeter? Thanks very much in advance for the help."}
{"id":"61851","title":"Faraday's law for a current loop being deformed","text":"I'm working through Jackson's Classical Electrodynamics text (3rd version, chapter 5.15) about Faraday's law: Faraday's law is pretty familiar: $\\int_c E \\cdot dl = -\\frac{d}{dt}(\\int_s B \\cdot n da)$ which states that the contour integral of the electric field around a bounded surface is equal to the negative time rate of change of the flux bounded by that surface. No problem. Here we have a total derivative and Jackson addresses 2 cases. \"The flux though the circuit may change because (a) the flux changes with time at a point, or (b) the translation of the circuit changes the location of the boundary\" In either of these cases, it is easy to show that: $\\frac{d}{dt}(\\int_s B \\cdot n da) = \\int_s(\\frac{\\partial B}{\\partial t} \\cdot n da) + \\int_c(B \\times v)\\cdot dl$ by using $\\frac{d}{dt} = \\frac{\\partial}{\\partial t} + v \\cdot \\nabla$ and then applying a \"product rule\", $\\nabla \\cdot B = 0$ and stokes theorem. i.e.: $\\frac{d}{dt}\\int_s B \\cdot n da = \\frac{\\partial}{\\partial t}\\int_s B \\cdot n da + (v \\cdot \\nabla) \\int_s B \\cdot n da$ **In the case where the surface is constant** , we can exchange the order of differentiation and integration: $\\frac{d}{dt}\\int_s B \\cdot n da = \\int_s \\frac{\\partial}{\\partial t} B \\cdot n da + \\int_s (v \\cdot \\nabla) B \\cdot n da$ I still have no problem with this. Now we can work on the second term on the right (apply a product rule, drop terms with $\\nabla \\cdot B$ and then apply stokes theorem) to get the $\\int_c(B \\times v)\\cdot dl$ term. which, when applied to faraday's law gives the nice result: $\\int_c [ E' - (v \\times B)] \\cdot dl = - \\int_s \\frac{\\partial B}{\\partial t} \\cdot n da$ The problem arises when you allow the shape of the bounded surface to change. (Consider a circular wire loop in a magnetic field that you then hit with a hammer causing a dent in one side). How does changing the surface influence the result that we derived above? It seems to me that this would prevent us from being able to exchange the order of differentiation and integration which seems like it leaves us in a pretty tight spot ..."}
{"id":"117396","title":"How does cryptochrome produce quantum entanglement?","text":"I found this video on the net, which explains how a bird called \"European robin\" uses a trick based on quantum entanglement to see the earth's magnetic field. I am just asking if some of you know any research paper or reference on this subject. What I want to know is how exactly the molecule Cryptochrome in the bird's eye produces quantum entanglement after being hit by a photon."}
{"id":"105769","title":"Magnetic flux and transformers","text":"I have two linked quesitons, 1. what is magnetic flux linkage's formal definitaiton and 2. when people say the flux flows better thru a soft iron core then the air what do they mean by flux flowing (in a transformer)???"}
{"id":"52094","title":"Exciton energy splitting in indirect semiconductor","text":"Let's take an indirect bulk semiconductor and imagine the exciton that build up at the indirect gap. Since we have a bulk semiconductor we should find three p-like orbitals: $p_x,p_y,p_z$. Would their energy levels be degenerate due to the mass anisotropy which we have at the indirect gap? My intuition screams yes. But if I try to bring it on the paper I actually find the opposite. Any suggestions, ideas or helpful tips?"}
{"id":"52099","title":"Lamb-Chaplygin dipole","text":"There is an interesting issue of hydrodynamics: whirlpools. I want to learn about the Lamb-Chaplygin dipole. Lamb-Chaplygin dipole corresponds to a steady solution of the two dimensional Euler equations. Can someone know where I can find a full development of Lamb-Chaplygin dipole?"}
{"id":"105762","title":"Could there be massive gravitons and in our spacetime?","text":"My question is: Could there in principle be sub-groups of gravitons (if they existed) that has mass (and in our spacetime dimensions) such as those affected by electric or by strong force or other forces which could be above the energy threshold of our current particle accelerators. (Perhaps they could be created, say, near pulsars due to high-energy collisions)?"}
{"id":"105764","title":"One Loop Higgs Mass Correction","text":"I am attempting to compute the one loop correction to the Higgs mass, which requires the evaluation of a scattering amplitude, namely $$\\require{cancel} \\mathcal{M} = (-)N_f \\int \\frac{\\mathrm{d}^4 k}{(2\\pi)^4} \\mathrm{Tr} \\, \\left[ \\left( \\frac{i\\lambda_f}{\\sqrt{2}}\\right) \\frac{i}{\\cancel{k}-m_f} \\left( \\frac{i\\lambda_f}{\\sqrt{2}} \\right) \\frac{i}{\\cancel{k} + \\cancel{p}-m_f}\\right]$$ which corresponds to the Feynman diagram: ![enter image description here](http:\/\/i.stack.imgur.com\/pX1RH.png) After combining constants, and rationalizing the denominators, I obtain, $$-\\frac{N_f \\lambda_f^2}{2} \\int \\frac{\\mathrm{d}^4 k}{(2\\pi)^4} \\frac{\\mathrm{Tr}\\left[ \\cancel{k}\\cancel{k} + \\cancel{k}\\cancel{p} +2m_f \\cancel{k} + m_f \\cancel{p} + m_f^2\\right]}{\\left(k^2-m_f^2\\right)\\left((k+p)^2 -m_f^2 \\right)}$$ Computing traces, via the relation $\\mathrm{Tr}[\\cancel{a}\\cancel{b}] = 4(a\\cdot b)$ yields, $$-2N_f \\lambda_f^2 \\int \\frac{\\mathrm{d}^4 k}{(2\\pi)^4} \\frac{k^2 +k\\cdot p + m_f^2}{\\left(k^2-m_f^2\\right)\\left((k+p)^2 -m_f^2 \\right)}$$ At this point, I employed dimensional regularization, followed by Feynman reparametrization to combine the denominators, and then completed the square, yielding $$-\\frac{2^{2-d}\\pi^{-d\/2}}{\\Gamma (d\/2)}N_f \\lambda_f^2 \\int_{0}^1 \\mathrm{d}x \\int_0^\\infty \\mathrm{d}k \\frac{k^{d-1}(k^2 +kp + m_f^2)}{\\left[ \\left(k-(x-1)p\\right)^2 +p^2(x-x^2 -1)\\right]^2}$$ **Additional Calculations (Edit)** I attempted to further simplify the integrand using a substitution in only the first integral, namely $\\ell = k-(1-x)p$ which implies $\\mathrm{d}\\ell = \\mathrm{d}k$, yielding (after several manipulations), $$-\\frac{2^{2-d}\\pi^{-d\/2}}{\\Gamma(d\/2)}N_f \\lambda_f^2 \\int_0^1 \\mathrm{d}x \\, \\int_{(x-1)p}^{\\infty} \\mathrm{d}\\ell \\frac{(\\ell + (1-x)p)^{d-1}[(\\ell + \\frac{1}{2}p(3-2x))^2 - \\frac{1}{4}p^2 + m_f^2]}{[\\ell^2 + p^2(x-x^2-1)]^2}$$ **N.B.** Mathematica evaluated the original integral over $k$, and outputted a combination of the first Appell hypergeometric series, which possess the integral representation, $$F_1(a,b_1,b_2,c;x,y) = \\frac{\\Gamma(c)}{\\Gamma(a)\\Gamma(c-a)} \\int_0^1 \\mathrm{d}t \\, t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b_1}(1-yt)^{-b_2}$$ with $\\Re c >\\Re a >0$, which has a structure similar to the beta function. If I can express the loop integral in a similar form, I may be able to express it in terms of these functions. At the end of the calculation, I will take $d \\to 4-\\epsilon$ to obtain poles in $\\epsilon$, using the usual expansion $$\\Gamma(x) = \\frac{1}{x} -\\gamma + \\mathcal{O}(x)$$ and a similar expansion should the final answer indeed contain the Appell hypergeometric series. **Passarino-Veltmann Reduction (Edit):** Based on my understanding of Veltmann-Passarino reduction, it is not applicable as the numerator contains an arbitrary power of loop momentum. I could plug in $d=4$, and impose a high momentum cut off, but this has already been done in many texts. As aforementioned, I would like a dimensionally regularized amplitude. I am stuck at this point, can anyone give some details as to how to proceed? In addition, I have a query regarding the hierarchy problem. If using a simple cut-off regularization, the one loop correction can be shown to be quadratically divergent. But why is this an issue that needs to be remedied, by for example, the minimally supersymmetric standard model? Can't the divergence be eliminated by a regular renormalization procedure?"}
{"id":"60183","title":"What is the term for heat generation by a flowing fluid?","text":"I would like to know more about the heat distribution over time in a flowing liquid. To this end, I consider the Navier-Stokes equation (where the coefficients may be temperature dependent) and the heat equation $$ \\rho c_p \\left(\\frac{dT}{dt} + u \\cdot \\nabla T\\right) = \\sigma + S(u) $$ where $T$ is the temperature and $u$ the flow velocity. The term $\\sigma$ may be some artificial heat source. The term $S(u)$ is now of interest to me: It corresponds to the heat generated by the friction of particles (I think), and I believe it must be proportional to the dynamic viscosity of the medium. I don't recall the specifics though. Can anyone help out on what $S(u)$ should be?"}
{"id":"69670","title":"Finding a discontinuity and jump (time independent Schrodinger eqn)","text":"A particle of mass $m$ is confined to move in a one-dimensional and Dirac delta-function attractive potential $$V(x)=-\\frac{\\hbar}{m}\\alpha\\delta(x)\\text{ $,\\alpha>0$ }.$$ Show that the function $\\psi(x)$ is discontinuous at $x=0$, and determine its jump at the point. Showing its discontinuous at zero is easy. It follow from the following: $\\frac{-\\hbar^2}{2m}\\int_{-\\epsilon}^{\\epsilon}\\frac{d^2\\psi}{dx^2}+\\frac{-\\hbar^2\\alpha}{m}\\int_{-\\epsilon}^{\\epsilon}\\delta(x)\\psi(x)=0\\implies \\lim_{\\epsilon\\to 0} \\frac{d\\psi}{dx}\\bigg|_-+\\frac{d\\psi}{dx}\\bigg|_+=2\\alpha\\psi(0)$ As for the jump, that's where I'm stuck. I assumed it was at $y=\\alpha$. But I believe I'm wrong."}
{"id":"69674","title":"Dimensions in lagrangian potential","text":"According to Mankowski flat space dimensions We can write, $$L= \\int \\text{dt} \\text d^d{x} \\left[ \\frac{1}{2} \\dot\\phi^2 - \\frac{1}{2} \\left(\\frac{\\partial \\phi}{\\partial r} \\right)^2 -V(\\phi)\\right] \\tag{1}$$ Where $V$ can be written as $$V = \\frac{1}{8} \\phi^2 (\\phi-2)^2$$ But the author wrote in his article in the equation (1) including dimensions. $$V= \\frac{1}{2} m^2 \\phi^2+ \\frac{\\lambda_3}{3!}m^\\frac{5-d}{2} \\phi^3+ \\frac{\\lambda_4}{4!}m^{3-d} \\phi^4$$ My question is how the dimensions incorporated with the potential?"}
{"id":"60188","title":"Are there functions of the metric that are scalars under spatial diffs up to total derivatives?","text":"Let $g_{\\mu\\nu}$ be a metric on a manifold with a time direction $x^0$ singled out. I'm wondering if there exists a function $F(g_{\\mu\\nu},\\partial_\\rho g_{\\mu\\nu},\\ldots)$ that transforms under spatial diffeomorphisms as \\begin{align*} F(g_{\\mu\\nu}'(x'),\\partial_\\rho g_{\\mu\\nu}'(x'),\\ldots)=F(g_{\\mu\\nu}(x),\\partial_\\rho g_{\\mu\\nu}(x),\\ldots)+ \\nabla_\\mu \\Lambda^{\\mu}(g_{\\mu\\nu},\\partial_\\rho g_{\\mu\\nu},\\ldots,x'), \\end{align*} where $\\Lambda$ is some functional of the metric and $x'$. This would imply that the integral \\begin{align*} \\int d^dx\\, \\sqrt{-g}F \\end{align*} is invariant under spatial diffs. Any ideas?"}
{"id":"1845","title":"Home experiments using wireless LAN or mobile phones about electromagnetism?","text":"Are there any nice experiments using wireless LAN access points or routers or mobile phones to demonstrate physical features of electromagnetic fields, especially em-waves? More precisely I am looking for experiments which use one of the devices above as the main tool and some materials that can be purchased at a reasonable cost from, say, a hardware store or electronic supply store only. I do not impose any restrictions on the cognitive level the experiments aim at; undergraduate level is fine as well as graduate level in em-theory. Please use separate answers for different experiments."}
{"id":"43607","title":"Calculate pressure of air trapped underwater","text":"Seal top end of straw, partially submerge straw in water, how do I calculate the air pressure inside the straw? Bonus points for compensating for any secondary factors like temp, humidity or flex of the straw. Application: measuring water level in 5 gallon aquarium to accuracy of about 5mm."}
{"id":"43602","title":"What process could produce two neutral pions only?","text":"Any examples? $$? \\rightarrow \\pi^0 \\pi^0$$ If such a process exist, could there be nonzero total orbital angular momentum in the final states of the two neutral pions? But then how to understand the interaction between the two neutral pions that make them rotate with each other to contribute the non-zero L?"}
{"id":"68741","title":"Online physics diagram tool","text":"What is a suitable tool for drawing physics diagrams online? I would love to know if there is a program like the excellent schematic drawing tool that is integrated into the electrical engineering stack exchange, only for physics. This would really help in answering questions more graphically and in an intuitive fashion, without having to rely on formulas. I am aware of the offline possibilities, but for most problems, I just need to draw simple shapes such as lines, circles, boxes, and squiggly lines. I work from several machines, so an online tool would be a lot more flexible."}
{"id":"81050","title":"Is there a derivation of all possible Bravais lattices in 2D and 3D?","text":"Do any of you know a (possibly elegant) derivation of all possible Bravais Lattice and the Crystallographic Group of Lattices in 2 and 3 dimensions? Group theory \/abstract nonsense approaches are welcome."}
{"id":"81056","title":"Why don't we normally see the Higgs boson?","text":"I am a physics student and my dad just asked me about the Higgs Boson. I've told him the little I know, that the Higgs field is a field that is supposed to give mass to elementary particles, and that finding the boson was crucial to see if this mechanism actually did exist. After telling that, a question has come into my mind. I have heard that it is very difficult to create the circumstances where we would have Higgs Bosons. I have been told also that this boson is the carrier of the field responsible of giving mass to other particles. So the question is quite natural to me, how come something which \"is not there\" in \"normal\" conditions happens to do its work? I mean, the carrier of the electromagnetic force are the photons and we have photons in \"normal conditions\"(by this I have meant not the conditions we have inside the LCH, for example) so it is natural to see their effects, but how is this possible with the Higgs?"}
{"id":"52638","title":"Theoretical power limit of nuclear bomb","text":"Is there any limits of power or power to mass (or any other) limits for nuclear bombs? I found this wiki article: Nuclear weapon yield: Yield limits. Is the information provided is correct? If yes, from where this limits come from?"}
{"id":"60723","title":"Spontaneous conversion of heat into work at negative temperatures","text":"Consider a heavy macroscopic object moving in a gas. Friction causes its kinetic energy to be converted into heat. Thermodynamically, there is (effectively) no entropy associated with the kinetic energy because all the energy is concentrated in a single degree of freedom. Therfore, if an amount $J$ of energy is converted from kinetic energy into heat, the total entropy change is $J\/T$, so we can see that this is a spontaneous process. But now consider an object moving relative to a gas with negative temperature. Such a thing has been created in the laboratory, so this is not just idle theoretical speculation. If an amount $J$ of kinetic energy gets converted into heat, the total entropy change is still $J\/T$, but now this is negative. This seems to mean that the opposite process - conversion of heat into kinetic energy, accelerating the object - would be spontaneous. This generalises to all other processes that convert work into heat. For example, performing the Joule heating experiment with a negative-temperature gas should cause the paddle to turn, and negative-temperature gas flowing through a pipe should experience an accelerating force rather than a decelerating one. Just as superfluids have zero viscosity, it seems that negative-temperature fluids must have negative viscosity. I realise that this does not lead to perpetual motion. As heat is converted into work the inverse temperature ($1\/T$) will increase until it reaches zero. But what does look odd is that in some ways the arrow of time appears to be reversed. I realise that experimentally we're very far from being able to produce the macroscopic quantities of negative-temperature fluids that would be required in order to observe these things. But is it possible in principle? And if it is, would we actually see the phenomena I described, or is there some fundamental reason why they wouldn't happen after all? And has such a connection between negative temperatures and the arrow of time been discussed or debated in the literature?"}
{"id":"94656","title":"Chiral Spin Liquid(CSL), Chern number, and the ground state degeneracy(GSD)","text":"Consider a 2D _gapped_ CSL with a _nonzero_ Chern number $m$, then is the GSD of the system on a torus directly related to the Chern number $m$? For example, see this article, in the last paragraph on page 7, the authors give the **4-fold GSD** from the Chern number $m=\\pm2$ for a CSL. I can not understand the explanation, can anybody present an intuitive illustration or a simple mathematical proof ? I will be very appreciated, thank you very much."}
{"id":"53324","title":"Is There a List of Physics Formulas?","text":"I am looking for a list of _simple_ physics formulas \\- like $F=ma$. Is there any website\/book that is a list of such equations?"}
{"id":"102230","title":"Does evaporation decrease entropy?","text":"I have a container of water in thermal equilibrium (there are no temperature fluctuations inside the container). Some molecules will evaporate out of the container of water thus decreasing its temperature. This is clearly a spontaneous process. But doesn't this mean that the entropy of the system is decreasing? The molecules that evaporated were clearly in thermal equilibrium with the other water molecules before. But now they gained some energy and the other water molecules lost energy. How can this be?"}
{"id":"75369","title":"Time travel and entropy","text":"I saw a post on reddit regarding immortality and how it would never be possible due to entropy. That said, assuming time travel was possible, would it not be possible to never reach this state of entropy if you could keep traveling back in time? Say for instance that you couldn't travel back in time within your own time curve, but you could travel to other time curves. You would merely jump to another curve when your current universe was about to reach its end. Is it too big of an assumption to think that your own initial entropy (by this I mean your entropy from your first timeline) would not 'follow you' to other time curves? I may be totally off-base in my assumptions, as I know next to nothing about physics. Any feedback\/criticism is welcomed. Here is the link to the post in reddit."}
{"id":"106444","title":"Weird resistor patterns","text":"I came up with the following question: > The vertices of tetrahedron ABCD are all connected by 1-ohm resistors. What > is the total resistance between A and B? I haven't the faintest idea how to solve it. Does it involve infinite path summations at some point? [edit: how about from A to C?]"}
{"id":"57806","title":"What's the electric field with a point charge not in the center of the sphere?","text":"That's a hollow conducting sphere link My charge is at P (10uC). R = 0.15m. PS = 0.05m. DS = 0.35m. What's the electric field at point D? ![enter image description here](http:\/\/i.stack.imgur.com\/UkYgu.png)"}
{"id":"96641","title":"Non-vanishing commutator of potential and mass matrices for Majorana fermions interaction theory","text":"Consider 2 different Majorana fermions $\\Psi_{L}, \\Psi_{R}$ (physically, neutrinos). In general case I can write the massive part of lagrangian of these fermions in the form $$ L_{m} = (\\bar {\\Psi}_{1} \\bar {\\Psi}_{2})\\begin{pmatrix} m_{11} & m_{12} \\\\\\ m_{21} & m_{22} \\end{pmatrix}\\begin{pmatrix} \\Psi_{1} \\\\\\ \\Psi_{2} \\end{pmatrix} = \\bar {\\Psi}^{a}\\hat {m}_{ab}\\Psi^{b}. \\qquad (1) $$ Then let's introduce matrix of some gauge interaction $$ \\hat {A}_{\\mu}^{ab} = A_{\\mu} = \\begin{pmatrix} A^{11}_{\\mu} & A^{12}_{\\mu} \\\\\\ A^{21}_{\\mu} & A^{22}_{\\mu} \\end{pmatrix}_{ab}. \\qquad (2) $$ and corresponding fermion's charge $g$. The full lagrangian with interaction and mass term takes the form $$ L = \\bar {\\Psi}(i\\gamma^{\\mu}\\partial_{\\mu} + g\\gamma^{\\mu}\\hat {A}_{\\mu})\\Psi - \\bar {\\Psi}\\hat {m}\\Psi . $$ I want to represent the lagrangian in terms of states with definite masses. Let's call them $\\Psi_{L}, \\Psi_{R}$. So by introducing mixing unitary matrix $$ \\begin{pmatrix} \\Psi_{L} \\\\\\ \\Psi_{R} \\end{pmatrix} = \\begin{pmatrix} cos(\\theta ) & -sin (\\theta ) \\\\\\ sin(\\theta ) & cos(\\theta ) \\end{pmatrix}\\begin{pmatrix} \\Psi_{1} \\\\\\ \\Psi_{2} \\end{pmatrix} $$ I can rewrite the mass term in a form of $$ L_{m} = (\\bar {\\Psi}_{L} \\bar {\\Psi}_{R})\\begin{pmatrix} m_{1} & 0 \\\\\\ 0 & m_{2} \\end{pmatrix}\\begin{pmatrix} \\Psi_{L} \\\\\\ \\Psi_{R} \\end{pmatrix}. $$ But I can't diagonalize the $A$-matrix in general case in this basis. So the $A$-matrix and $m$-matrix don't commute with each other. So there is the question: what is the physical meaning of this result in the case of Majorana fermions?"}
{"id":"99635","title":"emergent phenomena from microscopic equations of motion","text":"Is an emergent phenomenon like \"superfluidity\" or \"quantum Hall effect\" derivable from microscopic equations of motion? If No, why?"}
{"id":"114243","title":"The reason(s) of seasons on earth","text":"This maybe simple and usual question, although there is a lot of confusion about it over internet and even in some books, so I want from an astronomer \/ astrophysicist to fill the gaps for me accurately. Fixed axis tilt is the reason, this is the most common explanation, but with some research, you will find that it is not the only reason! Actually, earth distance from the sun playing its role: considering that the received energy from sun is inverse proportional to the square of earth distance, one can calculate that it's contribute to a change by 5-7% of the received energy. Moreover, it's effect is opposite, in senses that because we have summer when we are at the furthest point from sun, this actually reduces summer temperature due to axis tilt. Also I read a claim that mentions a third reason, that most of the texts ignores, it is the angle of earth orbit to the suns equator, anyway, I found no quantitative analysis. Your comments please."}
{"id":"619","title":"Why is it so hard to accelerate macroscopic objects?","text":"It seems all we're capable of accelerating currently are atomic particles. Why can't we, say, accelerate a clock to relativistic speeds?"}
{"id":"109323","title":"Numerical Tools to find Braiding Statistics of Quasiparticles","text":"While certain classes of systems that exhibit topological order can be solved exactly (such as the Toric Code, Abelian FQH Edges, etc.) there also exist systems (think of perturbed versions of the Toric Code or Abelian FQH Edges) that cannot be solved exactly. What are the common numerical tools that are used to extract the (not necessarily abelian) braiding statistics of quasiparticles from these models? What is the main idea of these techniques? For which systems do they work well and for which systems do they fail? Can one access the braiding statistics also from numerical exact diagonalization of a small system? (For say a perturbed Toric code I would guess so because at least in the unperturbed case the abelian braiding statistics do not depend on the system size) I would be very happy if someone could provide a brief overview. I am looking forward to your responses!"}
{"id":"614","title":"Where do atoms go after collision?","text":"[I edited the question according to Mark's and Grisha's answers.] Consider two point-like particles of equal mass colliding centrally in 2D. The final directions of the momenta of the two particles are not determined by conservation of energy and momentum: they can go anywhere. > Under which circumstances the direction of the two particles _is_ > determined?"}
{"id":"59402","title":"A charged sphere with pulsing radius","text":"![enter image description here](http:\/\/i.stack.imgur.com\/D0854.png) Radius increases and decreases periodically (as a pulse).And so does the charges on the surface of sphere. ![enter image description here](http:\/\/i.stack.imgur.com\/SqSv2.gif) I can't get what is gonna happen.the EM waves are produced perpendicularly to motion of the charges,but here where is the perpendicular ? Will the EM waves get produced into the space of sphere? Also, can two EM waves intersect each other or not? [as Electric fields lines can't intersect.]"}
{"id":"115094","title":"Standing sound waves phase difference","text":"This is a basic question but I don't fully understand what is going on. When we have a pipe in resonance we get standing waves, where displacement antinodes are at pressure nodes and vise versa. So we have a situation where pressure and displacement are 90 degrees out of phase, this means you can differentiate the displacement wave to recover the pressure wave. Does anyone have a nice explanation of this 90 degree phase difference?"}
{"id":"30804","title":"Frequency and wavelength of photons","text":"I try to better understand how electromagnetic radiation works. So I have some questions. If an antenna emits at 100MHz (the charges on the antenna oscillate at 100MHz) what frequency will have the emitted photons? Will they have the same 100 MHz? In the case of the photons what will this frequency represent in fact? Also what does the wavelength of the photons represent? I mean in the case of an electromagnetic wave , the wave represents the variation in intensity of the magnetic and the electric fields, and the wave-length is the distance between two crests of this variations (correct me if I am wrong). How about in the case of the photon wave-length?"}
{"id":"30802","title":"Do materials cool down in the vacuum of space?","text":"Do materials cool down in the vacuum of space? If yes, how does it really work?"}
{"id":"29662","title":"Moons of the planets","text":"What is the phenomena behind gas giants like Jupiter and Saturn having a large number of moons as compared to other planets. Were the present state moons a part of their parent planet long time ago ?"}
{"id":"26895","title":"Random Walk Randomly Reflected","text":"Hi I am not specialist in probability so I will not be surprised if the answer for this question is just a simple consequence of well known results from the random walk theory. In this case, I will be happy if you can tell me the \"magic words\" to find the material related to this problem. Let $\\sum_{i=1}^n X_i$ a random walk in $\\mathbb{Z}^d$ defined in the following way: In each step, the walker look at your current position and as long as its distance to the origin does not belong to the set $\\\\{d_1,d_2,\\ldots\\\\}$ ( we assume that $d_n\\in\\mathbb{N}$ and $d_i<d_{i+1}$ ) it jumps for one of its nearest neighbor with probability $1\/2d$. For the other hand if $\\|\\sum_{i=1}^n X_i\\|\\in \\\\{d_1,d_2,\\ldots\\\\}$ then the walker toss a biased coin if the result is head, which occurs with probability $p$, it goes back to the previous position, that is, $\\sum_{i=1}^{n-1} X_i$. If the result is tails it chose randomly one of its neighbors with probability $1\/2d$. I would like to know if there exist a non-trivial $p_c$ such that this random walk is transient for $p<p_c$ in dimension $d=3$. Of course, for this question to be well posed we have to define the sequence $(d_n)$. I would like to chose it randomly but I have no idea for what kind of distribution this problem is tractable. So at this time we can assume that $d_n$ is deterministic sequence and, for example, has a polynomial growth as $d_n=n^2$. Any references or comments are very welcome. PS: Could an user with more rep. place random walk tag for this question ?"}
{"id":"73800","title":"What is the shielding in nuclear reactors mainly against?","text":"I have a little knowledge about ionizing radiation and I have been confused over why nuclear reactors need these massive shields. So, if I am not mistaken, Alpha and Beta radiation are not that dangerous since they can be shielded with relatively light materials, the main problem is because Gamma and Neutron radiation. So now, which one requires that heavy shielding ? Also, if the reactor released either Gamma rays only or Neutrons only, which one of them would need less massive material to be reduced to a non-harmful level ? Any numbers on the density of radiation this shield has to stop will greatly help here too."}
{"id":"114945","title":"kinetic and potential energy","text":"I have 2 cubes where one mass is greater than the other. $M$ and $m$, where $M>m$, and there is a hill that is symmetrical on both sides, and has a friction factor of $k$ between the object and the surface. If we place both objects on one side of the hill, point 1, that has no slope, so it is flat, and give both of the masses the same velocities in the direction of the top of the hill so they finally land on the other side, on the flat area, point 2 (symmetrical to the point 1). How can I prove that the speed of the heavier cube is smaller than the lighter one in the point 2? (If I'm correct) I tried equating $E_{kinetic}$ and $E_{potential}$... I see there is work from the friction, but whenever I try anything I cross out the mass. Heres a diagram: http:\/\/www.wolframalpha.com\/input\/?i=sin%5E2(x)%2C+0%3Cx%3Cpi&x=0&y=0 Where point 1 is at (0, 0), and point 2 at (pi, 0). What I have so far: where $g$ is constant of gravity and $k$ friction factor, and $d$ distance. $E_{kinetic.start} - W_{friction} = E_{kinetic.end}$ $M(\\frac{v_0^2}{2}-gkd)=M(\\frac{v_1^2}{2})$ I could change $M$ with $m$ and nothing would change, they'd both have same speeds... So it's obvious that the masses cancel out and the energy is proportionally lower at both masses, the distance for friction is the same and starting velocities are the same, therefore the velocity should be the same for both bodies at the end. This confuses me as I believe the smaller body should be faster. (Cant noone help me with this? Im now doubting the solution i've provided first is right. I cannot prove otherwise.)"}
{"id":"34656","title":"How to recognize broken candies from whole ones","text":"Let's say I have a bag full of sugar candy. Some will be whole, some will be dent, some will be broken (in part, or half, etc). Let's say I have a device with an input box where I empty the bag, and the device has two output boxes where it start placing the whole candies in one box, and all the dented\/broken one in the other box. What principle of the physic could use the device, to be able to tell the whole candies from the dented\/broken ones?"}
{"id":"34652","title":"Is it best to look at light as a particle when trying to understand special relativity?","text":"So my course about special relativity explains time dilation using a moving train, where one sends up (i.e. perpendicular to the direction of movement) a light pulse which gets reflected etc. (a similar example is to be found here so you guys know what I'm talking about) Let's say the source of the light pulse is a flash light. In the moving frame of reference, it is pointed upwards, and thus only has an upward component. Everything fine so far. The thing is, that when you look at it from the outside (\"the train station\"), I picture the flash light to still be pointed directly upward, but somehow when the pulse leaves the flash light, it gets a horizontal component. This does not make sense if you think of it in terms of some stationary ether (which is also wrong as Michelson & Morley showed). The problem is that I can't seem to get my mind around the fact that it somehow seems that the flash light is pointed at an angle in the outside frame of reference. Except if I think of light as photons (particles), then it seems intuitively correct that they get a horizontal component (compare it to a skater throwing up a tennis ball). So, when trying to understand SR, do you have to think of light as particles? My course states this nowhere explicitly so I'm not sure. _I might have answered my own question again by formulating it in a more or less ordered fashion, as happens so often when I post it here :P_"}
{"id":"67843","title":"Help finding Delta T for a known position displacement with known accel\/decel, etc","text":"I'm looking for an equation to find the time it will take for a position displacement to happen, given known $V_{max}$, a (known) constant acceleration, deceleration, jerk, and displacement... and an initial velocity of 0. I am trying to estimate\/calculate the time it will take for a servo motor axis to travel a certain distance. Once I know the estimated time of that move, I can solve for jerk for the other axis. That way, I choose an optimally \"slow\" speed for whichever axis has a shorter move distance to make the maximum use of my time. FYI, this is a pick and place XY robot gantry."}
{"id":"112854","title":"Does the conversion of crude oil to greenhouse gases have any measurable effect on earth's gravitational pull?","text":"Oil underground is much denser than greenhouse gas in the atmosphere. Does the conversion in anyway effect the gravitational force from earth."}
{"id":"131751","title":"Making fire using a filled water bottle","text":"I have seen videos where a fire was started using a water bottle filled with water. I guess the bottle acts like a convex lens that converges light to a focal point to start a fire. But why does water have to be added?"}
{"id":"35383","title":"What is the origin of the many-body expansion?","text":"I'm looking for the original introduction of the many-body expansion (MBE) in the scientific literature. More specifically, I'm interested in a theoretical justification of the rapid convergence of the expansion, especially in the context of molecular physics. The MBE is often introduced in scientific papers without referring to foregoing work that introduces the concept. Instead, one often states something like: \"Following the well-known many-body expansion, one writes ...\". See for example http:\/\/pubs.acs.org\/doi\/abs\/10.1021\/ct600253j. (freely available at http:\/\/t1.chem.umn.edu\/Truhlar\/docs\/758FAV.pdf) The many-body expansion is a scheme to decompose the energy of a general system of $N$ particles as follows: $$ V = \\sum_{i = 1}^N V_i $$ with $$ V_1 = \\sum_{i=1}^N E_i $$ $$ V_2 = \\sum_{i=1}^N \\sum_{j=i+1}^N E_{ij} - E_i - E_j $$ $$ V_3 = \\sum_{i=1}^N \\sum_{j=i+1}^N \\sum_{k=j+1}^N \\biggl( (E_{ijk} - E_i - E_j - E_k) - (E_{ij} - E_i - E_j)\\\\\\ - (E_{jk} - E_j - E_k) - (E_{ki} - E_k - E_i)\\biggr) $$ and so on. In these equations, $E_i$ is the energy consisting only of particle $i$, $E_{ij}$ is the energy of a system containing only particles $i$ and $j$, $E_{ijk}$ is the energy of a system with three partilces, $i$, $j$ and $k$, and so on."}
{"id":"107314","title":"Does electric field cause changing magnetic field also?","text":"According to Faraday's law, changing magnetic field causes eletric field (imprecise wording, but generally accurate). My question is, can Faraday's law be interpreted in opposite way - that is, electric field causing changing magnetic field? Equation itself doesn't speak for the direction, so I ask."}
{"id":"131284","title":"Breathing Air Underwater From A Car Tyre","text":"I've seen in many movies and programs the event where a car plummets off the road and in to a body of water, sinking rapidly. The car slowly fills with water as it sinks and the occupant(s) don't break free until well under the water (50m say). At this point, when outside of the sinking vehicle, they then breathe air from the tyres and swim to the surface. Obviously it's not that easy to breath air from a tyre due to the valve set up (having to push the pin in to draw air under pressure out). But what I really want to know is, if a tyre is at 30psi and there is no pressure regulator used, and your under 50m of water pressure... Could you actually do this? Surely there'd be issues with the pressure at that depth, or even at atmospheric!"}
{"id":"61449","title":"Free spin (Curie) Paramagnetism","text":"I'm working through a derivation for Curie paramagnetism and hope someone could help clarify a couple of steps. The way that makes sense to me (although now I have seen the wikipedia derivation below I realise this way is pretty long) is to not take any high temperature approximations until near the end of the derivation where I have: $M=g_J\\mu_B[(J+1\/2)coth[g_J\\mu_B\\beta(J+1\/2)]-\\frac{1}{2}coth(g_J\\mu_B\\beta\/2)]$ now to get to the curie susceptibility it seems that when taking the high T limit of the above expression the leading $\\frac{1}{x}$ term of the coth expansion is ignored and the second $\\frac{1}{3}x$ term is considered (this pops out the correct answer $\\chi_{curie}=\\frac{n(g_J\\mu_B)^2}{3}\\frac{J(J+1)}{k_BT}$). I can't find or think or any sensible reason for this apart from when we take the limit of zero B the infinity is just a constant which has no temperature dependence (which is what we're interested in) so we happily ignore it. The wikipedia method below almost makes sense aside from the last equality where I can't follow how they've simplified the sums (I can see that you could drop the first parts of the sums as they have no H dependence so won't matter when it comes to finding $\\chi_{curie}$ but this still doesn't seem to work) $\\bar{m}=\\frac{\\sum\\limits_{M_{J}=-J}^{J}{M_{J}g_{J}\\mu _{B}e^{{M_{J}g_{J}\\mu _{B}H}\/{k_{B}T}\\;}}}{\\sum\\limits_{M_{J}=-J}^{J}{e^{{M_{J}g_{J}\\mu _{B}H}\/{k_{B}T}\\;}}}\\simeq g_{J}\\mu _{B}\\frac{\\sum\\limits_{M_{J}=-J}^{J}{M_{J}\\left( 1+{M_{J}g_{J}\\mu _{B}H}\/{k_{B}T}\\; \\right)}}{\\sum\\limits_{M_{J}=-J}^{J}{\\left( 1+{M_{J}g_{J}\\mu _{B}H}\/{k_{B}T}\\; \\right)}}=\\frac{g_{J}^{2}\\mu _{B}^{2}H}{k_{B}T}\\frac{\\sum\\limits_{-J}^{J}{M_{J}^{2}}}{\\sum\\limits_{M_{J}=-J}^{J}{\\left( 1 \\right)}}$ (Copied from Wikipedia paramagnetism article)"}
{"id":"79039","title":"How does an atomizer work?","text":"Will someone please explain how a basic pump atomizer works? I have a feeling it has something to do with pressure and the Venturi effect, but I'd really like to know precisely how they work as completely as possible. If you know, please feel free to use this image as a reference in explaining. EDIT: Please note that I'd specifically like to know how atomizers of the type pictured in the link I provided, the kind with a spring-based finger pump, work. I'm not talking about e-cigarette atomizers or atomizers with the bulb design."}
{"id":"61443","title":"How is the index of refraction dependence in Planck's law compatible with thermodynamics?","text":"In various formulae for black-body radiation where $c$ appears, there is an implicit index of refraction dependence, since $c=c_0\/n$, where $c$ is the speed of light, $c_0$ is the speed of light in vacuum, and $n$ is the index of refraction into which the blac kbody is radiating. This comes up, for example, in Planck's law, the Stefan-Boltzmann law, and Wien's displacement law.$^1$ I find this observation surprising, because it implies that black bodies radiating into $n\\neq 1$ materials emit a factor of $n^2$ more power than otherwise identical bodies emitting into vacuum. To me, it seems that this would violate the laws of thermodynamics. Consider the situation where two black bodies of equal area face one another. Suppose that the gap between the two black bodies is partly filled an index-$n$ material, with one black body in contact with the index-$n$ material, and the other surrounded by vacuum. Then, according to the argument of the preceding paragraph, one of the black bodies will radiate $n^2$ more power than the other. If the two black bodies are thermally isolated from the rest of the universe, then they will equilibrate to different temperatures. As I understand it, this violates the zeroth law of thermodynamics. There must be something wrong with the above argument. None of my books mention an $n$-dependence, so at first I thought that perhaps the $c$ in Planck's law actually indicated what I call $c_0$, the speed on light in vacuum. However, this does not seem to be the case; I've found at least one reference that explicitly mentions this $n$-dependence.: \"For radiation into a _medium_ within which the speed of light is not close to $c_0$, [Planck's law] must be modified by including an index-of-refraction multiplying factor.\" How is the index of refraction dependence in Planck's law compatible with thermodynamics? * * * $^1$ The $n$ dependencies in the Stefan-Boltzmann law and Wien's displacement law follow from the one in Planck's law, whereas the two former can be derived from the latter."}
{"id":"48596","title":"Calculating vertical velocity component of a particle with mass, given the hit point of parabolic motion","text":"Consider the following situation: ![enter image description here](http:\/\/i.stack.imgur.com\/ZhUhk.png) I have a particle with a given mass that at a given instant of time (let's say $t_{0}$) is placed at the system origin. The particle has a constant velocity component $V_{0x}$. I have to calculate the value of the vertical component ($V_{0y}$) to apply to that particle at time $t_{0}$, in order to provoke a parabolic motion that makes the particle hit the point $P$. The data I know are the following:   * $P$ coordinates   * $dx$   * $V_{0x}$   * $t$ (that can be computed from $dx$ and $V_{0x}$)   * $m$ (the mass of the particle) From the data above, is possible to calculate the value of $V_{0y}$? If yes then give some hints to solve it?"}
{"id":"24662","title":"What is displacement? Position relative to a reference point or change of position","text":"What is the \"official\" or most useful definition of displacement in the context of kinematics? There are two common ones:   1. Displacement is the length and direction of a line from a fixed reference point. (Basically position).   2. Displacement is the change in position. Textbooks using the first definition frequently define velocity as $v=\\Delta s\/\\Delta t$ (where $s$ is displacement) but then for acceleration give equations such as $s=ut+1\/2at^2$ (if they were consistent, they would have to use $\\Delta s$). What throws me off is that are the better textbooks. A similar confusion comes through in Wikipedia: A displacement vector is the straight path between the initial and the final position. But velocity is defined as $v=\\Delta d\/\\Delta t$. Background: I am writing on a learning tool for students but different textbooks require them to learn contradicting definitions and I would appreciate your help on which is the right definition to learn."}
{"id":"3841","title":"Dropping condition","text":"Imagine opening a water tap in order to have a smooth and cylindrical outflow and then slowly decrease the flow by adjusting the knob. At a certain moment, the side profile of the flow will become uneven, and even if you decrease the scale of the faucet, the water shall start to drip. It is possible to determine, based on the thermodynamic conditions of the system (pressure, temperature, and surface tension of water) the dropping condition?"}
{"id":"28137","title":"Behavior of shock waves at relativistic speeds","text":"Suppose I am in a spaceship traveling inertially at a velocity $v$ that is of the same order as $c$. As I pass by a metal bar that is oriented parallel to $v$, someone hits it with another metal bar, causing it to vibrate. The vibration waves travel down the bar with a velocity of $u$. Observing the bar from my spaceship, I see the waves travelling with velocity $u+v\\over 1+uv$.   1. Suppose that the waves are longitudinal, so that $v$ and $u$ are parallel. The atoms in the bar are individually moving back and forth, with a maximum velocity $w$ which is quite different from $u$. $w$ and $u$ are related in a way that depends on the material properties of the bar; I think on its elasticity its density. By comparing $u$ and $w$ I can calculate these properties of the bar. If I observe the motion of the individual atoms, I will see them moving back and forth with a maximum velocity of $v+w\\over 1+uv$ and by comparing this with the observed velocity of the travelling wave I will obtain a measurement of the stiffness and density of the bar. But this value will be different from the value that an observer would calculate at rest. What explains this discrepancy? Is it due to relativistic dilation of the bar's mass and volume, and hence its density? Is there some other effect?   2. Now consider transverse waves. Again $v$ and $u$ are parallel. But this time the individual atoms are moving back and forth _perpendicular_ to $u$ so the observed dilation of their velocity $w$ is different than it was for longitudinal motion. If I do the same calculations as in (1) I should get different results for the physical properties of the bar: I see the transverse and longitudinal waves moving at the same speeds in the bar, but when I look closely at the motion of atoms, and try to calculate material properties of the bar I get two different sets of results, because the atomic motion causing the longitudinal wave is relativistically dilated to a different degree than is the atomic motion causing the transverse wave. What is going on here? What does it really look like? Presumably there is no discrepancy and I see a consistent picture regardless of $v$. How do all these apparently conflicting measurements iron out in special relativity?"}
{"id":"28135","title":"Angular momentum operator and expectation values","text":"I was reading some notes and it says that $\\langle L_z^2\\rangle=\\langle L^2\\rangle$ IFF the system is radially symmetric. I can see that in order that the LHS of the statement implies that $\\langle L_x^2\\rangle=0=\\langle L_y^2\\rangle$. But I am not sure how to rigorously prove to myself that in order to make those 2 vanish we must have an isotropic system. Could someone please kindly explain this? Many thanks."}
{"id":"28134","title":"How to interpret the continuity conditions in the PDEs (for example, Maxwell equations) originated in physics?","text":"I am currently working on _PDEs_ in physics, mostly Maxwell equations. I am a mathematics graduate student, and this question has been haunting me for years. In PDE theory, or more specifically the _Sobolev space_ theory, the solution to the _Galerkin-type_ weak formulation of a PDE normally satisfies some continuity conditions across any interface in the domain of interest. From a mathematical point of view, the continuity conditions guarantees that the weak derivative of the function is well-defined. That is, whenever we do integration by parts in each subdomain and then sum them up, the results should be the same with doing the integration by parts on the whole domain. * * * For example, the Maxwell equations like the following I was working on: $\\newcommand{\\vect}[1]{\\boldsymbol{#1}}$ $\\newcommand{\\vH}{\\vect{H}}$ $\\newcommand{\\vE}{\\vect{E}}$ $\\newcommand{\\vB}{\\vect{B}}$ $\\newcommand{\\vD}{\\vect{D}}$ $\\newcommand{\\vJ}{\\vect{J}}$ $\\newcommand{\\vn}{\\vect{n}}$ $$ \\left\\\\{ \\begin{aligned} \\nabla \\times \\vE &=- \\frac{\\partial \\vB}{\\partial t} \\quad &\\text{ in } \\Omega\\times (0,T) \\\\\\ \\nabla\\times \\vH &= \\frac{\\partial \\vD}{\\partial t} + \\vJ \\quad &\\text{ in } \\Omega\\times (0,T) \\\\\\ \\nabla \\cdot \\vD &= q \\quad &\\text{ in } \\Omega\\times (0,T) \\\\\\ \\nabla \\cdot \\vB &= 0 \\quad &\\text{ in } \\Omega\\times (0,T) \\end{aligned} \\right. $$ If we have two different mediums, being labeled as $a$ and $b$, as in the following picture (shamelessly ripped off from the book Electromagnetic Theory and Computation: A Topological Approach): ![Continuity condition](http:\/\/i.stack.imgur.com\/NGDAV.png) the continuity conditions of the electric and magnetic fields\/fluxes are given as: $$ \\begin{aligned} \\vn \\times (\\vE^a - \\vE^b) &= 0 \\\\\\ \\vn \\cdot (\\vB^a - \\vB^b) &= 0 \\\\\\ \\vn \\times(\\vH^a - \\vH^b) &= \\vJ_S \\\\\\ \\vn \\cdot (\\vD^a - \\vD^b) &= q_S \\end{aligned} $$ where the superscripts denote the limiting value of the vector fields on the interface being approached from each side, and $\\vJ_S$ and $q_S$ are the surface current and surface charge respectively. Here $\\vE$ and $\\vB$ are continuous in the tangential and normal direction on the surface respectively. And these conditions play a pretty important role in the numerical simulation for Maxwell equations, for example, the use of Nédélec elements in computing $\\mathbf{curl}$-$\\mathbf{curl}$ type equation derived from time-harmonic Maxwell equations. (The introduction ends here.) * * *   * How do we interpret the physical meaning of these tangential\/normal continuity conditions in real life (analogous to the interpretation of Stokes theorem as the conservation of a certain quantity)? Are the real electric fields continuous in the tangential direction of a surface? Or are these continuity conditions governed by certain physical laws?   * From my understanding, most PDEs from physics are derived from conservation like Ampère's law and many others, and in the integral form of Maxwell equations no differentiabilities are assumed, hence no continuity conditions are enforced. Then are these continuity conditions artificial just because we would like to get some PDEs?"}
{"id":"79032","title":"Why the electric force does no work on the charge when perpendicular to the field?","text":"I was reading my physics textbook and came across this sentence: > _When a charge moves in an electric field, unless its displacement is always > perpendicular to the field, the electric force does work on the charge._ So it is saying that if the displacement is perpendicular to the field, the electric force does no work on the charge.   1. Why is it so?    2. By perpendicular to the field does it mean perpendicular to certain field line?    3. Is it implying that if the displacement is perpendicular to certain field line, some other force does work on it (but not the electric force)?    4. But how to be \"perpendicular to certain field line\" when field lines are normally curved?    5. Let's say it means literally the field but not field lines, then how to be \"perpendicular to the field\" when the field is not even a straight line but a 3 dimensional region in which electric force is effective regardless of the presence or absence of a material medium?"}
{"id":"38923","title":"Is \"Egyptian Year\" the same as a modern sidereal year?","text":"Copernicus uses the term \"Egyptian Year\" throughout his discussions of the movements of the Earth, and of his and other models of the movements of the planets; but is unclear from his text, or from the general definitions I've found, what this corresponds to in modern astronomical terms. What, precisely, is an \"Egyptian Year\"? Is it identical with a modern sidereal year; if not, what are the correct conversions between the two?"}
{"id":"22796","title":"Does the second law of thermodynamics tell me how the entropy changes?","text":"In thermodynamics I can e.g. compute the properties of ideal gases with certain energies $U_1,U_2$ in boxes with certain volumes $V_1$ and $V_2$. Say I have two such boxes and they have some specific parameter values, like say I know their temperatures $T_1,T_2$. Now I _put them together_ , the gases can interact and I thereby construct a new box with volume $V_1+V_2$ and the energy is $U_1+U_2$. Using the laws of thermodynamics, I can compute everything else again now. Say I find a new temperature $T_{\\text{new}}$. One says that the temperatures of the gases in the systems _changed_ by putting the boxes together. However, on the computational side what I did was just considering a system of new specifications. To get the new result, I didn't have to enter the real world staring conditions except for the variables which also were necessary to compute their respecive properties - effectively, the theory didn't have to tell me how the system changed, just what the restrictions are - and I basically entered into a new system with these values. For example, it's not relevant which gas had with specifications before. I just did a little trivial algebra and computed what has to be. When then talk about a _change_ of entropy and how the gas behaves, but that seems to be only decorative. If I say that the second law of thermodynamics tells me that heat flows from a system with high temperatures to a system of low temperature I'll be able to derive rules for the entropy say, how the combined system has to look like if I state their variables and insist on extensivity of certain variables and so on - but it's not about change in the sense that I compute how the system developes from one point to the other in details. Rather I just compute how the end configuration has to be. My question is > Is the starting configuration of a thermodynamics system ever relavant? And secondly, > In thermodynamics, if I compute \"the change a system\" in the sense of the > above example, do I always induce the necessarity for doing such a thing by > stating \"now we bring system one and system two in contact\"?"}
{"id":"75922","title":"When one can make telescope that can distuingish 1 mas?","text":"When one can make telescope that can distuingish(angular resolution) 1 mas object by angular diameter? If already possible, tell the name of telescope\/observatory. How to calculate angular resolution $\\theta$ of Telescope from the diameter $D$ of its mirror? Just in case I'm understood wrong; is it through Rayleigh equation; $\\theta \\approx 1,22 \\lambda\/D$?"}
{"id":"35524","title":"Interference, photon's phase, and the Hilbert space","text":"Dirac said that a photon can only interfere with itself. This is consistent with the tensor product of two photon spaces representation. On the other hand, it is known that there is interference between distinct sources, between two photons, between two photons again, and between two electrons. This is related to this SE question, and this one. Let $\\frac 1 {\\sqrt 2}\\left(|1\\rangle|2\\rangle + |2\\rangle|1\\rangle\\right)$ be a two-photon state. Let's consider interference between the two photons. If one \"rotates\" one of the wavefunctions with a phase factor, one expects that this will affect the interference, and the result will depend on the phase factor. So, the interference between $|1\\rangle$ and $|2\\rangle$ should be different than the interference between $|1\\rangle$ and $e^{i\\varphi}|2\\rangle$, and that between $e^{i\\varphi}|1\\rangle$ and $|2\\rangle$ (where $\\varphi$ is an arbitrary overall phase). This can be checked experimentally by using a phase shifter. Suppose that there is an algorithm to calculate the interference between the two photons, and that this algorithm receives as input the two-photon state. The state $\\frac 1 {\\sqrt 2}\\left(|1\\rangle|2\\rangle + |2\\rangle|1\\rangle\\right)$ is equivalent to $\\frac 1 {\\sqrt 2}\\left(|1\\rangle|2'\\rangle + |2'\\rangle|1\\rangle\\right)$, where $|2'\\rangle=e^{i\\varphi}|2\\rangle$. If the algorithm which gives us the interference depends only on the two-photon state, then it apparently gives the same result, even when we phase-shift the second photon. Apparently, the interference between photons depends on the way we decompose the two-photon state in two photons: in terms of $|1\\rangle,|2\\rangle$ we obtain different interference than in terms of $|1\\rangle,e^{i\\varphi}|2\\rangle$, respectively $e^{i\\varphi}|1\\rangle,|2\\rangle$. So, either the Hilbert space does not contain all the needed information, or I am missing something (which is by far more likely). Could you please explain me where I am wrong, and what is the correct description of interference using the two-photon state? _P.S. Let me emphasize that I am fully aware that QM has been tested successfully by so many experiments, and that most likely I am missing something. I just don't know what._"}
{"id":"35522","title":"Einsteins equation : Black hole solution","text":"Let Einstein's equations satisfy $ R_{\\mu \\nu } = 0 $. Suppose we solve it numerically with the aid of a computer. Can we know from the numerical solution if there is a black HOle in the solutions? For example, how can you know when you solve Einstein's equation if your solution will be a black hole or other particular non-smooth solution?"}
{"id":"35529","title":"Would Titan moon blow up if you light a match?","text":"Given that the atmosphere of Titan is mostly methane, and from a comment on this TED talk... Would it blow up from a lighted match-stick? Seems like it's at _least_ plausible. But, wouldn't the lack of oxygen hinder this event? Or perhaps I'm confusing the _blow up_ with **combustion**?"}
{"id":"101253","title":"Is strictly harmonic 2D lattice made of Hooke springs possible?","text":"If we connect a set of point masses in a 1D lattice with Hooke springs and consider longitudinal oscillations, we'll have a strictly harmonic system, for which there exist eigenmodes and the frequency of oscillations doesn't depend on amplitude. If we choose transverse oscillations, they cease to be harmonic: the potential for small oscillations appears to be $\\sim \\Delta x^4$. Now I'd like to generalize the first case, where the oscillations are linear, to 2 dimensions. However, it appears that if we make, e.g. a rectangular lattice with nearest neighbours connected by Hooke springs, then any displacement of a lattice site leads to longitudinal deformation of two springs _and_ transverse deformation of two others, thus making the returning force nonlinear function of displacement. Arranging the masses into a triangular lattice doesn't help to compensate the nonlinearity. So, the question is: is it possible to make a _strictly harmonic_ 2D lattice using just point masses connected by Hooke springs? What would be an example of such an arrangement?"}
{"id":"51753","title":"What does this quote about the four dimensional divergence of an antisymmetric tensor mean?","text":"> _In the beginning, God said that the four dimensional divergence of an > antisymmetric second rank tensor equals zero and there was light._ Can someone explain what is the meaning of this quote by Michio Kaku?"}
{"id":"51750","title":"Can the zeroth-component of a 4-velocity be negative?","text":"Is it allowed to have the zeroth-component of a four-velocity be negative? I presume the answer is yes, but I just want to make sure. Many thanks. * * * This is referring to $V^0$ for a curved space metric with signature $-+++$."}
{"id":"56766","title":"Other application of Liouville's theorem besides thermodynamics","text":"Are there any other important practical and theoretical consequences of Liouville's theorem on the conservation of phase space volume besides the calculation of the microcanonical potential in Thermodynamics?"}
{"id":"3014","title":"What Is Energy? Where did it come from?","text":"The simplistic undergrad explanation aside. I've never really understood what energy really is. I've been told that it's _something_ when converted from one kind of something to another kind does some \"work\", as defined by us, but what is that something? Moreover, if the total amount of energy in the universe is finite and we cannot create energy. Then, where did it come from? I've learnt from thermodynamics where it _goes_ , but where does it come from? I know this sounds as something trivially simple, but there is so much going on over here and I just can't grasp what it is. Of course, I lack the mathematical understanding to grasp the subtle things the universe is doing, but what is it doing and how do I get to point of understanding what it's doing? ( **Note:** What prompted me to ask this was this answer. I'm afraid that it just puzzled me further and I sat there staring at the screen for a good 10 minutes.)"}
{"id":"112439","title":"What is energy?","text":"What exactly is energy (in terms a non-physicist\/non-mathematician can understand)? I've been researching this topic, and it seems that energy is not a real physical state, i.e. it is only a scalar that is assigned to the state of physical systems for mathematical abstraction. In other words, energy appears to be a trait of mathematics used to describe traits of matter and non-matter fields. How accurate is this assumption? If energy is \"nothing,\" how can the breaking of a chemical bond catalyze another reaction (e.g. formation of a bond)? That energy that is transferred, what exactly is it?"}
{"id":"98514","title":"Why is electric charge conserved?","text":"We have long been taught that electric charges are neither created nor destroyed. But somehow it is okay to destroy two oppositely charged particles at once ! Why is that so? Let's just take a look at electron-positron annihilation; Two equally oppositely charged particle are bombarded into each other, both are instantly destroyed (lose their existence) and gamma ray photons emerge. It is certainly convenient to say that at $t$ second the net charge in system was 0 and at $t+1$ second it is also zero. But certainly it also raises the question what if you observe only a positron or an electron, in observed system certainly charge will not be conserved however overall it maybe. It jumps out and say that you cannot destroy individual charges, but if you get 2 equally and oppositely charged particles we can destroy both ! (kind of Romeo-Juliet) Then why is the fundamental postulate of conservation of charge that \/charge can neither be created nor destroyed\"? Since, clearly it can be destroyed, just in pairs (also created in pairs)"}
{"id":"128507","title":"calculate force between two magnets","text":"I have used a couple of online calculators to estimate the attractive force between two cylindrical bar magnets. Using d=1\"; t=1\"; x=0.1\"; Neo35 (Br=12,300 Gauss): source 1 gives 63.5 lbf   source 2 gives 29.6 lbf source 2 explains data is based on measured data, so this accounts for the difference from source 1. But in general, I think this is the right ballpark number. I then tried to reproduce these numbers myself, using a formula I found on Wikipedia. I could not get close to the right answer - my answer is about 4000 lbf. **Could someone please explain what I did wrong in my calculation?** ![formula](http:\/\/i.stack.imgur.com\/boNho.png) ![enter image description here](http:\/\/i.stack.imgur.com\/WUd0K.png) **EDIT** Tough crowd. Let me try and be more specific. The problem itself is pretty straightforward. I did not really mean to imply I need someone to check the numbers. I mean more on if I am applying the right approach to the problem:   1. Is the formula above valid when trying to replicate results in the calculators I linked? (i.e. \"Calculate the Pull and Repulsion Forces Between Two Disc Magnets\")   2. Perhaps \"real life\" is significantly different than theory for this application? In which case, the engineering calculators would apply a de-rating factor that is not represented in the Wikipedia formula. I did find any references to this, but perhaps someone with more knowledge could advise?   3. Assuming the formula is valid here, I think Magnetic Flux Density (B_0) is the most likely source of my error. It is my understanding that this is a physical property of the material. Is that correct? The value I found for Neo35 is Br = 12,300 Gauss (CGS units). I converted this to SI 1.23 Tesla using a 10^-4 conversion factor. Is this the correct interpretation of B_0?"}
{"id":"126844","title":"Simulation of fluid flow using Euler equation","text":"I have been looking on Euler's equations for a while and can't grasp one thing. Suppose we have initial system state with volumes of fluid \"hanging\" in air (time is frozen and equal to zero), each of them has its initial (x, y) coordinates and velocity vector (Vx, Vy). The space is divided in stationary volumes in which measurement takes place (Euler measurement model). ![2 dimension representation of segmented space and restrictions](http:\/\/i.stack.imgur.com\/AQBYW.jpg) As I understand Euler's equations impose mutual restrictions between functions of velocity ( u(x, y, t) ), mass density ( m(x, y, t) ), and pressure ( p(x, y, t) ) at any instance of time for stationary volume of space with coordinates (x, y) that contains volume of fluid (in that instance). So why each function has time parameter when equations must be solved for any instance of time (iteration process) ? We must have new set of these functions for any moment of time that satisfy equations. And it's unclear whether volume of space contains volume of fluid or not, how do we take this in consideration ? And the second question is how state of each volume of fluid (its coordinates and velocity vector) is related to these functions ? I need a system that has some initial state (array of fluid volumes with their coordinates and velocities) and uses Euler's equations to compute next state, which becomes initial for next iteration and so on. Something like this: ![scheme](http:\/\/i.stack.imgur.com\/DRe1Q.jpg) I don't need math details, I just want to grasp basic idea in context of computer simulation (how method of simulation can be bounded to standard analytic equations). Most papers are very complicated, with additional physical factors and user interaction, and they don't cover this \"link\". I want to consider the easiest case and how to start iteration process (just idea and how it is usually done). Sorry if some things sound ridiculous, please adjust my statements if they are wrong. If my model of understanding is completely wrong just point me in right direction, thanks."}
{"id":"104159","title":"In what ratio does the charge distribute if a charge and uncharged body touch each other?","text":"Lets say a large charged body has $n$ elementary charges and is brought in contact with a small uncharged body. When the charge distributes, does it distribute in the ratio of number of atoms or each body gets half of the charge?"}
{"id":"101922","title":"Significant figures vs. absolute error","text":"On NIST Avogadro's Number, $N_A = 6.022\\;141\\;29 \\times 10^{23} \\text{ mol}^{-1}$ has 9 significant figures and a standard uncertainty of $0.000\\;000\\;27 \\times 10^{23} \\text{ mol}^{-1}$. First, can it be written as the following? $$ N_A = 6.022\\;141\\;29 \\times 10^{23} \\pm 0.000\\;000\\;27 \\times 10^{23} \\text{ mol}^{-1} $$ If yes, consider the following quote from Rice's lab guide on error analysis (emphasis added): > Whenever you make a measurement, the number of meaningful digits that you > write down implies the error in the measurement. For example if you say that > the length of an object is 0.428 m, you imply an uncertainty of about 0.001 > m. To record this measurement as either 0.4 or 0.42819667 would imply that > you only know it to 0.1 m in the first case or to 0.00000001 m in the > second. You should only report as many significant figures as are consistent > with the estimated error. The quantity 0.428 m is said to have three > significant figures, that is, three digits that make sense in terms of the > measurement. Notice that this has nothing to do with the \"number of decimal > places\". The same measurement in centimeters would be 42.8 cm and still be a > three significant figure number. **The accepted convention is that only one > uncertain digit is to be reported for a measurement. In the example if the > estimated error is 0.02 m you would report a result of 0.43 ± 0.02 m, not > 0.428 ± 0.02 m.** and the following quote from A Project of Joliet Junior College (IL), Lee College (TX) and the National Science Foundation (emphasis added): > The absolute error places bounds telling you that the actual value is > expected to fall between x - D x and x + D x. The absolute error is an easy > way to report the error and makes it easy to check that you are not > reporting too many significant figures. **For example, suppose you had a > measurement that was reported to be 10.02 ± 0.3 m. Because the reported > value is uncertain in the tenths place, it is wrong to report it to the > hundredths place. It should be reported to the same decimal place as the > absolute error: 10.0 ± 0.3 m.** Another point is that the absolute error and > the reported value must have the same units. If there is such a convention (cf. Rice's lab guide) and it is wrong to report more significant figures than is warranted by the absolute error (cf. the joint project's guidelines), and NIST should have adhered to such convention and striven to be correct, is it true that NIST adheres and is correct because the standard uncertainty has two least significant digits (i.e., $27$)? Had the standard uncertainty been $0.000\\;000\\;2\\text{ mol}^{-1}$, would NIST have reported Avogadro's Number as $N_A = 6.022\\;141\\;3\\times 10^{23}$ with 8 significant figures?"}
{"id":"71334","title":"What is the smallest increment of space and the smallest increment of time?","text":"I believe that every point in space is a surface point. And if it was possible to break through or tear a point in space, it will create a black hole. I believe that's why a black hole can exist anywhere in space. Since every point in space is a surface edge then punching through a point in space will causes matter to get suck into a very powerful and vast vacuum (outside of our universe). Does any of this hold water? And what about the smallest increment of time and space? Where can I find supporting Theory and equations, if any?"}
{"id":"105205","title":"Mixing generators of different dimensionality","text":"Reading a paper about compactified manifolds used in Kaluza Klein theories the author discusses in which ways you can get $SU(2)\\times{}U(1)\\times{}U(1)$ as a subgroup of $SU(3)\\times{}SU(2)\\times{}U(1)$. At a certain point, it starts considering ways of getting $U(1)$. It says you can do it using the $\\lambda_8$ Gell-Mann's matrix as a generator of $U(1)$. The original $U(1)$ group as well can be, of course, used, whose generator he calls $Y$, and finally any of the $U(1)$ generators of $SU(2)$, which he calls $T_3$. Then he combines the three generators in a linear combination $p\\lambda_8+qT_3+rY$ Which is supposed to be a generator (via complex exponentiation) of a $U(1)$ group. How is this supposed to be understood? Being all three generators matrices of different dimensionality, how do we add them?"}
{"id":"33291","title":"Electrial Conductivity of Thin Metal Films","text":"What is the best way to find specific\/electric conductivity which is dependent of very thin film thickness?"}
{"id":"41450","title":"Is there a simple, accurate formula for calculating transit times from rise and set times?","text":"I have access to \"rise\" and \"set\" times for astronomical objects, and want to determine the corresponding times of culmination. Is there a reliable and accurate way to do this? It isn't clear to me from the sources of the data I have whether the times I have are for the geometric centers of the objects, or take into account refraction, or the angular extents of the objects, so I'm hoping there is an approach that is insensitive to these factors. In particular, I'm interested in transit times for the Sun."}
{"id":"41452","title":"constraint on scaling dimension","text":"How can we show that for any scalar operator $\\Delta\\geq1$ (where $\\Delta$ is the scaling dimension)? Where can I find a reference for reading where it comes from?"}
{"id":"41458","title":"What is a bulk phase transition?","text":"I have been able to google \"bulk phase transition\" and get plenty of results that verify that something called a bulk phase transition exists, however, I cannot seem to find a precise definition of what exactly a bulk phase transition is. Could someone please help me with a definition of _bulk phase transition_? I would very much appreciate it."}
{"id":"770","title":"What is the most efficient way to destroy the universe?","text":"Don't worry... I won't actually do it, just would like to know how."}
{"id":"75","title":"Mnemonics to remember various properties of materials","text":"I'm trying to figure out how to remember that   * hardness: how resistant it is to deformation   * toughness: how resistant it is to brittle failures   * stress: force on a surface area   * strength: ability to withstand stress without failure   * strain: measurement of deformation of a material Does anyone know of a mnemonic or easy way? I only know these from googling them, and I'm finding it tricky to remember them."}
{"id":"79","title":"Is quantum entanglement mediated by an interaction?","text":"You can get two photons entangled, and send them off in different directions; this is what happens in EPR experiments. Is the entanglement then somehow affected if one puts a thick slab of EM shielding material between the entangled photons? Have such experiments been made? According to EPR experiments measurements of the entangled states are at odds with SR, so based on that I'd assume the answer is \"no\"\/\"don't know\", but any citations would be appreciated!"}
{"id":"76649","title":"Time derivative to energy equates $0$","text":"Following up on a related question. This implies that, From the reference we can write \\begin{align*} E(\\lambda) &= I_F [g_\\lambda ] + I_K [f_\\lambda , g_\\lambda ] + I_V [f_\\lambda ] \\\\\\ & = \\lambda^{4-d} I_F [\\bar A] + \\lambda^{2-d} I_K [\\bar \\phi,\\bar A]+ \\lambda^{-d} I_V [\\bar \\phi]. \\end{align*} This must be stationary at $\\lambda = 1$, therefore we must have, \\begin{equation} 0= (4-d) I_F [\\bar A] + (d-2) I_K [\\bar \\phi,\\bar A]+ d I_V [\\bar \\phi].\\tag{1} \\end{equation} I think the equation (1) equates the time derivative to zero. Is it right? If this is true, then why are we taking this derivative and why does it need to be zero? * * * Source: Erick J. Weinberg, _Classical Solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics_ (p. 43)"}
{"id":"76644","title":"Different invariant gauge groups (IGG) on different lattices with the same form mean-filed Hamiltonian?","text":"Suppose that we use the Schwinger-fermion ($\\mathbf{S_i}=\\frac{1}{2}f_i^\\dagger\\mathbf{\\sigma}f_i$) mean-field theory to study the Heisenberg model on 2D lattices, and now we arrive at the mean-field Hamiltonian of the form $H_{MF}=\\sum_{<ij>}(\\psi_i^\\dagger u_{ij}\\psi_j+H.c.)$ with $u_{ij}=t\\sigma_z$($t>0$), where $\\psi_i=(f_{i\\uparrow},f_{i\\downarrow}^\\dagger)^T$, and $\\sigma_z$ is the third Pauli matrix. Now let's find the IGG of $H_{MF}$, by definition, the pure gauge transformations in IGG should satisfy $G_iu_{ij}G_j^\\dagger=u_{ij}\\Rightarrow G_j=\\sigma_zG_i\\sigma_z $ on link $<ij>$—(1) . Specifically, consider the IGGs on the following different 2D lattices: (a)Square and honeycomb lattices(unfrustrated): These two lattices can be both viewed as constituted by 2 sublattices denoted as $A$ and $B$. Due to Eq.(1), it's easy to show that for **both** of these two lattices the gauge transformations $G_i$ in **the same sublattice** are _site-independent_ while those in different sublattices differ by $G_A=\\sigma_zG_B\\sigma_z$ and $IGG=SU(2)$. (b)Triangular and Kagome lattices(frustrated):Due to Eq.(1), it's easy to show that for **both** of these two lattices the gauge transformations $G_i$ are _global_ (site-independent) and $G_i=\\bigl(\\begin{smallmatrix} e^{i\\theta }& 0\\\\\\ 0& e^{-i\\theta } \\end{smallmatrix}\\bigr)$ which means that $IGG=U(1)$. So my question is: The same _form_ mean-field Hamiltonian $H_{MF}$ may has _different_ $IGGs$ on _different_ lattices?"}
{"id":"18017","title":"How do I calculate the time it would take for a space craft launching from earth to reach another planet?","text":"I am making a matlab program the my final project for my programming class in college. I have chosen this problem, but I am bashing my head against the wall trying to figure out the equations and how they interact with each other. I have the user play with the thrust and the payload to carry to the other planet. The user also decides what planet to go to and what type of orbit to use (elliptical or parabolic). I know I have to usde the rocket equation, escape velocities have to be claculated, and time formula, but I still haven't figured out what to do."}
{"id":"108511","title":"Why does Gauss's law work for a charge off center in a spherical surface?","text":"CASE 1: Consider an enclosed spherical surface with a charge $q$ at its centre. From Gauss' law we can say that the flux through this sphere is $q\/\\epsilon_0$. CASE 2: The charge is inside but _off centre_. In this case the electric field will not be normal to the surface. So how can we use the formula $q\/\\epsilon_0$? It turned out that the answer is same for both cases. This is my confusion."}
{"id":"89672","title":"Why do degenerate normal modes appear as complex conjugate pairs?","text":"Can anyone prove this? THE DETAILS Suppose we have a system with n components, (i.e. $|~{\\psi}(t)\\rangle=\\sum \\langle x_i|~\\psi(t)\\rangle ~|~x_i\\rangle$) where our equation of motion is described by $$\\frac{d^2}{dt^2}|~\\psi(t)\\rangle=\\hat{M}^{-1}\\hat{K}~|~\\psi(t)\\rangle$$ And say we are looking for solutions $|~\\omega\\rangle$ such that $\\frac{d}{dt}|~\\omega\\rangle=\\omega|~\\omega\\rangle$, and thus our equation simplifies to $$\\hat{M}^{-1}\\hat{K}~|~\\omega\\rangle=\\omega^2|~\\omega\\rangle\\tag{1}$$ (Note that $\\hat{M}^{-1}\\hat{K}$ is Hermitian). Is there any way to prove that if $|~\\omega_1\\rangle,|~\\omega_2\\rangle$ are linearly independent and both satisfy (1), then $|~\\omega_1\\rangle=|~\\omega_2\\rangle^*$, i.e. the components of one vector are the complex conjugates of the components of the other vector? I ask this because this keeps on popping up in our class study of coupled oscillators and I want to know why this is happening Thanks!"}
{"id":"72811","title":"Is there an idealist rather than realist interpretation of QM?","text":"The many-worlds interpretation of QM is a realist explanation as it makes the wave function of the universe real. That is it makes the probabilities of outcomes real outcomes. One could argue that this view is biased that realist ideas of reality are better by definition. But philosophically speaking idealist ideas of realities have been mooted, for example by Berkely & the Cambridge Platonists, (or Brahman in Indian Philosophy). Is there is an idealist explanation which explains the probabilities inherent in the wave function as epistemological? That is states of knowledge of observers. Except here by observer I do not mean necessarily some one that is conscious, but simply a system that reacts to the information context around it."}
{"id":"527","title":"Why does evenly heating soup with a microwave take so long?","text":"My anecdotal observations (which could be incorrect, they're totally unscientific) indicate that it takes almost as long to evenly heat a big bowl of soup in a microwave as it does to heat it on the stove - 4-5 minutes in my most recent example. Yet a plate of meat or vegetables is reheated in 30 seconds or less."}
{"id":"132143","title":"Presence of planets in Milky Way and other galaxies","text":"Are there only planets in the Milky Way galaxy, or are there other planets in other galaxies? If planets are only in the Milky Way, why aren't there planets in others?"}
{"id":"5758","title":"Has anyone actually produced any braided anyons (quasi-particles) yet? (quantum computer)","text":"Quasiparticles theoretically can form braids on a 2D surface. The braids apparently are quite stable for reasonably long periods, allowing a superposition state more time before it decoheres through environmental interaction. This would be a big step in quantum computing. Has this actually been achieved or is it theory only at this point?"}
{"id":"132149","title":"Thermodynamics for Dummies: Entropy and temperature","text":"I do not study physics and I have never had a course in thermodynamics. I have no idea what it is about, but I am currently taking a course where we had something about entropy. Would be great if you could help me in easy words.. Wikipedia and other Websites did not help. So: We are looking at a box full of gas with N particles in it (see Maxwell's demon). The question is now to look at two different states. The first state ist when the whole box is used for the gas. The second state is when only one half is used. Now we consider the Transition. 1) -> 2): The second state has less states, hence less entropy. So we lose entropy. This Forces us to do work, to invest energy, otherwise this would not happen (or only with small probability), as the 2nd law of thermodynamics states. So what does this mean for the temperature in this box? Are we considering the temperature in this box or around this box, or both? Does it get warmer or colder? Where? 2) -> 1): This is more probable, and energy is set free. We have more entropy then. But also, this has to do something with temperature, so what are the consequences? I know how to calculate the energy that has to be invested to enforce 1)->2), but what I do not know is the consequences for the temperatures. So we talk about $\\Delta S \\geq - N k_B ln 2$ (why $\\geq$ and not $=$), the difference in entropy, so the temperature difference is $\\Delta Q = T \\Delta S$.. Sometimes I have read that more entropy means hotter. As if particles are hotter, they move faster, so there's more uncertainty. But then, I also read that if we go 2) -> 1) that the temperature gets colder. But the entropy grows. So I have no idea. My guess is, that sometimes one talks about the temperature in the gas, and sometimes one talks about the temperature outside this box. Thank you for your help. Hope I am right here."}
{"id":"130678","title":"Electron photon interaction potential in old fashioned perturbation theory (OFPT)","text":"In this PDF on old fashioned perturbation theory (OFPT) we find from equation (14) the potential describing the interaction between the electron and photon: $$ V = \\frac{1}{2}e \\int \\mathrm{d}^3x\\, \\psi_e(x)\\phi(x)\\psi_e(x) \\tag{14}$$ where $\\phi=photon. $ My question is why $V$ looks as it does and where exactly does it come from?"}
{"id":"128804","title":"Macroscopic Forces from QED","text":"In QED the carrier for electromagnetic interaction is a photon, while macroscopic forces are due to electromagnetic interaction (by macroscopic forces I mean: normal force, object collision, friction etc..). But for example, in the collision of two billiard balls, if 1 kgm\/s of momentum is exchanged between the balls, the amount of Energy required in the form of photon virtual particles would be enormous (3E8 J). And while they do not necessarily obey the mass-shell relation, still the farther off shell interactions are, the less they should contribute to the actual phenomena. For least action to be satisfied it seems like some other type of exchange interaction would occur. Is there some kind of larger scale decoherence type interaction going on?"}
{"id":"47148","title":"Can a nearly-extremal black hole be stable against Schwinger vacuum breakdown?","text":"I was doing some basic algebra to estimate the range of possible masses $M$ and electric charge $Q$ for a nearly extremal Reissner-Noström black hole. I want to see if the logic is correct the electric field near the event horizon of such a black hole would be $$ E= \\frac{1}{4 \\pi \\epsilon_0} \\frac{q}{r^2} = \\frac{1}{4 \\pi \\epsilon_0} \\frac{q c^4}{G^2 M^2} = 10^{66} \\frac{\\text{Volt Kg}^2}{\\text{m Coul}} \\frac{q}{M^2}$$ since Schwinger-pair vacuum breakdown occurs at $10^{18} \\frac{\\text{Volt}}{\\text{m}}$, it implies that $$ \\frac{q}{M^2} \\lt 10^{-48} \\frac{\\text{Coul}}{\\text{Kg}^{2}}$$ Would be a bound that guarantees electromagnetic vacuum stability **Question:** can a charged black hole that is stable against this breakdown still be nearly extremal? well, it seems so, because the extremality condition: $$ r_Q = \\frac{r_s}{2}$$ Implies that $$ Q \\sqrt{ \\frac{G}{4 \\pi \\epsilon_0 c^4}} = \\frac{GM}{c^2}$$ $$ \\frac{Q}{M} \\approx 10^{-16} \\frac{Coul}{Kg}$$ So the above two relations would imply that a black hole with a mass $$ M \\gt 10^{32} Kg \\approx 100 \\text{ Solar masses}$$ Can be both stable (against the specific Schwinger-pair decay) and extremal. In particular, if the mass is $10^{32}$ kilograms, a charge around $10^{16}$ Coulomb would make the black hole arbitrarily close to extremal as one might want, and is not at all clear that it would decay from the extremality state But probably i'm overlooking other ways than the extremality of the black hole can decay (in particular, i'm ignorant about how Hawking radiation will contribute in here, specially if the temperature is below 511 KeV). Any insights into this would be greatly appreciated"}
{"id":"132047","title":"Determining the moment of force","text":"A force $F = 3i + 2j$ passes through through a point $P$ with respect to an origin $O$. How do I determine the moment of the force at the origin."}
{"id":"23667","title":"How does Newton's 2-prism experiment help to explain why light does not get dispersed into 7-colors in a parallel glass slab?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/Tze2p.png) In a real parallel glass slide(with two prisms imagined to be touching each other to form a parallel glass slide),   The ray of light should pass through the Z in between without any dispersion or change in direction because the density in both of the prisms is the same(and they are touching each other). So Newton's experiment won't help in explaining this because in it, there is a vaccum(or air) in between and hence the change in direction of light. I hope that you are getting me, if not I'm ready to elaborate more. _[In case anyone is wondering that I did post a Q about dispersion earlier too, then that is true - but I did not get the answer of the question that I really meant to ask and hence this, I hope that you guys will help me better understand this.]_"}
{"id":"82746","title":"Why Don't the Ladder Operators Commute?","text":"I have two problems with ladder operators. The first is that I feel they should somehow result in measurable things. The asymmetry of applying the plus operator versus the minus operator is very strange to me. Second, I don't understand why applying the raising operator and then the lowering operator is different from applying the lowering operator and then the raising operator. The two are explained in terms of moving up and down energy levels, so shouldn't going up then down, or down then up, be the same operation? What is asymmetric about harmonic oscillators that causes this asymmetry?"}
{"id":"16557","title":"Quantum of Time","text":"I am not a String Theorist, but I just want to ask if there is anything which specifies a quantum of time, as per M-theory? Can a quantity such as $$ t_p = \\left(\\frac{\\ell_p}{c}\\right),$$ where $\\ell_p$ is Planck length and $c$ speed of light in vacuum, be called as time quantum?"}
{"id":"16550","title":"Why do mirages only appear on hot days?","text":"A previous question asked why the road sometimes appears wet on hot days. The reason is that when there's a temperature gradient in the air, it causes a gradient in the index of refraction, causing the light to bend. This diagram (from Lagerbaer's answer to the previous question) is ![enter image description here](http:\/\/i.stack.imgur.com\/8eZCm.png) I have two questions about this. First, Wikipedia says that in order for the effect to appear, there must be a temperature gradient on the order of a few degrees per hundred meters above the asphalt. To investigate this claim, I assumed the air near the surface of the asphalt is at constant pressure, so the density is inverse-proportional to the temperature (i.e. I assumed the density variation due to the weight of the air is small compared to the density variation induced by temperature gradient. Otherwise we would see mirages all the time.) Then I assumed the difference of index of refraction of air from one is proportional to the density, i.e. $n = 1 + a\\frac{\\rho}{\\rho_0}$ with $n$ the index of refraction of air, $a$ some dimensionless constant, and $\\rho_0$ some reference density. Looking up the index of refraction of air online, I guessed $a = 0.0003$. Then I assumed the temperature above the Earth is modeled by $T = T_0 + gy$ with $y$ the height and $g$ a temperature gradient in ${}^{\\circ}\\mathrm{C}\/\\mathrm{m}$. This completes the model. Fermat's principle gives a variational problem to solve for the path of the light. However, the resulting differential equation was hard to work with, so to first order I approximated that the light starts at a horizontal distance $x = -L$ at height $y =h$, slopes down towards the ground with some slope $m$ until it gets to $x=0$, then slopes back up at the same slope $m$ until it gets to $x = L$ and $y = h$ again. Then I chose $m$ to minimize the travel time of this path. To first order for small $h$, I found $$m = \\frac{agL}{2T_0}$$ The problem is that when I plug in $g = 5^{\\circ}\\mathrm{C}\/100\\mathrm{m}$, $a = 0.0003$, $T_0 = 300 \\mathrm{K}$, and $L = 100 \\mathrm{m}$ I get $m = 2.5*10^{-6}$, which is too small to explain the mirage. It would only allow light to dip a quarter millimeter in the path from a car $200 \\mathrm{m}$ away to me. In the Wikipedia image of a mirage, the light clearly dips down at least 1000 times that much (check out the blue car). ![enter image description here](http:\/\/i.stack.imgur.com\/eb2iN.jpg) So my first question is: what's wrong? Why are there mirages when this model predicts that there are not? My second is: why do we see them only when it's hot? This model depends only weakly on absolute temperature, and hot absolute temperatures actually decrease the effect. Evidently, there are only high temperature gradients on hot days, by why should that be? A temperature gradient over the asphalt comes from sun heating the asphalt directly more than it heats the air. Shouldn't that happen on cold days as well as on hot days?"}
{"id":"125953","title":"st. venant shallow water equations for pipe flow","text":"As far as I know from the fluid dynamics class, St. Venant equations (the shallow water equations) are derived by depth-integrating the Navier-Stokes equations. This depth-integrating is done with the assumption that the horizontal scale is much larger than the vertical scale, hence it would be used for overland flow or surface runoff modeling. But I was wondering if the St. Venant equations can also be used in open channel pipe flow, say to model the flow inside the drainage pipe. I imagine the flow would more or less be similar as that in a river channel, but in this case the horizontal dan vertical scale would be the same at some point of water depth, given the geometry of the pipe. So my question is, can St. Venant equations be applied correctly for modeling open channel pipe flow? If they can, will there be any constraint from the water depth condition? For example, if the water depth is 5cm in a 1m diameter pipe then perhaps SWE can be applied, but when the water depth is more than half of the pipe then wouldn't the horizontal to vertical scale ratio be the same?"}
{"id":"94095","title":"James Joule's Conservation of Energy","text":"Is there a manifest or documentation of the original Conservation of Energy as stated by James Joule word for word? **Edit 1.1:** I found Joule's Memoir on google, skimming through to find any conclusive evidence."}
{"id":"109897","title":"Does electricity flow on the surface of a wire or in the interior?","text":"I was having a conversation with my father and father-in-law, both of whom are in electric related work, and we came to a point where none of us knew how to proceed. I was under the impression that electricity travels on the surface while they thought it traveled through the interior. I said that traveling over the surface would make the fact that they regularly use stranded wire instead of a single large wire to transport electricity make sense. If anyone could please explain this for some non-physics but electricly incline people, I would be very appreciated."}
{"id":"100642","title":"Physical position eigenfunction normalisation","text":"We know that the Dirac function $$\\delta(a)=\\lim_{a \\rightarrow 0} \\delta_{a}(x)$$ can be written as an infinitesimally narrow Gaussian: $$ \\delta_{a}(x) := \\frac{1}{\\sqrt{2\\pi a^2}}e^{-x^2\/2a^2}$$ Our professor told us that for any value $a>0$, the physical position eigenfunction is $$\\psi_{x_0}(x)\\cong N_1\\delta_a(x-x_0).$$ How can I show that $\\psi_{x_0}$ is a physical position eigenfunction?"}
{"id":"21043","title":"M1 vs M2 on a frictionless surface","text":"I was wondering, if I had a frictionless surface and I had an M1 that was 100kg and an M2 that was 1kg and I pushed each separately with an equal force F would they move at the same speed or am I assuming incorrectly?"}
{"id":"21048","title":"Why does string theory (in case it is true ) have NO divergencies?","text":"Why is string theory in 10 or 26 dimension not divergent? Due to the high number of spacetime dimension (10 or 26) it should have a lot of UV divergencies of the form $ \\int k^{n}dk $ and gravity within the approach of the string theory should be non-renormalizable too, or shouldn't it?"}
{"id":"119925","title":"What is the theoretical\/intutive meaning of $x(t) = x_0 \\cos ( \\omega t + f )$?","text":"What is the meaning of $x(t)=x_0 \\cos(\\omega t+f)$, where $x_0$ is the amplitude, $\\omega$ the angular frequency, $t$ time and $f$ the phase constant? I know how to solve the mathematical problems which use this expression, but what I don't understand is how sine and cosine functions are used to express AC waves. I know the math but I do not understand the theory behind it. I do not have an intuitive understanding of these expressions. If you cannot explain at least guide me to a resource that can. i want to understand the math in a theoretical manner like are we going to take the sine\/cosine of the wt or the solved value of it, since sine\/cos is a ratio so basically are we multiplying the since\/cos fraction value to the variables, oh i am so confused please explain ."}
{"id":"81218","title":"Equation of state of a rubber band","text":"I have the following question that I attached in png format. ![enter image description here](http:\/\/i.stack.imgur.com\/v5MpZ.png) I have done part (a), but I am having difficulties in part (b) when I proceed according to the book. I have non zero tension at equilibrium length. I reached that conclusion by taking the derivate of entropy with respect to L and n at the same time. What is the physical principle underlying this?"}
{"id":"17689","title":"Difference between 1PI effective action and Wilsonian effective action","text":"What is the simplest ay to describe the difference between these two concepts, that often go by the same name?"}
{"id":"13595","title":"Where does the excess energy emitted by a microwave go?","text":"If there is nothing in the microwave, where does the excess radiation go? Why doesn't the radiation accumulate and blow it up? Should I cook two pieces of Canadian Bacon twice as long as I cook one?"}
{"id":"41844","title":"Where do the high order rainbows appear?","text":"This site shows locations of the high order rainbows up to sixth order. My thought is that, There should be even higher order bows like 7th, 8th, etc. (Though they would be practically impossible to observe.) So, Where would they appear on sky?"}
{"id":"53667","title":"Shoot object into the Sun using minimal energy","text":"Say I want to shoot a cannonball into the Sun with minimal energy (minimal initial velocity relative to Earth).   In which direction do I shoot it? Let's neglect Earth's gravity, if that would make things very complicated."}
{"id":"45541","title":"Help identifying an expression for the action","text":"I found the following expression for the action of a (free, I think) relativistic particle in my notes but I can't remember from what it came from: $$ S = \\int_{0}^{N} \\left [ \\frac{1}{4}\\eta_{\\mu\\nu}{\\mathrm{d} X^\\mu \\over \\mathrm{d} s}{\\mathrm{d} X^\\nu \\over \\mathrm{d} s} -m^2c^2 \\right]\\mathrm{d}s$$ I'm not even sure what some of the symbols mean; no idea what $N$ is for example. Big note-taking fail of my part! Any help is appreciated."}
{"id":"45540","title":"Solar System and the Plane of our Galaxy","text":"Why do all of the space probes launched travel within the plane of the solar system? Can we direct a probe perpendicular to the plane of the solar system and galaxy."}
{"id":"53122","title":"Filter out polarized light","text":"Unpolarized light enters the polarizer and gets polarized at one certain angle. If we place an analyzer behind the polarizer and align them, we observe that all light is transmitted. If we rotate the analyzer by an angle $\\theta$, then the intensity of light leaving the analyzer would fall as $\\cos^2(\\theta)$. What I am asking is: is there some device that when placed instead of the analyzer will transmit light as a delta function: max intensity for $\\theta = 0$ (perfect alignment) and zero intensity for all other angles ($\\theta$ not equal to zero)?"}
{"id":"53123","title":"Rolling a deformed ring","text":"Consider a ring rolling without slipping along a horizontal surface. Regardless of the speed of the ring, it is continuously in contact with the surface.   Let's deform the ring slightly so that it becomes an ellipse with small eccentricity of $e\\rightarrow 0$. Now, consider the deformed ring rolling without slipping along the surface. At low speed, the ring still keeps contact with the surface continuously. However, at some critical speed the ring will jump. I am interested in how to find this speed?"}
{"id":"58110","title":"Pendulum axes confusion","text":"Suppose I have a simple pendulum and I want to calculate its acceleration when the bob reaches the maximal angle. I usually choose my axes such that the y-axis will be parallel to the rope. Then the acceleration would be $g \\sin\\alpha$ (Because: $mg\\sin\\alpha=ma$). However, if I choose the y-axis to be parallel to the gravitational force ($mg$), the acceleration equals to $g\\tan\\alpha$ (Because: (1): $T\\cos\\alpha=mg$ and (2): $T\\sin\\alpha=ma$ where $T$ is the tension force (centripetal force) and $\\alpha$ is the given angle). Obviously, these two accelerations aren't the same. So my question - which one is right? And why the other one is wrong?"}
{"id":"15034","title":"How to calculate the resulting velocitys and rotation speed after two concave polygons collide in 2d","text":"so I've been searching google for how to do this, but all anyone seems to care about is collision detection =p, and no results come close to teaching me how to calculate **2d elastic collision between two concave polygons**. I know how to solve elastic collisions between two 2d circles, but I can't find anything more complicated than that. I'm also a very visual person, so it would be great if someone could show me how to do this or point me to a website! Thanks =)"}
{"id":"62568","title":"Alpha particle and helium nucleus","text":"The symbol for the alpha particle is α or $α^{2+}$, it can be written as $He^{2+}$. What I want to know is that, are they same? I mean alpha particle and helium nucleus are same or any subtle difference exists?"}
{"id":"53697","title":"Where does the energy produced by fusion come from?","text":"Fission, in layman's (or \"initiate's\") terms, is easy enough to understand; a large atom with a lot of protons and neutrons requires a large amount of force provided by the strong interaction to overcome electromagnetism and keep the nucleus together. The necessary energies were imparted to the atom during its formation, which we can replicate to some degree within a nuclear reactor; a combination of heat and fast-moving free protons\/neutrons recaptured by the atoms of the fuel turns uranium into plutonium, even as the actual atom- splitting produces much lighter krypton and barium isotopes (or a host of other possibilities, some more likely than others, as with many other types of reactions). That atom-splitting, by the way, reduces the size of the resulting nuclei, and therefore reduces the amount of strong interaction force required to contain them; the leftover energy is released as a high-frequency gamma photon. Fusion, however, is an odder beast. A fusion reaction requires a large amount of starting energy; enough to strip the electrons off of the deuterium\/tritium nuclei and form plasma. That heat is also enough to accelerate the particles fast enough that when they collide, the initial electromagnetic repulsion is overcome and the strong interaction binds them together. Now, that's a lot of energy required; the Sun's inner core where most of the fusion occurs is estimated to be about 15 million Kelvin. The reaction, however, doesn't seem to release any energy, based at least on this simplistic explanation. Now, obviously that's wrong; pretty much all the energy we have available to us right now is, however indirectly, a result of the Sun bathing us in the energy from nuclear fusion. The rest of it, such as from nuclear fission, is also star-based, via creation of superheavy elements in stellar nucleogenesis. But where is the source of this energy on the subatomic level? Surely more strong force is required to contain a helium atom than a hydrogen one, because the helium now actually has two protons and must actually overcome the Coulomb force (as opposed to hydrogen, which only has one proton and so the Coulomb force between the proton and its zero-charge neutrons is lesser). So, what imbalance does fusion induce within the nucleus that is then resolved with the release of a photon?"}
{"id":"2408","title":"Does the force of kinetic friction increase with the relative speed of the objects involved? If not, why not?","text":"Does the force of kinetic friction increase with the relative speed of the objects involved? I have heard and read that the answer is no. This is counter intuitive, and is a big part of why the \"Airplane on a Treadmill\" question is so interesting. What phenomena are at work with kinetic friction, and why does it not increase with relative speed?"}
{"id":"94714","title":"Question on measuring expectation value of spin with time variation","text":"I have a particle with the following wave function: $$\\psi(t) = \\frac12 |\\uparrow \\rangle e^{-i(\\omega_1+\\omega_2)t\/\\hbar} +\\frac12 |\\uparrow \\rangle e^{-i(\\omega_1-\\omega_2)t\/\\hbar} +\\frac{1}{\\sqrt{2}} |\\downarrow \\rangle e^{-i(-\\omega_1-\\omega_2)t\/\\hbar}$$ I am trying to calculate the expectation value of the spin. How should I go about doing that? My best guess is to calculate $\\psi ^* \\psi$ to find the time-dependent probability of both spin up and spin down and then multiply each probability by the appropriate eigenvalue ($\\pm \\hbar\/2$) and add them. However, when I did this I got a time-dependent probability for spin up but a time-independent probability for spin down, which seems impossible to me. $$P(\\uparrow)=\\frac12 + \\frac14 e^{i2\\omega_2t\/\\hbar} + \\frac14 e^{-i2\\omega_2t\/\\hbar}$$ $$P(\\downarrow)=\\frac12$$ I don't know if I messed up or if the answer is starting me in the face and I can't see it. Note: the two omega values of the time-dependent terms of the wavefunction come from two different particles; my question is only about the expectation value of spin for the first particle so I have not included other non- necessary information on the second particle."}
{"id":"122407","title":"Question about Atwood Machine and Changing Masses","text":"Say we have an object of X Newtons on the ground called object 1, and it is connected to another object called object 2. If we change the mass of object 2 in intervals until object 1 < object 2, how would the force the the ground exerts on object 1 change and how would it be if object 2 was greater than object 1. I know that the force would be less as object 2's mass increase, but I'm not sure how to display it in my problem with numbers. Would we have to do something with the tension as the mass increases? We can find the acceleration with the two masses but I don't understand conceptually how we could find how the force the ground exerts changes."}
{"id":"129008","title":"If virtual particles have negative mass why do they contribute positive mass to atoms?","text":"According to Lawrence Krauss, atoms containing in our body consists of merely 10% (if I remember correctly) of our total mass. The rest come from virtual particles popping in and out of existence from empty space. So if ~90% of our mass comes from virtual particles, and virtual particles have negative mass, shouldn't virtual particles mass be subtracted to ours? In other words, why we aren't getting lighter due to the effect of those particles?"}
{"id":"40825","title":"Physical meaning of some operators formed by $|Q\\rangle \\langle Q|$","text":"In Dirac's formulation of quantum mechanics, Suppose that $q$ represents position observable. About $|q\\rangle \\langle q|$: what does this operator mean? I do get that it results in an operator, but unsure of what physical meaning it has. The same with $|\\psi \\rangle \\langle \\psi |$. ($\\Psi$ representing wavefunction, and the corresponding eigenstate $| \\psi \\rangle$.)"}
{"id":"40824","title":"Is water considered a substance with low friction or high friction?","text":"Like, for example, rubber is a substance with high friction, as an object with neither high or low friction would easily stop on it, and wouldn't skid. Ice is considered an object with low friction, as an object with neither high or low friction wouldn't easily stop on it, and would skid. But, what about water?"}
{"id":"129001","title":"Efficiency of reversible engines","text":"I'm a little confused about something. All reversible engines have the same efficiency, or one could drive the other to move more heat in the reverse direction. Also, no engine has an efficiency greater than a Carnot engine. Finally, a Carnot engine is reversible. If all these statements are true, then isn't it true that all reversible engines operate at Carnot efficiency? Where am I misinterpreting the logic?"}
{"id":"129005","title":"Is it possible to design a large permanent magnet that creates a multi-Tesla field?","text":"Is it ever possible to create a LARGE magnet (in meters wide\/long) that could potentially create a powerful magnetic field?"}
{"id":"78819","title":"Why acceleration is not relative in General Relativity?","text":"I was thinking of it, If I say: \"I'm moving at a velocity $v_1$ relative to a reference frame $M$ then the acceleration will be the derivative of $v_1$ relative to the reference frame $M$.\" In other words, from the perspective of my brother at home, I'm travelling at a velocity $v_1$ and I have an acceleration $a_1$. But from my perspective, he is travelling at $v_1$ (and I'm standing still) and thus his acceleration is $a_1$. But General Relativity tell us that acceleration is not relative, so why?"}
{"id":"123756","title":"How to prove that a time-oriented spacetime possesses a nowhere vanishing timelike vector field?","text":"Penrose gave a very brief proof to this question. Since the spacetime is paracompact, there exists a positive definite metric called $h_{ab}$. Then, the nowhere vanishing time-like vector field $V^a$ is determined by the following equations: $(g_{ab}-\\lambda h_{ab})V^b=0$ with $\\lambda>0$, and $h_{ab}V^aV^b=1$. (By his convention, the signature of the Lorentzian metric is (+,-,-,-)) I totally cannot understand the logic. Although $g_{ab}V^aV^b=\\lambda h_{ab}V^aV^b=\\lambda>0$, so $V^a$ is indeed a time-like vector, how can we be sure there is such kind of vector field satisfying the conditions?"}
{"id":"59398","title":"What determines the end of space (universe)?","text":"What determines the end of our universe?   Is it defined by the farthest out physical object or is it defined by the farthest out form of energy like light?"}
{"id":"95044","title":"Does the presence of atoms hinder current flow?","text":"When current flows through a material, electric resistance is observed when the electrons collide with the material's vibrating atoms. My textbook says that when we cool the material till the point where the atoms reach a state of rest, the electrons will not collide with the atoms **at all** , and we can achieve **zero** resistance. But how is it possible that we can have absolutely zero collision? The analogy I have always used is to think about walking inside a train. When people are moving around inside the train, I can't move freely as I'll bump into them easily (high resistance). But even when no one is moving, I still can't move totally 100% freely, simply because their very _presence_ is blocking me. Doesn't the _presence_ of the atoms themselves hinder the flow of current, even when the atoms are not vibrating?"}
{"id":"111718","title":"Lowest Frequency De-excitation problem","text":"Before the formation of the molecule it is possible for the muonic hydrogen atom to be formed in an excited state and for radiation to be observed from a single transition to the ground state. Calculate the lowest frequency of the radiation that could be observed in this process and estimate the momentum of the recoiling atom. Muonic Hydrogen is 207 heavier than normal Hydrogen To find the highest frequency I would use E=hf and solve for E_2 - E_1=hf , because this is the biggest gap. I don't even have a clue on how to start this question"}
{"id":"59526","title":"What determines bubble locations in boiling water?","text":"Something a little different to our usual fare. I was boiling a pan of water for cookery the other day, and got to wondering what caused the location of the bubble streams from the bottom of the pan. The stream of bubbles seems to come from the same spot of the pan, is this a quirk of the fluid flow? Imperfections in the pan? These streams of bubbles appear to continue for several minutes until the water fully boils and it becomes too turbulent to follow anything. Any thoughts?"}
{"id":"1257","title":"Why do earphone wires always get tangled up in pocket?","text":"What is the reason? Is it caused by their narrow shape, the soft material, walking vibration or something else?"}
{"id":"127127","title":"How does the photon field operator change are understanding of the electromagnetic stress-energy tensor?","text":"Given the following photon field operators \\begin{align} \\mathbf{A}(\\mathbf{r}) &=& \\sum_{\\mathbf{k},\\mu} \\sqrt{\\frac{\\hbar}{2 \\omega V\\epsilon_0}} \\left(\\mathbf{e}^{(\\mu)} a^{(\\mu)}(\\mathbf{k}) e^{i\\mathbf{k}\\cdot\\mathbf{r}} + \\bar{\\mathbf{e}}^{(\\mu)} {a^\\dagger}^{(\\mu)}(\\mathbf{k}) e^{-i\\mathbf{k}\\cdot\\mathbf{r}} \\right) \\\\\\ \\mathbf{E}(\\mathbf{r}) &=& i\\sum_{\\mathbf{k},\\mu} \\sqrt{\\frac{\\hbar\\omega}{2 V\\epsilon_0}} \\left(\\mathbf{e}^{(\\mu)} a^{(\\mu)}(\\mathbf{k}) e^{i\\mathbf{k}\\cdot\\mathbf{r}} - \\bar{\\mathbf{e}}^{(\\mu)} {a^\\dagger}^{(\\mu)}(\\mathbf{k}) e^{-i\\mathbf{k}\\cdot\\mathbf{r}} \\right) \\\\\\ \\mathbf{B}(\\mathbf{r}) &=& i\\sum_{\\mathbf{k},\\mu} \\sqrt{\\frac{\\hbar}{2 \\omega V\\epsilon_0}} \\left((\\mathbf{k}\\times\\mathbf{e}^{(\\mu)}) a^{(\\mu)}(\\mathbf{k}) e^{i\\mathbf{k}\\cdot\\mathbf{r}} - (\\mathbf{k}\\times\\bar{\\mathbf{e}}^{(\\mu)}) {a^\\dagger}^{(\\mu)}(\\mathbf{k}) e^{-i\\mathbf{k}\\cdot\\mathbf{r}} \\right), \\\\\\ \\end{align} how do these photon field operators change our understanding of the classical electromagnetic stress-energy tensor \\begin{align} T^{\\mu\\nu} =\\begin{bmatrix} \\frac{1}{8\\pi}(E^2+B^2) && S_x\/c && S_y\/c && S_z\/c \\\\\\ S_x\/c && -\\sigma_{xx} && -\\sigma_{xy} && -\\sigma_{xz} \\\\\\ S_y\/c && -\\sigma_{yx} && -\\sigma_{yy} && -\\sigma_{yz} \\\\\\ S_z\/c && -\\sigma_{zx} && -\\sigma_{zy} && -\\sigma_{zz} \\end{bmatrix} ? \\end{align}"}
{"id":"114299","title":"How is foliation of manifolds' theory useful in General Relativity?","text":"I am interested on getting some hints on how Foliations Theory of Manifolds can be used fruitfully on General Relativity. I just started my Ph.D on Mathematics this semester focusing on studying Holomorphic foliations on projective manifolds. I just came across some texts which related the more general Foliation theory to Gen. Rel. Is there research going on that mixes both areas? Can someone direct me to introductory papers and textbooks?"}
{"id":"114294","title":"Movement of compressed air inside a rolling tire?","text":"Does it stay still wrt the axis? Does it rotate along with the tire? Is it drawn away from the center of rotation?"}
{"id":"55431","title":"Why one-dimensional strings, but not higher-dimensional shells\/membranes?","text":"One way that I've seen to sort-of motivate string theory is to 'generalize' the relativistic point particle action, resulting in the Nambu-Goto action. However, once you see how to make this 'generalization', it becomes obvious how to write down the action, not just for a string, but for manifolds of higher dimension as well. In fact, Becker-Becker-Schwarz (the main source I happen to be learning from) actually do this. But (as far as I have read), they merely write down the action and do nothing further with it. My question is: what happens when we proceed along the same lines as string theory, but when replacing a string with a 2-manifold, the simplest example of which would be the 2-sphere, a \"shell\/membrane\"? Assuming 3 spatial dimensions, this is the highest dimensional manifold we can consider (because we don't allow for non-compact manifolds). Furthermore, there is only one compact manifold of dimension 1; however, there are infinitely many compact manifolds of dimension 2, which could potentially make the theory much richer (and probably much more difficult). For example, in string theory, we are stuck with $S^1$ (if you insist upon having no boundaries), but if you allow for 2 dimensions, we could consider the sphere, the torus, etc. Because it seems that this path is not ever presented (in fact, I've never even heard of it), I would presume that something goes wrong. So then, what exactly does go wrong?"}
{"id":"55435","title":"How to choose a solution from all possible solutions of general relativity","text":"So there are so many solutions for general relativity - then how does one \"choose\" the solution that is right one? By checking with observation? (though I also know that it is currently unknown which one is the correct solution.)"}
{"id":"56160","title":"Scaling of quantum error correction","text":"I'm having a question regarding quantum error correction. Using a large number of imperfect (but already very good) quantum gates, it is in theory possible to build an equivalent, error-corrected gate. What I don't understand, however, is how it precisely scales when I want to do computation using algorithms with a large input space. To give a precise question: Let's say I can create many individual CNOT gates with an individual probability of success of $\\eta=99.99\\%$. How many of them do I need to implement Shor's algorithm to factor a $1024$ bits integer with an overall probability of success of $50\\%.$ ? Once I reach the point where I can build a single error-corrected gate between two qubits, have I won the fight against decoherence or will it just be harder and harder to correct the errors as the input space scales up ? Thanks."}
{"id":"110043","title":"Heat transfer in infrared","text":"Why does the most of the heat transferred on Earth come from infrared part of electromagnetic spectrum?"}
{"id":"110044","title":"Time dilation factor for the circular orbit at 3\/2 Schwarzschild radius","text":"What would the time dilation factor be if a massive (as in rest mass>0) point particle orbiting a Schwarzschild black hole in the photon sphere? If I understand correctly, this is the only possible orbit for photons, but it is also the _closest_ possible orbit for a massive particle. So this is the same thing as asking what the maximum physically possible time dilation for a circular orbit is. That point is 3\/2 times the Schwarzschild radius. Any closer than that, and no free-fall path goes to infinity or completes a full orbit. It's also unstable. However, an exactly timed maneuver could put a particle into orbit there for a large number of orbits, so I don't think the instability affects the meaningfulness of the question. One could even realistically transition from r=infinity to the outer edge of this orbit, complete several orbits, and then escape back to r=infinity. Reason for asking: a simplistic application of general relativity circular orbit time dilation tells me that the factor is infinity. In other words, time doesn't pass for a particle in this orbit. I can't even begin to rationalize that. How could the universe be frozen still for such an observer? I don't think that makes any sense."}
{"id":"114746","title":"How do you stop wobbling in a carbon nano tube elevator to space?","text":"If there was a carbon nanotube elevator that went to space how would you keep it from wobbling? As I picture it I see the structure being very strong, but would that prevent wobbling? It would be like a tall, thin building?"}
{"id":"96020","title":"What is difference between Acceleration due to gravity and Gravitational field intensity near the Earth Surface?","text":"I have read in so many standard books that Gravitational field intensity near the earth surface and acceleration due to gravity near the earth surface are Two Different physical quantities with same Direction, Magnitude and Units. So if All the units magnitudes and directions are same and also both mean the acceleration produced in a point mass near the earth surface What is difference between them?"}
{"id":"93054","title":"What is the amount of EM energy from the sun reaching the earth as a function of wavelength?","text":"The question stems from a desire to know if more light from the sun in the infrared wavelengths has more energy than light at say yellow or violet wavelengths - at the Earth's surface. I'm guessing a simple 2-D graph would illustrate the answer well."}
{"id":"118670","title":"How to prove R value for insulation?","text":"I ran some tests on different types of insulation. I heated water, and placed it into an insulated box. The temperature was recorded every minute, for the water, as well as the air. The insulation which is the best (high R value) will have the temperature drop the least. I have calculated the heat added using: $ Q=mc\\Delta T $ Is this right, or should I have down something different? Can the laws of Thermodynamics be used?"}
{"id":"61134","title":"What truly is mass, and is there a direct way to measure it?","text":"We know a mass of an object of one kilogram as an object that weighs W = mg = 9.8 N and we reference it to that, (when it should as a fundamental parameter describe weight not the opposite). But if we were to describe mass to an alien civilization on an alien planet we are exchanging knowledge with, by sending them a one kilogram object, according to their gravity they will measure it differently. Also their star could be curving space time in such a way, or their velocity according to SR will cause them to perceive mass on our planet -if they observe from away- differently. If we asked a crew on a space ship moving at a speed close to the speed of light wrt us, or moving in a gravitational field they don't know about, to measure the mass of our planet, they will get different results. On the same principle, we could be measuring the mass of far celestial objects like planets, differently. I perceive space time as full of curves and irregularities. We know the about some of these and we don't about others. I mean we know about the earth's gravity, the sun's but we ignore the effect of the galaxy and the cluster, dark matter, and who knows what else. Besides we change our position all the time with relation to these external factors that bend spacetime. So our measurements to an outside neutral observer, say these aliens could change. I've found definitions of mass like \"the quantity of matter in an object\" but that seems like the good old: mass = volume x density, but mass is the more fundamental quantity than either of them, with which theses parameters should be described by mass not the opposite, not to mention how relative these other quantities are, considering SR and GR. Or \"The resistance of an object to acceleration\" but again you have to describe how fast that object is moving and what spacetime it's in according to SR and GR. So what makes us so confident that mass is such a universal value, when we built everything on a concept referenced by our own gravity, and maybe our own reference frame? How do we describe mass to the aliens, who don't know about our (g)? How do we measure the mass of celestial objects say planets, by units like kilograms, and pounds, while they are not subject to the earth's gravity, (I'm well aware of the difference between weight and mass). I mean that's what they will weigh -ignoring their own gravity- if they were in our atmosphere on a huge scale? So what's the method? _**Bottom line_** Is there a direct way to measure mass like we do with other fundamental values like length and temperature, _other than_ using a scale or equations, i.e not depending on other physical parameters to describe it? \\--Forgive the length.."}
{"id":"61131","title":"Does a cross really have a smaller area moment of inertia than a square of the same area?","text":"I need to calculate the area moment of inertia of a cross for a homework assignment. The cross is a symmetric plus sign:                   _        | |     |==   ==|        | |         -      where the length\/width of each arm is 1\" (so the total area is 5 square in.). My understanding is that I can just separate the cross into three sections (left, middle, right), so the area moment of inertia would be $$ I = \\sum \\frac{bh^3}{12} = \\frac{1\\times 1^3 + 3\\times 1^3 + 1\\times 1^3}{12} = \\frac{5}{12}$$ This is surprising to me, because a 2.2\" x 2.2\" square would have the same area but a much larger moment of inertia: $$ I = \\frac{bh^3}{12} = \\frac{(\\sqrt{5})^4}{12} = 2.08$$ Since the arms of the cross span 3 inches, I would expect there to be more material further from the center, whereas the square only has a small amount of area in the corners that the cross doesn't have. Are these formulas correct or am I doing something wrong? Thanks in advance."}
{"id":"31394","title":"How to calculate Rest Mass practically with Standard Model?","text":"With relativistic physics, we can apply force to see resistance against acceleration. It'd give us relativistic mass and we have well established formula to get to the Rest Mass as long as we know the velocity. In Standard Model, we have Higgs Mechanism to assign something Rest Mass. I was curious how to apply that practically."}
{"id":"82267","title":"Dark Energy Explanation Found","text":"It's my sincere request to this Physics forum that if i am wrong (which has the least possibility atleast from my side) just reply \"wrong\" so that i can move on futher. I know personal theories are not allowed but i am not finding any place to get it checked. thanks DARK ENERGY The theory described below does not contain any self made false assumptions about anything but uses only one, most logical assumption about the potential energy of curved space. Let a particle of unit mass curve space according to general relativity.Now, if we suddenly evaporate it (i.e. convert it's mass into radiation), the curvature of space associated with it begins to die out.But what about the potential energy associated with the curved space after the mass is lost? We can better understand this with this example. We stretch a rubberband by applying a constant force (correspond to mg).Now, if we remove this force, the stretched rubberband tends to come to it's mean position.But, the potential energy due to initially stretched rubberband makes it go further from it's mean position making it vibrate along it's mean position. The same happens in space. Bodies cause the curvature of space which induces potential energy in space. So, when a body loses it's mass,the curvature associated with that mass does not just dies out but instead the space starts vibrating along it's mean shape i.e. the potential energy of the curved space associated with that mass makes the space vibrate along it's mean shape producing gravity - anti-gravity fields. But , if there is another body of mass m near it, the magnitude of anti-gravitational field is more in each oscilation as the other body tends to curve space (apply force i.e. mg) along it's direction. This is why the galaxies are moving away from each other.As the stars present in them lose their mass,anti-gravitational fields are produced along all directions making every galaxy move apart from each other. Therefore, the space is not expanding but instead the already curved space produces this effect as it starts vibrating along all directions when a body loses it's mass."}
{"id":"98352","title":"Qubit in Type 1.5 superconductor?","text":"I'm interested in Type 1.5 superconductors, first proposed by Egor Babaev in 2002 and found in the laboratory in 2009 (magnesium dibromide). Such conductors favor small bundles of vortices. The most stable number of vortices in the bundle varies depending on conditions but is typically from 2 to 12. The vortices have fractional quanta. The bundle is sheathed by larger vortex. Lets suppose that we have a bundle of three. Would each vortex be an anyon, and the bundle be a qubit?"}
{"id":"64410","title":"Charge inside conductor","text":"I know that the E field inside a conductor is zero. What happens if I put a source of charge inside the conductor? Say the conductor was spherical centered on the origin and there exists a charge at the origin. Then by Gauss's Law, there exists a non zero field inside the conductor. Or is it the case that even though this charge is within the sphere, it will redistribute evenly over the surface and hence the E field inside is zero? Many thanks"}
{"id":"117211","title":"Rocket towing an object","text":"I'm not sure if my question makes sense, but I don't know where to start. A crate of mass $m_2$ is connected by a massless rope of constant length $l$ to a rocket of mass $m_1$. We take into account gravity (the crate is initially hanging directly underneath), but the rocket has no vertical acceleration since it's acted upon by an upward force of $(m_1+m_2)*g$. We neglect air drag and treat the rocket and the crate it is towing as blocks of the same size. If a horizontal force $F$ acts upon the rocket, what happens to the crate? It's probably going to lag behind at a certain angle, but how can we find that angle? Is the situation different if there's no gravity? Edit: This link helped me answer my question: http:\/\/cnx.org\/content\/m14061\/latest\/?collection=col10322\/1.83"}
{"id":"117213","title":"Would a Nuclear bomb have any effect on the sun that could damage the earth?","text":"I was asked this recently and wasn't sure. The power of the sun is much more powerful than that of a Nuclear explosion but the question was what is the worst that could happen, we are assuming a situation where the nuclear bomb wasn't itself immediately destroyed by proximity of the sun for whatever reason. So, for example, if it was near the surface, could that cause a massive sun spot or similar which could disrupt electronics b.t.w. my background isn't a physics one :-) thanks"}
{"id":"110902","title":"Anodized aluminium reflectivity at 10.6 um","text":"I wonder how I can model anodized aluminium reflectivity in ray tracing-based optical simulation. I know that the parts my company is using are anodized to get covered with a ~20 um (as specified) thick corundum layer. I would like to be able to properly model angular dependence of such a surface reflectivity. I am using Zemax, but please do not limit yourself to it. Any hint is welcome. The two options I can think of are 1) Create a model coating in Zemax by specifying reflectivity vs. angle. Can be done if I know the values. I could not find them so far. Could be measured as well but that's a major effort which I'd like to spare. 2) Specify a thin layer of corundum on top of a bulk aluminium part in Zemax. However I could not find the dispersion data for Al2O3 at 10.6 um and I cannot find anything in Zemax' material catalogues. Besides, I assume such a layer of 20 um thickness would act as a thin film resulting in AR or HR properties. But I don't think the anodization process is precise enough in terms of layer thickness so that I can rely on this side effect. Any help or comment is highly appreciated."}
{"id":"2645","title":"What are the arguments towards the Life-and-Entropy relation?","text":"I've heard it from a few people, and I've seen it popup here in the site a couple of times. There seems to be speculation (and studies?) towards this idea, and this is what I've picked up so far:   * Living organisms are low entropy systems.   * The second law of thermodynamics is somehow strongly connected to the origin of life itself. This is a very interesting idea, unfortunately I've never seen anyone elaborate beyond that. I'd like to understand it a little better. Specifically:   * Why are the above statements true (neither is immediately obvious to me)?   * And what are the other arguments in favor of this theory? I'll accept both answers for and against this point of view, as long as it reasonably justifies _why_ the above statements are correct\/false. _Note: I believe the thermodynamic definition of entropy is the one commonly used with these arguments. And I'm not sure what definition they use for life. TheWikipedia page is pretty vague on this._"}
{"id":"105592","title":"Why cant Electrostatic field lines form closed loops?","text":"My physics textbook says \"Electrostatic field lines do not form closed loops. This is a consequence of the conservative nature of electric field.\" But I cant quite understand. Can anyone elaborate? PS:- This question is answered. Please refer both the accepted answer and also the answer and the comments in Robin Ekman's answer for a complete understanding."}
{"id":"37606","title":"Pseudo scalar mass and Pure scalar mass","text":"Since the only difference between pseudo scalar and a scalar term is just a change of sign under a parity inversion, is it possible that both of them be present in the same field and interact? For example let's say that the vector gauge field is copuling with some other field. And suppose I have a pseudoscalar mass term like $m\\varepsilon^{\\mu\\nu\\rho\\sigma}\\phi_{\\mu\\nu}\\phi_{\\rho\\sigma}$ and pure scalar mass term $\\frac{M}{16} \\phi_{\\mu\\nu}\\phi^{\\mu\\nu} $. In that case is it possible that both of them present in the same time in the field? Or there is any condition that they have to satisfy?"}
{"id":"102064","title":"What is Conductance?","text":"What is conductance? I am being said that conductance is inverse of resistance?I do know that resistance is the opposition to the flow of current but do not get conductance? > I do know that:it is ease of flow current but my brain doesn't accept > it,don't know why Ain't it so important!Couldn't even find it on wikipedia."}
{"id":"24043","title":"Is there a point interaction model of the electron?","text":"Is there a point interaction model of the electron? Is there a point interaction model of the electron? I imagine something like $\\propto(\\bar \\psi\\psi)^2$ (edited). Is such a thing in use? Since I expect some _but that's not renormalizable!_ -arguments: I think there are theories with scalar photons. Can you integrate out the massless photon completely? And that raises the question more generally: Does going from a renormalizable to an effective description theory change that property? * * * Edit: I just found the Four-fermion interactions. I guess that leaves the question if, or better how, these can result from swallowing the photon. Here is the question in that light: > Coming from QED, is there a point interaction model of the electron? PS: I like the last line of the article. Who would have seen that one coming :P"}
{"id":"20964","title":"How isolated must a system be for it's wave function to be considered not collapsed?","text":"As an undergrad I was often confused over people's bafflement with Schodinger's cat thought experiment. It seemed obvious to me that the term \"observation\" referred to the Geiger counter, not the person opening the box. Over time, I have come to realize that the Copenhagen interpretation actually _is_ ambiguous and that \"observer\" cannot be so easily defined. Nonetheless, an objective collapse theory (which is what I was unknowingly assuming) still seems to me the simplest explanation of wave collapse phenomena. I have read some of the objections cited in the Wikipedia article linked above, but it is still unclear to me why most physicists adopt the Copenhagen interpretation and reject objective collapse. For example, in this question on hidden observers, there was some discussion about the mechanism of wave collapse. It was suggested that perhaps the gravitational pull of a hidden observer would collapse the wave function. In response, it was pointed out that the gravitational pull would be negligible at the scales involved. Okay, then imagine the following: A hermetically sealed (i.e. isolated) box is balanced on a fulcrum. Inside the box is a radioactive isotope, a Geiger counter, and a trigger mechanism connected to a spring loaded with a mass on one side of the box. If the Geiger counter detects a decay, the trigger releases the spring and the mass shifts to the other side of the box. The shift in mass would, under observable conditions tilt the box on the fulcrum. According to the interpretation of Schrodinger's cat that I often hear (the cat is in a superposition) it seems that the box should slowly tilt over as the wave function of the system evolves with the half-life of the isotope. I can't imagine that anyone thinks this is a realistic expectation. I can see that people might object and say \"But the contents of the box are interacting gravitationally with the outside system and observer so it is not really isolated!\" Well, what of it? The same is true of the cat even if the interaction is less dramatic. The question, then, is: How isolated must a system be for it's wave function to be considered not collapsed?"}
{"id":"86090","title":"Massless spin 1\/2 particle","text":"Could a massless spin 1\/2 particle, or more generally massless half-integer spin particles exist? Does it make sense to say that they could be described for example by the Dirac equation by forgetting the mass term ?"}
{"id":"32263","title":"Why do wheels appear to revolve opposite to the direction they are rotating?","text":"When viewing cars that are driving along side of us, sometimes their wheels appear to be turning backwards even though they are traveling in the same direction as our car. Why do they look that way?"}
{"id":"34024","title":"Special Relativity Second Postulate","text":"That the speed of light is constant for all inertial frames is the second postulate of special relativity but this does not means that nothing can travel faster than light.   * so is it possible the point that nothing can travel faster than light was wrong?"}
{"id":"112038","title":"Minkowski metric -- why does it follow from the constancy of the speed of light","text":"In all the sources I’ve been able to find, the Minkowski metric appears ad hoc, or is defined analogously to the euclidean metric. I’d love to see an argument why this metric (time coordinates positive, space coordinates negative) must follow from the constancy of the speed of light. It is clear that the Minkowski metric is preserved under the hyperbolic transformation of space-time, but likely others are as well. Why this particular metric and not something else. Consider the determinant function of an n by n matrix. It has a god awful mathematical form involving the sum of n ! terms. Yet all you need to get the (unique) formula are a few postulates — the determinant of the identity matrix is 1, the determinant is a linear function of its rows (or its columns), interchanging any two rows of the determinant reverses the sign of the determinant, etc. etc. This basically determines the (unique) formula of the determinant. I’d really like to see the Minkowski metric come out of something like that."}
{"id":"71864","title":"Understanding the argument that local U(1) leads to coupling of EM and matter","text":"I'm trying to better understand the argument that U(1) local gauge invariance implies a coupling of EM and Dirac fields. I understand the math, but I'm not sure about the chain of logic. You start with $\\mathcal{L}_\\mathrm{Dirac} = \\bar\\psi ( i\\hbar \\partial_\\mu - m)\\psi $ and the following transformations: $$ \\psi \\rightarrow \\psi' = e^{i\\theta}\\psi \\qquad \\left( \\bar\\psi \\rightarrow \\bar\\psi' = \\bar\\psi e^{-i\\theta} \\right) \\qquad (1)$$ $$ A_\\mu \\rightarrow A'_\\mu + \\partial_\\mu \\chi \\qquad (2)$$ You often by applying (1) to $\\mathcal{L}_\\mathrm{Dirac}$, and note that to keep the form invariant, you have to change the partial derivative to the covariant derivative: $\\partial_\\mu \\rightsquigarrow D_\\mu = \\partial_\\mu - \\frac{i}{e} A_\\mu$. You do the calculation again, and then find $$\\mathcal{L}' = \\bar \\psi\\left( i \\hbar \\gamma^\\mu (D_\\mu - i \\partial_\\mu\\theta + i\\partial_\\mu\\chi) - m\\right)\\psi$$ Thus, to make $\\mathcal{L}$ invariant under the transformations, $\\theta = \\chi$ (up to factors of $e$ and $\\hbar$ that I missed). The phase of the electron wavefunction must in a way absorb the change of the four-potential given by the gauge transformation, or vice versa. I understand (2) comes from classical gauge freedom in electrodynamics. You are free to add a constant potential offset $\\Phi_0$ to $\\Phi$, or to add a divergence of a scalar field $\\vec\\nabla \\chi$ to $\\vec A$. Since $A_\\mu = (\\Phi, \\vec A)$, it follows that (2) leaves the physics invariant. But where does (1) come from? Is this just the regular freedom of phase change you have with any wavefunction? If so, why doesn't the U(1) symmetry show up everywhere? Or does this come from _postulating_ a U(1) symmetry for $\\psi$? If so, how is this motivated (other than \"oh, it gives the right result\")? For example, if I think (2) is better motivated, **is it possible to start with only (2) and possibly the covariant derivative, and get to (1)? In general, what assumptions can I put into the argument, and what can I get out of it?**"}
{"id":"104888","title":"Why dimensionality of the Electric Charge varies with the spacetime dimensions?","text":"The point is: We can find via dimensional analysis that the electric charge dimensionality varies with the dimension of space-time. $$[\\text{charge}] = eV^{(3-D)\/2}$$(You can see below the way I did it) The question is: **Why it occur? What is the meaning of this?** * * * I'm using the Heaviside-Lorentz Units with Natural Units ($\\hbar = c = 1$) so that everything has dimensions of energy (eV). $$[x] = [t] = eV^{-1}$$ $$[p] = [E] = eV$$ Using the action from Maxwell Theory $$ S = - \\frac{1}{4} \\int d^{1+D} x F^{\\mu \\nu} F_{\\mu \\nu}$$ And making a dimensional analysis, $$ [S] = [x]^{1+D} [F^{\\mu \\nu}]^2$$ $$ 1 = eV^{-D-1} [F^{\\mu \\nu}]^2$$ $$ [F^{\\mu \\nu}] = eV^{(D+1)\/2}$$ Then, using the inomogeneous equation $$ \\partial_\\nu F^{\\mu \\nu} = J^\\mu$$ Ang making an dimensional analysis $$ [\\partial_\\nu] [F^{\\mu \\nu}] = [J^\\mu]$$ $$ eV eV^{(D+1)\/2} = [J^\\mu]$$ $$ [J^\\mu] = eV^{(D+3)\/2}$$ Now lets look the charg dimension. The relation of the charge with the (1+D) density current is $$[J^\\mu] = \\frac{[\\text{charge}]}{[x]^D}$$ So, $$eV^{(D+3)\/2} = eV^{D} [\\text{charge}]$$ $$[\\text{charge}] = eV^{(3-D)\/2}$$ We get that only in 1+3 the charge is dimensionless. If we are at 1+1, and at 1+5 charge is just like time. * * * **Comment 1** At another place I was told that it (dimension of charge) have relation with the renormalizability of QED. And that the fact that charge is dimensionless only in 1+3 is related with Maxwell Equations being conformally invariant at 1+3."}
{"id":"78628","title":"How big would the impact of a nuclear warhead be?","text":"I am writing a post-apocalyptic science-fiction story which includes the detonation of a nuclear warhead. In this context, What scientific information is available about the impact of a modern day nuclear warhead?"}
{"id":"78629","title":"How are fictitious forces related to my feeling?","text":"This question arises when I am studying fictitious forces in an undergrad introduction to physics course. Suppose I am standing in an elevator with an acceleration $a$ directed upward. From the ground viewpoint, with $N$ being the normal force on me, we have $N - mg = ma $ and so $N = mg + ma$. From my viewpoint, I feel two force: an upward force of $m(g+a)$ and a total downward force (gravity + fictitious force) of $m(g+a)$ too. If I were standing somewhere on a planet with gravitational acceleration $G=g+a$, then I feel these two forces too. That's why in the elevator I have the feeling of being on the planet with $G = g+a$. OK, so far so good. Now let's say I'm freefalling. From my viewpoint, I feel two force: an upward fictitious force of $mg$ and a downward gravitational force $mg$. The situation is pretty the same as if I were standing on the ground. But clearly I wouldn't feel that normal when freefalling; in fact I feel weightless. Why? Thanks for your help."}
{"id":"3145","title":"Sensitivity of eye","text":"In which color human eye is most sensitive?"}
{"id":"37824","title":"Is quantum Hall current density local? (${\\bf j}({\\bf r}) = \\sigma_H {\\bf \\hat n \\times E}({\\bf r}) $)","text":"The good old Ohm's law $${\\bf j}({\\bf r}) = \\sigma_O {\\bf E}({\\bf r})$$ if translated into words would be \"the local current density is proportional to a local electric field.\" In a quantum Hall state (QHS), the linear response to an external electric field is given by the following $${\\bf j} = \\sigma_H \\bf \\hat n \\times E$$ where $\\bf \\hat n$ is the surface normal. _**Is it also a local relation between applied field ${\\bf E}({\\bf r})$ and response current ${\\bf j}({\\bf r})$?_** If the electric field is non-zero inside the bulk, it means there must be current propagating inside the gapped bulk. I tend to understand it in an intuitive way. As you see in the picture, for a uniform electric field in $x$-direction, the chemical potential is raised up linearly in $x$ and it intersects levels in the bulk. The current in the bulk comes from the chiral edge modes and sink into edge back again. I don't know whether my picture is correct or not. ![As you see, for a uniform electric field in $x$-direction, the chemical potential is raised up linearly in $x$ and it intersects levels in the bulk.](http:\/\/i.stack.imgur.com\/2edPK.png)"}
{"id":"106766","title":"Determination of displacement value in work formula","text":"A net force will always create displacement which approaches infinity if left undisturbed. So how to fix a value of displacement in work formula? Isn't a mention time interval necessary? Or do we measure displacement only while force is acting on the body i.e the moment we stop applying force we also stop measuring displacement."}
{"id":"23099","title":"How do we determine the temperature of a Black Hole?","text":"How do we determine the temperature of a Black Hole? Since we cannot see a Black Hole, which I presume, is because it absorbs light, would it not also prevent radiation from escaping, making temperature determination difficult."}
{"id":"109488","title":"Entropy and Stability","text":"What happens when you increase entropy in a system? Does it increase the instability? Is disorder the same as entropy?"}
{"id":"109482","title":"Jordan Wigner Transformation in 1d Majorana chain","text":"So, I was reading the paper by Fidkowski and Kitaev on 1d fermionic phase http:\/\/arxiv.org\/abs\/1008.4138. It explains the classification of 1d fermionic SPT phases with $\\mathbb{Z}_2^T$ symmetry for both the interacting and non- interacting cases. I understand why we had a $\\mathbb{Z}$ classification in the non-interacting case. Basically, it measures the number of dangling majorana modes when one cuts a TR-invariant majorana chain (see section II of the paper). But then, for the interacting case, apparently, in order to define the interacting invariants, and construct the matrix product states, we need to do a Jordan- Wigner transformation to a bosonic spin chain, so that the original hamiltonian is written in terms of the new bosonic variables, $\\sigma_j^x$, $\\sigma_j^y$,$\\sigma_j^z$ (end of section II). I don't understand why do we need to do this transformation and what good does it do to get the $\\mathbb{Z}_8$ classification in the interacting case. I don't see the connection between this transformation and the way he constructed the matrix product states in section III."}
{"id":"29145","title":"Is every QFT non-local in the U.V.?","text":"As much as I understand the renormalization group transformation and the concept of relevant\/irrelevant operators, I'd say that if we push the reasoning of only looking at relevant operators when we follow a \"renormalization group orbit\" (a curve in the space of the parameters) to the infra-red regime the other way, so going to the U.V., we should keep all the operators this time (as they are no more suppressed by huge factors), or there is a infinite number of them, depending on an infinite number of the derivatives, and we end up with a non-local theory. As this reasoning can be applied to every QFT, does this mean that they are all non-local in the U.V.? or the reasoning fooled somewhere (breakdown of the QFT framework somewhere for example...)?"}
{"id":"29141","title":"Time dilation at the speed of light","text":"Does a massless particle travelling at the speed on light in a vacuum (c) experience an infinite time dilation effect? i.e. Would the time dilation extend from the perspective of the massless particle past the measurable existence of the particle by a observer outside the dilation."}
{"id":"29140","title":"Is quantum entanglement an objective or subjective property?","text":"Imagine the following gedankenexperiment. Observer Alice is right here on Earth. Observer Bob is at say Alpha Centauri. A pair of maximally entangled qubits is formed with one qubit handed over to Observer Alice and the other sealed in a box on Pluto shielded from decoherence. It takes a few years for any signal to reach Observer Bob. Observer Alice measures her qubit and \"collapses\" it. To her, the pair of qubits is no longer entangled. However, for a few years, will Observer Bob think there is an entanglement? We have to be careful here. Once the qubit on Earth has been measured by Earthlings, the entanglement is \"shared\". By the well known monogamy of entanglement theorem, any correlation between both qubits by themselves will have become classical. However, Observer Bob, who has a very good working knowledge of quantum mechanics and relativity, knows that for a few hours, until light signals can reach Pluto, the system consisting of an expanding bubble around Earth and the Pluto qubit are in an entangled state. Alice, who lives inside this bubble, will beg to differ. Is quantum entanglement an objective or subjective property? Closely related to Wigner's friend and intersubjectivity in quantum mechanics and Given entanglement, why is it permissible to consider the quantum state of subsystems?."}
{"id":"67497","title":"What is the name of the equation which led to the Schrödinger one?","text":"What is the name of this equation: $$\\frac {d^2\\psi}{dr^2}+k^2\\psi=0?$$ (I want a Wikipedia link for this equation, but I don't know what its name is.) * * * Point: In this equation, the wave function is only function of space $\\psi (r)$, for example $\\psi=Ae^{\\mathrm i kr}$."}
{"id":"66726","title":"Varying the amplitude of a driven wave","text":"I' d like to know whether varying the amplitude of a system at resonance is possible or not and if it is, how? I've calculated the resonance frequency of a material and I'd like to know the displacement this frequency generates and whether I can control it by knowing the relationship between the amplitude (displacement) at resonance and power of driving source. It would make sense that increasing the power would increase the amplitude\/displacement of the oscillation, but I need formulas. Thanks!"}
{"id":"72742","title":"Why is this classical system seemingly non-deterministic?","text":"So, I was solving this problem1 and I realized that the system given seems to be non-deterministic when analysed with classical laws. ![enter image description here](http:\/\/i.stack.imgur.com\/Soh81.png) The situation is thus: A rope is wrapped for an angle of $\\theta_0$ (symmetrically) around a fixed pole (the circle in the diagram). One end is fixed to a wall, and the other end is pulled with a force\/tension $T_0$. The string is massless, but there is friction between the string and the pole of coefficient $\\mu$. There is no slipping (the system is static). If necessary, the length of the string and the radius\/relative position of the pole can be considered as givens. Now, if you follow my solution2, one gets that the tension in the string along the cylinder, where $\\theta=0$ is the part on the right where the string just touches follows the following constraints: $T_0e^{-\\mu\\theta}\\leq T(\\theta)\\leq T_0e^{\\mu\\theta}$. Extending this, you get that the tension $T_1$ in the part of the string connected to the wall follows the constraint $T_0e^{-\\mu\\theta_0}\\leq T_1\\leq T_0e^{\\mu\\theta_0}$. However, I don't see any way to _further_ analyse this system with Newtonian mechanics. Note that the inequality appears even if I consider a string with mass, though the calculation becomes more complicated. So basically I'm stuck with an inequality. Note that the system, as given, should be completely defined -- $\\theta_0$, $T$,$\\mu$, the radius of pole, and the relative positioning of pole are all that should be required to physically create this system. However, the macroscopic, measurable quantity $T_1$ somehow cannot be determined from Newton's laws. How does one resolve this apparent paradox? Classical systems are supposed to be able to fully determine measurable quantities. This doesn't seem to be the case here. 1\\. This was a course on Classical (Lagrangian & Hamiltonian) mechanics, and our professor gave us some Newtonian mechanics problems as a first assignment. We're not yet sure why, but most were quite interesting. The assignment due date is over, this is just a residual point that has been bothering me. 2\\. This is lifted from the document I submitted. I have blurred out the end because it contains my own resolution of the apparent paradox here. I may post it as an answer later, but for now I'd prefer to give everyone a fresh look at this situation."}
{"id":"76842","title":"The statistical nature of the 2nd Law of Thermodynamics","text":"Ok, so entropy increases... This is supposed to be an absolute statement about entropy. But then someone imagines a box with a 10 particle gas, and finds that every now and then all particles are in the left. Conclusion, the 2nd law holds only in a statistical sense. But then Szilard comes up with a thought experiment with only one particle and a piston that can compress to the left or right. The apparent loss of entropy when finding the particle in the left half is compensated by the very bit of information that indicates where the particle is. So perhaps the 2nd Law actually holds in an absolute sense, except that... > Is there consensus on the absolute vs statistical nature of the 2nd Law, or > is it subject to interpretation? Can the issue be settled within a classical > setting, or does one have to go quantum? **Addendum:** (as per Ben Crowell's request) Here is the paper Szilard, L., 1929, _“On the Decrease of Entropy in a Thermodynamic System by the Intervention of Intelligent Beings”_ , Zeitschrift fur Physik 53: 840–856. English translation in The Collected Works of Leo Szilard: Scientific Papers, B.T. Feld and G. Weiss Szilard (eds.), Cambridge, Massachusetts: MIT Press, 1972, pp. 103–129."}
{"id":"76843","title":"While finding the equivalent resistance why isn't the 8 ohm in series with the 5 ohm?","text":"Is there any way to tell if the circuit below is in series or parallel quickly and accurately? ![refer to image](http:\/\/i.stack.imgur.com\/uzR8Q.jpg)"}
{"id":"105867","title":"Modeling viscous fluid flow in the curved pipe","text":"I want to know if there is any simple (results don't have to be very precise) way of modeling viscous fluid flow throught the curved pipe with elastic walls. I need to gather preassure values and compare them to preassure of the fluid in the straight pipe with the same wall parameters."}
{"id":"92594","title":"Do eigenvectors of quantum operators span the whole Hilbert Space?","text":"I am trying to solve an exercise in Shankar's QM book (concretely 4.2.1), and I am asked the probability of each possible value for the operator $L_x$ when the particle is in a certain eigenstate of another operator $L_z$. I understand that I have to \"expand\" the state vector in the $L_z$ basis, which in this case is just the appropriate eigenvector, and then change the base to that of $L_x$, i.e to write the $L_z$ eigenstate in the $L_x$ base. The problem is that there is just no way of doing it, since the eigenvectors of $L_x$ don't span the whole Hilbert Space. I thought all quantum operators had a set of eigenvectors that spanned the whole space, I am wrong on that or am I wrong on my algebra?"}
{"id":"92593","title":"How to calculate Standing Waves in Electrical Cables?","text":"I have a 20 metre Coaxial Cable. I send digital signals down the cable ranging from 5 KHz to 50 KHz. I have noticed a pattern in the noise ratio, an oscillating wave. I predict this is to do with standing waves in the cable. How can I calculate at what Frequencies these standing waves would occur in the cable?"}
{"id":"6613","title":"Is there a chance for Earth to have an additional satellite someday?","text":"I am just an space enthusiast and not a physics professional so pardon me if this is not the right place to ask.   As title says, is there a chance for our Earth to have additional satellite(s) like our Moon? I tried to google it, without luck, however I know here there are very smart people that can perhaps give me some references to read or can give me an answer :)   Thanks"}
{"id":"6618","title":"Why is a suspended spinning body of liquid prolate?","text":"Instead of a flat disc one would expect the centrifugal forces to push it in.. The body is not in a container but suspended in air spinning horizontally (i.e. left to right or vice-versa) - how to do this could be challenging - but that is not the issue. **Answer:** It is not prolate: http:\/\/www.youtube.com\/watch?v=BxyfiBGCwhQ."}
{"id":"100330","title":"Hydrogen atom: potential well and orbit radii","text":"I happened to open up an old solid-state electronics book by Sah, and in it he says: \"it is evident that the electron orbit radius is half the well radius at the energy level En\" The orbit radius is $r_n=\\frac{4\\pi\\epsilon_0 ℏ^2 n^2}{mq^2}$ and the potential well $V(r_n)=\\frac{−q^4m}{(4\\pi\\epsilon_0)^2ℏ^2n^2}$ Of course the orbit has to be confined in the well, but it's not obvious to me why it should be exactly half the well radius? This isn't something I recall seeing before either in any other text. Thanks"}
{"id":"131127","title":"Physical interpretation related to a non-linear partial differential equation","text":"I am doctoral student in pure mathematics working on a particular problem. My question is if this problem has applications to real world phenomena. I will try to explain the direct problem starting from a simple case with known applications, and then generalising it to a nonlinear (actually, quasilinear) case, and then finally explain the inverse problem. I admit to not having studied physics at university level. Hence, the notation and language I use might be exotic, for which I apologise. **Conductivity equation** Suppose we have a conducting body, $\\Omega$, with conductivity (or conductance) $\\sigma$. I will be using terminology from electricity, but this might also be heat conductance. Conductivity is the inverse of resistance, which I will not write more about. We take the material to be isotropic, which corresponds to conductivity with scalar (real) values. By (differential) Ohm's law the current J is proportional to conductivity and electric field, which we write as the gradient of potential v: $$ J = -\\sigma \\nabla v$$ By Kirchhoff's law the current is solenoidal, or divergence-free: $\\nabla \\cdot J = 0$. All in all, we have the conductivity equation $$-\\nabla \\cdot \\sigma \\nabla v = 0$$ If the conductivity is constant, say one, this is simply the Laplace equation $-\\nabla^2 v = -\\Delta v = 0$. **p-Laplace equation** We consider a nonlinear Ohm's law: $$ J = -\\sigma |\\nabla v|^{p-2} \\nabla v $$ with finite parameter p > 1. The case p = 2 is the standard Ohm's law, and also the standard conductivity equation. Combining this with Kirchhoff's law we get the p-Laplace equation (with weight $\\sigma$): $$-\\nabla \\cdot (\\sigma |\\nabla v|^{p-2} \\nabla v) = 0$$ **Calderón's problem** The inverse problem of Calderón asks one to reconstruct the conductivity within the body by making boundary measurements: One prescribes either the current or the voltage on the boundary and measures the other one. These correspond to Neumann and Dirichlet boundary values for the equation - one sets the other and measures the other. This is a steady-state measurement, so one should wait for the current pattern to stabilise before making measurements. For the conductivity equation (p = 2 case) this problem has been studied, there are theoretical and computation results, and several applications, such as electrical impedance tomography which can be used to detect breast cancer, discover oil or image cracks in concrete. **Questions** Are there applications for the Calderón problem in the more general case where we need not have p = 2? That is, is there some medium on which the Ohm's law has the extra term $|\\nabla v|^{m}$ for some non-zero m, and where finding out the conductivity by non-invasive boundary measurements would be useful? Is there some other nonlinear Ohm's law that is more realistic in situations where electrical impedance tomography or similar imaging methods would be useful?"}
{"id":"100339","title":"How does the construction of a hydrophone differ from a microphone?","text":"Both devices serve the same function, but a hydrophone must do it in a denser medium. What's the major difference in design? (besides being water-proof!) I'm asking about the relevant physics driving design in this case. I think a physicist would be able to explain this so that a good mechanical engineer would be able to design microphones for use alternately in air, underwater, submerged in honey, or at 20,000 feet."}
{"id":"92049","title":"Generalized Unitarity cut of Scalar One-loop Box integral","text":"How does one perform the integrals in four particle cuts in generalized unitarity? It would be helpful how one finds solutions to the simplest case, the fully determined box integral given by: $\\int_{\\infty}^{\\infty} \\frac{d^4k}{(2 \\pi)^4} \\delta(k^2) \\delta((k+p1)^2)\\delta((k+p1+p2)^2)\\delta((k+p1+p2+p3)^2)$ Where the k and $p_i$ are 4-dimensional complex momenta, and the external $p_i$ are massless and on shell i.e., $p_i^2=0$. I know there exist two solutions to the problem, and it would be helpful to be expressed in terms of mandelstam variables s,t,u I'm sure you can take a jacobian of each delta argument expanded explicitly in terms of (k0,k1,...) but I feel there should be an easier solution"}
{"id":"22055","title":"What is the Quantum equivalent of chaos on a classical system? (if there's any)","text":"This is a question that bugging me around for some time now. It is not clear to me what is the meaning of a chaos if we consider a quantum system. What is the mathematical formalism (or the quantum analogy) to a orbit that is ergodic in the phase space? Can we still talk of a wave-function describing the particle? I mean, is it still well defined?"}
{"id":"86955","title":"Difference between a central potential that is a point and one that is a sphere?","text":"In quantum physics there is a special case known as a particle in a spherically symmetric potential. I have a problem which is similar to the case of a hydrogen atom in that there is one free electron, but the \"nucleus\" is actually a group of several atoms. So I'm not sure how to find the radial wavefunction for this situation? If I could say that it's a point charge I could use the same radial wavefunction as for the hydrogen atom (I think). But since the group of atoms takes up space, how does that affect the potential\/wavefunction?"}
{"id":"77716","title":"Moving the plates of a charged capacitor to calculate energy density - where's the flaw in my argument?","text":"For a charged air-dielectric capacitor, let the plates be parallel to the $xy$ plane, with the top carrying a positive charge $+Q$, the bottom a charge $-Q$. The force on an infinitesimal charge contained in a volume $dv$ inside the top plate is $$-\\vec E_z\\rho(x,y,z)dv$$ Integrating this over the volume of the plate gives the total force on its center of mass as $$-\\vec E_z Q$$ which becomes using Gauss's law $$-\\hat n_z E^2A\\epsilon_0$$ If the plates are moved apart a distance $dz$ by an opposing force $\\vec F$, the infinitesimal work done is $E^2\\epsilon_0Adz$, and integrating this from zero to their final separation $d$ gives the energy density as $E^2\\epsilon_0$. But this is different to the correct value of $1\/2E^2\\epsilon_0$, so where's the flaw in the above argument?"}
{"id":"72723","title":"Compressed Spring Dissolving in Acid","text":"This is just an interesting question a friend's uncle asked me that I was somewhat annoyed I couldn't answer. When a material dissolves in acid there is a chemical process that causes the changes in state\/composition of the materials in question. There will be a change in energy due to chemical bonds forming\/breaking and potentially a change in temperature. So we take a spring and compress it in a clamp. Obviously now there is a potential energy based on the compression of the spring as well as the chemical energy within the spring itself before compression. This is now placed into a bath of acid that will dissolve the spring but _not_ the clamp. There is a change of energy based on the dissolving of the spring, but **what happens to the potential energy due to it being compressed?** My thoughts were that in a realistic situation one (weak) point on the spring would be attacked, break, it would then probably just ping out of the clamp and lose the energy that way. But in the ideal scenario where is dissolves at a uniform rate, this wouldn't happen, so what happens at the instant that it is completely dissolved? This is obviously a massive assumption, so would be unlikely to be true. I thought maybe some kind of change of temperature would occur, due to the assumption that the compression would cause the atoms to be more tightly compacted causing a repulsion. As you remove layers this would gradually decrease, but there is still an inherent increase in energy for each atom, so this would most likely be expelled as something (like) heat. What are people's thoughts on this? **EDIT:** I'm mainly interested in the detail of the mechanisms for the energy tranfers on an atomic level as opposed to the broad picture. Specifically the transfer from coulomb repulsion to heat in the acid. The more low-level detail the better."}
{"id":"76267","title":"The conservation of energy when a spring is placed in acid","text":"Stretch a spring out and give it a bunch of potential energy. Lower the spring into a vat of acid, disolving the spring into nothingness. Where did the potential energy go?"}
{"id":"93314","title":"How is a standard unit divided into equally smaller or fractional units physically\/experimentally?","text":"Consider the standard unit of length: meter. How was it divided into decimeter, centimeter, millimeter, etc. when there were no shorter lengths than the standard? What is the physical\/experimental process involved?"}
{"id":"93316","title":"Circuit with inductor and lightbulb when switch is turned off?","text":"I want to know what happens with a lightbulb immediately after a switch is turned off. We're talking about a circuit with a battery (AC), an inductor, a lightbulb and a switch. I need to know it in case the inductor and lightbulb are in serie and in parallel. I know you can see the lightbulb as a resistor but i'm kinda lost on how to calculate this. Help would be much appreciated! I also need to draw the current in function of the time. Will this be a wave?"}
{"id":"93310","title":"What is the difference between the words transparent and translucent?","text":"Merriam Webster defines **_transparent_** as: > Having the property of transmitting light without appreciable scattering so > that bodies lying beyond are seen clearly. And **_translucent_** as: > Transmitting and diffusing light so that objects beyond cannot be seen > clearly. Now if you see any object through a lens or a bottle filled with water then most of the times whatever you see is not at all clear. What does this signify? Are these two transparent or translucent? Or is their behaviour conditional to how we see through them?"}
{"id":"93319","title":"Can Hubble red shift be interpreted as time dilation?","text":"Can we interpret the de Sitter universe as a spherical cosmic horizon null surface of finite radius, centered at Earth, and containing the Hubble volume of space where time is dilated and radial dimensions contract closer to the edge in such a way that objects closer to the edge do not recognize that they are radially contracted? Everything is attracted to the edge, but the total radius remains more or less constant and emits de Sitter radiation at finite temperature. ![enter image description here](http:\/\/i.stack.imgur.com\/xgtGt.png)"}
{"id":"5170","title":"What happens when you cut a biconvex lens in half?","text":"Specifically, does the focal length change? How can this be rationalized?"}
{"id":"122642","title":"One question about renormalization","text":"The idea of renormalization of \"naked\" perturbation theory is in principal possibility of addition counterterms which reduce infinity when calculating matrix elements. But I have met such concepts as exact propagator and the exact vertex part. So, the question: are exact propagator and vertex part collected all countreterms of our theory (i.e. all countreterms are convoluted into expressions which are called exact propagators and vertex)? I.e., do they (exact propagators and vertes) contain all information about renormalization of the given theory?"}
{"id":"122649","title":"Block on an inclined plane","text":"If you take moments about the centre of mass of a block positioned on an inclined plane so that the gravitational force can be drawn from the centre of mass of the block to one corner of the block, why is there no turning effect due to the reaction?"}
{"id":"26196","title":"Why is the universe so organized?","text":"If you think about the Big Bang and the flow of matter in all directions, you would think, how unorganized would this universe be? No matter how long it would take. The idea that matter or most of it will be organized in such a beautiful way is unthinkable. Check this for example. The Moon orbits around our planet in a perfect fashion, not one but millions of system are like this. Our planet circles the Sun, again there are millions of examples like it. Our star circle the galaxy center in a predefined path. Again billions of stars do this! Now if you look at the matter that is organized (has found fixed orbit) vs that that is not, almost 90% of the matter is organized. They have fixed trajectories and have a stable path. In fact, if you want to look at matter that is not unorganized, you will find very little (we can think of our own solar system). Now there is not one galaxy but billions of them, all of them similar shape, like or spiral, which has stars. Outside this galaxy, there are few stars or maybe a supernova somewhere. Why and how is our universe so organized? Is the law incredibly simple, since the law has been repeated billions of times in every galaxy, or is it the work of some phenomenon of which the details are not yet known?"}
{"id":"70298","title":"On Bell inequality and bound entangled states","text":"I have recently seen some presentation slides of Michał Horodecki (slide number 77) in which he discussed the following conjecture. > Bound entangled states satisfy all Bell inequalities The conjecture is not true for multi-partite systems, and the references are given in the talk itself. I believe by now, the bipartite case is also settled, one way or the other. Bell theorem has some applications in computer science areas, like cryptography (non-local games). Is there some similar application of the (falsification of) the above conjecture? I mean, what are the interesting applications which can take place by disproving the above conjecture? Advanced thanks for any help\/suggestions. **To the Moderators:** I do not know, whether I should have asked this question in Physics or in Cstheory. Considering the fact that there can be physical applications also, I am posted the question here. Please suggest and\/or take the required actions."}
{"id":"70292","title":"Organs & Oscillations: An Analysis on the Temperature Dynamics of Solids","text":"Does temperature have an influence on the frequency of an oscillating organ pipe?"}
{"id":"70291","title":"How do you calculate vortex shedding frequency?","text":"I am attempting to try to find out if there is any effect of Von Karman vortices on a group of wind speed readings where it is presumed that due to a mountain nearby the data collection spot Von Karman vortices were formed which would register in the data. Based on what I have read the vortex shedding frequency which should be present in the data is simply calculated from the formula $F= \\mathbb{St}\\cdot \\frac{V}{L}$ with $\\mathbb{St}$ as the Strouhal number, $V$ as the velocity and $L$ as the characteristic length. From what I understand the Strouhal number is essentially constant for a given flow, but when I looked up how to find the Strouhal number it was defined as $\\mathbb{St} = F\\cdot \\frac{L}V$, which is the exact same relationship as given in the first equation for the computation of the shedding frequency. My question then is in two parts: 1) How do I calculate the Strouhal number for a flow so that I can calculate the shedding frequency? 2) What should I use as my characteristic length for a landscape feature such as a mountain or cliff etc as everything online tells only in the case of uniform cylinders and using their diameter instead of something like a cone which can be used as an approximation of a mountain or other landmass. thank you for any help."}
{"id":"128311","title":"Rotational inertia of a ball","text":"This question refers to the solution of problem 12 here. It involves a spherical shell of mass $M$ filled with frictionless fluid of mass $M$ rolling down an inclined plane. (This is problem 12 of the 2013 F=Ma exam, a high school competitive physics exam). I understand the solution, but I don't understand how the given rotational inertia is obtained. We have a spherical shell of mass $M$ filled with frictionless fluid of mass $M$. The rotational inertia is apparently $$I_{contact} = \\frac{2}{3}MR^2 + MR^2 + MR^2$$ Why? In particular, why is the $MR^2$ term added twice?"}
{"id":"45839","title":"Why are Navier-Stokes equations needed?","text":"Can't we picture air or water molecules individually? Then, why are Navier- Stokes equations needed, after all? Can't we just aggregate individual ones? Or is it computationally difficult, or inefficient to generate the picture of whole flow?"}
{"id":"87330","title":"Do glass beads show any piezoelectric property?","text":"Do glass beads show any piezoelectric property? Since glass beads is mostly made out of SiO2 which is piezoelectric, will it show any piezoelectric property. Thanks. If anything wrong with my question kindly please tell me. Thanks."}
{"id":"45834","title":"How is the topological $Z_2$ invariant related to the Chern number? (e.g. for a topological insulator)","text":"This question relates to the $Z_2$ invariant defined e.g. for topological insulators: Is it correct to relate $Z_2$ = 1 to an odd Chern number and $Z_2$ = 0 to an even Chern number? If yes, is it also correct to think of an even or odd Chern number in terms of an even or odd number of band crossings across the Fermi energy? (If it's odd, there must be a band connecting the valence to the conduction band and therefore provide a topological protected surface state.) * * * Edit: These lecture notes* (under Point H) state: \"The formula (49) was not the first definition of the two-dimensional Z2 invariant, as the original Kane-Mele paper gave a definition based on counting of zeros of the “Pfaffian bundle” of wavefunctions. However, (49) is both easier to connect to the IQHE and easier to implement numerically.\" and furthermore: \"...and the Chern numbers of the two spheres are equal so that the total Chern number is zero. The above argument establishes that the two values of the Z2invariant are related to even or odd Chern number of a band pair on half the Brillouin zone.\"   * http:\/\/socrates.berkeley.edu\/~jemoore\/Moore_group,_UC_Berkeley\/Welcome_to_the_Moore_group,_Department_of_Physics,_UC_Berkeley_files\/mitcourse.pdf"}
{"id":"15521","title":"Can hair be used as an dielectric?","text":"I was letting my mind freewheel, and it occurred to me that hair, containing keratin, is a polymer. It is also capable of acquiring and maintaining a charge as happens, on occasion, when brushed. So ...   1. Can hair be used as a dielectric in a capacitor?   2. Do any capacitors\/manufacturers use hair as an dielectric already ?"}
{"id":"30664","title":"The meaning of Goldstone boson equivalence theorem","text":"The Goldstone boson equivalence theorem tells us that the amplitude for emission\/absorption of a longitudinally polarized gauge boson is equal to the amplitude for emission\/absorption of the corresponding Goldstone boson at high energy. I'm wondering what's the physical meaning of this theorem. Is there any relation between equivalence theorem and Higgs mechanism ?"}
{"id":"38094","title":"possible values of quantity in matrix mechanics","text":"Suppose that there is a quantity described by the matrix as the following: $M = \\begin{bmatrix} 3 & 0 & -i \\\\\\ 0 & 1 & 0 \\\\\\ i & 0 &3\\end{bmatrix}$ How we determine possible values of this quantity? (According to the book, it says that some possible values are 1,2 and 4, but it does not say how it determined this.)"}
{"id":"13806","title":"Why did my liquid soda freeze once I pulled it out of the fridge and opened it?","text":"This isn't a duplicate to \"Why did my liquid soda freeze once I pulled it out of the fridge?\". My question is why soda froze after it was opened. Opening a can or bottle seems to have a larger effect than just jostling it. Is it because of the disturbance noted in the previous question? Is it related to the pressure decrease? Is it because of the release of some CO2 when it was opened?"}
{"id":"89257","title":"Exact diagonalization by Bogoliubov transformation","text":"I am developing a model of multiple gaps in a square lattice. I simplified the associated Hamiltonian to make it quadratic. In this approximation it is given by, $$ H = \\begin{pmatrix} \\xi_\\mathbf{k} & -\\sigma U_1 & -U_2 & -U_2\\\\\\ -\\sigma U_1 & \\xi_{\\mathbf{k}+(\\pi,\\pi)} & 0 & 0\\\\\\ \\- U_2 & 0 & \\xi_{\\mathbf{k}+(\\pi\/2,0)} & 0\\\\\\ \\- U_2 & 0 & 0 & \\xi_{\\mathbf{k}+(0,\\pi\/2)} \\end{pmatrix} $$ And my Nambu operator is given by, $$ ψ_\\mathbf{k} = \\begin{pmatrix} c_{\\mathbf{k},\\sigma} \\\\\\ c_{\\mathbf{k}+(\\pi,\\pi),\\sigma} \\\\\\ c_{\\mathbf{k}+(\\pi\/2,0),\\sigma} \\\\\\ c_{\\mathbf{k}+(0,\\pi\/2),\\sigma} \\end{pmatrix} $$ I tried to diagonalized by making three Bogoliubov transformations, the first to diagonalize the upper right submatrix of H, and then the two others (a sort of nested transformations). But I get a lengthy result, what I would like to know if there is a smart transformation which allows me to write $$ H = A_1^\\dagger A_2^\\dagger A_3^\\dagger D A_3 A_2 A_1 $$ or simply $$ H = U^\\dagger D U $$ Or the only way is to use just brute force? My question may be rephrase as, is there any elegant representation or expression of the diagonalization as for example the one obtained in BCS Superconductivity? Thanks"}
{"id":"89250","title":"Equilibrium - uniform circular motion","text":"Maybe this is a bit of a silly question, but let us pretend we have a pendulum in a ideal universe with no friction, drag, or anomalous forces there to affect it. Additionally, our pendulum is subjected to an ideal gravitation force, the force vector of which is unaffected by the position of the ball (hence the responsible gravitating body is so massive that its own spherical geometry does not affect the forces on the pendulum-ball. Now we have our pendulum spin around so that it experiences uniform circular motion and some angle theta is formed between the resultant orientation of the spinning pendulum string and its previous stationary form. My question is, is this system in equilibrium? I know that no work is being done even though there is a force. And that from a relativistic (non-Newtonian) point of view, there is no centrifugal force to cancel the centripetal. So please, could you explain the \"answer\" from a Newtonian (high-school physics) POV and a relativistic POV."}
{"id":"11520","title":"How viable is reuse of nuclear waste at this point, and what are future prospects?","text":"I keep seeing documentaries describing the pile-up of nuclear waste from nuclear power stations in the US and in other countries. Countries such as Finland are working to deposit waste in old mines in geologically stable regions, and I was under the impression that the US was heading down that track as well, but this has now apparently been put on hold. A case of NIMBY I presume (not in my back yard). The media story is that the waste is piling up, and is being stored in pools at nuclear power stations all over the US - which sounds dangerous considering the prevalence of natural disasters. I wonder what the real truth is? I am not quite sure what the situation is in France and England, but I got the impression they are experimenting with methods of re-use of the waste products as well as storing some material under ground. What is the current state-of-the-art in terms of nuclear waste reuse? Are new reactors capable of burning discarded fuel from older reactors? Are there realistic expectations to be able to eliminate deposition of nuclear waste altogether in the foreseeable future?"}
{"id":"96590","title":"Are derivative indices summed in indicial notation?","text":"A paper I'm reading uses indicial notation and the convention that $u_{j,k}$ means the derivative with respect to $x_k$. Which one of these interpretations are correct? $$A_{ijk}u_{j,k} = \\sum_{j}A_{ijk} \\frac{\\delta u_j}{\\delta x_k}$$ $$A_{ijk}u_{j,k} = \\sum_{j,k}A_{ijk} \\frac{\\delta u_j}{\\delta x_k}$$"}
{"id":"89350","title":"When Venus is the brightest when viewed from Earth?","text":"Nowadays Venus is very bright. I can spot it during broad daylight without problem. It's because it's near Earth and appear as a crescent. This made me think: as it's reaching the inferior conjunction the crescent becomes slimmer, so I expect that it will fade a bit. So there should be an optimal phase angle for optimal brightness. So my question is at which phase angle Venus is the brightest when viewed from the Earth?"}
{"id":"96595","title":"Gravity and objects at 0K","text":"Let's say I put 2 objects at rest relative to each other with 0 Kelvin. Will they move towards each other because of gravity? Where they will get energy to move? I don't have education in this field, just wondering."}
{"id":"65129","title":"Intuition contradicting answer for calculating pressure","text":"I have a cylindrical container say of some base area $A$. The atmospheric pressure is $P_0$. The cylinder is moved at an acceleration $a$ horizontally right. I need to find which base would have to face more internal pressure. The cylinder is filled with some ideal gas. My intuition, even though I doubt it, says that in the frame of the cylinder the fictitious forces act towards left, so it should be the left face. But on some calculations using the laws of motion, I get equations as $$(P_0-P_1)A=ma$$ $$(P_2-P_0)A=ma$$   ($m$ is the mass of each base of cylinder) On solving I have $$P_1=P_0-{ma\\over A},P_2=P_0+{ma\\over A}$$ which means the right face faces more internal pressure. What is going wrong? P.S. just give me a hint. ![enter image description here](http:\/\/i.stack.imgur.com\/3euyA.png)"}
{"id":"87494","title":"Difference between primitive unit cell and the associated basis?","text":"As I understand it, the basis is the group of atoms whilst the primitive unit cell is the unit space that fits the total space without any gaps, and only containing one lattice point? How do the two relate to each other? Thanks."}
{"id":"82461","title":"Volume quotient in Carnot-cycle","text":"Problem: > One kilomole of an ideal, monatomic gas undergoes a reversible Carnot- > processes between temperatures 300 °C and 20 °C. The work done during one > cycle is 1500 kJ. > > a) Find the entropy-change in every process and show that total sum of > entropy-changes is zero > > b) What is the ratio between the largest and smallest volume that the gas > takes on during the whole process? I have already done a), it is b) I am struggling with. I realized that the greatest and smallest volume is assumed in the adiabatic processes of the Carnot-cycle, so I applied $T_1V_1^{\\gamma -1} = T_1V_1^{\\gamma -1}$ but to no avail. How can one do it?"}
{"id":"82460","title":"What limits laser pulse duration to single cycle?","text":"1. Why pulses shorter than single cycle cannot propagate in free space ? Is there fundamental reason or it is engineering ?   2. Is it possible for a laser pulse to have bandwidth that can support half cycle pulse and how it will propagate in free space ?"}
{"id":"103521","title":"Is the universe a perpetual motion machine?","text":"Is the universe a perpetual motion machine?"}
{"id":"64658","title":"Centripetal Force Acceleration","text":"In uniform circular motion, acceleration is $\\frac{v^2}{r}$ and time which it acts $\\rightarrow 0$. So $\\Delta v = 0$, but then why\/how does direction change, when the acceleration should be producing no effect at all ."}
{"id":"87107","title":"Why don't metals bond when touched together?","text":"It is my understanding that metals are a crystal lattice of ions, held together by delocalized electrons, which move freely through the lattice (and conduct electricity, heat, etc.). If two pieces of the same metal are touched together, why don't they bond? It seems to me the delocalized electrons would move from one metal to the other, and extend the bond, holding the two pieces together. If the electrons don't move freely from one piece to the other, why would this not happen when a current is applied (through the two pieces)?"}
{"id":"64650","title":"Physics of the point of contact for a spinning top","text":"I understand how spinning tops don't tip over, cf. e.g. this and this Phys.SE questions. What I'm more interested is in identifying the factors that determine the direction the spinning top moves to?"}
{"id":"121853","title":"Interpretations of quantum mechanics","text":"It is well known that there are many interpretations of quantum mechanics. I'm wondering if there is a specific reason why the Copenhagen interpretation is the most popular. Why is it that the wavefunction interpreted as a probability distribution is such a useful description. This question came to me when I was trying to write a paper and got a strange wavefunction (it wasn't normalizable, had singularities etc.). I'm a mathematical physics grad student so I was usually more concerned with mathematical elegance than physical interpretations, but this question has been bothering me."}
{"id":"98988","title":"Relativity of simultaneity and cause-effect","text":"Relativity says that two events simultaneous in one reference frame might not be simultaneous in another reference frame. Can we extend this idea a little and say that the order of two events might be be reversed in going from one reference frame to another? And if so, what if they have a cause-effect relation ship in the first frame. Isn't reversing the two in the second frame a contradiction?"}
{"id":"99316","title":"Tadpole symmetry factor","text":"Can someone help me with symmetry factor of one-loop tadpole diagram (one loop correction to one point Green function in phi-3 theory)?"}
{"id":"19959","title":"Am I talking about the correct units here?","text":"Paolo made a move I knew from dozens of townie soccer games played under lights on the fenced-in macadam pitch next to the elementary school in Badalucco, packed tight between Via Cristoforo Colombo and the bed of the Argentina River. Paolo twisted up his left leg; rotated that leg in an entirely autonomous and anatomically-impossible split-second burst of acetylcholine; power-rammed, with his left foot, the front bumper of our white cheese-tour van. A van, it now became clear, that Paolo had left precariously in neutral. With the parking brake disengaged. Tilted up along the road’s seriously steep grade. Its rear wheels inches from the verge, a sharp drop to the curve below. His cousin’s van, I remembered, as it lurched backward and gathered speed. Here’s the thing, often under-appreciated, about translational kinetic energy:   ET=(1\/2)mv2 I’m quite certain that Hugo failed to fully apprehend the ramifications of this formula, given our weighty cheese van and his own position vis-à-vis an initially lumbering, but definitely accelerating, object. Perhaps Hugo had forgotten his boyhood physics class with its repeated exhortation that v = v0 + at.   Gravity, Hugo, gravity. 9.8 m\/s² Hugo. But, most important: mass, Hugo. Mass. The van was heavy, no doubt about it, all by itself. Then you add all that luggage. Plus one 100 pound wheel of Parmigiano Reggiano. Plus Jack’s, or Lauren’s, or whoever’s six boxes of Canopic Cookie Jars. Plus whatever illicit items the Arbogasts had inserted into their collection. Plus nearly two weeks of accumulated detritus dumped on the floor by people who’ve spent way too many van-hours for the niceties of proper trash disposal to hold much sway. Plus Lauren. Maybe it was Lauren of whom Hugo was thinking when he jumped not away from but toward the rapidly-accelerating van, its front passenger door still gaping wide; a van that had by then about reached its tipping-point on the steep-shouldered roadside. Maybe it was Lauren. Or maybe Hugo had played linebacker for Texas A & M, back in the day, and still figured he was on top of his game, good to go, stop anything coming at him. But this was no 250 pound running back, Hugo. This was not even a steroidally-enhanced 350 pound running back. No. This was a stout door winged out from a 3000 kilogram white Fiat Ducato made weightier by its 20 gallon auxiliary tank full of used grease. So, let’s say three seconds elapsed between the moment when Paolo’s kick overcame the van’s inertia (and started ‘er rolling) and the instant when Hugo attempted to stop her. For ease of calculation, let’s put the inclined plane on which the Ducato rested at 45 degrees, i.e., pretty steep, and thus the parallel vector of gravity pulling her downhill at about 4.9 meters per second per second. After 3 seconds and change (we’ll ignore the rolling friction of axels and rubber tires) the Ducato would be chugging along at close to 15 meters per second. I think. Over 30 mph. Not so shabby. Which brings us back to ET=(1\/2)mv2. 3000 kg at 15 meters per second. At the instant of contact, a direct hit would work on a firmly-planted Hugo with something in the vicinity of 337,500 Joules. Fortunately for Hugo, he was not planted, immovable, like the high-security bollard he believed himself to be, but more along the lines of a wobbly bowling pin. The door dealt Hugo a split-second glancing blow. What we probably need to figure here, as in football, is impact force per unit area. Newtons\/m2. Stress. The specifics of this inelastic collision. Consult the law of cosines, maybe? You do the math, Hugo. Hugo was not going to be doing any math for quite a while. More or less simultaneously: I saw the open door plow into a Hugo determined to impede its progress. I heard the loudest noise I’d ever experienced. I saw the windshield explode in frost. I saw Paolo drop from sight. I heard Lauren scream. Then: I saw Hugo go down. I remembered Phil and Massimo. Under the backseat. Gliding toward oblivion.   Which must be when I made my own ill-advised lunge toward the driver’s seat, into the path of the open driver’s side door. A split-second later, as Pippa howled “Sybilla, don’t!” must be when I stumbled over a loose piece of rubble and myself went down. I’m told the bottom of the door just grazed my face. But, as noted, at 15 meters per second and rising, even a graze by a three ton vehicle can hit you with lots of Ns."}
{"id":"53716","title":"The things to consider when choosing a mesh for an electron detector (MCP detector)","text":"I am designing a electron velocity map imaging setup. I need a copper mesh just infront of the detector to accelerate the electrons towards the MCP detector. I am doing the instrument design for the first time. Can anybody please tell me how to choose a mesh and what are the things should I keep in mind while choosing a mesh for the detector (regarding the number of wires per inch, its transmission etc.)"}
{"id":"90335","title":"Conservation of Mathematical Constraints when deriving Energy and Momentum from $F=ma$","text":"**Background:**   * Starting from $F = ma$, integrating with respect to time, and using basic calc, one can derive $\\int Fdt = m (v_f - v_i)$   * Starting from $F = ma$, integrating with respect to distance, and substituting $a\\ ds = v\\ dv$ (from calculus), one can derive $\\int Fdx = KE_f - KE_i$ (Obviously I care about these results because, when combined with $F_{ab} = -F_{ba}$, they give conservation of momentum and a starting point for conservation of energy, but for this question I'm not going to consider this part of the derivation.) In 1-D, the two bulleted results make sense to me: the starting point is a 2nd-order ODE, and the two results form a coupled system of two 1st-order ODEs. This is precisely what math says should be possible - the number of mathematical constraints has been conserved. **Question:** A similar (slightly more involved) derivation is possible in 3 dimensions, but it's harder for me to classify the resulting mathematical constraints and reassure myself that the transformation doesn't add or remove constraints:   * Starting Point: $\\vec{F}=m\\ddot{\\vec{s}}\\ \\rightarrow$ 3 coupled second-order ODES    * Result 1: $\\int \\vec{F}dt = m(\\dot{\\vec{s}_f} -\\dot{\\vec{s}_i})\\ \\rightarrow$ 3 coupled first-order ODEs (viewing initial state as a boundary condition)   * Result 2: $\\int \\vec{F}\\cdot d\\vec{s} = \\frac{1}{2}m(\\dot{\\vec{s}_f} \\cdot \\dot{\\vec{s}_f}) - KE_i\\ \\rightarrow $ Unsure how to classify this      * Rewriting this as $\\int \\vec{F}(x,y,z)\\cdot d\\vec{s} = \\frac{1}{2}m(\\dot{x}_f^2 +\\dot{y}_f^2 +\\dot{z}_f^2 ) - KE_i $ makes it clearer that this is a single non-linear ODE with the first derivatives of 3 different dependent variables in it.  Math says that a system of 3 coupled second-order ODEs can be rewritten as a system of 6 coupled first-order ODEs, but I obviously only have 4 equations. **What type of ODE is Result 2?** A single first-order ODE? A 'triple' first- order ODE with 3 dependent variables? I'm ultimately looking to reassure myself that commuting the problem 'solve $F=ma$' to the problem 'solve conservation of momentum and conservation of energy' doesn't add or remove mathematical constraints. If anyone can refer me to a textbook which addresses this idea, I'd appreciate that as well. **Edit:** On further consideration, I think I've made a mistake. $F=ma$ is a differential equation with time as the independent variable, so integrating with respect to time (as is necessary to derive conservation of momentum) changes the order, and it's not actually reasonable to expect two coupled first-order differential equations as a result any more. In fact, we could integrate Conservation of Momentum wrt $t$ again, and then we'd no longer have a differential equation at all. Since integrating is a valid technique for solving differential equations, it now kind of appears as though Conservation of Momentum could be viewed as a [\/the] solution to $F=ma$, and could thus be expected to describes all the information which is present in the original ODE.   * That would make the $F=ma$-to-Conservation of Momentum derivation logical (from a Conservation of Constraints perspective) in both 1-D and 3-D, and then make the real question 'Where Conservation of Energy come from, from a constraints perspective?'   * On the other hand, the 1-D elastic collision problem can't be solved with Conservation of Momentum alone - Conservation of Energy is also required. Since Conservation of Energy can be derived from $F=ma$ and calculus alone, it must be representing a constraint which is in the original equation. Thus $F=ma$ holds more information than Conservation of Momentum. **Other things I've been thinking about:**   * This question may actually relate to integro-differential equations, which I know nothing about.    * There may be some subtleties related to $F$ which I'm missing, and writing it as $\\vec{F}$ instead of $\\vec{F}(\\vec{s},\\dot{\\vec{s}},t)$ may be sweeping them under the carpet."}
{"id":"54842","title":"Young's Double Slit experiment question","text":"> _Q-A beam of light consisting of two wavelenghts 600 nm and 450 nm is used > to obtain interference in Young's Double Slit experiment (YDSE). Find the > least distance from the central maximum where the bright fringes due to both > wavelengths coincide. The distance between the two slits is 0.4 mm and > screen is placed at a distance of 1.0 m from the slits._ Now my problem is not with the question but the solution given in my book. It says > _Let at distance $x$ from central bright maxima, the bright fringes due to > both the wavelengths coincide for the first time. It is possible only if > within the distance $x$ there are $n$ fringes for light of 600nm and $(n+1)$ > fringes for light of wavelength 450 nm._ Okay so my problem is that I am unable to figure out how have they made this argument about $n$ and $n+1$ fringes? It's isn't mention anywhere in the question. Can't it be ($n$+any number)? If there's any better solution than this I would be really thankful for that"}
{"id":"64386","title":"Does a guitar sound different in zero (or micro) gravity?","text":"Seeing a video of astronaut Chris Hadfield playing a guitar on the International Space Station made me wonder if a guitar or other stringed instrument played in zero-G would sound any different than on earth. It seems that on earth, the downward pull of gravity could cause an asymmetrical oscillation of the guitar string, with a larger amplitude as the string moves downward due to the pull of gravity. (of course, it wouldn't take zero-G to test this since a guitar could be held vertically) With the amount of tension that a guitar string is under, perhaps this effect is so minuscule so as to be unnoticeable?"}
{"id":"46808","title":"Design and use of the RLC model for signal processing systems and spatial acoustics","text":"Given a frequency response for a signal processing device, set of devices, or physical space it is possible to construct an equivalent (identical frequency response) series RLC circuit. 1) Does this offer information about the system that the frequency response in and of itself does not? 2) Is the series model the most appropriate model, or is some other configuration more informative? Many thanks in advance Joe"}
{"id":"68403","title":"Momentum of a particle?","text":"I really need help to understand what is momentum of a particle (of a photon, proton, an electron...) I see so many definitions! My main questions are: •What exactly is momentum •What are the differences between momentum and spin of a quantum particle. •What is the relation between momentum of a photon and it's wave. I just wanted simple explanations. Because after months hearing about momentum, I still don't know what is it."}
{"id":"13385","title":"Practical physics: where does this current come from?","text":"It's been a while since school and I don't remember the right laws and formulas. So I come here. :) I've got a power cord extender running from the wall outlet to my washing machine. It's about 3 or 4 meters in length. The outlet supplies 220V, but no grounding. The extender however does have grounding support, as does the washing machine. This means that the extender has three wires in it, and the grounding wire terminates at the outlet. When plug in one end of the outlet, and measure the AC voltage between the grounding (disconnected) and zero wires, it gives me about 60V. Which happens to be unpleasant enough when accidentally touching the washing machine (even when it's turned off). **Considering that the cable is so short, can this really be explained by induction?** To rule out a few other theories: the wall outlet does not connect anything to the grounding wire, it really does terminate there. The grounding wire does not touch any of the other wires, the resistance between it and other wires is infinity (or rather, beyond the range of my cheap multimeter)."}
{"id":"48180","title":"How is the wattage of a human larynx calculated?","text":"This site says the typical human voice has a wattage of about 1W. Another site indicates it is 80db with about 0.001W of energy. I'm confused. How does one compare 80db of 0.001W against 1W? Most importantly how is the wattage of the human larynx calculated?"}
{"id":"67445","title":"Is Biot-Savart law obtained empirically or can it be derived?","text":"Well, there's already a question like this here so that my question could be considered duplicate, but I'll try to make my point clear. Is there a way to derive Biot-Savart law from the Lorentz' Force law or just from Maxwell's Equations? The point is that we usually defined based on experiments that the force felt by a moving charge on the presence of a magnetic field is $F = qv\\times B$, now can this be used in some way to obtain Biot-Savart law like we obtain the equation for the electric field directly from Coulomb's Force law? And I wanted to know that because as pointed out in the question I've mentioned, although Maxwell's Equations can be considered more fundamental, those equations are obtained after we know Coulomb's and Biot-Savart's laws, so if we start with Maxwell's Equations to obtain Biot-Savart's having use it to find Maxwell's Equations then I think we'll fall into a circular argument. In that case, without recoursing to Maxwell's Equations the only way to obtain Biot-Savart's law is through observations or can it be derived somehow?"}
{"id":"44407","title":"How would I work this continuity problem?","text":"I am trying to design a system for a project and I need some input on if this is right or not. I have $10 \\frac{kg}{s}$ of fluid flowing through a pipe. It breaks up into $10$ 1 inch diameter pipes. I am giving a picture below. Does this mean that the feed pipe, the one supplying the $10 \\frac{kg}{s}$, should be just 10 inches in diameter instead of 1 inch? In case your wondering, the fluid is ethylene glycol, I dont think that matters ![enter image description here](http:\/\/i.stack.imgur.com\/90Omj.jpg)"}
{"id":"112971","title":"Understanding the energy transffered\/used from a power source?","text":"A source of power(from a generator,battery,etc...) is used to power a system of any kind. We can calculate the energy used via this formula $E$ = $IV$ x $t$ ? For example, a battery that is connected to a conductor, that conductor converts electrical energy to heat. $E_B$ = $P$ x $t$ E - energy used from the batter. t - time the battery is connected to the circuit and current flows. Is this the only way to model or analyze the energy transferred?"}
{"id":"20187","title":"How fast do electrons travel in an atomic orbital?","text":"I am wondering how fast electrons travel inside of atomic electron orbitals. Surely there is a range of speeds? Is there a minimum speed? I am not asking about electron movement through a conductor."}
{"id":"79115","title":"Neutrinos travels through our bodies: do they contribute for gravitational atractive force?","text":"Are neutrinos traveling through our bodies what really make us attracted to the Earth?"}
{"id":"106425","title":"SPEED of LIGHT as photons reach black holes","text":"As photons reach a black hole and begin fall in - is their speed increased as they go down (like a vacuum cleaner does to dust)."}
{"id":"54150","title":"Small car colliding with large truck","text":"A small car collides with a large truck. Why do both vehicles experience the same magnitude of force? Wouldn't the large vehicle experience less force than the small one?"}
{"id":"28783","title":"Is there a direct relationship between an isotope's neutron count and radioactivity?","text":"In my textbook, it lists isotopes of Carbon: C-12, C-13, and C-14. It noted that C-14 is radioactive (C-12 and C-13 are not). Is there a direct relationship between the number of neutrons and an element's radioactivity? In other words, since we know C-14 is radioactive, does that mean we also know that C-15 (if such a thing exists) would also be radioactive?"}
{"id":"135325","title":"Giant leap of the space goat","text":"This problem was inspired by this Math.SE post and similar problems involving tethered animals. A space goat is tethered to the north pole of a spherical asteroid of uniform density and radius $a$ by a rope of length $\\pi a$. Becoming bored with grazing the surface, the goat tests the limits of its freedom by making a series of leaps. What is the longest time that it can remain airborne? The answer should be expressed as a multiple of the orbital period of an object circling the asteroid just above its surface. Assume that the goat’s strength is not a limiting factor, the asteroid is very much more massive than the goat, atmospheric resistance and the weight of the rope are negligible, and solutions involving any possibility of the rope dragging across the surface are non-optimal. Note: this is an original problem and I don’t yet have a complete solution. It looks doable but don’t necessarily expect a result in closed form. [Edit: added assumption about relative masses.]"}
{"id":"3294","title":"Big Bang snuffed by a black hole?","text":"Wasn't the density of the universe at the moment after the Big Bang so great as to create a black hole? If the answer is that the universe\/space-time can expand anyway what does it imply about what our universe looks like from the outside?"}
{"id":"66523","title":"Do these equations apply to an object on a zip line or slope?","text":"I was looking at the conversion equations for Potential Energy to Kinetic Energy to Velocity here: http:\/\/www.ftexploring.com\/energy\/PE-to-KE.html Would these apply to an object on a zip-line? What, if any, adjustments need to be made."}
{"id":"27632","title":"About the definition\/motivation\/properties of the twisted chiral superfield in ${\\cal N}=2$ theories in $1+1$ dimensions","text":"The following is in the context of the ${\\cal N}=2$ supersymmetry in $1+1$ dimensions - which is probably generically constructed as a reduction from the ${\\cal N}=1$ case in $3+1$ dimensions.   * In the $\\pm$ notation what is the definition of ${\\cal D}_+$ and ${\\cal D}_{-}$, which I understand from context to be the gauge covariant superderivatives. (..It would be great if someone can relate them to the usual definition in the notation of say Wess and Bagger..)   * So what is the meaning\/motivation of defining a twisted chiral superfield as, $\\Sigma = \\\\{\\bar{{\\cal D}}_{+}, {\\cal D}_{-}\\\\} $ (..naively this looks like an operator and not a field - I guess there is some way of arguing that the derivative terms which are not evaluated on something actually go to zero..) I am guessing that in the above context it will be helpful if someone can explain as to what is meant by the following decomposition\/reduction of the gauge field from $3+1$ dimensions, $\\sum _ {\\mu = 0}^3 A_\\mu dx^\\mu = \\sum _{\\mu =0} ^1 A_\\mu dy^\\mu + \\sigma (dy^2-idy^3) + \\bar{\\sigma}(dy^2+idy^3)$ ?   * From the above (does it\/how does it) follow that one can write $\\Sigma$ as,  $\\Sigma = \\sigma + \\theta\\lambda + \\theta \\bar{\\theta}(F+iD)$ (..where I am not sure if $F,D,\\sigma$ are real or complex scalar fields...and $\\lambda$ is a Weyl fermion..)   * What is the R-charge of this twisted chiral super field? (..from some consistency conditions I would think that its 2..but I am not sure..) I guess that the R-symmetry transformations act as,   * The \"right\" R symmetry keeps $\\theta^-$s invariant and maps, $\\theta^+ \\mapsto e^{i\\alpha}\\theta^+$, $\\bar{\\theta}^+ \\mapsto e^{-i\\alpha}\\bar{\\theta}^+$   * The \"left\" R-symmetry keeps $\\theta^+$ invariant and maps, $\\theta^- \\mapsto e^{-i\\alpha}\\theta^-$, $\\bar{\\theta}^- \\mapsto e^{i\\alpha}\\bar{\\theta}^+$. Though I am not sure and like to understand as to why one wants to think of these two different R-symmetry groups as having two different origins - one coming from the rotation symmetry of the two spatial dimensions of the original $\\cal{N}=1$, $1+3$ theory and another coming from R-symmetry of the $\\cal{N}=1$, $U(1)$ gauge theory."}
{"id":"51522","title":"Put a sensor at the focal length, behind, or in front?","text":"My intuition tell me that at the focal length a convex lens all the light converges to a point. Following that logic, it would make sense to me that a camera chip would either need to go slightly in front or slightly behind the focal point in order to expose the entire chip to the incoming light. This would produce a normal or inverted image, respectively. Upon investigating this question, I read through these articles:   1) Wikipedia - Focal Length   2) Paragon Press - Focal Length   3) The Photo Forum That all agree that the sensor MUST be placed AT the focal length, not in front or behind. **How does a chip placed at the focal length not just produce a white spot of light at the center?**"}
{"id":"51523","title":"Translation Invariance without Momentum Conservation?","text":"Instead of the actual gravitational force, in which the two masses enter symmetrically, consider something like $$\\vec F_{ab} = G\\frac{m_a m_b^2}{|\\vec r_a - \\vec r_b|^2}\\hat r_{ab}$$ where $\\vec F_{ab}$ is the force on particle $a$ due to particle $b$ and the units of $G$ have been adjusted. Whenever the masses are unequal, the forces are not equal and opposite, violating Newton's third law and conservation of momentum in the process. As momentum conservation has been violated, my understanding is that translation invariance should be violated as well by this force. But the force law still depends only on separations rather than absolute coordinates, so the physics seems to be translation invariant. What am I getting wrong?"}
{"id":"129210","title":"What are the implications for the AdS\/Cft program if AdS is unstable?","text":"To my understanding recent progress in the study of the non linear stability of AdS spacetime suggest that $AdS$ might be unstable. If this is true, what are the physical and mathematical implications for the $AdS\/Cft$ approach? The stability of an spacetime warranties that that sufficiently small perturbations remain small. In the case of Minkowski the stability theorems proved by Christodoulou and Klainerman show that sufficiently small perturbations not only remain small but decay to zero with time in any compact region (this stronger type ofstability is called asymptotic stability). In the case of $AdS$ numerical and certain results in nonlinear partial differential equations suggest the posibility that the following conjecture might be true: The $AdS_{d+1}$ space (for d ≥ 3) is unstable against the formation of a black hole for a large class of arbitrarily small perturbations. Now, the $AdS\/Cft$ correspondence is a conjectured relationship between two kinds of physical theories. On one side of the correspondence are conformal field theories (CFT) which are quantum field theories, including theories similar to the Yang–Mills theories that describe elementary particles. On the other side are Anti-de Sitter spaces (AdS) which are used in theories of quantum gravity, formulated in terms of string theory or M-theory. In this program there are some examples that one can relate the formation of black holes in the bulk to certain thermodynamic properties of the conformal field theory. In particular,the presence of black holes can be seen as some thermalization of the field theory. How would the results on stability be interpreted in the dual conformal picture? Is there some meaningful thermodynamical process that explains the instability at the classical level?"}
{"id":"74604","title":"Cylindrical lens aberration fixed by roll","text":"I have two cylindrical lenses essentially forming a beam expander. The problem that I am having is that I am seeing a very strange aberration. The image becomes \"squashed\". Cylindrical lenses only focus in one direction unlike spherical lenses so with a circular input beam a crisp vertical line should form at the focal point of the converging lens (If I just use the converging lens...no beam expander). But, at the focal point of just the converging lens instead of seeing a vertical line, the line is slanted. Now looking at the beam expander system the image begins as a nice crisp expanded circular beam but as I continue looking at the progression of the beam the image looks as if it is folding over itself and squashing into a horizontal line. If I add in some relative roll between the converging and diverging lenses The squashing is improved but not perfect. I wanted to know if this kind of aberration is common or if there is a defect in the lenses I am using. In the Zemax simulations this should not be happening. It might be important to note that the converging lens is essentially saturated. The input beam is collimated in such a way that the maximum sized circular beam is sent through the lens without clipping. ![enter image description here](http:\/\/i.stack.imgur.com\/sEzv8.png)"}
{"id":"111694","title":"Normal modes of two wires fastened together","text":"The problem is to find the normal frequencies of the system formed by two fastened wires of length L, and different mass per unit length. I already wrote the boundary conditions, but I need to know if the joint has to be a node. I'm pretty sure it has to, otherwise it wouldn't be in a normal mode, but I can't set it up straight in a rigorous way. The boundary conditions I imposed are: $y_1(0,t)=y_2(2L,t)=0$ $y_1(L,t)=y_2(L,t)$ $\\frac{\\partial y_1}{\\partial x}(L,t)=\\frac{\\partial y_2}{\\partial x}(L,t)$ Where $y_1(x,t)=A_1e^{i(k_1 x-\\omega t)}+B_1e^{i(-k_1 x-\\omega t)}$ and $y_2$ is written similarly. Thanks!"}
{"id":"111692","title":"Earnshaw Theorem for ionic solids","text":"Is a single molecule of sodium chloride (say) or a cluster of molecules of NaCl unstable, although macroscopically NaCl is in fact, stable? How can I reason this based on Earnshaw's theorem?"}
{"id":"27183","title":"Could motives aid in the study of the Navier-Stokes equations?","text":"Recently, mathematicians and theoretical physicists have been studying Quantum Field Theory (and renormalization in particular) by means of abstract geometrical objects called motives. Amongst these researchers are Marcolli, Connes, Kreimer and Konsani. You can read about Marcolli's work here*. Now, in the wikipedia article on the Navier-Stokes Equations, there is a short paragraph on Wyld Diagrams. It is stated there that they are similar to the Feynman diagrams studied in QFT. Since motives and other algebraic approaches are currently used to study these Feynman diagrams, I was wondering if these approaches could also aid in studying the Navier-Stokes equations. If so, how? And to what extend could they potentially help solving the (in)famous Millennium Problem? If not, why not? Please note that I'm far from an expert in any of these fields. Thanks in advance. * _(I would like to include more hyperlinks to the books and articles of the respective scientists, but since I am a new user I am incapable of doing so. You can find a lot more literature simply by looking up the names of these researchers.)_"}
{"id":"27182","title":"Does it make sense to speak of amplitudes of finite closed boundaries in QFT?","text":"A example of amplitude in Relativistic Quantum Mechanics or specifically in QFT is the amplitude of a field configuration on a space-like hyper-surface of space-time to \"lead\" to another field configuration on another space-like hyper-surface of space-time. In the path-integral picture one simply integrates over all possible field configurations on the interior, giving each a weight in the normal way. Now if one wants to generalize this to finite closed boundaries, we would get an amplitude for each field configuration on a finite closed boundary of space-time. but how would we interpret this? This question relates to interpretations of quantum mechanics, has anyone investigated this line ?"}
{"id":"119064","title":"Determining diameter of effective image circle of large lens in a Camera Obscura","text":"I am in the process of gathering materials and designing a large lensed Camera Obscura. I would like to estimate the approximate diameter of the image circle and the approximate field of view. in degrees, but all the formulae I have encountered require parameters I do not yet know. I am under the impression that a free lens has an intrinsic image circle diameter\/FoV. **—The lens** **Diameter:** 304.8mm (12.0\"in) **Focal Length:** 1143mm (45.0\"in) **Type:** Plano-convex glass **—The Room:** An inflatable inhabitable sphere with a hanging translucent screen about a meter from the lens mounted on the wall. I am using 'clear' Polyethylene sheeting as my ground-glass surface. I have built a lensed Camera Obscura before with a 150mm diameter 2000mm FL glass lens. Documentation of project: https:\/\/vimeo.com\/94692036 **—The Question:** How does one go about determining the Image circle\/field of view of a single lens?"}
{"id":"92203","title":"Most particles that have ever been modeled","text":"Two billiard balls collide. Consider the two balls only, how many particles are in that system? Have particle systems of that size been modeled with each individual particle before? What is the largest object ever been modeled with pure particle physics?"}
{"id":"55603","title":"Work being done without distance","text":"I have this issue: If you push a 40.0 kg crate at a constant speed of 1.40 m\/s across a horizontal floor (µk=0.25 ), at what rate (a) is work being done on the crate by you and (b) is the energy dissipated by the frictional force? For starters, here is what I have so far: RN = mg = 40*9.8 = 392 N Ff = muRN = 0.25*392 = 98 N KE = 0.5mv^2 = 0.5*40*(1.40)^2 = 39.2 J At this point, I don't know how to answer part (a). If W = FD , how do I find the work being done without a distance?"}
{"id":"32201","title":"Question about entropy and \"smart materials\" (that remember shape)","text":"We all know that entropy is the measure of chaos in the system, and it's always increasing in the system. Now comes my question - how does entropy work in smart materials? Here is a youtube video about smart material that I'm talking about, it remembers its shape - you can distort it, but when you heat it, it returns back to its original shape. So effectively it reverses from more chaotic state to a more uniform state: http:\/\/www.youtube.com\/watch?v=fsBHF_j2FJ4 Therefore my question - how come the system became less chaotic just by adding heat? It looks like entropy decreased, but the 2nd law of thermodynamics says that entropy can only increase."}
{"id":"60960","title":"How can the derivative of this trace be constrained?","text":"I am studying for my exam on relativity and I am going through some problems sets including ones where I was not very successful in so I want to know how to do this problem. > (Convergence of particles): Consider a freely falling reference particle in > vacuum as in Exercise 1. Assume now in addition that the Hessian matrix of > the gravitational potential does not vanish initially in the vicinity of the > reference particle. Recall that the displacement of a nearby particle $y(t)$ > is given in terms of its initial displacement $y(0)$ (we assume that the > particle is intitially at rest relativie to the reference particle) by > $$y(t) = A(t)y(0)$$ NOTE: Just to give some context we say that $y(t)$ is a displacement between two freely falling particles positioned $x_{0}(t)$ and $x(t)$ so $y(t) = x(t)-x_{0}(t)$. We can say that a nearby freely falling particle satisfies to first order in y the tidal equations, mainly we can see that: $$\\frac{\\mathrm{d}^{2}y}{\\mathrm{d}t^2}=-\\nabla^2\\psi(x_0(t))\\cdot{y(t)}.$$ Back to the problem. > We call $A$ the deformation matrix. Show that there is an initial > displacement $y(0)\\neq0$ and a time $t^*>0$ such that $y(t^*)=0$. > > Hint. Recall the definition of the strain matrix: > $$\\theta=\\frac{\\mathrm{d}A}{\\mathrm{d}t}A^{-1}.$$ Now to solve this the question breaks it down to several small parts. I was able to solve the first part so I will skip it. But here is the part I am having trouble with: > (2) Deduce from the tidal equation the following ordinary differential > inequality: > $$\\frac{\\mathrm{d}}{\\mathrm{d}t}\\operatorname{tr}\\theta\\leq-\\frac{1}{3}(\\operatorname{tr}\\theta)^2$$ > Hint: Decompose $\\theta$ into its diagonal and its traceless part. NOTE: We deduced from lecture a long time ago that $\\frac{\\mathrm{d}}{\\mathrm{d}t}\\operatorname{tr}\\theta=-|\\theta|^2-\\operatorname{tr}M$ where $M=\\nabla^2\\psi(x_0(t))$ or in other words $M$ is the hessian of the gravitational potential of the particles that are freely falling. We also know that $\\operatorname{tr}M=\\Delta\\psi(x_0(t))$ and in a vacuum this is equal to 0. So, concerning this problem I figure if we have to break down $\\theta$ to a diagonal part and a traceless part that means we can call the diagonal $\\sigma$ and the traceless part $\\omega$, such that $\\theta=\\sigma+\\omega$. That means that: \\begin{align} &\\frac{\\mathrm{d}}{\\mathrm{d}t}\\operatorname{tr}(\\sigma+\\omega) \\\\\\ &\\frac{\\mathrm{d}}{\\mathrm{d}t}\\operatorname{tr}(\\sigma) \\end{align} Since the trace of the omega is 0 but honestly I am not sure what more to do. I think I actually have to move that derivative and maybe I will be able to get that inequality."}
{"id":"56551","title":"Fourier Transform on a Riemannian Manifold","text":"The question is quite simple: What would be the definition of Fourier Transform (and it's inverse) on a Riemannian Manifold? I've found that a similar question has been asked at Mathematics.SE but that question was asking for a FT on a Lorentzian Manifold. The answer given is actually very good but then it gets a little too technical and quite cryptic to me... The first comment on that question actually points at the right direction but could someone give me an example on a specific Riemannian Manifold (aside Euclidean), for example on a 2-cylinder with metric $g=dz^2+r^2d\\theta^2$? **PS:** Maybe I should have posted the question on Mathematics.SE but I was hopping for the physicists more direct approach to these kind of problems..."}
{"id":"52877","title":"Dark Matter 'Stars'","text":"I'm aware that the Milky Way has a dark matter 'halo' around it, presumably a spherically symmetric distribution. But I'm completely ignorant regarding the theories explaining dark matter... Is there any reason to not expect a star-sized object to also be made of dark matter? I know they'll be extremely difficult to detect, but I'm wondering if it's even physically possible to exist."}
{"id":"127715","title":"What's the corresponding symmetry of enstrophy conservation?","text":"In fluid mechanics, especially 2D turbulence study, people talk about conservation of enstrophy. But I can't really understand enstrophy very well, and what's the corresponding symmetry of enstrophy conservation?"}
{"id":"127718","title":"Are there more distinctive names of \"null curves\" with certain additional properties (absence of \"chord curves\")?","text":"In this answer (to the question \" _In general relativity, are light-like curves light-like geodesics?_ \", PSE\/q\/76170) a particular example of a curve is discussed whose \" _tangent is everywhere null_ \" and which is therefore called a \" _null curve_ \". I'll restate the example curve explicitly as $$\\nu : \\mathbb R \\rightarrow \\cal M,$$ together with a coordinate function $$\\mathbf r : \\cal M \\rightarrow \\mathbb R^{1,2}; \\qquad \\mathbf r := \\\\{~t,~x,~y~\\\\}$$ such that $$\\mathbf r \\circ \\nu[~\\lambda~] := \\\\{~t_{\\nu}[~\\lambda~],~x_{\\nu}[~\\lambda~],~y_{\\nu}[~\\lambda~]~\\\\} = \\\\{~\\lambda,~\\text{Cos}[~\\lambda~],~\\text{Sin}[~\\lambda~]~\\\\}.$$ The calculation of the corresponding \" _tangent_ \" value proceeds (roughly) via $$\\begin{align} \\left( \\frac{d}{d\\lambda}[~t_{\\nu}[~\\lambda~]~] \\right)^2 - \\left( \\frac{d}{d\\lambda}[~x_{\\nu}[~\\lambda~]~] \\right)^2 - \\left( \\frac{d}{d\\lambda}[~y_{\\nu}[~\\lambda~]~] \\right)^2 & = & \\\\\\ 1 - \\left( -\\text{Sin}[~\\lambda~] \\right)^2 - \\left( \\text{Cos}[~\\lambda~] \\right)^2 & = 0\\end{align}$$ for all values $\\lambda$ in the domain of \" _null curve_ \" $\\nu$. Now, interestingly, for any two distinct values $\\lambda := a$ and $\\lambda := b$ from the domain of \"null curve\" $\\nu$, there exist curves (which in the following are suggestively called \"chord curves\") $$\\kappa_{ab} : [~a, ~b~] \\subset \\mathbb R \\rightarrow \\cal M,$$ such that $$\\kappa_{ab}[~a~] = \\nu[~a~], \\qquad \\kappa_{ab}[~b~] = \\nu[~b~]$$ and: the \" _tangent_ of $\\kappa_{ab}$\" is everywhere **positive**. As one concrete case consider $$\\mathbf r \\circ \\kappa[~\\lambda~] := \\\\{~t_{\\kappa}[~\\lambda~],~x_{\\kappa}[~\\lambda~],~y_{\\kappa}[~\\lambda~]~\\\\} = $$ $$\\\\{~\\small{\\lambda,~\\text{Cos}[~a~] + \\left( \\frac{\\lambda - a}{b - a} \\right) \\left( \\text{Cos}[~b~] - \\text{Cos}[~a~] \\right),~\\text{Sin}[~a~] + \\left( \\frac{\\lambda - a}{b - a} \\right) \\left( \\text{Sin}[~b~] - \\text{Sin}[~a~] \\right)~}\\\\},$$ with the corresponding \" _tangent_ \" value $$\\left( \\frac{d}{d\\lambda}[~t_{\\kappa}[~\\lambda~]~] \\right)^2 - \\left( \\frac{d}{d\\lambda}[~x_{\\kappa}[~\\lambda~]~] \\right)^2 - \\left( \\frac{d}{d\\lambda}[~y_{\\kappa}[~\\lambda~]~] \\right)^2 = $$ $$\\small{1 - \\left( \\frac{\\text{Cos}[~b~] - \\text{Cos}[~a~]}{b - a} \\right)^2 - \\left( \\frac{\\text{Sin}[~b~] - \\text{Sin}[~a~]}{b - a} \\right)^2 = 1 - 4 \\left( \\frac{\\text{Sin}[~(b - a)\/2~]}{b - a} \\right)^2 \\gt 0.}$$ (As a sidenote: similarly, for given distinct values $\\lambda := a$ and $\\lambda := b$ from the domain of \"null curve\" $\\nu$, one may ask for \"chord curves\" whose \" _tangent_ \" value should be everywhere **negative** ; but I have not been able to construct a corresponding concrete example. In the following, it is not necessary to distinguish \"positive chord curves\" from \"negative\" ones, if cases of the latter exist at all. Relevant is only that the \" _tangent_ \" value of a \"chord curve\" exists everywhere and is **not \"null\" anywhere**.) **My question is:**   Is there some specific name (i.e. more distinctive than \"just another _null curve_ \") referring to \" _null curves_ \" which do **not have any** \"chord curves\" at all? (Are such special cases of \" _null curves_ \" perhaps called \" _null geodesics_ \"?)"}
{"id":"75555","title":"How to theoretically determine the angular momentum of an atom?","text":"To determine if an atom is a boson or a fermion I have to count the fermions that constitute the atom (protons, neutrons and electrons). My question is: How to theoretically (as opposed to experimentally) determine the total angular momentum of an atom? Can I just sum the spin of all the electrons and the nucleus spin, and the orbital angular momunta of the electron orbitals? For example: A Hydrogen atom in the ground state of can be considered like a spin 1 particle?"}
{"id":"114466","title":"Apparent violation of Newton's 3rd law and the conservation of angular momentum for a pair of charged particles interacting magnetically","text":"Consider a system of the two identical point positive charges situated in the space (isolated from influence of any other external fields) as shown in the figure.Particle1 is at (a,a,0) and Particle2 is at (0,a,0).Velocity of them ( at the instant considered ) are as shown in the figure. Now by applying Biot- Savart law we find that magnetic field due to Particle2 at the position of Particle1 is along +ve Y axis. And thus the force acting on Particle 1 is along +ve Z axis.(By Fleming’s left hand rule) Now similarly analyzing the force on particle 2 we find that no force is there on Particle 2. Now if we check net torque on the system of the two particles about Y axis then it is non zero and along the –ve Y axis…We have no external forces or torques acting on the two particle system considered , still net external torque on the system is non zero…WHY ? also NLM3 seems to be violted ...WHY? I know that the electromagnetic field itself takes away some momentum and moment of momentum about the considered axis but I think that if I consider only two charged particle as my system than the Abraham Lorentz force can be assumed to be acting upon the system and that is sufficient to make sure that we have considered the momentum being carried away by the electromagnetic field itself. Even after considering the action of the AbrahamLorentz force for the two particle system, both the Newton's third law, the angular momentum conservation rule and also the linear momentum conservation rule seem to be violated..... Because the Abraham Lorentz forces do not exactly counter balances the force and the torques on the considered two particle system due to simple magnetic field..... ![enter image description here](http:\/\/i.stack.imgur.com\/0weBR.jpg)"}
{"id":"114468","title":"How can things be chaotic on a quantum level, yet tangible on a classical level?","text":"This may seem basic, but I am wondering if anyone has any input on this topic. It doesn't make any sense to me (I mean I don't need to use the Schrödinger equation to find my cell phone...). I just do not understand how things can appear to be highly statistical when observing the rudiments of matter, yet when observing the large scale result, very consistent (for all practical reasons). How can things with no position make up something with a position? Is it possible that our detection methods are flawed beyond what Heisenberg suggested? Is there some intuition here that I could be missing?"}
{"id":"104297","title":"Wall stress of a hexagonal pressure vessel","text":"**Problem:** I want to calculate the stress in the walls of a hexagonal pressure vessel but I can't manage to get coherent results. For long vessels, cylinders are supposed to have the lowest hoop stress but I obtain out-of-wall stresses with close or lower magnitudes with my method, which is wrong... **Wrong method:** 1) I cut the cross section where I assume the max stress will be 2) drew a free body diagram where $\\sigma$ is the unknown and $F_p$ is the outward force that the pressure generates on each wall. I considered 2 options, one where the pressure is always normal to the walls (not represented), and one where it's omnidirectional coming from the center (represented). I am not sure about which one applies. 3) Respectively, $F_p=p*h*a$ and $F_p = 2\\int^{30°}_{0°}p*h\\frac{a}{2}cos(\\theta)d\\theta$ ; $F_{\\sigma}=\\sigma *h*t$ 4) When projecting on X and Y, $\\sigma$ appears to be out-of-wall (no Y component), and considering only $2*\\sigma_x*cos(30°)*h*t=2F_p$, I end up with 87MPa and 38MPa at the corners (lower elsewhere) compared to 75MPa for a cylinder (t=1mm,a=375mm)) - which does not sound right at all. a is the length of a side, h the height of the vessel (flat ends), p the internal pressure, t the thickness of the walls and $\\theta$ the angle between the pressure vector and the normal to the considered face. ![enter image description here](http:\/\/i.stack.imgur.com\/fKQUL.png) **Question** What is the right method?"}
{"id":"104298","title":"Will a sound composed of the frequencies 450Hz, 650Hz 850Hz have a clearly defined musical pitch? Why?","text":"According to my lecturer, the perceived pitch of a sound composed of the following harmonics: 750Hz, 1000Hz, 1250Hz is equal to the fundamental frequency which is the highest common factor of the harmonic frequencies; so 250Hz. He also said that the harmonic frequencies 450Hz, 650Hz 850Hz do not have a clearly defined musical pitch. How can this be true when the harmonic frequencies have a highest common factor and therefore a fundamental frequency of 50Hz?"}
{"id":"99870","title":"Notation for Propability Amplitudes","text":"I've recently stumbled upon a certain piece of notation that doesn't quite seem clear to me. When discussing the amplituhedron, my teacher mentioned the relation between the volume and the amplitude scattering, and then he wrote the following on the board: $$[1-,\\,1+,\\,3-,\\,3+]$$ I assume these are probability amplitudes, but I don't quite what the integers and the signs mean."}
{"id":"52125","title":"Effective Mass and Fermi Velocity of Electrons in Graphene:","text":"In graphene, we have (in the low energy limit) the linear energy-momentum dispersion relation: $E=\\hbar v_{\\rm{F}}|k|$. This expression arises from a tight-binding model, in fact $E =\\frac{3\\hbar ta}{2}|k|$ where $t$ is the nearest-neighbor hopping energy and $a$ the interatomic distance. But how does one know that the Fermi velocity is given by $v_{\\rm{F}} = \\frac{3ta}{2}$? Normally one would use $m_{\\rm{eff}}^{-1} =\\frac{1}{\\hbar^{2}} \\frac{\\partial^{2}E}{\\partial k^{2}}$ and $v_{\\rm{F}} = \\frac{\\hbar k_{\\rm{F}}}{m_{\\rm{eff}}}$, but in this case $m_{\\rm{eff}} = \\infty$."}
{"id":"113167","title":"Cavity dynamics from traveling waves to standing waves","text":"This is a question on mathematical physics. The conventional reasoning about the characteristic wave modes in a cavity is to apply the boundary conditions, namely, wave amplitude is equal to 0, and obtain discrete cavity-mode frequencies. My question here is, supposing we start by generating some arbitrarily shaped traveling wave inside the cavity. If this traveling wave contains both cavity-mode and non-cavity-mode frequencies, after bouncing around inside the cavity for while, how are those cavity-mode \"standing\" waves kept and other non-cavity-mode waves disappeared? In a math-physics setting, the dynamics obey a usual wave eq, with proper boundary (decaying) conditions. This is solved and the full dynamics in the solution should give us the answer. Has this problem already been fully treated and discussed in some math-physics books? if anyone knows such references, please kindly point me to them. Thanks! [Modification] \\--- Below is the revised question --- Let's consider a Fabry-Perot type wave filter, in which the input waves enter from one side and leave on the other end. Due to the constructing and destructing superposition at the output end, the output waves only take certain resonant frequencies. For example, $\\omega_f = n\\pi c\/d$, where d is the distance between the plates. In this set-up, input waves would be expanded over various non-resonant modes, which are in the functional space larger than the cavity-mode space. My question is: as the decay rate $\\gamma \\rightarrow 0$, how do non-resonant modes \"disappear\" at the output end? I have assumed the decay is so weak that it wouldn't be responsible for the \"disappearance\" of non-resonant modes. So my original intention of the question, was to look into a detailed mathematical solution to this process. By doing that, I might better understand the dynamics of \"disappearance\" of non-resonant modes. I believe the actual mechanism must be the conversion from non-resonant modes to resonant modes. Another interesting thing is, this dynamic process should be independent of the specific material that the reflecting plates are made of."}
{"id":"53838","title":"What is the significance of Planck force?","text":"I have been curious to find what could be the significance of Planck force? It is calculated by the formula $c^4\/G$, where $c$ is the speed of light; $G$ is the gravitational constant. Thus (the speed of light^4) \/ gravitational constant = 1.21031359 $\\times 10^{44}$ newtons."}
{"id":"113784","title":"Integration of Dirac propagator","text":"I've got confused with Dirac (or Spinor) Propagator. Everywhere in books I have examples of integration Klein-Gordon propagators, which are quite easy. But I don't understand how to integrate 4-component function. My second question - how to deal with a mix of Dirac and KG propagators in one equation? I know that if we have many scalar vertices we have to regularize our integral. And examples are shown in lot of books - Peskin, Maggiore, many more. But I couldn't find case with Dirac propagators. My equations were with: $$S_a(x_1-x)D(x-y)S_b(x-y)S_a(y-z)S_b(z-y_1)D(z-y_2)$$ Where $x_1$ were position of initial particle (it's 1->2 decay process) and $y_i$ are positions of final particles. $x,y,z$ coordinates are for internal vertices. We started from a-fermion particle (that's why there is $S_a$ propagator) and we ended with b-fermion and $\\phi$ - scalar particles. Interaction Hamiltonian contained only these 3 particles. If there is required, I can even post Feynman diagram here for process. EDIT: Here are the two Feynman Diagrams for decay process, in third order of perturbation. Described problem was for the first one: ![enter image description here](https:\/\/fbcdn-sphotos-h-a.akamaihd.net\/hphotos-ak- prn2\/v\/t34.0-12\/10363750_801409816536818_1693252894_n.jpg?oh=ef70b424d50f193856bc928c48909a19&oe=537F435A&__gda__=1400883615_ce6af1cbc461f014e8453ea50db92632) Ah: our theory is $\\mathcal{H}_I=k\\varphi\\bar{\\psi_b}\\psi_a$"}
{"id":"113785","title":"Why does strong interaction increase with distance?","text":"I read numerous times that strong interaction increases with distance. But how can one actually derive the force-distance relation from the lagrangian (quark field + gluon field + gauge coupling)? Since I can't use perturbation theory, I don't know what to do with it."}
{"id":"113783","title":"How does one refute a Machian mechanism for inertial emergence?","text":"**Introduction:** Consider the diagrams representing the duality between the weak and strong principles of equivalence. Now based on how these diagrams were taught to us (at least how they were taught to me); each of the two graphics are displayed for the sake of demonstrating how two separate situations are actually physically indistinguishable for the inside observer (Einstein himself). ![enter image description here](http:\/\/i.stack.imgur.com\/QwjKc.png) After careful consideration and analyzation, I claim that these diagrams in no way admits any information which can be used to suggest that in each diagram the two scenarios are in any way physically or _situation-ally nonequivalent_. That is; free fall may not be described as any sort of new class of inertial motion as the Wikipedia article on \"General Relativity\" suggests. Equivalently gravitational forces may not be thought of as a new class of accelerated motion. In my opinion this idea that free fall represents a new class of inertial motion is simply an artifact of the assumption that gravitational fields are not present at a distance far removed from a gravitational source. Essentially I believe all frames in which we don't feel inertial forces _must be free falling frames_ (such as orbit or flat space drift) Even though it initially doesn't seem that way (Since the gravitational source cannot be seen from a far distance). Perhaps this is already considered obvious to some but I know it is not to everyone (Since so much work is being done to understand inertia by other means...such as say radiation reaction forces(1)) I don't want anyone to misinterpret my meaning; I COMPLETELY agree with the principle of equivalence, in fact I believe it is far more powerful than most physicists give it credit for. I only claim that the diagrams are _misleading_ because each diagram uses two seemingly different yet _already fully equivalent_ scenarios as a means to describe how they are actually (surprise!) physically equivalent from the point of view of the inside of an elevator _but makes no claims about the equivalence on the outside as well_ ; In each diagram there is an indirect suggestion being made; that the two situations are _actually distinguishable_. It is as if Einstein pops his head out the door and says \"Oh I thought I was free falling, turns out my elevator was just drifting in locally Minkowskian space-time, silly me\". I say he was always free falling, just not in the traditional way. In my opinion all Minkowskian frames are free falling ones. The geodesic equation says that mass-energy follows its natural geodesic path through space-time but mass can only be measured via force probe if there is a resistance to that natural geodesic path. When you accelerate an object in a way which admits an inertial property of matter(One that is resisting its natural geodesic path through space-time) such as... the pushing force on our backsides as we accelerate in car, it MUST be the result of an objects coupling to the gravitational sources throughout the universe (and Earth too!) being resisted. This is essentially the meaning of a Machian principle of equivalence. $\\frac{d^{2}x^{\\lambda }}{d\\tau^{2} }+\\Gamma ^{\\lambda }_{\\mu \\nu }\\frac{dx^{\\mu }}{d\\tau}\\frac{dx^{\\nu }}{d\\tau }=0$ For the sake of clarity I will augment the geodesic equation to demonstrate how inertia may be calculated as an Machian effect of changing the natural geodesic path of an object. (The point here is to show that the connection coefficient is BUILT from all mass energy in the universe.) $m\\frac{d^{2}x^{\\lambda }}{d\\tau^{2} }+m\\Gamma ^{\\lambda }_{\\mu \\nu }\\left ( \\left ( T_{\\mu \\nu } \\right )_{Universe} \\right )\\frac{dx^{\\mu }}{d\\tau}\\frac{dx^{\\nu }}{d\\tau }=F^{\\lambda }_{inertial}$ Also when most of us speak of the geodesic equation, we refer to it's description of an object in an inertial frame since it is always equated to zero but the equation is equivalent to the four force equation which is not necessarily zero. This is precisely why inertia manifests. When an object is free falling. $F^{\\lambda }_{inertial}=0$ When an object is feeling inertial forces. $F^{\\lambda }_{inertial}\\neq 0$ (1) I intend no disrespect to the legitimacy of radiation reaction forces or to those undertaking research in radiation reaction forces, only that if there are radiation reaction forces then they are available here, raining upon us on the surface of the Earth all the time since we are always standing in an accelerated frame. I would even go as far to say that radiation reaction research may serve as a means to understand inertia and gravitation far better than we currently do. **Conclusions:** 1a) Weight and inertia are one-in-the-same as told by the strong principle of equivalence. 1b) Applied forces and gravitational normal forces are one-in-the-same as told by the strong principle of equivalence. 2) Free fall, free drift and orbit are one in-the-same as told by the weak equivalence principle. 3) An objects mass-energy content can only be defined by effective measurement of an obstruction (by a force probe) to the object's natural geodesic path through space-time. For this reason mass-energy MUST be defined this way. 4) The principle of equivalence is Machian by virtue of the augmented geodesic equation. (NOT IN THE SENSE THAT MACH ORIGINALLY PROPOSED ~ SE USERS: PLEASE DO NOT REFERENCE THIS VERSION OF MACH'S PRINCIPLE WHEN COMMENTING OR ANSWERING BECAUSE IT IS NOT THE CONCERN HERE) Yet most physicist claim any sort of Machian principle is erroneous. Additional note: When I approached my professor with this information; I originally suggested that if all objects with mass-energy in the universe (which share time-like separations) contributes to the natural geodesic paths of every other object in the universe. Then inertia can be described in terms of a Machian framework due to the principle of equivalence itself. He claimed that my argument was useless since that no one has been able to explain WHY mass energy contributes to space-time curvature. While I agree that the mechanism which somehow curves space-time due to the presence of mass energy is not well understood, I disagree with him that the information is pointless since these ideas must remain valid once we finally do figure out the mystery behind gravitational mechanism. The certainty of a Machian process for inertial emergence should not change once we understand gravitational mechanism. In my opinion, the understanding of gravitation in terms of a Machian principle of equivalence seems beneficial for solving new problems or setting new boundary conditions. **Questions:** 1) Where is my logic\/thought process flawed? 2) How can a Machian principle of equivalence of this type be refuted? (Since it is explicitly described by the geodesic equation and the principle of equivalence.) 3) If an object's mass energy content can only be defined by a measure of an obstruction to its natural geodesic path through space-time, then how can two separate interactions such as electromagnetism and gravitation both generate by experiment the exact same values for an objects mass-energy if the two interactions are not fundamentally one in the same? (Let me say that the lack of measurable gravitational forces and the availability of measurable electromagnetic forces at microscopic scales compared to macroscopic scales where the opposite is true seems all too artificial.)"}
{"id":"96446","title":"Interpretation Klein-Gordon equation","text":"What is the problem if we try to interpret KG equation as a single particle equation? Also I wish to know whether the born interpretation of wavefunction is applicable in relativistic quantum mechanics."}
{"id":"18263","title":"Why search for theoretical particle?","text":"Just out of curiosity, is there any special purpose in human life, the search for some theoretical particle for its uncertain benefit, to prove theory of everything which doesn't serve any advancement towards humanity (yet), using huge amount of resources which is better of allocated to fund research for alternative renewable energy? To enable warp drive? Anti-matter as alternate fuel? To know the ultimate answer of universe, life and everything?"}
{"id":"59936","title":"primary constraints for constrained Hamiltonian systems","text":"I would be most thankful if you could help me clarify the setting of primary constraints for constrained Hamiltonian systems. I am reading \"Classical and quantum dynamics of constrained Hamiltonian systems\" by H. Rothe and K. Rothe, and \"Quantization of gauge system\" by Henneaux and Teitelboim. Consider a system with Lagrangian $L(q,\\dot{q})$ and define the momenta $p_j = \\partial L\/\\partial{\\dot{q}_j}$, with $j$ from 1 to $n$ (for $n$ degrees of freedom), and the Hessian $W_{ij}(q,\\dot{q}) = \\partial^2 L\/\\partial{\\dot{q}_i}\\partial{\\dot{q}_j}$. Let $R_W$ be the rank of the Hessian $W_{ij}$. Assume that $L$ is singular and $R_W < n$. In order to ask the question, I'm now quoting how the Rothes describe the setting of primary constraints on pages 26 and 27 of their book. > Let $W_{ab}$, ($a,b = 1,...,R_W$) be the largest invertible submatrix of > $W_{ij}$, where a suitable rearrangement of the components has been carried > out. We can then solve the eqs. $p_j = \\partial > L(q,\\dot{q})\/\\partial{\\dot{q}_j}$ (1) for $R_W$ velocities $\\dot{q}_a$ in > terms of the coordinates $q_i$, the momenta $\\\\{p_a\\\\}$ and the remaining > velocities $\\\\{\\dot{q}_\\alpha\\\\}$: $\\dot{q}_a = > f_a(q,\\\\{p_b\\\\},\\\\{\\dot{q}_\\beta\\\\})$, with $a,b = 1,...,R_W$ and $\\beta = > R_W + 1,...,n$. > > Inserting this expression into the definition of $p_j$, one arrives at a > relation of the form $p_j = h_j(q,\\\\{p_a\\\\},\\\\{\\dot{q}_\\alpha\\\\})$. For $j = > a$ ($a = 1,...,R_W$) this relation must reduce to an identity. The remaining > equations read $p_\\alpha = h_\\alpha (q,\\\\{p_a\\\\},\\\\{\\dot{q}_\\beta\\\\})$. But > the rhs cannot depend on the velocities $\\dot{q}_\\beta$, since otherwise we > could express more velocities from the set $\\\\{\\dot{q}_\\alpha\\\\}$ in terms > of the coordinates, the momenta, and the remaining velocities. This is where the presentation from Rothe stops, and my concern is that equations of the form $p_\\alpha = h_\\alpha (q, \\\\{p_a\\\\},\\\\{\\dot{q}_\\beta\\\\})$ (2) with all $\\\\{\\dot{q}_\\beta\\\\}$ present can still be possible, and yet one cannot solve for more velocities from the set $\\\\{\\dot{q}_\\alpha\\\\}$ in terms of the coordinates, the momenta and the remaining velocities if the conditions stipulated in the Implicit Function Theorem are not met, for not all equations of the type (2) can be solved implicitly for $\\\\{\\dot{q}_\\alpha\\\\}$. Therefore it is not proven that there are $(n - R_W)$ primary constraints of the form $\\phi_\\alpha (q,p) = 0$. Henneaux and Teteilboim even state that these $(n - R_W)$ constraints of the form $\\phi_\\alpha (q,p) = 0$ are functionally independent, but give no justification to this statement. I would be most thankful if you could help clarify my above concern and also if you could clarify the statement by Henneaux and Teitelboin as to the fact that the constraints are functionally independent. Thank you!"}
{"id":"15090","title":"Is General Relativity applicable for all coordinate systems?","text":"My understanding was that relativistic physics can be expressed in any _inertial_ coordinate system, but not arbitrary systems. That is, no experiment can determine if we are \"still\" or \"moving\" at a constant velocity; but we can determine if we are accelerating, or moving in a circle (which by definition involves constant acceleration perpendicular to the current velocity). Thus, we can clearly state that the Earth is _orbiting_ , and can't view it as relativisticly stationary. But, to my shock, I recently came across this text http:\/\/books.google.com\/books?id=lWEmNBaHCJMC&pg=PA211&dq=einstein+infeld+physics+ptolemy+copernicus&hl=en&ei=dWZ_TubbKqn20gH8hNjSDw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q&f=false (which has Einstein as a coauthor) which, on page 212, seems to say that although special relativity requires an inertial coordinate system, general relativity does not! And that therefore we can state that the Earth is stationary and the Sun orbits it! I would reject this as pseudoscientific bunk, if not for the authors of the book."}
{"id":"30317","title":"Can a planet have a larger diameter than Jupiter?","text":"I have heard it said that if mass were added to Jupiter, then only its mass and density would increase but the diameter would stay about the same. Is this the case for Jupiter or some property of Jupiter specifically, or is 150,000 km pretty much the largest diameter that a non-fusing object can achieve? I am specifically interested in the case of a physically larger object (planet) orbiting a smaller but more massive object (star)."}
{"id":"39435","title":"How do I use the electro-optic tensor?","text":"I would like to calculate the performance of an optical phase modulator, in which a varying electric field across a crystal modulates the effective refractive index of light passing through the crystal. The effect depends on the crystal orientation, polarization (i.e. direction of the E-field of the optical beam), and the direction of the applied electric field. The dependence on these vectors is encoded into the electro-optic coefficients, which form the electro-optic tensor. My question: How do I apply these coefficients to give the magnitude of the electro-optical effect (in radians of optical phase per applied volt of potential difference) when the light is polarized in direction $i$, propagating in direction $j$, with an applied electric field in direction $k$? In short, how do I _use_ the electro-optical coefficients? For concreteness, I am interested in lithium niobate. A datasheet gives the coefficients as: ![enter image description here](http:\/\/i.stack.imgur.com\/2Tynp.png)"}
{"id":"68362","title":"Can someone explain to me inertia?","text":"I'm a little new to physics and I'm not quite getting the concept of inertia. Can someone help me?"}
{"id":"47402","title":"Normal modes of oscillation: how to find them","text":"Are normal modes the eigenvectors of the matrix $(\\omega ^2 T- V)$ where $T$ is the matrix of kinetic energy and $V$ is the matrix of potential energy? Is it the only way to express them? How can I express them using the coordinates that I have choosen at the beginning of the exercise?"}
{"id":"116830","title":"Experimental evidence for color-neutral gluons $( (r\\bar{r}−b\\bar{b})$ and $(r\\bar{r}+b\\bar{b}−2g\\bar{g}) )$","text":"Is there any experiment\/measurement that would have a different outcome if one of the following scenarios is applied:   1. The two color-neutral** gluons would not exist   2. Those gluons would have very large masses ** we are talking about the 2 gluons not exchanging color, conventionally written as $(r\\bar{r}−b\\bar{b})\/\\sqrt2$ $(r\\bar{r}+b\\bar{b}−2g\\bar{g})\/\\sqrt6$ EDIT: Since I am now aware that 'color-neutral' does not exist, the question only makes sense if stated as 'What would be the experimental evidence if color-symmetry was broken in some way? (such that 2 of the gluons were singled out)'"}
{"id":"35898","title":"Do I have the meaning of the property temperature correct?","text":"OKay my book just starts out talking about the vague definition we have for temperature and we ended up with the Zeroth law of Thermodynamics which states: > Two systems are in thermal equilibrium if and only if they have the same > temperature. So does that mean, the (brief) meaning of the property _temperature_ is **a measure of whether one system is in thermal equilibrium with another**?"}
{"id":"35899","title":"Photonics: Slab As a Lens","text":"The question can be found here: http:\/\/gyazo.com\/fc4d26cd35e6ce368ad2a8ed504f1dcc The refractive index it references can be found here: http:\/\/gyazo.com\/94fd2f3b5ea7da9226c3acd56b0024c1 I'm not really sure where to get started on the \"showing\" part of the question. I anticipate that Snell's Law will be used somehow. I'm not looking for a full answer, just a push in the right direction."}
{"id":"53059","title":"Thought experiment regarding an object approaching a mirror","text":"Here's a thought experiment I came up with in class today when my mind drifted (I however highly doubt I'm the first to think about this since it is pretty rudimentary) : **Let's say superman approaches a (flat) mirror at 200,000 km\/s. At what speed does superman's image approach him?** This has given me a big headache. I have pondered a lot of solutions, but none of them really convinced me. What is the correct solution to this thought experiment?"}
{"id":"76145","title":"Are there any modern textbooks on statistical mechanics which don't ignore Gibbs' analysis of the microcanonical ensemble?","text":"I have lately been reading Gibbs' book _Elementary Principles in Statistical Mechanics_ , and I'm surprised how much in that book seems to have been ignored by later textbook writers. In particular, Gibbs looks carefully into the question of whether in the microcanonical ensemble we can sensibly define entropy and temperature, and his answer, to paraphrase, is \"meh, kind of\". Gibbs pays close attention to exactly deriving properties of systems with any number of degrees of freedom (not just macroscopic systems), and shows that the canonical and grand canonical ensembles give perfect explanations for thermodynamic equations, _even extending them to small systems_. This is not so for the microcanonical ensemble:   * There are two valid definitions of microcanonical entropy (phase space \"surface\" entropy (Shannon\/Boltzmann entropy) and phase space \"volume\" entropy, to use the modern terminology). These are both valid since they become equivalent in the thermodynamic limit.   * Neither definition, when differentiated against energy, leads to a satisfactory value of temperature, because two microcanonical systems, when they are combined and allowed to exchange energy, do not maintain the same temperature that they had before. (Except in the thermodynamic limit.)   * For a system with 1 or 2 quadratic degrees of freedom (e.g. particle in a 1D or 2D box), these temperatures behave super strangely. As in, being always negative or always infinite, and the system becoming colder with increasing energy. Again, the thermodynamic limit escapes these problems. It seems like the only value of the microcanonical ensemble is in the thermodynamic limit where these two expressions of entropy and temperature become equivalent, and start to behave properly. On the other hand, in a lot of textbooks the microcanonical ensemble is taken to be somehow the \"fundamental\" ensemble, even though the ensembles become equivalent in the thermodynamic limit anyway! So, putting the question in a slightly different way, are there any modern textbooks on statistical mechanics which are as careful as Gibbs, or are learners simply expected to read Gibbs for the \"real thing\"? I would guess that proper treatment of small systems is a big deal for, e.g. nanotechnology."}
{"id":"35233","title":"Commonalities among string theories?","text":"So string theory is a developing theory with presumably different variants. What is the commonality among all these variants? What makes the next modification of a current string theory another string theory, as opposed to a completely new theory? More specifically, is there a general mathematical structure that is common to all the theories that are considered to be string theories?"}
{"id":"73460","title":"Decomposition of representations of the Virasoro algebra under $sl(2)$","text":"The Virasoro algebra has a finite $sl(2)$ sub-algebra generated by $L_{-1}$, $L_0$ and $L_{+1}$. Let's consider a unitary highest weight representation of the Virasoro algebra with conformal weight $h>0$ and central charge $c>1$. How does this representation decompose under the $sl(2)$ sub-algebra? It is clear that there is an invariant $sl(2)$ sub-module of weight $h$ consisting of the highest weight state $|h\\rangle$ and the descendants $L_{-1}|h\\rangle$, $L_{-1}L_{-1}|h\\rangle$, $\\dots$ However, there are many other descendants, such as $L_{-2}|h\\rangle$, $L_{-3}|h\\rangle$, etc. What $sl(2)$ modules will these states sit in? In particular, is the full $sl(2)$ representation completely reducible, so that it can be written as a sum of irreducible $sl(2)$ representations? Can anything general be said about the multiplicity of the $sl(2)$ representations appearing in the decomposition?"}
{"id":"101767","title":"Writing Dirac Mass Term For Massive Neutrinos","text":"How does one write down the following Dirac mass term for a collection of \"massive\" neutrinos? \\begin{equation} -[\\overline{(\\psi_R)}M_D\\psi_L+\\overline{(\\psi_L})M^\\dagger_D\\psi_R] \\end{equation} I can write down the mass term as $-\\bar\\psi M \\psi$, where M is a diagonal matrix with mass eigenvalues, $\\psi=(\\psi_1,...,\\psi_n)^T$ is a vector in flavor space and defining adjoint as $\\psi=(\\bar \\psi_1,...,\\bar\\psi_n)$ and any $\\psi_i=\\begin{pmatrix} \\psi^i_L\\\\\\ \\psi^i_R \\end{pmatrix}$. My confusion is, with these I can show the above expression except I get only $M_D$ and not $M_D^\\dagger$, but in the literature of neutrino mass people write it this way."}
{"id":"87969","title":"Anomaly cancellation and fermion number violation","text":"In the standard model, an axial $SU(3)$ currents has anomaly which after quantization leads to the fermion number violation. However, taking all the fermions into account we note that the anomalies cancel. Does the cancellation imply that the net fermion number violation is zero?"}
{"id":"43878","title":"Lorentz model and energy exchange","text":"The Lorentz model, describing the electron of the atom as an harmonic oscillator forced by an oscillating electric field $\\vec{E}$, shows that the dipole moment $\\vec{D}$ obeys the following equation in the stationary regime: $$\\vec{D}=\\frac{q^2}{2m\\omega_0}\\frac{\\vec{E}}{\\omega_0-\\omega-i\\gamma_d}$$ where $\\omega_0$ and $m$ are the natural frequency and mass of the electron, $\\gamma_d$ is the dissipation introduced by hand to take account of the energy emitted by the electron when accelerated (and also its collisions). My trouble is that one finds that the energy transmitted by the electron to the field by unit of time, given by: $$P=-\\vec{E}.\\frac{d\\vec{D}}{dt}$$ is always negative, implying that the electron cannot \"feed\" the field, however how much it is radiating. Can it be attributed to the fact that the electron energy is in first place coming from the field, and thus, our electron cannot give more than it receives? In what cases does the electron sources the background field?"}
{"id":"43875","title":"Finding the acceleration of Block attached using tricky string setup","text":"Below shown is a setup, and block B starts from rest and moves towards right with a constant acceleration. Does the acceleration differ for the blocks ? I am a bit confused because of the tricky string setup. If after time t, the velocity of A with respect to B becomes v, then what would be the acceleration of A ? My understanding is : After time t, Block B will be brought to rest instantaneously resulting in non-zero relative velocity which otherwise will continue to be 0. ![Block String Setup](http:\/\/i.stack.imgur.com\/ErLPK.png)"}
{"id":"43876","title":"Has BCS Cooper pair condensate been observed in experiment?","text":"Feshbach resonance in s-wave scattering states a BCS Cooper pair condensation at B-field just above the resonance where the scattering length a <0\\. Just wondering if the condensation has been observed in an experiment in any group? I was looking at a notes written in 2005 which claimed that this observation was the holy grail, and I seem not to be able to find a paper claiming that particular observation in experiment.. Many thanks :) Simon"}
{"id":"43871","title":"Fresco in his \"Future By Design\" introduces drawing of underwater vessel and its front system of generating air bubbles. Is idea energy efficient?","text":"I recently saw Fresco's Future By Design and noticed something miniature to investigate. My notice regards about one of his illustration he describes in the documentary. Here is the link on youtube, timecode added where it starts with stuff that cought my eye. http:\/\/www.youtube.com\/watch?feature=player_detailpage&v=5zn8MRKOskw#t=2812s He introduces front system of generating air bubbles in one of his underwater vessel. It's usage would be for reducing resistance while the vessel is moving thru water. I recon he invested some time to devise this illustration and this system of air bubbles fits his high tech vision perfectly and my intention with this question is to kinda have his work evaluated xD Now my question: can this system be constructed to be energy efficient? i'am no engineer but i have an opinion that in reality, its invested\/returned energy ratio has to be high and therefore ridiculous to be used."}
{"id":"123712","title":"Calculate TEC efficiency","text":"I am currently doing a small experiment using a Peltier to cool 250ml of water. My aim is to achieve the cooling of 250ml from 23degC to 8degC in under 20minuets which I have done by strapping the Peltier to the underside of an aluminium cup and a large air cooled heat sink and fan to the hot side of the Peltier. The Peltier was consuming 4A at 6V The heatsink reached 31degC (average from top to bottom) I am now looking at the theory behind the efficiency of it. I have read that (currently) peltiers can only achieve a maximum efficiency of 15% to me that means for every watt of power in you get 0.15watts cooling power, the rest is wasted in heat and work. When I researched how they got to that figure it came up with co-efficiency of performance (COP). I took the figures from my experiment and applied the COP formula and got somewhere around 0.47COP as COP is a ratio that means I should be getting somewhere close to 50% efficiency. Can someone explain this to me please?? I would also like to rerun the experiment after tweaking some values to find its optimum performance, but I need to address the efficiency thing first."}
{"id":"101493","title":"Divergent path integral","text":"What does it mean to have a divergent path integral in a QFT? More specifically, if $$\\int e^{i S[\\phi]\/\\hbar} D\\phi (t)=\\infty $$ What does this mean for the QFT of the field $\\phi $? The field $\\phi$ has action $$S[\\phi]=\\int\\left(\\frac{1}{2}\\partial_\\mu\\phi\\partial^\\mu\\phi-V(\\phi)\\right)\\mbox{d}vol$$ where we use Minkowski signature $(+,-,-,-)$."}
{"id":"101496","title":"Energy difference between symmetric and antisymmetric wavefunctions","text":"Is there any energy difference between a particle in a symmetric wavefunction and an identical particle in an identical potential but in a state with an anti-symmetric wavefunction? Or is it case-dependent? The full problem statement (from this assignment) is ![problem](http:\/\/i.stack.imgur.com\/QAzs5.png)"}
{"id":"128878","title":"Why do people say \"Bosons are either gapped or condensated, except physical principle protected cases (Goldstone boson, photon).\"?","text":"Bosons are either gapped or condensated, except physical principle protected cases (Goldstone boson, photon, etc.). I read this in a paper (version1 of http:\/\/arxiv.org\/abs\/1404.3728v1, 1st page 1st column) and in other lecture slides (page 6 of http:\/\/discor2013.files.wordpress.com\/2013\/12\/lee.pdf). This slide appears to talk about some mean field result of spin liquid state. This statement is not really pertinent to the mainstream of that paper. But it sounds rather intriguing and necessary to be further clarified. So could anyone make it clearer? Thanks!"}
{"id":"117418","title":"How to transfer mechanical power from the inside of a vacuum chamber to the outside while maintaining a seal?","text":"In a vacuum chamber how would one transfer mechanical power (either rotation or linear) from inside to the external environment? I'm working on an idea for a new\/different type of motor that would require an evacuated internal atmosphere and am wondering how to transfer the generated motion outside the case. I have thought about using rotating magnetic coupling on either side of a thinned wall section but don't think it would scale well for larger versions of the motor (perhaps powering a bike or car). I expect with a high enough budget (which I don't have) that an extremely high-tolerance mechanical seal could be machined but am wondering if there are any solutions that could sidestep the high tolerances by thinking laterally."}
{"id":"111161","title":"Momentum in Free Scalar Field","text":"I have a few issues with making the transition between these: $\\phi(x)=\\int{\\frac{d^3p}{2\\pi^3}\\frac{1}{\\sqrt{2\\omega_{\\vec{p}}}}(a_{\\vec{p}}e^{i \\vec{p} \\vec{x}}+ a^{\\dagger}_{\\vec{p}}e^{-i \\vec{p} \\vec{x}}})$ $\\pi(x)=\\int{\\frac{d^3p}{2\\pi^3}(-i)\\sqrt{\\frac{\\omega_{\\vec{p}}}{2}}(a_{\\vec{p}}e^{i \\vec{p} \\vec{x}}- a^{\\dagger}_{\\vec{p}}e^{-i \\vec{p} \\vec{x}}})$. So from the Lagrangian of the real scalar field, we get $\\pi(x)=\\partial_0 \\phi(x)$ which just means differentiating our $\\phi(x)$ w.r.t time. But where does the $-i$ come from (as in the minus sign)? Also aren't the exponentials independent of time as $\\vec{p} \\vec{x}$ isn't a four vector? So now I'm wondering where the $\\omega_{\\vec{p}}$ has come from as well!"}
{"id":"128297","title":"How to calculate the partial entropy in a fully connected ising system","text":"I'm trying to reproduce a calculation that should lead to the partial entropy in a fully connected ising model for the high-temperature range ($\\beta < 2$) in the thermodynamic Limit ($N \\rightarrow \\infty$). Although this is a very tempting goal -- the partial entropy would lead to an analytic expression for the mutual information -- I'm still failing and after a very frustrating week my last hope is to find help this way. The model is described by the hamiltonian $H=-\\frac{S^2}{N}$. Here $S:=\\sum_{i=1}^N s_i$ is the sum of all spins and $N$ is the number of spins in the model. The partial entropy then is $H_A = - \\sum _{p_A=0}^{N_A} \\binom{N_A}{p_A} R(p_A) \\log_2 R(p_A)$ with $R(p_A):= \\sum_{p_B=0}^{N_B} \\binom{N_B}{p_B} \\frac{\\exp \\left[ N \\beta \\left( \\frac{p_A}{N} + \\frac{p_B}{N} - \\frac{1}{2} \\right)^2 \\right]}{Z}$ where $N_A$ and $N_B$ are the number of spins the subsystems $A$ and $B$ respectively and $p_i$ are the number of spins of the subsytsem $i$ pointing in the $-x$ direction. What i'm failing now is to reproduce that this leads to $H_A = \\tau N - \\frac{1}{2 \\log 2} \\left\\\\{ \\frac{\\beta \\tau}{2 - \\beta} + \\log \\left[ \\frac{2-\\beta}{2-\\beta(1-\\tau)} \\right] \\right\\\\} + O(1\/N), \\quad N \\rightarrow \\infty,\\ \\beta < 2, \\tau := \\frac{N_A}{N}$. My last attempt was to apply the replica trick to get rid of the $\\log_2$ in $H_A$ and then use some kind of saddle-point method for the resulting $n+1$-fold Integral to solve it in the thermodynamic limit. This looked promising on the first sight but i can't find a closed analytic expression for the determinant of the Hesse-Matrix. The analytic expression is formula #26 in \"J. Wilms, J. Vidal, F. Verstraete, and S. Dusuel. Finite-temperature mutual information in a simple phase transition. _Journal of Statistical Mechanics: Theory and Experiment_ , 2012:P01023, 2012.\""}
{"id":"45111","title":"Application of diffraction problem!","text":"Here is a problem that I am working on, which is the applying the concepts of diffraction to the setting of the sun: > Air has a small, usually negligible index of refraction. It is 1.0002926. > This causes the Sun to actually be below the horizon when it appears to be > just on the verge of sinking below it. > > Suppose you are on the sea-shore watching the Sun apparently sinking into > the ocean. When only its upper tip is still visible, by what fraction of the > diameter of the Sun is that tip actually already below the surface? > > As an approximation, take the earth's atmosphere as being of uniform density > out to a thickness of 8.600 km, beyond which there is no atmosphere. This > means that, with the Earth's radius being 6400. km, your line of sight due > West along the ocean surface to the horizon will intersect this \"upper > surface\" of the atmosphere at about 331.9 km from your eye. (The diameter of > the sun subtends 0.5000 degrees at your eye). This is how I've attempted to model the problem, but I'm not sure if it's correct... ![problem](http:\/\/i.stack.imgur.com\/mOsYQ.png) I'm not really sure how to approach this problem.... Can anyone please help me?"}
{"id":"44247","title":"Can I take heat from the air and convert it to electricity?","text":"Its a summer day and the air in my house has been heated up. I could switch on my air conditioning, but then I'd be using energy from the grid in order to reduce the amount of energy in my house. What I'd much rather do is capture the heat energy from the air, so cooling it down, and turn that heat energy into electricity. (So I could store it in a battery or sell it to the grid or use some other means of disposing of it.) Is such an apperatus possible within our present understand of nature? Why or why not?   (I presume not, as people would be selling kits.) Just to pre-empt a possible answer: I've seen similar conversations in the past and \"inefficient\" has been given as the answer as to why not. I don't understand this as an objection, because the heat energy is effectively free. The sun rudely went and gave me an excessive amount of heat energy without being asked to. Is converting heat to electricity really so ineffiecient that I'd be better off running air conditioning?"}
{"id":"93982","title":"Understanding Calculus Notation in Physics","text":"I have just started a first-year calculus-based physics course about electromagnetism and waves. I am having trouble understanding what calculus notation means in the context of physics. Here is a paraphrased example from Fundamentals of Physics 9th ed. by Halliday et al. that considers finding the electric field at point P (refer to the figure below): Let $ds$ be the arc length of any \"differential\" element of the ring. $dq=\\lambda~ds$ is the total charge of the ring, where $\\lambda$ is linear charge density. The magnitude of the field is given by $$dE=\\frac{1}{4\\pi\\epsilon_0}\\frac{dq}{r^2}=\\frac{1}{4\\pi\\epsilon_0}\\frac{\\lambda~ds}{r^2}$$ Since the x-components of the field are balanced at point P, after the application of geometry, the equation becomes $$dE\\cos{\\theta}=\\frac{z\\lambda}{4\\pi\\epsilon_0 (z^2+R^2)^{3\/2}}ds$$ The author then puts an integral sign in front of both sides and integrates around the circumference of the ring: $$E=\\int dE~\\cos{\\theta} = \\frac{z\\lambda}{4\\pi\\epsilon_0 (z^2+R^2)^{3\/2}}\\int_0^{2\\pi R}ds =\\frac{z\\lambda(2\\pi R)}{4\\pi\\epsilon_0 (z^2+R^2)^{3\/2}}$$ In math, I was taught that Leibniz's notation ($dy\/dx$) is not a fraction. After a little bit of research, I found that it sometimes _acts_ as a fraction, even though it is _not_ a fraction when defined with limits. Nevertheless, what should I think of when I see $dq$, $ds$, and $dE$? What does \"differential\" mean as used in the text above? So far it has made sense for me to think of $dq$ as an \"infinitesimal\" charge (for example), even though I was taught calculus using limits (I know very little about infinitesimals). Secondly, the way the integral is taken seems informal and confuses me. I understand that it represents the sum of the electric field caused by all the charge around the ring, but how can you just put an integral sign in front of both sides and take the integral with respect to whatever $dE$ or $ds$ that happens to be there? Also, the integral on the left side is indefinite whilst the one on the right is definite. How should I view this kind of material? ![figure](http:\/\/i.stack.imgur.com\/SesBB.png) I am unsure if this question belongs on the math stack exchange or this one."}
{"id":"71282","title":"why is a resistor frequency independent","text":"I had a doubt that why is a resistor, frequency independent? Since, as frequency increases the movement of electrons increases so heat increases which causes change in resistance. So my question is why a resistor offers same resistance to every frequency. Any Clarification on this is appreciated. Thanks."}
{"id":"98","title":"Obtaining isotope stability","text":"For a given isotope, one can obtain the binding energy using the semi- empirical mass formula. For example, ![From Codecogs](http:\/\/i.stack.imgur.com\/U4is7.png) has a binding energy of 1782.8 MeV. From this information, how can the likelihood of the isotopes stability be found?"}
{"id":"36157","title":"Why is the critical density of the Universe non-zero?","text":"I'm missing something and I wondered if someone could enlighten me. As I understand it, Einstein's equations say, among other things, that matter curves spacetime. I also understand, the Universe has a 'critical density', defined to be the mass (or energy) density required for space to have zero curvature (found by solving Friedmann's equations). Why does the Universe need a non-zero density (the critical density) for zero curvature? Many thanks."}
{"id":"36150","title":"What is the meaning of following expresion $C=\\frac{\\delta Q}{dT}$ mathematicly","text":"Our professor raised the following question during our lecture in Statistical Physics (even so it's related to Thermodynamics): Many text books (even wikipedia) writes wrong expressions (from mathematical point of view) for Heat Capacity Coefficient, and the right way to wright it is as following: $$C=\\frac{\\delta Q}{dT}$$ But as we see it is neither usual differential, nor a functional derivative, so the question what is this? I couldn't find the answer in math books, and it is true that many text books writes it in very different ways mixing exact and inexact differentials, so anybody have a clue what is the right expression for c and why from mathematical point of view?"}
{"id":"108828","title":"Curvilinear coordinate system around body of revolution","text":"In _Boundary-Layer Theory_ by Schlichting he gives the boundary-layer equations for a body of revolution according to the paper by Boltze$^1$. Unfortunately, this paper is in German. He apparently uses the following local curvilinear coordinate system to derive the boundary-layer equations: ![Fig. 11.6 from *Boundary-Layer Theory*](http:\/\/i.stack.imgur.com\/zPgH8.jpg) I'm not specifically interested in deriving the boundary-layer equations for a body of revolution, however, I am working on a problem with a very similar geometry and am having trouble wrapping my head around the coordinate system. **Question:** For the locally orthogonal basis vectors $\\mathbf{e}_x$, $\\mathbf{e}_y$, and $\\mathbf{e}_\\theta$, what are the _gradient_ , _divergence_ , and _Laplacian_ operators? This is what I have tried: $$x = \\hat{x}$$ $$y = \\hat{y}$$ $$r(x)\\cos\\theta = \\hat{z}$$ where $\\hat{x}$, $\\hat{y}$, and $\\hat{z}$ are coordinates in a Cartesian system with the same origin. Lamé's coefficients are then: $$h_x = \\left\\vert\\left(\\frac{\\partial \\hat{x}}{\\partial x}, \\frac{\\partial \\hat{y}}{\\partial x}, \\frac{\\partial \\hat{z}}{\\partial x}\\right)\\right\\vert = 1$$ $$h_y = \\left\\vert\\left(\\frac{\\partial \\hat{x}}{\\partial y}, \\frac{\\partial \\hat{y}}{\\partial y}, \\frac{\\partial \\hat{z}}{\\partial y}\\right)\\right\\vert = 1$$ $$h_\\theta = \\left\\vert\\left(\\frac{\\partial \\hat{x}}{\\partial \\theta}, \\frac{\\partial \\hat{y}}{\\partial \\theta}, \\frac{\\partial \\hat{z}}{\\partial \\theta}\\right)\\right\\vert = \\left\\vert r(x)\\sin\\theta\\right\\vert$$ With these coefficients it is straightforward to define the operators since this system is orthogonal (for example here are the formulas). I do not think that the results above are correct though since it compares to a local Cartesian system at each point. What are the correct transformations?   1. Boltze, Ernst. Grenzschichten an Rotationskorpern in Flussigkeiten mit kleiner Reibung. Georg-August-Universitat zu Gottingen., 1908. **UPDATE:** For any future Googlers, the correct transformation is: (based on Fig. 4 of @Floris's answer, $\\xi$ is aligned with $U_\\infty$ in the picture above) $$\\xi = x\\cos\\phi - y\\sin\\phi$$ $$\\eta = -(r(\\xi) + y\\cos\\phi + x\\sin\\phi)\\sin\\theta$$ $$\\zeta = (r(\\xi) + y\\cos\\phi + x\\sin\\phi)\\cos\\theta$$ where $\\phi$ is the local angle of the body with respect to the axis of rotation, i.e. $\\phi = arctan(\\frac{dr}{dx})$. The paper by Boltze assumes this angle is small. The Jacobian is then: $$J = r(\\xi) + y\\cos\\phi + x\\sin\\phi$$ which is equivalent to Boltze's value for small $\\phi$ and $y \\ll r(\\xi)$. And the Lamé coefficients are: $$h_x = 1$$ $$h_y = 1$$ $$h_\\theta = r(\\xi) + y\\cos\\phi + x\\sin\\phi$$"}
{"id":"71750","title":"Relationship between lightlike and spatial compactification","text":"The compactification of a spatial dimension, say $x^1$ given by the identification $x \\sim x^1 + 2\\pi R$ is said to be related to the lightlike compactification by a Lorentz boost : $$ \\left( \\begin{array}{c} x^0 \\\\\\ x^1 \\\\\\ \\end{array} \\right) \\sim \\left( \\begin{array}{c} x^0 \\\\\\ x^1 \\\\\\ \\end{array} \\right) +2\\pi \\left( \\begin{array}{c} -R \\\\\\ R \\\\\\ \\end{array} \\right) $$ What exactly is the relationship between them? How can I see this? And what does it mean from a physics point of view?"}
{"id":"17823","title":"Heisenberg picture saves locality in QM?","text":"David Deutsch recently released a paper titled \"The vindication of locality\" where he argues that by using the Heisenberg picture rather than the Schroedinger picture, you can \"save\" locality. The paper is here: http:\/\/arxiv.org\/abs\/1109.6223 Is Deutsch wrong or has he indeed \"proven\" locality?"}
{"id":"68263","title":"Decomposition of this wave function in eigenfunctions","text":"I have this wave function of a system on a central potential: $V(r)$: $$\\Phi(x,y,z)=C(x+y+z)e^{-\\alpha r^2}.$$ And I'm asked a few things about probabilities. I don't have problems with that, because I understand how it's done. The main part is decomposing as a sum of eigenfunctions of the form. Changing the variables $x,y,z$ by their spherical forms, we have: $$\\Phi = Cr(\\sin\\theta\\cos\\phi+\\sin\\theta\\sin\\phi+\\cos\\theta)e^{-\\alpha r^2}.$$ The part that depends on $\\theta,\\phi$ can be expressed as: $$\\sin\\theta\\cos\\phi+\\sin\\theta\\sin\\phi+\\cos\\theta=\\sum_{\\ell=0}^{\\infty}\\sum_{m=-\\ell}^{\\ell} a_{\\ell m}Y_\\ell^m(\\theta,\\phi).$$ Being $a_{\\ell m}$ the scalar product of our function with the eigenfunctions. What I don't understand is that when we did it in class the teacher said that it was necessary to do it only for $\\ell=0,1$, and that we didn't need to go any further. I don't remember why this was possible to say before making any calculations. Why is it?"}
{"id":"87101","title":"How many kinds of topological degeneracy are there?","text":"Here I want to summarize the various kinds of topological ground-state degeneracy in condensed matter physics and want to know whether there exists any other kind of topological degeneracy. For convenience, let's consider a 2D lattice system with $N$ lattice sites, and we may consider the finite system with open boundary condition(OBC) or periodic boundary condition(PBC). Now there are two kinds of topological degeneracy: (1)The topological degeneracy is **approximate** for finite $N$ no matter with OBC or PBC, and it becomes **exact degenerate** only under the thermodynamic limit($N\\rightarrow \\infty $). **Example:** FQHE. (2)The topological degeneracy(>1) is **exact** for **any finite $N$** with **PBC(on a torus)** , and it's **nondegenerate** for **any finite** $N$ with **OBC**. **Example:** Kitaev's toric code model. Is there any other kind(in the above sense) of topological degeneracy?"}
{"id":"87100","title":"Free path distribution","text":"I'm studying statistical mechanics, and I'm trying to resolve some problem known from my thermodynamics course. So I want to _calculate mean free path_ for particles with a concentration $n$ and scattering cross section $\\sigma$ using **only statistical approach**. I understand that for the first step I need to find the probability distribution of the free path of the molecules of gas between successive collisions and then calculate the average value. But here I have some troubles, because I have never solved something like this. I would be grateful if someone would give me some links or give some ideas."}
{"id":"87105","title":"Can we show that time is orthogonal to space?","text":"It's easy to show that the time we measure is \"in a different direction\" from the space directions we measure. However, it's not immediately obvious to me that these directions are **orthogonal**. How do we prove that any directions are orthogonal? What I came up with is the following. [For short distances\/does this cause trouble?], we can use the pythagorean theorem. If I have two directions in space, I can simply travel 1 meter in each of the two directions, and then measure the distance between the two endpoints. The result is sqrt(2) meters if and only if your two directions were perpendicular. ![Finding whether two directions are orthogonal](http:\/\/i.stack.imgur.com\/I4rIu.png) Easy! My question is: **can this method be used to prove that our concept of time is orthogonal to our concept of space?** I have an idea of where to start. We would need to move in space by 1 meter, while moving in time by 1 meter (converting time into meters using c). Then we would need to measure whether the distance traveled was sqrt(2) meters. I'm not sure I know what that would mean. Hopefully this question makes sense; thanks for your assistance!"}
{"id":"87106","title":"What happens to the physical properties of electrons after diffraction?","text":"Particle Wave duality shows us that waves and particles are the same thing. Therefore electrons can be viewed as both particles and waves. The wave properties of electrons can be seen in the double split experiment where a diffracted electron interacts with itself. If a single electron was diffracted, what would happen to it's physical properties? How could the mass spread out like the wave? Similarly how would the charge and spin spread out? These properties are usually considered to be contained in particles, yet if a single electron was diffracted, it surely could not become many particles spreading out. Also, what would happen if something interacted with this 'electron wave' causing part of it to act like a particle again? The wave can't be able to turn back into a particle after spreading out so much, yet it must still be able to act like a particle for certain interactions."}
{"id":"83618","title":"Rotational inertia of a sphere and a cube?","text":"Can someone explain which would have more rotational inertia, a sphere or a cube, and why? Suppose they have the same mass, the side of the cube is equal to the diameter of the sphere, and the cube's rotation axis is perpendicular to two of its faces. Which one would have more rotational inertia about an axis through the center of mass?"}
{"id":"123389","title":"i can't detect what wrong have i done in a kinematics problem","text":"a question was given to us in assignment which was as follows > A person moves 20 metres towards north then 30 metres towards east and then > moves 30 x 1.414 metres towards south west. The whole trip takes 5 seconds. > What is the person's average velocity for this whole journey? i just squared every magnitude of velocity in all the three directions and then I took square root of that sum doing so what i got is this number `55.735267112` so this will be the total displacement of that person and dividing this with the given time [5in this case] we get the answer as 11.1 but the answer is 2 m\/s can any one suggest me where is my concept going wrong"}
{"id":"8013","title":"What does \"soft\" in \"soft symmetry breaking\" mean?","text":"For example it is stated that if supersymmetry breaking is _soft_ then stability of gauge hierarchy can be still maintained."}
{"id":"44932","title":"Born rule and unitary evolution","text":"Is the Born rule a fundamental postulate of quantum mechanics, or can it be inferred from unitary evolution?"}
{"id":"95948","title":"Why are these specific stress invariants chosen?","text":"I've seen these invariants of the Cauchy stress tensor $S$ defined in multiple places: $$J_1 = \\lambda_1+\\lambda_2+\\lambda_3 = Tr(S)$$ $$J_2 = \\lambda_1^2+\\lambda_2^2+\\lambda_3^2$$ $$J_3 = \\lambda_1\\lambda_2\\lambda_3 = \\det{S}$$ where $\\lambda$ are the eigenvalues of the stress tensor. But I suppose we could use almost any three functions of the eigenvalues. Is there any specific reason to choose these, other than the fact that they can be calculated without first finding the eigenvalues?"}
{"id":"95949","title":"How do photons 'connect' during wireless connection","text":"So wireless router broadcasts a signal and then your device searches. So what actually happens when the photons 'meet' it's kind of like saying, 'ah your one of us, so we will follow you, show us the way' It's so bizarre, how do photons connect during wireless connection?"}
{"id":"65084","title":"In coordinate-free relativity, how do we define a vector?","text":"Relativity can be developed without coordinates: Laurent 1994 (SR), Winitzski 2007 (GR). I would normally define a vector by its transformation properties: it's something whose components change according to a Lorentz transformation when we do a boost from one frame of reference to another. But in a coordinate-free approach, we don't talk about components, and vectors are thought of as immutable. For example, Laurent describes an observer using a timelike unit vector $U$, and then for any other vector $v$, he defines $t$ and $r$ uniquely by $v=tU+r$, where $r$ is orthogonal to $U$. The $(t,r)$ pair is what we would normally think of as the coordinate representation of $v$. In these approaches, how do you define a vector, and how do you differentiate it from things like scalars, pseudovectors, rank-2 tensors, or random objects taken from something that has the structure of a vector space but that in coordinate-dependent descriptions would clearly not transform according to the Lorentz transformation? It seems vacuous to say that a vector is something that lives in the tangent space, since what we mean by that is that it lives in a vector space isomorphic to the tangent space, and any vector space of the same dimension is isomorphic to it. [EDIT] I'm not asking for a definition of a tangent vector. I'm asking what criterion you can use to decide whether a certain object can be described as a tangent vector. For example, how do we know in this coordinate-free context that the four-momentum can be described as a vector, but the magnetic field can't? My normal answer would have been that the magnetic field doesn't transform like a vector, it transforms like a piece of a tensor. But if we can't appeal to that definition, how do we know that the magnetic field doesn't live in the tangent vector space? Bertel Laurent, Introduction to spacetime: a first course on relativity Sergei Winitzki, Topics in general relativity, https:\/\/sites.google.com\/site\/winitzki\/index\/topics-in-general-relativity"}
{"id":"88709","title":"About partially polarized light and the degree of polariztion","text":"When I was taking Optics course, I found there were several questions about polarization of light. I use the textbook of Hecht.   1. It seems that the definition of degree of polarization may be not so well-defined if $V=\\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$. For a elliptical polarized light, there is no natural polarized part, but still $V\\neq 0$.   2. I found it hard to deal with partially polarized light. First the definition. What is the definition of partially polarized light? Light with $0<V<1$?   3. Is all partially polarized light can be presented by the superposition of a plane polarized light and a natural light? I suppose it is true but why? Is there any formal explanation?   4. Consider a real problem. Suppose there is a beam of light incident on an air-glass interface with $n_{ti}=1.5$ at a certain degree, say $30°$. Then how to characterize the reflected light or the transmitted light since they are all partially polarized. What is $V$ for these lights? I gauss the answer may be $V=\\frac{|r_p|^2-|r_s|^2}{|r_p|^2+|r_s|^2}$. But I can't convince my self why this correspondes to the definition above. (This is essentially a problem in Hecht.) Thanks a lot! I am really confused with that."}
{"id":"119456","title":"Can a charged particle pass through a superconductor?","text":"Can a charged particle pass through a superconductor? I think it is impossible, because the superconductor will induce an infinite electric field which keeps the particle away."}
{"id":"119453","title":"Proof involving the fine-structure Hamiltonian of the Hydrogen atom","text":"Given the perturbed Hamiltonian of the Hydrogen atom: $$ H=H_0+H_f=\\frac{\\bar{p}^2}{2m}+V(r)-\\frac{\\bar{p}^4}{8m^3c^2}+\\frac{1}{2m^2c^2}\\frac{1}{r}\\frac{dV(r)}{dr}\\bar{L}\\cdot\\bar{S}+\\frac{1}{8m^2c^2}\\nabla^2V(r), $$ where the last three terms are the relativistic perturbation, the spin- orbit coupling and the Darwin term and they make up the fine structure Hamiltonian $H_f$; assuming we are in the case $n=2$, I'm asked to prove that $H_f$ doesn't link the eigenvectors $| n=2,l=0,m_l=0,m_s=\\pm1\/2\\rangle$ with the eigenvectors $| n=2,l=1,m_l=-1,0,+1,m_s=\\pm1\/2\\rangle$. The only thing that comes to mind is to apply $H_f$ to each of the first two vectors and check that the output kets are only linear combinations of those two. I applied the separate parts of $H_f$ to those two vectors and found immediately that the relativistic part (the one with $\\bar{p}^4$) doesn't mix the subspaces. But I'm stuck with the spin-orbit term: my guess is that I should use the Clebsch-Gordan coefficients to change to the $|n,l,j,m\\rangle$ basis, but I'm not sure how to do this since I've never calculated CG coefficients. My questions are:   * Am I on the right track? If so, could somebody point out how to calculate the CG coefficients? If I'm not on the right track, what could I do?   * Also, the Darwin Hamiltonian ends up being proportional to $\\delta(\\bar{r})$. Is the proof that it doesn't link the aforementioned kets, as trivial as it looks? It just multiplies the kets and is thus its own eigenvalue? PD: I forgot to mention earlier that potential is Coulombian: $V(r)=-e^2\/r$, which gives the delta in the Darwin term."}
{"id":"2300","title":"Relating the power to weather variables","text":"Could you please help me in this problem? I have 3 independent variables, (T,H,t) , as inputs and one output P ( I have all data for these inputs and the output, done experimentally measured every hour during one year). I want to find a formula of this form: P=f(T,H,t) where t is the time in hours and it is always in the x -axis (index), T is temperature, H is humidity, and P is power. I have all the data, and when I draw them in the same graph during one year, meaning that P , T , and H vs. hours. I found that the behavior of P is oscillating, making a sinusoidal shape over the entire year: ![](http:\/\/www10.0zz0.com\/2010\/12\/27\/18\/159867792.jpg) So, if I make a zoom view to this figure, for example from the 2000 th hour of the year to the 3000 th hour, it is clear that it almost has the same shape but it is oscillating. So, it keeps oscillating and increasing up to the peak point and then it starts decreasing till the end of the year. So, how can I predict the structure of the formula that relates P with T , H , and t ? What is the effect of T,H and t on P . Is there any approach that you advise me to follow? Sorry for this long question and any help from you is highly appreciated. I read many papers but I could not know how to solve the problem."}
{"id":"116337","title":"Electron indistintinguishability and handedness","text":"I just learned that right ad left handed electrons behave in a remarkably different way under the weak interaction. Up till now I have been told that all the electrons are exact copies of one another and all this fermi-dirac statistics story. Nonetheless, since right handed and left handed electrons DO have fundamental differences, shouldn't this remove their indistinguishability?"}
{"id":"90818","title":"What coefficients may be used for evaluation the heat keeping?","text":"The final goal is to compare two objects (similar to boxes) from the standpoint of keeping the heat inside? May be there is numerical coefficient or any approaches?"}
{"id":"19560","title":"How thin a filter to filter particles out of salt water to make salt for human consumption?","text":"I have gathered some salt in the Death Valley and for the fun of it I would like to clean it and make it edible. The process I plan to follow is simple:   1. Dissolve the salt in water.   2. Boil water for 30 min to kill bacteria.   3. Filter water to remove foreign particles.   4. Evaporate water to get clean salt. At step 3, I'm planning to buy paper filters with pores of 3 µm to filter fine particles. Is it sufficient for the purpose of human consumption? I'm mostly wondering if there is a key pore size threshold beyond which it is significantly less useful to filter, or if it's just a matter of degree, the smaller the better at an consistent size\/benefit ratio."}
{"id":"19565","title":"What is a gauge transformation of the metric in GR?","text":"I'm confused by this question: Why can't General Relativity be written in terms of physical variables? I can't quite see how you can make any change to the metric without either: (a) changing the physics; or (b) changing which point in your manifold represents which physical point in spacetime. Either of these would, it seems to me, disqualify the change as being a gauge transformation, at least in the usual sense. So is there some cunning way of changing the metric that avoids either (a) or (b), or is there some sense in which (b) can still be thought of as a gauge transformation?"}
{"id":"19567","title":"How did Enrico Fermi calculate the classical Fermi Problem?","text":"From Wikipedia: Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name. One example is his estimate of the strength of the atomic bomb detonated at the Trinity test, based on the distance travelled by pieces of paper dropped from his hand during the blast. Fermi's estimate of 10 kilotons of TNT was remarkably close to the now-accepted value of around 20 kilotons, a difference of less than one order of magnitude. I have not been able to find any references explaining how he made this calculation. Please provide a reference or an example calculation. See also: http:\/\/en.wikipedia.org\/wiki\/Fermi_problem"}
{"id":"63904","title":"When a variation of a tensor is not a tensor?","text":"In a comment about variation of metric tensor it was shown that $$\\delta g_{\\mu\\nu}=-g_{\\mu\\rho}g_{\\nu\\,\\sigma}\\delta g^{\\rho\\,\\sigma}$$ which is contrary to the usual rule of lowering indeces of a tensor. That means $\\delta g^{\\mu\\nu}$ is not a tensor, while $g^{\\mu\\nu}$ is a tensor quantity, so taking a variation leads to loss of tensor nature here. How often does that happen to other tensor quantities? Are there general rules or guidelines to know if their variations are tensors or not? **Update:** Misunderstanding clarified. I should not interpret $\\delta T^{\\mu\\nu}$ as $(\\delta T)^{\\mu\\nu}$ and try to apply tensor rules to indices immediately. Instead, I should take it as $\\delta(T^{\\mu\\nu})$, and if I wish to lower indices, I should write $\\delta(g_{\\mu\\rho}T^{\\rho\\nu})$ and then apply Leibniz rule to the variation as $\\delta(AB)=(\\delta A)B+A(\\delta B)$. That would generally add some terms in case $\\delta g^{\\mu\\nu}\\ne 0$."}
{"id":"64026","title":"Error in Sear's and Zemansky's University Physics with Modern Physics 13th Edition (Young and Freeman)?","text":"I was reading up on the Ideal Gas Equation in University Physics with Modern Physics by Young and Freeman when I chanced upon a seemingly illogical mathematical equation. Can anyone rectify this error? Or is it misunderstanding on my part? Here is the portion (Page 600, Chapter 18, Equation 18.12): > $$pV = \\frac{1}{2}Nm(v^2)_{av} = > \\frac{1}{3}N\\biggl[\\frac{1}{2}m(v^2)_{av}\\biggr]$$ It should be clear that $\\frac{1}{3} \\neq (\\frac{1}{3} \\times \\frac{1}{2})$."}
{"id":"21282","title":"How was Be-8's Half-Life of 7E-17 Second Determined?","text":"Radionuclides occur with half-lives in a vast range of over 37 magnitudes as listed in this site. In question 7584, Lubos Motl explained how Gyr half-lives were determined. This method doesn't appear applicable to evaluate sub- microsecond half-lives. How are extremely short half-lives determined?"}
{"id":"21280","title":"Stokes' theorem in GR","text":"I read this formula in Sean Carroll's book of GR: $$\\int_{\\Sigma}\\nabla_{\\mu}V^{\\mu}\\sqrt{g}d^nx~=~\\int_{\\partial\\Sigma}n_{\\mu}V^{\\mu}\\sqrt{\\gamma}d^{n-1}x$$ where n is the 4-vector orthogonal to the hypersurface $\\partial\\Sigma$ and $\\gamma$ the induced metric defined by $$\\gamma_{ij} = \\partial_i X^{\\mu}\\partial_jX^{\\nu}g_{\\mu\\nu}.$$ The only trouble I have is with the appearance of this induced metric, because I have the formula $$\\int_{\\Sigma}\\nabla_{\\mu}V^{\\mu}\\sqrt{g}d^nx=\\int_{\\Sigma}\\partial_{\\mu}(V^{\\mu}\\sqrt{g})d^nx$$ which then should give me straightforwardly $$\\int_{\\partial\\Sigma}n_{\\mu}V^{\\mu}\\sqrt{g}d^{n-1}x$$ by the classical Stokes' theorem. I was thinking that it may be something with the differential element: going from $d^nx$ to $d^{n-1}x$ forced the apparition of this indeuced metric but I didn't manage to figure out how so far."}
{"id":"21286","title":"Amount of material required for a pressure tank","text":"I read the answer for the question Why is a hot air balloon “stiff”? and thought something sounded ridiculous. My engineering requirement is that the walls be strong enough. Here $T$ will be the tension (for a surface, not sure about those units) and $R$ is radius of curvature of the wall. Requirement is: $$\\Delta p < 2 \\frac{T}{R}$$ Let $d$ be the thickness of the wall and $\\sigma$ the material tensile strength. $$T \\propto d \\sigma$$ This would indicate that **thickness increases linearly with scale**. That sounds ridiculous. _Why it sounds so silly_   * Volume scales as $R^3$ and surface area as $R^2$. SA x (thickness) = material volume = constant, so that implies there are no economies of scale for pressure tanks in terms of pressurized volume divided by structural materials. That sounds nonsensical. That means a chemical plant wouldn't save any materials by buying a large tank as opposed to 1,000 tiny tanks.   * Say that I have a tank shape in mind. If I build a small tank and a large tank, they will be geometrically congruent. That is, if thickness if 5% of the diameter of the small one, it will be 5% of the diameter of the large one. Please prove me wrong. And if you can't prove me wrong, please establish a physical intuition as to why this should be the case."}
{"id":"106163","title":"PN Junction. Germanium VS Silicon","text":"As far as I understood, when we connect P and N materials, electrons from N drift to P, thus create negative ions in P and positive ions in N. => electric field (barrier potential) must depend on an amount of electrons recombined with holes in P region. => The more electrons recombined, the more ions created, the more barrier potential. BUT! Silicon has 14 electrons and 4 on the 3d level. Germanium has 32 electrons and 4 on the 4th level. => at 25C degrees, electrons in N region in Germanium must have more energy than electrons in Silicon, because less energy required to remove electron from an atom => more electrons can drift to P region, thus barrier potential for Germanium should be higher, however Vbp Germanium = 0.3 V Vbp Silicon = 0.7 V why? P.S. I am talking about P doped and N doped silicon compared to the junction between P doped and N doped germanium"}
{"id":"99509","title":"Thermodynamics - convection current","text":"please could someone help explain the mechanism of a convection current as I keep finding conflicting explanations. Is a convection current caused by differences in density or differences in pressure? Most seem to say it is caused by differences in density (more specifically buoyancy) since you cannot get a convection current in zero gravity. However, could someone help explain the following: If you take a house on a cold winters day then hot air in the house will rise due to differences in density and some of this air will find some way of escaping out the top of the house. When this warm air escapes, as far as I know, this causes cooler air from outside to be drawn in neat ground level (e.g. under your door). Surely this mechanism is caused by pressure differences? Why else would the cold air come inside the house? If it didn't then you would eventually create a vacuum no? I know it's not possible but if you completely stopped hot air leaving the top of your house would this completely stop drafts at the bottom of the house since a draft is just cold air coming in to replace the escaping hot air? Or vica versa, if you completely blocked cold air coming into your house at ground level would this have the knock on effect of stopping hot air escaping near the top? Thanks to anyone who can help out!"}
{"id":"91089","title":"Why isn't the gravitational acceleration taken in to account when determining the net acceleration of a person riding a Ferris wheel?","text":"Say for example you have a person riding a Ferris wheel. What is the difference in his acceleration (towards the center) for when he is at the highest point to when he is at the lowest point? So the answer is that his acceleration is the same every where, that is it is the centripetal acceleration. My question is, why isn't his net acceleration at the top g+Ca, towards the ground, and his acceleration at the bottom g-Ca, towards the sky?"}
{"id":"10574","title":"If the multiverse and many-worlds don't exist, how should we interpret probabilities?","text":"If the multiverse of eternal inflation and the many-worlds of quantum mechanics don't exist, how should we interpret the meaning of probabilities? If there is only one copy of the universe out there, what does it mean to say a radioactive nucleus has a 1\/2 probability of decaying? Does it mean if we have 1,000,000 similar radioactive nuclei, 500,000 $\\pm$ 1,000 of them will decay? How do we make sense of the $\\pm$ 1,000 error bar? Thanks"}
{"id":"10570","title":"Paper stiffness","text":"As piece of paper is folded and unfolded, the stiffness of the sheet may seem to be greatly increased. To those of you who don't recognise it: take a sheet of A4 paper, grasp one of the edges and move your hand vertically. Now fold the paper in half, unfold it and repeat. Is anyone aware of any formal or heuristic explanation of what changed?"}
{"id":"10571","title":"1D Topological insulator with PT symmetry","text":"Assume I have the Hamiltonian for a 1D topological insulators as: $H=sin(P_x) \\sigma_x+i \\Delta \\sigma_{y}+(1-m-cos(P_x)) \\sigma_z $ where $m$ is the mass term, $P_x$ is the momentum and $\\Delta$ is an extra term to make the Hamiltonian non-Hermitian. If I choose the time reversal operator $T=i \\sigma_y K$ and the parity operator $P=\\sigma_{y}$, will the Hamiltonian be PT symmetric?"}
{"id":"21425","title":"What meaning do changes in the absolute value of Gibbs free energy have in a simple expansion process?","text":"Below is a simple representation of the thermodynamics of a steam turbine. Stream kinetic and potential energy changes are neglected and no other type of non-PV work is done besides shaft work. Enthalpy and entropy are straight from the steam tables. Gibbs free energy is calculated using $G=H-TS$. Outlet pressure was arbitrarily chosen as 400 psig. Steam quality is 1.0 in both inlet and outlet conditions. What physical significance, if any, does the _change_ in Gibbs free energy have? In this example, note that Gibbs free energy _increases_ which seems counterintuitive for a supposedly spontaneous process. Indeed, if I use an isentropic efficiency of 0 in this model to represent a simple throttle valve, I get a _decrease_ in Gibbs free energy to -798 Btu\/lb which makes sense as a spontaneous process. I am familiar with the availability approach. I would just like to understand what significance the change in absolute values of Gibbs free energy have, if any. ![Turbine Thermodynamics](http:\/\/i.stack.imgur.com\/3JpJU.jpg)"}
{"id":"78478","title":"Confusion With How Dimensions Work","text":"Form what I understand if you have an equation such as: $$v = v_0 + at$$ then the dimensions must match on both sides i.e. $L\/T = L\/T$ (which is true in this case), but I have seen equations such as 'position as a function of time' $x(t) = 1 + t^2$, and obviously time is in $T$, but apparently the function gives you position which is $L$... so what happens to $T$ and where does the $L$ come from? I thought dimensions must always match... Also, let us say that you know the time to reach a destination is proportional to distance i.e. double the distance and you get double the time, now this makes sense to me, but as I said earlier I thought that dimensions must always be consistent or else you can not make comparisons in physics, so if you are giving me $L$ (the distance), how can that become $T$ (time) all of a sudden?"}
{"id":"94332","title":"x-rays from dark matter accretion into a supermassive black hole?","text":"I understand that accretion of normal matter into a super-massive black hole leads to x-ray emissions. Is the same effect expected to occur for dark matter accretion into a supermassive black hole? Are there any good articles ( _e.g._ on arXiv) I could read to learn more about this?"}
{"id":"8848","title":"Math and Wormholes","text":"Hopefully this is the correct forum for this. I felt that Physics Overflow may not be the correct place. I had a student approach me ask me what kinds of mathematics goes into the study of wormholes. She specifically asked whether there is any amount of topology involved, but I'd appreciate it if I could give her a more full answer. So, topology answers are the best but other areas of math would be nice. I am a topology Ph.D. student but know almost zero physics, just in case my background helps with your answer. I didn't know which tag(s) to add. Thanks."}
{"id":"8843","title":"Is it possible that Atomic Electron Probability Density is a result of Heat?","text":"The Schrödinger Equation provides a Probability Density map of the atom. In light of that, are either of the following possible:   1. The orbital\/electron cloud converges to a 2d surface without heat (absolute zero)?   2. heat is responsible for the probability density variation from the above smooth surface? I have taken two calculus based physics, and Modern Physics with the Schrödinger equation, Heisenberg Uncertainty Principle, Etc."}
{"id":"67591","title":"How is the Geometric Phase measured in the experiment?","text":"I had read some papers that have mentioned the geometric phase (Berry phase) can be used to detect the quantum phase transitions in a quantum many-body system. My question is: How is it measured in the experiment?"}
{"id":"115011","title":"How much power would it take to stop a bullet with a magnet?","text":"If a 7.5 gram copper-jacketed lead bullet (say, a 9x19mm Parabellum) was travelling at 360 m\/s, how much power would it take to diamagnetically stop it in the space of one meter? This question comes from discussion about this question on the SF&F SE, about the scene in the _X-Men_ movie where Magneto stops a bullet before it hits someone in the face. I'm aware that a strong enough magnet to cause this effect is going to have a whole heap of other side effects. I'm mainly interested in the ideal case of simply stopping a bullet, rather than the complication of doing it within an inch of someone's face."}
{"id":"115017","title":"Beginner Quantum Mechanics Questions and Theories","text":"So, I recently read the following article: http:\/\/www.preposterousuniverse.com\/eternitytohere\/quantum\/ and I have some questions on some of the things it discussed, as well as some theories of my own. ### My Theories   * I also was starting to think about entanglement, and I eventually came up with the consensus that entanglement is when one thing's superposition is dependent on another thing's superposition or (collapsed) wave function.   * If that is correct, then wouldn't every ~~atom~~ elementary particle be entangled with every other elementary particle in the universe because of the fact that two things can't be in the same space. As soon as the wave function of one of the elementary particles collapses, we are given information about every other particle in existence, because we know that all other particles can't occupy that space anymore. If this keeps happening we would get some sort of ripple effect among the particles. Does this make sense or am I completely off my rockers here? Now the way I eventually came to thinking about \"quantum observation\" is that if information about a thing's position\/status is persistent (it survives from at least one _instance_ of time to the next), then it has been observed. It's kinda like the example in the article where if we don't know where the cat is, we can work off the assumption that it's everywhere at once. As soon as information is recorded about that cat's position, that assumption is no longer valid because information has been provided to narrow down the field of possibilities. (Basically Shrödinger's cat, with a quantum twist). It all boils down to human uncertainty, and the equations we work with are tailored to work with the most specific information, even if that information isn't necessarily one value. Again, does that make any sense at all? (I hope so haha) ### Other Info   * Overall, I'm brand new to quantum mechanics, and have almost no physics background. (You'll know why in a moment.) Some of these are probably crazy to anyone who knows what they're talking about, but it's my first try at understanding these concepts. I'd love to get guidance and get better with this topic as I go on.   * I learn best when I'm given concrete examples and analogies, and then am allowed to branch off from there to understand the more abstract topic, once I have the concrete stuff under my belt.   * I'm only a freshman in high school, so I don't speak calculus yet. Please avoid throwing big equations at me, but if you think I'd benefit from seeing them, please bear that in mind and provide some explanation along with them."}
{"id":"26817","title":"6d Massive Gravity","text":"Massive gravity (with a Fierz-Pauli mass) in 4 dimensions is very well- studied, involving exotic phenomena like the vDVZ discontinuity and the Vainshtein effect that all have an elegant and physically transparent explanation in terms of an effective field theory of longitudinal modes, as explained by Arkani-Hamed, Georgi, and Schwartz. Is there any analogous work on six-dimensional massive gravity? (The right mass term would still be of Fierz-Pauli form, but the little group is bigger and so I would expect a more complicated set of longitudinal modes to think about.) EDIT: added a bounty to renew interest."}
{"id":"26814","title":"Minimal strings and topological strings","text":"In http:\/\/arxiv.org\/abs\/hep-th\/0206255 Dijkgraaf and Vafa showed that the closed string partition function of the topological B-model on a Calabi-Yau of the form $uv-H(x,y)=0$ coincides with the free energy of a certain matrix model. Then, after taking the double-scaling limit, they get an identification between the B-model partition function and the minimal string partition function. The latter is a minimal model coupled to the Liouville theory, and the equation $H(x,y)=0$ corresponds to what is known as the minimal string Riemann surface (see http:\/\/arxiv.org\/abs\/hep-th\/0312170). For the $(p,q)$ minimal model (without any insertions) one gets $H(x,y)=y^p+x^q$. There are two kinds of branes in the Liouville theory: FZZT and ZZ, where the FZZT branes are parametrized (semiclassically) by the points on the Riemann surface $H(x,y)=0$. What are the equivalents of the FZZT and ZZ open string partition functions in the B-model?"}
{"id":"68615","title":"Holographic Field Theory","text":"I am trying to read this paper http:\/\/arxiv.org\/abs\/1204.1780 and I don't understand how to get from eqn 91 which is, $$S_{2} = N^{2} \\\\{V[P^{(1)}_{m}] + (J^{(1)m} - \\mathcal{J}^{m})P_{m}^{(1)}\\\\} -NJ^{(1)2'} tr(\\phi^{2}) - N m^{2} tr(\\tilde{\\phi^{2}}) - N \\sum_{n=3}^{\\infty}tr(\\phi + \\tilde{\\phi})^{n}).$$ Here $\\phi$ is a low energy mode and $\\tilde{\\phi}$ is the high energy modes. The partition function is $$Z = \\int d\\phi d\\tilde{\\phi} dJ^{(1)n}dP_{n}^{(1)} e^{iS_{2}}.$$ I am interested to remove the high energy fields $\\tilde{\\phi}$ to obtain a low energy effective action for $\\phi$. Which is given in the eqns 92 and 93. But this integral is cannot be evaluated, so if someone knows a technique\/method, it would great."}
{"id":"68612","title":"Is it possible to derive the brane action in pure supergravity?","text":"The branes that source the RR fields of supergravity are described by the DBI action plus a CS term. I know this only from superstring considerations. Is there a way to find this result without requiring superstring arguments? It should, but I am not sure. I would say that one consider deformations of a static, flat brane, and asking for the deformation to be again solution of sugra equations would impose conditions on the deformations, themselves equivalent to equations of motion derived from the brane action, but I am not sure. Does someone have seen something like this in the literature? Many thanks for your help."}
{"id":"50082","title":"Finding and plotting the current graph from a voltage graph (piecewise function)","text":"Problem description: ![Problem description:](http:\/\/i.stack.imgur.com\/tHj5i.jpg) My attempt at making the piecewise function: ![enter image description here](http:\/\/i.stack.imgur.com\/7qsm4.gif) Then, in order to find i(t), I took the derivative of 80x\/3 and -80x\/3, with respect to x, and multiplied C. Obviously, I'm left with a unknown variable C, so I'm not sure how to proceed from here. Am I supposed to find C first and then take the derivative? (how do I do that? I've been searching the web for this possibility and can't seem to find anything)"}
{"id":"1541","title":"An iPhone falling on carpet is fine, is it true?","text":"I heard that an iphone falling on carpet would not cause damage to it, both internal and external. Any physical explanation for this?"}
{"id":"101016","title":"Partition Function of a three state particle system","text":"I've just finished studying the partition function of a two-state particle system, where particles can have a 0 energy value or E energy value . That is: ![enter image description here](http:\/\/i.stack.imgur.com\/LXZiV.png) ![enter image description here](http:\/\/i.stack.imgur.com\/aTpjV.png) Where $t_j$ is a variable of value 0 or 1. In this case the set $\\\\{t_j\\\\}$ determines a microscopic state of the system and at the same time determines the energy of the respective microscopic state. Having this form of the partition function i can easily get a simple expression that will give me thermodynamics properties of the system. My problem is: Having a N localized non interacting atoms in which each atom can be in one of three quantum states characterized by the quantum number k=-1,0,1 ,an atom has the same energy E>0 in k=1 or k=-1 state, and an atom in k=0 state has E=0 energy. What is the partition function of this system ?"}
{"id":"55109","title":"What is light, and how can it travel in a vacuum forever in all directions at once without a medium?","text":"I know there are many questions that are similar (maybe identical?). I am not a physicist nor a student - I am just interested in physics and have been watching many physics channels on youtube recently in my spare time. Here is a (nearly) identical question: Why don't electromagnetic waves require a medium? Here is one response to this question : > Well, I would say the electromagnetic field is the medium. I don't understand this - How can something be the medium for itself? The answers to these questions I don't really think address specific points of confusion I am having - so I am going to try to ask the question a little differently How can light (or electromagnetic radiation) travel through a vacuum when there is nothing there to act as a medium, and do so forever in all directions? For example the light coming from a star millions of light years away. With water, for example, it is easy for me to see how a wave can disperse over the surface in all directions and travel for as long as the energy in the wave doesn't dissipate. The molecules in the water act as the medium, and are oscillating and pushing all neighboring molecules causing a domino effect outwards. But what is the light doing? One thought I had is, if light is made of something then it just moves outwards from the point of origin in whatever direction the light was headed. Which leads me to photons... Here is the first paragraph from Wikipedia : http:\/\/en.wikipedia.org\/wiki\/Photon > A photon is an elementary particle, the quantum of light and all other forms > of electromagnetic radiation, and the force carrier for the electromagnetic > force, even when static via virtual photons. The effects of this force are > easily observable at both the microscopic and macroscopic level, because the > photon has no rest mass; this allows for interactions at long distances. > Like all elementary particles, photons are currently best explained by > quantum mechanics and exhibit wave–particle duality, exhibiting properties > of both waves and particles. For example, a single photon may be refracted > by a lens or exhibit wave interference with itself, but also act as a > particle giving a definite result when its position is measured. So... maybe photons make up light, and they are just shooting through space _like_ a wave, but it really isn't a physical wave (like a sound or water wave), it's just \"acting like one\"? Here is a video from minutephysics : https:\/\/www.youtube.com\/watch?v=Q_h4IoPJXZw The video talks about the particle-wave duality of an electron. So a single electron can act as a wave, but when it hits something it hits only one place. Is that the same for photons? **Thought Experiment** : Out in the vacuum of space there is a huge hollow sphere 1 light year in radius. Inside is just vaccuum, and at the exact center a flash of light is emitted in all directions from a point source. 1 year later the light hits the sphere at all places at once. That's 12.6 (or 4 times pi) light years squared of surface area to cover! How can there be enough photons to hit every spot inside the surface area of this huge sphere? I understand that very little light would hit this far away, but even if non-zero at least some light is hitting. Or maybe does this depend on how big the flash was? Maybe it doesn't hit every spot? If light really is made of photons and acts like a wave but each photon hits only one place, then it seems to me there must be an infinite amount of photons in that flash of light to hit so many places inside the sphere. But that must be impossible, so there must be a flaw in my reasoning."}
{"id":"114056","title":"Why is a dipole moment called a dipole moment","text":"The General Formula for a moment is the following one: $$ \\vec{M} = \\vec{r} \\times \\vec{F} $$ However the formula for a dipole moment is this one: $$ \\vec{p} = Q \\vec{d} $$ How comes this is still called a \"moment\"? A charge is not the same as a force, so if you strictly follow the definition it is not a moment, or am I mistaken?"}
{"id":"110578","title":"Dropping a ball in a train moving close to the speed of light?","text":"Suppose a train is moving very close to the speed of light, say 0.999c relative to a stationary observer on Earth. Now a stationary observer on Earth will observe clocks on the train to tick slower than usual. Now suppose a boy within the train drops a ball. The stationary observer measures how long it takes for the ball to hit the ground. Will the result simply be $\\sqrt{\\frac{2s}{g}}$ or will the result be a greater number due to time dilation?"}
{"id":"115199","title":"A question about sign in Euclidean path integral","text":"I have a question about the sign in the Euclidean path integral in Polchinski's string theory vol I, p 337. In page 335, Polchinski introduced path integral in Euclidean space > $$ \\langle q_f, U| T \\left[ \\Pi_a \\hat{q}(u_a) \\right] |q_i, 0 \\rangle_E = > \\int [dq]^{q_f,U}_{q_i,0} \\exp \\left[ \\int_0^U du L(q, i \\partial_u q) > \\right] \\Pi_a q(u_a) \\tag{A.1.30} $$ The weight in the Euclidean path > inegral is conventionally defined to be $\\exp(-S_E)$, so the Euclidean > action contains a minus sign $$ S_E = \\int du L_E(q, \\partial_u q) = - \\int > du L(q, i \\partial_u q) \\tag{A.1.31} $$ If I used the sentence above (A.1.31) and (A.1.31), I got $\\exp(-S_E)=\\exp(\\int du L(q, i \\partial_i q))$ Later in page 337, it is said > Begin with the free action $$ \\frac{1}{2} \\int d^d \\phi(x) \\Delta \\phi(x) > \\tag{A.1.38}$$ ... $$ Z[J] = \\left\\langle \\exp\\left[ i \\int d^d J(x) \\phi(x) > \\right] \\right\\rangle $$ $$ = \\int [d\\phi] \\exp \\left[ -\\frac{1}{2} \\int > d^dx \\phi(x) \\Delta \\phi(x) + i \\int d^d x J(x) \\phi(x) \\tag{A.1.39} > \\right]$$ It seems the action (A.1.38) is in Euclidean space, otherwise the minus sign in (A.1.39) won't appear. However, Polchinski's book is using $(-+++)$ metric. Under this metric, the action free scalar field is $$\\int d^dx - \\frac{1}{2} \\partial_{\\mu} \\phi \\partial^{\\mu} \\phi -\\frac{1}{2} m^2 \\phi = \\int d^dx + \\frac{1}{2} \\phi \\partial^{\\mu} \\partial_{\\mu} \\phi -\\frac{1}{2} m^2 \\phi $$ I got a **plus** sign in front of the kinetic term after integrating over variables. In (A.1.39) it is a **minus** sign. My question is, why I got a different sign?"}
{"id":"89113","title":"Dimension of the space of solutions in an electric circuit","text":"Consider an electric circuit with dc sources ( voltage and current) and resistors. Write down the equations. In the most general case, the solution of the system is not unique. The set of solutions can be empty or positive dimensional (simple example: 2 points in the graph and two batteries in parallel joining the points). The dimension of the space of solutions can be computed with two different methods: 1) Mathematically : compute the determinants and the compatibility conditions 2) Physically. Give conditions on the graph to give a meaning for the dimension. For instance a cycle of batteries give an empty set of solutions or make the dimension increase by one. I am interested in second method and I am looking for references in the litterature. What I found is the following. In the student books, the unicity is always assumed to be true, eg. in the standard book by nilsson and riedel. In more advanced books, I have found discusions only for particular cases, and with very technical tools. For instance, in Frankel (The geometry of physics), only purely resistive circuits are considered with source of currents in the nodes. And the proof uses ( a simplified version of) Hodge theory. Now my question: \\- Is there a book or an article where the very general case ( any graph with dc sources and resistors ) is considered and the dimension of the system described in terms of the graph ? I am interested both in sophisticated answers as above and in answers with basic tools of linear algebra. All references welcome. I am looking for references, not for the solution. I have already written a solution for my students (an elementary one, with basic linear algebra). I want to compare my solution with the existing litterature. If this is useful and not a waste of time, I will make public my personal notes."}
{"id":"74038","title":"Radius of curvature of lens: Newtons Ring method vs spherometer","text":"I have to do an experiment to find out the radius of curvature of a lens using the Newtons ring method given that you know the wavelength of the monochromatic light used in the experiment. What the question is that why we are using the Newtons ring when we have a spherometer available which expects the answer that the measurement by one of them is more accurate than the other( Internet tells me that the newtons ring is more accurate than the spherometer without citing a reason) So in short what I want to know about is **_Why is using Newton's Ring method to calculate the radius of curvature of the lens more accurate than the spherometer measurement_**. Some of my observation so far   1. I have compared the measurements of the radius of curvature of my friends measurement, and found that the newtons ring and spherometer measurement deviates upto 3cm.   2. When I have looked at the principle behind both of the experiments I have found no approximation in the formulas making the question puzzling"}
{"id":"130323","title":"A clock in freefall","text":"Time dilation calculated using Schwarzschild metric for a non rotating spherical body is: $$t_0=t_f\\sqrt{1-\\frac{2GM}{rc^2}}$$ For such a non rotating spherical body, what would the time dilation of a clock in vacuum free-falling from infinity be? (If the answer is non-trivial; a high level outline of the calculation would suffice \/ be appreciated) **Edit:** I am currently working on an iOS app that is trying to model the mechanism underpinning relativity. So, far the mechanism that I have created is shockingly simple and shockingly good at conforming to Relativity. However, I am trying to break it. I am trying to find any possible areas where the two may diverge. I have noted that using my model a clock in freefall will experience no time dilation, i.e. $t_0=t_f$ and I want to make sure Relativity agrees. I have noted the gravitational component of time dilation above. Since my clock is moving one might also expect a kinematic time dilation. I can calculate the velocity of my clock: $$E_k=\\frac{1}{2}mv^2$$ $$E_p=\\frac{-GMm}{r}$$ $$v=\\sqrt{\\frac{2GM}{r}}$$ Plugging this velocity into the kinematic time dilation equation: $$\\Delta t'=\\frac{\\Delta t}{\\sqrt{1-\\frac{v^2}{c^2}}}$$ $$\\Delta t'=\\frac{\\Delta t}{\\sqrt{1-\\frac{2GM}{rc^2}}}$$ At this point one might make the observation that the kinematic dilation is the inverse of the gravitational dilation and therefore conclude that: $$t_0=t_f$$"}
{"id":"98666","title":"water in vacuum(or space) and temperature in space","text":"So, water in vacuum will boil first and then freeze. I don't know how the freeze happens. As pressure lowers to zero, what happened to freezing point?(I know heat taken by vapor, and the water cool down, but I don't think it will be that cold, will it? In vacuum, boiling point is so low that water shouldn't need so much heat as it does in normal pressure, which means vapor actually takes more heat away under normal pressure than in vacuum, so water under normal pressure would be cooler??(I'm guessing) And temperature comes from heat generates by motion of molecules(I guess so), so in vacuum, there is no temp??? What happen when I heat up a vacuum tube? Does heat need a medium to “travel\"?"}
{"id":"9969","title":"Does a magnetic field arise from a moving charge or from its spin, or both?","text":"I learned that a moving charge creates a magnetic field perpendicular to its direction of motion. I also learned that charged particles like electrons have spin and they also create a magnetic field because of their magnetic dipole moment. I don't understand what magnetic dipole moment is, and I can't find a decent explanation for it by using Google. First, Can someone please explain magnetic dipole moment arising from spin in an effective way? Second, is my main question; Does a magnetic field arise from a moving charge or from its spin, or both? Third, I am very confused. I think I am mixing together the classical electrodynamics description of magnetic field, (moving charge), with quantum mechanics description of magnetic field, (magnetic dipole moment due to spin?)"}
{"id":"55817","title":"Observing visible light itself by our eyes","text":"We know that for any object to be seen by a human eye, the light from any source must be reflected from that object so that it reaches our eye and we can detect it.....but how do we see light itself? Bcoz obviously light can't bounce off light itself, it just pass right through each other, do our eyes absorb the light for it to be detected??"}
{"id":"26040","title":"What is the official difference between a planet and a dwarf planet?","text":"I'm trying to understand how objects are classified as planets, moons, or dwarf planets. Can someone please explain the differences between them? I'm really curious about why Pluto is a dwarf planet, for example."}
{"id":"26044","title":"Population I and II stars","text":"I have been thinking about the formation of the galaxy. I can easily understand that old, low-metallicity stars are in the halo, but I'm missing something when it comes to the disk. If slowly-rotating gas settled into the disk, formed stars, then did the disk at one time consist of Population II stars which have since exploded? Why did not the same processes happen in both the disk and the halo? That would mean that a lot of Population II halo stars will have exploded and formed plenty of Population I halo stars . . . but without disk dynamics perhaps not many would form. How about the missing Population II disk stars, why are they not still around like the halo clusters?"}
{"id":"3531","title":"Photon wave packets from distant stars","text":"A distant star like the sun, thousands of light years away, could be so faint that only one photon might arrive per square meter every few hundred seconds. How can we think about such an arriving photon in wave packet terms? Years ago, in a popularisation entitled “Quantum Reality”, Nick Herbert suggested that the photon probability density function in such a case would be a macroscopic entity, something like a pancake with a diameter of metres in the direction transverse to motion, but very thin. (I know the wave packet is a mathematical construct, not a physical entity). I have never understood how such a calculation could have been derived. After such a lengthy trip, tight lateral localisation suggests a broad transverse momentum spectrum. And since we know the photon’s velocity is c, the reason for any particular pancake “thickness” in the direction of motion seems rather obscure. (Herbert then linked the wave packet width to the possibilities of stellar optical interferometry)."}
{"id":"75074","title":"Is the \"pressure\" as we know a kind of \"instantaneous action at a distance\"?","text":"This may be a very simple question, but I still dont know how that work in the essence (quantic\/atomic level) Consider the image bellow: ![enter image description here](http:\/\/i.stack.imgur.com\/qcggt.png) How the \"pressure (air atoms) \" known that in the other side of the barrier there is nothing and pull the bairrer to the \"low pressure\". There is some sort of instantanious action at a distance to the particles knowing where are holes to go ? Where does the pressure comes from ? In a practical way, how the water in the ocean know that there is air inside the iron ship to pull it upwards out of water ?"}
{"id":"106392","title":"Internal and spacetime symmetries?","text":"I am trying to understand Wiki's explanation about correlation. Part of this article talks about internal and spacetime symteries: > If the probability distribution has any target space symmetries, i.e. > symmetries in the space of the stochastic variable (also called internal > symmetries), then the correlation matrix will have induced symmetries. If > there are symmetries of the space (or time) in which the random variables > exist (also called spacetime symmetries) then the correlation matrix will > have special properties. And then it follows with two examples about symmetries. However I didn't quite understand the definition nor I know any intuitive example for each case? Can somebody help me regarding this; first an explanation of definition and secondly some examples. What does it mean to say \"target space\" symmetries in space of stochastic variable and what is the difference of that to \"symmetries of the space (or time) in which the random variables exist\"?"}
{"id":"53975","title":"comparing torque and 0-60 speed in cars","text":"Im looking a two cars, both are the same model but ones got a 2.0L turbo petrol engine and ones got a 3.0L turbo diesel engine. These are the full specs **2.0L Petrol** ![enter image description here](http:\/\/i.stack.imgur.com\/herd0.png) **3.0L Diesel** ![enter image description here](http:\/\/i.stack.imgur.com\/HopJW.png) what im wandering is that although the 0-60 speed is similar the torque is almost 50% more. Ive driven the diesel and what i really liked about its 'effortless' acceleration, is this down to the torque, rather than the speed - if that makes sense ? sorry if thats a very obvious question my knowledge of physics is 0."}
{"id":"126910","title":"How can I get average numbers of days wind flow from 16 direction?","text":"Wind flow from North direction a nos. of days, Similarly from North East b nos. of days, East c nos. of days, South East d nos. of days, South e nos. of days, South West f nos. of days, West g nos. of days, North West h nos. of days. Then how can I get average numbers of days wind flow from 16 directions ? 16 directions are N,NNE, NE, NEE, E, SEE,SE, SSE,S, SSW,SW, SWW, W, NWW, NW, NNW."}
{"id":"70644","title":"What are renormalons from a physics point of view?","text":"This is again a question in the context of this paper about the Exact Renormalization Group. On p 23 and the following few pages, it is explained that for a $\\lambda \\phi^4$ bare action at the bare scale $\\Lambda_0$, after integrating out degrees of freedom and assuming a small coupling $\\lambda$, the effective action at the larger scale $\\Lambda$ can be written as the sum of a perturbative series plus nonperturbative power corrections (Eq. 2.6) $$ S_{\\Lambda,\\Lambda_0}[\\phi] = \\sum\\limits_{i=0}^{\\infty}\\lambda^{i-1}S_i[\\phi] +O(\\Lambda\/\\Lambda_0) $$ Naively taking $\\Lambda_0 \\rightarrow \\infty$ makes the second part disappear and leaves the first part which is self-similar which means that the theory is renormalizable. However, as stated in the paper, UV renormalons can make the perturbation series ambiguous, such that the power corrections and therefore the bare scale can not be removed and the theory would then not be renormalizable. The issue is explained by the mathematical argument that in order to obtain a unique finite value for the perturbative series, one makes use of the so- called Borel transformation to define a function which has the same power series expansion defined as an integral in the complex Borel plane. This integral exists only of there are no poles on the real axis, otherwise the contour of integration around these poles can be deformed which makes the integral and the function defined by it ambiguous. If this is the case, the power correction terms have to be kept to restore the uniqueness of the integral which means that the bare scale can not be removed. In the paper it is said, that the poles on the real axis are for example due to UV renormalons, that arise from large loop momenta in certain Feynman diagrams. My question now is: What are these renormalons from a physics point of view? How do they enter the Lagrangian of the theory? Are they some kind of unphysical auxilliary fields that appear in the mentioned Feynman diagrams? And what do the Feynman diagrams that contain them look like?"}
{"id":"38623","title":"Quantizing first-class constraints for open algebras: can Hermiticity and noncommutativity coexist?","text":"An open algebra for a collection of first-class constraints, $G_a$, $a=1,\\cdots, r$, is given by the Poisson bracket $\\\\{ G_a, G_b \\\\} = {f_{ab}}^c[\\phi] G_c$ classically, where the structure constants are functions of the dynamical degrees of freedom, $\\phi$. When quantizing a gauge theory, a physical state $|\\psi\\rangle$ has to satisfy the first-class constraints $\\widehat{G}_a |\\psi\\rangle = 0$. From this, one can easily see $[\\widehat{G}_a, \\widehat{G}_b]|\\psi\\rangle = 0$. In the quantum version of the theory, the Poisson equation has to be replaced by an operator commutator equation. In general, ${\\widehat{f}_{ab}^c}[\\widehat{\\phi}]$ doesn't commute with $\\widehat{G}_c$. One possibility is the right hand side of the equation for the commutator of two constraints is ordered so that the constraint $\\widehat{G}_c$ is always on the right in the operator product. However, the resulting product will be nonhermitian in general due to noncommutativity. The commutator of two Hermitian operators is always antihermitian. So, this means the first-class constraint operators have to be nonhermitian. If we want the constraint operators to be hermitian, we require $[\\widehat{G}_a, \\widehat{G}_b] = i O(\\widehat{f}_{ab}{}^c[\\widehat{\\phi}]\\widehat{G}_c)$ where $O$ is some form of operator ordering. However, this operator ordering will in general contain some terms which don't annihilate $|\\psi\\rangle$ in general because $\\widehat{G}_c$ won't always be on the right. How does one get around this?"}
{"id":"38624","title":"Determining the center of mass of a cone","text":"I'm having some trouble with a simple classical mechanics problem, where I need to calculate the center of mass of a cone whose base radius is $a$ and height $h$..! I know the required equation. But, I think that I may be making a mistake either with my integral bounds or that $r$ at the last..! $$z_{cm} = \\frac{1}{M}\\int_0^h \\int_0^{2\\pi} \\int_0^{a(1-z\/h)} r \\cdot r \\:dr d\\phi dz$$ 'Cause, once I work this out, I obtain $a \\over 2$ instead of $h \\over 4$...! Could someone help me?"}
{"id":"109130","title":"Why so much geometry in principia and others","text":"I was recently reading 'On the the shoulders of giants' by stephen hawking and looked at many physics(mechanics) problems solved by Copernicus,Newton etc. why is there so much of geometry used by them. I mean, most of the problems are solved using pure math and not much physics. Why is this?"}
{"id":"134897","title":"Why are Lagrangian subspaces called 'Lagrangian'?","text":"I am wondering what the special role of Lagrangian subspaces (or submanifolds) are in mechanics. Do these subspaces have some sort of special property for which we have some sort of `Lagrangian property?' The definition I know is, a subspace $Y$ ( submanifold $Y\\ $) of a symplectic vector space $(V,\\omega)$ (symplectic manifold $(M,\\omega)$) is called Lagrangian if $\\omega|_{Y\\times Y}\\equiv 0$ and $\\dim Y=\\frac{1}{2}\\dim V$."}
{"id":"133324","title":"Could we tell the difference between population I and II (or even III) neutron stars?","text":"This question is related to thoughts I was having about the mass-radius relationship for neutron stars. Is it unique? Is there a single relationship between $P$ and $\\rho$ or is there any chance that the NS \"remembers\" the composition or type of star from which it was born? So, suppose I were able to measure the masses and radii of a number of (isolated) neutron stars in the solar neighbourhood. Most of these would presumably be population I, but would I be able to spot a population II\/III interloper?"}
{"id":"113334","title":"Exception to weight of hot vs cold water?","text":"I have seen the various posts regarding the comparison of weight of heated and cold water.But is there any contradiction?I live in really hot conditions and as such tap water literally 'boils' here.What we do is to let the water flow to release the initial hot water and later obtain the cold water. Now , as water storage tanks have the supply ducts at the bottom and the hot water at the top,how is it that one gets hot water first? Could someone please ,explain this?"}
{"id":"83450","title":"Is it possible to derive Schrodinger equation in this way?","text":"Let's have wave-function $\\lvert \\psi \\rangle$. The full probability is equal to one: $$\\langle \\Psi\\lvert\\Psi \\rangle = 1.\\tag{1}$$ We need to introduce time evolution of $\\Psi $; we know it in the initial moment of time. So it's naturally set $$\\lvert \\Psi (t) \\rangle = \\hat {U}|\\Psi (0) \\rangle ,$$ and from $(1)$ it follows that $$\\hat {U}^{\\dagger}\\hat {U} = \\hat {\\mathbf E}.$$ So it may be represented as $U = e^{i \\alpha \\hat {H}t}$ (we suppose that $\\hat {H}^{\\dagger} = \\hat {H}$ and $\\hat {H} \\neq \\hat {H}(t)$ for simplifying derivation). Thus it is possible to write $$\\partial_{t}\\lvert\\Psi (t) \\rangle = i\\alpha \\hat {H}| \\Psi\\rangle.$$ But how to get the physical interpretation for $\\hat {H}$?"}
{"id":"41349","title":"How to prove equations for energy of a wave","text":"In my textbook, it says that the energy (I) of a wave, determined by the power of a wave (P) divided by area (A), is determined by the following formula: $$I = \\frac{1}{2}\\rho v\\omega^2A^2 = 2\\pi^2 \\rho vf^2A^2$$ where $\\rho$ is density, $v$ is propagation speed, $\\omega$ is angular speed ($\\omega = 2\\pi f$ and $A$ amplitude. Can anyone show how it is like this?"}
{"id":"15118","title":"Calculating the required amount of acid\/base to reach a pH","text":"Bit stuck on working this out - or if it is even possible. I have a reservoir of solution (water\/plant feed\/dead flies) that is pH 6 and I want to get it to pH 6.25. Here is the information I have:   * Size of reservoir: 50 liters   * pH of reservoir: 6   * Required pH: 6.25   * Alkali\/Base Stuff: Potassium Hydroxide at 50% Strength Is there any way I can work this out to say, 15 or 20%? Many Thanks"}
{"id":"15113","title":"Electromagnetic-Radiation","text":"1. describe how radiation from a resonant LC circuit is increased by opening out the capacitor plates?   2. Explain that a straight wire provides reactive elements of a LC circuit."}
{"id":"107115","title":"Freshman year forces problem","text":"I have been sitting with this problem for about 2 hours now. I know it involves simultaneous equations of some sort... I am trying to figure out maybe if there was not enough information given .. maybe the community could help me out :) Question: _It Is Question 2_ ![Image of question 2](http:\/\/i.stack.imgur.com\/UDhJC.png) My Approach   ![enter image description here](http:\/\/i.stack.imgur.com\/Q6QNS.png)"}
{"id":"108590","title":"Conflict between Lippmann–Schwinger equation and Gell-Mann and Low theorem about energy","text":"Lippmann–Schwinger equation states that scattering state will have the same energy as free state, while Gell-Mann Low theorem says that they have different enery. Lippmann–Schwinger equation says: That is, if $|f\\rangle $ is the eigenstate of interacting picture's free Hamiltonian $\\hat{H}_0^{(I)}$ with eigenvalue $E$, then $$|F \\rangle := \\hat{U}(0,-\\infty)|f\\rangle$$ is the eigenstale of the full Hamiltonian $\\hat{H}^{(I)}(0)$ at $t=0$ with the same eigenvalue $E$, where $\\hat{U}(t,t_0)$ is the evolution operator in interacting picture. On the other hand, Gell-Mann and Low theorem states that : Let $|f\\rangle$ be an eigenstate of $H_0$ with energy $E_0$ and let the 'interacting' Hamiltonian be $H=H_0 + gV$, where $g$ is a coupling constant and $V$ the interaction term. We define a Hamiltonian $H_\\epsilon=H_0 + e^{-\\epsilon |t|}gV$ which effectively interpolates between $H$ and $H_0$ in the limit $\\epsilon \\rightarrow 0^+$ and $|t|\\rightarrow\\infty$. Let $U_{\\epsilon I}$ denote the evolution operator in the interaction picture. The Gell-Mann and Low theorem asserts that if the limit as $\\epsilon\\rightarrow 0^+$ of $|F_\\epsilon \\rangle = \\frac{ U_{\\epsilon I} (0,-\\infty) |f\\rangle}{\\langle f | U_{\\epsilon I}(0,-\\infty)|f\\rangle}$ exists, then $|F_\\epsilon \\rangle$ are eigenstates of $H$. But the energy $E$ will not be same as $E_0$ and will have a shift. Here is a proof of my first state that energy will be same ![enter image description here](http:\/\/i.stack.imgur.com\/rhih5.png) ![enter image description here](http:\/\/i.stack.imgur.com\/rH5wk.png) ![enter image description here](http:\/\/i.stack.imgur.com\/onN6Q.png) ![enter image description here](http:\/\/i.stack.imgur.com\/zbIlB.png) ![enter image description here](http:\/\/i.stack.imgur.com\/QZ3RB.png) ![enter image description here](http:\/\/i.stack.imgur.com\/ArOis.png) ![enter image description here](http:\/\/i.stack.imgur.com\/9kv4n.png) ![enter image description here](http:\/\/i.stack.imgur.com\/7TjRL.png) ![enter image description here](http:\/\/i.stack.imgur.com\/GkYAw.png)"}
{"id":"31740","title":"“Relativistic Baseball”","text":"On Randall Munroe’s new blog “what if”, he answers the question: > “What would happen if you tried to hit a baseball pitched at 90% the speed > of light?” http:\/\/what-if.xkcd.com\/1\/ He concludes: > … the air molecules in front of this ball don’t have time to be jostled out > of the way. The ball smacks into them hard that the atoms in the air > molecules actually fuse with the atoms in the ball’s surface. Each collision > releases a burst of gamma rays and scattered particles. > > These gamma rays and debris expand outward in a bubble centered on the > pitcher’s mound. They start to tear apart the molecules in the air, ripping > the electrons from the nuclei and turning the air in the stadium into an > expanding bubble of incandescent plasma. > > … Suppose you’re watching from a hilltop outside the city. The first thing > you see is a blinding light, far outshining the sun. This gradually fades > over the course of a few seconds, and a growing fireball rises into a > mushroom cloud. Then, with a great roar, the blast wave arrives, tearing up > trees and shredding houses. > > Everything within roughly a mile of the park is leveled, and a firestorm > engulfs the surrounding city. The baseball diamond is now a sizable crater, > centered a few hundred feet behind the former location of the backstop. Is that all actually reasonable, or is it all just hyperbole?"}
{"id":"88544","title":"Capillary tube of insufficient length","text":"I was wondering if we have a very thin glass tube placed in a tub of liquid and the portion of the tube outside the liquid is lesser than the height to which the liquid can rise because of capillarity, what will happen? Will the liquid overflow?"}
{"id":"72279","title":"Wire forces in an inductor","text":"For an inductor, the electromotive force comes from Faraday's law, as I understand it. The governing equation includes the number of turns, the magnetic flux (field and area), and a time differential. $$ \\text{emf} = n \\frac{ d \\phi }{ dt } = n \\frac{ d ( B A) }{ dt } $$ The idea is that $B$ changes, because that comes directly from the current value. So does $A$ want to change too? Since increasing area would decrease the flux, I picture a natural tendency to close or open the area in the same way as the inductor does for the current, because they're contained in the same derivative. Does that imply that the wire experiences an outward or inward force? Inductors you can buy tend to be fairly well secured, but if they weren't, is there are combination of $I$ and $dI\/dt$ that would make the loop balloon out, and also contract in? I ask because I'm curious if this is used as artificial muscle fibers. Make a connected series of loops like a chain, and then alter the current to contract and expand it. Other examples of such artificial muscle fibers I've seen use less direct material properties. But I've not seen the idea mentioned before, and in fact, I've never seen it mentioned that inductors experience wire forces. I'm not really sure how to formalize that in the first place."}
{"id":"35911","title":"Is thermal noise \"quantum random\"?","text":"Is the randomness that can be extracted from thermal noise \"as random\" (that is, even theoretically inaccessible to measurement according to our knowledge of quantum mechanics, and not just random for practical purposes) as the one that comes from \"true\" quantum phenomena, like radioactive decay of nuclei? Due to the uncertainty principle, we can't predict both a particles position and speed infinitely accurate, and therefore we shouldn't be able to predict thermal noise, so it would seem that there is at least some level of quantum mechanics involved, but is that the same randomness that dictates the decay of nuclei, tunneling and other quantum phenomena? So, is there a true distinction between thermal and \"quantic\" noise on a theoretical level, or are they basically the same? In other words, is the thermal movement of atoms and molecules fundamentally deterministic or indeterministic?"}
{"id":"133817","title":"Degeneracy in quater-wave stack","text":"Consider a 1D photonic crystal, the quarter-wave stack, and its band structure. A famous conclusion is that there's no gap at Brillouin zone's center. In other words, successive bands are degenerate at k = 0. It's this degeneracy I try to understand. My current understanding is that, at k = 0, the quarter-wave stack supports standing waves and since each layer is exactly a quarter-wavelength in thickness, the energy densities of a certain pair of successive bands correspond to the same energy level. Is there any alternative explanation of this fact? Can one predict this degeneracy from group theory?"}
{"id":"133818","title":"What is the physical meaning of the norm of the electromagnetic four-potential?","text":"In SI units this would be $\\frac{1}{c^2}\\phi^2 - A_x^2 - A_y^2 - A_z^2$. Is this just not a physically meaningful quantity at all because it's not gauge invariant?"}
{"id":"99511","title":"Does space curvature automatically imply extra dimensions?","text":"Total newbie with basically no physics knowledge here :) I would welcome any correction to the steps of my reasoning that lead to my question, which could easily turn out to be invalid :) My current understanding is that General Relativity is definitely accepted by the vast majority of scientists, and that according to General Relativity spacetime is curved by the presence of mass. I also seem to understand that not as many scientists (although probably still the majority) agree that extra dimensions of space exist. But if we agree that spacetime can \"curve\", aren't we automatically saying that extra dimensions of space exist? I mean, if I grab a stick and I bend it, it becomes curved with respect to our good old 3D space. The stick needs to be in a 3D space, with respect to which it can be straight or curved. So, if the thing that gets curved is not an object in space but space itself, doesn't space need to be in \"another\" space in order to be curved? I apologize in advance if I used inaccurate terms here, which I most likely did :)"}
{"id":"91610","title":"What happens when a charged (negative or positive) object touches an insulator?","text":"I know that insulators do not conduct electricity because they do not allow the free movement of electrons. Let's assume the object that's going to touch the insulator is negatively charged. Does some charge get transferred to the the insulator or does no charge get transferred at all? I want to understand this more conceptually. I don't really get why insulators are used in electrostatic experiments."}
{"id":"43449","title":"Trying to link Eulerian and Lagrangian perturbations","text":"I'm trying to make sense of the link between a (linear) Eulerian (i.e., at a given point) and Lagrangian (following a fluid element) perturbation. I will here express not only where I'm stuck, but also my current understanding of the topic, so I beg that if you find any misconception, please point it out! **>> Background** So far, I (think I) know that in the **Eulerian** perturbation, if $h(\\boldsymbol{r},t)$ is the perturbed quantity, where $\\boldsymbol{r}=\\boldsymbol{r}_0+\\boldsymbol{\\delta r}$ is the new position vector and $\\boldsymbol{\\delta r}$ is a small displacement around $\\boldsymbol{r}_0$, then $$h(\\boldsymbol{r},t)=h_0(\\boldsymbol{r})+h'(\\boldsymbol{r},t),\\ \\ \\ \\ \\ \\ \\ \\ (1)$$ where $h_0(\\boldsymbol{r})$ is the quantity at an equilibrium state evaluated at $\\boldsymbol{r}$ and $h'(\\boldsymbol{r},t)$ is a small perturbation. On the other hand, the **Lagrangian** perturbation $\\delta h(\\boldsymbol{r})$ can be written as $$\\delta h(\\boldsymbol{r})=h(\\boldsymbol{r}_0+\\boldsymbol{\\delta r})-h_0(\\boldsymbol{r}_0)=\\left[h(\\boldsymbol{r}_0)+\\boldsymbol{\\delta r}\\cdot \\nabla h_0(\\boldsymbol{r}_0)\\right]-h_0(\\boldsymbol{r})$$ or, by using the relation found for the Eulerian perturbation, eq. $(1)$, evaluated at $\\boldsymbol{r}_0$, $$\\delta h(\\boldsymbol{r})=h'(\\boldsymbol{r}_0)+\\boldsymbol{\\delta r}\\cdot \\nabla h_0(\\boldsymbol{r}_0).$$ **>> The question** What I have trouble relating is $\\delta h(\\boldsymbol{r})$ and $h'(\\boldsymbol{r},t)$. Both are small perturbations but, in this case, there is a explicit dependance on $\\boldsymbol{r}_0$ by $\\delta h(\\boldsymbol{r})$. However, if I fix $\\boldsymbol{r}_0$, both represent a perturbation about that point so...why are they different? My intuition (probably messed up by know; thanks QM) tells me that they are equal, but the notation is different in almost every book I've read on the topic. Can anyone shine some light here?"}
{"id":"4749","title":"Is there any physics that cannot be expressed in terms of Lagrange equations?","text":"A lot of physics, such as classical mechanics, General Relativity, Quantum Mechanics etc can be expressed in terms of Lagrangian Mechanics and Hamiltonian Principles. But sometimes I just can't help wonder whether is it ever _possible_ (in the future, maybe) to discover a physical law that can't be expressed in terms of Lagrangian Equations? Or to put it in other words, can we list down for all the physical laws that can be expressed in terms of Lagrangian equations, what are the mathematical characteristics of them( such as it must not contain derivatives higher than 2, all the solutions must be linear etc)?"}
{"id":"4740","title":"Why can't massless particle exceed speed of light?","text":"Why massless particle can't exceed speed of light?"}
{"id":"17601","title":"What does an electron's wavevector mean inside of a crystal?","text":"With a plane wave, I always took the direction of the wavevector, $k$, as the direction of propogation (magnitude proportional to the inverse wavelength). Alternatively, it could represent the momentum (minus a factor $\\hbar$) of a particle. However inside a crystal, the electron wavevector and the electron velocity are not necessarily in the same direction. I'm thinking here of a 2D material with a cylindrical Fermi surface where the momentum may have a z component, but the Fermi velocity does not. In everyday cases you would expect momentum and velocity to be in the same direction, moreover I considered the propogation of the wave to be in the same direction as its particle analogue. I realise that inside a crystal the electrons are no longer simple plane waves, but what then does the $k$ vector mean?"}
{"id":"101513","title":"Derivation of cylindrical line heat source problem?","text":"I have a line heat source embedded inside a cylinder and I am trying to find the temperature distribution T(r,t). By using the similarity variable, the solution to the differential equation is $$\\theta(\\eta)=C_{1}\\int_0^\\eta \\\\! \\frac{e^{-u}}{u} \\, \\mathrm{d}u + C_{2}$$ and my boundary conditions are $-4\\pi kh\\frac{\\partial\\theta}{\\partial \\eta}|_{\\eta=0}=q_{0}$ and $\\theta\\to0$ as $\\eta\\to\\infty$. When I apply the first boundary condition using the derivative I get $1\/0$ which is indeterminate so what do I do there? Also, the second boundary condition is supposed to give the exponential integral function, but I am not sure where that comes from. I guess I am stuck on applying the boundary conditions to this. The derivation says the final answer should be $$\\theta(\\eta)=\\frac{-q_{0}}{4\\pi h k}\\int_\\eta^\\infty \\\\! \\frac{e^{-u}}{u} \\, \\mathrm{d}u$$"}
{"id":"101519","title":"Calculating the velocity needed to make a distance in the horizontal with a given height and no upward motion","text":"In 1780, in what is now referred to as \"Brady's Leap,\" Captain Sam Brady of the U.S. Continental Army escaped certain death from his enemies by running over the edge of the cliff above Ohio's Cuyahoga River, which is confined at that spot to a gorge. He landed safely on the far side of the river. It was reported that he leapt 22ft across while falling 20ft ![enter image description here](http:\/\/i.stack.imgur.com\/7MH00.jpg) Now how do I calculate the minimum speed needed to make the \"jump\"?"}
{"id":"128626","title":"How much more efficient is a road bike than a mountain bike?","text":"What would be a good estimate on the difference of efficiency between a road bike and a mountain bike? A number of links cite all the usual reasons: thinner tires, better aerodynamics... But I'm not able to find one where they quantify this efficiency"}
{"id":"116973","title":"Simultaneous conservation of linear and angular momentum","text":"Suppose there is a ring in the space where there is no gravity. The width of the ring is $r$ which is negligible compared to its inner radius $R$. The ring is in horizontal position. Now imagine a human starts to move on this ring. Before the start of the motion angular and linear momentum are zero and system is isolated. When the human walks on the ring linear and angular momentum must remain zero. $$m_{human}*R^2*\\omega_{human} = -M_{ring}*R^2*\\omega_{ring} \\rightarrow \\omega_{human}=-\\frac{M_{ring}}{m_{human}}*\\omega_{ring}$$ There is no problem with this part. Now we consider the linear momentum. The linear momentum of the ring after the beginning of motion is zero because for every element of mass there is another element of mass moving in the opposite direction. However, the linear momentum of the human which can be considered a point mass is not zero. The magnitude of the linear momentum of the human is $ m_{human} \\cdot \\omega_{human} \\times R$. So the total linear momentum is not zero despite the fact that the initial linear momentum was zero and the system is isolated. If there is not anything wrong I have to conclude that $\\omega_{human} = 0$. What am I missing here?"}
{"id":"27066","title":"Dual Pairs in Four Dimensions","text":"Following the conversation here, I am wondering if anyone knows of an example of dual pair with 4-dimensional N=1 SUSY which relates a non-Abelian gauge theory on one side to a theory with a Lagrangian description but no non- trivial gauge group. Cannot think of one off the top of my head, which doesn't mean it does not exist or even is well-known..."}
{"id":"134138","title":"How to calculate the kinetic energy of supernova ejecta?","text":"I believed I could use $0.5\\times M_{ej} V_{ej}^2$, with $M_{ej}$ being the ejected mass and $V_{ej}$ being the velocity of the ejected mass. But I noticed in this and this that the the mean velocity is somehow calculated as following : $V_{mean}=(3\/5)^{0.5}\\times V_{ej}$. And so the kinetic energy was calculated as : $(3\/5)(0.5)\\times M_{ej} V_{mean}^2 =(3\/10)\\times M_{ej} V_{mean}^2$. So is there any explanation for how this factor of $0.3$ or the mean velocity was calculated ?"}
{"id":"7232","title":"Does GR provide a maximum electric field limit?","text":"Does GR provide a limit to the maximum electric field? I've gotten conflicting information regarding this, and am quite confused. I will try to quote exactly when possible so as not to confuse things more with my paraphrasing. The author of the Motion Mountain physics textbook claims in his book there is a limit, and clarifies on his site that ( http:\/\/www.motionmountain.net\/wiki\/index.php?title=Dislike_Page ): > \"electromagnetic fields are limited in magnitude. Now, every electromagnetic > field contains energy, and energy density is limited by general relativity: > if energy density is too high, a black hole appears. The smallest possible > black hole then leads to a field limit. If you deny an upper field limit, > you deny general relativity. However, general relativity has been confirmed > in every experiment so far.\" This sounds very obvious and intuitive to me. However one of my physics TA's got very upset when I used this in a thought experiment when discussing some limits in physics. When I told him the textbook I got that from, he looked it up, and commented on the Motion Mountain wiki website his argument: > \"I'd like to add something here. Suppose for a moment the energy density > limit is correct, then if an object is sped up until it length contracts > enough such that the energy density is greater than this limit, does it turn > into a black hole? No, quoting John Baez \"The answer is that a black hole > does not form\"* > http:\/\/math.ucr.edu\/home\/baez\/physics\/Relativity\/BlackHoles\/black_fast.html > *The claim that \"if energy density is too high, a black hole appears\" is an > incorrect oversimplification. The issue is that in GR, gravity depends on > more than just the energy density (component T^00 of the stress energy > tensor). So we can't just look at the \"relativistic mass\" (E\/c^2) to judge > whether a black hole forms. It actually isn't even enough to look at the > \"invariant mass\", for while the trace T of the stress energy tensor for a > particle is just its invariant mass, for an electromagnetic field it is > identically zero even though electromagnetic fields curve spacetime in GR. > So none of these concepts of mass are sufficient when discussing gravity > using GR (especially when considering electromagnetic fields), because > gravity couples to the entire stress energy tensor. I hope this was > helpful.\" This makes much less sense to me, and I don't understand how energy density tending to infinity could EVER avoid being a black hole. No offense to my TA, but I'm skeptical as he's disagreeing with a textbook author. Plus, the author's response was that my TA is another Einstein denier, and not worth responding to. So I'd like a third party's answer on this. Does GR provide a theoretical limit to the strength of an electric field? Is it best to just ignore my TA on this one?"}
{"id":"7231","title":"Does Heisenberg's uncertainty under time evolution always grow?","text":"Recently there have been some interesting questions on standard QM and especially on uncertainty principle and I enjoyed reviewing these basic concepts. And I came to realize I have an interesting question of my own. I guess the answer should be known but I wasn't able to resolve the problem myself so I hope it's not entirely trivial. So, what do we know about the error of simultaneous measurement under time evolution? More precisely, is it always true that for $t \\geq 0$ $$\\left<x(t)^2\\right>\\left<p(t)^2\\right> \\geq \\left<x(0)^2\\right>\\left<p(0)^2\\right>$$ (here argument $(t)$ denotes expectation in evolved state $\\psi(t)$, or equivalently for operator in Heisenberg picture). I tried to get general bounds from Schrodinger equation and decomposition into energy eigenstates, etc. but I don't see any way of proving this. I know this statement is true for a free Gaussian wave packet. In this case we obtain equality, in fact (because the packet stays Gaussian and because it minimizes HUP). I believe this is in fact the best we can get and for other distributions we would obtain strict inequality. So, to summarize the questions >   1. Is the statement true? >   2. If so, how does one prove it? And is there an intuitive way to see it > is true? >"}
{"id":"121283","title":"Who of these cyclist travel takes less time?","text":"There are two cyclist. They start off with equal velocity $v_0$. The first one bikes a straight path, while the other bikes through a valley, something like this: ![Like this ______\/](http:\/\/i.stack.imgur.com\/pQ8pp.jpg) We assume friction doesn't affect either cyclist, so only gravitational potential energy plays a role here. Who should arrive first? I think its about to find the angles of the second way where the time its a minimun. Someone said I should use the conservation of mechanical energy, but I don't understand how could add the two different velocities (one simply contant, while the other is related to the gravitational potential)."}
{"id":"122959","title":"Why are carbon nanotubes are so difficult to create?","text":"I have heard that the future of the techonolgy are the carbon nanotubes due to their resistance and conductivity, but why are they so difficult to create?"}
{"id":"70366","title":"Light dispersion in water","text":"Simple question: Water is clearly not disspersive if we have it, for example, in a glass of water. We don't see the rainbow through a glass of water. However drops of water disperse light in the atmosphere and allow us to see the rainbow, am I right? Why is this? In which conditions is water disspersive and in which conditions it's not?."}
{"id":"70365","title":"Deformation due to isostatic compression","text":"When loading a solid material in the shape of a cylinder in isostatic compression, what is the relationship for the change in length versus change in diameter? For example. If I know that when I apply an isostatic load of 6000 psi to a cylindrical core of rock that is initially at a volume of 11,351 cc the deformation will be ~10.8 cc. Knowing the initial length and diameter of my cylinder, how do I predict the new length and diameter?"}
{"id":"121754","title":"How exactly to show that s-matrix elements diverges because time-ordering is not well determined?","text":"Let's have s-matrix: $$ S_{\\alpha \\beta} = \\langle \\alpha | \\hat {S} | \\beta \\rangle , $$ $$\\hat{S} = \\hat{T}e^{-i\\int \\hat{L}(x)d^{4}x}, \\quad \\hat{T} \\left( \\hat{\\Psi}(t) \\hat{\\Psi}(t') \\right) = \\theta (t - t')\\hat{\\Psi}(t)\\hat{\\Psi}(t') \\pm \\theta (t' - t)\\hat{\\Psi}(t')\\hat{\\Psi}(t). $$ How to show that matrix elements diverges because time ordering is not well-defined operation when $t=t'$?"}
{"id":"817","title":"What are some approaches to discrete space-time used in modern physics?","text":"This thought gave rise to some new questions in my mind. What are the consequences for:   1. How would it affect duality i.e. particle, wave property of photons?   2. How does this statement affect the information theoretical aspect (entropy) of the universe? **Update:** Given a volume V of space, is the entropy (maximum information that can be store) in this volume changed when this statement is applied?   3. How is a black hole affect by this statement? **Update** How is entropy changed inside the black hole?   4. Could one consequence be that the universe is hologram, since the construction isn’t continues?   5. Would the smallest quantified space be planck's constant? Is there an equivalent constant for time? I hope to get some of your feedbacks regarding this statement."}
{"id":"119238","title":"Does mutual gravitational attraction imply a negative pressure?","text":"Imagine a small ball of gravitating dust particles initially at rest at the center of a large volume $V$. Following John Baez, Einstein's field equations say (in units of $c=8 \\pi G=1$): $$\\frac{\\ddot{V}}{V} = - \\frac{1}{2} (\\rho + 3 p)$$ Now traditionally dust particles have a pressure $p=0$. However the dust particles have a mutual gravitational attraction for each other. Could this mutual attraction be described within the framework of GR by assigning a negative pressure $p$ to the ball of dust particles? Another way of looking at it is to say that the ball of dust particles has a certain amount of negative gravitational self-energy which counteracts some of the gravitational effects of its positive rest mass energy. Apparently the concept of gravitational energy in GR is complex so maybe negative pressure is a better way to think about it."}
{"id":"80834","title":"Time reversal invariance and statistics","text":"To what extend does the behaviour of time reversal invariance depend on the statistics of the particle under consideration? More explicitly: To what extend does the action of the time reversal operator depend on the commutation relations of say fermions, bosons, etc.? Given a certain Hilbert space (say $\\mathbb{C}^N$) with $N>2$ (so we are not talking Spin 1\/2). What is the general way to construct a Time reversal symmetry operator? I am looking forward to your response."}
{"id":"80836","title":"the physical properties of leak testing.","text":"Description: To test for leaks a pressure bubble test is performed. This is adding a specified pressure and submerging the test object under water to see if air leaks emerge. The requirement is that no leaks shall be detected when a certain pressure 1 is applied during a predefined time 1. I’m currently looking for a way to speed up this process. When decreasing the time, pressure should logically increase to draw the same conclusions. However what I would like to know is if there is more know about this relationship or\/and if it is possible to superimpose this if this relationships is too complicated."}
{"id":"5266","title":"Ski Jumper's vertical velocity after $246.5m$ record?","text":"What would be the vertical velocity of this ski jumper (ski flyer), after he first touches down, after he breaks the record with a $246.5m$ jump? What g force would he experience as he slows down? Video (the last video) I think he gets at least 6 seconds of airtime from $26$ to $32$ second mark. The ramp gradient at take off is listed at $11$ degrees and the take off speed is about $105km\/h$. Some statistics] are here. [1] [2]"}
{"id":"5268","title":"Covariant Quantisation and the Time Operator in String Theory","text":"Covariant quantisation in string theory is accomplished by giving the commutator relations $[X^\\mu(\\sigma,\\tau),P^\\nu(\\sigma',\\tau)] = i \\eta^{\\mu\\nu} \\delta(\\sigma - \\sigma')$. Although string theory is not a field theory this approach is identical to the quantisation of fields $X^\\mu$ in 1+1 dimensions. From this point of view it is also clear how to proceed from here (e.g. getting rid of the negative norm states due to the \"wrong\" sign of the zeroth component). However, there are two issues that I do not quite understand. Physical states are supposed to have positive energy, i.e. $P^0 \\geq 0$. Especially when supersymmetry comes into the game, the algebra so to say dictates positive energies. But when the energy is bounded from below it is not possible to have a time operator obeying the commutator relation $[x^0,p^0] = -i$ where $x$ and $p$ are the above Operators $X$ and $P$ integrated over $\\sigma$. For the case of quantum mechanics this has been proved by Pauli in 1933. The prove goes roughly like this: For an operator $H$ which is bounded from below let there be a Hermitian time operator $T$ with $[T,H] = i$ It is possible to shift the spectrum of H because of $e^{-i\\epsilon T} H e^{i\\epsilon T} = H - \\epsilon$ for arbitrary $\\epsilon$. That means for an eigenstate $|\\psi\\rangle$ with $H|\\psi\\rangle = E|\\psi\\rangle$ there is also an eigenstate $|\\phi\\rangle = e^{i\\epsilon T}|\\psi\\rangle$ with $H|\\phi\\rangle = (E-\\epsilon)|\\phi\\rangle$ For $\\epsilon > E$ one has to either conclude that H can not be bounded from below or that T is not a Hermitian operator because it would generate states that lie outside of the Hilbert space. One possible solution could be to allow negative $p^0$ and to apply some kind of Feynman-Stueckelberg approach by reversing the sign of the world-sheet coordinate $\\tau$. In this sense the decay of the vacuum into a positive and negative energy state is simply the propagator of a free state. However, this would not be possible in the supersymmetric case where the states have $p^0>0$. The other issue related to defining the zeroth component of the commutator is the construction of the Hilbert space of the physical states. In his book String Theory Polchinski starts with a \"vacuum\" state $|0;k\\rangle$ with the norm $ \\langle 0;k|0;k'\\rangle \\sim \\delta^{26}(k-k')$ However, when integrating over $k$ one ends up with a superficial delta function because of the additional mass shell constraint. Polchinski solves the issue by introducing a reduced product $ \\langle 0;k||0;k'\\rangle \\sim \\delta^{25}(k-k')$ This seems to be a logical step because selecting the physical states by $L_0|\\psi\\rangle = \\alpha'(p^2+m^2)|\\psi\\rangle = 0$ yields a Hilbert space whose functions depend on 25 momentum components. Still, the full Hilbert space is not normalisable (this differs from what is usually done in QFT during gauge fixing, where the Hilbert space is constructed via an equivalence relation for the gauge and not by selecting a subspace). The question is whether the Hilbert space constructed in such a way does exist at all. Also it seems as if the number of spacetime operators is reduced by one, thus rendering the steps taken for the covariant quantisation needless. (REMARK: I edited my question to make the point a bit clearer. This is not meant to be a criticism against string theory. It is just that during studying string theory I was confronted by someone with this issue. I just want to understand this detail of the basics because I haven't yet been able to resolve it. I hope somebody can help me.)"}
{"id":"49788","title":"Dyson Series Second Order Term Importance vs. Time","text":"Given a time ordered Dyson series expansion of $$H_I=e^{-\\frac{i}{\\hbar}\\sigma_3t}V_0\\sin(\\omega t)\\sigma_1e^{\\frac{i}{\\hbar}\\sigma_3t}$$ $${\\cal T}\\exp\\left[-\\frac{i}{\\hbar}\\int_0^tH_I(t')dt'\\right]=I-\\frac{i}{\\hbar}\\int_0^t dt' V_0\\sin(\\omega t')\\sigma_1e^{\\frac{2i}{\\hbar}\\Delta\\sigma_3t'}+\\ldots.$$ where $\\sigma_i$ are the Pauli matrices, what is the condition for the second or higher order terms to become non-negligible? I know it scales like $V_0>V_0^2$ for the first to second order, but what is the role of the time? at very long times, the second term must become more and more important, what is that time scale? (based on this question: Rigorous justification for rotating wave approximation )"}
{"id":"14323","title":"3D Quantum harmonic oscillator","text":"For an _isotropic_ 3D QHO in a potential $V(x,y,z)={1\\over 2}m\\omega^2(x^2+y^2+z^2)$. I can see by independence of the potential in the $x,y,z$ coordinates that the solution to the Schrodinger equation would be of the form $\\psi(x,y,z)=f(x)g(y)h(z)$. Explicitly, what would it be? Is it $\\psi(x,y,z) = cH_{n_x}H_{n_y}H_{n_z}e^{-{m\\omega\\over2\\hbar}(x^2+y^2+z^2)}$, where $H_{n_i}$ are the $ith$ Hermite polynomial? (A side query, surely since the potential is radial, there is a polar coordinate form of solution which might be better? But this is not asked for in the question. Also, does _isotropic_ just mean that the potential is spherically symmetric?) How many linearly independent states have energy $E=\\hbar\\omega({3\\over2}+N)$? Am I supposed to be counting the number of combinations of $n_x,n_y,n_z$ s.t. $n_x+n_y+n_z = N$? I vaguely remember some notion $(n,l)$ mentioned once, but I can't remember what it is nor find the bit of notes on this. Thanks in advance."}
{"id":"94976","title":"Berry curvature of Landau levels","text":"If we consider an electron on a two dimensional surface with a magnetic field normal to the surface, we know the states the electron can occupy are Landau levels. If we additionally impose periodic boundary conditions, i.e., that these states are eigenstates of the magnetic translation operators, then our surface becomes a torus, and the eigenstates are generalized Bloch states $\\psi_{\\mathbf{k}}(x,y) = e^{i\\mathbf{k}\\cdot\\mathbf{r}}u_{\\mathbf{k}}(x,y)$, with $k_i$ taking values in $\\frac{2\\pi}{L_i}n, 1 < n < N_{\\Phi}$, where $L_i$ are the system dimensions, and $N_{\\Phi}$ is the number of flux quanta passing through the area $L_x L_y$. For this boundary condition to be meaningful, we further require that the magnetic translation operators in either direction commute, so that $L_x L_y = 2\\pi N_{\\Phi} \\ell^2$ ($\\ell$ is the magnetic length). We can write down wavefunctions $\\psi_{\\mathbf{k}}(x,y)$ that statisfy these boundary conditions, as done, for example, by Yoshioka (http:\/\/prb.aps.org\/pdf\/PRB\/v29\/i12\/p6833_1) and Haldane and Rezayi (http:\/\/prb.aps.org\/pdf\/PRB\/v31\/i4\/p2529_1). To calculate the Berry curvature, one could simply calculate the Berry connection in some gauge, $$ \\mathbf{A} = \\int d^2x \\,\\, u_{\\mathbf{k}}^*(\\mathbf{r}) \\nabla_k u_{\\mathbf{k}}^*(\\mathbf{r}) $$ and then the curvature $$ F_{xy} = \\partial_{k_x}A_y - \\partial{k_y}Ax $$ We know the Berry curvature for the Landau levels should be constant in $\\mathbf{k}$. My question is, how does one do this in practice? It seems that the calculation of the Berry connection (in the Landau gauge $A = -Bx\\hat{k_y}$) reduces to calculating the expectation value of the x position $\\left<x\\right>$. However, while this calculation would be trivial in an infinite plane, with periodic boundary conditions the wavefunctions become much more complicated."}
{"id":"17984","title":"Why do balls in a spinning ellipsoid move to the minor axis plane?","text":"There is a question concerning the Physics of a small child's tall that has been bothering me for some time now. I have investigated this to a small degree, but I have not been able to find a satisfactory answer and I believe the resulting answer must be of an elementary nature. My newborn son has a small plastic toy that is an ellipsoid filled with small, plastic balls (approximately 10-15 balls). The ellipsoid can be revolved around an axis of revolution (the axis of revolution is the major axis of the planar ellipse) and when one spins the ellipsoid, the small, plastic balls find there way (almost immediately) to the plane that is perpendicular to the axis of revolution and contains the minor axis of the planar ellipse where they continue to spin in said plane as long as their speed is sufficiently large. ![Spinning ellipsoid](http:\/\/i.stack.imgur.com\/popDF.png) In the image, the red curve is a metal rod (elephant's arms) that keep the ellipsoid in suspension, the green line is the axis of revolution, and the blue curve is the curve that the small, plastic, balls move to (almost immediately) and rotate along as long as the speed is sufficiently fast enough. Is there an elementary explanation (elementary = 2 undergraduate courses in Physics) for what I am observing? I am also interested in any explanation of what I am observing that would lead me to learning something new."}
{"id":"122229","title":"What is the relationship between Force and Kinetic Energy?","text":"I am an Middle-School (Grade 8) student, who is self-learning classical mechanics. I saw this equation from Classical Mechanics: $\\vec F=m \\vec a$, the following equation is pretty straight forward Force equals to mass x acceleration but I as a newbie to this subject am left very confused. As you may see the $a$ in the formula is an vector and so is the $F$. That in mind, it must mean force & acceleration is an direction we apply to an object. That seems extremely familiar to Kinetic Energy as Kinetic energy tends to be applied to an object in an direction, yet its equation is $E = \\frac{1} {2}mv^2 $ is not too similar except from it pertaining to do with mass and movement. That in mind, my question is what is the relationship between Kinetic Energy & Force?"}
{"id":"94979","title":"What is the energy distribution of light if it has an infinite length?","text":"What is the energy distribution of light if it has an infinite length? ![Electromagnetic wave](http:\/\/i.stack.imgur.com\/wgZiU.png) I have read in one of the answers here on phys.SE that light has actually an infinite length. But then what is the energy distribution of that electromagnetic wave?"}
{"id":"111020","title":"Does Feynman path integral include discontinuous trajectories?","text":"While reading this derivation of relation of Schrödinger equation to Feynman path integral, I noticed that $q_i$ can differ form $q_{i+1}$ very much, and when the limit of $N\\to\\infty$ is taken, there remain lots of paths, which are discontinuous (almost) everywhere — i.e. paths consisting of disconnected points. Do I understand this wrongly? How do such discontinuous paths disappear on taking the limit? Or maybe they have zero contribution to integral?"}
{"id":"129296","title":"Statics with ladder and wall","text":"> _A uniform ladder of mass m and length L rests against the wall. The > coefficients of static friction between the floor and the ladder and between > the wall and the ladder are equal to each other (μ). What is the maximum > value of angle θ that the ladder can make with the wall without sliding._ My way to solve this: since ladder is not moving Torque on the ladder $$ -\\frac{mgL}{2}\\sin\\theta-\\mu N_{f}L\\cos\\theta+N_{f}L\\sin\\theta=0$$ Horizontal forces $$ N_{w}-f_{f}=0$$ Vertical forces $$ N_{f}+f_{w}-mg=0$$ where $N_{f}$ and $N_{w}$ is normal forcé due to floor and Wall, $f_{w}$ and $f_{f}$ friction due to Wall and floor. Now I solve for $N_{f}$ the first equation, to get: $$N_{f}=\\frac{mg\\tan\\theta}{2(\\tan\\theta-\\mu)}$$ Sincé the angle is the situation where the ladder is about to start sliding, friction forces will have their máximum value, so: $$f_{f}=\\mu N_{f}$$ from 2º equation: $$N_{w}=f_{f}$$ and for same reason like befor $$f_{w}=\\mu N_{w}$$ substituting all this values in the 3º equation I get: $$\\frac{\\tan\\theta}{2(\\tan\\theta-\\mu)}(1+\\mu^{2})=1$$ and solving this for $\\tan\\theta$: $$\\tan\\theta=\\frac{2\\mu}{1+\\mu^{2}}$$ but aparently this is wrong. Could someone check this and point where is the mistake. Ty"}
{"id":"111029","title":"Probability of spontaneous Boltzmann brain formation","text":"I was reading through: http:\/\/www.scottaaronson.com\/papers\/giqtm3.pdf But I can't make sense of page 61 discussing Boltzmann brains. Specifically the fact that it says: > But the problem is worse. Since in an eternal universe, you would have > infinitely many Boltzmann-brain doppelgängers, any observer with your > memories and experiences seems infinitely more likely to be a Boltzmann > brain, than to have arisen via the “normal” processes of Darwinian evolution > and so on starting from a Big Bang! I don't understand what this last statement follows from and it seems very unintuitive. A Boltzmann-doppelgänger brain arising by chance seems very improbable since that would require the formation of a brain with a full brain state, i.e. essentially all the individual atoms\/molecules appearing in the exact (or at least close enough) configuration. In other words, a very high level of organization. Whereas the Big Bang scenario merely assumes a hydrogen\/helium cloud and that the basic laws of nature are in place. This seems to be a much simpler object as a starting point and hence much more probable. This starting point, once it has occurred (maybe due to chance) then evolves (through the relatively simple laws of nature) into the state we have now, and gives rise to processes like natural selection etc. which eventually bring about intelligent life and the very high level of organization in the brain. Evolution through natural selection, while fundamentally also depending on stochastic events, seems much more probable since natural selection guides the process, allowing it to \"tune in\" on the target state, rather than there being a requirement of the target state just emerging in one step as with the Boltzmann-doppelgänger. Another argument is that the human brain, while seemingly a very complex structure, is probably the result (as part of fetus\/child development) from a much simpler process, described in the human DNA. Like how a seemingly complex fractal can emerge from a simple equation. Hence, it seems a much easier way of producing a brain by chance is to produce this simpler mechanism by chance \"and let it run\" until it forms a brain, rather than demanding that a full brain itself appears by chance as in the Boltzmann case. But given Scott Aaronson is very smart there must be something about this I'm missing. Because I'm reading the statement as saying that the Boltzmann- doppelgänger are more probable than the big bang, but to me it seems the converse is true."}
{"id":"8735","title":"Question on Conformal Field Theory","text":"Since every question has to be asked in a seperate topic, I'm asking a question refering to the following topic: Beginners questions concerning Conformal Field Theory In particula I'm refering to the subsection \"Implementing a symmetry on operators\" of the answer given by Lubos Motl. It is clear to me that we obtain $\\delta\\phi=i\\epsilon [L_{m},\\phi]=i \\epsilon z^{m+1} \\partial_{z} \\phi$ and that $z^{m+1} \\partial_{z}$ are the generators of conformal symmetry. What confuses me though is that these generators fulfill the Witt-Algebra and not the Virasoro Algebra. Since we're in the quantum case and we want to show that the theory is invariant on a quantum level, shouldn't we get generators that fulfill the Virasoro Algebra? Thanks in advance for the responses. Best regards."}
{"id":"8733","title":"Give me examples of crackpots who were right after all","text":"I am interested in examples of crackpots coming up with correct results in physics. Why do mainstream physicists look down so much upon \"crackpots\"?"}
{"id":"105741","title":"Fuel and heated iron?","text":"If I have a pile full of fuel (gasoline) - and I also have heated iron ( very very hot - color orange) and I put the Iron in the pile , Would the gasoline will burn ? I mean - does gasoline need heat or actual fire in order to burn ? nb - I can't test it since I don't have the means to do it."}
{"id":"78943","title":"Why doesn't sound infinitely loop?","text":"Let say we have a microphone and a speaker. The microphone will receive the surrounding sound, and the speaker will amplify it. Assume we have unlimited power for the microphone and speaker, so they can work forever. When we speak to the microphone, the sound should be repeated by the speaker. While the speaker generates sound, the sound should input to the microphone, and the microphone should repeat to the speaker. So, the sound should never stop, in a loop. But in reality, we don't have this issue. What's going on? Do my assumptions have something missing? Thanks."}
{"id":"66954","title":"Electric Field Lines between two non parallel plates","text":"**A potential difference is applied between two metal plates that are not parallel. Which diagram shows the electric field between the plates?** ![enter image description here](http:\/\/i.stack.imgur.com\/lFmHv.png) A is the answer BUT WHY explanation needed !"}
{"id":"126837","title":"Constructing Ward identity associated with conserved currents","text":"Consider constructing the Ward identity associated with Lorentz invariance. It is possible to find a 3rd rank tensor $B^{\\rho \\mu \\nu}$ antisymmetric in the first two indices, then the stress-energy tensor can be made symmetric. Once done, the conserved current coming from the classical analysis is of the form $$j^{\\mu \\nu \\rho} = T_B^{\\mu \\nu}x^{\\rho} - T_B^{\\mu \\rho}x^{\\nu}$$ This ensures the symmetry of the conserved current which can be seen most easily be invoking the conservation law $$\\partial_{\\mu}j^{\\mu \\nu \\rho} = 0 $$ and $$\\partial_{\\mu}T_B^{\\mu \\nu} =\\partial_{\\mu} (T^{\\mu \\nu}_C + \\partial_{\\rho}B^{\\rho \\mu \\nu}) = 0.$$ Let $X$ denote a set of $n$ fields. The Ward identity associated with Lorentz invariance is then $$\\partial_{\\mu} \\langle (T^{\\mu}x^{\\rho} - T^{\\mu \\rho}x^{\\nu})X\\rangle = \\sum_i \\delta(x-x_i)\\left[ x^{\\nu}_i \\partial^{\\rho}_i - x^{\\rho}_i\\partial^{\\nu}_i\\langle X \\rangle - iS^{\\nu \\rho}_i \\langle X \\rangle\\right].\\tag{1}$$ This is then equal to $$\\langle (T^{\\rho \\nu} - T^{\\nu \\rho})X \\rangle = -i\\sum_i \\delta (x-x_i)S^{\\nu \\rho}_i\\langle X \\rangle,$$ which states that the stress tensor is symmetric within correlation functions, except at the position of the other fields of the correlator. My question is: how is this last equation and statement derived? I think the Ward identity associated with translation invariance is used after perhaps splitting (1) up like so: $$\\sum_i^n x^{\\nu}_i \\sum_i^n \\delta(x-x_i)\\partial^{\\rho}_i \\langle X \\rangle - \\sum_i^n x^{\\rho}_i \\sum_i^n \\delta(x-x_i)\\partial^{\\nu}_i \\langle X \\rangle - i\\sum_i^n\\delta(x-x_i)S^{\\nu \\rho}_i\\langle X \\rangle $$ and then replacing $$\\partial_{\\mu}\\langle T^{\\mu}_{\\,\\,\\,\\rho}X \\rangle = -\\sum_i \\delta (x-x_i)\\frac{\\partial}{\\partial x^{\\rho}_i} \\langle X \\rangle$$ for example. The result I am getting is that $$\\langle ((\\partial_{\\mu}T^{\\mu \\nu})x^{\\rho} - (\\partial_{\\mu}T^{\\mu \\rho})x^{\\nu} + T^{\\rho \\nu} - T^{\\nu \\rho})X \\rangle = \\sum_i x^{\\nu}_i \\partial_{\\mu}\\langle T^{\\mu \\rho}X \\rangle + \\sum_i x^{\\rho}_i \\partial_{\\mu} \\langle T^{\\mu \\nu} X \\rangle - i\\sum_i\\delta(x-x_i)S^{\\nu \\rho}_i\\langle X \\rangle$$ To obtain the required result, this means that e.g$$ \\sum_i x^{\\nu}_i \\partial_{\\mu} \\langle T^{\\mu \\rho}X \\rangle = \\langle(\\partial_{\\mu}T^{\\mu \\rho})x^{\\nu} X \\rangle,$$ but why is this the case? Regarding the statement at the end, do they mean that when the position in space $x$ happens to coincide with one of the points where the field $\\Phi_i \\in X$ takes on the value $x_i$ (so $x = x_i$) then the r.h.s tends to infinity and the equation is then nonsensical?"}
{"id":"126831","title":"Is this theory about Universe information true?","text":"I recently saw this video about information and randomness. At some point, it states that a completely predictable universe would infringe the second law of thermodynamics, because it would imply that the overall information of the Universe would be conserved throughout time, thus leading to a costant entropy Universe. Also, it states that the additionally information that is increasing entropy is coming from QM randomness. I recognize it's a really fascinating theory, but is it true? Is the equivalence between information and entropy sensible?"}
{"id":"126833","title":"Action of the Poincare Group on a Scalar Function","text":"Let $F(x^\\mu)$ is a scalar function; i.e. $F(x^\\mu): \\mathbb{R}^{1,3} \\rightarrow \\mathbb{R}$. How the Poincare Group $P(1,3)$ will act on it; i.e., by which formula I can calculate it for a specific function $F(x^\\mu)$? Edit 1: We know that a subgroup of Poincare group $P(1,3)$ leaves the hypersurface $\\Sigma: x^+ = 0$ invariant where $x^+ \\equiv x^0 + x^3$. To show this a formula is needed."}
{"id":"12444","title":"Feynman diagrams in effective theories","text":"I've been seeing many feynman diagrams lately that I can't quite interpret yet. I've heard a basic Quantum Field Theory lecture and so to me, a Feynman diagram is simply a mnemonic picture to quickly write down (and remember) all of the possible terms in the perturbation series of the matrix element in scattering. But they seem to be much more than that. Consider for example this diagram of proton-neutron scattering via pion exchange. The picture seems to have an intuitive meaning, but how can it be a valid Feynman diagram? Wouldn't that imply that there is an unterlying theory with a proper Lagrangian from which Feynman rules can be derived that would assign a number to this diagram? **Question: When describing nucleon-nucleon interactions in this way, do people actually write down a Lagrangian and derive those rules?** (I am confused because I've seen diagrams like that in several books and they usually write down the cross sections simply from analogy arguments to other theories.) Any clarification will be greatly appreciated, David"}
{"id":"8194","title":"Has somebody ever tried to measure the gravity oscillations of nearby rotating masses?","text":"I'd like to extend this question: Are Newton's gravity waves detectable by a laser interferometer? but with some changes. Has somebody ever tried to measure the gravity oscillations nearby (< some meters) of such rotating masses (as in Carl Brannens question)? Say some kgs or grams rotating on a beam with the highest rpm technically possible? (In a vacuum chamber of course!) I dont know what will \"radiate\" stronger , some grams at 100 000 rpm or some kg at say, 5000 rpm (and some cm distance), therefore, has anybody ever tried something like that in lab? I am aware, that such a experiment is plagued by mechanical (elastic) coupling through air, supporting structures etc, but I think that one can make use of mechanical resonance in the detecting device, helping a lot for selectivity. The fact that the oscillations come from a quadrupole(frequency doubling) , whereas mechanical coupling is not at double frequency, might help to \"see\" something. As often, its a question of signal\/noise ratio :=("}
{"id":"92980","title":"What is the intrinsic colour of the star?","text":"Hey I have this question and I am wrestling with it all day and I am completely lost. The V magnitude of a star is 15.1, B-V =1.6, and absolute magnitude Mv= 1.3 The extinction in the direction of the star in the visual band is av = 1 mag\/kpc. What is the intrinsic colour? Can somebody help me? Edit: Ok, apparently I have to calculate first the distance and after that the excess colour and then I can calculate the intristic colour. I guess I need this formula for the distance: V = Mv + 5log(r\/10pc) + Av But if i fill this in and get the R I get a different r and I dont know what the next step if to go to the excess colour."}
{"id":"19244","title":"From spectrum\/dispersion relation to the partition function","text":"I know the spectrum\/dispersion relation for a bosonic system. $$E \\left( \\mathbf{k} \\right) = \\cdots$$ Is there a general method for writing down the partition function when the spectrum of the system is known? Thanks in advance!"}
{"id":"19246","title":"Putting a spoon in boiling tea brings the froth down. Why?","text":"Two questions really: 1\\. Why does putting a spoon in boiling tea bring the froth down? 2\\. Also, the tip of spoon (outside tea) never heats up. I cannot imagine that normal air can have so much of a cooling effect. The heat conduction from spoon inside tea should be far more than cooling effect of air. Isnt it?"}
{"id":"96609","title":"Spring Force in a Dynamic Equation","text":"I am working on dynamic simulation of a bipedal robot, I have dynamic equations of a preliminary structure. But Now I have to add some springs to the dynamics. I am having a problem of how to account for the forces that will be produced by spring. I have solved the ODE by calculating the force vectors and it runs good, but now I want to solve using Lagrangian equations. The problem is I am able to compute the Kinetic Energy using the Jacobians and generalised variables but I am having the problem accounting the spring Forces in the Potential Energy. ![enter image description here](http:\/\/i.stack.imgur.com\/elaSl.png) As You can see in the image, My question, How to account for the Force $F$ on $M_1$, $M_2$ and $M_3$, in a Dynamic equation using Lagrange's method."}
{"id":"96600","title":"Measuring radon level","text":"In theory what are the techniques used to measure the levels of radon in air? On which principle\/law do they rely?"}
{"id":"117292","title":"Why is initial position negative?","text":"> A student on ground throws a ball straight up, the ball leaves students hand > with speed of 15m\/s when hand is 2.0mm above ground How long is the ball in > the air before it hits the ground? I was able to figure out the answer to this question using $V_0 = 15\\text{ m\/s}$, $t_0 = 0\\text{ s}$, $Y_0 = 0\\text{ m}$, $Y_1 = -2\\text{ m}$, $a = -9.8\\ \\mathrm{m\/s^2}$ and using $S_f = S_i + V_i t + \\frac{1}{2} a t^2$ I could find $t_1$. My question is, why is $y_1$ negative? I also remember learning that $a$ is always negative, at least in these kind of problems."}
{"id":"117293","title":"Can all laws of physics derived by a single or lists of more general laws?","text":"(This is actually my first question - please understand my lack of knowledge and bad English..) I have always been curious about deriving thousands of laws from more general ones.. Can general laws or models of physics explain minor concepts completely? For instance, can Standard Model explain, well, F=ma?"}
{"id":"107457","title":"Is this expression of electric field correct?","text":"I know that the electric field component of electromagnetic wave should be written as the first equation shown, but some times it can be expressed in terms of complex amplitude of electric field as in second equation. I later found integration in the exponent term and I didn't know why this term appeared. Which one is correct, or more general? $$ \\mathbf{E}(\\mathbf{r},t)=\\mathrm{Re}\\left[\\hat{\\mathbf e}F(x,y) A(z,t)\\exp(i\\beta_0-i\\omega_0t)\\right], $$ $$ \\mathbf{E}=\\exp\\left(-j\\int_0^z\\beta_\\text{eq}\\mathrm dz\\right)\\mathbf{A}. $$"}
{"id":"113616","title":"Why does sound need air in order to travel?","text":"I know that in space, because of vacuum that we can't hear voices or sound to be specific, why is that? Why does sound need a medium to go through?"}
{"id":"43164","title":"If 1 screw can support 120 lbs, how much weight can 25 screws support?","text":"While this is mainly about my personal home improvement project, I figure this would be a good opportunity for some of you to apply your physics skills to a real-world situation. My situation is this: I am hanging 500 lbs of drywall (two layers weighing about 250 lbs each) on 5 rows of metal furring channels (hat channels) which are mounted onto RSIC (sound insulation clips) which are screwed into studs with drywall screws. There are 25 total clip assemblies across all 5 rows of channels. This wall is suspended (ie, mounted solely on channels and none of the edges touch the walls, floor, or ceiling). Here's a diagram of a single clip assembly: ![digaram](http:\/\/i.stack.imgur.com\/7AywV.jpg) **Edit 11\/1\/2012 4:56AM EDT** Below is a diagram of my actual clip assembly array. The wall is 13' 10\" long and 8' tall. The vertical black lines represent the studs and are spaced apart at about 16\" (it's not exact because I had to add a couple studs and the frame joins up with another frame at one point in the wall), the red dots are the clip assemblies, and the horizontal grey lines are the hat channels that are mounted onto the clips. Ignore the green dots and yellow arrows. ![clip array](http:\/\/i.stack.imgur.com\/d8Uwc.jpg) The problem: Since drywall screws are not as strong as wood screws which I should have used, I am worried that the 25 screw\/clip assemblies may not be strong enough to bear the weight of 500 lbs. To test the strength of a single screw, I mounted a single clip into a dummy stud and ended up being able to hang 120 lbs of dumbells on it. This mock assembly has been up for over a month and shows no signs of being on the verge of snapping. I could probably hang another 30 lbs on it before it gave away. So, using simple math, it seems I would be able to say, \"Well, if one screw can support 120 lbs then 25 screws can support 3,000 lbs!\" Of course, I am sure the math is not so simple. I am sure there is a curve there somewhere where adding more screws certainly allows for greater weight capacity, but the weight capacity deteriorates the more screws and more weight you add even if the number of screws and amount of weight was proportionate. So is it really as easy as using simple math to solve this problem or does it require something more advanced? BTW, I just took stabs at tags here. I am hopinq this is an actual physics question and that it will not get deleted. I've already posted a similar question as this on SE DIY site, but I think it's not so much related to handiwork as it isnecessarily"}
{"id":"2532","title":"What evidence exists for string theory viability?","text":"I know that string theory is still under heavy development, and as such it still can't make predictions (or not that many predictions anyways). On the other hand, it is clear from the number of years this theory has been under development and from the large number of theoretical physicists studying it, that it is considered a good and viable candidate as a quantum gravity theory. > So, what is the evidence that this is true? Why is it considered such a good > candidate as the correct quantum gravity theory? Without wanting to sound inflammatory in the least, it has been under heavy development for a very long time and it's still not able to make predictions, for example, or still makes outlandish statements (like extra dimensions) that would require a high amount of experimental evidence to be accepted. So - if so many people believe it is the way to go, there have to be good reasons, right? What are they?"}
{"id":"46534","title":"Showing that the probability density of a linear harmonic oscillator is periodic","text":"The complete question I am trying to answer is the following: Show that the probability density of a linear harmonic oscillator in an arbitrary superposition state is periodic with period equal to the period of the oscillator. I have: $$\\Psi(x,t)=\\sum\\limits_n c_n \\psi_n(x) \\exp\\left(-\\frac{iE_nt}{\\hbar}\\right)$$ Therefore, the probability density is: $$P(x,t)=\\Psi^*(x,t)\\Psi(x,t)=\\sum\\limits_n c_n^2 \\psi_n^2(x) $$ (The exponential term should cancel out because of the conjugation). So I get the probability density as being time-independent, which seems wrong, according to how the question is stated. In the solution in the book, we have:$$P(x,t)=\\Psi^*(x,t)\\Psi(x,t)=\\sum\\limits_m \\sum\\limits_n c_m^* c_n\\psi_m^*(x)\\psi_n(x) \\exp\\left(\\frac{i(E_m-E_n)t}{\\hbar}\\right)$$ I don't understand why, in the solution, we have the conjugate of the wavefunction have the subscript $m$, while the wavefunction itself has the subscript $n$. It changes the whole thing, when we introduces different subscripts. Moreover, the formula for probability density is $$P(x,t)=\\Psi^*(x,t)\\Psi(x,t)$$not $$P(x,t)=\\Psi_m^*(x,t)\\Psi_n(x,t)$$ What is going on here?"}
{"id":"22839","title":"What is \"surface fluid adhesion energy\"?","text":"This is related to my previous question. Pardon me for asking so many questions recently. My physics knowledge is not that good, and some answers are hard to find. In the question in the link, I asked what was the correct interpretation for $\\beta_i$ in the energy formulation: $$\\mathcal{F}(S_1,S_2,S_3)= \\sum_{1\\leq i<j\\leq 3}\\sigma_{ij}P_\\Omega(S_i \\cap S_j)+\\sum_{i=1}^3 \\beta_i P_{\\partial \\Omega}(S_i)+\\sum_{i=1}^3 g \\rho_i\\int_{S_i} z dV$$ I got an answer which said that the correct interpretation for $\\beta_i$ is **surface fluid adhesion energy**. I tried googling the term, but I did not find anything clear. My question is > What is **surface fluid adhesion energy** and how is this related to the > fluid-solid interfacial tension? What references are there on this subject? * * * Here is the context: So, in my hypothesis the container $\\Omega$ is partitioned in $S_1,S_2,S_3$, the three fluids, with prescribed volumes $v_i$ and prescribed densities $\\rho_i$. I took into account in the formulation of the energy the interfacial tensions, the gravity, and the contact of the fluids with the container $\\Omega$ (these are not my ideas; they are taken from other similar mathematical articles). I will denote $P_\\Omega(S)$ the perimeter of $S$ situated in the interior of $\\Omega$ and $P_{\\partial \\Omega}(S)$ the perimeter of $S$ which is situated on the boundary of $\\Omega$. I will not be very rigorous in what I'm about to write: I will write, for example $P_\\Omega(S_i\\cap S_j)$ the perimeter of the intersection $S_i\\cap S_j$ even if as set theory intersection, this is void. Still, I think that the idea will be clear. * * *"}
{"id":"17164","title":"Unconstrained action in Yang-Mills theory","text":"Is there any gauge in which Yang-Mills theory (4d, non-SUSY) can be written as a local action containing only the propagating modes?"}
{"id":"68136","title":"In turbulence theory, what happens if i take space average of fluctuating part?","text":"According to Reynolds decomposition, velocity field is split into two time average and fluctuating parts: $$u_{({\\bf x},t)}=\\overline u_{(\\bf x)}+u'_{({\\bf x},t)}$$ we know that time average of fluctuating part is zero $\\overline u'=0$, but what if i take space average of fluctuating part?"}
{"id":"68134","title":"Casimir effect when the zeta function has a pole","text":"let us suppose the Casimir force\/ Vaccumm energy of a certain system $$ E_{casimir}= \\sum _{n} \\omega _{n} $$ which is formally equal to $ E_{casimir}= \\zeta _{spec}(-1) $ with $ \\zeta (s)_{spectral}=\\sum_{n}\\omega _{n} ^{-s}$ however what would happen if the spectral function has a pole at $ s=1 $ what would be then the value of $ E_{casimir} $ can this pole problem be overcome by regularization ?? perhaps in this case we could take the finite part $ FP \\zeta (-1)_{spec}= \\sum_{n}\\omega_{n}-\\int_{0}^{\\infty}\\omega (x)dn(x)$ the secon integral would be the SEMICLASSICAL approximation ot the frecuencies energies $ E_{n}= \\hbar \\omega _{n} $"}
{"id":"17168","title":"What is the rationale behind representing a state function by a complex valued function in QM?","text":"What is the rationale behind representing a state function of an electron with a complex valued function $\\Psi$. If only the probabilistic argument was required then why not represent it with just a real valued which directly gives the probability. My question is why should we use complex valued state functions? There must be some strong non-mathematical reason which I am not able to find in books I have referred?"}
{"id":"94110","title":"Error propagation estimations for sine and cosine","text":"My lab manual gives this: $B$ is a function of $A$, Greek are uncertainties... $$B + \\beta = \\sin(A + \\alpha) = \\sin(A)\\cdot\\cos(\\alpha) + \\sin(\\alpha)\\cdot\\cos(A)$$ \\--> because $\\alpha$ is taken to be (at least relatively) small, $\\cos(\\alpha) \\to 1$, and $\\sin(\\alpha) \\to \\alpha$, measured in radians. I see that $\\cos(\\alpha) \\to 1$, but I would have expected that $\\sin(\\alpha)$ for a small would, by the same logic, go to 0. Has it got something to do with the rate of change of either function near zero? Why is $\\cos(\\alpha)$ of small $\\alpha$ not also proportional or written by relation to $\\alpha$?"}
{"id":"9569","title":"Cassini 's and Pioneers RTG power output datafile","text":"I'd like to have access to the datafile of the electrical output of the RTG (Radioisotope Thermoelectric Generator) of the Pioneer's and Cassini missions, as can be seen in the figures 1 of the papers of Anderson,2002 and Peter S. Cooper, publ 2009.   I know that I can grab the aprox values directly from the published graphs with the help of software, but accessing the original data is better. I g-searched but I did not found them. Can someone give advice, pls? The question is specific but the answers can say more about the difficulties and solutions of accessing scientific data."}
{"id":"90605","title":"Stability of square of masses on strings under rotation","text":"Imagine we have a square of masses, $m$, connected by light inextensible strings, length $l$, rotating around it's centre at angular speed, $\\omega$. It's easy enough to show that there must be a tension in these strings, $T=\\frac{ml\\omega^2}{2}$. However if we then increase the mass of any two opposite corners resolving forces seems to show that there is no solution for continued circular motion except when the two heavier masses are a distance $l$ from the c.o.m and the lighter ones, a distance 0. I.e. the problem reduces to a line. Is this true, or has my logic just broken down at some point? Also, the same seems to happen if you keep all the masses equal and change any angle just a little. Does this mean that a rotating square (or any other regular shape) of equal masses such as this is the only equilibrium position for the family of systems of shapes and masses such as this? Here's a diagram if anyone's having trouble picturing it, just a random one from the net that had most of the right features (ignore the labels), it is colourful though... ![](http:\/\/i.stack.imgur.com\/20dec.png)"}
{"id":"104803","title":"Ideal gas system with diathermic piston externally manipulated","text":"I wanted to ask about the following problem: An ideal monatomic gas is separated into two volumes $V_{1}$ and $V_{2}$ through a diathermic piston, such that each volume containing $N$ atoms and the two sides are at the same temperature $T_{0}$. The entire system is isolated from the outside by means of insulating walls. The plunger is reversibly externally manipulated until the two gases are in thermodynamic equilibrium with each other. After the problem has different questions, however, I have not been able to answer the following: How is called the type of process? Ie is isothermal, adiabatic, etc. Show that $\\Delta S_{1}$ = $-\\Delta S_{2}$, where $\\Delta S_{1}$ and $\\Delta S_{2}$ are the changes in entropy of the two gases."}
{"id":"104808","title":"Electricity from lightning","text":"1. According to the internet, a lightning strike contains about 5 billion joules or 5 GJ. How was this calculated?    2. Another thing: Once the lightning strike a metal grounded rod, a current will flow. What type of that current and how much current? I think it will be a 325 MV current which is the voltage of the lightning. Is that correct? Edit: after i read a little bit about lightning , i found it was such a big scale , so how about a small electrostatic charge , say from a 1 MV van de graaff generator, i have a spark source ( my CRT TV :D ) and i will buy a multimeter and conduct some experiments , but i need to know what to expect"}
{"id":"80445","title":"Wave functions for 2D potential with spin interactions","text":"So consider a 2D system with a circular potential and a spin-orbit interaction: $V(r) = V_0 \\theta(r_0 - r) + c r_0 V_0 L_z S_z \\delta(r-r_0)$ where $\\theta$ is step function. So the operators $L_z$ and $S_z$ commute with the Hamiltonian are are therefore conserved quantities. For the same reasons we can write the wave function as a product of radial and orbital parts (and spin parts too): $R(r) e^{i l \\theta}$ where $\\theta$ here is the polar angle and $l$ is the orbital quantum number. A spinor can be affixed to the wave function but seems unnecessary as no transitions can occur for the spin. My question regards adding another spin interaction to $V(r)$ of the type $b_z S_z\\theta(r_0-r)$ that only acts within the circular potential. Will the form of the wave functions change as a result of this addition? My thought is that the wave functions remain the same since once again spin should be preserved so the spinors will not see any further structure. The only purpose of this new interaction will be to alter the effective potential of the circular well - the potential will be shifted either up or down depending on the direction of the spin ($m_s$ = up or down). So is my reasoning correct? I understand that this problem becomes much more difficult in 3D when the full spin-orbit interaction is used since then you will have a lack of commutation."}
{"id":"125918","title":"What happens to proteins under x ray\/gamma ray radiation?","text":"A simple question to which I am yet to find an answer. I have not found any papers which would go into detail about it, there is some stuff on protein damage in x-ray crystallography but they are looking at the crystallized proteins, I am interested in \"natural\" proteins..."}
{"id":"48073","title":"Higher dimensions","text":"1) How we determine whether the higher dimensions are Unstable or Unpredictable? Or on the basis of what assumption we make this prediction? ![Dimensions](http:\/\/i.stack.imgur.com\/0jX6u.gif) (Source of Image: Max Tegmark. See also this Wikipedia article.)"}
{"id":"99267","title":"If two ends were a certain \"length\" apart were they therefore at rest (or at least rigid) to each other?","text":"Considering the definition of the SI unit of \"length\" [1] and [2 (\" method a.\")] I'm missing any requirements about the two \"ends\" of the required \" _path travelled by light_ \" being \"at rest to each other\", or at least \"rigid to each other\". Are such requirements perhaps presumed to be understood implicitly? Accordingly, if some particular pair of \"ends\", $A$ and $B$, in some particular trial, were characterized as having been a certain \"lenght of $x \\, \\text{m}$ apart from each other\", where \"$x$\" is some particular positive real number and \"$\\text{m}$\" denotes the SI base unit \"metre\", is it then understood:   * that ends $A$ and $B$ had observed exchanging signal pings between each other; and not only once for each trial, but for any of their signal indications throughout a sufficiently extended trial?,   * that for any two (distinct) signal indications $A_J$ and $A_K$ of end $A$ during this trial the corresponding ping durations of end $A$ were equal to each other:   $\\tau_A[ \\,_J, \\,_{\\circledR}^{B \\circledR AJ} ] = \\tau_A[ \\,_K, \\,_{\\circledR}^{B \\circledR AK} ]$ ?,   * that for any two (distinct) signal indications $B_P$ and $B_Q$ of end $B$ during this trial the corresponding ping durations of end $B$ were equal to each other:   $\\tau_B[ \\,_P, \\,_{\\circledR}^{A \\circledR BP} ] = \\tau_B[ \\,_Q, \\,_{\\circledR}^{A \\circledR BQ} ]$ ?, and   * that for any signal indications $A_J$ of end $A$ and any signal indication $B_P$ of end $B$ during this trial the corresponding ping durations were equal to each other:   $\\tau_A[ \\,_J, \\,_{\\circledR}^{B \\circledR AJ} ] = \\tau_B[ \\,_P, \\,_{\\circledR}^{A \\circledR BP} ] = \\frac{2 x}{c} \\text{m} $ ?. References:   [1] SI brochure (8th edition, 2006), Section 2.1.1.1; http:\/\/www.bipm.org\/en\/si\/base_units\/metre.html (\" _The metre is the length of the path travelled by light in vacuum during a time interval of 1\/299 792 458 of a second._ \"). [2] \"the mise en pratique of the definition of the metre\"; http:\/\/www.bipm.org\/en\/publications\/mep.html"}
{"id":"9387","title":"Lawrence Krauss' Theory of Origin","text":"I don't understand how a vacuum, the absence of matter, can hold energy. How can it hold energy when Einstein proved that matter is energy? And a second related question; how does the energy in a vacuum allow the Universe to produce itself? And how does that account for why the Universe is expanding? I saw Krauss' lecture on YouTube called Universe from Nothing and only understood some of it, even though he was fantastic at putting things in layman's terms and making these concepts easy to visualize. However, he had a bunch of \"asides\", so it was a bit confusing. He mentioned that the curvature of the Universe was flat, but then also went on to say how that was the worst possible scenario. Why is a flat Universe such a bad thing?"}
{"id":"57921","title":"Rotational Constant and Moment of Inertia of Fluorine gas","text":"I have come across some homework question on thermodynamics which needs me to calculate $B$ of $F_2$ My attempt: $B= \\frac{h}{8\\pi^2cI}$ where $I=\\mu r^2=\\frac{m_1m_2}{m_1+m_2} r^2$ Atomic mass of fluorine atom is $18.998$ $g$ ${mol}^{-1}$ and Bond length is $141.8\\times10^{-12}$ $B=\\frac{6.626\\times 10^{-34}}{8 \\pi^2 (3\\times10^{10})\\frac{\\frac{18.998^2}{18.998\\times2}}{6.02\\times 10^{23} \\times 1000}(141.8\\times10^{-12})^2}\\approx0.882$ which I think it is wrong because when I continue to calculate the rotational temperature $\\Theta_R$, the result is way too low. $\\Theta_R=\\frac{hcB}{k_B}=\\frac{(6.63\\times10^-34)(3\\times10^{10})(0.882)}{1.38\\times10^{-23}}=1.27 K$, $K$ for kelvin. Please help me find the mistakes in the calculations..."}
{"id":"68868","title":"Electromagnetic wave propagation through two lossless dielectrics","text":"In Elements of Electromagnetics (Sadiku, 3rd edition, Section 10.8), the author says to consider two lossless dielectric materials joined at an interface $z=0$. Here two lossless dielectric materials can be translated to mean that the conductivity of medium 1 (located to the left of $z=0$, i.e. for $z<0$) is equal to the conductivity of medium 2 (located to the right of $z=0$, i.e. for $z>0$) which is equal to 0 Siemens\/meter. Recall that intrinsic impedance is given by $$\\eta = \\sqrt{\\frac{j\\omega\\mu}{\\sigma+j\\omega\\varepsilon}}$$So if $\\sigma=0$, then $$\\eta = \\sqrt{\\frac{j\\omega\\mu}{j\\omega\\varepsilon}} = \\sqrt{\\frac{\\mu}{\\varepsilon}}$$This means that the intrinsic impedances $\\eta_1$ and $\\eta_2$ are real numbers (with units) since $\\sigma_1=\\sigma_2=0$ S\/m and $\\mu$ and $\\varepsilon$ are taken to be just real constants (with units), i.e. they do not change with space or time, except at $z=0$ where there is a transition from one lossless dielectric to another of course. Also note that the reflection coefficient $\\Gamma$ is given by $$\\Gamma = \\frac{E_{r0}}{E_{i0}} = \\frac{\\eta_2 - \\eta_1}{\\eta_2 + \\eta_1}$$ where $E_{r0}$ is the amplitude of the reflected wave at $z=0$ (the interface) and $E_{i0}$ is the amplitude of the reflected wave at $z=0$ (the interface). The author considers the case where $\\eta_2 > \\eta_1$ (meaning $\\Gamma > 0$). A normally incident electromagnetic wave (normal to the interface) reaches the interface and sends some EM energy through to medium 2 in the form of a transmitted electromagnetic wave and reflects some EM energy in the form of a reflected electromagnetic wave. The incident and reflected waves have different amplitudes. The electric field in medium 1 is the sum of the incident and the reflected waves (described mathematically below) $$ \\vec{E}_{1} = \\text{Re}\\left[\\underbrace{\\left[\\vec{E}_{i0}e^{-\\gamma_1z} + \\vec{E}_{r0}e^{\\gamma_1 z}\\right]}_{\\text{Spatial variation}}\\underbrace{\\left[e^{j\\omega t}\\right]}_{\\text{Temporal variation}}\\right]$$ Here, $\\gamma_1$, is the propagation constant (most generally a complex number) in medium 1 given by $$ \\gamma = \\underbrace{\\alpha}_{\\omega\\sqrt{\\frac{\\mu\\varepsilon}{2}\\left[\\sqrt{1+\\left[\\frac{\\sigma}{\\omega\\varepsilon}\\right]^2}-1\\right]}} + j\\underbrace{\\beta}_{\\omega\\sqrt{\\frac{\\mu\\varepsilon}{2}\\left[\\sqrt{1+\\left[\\frac{\\sigma}{\\omega\\varepsilon}\\right]^2}+1\\right]}} $$ But since $\\sigma_1=0$ S\/m, $\\gamma_1=j\\beta=j\\left[\\omega\\sqrt{\\mu\\varepsilon}\\right]$. Since $e^{j\\theta}=\\cos\\theta+j\\sin\\theta$, taking the real part of a complex exponential is a compact way of writing a sinusoidal wave, which is very relevant here as there are three sinusoidal waves involved here -- the incident wave representing the incident electromagnetic wave's electric field, the reflected wave representing the reflected electromagnetic wave's electric field, and the transmitted wave representing the transmitted electromagnetic wave's electric field. With this background, here is what I do not understand. **Problem** : The author claims, just before Eqs. 10.88 and 10.89, that the amplitude of $\\vec{E}_{1}$, $\\left|\\vec{E}_{1}\\right|$ is maximum for when $$-\\beta_1 z_{\\text{max}} = n\\pi$$Further, the author claims the amplitude of $\\vec{E}_{1}$ is minimum for when $$-\\beta_1 z_{\\text{min}} = \\left(2n+1\\right)\\frac{\\pi}{2}$$I cannot seem to replicate this result. Here is my attempt at a proof. **Partial proof** : Assuming the electric field oscillates in the $\\mathbf{a}_x$ direction (alternately called $\\mathbf{e}_x$ or $\\hat{x}$ direction) and $E_{r0}=\\left|\\vec{E}_{r0}\\right|$, $E_{i0}=\\left|\\vec{E}_{i0}\\right|$ ![Proof up to phasor envelope representation](http:\/\/i.stack.imgur.com\/5aUGr.png) To determine the magnitude of this, it may be helpful to consider the geometry of this situation by using a phasor diagram. The phasor diagram would plot two vectors (phasors) – one of magnitude $E_{i0}$ and the other of magnitude $\\Gamma E_{i0}$ (remember $\\Gamma>0$ in this case but $0\\le\\left|\\Gamma\\right|\\le 1$ holds in general). If the reference phasor is $\\cos\\left(\\beta_1 z\\right)$, then $\\cos\\left(\\beta_1 z-\\omega t\\right)$ will be located a certain distance in the clockwise direction away from the reference and $\\cos\\left(\\beta_1 z\\right)$ will be located the same distance in the counterclockwise direction. The resulting vector (or phasor) sum of the two will have a magnitude of $\\sqrt{E_{i0}^2 + \\left(\\Gamma E_{i0}\\right)^2}$. I am trying to figure out the phase angle for this resulting vector. I am currently stuck here, hoping someone can help me out. Thanks in advance! EDIT: Here's the picture I drew.![Phasor diagram](http:\/\/i.stack.imgur.com\/PN8Ox.png) It seems like this all simplifies to finding $\\phi$ using the law of sines \/ law of cosines. The goal is to see for what values of $\\phi$ will $\\left|\\vec{E}_1\\right|$ be maximum. Hopefully, it'll end up validating what the book says."}
{"id":"109400","title":"Ultrarelativistic limit of helicity values","text":"Let's have helicity operator: $$ \\hat {h} = \\frac{(\\hat {\\mathbf S} \\cdot \\hat {\\mathbf p})}{|\\mathbf p|}. $$ For massive case spin $s$ representation it has $2s + 1$ eigenvalues $s, s - 1,..., -s$, for massless one it has one or two eigenvalues $\\pm s$. How to show that ultrarelativistic limit of massive case gives massless one? I mean that effect like relativistic aberration leads us to the result that in ultrarelativistic case we will see only two possible values of projection of spin on motion direction."}
{"id":"109407","title":"Schroedinger equation. Mass. Charge","text":"The mass of a particle appears in Schroedinger equation but it does not appear its charge, although both terms have their effect on movement.Why?"}
{"id":"68863","title":"What is a geometrical object?","text":"From the Wikipedia link for Geometry: > Geometry (Ancient Greek: γεωμετρία; geo- \"earth\", -metron \"measurement\") is > a branch of mathematics concerned with questions of shape, size, relative > position of figures, and the properties of space. Up to the work of Riemann and Gauss, this definition would have made clear to me examples of geometrical objects: a line, square, cube, hypercube; each of these possessing geometrical properties such as the number of sides, angles between faces, the dimension of space that contains them etc. Hence a geometrical object was a set of measurements associated with an object using a ruler. After the work of Einstein and Minkowski who showed that time and space were a part of one another, would it be correct to say:   1. a geometrical object is a set of measurements associated with an object of distance _and time_ using a ruler and a clock ?   2. Geometrical objects includes an interval of time, the invariant space-time interval?"}
{"id":"72076","title":"Energy of an EM wave compared to energy of a photon","text":"Several posts on this forum ask the question about the role of amplitude in calculating the energy of an Em wave. This struck me as odd since I learned that E=hv. There is no amplitude in the Planck relation. But I see why this might arise as a question since every other description of waves includes the amplitude in energy calculation. This posts on this forum highlight the question: The energy of an electromagnetic wave Energy in an EM wave should depend on frequency Where is the amplitude of electromagnetic waves in the equation of energy of e\/m waves? In one post I read that 'The macroscopic electric field of a wave consisting of photons does have an amplitude which is statistically built up from the individual photons.' My interpretation of the answers in these posts is that the photon is a fundamental unit with a fixed amplitude. In the same way that the speed of light is a fixed velocity, it appears that the amplitude of a photon is also fixed. So this leads to the question of what is the amplitude of a photon Amplitude of an electromagnetic wave containing a single photon I cant seem to determine whether the question was actually answered in that post. But why is it not valid to just use the fact that in a wave, energy is proportional to amplitude squared Why is energy in a wave proportional to amplitude squared E=hv E~A^2 so the amplitude of a photon should be a~(hv)^1\/2"}
{"id":"72073","title":"Black magic \"Hartree\" approximation","text":"The question is about an unusual looking version of the Hartree or mean field approximation. The context is several papers I've been reading recently about the out of equilibrium dynamics of phase transitions in the early universe [1-3]. The procedure is to shift a (scalar) field $\\Psi \\to \\psi + \\langle \\Psi \\rangle$, where $\\psi$ are fluctuations and $\\langle \\Psi \\rangle$ is a background field. This is perfectly fine. Then you simplify the dynamics of the fluctuations by replacing cubic and quartic terms by quadratic terms like so: $$ \\psi^4 \\to 6 \\langle \\psi^2 \\rangle \\psi^2 - 8 \\langle \\psi \\rangle^3 \\psi + 6 \\langle \\psi \\rangle^4 - 3 \\langle \\psi^2 \\rangle^2, $$ $$ \\psi^3 \\to 3 \\langle \\psi^2 \\rangle \\psi - 2 \\langle \\psi \\rangle^3. $$ See footnote 11 of [3] for these full formulas; the other refs simplify these by using $\\langle \\psi \\rangle = 0$. Also note this is supposed to work for _bosonic_ fields. In the papers these substitutions just come out of the blue. I get that the idea is to make the problem solvable by reducing the order of the interaction terms (and then eventually determining a self-consistent $\\langle\\Psi\\rangle$), but I'm very confused about the numerical coefficients. Where do they come from? Is there a systematic way to derive them, or consistency condition they must satisfy? Why the alternating signs? What is the relation of this \"Hartree\" approximation to _the_ Hartree approximation which has to do with minimizing the energy for separable wavefunctions or summing up a particular class of Feynman diagrams? I've tried the seemingly likely stories:   * It looks at first like maybe something to do with Wick contractions, but the signs are a problem even if all the numbers came out the right size - which they don't.   * Repeatedly applying the mean field theory rule $AB \\to A\\langle B\\rangle + \\langle A\\rangle B - \\langle A\\rangle\\langle B\\rangle$ doesn't seem to work out either.   * Neither does writing $\\psi^4 = (\\psi - \\langle\\psi\\rangle)^4 + \\cdots$ and crossing off the $(\\psi - \\langle\\psi\\rangle)^4$ term, or same thing for $\\psi^4 = (\\psi^2 - \\langle\\psi^2\\rangle)^2 + \\cdots$   * It looks similar to the cumulant expansion but I can't get it out of that either.   * The online encyclopedia of integer sequences was completely useless! A literature pointer would be acceptable if the derivation is long, but please give me an article that _explains_ the result! So far I've only found things which _quote_ it and act as if it's the most obvious thing in the world. Sorry if it is... I feel like I'm missing something very basic. :)   1. Boyanovsky, D., Cormier, D., de Vega, H., & Holman, R. (1997). Out of equilibrium dynamics of an inflationary phase transition. Physical Review D, 55(6), 3373–3388. doi:10.1103\/PhysRevD.55.3373   2. Boyanovsky, D., & Holman, R. (1994). Nonequilibrium evolution of scalar fields in FRW cosmologies. Physical Review D, 49(6), 2769–2785. doi:10.1103\/PhysRevD.49.2769   3. Chang, S.-J. (1975). Quantum fluctuations in a φ^{4} field theory. I. Stability of the vacuum. Physical Review D, 12(4), 1071–1088. doi:10.1103\/PhysRevD.12.1071"}
{"id":"26996","title":"Infinity of running couplings","text":"A Landau pole \\- an infinity occurring in the running of coupling constants in QFT is a known phenomena. How does the Landau pole energy scale behave if we increase the order of our calculation, (more loops) especially in the case of Higgs quadrilinear coupling?"}
{"id":"79575","title":"Determing time to complete known distance with constant acceleration","text":"**Answer: $a=v^2\/d$ is the formula I needed.** This is a problem in a programming assignment, I haven't taken physics. I have a starting speed (0 m\/s), an final speed (208.33m\/s) and the distance it took to reach that speed (200km). I need to get the time it will take to travel any arbitrary distance lower than 200km. From what I remember in highschool, I used basic calculus to get $a\/2(t^2)=distance$ or for what I have $t=\\sqrt(400000(m\/s)\/a)$. what I don't know is the acceleration, usually to calculate acceleration I'd need time. I'm not sure where to go from here."}
{"id":"79576","title":"Projectile motion of the water surface from water discharging from a pipe","text":"I am looking at pipe flow into a pit. How do I calculate the outside water surface path. I understand how to calculate projectile motion from calculus in terms of a velocity, point object and projectile angle however I do not know how to add thickness and calculate the outside path. We can ignore drag force unless someone can provide suggested values based on field or lab validation. Also I can accept that it will be hard to calculate the expansion of water discharging from the pipe into the pit so unless someone knows suggested values from field or lab validation then this can also be ignored."}
{"id":"92516","title":"If the moon was twice as big but twice as far away, would there be any difference?","text":"I'm just going to go ahead and steal this question question directly of Reddit since I have more trust in the answers I get on this site. So, if the moon was twice as big but also twice as far way, would there be any difference in the effects that the moon now has on earth, like tides and such? Also, would there be any difference in the moons orbital motion apart from the fact that the time it takes for the moon to rotate around earth would be longer? What will increase is the mass and diameter(by a factor of 2) of the moon. The density of the moon will stay the same."}
{"id":"107050","title":"What is the smallest particle exhibiting gravitational properties?","text":"I've long been taught that all matter having mass, possesses attractive forces somewhat akin to gravity. As such, imagine we can 'teleport' a gravitonic detection device that can accurately measure the gravitational forces regardless of strength present at each location, it can detect all gravitational forces ordered by direction of pull within a given radius of up to let say [through a range of settings\/switches of 0.001 nanometer to 100,000 km. We teleport these detectors equi-distantly apart in the earth, from troposphere to molten mantle to core's center of mass. measure and plot all the forces detected. Supposedly, there is no gravity at earths center of mass...that means gravity can be concentrated and nullified. Logic tends to indicate 'no gravity=no weight. Therefore why does all reference material say massive objects have tremendously large core pressures and squeeze atoms to extremely dense masses. If gravity is nullified, what is source of this crushing, without usurping all gravitational common sense?"}
{"id":"107059","title":"Lamination on a conductor?","text":"How can lamination help reduce eddy currents? I understood that it reduces them, by making their circulation incomplete, or something like that. Can anyone explain this please?"}
{"id":"23016","title":"Helium plasma in space and its properties","text":"It is said, that, \"Most of the Helium in the Universe, is in a plasma state\". Plasma's are now talked of, as the forth state of matter, but this does not seem to be a majority opinion. Plasma's are also normally regarded, to be at a high temperature, I believe because they are formed from elements that would normally be solids at Earth temperatures. Since the mass of Helium in the Universe is > 20% of the total mass, and the average temperature (CMB), of the Universe, is 2.7K, there must be a lot of Helium Plasma in the Universe at, or below this temperature. My questions are:   1. 2.7K is very close to the Lambada point of atomic Helium. Does Helium plasma, still have the same Lambada properties as atomic Helium? If so, will Helium plasma, below it Lambada point, be both superfluid and supercondutive?   2. Is there a point, that Helium plasma reverts to atomic Helium and what causes it to revert? I assume there is a point, since some of the small amount of Helium we still have on Earth, trapped in pockets, and which dates from the Earth's birth, would be of the plasma verity, because it origins, are in the Universe.   3. The majority of the Helium in the Universe, is said to be Helium-4. I presume, this is Helium-4 plasma. Apart from the properties mentioned in question 1, are there any other major differences between Helium-4 plasma and atomic Helium-4. If the answer to this question is yes, and the answer to questions 1 & 2 is no, then the differences between the two states, seems so great, that maybe, there is a case for the plasma, to have a different classification, and a different name, even though, their atomic structure, is very similar."}
{"id":"23019","title":"Time-reversal symmetry","text":"For a quantum system with time-reversal symmetry, other than the absence of a magnetic field, can we infer anything else about the system?"}
{"id":"66234","title":"Proof that the Earth rotates?","text":"What is the proof, without leaving the Earth, and involving only basic physics, that the earth rotates around its axis? By basic physics I mean the physics that the early physicists must've used to deduce that it rotates, not relativity."}
{"id":"28677","title":"RC circuit theory and voltage in the capacitor vs Ohm's Law","text":"Consider a simple series RC circuit at steady state (capacitor is full). I've been told that once the capacitor is full we can literally \"cut\" the circuit because no current can flow. That argument makes sense, but because the capacitor is full, doesn't that also mean it has a potential difference across it? In other words, the capacitor is a new battery? Then we should apply Kirchhoff Law in the circuit if we want to find the current? (even though I just said it was zero). What is going on? What is this contradiction? There is no current in the circuit yet clearly the potential difference in the capacitor is NOT zero. **EDIT** ![enter image description here](http:\/\/i.stack.imgur.com\/ICDhG.jpg) For part (c), why do I have to use $I = I_0 e^{-t\/ \\tau}$? Clearly at steady state, the current of the right circuit is just the potential difference across the capacitor (which now can be treated as a battery) and the resistance. As pointed out before I can only apply Ohm's Law to resistors and batteries, but at steady state the capacitor _can_ be thought of as a battery. So the contradiction lies here. On the left circuit the current is simply $10V = I_1(50k \\Omega)$, and on the right it should be $10V = I_2(100k \\Omega)$ **only because this is steady state** **EDIT2** Let me clarify my question again. I am also confused why $I = I_0 e^{-t\/ \\tau}$ is used? Why does the capacitor discharge? Yes it's considered an isolated circuit once the switch is shut, but does that mean even though part of the circuit is still hooked up to the battery, that the current doesn't continue to build to prevent discharge?"}
{"id":"14401","title":"Could the Schrödinger equation be nonlinear?","text":"Is there any specific reasons why so few consider the possibility that there might be something underlying the Schrödinger equation which is nonlinear? For instance, can't quantum gravity (QG) be nonlinear like general relativity (GR)?"}
{"id":"34203","title":"Mars Curiosity Power System","text":"Curiosity's power source comes from a radioisotope thermoelectric generator (RTG) rather than from solar panels like the two Mars Rovers. 2,000 watts of thermal power from alpha particle emission in 4.8 kg of Pu-238 (half-life 87.8 years) yields about 120 watts of electric power via the Seebeck effect (thermocouples). This source slowly declines in power output (~1%\/year), and is expected to provide a reliable power source for years. > It appears that the efficiency of Curiosity's RTG is only 120\/2000 = 6%. Why > is the RTG efficiency so small? Is there a problem maintaining greater > hot\/cold junction delta temperatures or are there other issues?"}
{"id":"35649","title":"Mass vs. weight on a spring-loaded bathroom scale","text":"You stand on a spring-loaded bathroom scale in a bathroom. The scale \"reads\" your mass. What is the scale actually _measuring_? Similarly, you stand on a spring-loaded bathroom scale in an elevator that is accelerating upward at 2.0 m\/s^2. The scale \"reads\" your mass. What is the scale _measuring_? A) Your mass B) Your weight C) The force of the scale pushing up on your feet D) The force of your feet pushing down on the scale To me, C seems correct for both of them, but B also seems correct for the first part, as weight is equal to the force of the scale on your feet for that part. Can anyone corroborate or deny my assumptions?"}
{"id":"73035","title":"Opening the fridge door to cool a room","text":"I'm well aware that the default answer to this textbook default question is \"it doesn't work\", but still, I believe it does. To cool the insides of the fridge, the compressor must do work, and since the efficiency isn't 100% you are constantly warming the whole room to cool it's insides, the winning move here is simply turning the fridge off. However, let's suppose the fridge must stay on, wouldn't it be better to open the door? In other words: Isn't opened fridge turned on better than closed fridge turned on for the whole room temperature?"}
{"id":"32913","title":"Question about units in Lagrangian dynamics (inertia matrix)","text":"I have a 3 degree of freedom system and my equation of motion is like this: $$M(q)q_{dd} + C(q,q_d)q_d+G(q)~=~0$$ $M(q)$: inertia matrix $C(q,q_d)$: Coriolis-centrifugal matrix $G(q)$: potential matrix. where $q_d$ is the first derivative of q etc and q is a vector of my variables. $q=[θ,γ,a]^T$ \"θ\" and \"γ\" are angles in (rad) and \"a\" is length in (m). Now my question is this: I have 3 equations with 3 unknowns but 2 variables are in terms of angles and 1 in terms of distance, so the elements of my inertia matrix don't have the same units. Is that wrong? I mean each equation is consistent in units. The first 2 have units of kg(m\/s)^2 while the 3rd has units of kg(m\/s^2). I can make the whole thing dimensionless but I read somewhere that the inertia matrix should be symmetric(which it is symmetric at the moment). If I make my equation of motion dimensionless then inertia matrix won't be symmetric anymore. What shall I do? Any suggestions are welcome"}
{"id":"109344","title":"Effect of CO2 on the environment, besides greenhouse effect and ocean acidification","text":"What effects does the raising level of CO2 in the atmosphere have, besides being a greenhouse gas and responsible for the increased acidity of the oceans? What other environmental damages does it cause?"}
{"id":"130219","title":"The Gluon - Does It Exhibit Wave Properties?","text":"Do Gluons have frequencies and wavelengths? I assume that they do, but have been unable to find anything on point in SE or Wikipedia. Just beginning to study university-level physics here."}
{"id":"101064","title":"Stress-energy tensor explicitly in terms of the metric tensor","text":"I am trying to write the Einstein field equations $$R_{\\mu\\nu}-\\frac{1}{2}g_{\\mu\\nu} R=\\frac{8\\pi G}{c^4}T_{\\mu\\nu}$$ in such a way that the Ricci curvature tensor $R_{\\mu\\nu}$ and scalar curvature $R$ are replaced with an explicit expression involving the metric tensor $g_{\\mu\\nu}$. Using the equations $R=g^{\\mu\\nu}R_{\\mu\\nu}$ (relating the scalar curvature to the trace of the Ricci curvature tensor) and $R_{\\mu\\nu}=R^\\lambda_{\\mu\\lambda\\nu}$ (relating the Ricci curvature tensor to the trace of the Riemann curvature tensor), would anyone be willing to give recommendations on how to proceed, or already know the equation?"}
{"id":"101067","title":"Maxwell's Theory Of Wave Propagation","text":"What is Maxwell's theory of Wave propagation and what is its physical interpretation?"}
{"id":"93391","title":"Why water from upper level not goes into lower level in this drainage system","text":"In the following typical water drainage system, I'd like to know why waste water from upper floor (let say, from 2nd floor's toilet) does not come to the lower floor's equipment via the common pipes (the 1st floor's sink), according to the equal level principle. In what case would this happen? If the water tends to go through the main pipes but not the branches, is there chance for it to go to the sink? ![drainage system](http:\/\/i.stack.imgur.com\/QqKcF.jpg)"}
{"id":"91463","title":"Unbalanced wheatstone bridge","text":"Suppose I have an unbalanced Wheatstone bridge (resistances $R_1, R_2, R_3, R_4, R_g$), to which I apply some voltage $V$. I need to find the voltage across the the galvanometer. Solving the system of equations obtained by Kirchoff's Law would yield the answer, but it will definitely be enormous. I recall seeing somewhere it could be written as the determinant of a somewhat elegant matrix, but I can't see how I would find it (Cramer's rule applied to the system is way too ugly, and solving first, then backtracking to finding the determinant is unpractical). Does anybody know how to approach this problem?"}
{"id":"36340","title":"How does one solve the Schroedinger equation for a 2D, time-dependent harmonic potential?","text":"This is the Schroedinger equation with a particular 2D harmonic potential: $$\\begin{multline}i\\hbar\\frac{\\partial}{\\partial t}\\Psi(x_1,x_2,t) = \\\\\\ \\Biggl[-\\frac{\\hbar^2}{2m}\\nabla^2 + \\frac{1}{2} m\\omega^2\\Biggl(\\biggl(x_1 - \\frac{x_0(t)}{2}\\biggr)^2 + \\biggl(x_2 + \\frac{x_0(t)}{2}\\biggr)^2\\Biggr)\\Biggr]\\Psi(x_1,x_2,t)\\end{multline}$$ Can anyone please tell me what the upside down triangle means? I know its the second derivative, but since the problem I have is of variables $x_1$, $x_2$, and $t$, do I take the second derivative of time? Or does that upside down triangle only does it for $x_1$ and $x_2$? $x_0$ is the separation distance between $x_1$ and $x_2$ and its a function of time because the two double HO potential wells are moving apart from each other. Also, how would I go about doing separation of variables here? Would I move all the time pieces to one side and all the x pieces to the other side? Won't I have to do separation of variables twice? Where the second time is when I have to separate $x_1$ and $x_2$?"}
{"id":"128395","title":"The Aharonov-Bohm effect is purely classical, right?","text":"Every discussion I've ever seen of the Aharonov-Bohm effect makes a big deal of its being a quantum effect with no classical analogue. But as far as I can tell it is present already at the classical level in QED. It also seems to have a close analogue in GR: the Riemann curvature outside an infinite straight cosmic string is identically zero, but an interferometer encircling it will see a phase shift that depends on its mass density. Is there something I'm missing? If not, can someone point me to a reasonably trustworthy textbook or paper that makes the point that it's classical, especially one that also mentions the GR analogy? (edit: By \"present at the classical level\" I mean that if you take the QED Lagrangian and derive classical equations of motion from it, you get a classical theory of Maxwell's electromagnetism coupled to a charged wave in which the A-B effect apparently exists just as in QED. This theory was never investigated before the quantum era, but it could have been, and the A-B effect could have been found then, as far as I can tell. I'm hoping for a published paper by a well-known author that says the above explicitly, in part because I'd like to add it to Wikipedia. I'm not interested in attempts to explain away the effect as being due to an external electric field, at least not for the purposes of this question. This is a theory question and there's no electric field in theory.)"}
{"id":"4882","title":"Low frequency electromagnetic waves in General Relativity","text":"I am becoming familiar with the Geometric Optics approximation in General Relativity which (to summarise) says that EM waves follow null geodesics under the geometric optics approximation. In the context of General Relativity the Geometric Optics approximation amounts to the assumption that: $\\lambda << R$ where $\\lambda$ is the wavelength and R is some measure of curvature (~ 100 meters near Black Holes). So under this approximation EM waves follow null geodesics. Additionally the \"photon\" model can make sense too, and we can say \"photons follow null geodesics\". But what about the \"general case\"? In the general case the order of curvature is the same as (or less) than that of the wavelength. So the usual statements do not obviously apply. Where is the full case discussed and what implications are there? To give some extreme examples for those who have not thought about this before:   1. In a Schwarzchild model at R=2M photons can get into circular null orbits. But clearly this provides an IR cut-off as they would be in a Bohr-like orbit with a max wavelength $\\lambda_{max}=4\\pi M$.   2. The usual \"equivalence principle\" may fail as the local trivialisation cannot extend far enough into the curvature region (and $\\lambda$ is larger still). So the equivalence laboratory can detect such waves, but cannot measure their behaviour and prove that they follow geodesics (local straight lines). I am struggling to find any coherent account of this, just odd facts and claims. **EDIT** In large part this question is answered by the comment from Genneth, and the links related to this Wikipedia article: Maxwell's equations in curved spacetime."}
{"id":"113275","title":"Relationship between amplitude and frequency of a wave","text":"I wanted to ask what is the relationship between amplitude and frequency of a wave. Some say there isn't , some say that there is , but from their answers the relationship is still unclear."}
{"id":"113273","title":"Why formation of rain is adiabatic?","text":"I recently went through the wikipedia page for Rain. It said that > There are four main mechanisms for cooling the air to its dew point: > adiabatic cooling, conductive cooling,.... It proceeds as > Adiabatic cooling occurs when air rises and expands. Why is this a adiabatic process?"}
{"id":"134448","title":"kinematics question about runner","text":"Here's the question: Your goal is to complete your 10,000-m run in less than 30.0 min. Apparently you've started too slowly because at 27.0 min, you still have 1100 m to go! You can accelerate at 0.20 m\/s2 . For how many seconds should you accelerate (and after this have constant velocity) in order to complete the run on time? So i can only think of using the equation x=xo+vot+0.5at^2 to solve for vinitial, but after that idk what to do, not sure if my first step is even correct"}
{"id":"86465","title":"Free energy of the critical U(N) model","text":"* Can someone help explain how the equations 30, 31 and 34 were obtained in this paper.  At a conceptual level I am wondering looking at equation 34 as to if they mean that $\\lambda$ is somehow the anomalous dimension of the single-trace operators in this critical U(N) model - although looking at equation 30 it seemed that $\\lambda$ is just the Hubbard-Stratonivich auxiliary field (though its not clear to me as to how is equation 30 derived). That factor of $(l + \\frac{1}{2})$ is confusing me everywhere. Also what they call as $1\/g$ in equation 30 is what I would think of as the threshold value of the translationaly invariant condensate at which the symmetry breaks. So it is like an order parameter and I can't see that as something that was there in the UV lagrangian as a coupling constant as they seem to imagine it here.   * Although I understand how equation 1 in this paper was derived but I still don;t see how equation 35 somehow follows immediately from it."}
{"id":"132846","title":"How can metal objects become electrically charged, if current flow means that an equal number of electrons enter\/leave the object?","text":"I am trying to answer to the question in the title. I am restricting my question to metal objects only. Here is my logical reasoning:   1. Current is the flow of charge over time.   2. In a circuit (simple series circuit, let's say), the flow of current is the same at every point in the circuit.   3. Therefore, the same # of coulombs of charge is flowing at every point in the circuit.   4. Electrons are the \"material\" of the charge that is flowing.   5. Therefore, equal flow of charge at every point in the circuit must mean equal flow of electrons at every point in the circuit.   6. Therefore, current can never cause a metal object to become positively or negatively charged, because the net number of electrons in the metal object will never change due to current flow. (!) Of course, objects CAN become electrically charged, gaining or losing electrons. So something is wrong with my reasoning or my premises. I just don't know what it is. Where am I going wrong?"}
{"id":"132847","title":"How does Ptolemy's table, of angles in air and angles in water, fit perfectly on a parabola?","text":"In \"The Feynman Lectures on Physics\" Vol.1 Optics Ch.26, Table 26-1 Ptolemy's list of angle in air and angle in water are given, in connection with Snell's law of refraction. I quote below values from that table: The angles are in degrees. Angle in air, Angle in water: (10, 8); (20, 15.5); (30, 22.5); (40, 29); (50, 35); (60, 40.5); (70, 45.5); (80, 50). In table 26-2 the corresponding angles according to Snell's law are given, to show the remarkable agreement with Ptolemy's list. In the text above the tables, a statement says all these values 'fit pefectly on a parabola'. I am unable to see the how these values fit on a parabola. How do you get the parabola from the listed values in the table?"}
{"id":"38019","title":"Conservation Laws and Symmetries","text":"Usually, in Quantum Mechanics, an observable is an operator on the space of the possible quantum states (labelled as $|\\psi\\rangle$). If this quantity is conserved, in the meaning that the associated operator $\\hat D$ is constant: $$\\frac{d\\hat D}{dt}=0$$ it has to commute with the Hamiltonian operator $\\hat H$. To prove this, in the Heisenberg picture of Classical QM, we use the Schroedinger Equation $$i\\hslash\\frac{\\partial}{\\partial t}|\\psi\\rangle =\\hat H|\\psi\\rangle $$ On the other hand, the Noether theorem states that if a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time. And this is true either for classical, relativistic and quantum systems. Therefore, first question:   * Is it true, even in the relativistic case, when the Schroedinger eq. doesn't apply, that a conserved quantity has to commute with the hamiltonian to be conserved during the motion?  Then, one usually studies the transformations generated by $\\hat D$, for instance if $\\hat D$ is the angular momentum, the transformation $U(\\theta)=e^{-i\\hat D \\theta}$ (a rotation of $|\\psi\\rangle$ with angle $\\theta$). And assuming the angular momentum conserved, or verifying this by means of the commutation properties, we can construct the space of the eigenstates of $\\hat H$ knowing that they are also eigenstates of $\\hat D$. For instance we construct the eigenstates of a spin-1\/2 particle using $\\hat D = \\hat S$, and knowing that $[S^2, S_z]=0$. Since it exists only two states: spin up and spin down, $|\\psi\\rangle$ will be a two-components spinor, and $\\hat S$ will be a 2x2 matrix. We see that the Lie Group of the $\\hat U$ transformation is generated by the Pauli's matrices throught the expressions $$\\hat S_i=\\hslash\\sigma_i\/s$$. So, I understand that, wanting to impose the conservation of $S$, we get into the $SU(2)$ Lie-group, because of the commutation relations satisfied by the $S_i$, and the infinitesimal decomposition of each $\\hat U$. Final questions:   * If we want to conserve something else than the spin, as it happens in the strong interaction study, when we have the Isospin and the Strangeness to be conserved, we use $SU(3)$. Why?    * Who is the corresponding Casimir operator? (the analogous of $S^2$)   * Why in this case the generators should be 8, and not 3?   * What is the relationship between a normal rotation in the physical space, for which the generator is $\\hat L$, and the rotation in the total space $\\hat J$?    * Why in the case of the strong interaction we don't consider any spin or orbital angular momentum in the labeling and classification of the eigenstates?    * At last, is there any good textbook you would suggest to have a global view of this subject? Because I've been studying this problems in several courses during my undergrad courses, in QM classes, RQM classes, Nuclear Physics, and so on, but every time they've just mentioned some concepts without getting deeper, and this causes me a lot of confusion..."}
{"id":"38013","title":"Cramer's rule, Origin of Quarks Fractional electric charge?","text":"In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns. 2u+1d=1 1u+2d=0 $$a_1d+b_1u=c_1$$ $$a_2d+b_2u=c_2$$ $$u=\\frac {c_1b_2-c_2b_1}{a_1b_2-a_2b_1}$$ $$d=\\frac {a_1c_2-a_2c_1}{a_1b_2-a_2b_1}$$ u=+2\/3 d=-1\/3 there ain't no experiment that could be done, nor is there any observation that could be made, that would say, \"You guys are wrong.\" The theory is safe, permanently safe. Is that a theory of physics or a philosophy? I ask you."}
{"id":"116679","title":"Mathematical Formulation for Accelerating Charges Generating Light?","text":"I haven't been formally taught the fact that accelerating charges generate light, and I have seen the picture of an accelerating charge sending ripples down its field lines, but is there a mathematical formulation someone can provide that show that an acceleration on a charge produces E&M waves?"}
{"id":"102806","title":"Density as a function of Temperature?","text":"I know that volume = mass\/density....and density is the function of temperature. Is there any standard equation that describes density as a function of temperature? The system I am interested in is like a liquid tank or reservoir"}
{"id":"103031","title":"Actions at a distance vs. contact interaction","text":"The ancestors could not imagine an action at a distance (in German: \"Fernwirkung\"). Today physicists don't take serious its _opposite_ anymore (in German: \"Kontaktwechselwirkung\"). So my first question is: > What is\/was the official _english_ name of the opposite of an \"action at a > distance\"?   >  Something like \"contact interaction\"? Next to terminology there is the matter itself: The prototypical _repelling_ \"contact interaction\" has always been collision. > But has there ever been a prototypical _attracting_ \"contact interaction\"?   > Something like agglutination? What have been prototypical examples, eventually?"}
{"id":"47062","title":"Why does the monitor make a cracking noise?","text":"After switching off the monitor, there's a single crack after a while. I wonder where exactly it comes from. I know that this is normal and not an indicator of being defect. This sound is also not related to a specific type of monitor. I've heard this sound from my CRT as well as from my TFT. I guess it possibly has to do with cooling down. I found a lot of statements that second this thought but there's no reliable source confirming this theory though. However, an explanation that only says that it has to do with the cooling process is not sufficient at all. I'm interested in what exactly makes this sound? Is it, for example, a capacitor which has fully unloaded (if this is the proper term). I really haven't the slightest inkling what exactly can make this crack noise."}
{"id":"114041","title":"How intense a magnetic field would it take to keep an hypothetical iron-made moon orbiting around it?","text":"The intention of the question is to provide an example of the weakness of gravity. I imagine a horseshoe magnet located at the Earth's centre (remove the Earth), and a ferromagnetic moon. How intense would the magnetic field need to be to keep such a moon in orbit at the same distance that the moon is from the Earth?"}
{"id":"6415","title":"Axion related questions","text":"I have several question regarding axion. Could anyone give me some brief introduction to what is a axion string, axion field and how is this related to fermion zero mode and chiral zero mode?"}
{"id":"87416","title":"How irreversible processes are possible?","text":"Susskind says that all laws of mechanics are reversible and any valid mechanic law most be reversible: you can always determine the previous state of any physically valid system. However, the simplest operations in computation, _reset_ as well as the binary AND and OR operators, are irreversible. You may say that \"this is only a virtual simulation\" and everything is reversible in reality. However, because of the dissipation, almost all processes in the world are irreversible. So, despite irreversibility is completely forbidden, everything seems to be irreversible and entropy only increases. I want to understand how this contradiction is resolved (at least in one case). I also do not understand why do you represent a system configuration space by a box. You say that the state is a point in that box and its trajectory represents the state evolution. However, reversibility requirement demands that the trajectories do not intersect. This means there must be only one trajectory. So, you should rearrange all the dots of the configuration box into a line (it might be a segmented into unconnected circles). However, that might be another question."}
{"id":"19483","title":"How is this classical \"paradox\" resolved in electromagnetism?","text":"A magnet and a coil move relative to each other. In the frame of reference of the magnet, there is a magnetic field and consequently a force acting on the charges in the coil according to the Lorentz force $F=qv\\times B$ but there is no net electric field. In the frame of reference of the coil, there is a magnetic field and also an electric field, induced by the magnet, $E'$ that moves the charges in the coil, producing a current. But, in the first case no work is done on the charges, since the force is perpendicular to the velocity. In the second case, the force $qE'$ does work on the charges. How is this \"paradox\" resolved in classical electromagnetism?"}
{"id":"127988","title":"Why point mass leaves it's own gravitational well?","text":"I suppose that point mass has its own gravitational well. Why this point mass is still attracted by other gravitational waves. I expect that this point mass is surrounded by \"gravitational hills\". Why it is not blind to other weaker gravitational waves? If we think about gravity in terms of shape of spacetime. I mean that point mass (left chart) creates \"sharp\" gravity potential. Lowest gravity potential is exactly where point mass is located. Point stays in place (?). When mass is spread across spacetime (right chart) this is different. Lowest gravity potential is moved from the centre of the object. Object accelerates. Does point mass exist in real world? ![enter image description here](http:\/\/i.stack.imgur.com\/A5RFM.png)"}
{"id":"111129","title":"Compute and plot satellite position on 2D Earth map","text":"I would like to create a simple satellite tracking software, that will compute the position of a satellite at any time, and show its location on a 2D map of Earth. Let's assume it is possible to find online position of a satellite in the Earth-centered inertial (ECI) coordinate frame ( in form of TLE for example ) and use some method to propagate its state to a moment in future. Then if I want to plot the position on Earth map, I will need to know orientation of Earth at a given moment in relation to the original ECI frame, probably with the help of ECEF (Earth-Centered, Earth-Fixed). And here come the questions, how do I :   * transform from ECI to ECEF    * take in account Earth precession and nutation   * do the proper time conversion, to also involve leap years, leap seconds, added seconds...  **EDIT :** During my online searches, I came across this article. It seems to be describing what I am looking for. However it uses a lot of terminology that I am unfamiliar with. It also doesn't tell a lot about precision of this technique and where the equations come from ( I need to be confident that I am implementing the right procedures ). I would still like to address all of the issues I listed in the question above."}
{"id":"113007","title":"What properties does the conductor making the plates for Casimir effect have?","text":"Reading this explanation, I've understood that the divergence in computation of Casimir force on two parallel conducting plates is because of an unphysical model of ideal conductor, which makes EM field vanish on its boundaries. This looks much like unphysical model of rigid body in relativistic mechanics. Still, zeta regularization appears to somehow correctly remove the divergence and give sensible results. This must mean that effectively, it replaces our \"too ideal\" conductor with a \"moderately ideal\" one. So, the boundary conditions for EM field are no longer as simple as zero. So, my question is now: what boundary conditions does zeta-regularization implicitly impose on the EM field instead of the original ones? Does such change of boundary conditions in fact mean that the force will depend on which conductor the plates are made of?"}
{"id":"73959","title":"Is the electromagnetic spectrum discrete?","text":"I'm just starting to learn physics and I have a question (that is probably stupid.) I learned that energy levels that the bound electron can have are discrete. I also learned that when an electron transitions from one level to another, a photon with a specific wavelength (energy) is created and released into the wild. My question is this: are there holes in the electromagnetic spectrum, values of frequency that the created photon can't possibly have?"}
{"id":"39051","title":"How can such a high exponent arise in this physics equation?","text":"This site http:\/\/what-if.xkcd.com\/14\/ states that during a helium flash, \"the reaction rate is proportional to the 40th power of the temperature\". Taking for granted that this is true, how can such a large exponent arise in a physic equation? It just makes no sense to me that some physical process could give rise to this phenomenon. An explanation of how it can arise for a helium flash would be satisfactory, but the question is really more general. How can a physical process accumulate enough factors to get up to x^40? Thermodynamic processes tend to be in the x^2 or x^3's and geometry can only add a few more x^3's. Note that we are not talking about the scale of things (obviously the 40th power is not a big deal in the length or time scales), we are talking about a physics \/equation\/."}
{"id":"217","title":"Why can you \"suck in\" cooked spaghetti?","text":"We all know that there is no \"sucking\", only pushing. So how are cooked spaghetti pushed into your mouth? The air pressure applies orthogonal on the spaghetti surface. Where does the component directed into your mouth come from?"}
{"id":"12076","title":"Returning Light","text":"What happen if someone is just close enough to a black hole, and he emits light tangentially out, and the light got bent and circles around, what will he see, can he see the light source from his back?                            +---------->                 \/            \\        light   \/              \\               ^                \\     observer  +   O black hole \/     sees light\\               \/                \\             \/                  <----------- light path"}
{"id":"88087","title":"Newton's second law, gravity and buoyancy","text":"> A body of mass $m$ and volume $V$ is immersed into the sea. The body moves > under the action of two forces: gravity and buoyancy (Archimedes). > > Gravity force has a magnitude $mg$, where $g$ is a constant acceleration due > to gravity. Buoyancy force has a magnitude $\\rho V g$, where $\\rho$ is the > water density, and acts in the direction opposite to the gravity force. > > Part 1. > > Assume that the water density is a constant, $\\rho V >m$ always holds, and > at the initial time moment the body is released from rest at the depth of > $h>0$ metres below the sea surface. Assume that the origin is at the sea > surface and the positive direction of motion is towards the centre of the > Earth. > > (a) Complete the diagram by showing all forces acting on the body, and the > initial displacement of the body. > > (b) Write down Newton's second law of motion. Solve this equation, i.e. > write down the expressions for the velocity and the displacement as > functions of time. > > (c) Show that the body will move towards the sea surface, and when it > reaches the surface it has the velocity $v= -\\sqrt{\\frac{2gh(\\rho > V-m)}{m}}$. 1a) I can do 1b) I know that Newton's second law is $mx=F$ with $F$ being the forces. I have written the formula as $mx=\\rho Vg - mg$ but I am not sure if this is correct. Then I've divided both sides by $m$ to get $x$ on its own and integrated with respect to t (time) to get $x$ as a function of time. Giving me $x=-gt+\\frac{\\rho Vgt}{m} + c$ 1c) I'm not sure where the $h$ has come from, or how I could input it into my previous equation"}
{"id":"81157","title":"relationship between kinematics and area under curve","text":"My teacher is giving us an honors pre-calc\/calculus introduction to physics and the kinematics learning target he has given us is to understand the \"relationship between area under a curve and the equation that calculates motion\". I understand that the area under a curve (from the curve to the x-axis) will calculate the slope of the curve (which makes absolutely no mathematical sense to me), but I don't know how that relates to calculating motion. Please understand that my teacher is a very strict, high-standards, superfluous teacher who works at a university speed (he says so himself, and I'm only in HS) so there may be gaps in my understanding. I know what $\\Delta$ means and I have a very, very loose understanding of the relationships between acceleration, velocity, and distance."}
{"id":"100073","title":"How to get a function for the voltage across a capacitor connected to an AC voltage source?","text":"I am looking for the way of obtaining a solution for $V_{c}$ ,as a function of $t$ depending of $\\omega$, of the following differential equation related to an electrical circuit involving a low-pass filter : $ \\frac{ d V_{c}(t)}{d{t}} + \\frac{V_{c}(t)}{\\tau} = \\frac{V_{s}(t)}{\\tau} $. where, $V_{c}$ is the voltage across the capacitance, $V_{s}$ is the voltage given by the AC voltage source, $\\tau$ is the time constant, considering that, $\\tau = R C$ , $V_{s} = V_{in} \\sin{( \\omega t )}$ , $I_{R} = I_{C} = \\frac{V_{s}(t) - V_{c}(t)}{R} = C \\frac{dV_{c}(t)}{dt}$. I approached the problem by first solving the homogeneous part ( $ \\frac{ d V_{c}(t)}{d{t}} + \\frac{V_{c}(t)}{\\tau} = 0$ ) and for which I get the following solution : $V_{c}(t) = K e^{\\frac{-t}{\\tau}}$ (where $K$ is a constant). I now need find the particular solution (to get the general solution : $S_{general} = S_{homogeneous} + S_{particular}$ ). I think the particular solution might be of the type : $V_{c}(t) = A \\cos{(\\omega t + \\phi)}$. Edit : Once I replace the particular solution in the equation, I come to something depending on $\\omega$ , $\\phi$ and $\\frac{A}{V_{in}}$ , but I don't see how to continue using the sum function sum formulas."}
{"id":"119803","title":"Single-mode fibers and ray-theory of light","text":"I have read ray-theory cannot be applied to single-mode fibers. Why? I know rays are kind of an approximation to describe the real behavior of light, and electromagnetic theory and photonics are closer to nature, but are there no \"rays\" in single-mode fibers? Thank you."}
{"id":"119801","title":"How is a vacuum formed and measured by this pneumatics attachment?","text":"Please excuse me if this question is misplaced and also for my narrow understanding of physics. Firstly, I'm asking simply because the robot I've now been set to work with uses an air pneumatics system. By way of a small vacuum it can move about objects from point A to point B. This vacuum is somehow created from a small attachment that is connected to a steady supply of pressurised air. This crude illustration is made on MS Paint shows how the vacuum is made. I'm still puzzled as to why the input of air doesn't simply flow out of **both** of the other openings. ![crude illustration](http:\/\/i.stack.imgur.com\/CwyCl.png) **EDIT** I can intuit that the volume of air from the exhaust is the sum of the input and the additional air taken from the vacuum but **what is the force of the vacuum and how is it measured?** (Side note) > Does the surrounding environment (atmospheric presure) have any impact on > the vacuum? Any response is much appreciated."}
{"id":"43762","title":"Why are Topological Superconductors hard to make?","text":"Topological insulators (TI) have already been made in lab. Topological superconductors (TSC), being close cousins of TI, seem harder to make. Why is that? It seems that materials in connection with majorana zero modes all have bad fate in lab. Are there any deep reasons behind?"}
{"id":"21679","title":"What is a boundary condition for capacitors\/dielectrics?","text":"I am extremely confused about what boundary conditions are. One minute ago I was solving easy capacitor questions and the next minute I am being asked boundary condition questions and there is no such section in my textbook on boundary condition. I have some questions below if you could please explain from scratch\/beginner level what boundary condition is and what the goal is. I have some questions below you could use to explain what these conditions are… I have looked all over google and search various videos and I still cant find any information on them. * * * ![enter image description here](http:\/\/i.stack.imgur.com\/enZ48.png) * * * ![enter image description here](http:\/\/i.stack.imgur.com\/qNbaO.png)"}
{"id":"21673","title":"How can I subtract Spectra?","text":"I am writing an application that should allow to view and manipulate spectra ( **Raman** ). I am only given one normal spectrum consisting of 1024 coordinate points. Plus I have a background spectrum that is supposed to be subtracted from the first spectrum. f.e.: The first point in spectrum_1 is: **-55.988** , **9373** The according point in background spectrum: **-55.988** , **9382**. After subtracting the two in a third-party software, the first point of the resulting spectrum is: **-55.988** , **-641.968** So I asked myself how this is calculated. According to the Documentation of this software, it is using the following formula: > result spectrum(n) = multiple component spectrum(n) > > – Factor1 * single component spectrum (n) > > – Factor2 * single component spectrum (n) Unfortunately I can't do anything with it. What is meant by \"component spectrum(n)\"? Is it the Spectrum equation? Also what could the both factors be and what does the (n) mean? Here is an image of all three spectra that are involved in the calculation. Here is an image of all three spectra that are involved in the calculation. http:\/\/imageshack.us\/photo\/my-images\/546\/spectrumsubtractionresu.jpg\/ From Top to bottom:   1. First spectrum   2. Background spectrum   3. The resulting spectrum after the subtraction I simply can't imagine or reproduce the calculation that leads to this result. Any help and\/or advice is welcome. I am highly willing to learn and understand so please don't hesitate to share some links or other references that could help. Thanks in advance, BC++"}
{"id":"80242","title":"Photonic. vs Plasmonic","text":"I hear a lot about photonic and plasmonic structures being the future in many different applications, but what are the differences between them? Why would one want a material with both as opposed to just one?"}
{"id":"90440","title":"How is it possible the universe expanded faster than the speed of light during inflation?","text":"In a documentary written in collaboration with Stephen Hawking, the narrator (supposedly Stephen Hawkings) says that by the time the cosmos was 10 minutes old, it had already expanded thousands of light years in diameter. How is it possible that matter was traveling faster than light?"}
{"id":"38830","title":"Quantum Mechanics in terms of *-algebras","text":"I'm currently trying to find my way into the geometric description of Quantum Mechanics. I therefor started reading: Geometry of state spaces. In: Entanglement and Decoherence (A. Buchleitner et al., eds.). Lecture Notes in Physics 768, Springer Verlag, Berlin, New York, 2009, 1-60. A document that can also be found as a manuscript via: http:\/\/www.physik.uni- leipzig.de\/~uhlmann\/PDF\/UC07.pdf Even though I thought that I have a solid background in abstract algebra I somewhat got lost in Chapter 2 when he's trying to classify all the *-algebras that represent actual physical systems (starting at page 24 in the document). Do you have some recommendations for texts that introduce the *-algebra language in Quantum Mechanics in a more 'detailed' way. Because I kind of have the feeling that at a certain point Uhlmann just keeps skipping steps and I also lack some of the physical intuition concerning partial traces, canonical traces, purification and all that. From time to time I'd also be happy to see a concrete example. I'm looking forward to your responses. Best regards."}
{"id":"101307","title":"Dirac equation in QFT vs relativistic QM","text":"How does the Dirac equation in quantum field theory solve the existing problems in the interpretation Dirac equation (as a single-particle wave equation) in relativistic quantum mechanics? EDIT: The question was not clear. So I ask again. Why is the Dirac equation not meaningful as a single particle wave equation? Does many-particle interpretation of Dirac equation make it meaningful? If yes, how?"}
{"id":"101301","title":"Toroidal Planets","text":"I have read online in numerous places about the possible existence of toroidal planets, and I most people seem to believe that they could exist, but they also have no evidence to support this claim. I recently did a project in which I calculated the gravitational field around one-such planet, and I would like to continue working on it to either prove or disprove the possibility of these planets. Before I continue, I would like to know if any mathematically rigorous investigations into this question have already been conducted. I have searched, but I haven't found anything on planets, only black holes."}
{"id":"80280","title":"Lorentz force from velocity-dependent potential and Lagrangian","text":"There is something i'm missing. I am at page 22-23 of Goldstein Classical Mechanics 3rd ed. Lorentz force can be derived from a potential $$U=q\\phi-q\\mathbf{A}\\cdot\\mathbf{v}$$ Where $\\phi(t,x,y,z)$ is a scalar potential and $\\mathbf{A}(t,x,y,z)$ is a vector potential. The Lagrangian $L=T-U$ Where $T$ is the potential energy. I'm trying to solve this with Lagrange equations for the x coordinate, and for the term $$\\frac{d}{dt}\\big(\\frac{\\partial{L}}{\\partial\\dot{x}}\\big)$$ I'll show you how I'm trying to compute this with detail: $$\\frac{d}{dt}\\big(\\frac{\\partial{L}}{\\partial\\dot{x}}\\big)=m\\ddot{x}-q\\frac{d}{dt}(\\frac{\\partial{}}{\\partial{\\dot{x}}}\\phi)+q\\frac{d}{dt}(\\frac{\\partial\\mathbf{A}}{\\partial{\\dot{x}}}\\cdot\\mathbf{v}+\\frac{\\partial\\mathbf{V}}{\\partial{\\dot{x}}}\\cdot\\mathbf{A})$$ Now, here's my big problem, I'll may have some trouble understanding the condition in wich i can interchange the differentiation with respect to $\\dot{x}$ and $t$, i.e for example, $$q\\frac{d}{dt}(\\frac{\\partial{}}{\\partial{\\dot{x}}}\\phi)=q\\frac{\\partial{}}{\\partial{\\dot{x}}}(\\frac{d}{dt}\\phi)$$ This is done some pages before when Goldstein is deriving the lagrange equations, and I solved a problem in this chapter usig this. So now I have: $$q\\frac{\\partial{}}{\\partial{\\dot{x}}}(\\frac{\\partial{}}{\\partial{x}}\\phi\\frac{dx}{dt}+\\frac{\\partial{}}{\\partial{y}}\\phi\\frac{dy}{dt}+\\frac{\\partial{}}{\\partial{z}}\\phi\\frac{dz}{dt}+\\frac{\\partial{}}{\\partial{t}}\\phi)$$ Now I have looked now for many hours for many ways in wich this derivation is made and the the expression above vanishes, but i don't understand why, is it because I'm assuming x,y,z in the scalar potential are functions of time themselves? If not i can see why it would vanish. Also this vanishes because i can't exchange the order of differentiation? * * * I totally see what you mean, I was applying it wrong, after you answered me I encountered that final term $\\frac{\\partial \\phi}{\\partial q_i}$, I had to substract it because it was actually: $$q\\frac{d}{dt}(\\frac{\\partial{}}{\\partial{\\dot{x}}}\\phi)=q[\\frac{\\partial{}}{\\partial{\\dot{x}}}(\\frac{d}{dt}\\phi)-\\frac{\\partial f}{\\partial q_i}]$$ Thanks a lot you're awesome!"}
{"id":"62352","title":"Is light red shifted in optical tweezers?","text":"This is a question I put to my supervisor during my PhD many years ago, and never really got a satisfactory answer to. In an optical tweezers, assume that a beam of light is used to move a glass bead. My question is whether the outgoing light is red-shifted. If it is not I cannot reconcile how energy is conserved in the system, as work has been done to move the bead? Answers appreciated as always..."}
{"id":"14826","title":"Moving \"virtual\" singularity?","text":"Imagine two close, really big black holes rapidly spinning around each other. That setup would emit a terrible amount of gravitational waves. My question is, could those gravitational waves, if big enough, temporarily collapse space around them into a singularity? That is, not the black holes themselves forming a singularity, but the ripples they form on space-time, constructively interfering to form a singularity in a place where there is no actual mass."}
{"id":"48672","title":"Question on inflation","text":"I have two particular questions regarding the inflationary scenario. They are: 1.) What is the physical origin of the inflaton field? 2.) Why has the potential of the inflation field its particular form?"}
{"id":"45283","title":"How to find the distance of the source of sound in this problem?","text":"On a cloudy day, the sound of thunder was heard 4.5 s after the flash of light was seen. How far was the cloud? Given that speed of sound = 340 m\/s **This is what I've tried:** speed of light(c) = 10^6 * speed of sound( _v_ ) therefore, speed of light=340*10^6 m\/s If _x_ is the distance of the cloud, then               x\/(340*10^6)  -  x\/340 = 4.5          =>(340*10^6  -  340)\/x =  1\/4.5     => x=1529998470`      but the correct answer is 1530m . Where am I going wrong?"}
{"id":"44777","title":"What is the maximum mass of planet HD 40307 g","text":"Recently discovered planet HD 40307g around HD40307 Star system is claiming that its inside a habitual zone of a solar system. Besides It has water where life could be evolved. I tried to calculate `g` of that planet. But failed! This is because the true mass of the planet could not be found. What we have is Minimum Mass which is `7.1` times Earth mass. My question is how do I know the Maximum Mass?"}
{"id":"44774","title":"Force on a bar magnet","text":"A bar magnet is freely falling vertically inside a conducting ring placed in horizontal plane. Will it fall with an acceleration equal to g, less than g or greater than g? ![magnet](http:\/\/i.stack.imgur.com\/KsRlC.png)"}
{"id":"111614","title":"Toroidal magnet in Tonomura et al. demonstration of Aharonov-Bohm effect","text":"What does the magnetic field look like in Tonomura's toroidal magnet coated with a superconductor? ![Tonomura's toroidal magnet coated with a superconductor](http:\/\/i.stack.imgur.com\/Q0e6x.jpg) See this paper to find results showing conclusive proof of the A-B effect and flux quantization. Context: For the A-B effect to work, there needs to be a magnetic flux in the area between the alternate paths which will produce the interference fringe patterns (but not necessarily along the paths themselves). The paper shows fringe patterns due to interference between the center of the toroid and outside the toroid, which means that there needs to be a flux in the plane of the toroid for A-B effect to be observed. Doesn't the superconductor coating quantize the flux penetrating the center of the toroid rather than the in- plane flux?"}
{"id":"111615","title":"How to find the electric field due to a point charge in 3 dimensions?","text":"A point charge with charge $+q$ is situated at $(x,y,z)$. How do I find the electric field at $(p,q,r)$? $E=k\\frac{q}{r^2}$, right? So why isn't the electric field, $E=k\\frac{q}{(x-p)^2+(y-q)^2+(z-r)^2}$ ??? Question: A charge $q=1\\,\\mu C$ is placed at a point (1m, 2m,4m). Find the electric field at point P (0m,-4m,3m). I end up with $E$ = 2.38x10$^2$ N\/C; the given answer is 1.46x10$^3$ N\/C"}
{"id":"126112","title":"Experiments which show the phenomenon of closed electric field lines","text":"In german physics introductory texts often the following experiment is cited which \"shows\" that a changing magnetic field can induce an electric field with has closed field lines. Here is a picture of the setup: ![](http:\/\/i.stack.imgur.com\/iUV4v.jpg) A high frequency ac-voltage is applied to a coil around a glass sphere which is filled with a gas. Then a circular lighting effekt occurs as shown on the image. I have some questions about this:   1. How is this experiment called in english? Do you have references of textbooks or papers which discuss this in detail?   2. Why is it reasonable to say that this experiment \"shows\" that there \"must\" be a circular electric field? Let general epistomology aside, I think that the rough theory about this is, that the electrons wiggles back and forth because of the changing induced electric field. Since the magnetic field changes in sign the mean lorentz force due to the magnetic field is zero, so the lorentz force doesn't matter here. But if you don't know that the field lines are circular, especially that the field vectors are tangential to the circle, how conculde this from the experiment? For example it could be conceivable, that the field changes from radially invards to radially outwards and we get a radial wiggling of the charges which produce the lighting!? What (and what not) can one reasonable \"conclude\" from this experiment about the nature of the induced electric field and why?   3. How can one describe this experiment mathematically in detail?   4. Are there other, perhaps better or clearer experiments which demonstrate the existance of closed electric field lines produced by induction?"}
{"id":"1112","title":"Modified Maxwell's equations","text":"In 1985, Harmuth wrote that Maxwell's equations are incompatible with causality, and overcame the problem by adding a term for magnetic dipole currents, and as a consequence the problem of infinite zero-point energy and renormalization disappears. At least according to Harmuth's book: Calculus of finite differences in quantum electrodynamics By Henning F. Harmuth, Beate Meffert The foreword is readable at http:\/\/books.google.com\/books?id=EQFENquY6fwC Are Harmuth's modifications generally accepted by the physics community as a more accurate description of reality than the unmodified equations ?"}
{"id":"22611","title":"advantage of string theory over other theory-of-everything candidates","text":"I am getting curious over why string theory, especially M-theory, is the most popular candidate for the theory of everything. It seems that all candidates of the theory of everything lack substantial experimental supports, so why is string theory so special? Thanks."}
{"id":"26507","title":"Can neutrino detectors tell what direction the neutrinos came from?","text":"I was reading this question and got to thinking. Can neutrino detectors give us any clue where the neutrinos came from or when a supernova may occur? I was unsure and decided to ask that here."}
{"id":"109308","title":"Electrons Orbiting Sphere Magnet in Crooke's Tube","text":"Were I to assemble a Crooke's Tube and insert a spherical magnet, with the poles of the magnet perpendicular to the electron stream, would the electrons begin to orbit the poles of the magnet as in the images of Terrella experiments? I am pretty sure yes. If so, are there equations to deduce the ideal distance from the electron emitter that the spherical magnet should sit? To determine how strong my magnet should be? What should I be reading to become more fluent in these sorts of experiments? Thank you for any assistance."}
{"id":"95871","title":"Converse of the Lagrangian form-invariance","text":"The form-invariance of the Lagrange equations implies the existence of a function $\\ A( q_k, t)$ so that $\\ \\begin{equation} L' (q_k, v_k, t) -L(q_k, v_k, t) = \\frac d {dt} A( q_k, t) \\end{equation}$ Is there any way of proving the converse? That is, asuming the existence of the funcion $A$ prove the form-invariance of the Lagrangre equations."}
{"id":"61075","title":"Can virtual image be photographed","text":"Is there any instrument that can photograph a virtal image? How would such a device work?"}
{"id":"61074","title":"Can acceleration feel like constant gravity for indefinitely long?","text":"So here's the setup: I'm in a spaceship, without windows as always, and the ship is accelerating upwards at a constant rate of $1\\,\\text{g}$. So inside the spaceship it feels like I'm being pulled down by gravity as normal. But this constant acceleration can't last forever: I have to approach $c$ asymptotically. I believe the equation for my speed is: $$v\/c = \\tanh(at\/c).$$ That's what an outsider would observe. Correct? Not really my question though. What I want to know is, as my speed approaches $c$ and the outsider observes my acceleration slowing down, do I feel that? Or does time dilation and length contraction mean that I would feel the exact same acceleration\/gravity inside, forever?"}
{"id":"61076","title":"Simple harmonic oscillator system and changes in its total energy","text":"Suppose I have a body of mass $M$ connected to a spring (which is connected to a vertical wall) with a stiffness coefficient of $k$ on some frictionless surface. The body oscillates from point $C$ to point $B$ and $CB=d$. Its motion is harmonic. The total energy of such a system is simply $\\frac{1}{2} k \\left(\\frac{d}{2} \\right)^2=\\frac{1}{8} k d^2$ (because the equilibrium point is at $d\/2$ and $d\/2$ is the amplitude). Now, suppose we drop vertically some plasticine with the same mass $M$ from some height $h$. After it hits the oscillating object, it just sticks to it. My question is - why the total energy of the system doesn't change if the plasticine hits the object at point $C$ but it changes when it hits the object in the middle of $CB$ (it equals there to $\\frac{1}{16} k d^2$)? Intuitively, I do understand that such a plastic collision contributes to the loss of energy, but I'm not sure how exactly it fits here and how the energy is lost. And even so, I do not understand how the position of the body influences the change in energy. * * * **Proposed solution:** The total mechanical energy at point $d\/2$ is $E_{k,max}=P_{total}=\\frac{1}{8} k d^2$: $E_k=\\frac{M}{2}V^2\\\\\\ V=\\sqrt{\\frac{2E_k}{M}}=\\sqrt{\\frac{2P_{total}}{M}} $ Due to the conservation of the horizontal momentum we get: $MV=2MV'\\\\\\ \\frac{2E_k}{M}=4V'^2\\\\\\ V'=\\sqrt{\\frac{E_k}{2M}}=\\sqrt{\\frac{\\frac{1}{8}kd^2}{2M}}=\\sqrt{\\frac{kd^2}{16M}}$ Therefore, the momentary kinetic energy (which is the total energy) after the hit is: $$E_{k, after}=\\frac{2MV'^2}{2}=M \\frac{kd^2}{16M}=\\frac{kd^2}{16}$$"}
{"id":"116820","title":"Coarse-graining on a second channel decreases mutual information?","text":"Let $X_1,B_1,X_2,B_2$ and $Y_1,A_1,Y_2,A_2$ and $C_1$ and $C_2$ be binary random variables. Suppose: $I(X_2:B_2|C_2=0)+I(Y_2:A_2|C_2=1) \\leq 1$. This can be thought of as a bound on the capacity of a quantum channel called 2 (for example, you either perfectly correlate input $B_2$ with output $X_2$ (measurement result), or input $A_2$ with output $Y_2$, depending on the value of a bit $C_2$). Do we have: $I(X_1:B_1|C_1=0) \\geq I(X_1\\oplus X_2:B_1\\oplus B_2|C_1=0, C_2=0)+ I(X_1\\oplus Y_2:B_1\\oplus A_2|C_1=0, C_2=1)$ ? $I$ is the mutual information and $\\oplus$ the sum modulo 2. If this inequality is true, how would you show it?"}
{"id":"82349","title":"Hyperkahler manifolds and their use in theoretical physics","text":"Just as the title says:   1. What is the easiest definition of a Hyperkahler Manifold?   2. Could you give some examples of Hyperkahler manifolds, and manifolds which fail to be hyperkahler?   3. Why are such manifolds considered to be interesting in physics, and how do they arise in the study of supersymmetric gauge theories?   4. Apart from SUSY, are there any other branches of physics in which Hyperkahler manifolds appear?"}
{"id":"82345","title":"Normal force and reaction","text":"It is said that the normal force comes into play any time two bodies are in direct contact with one another, and always acts perpendicular to the body that applies the force. This force is a reactionary force because its value depends on the force the body from which the Normal force is emanating experiences and changes with change in the force applied. When I apply force on a body at an angle to the normal, even then I am getting a reaction equal to my force. This, too, is emanating from the body that is reacting. So why is the normal special?? When we discuss the free body diagrams of forces applied on bodies, we do not always show this normal - harness attached to a horse, rope on a pulley, wire attached to picture etc.?"}
{"id":"64513","title":"Is light's path a wave?","text":"In a lot of textbooks I see a schematic of light drawn as a squiggly line. I have even heard that some things are too small to be seen because they are smaller than the wavelength of light (and therefore light wiggles around them.) But now I think that light actually travels in a perfectly straight line and that no particles will feel any force from this light when they are not on this line. My friend says I'm wrong, and that if you increase the amplitude of the light, the ray will wiggle into the particle. To back it up, he linked me to a video about the single photon double slit experiment, where a single photon seems to interfere with itself. I'm not sure that this experiment proves him right, but it does seem to say that light wiggles around."}
{"id":"64516","title":"Absorption of Water","text":"1. I was just watching the videos Chris Hadfield put on youtube from space, and was wondering why water doesn't get absorbed as well onto his toothbrush in space? And what characteristic of toothbrushes and wash clothes causes them to absorb water?   2. What exactly is water absorption?    3. Also, when he put water on his eye, it spread out over the surface. Why would water still want to spread out over his eye's surface without the force of gravity pulling on it?"}
{"id":"21992","title":"Function to determine Earth-Moon distance vs. time","text":"I had no luck trying to find a function that determines the distance between Earth and Moon versus time.   (Moon ground level to Earth sea level) _Accuracy:_   * the acceptable time range would be for years in [ 1900, 2100 ]   * 1 km error margin (or better)"}
{"id":"105166","title":"Symmetry Breaking And Phase transition","text":"1. Is every phase transition associated with a symmetry breaking? If yes, what is the symmetry that a gaseous phase have but the liquid phase does not?   2. What is the extra symmetry that normal $\\bf He$ has but superfluid $\\bf He$ does not? Is the symmetry breaking in this case a gauge symmetry breaking?"}
{"id":"56338","title":"how to determine the direction of a wave propagation?","text":"In the textbook, it said a wave in the form $y(x, t) = A\\cos(\\omega t + \\beta x + \\varphi)$ propagates along negative $x$ direction and $y(x, t) = A\\cos(\\omega t - \\beta x + \\varphi)$ propagates along positive $x$ direction. This statement looks really confusing because when it says the wave is propagating along $\\pm$ x direction, to my understand, we can drop the time term and ignore the initial phase $\\varphi$ while analyzing the direction, i.e.$y(x, 0) = A\\cos(\\pm\\beta x)$, however, because of the symmetry of the cosine function, $\\cos(\\beta x)\\equiv \\cos(-\\beta x)$, so how come we determine the direction of propagation from that? I know my reasoning must be something incorrect but I don't know how to determine the direction. So if we don't go over the math, how to figure out the direction of propagation from the physical point of view? Why $-\\beta x$ corresponding to the propagation on positive x direction but not the opposite? Thanks."}
{"id":"98161","title":"What is $p$ in this candidate solution for the motion of N masses?","text":"In this problem: > Consider the system of $N$ masses shown in the figure (see picture 1). > > a) Using the approximation for small angles, write the equation of > transverse movement for the n-th particle. > > b) Propose a solution of the form: > ${ψ_n}^{(p)}(t)=A^{(p)}\\cos(nk^{(p)}+α^{(p)})\\cos(w^{(p)}t+\\phi^{(p)})$. > Find the dispersion relation and decide whether it depends on the contour > conditions. > > **Figure 1:** > > ![picture 1](http:\/\/i.stack.imgur.com\/MuQu5.jpg) > > **Figure 2:** > > ![picture 2](http:\/\/i.stack.imgur.com\/NLmaK.jpg) With regard to part b), I don't understand the notation of the proposed solution, to be more specific, I don't understand what $p$ is, is it an index? Could someone explain me what $\\ddot{y_n}$ would be just to see what $p$ stands by?"}
{"id":"16403","title":"What formulae for calculating fluid pressure change involves these parameters?","text":"Okay... this is a bit desperate... I am reverse engineering an excel program that is used for calculating outputs for a pressure vessel used for industrial process fluid heating. There is a part in the excel sheet where a pressure drop is being calculated. But there are too many unit conversions and other magic constants in intermediate steps i cannot regonise. I have some physics background but not enough in fluid mechanics to decipher. For starters, I have intermediate step formulas like:               (fluid flow rate[lbs\/h]]) * (1545) * ((inlet plus outlet temp[deg c] \/ 2)*9\/5+460) \/          ((molecular weight[??])*((pressure[psig])+14.5)*144)      ## The formula looks like this: $\\frac{\\dot{m} \\times 1545 \\underbrace{\\left[\\frac{(T_\\text{in} + T_\\text{out})}{2}\\times(\\frac{9}{5}) + 460 \\right]}_\\text{Degrees C to Deg F to Rankine}}{M_w \\times (P_g + 14.5) \\times 144}$ I am hoping someone who deals with fluid mechanics will be familiar with many formulae to tell what it might be. The inputs to the entire calculation steps consists of flow rate, inlet and outlet temperatures, molecular weight, viscosity and thermal conductivity; and also dimensions of the vessel. Greatly appreciate everyones help! Thanks. **EDIT** : The actual problem is here http:\/\/www.jakesee.com\/se\/physics_se_16403.png and the birthplace of the image is here http:\/\/stackoverflow.com\/questions\/7647478\/generate-a-flat-list-of- all-excel-cell-formulas"}
{"id":"93841","title":"Entropic force in rubber bands","text":"The formulation for entropic force of stretching a rubber band is now known. Are there any other such daily life examples that also have mathematical formulations of entropic forces associated with them? Or can the idea and theory be extrapolated in such a way that more physical phenomena can be explained by it?"}
{"id":"9481","title":"Distance increase between Sun and Venus","text":"Is the distance between the Sun and Venus increasing?   What is the measurement?   May I request for a formula?"}
{"id":"93842","title":"Satellites and gravitation","text":"A satellite with mass $m$ orbits a planet of mass $M$ in a circular path with radius $r$ and velocity $v$. Due to some internal technical failure, the satellite breaks into two, similar parts with mass $m\/2$ each. In the satellite's frame of reference, it appears the two parts move radially, in opposite directions, along the line connecting the original satellite and the planet's center, each with velocity $v_0\/2$. I am expected to show that right after the technical failure, each of the two parts has a total energy equal to $-3GM\/(16r)$ and angular momentum equal to $(m\/2)√(GMr)$, wrt the planet's center. The total energy of each of the two parts should be, I believe: $E_{tot} = mv_0^2\/16 - GmM\/(2r)$. Now, isn't angular momentum preserved despite the failure? However, why isn't the angular momentum zero if the two parts are moving in opposite directions?"}
{"id":"9484","title":"Can you use a laser to measure the speed of light with a rotating mirror?","text":"I have learned that the classical measurement of the speed of light with a rotating mirror does not work with a laser (as opposed to, say, a mercury- vapor lamp). Can you tell me if and how coherency of light is necessary\/contraindicated for the success of this experiment?"}
{"id":"61926","title":"Two-body problem questions","text":"I am self studying the two body problem and I'm stuck on the following: I have given $$\\ddot{\\vec{x}}_1= - G m_2 \\frac{\\vec{x}_1-\\vec{x}_2}{|\\vec{x}_1-\\vec{x}_2|^3}$$ and $$\\ddot{\\vec{x}}_2= - G m_1 \\frac{\\vec{x}_2-\\vec{x}_1}{|\\vec{x}_1-\\vec{x}_2|^3}$$ with masses $m_1,m_2$ at the position $x_1,x_2$ and gravitational constant $G$. How can you show only with these two equations that total momentum $m_1\\dot{\\vec x_1}+m_2\\dot{\\vec x_2}$ is constant in time? And why is the total mechanical energy $$\\frac12m_1|\\dot{\\vec x_1}|^2+\\frac12m_2|\\dot{\\vec x_2}|^2-\\frac{Gm_1m_2}{|\\vec{x}_1-\\vec{x}_2|}$$ conserved?"}
{"id":"79417","title":"Why does a laser beam diverge?","text":"I was wondering why a laser beam diverges. If all the photons are in the same direction, I would imagine that it would stay that way over a long distance. I am aware that a perfectly collimated beam with no divergence cannot be created due to diffraction, but I am looking for an explanation based on photons rather than wave physics."}
{"id":"53215","title":"Minimal Kinetic energy for particle in a box","text":"This is driving me crazy! The question goes as follows: A proton is enclosed in a zone of length 2pm along the x-axis. The minimal kinetic energy of the proton lies closest to:   * 5000eV   * 0.5eV   * 50eV   * 500eV   * **5eV** (this is supposed to be the correct answer) So first I calculate the energy level for a particle in a box which is given by the equation $E_n = p_n^2\/(2 m) = (n^2 h^2)\/(8 m L^2) $ Here $L$ is the length of the zone. So I find $E_1=94007.53 eV$.. So.. How do I relate this to the minimal kinetic energy? \\-- EDIT 1 -- After the first hint from someone, I changed my calculations (I used the mass of an electron instead of a proton..) NOW I get $E_1 = 51eV$, which is still wrong.."}
{"id":"53212","title":"Special relativity, acceleration and being in two places at once (How to resolve this paradox?):","text":"Imagine I am on Earth I have a clock which measures time $t$. When my Earth clock reads time $t$, then when I look at a moving spacecraft's clock I see the time $t'$. Let us assume time is measured in years. Let us suppose the clocks were initially synchronized so that when $t=0$, $t'= 0$. Now let us suppose that the speed of the rocket continues to increase in the following way (Note that I have increased the speed in discrete time steps for convenience, but I'm sure a continuous speed function could be obtained but just mathematically tedious): At $t = 0$, $v = c\\sqrt{3\/4}$. So when $t = 1$, $\\delta t'= \\frac{\\delta t}{\\gamma} = 0.5.$   At $t = 1$, $v = c\\sqrt{15\/16}$. So when $t = 2$, $\\delta t'= \\frac{\\delta t}{\\gamma} = 0.25.$   At $t = 2$, $v = c\\sqrt{63\/64}$. So when $t = 2$, $\\delta t'= \\frac{\\delta t}{\\gamma} = 0.125.$ and so on... If $v$ continues to increase over time in this way, then $t'$ will approach 1, but will never reach 1. It appears from the perspective of the Earthling, that the person in the rocket never ages by over 1 year. From the person in the rocket however, he will experience time as normal. Now let us assume that after the person in the rocket has aged 10 years, he leaves his rocket ship, and travels back to Earth, leaving his rocket to continue accelerating in the same pattern. Now when the man reaches Earth, will he not see himself in the rocket, where he sees himself not having aged 1 year? (Ignore optical effects for the purpose of this question.)"}
{"id":"21337","title":"Air velocity in a double-skin facade","text":"I was wondering how the air flows in a double skin facade? Say I have a glass facade with a depth of 1m, a height of 200m and a width of 10m. It is a single vertical shaft all the way up with openings at the bottom to let air in and at the top to let air out: ![](http:\/\/i.stack.imgur.com\/w4Xrc.png) There is a solar radiation of 100W\/m2 hitting the outside of the facade and the air velocity at ground level is 1m\/s. Can I expect the velocity of the air to increase as it travels through the shaft? and to what degree?"}
{"id":"21335","title":"size and distance of mirror used in projected \"smart\" whiteboard","text":"Apologies in advance for complete ignorance of optics (beyond conic sections) How is the size, distance, and angle of the mirror used here calculated? The mirror is I believe the open flap in front of the projector. ![enter image description here](http:\/\/i.stack.imgur.com\/BNEiv.jpg)"}
{"id":"63562","title":"What are the rainbow and ladder approximations in a solid state physics context?","text":"All references I find talk about quarks and gluons, where I have only very limited knowledge about. From it's name (rainbow) I guess it applies to fermions coupled to bosons and we're interested in the single particle Green function for the fermion, and we approximate the sum over all diagrams by only summing those diagrams where no phonon lines cross (so that Feynman diagrams look like rainbows). Is that correct? I also noted that all search results always talk simultaneously about the rainbow and the ladder approximation, as was also pointed out in a comment below."}
{"id":"22978","title":"Flow rate of a syringe","text":"Suppose a syringe (placed horizontally) contains a liquid with the density of water, composed of a barrel and a needle component. The barrel of the syringe has a cross-sectional area of $\\alpha~m^2$, and the pressure everywhere is $\\beta$ atm, when no force is applied. The needle has a pressure which remains equal to $\\beta$ atm (regardless of force applied). If we push on the needle, applying a force of magnitude $\\mu~N$, is it possible to determine the medicine's flow speed through the needle?"}
{"id":"112251","title":"Optical Theorem ,how can experiment distinguish the unscattered wave from the forward scattered wave?","text":"How can experiment distinguish the unscattered wave from the forward scattered wave? The Optical Theorem says the imaginary part of the forward wave determines the cross section for an initial two-particle state $|i\\rangle$ in the center of mass frame through the equation $$2E_Tp_{in}\\sigma=Im{\\cal M}_{ii}$$"}
{"id":"79411","title":"Simularity transformation of Heisenberg XXZ Hamiltonian","text":"I am considering the Heisenberg XXZ Hamiltonian: $$ H(\\Delta, J) = J\\sum_{i=1}^L\\left(\\sigma^x_i\\sigma^x_{i+1} + \\sigma^y_i\\sigma^y_{i+1} + \\Delta \\sigma^z_i\\sigma^z_{i+1} \\right) $$ Apparently, one can show: $$-H(-\\Delta) = UH(\\Delta)U^{-1}$$, where $U=\\Pi_{m=1}^{M\/2}\\sigma_z^{2m}$ is a unitary transformation. I am having some trouble showing this. Of course the transformation commutes with the z-Pauli matrices and using $(\\sigma^z)^{\\dagger} = \\sigma_z$ and $(\\sigma_z)^2 = \\textrm{id}$ this term is unchanged. However, I am left with mix terms for the first and second term. Furthermore, my reference claims that because of this simularity transformation, $J$ can be omitted if $-\\infty<\\Delta<\\infty$. I don't really see how this follows from the simularity transform. Any help is much appreciated!"}
{"id":"22973","title":"What is the interaction with Higgs field(s) that give the quarks so much different masses?","text":"The masses of quarks are:                mu 2∼3 MeV    md 4∼6 MeV      mc 1.3 GeV    ms 80∼130 MeV      mt 173 GeV    mb 4∼5 GeV"}
{"id":"71949","title":"Quantum XOR: How do you generalize it?","text":"Consider the classical XOR Gate: Given a 2 bit system: $G = [u_1, u_2]$ $$XOR(G) = (u_1 + u_2) \\ mod \\ 2$$ Is the following a good generalizaiton of a Quantum XOR Gate: Given a 2-qubit system: $Q = a_1|00] + a_2|01] + a_3|10] + a_4|11]$ We now write: $$XOR(Q) = a_1|0] + a_2|1] + a_3|1] + a_4|0] = (a_1 + a_4)|0] + (a_2 + a_3)|1]$$ Notice that if we set $a_i = 1$ and the rest to $0$ that the quantum XOR behaves exactly like a classical XOR but it also has the ability to generalize to fractional and complex $a_i$"}
{"id":"28536","title":"Lightbulbs in circuits. Why does the current go UP?","text":"> **Possible Duplicate:**   >  Why does the light bulb's brightness decrease? ![enter image description here](http:\/\/i.stack.imgur.com\/uZeX6.jpg) So I have a circuit in the picture with three resistors. If I were to remove\/disconnect $R_2$ from the circuit, what would happen to the overall brightness of $R_3$ and $R_1$? Now the answer I already have is that $R_1$'s brightness will go down and $R_3$ will go up. I understand that $R_1$'s brightness goes down because by removing the $R_2$, the net resistance goes up (destroying a parallel combination) But I don't understand why $R_3$ would go up? Conceptually this makes no sense to me. How do I know that the new current would always be larger than when the current splits at the junction? I would like a conceptual **physics** answer and not equations comparing inequalities."}
{"id":"102301","title":"Reissner-Nordström Black Holes","text":"The Reissner-Nordström black holes are described by the metric, \\begin{align} ds^2 = -\\left(1-\\frac{2M}{r}+\\frac{Q^2}{r^2}\\right)dt^2 + \\frac{1}{1-\\frac{2M}{r}+\\frac{Q^2}{r^2}}+r^2d\\Omega^2 \\end{align} where $M$ is the mass of the black-hole and $Q$ is the charge of the black hole. I am using the convention of $(-,+,+,+)$ here. The minimum mass horizon, which one can calculate, is given by $M_{min} = Q^2$. Part of the derivation is at: http:\/\/aesop.phys.utk.edu\/ads-cft\/L3.pdf. (Notes on AdS\/CFT Correspondence from University of Tennessee, notes 3). Suppose we approach the horizon, the metric begins to approach the $AdS_2 \\times S^2$ metric, given by, \\begin{align} ds^2 = R_1^2\\frac{-dt^2}{z^2}+R_1^2 \\frac{dz^2}{z^2} +R_2^2 z^2 d\\Omega^2. \\end{align} Note that in the link I provided, there is a sign switch (since the note writes for a Euclidean AdS). I was wondering, what would be the overall generators of the group? Is it just the generators of $AdS_2$ and the generators of $S^2$ (which are from $SO(3)$)? Are there not some more complicated generators that mix the two spaces?"}
{"id":"132645","title":"Particle-hole symmetry of Bogoliubov - de Gennes equation","text":"Bogoliubov - de Gennes equation reads, $$\\left( \\begin{array}{cc} H_{0} - E_{F} & -i\\sigma_{y}\\Delta \\\\\\ i\\sigma_{y}\\Delta^{*} & E_{F} - H_{0}^{*} \\end{array}\\right) \\left( \\begin{array}{c} \\psi_{e}^{\\uparrow} \\\\\\ \\psi_{e}^{\\downarrow} \\\\\\ \\psi_{h}^{\\uparrow} \\\\\\ \\psi_{h}^{\\downarrow} \\end{array}\\right) = \\mathcal{E} \\left( \\begin{array}{c} \\psi_{e}^{\\uparrow} \\\\\\ \\psi_{e}^{\\downarrow} \\\\\\ \\psi_{h}^{\\uparrow} \\\\\\ \\psi_{h}^{\\downarrow} \\end{array}\\right) $$ The above Hamiltonian which is $4$x$4$ obeys particle-hole symmetry, $$H = -\\mathcal{C}H\\mathcal{C}^{-1}$$ where $\\mathcal{C} = \\tau_{x}\\mathcal{K}$, with $\\tau_{x} = \\sigma_{x} \\oplus \\sigma_{x}$ and complex conjugation operator $\\mathcal{K}$. This symmetry implies that if You have solution $\\Psi$ with energy $\\mathcal{E}$, then You also have solution $\\mathcal{C}\\Psi$ with energy $-\\mathcal{E}$. I always thought that if Hamiltonian posseses some symmetry then You have solution with the same energy. I am confused, cause I also read that holes are time-reversed electrons, $$\\psi_{h} = \\mathcal{T}\\psi_{e}$$ with $\\mathcal{T} = i\\sigma_{y}\\mathcal{K}$, but I can't show this from above BdG equation."}
{"id":"27477","title":"Cosmic bubble collision","text":"I'm reading this review right now. The claim seems to be that when you have an expanding \"false vacuum\", finite size bubbles form due to phase transition (cause by finite action instantons) (which, in itself might then evolve, etc). My questions is, if I imagine these bubbles to be moving away from each other due to exponential expansion of the false vacuum, then how will they come near each other and collide?"}
{"id":"91140","title":"on quantum steering","text":"I have become interested in quantum steering after listening a talk and tried to read more about it. I think I am more confused now. My understanding is as follows: > Sharing a (entangled) state, Bob wants to make a series of measurements and > Alice wants to make a guess of the measurement outcome. After which I am totally at loss, as a few papers which I have seen refers to a steering inequality. However it is heavily state dependent, and not like Bell (or Bell type) inequalities. Can someone point out some basic references or clear up my confusion. I am sorry for not writing it more clearly. May be by the questions, I shall be able to understand. Advanced thanks for any help."}
{"id":"100000","title":"Difficult to visualize Franck-Condon overlap","text":"The franc condon principle says about the vertical transitions, means the coordinates of the nuclei remains unchanged during an electronic transition. But, what is the physical meaning or how to visualize 1) Franck Condon overlap or Franck Condon Factor? 2) I specifically came across the vibrationally mediated photodissociation (VMP). Here it says VMP is effective when there is a favorable Franck Condon overlap between vibrationally excited states of the ground electronic states and those of the electronically excited dissociative or predissociative states. Does this mean the potential energy surfaces of the said states really overlap? if anyone can give me a pictorial explanation, that would be nice."}
{"id":"106531","title":"Pendulum (highschool)","text":"I have a question concerning pendulum. On our physics lessons we built simple pendulum consisting of weight and a thread ($\\ell=0.31$ m). We tried to calculate $g$ from the well known formula $$T_m = 2\\pi\\sqrt{\\frac{\\ell}{g}}$$ which leads to $$g = \\frac{4\\pi^2 \\ell}{T_m^2}$$ Of course, this formula comes from conclusion that $\\sin(\\alpha)-\\tan(\\alpha)$ is neglectable for \"small\" $\\alpha$. Hence our target was to find formula, which works for larger $\\alpha$ as well. According to this two sources, this should do the trick: $$T_m = 2\\pi\\sqrt{\\frac{\\ell}{g}}(1+\\frac{1}{4}\\sin^2(\\frac{\\alpha}{2}))$$ from which I came to this conclusion: $$g=\\frac{\\ell\\pi^2(\\cos(\\alpha)-9)^2}{16T_m^2}$$ Here is my problem: After using this formula and our measured data, I got $\\overline g = 8.9085$, which is obviously not very accurate (for ~48°N). Any ideas where I've made a mistake?"}
{"id":"132531","title":"Can you gauge a $U(1)_L$ symmetry?","text":"I recently calculating the one loop correction for the propagator of a gauge boson, $\\hspace{5cm}$![enter image description here](http:\/\/i.stack.imgur.com\/gk0dJ.png) I assumed arbitrary left and right couplings, $ g _L $ and $ g _R $. I found that the one loop correction was, \\begin{equation} = \\frac{ - 4i }{ ( 4\\pi ) ^{ d \/ 2}} \\int dx \\frac{ \\Gamma ( 2 - d \/ 2 ) }{ \\Delta ^{ 2 - d \/2 }} \\left[ ( g _L ^2 + g _R ^2 ) x ( 1 - x ) \\left( g _{ \\mu \\nu } - \\frac{ p _\\mu p _\\nu }{ p ^2 } \\right) + g _{ \\mu \\nu } ( g _L ^2 - g _R ^2 ) m ^2 \\right] \\end{equation} Now we also know that for a $ U(1) $ invariant theory we should have the Ward identity and hence this one loop correction would be of the form, \\begin{equation} \\Pi _{ \\mu \\nu } = \\Pi ( p ^2 ) \\left( g _{ \\mu \\nu } - \\frac{ p _\\mu p _\\nu }{ p ^2 } \\right) \\end{equation} So it appears that in order for this propagator to arise from a $U(1)$ invariant theory, we must have $ g _L = g _R $ (unless I made a mistake). I found this very strange. Is it not possible to gauge a $ U(1) _L $ symmetry and if so why? As a working example I invented the theory below which seems like it could be gauged: \\begin{equation} {\\cal L} = i \\psi ^\\dagger \\bar{\\sigma} ^\\mu \\partial _\\mu \\psi + i \\chi \\sigma ^\\mu \\partial _\\mu \\chi + \\phi ^0 \\chi \\chi + \\phi ^{+ + } \\psi \\psi + \\phi \\mbox{ terms} \\end{equation} where $ \\psi $ is a left chiral spinor and $ \\chi $ is a right chiral spinor. The particles transform under the symmetry as, \\begin{equation} \\psi \\rightarrow e ^{ i \\alpha } \\quad \\chi \\rightarrow \\chi \\quad \\phi ^0 \\rightarrow \\phi ^0 \\quad \\phi ^{ + + } \\rightarrow e ^{ - 2i \\alpha } \\phi ^{ + + } \\end{equation} This seems like a perfectly fine theory, whose symmetry can be gauged but if my conclusion above is correct for some reason I shouldn't be able to gauge this $ U(1 ) _L $ symmetry. Did I make a mistake or is there some deep reason this can't be done?"}
{"id":"111933","title":"Order of Monte Carlo integration and frequency summation","text":"I am currently trying to calculate an integration formula of a linear response function by Monte Carlo method. It is a multiple integration over three 3D vectors, i.e., nine dimensions in all. And because of green function, there will also be an infinite discete frequency summation. This can be done either inside the integral, i.e., when evaluating the integrand, or outside the integral, i.e., summing up the integral result for each discete frequency point. My question is that whether this order issue matters or not."}
{"id":"103049","title":"Electric field near surface of a conductor?","text":"> If the electric field on a positive charge is non-zero, then the charge > accelerates in the direction of the field. The field at the surface of a conductor is perpendicular to the surface. Why don't the surface charges accelerate away from the surface of the conductor then?"}
{"id":"111939","title":"A question about a complex integration in Peskin's QFT textbook","text":"In page 27 (2.52), the integration is $$\\int_{-\\infty}^{\\infty}dp \\frac{p e^{ipr}}{\\sqrt{p^2+m^2}}$$ He says that there are two branch cuts starting from $\\pm im$ ![enter image description here](http:\/\/i.stack.imgur.com\/pyO89.png) But I learn in complex analysis that $\\sqrt{z^2+m^2}$ has only one branch cut from $-im$ to $im$, because point going around $im$ or $-im$ only will gets a minus, but point going around $\\infty$ only will keep the sign. Therefore $\\infty$ point is not the branch point and the branch cut is from $-im$ to $im$. So who is wrong?"}
{"id":"23206","title":"Information loss in a black hole","text":"How does the Holographic Principle help to establish the fact that all the information is not lost in a black hole?"}
{"id":"66001","title":"Non-Hermiticity when Fourier transforming onto a finite lattice","text":"I'm doing numerical simulations. I have the Haldane model in a honeycomb lattice where $$ H = \\sum \\limits_{<ij>}a^\\dagger_i b_j + h.c $$ Where $i$ belongs to sublattice $A$, and $j$ to sublattice $B$, which comes from the fact that there are two atoms per unit cell. $a$ is the annihilator on sublattice $A$ and $b$ on sublattice $B$. This operator is Hermitean. I can use its Fourier representation $$ H \\sim \\sum \\limits_k H(k) e^{ikr} $$ Where $r$ denotes lattice vectors and $k$ the corresponding lattice momentum. The amplitudes $H(k)$ are not Hermitean but they satisfy $H(k)^\\dagger = H(-k)$ obviously, so the above sum remains Hermitean if I take out pairs of $(k,-k)$. I cannot figure if by discretizing the elementary cell in $k$-space, thus effectively taking out some of the $k$'s in the above sum, the Hermiticity of this truncated Hamiltonian would get lost."}
{"id":"66005","title":"Young's Double slit experiment (Finding the value for slit separation)","text":"The question illustrates mono-chromatic red light passing through 2 slits and forming an interference pattern on the screen. The question then asks to suggest a suitable value for the slit separation. How do I find a value for this or what basis should I approximate my answer?"}
{"id":"66007","title":"Super High Frequency Electromagnetic Radiation - String Theory","text":"I am a serious high school student with one year of physics class experience, so please point out if there are any flaws in my question or reasoning. Thanks! Gamma ray radiation possesses a wavelength below 10 picometers ($10^{-11}\\: \\mathrm{m}$) and is capable of ionizing atoms. If a photon was endowed with sufficient energy ($~1.24 \\times 10^{10}\\: \\mathrm{J}$) to yield a wavelength on the scale of a Planck length ($~1.62 \\times 10^{-35}\\: \\mathrm{m}$), the theorized length of strings, would it be capable of producing interference or some other influence on strings, if they exist? From what I have learned, string theory postulates that all matter and energy is composed of tiny vibrating strings that require 11 dimensions (please correct me if I am wrong). Would EM radiation with a wavelength on the scale of a string's length then be able to cause some sort of interference or influence on the string's properties or vibrational pattern? Considering that an electron is so much larger than a string, such an experiment seems like trying to cause a ripple through a floating one inch piece of string by rapidly shifting an adjacent mountain one inch back and forth. At such high energy levels, would an electron even be able to remain stable, or would it dissociate into its constituent particles? If so, would there be some amount of sufficient energy that would dissociate even those particles? Here is the equation that I used to approximate the energy required to endow an electron with a wavelength of $~1.6 \\times 10^{-35}\\: \\mathrm{m}$: $$E=\\frac{hc}{\\lambda} = \\frac{6.63 \\times 10^{-34}\\: \\mathrm{Js} \\cdot 3.00 \\times 10^8\\: \\mathrm{m\/s}}{1.62 \\times 10^{-35}\\: \\mathrm{m}}= 1.23 \\times 10^{10}\\: \\mathrm{J}$$"}
{"id":"109211","title":"How to reconcile angular momentum measurements?","text":"Imagine a particle tracing a counter-clockwise circular path on a flat table with a certain speed. The particle is tied with a massless string of length $R$ to a point $P$ at the center of the circular path. Will the particle rotate about $P$ forever at constant speed in the absence of any external force? Consider using different origins to measure the physical quantities. **Measurement Case 1 (origin $O$ at $P$):** If I choose the origin $O$ of a Cartesian coordinate system to be at $P$, the flat table is the xy-plane and the particle rotates about the point $O$ as described. Specifically, the angular velocity $\\vec{\\omega}$ of the particle points in the direction of the positive z-axis, the position of the particle at any time is specified by the position vector $\\vec{r}$, the angle between $\\vec{\\omega}$ and $\\vec{r}$ is always $\\frac{\\pi}{2}$, and the angular momentum $\\vec{L}$ is as follows: $\\vec{L} = \\vec{r}\\times\\vec{p}\\;\\ldots\\text{ definition} \\\\\\ \\hphantom{\\vec{L}} = m\\,(\\vec{r}\\times\\vec{v})\\;\\ldots\\text{ definition of linear momentum }\\vec{p} \\\\\\ \\hphantom{\\vec{L}} = m\\,(\\vec{r}\\times(\\vec{\\omega}\\times\\vec{r}))\\;\\ldots\\text{ definition of linear velocity in terms of angular velocity} \\\\\\ \\hphantom{\\vec{L}} = m\\,(\\vec{\\omega}\\,(\\vec{r}\\cdot\\vec{r}) + \\vec{r}\\,(\\vec{r}\\cdot\\vec{\\omega}))\\;\\ldots\\text{ scalar triple product}\\\\\\ \\hphantom{\\vec{L}} = m\\,(\\vec{\\omega}\\,(r^2) + \\vec{r}\\,(0))\\;\\ldots\\;\\vec{r} \\perp\\vec{\\omega}\\\\\\ \\hphantom{\\vec{L}} = (m\\,r^2)\\,\\vec{\\omega}$ That is, $\\vec{L}$ and $\\vec{\\omega}$ have the same direction, and $\\vec{L}$ does not change direction and magnitude while the particle is rotation about $P$. Therefore, the particle will rotate about $P$ forever at constant speed in the absence of any external force because the angular momentum $\\vec{L}$ is conserved (the presence of a centripetal force does not affect the angular momentum $\\vec{L}$ in any way because a centripetal force is always parallel to the moment arm, and therefore, does not give rise to a torque $\\vec{\\tau}$). **Measurement Case 2 (origin $O'$ vertically directly beneath $P$):** Now if I choose to describe the same problem by choosing an origin $O'$ of a Cartesian coordinate system to be vertically directly beneath $P$, the particle's position vector $\\vec{r}'$ makes an angle $\\phi$ with the z-axis and the particle's angular momentum $\\vec{L}'$ also makes a $\\phi$ angle with the z-axis. As the particle rotates, however, $\\vec{r}'$ also rotates about the z-axis, and therefore, the angular momentum $\\vec{L}'$ keeps changing direction as shown below: $\\vec{L}' = \\vec{r}'\\times\\vec{p}'\\;\\ldots\\text{ definition} \\\\\\ \\hphantom{\\vec{L}'} = m\\,(\\vec{r}'\\times\\vec{v}')\\;\\ldots\\text{ definition of linear momentum }\\vec{p}' \\\\\\ \\hphantom{\\vec{L}'} = m\\,(\\vec{r}'\\times(\\vec{\\omega}\\times\\vec{r}'))\\;\\ldots\\text{ definition of linear velocity in terms of angular velocity} \\\\\\ \\hphantom{\\vec{L}'} = m\\,(\\vec{\\omega}\\,(\\vec{r}'\\cdot\\vec{r}') + \\vec{r}'\\,(\\vec{r}'\\cdot\\vec{\\omega}))\\;\\ldots\\text{ scalar triple product}\\\\\\ \\hphantom{\\vec{L}'} = m\\,(\\vec{\\omega}\\,(r'\\,^2) + \\vec{r}'\\,(r'\\,\\omega\\,\\cos(\\angle(\\vec{r}', \\vec{\\omega})))) \\\\\\ \\hphantom{\\vec{L}'} = (m\\,r'\\,^2)\\,\\vec{\\omega} + (r'\\,\\omega\\,\\cos\\phi)\\,\\vec{r}'$ Because the angular momentum $\\vec{L}'$ is not conserved due to changing direction, and a changing $\\vec{L}'$ requires the presence of an external force to give rise to a net torque $\\vec{\\tau}'$ about the point that is used to measure $\\vec{L}'$, which is the origin $O'$, the particle will not rotate about $P$ forever in the absence of any external force. But then, a contradiction arises: **the same phenomenon has a different outcome depending on the choice of origin $O$ or $O'$!** That is _unacceptable_ in Physics! How to resolve this contradiction?"}
{"id":"493","title":"Notation of plane waves","text":"Consider a monochromatic plane wave (I am using bold to represent vectors) $$ \\mathbf{E}(\\mathbf{r},t) = \\mathbf{E}_0(\\mathbf{r})e^{i(\\mathbf{k} \\cdot \\mathbf{r} - \\omega t)}, $$ $$ \\mathbf{B}(\\mathbf{r},t) = \\mathbf{B}_0(\\mathbf{r})e^{i(\\mathbf{k} \\cdot \\mathbf{r} - \\omega t)}. $$ There are a few ways to simplify this notation. We can use the _complex field_ $$ \\tilde{\\mathbf{E}}(\\mathbf{r},t) = \\tilde{\\mathbf{E}}_0 e^{i(\\mathbf{k} \\cdot \\mathbf{r} - \\omega t)} $$ to represent both the electric and magnetic field, where the real part is the electric and the imaginary part is proportional to the magnetic. Often it is useful to just deal with the _complex amplitude_ ($\\tilde{\\mathbf{E}}_0$) when adding or manipulating fields. However, when you want to coherently add two waves with the same frequency but different propagation directions, you need to take the spatial variation into account, although you can still leave off the time variation. So you are dealing with this quantity: $$ \\tilde{\\mathbf{E}}_0 e^{i\\mathbf{k} \\cdot \\mathbf{r}} $$ My question is, what is this quantity called? I've been thinking _time- averaged complex field_ , but then again, it's not really time-averaged, is it? _Time-independent_? Also, what is its notation? $\\langle\\tilde{\\mathbf{E}}\\rangle$?"}
{"id":"105946","title":"Dimensional Analysis to Determine a Formula","text":"The kinetic energy of a particle confined to a spherical region with a uniform internal potential depends on its mass, the radius of the sphere, and the Planck constant. An electron, confined to such a region of radius $1$ nm, has kinetic energy $0.38$ eV. Use dimensional analysis to find the kinetic energy of a proton confined to a region the size of an atomic nucleus (radius ≈ $5 × 10^{−15}$ m). I cannot produce a formula which results in $0.38$ eV as in the example."}
{"id":"105941","title":"Commutator summation notation","text":"I have the relation $ e^L M e^{-L}=\\sum_{n=0}^\\infty \\frac 1{n!} [L,M]_{(n)}$ where $L$ and $M$ are operators. What does the subscript $n$ after the commutator bracket denote?"}
{"id":"113091","title":"Find the points where potential is null","text":"Let's say we have two charges called $q_1$ and $q_2$, respectively $20 \\, C$ and $-40\\,C$, at a distance $d=1\\,m$ We want to find all the points where electric potential is null. I solved the equation $$\\frac{q_1}{4\\pi\\epsilon_0r_1} + \\frac{q_2}{4\\pi\\epsilon_0(d-r_1)}=0$$ For $r_1$ (distance from $q_1$), and found $r_1=\\frac13\\,m$ However this is not the only solution: there is another point not in-between the charges, but $1\\,m$ left from $q_1$ ($r_1=-1\\,m$) How can I set up an equation giving me both the solutions?"}
{"id":"95785","title":"Maximising entropy when energy is shared between systems","text":"![enter image description here](http:\/\/i.stack.imgur.com\/9kYAR.png) This is a problem to do with statistical physics, and the exchange of energy when we have two microcanonical ensemble. I don't understand why there should be a minus sign in the middle, if Energy* is the energy of system 1 , Energy total - E* is the energy of another system, Two system were put together and interact, I was told this equation can maximise entropy, but I can't really see why? I thought S = (S1(E*))+S(E total - E*) is the total entropy. therefore it should be a positive sign when you differentiate.. Just to clarify, S(E*) and S(E total - E*) , they are just labels, not function of E* and E total - E*"}
{"id":"6710","title":"Angular momentum and EM wave","text":"Is there any sense in saying that circularly polarized EM waves have angular momentum?"}
{"id":"92726","title":"Some questions about spacetime topology, causality structures and other GR businesses","text":"1) What are the exact conditions required for the canonical transformation? Most papers just assume away with global hyperbolicity, but is there a more general condition for it? \"Quantum gravity in 2+1 dimensions\" does contain an ADM description of the 2+1 D equivalent of the Kerr black hole, which also contains CTCs (but no event horizon since vacuum solutions in 2+1D have a vanishing Riemann tensor). And some spacetimes with CTCs (or non-time orientable) do have a well-enough behaved Cauchy problem, so I am wondering how far the causality can be weakened until you cannot define a hamiltonian. 2) Why is De Sitter space always assumed to have the topology RxS^3? I know it cannot be S^4, which would be the most natural (just a Wick rotation of the sphere metric), since S^4 does not admit a Lorentz structure, and I guess that it cannot be R^4 because I think it would suffer from geodesic incompleteness (missing pole and all), but why RxS^3, and is there any other manifold that would admit that metric? I guess that SxS^3 would work, and whatever other terrible quotient manifold you could do with S^3."}
{"id":"92725","title":"Is the sun's solar radiance spectrum matching up with water's absorption spectrum just coincidence?","text":"People frequently point out that water has a pretty narrow range in which it isn't very absorbing of light, reaching a minimum at a wavelength of about 500nm: ![h2o](http:\/\/i.stack.imgur.com\/973K4.png) And that our eyes have made good use of this tiny available range, through evolution: ![eye](http:\/\/i.stack.imgur.com\/s9Sfa.jpg) That part makes sense to me, because the eye changed to match the physical parameters available. But if you look at the solar radiance spectrum: ![solar](http:\/\/i.stack.imgur.com\/o4zaF.jpg) You'll see that its peak matches VERY closely with the range water lets light transmit through. But the sun's radiance is just roughly a black body, and the absorption of water is just something inherent to the chemical structure of water. So is it coincidence that they match up? Thanks!"}
{"id":"92724","title":"The common major difference in thermal conducting properties between a random rock sample and metal?","text":"what are some common major differences in thermal conducting properties between a random rock sample(non-metal) and metal?"}
{"id":"92720","title":"NPT simulations: what is the inertia of volume fluctuations (using periodic boundary conditions)?","text":"NPT molecular dynamics (MD) are most often implemented by adding some kind of piston that regulates the pressure. The inertia of this piston determines the dynamics of volume fluctuations in the system, i.e. how fast can the volume increase\/decrease. When periodic boundary conditions are imposed, the system is not in contact with such pistons. What would be the intrinsic (not chosen by the user) inertia of the volume fluctuations? If I imagine an infinite periodic system, any volume change of the periodic box seems impossible, as it would make the unit cells at large distance sweep back and forth over very large distances, which would require an enormous kinetic energy. So, should I conclude that the unit cell can not change its shape\/volume by itself under periodic boundary conditions? The application I have in mind is an extension of the \"normal mode analysis\" which is used for computing vibrational frequencies, by diagonalizing the mass-weighted Hessian matrix. The Hessian matrix $H$ contains the second derivatives of the energy with respect to atomic positions, while the mass matrix $M$ is a diagonal matrix with the masses on its diagonal. The mass- weighted Hessian is then given by $\\tilde{H}=M^{-1\/2}HM^{-1\/2}$. Now I want to extend this principle to the volume degrees of freedom. The second derivatives of the energy with respect to shape and volume deformations of the unit cell are given by the elastic tensor $C$. So what I am now wondering about, is what is the corresponding \"mass\" matrix $M_C$? If I then diagonalize $\\tilde{C}=M_C^{-1\/2}CM_C^{-1\/2}$, I will find the frequency of the volume fluctuations. Normal mode analysis can also be derived by applying the Lagrangian formalism on the total energy (kinetic plus potential) of the system after first approximating it by a Taylor expansion up to second order in the displacements. That is why I wonder about the \"correct\" MD equations of motion (without pistons attached to the system, but rather the system by itself). I hope to recognize the inertia $M_C$ in these equations."}
{"id":"24460","title":"The Difference between Thomas-Fermi Screening and Lindhard Screening","text":"Assuming the general theory of screening related to electron-electron interactions, I was wondering if anyone could provide a clear, yet _conceptually_ complete explanation of the differences between the Thomas- Fermi and the Lindhard theories? Following the treatment by Ashcroft and Mermin (1976),I get the impression that the main difference is that the Thomas-Fermi model assumes that the electrostatic potential is slowly-varying in $\\vec{r}$: $$E_i(\\vec{k}) = \\frac{\\hbar^2k^2}{2m}-e\\phi(\\vec{r})$$, but I'm having trouble grasping the full physical significance of this statement."}
{"id":"24469","title":"Why are do neutral atoms shrink as their valence shells approach 8 electrons?","text":"Why do neutral, unbonded atoms shrink in size as they approach having 8 electrons in their valence shells? A good example is elements 3 through 10 in this table, that is, lithium (1 valence electron) through neon (8 valence electrons). * * * For posterity, below is my earlier _incorrect_ version of my question. It's incorrect because it based on a table that used the atomic radii of _bonded_ atoms to describe their sizes. That is a very different measurement from the sizes of neutral atoms, since for bonded atoms you are tossing in additional electrons via the bonds. Very small atomic radii for halogens are easy to explain in that case, since the incomplete shells must fight very hard over the shared electrons to complete their own octets, making their bonding radii anomalously short. This question had an interesting back-and-forth, and I thank the two contributors both who both caught my chart interpretation error and answered the real and interesting question that lurked beneath my incorrect one. * * * If you look here at the relative sizes of charge-neutral atoms, you can see something I never noticed or even thought about until I tried to think through this question in the new Chemistry beta. Why are nearly-complete atomic shells, such as the seven valence electrons of fluorine, so compact in comparison to the bloated-but-very-stable situation created by adding one more electron to make neon? I thought I could answer that in terms of Pauli exclusion and wound up completely baffled. Can anyone help on this one? Why does \"almost complete\" equal \"very small\" in a charge-neutral atom?"}
{"id":"24468","title":"Rotational kinetic energy during vertical circular motion of a particle","text":"Why is it not necessary to take into account rotational kinetic energy when using the Law of Conservation of Mechanical Energy to solve vertical circular motion problems? After all, the particle is rotating about the centre of the circle and does have rotational KE, doesn't it? All the examples I have seen just use KE= 1\/2 $mv^2$, e.g. here: http:\/\/www.physicsforums.com\/showpost.php?p=2312566&postcount=4"}
{"id":"55520","title":"Why is there no time dilation for frequency of a wave?","text":"Since frequency of a wave is a function of time, then for a particular ray of the wave, why would the frequency remain the same when observed from a moving reference frame? The frequency should change according to time dilation. No?"}
{"id":"21606","title":"Assumptions in Bell's Theorem","text":"It is often Stated that Bell's Theorem is equivalent to the statement: No theory of Local Hidden Variables can reproduce all of the predictions of quantum mechanics. I see nowhere in Bell's Theorem the assumption of hidden variables. As far as I can tell, there is one assumption: locality, which is shown to conflict with quantum mechanics. If I am wrong on this, could someone demonstrate exactly how Bell failed to demonstrate that quantum mechanics is non-local by showing how the assumption of hidden variables is used in his theorem? To me -- and by the way Bell -- the problem is locality."}
{"id":"77458","title":"About vector form of friction","text":"I read a text on mechanics and in the chapter on friction, there written that the kinetic friction is in the form $$f_k = \\mu_k F_N$$ where $f_k$ is the kinetic friction, $\\mu_k$ is the kinetic friction coefficient, $F_N=mg$ is the normal force due to the weight of the object. In another chapter, there mentioned the vector form of the force. But it is confusing that if $f_k = \\mu_k F_N$, should the vector form written as $\\vec{f}_k = \\mu_k\\vec{F}_N$? It doesn't look right to me since $\\vec{F}_N$ is downward but $\\vec{f}_k$ is along horizontal. So what's the right way to write the vector for, of kinetic friction? How does people know that $f_k = \\mu_k F_N$? From experiment? I am guessing that if the object is moving at the direction $\\hat{\\mathcal{d}}$, so the vector form of kinetic friction should be $$ \\vec{f}_k = -\\mu_k(\\vec{F}_n\\cdot\\hat{\\mathcal{d}})\\hat{\\mathcal{d}} $$ Is that correct?"}
{"id":"77450","title":"How the Lorentz transformation affects the metric tensor?","text":"After performing a Lorentz transformation, the orthogonal coordinates will become askew, as in the following figure: ![enter image description here](http:\/\/i.stack.imgur.com\/5NiiK.gif) and in such coordinate system, according to this Wikipedia article, the metric will have off-diagonal non-zero elements: $$g_{ij}=\\mathbf{e}_i.\\mathbf{e}_j$$ which is not that of a flat spacetime: $$ g = \\begin{bmatrix} 1 & 0 & 0 & 0\\\\\\ 0 & -1 & 0 & 0 \\\\\\ 0 & 0 & -1 & 0 \\\\\\ 0 & 0 & 0 & -1 \\end{bmatrix}$$ What's the problem?"}
{"id":"100894","title":"is centrifugal release equal to explosive release","text":"**Scenario # 1** I put gunpowder and then a ball bearing in an old musket and fire the bullet. **Scenario # 2** Lets imagine I had a motor with a disk on it and there was the same ball bearing stuck on the edge of the disk somewhere. If I spin the motor up fast and then release the ball bearing somehow, it will fire off perpendicular to the tangent line of the release point. Lets imagine for the purposes of comparison, I spin up the motor with enough rpm such that the ball bearing’s velocity equaled the velocity of the musket shot in scenario 1. Newton's 3rd law of motions states that \"for every action there is an opposite and equal reaction\". With that in mind, it shouldn’t really matter how I expelled the hunk of lead, would both systems react in the opposite direction the same at the moment of release? (I’m aware that the spinning system would have a torque while it was spinning, but lets ignore that) The reason I’m asking that is; intuitively the ball bearing is desperate to escape the faster the disk is being spun. It would seem weird that it would suddenly kick the system backwards when it was already straining to leave the system the whole time. Also, there seems to be an intuitive difference because the expelling force in the spinning system is 90 degrees to the applied force."}
{"id":"73206","title":"A question about the Weyl transformation for the vertex operator of the closed-string tachyon","text":"I met a problem of derving the Weyl transformation on the closed-string tachyon, Eq. (3.6.8) in Polchinski's string theory, vol 1, p 103. Given the vertax operator of the closed-string tachyon > $$V_0 = 2g_c \\int d^2 \\sigma g^{1\/2} e^{ik \\cdot X} \\tag{3.6.1}$$ It is said the Weyl transformation of Eq. (3.6.1) is > $$\\delta_W V_0 = 2g_c \\int d^2 \\sigma g^{1\/2} \\left( 2 \\delta \\omega( > \\sigma) - \\frac{k^2}{2} \\delta_W \\Delta (\\sigma, \\sigma) \\right) [ e^{ik > \\cdot X(\\sigma)} ]_r \\tag{3.6.8}$$ here renormalized operator $[]_r$ and $\\Delta (\\sigma, \\sigma)$ are defined as > $$ [ \\mathcal{F}]_r = \\exp \\left( \\frac{1}{2} \\int d^2 \\sigma d^2 \\sigma' > \\Delta(\\sigma, \\sigma') \\frac{ \\delta}{\\delta X^{\\mu}(\\sigma)} \\frac{ > \\delta}{\\delta X_{\\mu}(\\sigma')} \\right) \\mathcal{F} \\tag{3.6.5} $$ $$ > \\Delta(\\sigma,\\sigma') = \\frac{ \\alpha'}{2} \\ln d^2(\\sigma,\\sigma') > \\tag{3.6.6} $$ where $d(\\sigma,\\sigma')$ is the geodesic distance between > points $\\sigma$ and $\\sigma'$. I can get the first term in the big parathesis $()$ of Eq. (3.6.8), namely consider $$ \\delta g^{1\/2} = \\frac{1}{2} \\frac{1}{g^{1\/2}} \\delta g= \\frac{1}{2} \\frac{1}{g^{1\/2}} g g^{ab} \\delta g_{ab} = \\frac{1}{2} \\frac{1}{g^{1\/2}} g g^{ab} 2 \\delta \\omega g_{ab} = 2 g^{1\/2} \\delta \\omega $$ I try to get the second term in the big parathesis of Eq. (3.6.8) $$ \\delta e^{i k \\cdot X} = e^{i k \\cdot X} i k^{ab} X^{ab} \\delta g_{ab} = e^{i k \\cdot X} i k^{ab} X^{ab} 2 \\delta \\omega g_{ab} = 2 i k \\cdot X e^{ik \\cdot X} \\delta \\omega,$$ but where do the $\\Delta(\\sigma,\\sigma)$ and renormalized opeator $ [ e^{ik \\cdot X(\\sigma)} ]_r $ in (3.6.8) come from?"}
{"id":"1048","title":"What if the universe is rotating as a whole?","text":"Suppose in the milliseconds after the big bang the cosmic egg had aquired some **large** angular momentum. As it expanded, keeping the momentum constant (not external forces) the rate of rotation would have slowed down, but it would never reach zero. What implications would that have to measurments to distant supernova and CMB radiation? Do we have any experimental data that definitely rules out such as scenario? And to what confidence level? **Edit** A Recent Article suggests that the universe might indeed be spinning as a whole. Anyone care to poke holes at it?"}
{"id":"130394","title":"What restores balance to a repulsive electric force on an electroscope?","text":"I am an amateur physics enthusiast (during the day I am a police officer), and recently, I learned how to build an electroscope. I regret that I am learning how wonderful our universe is so late in my life, and I suppose it is hard to teach an old dog new tricks ( I do not know how to properly research a question, or really how to locate a correct answer surrounded by a lot of other information). My question is this: On a simple electroscope, when one applies a charged object, the thin pieces of foil will separate, due to a separation of charges. However, I am curious as what factor balances the electric force of repulsion, so that the leaves do not continue to separate any further."}
{"id":"130395","title":"Free vs bound vectors and torque","text":"When considering basic Newtonian mechanics, we can treat vector as free and move their point of application at will. This is consistent with the affine nature of Euclidean space. However, when calculating torque on a body, we must treat forces as bound to their point of application. What is the mathematical reason for this? What does it imply for affine structure of Euclidean space*"}
{"id":"130399","title":"Determine the normalisation constant of a piecewise wavefunction","text":"I'm trying to find the normalisation constant $N$ for the following wavefunction: $$ \\psi\\left(x\\right) = \\left\\\\{ \\begin{array}{lr} N \\left(x^2 - l^2\\right)^2 &\\: \\left|x\\right| \\le l \\\\\\ 0 &\\: otherwise \\end{array} \\right. $$ Using: $$ \\int_{-\\infty}^{\\infty} \\left|\\psi\\left(x\\right)\\right|^2 \\, dx = 1 $$ The answer should be: $$ N = \\sqrt{\\frac{315}{256}} \\frac{e^{i \\phi}}{\\sqrt{l}} $$ However I get: $$ \\int_{-\\infty}^{\\infty} \\left|\\psi\\left(x\\right)\\right|^2 \\, dx \\ = \\int_{-l}^{l} N^2 \\left(x^2 - l^2 \\right)^4 \\, dx \\ = \\ \\frac{N^2}{10} \\left[\\frac{\\left(x^2 - l^2\\right)^5}{x}\\right]_{\\,-l}^{\\,l} = 0 $$ Which is clearly wrong, and I do not understand where the phase could have come from. Am I approaching this completely the wrong way? * * * I have now corrected the integration, however I get (double checked with Mathematica): $$ N = \\sqrt{\\frac{315}{256}} \\frac{1}{l^\\frac{9}{2}} $$ Which is the wrong power of $l$ (it should be $\\frac{1}{2}$). Substituting either the answer or my answer into the original integral does not yield $1$ either. (I am working though these quantum physics notes: http:\/\/ocw.mit.edu\/courses\/physics\/8-04-quantum-physics-i-spring-2013\/lecture- notes\/MIT8_04S13_Lec04.pdf)"}
{"id":"93455","title":"Recovering 4-vector Lorentz transformation from spinor formalism","text":"I'm trying to recover the 4-vector transformation laws using spinors. I have defined $$v^{\\dot{a}b} = v^{\\nu} \\sigma_{\\nu}^{\\dot{a}b}$$ as usual, with $\\sigma_0=1$. Now with the rules for dotted and undotted spinor indices i get the transformed v for a boost in the z-direction $$v^{\\dot{a}b} \\rightarrow v^{\\dot{a'}b'}= \\left({\\mathrm{e }}^{ - \\theta \\frac{\\sigma_3}{2}} \\right)^{\\dot{a'}}_{\\dot{a}} \\left({\\mathrm{e }}^{ \\theta \\frac{\\sigma_3}{2}} \\right)^{b'}_{b} v^{\\dot{a}b} = \\begin{pmatrix} {\\mathrm{e }}^{- \\frac{\\theta}{2}}&0 \\\\\\ 0&{\\mathrm{e }}^{ \\frac{\\theta}{2}} \\end{pmatrix} \\begin{pmatrix} v_0+v_3&v_1-iv_2\\\\\\v_1+iv_2&v_0-v_3 \\end{pmatrix} \\begin{pmatrix} {\\mathrm{e }}^{ \\frac{\\theta}{2}}&0 \\\\\\ 0&{\\mathrm{e }}^{ - \\frac{\\theta}{2}} \\end{pmatrix}$$ where i have used the fact that $\\sigma_3$ is diagonal and that ${\\mathrm{e }}^A= \\begin{pmatrix} {\\mathrm{e }}^{A_{11}}&0\\\\\\0&{\\mathrm{e }}^{A_{22}} \\end{pmatrix}$ holds for every diagonal matrix A. This gives me the wrong transformation! It would give me the correct transformation if I had $$v^{\\dot{a}b} \\rightarrow v^{\\dot{a'}b'}= \\left({\\mathrm{e }}^{ - \\theta \\frac{\\sigma_3}{2}} \\right)^{\\dot{a'}}_{\\dot{a}} \\left({\\mathrm{e }}^{ \\theta \\frac{\\sigma_3}{2}} \\right)^{b'}_{b} v^{\\dot{a}b} = \\begin{pmatrix} {\\mathrm{e }}^{- \\frac{\\theta}{2}}&0 \\\\\\ 0&{\\mathrm{e }}^{ \\frac{\\theta}{2}} \\end{pmatrix} \\begin{pmatrix} v_0+v_3&v_1-iv_2\\\\\\v_1+iv_2&v_0-v_3 \\end{pmatrix} \\begin{pmatrix} {\\mathrm{e }}^{ - \\frac{\\theta}{2}}&0 \\\\\\ 0&{\\mathrm{e }}^{ \\frac{\\theta}{2}} \\end{pmatrix}$$ But I can't figure out why this should be the case. The only possibility I can think about would be if a relation like $\\ A'_{\\nu\\mu} = M_{\\mu}^{\\ \\rho}(M_{\\nu}^{\\ \\theta})^{-1}A_{\\rho\\theta }$ $\\rightarrow$ $A'=MAM $ would hold, but i can't find a formula like this. Doing the summation by hand i get the same result as with normal matrix multiplication without using the inverse matrix on the right hand side. Any tip or help would be much much appreciated!"}
{"id":"5032","title":"Why do stars have absorption spectra?","text":"Absorption spectra are a result of light of a certain wavelength exciting an atom from a lower energy level to a higher one and at the same time being absorbed. However, the atom should eventually go back down to its lower energy state, and at the same time emit a photon of same frequency it absorbed earlier. Overall, no changes to the star's spectrum should occur."}
{"id":"38584","title":"Supermarket refrigerator - why is it noticebable colder in this shop area?","text":"As we know fridge can't cool room in which it is (according to Second Law of Thermodynamics, heat emitted by fridge is greater than heat absorped). However, when we go next to the fridge in supermarket we can feel coldness. Why? Thanks!"}
{"id":"130023","title":"Force and torque on rigid body in space","text":"![enter image description here](http:\/\/i.stack.imgur.com\/kAclJ.jpg) Assume that we have two disks in space. If we act same force F on each disk like above figure, how they react? I first thought that since $F_{ext}=\\frac{dP_{cm}}{dt}$, the CM(center of mass) momentum will become same. But what about angular momentum thus energy? In first case, there will be no torque, so no angular momentum. But for second case, there exists external torque about CM, so the ball will rotate. Then, it will have spin motion, so there will be additional kinetic energy. Is this correct? How same force can give different total energy change?"}
{"id":"70155","title":"Time to immerse in a fluid","text":"Whenever you drop an object in water, it takes some time to get fully immersed. I was wondering if this depends upon the buoyant force of the water on the object, slowing it down. However, I was not able to come up with a general formula for this. I wanted to know, what would be the time taken for an object of density $d_1$, dropped from a height $h$ to be fully immersed in a liquid of density $d_2$. Take gravity as $g$. Part of the reason I was not able to do this was because I thought it might involve calculus (since the volume of water displaced changes over time and so does the buoyant force). I'm not competent enough with calculus to be able to attempt this sort of problem. I know friction will affect the time, but what if there were $no$ friction? Can this be further generalized to gases as well?"}
{"id":"128215","title":"Dilation operator in CFT viewed as 'hamiltonian'?","text":"From the commutation relations for the conformal Lie algebra, we may infer that the dilation operator plays the same role as the Hamiltonian in CFTs. The appropriate commutation relations are $[D,P_{\\mu}] = iP_{\\mu}$ and $[D,K_{\\mu}] = -iK_{\\mu}$, so that $P_{\\mu}$ and $K_{\\mu}$ are raising and lowering operators, respectively, for the operator $D$. This is analogous to the operators $\\hat a$ and $\\hat a^{\\dagger}$ being creation and annihilation operators for $\\hat H$ when discussing the energy spectra of the $n$ dimensional harmonic oscillator. My question is, while $\\hat a$ and $\\hat a^{\\dagger}$ raise and lower the energy by one unit $( \\pm \\hbar \\omega)$ for each application of the operator onto eigenstates of $\\hat H$, what is being raised and lowered when we apply $P_{\\mu}$ and $K_{\\mu}$ onto the eigenvectors of $D$? Secondly, what exactly do we mean by the eigenvectors of $D$? Are they fields in space-time? Using the notation of Di Francesco in his book 'Conformal Field Theory', the fields transform under a dilation like $F(\\Phi(x)) = \\lambda^{-\\Delta}\\Phi(x)$, where $\\lambda$ is the scale of the coordinates and $\\Delta$ is the scaling dimension of the fields. Can I write $F(\\Phi(x)) = D\\Phi(x) = \\lambda^{-\\Delta}\\Phi(x)$ to make the eigenvalue equation manifest? Thanks for clarity."}
{"id":"41294","title":"Formation of a black hole and Hawking radiation","text":"From the perspective of an outside observer it takes infinitely long for the black hole to form. But if the black hole is no extremal black hole, it emits Hawking radiation. So the outside observer can detect the Hawking radiation of a black hole that hasn't yet formed from his point of view. How can this be?"}
{"id":"122079","title":"Falling off a chair, how best to save yourself","text":"If I consider a man sitting on an office chair that reclines backwards iff you lean backwards. What could be done to prevent hin from falling? a) raising his legs till they are parallel to ground. b) bring the feet closer to himself(as close as possible) and press them down on the ground. Please don't suggest any other way. I want to compare these two. Ie, which is better. Argument in favor of a) The center of mass moves away from the rear end of the chair towards the forward end of the chair so torque of weight will restore the chair. (This works if you are in an armchair, and are rocking to and fro). However, intuition says contrary to this. I feel scared of falling If I do this. Argument in favour of b) Intuition favours this. Normally when we talk about balancing our body, what \"feels right\" is better for the safety. So, I can not say what is right. Please explain what your views are in this case. Again, the purpose of this question is not to ask what should be done, only what is better to do out of these two choices."}
{"id":"41290","title":"A question from Hilbert and Courant's Vol II of Methods of Mathematical Physics (I might have spotted an error)","text":"In page 751 (I hope some folks have a copy of it, legal or otherwise, I have a legal one :-D), I am attaching scans of pages 750-751. ![enter image description here](http:\/\/i.stack.imgur.com\/9S24K.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/1TQ97.jpg) Anyway, I don'tunderstand two things, the equation in page 751, the RHS of $u(x_0,0,0,t_0)$, in $\\psi$, shouldn't the x-component be $\\alpha t_0\/2 + x_0$? and in the second term shouldn't it be just $\\alpha t_0 \\psi_x$? Here is my reasoning, the x-component should be calculate according to Lorentz transformation in page 750: $$x = \\frac{t_0}{\\sqrt{t_0^2-x_0^2}} (\\frac{\\alpha}{2}\\sqrt{t_0^2-x_0^2})+\\frac{x_0}{\\sqrt{t_0^2-x_0^2}}\\sqrt{t_0^2-x_0^2}=t_0 \\alpha\/2 +x_0$$ and as for the second term there, I might be wrong but: $$\\chi_{x'} = \\psi_{x'} = \\psi_x \\frac{\\partial x}{\\partial x'} = \\frac{t_0}{\\sqrt{t_0^2-x_0^2}} \\psi_x $$ So shouldn't it be $\\alpha t_0 \\psi_x$, instead of $(\\alpha t_0+x_0) \\psi_x$. I guess I missed something. Thanks in advance..."}
{"id":"90490","title":"Name for thermodynamic derivative $dP\/dT|_V$?","text":"While trying to express the isoentropic sound speed as partial derivatives of $V$ and $T$ only I end up, as part of the longer expression with $dP\/dT|_V$ (which according to a Maxwell relation is the same as $dS\/dV|_T$). Does such quantity $dP\/dT|_V$ have a given name? or it is simply expressible as a combination to other quantities (like $V dP\/dT|_V$) with a given name? (By \"given name\" I mean, for example that $-V dP\/dV|_T$ is the Isothermal Bulk Modulus)"}
{"id":"122070","title":"Chandrasekhar Limit","text":"A white dwarf is essentially a degenerate electron gas, in which pressure of degenerate electrons supports gravitational pressure. As a simplified model of such an object, consider a spherical star of ideal gas consisting of electrons and completely ionized hydrogen atoms, and of constant density. Radius of star is $R$, Mass of star is $M$, $N$ = number of electrons. A) Consider the electrons to be ultra relativistic. Density of star is given to be ${10}^{12}\\, \\mathrm{kg \\cdot m}^{-3}$ . What is the critical mass for which the dwarf can be in equilibrium? I am trying to find the gravitational pressure and the fermi pressure due to the electrons, and then I will equate them. I have successfully calculated the gravitational pressure, but to calculate Fermi pressure, I need total energy to Pauli Exclusion principle applied to electrons. The total energy is coming to be a function of N, and due to some mistake (which I do not know , where am I making) , the radius term is not cancelling on both sides (which should). How to calculate total energy due to pauli exclusion principle?"}
{"id":"41298","title":"Would two astronauts one in a satellite, one on top of a tower have the same experience?","text":"Imagine two astronauts. The first is in a satellite in geostationary orbit around the earth. The second is on top of a 35,786 km tall tower that has been built from the equator to space. This means that that the top of the tower is at the same height as the geostationary satellite. Naive logic would seem to indicate that the astronaut in the satellite would experience free-fall (as do all satellites in orbit), whereas the astronaut on top of the tower would feel the effects of gravity (albeit at a reduced level due to being 35,786 km above the surface of the earth. However, I can't believe that this would be the case. What's the flaw in my logic? Is the acceleration due to the earth's gravity zero at 35,786 km above the earth?"}
{"id":"87272","title":"Calories burned doing 1 squat","text":"To calculate the calories burned doing 1 squat, wouldn't it simply involve the distance that your center of mass is moving and your weight? I read some other article that said your legs act as levers, so the calculation has to involve that. Then of course you have to come up with the efficiency of your body."}
{"id":"133612","title":"Vorticity and circulation in fluid mechanics","text":"Given a fluid element whose circulation was determined counter-clockwise, how can I calculate the Vorticity of the element? * * * The circulation is $$\\Gamma = (\\frac{\\partial u_2}{\\partial x_1}-\\frac{\\partial u_1}{\\partial x_2})\\delta x_1 \\delta x_2$$ I thought this might give a better idea about the question. Here $u_1$ is the velocity along $x_1$ and $u_2$ is the velocity along $x_2$."}
{"id":"133610","title":"What is supersymmetry?","text":"1. What is supersymmetry?    2. How is it useful in unifying physics? I have heard that supersymmetry states that every fermion is associated with a boson and vice versa. But I don't get hope this can help resolving problems in unifying physics."}
{"id":"133616","title":"Question on spin-orbit interaction","text":"When you study the spin-orbit interaction in quantum mechanics, even for a simple hydrogen atom, you find only the electric field in the nucleus reference system, while in the electron reference system you find the magnetic field due to the relative orbit of the nucleus around it (through Lorentz Transformations). Why is that? Shouldn't there be a magnetic field in the nucleus reference system due to the orbit of the electron as well? Is it just a convention, since electric and magnetic field are actually just one field?"}
{"id":"133617","title":"Why is the diffusion capacitance greater than the depletion capacitance in a p-n junction?","text":"In a p-n junction the total capacitance is the sum of depletion capacitance and diffusion capacitance and it is mentioned that the diffusion capacitance is almost a million times greater than the depletion capacitance. Why is it so? Is this true that in a p-n junction diffusion capacitance dominates at all situations?"}
{"id":"83699","title":"Does unitarity imply conservation of energy?","text":"Not too long ago, someone began to discuss the thinking and motivation behind the Lagrangian and its formalism for the Newtonian framework and an intuitive understanding of such formalism. Somehow, it ended in the case that the Lagrangian can be understood in terms of conservation of information (largely) without having to rely on other laws. Is this line of argument correct? And, more importantly, would this mean that unitarity is a stronger statement than conservation of energy?"}
{"id":"71206","title":"Determination of mass density distribution of an object","text":"This is a follow-up to a previous question How can you weigh your own head in an accurate way?. My purpose is not to restart the flurry of more or less humorous jokes (which are not such a bad thing when infrequent) but to try to draw some conclusions, as I found the event instructive, and hopefully to get a real answer. Beyond the jokes, I think the question was also interesting from a more social point of view: how fallacies operate, how people vote, what is the effect how the fast answer inducement by site rules. It is clear now, at least to me, that the real physical problem was to determine the mass density distribution of a (partly ?) solid object without using destructive means. Of course the concept of destructiveness may depend on the concerned body, and that was one source of jokes. Still, it is a relevant topic to review the means, if any, for achieving such a purpose, as it seems that there are not that many. This is actually the question I am asking. But I would like also, at the same time, to review some of the answers given to the previous qiestion. One that attracted my attention was the micro-satellite solution. Does it really work ? So we could precisely ask: given a precise knowledge of the gravity field created by a solid object, can we deduce from it the mass density distribution in the solid. The answer seems to be no. Take the trivial case of a sphere with a uniform radial density distribution. The gravity outside the sphere depends only on the total mass, so that it tells us nothing about the internal density distribution. Is that an exceptional degenerate case, or is it a strong hint that the analysis of the gravity field is not enough ? The slicing solution I proposed was an interesting mistake, certainly a silly one. But it is ever so tempting to believe the solution is near when you have a set of equations, and easy ones to boot. A cute trap. It becomes more obvious when you try to do it in continuous rather than discrete form. It should have been obvious, as we know that the torque produced by weighing a massive object depend only on total mass and distance of the center of mass to the axis. What was less obvious, at least to me (but I have not done much physics since college), was the use of the moment of inertia. One could also think of measuring moments of inertia after virtual slicing of the object as previously with torque. Unfortunately the moment of inertia depends only on 3 values, the previous two and a reference moment of inertia (for a given direction of the rotation axis). It is a useful remark for computing moments of inertia, but shows that there is no hope there for solving our problem.   This was remarked by @Ben Crowell in another question (see below). More generally, it seems there is no hope from any discretized measurement of a quantity that depends polynomially on the distance. One cannot get more unknowns than the degree of the polynomial. I will not comment on Compton scattering, and other uses of indirect physical phenomena, if only for lack of competence. There is also the fact that I would like to know of a solution involving only mechanics. One technique I heard of is measuring wave propagation across the object. I hinted to that in a comment on the Compton scattering solution. I have no expertise on that, but I think that is how the structure of the planet is analyzed by geophysicists (earthquake waves). I also heard of underground analyses by similar means, using explosives. But I am not sure about the type of information that can be obtained in this way. Does one get density distribution. Does it always work ? The other technique I thought of is ultrasound based medical imagery. Could it measure densities. Now since we are considering waves, could we get something out of the mesurement of gravity wave propagation, assuming it is possible to do the necessary measurements. But I have no idea what this actually entails or means. Interestingly, a simpler version of the problem has already been discussed to some extent two months ago in How can I determine whether the mass of an object is evenly distributed? Some ideas were repeated for the recent question, but new ideas emerged too (gravity field analysis) even if they have weaknesses, and are somewhat hard to use in most situations. Then, what techniques have been used to know the density distribution of Earth, the Sun or possibly other bodies. Is there anything systematic that we overlooked. So, are there means of solving the problem purely with mechanics and gravity? A related question is whether it is easier, or as hard, to determine whether the mass density distribution is uniform. More generally, I am wondering whether there is a way of characterizing the properties of phenomena that can help determine the density distribution inside a solid object. Alternatively, could one prove it is not achievable by purely mechanical means."}
{"id":"7101","title":"Alpha Beta Gamma Biological Impact","text":"Assume that alpha & beta particles and gamma photons each reach skin at the same energy. It's known that they penetrate most deeply in order by gamma, beta and alpha. How would they compare in terms of total biological impact? In other words, which would be most harmful?"}
{"id":"123308","title":"Principle of superposition for capacitor","text":"In a capacitor the parallel plates of opposite charge create equal electric fields in opposite directions. We know field outside the capacitor is zero but inside the capacitor it is non zero. My question is if we assume zero field because of superposition of the two electric fields, why can't we do the same in between the two plates, because here also the fields are opposite to each other and hence will cancel? Basically, I want to know when we can use superposition principle and when we can't."}
{"id":"86580","title":"Proof of diffraction equations","text":"If light incident on a grating makes an angle φ with respect to the normal to the grating, show that dsin(θ)=mλ becomes d[sin(θ-φ)+sin(φ)]=mλ. I began by showing:               d[sin(θ-φ)+sin(φ)]=mλ, θi=φ=θ at m=0      (see my note after about why I think this line is wrong) then:               d(sin(θ-θ) + sin(θ))=mλ     d(sin(0)+sin(θ))=mλ     d(0+sin(θ))=mλ     dsin(θ)=mλ      It decomposes nicely to what I'm asked for but now that I look at it again I see that I've also declared m=0 which means that:               dsin(θ)=0λ     dsin(θ)=0      which implies that either d=0 or sin=0, both of which I don't find probable. I suppose I'm stuck on where or how to introduce φ to the equation. Any suggestions welcome."}
{"id":"86581","title":"Vector Maths to Simulate Car Physics","text":"I'm working on a 3D program that involves controlling a Cube on-screen (which represents a car). I use C++, OpenGL and GLM (for vector and matrices etc, if you're familiar with programming; Once the car rotates (or steers), I wish to make its direction change and follow the new path... ). I would like to know the equations to simulate steering (one force at a time, starting with the basic)... I've read lots about velocity vectors and multiplying by the sin\/cos of the angle to find it's new path... My question is, how can this be represented using vectors? And what are the equations\/vectors need to simulate basic car movement?"}
{"id":"30053","title":"Does an airplane's speed include the speed of the Earth?","text":"After take off, does airplane's speed include Earth's movement\/speed? Do airplanes turn with Earth movement\/rotation?"}
{"id":"14777","title":"Why isn't V. A. Fabricant given the credit for the discovery of the laser?","text":"In another question here I asked for help in finding old Russian papers in physics and as an example I gave the founding papers on the laser by Basov and Prokhorov (from 1945 and 1954). In an answer @voix was so kind as to post this link http:\/\/www.lebedev.ru\/data\/50laser\/Beginning_of_the_Laser_Era_in_the_USSR.pdf with a collection of some of the founding papers that have shaped the area of quantum electronics and, in particular, the idea of the laser. In this collection I found with astonishment that it was not Basov and Prokhorov, neither was it Townes who discovered the laser but the discovery of reverse absorption which is at the basis of the laser action is in fact due to V. A. Fabrikant in 1939, years before the ones I have always known to be the discoverers and who got the Nobel prize for it. Fabrikant even has a 1951 patent and a diploma No. 12 for a discovery in USSR regarding the phenomenon of reverse absorption.The astonishing fact is that Basov and Prokhorov do not even cite Fabrikant in their 1954 paper. Neither do Townes et al in their 1954 paper. What phenomenon, different from what Fabrikant has discovered did Basov, Prokhorov and Townes find to deserve all the credit for the discovery of the laser and have Fabrikant sent to obscurity?"}
{"id":"14778","title":"What is a proton-rich atom?","text":"http:\/\/wiki.chemprime.chemeddl.org\/images\/e\/e4\/Plot_of_Neutron_Number_vs._Proton_Number_.jpg The above graph shows that all elements have more neutrons than protons in this nucleus. So how is there such a thing as a proton rich atom? If all the heavy unstable atoms are attempting to balance the number of protons and neutrons (this is what I was taught) why do we get protons decaying to (eventually\/after intermeditary particles) neutrons (and emitting a positron?), of which there are excess?"}
{"id":"47925","title":"Could the Bernoulli effect be causing my ceiling to come down?","text":"Very 'applied' question, but I have nowhere else to turn, so I'm asking the physics experts here: I have a carport whose ceiling is made of very lightweight paneling. I've had several times now that those panels have fallen down, without an evident reason. So I'm wondering: the wind (because of the placement of other buildings in the area) blows very strong underneath the carport. Could it be that strong wind gusts, through the Bernoulli principle, are causing 'pull' on the panels and tearing them from the staples they are attached with? Stated more generally, does a fluid flowing fast beneath a surface cause a force on that surface? Thanks."}
{"id":"5597","title":"Spinning Tachyons","text":"In all examples that I know, tachyons are described by scalar fields. I was wondering why you can't have a tachyon with spin 1. If this spinning tachyon were to condense to a vacuum, the vacuum wouldn't be Lorentz invariant---seems exotic but not a-priori inconsistent. Is there some stronger consistency requirement which rules out spinning tachyons? If someone could provide a reference that would be helpful too! Here's another confusion: I was reading Wikipedia, which claims that tachyons should be spinless and obey Fermi-Dirac statistics(?). (They reference an original paper by G. Feinberg which unfortunately I am not wealthy enough to download). The claim about Fermi-Dirac statistics is baffling---isn't the Higgs field a boson? Does anyone understand what they're talking about?"}
{"id":"89784","title":"The Goos Hanchen shift mechanism","text":"When the light is totally reflected in the interface between dense and less- dense medium, we know that the reflected beam will shift a little. Currently I have known the reflection coef r, will be a complex number and its phase angle will vary with the incident angle theta. Some paper explained this phenomenon as the light penetrates the less-dense medium a little, and re-emerge again, just like it is reflected by some virtual plane in the less-dense, but how can this be explained? Or by what mechanism?"}
{"id":"90026","title":"Centrifugal force in inertial reference frame","text":"Take for example the case of a rod rotating about an axis passing through its centre of mass and perpendicular to it. It has a ring hung from one of its sides. The rotation of the rod causes the ring to move out wards and ultimately fall off the rod. In the ground frame there is no real force acting on the body, but the ring still moves. If I am right, centrifugal force causes the motion of the ring. So does that mean that centrifugal force acts even in inertial reference frames? Also, I am finding it difficult to imagine the ring's reference frame? Is it a rotating reference frame? How exactly will u visualize motion in this frame of reference?"}
{"id":"72963","title":"Constrained motion in a parabolic tube","text":"> _A smooth parabolic tube is placed with vertex downwards in a vertical > plane. A particle slides down the tube from rest under the influence of > gravity. Prove that in any position, the reaction of the tube is equal to > $$2w({h+a})\/\\rho$$ where $w$ is the weight of the particle, $\\rho$ the > radius of curvature of the tube, $4a$ the latus rectum and $h$ the initial > vertical height of the particle above the vertex of the tube._ I am not able to figure out how to write the equations of motion for motion in a parabolic tube. Have read constrained motion for a vertical cycloid. Is a tubular cycloid and parabolic tube the same? Please help!"}
{"id":"38208","title":"Importance of phase in probability amplitude in QFT","text":"I have started teaching myself QFT from the textbook by A. Zee. From reading that book, my idea of a path integral in field theory is the probability amplitude to go from a given field configuration to a different field configuration. At a point, he adds a source function $J(x)$ in the free field theory Lagrangian and evaluates the path integral. It turns out that $Z(J)=Ce^{iW(J)}$ where $W(J)$ is a term containing the integral of source function and the propagator. Then, he says > The overall factor C does not depend on $J$ and, as will become clear in the > discussion to follow, is often of no interest to us. Then, he goes on to consider $W(J)$ for some special localized functions $J$ and that leads to the interpretation of particles in field theory and a derivation of inverse square law. But, I am quite disturbed by this. QM teaches us that only probabilities are something that are physical and the absolute phase factors in the probability amplitude are of no interest to us; that we should just concentrate on $C$ above. Instead, Zee says that it is of no importance to us and such spectacular stuff comes out just by examining the phases. So, why is it that phase is so important in QFT? Why did people even try to analyze the phase and interpret the results in terms of particles and forces that can physically be observed?"}
{"id":"13694","title":"Second Law of Black Hole Thermodynamics","text":"I've been looking for a satisfying proof of this, and can't quite find it. I read the brief proof of the black hole area theorem in Wald, which is similar, but doesn't quite come down to the actual statement $\\frac{dA}{dt} \\geq 0$. I had pictured something fairly succinct relating $T_{ab}k^ak^b \\geq 0$ (weak energy condition) directly to $\\frac{dA}{dt}$, and I tried but I couldn't get anything to quite work out. Anyway- can anyone point me to a good source, or even just give the proof?"}
{"id":"13695","title":"a priori validity of $W=\\int Fdx$ in relativity?","text":"There are lots of different ways of arriving at the relativistic relations involving mass, energy, and momentum such as $E=mc^2$ and $m^2=E^2-p^2$ (the latter with $c=1$). One that I've seen in some textbooks is to start with the equation $W=\\int F dx$ for mechanical work. Here is a particularly careful and rigorous version, which makes it explicit that this is a nontrivial assumption, along with $F=dp\/dt$: http:\/\/www.physicsforums.com\/showthread.php?p=2416765 Historically, Einstein made use of $W=\\int F dx$ in section 10 of his 1905 paper on SR http:\/\/www.fourmilab.ch\/etexts\/einstein\/specrel\/www\/ . This was before the \"Does the inertia of a body...\" paper, in which he derived $E=mc^2$ and gave it its full relativistic interpretation. Einstein later decided that force wasn't a very useful concept in relativity and stopped appealing to it. What I have never seen in any of these treatments (neither the careful ones above nor the sloppy ones in some freshman physics texts) is any argument as to why the nonrelativistic relation $W=\\int F dx$ should be expected _a priori_ to hold without modification in SR. If one has _already_ established the form and conservation of the energy-momentum four-vector, then I don't think it's particularly difficult to show that $W=\\int F dx$. But what justification is there for assuming $W=\\int F dx$ _before_ any of that has been established? Is there any coherent justification for it? (A secondary problem with some of these treatments based on $W=\\int F dx$ is that they need to establish the constant of integration.)"}
{"id":"43354","title":"Does wavefunction reach its largest peak near(not in) the classical forbidden region?","text":"As we can see in the picture in this website: http:\/\/ctz116.ust.hk\/xyli2\/images\/animation\/quchem73.html It's strange that the bound state wavefunction always reach its largest peak near the boundary of its classically forbidden region(not in the region). Is it true that this phenomenom holds for all bound state wavefunction? I think that the reflected wave may interfere with the original one, thus creating the peak near the forbidden region.But I can't explain why it is the largest peak or there is no peak inside the classically forbidden region,Thanks for your attention."}
{"id":"65009","title":"Problem with a rotating frame of reference on the South pole","text":"Consider this problem: > _A high-speed train is traveling at a constant 150 m\/s (about 300 mph) on a > straight horizontal track across the south pole. Find the angle between a > plumb line suspended from the ceiling inside the train and another inside a > hut on the ground. In what direction is the plumb line on the train > deflected?_ I'm positively perplexed by it. I think that what we're looking for is a relationship $\\tan(\\theta)=\\frac{a_n}{a_t}$, because for the plumb line in the hut $a_t=1$ but not for the plumb line in the train due to the Coriolis force. However,   1. What is the frame of reference here? Is it rotating, fixed so that the train is standing still? Is it rotating with the earth?   2. Since the train is on the south pole, isn't that the same to say that the earth's rotation doesn't affect it?   3. Can you show me how to get the right answer? The right answer is supposed to be 0.13 degrees. In which direction?"}
{"id":"88786","title":"Lower limit of the size of the Universe? (WMAP)","text":"The measurement of the WMAP satellite resulted a planar geometry of the universe with a 0.4% uncertainity (http:\/\/en.wikipedia.org\/wiki\/Shape_of_the_universe). If there is a little deviation from the measured zero curvature, I think it could give a lower limit to the size of the universe (in case of positive curveture and spherical geometry). How big is it? How could it be calculated?"}
{"id":"65001","title":"How can a car's engine move the car?","text":"Newton's First Law of Motion states that an object at rest or uniform motion tends to stay in that state of motion unless an unbalanced, external force acts on it. Say if I were in a car and I push it from the inside. It won't move. So how is the engine of a car capable of moving the car?"}
{"id":"103200","title":"Human and Ultraviolet light","text":"As I know, ultraviolet light can be created from burning something at high temperature. So I have a question: Can a human body become a source of ultraviolet light at high temperature? If the answer is yes, at what temperature does this happen in a human?"}
{"id":"65003","title":"Finding Hamilton's equations given a Hamiltonian","text":"I am trying to find Hamilton's equations for a general Hamiltonian given by $$H[u]=\\int_\\mathbf{R} \\phi(u,u_x)dx$$ Suppose $$\\frac{\\delta f[u]}{\\delta u(x)}\\equiv \\frac{\\partial f}{\\partial u}-\\frac{\\partial }{\\partial x}\\frac{\\partial f}{\\partial u_x}+\\left(\\frac{\\partial }{\\partial x}\\right)^2\\frac{\\partial f}{\\partial u_{xx}}+...$$ then Hamilton's equations are given by $$\\frac{\\partial u}{\\partial t}=\\\\{u,H[u]\\\\}=\\frac{\\partial }{\\partial x}\\frac{\\delta H[u]}{\\delta u(x)}$$ But I don't know how to evaluate the last quantity. Would someone mind teaching me, please? * * * My guess would be $$\\frac{\\partial }{\\partial x}\\int_\\mathbf{R} \\frac{\\partial \\phi}{\\partial u}-\\frac{\\partial }{\\partial x}\\frac{\\partial \\phi}{\\partial u_x}dx$$ Then $$\\left[\\frac{\\partial \\phi}{\\partial u}-\\frac{\\partial }{\\partial x}\\frac{\\partial \\phi}{\\partial u_x}\\right]_{-\\infty}^{\\infty}$$ Would that be right?"}
{"id":"65005","title":"Estimating the Kolmogorov Complexity of the Standard Model","text":"The Kolmogorov complexity of a hypothesis \/ theory \/ model is the shortest computer program that simulates it, regardless of how inefficient executing that program may be in terms of memory and time. I'm interested in how complex the standard model is, by this measure. For example, this MinutePhysics video notes that the standard model is (almost) one equation. That's pretty short (less than 50 characters), but of course in order to turn it into a computer program you also need to encode how to perform the underlying math. On the other end of the spectrum: teaching a human physics via text books can be done with millions of characters, but the majority of that \"millions\" is presumably due to the constraints of communicating to a human. I guess I expect the answer to be less than a million bytes, and _maaaaaaybe_ less than a kilobyte, but that's not really based on much except intuition. I searched google, and google scholar, and was surprised to fail to quickly find even a loose upper bound on the complexity of the known laws of physics. Has such an estimate been done? How difficult is it to do one? _How complicated is the standard model?_"}
{"id":"103204","title":"Challenge: Answer this gedanken (PFP - Perpendicularly Fired Photon)","text":"I'm challenging anyone who can answer the following question objectively: As usual, imagine a railway station and trains which are equipped with single photon sources (one each in the platform and inside the trains) and a narrow strip of detector which is positioned right above the photon sources. The photons are fired vertically upwards, perpendicular to the train’s direction of travel. In addition, let us assume that apart from the photo detectors on the train and the platform, the roof of the train is open, and that the ceiling of the station above the train also has photo detectors, corresponding to the position of the photo detectors on the platform. Now, the photon sources in the station and the trains are set to trigger at the same moment – i.e. when the train’s photon source passes in front of **Alice**. Now, **Bob** is sitting in front of the photon source in the train which is travelling at approximately half the speed of light, towards the station. When the train in which **Bob** travels crosses **Alice** , both photon sources emit a photon (on platform and on the train). Hence, one photon is emitted towards the roof of the railway station by the photon source stationed in the platform; another is emitted inside the train above which is the photo detector on the train (moving w.r.t. Alice) and the photo detector on the station’s roof (stationary w.r.t. Alice) above the train. Which means, **Alice** will (? might) record two photons hitting the photo detectors on the ceiling of the station (the photon from the platform and the photon from the train). The photon emanating from inside the train should hit the station’s ceiling since the train’s detector would have moved away by the time the photon travels upwards. Now from **Bob’s** view point, the photon emitted inside the train will be detected by the sensor IN the train; the photon emitted by the emitter on the platform will miss the detector on top of it (since the station is moving relative to Bob’s train). Either way, SR would not work for both observers at the same time. In case you are thinking that there is some trickery with which this question can be answered in a manner similar to the barn door paradox, let Charlie travel in the track parallel to Bob, in the opposite direction (both Bob and Charlie are equidistant from Alice’s view point and are approaching with the same speed). Now we will have three different observers with different predictions; but due to law of conservation of energy, the photons can be detected by only one of the sensors, which should settle the confusion (or if they do separate, we would be able to tell because of the lesser energy liberated by the photon). In general, the experiment suggests that depending on the relative speed of the train to the station, there are an infinite number of positions in which the photon can be found, which could either support the many worlds theory, or is at least at odds with the Special relativity. I have been digging around this for almost 13 months; still no solution. Any ideas? Would theory of General relativity help? Should I rephrase my question -or- add details?"}
{"id":"65007","title":"How do particles become entangled?","text":"A person asked me this and I'm just a lowly physical chemist. I used a classical analogy (how good or bad is this and how to fix?) Basically, light has a net angular momentum of zero, insofar as it is not polarized into its left and right plane polarized forms until it hits a crystalline structure. However, once it does hit such a structure, we have left and right plane polarized light--that is left and right photon beams. Since the original light was not polarized, this polarization (left right) be inherent in the light. The original light is a superposition of left and right polarized light, each with a total angular momentum of -1 and 1, so that they result in the total zero polarized incident light. Thuse they are entangled to lower their spin (quantum angular momentum). Once we measure one of the particles in the superposition, we know the other by conservation of angular momentum. Is this close?"}
{"id":"60239","title":"Minimum atomic clearance permitting motion","text":"Suppose you were to build the piston and cylinder in a car engine atom-by- atom. Let's just say carbon, since you can make a lot of different shapes due to it's high valence. So assuming you make the most perfect piston possible, a perfect cylinder accurate to within an atom; and you make a perfect cylinder wall to go around the piston, again accurate to within an atom; and the diameter of piston has a single atom clearance from the cylinder wall. What would happen if you tried to move the piston up and down inside the cylinder? Would there be significant force due to the Casimir effect? Would there be friction? The two objects aren't strictly touching, but then again the atoms in my fingers aren't strictly touching the atoms in the keys on my keyboard. It's all just the electro-magnetic force. What is the minimum distance required between the piston and cylinder to allow smooth motion with no friction? (Assume we have a way of moving the piston up and down perfectly along the common axis, and it's not \"wobbling\" about)"}
{"id":"64753","title":"Grover algorithm $R_D$ Circuit","text":"I need sketch two circuits to understand Grover algorithm. The first is the operator $R_f$ and another is the operator $R_D = H^{\\otimes n}(2|0\\rangle\\langle0|-I)H^{\\otimes n}$. I get the first operator. How I will be able to sketch the operator $R_D$ using Toffoli and Hadamard gates? I need a hint to solve this exercise please"}
{"id":"120024","title":"What is the current rate of change of the volume of the Universe?","text":"Is it possible to know the current rate of change of the volume of the Universe? If we can't see past the cosmic horizon, can we still get a value for this rate of change, and if so, how would it be calculated?"}
{"id":"60232","title":"covarient derivative of electromagnetic field tensor","text":"I'm trying to prove the energy momentum tensor in curved spacetime for Electromagnetic field is **Divergence-less** directly(Without using general lie derivative method which can prove **any** energy momentum tensor obtained from varying $g^{ab}$ . From free lagrangian $$ L = -1\/4 F_{ab}F_{cd}g^{ac}g^{bd} $$ We get $$ T_{ab} = 1\/4(F_{ac}F_{bd}g^{cd} -1\/4g_{ab}F_{mk}F_{nl}g^{mn}g^{kl})$$ Now what we have in hand is $$\\nabla_aF^{ab} = 0$$ But To prove $\\nabla_aT^{ab} = 0 $ from the above expression of $T_{ab}$ We have to compute stuff like $\\nabla_aF_{bc} $ But I'm actually getting no where, i have started like following: $$ \\nabla_a F_{bc}$$ $$ = g_{bn}g_{cn}\\nabla_a F^{bc}$$ $$= g_{bn}g_{cn}( \\partial_aF^{bc} + \\Gamma^b_{af}F^{cf} + \\Gamma^c_{af}F^{bf})$$ From the divergence relation I can get $$ \\partial_aF^{ab} + \\Gamma^a_{af}F^{bf} + \\Gamma^b_{af}F^{af} = 0 $$ $$ \\Rightarrow \\partial_aF^{ab} + \\Gamma^a_{af}F^{bf} = 0$$ What should i do from here?"}
{"id":"60233","title":"Does an Ising lattice that returns to equilibrium create a current by induction?","text":"Consider you have an Ising lattice with a dominant up component out of thermal equilibrium, that's your initial state. The down spins want to flip up and align with the ups, and they'll do so until a thermal distribution is reached. While they flip, the overall magnetization increases. Does that create a current by induction? I haven't been able to find a good reference."}
{"id":"64754","title":"Two dimensional elastic collisions with varying angle of incident","text":"If in an elastic collision I know all initial values and that mass for each object remains constant throughout the collision (but different from one another) how can I determine their final velocity vectors when the angle of incident is also a variable? I've tried decomposing the vectors and have determined that for perpendicular collisions the following formula will work: $$v_1=\\frac{u_1(m_1−m_2)+2m_2u_2}{m_1+m_2}.$$ Will this work for variable angles as well?"}
{"id":"80670","title":"What are natural sources of longwave radiation","text":"I am studying now the Earth radiation balance and I came across the question regarding the sources of the long-wave radiation on earth surface. The only source of theses wavelengths indicated in the book are clouds, which have high reflectivity and reflect a part of the long-wave radiation. I am looking for some other sources of it. I was thinking that CO2 can be another source because of this greenhouse effect. Are there any other major global sources which can bring long-wave radiation into earth system>"}
{"id":"80671","title":"What exactly is $\\hat{\\psi}^\\dagger(x)$? An operator or a function?","text":"I've recently read Cohen-Tannoudji on quantum mechanics to try to better understand Dirac notation. A homework problem is giving me some trouble though. I'm unsure if I've learned enough yet to understand this. The creation operator is defined as: $$\\hat{a}_k^\\dagger = (2 \\pi)^{-1\/2}\\int dx\\,e^{ikx} \\hat{\\psi}^\\dagger(x)$$ From my notes, I have $\\hat{\\psi}(x)$ defined (in one dimension) as: $$\\hat{\\psi}^\\dagger(x) \\lvert 0\\rangle = \\lvert x\\rangle $$   * Cohen-Tannoudji says linear operators take a ket vector and associate them with another ket vector within $\\mathscr{E}$ space. $\\hat{\\psi}^\\dagger(x)$ does not look like a linear operator to me. It looks like an operator-function hybrid. What do you call this thing? How is it rigorously defined? Why is $x$ not a subscript? Assuming it's a linear operator and proceeding anyway, let's apply $\\hat{a}_k^\\dagger$ to some ket vector: $$\\hat{a}_k^\\dagger = \\left( (2 \\pi)^{-1\/2}\\int dx\\,e^{ikx} \\hat{\\psi}^\\dagger(x) \\right)\\lvert\\phi\\rangle$$ I assume the order of operation is to apply $\\hat{\\psi}^\\dagger(x)$ first to $\\lvert\\phi\\rangle$, then multiply by $e^{ikx}$, then integrate, then multiply by $(2\\pi)^{-1\/2}$   * How do I integrate over the ket vector $e^{ikx}\\hat{\\psi}^\\dagger(x)\\lvert\\phi\\rangle$? Such an operation does not seem to be defined. I can integrate over complex numbers, but not kets. I still don't understand why $k$ is a subscript but $x$ is an operator-function-hybrid variable. What am I missing here?"}
{"id":"95474","title":"What happens if the energy has no lower bound?","text":"In almost all theories that we investigate there is an lower bound on the energy. But is this necessary for a theory to be a good description of reality? I know from quantum field theory, that in its early days there was the idea of a Dirac sea of negative energy particles. This interpretation was replaced and is not used today, was this also due to the fact that there has to be a lower bound to the energy?"}
{"id":"53050","title":"Why does gravity assist transfer twice the planet's velocity?","text":"In orbital mechanics and aerospace engineering a gravitational slingshot (also known as gravity assist manoeuver or swing-by) is the use of the relative movement and gravity of a planet or other celestial body to alter the path and speed of a spacecraft, typically in order to save propellant, time, and expense. Gravity assistance can be used to accelerate (both positively and negatively) and\/or re-direct the path of spacecraft. Over-simplified example of gravitational slingshot: the spacecraft's velocity changes by up to twice the planet's velocity. ![SHOOOT](http:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/3a\/Gravitational_slingshot.svg\/200px- Gravitational_slingshot.svg.png) Where does the factor of $2$ come from in this relation?"}
{"id":"10641","title":"How can I physically interpret voltage and current?","text":"My understanding of an electrical current is that it is the flow of electrons through a material. The only magnitude I can fathom for this process is the number of electrons which are flowing. I know ohms law, and how some of the basic components of a circuit work (resistors and capacitors, specifically). If there is a circuit attached to a 120V power source, the voltage at the beginning of the circuit will be 120V, and at the end of it will be 0V. When the current experiences resistance, voltage is lost, so more current has to be supplied to the circuit so that exactly all of the voltage is lost by the end of the circuit. My question is: what is happening when voltage is lost? When current is increased? When learning about capacitors in university, I understood that voltage is built up on each plate by electrons flowing in and spreading out across the surface of the plate. But if voltage is the number of electrons in a circuit, what is current? And if all voltage is dropped at the end of a circuit, where do the electrons go? I figured that since electrons won't build up ad infinitum throughout the circuit (obviously since everything would build up an increasingly negative charge), the magnitude of the current must be the number of the electrons. But then what is voltage? What is lost? How should I interpret the stored charge on a capacitor?"}
{"id":"113115","title":"Symmetry factor of tree diagram","text":"In Mark Srednicki's Quantum field theory(page 89) it says > This is a general result for _tree diagrams_ (those with no closed loops): > once the sources have been stripped off and the endpoints labeled, each > diagram with a distinct endpoint labeling has an overall symmetry factor of > one. The tree diagrams for a given process represent the lowest-order (in g) > nonzero contribution to that process. So the question is why the symmetry factor of the tree diagram is one with each endpoint labeled? Could you kindly provided a proof or some links including the proof?"}
{"id":"20806","title":"Laws of Motion: Acceleration to be applied on a free falling object to reduce velocity to 0","text":"Assume that an object of mass 'm' is falling to the earth.      The force acting on the same would be F = m*g = 10m (assuming g = 10m\/s^2).   In this case the velocity of the object at time 't' = 10 seconds will be 100 m\/s (v = u + at).   At this point, if an opposing force of F' = -10m is applied on the falling object, how long would it take for the velocity of the falling object to reduce to 0?      Would it be valid to say that the opposing force of F' = -10m has to be applied for time 't' = 10 seconds to reduce the velocity of the falling object to 0?"}
{"id":"20800","title":"Acceleration: Value Disparity?","text":"If we consider a ball moving at an acceleration of $5ms^{-2}$, over a time of 4 seconds, the distance covered by the ball in the first second is $5m$. In the 2nd second will $5 + 5 = 10m$. In the third second it will cover a distance of $5 + 5 + 5 = 15m$ and so on and so forth. Now, when we substitute this answer in the equations of motion derived from the area under velocity- time and distance-time graphs, we see a variation: $$s = 1\/2at^2$$ $$ = 1\/2*5*4^2$$ $$ = 1\/2*5*16$$ = 40 metres is the distance covered. Now if we go back to our initial description of acceleration we that in the 1st sec = 5m 2nd sec = 10 m 3rd sec = 15 sec 4 sec = 20 sec. Total distance covered in this case is 5 + 10 + 15 + 20 = 50 metres? 40 != 50? Why this disparity between the values? Can someone please explain?!"}
{"id":"80098","title":"Simple circuit with resistances","text":"I have a question about a circuit that I made in class! So, we had a circuit with 2 resistors, one of $R_1 = 330 \\Omega$ and one of $R_2 = 470 \\Omega$. The tension applied to the circuit should be $10V$ but we applied $9.99V(V_f)$. When measuring the resistances with a multimeter the value for the $R_1$ was $328 \\Omega$ and for the $R_2$ was $470 \\Omega$. The value for both $R_1$ and $R_2$ in series was $R_1 + R_2 = 799 \\Omega$. The tension measured in $R_1$ was $4.02V(V_{r1})$ and in $R_2$ was $5.74V(V_{r2})$ The intensity of the current ($I$) was $12.34 mA$ With that said, we were asked to calculate $R_1$, $R_2$ and $R_1+R_2$ theoretically using the values of $V_{r1}$, $V_{r2}$ and $V_f$. I used Ohm Law: $R = \\frac VI$ $R_1 = 326 \\Omega$ and $R_2 = 465 \\Omega$ **My question is: should be $R_1+R_2$ be calculate using $V_f (9.99\/0.01234)$ as it was asked or calculating an equivalent resistance? By the first method the result is $810$ and be the second is $791$. The ammeter's resistance is $11 \\Omega$ at the used scale $(20mA)$.** **I'm asking cause I found it weird that $810$ is bigger than $799$(experimental value) given that the experimental should be bigger because of ammeter's resistance.** Sorry if the question is dumb, but I'm not a Physics student though I attend Physics classes."}
{"id":"17593","title":"Action principle for the Electromagnetism and Gravity","text":"Here is the formula for the stress energy tensor: $$ T_{\\mu\\nu} = - {2\\over\\sqrt{ |\\det g| }}{\\delta S_{EM}\\over \\delta g^{\\mu\\nu}} $$ (This follows from varying the total action $S = S_H + S_{EM}$, where $S_H={c^4\\over 16\\pi G} \\int R \\sqrt{ |\\det g_{\\mu\\nu}| } d^4 x$ is the Hilbert action and it gives the Einstein's equations, and $S_{EM}$ are other terms in the Lagrangian, that contribute to the right hand side of the Einstein's equations in form of the $T_{\\mu\\nu}$ above.) The Lagrangian of the electromagnetic field is: $$ S_{EM1} = -\\int {1\\over 4\\mu_0} F_{\\alpha\\beta} F^{\\alpha\\beta} \\sqrt{ |\\det g| } d^4 x $$ and using the formula above, we get for the stress energy tensor: $$ T_{\\mu\\nu} = {1\\over \\mu_0} \\left( F_{\\mu\\beta} F_\\nu{}^\\beta -{1\\over 4} F_{\\alpha\\beta} F^{\\alpha\\beta} g_{\\mu\\nu} \\right) $$ which is the correct elmag. stress energy tensor. However, the interaction part of the elmag. Lagrangian is $$ S_{EM2} = -\\int j_\\mu A^\\mu \\sqrt{ |\\det g| } d^4 x $$ and if we interpret this as a function of $g^{\\mu\\nu}$ it would also contribute to the stress energy tensor like this: $$ \\delta S_{EM2} = -\\delta\\int j_\\mu A^\\mu \\sqrt{ |\\det g| } d^4 x =-\\int \\delta (g^{\\mu\\nu} j_\\mu A_\\nu \\sqrt{ |\\det g| }) d^4 x = $$ $$ =-\\int (\\delta g^{\\mu\\nu}) (j_\\mu \\sqrt{ |\\det g| })) A_\\nu d^4 x $$ where the $j_\\mu \\sqrt{ |\\det g| }$ is treated as the current _density_ (and thus not depending on $g^{\\mu\\nu}$ when varying), however, clearly the stress energy tensor corresponding to this would be: $$ T_{\\mu\\nu} = 2 j_\\mu A_\\nu $$ (Possibly only the symmetric part contributes, because the antisymmetric part cancels with $g^{\\mu\\nu}$, so we would get $T_{\\mu\\nu} = j_\\mu A_\\nu + j_\\nu A_\\mu$.) In either case, such terms should then appear on the right hand side of the Einstein's equations. However, I don't think that this is correct. Does anybody know what is wrong here?"}
{"id":"17592","title":"Guessing what a simple partial differential equation is describing physically","text":"Is there an easy way to look at a partial different equation and get a sense of what kind of phenomena it is physically describing? I have an equation that looks like this: $\\partial_tA=C_3\\partial^2_{xx}A+C_2\\partial_x\\left(vA\\right)+C_1A+C_0$ From inspecting it and doing some searching, it seemed like it might be trying to describe a phenomena that had advection and diffusion occurring at the same time. Is this a correct interpretation? Note that $C_3>0$, $C_2<0$, $C_1<0$, $C_0>0$ and $v(x,t)$ is velocity. I'm not sure, but it seems reasonable to assume velocity can be either negative or positive. Also, $A(x,t)$ should be positive. Note that I'm not looking for a solution to the equation, just the physical interpretation of what kind of physics would be involved."}
{"id":"44527","title":"Meaning of the word \"Moment\"?","text":"This question is more of a question about the origin of a physical term moment used in many contexts. My question is about the linguistic or historical meaning of the word \"moment\". Please don't provide the mathematical equations for the moment of inertia or the electric of magnetic dipole moment. My question is about the essence of the word. I'm asking this question because as a non-native English speaker, I find it confusing as how did this word got to be used in the different contexts."}
{"id":"134882","title":"Is scattering cross section determined by mass?","text":"This is just for visible light\/photon scattering. Let's say I have small spherical particles in water, if I keep the weight concentration of glass the same but change the diameter of glass sphere from 0.5$\\mu$m to 1$\\mu$m, how will the scattering power, or the total scattering cross section change for visible light, e.g., at 632nm? I was told as long as the mass doesn't change, the scattering cross section won't change, the size of diameter only modifies the phase function of the scatter. is that true?"}
{"id":"134883","title":"Thought Experiment: Force on magnets in a Stern Gerlach Experiment","text":"Background: In the SG experiment, an inhomogenous magnetic field affects a force on particles passing between two magnets. \"Measurement\" takes place when a screen is placed on one end, blocking one of the states. The final measurement we perform (spin state of particles exiting) is heavily dependent on whether we perform a measurement or not, whether by blocking particles exiting the device or by blocking particles by setting up a screen within a modified SG device ![Modified SG device](http:\/\/i.stack.imgur.com\/motIz.png) The interaction between the particles and magnets should also exert a force on the magnets, causing them to move slightly. The motion of these magnets seems to not affect the outcome of the experiment significantly, yet that would imply that the motion of the magnets could be measured without affecting the outcome of the experiment. This seems at odds with Heisenberg's uncertainty principle, that we are getting information for free. My Professor tells me that the force exerted on the magnets by the particles is so minuscule that it cannot be measured (I assume it is similar to how we can't measure particles' momentum and position simultaneously, because any photons we shoot at the particle will end up imparting change?) What's going on here? I wanted to double check here and see where I went wrong with my assumptions."}
{"id":"90186","title":"Do I need to take elastic potential energy into account?","text":"> Let's say I have a vertical spring with end points $A$ and $B$ and length > $a$ and a mass attached to the endpoint $B$. The mass is dropped from the > point $A$ and I need to find the kinetic energy of the mass at a point $ak$ > where $k>1$. Do I need to include the Elastic Potential Energy to find the kinetic energy? If yes, why can't I just use the Kinetic Energy and Gravitational Potential Energy? I know $\\text{Work Done}=mgka$ by the particle and the change in GPE is $mg(ka-a)$ but I don't get what else am I missing?"}
{"id":"126238","title":"Why is an electric and magnetic field present in all universe?","text":"the Earth has its fields due to the motion of molten metals in the core but how does this whole universe has its fields without any discontinuity? And we can say for sure that fields are present in whole universe as we can see light traveling thousands of light year and as light is an EM wave it can't travel without the presence of fields."}
{"id":"94404","title":"Observations used to derive orbital and timing information required for GPS satellites","text":"Although I have studied GPS receiver position finding procedure in some detail, I am struggling to find enough information to allow me to deduce how the essential highly accurate orbital and time information required for each GPS satellite is actually obtained. I can only surmise that the information is somehow slowly and iteratively derived, for each satellite, from all the satellite signals received (probably receiver time-tagged) by the earthbound GPS monitoring stations, that the position information is resolved in an earth-centred inertial frame-of-reference (ECIF) and that the time information (in the ECIF) is synchronized to that from the US Navy master time clock value referred to the earth's geoid. I realise that relativistic corrections are also required at the receiver for orbit ellipticity and to refer the satellite positions to the earth-centred, earth-fixed (ECEF) WGS84 frame of reference. I am particularly wondering whether any other types of observations of the GPS satellites are also involved and I'm hoping someone could kindly give me a rigorous-descriptive rather than mathematical explanation."}
{"id":"122538","title":"Invariance of a tensor under coordinate transformation","text":"I know, that a tensor is a mathematically entity that is represented using a basis and tensor products, in the form of a matrix, and changing a representation doesn't change a tensor, is kind of obvious. So does the invariance of a tensor under coordinate transformation mean what I stated above or does it mean that under a set of particular transformation the representation of a particular tensor also doesn't change. Quoted from Wikipedia: > _A vector is invariant under any change of basis, so if coordinates > transform according to a transformation matrix $L$, the bases transform > according to the matrix inverse $L^{−1}$, and conversely if the coordinates > transform according to inverse $L^{−1}$, the bases transform according to > the matrix $ L$._ Can someone please shed some light on this?"}
{"id":"20373","title":"Is there any idea why the electric charges of electron and muon are equal?","text":"Is there any idea explaining why the electric charges of electron and muon are equal? Edit: The total charge of a particle is proportional to the integral of its own electric field flow through the sphere of a big radius surrounding the particle at rest. The free Dirac equation describes charged fermion. It contains the mass term $m$. If $m$ tends to zero, Dirac equation tends to the pair of Weil equations that describe electrically neutral particles. Does it mean that charge somehow depends on mass? If yes, why do the electron and muon (both described by Dirac equation, but with different mass terms) have the same electric charges?"}
{"id":"20372","title":"Statics software for structural engineering","text":"I'm attempting to expand my knowledge of engineering software. I've found comsol and ansys for acoustics and thermodynamics\/fluid dynamics (not necessarily in that order), now I'd like to see if I can find something for architectural statics\/load calculations. Which packages are in common industry use?"}
{"id":"67515","title":"Newton's third law of motion, and the collapse of objects under heavier weight","text":"Suppose a table is placed on earth. The weight of the table exerts a force on the earth say $10N$, then the earth will also exert a force of $10N$ on the table. If we place a truck on the table weighing $4000N$, then the total force exerted by the table and the truck is $4010N$ on the earth. The earth would also exert a force of $4010N$ on the table. Therefore, the net force on the table is $10N$ and the table colapses. Is this what happens to a table scientifically when it gets broken by something having a relatively a much greater weight? Whereas, my textbook says: > Consider a book of weight $W$ which is placed on a horizontal table, it > pushes the table with a force in downward direction in response to this > action the table also pushes the book in the upward direction the resultant > force acting the book must be zero that is, > > $F = 0 => F = 0$ > > Two forces are acting on the book, > > (a) the force of gravity acting downward which we call the weight of book. > > (b) the other force is the push of table on the book in the upward > direction. > > These two forces are equal and opposite to keep the book in the stationary > position. If this is so, then the truck should also have remained in the stationary position. However, what we normally see is that the truck will crush the table."}
{"id":"1094","title":"How can I create hindrances to radio waves?","text":"How can I create hindrances to radio waves?"}
{"id":"92805","title":"What is the equation for the scale factor of the universe, a(t), for the best fit of data to the $\\Lambda CDM$ Model of Cosmology?","text":"Ideally I like a single formula or multiple formulas for different time ranges that would cover the time from the end of inflation through 100+ billion years after the big bang using the $\\Lambda CDM$ Model. I know that from the end of inflation back to the time of the big bang would be much more speculative, but some wild estimate would be appreciated for that time range also! The Friedmann–Lemaître–Robertson–Walker metric defines the scale factor, a(t), from the metric: $$-c^2d\\tau^2 = -c^2dt^2+a(t)^2d\\Sigma^2$$ where $d\\Sigma^2$ ranges over the 3 dimensional space of the universe and does not depend on time. Usually the scale of the scale factor is set by defining $a(t_{now}) = a(13.78 B yr) \\equiv 1$"}
{"id":"1320","title":"Why does breaking one bulb on a string of Christmas Lights cause all the rest to break in some lights?","text":"If I break one bulb on a string of Christmas lights, or the filament goes out, then all of the rest seem to stop working. Why is this? However, I've also seen the opposite happen in some newer lights -- when one breaks, the others stay on. What makes this one different than the others?"}
{"id":"59266","title":"About the work done by gravity to pull the object down to the earth surface","text":"The following question is from an physics exercise and I know the answer and the way to solve the problem but just curious why my own way doesn't work. The question is asking \"A 2500kg space vehicle, initially at rest, falls vertically from a height of 3500km above the Earth's surface. Determine how much work is done by the force of gravity in bringing the vehicle to the Earth's surface.\" The way to solve that is to integrate the differential work from the force $F_g = GM_\\text{earth}m\/r^2$ such that $$W = \\int_{6400 \\times 10^3}^{(6400+3500)\\times 10^3}F_g dr$$. It does solve the problem but my question is why can't I use the $mgh$ to estimate the work done by gravity?"}
{"id":"59261","title":"Ask for help: the Reaction-Diffusion related problem","text":"In container there are two kinds of molecules A and B which are distributed uniformly. Initially the quantities of A and B are $N_A$ and $N_B$ respectively A and B are distributed uniform, i.e., in any small space (of cause the space should be large enought that there are thousands of molecules in the space), the ratio of number A and B is equal to the ratio of total number of A and B. Assume the molecules are fixed all the time (i.e., we don't consider diffusion) and in every peroid, there is at most one molecule $B_j$ around molecule $A_i (i=1,2,...,N_A)$ be transformed to molecule A: if the molecule $B_j$ is on the boundary\/neighbor of molecule $A_i$, the molecule $B_j$ would be transformed to A, otherwise (i.e., all molecules neighboring $A_i$ are molecule A) no molecule around $A_i$ be transformed. In addition, if there are several $A_k$ and $B_j$ on the neighbor of molecule $A_i$, the closest $B_j$ would be transformed to A. Now the problem is: Is there any analytical expression to show how many molecules (approximately) would be transformed to A in every peroid until all molecule B are transformed to molecule A? Thanks, Tang Laoya"}
{"id":"119381","title":"Spin matrix for various spacetime fields","text":"Let $V^{\\mu}$ be a vector field defined in a Minkowski spacetime and suppose it transforms under a Lorentz transformation $V'^{\\mu} = \\Lambda^{\\mu}_{\\,\\,\\,\\nu}V^{\\nu}$. We can write this like $V'^{\\mu} = (e^{i\\omega})^{\\mu}_{\\,\\,\\,\\nu}V^{\\nu}$ I think where $\\omega$ denotes a rotation in some plane spanned by indices $\\left\\\\{\\rho \\sigma\\right\\\\}$, say. In 2D Euclidean space time, we can write the matrix representation of $\\Lambda$ as $$\\begin{pmatrix} \\cos \\omega & \\sin \\omega\\\\\\\\-\\sin \\omega&\\cos \\omega\\end{pmatrix}$$ and in Minkowski space this changes to the 'hyperbolic' rotation. Linearising the above yields $$\\begin{pmatrix}1&\\omega\\\\\\\\-\\omega&1\\end{pmatrix} = \\text{Id} + \\begin{pmatrix} 0&\\omega\\\\\\\\-\\omega&0\\end{pmatrix} = \\text{Id} + \\omega \\begin{pmatrix} 0&1\\\\\\\\-1&0\\end{pmatrix}$$ Now compare with the more general treatment: $V'^{\\mu} = \\Lambda^{\\mu}_{\\,\\,\\,\\nu}V^{\\nu} \\approx (\\delta^{\\mu}_{\\nu} + \\omega^{\\mu}_{\\,\\,\\,\\nu})V^{\\nu}$, where $\\omega^{\\mu}_{\\,\\,\\,\\nu} \\equiv (\\omega^{\\rho \\sigma} S_{\\rho \\sigma})^{\\mu}_{\\,\\,\\,\\nu}$ In 2D, the spin matrix $S$ when acting on vectors in 2D Euclidean space time is therefore the matrix multiplying $\\omega$ above, which agrees with the single generator of the SO(2) group. If we continue with the general analysis, we obtain $V'^{\\mu} - V^{\\mu} = \\omega^{\\mu}_{\\,\\,\\,\\nu}V^{\\nu} = \\eta^{\\mu \\rho}\\omega_{\\rho \\nu} V^{\\nu} = \\omega^{\\rho \\sigma} \\delta^{\\mu}_{\\rho} \\eta_{\\sigma \\nu} V^{\\nu}$ Now use the antisymmetry of $\\omega_{\\rho \\sigma}$ gives $$2(V'^{\\mu} - V^{\\mu}) = \\omega^{\\rho \\sigma}(\\delta^{\\mu}_{\\rho} \\eta_{\\sigma \\nu} - \\delta^{\\mu}_{\\sigma} \\eta_{\\rho \\nu})$$ from which we can identify $S$ to be $\\delta^{\\mu}_{\\rho} \\eta_{\\sigma \\nu} - \\delta^{\\mu}_{\\sigma} \\eta_{\\rho \\nu}$. I am wondering how this agrees with the matrix I obtained above. Many thanks."}
{"id":"119388","title":"Why must phonon-phonon interaction be anharmonic?","text":"I'm learning about phonons and this is really interesting. Why can't I describe two phonons colliding as a harmonic oscillation? Why does it have to be anharmonic for the phonon model to work?"}
{"id":"92128","title":"Edge states at high magnetic field (Quantum ballistic transport)","text":"I am learning about edge states at high magnetic field (Quantum ballistic sample). What I understand so far from it is that at high magnetic field, Landau states arise. Then (I don't know exactly why?), edge states are formed. To enhance my understanding, can someone help me with these questions: *Why must the chemical potential cross a Landau level? *Why is the net current carried solely by these edge states? (I am only interested in intuition, no derivations)"}
{"id":"126320","title":"Where are the time dilatational effects of orbital motion and gravitational acceleration equal?","text":"Nearly four years ago, upon hearing of the observation of time dilation in two optical atomic clocks at an elevation one metre apart, due to acceleration towards earths centre of gravity by Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. I wondered where the point would be, at which time dilation caused by acceleration toward the earth gives way to time dilation caused by acceleration around the earth, via orbital velocity. (I mean the point at which the two effects are at equilibrium with each other.) I asked a friend who is a physicist, he got back to me almost immediately, fascinated by his back of an envelope calculations that this point is the Clarke orbit, or the geosynchronous orbit. If for instance the rotation velocity of earth were a slightly different speed, the result would be different, and no relationship apparent. I think there may be a very interesting relationship waiting to be revealed. Is there a physical or mathematical relationship? And if so, what is it? Not being a mathematician (I'm an artist) I'm not capable of doing the math, but this has fascinated me ever since."}
{"id":"126323","title":"Curious about this perpetuum concept","text":"I know it's not possible to build a perpetual motion machine, but I still got this concept running in my mind.   If you take a (longer)copper coil, and insert an axis through it(right-angled) with a magnet fixed on it,   it would generate electricity when you spin the axis. The axis and magnet rotate, thus alteration of the flux, thus electricity. Of course normally there is a soft-iron core inserted in the coil to achieve better efficiency,   but it also brakes the movement. If you don't use a core and insert the axis+magnet into the coil,   you'd still generate a certain amount of electricity. The magnet is **only** braked by the friction of the axis and the air, but this friction is almost insignificant. So if you use good bearings to reduce this friction to a minimum, and you take a coil with as much as possible windings, and the strongest magnet you can find, you could _in theory_ build a perpetuum mobile right? I'm almost sure I'm not right but I'm asking anyway. Just curious.   Any comments or responses are welcome."}
{"id":"22426","title":"What does the appearance of a classical particle fundamentally reduce to?","text":"I've been reading an article that describes what seems to be a classical particle as a regularity in the global wavefunction over a quantum configuration space: > When you actually see an electron trapped in a little electron trap, what's > really going on, is that the cloud of amplitude distribution that includes > you and your observed universe, can at least roughly factorize into a > subspace that corresponds to that little electron, and a subspace that > corresponds to everything else in the universe. So that the physically real > amplitude distribution is roughly the product of a little blob of amplitude- > factor in the subspace for that electron, and the amplitude-factor for > everything else in the universe. ...which looks sensible enough, but I cannot figure out what is actually meant by this configuration space; specifically, what is the identity of an individual point in that space? A simplified picture - mentioned in the above article - tells us that an individual point corresponds to something like \"a photon here, a photon there (etc., for every photon), an electron here, an electron there, etc. for every species of elementary particles\". But this can't be quite right: if an electron corresponds to a blob of amplitude in the configuration space, a single point in that space can't correspond to a position of this blob... and besides, what would a dimension of that space even be, since the number of particles in the universe isn't fixed. An \"explanation\" offered in the article is as follows: > A single point in quantum configuration space, is the product of multiple > point positions per quantum field; multiple point positions in the electron > field, in the photon field, in the quark field, etc. But what does this actually _mean_? As I'm given to understand, in quantum field theory one has a wavefunction(al) defined over all configurations of a specific field, so if one jumps over to the real world where space-time is filled with many fields, is it possible to speak of the quantum configuration space over which the universal wavefunction is defined as a space where every point corresponds to some configuration of every quantum field, and if so, can this picture still be used to see an electron as a roughly independent part\/blob of this giant amplitude distribution or not? I've been gnawing at this for two days now and I'd really appreciate some help here..."}
{"id":"131492","title":"Decoherence without time?","text":"Decoherence is a phenomenon that provides a part of the explanation of why quantum systems and classical systems behave differently. What I understood from decoherence so far is that it requires time because it simply takes a (somehow very short) while for a large quantum system to transit to classical behaviours through decoherence. Now, the status of time in quantum gravity is not clear, and it could very well occur that time is an emergent quantity. So I thought decoherence should occur at time scales where time has already emerged. Yet I found this assertion on www.decoherence.de > Decoherence explains also how the Schrödinger equation of general relativity > (the Wheeler-DeWitt equation) may describe the appearance of time in spite > of being time-less\" I would like to understand how can decoherence be active in the timeless domain of the Wheeler-deWitt equation to allow the emergence of time, as decoherence seems to require time itself to happen. (Handwaving arguments are OK, but references are welcome!)"}
{"id":"26334","title":"Why are binary objects so critical to astronomy?","text":"There are a lot of astronomical objects out there to study but binary objects seem to get more attention. Why is this? What makes binary objects (stars, Kuiper Belt Objects, black holes, galaxies, etc.) more important or special than lone objects?"}
{"id":"17787","title":"Would a solution to the Navier-Stokes Millennium Problem have any practical consequences?","text":"I know the problem is especially of interest to mathematicians, but I was wondering if a solution to the problem would have any practical consequences. Upon request: this is the official problem description of the aforementioned problem."}
{"id":"116988","title":"Quick question on perturbation theory","text":"Suppose we have a particle in an infinite potential well, with $V(x) = 0,\\space 0< x < a $ and infinity everywhere else. Now suppose we have a perturbation on the LHS of the well: $V_1(x) = v, \\space 0 < x < \\frac{a}{2}$. By first order perturbation theory, all energy levels are shifted by the same amount: $$\\Delta E = \\langle E_n|v|E_n\\rangle = \\frac{v}{2} $$ Ok all is good. By considering the perturbed wavefunction: $$ \\psi_n = \\phi_n + \\sum_{n \\neq k} \\frac{\\langle \\phi_k|V_1|\\phi_ n\\rangle}{E_n - E_k} $$ How do I make use of the condition that $v << E_2 - E_1 $ to show the results above is correct?"}
{"id":"116985","title":"Can the Berry Connection be derived from a metric?","text":"The Berry Connection is $$A_\\mu(R)=-i \\langle \\Psi(R) |\\partial_\\mu \\Psi(R) \\rangle$$ which allows us to parallel transport a state indexed by $R$. We can integrate the Berry Connection to get the Berry Phase, and we can differentiate the Berry Connection to get the Berry Curvature. Can the Berry Connection be derived from a metric? As a prototypical example, I'm thinking about how the Christoffel Symbols in General Relativity (GR) can be derived from the metric tensor. Also, I think for every connection, there exists a metric for which the connection is a Levi-Civita connection. However, is there a natural and physical metric that induces the Berry Connection? In this presentation, the great and powerful Haldane relates \"quantum distance\" to the Berry Curvature, but it doesn't look like you can derive the Berry Curvature from the quantum distance in the same way curvature and the metric are related in GR."}
{"id":"116984","title":"Calculation of the differential of the entropy","text":"In this review (for those who wants a precise reference see page 8 eq 21), the Author says that: \\begin{equation*} S=-\\sum_{i}P\\left(i\\right)\\ln P\\left(i\\right) \\end{equation*} and using the Maxwell-Boltzmann distribution \\begin{equation*} P\\left(i\\right)=P_{MB}\\left(i\\right)+dP\\left(i\\right) \\end{equation*} \\begin{equation*} dS=-\\sum_{i}dP\\left(i\\right)\\ln P_{MB}\\left(i\\right)=-\\sum_{i}dP\\left(i\\right)\\ln e^{-\\beta E\\left(i\\right)}=\\beta\\sum_{i}dP\\left(i\\right)E\\left(i\\right) \\end{equation*} However I tried to do the detailed calculation but here is what I get: \\begin{equation*} dS=-\\sum_{i}dP\\left(i\\right)\\ln\\left[P\\left(i\\right)\\right]-\\sum_{i}P\\left(i\\right)\\frac{1}{P\\left(i\\right)}dP\\left(i\\right)=-\\sum_{i}dP\\left(i\\right)\\ln P\\left(i\\right)-\\sum_{i}dP\\left(i\\right)= \\end{equation*} \\begin{equation*} =-\\sum_{i}dP\\left(i\\right)\\ln\\left[P_{MB}\\left(i\\right)+dP\\left(i\\right)\\right]-\\sum_{i}P\\left(i\\right)+\\sum_{i}P_{MB}\\left(i\\right) \\end{equation*} Can someone help me to get the right result? Thanks!"}
{"id":"57766","title":"Mirror problem of radiation pressure","text":"If two perfect mirrors are placed facing one another and they are in proximity, and photons (don't ask me how) are traveling between them and toward one of them, what is to keep the radiation pressure from reaching incredible amounts? I might be way off of base here because I am new to this field. I heard that the radiation pressure doubles on a mirror. If this is the case, a laser beam focused on a mirror at any angle will cause the mirror to accelerate one of two ways-- no more. This also violates energy conservation and thus isn't valid."}
{"id":"99933","title":"Infinite and Finite Square Wells","text":"For the **infinite square well** in the first region, **outside the well** : $$\\frac{-\\hbar^2}{2m}\\frac{d^2 \\psi}{dx^2} + V(x) \\psi (x) = E \\psi (x),$$ where you set **$V = 0$**. Rearranging gives $$\\frac{d^2 \\psi}{dx^2} = -\\frac{2mE}{\\hbar^2} \\psi.$$ We identify $$k = \\sqrt{\\frac{2mE}{\\hbar^2}}.$$ For the **finite square well** we have the same situation (for bound solutions) but we set: $$\\alpha = \\sqrt{\\frac{-2mE}{\\hbar^2}}.$$ **Why do we absorb the minus sign into the square root now and not before?** How does this tie in with bound\/unbound solutions and parity?"}
{"id":"99932","title":"Why does a car bonnet (hood) rise when you connect the clutch with a brake on?","text":"Is the rotational force to overcome the brakes moved to the opposite effect of moving the car chassis, until the brake is released?"}
{"id":"99930","title":"Where \/ what is the source of the force that causes refraction?","text":"So we know that as light moves from, say, a glass block to air, it speeds up and refracts, but if there is to be acceleration then surely there must be a resultant force to cause it? So where does that come from? Secondly as the light enters the glass block it slows down, but where does the excess energy go?"}
{"id":"104403","title":"Band gaps: are they at the centre or at the edge of the Brillouin zone?","text":"Reading about electronic band structures, I came across the following: > Band gaps open at the edges of the Brillouin zone (BZ), since that is where > the Bragg scattering occurs. I am slightly confused, as most band structures show a gap opening at $\\Gamma$ which is at the centre of the BZ. What am I missing?"}
{"id":"52042","title":"The viscous force between the layers of liquid is same, then why there is variation in the velocities of its layers?","text":"I have learned in my textbook that when the liquid flows the bottom layer of the liquid never moves because of friction, but the upper layers move with increasing velocities how it is possible if the viscous force between all these layers is same"}
{"id":"102809","title":"Why is second moment of area often referred as moment of inertia?","text":"In many references, mostly civil engineering, the second moment of area is referred as Moment of Inertia. Is that \"really\" correct? From my understanding, moment of inertia is analogous to mass in rotational motion, not the second moment of area."}
{"id":"99376","title":"Electrostatics kinematics equations (Coulombs Law)","text":"If there are two similarly charged particles $Q$ and $q$ of masses $M$ and $m$, a Coulomb force is applied on each one (action-reaction). Suppose that $Q$ is stationary and we are shooting $q$ with an initial velocity $\\vec{v_0}$ from $\\infty$ into the field. I have, as well, figured out that by rearranging Coulombs Law and the Fundamental Law of Mechanics we end up with the following differential equation: $$x''(t) [x(t)]^2 = \\frac{kQq}{m}$$ because: coulombs law states that $$F = \\frac{kQq}{[x(t)]^2}$$ and 2nd Newton's Law: $$F = ma = m x''(t)$$ By rearranging we get: $$m x''(t) = \\frac{kQq}{[x(t)]^2} \\Leftrightarrow x''(t) [x(t)]^2 = \\frac{kQq}{m}$$ We also know that its velocity is zero when $x = \\frac{2kQq}{m{u_0}^2}$ (from Work-Energy Thm) Is the general solution of this differential equation (2nd order non-linear ODE) the solution of the displacement over time function? If yes, what is it!?"}
{"id":"57497","title":"What causes acceleration of particles in the expansion section of a De Laval nozzle?","text":"A De Laval nozzle has a compression section, where the propellant is compressed (and thereby accelerated) as it moves towards a narrow section (the throat). After the throat, the nozzle widens out again. Here's a reference image. ![](http:\/\/i.stack.imgur.com\/6Gv6t.png) According to what I've been reading, the propellant is moving at subsonic speeds before the throat, and is accelerated to supersonic speeds as it passes the throat. Furthermore, the propellant _continues to accelerate_ as the nozzle expands again. What I don't understand is: what force continues to accelerate the propellant after it passes through the throat? I've been trying to imagine a single molecule of propellant and thinking about the forces from the particles around it, but can't come up with a force \"pushing\" from upstream (because it's moving at supersonic speeds), or \"pulling\" from downstream (because at the minimum there would be 0 opposing force in a vacuum, but certainly not a negative attractive force from downstream). Any ideas?"}
{"id":"57495","title":"Einstein gravity versus Newton's gravity","text":"What's the basic difference between the gravity as seen by Einstein, and that by Newton?"}
{"id":"24863","title":"Is there a good explanation for the observation of Martian canals?","text":"Martian \"canals\" have been observed by independent observers after their first description. Now, they are attributed to \"optical illusion\", but I think that this is not a good choice of word, because an optical illusion should be visible today as well. It would have to be a psychological effect, but it is rather astonishing to have people draw whole maps of non-existent canal systems. Is there a good explanation (optical or psychological) for the observation of Martian canals?"}
{"id":"21814","title":"\"Dear radioactive ladies and gentlemen\" - Letter by Wolfgang Pauli","text":"In 1930, **Wolfgang Pauli** wrote a letter to Lise Meitner for a convention in Tübingen, considering the problem of beta decay. Does anybody know, where to find the original letter online ?"}
{"id":"24869","title":"What is Dark Energy, and how was it discovered?","text":"I've heard a lot about Dark Energy, and how it's supposed to be one of the most powerful forces in the universe. What is it, and how was it discovered?"}
{"id":"110297","title":"What is a proportionality constant? (Planck's constant)","text":"I understand that Planck's constant is essentially the ratio between the energy of a photon and its frequency. There are 2 things that im trying to verify:   1. isn't the number that Planck's constant represents just an arbitrary result of the measurement system we use? As in, Planck's constant represents the ratio between 1 Joule and 1 Hertz? So if i assume correctly, Planck's constant is the increase in energy (in joules) that you would get from an increase in frequency of 1hz? Thus if i wanted to I could make an alternative unit for energy which had a 1 to 1 correspondence with hz, and no proportionality constant would be required?   2. So regardless of units used to measure, a fixed percentage increase in one quantity results in an equal percentage increase of the other, correct?"}
{"id":"21818","title":"The Dual Cloud Chamber Paradox","text":"**2012-04-07 Addendum: The Dual Cloud Chamber Paradox** Two 10m diameter spheres $A$ and $B$ of very cold, thin gas have average atomic separations of 1nm. Their atoms are neutral, but ionize easily and re- radiate if a relativistic ion passes nearby. Both clouds are seeded with small numbers of positive and negative ions. The clouds collide at a relative speed of $0.994987437c$, or $\\gamma=10$. Both are sparse enough to minimize collisions and deceleration. Both show passage of each other's ions. (a) What do distant observers moving parallel to $A$ and $B$ observe during the collision? (b) Are their recordings of this event causally consistent? The answer to (a) requires only straightforward special relativity, applied from two perspectives. For the traveler moving in parallel to cloud $A$, cloud $B$ should appear as a Lorentz contracted oblate spheroid with one tenth the thickness of cloud $A$, passing through cloud $A$ from right to left. If you sprinkled cloud $A$ with tiny, Einstein-synchronized broadcasting clocks, the $A$-parallel observer would observe a time-stamped and essentially tape- recording like passage of the $B$ spheroid through cloud $A$. The $B$-parallel observer sees the same sort of scenario, except with cloud $A$ compressed and passing left-to-right through cloud $B$. If $B$ is sprinkled with its own set of Einstein synchronized broadcasting clocks, the $B$p-parallel observer will record a time-stamped passage of the compressed $A$ through $B$. So, am I the only one finds an _a priori_ assumption causal self-consistency between these two views difficult to accept? That is, while it may very well be that a recording of a flattened $B$ passing through cloud $A$ in $A$-sequential time order can always be made causally self-consistent with a recording of a flattened $A$ passing passing through through cloud $B$ in $B$-sequential order, this to me is a case where a mathematically precise proof of the information relationships between the two views would be seem like a good idea, if only to verify how it works. Both views after all record the _same_ event, in the sense that every time a clock in $A$ or $B$ ticks and broadcasts its results, that result becomes part of history and can no longer be reversed or modified. It's tempting to wonder whether the Lampa-Terrell-Penrose Effect might be relevant. However, everything I've seen about L-T-P (e.g. see the video link I just gave) describes it as an optical effect in which the spheres are Lorentz contracted at the physical level. Since my question deals with fine-grained, contact-like interactions of two relativistic spheres, rather than optical effects at a distance, I can't easily see how L-T-P would apply. Even if it did, I don't see what it would mean. So, my real question is (b): Does an information-level proof exist (not just the Lorentz transforms; that part is straightforward) that the $A$-parallel and $B$-parallel recordings of dual cloud chamber collision events will always be causally consistent? * * * **2012-03-03: This was my original version of the question, using muonium clocks** My SR question is how to predict length contraction and time dilation outcomes for two interacting beams of neutral particles. The two beams are:   1. In different inertial (unaccelerated) reference frames for the duration of measurement part of the experiment.   2. Intermixed at a near-atomic level so that frame-to-frame photon exchange times are negligible. An example would be two intersecting beams of relatively dense, internally cold muonium atoms.   3. Clock-like even at the atomic level (e.g., decaying atoms of muonium).   4. Part of a single causal unit. By this I mean that there is a sufficient short-range exchange of photons between the two beams to ensure that state changes within each beam have an entropic effect, however small, on nearby components of the other beam. This makes their interactions irreversible in large-scale time. An example would be decay of an anti-muon in one frame transferring energy to nearby muonium atoms in the other frame. Simply illuminating the intersection region with light frequencies that would interact with muonium in both of the beams would be another option.   5. Observed by someone who is herself isolated from the rest of the external universe. Muons generated by cosmic rays and traveling through earth's atmosphere provide a helpful approximation the above experiment. The muons provide the first beam, and the atmosphere forms the second one, which in the case of muons is shared by the observer. Such muons have tremendously extended lifespans, which is explained by invoking time dilation (but not length contraction) for the muon frame as viewed from the atmosphere frame. Conversely, length contraction (but not time dilation) is invoked to describe the view of the atmosphere from the perspective of the muon frame. Since this results in the atmosphere appears greatly compressed in the direction of travel, the muons simply travel a shorter distance, thereby ensuring the same decay result (causal symmetry) for both views of the interacting frames. My question then is this: _For the thought experiment of intersecting, causally linked beams of muonium, what parameters must the observer take into account in order to predict accurately which of the two intersecting muonium beams will exhibit time dilation, and which one will exhibit length contraction?_ **2012-03-04 Addendum by Terry Bollinger (moved here from comment section):** Sometimes asking a question carefully is a good way to clarify one's thinking on it. So, I would now like to add a hypothesis provoked by own question. I'll call it the local observer hypothesis: _Both beams will be time dilated based only on their velocities relative to the observer Alice; the beam velocities relative to each other are irrelevant._ Only this seems consistent with known physics. However, it also implies one can create a controllable time dilation ratio between two beams. I was trying to avoid that. So my second question: Are physical time dilation ratios ever acceptable in SR? **2012-03-06 Addendum by Terry Bollinger:** Some further analysis of my own thought problem: A $\\phi$ _set_ is a local collection of clock-like particles (e.g. muons or muonium) that share a closely similar average velocity, and which have the ability to intermix and exchange data with other $\\phi$ sets at near-atomic levels, without undergoing significant acceleration. A _causal unit_ $\\chi = \\\\{\\phi_0 ... \\phi_n\\\\}$ is a local collection of $(n+1)$ such $\\phi$ sets. By definition $\\phi_0$ contains a primary observer Alice, labeled $\\aleph_0$, where $\\aleph_0 \\subset \\phi_0$. Each $\\phi_i$ has an associated XYZ velocity vector $\\boldsymbol{v_i} = (\\phi_0 \\rightarrow \\phi_i$) that is defined by the direction and rate of divergence of an initially (nominally) co-located pair of $\\phi_0$ and $\\phi_i$ particles, with the $\\phi_0$ particle interpreted as the origin. The vector has an associated magnitude (scalar speed) of $s_i=|\\boldsymbol{v_i}|$. Theorem: If $\\phi_i$ includes a subset of particles $\\aleph_i$ capable of observing Alice in $\\phi_0$, then $\\aleph_i$ will observe $\\phi_0$ as length contracted along the axis defined by $\\boldsymbol{v_i}$. Conversely, Alice ($\\aleph_0$) will observe each $\\phi_i$ as time dilated (slowed) based only on its scalar speed $s_i$, without regard to the vector direction. This dependence of time dilation on the scalar $s_i$ means that if $\\phi_a$ has velocity $\\boldsymbol{v_a}$ and $\\phi_b$ has the opposite velocity $\\boldsymbol{-v_a}$, both will have the _same_ absolute time dilation within the causal unit (think for example of particle accelerators). Analysis: This $\\chi = \\\\{\\phi_i\\\\}$ framework appears to be at least superficially consistent with data from sources such as particle accelerators, where time dilation is observable in the variable lifetimes of particles at different velocities, and where absolute symmetry in all XYZ directions is most certainly observed. Applying observer-relative time dilation to fast particles (e.g. in a simple cloud chamber) is in fact such a commonly accepted practice that I will assume for now that it must be valid. (It is worth noting that while particle length contraction is typically also assumed to apply to fast-moving particles, serious application of length contraction in a causal unit would cause the particles to see longer travel paths that would shorten their lifespans. This is the same reason why muons are assumed to see a shorter path through a length-contracted atmosphere in order to reach the ground.) My updated questions are now: (1) Is my $\\chi = \\\\{\\phi_i\\\\}$ framework for applying SR to experiments valid? If not, why not? (2) If $\\chi = \\\\{\\phi_i\\\\}$ is valid, _what property makes the observer $\\aleph_0$ in $\\phi_0$ unique from all other $\\aleph_i$?_"}
{"id":"134812","title":"Angular acceleration and linear acceleration","text":"I have a small confusion. I learned very recently that all particles of a rotating body have the same angular acceleration but different linear acceleration (same for velocity as well). But how is this possible? I find difficulty understanding this though I know that it is almost simple. Can you please help me understanding this?"}
{"id":"116445","title":"Why does the Hubble parameter keep getting smaller if the expansion of the universe accelerates?","text":"Suppose a region of space at a distance D from Earth is escaping from us with the velocity v. Since it seems like the expansion of the universe is accelerating, things at D from Earth should be receding faster and faster. It would seem from the v = HD equation, that H is then getting bigger in time. However, in other sources we can read, that the Hubble \"constant\" it's getting smaller with time. How come?"}
{"id":"27884","title":"Why is force described as rate of change of momentum?","text":"> momentum = mass * velocity Differentiating both sides leads to > force = mass * acceleration since the mass doesn't participate in the differentiation as it is constant. * * * **Is this a sound definition** , or a pointless use of differentiation? Or are there systems (like rockets) where the mass is also a variable wrt time?"}
{"id":"112063","title":"Super-String and Anti-Particle","text":"I'm afraid this is quite basic question . What I learned from the String theory is that the theory unify the Path-integral, Superposition Principle, and commutation relation .   In that context , I wondered whether the Anti-Particle can be described by the similar way the Superposition Principle is expressed by the string theory . How can the anti-particle be described by the String Theory ?"}
{"id":"112357","title":"Confusion about Lie derivative on metric","text":"According to this site, the Lie derivative of a $(0,2)$-tensor is $$ \\mathcal{L}_XT_{ab}=\\partial_XT_{ab}+T_{cb}\\partial_aX^c+T_{ac}\\partial_bX^c $$ However, according the same website, the Lie derivative of the metric is $$ \\mathcal{L}_Xg_{ab}=\\nabla_Xg_{ab}+g_{cb}\\nabla_aX^c+g_{ac}\\nabla_bX^c $$ and if the connection is metric compatible, the first term vanish. My question is, why do we change $\\partial$ into $\\nabla$ here? In fact, in a note, I see both two expressions for the Lie derivative of the metric. Are they really equivalent? * * * Update: I tried to expand the second equation, and turns out it indeed is the first one. In fact, $$ \\begin{align} &\\nabla_Xg_{ab}+g_{cb}\\nabla_aX^c+g_{ac}\\nabla_bX^c\\\\\\ =&X^c(\\partial_cg_{ab}-\\Gamma^{d}_{ac}g_{db}-\\Gamma^d_{cb}g_{ad})+g_{cb}\\partial_aX^c+g_{ac}\\partial_bX^c+g_{cb}\\Gamma^{c}_{ad}X^d+g_{ac}\\Gamma^c_{ad}X^d\\\\\\ =&\\partial_XT_{ab}+T_{cb}\\partial_aX^c+T_{ac}\\partial_bX^c \\end{align} $$ All the $\\Gamma$'s cancel out. But now my question is, is it generally true that when expanding $\\mathcal{L}_XT$ for some tensor $T$, we can use either partial derivative or covarient derivative?"}
{"id":"105469","title":"String breaking in QCD","text":"I have trouble understanding string breaking in QCD. I have read an article on arxiv (http:\/\/arxiv.org\/abs\/hep-lat\/0505012), and I still don't understand what truly happens. My understanding of string breaking (of a meson - let's say bottomonium) is as follows: We have a quark $b$ and an anti-quark $\\bar b$ on distance of $R$ at temperature $T=0$ (this is probably to limit the effects of quark-gluon plasma?). As we \"pull\" them apart the potential energy increases. At some crytical distance $R_c$ it is more favourable that a pair of quark-antiquark is created from vacuum and thus the final state are two mesons. For calculating the static potential between a quark and antiquark we use correlation function for creating a meson at $t=0$ and destroying it at $t=\\tau$. We then use Wick rotation $t \\rightarrow -it$ which then transforms oscilating functions into exponentially decaying ones: $$C(R,t)=a_0e^{-\\frac{E_0}{\\hbar}t}+a_1e^{-\\frac{E_1}{\\hbar}t}+\\ldots \\hspace{2 mm}.$$ We interpolate ground state with $e^{-\\frac{E_0}{\\hbar}t}$ which is just the static potential $V(R)$, since we neglect kinetic energy. My first question is: Does the potential static potential $V(R)$ at temperature $T=0$ saturate or not. Many articles say it follows a Cornell potential, which is $$V(r)=-\\frac{e}{r}+\\sigma r\\hspace{1mm},$$ which grows linearly with distance (and doesn't saturate and thus doesn't show string breaking). My second question is about the first excited state, which should be a final state of two mesons (in this case probably $B$ and $\\bar B$ meson). Can we get the energy of this excited state with simple correlation function which I mentioned (energy $E_1$), or must we use the correlation function (which is a matrix) in the article (defined on the first page). They also calculate mixing of states which I don't understand fully. Where do they get the energy of eigenstates? The eigenstates they calculate are:$$|1\\rangle=\\cos(\\theta)|b\\bar b\\rangle + \\sin(\\theta)|B\\bar B\\rangle \\\\\\ |2\\rangle=-\\sin(\\theta)|b\\bar b\\rangle + \\cos(\\theta)|B\\bar B\\rangle\\hspace{1 mm}.$$ PS The picture of string breaking is showed on page 19 in the article. I would be extremely thankfull if someone showed me an article which explains (qualitatively and quantitatively) string breaking in simpler language."}
{"id":"30210","title":"How can a body be transparent?","text":"How can a body be transparent? I need the theoretical explanation for the same. Has anybody succeeded in doing that practically?"}
{"id":"109024","title":"Can \"state\" be considered a 5th dimension?","text":"I searched for an answer to this question on Google but just found articles that mention either string theory or a 5th dimension in passing (such as Maxwell equations as they relate to Riemann curvature tensor.) I stumped myself while driving to school today thinking about this... We can explain an objects position in the universe by describing its spacial and time locations, such as _at 2nd and 3rd street on the fifth floor at 10:00am_ But is that enough to fully describe an object when you take into account Schrödinger's cat? For example, what if, Schrödinger's cat is the object that is at that location and time. It seems like you would have to have a 5th _state_ dimension to fully explain its position, as in _at 2nd and 3rd street on the fifth floor at 10:00am with a probability of .5_ It seems like if we had _state_ as an additional dimension it would help explain things like quantum entanglement as the particles could be moving away from each other in space-time but standing still in state. Am I merely relating things that have no business being related, or is there a connection between state and dimension?"}
{"id":"60918","title":"Accessible microstates of harmonic oscillator in microcanonical enemble","text":"While reading up on statistical physics, I am going through the calculation of the partition function of the harmonic oscillator in the microcanonical ensemble. The result for the partition function is: $\\Omega(E) = \\frac{E_o}{\\hbar \\omega}$, i.e. all one-dimensional harmonic oscillators of the same frequency $\\omega$ have the same number of accessible microstates. Consequently, as $\\omega$ grows, $\\Omega(E)$ goes to zero. This is somewhat counter-intuitive to me. I would have expected that an oscillator with a higher frequency $\\omega$ would have more accessible microstates than one with a lower frequency. What is wrong with my intuition? Note that I am going through the classical calculation, the partition for the harmonic oscillator is then given as: $\\Omega(E) = \\frac{E_0}{h}\\int dp \\int dx \\delta \\left( \\frac{p^2}{2m} + \\frac{1}{2}k x^2 - E\\right) $"}
{"id":"107191","title":"Why do rocket engines have a throat?","text":"Diagrams of rocket engines like this one, ![rocket engine](http:\/\/i.stack.imgur.com\/EZWGCm.png) (source) always seem to show a combustion chamber with a throat, followed by a nozzle. Why is there a throat? Wouldn't the thrust be the same if the whole engine was a U-shaped combustion chamber with a nozzle?"}
{"id":"68080","title":"Short term track of a hand held sensor (gyro, accelerometer) data with rough precision","text":"I have acceleration and attitude data for the movement of a device in 3d space. what transformations, before numerical integration, are required, in order to produce a 3D track of the the movement in the earth's frame of reference. I know about the precsion and error issues. And I am not looking for commercial or mil spec accuracy. I want to try. I have seen the transformations in various papers. Can anyone suggest links or code that could accelerate my efforts?"}
{"id":"131947","title":"Possible Error in deriving conformal generator","text":"My professor gave me the following derivation for the full generator of the Lorentz transformations. The starting point is to consider a subgroup of the conformal group that leaves the origin fixed (as done so in Di Francesco, for example). Denote by $J_{\\mu \\nu}$ the full generator and so $J_{\\mu \\nu}\\Phi(0) = S_{\\mu \\nu}\\Phi(0)$, where $S_{\\mu \\nu}$ is the component of the full generator transforming only the field spin components. Infinitesimally, $\\Phi'(0) = (1 + \\epsilon S_{\\mu \\nu})\\Phi(0)$ but also $\\Phi'(0) = e^{-ix' \\cdot P}\\Phi'(x')$. This I understand. The next line is the first one I am not so sure of. He then writes $$\\Phi'(0) = e^{-ix' \\cdot P}(1+\\epsilon J_{\\mu \\nu})\\Phi(x)$$ How is this true? He seems to be using the transformation $\\Phi'(x') = D \\Phi(x)\\,\\,\\,(1)$, but using $D$ as the **_full_** generator. If I had read that statement a few months back, I would have accepted it fine, but I posted a question here earlier that discussed the fact that in $(1)$, $D$ is the **_spin_** component of the generator because $x'$ and $x$ represent the same physical point in the space, they are just two different representations of the same point relative to two different coordinate systems. So the statement seems to be contradicting what I thought I understood. That was my first concern. My second concern is less fundamental (more of a computational problem) and I can edit it into my answer later. Many thanks! Edit: So assuming the last line to be correct, we proceed: $$\\Phi'(0) = e^{-ix' \\cdot P}(1 + \\epsilon J_{\\mu \\nu})e^{i x \\cdot P}\\Phi(0) \\approx (1 + \\epsilon \\underbrace{e^{-ix \\cdot P}J_{\\mu \\nu}e^{i x \\cdot P}})\\Phi(0)$$ using the fact to zeroth order $x' \\approx x$. The brace can be evaluated using the Hausdorff formula: \\begin{align}e^{-ix \\cdot P}J_{\\mu \\nu}e^{i x \\cdot P} &\\approx J_{\\mu \\nu} + [J_{\\mu \\nu}, ix^{\\sigma}P_{\\sigma}] \\\\\\&= J_{\\mu \\nu} + i\\left[x^{\\sigma}[J_{\\mu \\nu},P_{\\sigma}] + [J_{\\mu \\nu}, x^{\\sigma}]P_{\\sigma}\\right] = J_{\\mu \\nu} + x_{\\mu}P_{\\nu} - x_{\\nu}P_{\\mu},\\end{align} where the second commutator on the RHS there vanishes. Now compare with $\\Phi'(0) = (1+\\epsilon S_{\\mu \\nu})\\Phi(0)$ as written at the top and we have the result $J = L + S$. My question is: why does this second commutation relation vanish? I should say I am going through 'Conformal field theory' Di Francesco section 4.2.1. The above argument was an alternate argument my professor posed when we sat down and agreed that equation (4.26) in Di Francesco is simply wrong."}
{"id":"126433","title":"Antipodes are mostly ocean - so what happens after digging that hole through earth?","text":"Digging a hole through earth is a common **thought experiment** , often used to explain effects of gravity. But what would happen if someone finally **dug the hole**?   Sure, he took care to stabilize and isolate it inside with two and a half layer of unobtanium. But he forgot to check what exactly is on the other side of the earth, the antipode.   And, just like surprising 85% of earth [travel.SE: antipodes], it turns out to be ocean. The tunnel is under vacuum, and about the diameter of a two track railway tunnel. In the last step building it, obviously the ocean floor broke, and water is streaming in.   What happens then? I'm aware that a body would **oscillate** through the core - the water can not, if the ocean does not run dry assume - at least if the water coming in can mostly fill the diameter. First, I thought the water will just fill the hole, and directly end up in an **equilibrium regarding gravity and pressure**. But that assumes the water, filling the diameter at the ocean ground will do so in the whole tunnel; But doesn't the **stream** of water **get thinner** , as it accelerates up to the center of the earth, and thicker from there? So will it oscillate for a while?   Would it change if we assume the hole in the ocean floor is much smaller than the tunnel's diameter?"}
{"id":"35997","title":"What is the difference between Quantum teleportation and Radio waves?","text":"Quantum teleportation over 143 kilometres using active feed- forward What is the difference between Quantum teleportation and radio waves?"}
{"id":"68557","title":"Must string models that describe 4d effective field theories always have D-branes that extend in the 4 non-compact spacetime dimensions?","text":"In string theory the D-branes give those directions that the strings are allowed to move along. The string excitations give the fields that we detect. Is it correct to think of a particle propagating in 4d spacetime as a string propagating in the same 4d spacetime (in other words if we could zoom in sufficiently far a photon for example would reveal itself to be a string). If so, this implies that a realistic model should always have D-branes that extend in these 4 spacetime directions. Otherwise how would the strings, and their associated excitations, propagate in the 'normal' spacetime we see at low energies?"}
{"id":"108377","title":"fusing two air containers with fixed pressures $p_1,p_2$ and temperatures $T_1, T_2$, what end-result $T$ and $p$ will be?","text":"Let's say that I have two boxes with both volumes equal $V$, filled with an air. Air in first box have pressure $p_1$ and temperature $T_1$, likewise air in second box have pressure $p_2$ and temperature $T_2$. Suppose this is ideal gas, then from Clapeyron equation we have: $$ p_1 V = n_1 R T_1 $$ $$ p_2 V = n_2 R T_2 $$ Thus, after simple transformation: $$ \\frac{p_1 V} {R T_1} = n_1 $$ $$ \\frac{p_2 V} {R T_2} = n_2 $$ Now, let's say that we want to fuse both boxes and remove bulkhead, so we will have instead 1 box with volume equal $2V$. What will be temperature $T$ and pressure $p$ of air in the new box? We can use the same law as above to get: $$ 2 p V = ( n_1 + n_2 ) R T = \\left( \\frac{p_1 V} {R T_1} + \\frac{p_2 V} {R T_2} \\right) R T $$, thus: $$ p = \\frac { T } { 2 } \\left( \\frac { p_1 } { T_1 } + \\frac { p_2 } { T_2 } \\right) $$ **And there's my question:** If I had heated air after fusion of boxes to temperature $T'$, in other words I had known temperature, I could say what pressure is. Likewise, knowing pressure $p'$ I could say the temperature. But if I just fuse boxes of air and \"let it be\" (wait very long for air to mix), what the end result should be? Shouldn't (statistically) the pressure and temperature always fix to some certain, unambiguous state? If yes, how I can find them? I mean that I have one equation $ \\quad $ $ p = \\frac { T } { 2 } \\left( \\frac { p_1 } { T_1 } + \\frac { p_2 } { T_2 } \\right) $, $ \\quad $ and two variables: $T, p$, so how can I find both of them? Did I missed something? Or I must to make some more assumptions?"}
{"id":"91339","title":"Wavefunctions of different observable values are orthogonal because for any state one of them equals zero. Right?","text":"I'm trying to understand correspondence between observables and quantum operators. Let's assume that $\\phi_1$ and $\\phi_2$ are wavefunctions that can be obtained after the measurement of some observable (in other words - quantum states of the system immediately after measurement). $\\phi_1$ is a quantum state if the measurement result equals $\\lambda_1$, and $\\phi_2$ is a quantum state if the measurement result equals $\\lambda_2$ ($\\lambda_1 \\neq \\lambda_2$). Let's assume that $x$ is a classical state of the system, i.e. point in position and momentum space of the system ($x = \\\\{p_i,q_i\\\\}$). And $f(x)$ is a value of the aforementioned observable in state $x$. If $\\phi_1^*(x) \\phi_1(x) \\ne 0$ than $f(x) = \\lambda_1$ (i.e. there is non- zero probability for the system in quantum state $\\phi_1$ to be in point $x$). If $\\phi_2^*(x) \\phi_2(x) \\ne 0$ than $f(x) = \\lambda_2$ (i.e. there is non- zero probability for the system in quantum state $\\phi_2$ to be in point $x$). It means that, for any $x$ either $\\phi_1^*(x) \\phi_1(x) = 0$ or $\\phi_2^*(x) \\phi_2(x) = 0$ (or both). So we can suppose (can we?) that for any $x$ either $\\phi_1(x) = 0$ or $\\phi_2(x) = 0$ (or both). Therefore for any $x$ $\\phi_1^*(x) \\phi_2(x) = 0$ , and therefore $\\langle\\phi_1 | \\phi_2 \\rangle = 0$. The last equality means orthogonality of $\\phi_1$ and $\\phi_2$. Having full set of possible observable values and corresponding orthogonal wavefunctions by spectral theorem we can build an operator $A$ for this observable as following: $$ A = U\\Lambda U^*$$ where $\\Lambda$ - diagonal matrix with $\\Lambda_{ii} = \\lambda_i$ and $U_{ij} = \\phi_i(x_j)$. Is this derivation correct?"}
{"id":"111875","title":"What are particle multiplets in the Standard Model?","text":"The particles of the standard model are often displayed in groupings known as multiplets. I know that this somehow relates to the underlying symmetries of the standard model, which can be viewed as tranforming these particles into each other. I have still struggled to pin down exactly what is meant by these multiplet groupings however, and to get a good 'feel' of the concept. Could somebody elaborate please?"}
{"id":"103443","title":"Order of magnitude estimation for some intriguing questions","text":"The physics TA showed us a few examples in which one can estimate many things from first principles and sound logic based and scaling arguments. This led usually to understanding of why some numbers have the values they do. I asked myself a few questions and after struggling with them, I could not get a clear answer: why our body temperature is about 30 C, is there a way to understand where that number come from? why pregnancy period in humans is about a year, not 10 years, not a month? why our life time is about 100 years, not 1000, not a year? Any help will be really appreciated"}
{"id":"73503","title":"Are locality and separability two distinct notions?","text":"Is there any difference between locality and separability in quantum mechanics, or do they mean the same thing? It seems authors do not always agree."}
{"id":"109397","title":"Problem in the Ladder paradox in relativity?","text":"The ladder paradox consists of a ladder rushing towards a garage with two open doors. In the actual paradox, proper length of the ladder is greater than that of the garage, but in this case lets consider them to be of the _same proper length_. In the frame of the garage, the ladder is Lorentz contracted, and hence it is possible to \"trap\" the ladder inside the garage. By \"trap\" I mean that both doors of the garage must be closed while the ladder is completely inside the garage. But in the frame of the ladder, the garage is Lorentz contracted, and hence it is never possible to \"trap\" the ladder inside the garage. If I am able to trap the ladder in one frame, I should be able to do it in the other frame too right? Why then is there this paradox?"}
{"id":"109390","title":"Rotating frames","text":"A bird of mass $m$ is on a merry-go-round of radius $a$ which rotates at constant angular velocity $-\\omega_b$ in the $y$ direction. A woman of mass $M$ is on a second merry-go-around of radius $b$ which rotates at constant angular velocity $\\omega_s$ in the $z$ direction around the same origin. At $t=0$ the bird is at $(a,0,0)$ and the woman at $(b,0,0)$. What is the force of friction acting on the bird as seen by the woman?"}
{"id":"131491","title":"Smallest view-able object by radio wave or UV magnification with metamaterial?","text":"**What is the theoretical and in practice microscopic maximum\/best resolution of a meta material radio wave XOR UV lens (assuming diffraction or beyond diffraction techniques)?** I'd also like to use diffraction or beyond diffraction as described in this paper: A three-dimensional self-supporting low loss microwave lens with a negative refractive index. Given the Theory of Optical instruments defines the maximum resolving power of a light based microscope as: ![enter image description here](http:\/\/i.stack.imgur.com\/vRmVh.png) ![enter image description here](http:\/\/i.stack.imgur.com\/2cLag.png) Diffraction based analysis allows for an increased resolution but I have not found those formulas. If a meta material lens and diffraction alone do not result in a sub cellular wave what additional tools could be used to generate the tiny subwave? Would UV metameterials and defection techniques result in a small enough wave? I also have not found a radio wave scanning device consistent with an A4 scanner. These are the numbers I know: 11Ghz or wavelen 27.27mm, the focal length between 6 and 10cm, a refraction index of -1, a lens of 255mm by 255mm which is 40mm thick."}
{"id":"25901","title":"Why do rockets need a cosmic ray detector?","text":"I was watching the video _Video camera installed on rocket that reaches 121,000 ft._ , and this rocket has a cosmic ray detector. Why is this needed in a rocket?"}
{"id":"78121","title":"Textbook Problem: Free energy expression","text":"> In Klotz, Introduction to Chemical Thermodynamics, Ex. 8.2 requires me to > derive $$ dG = V \\left( \\frac{\\partial p}{\\partial V} \\right)_V dV + \\left[ > V \\left( \\frac{\\partial p}{\\partial T} \\right)_V - S \\right]dT, $$ where $G$ > is the Gibbs free energy, $S$ is the entropy and all the other variables > have their ordinary common-sense meaning. I'd appreciate feedback on my solution, which I find just very simple, maybe too simple to be right. First of all, when recognizing that $G$ is a function of $V$ and $T$, we can write the total differential as $$ dG = \\left( \\frac{\\partial G}{\\partial V} \\right)_T dV + \\left( \\frac{\\partial G}{\\partial T} \\right)_V dT $$ In Eq. 8.19, we learn that $$ dG = V dp - S dT $$ and so, when multiplying both sides by $1\/\\partial V$ at constant temperature, we find $$ \\left( \\frac{\\partial G}{\\partial V} \\right)_T = V \\left( \\frac{\\partial p}{\\partial V} \\right)_T - S \\left( \\frac{\\partial T}{\\partial V} \\right)_T $$ where $(\\partial T \/\\partial V)_T$ is zero, because we're at constant $T$. A similar procedure yields the bracket expression, this time instead we just multiply both sides with $1\/\\partial T$ and using constant $V$, we arrive at $$ \\left( \\frac{\\partial G}{\\partial T} \\right)_V = V \\left( \\frac{\\partial p}{\\partial T} \\right)_V - S \\left( \\frac{\\partial T}{\\partial T} \\right)_V = V \\left( \\frac{\\partial p}{\\partial T} \\right)_V - S $$ and from comparison with the total differential expression, one immediately recognizes the above very first expression. Is this a valid solution\/Ansatz?"}
{"id":"25909","title":"Missing radio source coordinates in G\/T measurement technique ITU-R S.733.2","text":"Section 3 ‘Suitable radio stars’ of this recommendation lists the different stars\/nebula and their corresponding flux-densities that can be used in the G\/T measurement. There is no reference to the coordinates these flux densities were measured at. Some of the Nebula have large angular extension and I’m not sure where to point my antenna. > Is there a list of celestial coordinates of the radio stars used in this > recommendation? http:\/\/www.itu.int\/dms_pubrec\/itu-r\/rec\/s\/R-REC-S.733-2-200001-I!!PDF-E.pdf"}
{"id":"23432","title":"Maximum efficiency for a counter-current heat exchanger (double flux controlled motorized ventilation)","text":"I am not sure if I can explain the question correctly because I don't know the name of this mechanism in English. This is my explanation attempt: In a house, a tube is expelling the air from the inside to the outside, and a tube is aspiring the air from the outside to the inside. The 2 tubes are interlaced in order to transmit the heat between their contents. The goal is to have a good thermal isolation between inside the house and outside:   * When the house is more hot than the outside, the outgoing air is warming the incoming air.   * When the house is more cold than the outside, the outgoing air is cooling the incoming air. Question: **In theory** , what is the maximum efficiency of such thermal exchange? Could it be possible to reach something close to 1 or the upper bound of the efficiency is well under?"}
{"id":"101591","title":"Determining if a particle is bounded or unbounded with motion in polar coordinates","text":"![a](http:\/\/gyazo.com\/5be8000bab6937ba21923ffa75cf0997.png) Firstly, I determined the equation for the particles path for $k <-mV^2a^2$ as $u = {Ae^{\\sqrt{\\alpha}\\theta}+Be^{-\\sqrt{\\alpha}\\theta}}$ where $ \\alpha = 1 + \\dfrac{k}{mV^2a^2},$ $u = 1\/r$ I have two questions; (1) Could I solve for the constants $A,B$ with the information given? (It is not given that at r = a, $\\theta = 0$, so I figured no) (2) I'm not sure how to determine if the motion is bounded or unbounded in polar coordinates - For instance, if we were dealing with $r(t)$ then I'd take $t \\to \\infty$ and see what happens, but we have theta here, how can we deal with theta? In this case with $k <-mV^2a^2$ it is bounded (if we take $\\theta \\to \\infty$)"}
{"id":"76930","title":"What are the good introductory resources for M-theory towards AdS\/CFT?","text":"I see a list here with a section titled M-theory - http:\/\/www.superstringtheory.com\/links\/reviews.html In there these two look promising, http:\/\/arxiv.org\/abs\/hep-th\/9607201 and http:\/\/arxiv.org\/abs\/hep-th\/9802051 I would like to know if people have opinions of good starting points towards understanding M-branes and how they are used for constructing exact examples of AdS\/CFT. [...I guess all exact AdS\/CFT scenarios are M-theory constructions..]"}
{"id":"117116","title":"Principle of locality and forces","text":"I have a silly confusion about the statement written in the link Einstein and Locality _''external influence on A has no direct influence on B; this is known as the Principle of Local Action.''_ Since GTR and Electromagnetism obey principle of locality. And in both cases change in A position affect the B and vice versa. Then why this contradiction come. Please correct me."}
{"id":"118714","title":"What does it mean for electrons to be \"diffusive\"?","text":"I'm reading this paper and it has the line (end of 3rd paragraph, page 2): > It turns out that the simple fact that electrons are diffusive instead of > freely propagating leads to a profound modification of the traditional view > based on the Fermi-liquid theory of metals. So it kind of suggests it means something that's _not_ freely propagating, like bound I guess?"}
{"id":"74578","title":"Why does a larger sprocket need more power to accelerate at the same rate?","text":"Why does a large drive sprocket require more power to accelerate at the same rate as a smaller drive sprocket if the overall gear ratio is the same (with all other components such as the chain the same between systems)? As an example, we could compare 11\/33 to 13\/39 (same 1\/3 ratio). Why does the 13 tooth sprocket need more power to accelerate at the same rate as the 11 tooth setup? ![enter image description here](http:\/\/i.stack.imgur.com\/QmYrH.jpg) Can we agree that the top sprocket will need more torque to accelerate an atv at the same rate as the same atv with the bottom sprocket? And assuming the overall gear ratio is the same. The moment of inertia differences of the sprockets are insignificant. Assume the moments of inertia are equal."}
{"id":"74577","title":"How do I calculate integral analytically for small $k$?","text":"In a Heisenberg antiferromagnet, the dispersion relation is \\begin{equation} \\omega_{\\mathbf{k}} =JSz\\sqrt{ 1-\\gamma_{\\mathbf{k}}^2} \\end{equation} where $\\gamma_{\\mathbf{k}}=\\frac{1}{d}\\sum_{i=1}^d \\cos k_ia$, where $d$ is the dimensionality of the cubic lattice. In staggered magnetisation calculation, we get an integral which is given by \\begin{equation} \\delta m =\\int_{BZ} \\frac{d^dk}{(2\\pi)^d}\\frac{1}{\\sqrt{ 1-\\gamma_{\\mathbf{k}}^2}}. \\end{equation} This integral is dominated by small $k$, since we have Goldstone modes at $k=0,(\\pi,\\pi,\\dots)$. Using this property, we can extend the limits of this integral from Brillouin zone to infinite $k$-space and we replace $\\gamma_{\\mathbf{k}}$ by its small $k$ limit: \\begin{eqnarray} \\delta m=\\frac{\\sqrt{d}}{a}\\int_0^\\infty \\frac{d^dk}{(2\\pi)^d}\\frac{1}{k} \\end{eqnarray} In 1D, its gives the logarithmic divergence which is expected. But, in 2D and 3D we get \\begin{eqnarray} \\delta m &\\sim& [k]_0^\\infty, \\quad \\text{in}\\, 2D\\\\\\ &\\sim&[k^2]_0^\\infty, \\quad \\text{in}\\, 3D \\end{eqnarray} which still show divergence. This is not correct as if we solve the Brillouin zone integral we get some finite value. I do not understand what is wrong with the infinite k space integral? Could anybody please tell me how to solve such integral analytically for small $k$?"}
{"id":"121201","title":"How is graphene a major breakthrough?","text":"What is graphene from a physics standpoint? Why do I keep hearing that graphene is considered to be such a major breakthrough? How is graphene going to transform the world?"}
{"id":"81872","title":"Quantum Regime of Particles in Solids","text":"On my midterm today, I read that when the deBroglie wavelength of a particle exceeds the spacing between the particles in a solid or liquid, the particles begin to behave quantum dynamically. Why is this? I thought a larger deBroglie wavelength implied a less quantum mechanical behavior."}
{"id":"81870","title":"Truck Mass & \"4th power law\"","text":"Is it true that the damage caused to a road surface is proportional to the 4th power of vehicle axle weight, and if it is, what factors cause it to be related by a power of 4, or is it just the results of some measurements? I'm chiefly looking for the answer to \"why?\", although it'd be handy to know if it is indeed the 4th power, and if not the 4th power, then what?"}
{"id":"75622","title":"Einstein Equation at the Singularity. Understanding the Dr. Michio Kaku's explanation","text":"I'm trying to understand what Dr. Michio Kaku is exaplaining in this video: http:\/\/www.youtube.com\/watch?v=hydDhUNvva8 I'm just able to rewrite this two equations. I think I've probably made ​​some mistake in the process of copying. Otherwise, I don't understand the meaning of the two $\\mathrm O$ and the $I^{'}$ in the second equation. In the video the Dr. Michio Kaku is solving the first equation right? But I saw that is slightly different from the classical field equation, why? What's the difference? $As^2$ is the solution of the first equation? Is possible understand something from the video or is it too cutted? \\begin{equation} R_{\\mu\\nu}-\\frac{1}{2}g_{\\mu\\nu}R=T_{\\mu\\nu} \\end{equation} \\begin{equation} As^2=\\left(_{\\mathrm O}\\left(1-\\frac{2MG}{r}\\right)^{I^{'}\\mathrm O}\\right) \\end{equation}"}
{"id":"38460","title":"Is the minimum radius of a positronium system of the order of compton wavelength or less than that?","text":"Since from electron-positron annihilation energy and uncertainty principle,the minimum radius of positronium comes out as half of the compton radius."}
{"id":"38462","title":"Units in cgs system","text":"How do I find the dimensions of this quantity (in $cgs$)... $$\\frac{4\\pi me^2}{h^2n_o^{1\/3}}$$ where $m$ is the mass of electron $e$ is the magnitude of electronic charge $h$ is the Planck's constant & $n_o$ is equilibrium number density of electrons"}
{"id":"88555","title":"Determining distance from object in concave lens","text":"so I was studying for my Yr11 Physics test tomorrow and I came across a question that I got the wrong answer on, all of my friends are getting the same answer. Question: You wish to project the image of a lamp, magnified three times, onto a screen 5.0m from the lamp. How far from the lamp should the mirror be placed? I though that we would just use $M=\\frac{-d_i}{d_o}$ where $M=3$ and $d_i=5$, which will work out that $d_o\\approx -1.7$ which has a magnitude of 1.7m from the lamp However, our answers book says that the answer is 2.5m What have I done wrong? Also, please keep in mind that I am in year 11 and may not understand anything beyond that level"}
{"id":"36052","title":"Why does letting air into a box of wine help the flow?","text":"I've notied that when I get down to the last part of a box of wine, the flow slows to a trickle. If I open the nozzle, tilt the box back (letting air gurgle into the bladder) and then tilt it back forward the flow is greatly increased. Why does letting air into the bladder help increase the flow?"}
{"id":"56376","title":"Falling into a black hole emitter vs observer","text":"Let's say we are working with the Schwarzschild metric and we have an emitter of light falling into a Schwarzschild black hole. Suppose we define the quantity $$u=t- v$$ where $$dv\/dr= 1\/(1-r_{s}\/r)$$ where $r_s$ is the Schwarzschild radius. What is the $u$ as observed by the emitter? I just need a _definition of $u_e$_. I have problems identifying the quantities as measured by an observer at large $r$ and that of the emitter. Would I be right at least to say that $$t_{e}=\\tau$$ the proper time? Many thanks. * * * In fact, I've been told that $$du_o\/d\\tau=du_e\/d\\tau$$ Why is it? * * * My apologies for leaving out a subscript $e$ in the question. It has been added."}
{"id":"71019","title":"Apart from Alfvenic turbulence, what is the most important unsolved problems in plasma physics?","text":"**Apart fromAlfvénic turbulence**, what is the most important unsolved problems in plasma physics? Is there a review article describing some important unsolved problems in plasma physics, I repeat, apart from turbulence, which is an unsolved matter in all areas of physics."}
{"id":"71011","title":"Poincare recurrence and the multiverse","text":"In this paper Susskind claims that a stable de Sitter universe is problematic (among other things) due to the existence of Poincare recurrence, which happen because of finite entropy. I disagree that this is a problem for reasons similar to those given here. In any case, he also proposes a variant of eternal inflation driven by false vacuum decays. These decays, as far as I know, are driven by Coleman - de Luccia instantons and are probabilistic in nature, that is to say there is always some probability of instanton creation via quantum fluctuation. My questions are:   1. Since false vacuum decay is probabilistic is it possible that in this bubble cosmology setup there are casual patches which live beyond their Poincare recurrence time?    2. Is there any possibility in string theory that our Universe, assuming it has finite entropy, will eventually be stabilized for an infinite amount of time or is this question settled beyond all reasonable doubt?  Thanks!"}
{"id":"119037","title":"Velocity and acceleration with displacement","text":"Find the velocity and the acceleration for a moving object with displacement $= te^{-t}$. The $e$ is the exponential function."}
{"id":"41485","title":"Holstein-Primakoff and Dyson-Maleev representation","text":"In Holstein-Primakoff and Dyson-Maleev representation, spin operators are represented by bosonic operators. Roughly speaking, a state with $S^z=S-m$ corresponds to a state containing $m$ bosons. In the Dyson-Maleev representation, $(S^+)^\\dagger\\neq S^-$, so we have a non-unitary transformation. Are eigenvalues and matrix elements invariant under such a transformation? In matrix form, what exactly are the two transformations that will give rise to the two representations, say for spin $S=3\/2$?"}
{"id":"135210","title":"Time derivative of angular velocity in rotating reference frame","text":"I am going through a section in a textbook regarding the Newton Euler equations for a system of rigid bodies (robotics text). There is a particular line in the derivation I don't understand, I've attached the image to make it clear: ![enter image description here](http:\/\/i.stack.imgur.com\/scZIL.png) I think the derivative of the second term on the right-hand side of 3.28 is using the product rule to get to 3.32, but I don't understand where the cross- product comes from. Could someone explain this particular part of the derivation? Thanks."}
{"id":"44056","title":"What is the weakest magnetic field that can practically be measured?","text":"I would like to be able to measure magnetic fields that are generated by nerve impulses in extremities like in fingers. I know that they are very weak: about 100 mV electric potential would give us about current 10 mA if resistance = 1\/10 Ohm, but interested in measuring magnetic fields. **How many rounds should you have in a coil to measure 10 mA current?** The jumping of electrons from one level to another may become a problem in this case. However, I am not sure where and how. Assume that there is a steady current in the situation so Biot-Savart law. The radius of the nerve is r = 0.003m."}
{"id":"12996","title":"Virtual particles contributing to universal expansion.","text":"> **Possible Duplicate:**   >  Virtual particle production in space-time. Does virtual particle production contribute to observed universal expansion?"}
{"id":"41734","title":"Find magnetic scalar potential for superconducting sphere","text":"In regions where $J = 0$, the curl of the magnetic field $B$ is necessarily zero (since $\\nabla \\times B = \\mu_0 J$). Therefore $B$ can be written as $B = -\\nabla V_m$, where $V_m$ is a scalar function of position.   A superconducting ball of radius $a$ is placed in an otherwise uniform magnetic field $B = B_0 \\hat {z}$. Since the $B$ field is zero inside the ball, it must hold that $B_{\\perp} = 0$ just outside the surface of the ball (by magnetic boundary conditions).   (a) Find the magnetic _scalar_ potential $V_m (r, \\theta)$ everywhere.   Far from the sphere the field is $B_0 \\hat {z}$, and hence $V_m \\rightarrow -B_0 z$. So in spherical coordinates, $V_m \\rightarrow -B_0 r \\cos \\theta$ for $r \\gg a$. Is the following true? $$ \\frac {\\partial V_{\\text{above}}}{\\partial n} = \\mu_0 K $$ Since $V_{\\text{below}} = 0$ due to the sphere being a superconductor. Using separation of variables, the first condition yields $$ \\sum_{l=0}^{\\infty} A_l r^l P_l (\\cos \\theta) = -B_0 r \\cos \\theta, $$ So, only one therm is present, $l = 1$. Then $A_1 = -B_0$ with all other $A_l$'s zero. From the second condition, $$ \\frac {\\partial V}{\\partial n} = A_1 \\cos \\theta - \\sum_{l =0}^{\\infty} (l+1)\\frac {B_l}{R^{l+2}} P_l (\\cos \\theta) = \\mu_0 K. $$ Now I'm stuck as to how to proceed and remove some of the $B_l$ terms. Perhaps my boundary conditions are not correct."}
{"id":"54793","title":"Find work done by force along a path - is parameterization the only way?","text":"$F = x^3y^4 \\hat i + x^4y^3 \\hat j$ from $(0,0)$ to $(1,1)$. I am given different paths. For example, \"first along x axis and then along the y axis\" is one of the paths. Is this problem solvable only using parameterization or is there another way to handle\/integrate it?"}
{"id":"123170","title":"Kinematics - moving body","text":"If I have an inclined plane, which has acceleration $\\alpha$, will a block placed on the inclined plane slip back with the same acceleration? (considering that the surface is frictionless)"}
{"id":"127695","title":"Why is the pressure inside a soap bubble higher than outside?","text":"Apparently, the air inside a soap bubble is under higher pressure than the surrounding air. This is for instance apparent in the sound bubbles make when they burst. Why is the pressure inside the bubble higher in the first place?"}
{"id":"65639","title":"Gauge fields and strings: Loop equations","text":"I am trying to derive Eq. (7.25) (p. 117) of Polyakov's book: $$ \\delta \\Psi (C) = \\int_{0}^{2\\pi} {\\rm P} \\left(F_{\\mu\\nu}(x(s)) \\exp \\oint_C A_\\mu dx^\\mu \\right)\\dot{x}_\\nu \\delta x_\\mu(x) \\, {\\rm d} s $$ where the non-abelian phase factor around a closed loop $C$ is defined as $$ \\Psi(C) = {\\rm P}\\exp \\left(\\oint A_\\mu dx^\\mu \\right) = {\\rm P}\\exp \\left(\\int_{0}^{2\\pi} A_\\mu \\dot{x}_\\mu\\, {\\rm d}s \\right)$$ It seems that he is using the relation given on p. 116: $$ \\delta \\, {\\rm P} \\exp \\int_{0}^{2\\pi} M(\\tau) {\\rm d}\\tau = \\int_{0}^{2\\pi}{\\rm d}t\\,{\\rm P} \\left(\\delta M(t) \\exp \\int_{0}^{2\\pi}M(\\tau){\\rm d}\\tau\\right)$$ Matching with (7.25) I find $\\delta A_\\nu = F_{\\mu\\nu} \\delta x_\\mu$. This relation seems to be saying that if I change the position of the loop at the parameter $s$ by $\\delta x_\\mu(s)$ then the vector potential changes by $\\delta A_\\nu(x(s)) = F_{\\mu\\nu}(x(s)) \\delta x_\\mu(s)$. I don't know how to derive this relation. Is it legitimate?"}
{"id":"65638","title":"Will a spinning object come to rest?","text":"Will a sphere spinning on its own axis come to rest given enough time, provided no other forces act upon it? I know that if you have _two_ spinning spheres in the depths of space and orbiting each other they'll eventually stop spinning -- they'll tidally lock each other. The internal friction of the shifting tidal bulge eventually overcomes the rotation. But I don't see what forces would stop a sphere from rotating around its own axis, in absence of other forces. And yet, each point on the sphere (except points along the axis) is experiencing accelerated motion, which implies a force. My intuition says that something must maintain the acceleration of each point, which implies that the acceleration causes internal stress. I know this to be true macroscopically (the sphere will have an equatorial bulge), but what forces maintain the acceleration of each particle, and will the stresses ever overcome the rotation of the sphere? **Clarification** : I am _not_ considering infinitely rigid spheres or point particles. I'm wondering about actual non-rigid macroscopic spheres, such as planets."}
{"id":"118984","title":"Could this suggest that there is a wavelength smaller than Planck's?","text":"Suppose the earth receives a photon with a wavelength $\\gamma_1$. Since spacetime is expanding, we know that this photon had an original wavelength $\\gamma_2$, such that $\\gamma_2\\lt\\gamma_1$. This is phenomenon is known as redshift. Nothing special. Now, here's the thing, what if the earth receives from far away a photon whose wavelength is equal to Planck's length $\\ell_p$. This means that the photon before traveling all this distance a wavelength smaller than $\\ell_p$. But this is impossible, since $\\ell_p$ is the smallest possible wavelength. So would this suggest that there is a wavelength smaller than Planck's?"}
{"id":"69930","title":"Deuterium density in seawater","text":"I heard today that deuterium for nuclear fusion could be extracted from the Oceans, in which it is present in \"heavy water.\" I heard this claim: > _\"Deuterium is distributed uniformly with Ocean depth.\"_ This contradicted my naive expectation that deuterium density would _increase_ with depth, on account of it being in \"heavy water.\" For example, salinity increases with depth. I asked, and received two contradictory explanations. The first was that because powerful waves and Ocean currents mix Ocean water, gravity is negligible. This surprised me; I thought that Ocean's were stratified, with predominantly tangential, rather than vertical, currents. The second explanation was that my view of seawater is naive - that I should think of seawater not as heavy and regular water molecules, but as a soup of weakly interacting deuterium, the normal hydrogen isotope and oxygen, forming and breaking water and heavy water molecules with a short time-scale. If so, apparently, deuterium density and the normal hydrogen isotope would be distributed evenly with depth. This did not help me, because, I then naturally wondered why _deuterium's_ density doesn't increase with depth, since it is heavier than hydrogen-1? Is the claim correct? and if so, what is really going on? I have the feeling that Ocean behaviour with depth might be non-trivial..."}
{"id":"13715","title":"If LHC is the most powerful microscope in the world, what does it magnify?","text":"Some experts (and non experts) have called LHC the most powerful microscope in the world. I am wondering, what does the LHC magnify? Or is calling it a microscope just wrong and misleading?"}
{"id":"13714","title":"Embrace Physics","text":"Does anyone have any suggestions as to what is a good topic for a short talk on theoretical physics to a bunch of Math and Physics undergrads that might make them \"embrace\" theoretical physics? Thanks. (Brain storm: String theory, quantum tunneling, etc...?) Also it would be great if someone could suggest some resources."}
{"id":"96356","title":"How does water falling down a waterfall gain energy?","text":"I'm kind of a noob in the physics area. But I know that energy cannot be created, it can only be transferred. So how and what \"type\" of energy does water get when free-falling?"}
{"id":"116478","title":"Finding out the Potential difference","text":"> _Find the potential inside and outside a uniformly charged solid sphere > whose radius is $R$ and whose total charge is $q$. Use infinity as your > reference point. Compute the gradient of $V$ in each region, and check that > it yields the correct result._ The above one is a question cropped from Griffiths page no 82. When I try to solve the problem they used a potential equation inside the sphere ![enter image description here](http:\/\/i.stack.imgur.com\/ctSoI.png) My problem is in the right part. Why do we need to consider the next part? we have been asked to solve potential inside a charged sphere. so can't I use only the limit $\\infty $ to $ R$? I am saying this because, i have considered the infinity as the reference point and R as the point where I will find the potential. > Is it because we need to add the potential outside of the sphere and inside?"}
{"id":"60374","title":"Conversion of energy to mass with radioactive decay","text":"Consider the following problem: _The nuclei of Am-241 decay by the emission of $\\alpha$-particles with a kinetic speed of $8.8 \\cdot 10^{-19} J$. In a certain source of Am-241 there are $ 4.0 \\cdot 10^3$ decaying nuclei per second. Caculate the mass this source loses per year._ So what the correction sheet tells me is that you basically just have to find the mass of an $\\alpha$-particle in $kg$ and then just multiply that number until you get the amount of decaying $\\alpha$-particles in a year. However, I think this is wrong, since the $8.8 \\cdot 10^{-19} J$ isn't accounted for. I'd say you have to take the mass of the $\\alpha$-particle, then add to that the $8.8 \\cdot 10^{-19} J$ after you've converted it to $kg$ by using $E = mc^2$ and THEN multiplying that number in order to get the amount of decaying $\\alpha$-particles in a year. Which is correct, or perhaps better: which is less wrong?"}
{"id":"60372","title":"Question about a finite time interval step in the derivation of the Feynman path integral in Sakurai","text":"This may be a possible errata but Sakurai (pp 126 in the 2nd Edition) states that starting with $$S = \\int dt \\,\\,\\scr{L_{\\mathrm{classical}}}$$ Looking at a finite-time-interval of the action: \\begin{equation} S(n,n-1) = \\int\\limits_{t_{n-1}}^{t_n} dt \\left( \\frac{m \\dot{x}^2}{2} - V(x)\\right)\\,\\,\\,\\,\\,\\,(1) \\end{equation} Now using a straight-line approximation for points $(x_{n-1},t_{n-1})$ and $(x_n,t_n)$, which implies $$\\dot{x} = \\frac{x_n-x_{n-1}}{t_n-t_{n-1}} = \\frac{x_n-x_{n-1}}{\\Delta t}$$ Sakurai then has Eq (1) become: $$S(n,n-1) = \\Delta t\\left( \\left(\\frac{m}{2}\\right) \\left(\\frac{x_n-x_{n-1}}{\\Delta t}\\right)^2 - V\\left(\\frac{x_n+x_{n-1}}{2}\\right)\\right)$$ My question is why is the potential now dependent on $ V\\left(\\frac{x_n + x_{n-1}}{2}\\right) \\mathrm{\\,\\,instead \\,\\,\\,of\\,\\,\\,} V\\left(x_n- x_{n-1}\\right)$? I checked the most recent errata (4\/5\/13) posted by J. Napolitano ( http:\/\/homepages.rpi.edu\/~napolj\/ErrataMQM.pdf ) and there is none for this page. Could anyone clarify this step for me? Thanks."}
{"id":"94196","title":"How can laser communication be faster than microwave?","text":"My textbook states that \"Laser communication is much faster than microwave communication.\" But, how can that be? Both are electromagnetic waves with different frequencies but, how can the speed be different? Or Is the statement referring to any kind of processing speeds involved?"}
{"id":"81115","title":"About Dirac equation in curved spacetime (spherical)","text":"I would like to ask you about the separation of variables of the Dirac equation in curved space-time. The metric is given by $$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\\theta^{2}+\\alpha^{2}r^{2}\\sin^{2}\\theta d\\phi^{2},$$ where $\\alpha$ is a positive constant. The Dirac equation(a part of) is given by $$i \\gamma^{\\mu}(x)(\\partial_\\mu+\\Gamma_{\\mu}(x))\\psi(x)=0$$ where $\\Gamma_{\\mu}(x)$ is spinor affine connections and $\\gamma^{\\mu}$ are the general Dirac matrices. If we define the tetrad base as $$e^{\\mu}_{(a)}=\\left( \\begin{array}{cccc} 1 & 0 & 0 & 0 \\\\\\ 0 & \\sin\\theta \\cos\\phi &\\sin\\theta \\sin\\phi & \\cos\\theta \\\\\\ 0 & \\frac{\\cos\\theta \\cos\\phi }{r}&\\frac{ \\cos\\theta \\sin\\phi}{r} & -\\frac{\\sin \\theta}{r} \\\\\\ 0 & -\\frac{\\sin\\phi}{\\alpha r \\sin\\theta} & \\frac{\\cos\\phi}{\\alpha r \\sin\\theta}& 0\\\\\\ \\end{array} \\right) ,$$ using the tetrad base we obtain(But I cannot!): > Blockquote $\\gamma^{\\mu}(x)\\Gamma^{\\mu}(x)=\\frac{1}{2r}(1-\\frac{1}{\\alpha})\\csc\\theta(\\gamma^{(1)}\\cos\\phi+\\gamma^{(2)}\\sin\\phi)$ > Blockquote . Here, $\\gamma^{(a)}$ are flat Dirac matrices. Moreover, $$\\gamma^{\\mu}(x)\\Gamma^{\\mu}(x)=\\frac{1}{2}\\gamma^{(a)}[(\\partial_{\\mu}e_{(a)}^{\\mu}(x)+e_{(a)}^{\\rho}(x)\\Gamma^{\\mu}_{\\rho \\mu})+i\\gamma^{(5)}\\varepsilon_{(a)(b)(c)(d)}e^{(b)\\mu}e^{(c)\\nu}\\partial_{\\mu}e^{(d)}_{\\nu}]$$ Now my question is: how one can obtain $\\gamma^{\\mu}(x)\\Gamma^{\\mu}(x)$ expansion above, which is $$\\gamma^{\\mu}(x)\\Gamma^{\\mu}(x)=\\frac{1}{2r}(1-\\frac{1}{\\alpha})\\csc\\theta(\\gamma^{(1)}\\cos\\phi+\\gamma^{(2)}\\sin\\phi),$$ because I couldn't obtain it. I'm not used to this notation :("}
{"id":"90687","title":"To what degree does guitar construction affects the vibration of the strings?","text":"There's an old debate going on in the guitar community about how much does wood choice and body shape affect the sound of an electric guitar. No one denies that there's a difference acoustically (how the guitar sounds unplugged) because in this situation it's the wood and body shape that amplify the sound made by string vibrations, but when we're talking about the sound as it comes from the pickups, things get much more uncertain because there are a lot of variations even in supposedly identical guitar parts, and accounting for all of them for the purpose of doing an experiment where the only difference between two guitars is the wood is difficult (I certainly haven't heard of an experiment that was satisfying enough, but feel free to prove me wrong). I'm interested in a way to circumvent these practical difficulties using a theoretical explanation: Since the pickup only sees the string's vibrations, the question basically becomes \"does body shape and wood make enough difference in the way a string vibrates that it changes the sound in a noticeable way\", and this sounds like something that may be possible to figure out mathematically, or at the very least should be much easier to test because there are less variables involved. My question is: Is what I'm describing possible to calculate\/test, and has this been done before?"}
{"id":"82845","title":"Could velocity be taken as fundamental instead of time?","text":"In physics time and length are taken as fundamental in the SI system and, as it seems, in the thinking of physicists. Could one instead take velocity, with c as its unit, together with length as fundamental and then understand time by dimensional analysis in terms of l\/v (length divided by velocity)? If not, why not?"}
{"id":"46023","title":"about wavefunction and vector entries","text":"I am beginer of physics and I am studying some very fundamental idea of quantum mechanics by myself. In the introducing book I am reading, there is an example to show a particle diffraced by a slit or something small then there show diffraction pattern. So if the wavefunction of the particle is represented in momentum space such that it has n entries in the wavevector so integer n stands for the nth diffraction order. So what's the significance of the square of the nth entry? Does it mean the probabilty of finding the particle with momentum diffracted into the nth order of the diffraction pattern? Let's say I have a source of particles, each particle emitted at different speed towards the slit so they will be diffracted independently. How does the intial speed of each atom affect the diffraction pattern? If we look at the nth diffraction order, how will the speed of the particle change the probablity of finding the particle in the nth order? Thanks a lot"}
{"id":"64275","title":"Row of pivoted magnets and energy scale","text":"This question is about a system involving a horizontal row of length **L** of equally spaced pivotable magnets, each with a pole at either end. These magnets will often be referred to as units. So each unit when rotated causes it's neighbours to rotate in the opposite direction to itself. When the first unit is quickly rotated 1\/4 of a turn and fixed in place rotationally, each will likely turn less than the previous at first, taking longer to complete the full 1\/4 of a turn. They will all eventually rotate the full 1\/4 as the far end of the line is still not fixed. . > Turning the first magnet will require energy. For a line of 1000, while this > will be more energy than that for a line of for example 2, it will be less > than 1000 times the energy required for a line of 1. This is because: > >>   1. Momentum means that every unit doesn't need to be rotated the full distance to turn the first unit, it is not as if they are connected together by rods, the first can be turned all the way before the others have had time to rotate much at all. >> >>   2. More distant magnets contribute less force, so most units are contributing very little force to the first unit directly. Another way to think about this is that the line of separated magnets forms a compound magnet that will result in more force than each component magnet has individually, but not as much as 1000 times more, for example if metal becomes attached to an end it will take less than 1000 times the force of an individual magnet to remove. >> >> . Coils around each unit slow rotation but produce **N** electricity after 1\/4 of a turn, however long the 1\/4 turn takes. The amount of electricity produced appears to be L * N, for an input power that is increment by a diminishing amount for the growth of L, an example of this sort of growth could be sqrt(L). . > **Question:** This has the potential not to add up energy wise for long > lines, what's going on?"}
{"id":"45593","title":"Chern number in condensed matter physics","text":"In mathematics, the Chern number is defined in terms of the Chern class of a manifold. What is the exact definition of Chern number in condensed matter physics, i.e. quantum hall system?"}
{"id":"133796","title":"Relationship of multiple particles under collision","text":"Consider 3 particles. All 3 particles travel along the x-axis.   * The 1st particle possesses some mass, m, and its initial position is somewhere on the negative x-axis. It contains some (positive) velocity v.   * The 2nd particle possesses some mass, A*m, and its initial position is at the origin of the x-axis (0). It contains no velocity (at rest).   * The 3rd particle possesses some mass, B*m, and is initially situated somewhere on the positive x-axis. Like particle 2, it contains no velocity (at rest). Find the relation between A and B, under which the 1st and 2nd particles will collide with each other more than once. note: it is assumed the collisions are elastic as well as there being no external forces considered. also the collisions are such that all particles remain on the x-axis."}
{"id":"16877","title":"Chiral anomaly and decay of the pion","text":"I am told that if all classical symmetries were reflected as quantum symmetries, the decay of the neutral pion $$\\pi^0 ~\\longrightarrow~ \\gamma\\gamma$$ would not happen. Why would the conservation of the axial current in QED prevent the decay of the pion? What is the non-conserved charge in this decay?"}
{"id":"12147","title":"Does my model aircraft propeller beat the sound barrier?","text":"I fly model aircraft. Recently I changed from a \"1300kv\" motor to a \"1500kv\" motor. The difference is in speed; the 1300kv can go about 16,400rpm full- throttle and with a fully charged battery and the 1500kv can go nearly 19,000rpm. (This is no load speed, but the prop adds very little load when the aircraft is not flying.) I noticed that when I increased the throttle beyond about 75% the motor and propeller became incredibly loud. Below that they were loud, but no where near as much. So I did some ballpark calculations. I'm using a 7x3.5\" propeller - which means it has a 7\" diameter and 3.5\" pitch. In this case we are only interested in diameter. 7\" diameter means 3.5\" radius, or 8.86 cm. Doing 19,000 rpm means than I'm doing about 317 rps. According to 2 * pi * r the circumfrence of the prop's tip is 55.64 cm. Plug this into 317 rps and we get 17,638 cm\/s or 176.4 m\/s. Which is about half the speed of sound. Although this is not actually the speed of sound it's very close to it, and much higher than I expected. So what does that mean? I figure that although the propeller itself isn't traveling faster than the speed of sound, the air it pushes out of the way is. My physics teacher showed that if you take a metre stick and swing it through the air (avoiding hitting anyone) it makes a wooshing sound - the air is moving faster than the ruler was. So the prop is pushing the air away faster than this and causing this very loud noise. Is my reasoning correct? Is my prop breaking the sound barrier?!"}
{"id":"12146","title":"Is the Keppe Motor \"real\"?","text":"Its makers say that it's a highly efficient motor because it resonates with (what they call) \"Essential Energy\", which is free energy that is everywhere in space. (I think that’s just a different name for Dark Energy which somehow they are managing to disturb with the coils of the motor, and when it disturbs back, a bit of power is generated in the coil). I’ve listened to a radio interview of the makers, and they sad that a small motor powered by a 9v battery ran for about 2 hours with enough torque that they couldn’t stop it with their own hands. Is that what is really happening? Is it revolutionary? Or just bull… just a motor that consumes less energy since its coils are turned off half of the time? Also, strange things happen on the proximities of a running motor (a similar motor will start running even though it has no power, but it will run slower than the first one). Caused by the same thing that wireless cell phone chargers use to work?"}
{"id":"90023","title":"Speed of ejection of liquid from revolving tank","text":"Consider a cylindrical tank filled with a liquid. Suppose it has an orifice somewhere at the bottom and the liquid is spurting out of this hole at a certain velocity. Now, if you rotated this cylinder about an axis passing through its centre, will the speed of ejection of fluid change? I feel it should, because of a centrifugal force, but how do I prove this mathematically?"}
{"id":"53620","title":"How is space ship's acceleration perceived if the acceleration is perpendicular to the velocity?","text":"Spacecraft in orbit around the Earth are constantly accelerated by the gravitational field of Earth. That's why the spacecraft ($m \\ll M$) is in an (elliptical) orbit around the centre of gravity of the Earth, accelerated by $ \\mathbf{g}=-{G M \\over r^2}\\mathbf{\\hat{r}} \\, .$ Plugging in the numbers for a spacecraft orbiting at roughly 420 km (such as the International Space Station), this gives: $ \\mathbf{g} = {- 3.986 \\cdot 10^{14} \\over (6371 + 420) \\cdot 10^3 m} = - 8.6 \\, \\mathrm{m}\/\\mathrm{s}^2$ So, an astronaut on-board the ISS is in a reference frame that is constantly accelerating at an acceleration of $8.6 \\, \\mathrm{g}\/\\mathrm{m}^2$. Yet unlike the astronauts featured in this related question, they do in fact feel no gravitational acceleration at all; at most they may feel some centrifugal pseudoforce, but this is considerably less than the gravitational acceleration, and at most at microgravity levels. Is acceleration only perceived when it changes the magnitude of the velocity, as opposed to the direction? What is the fundamental reason for this? Or am I misunderstanding something?"}
{"id":"53627","title":"Finding the electric field","text":"The charge per unit length on a long, straight filament is -91.9 µC\/m. > (a) Find the electric field 10.0 cm from the filament, where distances are > measured perpendicular to the length of the filament. (Take radially inward > toward the filament as the positive direction.) MN\/C > > (b) Find the electric field 50.0 cm from the filament, where distances are > measured perpendicular to the length of the filament. MN\/C > > (c) Find the electric field 150 cm from the filament, where distances are > measured perpendicular to the length of the filament. $λ_q=\\frac{dq}{dl}→dq=λ_qdl$ $\\vec{E} =k_e\\frac{q}{r^2}\\hat{r}$ $||\\vec{E}||=\\frac{λ_qdl}{r^2}$ How can I solve this without knowing the length of the filament? Supposedly this is the suggested way of solving this problem--and I'd like to understand this method very much. Could someone help me? Also, in addition to solving it by this expedient, is it possible to solve this problem by employing a Gaussian surface?"}
{"id":"63300","title":"Capacitor Charging and Discharging when connected to the ground","text":"When we charge a capacitor using a battery and then remove the battery, the plates of capacitor becomes charged. One holds positive charge and the other one gets equal negative charge. o. k. ? Now if we attach a wire to the positive plate and connect it to the ground , will the electrons from ground climb on the positive plate and make it neutral ? No. But if we connect positive plate to the negative plate then the capacitor will get discharged. Now consider a situation when we connect 4 capacitors A,B,C,D of equal capacitance in series and connect them to a 10 Volt battery. Now the P. D. between positive and negative plate of capacitor A will be (10- 7.5) i.e. 2.5 . For B it will be (7.5 -5 ) i.e. 2.5 , For C it will be (5- 2.5 ) i.e. 2.5, For D it will be (2.5-0) i. e. 2.5. So potential at positive plate of A is 10 and potential at negative plate of D is 0 and the Potential Difference is 10 volts, which is the potential difference of the battery. Now connect the wire joining C and D capacitor to ground and now record the potential difference at A, you will find it 7.5 and at positive plate of D it will be 0, and at negative plate of D it will be -2.5. This happens because negative charge from ground climbs on the positive plate of capacitor D and makes it neutral. My question is why in this case negative charge climbs on this positive plate of D and makes its potential zero ? But such thing does not happen when we connect positive plate of a charged capacitor to the ground."}
{"id":"127722","title":"Pellet of lithium in a vacuum","text":"What would happen to a grain of sand sized pellet of lithium in a vacuum? Because there is no pressure, would it become more like a sticky liquid?"}
{"id":"49962","title":"Is infrared light diffused on walls?","text":"Lamps that produce visible light can light up a whole room because the ceiling and the walls diffuse light. This way, when there is a single lamp in a room, something that is in the shadows is not completely dark. If I set up an infrared lamp in a room, will the whole room be lit by infrared light like with a regular lamp, or will it be absorbed by the walls ? If it is not, will the wavelengths be the same as the ones emitted by the lamp ? Thanks !"}
{"id":"68715","title":"What is the $\\delta$ in fluctuation model","text":"I found this formula : $G-G_0=\\frac{-nkT}2\\delta^2$ on this Site and I'd like to ask what does the $\\delta$ stand for?"}
{"id":"24214","title":"Why must the excitation of closed strings in String Theory be spin-2?","text":"In String Theory it is predicted that as a result of the closed strings we have spin-2 gravitons. 1) How do we know there must be an excitation of spin-2 particles? 2) Why does a spin-2 particle HAVE to be a graviton?"}
{"id":"62265","title":"Firing machine question","text":"Suppose we have a firing machine on a frictionless surface at point $x=0$. It fires a bullet of mass $m$ every $T$ seconds. Each bullet has the same constant velocity $v_0$. There's a body of mass $pm$ ($p$ times larger than the bullet's mass) at point $x=x_0$. I know, according to the law of momentum conservation (inelastic collision) that the velocity of the body as a function of bullets inside of it will be: $v(n)=\\frac{v_0 n}{n+p}$. How can I know the elapsed time $\\Delta t$ from the $n-1$ bullet hit to $n$ bullet hit? Is there any way to solve this without a recursion? What I mean by recursion is this (the answer to the question): $ \\left\\\\{ \\begin{array}{l} a_1=\\frac{x_0}{v_0}\\\\\\ a_n=a_{n-1}+ T \\cdot \\frac{{p-1+N}}{p} \\end{array} \\right. $ This series represent the time when nth bullet hits the object. I do not quite understand why $a_n-a_{n-1}= T \\cdot \\frac{{p-1+N}}{p}$."}
{"id":"62264","title":"How is it possible to equate the internal energy at constant volume with the internal energy of an adiabatic process?","text":"I hope my question makes sense. My problem is that, I have read through numerous textbooks that nC(cons. volume)dT = -PdV when deriving the relationship between T and V for an adiabatic process, though I can't seem to understand why this relation has been made. Surely, at constant volume, work done by the system should be zero? I am lead to believe that, at constant volume, no work is done by the system, but for an adiabatic process, work is only done by the system, not on it. So why is it possible to equate the two internal energies together when they describe two different systems?"}
{"id":"79065","title":"Why uniform ensemble and microcanonical ensemble are the same in thermodynamic limit?","text":"So uniform ensemble is the ensemble of all possible states with energy less than E, and since entropy derived from both are equivalent with each other in the thermodynamic limit, many functions derived from them are the same, however, I am wondering why that is the case since apparently the uniform ensemble contains more states and thus more information than the microcanonical ensemble."}
{"id":"77313","title":"How to name different approaches to relativistic quantum theory","text":"In the introductory chapter of the QFT book by Mark Srednicki the author notes that > [p. 26] So now we have two different approaches to relativistic quantum > theory [...] Which [one of those two] we use is a matter of convenience and > taste. And _quantum field theory_ , the formalism in which position and time > are both labels on operators, is much more convenient and efficient for most > problems. The other approach is described as follows: > [p. 25] Let us discuss the second option [...] to promote time to an > operator. [...] We can use the _proper time_ $\\tau$ of the particle [...] as > the time parameter. The coordinate time $T$ [...] is then promoted to an > operator. [...] Relativistic quantum mechanics can indeed be developed along > these lines, but it is surprisingly complicated to do so. Does this \" _second option_ \" have an own specific name (for distinction from _quantum field theory_ , in Srednicki's sense)?"}
{"id":"106901","title":"Internal potential energy and relative distance of the particle","text":"Today, I read a line in Goldstein Classical mechanics and got confused about one line. > _To satisfy the strong law of action and reaction, $V_{ij}$ can be a > function only of the distance between the particles: $V_{ij} = V_{ij}(|{\\bf > r}_i-{\\bf r}_j|)$._ What confuses me is that I can't see the logic between these two statements. Obviously, I understand strong law of action and reaction and Internal energy. But why the strong law of action and reaction leads to internal energy only depending on relative distances? I prefer to receive mathematical proof (not thorough, but provide a direction so that I can know where I'm going); yet, intuitive illustration is also welcome."}
{"id":"106909","title":"Angles between axes after Lorentz transformations","text":"Consider frame $K_1, K_2, K_3$ such that $K_1$ moves along the $y$ axis of $K_2$ with speed $v$ and $K_2$ moves along the $x$ axis of $K_3$ with speed $u$. Find the angle between the $x$ axis of $K_1$ and that of $K_3$. My question is that I can't see intuitively why the $x$ axis of $K_1$ rotates after Lorentz transformations. And moreover, if one calculates the Lorentz transformations connecting $K_1$ and $K_3$, one may have the following: $$t=\\gamma\\gamma't'+\\gamma\\beta x'+\\gamma\\gamma'\\beta'y'$$ $$x=\\gamma'\\gamma\\beta t'+\\gamma x'+\\gamma\\gamma'\\beta\\beta' y'$$ $$y=\\gamma'\\beta't'+\\gamma'y'$$ where the $'$ coordinates are those of $K_1$ and the others are coordinates of $K_3$. However, if we consider transformations of the origin and a point on the $x$ axis, we may get $\\Delta y=0$, that is, the x axis does not rotate."}
{"id":"4754","title":"What happens in string theory beyond the Hagedorn temperature?","text":"What happens in string theory when the temperature exceeds the Hagedorn temperature? Is that even possible? If yes, what is the nature of the phase transition and the phase beyond that? What happens to spacetime in the new phase?"}
{"id":"93648","title":"Geometric quantization in Kepler problem in hydrogen atom","text":"Why in the usual geometric quantization calculation the dimensions of eigenspaces is wrong (we can see this obstacle for Kepler problem in hydrogen atom). Here is a refference see"}
{"id":"61760","title":"Basic energy calculation for N identical spin system","text":"We have a system that has N identical spins $n_i$, and each spin can be in state 1 or 0. The overall energy for the system is $\\epsilon\\sum_{i=1}^{N}n_i$. My understanding: There is only one distinct way in which we could have 3 spins in state 1, because the spins are identical. Likewise for 4,5 or 6... spins in state 1, there will only ever be one way of choosing. So I think that the multiplicity is $N$, therefore entropy $S(E,N)=k_blog(N)$. I need to calculate the $E(T,N)$, I know $(\\frac{\\partial S}{\\partial E})_{V,N}=1\/T$ but my entropy doesn't have an E in it. How can I calculate the energy and is what Ive got so far correct?"}
{"id":"61768","title":"Energy density of a quantum mechanical ensemble","text":"How do we determine the energy density of a given system? I have seen that the density operator $$\\rho~=~\\frac{\\exp(-\\beta \\hat{H})}{\\text{tr}(\\exp(-\\beta \\hat{H}))}.$$ What does this mean exactly and how does it relate to the pure and unpolarized states of a system? For example, given a system of relativistic spin-1\/2 particles, the completely unpolarized beam density is $\\sigma~=~\\frac{1}{2} \\left| \\uparrow\\right>\\left<\\uparrow \\right|+\\frac{1}{2} \\left| \\downarrow\\right>\\left<\\downarrow \\right|$ or $$\\sigma~=~\\left( \\begin{array}{cc} \\frac{1}{2} & 0 \\\\\\ 0 & \\frac{1}{2} \\end{array} \\right)$$ Are the diagonal elements the probabilities that within the ensemble a particular particle will be found to be in that spin state? If so, how does this affect the energy density of the system? I am not entirely sure what the Hamiltonian would be in this case. But I think that spin-up and spin-down are eigenstates with eigenvalues $ \\lambda_{\\pm}=\\pm \\frac{\\hbar}{2}$, so as a guess, since we are talking about spin-projections, that the energy eigenvalues are going to be proportional to $\\lambda_{\\pm}$ $$\\displaystyle \\rho~=~\\frac{1}{\\text{e}^{-\\frac{\\beta\\hbar}{2}} + \\text{e}^{\\frac{\\beta\\hbar}{2}}} \\left( \\begin{array}{cc} \\text{e}^{\\frac{\\beta\\hbar}{2}} & 0 \\\\\\ 0 & \\text{e}^{-\\frac{\\beta\\hbar}{2}} \\end{array} \\right) $$ where $\\beta$ is the inverse of temperature given by $\\beta=\\frac{1}{kT}$, however I am not sure what the constant of proportionality will be since I am assuming that the Hamiltonian will be proportional to $\\sigma_z$ and not equal to it. Would it be possible to see an example or have an explanation of what is happening? I am referring to Sakurai's _Modern Quantum Mechanics_ and _Quantum Mechanics_ by Bransden and Joachain."}
{"id":"73744","title":"Difference between photo electron spectrum and photoelectron angular distribution","text":"I am trying to learn the Photoelectron velocity map imaging. While I was going through the article \"Chem. Soc. Rev., 2009,38,2169-2177\", it is said that the \"photoelectron spectrum reflects the energy eigenvalues of the parent atomic or molecular system, while the phtoelectron angular distributions reveal the characteristics of the corresponding wave functions\" (in the page 2170 of the said reference). I didn't understand what it is mean by the statement \"photoelectron angular distributions reveal the characteristics of the corresponding wave function\"!. Can anybody please explain me this?"}
{"id":"73747","title":"Work done on a moving particle in electric field","text":"This is one of the \"fast answer\" exercises I've been given to train (should be answerable in around 6-7 minutes). I can only think of a very long-round way to solve this. The question is as following: > Two point charges 2Q and Q are located at the opposite corners of a square > of length l (2Q at the top right corner). The point A is in the lower left > corner and the point B is located halfway the right side of the square. > Determine the work $W_{AB}$ required to move a particle with charge q from A > to B. There are several methods I can think of, though non come to an answer. My best guess is to calculate the magnitude of the force acting on the particle in A, and in B. And then substract these... However to simply calculate the effective force in position B proves itself to be lengthy task. So there must be a simpler mechanic?"}
{"id":"22323","title":"Unknown isotope","text":"Problem description: The most common isotope of a single nucleus stripped of its electrons is accelerated through a potential difference of 1225V and fired horizontally into a B-field directed perpendicularly into the page (flat page lying on desk). The nucleus is observed to travel in a circular path of radius 9.973 cm. Afterward an alpha particle is accelerated through the same potential difference and fired in the same manner into the magnetic field and observed to travel in a circular path with radius 8.980 cm. What element is the unknown nucleus? (For this problem determine the masses of the nuclei--unknown and alpha--by summing the individual masses of the protons and neutrons. Also, include a diagram with your solution showing all aspects of the problem description.) This is what I have so far: ![Valid XHTML](http:\/\/i44.tinypic.com\/ht6zad.jpg). ![Valid XHTML](http:\/\/i40.tinypic.com\/qytrba.jpg). This is the point where I am confused as to how to find the mass and charge of the unknown particle, or where exactly I'm supposed to go from this point to solve the problem. (I have heard from other students that there is some guessing involved (from the periodic table)) Help would be appreciated. Thank you."}
{"id":"22327","title":"How To Stop A Heat Ray Gun?","text":"What material would be best suited to create a shield to protect from the new heat ray gun?"}
{"id":"25678","title":"How can we know, today, that there's something from 100 light-years from here?","text":"In my understanding, to take a picture of something that is 100 light-years from here, our \"camera\" would have to travel 100 years at light speed, take the picture, send to us, and 100 years later we would receive it. So, how do we have pictures of things that are 100 light-years from here?"}
{"id":"130291","title":"Magnet substitution","text":"I wonder if the two magnet configurations below are equivalent. If they are not - what will be the effects on the flux? ![enter image description here](http:\/\/i.stack.imgur.com\/Zqble.png)"}
{"id":"9887","title":"Why do 'dead' batteries work again after exchanging the places of the batteries in an electronic device?","text":"My camera, which is powered by two AA batteries in series, would not power on. I removed the batteries, exchanged their locations, and the device worked again - for another 15 minutes or so. The temperature of the batteries did not change, and only about few minutes passed before I discovered this. Is there a physical explanation for this phenomenon?"}
{"id":"9884","title":"Is relativistic motion equivalent to fluctuating gravitational fields?","text":"The theory of relativity makes very precise predictions about how an object's motion through space-time affects the passage of time for both the object and observers in other frames of reference. I am curious, however, whether this motion is equivalent to changing \/ fluctuating gravitational fields? What if instead of an object moving through space\/time we instead had the same object exposed to shifting gravitational fields, similar to what it would experience if it were actually moving relative to other masses? Can it be shown that the same set of effects would still exist from the object's frame of reference (eg time dialation, length- contraction, etc)?"}
{"id":"9881","title":"In GR, how do particles know how to fall in instead of out of a gravitational well?","text":"The geodesic equation (let's suppose that we're talking about massive particles, so I'll parameterize the path by proper time $\\tau$) $\\frac{d^2 x^\\mu}{d \\tau^2} + \\Gamma^\\mu_{\\rho \\sigma}\\frac{d x^\\rho}{d \\tau} \\frac{d x^\\sigma}{d \\tau}=0$ is invariant under $\\tau \\rightarrow -\\tau$. However, falling particles clearly have a direction, they always fall in instead of out. Formally, how does a falling particle 'know' which way to move in $\\tau$?"}
{"id":"9889","title":"If dark matter is a new type of particle, what does that imply?","text":"My understanding is that dark matter cannot be (or is at least highly unlikely to be) an exotic form of any known particle. On the other hand, articles about particle accelerators seem to say that the Higgs is the last piece missing in the Standard Model jigsaw puzzle. _If_ dark matter is determined to be some form of new particle, what are the certain implications? Might such a discovery \"stand to the side\" of the Standard Model or would it certainly change the foundations? (Forgive my extremely lay understanding and vocabulary -- please feel free to correct my mistakes.)"}
{"id":"3759","title":"Do Category Theory and\/or Quantum Logic add value in physics?","text":"I know they have their adherents, but do more or less esoteric branches of mathematics such as Category Theory and\/or Quantum Logic provide powerful tools for new theory development or are they just occasionally-useful presentational frameworks? More particularly, is it worth investing the time to study these formalisms?"}
{"id":"100937","title":"Calculate gravitational force between a sphere and a uniform rod","text":"I want to calculate the gravitational force between a uniform rod of mass 2 kg, length 1 meter and a uniform sphere of mass 0.2 kg. Distance between the sphere and the rod is 0.1 meter ( The sphere is in the same line as the rod I.e. sphere is one meter from the near end of the rod). I thought of integrating $(G*m*M)\/x^{2}$ from 0.1 to 1.1 meter which does yield the correct result but the solution in the textbook has a different procedure."}
{"id":"55918","title":"Newtonian Gravity on a Riemannian $3$-Manifold","text":"To solve the Poisson equation for the Newton Potential, say $\\phi$, one can use the divergence theorem, such that $$\\int_U \\nabla^2 \\phi \\sqrt{g}~ dV= \\int_{\\partial U} <\\nabla \\phi,n> \\sqrt{g_\\Sigma}~ d\\Sigma=1,$$ where $\\sqrt{g_\\Sigma}$ represents the induced metric on the border. Considering that $\\phi$ is classic it will only depend on the relative position between the source and the point where we want to compute the potential, so, in order to use the divergence theorem, we need to find the set of points that are at the same distance of a given point (where we consider the source of the field to be) and that will be the border, $\\partial U$, of the considered volume that contains the source. Well, if space is the three dimensional Euclidean space then the volume that contains the source is straightforwardly the 3-ball whose center is at the source position and the border $\\partial U$ the corresponding 2-sphere. But if space is a different 3 Riemannian Manifold things are a \"little\" bit harder and finding the set of points that are at the same (geodesic) distance from the source is in general quite difficult or even impossible... But when searching for the geodesic curves on a manifold one can, in some circunstances, instead of searching for the curve that minimizes the distance $$d~=~\\int_a^b [g(\\dot{c}(s),\\dot{c}(s))]^{1\/2} ds,$$ search for the curve that minimizes the energy functional $$E~=~\\int_a^b [g(\\dot{c}(s),\\dot{c}(s))] ds,$$ since in those circunstances the curves that minimize the energy are the same as the curves that minimize the distance. Well, finding the set of points that are at the same energy is much easier...I would then ask: what is the significance of the energy defined above and would it make sense, physically, to say that equipotetial surfaces are not the set of points that at the same distance from the source but the set of points that have the same energy. **PS:** Note that in the Euclidean Space, the set of points that are at the same distance is the same as the set of points that are at the same energy."}
{"id":"32338","title":"What physical forces give rise to the peculiar bond angle of hydrogen peroxide?","text":"In one of the Periodic Table videos, Prof. Poliakoff shows a model of a hydrogen peroxide molecule and claims that the H-O bonds will always be at right angles to each other. I have a rudimentary intuition for how molecules arrange themselves based on mutual attraction and repulsion - I've solved some Foldit puzzles - but the H2O2 defies that kind of common sense explanation. I get that there are forces pushing the hydrogens apart and forces pulling them together, but how does that fragile equilibrium of forces result in an _exact_ right angle?"}
{"id":"32336","title":"Optical Drive Physics","text":"I have been recently wondering how is data stored on and retrieved from optical devices like CDs, DVDs, and Bluray. What makes these different storage types different from each other?"}
{"id":"55917","title":"Is there any way to prove\/disprove we are in a computer simulation?","text":"Is there any way to prove\/disprove we are in a computer simulation in some transcendental reality? Even if we are \"really\" not in a simulation, can we ever prove so? Even if some messages\/evidence flashing \"You are in a simulation by an Overlord\" pops up, can we always dismiss it as a naturalistic hallucination? Can quantum mechanics apply to closed systems? MWI and consistent histories are candidates, but run into the preferred basis\/choice of consistent histories problem. Copenhagen requires a transcendental observer and a Heisenberg cut, doesn't it? If there is a transcendental observer, doesn't that imply there has to be a transcendental realm? So, if our universe is a subsystem in some transcendental realm which can perform, and only perform, Turing computable computations, isn't that equivalent to saying our universe is a computer simulation in some transcendental realm? This doesn't necessarily imply there is some Overlord who is an Intelligent Designer Programmer. It could be an Overlord who is a \"Blind Watchmaker\" transcendental observer. This Overlord might be deist or an occasional interventionist. In Copenhagen, in order for something to exist, it has to be observed, right? That means we need an \"all-seeing eye in the sky\" transcendental Overlord, right? If this Overlord forgets about us, we die, and pop out of existence, right?"}
{"id":"32335","title":"Question about laws of conservation in electrodynamics","text":"One way of deriving laws of energy, impulse and angular momentum of electromagnetic field conservation is following:   1. Introduce two values below: $$ \\mathbf P = \\frac{c}{4 \\pi}[\\mathbf E \\times \\mathbf B], \\quad W = \\frac{1}{8 \\pi}(\\mathbf E^{2} + \\mathbf B^{2}). $$   2. Using Maxwell's equations and time derivative of these values, get the laws. But how to argue an introduce of values from it. 1? Can Noether's theorem help to argue it?"}
{"id":"22589","title":"What is a qualitative explanation for order of values of standard entropies of formation for isomers of hexane?","text":"I have literature values for the standard enthalpy and Gibbs energy changes of formation at 298 K, $\\Delta H_\\text{f}^o$ and $\\Delta G_\\text{f}^o$, _for product in the ideal gas state_. Since $G=H-TS$ and by definition, the reactions are carried out at constant temperature I can solve for $\\Delta S_\\text{f}^o$ via: $$\\Delta S_\\text{f}^o=\\frac{\\Delta H_\\text{f}^o-\\Delta G_\\text{f}^o}{T}$$ I've done this for three hexanes, n-hexane, 2-methylpentane, and 2,3-dimethylbutane. Below are the results: ![hexanes thermochemistry values](http:\/\/i.stack.imgur.com\/Fzrd4.jpg) I guess the sign makes sense to me. It says that all three molecules represent more order than their reactants (C and H2). What is strange to me is the order of the standard entropies of formation. Since these are values in the ideal gas state, entropy effects due to intermolecular forces should not be included, correct? Hence, I would have thought that n-hexane would be the most \"ordered\" of the three molecules. Surely a linear molecule is less likely (requires higher order) than a branched molecule. What have I missed?"}
{"id":"22580","title":"What is 'tilt locking' of a laser cavity?","text":"In another question, jasonh mentions a scheme for locking a laser to a resonant cavity called 'tilt locking': > The signal is used for locking the frequency of a laser to a high finesse > cavity. The lock has to be stable enough for intensity and phase > fluctuations resulting from e.g. acoustic vibrations of the cavity to be > strongly suppressed. The error signal is created via the tilt locking > scheme, i.e. it is the difference of intensities on two halves of a split > photo diode. I'm familiar with Pound-Drever-Hall locking of a laser cavity, but I've never heard of 'tilt locking' before. How does it work?"}
{"id":"100997","title":"Is vapour pressure based on partial pressures or just total pressure on the liquid?","text":"The explanation for the boiling point of water is that at 100C, the vapour pressure becomes greater than atmospheric pressure. But say you had a a jar of water sealed in argon at 1atm, which is larger than the vapour pressure of water, wouldn't some water evaporate? I'm guessing yes, because there are no water molecules in the argon initially, so wouldn't some randomly leave the liquid, making the partial pressure on average non-zero? Another reason I can think of it being based on partial pressure is that an open jar of water in a windless dark, dry room will still eventually evaporate. So why is it that at 100C there is a sudden dramatic change? Isn't the vapour pressure at e.g. 95C more than the partial pressure of water in the atmosphere as well?"}
{"id":"100995","title":"Gauge fixing of an arbitrary field","text":"How to count the number of degrees of freedom of an arbitrary field (vector or tensor)? In other words, what is the mathematical procedure of gauge fixing?"}
{"id":"24981","title":"Why are there more lunar maria on the near side than on the far side?","text":"It is widely known that the side of the moon that faces us has a particularly large number of dark patches, which resemble seas of liquid, hence, \"maria\". It is strange, though, that the far side of the moon has a great lack of these features. Why does the far side of the moon have so few maria? If they maria formed due to impacts, is there some reason that the far side would have less collisions?"}
{"id":"24987","title":"What objects look best in an O-III filter?","text":"I've heard that an O-III (Oxygen III) filter is great for planetary nebulae. Is this true for all planetary nebulae, or just some or most? What other target types are often improved with an O-III filter?"}
{"id":"63853","title":"Ising Hamiltonian for relativistic particles","text":"An Ising system is described by the simple Hamiltonian: $$H = \\sum\\limits_{i} c_{1i} x_{i} + \\sum\\limits_{i,j} c_{2ij} x_i x_j \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,(1)$$ Here the $x_i$ are spins (+1 or -1 in units of $\\hbar$), the $c_{1i}$ are local magnetic fields and the $c_{2ij}$ are (nearest neighbor) interaction constants. So one usually only keeps constants $c_{2ij}$ if $x_i$ and $x_j$ are located close enough in space; after a certain spatial range the $c_{2ij}$ are supposed to become negligible. Note that (1) can be interpreted as two low-order terms of a general Taylor expansion of a general Hamiltonian H($x_1$,$x_2$,…,$x_n$): $$H = c_0 + \\sum\\limits_{i} c_{1i} x_{i} + \\sum\\limits_{i,j} c_{2ij} x_i x_j + \\sum\\limits_{i,j,k} c_{3ijk} x_i x_j x_k + … \\,\\,\\,\\,\\,\\,(2)$$   My questions are the following:   Q1) Does (1) also hold if the particles are moving ? Looking at Feynman’s book on stat mech it seems that (1) is then also valid: the form (1) arises according to Feynman due to magnetic dipole interaction or due to a “Coulomb potential combined with Pauli’s exclusion principle”; nothing in the derivation demands that the particles are static.   Q2) Suppose all particles are on a 2D lattice, except two particles (1 and 2) which are emitted from a source in the middle of the lattice and move in opposite direction through the lattice at relativistic speeds. Can we apply (1) here or do we need a corrected version ? (One way to correct (1) if some particles are moving might be to include higher order terms as in (2).)"}
{"id":"4805","title":"Effects of space mining on Earth's orbit","text":"I was reading a post about space mining, specially lunar mining. I was thinking about what would change in Earth's orbit if we start bringing tons of rocks to it? I mean, in a huge scale. So, would space mining change Earth's orbit in a way to produce any dangerous effects? If the system in question is not a planet\/satellite like Earth\/Moon, would that change anything? Would bringing tons of rocks from Mars be any different than Moon?"}
{"id":"57515","title":"Can you create mass with $E=mc^2$?","text":"If you use the equation $E=mc^2$ could you make matter by dividing the $c^2$? I'm sorry if this is a really stupid sounding question or if it shouldn't be asked here."}
{"id":"14128","title":"When is many-body perturbation theory valid?","text":"I'm calculating expectation values (thermal, time-independent) using many-body perturbation theory, but I'm unsure how to work out what values the parameter I'm expanding the perturbation series in can take. I read that it's when the matrix elements, $\\langle i | H_{pert} | j \\rangle$ where $H_{pert}$ is the perturbative term of the Hamiltonian and $| i \\rangle$ and $| j \\rangle$ are the eigenvectors of the unperturbed Hamiltonian, are much smaller than the energy difference between $i$ and $j$. But I don't really understand what that means, or how it helps me to calculate what values my perturbation parameter can take - Is there a method I can use to figure it out? **Edit:** As requested, to make this concrete, I have a one dimensional fermionic Hubbard model with Hamiltonian \\begin{equation} H= -t \\sum_{\\langle l,m \\rangle} (c^\\dagger_l c_m + h.c.) + U \\sum_l (n_{l\\uparrow} - 1\/2)(n_{l\\downarrow}-1\/2) \\end{equation} I have a special case where I know that $U$ is very small and I want to use many-body perturbation theory to see its effects on correlation functions. I calculate the correlation functions using the functional integral method (i.e. calculating a functional partition function). For this case, how would I go about finding out how small $U$ has to be in order for perturbation theory to be valid? Secondly, (if this should be a separate question, please let me know!) if instead I have a random $U$, dependent on its position in the lattice, \\begin{equation} H= -t \\sum_{\\langle l,m \\rangle} (c^\\dagger_l c_m + h.c.) + \\sum_l U_l (n_{l\\uparrow} - 1\/2)(n_{l\\downarrow}-1\/2) \\end{equation} I can then use a similar functional integral technique, but take an average over the functional partition function (e.g. over a Gaussian distribution). This average removes the $U_l$ and leaves $\\Delta$, the variance of the distribution we've averaged over. In this case, it is $\\Delta$ which the perturbation series is expanded in. How would I go about finding how small $\\Delta$ has to be for the perturbation series to be valid? I don't want an answer that's true for any system, I just want to understand how to go about finding it for any system. So if anyone knows of another system where it is shown how small the expansion term has to be, please let me know. Thanks."}
{"id":"134997","title":"Can a harmonic drive reducer\/strain wave gear be used to gear up instead of down?","text":"I wish to be able to gear up from a rotating bicycle hub with a revolutions per minute range of 150 to 850 to a ratio of 200:1. Can I do this by simply reversing the direction of a harmonic drive reducer? To clarify could I apply torque through the flex spline and use what is normally the torque in put shaft as an output shaft?"}
{"id":"106121","title":"Why does grinding steel create sparks but aluminium doesn't?","text":"Grinding steel produces sparks. Grinding aluminium does not. Why? I found this article about why grinding steel produces sparks but would like to confirm if it is true. I would also like to know why does it behave so on a molecular level if anyone can explain that."}
{"id":"102630","title":"Saturation of an iron core?","text":"Lets assume we have a 1 kg iron core, and a 100 kg iron core. Now saturation is defined as how much that core can absorb the magnetic field, since they are different sizes, don't they saturate at different fields? Lets assume the 1kg core saturates at 1 Tesla, shouldn't the larger core saturate at a higher magnetic field?"}
{"id":"83247","title":"Fluid mechanics Grashoff number","text":"When I derived the Grashof number (for natural convection at a vertical plate) starting with the ratio of buoyancy forces to viscous forces, I realized that I had set the Reynolds number to 1 (ie I had to set velocity*length\/kinematic viscosity = 1) to get the accepted equation. Why is that?"}
{"id":"5739","title":"What is Fermi surface and why is this concept so useful in metals research?","text":"What is Fermi surface and why is this concept so useful in metals research? Particularly, I can somewhat appreciate the Fermi energy idea - the radius of Fermi surface which is a sphere. But is there any quantitative use of more complicated Fermi surfaces?"}
{"id":"54709","title":"What happens to the time it takes for my noodles to cook, and the temperature of the boiling water?","text":"I have a boiling pot of water that I am going to cook my Sponge Bob Noodles in. I am very hungry and want the noodles to cook faster, so I turn up the heat from medium to high.( Note, the water was already boiling at medium).What happens to the time it takes for my noodles to cook, and the temperature of the boiling water?"}
{"id":"133174","title":"Why is it so hard to mix spaghetti with meatballs?","text":"When I cook spaghetti with a sauce that includes meatballs or chopped vegetables, those stay in chunks and won't get uniformly mixed in the pasta. Is there a simple reason for that? Has it been studied? you know some people study piles of sand with various grain sizes?"}
{"id":"71017","title":"How can an object with zero acceleration move?","text":"My physics text has a problem in which it is said that a person moves a block of wood in such a way so that the block moves at a constant velocity. The block, therefore, is in dynamic equilibrium and the vector sum of the forces acting on it is equal to zero: $\\sum{F} = ma = m \\times 0 = 0$ This is where I get confused. If the person is moving the block in such a way so that the sum of the forces acting on it is equal to zero, how can he be moving it at all? I realize that   1. even if there is no force acting on an object it can still have velocity (Newton's First Law)   2. force causes _acceleration_ (a change in velocity--not velocity itself) but for some reason I don't understand how it is possible for the object to move with constant ~~acceleration~~ velocity. Do I have all the pieces? I really feel like I am missing something. **EDIT** The block was at rest to begin with, and the person is moving the block through the air **EDIT: THE ACTUAL PROBLEM** For the sake of clearing up ambiguity, here is the actual problem. (I thought it was wooden blocks at first, sorry) > Two workers must pick up bricks that like on the ground and place them on a > worktable. They each pick up the same number of bricks and put them on the > same height worktable. They lift the bricks so that the bricks travel upward > at a constant velocity. The first gets the job done in one-half the time > that the second takes. Did one of the workers do more work than the other? > If so, which one? Did one of the workers exert more power than the other? If > so, which one? Basically, **_the intent of this question is to ask if the above problem is entirely hypothetical._** My text does not indicate that it is, which is what I find confusing."}
{"id":"15652","title":"What's the relation between virtual photons and electromagnetic potentials?","text":"Given that: 1) virtual photons mediate the electric and magnetic force fields 2) the magnetic field is the curl of the magnetic vector potential 3) the electric field is the negative gradient of the electric scalar potential How do we understand the vector and scalar potentials in terms of virtual photons? Specifically, I mean curl-free magnetic vector and gradient-free electric scalar potential fields, where no electromagnetic waves or physical forces are present. For instance, what's actually happening outside of the solenoid in the Aharonov-Bohm experiment that alters the phase of the passing electron? Are virtual particles involved there? This question has really stumped me because I only have an undergrad level of understanding in physics. Thanks for any help."}
{"id":"78071","title":"Optical Raytracing by using Adiabatic Hamiltonian Method","text":"I'm looking into raytracing a Lüneburg Lens which is a gradient index (GRIN) optical element with a radially varying refractive index: $$ n(\\rho)=n_0\\sqrt{2-\\left(\\frac{\\rho}{R}\\right)^2}, 0<\\rho<R $$ The good paper by Hanhong Gao et.al, \"Design of thin-film photonic metamaterial Lüneburg lens using analytical approach\" describes the theoretical background and applies it to the Lüneburg lens, but does not indicate any code which can be used to carry out the raytracing. I'm considering to try out an implementation in Mathematica, however if there is any comparable example of the application of Hamiltonian optics raytracing this would give me a head start. In the book \"Geometric Optics\" by Antonio Romano has provided an online site of Mathematica notebooks which were offering some implementation of the Hamiltonian formulations but the Link to these notebooks seems to be broken. Any info about already existing implementations on Optical Raytracing with Hamiltonian Method would thus be most welcome!"}
{"id":"78076","title":"Bug in linear thermal expansion, $L_0$ must be $0$","text":"Assume we change the temperature of an object with negligible size in $2^{nd}$ and $3^{rd}$ dimensions from $T_0$ to $T_1$ to $T_2$, with all of them pairwise different. We choose a substance with coefficient $\\alpha \\ne 0$. $\\alpha L_0(T_2 - T_0) = L_2 - L_0 = (L_2 - L_1) + (L_1 - L_0) = \\alpha L_1(T_2 - T_1) + \\alpha L_0(T_1 - T_0)$   $\\Leftrightarrow L_1T_2 - L_1T_1 + L_0T_1 - L_0T_2 = 0$   $\\Leftrightarrow (T_2 - T_1)(L_1 - L_0) = 0$   $\\Leftrightarrow \\alpha L_0(T_2 - T_1)(T_1 - T_0) = 0$   $\\Rightarrow L_0 = 0$ Where is my mistake?"}
{"id":"108494","title":"Franck and Hertz graph function","text":"I need a function whom graph is the following (the Frank and Hertz experiment graph): > ![](http:\/\/i.stack.imgur.com\/WifSY.png) Currently I wrote the following function:               function comptuteValue (value) {              var period = 4.9           , low = 6           ;              function addOrSubstract (x) {                  if (x < low) {                 if (x <= period) {                     return Math.pow(x, 3\/2);                 } else {                     return Math.pow(period + low - x, 3\/2) - x * 1.1;                 }             }                  return addOrSubstract (x - period);         }              return (2 * (Math.floor(value \/ period + 0.1)) + addOrSubstract(value) * 1.1989).toFixed(2);     }      And the graph looks like: > ![](http:\/\/i.stack.imgur.com\/gPPca.png) As you can see there are differences between the two graphs. What's the function of the Franck and Hertz experiment graph? $f(x) = ???$ For example: $f (0) = 0$ $f (4.9) = 12$ $f (5) = 6$ And so on."}
{"id":"78078","title":"Why do nucleons feel a repulsive force when less than 1 fm?","text":"My Modern Physics textbook by Taylor states that when nucleons are less than 1 fm apart, there is a strong repulsive force between them. I am fairly certain that it is not the Pauli Exclusion Principle that the book is talking about. So if this is true, what is the mechanism for such a force?"}
{"id":"45901","title":"What gonna happen to beer on the moon?","text":"Let's assume that we have bottle of beer on the Moon. What would happen to it in sunlight? Some cases:   1. Beer can be dark or lager.   2. It can be contained in glass bottle (transparent or dark) or aluminium can. Will it freeze or vaporize or what?"}
{"id":"15388","title":"Which device amplifies the optical signal in fiber optics?","text":"My teacher ask me one question. How the optical signal or information was transfered in fiber optics communication? Then i explained it and use the word optical amplifier then teacher ask me what is optical amplifier i said which amplifies optical signal then again which device amplifies the optical signal in fibre optics communication? I was dump at that moment i dont have the answer for it and i google it then i read about Doped fiber amplifier(DFA) but i didn't get it so can you give me any suggestion to understand it like any website or explain it."}
{"id":"27699","title":"Orbits of maximally entangled mixed states","text":"It is well known (Please, see for example Geometry of quantum states by Bengtsson and Życzkowski ) that the set of $N-$dimensional density matrices is stratified by the adjoint action of $U(N)$, where each stratum corresponds to orbits with a fixed type of degeneracy structure of the density matrix spectrum. These orbits are flag manifolds, for example the orbit of density matrices of spectrum $(1, 0, 0, 0)$ (pure states) is $\\mathbb{C}P^3$ and the orbit of density matrices of spectrum $(0.5, 0.5, 0, 0)$ (and also $(0.4, 0.4, 0.1, 0.1)$) is the complex Grassmannian $Gr(4,2, \\mathbb{C})$. These spaces - being coadjoint orbits - are known to be Kahler-Einstein. An observation by Ingemar Bengtsson which was stated and proved in the following article, asserts that in the composite system of two $N-$ state quantum systems $\\mathcal{H}^N \\otimes \\mathcal{H}^N$, whose orbit of pure states is $\\mathbb{C}P^{N^2-1}$, the orbit of maximally entangled pure states is $\\mathbb{R}P^{N^2-1}$, which is a minimal Lagrangian submanifold.This observation was also mentioned in the first reference. My questions:   1. Does this result generalize to non-pure states, for example is the orbit of maximally entangled states in $Gr(N^2,2, \\mathbb{C})$ (the orbit of density matrices of the type $(0.5, 0.5, 0, 0, . . .)$ in $\\mathcal{H}^N \\otimes \\mathcal{H}^N$) isomorphic to $Gr(N^2,2, \\mathbb{R})$.   2. Is there a physical interpretation of this result (a question left open by Bengtsson). Update: This is a clarification following Peter Shor's remark. Among the biparticle pure states, the maximally entangled states, have the property that their partial trace with respect to one system is totally mixed (please, see the discussion following equation 22 in the second reference). As a generalization I wish to know the local orbits of the states within a fixed biparticle density matrix orbit whose partial trace has the maximal Von Neumann entropy relative to all other states in the same biparticle orbit. My motivation is that if the local orbits will happen to be the real flag Lagrangian submanifolds of the complex flag manifolds defining the biparticle state orbits (which is the case for the pure states), these manifolds have well known geometries and this can contribute to our understanding of mixed state entanglement."}
{"id":"39323","title":"Complete set and Klein-Gordon equation","text":"In http:\/\/www.physics.ucdavis.edu\/~cheng\/teaching\/230A-s07\/rqm2_rev.pdf, it says that when there is some external potential, the Klein-Gordon equation is altered, and it says the following: > The solution $\\Psi$ can always be expressed as a superposition of free > particle solutions, provided that the latter form a complete set. They from > a complete set only if the negative energy components are retained, so they > cannot be simply discarded. Can anyone explain what this is talking about? First of all, what is a complete set? And why does forming a complete set require negative energy components being retained? Why is the equation (solutions, I guess) need to be a complete set?"}
{"id":"39328","title":"Graph Invariants and Statistical Mechanics","text":"Many intuitive knot invariants including Jones' polynomial are inspired by statistical mechanics. Further profound connections have been explored between knot theory and statistical mechanics. I was looking if similar connection have been found for graph invariants. I am interested to know if any attempts have been made form statistical physics community to solve graph isomorphism problem. Since both knot problem and graph isomorphism are problem about testing structural equivalence there might be connection between them."}
{"id":"91108","title":"Quantum Eraser under Lorentz Boost","text":"Suppose I am conducting the Quantum Eraser experiment. The results of this experiment are easy to understand with the traditional quantum mechanical interpretation of a pair of entangled photons. Specifically, suppose that I am performing the \"eraser\" part of the experiment in which one photon is diagonally polarized so that when the entangled partner passes through the circular polarizers of the double-slit apparatus it cannot be determined through which slit it passed (thus reproducing an interference pattern). Now suppose (It's like supposition _Inception_! Supposings inside supposings inside supposings.) that I Lorentz boost into a reference frame in which the particle passing through the double-slit apparatus is received at the detector (and consequently has its polarization measured) **before** its entangled partner gets diagonally polarized. It appears that there are two possible outcomes: either (a) I disagree with the results recorded by an observer in the commoving frame of the experiment (i.e. I see no interference pattern.) or (b) I must conclude that the **future** polarization of the entangled partner somehow reached back in time and changed the outcome of my measurement. So which is it? Do I get different results or do I need to embrace a further layer of quantum non-locality? This is a tough one for me to understand because of its complementary experiment: Imagine putting the polarizer much farther away than the double slit apparatus so that there is no interference pattern created (because the entangled partner does not get diagonally polarized until **after** the measurement has already occurred). In that case, I could boost into a frame where an interference pattern _should_ occur because the partner gets polarized first. (Or am I missing something? Will there be - or not be - an interference pattern regardless?) If the answer is that I get the same results as the commoving measurement, then why is the commoving frame preferred? Why is it _its_ results that are maintained in all reference frames and not the (expected) results from my boosted frame (which should be just as valid)? Also, I think that this version of the experiment is fundamentally different from the Delayed-Choice Quantum Eraser experiment and Wheeler's thought experiment since both of those experiments are easily explained by representing the particle in a different eigenbasis (namely, as being in a superposition of the interfering and the non-interfering states - instead of being in a \"collapsed\" state of another observable). However, I am also interested to know whether I am wrong in this estimation. Are these experiments basically the same? Can conclusions about or explanations of one of them be generalized to the others? **Edit** : Because of the question in the comments, I've added a little additional explanation for why I think the experiments are basically different. Primarily, the difference comes from the fact that both Wheeler's thought experiment and the Delayed-Choice Quantum Eraser can be explained in such a way that apparent \"cause\" always precedes \"effect.\" Causality is still violated, but at least its not giving us information about the future. For instance, if we didn't know whether or not the entangled partner beam would get polarized a year from now (in the quantum eraser experiment), we could predict whether it does or not **today** by determining whether or not there was an interference pattern at the double-slits. If there exists any boosted frame in which the beam gets polarized first, then that result must carry over to the frames where it gets polarized after. So \"future-telling\" can happen in the relativistic quantum eraser, but not in the other two experiments. For example, in the Wheeler experiment, we polarize the photon and then _after the fact_ decide at random whether to measure its polarization or not. If we measure the polarization, we get no interference; if we don't measure the polarization, we do get interference. Often this is ascribed to a sort of \"predictive\" non-locality, but just as easily, we could say that the polarized state of the photon is actually a superposition of states (namely, interfering and non-interfering) and when we measure its polarization we collapse it into the non-interfering state (and if we don't, we collapse it into interference). Likewise for the delayed-choice quantum eraser: instead of thinking that the beam-splitter in the future decided the interference in the past, you could just as easily conclude that the interference (or not) in the past decided the outcome of the beam-split in the future."}
{"id":"47577","title":"Is it possible to enlarge the output angle of a projector with outside lens?","text":"I have a very small projector, with a small output angle (30 degree). In my current demand I want a bigger output angle(90 degree), so that I can get a bigger image within a given distance. Is it possible to get it by outside lens? The projector: ![enter image description here](http:\/\/i.stack.imgur.com\/fp4XE.jpg) What I expect: ![enter image description here](http:\/\/i.stack.imgur.com\/LOb0F.jpg) If it's possible, how to design the lens? If not, why? What would I lose? (resolution,brightness,distortion etc)"}
{"id":"47571","title":"Path integral with zero energy modes","text":"Consider the field integral for the partition function of a free non- relativistic electron in a condensed matter setting, i.e. $$ Z = ∫D\\bar\\psi D\\psi \\exp\\left(-\\sum_{k,ω} \\bar\\psi_{k,ω} (-iω + \\frac{k^2}{2m} - \\mu) \\psi_{k,ω}\\right) $$ where the action is written in Fourier representation and $\\mu$ denotes the chemical potential. Now, this integral is well known to be the determinant $$ Z = \\det\\left(β\\left(-iω+\\frac{k^2}{2m} - \\mu\\right)\\right) $$ which is equal to the product of all eigenvalues of the quadratic form in brackets. But here is my problem: > How to calculate a quadratic path integral if the quadratic form has some > _eigenvalues_ that are equal to _zero_? If the chemical potential $\\mu$ is positive, then all all momenta with $\\frac{k^2}{2m} = \\mu$ (the Fermi surface) will represent an eigenvalue equal to zero and would force the determinant to become zero. Of course, I could just drop the problematic eigenvalues from the determinant and call it a day, but unfortunately, I would like to understand quantum anomalies a bit better and > zero energy eigenmodes are important for understanding the axial quantum > anomaly for example of the $1+1D$ Schwinger model. Fujikawa's book on quantum anomalies argues that the axial anomaly comes from an asymmetry of zero modes of the Dirac operator, but I am very confused because a zero mode would make the determinant of the Dirac operator and hence the path integral vanish. How to make sense of this?"}
{"id":"127909","title":"Resonance of a tube of air in case of more complex shapes","text":"I've been thinking about posting this question on Music Performance stack, but finally I think it fits more here since I'm interrested in technical details. The subject of resonance of a tube of air is pretty well documented for simple shapes, like cylinders (open of closed), cones, closed rectangular boxes, spheres et cetera. I even found formulas in my high school books. But I am interested in much more complex case. Let's imagine we have a solid tube with a certain shape with holes in it filled with vibrating air of frequency $f_i$. How can we predict the frequency $f_o$ of air that come outside of the tube? For now I can imagine only two ways.   * Decompose the problem into sub-problems, in this case complex shape into much simpler shapes (cylinders, cones, rectangular boxes, spheres etc). But still I have no idea how a whole in the tube would affect the frequency, I suppose we would need to take in count the placement of this whole and its size.   * Run a computer simulation of such system. This would require knowledge how does the frequency propagade through air in such solid tube. To simplify a little bit this complex problem let's forget conditions like air temperature, pressure and humidity. I'm more interrested in second approach, to create a tool that works a little bit like fuild mechanics simulation software, that help for example design heating installations etc cetera, except that this tool would be usefull for acoustic installations, architecture or instrument design. If anyone could share some knowledge usefull to understand this phenomena I would be greatfull. What I have found in school and university books or wikipedia is not enough, or I'm not searching for right words."}
{"id":"127908","title":"Are operators in quantum mechanics linear transformations?","text":"Observables in quantum mechanics correspond to self-adjoint linear operators. If $\\psi$ is an eigenvector of $\\hat A$, then $\\hat A\\psi=\\alpha\\psi$ where $\\alpha$ is the eigenvalue of $\\psi$. Otherwise, if the initial state $\\psi$ is not one of the eigenvectors of $\\hat A$, then the terminal state will be one of $\\hat A$'s orthonormal eigenbasis with probability determined by the projection.   1. Then what is the physical meaning of $\\hat A\\psi$ in this situation?    2. Shouldn't observables be more appropriately described by maps from the Hilbert space to the set of probability distributions on that space?"}
{"id":"127904","title":"Can an elliptical orbit take the same time as a circular obit?","text":"In the picture below you can see two orbits of potential pbjects. The main aspect of the orbits is that they have a collision point at the maximum of the elliptical orbit. My question is, could the object travelling elliptical orbit take the same amount of time as the object taking the circular orbit? The reason behind this question is that I am doing an analysis on the movie called Gravity. The only information provided in the movie is that the debris cloud will collide every 90 minutes (with the Hubble telescope) and that it is going at 50000 miles per hour (which is much faster that the Hubble telescope). Obviously, since they are going at different speeds, they have to be in different orbits. So this is the only possibility I have come up with where they could possibly collide every 90 minutes, or that the orbit time of the elliptical orbit would take 45 minutes, or 22.5 minutes (any factor of 90). ![enter image description here](http:\/\/i.stack.imgur.com\/g99H4.png)"}
{"id":"4395","title":"asymptotic curvature of the universe and correlation with local curvature","text":"There is not-so-rough evidence that at very large scale the universe is flat. However we see everywhere that there are local lumps of matter with positive curvature. So i have several questions regarding this: 1) Does the fact that a manifold with a) asymptotic (space) curvature zero and b) local inhomogeneities with positive (space) curvature imply that there will be regions with negative (space) curvature? 2) a Region of negative (space) curvature implies dark energy in that region? 3) assuming answer to both 1) and 2) are true: does this represent an independent confirmation of dark energy? or there is somehow an geometric relationship relating asymptotic flatness to accelerated expansion (the traditional reason to introduce dark energy in the first place)? EDITED: to reflect distinction between space and space-time curvatures."}
{"id":"19801","title":"Origins of the principle of least time in classical mechanics","text":"Is it possible to derive the principle of least time from the principle of least action in lagrangian or hamiltonian mechanics? Or is Fermat's principle more fundamental than the principle of least action?"}
{"id":"4022","title":"Renormalization and Infinites","text":"Measuring a qubit and ending up with a bit feels a little like tossing out infinities in renormalization. Does neglecting the part of the wave function with a vanishing Hilbert space norm amount to renormalizing of Hilbert space?"}
{"id":"100388","title":"Kolmogorov entropy and noise","text":"There are various ways of calculating the Kolmogorov entropy (KE)of dynamical system. According to Pesin's theorum, it is the summation of the lyapunov exponents; frominformation theoretic concet, it is the supremum of the Shannon entropy defines for all the partitions. I am using the formula given here. > (Jaap C. Schouten, Floris Takens, and Cor M. van den Bleek , \" Maximum- > likelihood estimation of the entropy of an attractor\" Phys. Rev. E 49, 126) The desired signal $x(t)$ whose entropy is to be measured is corrupted with additive white gaussian noise. Say: $$x(t)= ax(t-1) + bx(t-1) + \\text{Chaos_Signal}+\\text{White noise}$$ I want to calculate the KE of $x$. Assuming, the chaotic signal to be logistic map, my question are (Q1) for different values of the parameters $(a_1,b_1),(a_2,b_2),\\ldots$ will the KE measure be same for a particular signal-to-noise ratio (SNR)? (Q2) Will KE decrease with increase of SNR? Intuitively, entropy is the measure of randomness. So, higher the level of noise (low SNR) then greater is the entropy. Again, noise also changes the value of the Lyapunov exponents of the system. I guess, Lyapunov exponent increases. Is there any proof that shows relationship between KE of time series increasing or decreasing with noise?"}
{"id":"130885","title":"Particles Associated With Gravitational Waves","text":"I've been reading about linearized GR and the study of gravitational waves, and an odd thought popped into my head. According to wave-particle duality (admittedly, usually used in quantum mechanics!), particles can act like waves, and waves can act like particles. My logic was that, therefore, there should be a particle associated with a gravitational wave. There is one thing, though, that might interfere with this: gravitational waves are plane waves, not longitudinal waves - which is how we describe, for example, photons. But my original question still stands: Is there a particle associated with a gravitational wave? If not, why not? If so, is there a separate mathematical structure associated with it (aside from the metric for the wave)? FYI: Nope, I'm not talking about gravitons. I know those are completely different and unrelated (and, at the moment, hypothetical)."}
{"id":"86988","title":"Anti-matter as matter going backwards in time? (requesting further clarification upon a previous post)","text":"I understand this question has already been asked here, however, I don't have enough reputation points to place a comment (I suppose that's the reason) on a specific answer to request a reference. A user has answered (I'm not sure I can cite his name here, so I won't): > There is one technical inaccuracy in saying that antimatter moves back in > time (whatever it might mean). In quantum field theory we get positive > energy solutions (usual particles) and negative energy solutions. Negative > energy solutions behave in time as if they were propagating backward in > time. But they are not the antiparticles, they are just the \"negative-energy > particles\". Antiparticles are positive energy solutions, and they are > obtained by acting with charge conjugation operator on the negative-energy > solutions. So, antiparticles move forward in time, as usual particles. and, even though someone else comments > This is interpreting \"going back in time\" differently than anyone else does. I'd like a reference for (or further clarification on) that, since I have already heard that before (also informally). I understand this is a repost but if someone could at least give me some reference before closing it, I'd be thankful. PS: My first intention was to ask how to proceed on this case in \"meta\", but I don't have reputation points to place a question there. **EDIT:** To clarify further my question: I had originally the same question as the OP from the post I quoted. At first, I thought that an explanation of what are anti-particles and the interpretation as particles going back in time could be found in quantum mechanics alone, without more advanced quantum field theory. The argument I knew is that anti-particles are really related to the originally negative-energy solutions, that can be reinterpreted as positive- energy solutions, as long as one invert the sign of time in the unitary time- evolution operator. This would hence lead to the reinterpretation of such states as positive-energy anti-particles _going back in time_ or, at least, with _anomalous time dependency_. The particular answer I quoted, seems to contradict this interpretation, saying that it is a mistake. I'd like a further clarification here. Is this interpretation really wrong? From what is said on Wikipedia, looks like the _Feynman–Stueckelberg interpretation_ relies on the fact that anti-particles are to be understood as particles going backwards in time and then related to the latter by charge conjugation. Where does this interpretation primordially comes from? Where is it primordially needed if their association with the negative energy solutions of Dirac's equation (say) is a mistake? I understand that in quantum field theory one uses complex fields to describe both particles and anti-particles, relating them by charge conjugation. But isn't this done this way to accommodate Feynman–Stueckelberg's interpretation?"}
{"id":"57542","title":"How much time has passed for Voyager I since it left the Earth, 34 years ago?","text":"34 years have passed since Voyager I took off and it's just crossing the solar system, being approximately at 16.4 light-hours away. How much time have passed for itself, though?"}
{"id":"43984","title":"Two masses with interacting forces and an external force","text":"Two masses in 3d space attract each other with a potential relative to the distance between them. There is also an external force on each particle based on the distance from a origin. I want to find the equations of motion. I thought about reducing it to the 1 body problem or use Lagrangian formalism to find the equation of motion, however, the external force kind of ruins that approach. Any ideas on how to approach this?"}
{"id":"135108","title":"Earth-Moon system","text":"While learning about motion of centre of mass, I came across this statement online, > _It is also more accurate to say that the Earth and Moon together revolve > about their common center of mass._ I am not able to understand this. Can anyone please explain this statement?"}
{"id":"135109","title":"clear picture about charge and flux","text":"What is the difference between charge and electric flux?? According to my knowledge electric flux is nothing but charge enclosed by the closed surface."}
{"id":"135105","title":"Oscillation and wave","text":"A box of mass $0.2 \\,\\text{kg}$ is attached to one end of a spring whose other end is fixed to a rigid support. When a mass of $0.8 \\, \\text{kg}$ is placed inside the box, the system performs four oscillations per second and the amplitude falls from $2 \\, \\text{cm}$ to $1\\, \\text{cm}$ in $30 \\, \\text{s}$ . Calculate the force constant, the relaxation time, and the Q-factor."}
{"id":"46669","title":"Do wide-angle videos make the first-person view seem slower than perceived in real life?","text":"I considered posting this on other SE sites such as _Audio-Video Production_ and _Photography_ , but I didn't feel I'd get the definitive, fact-based (rather than experience-based) answer I'm seeking. **Background** I used a GoPro HD Hero (which has a 170-degree view) to record a track day (driving a car) and a snowboarding trip. Playing back the videos after each event, I sensed immediately that they seemed _slow_. However, since clocks and speech seemed to progress at a normal rate, I dismissed my senses, figuring adrenaline and wind must've made me perceive _real life_ faster than it actually was. And there's probably some truth to this adrenaline-and-wind explanation, but watching the videos again today (more than a year later), I sensed the slowness as I always did. I came up with a potential, partial explanation, but still lacking; I was hoping someone might be able to shed some light as to whether this \"slow video\" phenomena is merely psychological, or an expected effect of optics. **Partial Explanation** I think there _may_ be an effect, due to _depth perception_ being altered by wide-angle lenses. Suppose we were to record a skateboarder from the side, _i.e._ so the skateboarder would skate from left to right in our camera's field of view. In this case, because the axis of movement (the skateboarder's \"line\") would be perpendicular† to the axis of distortion (the camera's \"line\", or depth), there would be no distortion of perceived speed: the skateboarder moves relative to an undistorted frame of reference (the ground). However, if we were to mount the camera on the skateboarder's helmet, _i.e._ record in a first-person view, then the axes of movement and distortion would be one and the same. From here though, I'm not sure how to explain why this translates to slower-seeming playback. **Question** In layman's terms, my problem is, \"The ground and the trees pass by me slower,\" when playing back a wide-angle video.   1. Is this psychological, or is there a physical explanation?   2. Is there a mathematical formula that can tell by how much one must speed up playback in order for \"the ground and the trees\" to pass by as they seemed to in real life? **Footnotes** † Technically, I guess \"perpendicular\" would be a curve, _i.e._ staying equidistant from the camera, but for simplicity, suppose the skateboarder is far enough from the lens that we may consider the curve a \"line\"."}
{"id":"82767","title":"How does velocity cause time dilation?","text":"I've been trying to learn about the speed of light and time dilation, but I'm at an impasse. The presented facts say that if I travel around the solar system at 50% the speed of light and then come back to earth I will have experienced less local passage of time than them. I will effectively have traveled to their future. I've also read that gravity causes time dilation too. I understand that space-time is a sort of unified thing and that affecting space affects time. It makes sense then that gravitational forces bending space will also bend time. But velocity? I can't wrap my head around it and I can't find a good explanation for it. People cite orbiting craft and planes as proof of time dilation, since their clocks will go slower than those in the surface (and vice-versa). This certainly explains gravitational time-dilation, but not necessarily velocity. Can't the change in local time passage be caused solely by the gravitational bending of space-time? Trying to find an answer, I came to a very recurring and frustrating example in texts that seek to explain time dilation. A man on a moving train throws a ball forward. Since he's moving with the train and the train is his point of reference, the ball to him only moves at the speed he threw it. But to a woman on the station the ball is moving at the speed of the train plus the speed it was thrown with. To some authors, this seems to open the mind to time-dilation understanding. To me it only explains the relative nature of motion. It says nothing of time. Another example I've found: if person A speeds away from person B very quickly, A's clock will seem to advance slower from B's point of view. How is this time dilation, though? The difference can be explained by the longer time it takes light to get to B, can't it? I realize I can't be right against the fine physicists out there, so I was hoping someone here could enlighten me. Where does the notion that velocity causes time-dilation come from?"}
{"id":"10140","title":"Are many-worlds and the multiverse really the same thing?","text":"Are many-worlds and the multiverse really the same thing? Not too long ago, Susskind and Bousso uploaded the article \"The Multiverse Interpretation of Quantum Mechanics\" with the thesis that the many-worlds interpretation and the multiverse of eternal inflation are one and the same thing. The parallel worlds of one are exactly the same thing as the parallel worlds of the other. First, they claim decoherence can't happen over a complete description of the future light-cone of the measurement. Then, they apply that principle to eternal inflation. Without decoherence, superpositions of nucleating bubbles and metastable vacua can't decohere. According to the anthropic principle, most bubbles have no conscious observers, but an exponentially small minority do. Apply black hole complementarity to causal horizons. Then, somehow, in a way I can't follow, they combine causal diamond worlds into a global multiverse. Then they claim decoherence is reversible. My head is spinning. What are your opinions on this paper?"}
{"id":"40983","title":"Is the total energy of the universe constant?","text":"If total energy is conserved just transformed and never newly created, is there a sum of all energies that is constant? Why is it probably not that easy?"}
{"id":"45208","title":"Would a gauss rifle based on generated magnetic fields have any kickback?","text":"In the case of currently developing Gauss rifles, in which a slug is pulled down a line of electromagnets, facilitated by a micro-controller to achieve great speed in managing the switching of the magnets, does the weapon firing produce any recoil? If so, how would you go about calculating that recoil?"}
{"id":"80978","title":"Moment of Inertia","text":"Let $f(x) = \\frac{1}{L}$ be a probability function, where $L$ is constant. Find the mean and variance. Discuss your results by making a connection to the moment of inertia definition."}
{"id":"81589","title":"Why isn't absolute $0 K$ temperature possible?","text":"So $T$ is defined as $$T = \\left(\\frac{\\partial E}{\\partial S}\\right)$$ and $S$ is defined as $$S = k_B \\ln \\Omega$$ where $\\Omega$ is the number of accessible states of the system for a given $E$. I suppose you could write it as $\\Omega(E)$. Hence, for $T$ to be zero, it must be possible to prepare a system that has an equivalent number of possible $|\\psi\\rangle_E$ for different $E$. It seems to me like you could find some weird system for which this applies. Why can't this be true?"}
{"id":"20097","title":"How Does Dark Matter Form Lumps?","text":"As far as we know, the particles of dark matter can interact with each other only by gravitation. No electromagnetics, no weak force, no strong force. So, let's suppose a local slight concentration of dark matter comes about by chance motions and begins to gravitate. The particles would fall \"inward\" towards the center of the concentration. However, with no interaction to dissipate angular momentum, they would just orbit the center of the concentration and fly right back out to the vicinity of where they started resulting in no increase in density. Random motions would eventually wipe out the slight local concentration and we are left with a uniform distribution again. > How does dark matter form lumps?"}
{"id":"5181","title":"BCS theory, Richardson model and Superconductivity","text":"I'm studying Richardson Model in second quantization. There are many initial points that I don't understand:   * We supposed that an attractive force between 2 electrons exists, due to electron-phonon interaction: an electron generates a phonon that is absorbed by the other electron, and we prove that the resulting potential is attractive. But is this really stronger than the Coulomb repulsive potential? How can one estimate this?   * Why does it neglect the attractive force between electrons with opposite spin? ($\\pm$) They have an opposite magnetic momentum too...   * Why do we only consider the attractions between electrons with opposite spin?   * Why do we suppose $k_f ~R ~\\ll ~1$ , where $k_f$ is (Fermi momentum)\/$\\hbar$, and $R$ the interaction distance?"}
{"id":"134764","title":"If one is talking about antineutron production, what is the charge exchange?","text":"In this question was given the antineutron production in the way $$ p + \\bar{p} \\rightarrow n + \\bar{n}. $$ Internet search gives one source only where the charge exchange is mentioned. They talk about the \"value for the $ \\sigma (p + \\bar{p} \\rightarrow n + \\bar{n}) $\". So I couldn't understand how does charge exchange works. Where comes the missing energy from? What is the full equation? Can one give later primary sources?"}
{"id":"133238","title":"Unsure how resistors are connected? (Series\/Parallel)","text":"![enter image description here](http:\/\/i.stack.imgur.com\/vDRnq.jpg) I am currently attempting this practice question in my tutorial and while I am able to reduce the circuit below to ![enter image description here](http:\/\/i.stack.imgur.com\/ZCFK4.png) I do not know whether the 6\/6\/6\/6 pane is connected in series or parallel, it should not be parallal else R(Total) = 3+3+3 and Vo will be 2\/3 * 12 = 4 The answer im expecting is 4\/3V so my current attempt is wrong. I also doubt it is a Wye-Delta transformation. Can anyone explain how the resistors are connected and why? Much thanks in advance."}
{"id":"29962","title":"How can the Huygens-Fresnel principle be derived from the Maxwell equations?","text":"The Huygens-Fresnel principle states that every point to which a luminous disturbance reaches becomes a source of a spherical wave. I have been trying to understand this considering a infinite screen with a microscopic hole $dS$ on which a plane wave is incident, but I see no obvious way to describe the resulting propagating wave. How can it be derived from the Maxwell equations?"}
{"id":"94830","title":"Accelerometer on a wheel's rim","text":"If I put a 3-axis accelerometer on a vertical wheel's rim which acceleration is going to give to me? Will the gravity interfere my results?"}
{"id":"83721","title":"How are the finite speed of light and the atomic nature of mater related to the end of Moore's law?","text":"In this article from 2007, Moore talks about the end of his Law. Can someone throw more light as to how the finite speed of light and the atomic nature of mater are related to the end of Moore's law?"}
{"id":"127248","title":"Experiment with spring and two light sources emitting light in opposite directions","text":"Two (almost massless) light sources connected with a (almost massless) spring emit light beams in opposite directions. What is the speed at which the spring extends?"}
{"id":"127240","title":"What does the term 'hyperbolic model' mean?","text":"I am reading this non-linear discrete dynamical system paper. The authors mention the term hyperbolic model. What does that mean?"}
{"id":"80725","title":"How can I relate the energy of a gamma particle to electromagnetic radiation power?","text":"I have a distinct feeling that dimensional analysis says these two (let's pick 1 MeV and +30dBm) can not be directly compared. What is the missing link between the two? I know one is energy and one is power. I suppose I'll have to get the energy of +30dBm radiated by an ideal +3dBm antenna at a distance of 2cm impacting an area of $10\\times10\\,mm$. $1.6\\times10^{-13}$ joules for the 1MeV gamma [unknown] for the radio wave (for reference it will be in the low GHz frequency)"}
{"id":"110124","title":"Evaluating the Einstein-Hilbert action","text":"The Einstein-Hilbert action is given by, $$I = \\frac{1}{16\\pi G} \\int_{M} \\mathrm{d}^d x \\, \\sqrt{-g} \\, R \\, \\, + \\, \\, \\frac{1}{8\\pi G}\\int_{\\partial M} \\mathrm{d}^{d-1}x \\, \\sqrt{-h} \\, K$$ including the Gibbons-Hawking-York boundary term. A well-known derivation of the entropy of the Schwarzschild metric requires the evaluation of the boundary term. However, one must introduce a radial regulator $R$, and subtract off a counter-term which is the Gibbons-Hawking action of emtpy space with the same boundary. The final result is finite as $R\\to\\infty$. I am attempting to calculate the full action for a solution for which $R\\neq0$, hence I need to also compute the pure Einstein-Hilbert piece. However, I must introduce a regulator, and the final action is not finite as I take it to infinity. My question: **is there an analogous procedure to \"tame\" the infinity of the pure Einstein-Hilbert piece, perhaps similar to the treatment of the Gibbons-Hawking term?** * * * I actually have to introduce two regulators. For my solution, the Ricci scalar is independent of a particular coordinate, $x_1$, so I get a factor of $x_1$ after integration evaluated at $\\pm\\infty$, so I introduced regulators, such that $L^{-} < x_1 < L^{+}$. As I take $L^{\\pm}\\to \\pm \\infty$, of course it is divergent."}
{"id":"7526","title":"Gravity and the Standard Model","text":"Gravity is ignored in the SM. The proton rest mass is ~0.938 GeV\/$c^2$. LHC protons will move with 7 TeV energy, presumably with a relativistic mass about 7,450 times rest mass. A cosmic ray with the highest energy was detected at about $6\\times 10^{21}$ eV. If it was a proton, its relativistic mass would have been about $6.4\\times 10^{12}$ times rest mass. My question is, at what energy levels would it be necessary to include gravity in the SM? I recognize this isn't an issue for forseeable particle accelerators, but astronomical events like supernova are capable of generating extremely high energy particles."}
{"id":"110120","title":"Fate of light ray","text":"![enter image description here](http:\/\/i.stack.imgur.com\/VQWpx.png) This question has been on my mind since I first learned optics. Where will the light ray go? First, I thought of it taking a U turn. But, I don't think that will happen. Now I have studied a bit about wave optics and I think that the light will go... everywhere? Can someone please explain it in realm of ray optics? I wouldn't mind a bit about wave optics. Just don't go deep down there. Thanks."}
{"id":"19364","title":"Can Black Holes be the Dark Matter?","text":"Seems to fit the definition: interacts with gravity, doesn’t radiate energy (except Hawking Radiation) and could create gravity lensing without absorbing very much of the light. Could 80% of the original matter created in the Big Bang have ended up in lots of black holes thereby giving us the amount of mass to balance the universes gravity? Kent"}
{"id":"96520","title":"Complex Representation of a gauge group and a Chiral Gauge Theory","text":"In this John Preskill et al paper, a statement is made in page 1: > We will refer to a gauge theory with fermions transforming as a **complex > representation of the gauge group** as a **chiral gauge theory** , because > the gauged symmetry is a chiral symmetry, rather than a vector-like symmetry > (such as QCD). But my question is: why does a **Complex Representation of gauge group** imply a **Chiral Gauge Theory**? If fundamental representation of SU(3) is a complex representation (with complex conjugate anti-fundamental Rep), then isn't QCD with fundamental representation of SU(3) a perfect counter example where the gauge symmetry is vector-like, instead of chiral??? ps. See this page, or learn that from Wiki: > In physics, a complex representation is a group representation of a group > (or Lie algebra) on a complex vector space that is neither real nor > pseudoreal. In other words, the group elements are expressed as complex > matrices, and the complex conjugate of a complex representation is a > different, non-equivalent representation. For compact groups, the Frobenius- > Schur indicator can be used to tell whether a representation is real, > complex, or pseudo-real. > > For example, the N-dimensional fundamental representation of SU(N) for N > greater than two is a complex representation whose complex conjugate is > often called the antifundamental representation."}
{"id":"88635","title":"Centripetal issue when considering gravity","text":"Forgive me if my question seems silly, but I am quite baffled. Suppose you have a satellite orbiting a horizontal swing planted into the ground and we want to find the velocity with which the satellite must be moving in order to succeed in taking a fully circular path around the swing (no failure followed by parabolic fall). Indeed if we solve for F = m(v^2)\/R where F can be equated to g, we can also solve for the velocity; all nice and good, and the nature of this method makes some intuitive sense but there is a problem (at least for me): the force that the satellite experiences is an inward force towards the center, and therefore the force described in the aforementioned equation is the centripetal force. However, this conflicts with the method of solving the question, since at the top of the path, the centripetal Force and the gravitational force will be both pointing downwards. Therefore shouldn't the satellite actually be \"doubly\" compelled to accelerate to the ground? But clearly when we for for v with g, we assume that they cancel each other out - hence they point in opposite directions. The idea that the cancel also \"explains\" the intuition that at the top of the circular path, the satellite will be feeling the most \"weightless\" it will throughout its revolution. Additionally from a different reference frame \/ non centrifugal one, it makes sense that at the top, the most force will be on the satellite to turn its path downward as it is after that instant that the satellite sees its downward acceleration. I am, essentially confused with how to handle centripetal\/centrifugal \"fictitious\" forces in this worked example."}
{"id":"88631","title":"Two-Person Problem","text":"The following is a thought experiment I've been stuck on. (No, this isn't homework. I made it up.) Here it is:   **Two-Person Problem**   Let's say we have a superposed particle going through a tube. There are two people on each side of the tube, as well as two detectors, one 1 yard away from the start of the tube, and the other 2 yards away from the start of the tube, each on different sides. Also they are two people on each side. Now, when the particle reaches the first detector, it is collapsed to person A, but still superposed to person B. Let's say that the result was $|0 \\rangle$. My question is, is it possible to be collapsed and superposed at the same time? Even worse, when the particle reaches the second detector, it is possible it could have a result of $|1 \\rangle$? And no, in the experiment they are not allowed to communicate with each other.      What's happening here!"}
{"id":"10971","title":"Relation between classical and quantum information","text":"It is known that for a classical system the amount of information needed to store its state is the same as the amount of information that can be stored in that system. This amount is equal to Shannon's entropy or its classical generalizations. On the other hand the amount of information needed to describe the state of a quantum system is greater than the amount of information that can be stored in such quantum system. My impression is that to describe the state of a quantum system of $N$ qubits one needs $2^N$ bits of information, on the other hand one can only store $N$ bits of information inside such system, which is in my impression equal to von Neumann's entropy of the system. Thus I have two possible explanations of such discrepancy. a) The true entropy of a quantum system is exactly von Neumann's entropy. The difference is only due the fact that a classical storage can only independently operate with digits not less than one bit, thus to store, say 4 variables of 1\/4 bit each a classical system needs 4 bits (at least one bit for each variable) while a quantum system can use parts of a digit independently and thus store all 4 variables in one binary digit. b) The true entropy of a quantum system of $N$ qubits is exactly $2^N$ bits but the amount of **extractable** information is only $N$ bits. Which explanation do you consider more correct or are there possible other explanations?"}
{"id":"10979","title":"Superposition and particle mediated interactions","text":"Suppose particle A interacts with particle B by means of an exchange particle E. Presumably, since this is quantum physics, the exchange particle emission and receipt are just one of many possible paths by which the two particles can interact. Does that mean that A is in a continual state of superposition of having emitted an exchange particle in every direction? And that B then is in a continual state of superposition of having absorbed it (and thus moved if there was a force involved) and not? Somehow that seems like the most sensible option, because the alternative is that A knows where B is and sends it E at some frequency determined by the strength of the fields. The process above would also entangle A and B as they cannot disagree over whether they exchanged E."}
{"id":"64842","title":"The potentiality of the electric field","text":"Could you please explain using just words why electric the field is potentially? I know the proof using integral: $$A = \\int_{12}q\\vec{E}\\cdot{d}\\vec{r} = qQ\\int_{12}\\frac{\\vec{r}\\cdot{d}\\vec{r}}{r^3} = qQ\\int_{r_1}^{r_2}\\frac{dr}{r^2} = qQ\\left(\\frac{1}{r_1} - \\frac{1}{r_2}\\right)$$ (so the work depends only on the coordinates of the beginning and the end). I can hardly understand it and I want to know how to prove it using only words (maybe with simple school math)."}
{"id":"64846","title":"Optical depth in astrophysics","text":"How can be the exponential decay of the intensity $I$ via the optical depth $\\tau$ be derived? $$I(\\tau)=I(0)e^{-\\tau} $$"}
{"id":"100844","title":"Representations of the Poincare group","text":"Which type of states carry the irreducible unitary representations of the Poincare group? Multi-particle states or Single-particle states?"}
{"id":"52891","title":"What if $F\\neq \\frac{dp}{dt}$?","text":"I was thinking of this idea that maybe there are esoteric cases where the force is not given in classical mechanics as $F=dp\/dt$ but as some function of $F=F(p,q,\\dot{p},\\dot{q})$ E.g, something like: $k\\cdot \\frac{dp}{dq}$ with a suitable constant $k$, or any other sort of function of p,q and its time derivatives. Are there any toy models that theorists suggest on this idea?"}
{"id":"119521","title":"Does the Earth's core have tides?","text":"Does the moon produce a measurable tidal-effect on the Earth's (liquid) core? If so, how strong is it? Would it play a factor in other geological effects like earthquakes, volcanoes, etc?"}
{"id":"69749","title":"Cross-section of a wave packet","text":"In text books, wave packets are one-dimensional drawings. But we live in a three-dimensional world. Suppose a wave packet from a HI-cloud (frequency 1420 MHz) is approaching the earth, distance about 10 million miles and it will arrive here in a few seconds. How long is this packet (FWHM)? How large is the cross-section? Is the diameter zero? Or 1\/10 of a wavelength or three time the wavelength? Or even bigger? Any good ideas? (This question is partly inspired by this Phys.SE answer.)"}
{"id":"11590","title":"Partially polarized light in laboratory","text":"Does anybody know a way to create (in laboratory, but with no super- sophisticated equipement) a beam of light with partial linear polarization whose degree of polarization can be adjusted (at least over the range 0-0.5)?"}
{"id":"56914","title":"Ground states of the Hamiltonian of a two spin system","text":"For the spin system shown in this graph (http:\/\/i.stack.imgur.com\/3lg1R.png), the Hamiltonian is $$S^{(1)}_z\\cdot S^{(1)}_z=\\frac{1}{4}\\begin{pmatrix} 1 & 0 &0 &0 \\\\\\ 0&-1 &0 &0 \\\\\\ 0 &0 &-1 &0 \\\\\\ 0 &0 &0 &1 \\end{pmatrix}.$$ The eigenvalues of this Hamiltonian are $-\\frac{1}{4}$ when the eigenstate is $$\\begin{pmatrix} 0 \\\\\\ 1 \\\\\\ 0 \\\\\\ 0 \\end{pmatrix}$$ or $$\\begin{pmatrix} 0 \\\\\\ 0 \\\\\\ 1 \\\\\\ 0 \\end{pmatrix}$$ and $\\frac{1}{4}$ when the eigenstate is $$\\begin{pmatrix} 1 \\\\\\ 0 \\\\\\ 0 \\\\\\ 0 \\end{pmatrix}$$ or $$\\begin{pmatrix} 0 \\\\\\ 0 \\\\\\ 0 \\\\\\ 1 \\end{pmatrix}.$$ Moreover, the expectation value of the Hamiltonian for $$\\begin{pmatrix} 0 \\\\\\ 1 \\\\\\ 0 \\\\\\ 0 \\end{pmatrix}$$ and $$\\begin{pmatrix} 0 \\\\\\ 0 \\\\\\ 1 \\\\\\ 0 \\end{pmatrix}$$ is $-\\frac{1}{4}$ and for $$\\begin{pmatrix} 1 \\\\\\ 0 \\\\\\ 0 \\\\\\ 0 \\end{pmatrix}$$ and $$\\begin{pmatrix} 0 \\\\\\ 0 \\\\\\ 0 \\\\\\ 1 \\end{pmatrix}$$ is $\\frac{1}{4}$. Based on these information can we say that $$\\begin{pmatrix} 0 \\\\\\ 1 \\\\\\ 0 \\\\\\ 0 \\end{pmatrix}$$ and $$\\begin{pmatrix} 0 \\\\\\ 0 \\\\\\ 1 \\\\\\ 0 \\end{pmatrix}$$ are the ground states of the Hamiltonian while $$\\begin{pmatrix} 1 \\\\\\ 0 \\\\\\ 0 \\\\\\ 0 \\end{pmatrix}$$ and $$\\begin{pmatrix} 0 \\\\\\ 0 \\\\\\ 0 \\\\\\ 1 \\end{pmatrix}$$ are the first excited states?"}
{"id":"92453","title":"Without apparatus can we say that the system is measured(decohered) by the environment?","text":"\"Einselection\" and \"tridecompositional uniqueness theorem\" seem to resolve the preferred basis problem. But the premise is that there are three parts in discussion.(system, apparatus, environment) However, it seems that in many situations we don't have the role of apparatus, and thus there are just system and environment. For instance, we usually say that the system is monitored by its environment and thus in a state with determinate classical physical value. In these situations, in which there are just two parts(system and environment) and the Schmidt form of the total system is not unique, can we say that the system is measured(decohered) by the environment?"}
{"id":"17288","title":"Proof of the Hydrostatic weighing equation?","text":"I've been trying to derive this equation for about two hours: ![enter image description here](http:\/\/i.stack.imgur.com\/VoM8x.png) But I can't seem to get it. In my class, we are using it to accurately measure the volume of irregulary-shaped objects, but I completely missed the derivation. I'm not looking to leech off of somebody's proof, but any help with this would be greatly appreciated. Thank you in advance! * * * Just in case anyone would like to see my implementation of Mark's answer: $$ w_{dry} = m_{dry}g $$ $$ m_{dry}g = \\rho_{o}Vg $$ $$ \\rho_{o} = \\frac{m_{dry}}{V} $$ The forces acting on the same object, but submerged fully in a fluid are: $$ w_{wet} = m_{dry}g - B $$ $$ = m_{dry}g - \\rho_{f}gV $$ $$ m_{dry} - m_{wet} = \\rho_{f}V $$ $$ \\rho_{f} = \\frac{m_{dry} - m_{wet}}{V}$$ The specific gravity of the object in relation to the fluid it is submerged in is therefore: $$ S_{g} = \\frac{\\rho_{o}}{\\rho_{f}} $$ $$ = \\frac{\\frac{m_{dry}}{V}}{\\frac{m_{dry} - m_{wet}}{V}} $$ $$ = \\frac{m_{dry}}{m_{dry} - m_{wet}} $$"}
{"id":"9400","title":"Induced EMF of a spinning metal rod","text":"Hey everyone, I'm currently taking an Eletricity and Magnetism course and I'm having dificulty answering this problem. I've gone to my tutor, but sadly, he hasn't taken E+M for a while now and couldn't recall off the top of his head how to go about solving this. We went through my book together, but couldn't find any examples pertaining to this type of problem. (We did find examples of induced EMF's obviously, but making the leap to a spinning rod seems to be too great a task) Anyways, here is the problem I'm working on: A metal rod is spinning on an axis that is located in the middle of the rod, and perpendicular to the rod's length. If the rod is in an external magnetic field that is parallel to the axis the rod is spinning on, how would I go about finding the induced emf? The rod's thickness is negligible but has length L. It's spinning at 1500 rev\/s and the magnetic field is 7.5T Any pointers would be appreciated!"}
{"id":"9407","title":"Why do bar magnets have least attraction in its center?","text":"Why do bar magnets have the least attraction in its center?"}
{"id":"57062","title":"How is parity relevant to determining angular momentum?","text":"# Question: > Particle A, whose spin $\\mathbf{J}$ is less than 2, decays into two > identical spin-1\/2 particles of type B. > > What are the allowed values of the orbital angular momentum $\\mathbf{L}$, > the combined spin $\\mathbf{S} = \\mathbf{s}_1+\\mathbf{s}_2$ (where > $\\mathbf{s}_1$ and $\\mathbf{s}_2$ are the spin vectors of the B particles), > and the total angular momentum $\\mathbf{J}$ of the decay products? Give the > results from two cases: first if particle A has odd parity and then if A has > even parity. # My thoughts: Particle A can be spin-1\/2, spin-1, or spin-3\/2. Since $\\mathbf{J}<2$, we see that there are four possibilities for A: $$ \\begin{align*} &(1): \\;\\;\\mathbf{S}_A = 1\/2 \\quad\\quad \\mathbf{L}_A = 0 \\quad\\Rightarrow\\quad \\mathbf{J} = 1\/2 \\\\\\ &(2):\\;\\;\\mathbf{S}_A = 1\/2 \\quad\\quad \\mathbf{L}_A = 1 \\quad\\Rightarrow\\quad \\mathbf{J} = 3\/2 \\\\\\ &(3):\\;\\;\\mathbf{S}_A = 1 \\quad\\quad\\quad \\mathbf{L}_A = 0 \\quad\\Rightarrow\\quad \\mathbf{J} = 1 \\\\\\ &(4):\\;\\;\\mathbf{S}_A = 3\/2 \\quad\\quad \\mathbf{L}_A = 0 \\quad\\Rightarrow\\quad \\mathbf{J} = 3\/2 \\\\\\ \\end{align*} $$ The total spin of the B particles can be either $1$ or $0$, and each particle can individually have an orbital angular momentum, along with the angular momentum of the particles as a system. With this thought, cases 1,2, and 4 are impossible because the orbital angular momentum of the B particles is an integer, as is their total spin (and therefore their total angular momentum too). Thus we find that only case 3 is allowed, so the total angular momentum of the B particles is $1$ and their orbital angular momentum is $0$ (so $\\mathbf{J}=1$). I have a strong feeling that this is incorrect, because the question asks for the cases when A has odd parity and even parity (what does that even mean?!) so I suspect there should be more than one possible answer. Where did I go wrong?"}
{"id":"64337","title":"How to understand topological order at finite temperature?","text":"I have heard that in 2+1D, there are no topological order in finite temperature. Topological entanglement entropy $\\gamma$ is zero except in zero temperature. However, we still observe some features of topological order in fractional quantum Hall effect, such as fractional statistics, symmetry factionalization. So what is the meaning when people say \"there is no topological order at finite temperature in 2+1D\"? And what is the situation in 3+1D?"}
{"id":"17750","title":"Why do or don't neutrinos have antiparticles?","text":"This was inspired by this question. According to Wikipedia, a Majorana neutrino must be its own antiparticle, while a Dirac neutrino cannot be its own antiparticle. Why is this true?"}
{"id":"27961","title":"Interaction speed between electric charges and magnetic materials","text":"Einstein said that the speed of a matter in universe cannot exceed the speed of light.   1. Is it correct for electric force transmission speed from one electric charge to other one?    2. What is transmission speed between each other?    3. Is the reaction delay of one of them speed of light time when we move another one?   4. Also same question for magnetic force interaction speed with each other.   5. And is there any particule theory about the subject as gravitation can be created by gravitons? Thanks a lot for answers"}
{"id":"60653","title":"Is it possible to \"add cold\" or to \"add heat\" to systems?","text":"> Amanda just poured herself a cup of hot coffee to get her day started. She > took her first sip and nearly burned her tongue. Since she didn't have much > time to sit and wait for it to cool down, she put an ice cube in her coffee > and stirred it with a metal spoon. After a moment, she felt the spoon warm > up, but when she took another sip, the coffee was cooler. She was pretty > sure, the ice added cold to her coffee, and the coffee added heat to her > spoon. Would you agree?"}
{"id":"60657","title":"It would take an elephant, balanced on a pencil, to break through a sheet of graphene the thickness of Cling Film","text":"I'm currently doing some work on a presentation about graphene, and have come across numerous articles which claim something along the lines of > It would take an elephant, balanced on a pencil, to break through a sheet of > graphene the thickness of Saran Wrap \/ Cling Film. My question is, is there any proof \/ calculations to back up this claim? Every article I come across seems to be very similar and I cannot find the original source which might contain proof, or a link the relevant paper\/study **Sources**   * http:\/\/www.dailymail.co.uk\/sciencetech\/article-2045825\/Graphene-strong-sheet-clingfilm-support-elephant.html   * http:\/\/www.scientificamerican.com\/article.cfm?id=balancing-act"}
{"id":"80561","title":"Thermodynamics shell-and-tube heat exchanger","text":"A shell-and-tube heat exchanger is used to cool a flow of steam using a stream of cooling water cooling water. The steam enters the heat exchanger at $0.1 ~bar$ and a quality of $91$% , and exits as a saturated liquid at the same pressure. The mass flow rate of the steam is $2.6 ⋅ 10^5 \\frac{kg}{hour}$. The cooling water enters the heat exchanger at $5 ~MPa$ and $15^ \\circ{C}$ and exits at $38^ \\circ{C}$at the same pressure. Neglect stray heat transfer and neglect kinetic and potential energy effects. As well, assume the system is operating under steady-state conditions. Draw a schematic of the system and locate the state points on a T-v diagram. Determine the mass flow rate of cooling water in kg\/s and determine the rate of heat transfer between the steam and the cooling water in MW. ..... I know by neglecting PE and KE and rearranging we have, $$m_3 = m_1 ⋅ \\frac{h_1-h_2}{h_4-h_3}$$ which i solved for using some table values for enthalpy to get $m_3 = 1634.44 \\dfrac{kg}{s}$ now how do I determine the rate of heat transfer between the steam and the cooling water? and if someone could show me the T-v diagram, would be awesome!"}
{"id":"39299","title":"Why are some materials diamagnetic, others paramagnetic, and others ferromagnetic?","text":"1. Why are some materials diamagnetic, others paramagnetic, and others ferromagnetic?   2. Or, put another way, which of their atomic properties determines which of the three forms of magnetism (if at all) the compound will take?   3. Is paramagnetism to ferromagnetism a continuous spectrum, or is there no grey zone in between?"}
{"id":"62654","title":"How to define the mirror symmetry operator for Kane-Mele model?","text":"Let us take the famous Kane-Mele(KM) model as our starting point. Due to the time-reversal(TR), 2-fold rotational(or 2D space inversion), 3-fold rotational and mirror symmetries of the honeycomb lattice system, we can derive the intrinsic spin-orbit(SO) term. Further more, if we apply a spatially uniform electric field perpendicular to the 2D lattice(now the mirror symmetry is broken), a (extra) Rashba-type SO term will emerge. To present my question more clearly, I will first give a more detailed description of the above symmetry operations in both first- and second- quantization formalism. In the follows, a 3D Cartesian coordinate has been set up where the 2D lattice lies in the $xoy$ plane. **First-quantization language:** (1) TR symmetry operator $\\Theta:$ $\\Theta\\phi(x,y,z)\\equiv \\phi^*(x,y,z)$, hereafter, $\\phi(x,y,z)$ represents an **arbitrary wave function** for single electron. (2) 2-fold rotational operator $R_2:$ $R_2\\phi(x,y,z)\\equiv \\phi(-x,-y,z)$, where we choose the middle point of the nearest-neighbour bond as the origin point $o$ of the coordinate. (3) 3-fold rotational operator $R_3:$ $R_3\\phi(\\vec{r} )\\equiv \\phi(A\\vec{r})$, where $A=\\begin{pmatrix} \\cos\\frac{2\\pi}{3}& -\\sin\\frac{2\\pi}{3}& 0\\\\\\ \\sin\\frac{2\\pi}{3}& \\cos\\frac{2\\pi}{3} & 0\\\\\\ 0& 0 & 1 \\end{pmatrix}$ $\\vec{r}=(x,y,z)$ and we choose the lattice site as the origin point $o$ of the coordinate. (4) **Mirror symmetry operator $\\Pi:$ $\\Pi\\phi(x,y,z)\\equiv \\phi(x,y,-z)$.** **Second-quantization language:** (1) TR symmetry operator $T:$ $TC_{i\\uparrow}T^{-1}=C_{i\\downarrow}, TC_{i\\downarrow}T^{-1}=-C_{i\\uparrow}$, where $C=a,b$ are the annihilation operators referred to the two sublattices of graphene. (2) 2-fold rotational operator $P_2:$ $P_2a(x,y)P_2^{-1}\\equiv b(-x,-y), P_2b(x,y)P_2^{-1}\\equiv a(-x,-y)$, $P_2$ is unitary and we choose the middle point of the nearest-neighbour bond as the origin point $o$ of the coordinate. (3) 3-fold rotational operator $P_3:$ $P_3C(\\vec{x})P_3^{-1}\\equiv C(A\\vec{x}), \\vec{x}=(x,y),C=a,b$, where $A=\\begin{pmatrix} \\cos\\frac{2\\pi}{3}& -\\sin\\frac{2\\pi}{3}\\\\\\ \\sin\\frac{2\\pi}{3}& \\cos\\frac{2\\pi}{3} \\\\\\ \\end{pmatrix}$ and $P_3$ is unitary, we choose the lattice site as the origin point $o$ of the coordinate. (4) **Mirror symmetry operator $M:$ ????** as you see, that's what I want to ask: how to define the mirror symmetry operator $M$ **in terms of second-quantization language** for this 2D lattice system? Or maybe there is no well defined $M$ for this model? Thanks in advance. **Remarks:** (1) A direct way to verify your definition of $M$ being correct or not is as follows: The intrinsic SO term $i\\lambda\\sum_{\\ll ij \\gg }v_{ij}C_i^\\dagger\\sigma_zC_j$ should be invariant under $M$ while the Rashba term $i\\lambda_R\\sum_{<ij>}C_i^\\dagger \\left ( \\mathbf{\\sigma}\\times\\mathbf{p}_{ij}\\right )_zC_j$ will not be. (2)Here **mirror** operation is just reflection in one of the three spatial axes (i.e. $(x,y,z)\\rightarrow (x,y,-z)$), not the “parity” operation in the context of \"CPT symmetry\" in field theory."}
{"id":"20467","title":"Lightning Power?","text":"What is the power source of a lightning? It doesn't consume fuel, nor an atomic fission\/fusion, but it discharges a large amount of energy. Is it feasible to create an artificial lightning using the same method of the natural lightning? If it is, can we create a power plant from the same source?"}
{"id":"78271","title":"Taylor expansion in classical 1D harmonic chain (classical field theory)","text":"I'm struggling with something apparently really easy but not entirely straightforward. In \"Condensed Matter Field Theory\" of Altland and Simons the classical 1D harmonic chain is treated as introductory example to classical field theory. Consider an harmonic chain of N particles on a periodic 1D lattice (N+1=1). The position of the I-th particle is $R_I=Ia$. The Lagrangian reads $L=\\sum_{I=1}^N[\\frac{m}{2}\\dot{R_{I}^2}-\\frac{k_s}{2}(R_{I+1}-R_{I}-a)^2]$ Now he parametrizes $R(t)=R_I+\\phi(t)$ where the deviation from equilibrium is small, so that $L=\\sum_{I=1}^N[\\frac{m}{2}\\dot{\\phi_{I}^2}-\\frac{k_s}{2}(\\phi_{I+1}-\\phi_{I})^2]$ Now he goes to the continuum limit, that is, he introduces continuum degrees of freedom $\\phi(x)$ and applies a Taylor expansion such that: $\\phi_I\\rightarrow a^{1\/2}\\phi(x)\\left.\\right|_{x=Ia} \\hspace{30pt} \\phi_{I+1}-\\phi_I\\rightarrow a^{3\/2}\\partial_x\\phi(x)\\left.\\right|_{x=Ia} $ I can't figure out how this substitution and Taylor expansion works. If you look to the first, you should get for the second one: $\\phi_{I+1} - \\phi_I\\rightarrow a^{1\/2}\\left[\\phi(x+a)-\\phi(x)\\right]\\left.\\right|_{x=Ia}$ How is this now equal to the derivative, should i take a limit for $a\\rightarrow 0$? And where does the prefactor $a^{3\/2}$ come from? Cheers"}
{"id":"68909","title":"Neutrinos: how can they carry information about universe?","text":"I know that neutrinos are particles with a very small mass and no electric charge. They infrequently interact with matter and so they can give us information about the \"old\" universe. But how can they do it?"}
{"id":"109526","title":"How to calculate force generated from a magnet coil to a magnet?","text":"I go to a website http:\/\/easycalculation.com\/engineering\/electrical\/solenoid- force.php They have the calculator for the force between a solenoid coil and another piece of ferromagnetic material. What if I have a strong magnet (max pull force is 50lb) instead of ferromagnetic material? How does the force change?"}
{"id":"63305","title":"Energy conservation in nuclear reactions and radiactive decay","text":"Reading \"Fundamentals of Nuclear Physics\" by Atam P. Arya, I understand that in a nuclear reaction, let say $x+X \\to y+Y$ meaning that \"when a particle $x$ strikes a target nucleus $X$, the outcome of the nuclear reaction is a recoil nucleus $Y$ and a particle $y$. In many cases more than one type of particle may be given out.\" Now, Arya applies the energy conservation to the nuclear reaction, writing $$E_i=E_f$$ where $$E_i=K_x+m_xc^2+K_X+M_Xc^2$$ and $$E_f=K_Y+M_Yc^2+K_y+m_yc^2$$ being $K_x, K_X, K_y, K_y$ the kinetic energies of the particle $x$, the target nucleus (parent), the recoil nucleus (daughter) and the particle $y$, respectively. And $M$ & $m$ its respective masses. Rewriting energy conservation, he defines a quantity $Q$ (the disintegration energy): $$Q:=(K_Y+K_y)-(K_X+K_x)=(M_X+m_x)c^2-(M_Y+m_y)c^2 \\qquad (1)$$ Now, my **question** is whether or not the equation (1) (and its interpretation) can be applied to alpha and beta decay (where there are no particle colliding with the nucleus)."}
{"id":"37946","title":"Why is my speed imaginary?","text":"A particle is released from a dense atmosphere free planet of radius r at a distance R (R >> r ). find the speed on impact. $$F = \\frac{GmM}{(r + R)^2} = m \\frac{dv}{dt} = mv\\frac{dv}{dR}$$ $$GM\\int^{R_f}_{R_0}(r + R)^{-2} dr = \\int^{V_f}_{V_o}vdv$$ But $R_f$ = 0 when the object strikes the planet $$-GM[r^{-1} - (r +R_0)^{-1}] = 1\/2[v_f^2 - v_0^2]$$ $$-2GM[\\frac{R_0}{r(r + R_0)}] + V_0^2 = V_f^2$$ why is it imaginary?"}
{"id":"37949","title":"Stroboscope-and-telegraph problem","text":"**Narrative:** Consider, in a suitably flat region, two straight lines which don't necessarily intersect. Let vector $\\mathbf{x}$ point along one line, and vector $\\mathbf{y}$ point along the other. Let $X$ and $Y$ be the two points closest to each other on these two lines, or otherwise let $P$ be their intersection point. The real number $\\phi := \\text{ArcCos}\\bigl( \\frac{\\mathbf{x} \\, \\cdot \\, \\mathbf{y}}{|\\mathbf{x}| \\, |\\mathbf{y}|}\\bigr)$ is supposed to be known as part of the description of how the two lines are related to each other. Now consider a stroboscope \"$B_x$\" moving uniformely along one of these lines (in direction of vector $\\mathbf{x}$, passing point $X$), and a telegraph \"$A_y$\" moving uniformely along one of these lines (in direction of vector $\\mathbf{y}$, passing point $Y$). The case of intersecting lines may be dealt with later on. Stroboscope $B_x$ is supposed to _blink_ ceaselessly (therefore its name: \"$B_x$\") with equal durations $\\tau_B$ from any one \" _blink_ \" ($b_j$) to the next one ($b_{j+1}$); and telegraph $A_y$ is supposed to re-shuffle its \" _arms_ \" all the time (... cmp. \"$A_y$\") with equal durations $\\tau_A$ from settling into any one signal state ($a_k$) to settling to the next ($a_{k+1}$). The real \"ratio\" number $r := \\tau_B \/ \\tau_A$ is supposed to be known, too. Let $a_0$ be the state of telegraph $A_y$ while passing point $Y$, and $b_0$ be the state of stroboscope $B_x$ while passing point $X$. Let integer $p_{BA}$ be given as the number of \" _blinks_ \" stated by stroboscope $B_x$ after having passed point $X$ (i.e. after having been in state $b_0$) and before observing telegraph $A_y$ passing point $Y$ (i.e. before observing that telegraph $A_y$ had been in state $a_0$). If the described order of observations is inverted (i.e. if stroboscope $B_x$ first observed that telegraph $A_y$ had passed point $Y$ and only later passed point $X$) then integer $p_{BA}$ is obtained as negative value (or as value $0$). Finally suppose that stroboscope $B_x$ is (partially) surrounded by a shiny case such that telegraph $A_y$ can see its (own) reflection and recognize its states in the mirror image; and suppose that telegraph $A_y$ is (perhaps rather diffusely) reflective as well such that stroboscope $B_x$ would be able to see reflections of its \" _blinks_ \". Now: For any particular signal state $a_k$ telegraph $A_y$ can count how many subsequent times it went on to re-shuffle its arms before recognizing the reflection (from $B_x$) corresponding to state $a_k$; lets call this number $\\alpha_{Ak}$. Along with that, again for any particular signal state $a_k$, telegraph $A_y$ can count how many subsequent times it saw stroboscope $B_x$ _blink_ subsequently before recognizing the reflection (from $B_x$) corresponding to state $a_k$; lets call this number $\\alpha_{Bk}$. Correspondingly, stroboscope $B_x$ may determine the numbers $\\beta_{Bj}$ and $\\beta_{Aj}$ ...   (No!, too bad: stroboscope $B_x$ may not readily determine these numbers because it cannot easily tell the reflection of any one _blink_ from that of any other. But this may lead to yet another question ... In order to fix the narrative let's here just assume that \"stroboscope $B_x$\" is in fact a \"telegraph\" as well, with distinct states $b_j$ whose reflections from $A_y$ can be individually recognized, too.) **Question (1):** Do the numbers $\\alpha_{Ak}$ have a minimum, $\\alpha_{\\text{min}}$ ?,   in particular, if the two lines don't intersect.   (And similarly: Do the numbers $\\beta_{Bj}$ have a minimum, $\\beta_{\\text{min}}$ ?) And if so: **Question (2):** Can the \"ratio\" number $r$ be calculated from the (given) numbers $\\phi$, $p$, $\\alpha_{Ak}$, $\\alpha_{Bk} $, $\\beta_{Bj}$ and $\\beta_{Aj}$ ?, or perhaps at least approximated for sufficiently large numbers $\\alpha_{\\text{min}}$ and\/or $\\beta_{\\text{min}}$ ?"}
{"id":"34417","title":"Experimental observation of elementary particles?","text":"I posted a similar yet totally unrelated question recently, and got really satisfying responses to it. Thus, on the same theme... **How have we come to realize the existence of elementary particles in general? What evidence have we accumulated over the years that proves their existence?** Once again, please note that I'm not _doubting_ the existence of elementary particles, but am just curious as to how we have found out about them."}
{"id":"79634","title":"Does diffraction occur before interference?","text":"In the double slit experiment, light passes through two slits and the waves interfere and form an interference pattern. A single slit is required for diffraction. So, I was thinking about whether the light in the interference setup undergoes diffraction first at the two slits? I think that's the case, because light waves spread due to diffraction at the two slits. The two waves interfere to form the interference pattern. Is my thinking correct?"}
{"id":"92411","title":"What exactly is regularization in QFT?","text":"**The question.** Does there exist a mathematicaly precise, commonly accepted definition of the term \"regularization procedure\" in perturbative quantum field theory? If so, what is it? **Motivation and background.** As pointed out by user drake in his nice answer to my previous question Regulator-scheme-independence in QFT , it is often said that in a renormalizable quantum field theory, results for physical quantities (such as scattering amplitudes) when written in terms of other physical quantities (like physical masses, physical couplings, etc.) _do not depend on the regularization procedure_ one chooses to use. In fact, user drake takes this desirable property as part of the definition of the term \"renormalizable.\" In my mind, in order for such a statement or definition to be meaningful and useful, it helps to have a precise, mathematically explicit notion of what constitutes a regularization procedure. That way, when one wants to compute something physical, one can use any procedure one wishes provided it satisfies some general properties. **My current understanding in a (small) nutshell.** When we perturbatively compute, say, correlation functions for some theory pre-regularization\/renormalization, we obtain formal power series' in the bare parameters that characterize the theory. Such power series' contain expressions for loop integrals that generically diverge, often due to high- momentum (UV) effects, so we \"regularize\", namely we implement some procedure by which these integrals are made to depend on some parameter, call it $\\Lambda$, and are rendered finite provided $\\Lambda$ doesn't take on a certain limiting value $\\Lambda_*$ (that could be $\\infty$) corresponding to the physical regime (like the UV) that led to the original divergence. We then renormalize and find (in renormalizable theories) that $\\Lambda$ drops out of physical results. **What sort of an answer am I looking for?** I am looking for something like this. > A _regularization procedure_ is a prescription by which all divergent > integrals encountered in perturbation theory are made to depend on a > parameter $\\Lambda$ and which satisfies the following properties: (1) All > divergent integrals are rendered finite for all but a certain value of > $\\Lambda$. (2)... I know that there are other properties, but I don't know what constitutes a sufficiently complete list of such properties such that if you were to show me some procedure, I could say \"oh yes, that counts as a valid regularization procedure, good job!\" Surely, for example, the putative regularization procedure cannot be too destructive; I cannot, for instance, simply replace every loop integral with $3\\Lambda$ and call it a day because that would completely destroy all information about how many loops the corresponding diagrams contained. How much of the \"formal structure\" of the integrals does the procedure need to preserve? As far as I can tell, there is no discussion of this in any of the standard QFT texts who simply adopt tried-and-true procedures like a hard UV cutoff, dim reg, Pauli-Villars, etc. without commenting on general conditions sufficient to guarantee that these particular procedures count as \"good\" ones. There is, of course, a lot of discussion of whether certain regulators preserve certain symmetries, but that's a distinct issue. **Edit. (January 8, 2014)** Discussions with fellow graduate students have led me to believe that the appropriate definition proceeds by appealing to the idea of effective field theory. In particular, if we view our theory as an effective low-energy description of some more complete theory that works at higher energy scales, then imposing a high momentum cutoff has a conceptually privileged position among regulators; it is the natural way of encoding the idea that the putative theory only works below a certain scale. This can then be used to define a regularization procedure that, in some sense, reproduces the same structure encoded in regularizing with a cutoff. Unfortunately, I'm still not entirely sure if this is the correct way to think about this, and I'm also not sure how to formalize the notion of preserving the structure the comes out of cutoff regularization. My inclination is that the most important structure to preserve is the singular behavior of regularized integrals as the cutoff $\\Lambda$ is taken to infinity."}
{"id":"67680","title":"What do the components of light velocity look like in polar coordinates?","text":"The Schwarzschild solution makes use of polar coordinates, and I'm wondering how the different components of velocity of light change with the position. Might I get some examples of light velocity vectors (in units of $c$ such that $\\langle 1,0,0,0\\rangle$ is the velocity vector for a light ray moving radially away from the black hole at the speed of light)?"}
{"id":"61683","title":"Bremsstrahlung: why is electron slowed\/stopped by the positive nucleus?","text":"I can't understand why the electron is slowed\/stopped by the nucleus. The electron is a negative charge and the nucleus is positive... they should attract each other..."}
{"id":"61684","title":"What is mathematical definition of a strong gravity?","text":"Mathematical definition of a weak gravity is simple $g=\\frac{GM}{r^2}$ but what is mathematical definition of a strong gravity? (blackhole-like or close to a blackhole-like object)"}
{"id":"100075","title":"How do wind turbines cause lightning?","text":"I saw this paper about Lightning discharges produced by wind turbines More lightning strikes wind turbines than comparable non-moving structures. What would be the physical cause of this? Does it depend upon the motion of the wind turbine blades or is it because these are connected to the electric power grid?"}
{"id":"77360","title":"How to get Gell-Mann matrices?","text":"How to get Gell-Mann matrices $f_{i}$ (more or less strictly)? What are the requirements for getting them, excluding $||f_{i}|| = 1$, commutational law $[f_{i}, f_{j}] = if_{ijk}f_{k}$ and hermitian nature? Is that enough?"}
{"id":"107263","title":"How an accelerated object sees another accelerated body in special relativity?","text":"Assume two objects are moving with a constant acceleration $a_1$ and $a_2$, which are the measured accelerations by respective object (or constant force being applied to each of the objects). My problem is how one object sees the motion of the other one. I know that the accelerations are transformed according to the relation (Why proper acceleration is $du\/dt$ and not $du\/d\\tau$?), $$ a = \\frac{a'}{\\gamma^3} $$ where $a'$ is the measured acceleration and $a$ is the acceleration in a inertial frame."}
{"id":"134615","title":"Modeling magnetic field shielding","text":"I am wondering what is the best way to model magnetic and electric field shielding from a magnetic dipole in the near field? For example, let's say you have a coil of current carrying wire in the x-y plane which creates an AC magnetic field along the z-axis. I am interested in a model in which I could put in material parameters (such as $\\sigma$, $\\epsilon$, $\\mu$) for a shielding plate located distance $r$ from the source. I tried using section 5.4 of an online electromagnetic book which talks about reflection, and transmission of of EM waves. I would estimate wave impedance as $Z=\\omega\\mu_0r$ which is the equation for near field magnetic source. I determine the intrinsic impedance of the shielding material as $\\eta=\\sqrt {\\frac {j\\omega\\mu} {\\sigma + j\\omega\\epsilon}}$. I want to try good conductors as well as ferrite so I'm not making any assumptions such as good conductor or loss-less medium. I implemented this but the result was not what I expected. I would expect highest energy reflection with the ferrite because it has low reluctance to magnetic fields and would guide the flux through it and return to source. However, a good conductor (e.g aluminum) had almost 100% energy reflection, but I was expecting the shielding mechanism of conductors would be all energy lost as absorption from eddy currents. Let me know your thoughts of whether this is a legit model; I can give more info about parameters\/equations used if that's helpful."}
{"id":"66083","title":"Newton's First Law of Motion; Empirical Aspects","text":"Newton's first law states that in an inertial frame, a body at rest continues to be at rest, and a body in constant rectilinear motion continues its motion, unless an external force is applied upon the bodies. My question is how far have these assertions been tested experimentally? More precisely, is there any need to test this statement, or is it taken as a \"definition\" of sorts? I assume the body at rest is a trivial case. On the other hand the case of the body in constant motion would be considerably harder to test, since no external force on a body is an idealization. But I am still curious whether there have been any attempts to test the case of the moving body."}
{"id":"23280","title":"Why does elastic energy only depend on first derivatives?","text":"Say we have an elastic material that is deformed with displacement function $u : \\mathbb{R}^n \\rightarrow \\mathbb{R}^n$. It is reasonable to assume that the energy required for such a displacement is an integral over local properties of $u$: $$E = \\int_S F(u_i, u_{i,j}, u_{i,jk}, \\cdots)$$ Where $u_{i,jk\\cdots z} = \\frac{\\partial^m u_i}{\\partial x_j \\partial x_k\\cdots \\partial x_z}$. It is clear that $F$ cannot depend on $u_i$, because the energy is translation invariant. As it turns out, for small deformations $F$ depends on all $u_{i,j} = \\frac{\\partial u_i}{\\partial x_j}$ and does not depend on higher derivatives. Furthermore, $F$ is a quadratic in $u_{i,j}$, namely $F = \\sum T_{ijkl} u_{i,j} u_{k,l}$ where $T_{ijkl}$ depends on the material. Is there a reason for this, or is this just an empirical fact? Is there a set of intuitive assumptions that lead to this form of $F$?"}
{"id":"23286","title":"What's the mathematical background to the representation for Gaussian beams?","text":"# Background A general optical system (not necessarily having an axis of rotational symmetry) can be represented, for small deviations from a base ray, by the matrix transfer equation, $$ \\left[\\begin{array}{c} n'L'\\\\\\ n'M'\\\\\\ x'\\\\\\ y' \\end{array}\\right] = \\left[\\begin{array}{cc} B & -A\\\\\\ -D & C \\end{array}\\right] \\left[\\begin{array}{c} nL\\\\\\ nM\\\\\\ x\\\\\\ y \\end{array}\\right] = D(g) \\left[\\begin{array}{c} nL\\\\\\ nM\\\\\\ x\\\\\\ y \\end{array}\\right] $$ where $L,M,N$ are the direction cosines of a ray, $n$ is refractive index, $x,y$ are coords on a reference plane, $A,B,C,D$ are $2\\times 2$ matrices and a prime (') denotes the image space and unprimed quantities are in the object space. The sign convention for the matrix sub- blocks is from ``The Ray and Wave Theory of Lenses'' by A. Walther. The $4\\times 4$ transfer matrices $D(g)$ are the defining represention of the symplectic group $Sp(4,R)$ carried on a 4-d vector space $V_{4}$ . They obey, $$ D(g)^{T} \\left[\\begin{array}{cc} 0 & 1\\\\\\ -1 & 0 \\end{array}\\right] D(g) = \\left[\\begin{array}{cc} 0 & 1\\\\\\ -1 & 0 \\end{array}\\right] $$ where the $0$ is $2\\times 2$ and $1$ is the unit $2\\times 2$ matrix. Thus one can say that geometric optics is the study of a finite dimensional representation of $Sp(4,R)$ and each optical system is a $g\\in Sp(4,R)$. Furthermore, two optical systems $g$ followed by $h$ are equivalent to the single optical system $hg$. It turns out (see Guillemin and Sternberg ``Symplectic Techniques in Physics''), provided the optical system is lossless, that diffractive optics in the Fresnel approximation is the study of an infinite-dimensional projective representation of $Sp(4,R)$ carried on a Hilbert space. In this case, the complex amplitude $u(x,y)$ on a reference plane is a vector $|u\\rangle$ and the transfer equation between object and images reference planes is now, $$ |u'\\rangle=K(g)|u\\rangle $$ where $K(g)$ is a kernel which is actually a projective representation of $Sp(4,R)$ carried on the Hilbert space. Note that, in a book on diffraction, like Walther's, the transfer equation would be appear as, $$ u'(x',y')=\\int K(x',y'|x,y)u(x,y)dxdy $$ and the kernel $K(x'y'|x,y)$ depends on the matrices $A,B,C,D$ via a function called the eikonal. The details are not important for this question. # Question If one studies the diffraction of a Gaussian beam, $$ u(x,y)=\\exp{\\left\\\\{\\frac{i}{2} \\left[\\begin{array}{cc} x & y \\end{array}\\right]\\left(\\frac{q}{n}\\right)^{-1} \\left[ \\begin{array}{c} x \\\\\\ y \\end{array} \\right]\\right\\\\}} $$ by an optical system in the Fresnel approximation, then the output beam is also Gaussian and the relation between the $2\\times 2$ input and output complex beam parameter matrices is: $$ \\frac{q'}{n'}=\\left(C\\frac{q}{n}-D\\right)\\left(B-A\\frac{q}{n}\\right)^{-1} $$ This is another representation (realization) of $Sp(4,R)$ carried on the space of symmetric $2\\times 2$ matrices like $q\/n$. It's some sort of conformal transformation because, for cylindrical lenses, $q\/n$ is a complex number and $A,B,C,D$ are real numbers, so the above equation is a conformal transformation. The existence of this representation was a complete surprise for me; it's clearly related to the infinite-dimensional projective representation of $Sp(4,R)$ because the derivation proceeds via Fresnel theory, but it's carried on the 3-d space of symmetric $2\\times 2$ matrices whilst the projective representation has no irreducible subspaces other that the even and odd parity irreps. My question is, ``What is the mathematical background to the representation for Gaussian beams?''"}
{"id":"91854","title":"What groups of symmetry are most suited for filling uniformely a spherical 3D space, whilst possessing the lowest possible surface-to-volume ratio?","text":"I am looking for the closest known approximate solution to Kelvin foams problem that would obey a spherical symmetry. One alternative way of formulating it: I am looking for an equivalent of Weaire–Phelan structure that would be able to fill in a sphere, while stacked in layers around the central core. Another alternative way of formulating it: I am looking for a crystalline structure that would result in growth of spherical or (almost) spherical crystals. In case it is impossible, what would be the closest quasicristaline structure for this? Any pointers towards textbooks or publications or just intuitive explanations would be more then welcome."}
{"id":"32816","title":"How do I solve these integrals of wave function and operator?","text":"1. First integral $$\\int \\Psi^*({\\bf r},t)\\hat {\\bf p} \\Psi({\\bf r},t)\\, d^3r,$$ where the $\\Psi({\\bf r},t)=e^{i({\\bf k}\\cdot{\\bf r}-\\omega t)}\\,\\,\\,$ and $\\hat {\\bf p}=-i\\hbar \\nabla$.   2. Second one $$\\int \\Psi^*({\\bf r},t)\\hat {\\bf p}^2 \\Psi({\\bf r},t)\\, d^3r,$$ where the $\\Psi({\\bf r},t)=e^{i({\\bf k}\\cdot{\\bf r}-\\omega t)}\\,\\,\\,$ and $\\hat {\\bf p}^2=-\\hbar^2 \\nabla^2$."}
{"id":"51597","title":"Is color confinement detected?","text":"I'm a graduate student studying QFT. I'm quite interested that is color confinement detected or proved? (both directly and indirectly) Or it is just an assumption?"}
{"id":"107841","title":"Sign convention for EMF","text":"When we define the field generate by EMF, why there is not negative sign in $\\mathcal{E} = \\oint \\vec{E} \\cdot d\\vec{l}$? Usually we talk about potential, there should be a negative sign, right?"}
{"id":"107847","title":"Gravitational Anomalies and Topological Order","text":"I wonder the relation of gravitational anomaly and topological order. Specifically: 1) What is the definition of gravitational anomaly here? 2) How are they related?"}
{"id":"100813","title":"How do we measure the physical field of a particle?","text":"In the renormalization procedure of quantum field theories, say $\\lambda \\phi^4$ theory for simplicity, we use the physical mass $m$, the physical coupling constant $\\lambda$ and the physical field $\\phi$ to fix the bare quantities $m_0$, $\\lambda_0$ and $\\phi_0$. It is clear that the physical mass $m$ and the physical coupling constant $\\lambda$ can be found from experiments (and using the S-matrix). But how does one \"measure\" the physical field $\\phi$? **Edit:** I would like to (hopefully) clarify my question. My question is related to field renormalization $\\phi_0(x)=\\sqrt{Z_\\phi}\\phi(x)$, and how we decide what value $\\phi$ should be. For the physical mass and coupling constant, we can perform experimental measurements to determine their fixed values but we don't seem to do this for the physical field (as far as I understand; I still need to properly interpret JeffDror's answer). **Are we basically looking at the divergence of Feynman diagrams and choose $Z_\\phi$ such that we can absorb this divergence?** If this is true, then this would mean that the \"physical\" field is a badly chosen word and it would make more sense to call it the \"renormalized\" field?"}
{"id":"73282","title":"Why does the guy moving on spaceship look younger in twin paradox?","text":"If there is no particular absolute choice of frame of reference, the guy who sits on Earth is also moving away from the guy on spaceship perspective and hence time on Earth should also dilate when viewed from the guy on spaceship perspective. But why does the guy moving on spaceship look younger in twin paradox? What am I misunderstanding terribly? ADDED:: Is time dilation symmetric? If one frame of reference are moving with constant velocity w.r.t other, we have to transformation relations $$\\Delta t = \\frac{\\Delta t'}{\\sqrt{1 - \\frac{v^2}{c^2}}}$$ Does each one of them see clock on other tick slowly than their own?"}
{"id":"51627","title":"Quantum mechanics, whats possible?","text":"There is a thread in Physicsforums.com which states due to Quantum Mechanics, if you wait long enough diamonds will appear in your pocket, it also states its possible for all your atoms to spontaneously re-arrange themselves so you turn into a Boeing airplane. Surely this is fiction?"}
{"id":"51620","title":"What's the difference between divergences that can be corrected and those that can't","text":"I'm confused by renormalization . If a lagrangian has a term with negative mass dimension , why can't the divergences be absorbed into lagrangian coefficients? What's the difference between divergences that can be corrected and those that can't"}
{"id":"65857","title":"Why is a degree Celsius exactly the same as a Kelvin?","text":"How on earth is it possible that the difference between two temperatures in Celsius and Kelvin is exactly the same. Given the historical definition of Celsius, I find it hard to believe that this is pure coincidence."}
{"id":"27156","title":"What is the importance of studying degeneration on $M_g$","text":"Let $M_g$ be the moduli space of smooth curves of genus $g$. Let $\\overline{M_g}$ be its compactification; the moduli space of stable curves of genus $g$. It seems to be important in physics to study the degeneration of certain functions on $\\overline{M_g}$ as you approach its boundary. For which functions on $M_g$ is it interesting to physicists to study their degeneration as you approach the boundary? I know of theta functions, Green functions and delta invariants. Are there others? As in my other question, here is an article by Wentworth which I find interesting."}
{"id":"25422","title":"Where does a star's angular momentum go as its spin slows down?","text":"So we know that stars slow down as they age. But total angular momentum must be conserved. Where does that angular momentum go? The dissipation of Earth's tides _somehow_ transfers Earth's angular momentum to the moon (as shown in my answer at http:\/\/www.quora.com\/Has-the-Earths- rotational-period-always-been-24-hours-If-not-what-was-it-before-and-what- caused-the-change). But where does a star's dissipation go?"}
{"id":"99887","title":"Can you equate the diffusivity constant in random walks with the one in Brownian motion (Einstein relation)?","text":"In an unbiased random walk in one dimension, the coefficient of diffusion is $D = l^2\/2\\tau$, where $l$ is the size of the jump and $\\tau$ is time taken for that jump. In simple Brownian motion, Einstein's relation says that $D = \\mu kT$ where $\\mu$ is the mobility, $k$ is the Boltzmann constant and $T$ is temperature. In both cases, it turns out: $$ \\langle x^2 \\rangle = 2Dt$$ So my question is, can we equate the two constants? So, can we say that: $l^2\/2\\tau = \\mu kT$?"}
{"id":"99884","title":"Where does energy in a field come from?","text":"Let us consider for example Earth's gravitational field. If we put a ball somewhere in this field, the ball starts to accelerate due to the gravitational force exerted on it. I understand the explanation for this : \"Gravitational potential energy (GPE) is converted into kinetic energy\". But why is it that this GPE seems to exist only when I introduce the ball into the gravitational field? Am I creating this GPE somewhere? Also, for my second question : If the gravitational field does work in accelerating the ball, then won't it have to lose energy in order to do all this work? Where does all this energy required come from? From all this thinking I come to the conclusion that a field is some sort of wizardy with an infinite amount of energy that can be transferred to objects placed in it. But I am perpetually confused because for all my life, I've been taught that energy cannot be created. So where do all these energies come about, really?"}
{"id":"57307","title":"Fermi and Boltzmann distribution of carriers in semiconductor","text":"What are the requirements for describing charge carriers (e.g. electrons) in a semiconducting material by \\- Fermi distribution \\- Boltzmann distribution When do we apply the one or the other? If the explanation to this question is in Ashcroft\/Mermin, reference to the relevant chapter would be appreciated. I don't agree with both commentators so far. * * * Edit Consider Eq. 21 in Sze, PHysics of Semiconductor Devices: $$n = N_C exp\\left(-\\frac{E_C-E_F}{kT}\\right)$$ or Eq. 2.16 in Pierret, Semiconductor device fundamentals."}
{"id":"36482","title":"Ideal gas law, pressure increase and temperature","text":"If I had a container, full with air, and I suddenly decreased the volume of the container, forcing the air into a smaller volume, will it be considered as compression, will it result in an increase in temperature, and why? P.S: the container is strong enough to bear high pressure and not to explode."}
{"id":"33053","title":"How is a cathode ray tube different from beta minus radiation?","text":"In beta minus the result is one neutron in the nucleus changing to a proton, plus an electron and an anti-neutrino being sent off. The antineutrino is indifferent to our health. So I guess what makes a beta source dangerous compared to a cathode ray tube must be a difference in the kinetic energy of the emitted electrons?"}
{"id":"41216","title":"Qualitative discussion about entropy and disorder","text":"Many discussions about entropy and disorder use examples of decks of cards, pages of books thrown in the air, two gases being mixed in a container, even the state of a nursery at the end of the day compared to the beginning of the day in order to explain the idea of order (and disorder). In all these examples it is pointed out that the disorder of the system increases, or that the system is in an ordered state and finishes in a disordered state after something has happened. Take the case of throwing the pages of the book in the air. You start with the pages numbered in sequence (I didn't want to use the for \"order\"), you throw them in the air, they land on the floor, you collect them up and notice the pages are not in sequence anymore. ANd the point is \"They are not in the sequence _I_ call ordered. Nonetheless, they are in a new sequence.\" And, it appears to me that the probability to find the pages in this precise new sequence is equal to find them in the original sequence\". In that sense, 'order' seems to be something that us humans define and it doesn't appear to be a property of the system. On the other hand, I can see that in the case of two gases mixing, empirically we find more states where the two types of molecules are occupying the entire volume of a container than one type of molecule in the left side and the other type in the right side of the container. Nonetheless, the precise state of each molecule, its position and therefore the entire state of the mixed up system is qualitative the same, isn't it? Isn't it equally difficult to make each molecule occupy that precise position in the mixed up state as in the unmixed state? Does this make sense? Thanks, Marcio"}
{"id":"41214","title":"Action on Lard Oil","text":"If water is mixed with lard oil and heated (creating some _super-critical_ liquid with water), how does this affect the volatility of the mixture in comparison with its _purity_..? So, My question is: What happens when a mixture of lard oil and water is compressed and then heated to over some $\\approx800^o\\text{C}$ almost instantaneously? **Note:** I'm no physicist or chemist, so simplicity would be appreciated :-)"}
{"id":"53221","title":"Difference between spinor and vector field","text":"How do we distinguish spinors and vector fields? I want to know it in terms of physics with mathematical argument."}
{"id":"70730","title":"Sign convention in optics","text":"Why is the sign convention used in the derivation of the lens formula and yet used again when it is applied in numerical problems? Won't the whole idea of sign convention be eliminated if it is used twice?"}
{"id":"70732","title":"What if exactly half the Earth's population jumped at one instant? + Secondary Question","text":"I read somewhere that when you jump, the sole effect caused by your jump on the earth moves it about $10^{-18}m$ (I don't remember the figure exactly, but I think it was that). However - obviously - with so many people running, jumping, etc. on the whole surface of the earth, the whole effect is canceled out, so it's safe to say there is no net displacement caused by us. This got me to wonder, what if the whole population of one longitudinal hemisphere of the earth jumped at the exact same time - while the other half remained stationary - **_would we be able to notice any discernable displacement in the short time before we all land back to the ground?_** Let's consider the earth to have a perfect distribution of people on both hemispheres, and let's consider the people of the Eastern Hemisphere to jump. Now the people jumping along the Prime Meridian and the International Date Line (these two lines together form the circle that divides the earth into the West and Eastern Hemispheres) would produce no effect, since they would simply cancel each other out. However, the closer you go to $0°N, 90°E$ (the centre of the East Hemisphere), the component of force which adds up will keep increasing. So there will be a net force in the direction of the Western Hemisphere. I was wondering if it would be noticeable. ![enter image description here](http:\/\/i.stack.imgur.com\/6Pk9u.gif) * * * Another question which just struck me: Out of all the satellites\/rockets\/space-shuttles we've sent out, have **_they_** caused any displacement on the earth? They are quite massive in comparison to _us_ (not the earth, of course), and have large speeds. Many of them are continuously moving further and further away from the earth - so there's no question of them falling back to earth and thus shifting it to its original position again. Will the momentum be enough to displace the earth?"}
{"id":"43251","title":"What bends fabric of space-time?","text":"I know that mass can bend fabric of space-time, which causes gravity by making an object curve around a planet or star but is there anything else that can bend it? Other energy sources, forces anything?"}
{"id":"76770","title":"Question about the Dirac notation for partial trace","text":"I saw the following definition for the partial trace operator: > $\\rho_A=\\sum_k \\langle e_k|\\rho_{AB}|e_k\\rangle$, where $e_k$ is basis for > the state space of system $B$. From what I know, in the Dirac notation, the meaning of $\\langle v|A|u\\rangle$ is the inner product of the vectors $|v\\rangle$ and $A|u\\rangle$, so I have two problems with this notation of the partial trace. First, how can an inner product be an operator? An inner product should be a complex number, so I guess that inner product represents an operator somhow. Second, what is the meaning of $\\rho_{AB}|e_k\\rangle$? $\\rho_{AB}$ is a mapping on the space $A\\otimes B$, but $e_k$ is a vector from the space $B$. So, I don't really know how to interpret the meaning of this notation."}
{"id":"38288","title":"how is the reference for the star maps in google determined?","text":"If I go to google sky - http:\/\/www.google.com\/sky\/ Ihere is a longitude and latitude measure at the bottom of the page. Iow are these coordinates determined and who sets it? Do those coordinates move at all over time?"}
{"id":"103176","title":"Power counting with a cutoff","text":"In Effective Field Theory video lectures found here, the professor explained power counting in effective field theories and the difficulties of power counting associated with loop diagrams. He then mentions that introducing a cutoff ($\\Lambda_{UV}$) to regulate our divergences does not preserve power counting due to the new scale that we are introducing. To see this he uses four-fermi theory with the diagram, $\\hspace{6cm}$ ![enter image description here](http:\/\/i.stack.imgur.com\/VwG04.png) We do our power counting (i.e., Taylor expansions) in powers of $m^2\/M^2$ and then go on to consider the mass correction through, $\\hspace{6cm}$ ![enter image description here](http:\/\/i.stack.imgur.com\/xkHEw.png) Using a cutoff this gives a mass correction, \\begin{align} a\\frac{ m }{ M ^2 } \\int _0^{\\Lambda_{UV}}\\frac{ \\,d^4k _E }{ (2\\pi)^4} \\frac{1}{ k _E ^2 + m ^2 } & = a\\frac{ m }{ ( 4 \\pi ) ^2 } \\left[ \\frac{ \\Lambda _{ UV } ^2 }{ M ^2 } + \\frac{ m ^2 }{ M ^2 } \\log \\frac{ m ^2 }{ \\Lambda _{ UV } ^2 } - \\frac{ m ^4 }{ M ^2 \\Lambda _{ UV } ^2 } + ... \\right] \\end{align} If I understand correctly this breaks the power counting because even if $\\Lambda_{UV} \\sim M$, the first term is an order 1 correction since its not proportional to $m^2\/M^2$. So far so good. However, then the professor says that you can still use power counting with a cutoff if you fix the power counting order by order and that this can be done by introducing an intermediate scale, $\\Lambda$. But I don't how this fixes anything... With an intermediate scale ($\\Lambda$) we have, \\begin{equation} a\\frac{m}{M^2}\\int _{ \\Lambda } ^{ \\Lambda _{ UV }} \\frac{ \\,d^4k _E }{ (2\\pi)^4 } \\frac{1}{ k _E ^2 + m ^2 } = \\frac{a\\,m}{ (4\\pi)^2M^2 } \\left\\\\{ \\left(\\Lambda ^2 + m ^2 \\log \\frac{ m ^2 }{ \\Lambda ^2 + m ^2 } + ... \\right) + \\left( \\Lambda _{ UV } ^2 - \\Lambda + m ^2 \\log \\frac{ \\Lambda ^2 + m ^2 }{ \\Lambda ^2 _{ UV }} \\right) \\right\\\\} \\end{equation} but how does this fix anything? For more context see my lecture notes here under Effective Field Theory (starts around equation 4.6)"}
{"id":"47303","title":"A Short way to show Conservation of Quantum Laplace–Runge–Lenz Vector?","text":"I had been asked to prove the conservation of Quantum Laplace–Runge–Lenz Vector: $$\\hat{A}=\\frac{i}{\\hbar}\\left[\\hat{p},\\,\\frac{1}{2}\\hat{L}^{2}-k\\left|\\hat{r}\\right|\\right]=\\frac{1}{2}\\left(\\hat{p}\\times\\hat{L}-\\hat{L}\\times\\hat{p}\\right)+k\\frac{\\hat{r}}{\\left|\\hat{r}\\right|}.$$ But unlike the case of classical mechanics, even after a lot of tries I found no beautiful\/short way to show it's commutation with Hamiltonian, the calculation are quit lengthy opposite to my first expectations, but something tells me that there is a shorter way, so dose anybody managed to find a simple way or trick to show that $$\\left[\\hat{H},\\hat{A}\\right]~=~0?$$"}
{"id":"76196","title":"Why Counterclockwise starts from East?","text":"In vector lessons, we have counterclockwise angle of rotation of the vector starts from East. \"The direction of a vector is often expressed as a counterclockwise angle of rotation of the vector about its \"tail\" from due East\" http:\/\/www.physicsclassroom.com\/class\/vectors\/u3l1a.cfm Why doesn't start from north or south or any other direction? Or is it just one of common rule between physicist."}
{"id":"1906","title":"Kubo Formula for Quantum Hall Effect","text":"I'm trying to understand the Kubo Formula for the electrical conductivity in the context of the Quantum Hall Effect. My problem is that several papers, for instance the famous TKNN (1982) paper, or an elaboration by Kohmoto (1984), write the diagonal entries of the conductivity tensor in the form $$ \\sigma_{xy}(\\omega \\to 0) = \\frac{ie^2}{\\hbar} \\sum_{E^a < E_F < E^b} \\frac{\\langle a|v_x|b \\rangle \\langle b|v_y|a \\rangle - \\langle a|v_y|b \\rangle \\langle b|v_x|a \\rangle}{(E^a - E^b)^2} .$$ This is the static limit $\\omega\\to 0$ and low temperature $T\\to 0$. The sum goes over all eigenstates $|a\\rangle$ and $|b\\rangle$ of the single-particle Hamiltonian. $E_F$ is the Fermi energy. $v_x$ and $v_y$ are the single- particle velocity operators. However, these papers don't derive this equation, which is unfortunate because the Kubo formula is usually not presented in this form. I have found (and succeeded in rederiving) the following variation instead $$ \\sigma_{xy}(\\omega+i\\eta) = \\frac{-ie^2}{V(\\omega + i\\eta)} \\sum_{a,b} f(E^a) \\left( \\frac{\\langle a|v_x|b \\rangle \\langle b|v_y|a \\rangle}{\\hbar\\omega + i\\eta + E^a - E^b} + \\frac{\\langle a|v_y|b \\rangle \\langle b|v_x|a \\rangle}{-\\hbar\\omega - i\\eta + E^a - E^b} \\right).$$ This is formula (13.37) from Ashcroft, Mermin, though they don't actually prove it. $f(E)$ is the Fermi distribution. A nice derivation is given in Czycholl (german). Now, my question is, obviously > How to derive the first formula from the second? I can see that the first equation arises as the linear term when writing the sum as a power series in $\\omega$, but why doesn't the constant term diverge?"}
{"id":"87751","title":"Do low frequency sounds really carry longer distances?","text":"It is a common belief that low frequencies travel longer distances. Indeed, the bass is really what you hear when the neighbor plays his HiFi loud (Woom Woom). Try asking people around, a lot of them believe that low sounds carry longer distances. But my experience isn't as straightforward. In particular:   * When I stand near someone who's listening loud music in headphones, it is the high pitched sounds that I hear (tchts tchts), not the bass.   * When I sit next to an _unamplified_ turntable (the disc is spinning but the volume is turned off), I hear high pitched sounds (tchts tchts), not the bass. So with very weak sounds, high frequencies seem to travel further? This makes me think that perhaps low frequencies do not carry longer distances, but the very high amplitude of the bass in my neighbor's speakers compensates for that. Perhaps also the low frequencies resonate with the walls of the building? Probably also the medium the sound travels through makes a difference? Or perhaps high frequencies are reflected more by walls than low frequencies? I found this rather cute high school experiment online, which seems to conclude that low and high frequencies travel as far, but aren't there laws that physicist wrote centuries ago about this?"}
{"id":"109880","title":"DC current in ideal conductor and skin effect?","text":"I know that the skin depth derived for AC current goes as $$\\delta \\propto \\sqrt{\\frac{1}{\\omega \\sigma}}$$ where $\\omega$ is the angular frequency of the field and $\\sigma$ is the conductivty. Now: for a conductor with **infinite conductivity** but with **DC current** , we find a $0\\times\\infty $ in the denominator and I do not know how to treat it. That said: 1) In a conductor with infinite conductivity and DC or AC current, does the current only flow on the surface? 2) What is keeping charge inside a real conductor? Shouldn't they always try to eliminate the electric field inside?"}
{"id":"132070","title":"Find a central force given the orbit","text":"I've been trying to solve the following problem for a long time. Let's consider a particle of mass $m$ in $\\mathbb{R}^3$ with polar coordinates $(r,\\theta,\\phi)$. The particle moves on the orbit described by the equation $r=2a\\sin\\theta,\\,\\,a>0$, with velocity $|\\mathbf{v}|=k\/\\sin^2\\theta,\\,\\,k>0$. The task is to show that the force is central and to determine such a force. My idea has been to use the two equations given to prove the two components \\begin{equation} F_\\theta=m\\left( r\\,\\ddot\\theta + 2\\dot{r}\\,\\dot\\theta - r\\,\\dot\\varphi^2\\sin\\theta\\cos\\theta \\right)\\\\\\ F_\\phi=m\\left( r\\ddot\\varphi\\,\\sin\\theta + 2\\dot{r}\\,\\dot\\varphi\\,\\sin\\theta + 2 r\\,\\dot\\theta\\,\\dot\\varphi\\,\\cos\\theta \\right) \\end{equation} to vanish; unfortunately, I wasn't able to get to any interesting result. Can anybody give me some help on this?"}
{"id":"15021","title":"The bar tender says we don't serve Tachyons around here?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/iEOWe.jpg) What's the physics behind this joke?"}
{"id":"77824","title":"Global Chern-Simons forms and topological gauge theories","text":"I am reading the classic Dijkgraaf and Witten paper on topological gauge theories and something struck me that I didn't understand. For a trivial bundle $E$ on smooth 3-manifold $M$ with compact gauge group $G$ the Chern- Simons form is of course $$S(A)=\\frac{k}{8\\pi^2}\\int Tr (A\\wedge dA + \\frac{2}{3}A\\wedge A\\wedge A).$$ The confusing sentence is the following: \"If the bundle $E$ is not trivial, the formula for the action does not make sense, since a connection on a non- trivial bundle cannot be represented by a Lie algebra valued 1-form as in that formula.\" My concern is that this is wrong - of course we can represent a connection in a non-trivial bundle with a Lie-algebra valued 1-form! In fact, some texts define the connection in exactly that way (my trusty Choquet-Bruhat and DeWitt-Morette has that as one of their three equivalent definitions for a connection)- locally, a connection is represented as a linear map $\\omega_p:T_p(E)\\to \\mathfrak{g}$. So are we supposed to read the \"as in that formula\" to mean \"globally\"? Since of course, if the bundle is not trivial there isn't a _global_ Lie-algebra valued 1-form that we can use for the Chern-Simons action. So that seems like a fine argument, but in QFT we are _always_ writing down actions with sections of bundles (fields $\\phi(x)$, say) which are not trivial - we just make sure that the action is invariant under the gauge transformations. So, what is going on with that statement? Is it really saying we cannot represent a connection in such a way, or is it just stating the action is now local, and the rest of the paper is going to go on and tell me how to fix this with an element of $H^3(BG,\\mathbb{R})$, etc.?"}
{"id":"39485","title":"Path integral on matrix model","text":"I was looking at a 0-dimensional matrix model, where the variables are $N\\cdot N$ Hermitean matrices. It had a gauge symmetry, e.g. $U(N)$. And in the path integral, the Faddeev-Popov trick was used. So instead of integrating over the set of matrices, one was reduced to integrating over the $N$ (real) eigenvalues. Or, I tried counting the number of degrees of freedom and I as puzzled:   1. N by N Hermitean matrices have $N^2$ real independent variables ($N(N-1)$ values for the triangular part (non counting the diagonal ones), plus _$N$_ real variables for the diagonal ones, giving us a total of $N^2$   2. $U(N)$ has real dimension $N^2$, classic result from group theory   3. At the end, I arrive at $N^2 - N^2 = 0$ variables instead of $N$ real eigenvalues. I tried seeing it from another point but I can't derive what I'd like to. What I am thinking\/doing wrong?"}
{"id":"77822","title":"Simultaneous diagonalization of Hamiltonian and momentum operator","text":"I'm looking at a translationally invariant problem with 3 atoms arranged in a circle each with one valence electron capable of tunelling to either of its two neighbors. With a tunelling rate of $-|A|\/\\hbar$, we have the Hamiltonian $$H = \\begin{pmatrix} E_a & \\- |A| & \\- |A|\\\\\\ \\- |A| & E_a & \\- |A|\\\\\\ \\- |A| & \\- |A| & E_a \\end{pmatrix}$$ which can be shifted by $+|A|$: $$\\begin{pmatrix} E_a + |A| & 0 & 0\\\\\\ 0 & E_a + |A| & 0\\\\\\ 0 & 0 & E_a + |A| \\end{pmatrix}.$$ Because of translational symmetry, $p$ is a good quantum number and $[H,\\hat{p}]=0$. This means that we can diagonalize $H$ as well as $\\hat{p}$ simultaneously and construct momentum eigenstates out of the Hamiltonian eigenstates. How do I do this? **EDIT:** Okay so I've shifted the matrix as follows (simply $-E_a$ on the diagonal): $$\\begin{pmatrix} 0 & -|A| & -|A|\\\\\\ -|A| & 0 & -|A|\\\\\\ -|A| & -|A| & 0 \\end{pmatrix}.$$ Eigenvalues are $-2|A|$ corresponding to the eigenstate (1,1,1), $|A|$ corresponding to (-1,0,1) and |A| corresponding to (-1,1,0). So $|A|$ is degenerate. Now I need to diagonalize $p$. I guess what my problem is, is that I'm not quite sure of the matrix representation of $p$ here. How do I then diagonalize it, if I can't find its matrix representation? I know each matrix element is $\\langle \\psi_i | p | \\psi_j \\rangle$, but I don't know what the wave functions look like. How do I proceed?"}
{"id":"91193","title":"Precision of spectroscopy for astronomy","text":"How precise can the measurements be when looking at spectral lines in astrophysics? For example, suppose I have a telescope in orbit, and I am looking at $H_\\alpha$ lines coming from a star at 613 pc. Is it realistic to observer the lines move from 626.282 nm to 656.282002854 nm? this would correspond to a radial velocity of about 1.3 m\/s (I am not an astronomer, and I just want to get a feel for whether this is realistic to measure, or out of the question)"}
{"id":"91195","title":"In a non auto-ranging digital multimeter, what do the increasing numbers in the Ohm section represent?","text":"For my science fair project I am measuring the resistance of nichrome wires under different conditions. I am using an Innova 3300 digital multimeter to measure the resistance. Other websites have told me to move the pointer to the Ohms section. However, there are multiple numbers in that section. What number should I use and how does changing the number affect the results?"}
{"id":"35863","title":"What does adding a scalar field component to the Einstein field equations mean for black holes and string theory?","text":"If a scalar field component has to be added to the Einstein field equations (see below) to solve dark matter\/energy, then how would string theory need to be modified and do black holes still exist? The proposed modified equations are (ignoring physical constants) $$ R_{ij} - \\frac12 R g_{ij} = T_{ij} + \\nabla_i\\nabla_j \\varphi $$ where $\\varphi$ represents some kind of scalar potential. The conservation law for energy- momentum is proposed to be $$ \\nabla^i(T_{ij} + \\nabla_i\\nabla_j\\varphi) = 0 $$ Refs:   * Dark Matter, Dark Energy, and the Fate of Einstein’s Theory of Gravity http:\/\/mathgradblog.williams.edu\/dark-matter-dark-energy-fate-einsteins-theory-gravity   * UNIFIED THEORY OF DARK ENERGY AND DARK MATTER, TIAN MA, SHOUHONG WANG http:\/\/www.indiana.edu\/~fluid\/paper\/report.pdf"}
{"id":"74846","title":"QM -group reps and transforming wavefunctions","text":"QM texts seem to have many ways of motivating the angular momentum operators and deriving the l and m quantum numbers . But the connection between physical rotaions in 3 dim space and an operator in hilbert space that is called \" angular momentum\" is still somewhat unclear to me I found a set of notes on Lie Groups in Physics by Nobel prize winner Veltman - that seems to be particualry clear explanation of irreducible reps of SO(3) and angluar momentum operatiors see on page 10 of www.phys.uu.nl\/~thooft\/lectures\/lieg07.pdf He begins with discussion of general linear diffrential equations that have a set of solutioins . He then says that when a 3 dim rotation takes place a solution of the DE is then transformed into a linear mixture of other solutions Is this true for all linear diffrential equations -?? or are these type of transformations amoung solutions in response to 3 dim rotations an assumption that is made about wavefunctions or about states in a Hilbert space as part of the assumptions of QM ? THanks Len"}
{"id":"74845","title":"Question about streamline in boundary layers","text":"I was wondering what the streamlines look like in a boundary layer. I found a picture that shows this: ![Streamline in boundary layer](http:\/\/i.stack.imgur.com\/pgnq0.png) I would like to know if this presentation of the streamline is correct. I don't think that it is correct in the turbulent region, but I'm more interested in the laminar region. The streamline seems horizontal to me in that region, so it looks like there are no velocities in the y-direction, so $v_y=0$. I wonder if this is correct, because this seems to contradict the boundary-layer equations in which they use a $v_y$. I feel like that the picture isn't correct, but that the streamlines tend to follow the angle of the dotted line. So I wonder: is that feeling correct and do the streamlines have more or less the same slope as the dotted line? I also feel that the more we go in the y-direction the more the tendency to follow the dotted line becomes, so close to the plate they are horizontal, can anyone confirm or contradict this?"}
{"id":"131525","title":"Is there a difference in handwritten nabla $\\vec{\\nabla}$ with an overset arrow and typeset nabla $\\nabla$?","text":"According to some physicist at KIT it is usual to write the following when using pen and paper: ![enter image description here](http:\/\/i.stack.imgur.com\/cDBVO.png) whereas in typeset texts you write $\\nabla$. Is that true? Are there sources for this convention?"}
{"id":"96686","title":"What would happen to wet laundry in the vacuum of space?","text":"Imagine for a second that I was in a college dorm room frustrated that all the dryers were broken or in use. I'm impatient and not a jerk so I don't wait or take out someone else's laundry, I shoot it into space with the ability to get it back and land it successfully on my campus. Ignoring practicality of course: How would my clothes react? Would the water in them freeze? Would the water just dissipate into the rest of space? Would it boil if it were in the light of the sun? I'm curious about three cases: One, where the clothes are in a general vacuum regardless of the effect of the sun. Two, where my clothes are specifically behind the Earth and so they are eclipsed by the Earth. Three, where my clothes are specifically between the Earth and the sun. For those willing to make the argument that this is a silly topic: would it or would it not have implications with space suits? Thanks!"}
{"id":"90273","title":"Resonance Raman spectroscopy vs fluorescence","text":"In Resonance Raman Spectroscopy we often want to avoid the fluorescence. But what is the problem with fluorescence . What we want is a shift between exciting line and emitted radiation and both can show the vibration energy difference?? also from \"What is the difference between Raman scattering and fluorescence?\" I come to know the difference between Raman and fluorescence is that of lifetime of the molecule in the excited state, but how does a molecule come to know that it is studied for Raman spectroscopy so its lifetime in excited state is small or it is studied for fluorescence."}
{"id":"101461","title":"Is there a general law for physics?","text":"Here is why I pose this question, as I advance in physics more and more this idea haunts me and in fact helped me figure out and memorize many physical laws. It seems to me that there is a relation between all the laws of physics, something that is a general law that determines how should all other laws be _generally_ stated. I can't express this law in a complete sentence and I'm not able to verify if it has an exception or not but I will try to explain this by examples to show what I really mean: ## Mechanics: Oscillations: It seems that the whole process happens because the object is trying to get back into its **initial state** i.e the origin. ![oscillator](http:\/\/i.stack.imgur.com\/I7qc4.png) ## Electricity: Lenz's law: An induced electromotive force (emf) always gives rise to a current whose magnetic field opposes the original change in magnetic flux. in other words: the current is trying to get its **initial state** by resisting the change of current. ## Atoms Radiation and Nuclear reactions happen because the atom is trying to be **stable.** Of course there are many other examples but I'm just stating few to make my point. The question is: is there something like that in physics? if not can you give me any counter example on this?"}
{"id":"18207","title":"How to integrate over rapidity for Parton Luminosities in LHC?","text":"I want to make a comparison of parton luminosities between Tevatron and LHC. According to Factorization theorem the cross section in hadron colliders, as long as the partonic cross section has $\\hat \\sigma \\sim \\frac{1}{\\hat s}$ dependence, could be written like that : $$\\sigma =\\sum_{12}\\int dx_1 dx_2 f_1 \\left(x_1,\\mu_F^2\\right) f_2\\left(x_2,\\mu_F^2 \\right) \\hat{\\sigma}_{12} \\left(x_1 x_2 s,...\\right)$$ here $f_{i}(x_{i},\\mu_F^2)$ represents the parton distribution function for $x_i$ fraction of the incoming hadron, $i$ represents the parton flavor and $\\mu_F$ is the factorization scale. I wanted to reproduce the Figure 4.4 (page 82) in book Physics at The Terascale or the Figures in LHC Physics Potential vs. Energy. **My Questions?** I appreciate for any reply.   1. When I am scanning over the $\\sqrt{\\hat s }$ (partonic-center-of-mass energy) what value should I set the factorization scale? $\\mu_F = \\sqrt{\\hat s } $ or $\\mu_F = \\sqrt{s}$ (center-of-mass energy)?   2. How could I integrate over the rapidity? What I mean is on page 82 it says that \"The Luminosity $dL_{ij}\/d\\hat s$ integrated over y is shown in Figure 4.4a\" and I don't understand what changes after integration over y. In Equation 4.11 the parton distribution functions are not function of rapidity thus I could simply move $dy$ to the right side and carry out the integration and it will contribute as a factor (though I don't know the bounds $-\\infty<y<\\infty$ or something specific for the collider $-5<y<5$ ) however there is relation between the rapidity($y$) and $x$ like this $y=\\ln(x1\/x2)\/2$. Thus I don't understand that line?   3. If I just ignore the rapidity and use the Equation (1) in \"LHC Physics Potential vs. Energy\" I got some results but I am still off compared to the Figure 1 in the same paper, so I concluded that I don't know how the integration over rapidity affects the over all Parton Luminosity. Equation 4.11 $$\\frac{dL_{ij}}{d \\hat{s} dy}=\\frac{1}{s}\\left(x_1,\\mu\\right) f_2\\left(x_2,\\mu \\right)$$"}
{"id":"62417","title":"Harmonic oscillator with light damping","text":"My textbook gives the following for x as a function of time for a lightly damped harmonic oscillator: $$ x = Ae^{- \\gamma t} \\cos (\\omega \\, t)$$ for $\\gamma = \\dfrac b {2m}$. It says this implies the amplitude follows the relationship: $$x_{max} = Ae^{-\\gamma t}$$ I am not understanding where this second equation comes from. Why does the cosine term disappear if $\\omega$ is a constant?"}
{"id":"81947","title":"When can a surface charge density exist?","text":"In my syllabus about electromagnetism, they state: \"This surface charge density will not always be present, e.g. when considering two non-conducting dielectrics such surface charge density remains absent. However, at a perfect conductor, a surface charge density will be present. If one of the media (or both) carry a conduction current, a surface charge can also be present (explain why!)\" I don't really see though, how this can be explained. They seem to imply that there is a relation between the conductivity and the possibility of a surface charge, but I can't figure how they are related. They also state somewhere earlier that there can't be a surface current between two lossy dielectrics. This seems to be analogous to my previous question. I thought that this was due to the fact that dielectrics don't conduct current, but I wonder if someone knows a better explanation here too (that perhaps explains why they explicitly mention that the dielectrics are lossy)."}
{"id":"51886","title":"Hamilton's equations for a simple pendulum","text":"I don't get how to use Hamilton's equations in mechanics, for example let's take the simple pendulum with $$H=\\frac{p^2}{2mR^2}+mgR(1-\\cos\\theta)$$ Now Hamilton's equations will be: $$\\dot p=-mgR\\sin\\theta$$ $$\\dot\\theta=\\frac{p}{mR^2}$$ I know one of the points of Hamiltonian formalism is to get first order diff. equations instead of second order that Lagrangian formalism gives you, but how can I proceed from here without just derivating again wrt. $\\dot\\theta$ and substituting $\\dot p$ to get the same equation that I get with the Lagrangian formulation? Or is that the way to do it? And how could I get the path of the system on the phase space with those equations?"}
{"id":"20665","title":"Can x-ray radiation be compared to background radiation?","text":"I've been trying to learn about the possible effects of x-ray radiation from dental x-rays and most of the resources I come across compare the exposure to that of natural background radiation. Here's an example (not specific to dental x-rays): http:\/\/www.hpa.org.uk\/webw\/HPAweb&HPAwebStandard\/HPAweb_C\/1195733826941?p=1158934607708 I know electro-magnetic-radiation comes in different wavelengths and some are ionizing and some are not. Is this a valid comparison? **Update:** Thanks for the helpful responses so far. Looking at the chart here from @anna v, is comparing x-rays with background radiation comparing EMR that is 10^-10 with wavelengths that are longer than infrared? That's what I still don't understand. Wouldn't those different types of waves have different properties?"}
{"id":"118940","title":"Most natural tensor structure for a quantum field","text":"A quantum field is described by a Hilbert space. In many instances, the chosen tensor structure on this Hilbert space corresponds to that of space-like separated regions of space-time. The justification to this is that we can only perform local measurements, so the tensor structure should correspond to what subsystems operationally are, e.g. space-like separated regions of space-time. Does any one know a paper where the choice of a tensor structure for quantum fields is better motivated, and alternative tensor product structure proposed (for example in the context of relativistic quantum information, where space- like separated regions should not be the fundamental unit anymore since communication is generally allowed)?"}
{"id":"94730","title":"Boltzmann entropy and phase space volume","text":"In Huang's book on Statistical mechanism and Statistical interpretation of Entropy, it is not mentioned that $\\Omega$ is the phase space volume, but it is the states of the system. So, how does entropy be called as logarithm of phase space volume? A detailed explanation and reference where it is shown that Entropy as phase space volume will be appreciated. Where can I find the derivation that entropy is the phase space volume?"}
{"id":"51880","title":"Covariant derivative with upper index","text":"I just need clarification, that is, to see that I'm doing the right thing. When calculating central charge for certain metric, I need to solve an integral that contains Lie brackets etc. And I have some expression, that contains terms like: $$\\xi_\\sigma D^\\nu h^{\\mu\\sigma}$$ Where $D_\\nu$ should be covariant derivative. Now, can I just say: $$D^\\nu=g^{\\nu\\mu}D_\\mu$$ And similarly for the $h^{\\mu\\sigma}$, and use the standard definition of covariant derivatives?"}
{"id":"134047","title":"Possible Deflection Distance For Falling Object","text":"I work for a company selling tethers that prevent tools on building sites from falling away from the user. One of the problems that has come up is that objects can deflect horizontally when they strike an object whilst falling. You can imagine that they've fallen and then hit a steel girder or something on the way down. Given that the weight of the object, the height dropped from and the impact height are known, how would I calculate the _possible_ deflection distance?"}
{"id":"134042","title":"Lie algebra - basis for adjoint matrix products in SU(N)?","text":"In $SU(N)$, the set of matrices in the fundamental representation plus the identity: $$ \\left\\\\{ \\mathbf{1}, t^a \\right\\\\} $$ acts as a basis for generator products under matrix multiplication, such that any combination of the elements in the set can be written as a linear combination, i.e. for any $$ X_F = t^{a_1}\\dotsm t^{a_n} $$ we can write $$ X_F = c^0 \\mathbf{1} + c^a t^a \\, .$$ It is easy to show that $$ c^0 = \\frac{1}{N} \\text{tr} (X_F) \\, , \\quad c^a = - 2 i\\, \\text{tr} (X_F t^a)\\, . $$ We cannot do the same for the adjoint representation because the set $$ \\left\\\\{ \\mathbf{1}, T^a \\right\\\\} $$ doesn't span the full product space, e.g. $T^a T^b$ cannot be expressed solely in terms of $\\mathbf{1}$ and $T^c$. So my question is: can we add - and if yes which - a minimal number of $n^2\\\\!-\\\\!1$ matrices $\\\\{K^a\\\\}$ to $ \\left\\\\{ \\mathbf{1}, T^a \\right\\\\} $ in order to make it a closed set? **EDIT** Where it comes from is that because in the fundamental representation we have additional anti commutation rules $$ \\left\\\\{ t^a, t^b \\right\\\\} = \\delta^{ab} \\frac{\\mathbf{1}}{N} + d^{abc} t^c \\, , $$ we can take the mean of these and the standard commutation rules to write $$ t^a t^b = \\frac{1}{2} \\delta^{ab} \\frac{\\mathbf{1}}{N} +\\frac{1}{2} h^{abc} t^c \\,,$$ where $h^{abc}=d^{abc}+i f^{abc}$. We chain this relation to write for any product of fundamental generators $$ t^{a_1}\\cdots t^{a_n} = A^{a_1\\cdots a_n} \\frac{\\mathbf{1}}{N} + B^{a_1\\cdots a_n b} t^b \\, ,$$ with $A$ and $B$ some constants. This is the same statement as above. However we cannot do the same in the adjoint representation, because then the anti commutation rules don't close, i.e. they cannot be written as a linear combination of $\\mathbf{1}$ and $T^a$. So the question is, can we add some extra matrices $\\\\{K^a\\\\}$ such that we can write the anti commutation rules as $$ \\left\\\\{ T^a, T^b \\right\\\\} = \\delta^{ab} \\frac{\\mathbf{1}}{N} + d^{abc} T^c + c^{abc}K^c\\, , $$ which would give $$ T^{a_1}\\cdots T^{a_n} = A^{a_1\\cdots a_n} \\frac{\\mathbf{1}}{N} + B^{a_1\\cdots a_n b} T^b + C^{a_1\\cdots a_n b} K^c\\, .$$"}
{"id":"102284","title":"2nd - 3rd Year Physics Labs","text":"I'm looking to propose a new physics laboratory experiment, does anyone have any particularly fond memories of an experiment they carried out whilst at University? I'd like to include a section that makes use of Mathematica, Arduino or Python. The current labs are lacking computational aspects. The lab would ideally run over three 4 hour sessions."}
{"id":"48448","title":"Equation of motion in quantum system","text":"The commutation relation between H and $L_i$ is $[H,L_i]$. I want to show the equation of motion. **Is this right: $ [H,L_i] = -i \\frac{dL_i}{dt}?$**"}
{"id":"121594","title":"Why must these Spinors be normalized?","text":"I have just begun studying spin and there are two spinors mentioned: The main spinor $\\chi $ and the spin-up spin down spinors (eigenspinors) $\\chi_+ ,\\chi_- $. I learned that the main spinor is a linear combination of the two spin spinors and I understand that the main spinor must be normalized. I don't understand however why the spin-up and spin-down spinors must also be normalized. As far as I understand, they are not involved in probability calculations for example: $\\chi_+^{(x)} = \\left( \\begin{array}{ccc} \\frac{1}{\\sqrt2} \\\\\\ \\frac{1}{\\sqrt2} \\\\\\ \\end{array} \\right)$ instead of just $\\chi_+^{(x)} = \\left( \\begin{array}{ccc} 1 \\\\\\ 1 \\\\\\ \\end{array} \\right)$ What am I missing?"}
{"id":"127540","title":"Space-time curvature creates gravity or is it (could it be) vice-versa too?","text":"Mass (Energy) creates space-time curvature and thus it forms the reason for gravity. Can it be vice-versa too? Like, mass created gravitational field, gravitational field created space-time curvature? I would like to agree, because, in that sense, I can say electromagnetic field also produces space-time curvature! This can lead to a unification between them! (that's worth a question)"}
{"id":"54504","title":"Is cosmic background radiation dark-matter and\/or dark-energy?","text":"Dumb question alert: Is it possible that the cosmic background radiation might be the source of dark-matter and\/or dark-energy? What is the mass of the background radiation in the known universe?"}
{"id":"8607","title":"Temperature vs AC energy consumption","text":"I'm need to understand the following: to keep the room at confortable temperature (70 degree, for example), how does the amount of energy consumed by the AC grow as the outside air temperature rises in the summer over the same length of time (a day, say)? Is it more or less linear? quadratic? exponential? or anything else? I posted the same question on another StackExchange site, but I figured that physicist may provide more help because it seems to me that it is related to both heat conduction and the working mechanism of AC. Thanks."}
{"id":"15923","title":"why isn't the variance of Fano noise proportional to the number of photon arrivals?","text":"I've been reading \"Photon Transfer\" by James Janesick and in Chapter 3 (http:\/\/spie.org\/samples\/PM170.pdf) he describes the various sources of noise in a CCD. Basically, incoming photons interact with the CCD and cause some electrons to jump the bandgap. Both of these processes have some inherent randomness and must be described statistically - the photon arrival rate's variability gives rise to the shot noise, while the variability in electrons produced from a given photon impact gives rise to the Fano noise. The photon arrival is a Poisson process, so if m is the expected number of photon arrivals during a given exposure, the actual number of arrivals is a Poisson distributed random variable N(m), and var(N) = m. Now here's the part I don't understand. I quote from Janesick, editing slightly for clarity (p. 26): > If all the energy of an interacting photon was spent in the production of > electron-hole (e-h) pairs, then there would be no variation in the number of > e-h pairs produced. On the other hand, if the energy was partitioned between > breaking covalent bonds and lattice vibrations, or if phonon production was > uncorrelated, Poisson statistics would apply. But neither is the case in > nature. The variance in multiple electron-hole charge generation, called > Fano noise, is empirically described by > > $$ \\sigma_{FN}^2 = F_F \\eta_i = F_F \\frac{h_v}{E_{e-h}} $$ > > where [$\\sigma_{FN}^2$ is the variance of the Fano noise], and $F_F$ is > referred to as the Fano factor, which is defined by the variance in the > number of electrons generated divided by the average number of electrons > generated per interacting photon. [notation: $h_v$ and $E_{e-h}$ were > defined to be the photon energy in eV and the energy required to create an > electron-hole pair, and $\\eta_i$ is the ratio of the two. So $\\eta_i$ is the > expected number of electrons per incident photon, if all photon energy goes > into production of electrons.] Based on this description of the process, I would assume that each incident photon liberates an uncertain number of electrons, and therefore each incident photon should contribute a constant amount of variance. So I think the Fano noise variance should be **proportional** to the incident number of photons. But according to Janesick, the Fano noise contributes a constant variance **independent** of the number of photon arrivals. To be more mathematically precise, I think that the whole process should be modeled as a compound Poisson process, since both the number of arrivals and the number of electrons liberated per arrival are stochastic quantities. The wikipedia article on the compound Poisson process shows that the variance of the compound process has two components (one for the randomness of arrivals and one for the random size of the arrivals), and both components are proportional to the mean number of arrivals. So can anyone explain the discrepancy between my expectations and the book's equations? I feel I must have misunderstood what Fano noise is -- because if it's what I think it is, the natural model of it contradicts Janesick's equation. By the way, I know that $\\sigma_{FN}^2$ is **not** referring to the variance per incident photon. If it were, then Janesick would multiply it by the mean photon arrivals when calculating the total image noise, but he doesn't. Here is his calculation of the total noise variance (equation 3.17, p. 34, irrelevant flat field term removed): $$ \\sigma_{TOTAL}^2 = \\sigma^2_{READ} + \\eta_i F_F + \\eta_i S $$ where $S$ is the expected number of electrons from the exposure. The first term is from shot noise, the second from Fano noise, the third is from shot noise. Note that the shot noise term scales linearly with $S$ but the Fano noise term is independent of $S$."}
{"id":"126017","title":"Should a radiation-filled Universe be scale invariant?","text":"Imagine a spatially flat Universe, without cosmological constant, filled only with EM radiation. As Maxwell's equations without charges or currents are scale invariant then should this Universe be scale invariant as a whole? i.e. should there be no length scale associated with such a Universe? Friedmann's equation for such a Universe is conventionally given by: $$\\left(\\frac{\\dot a}{a}\\right)^2 \\propto \\frac{1}{a^4}$$ The solution is: $$a(t) = \\frac{t^{1\/2}}{t_0^{1\/2}}$$ where $t_0$ is the current age of the Universe. Now let us calculate the particle horizon distance $D$ given by: $$D = \\int_0^{t_0} \\frac{dt}{a(t)}$$ We find that: $$D = 2 t_0$$ Therefore conventional theory says there is a length scale associated with this Universe. But is that reasonable? PS If for some reason we had $\\rho \\propto 1\/a^2$ so that $a(t) = t\/t_0$ then we could have a truly scale invariant cosmology where both the particle horizon and the cosmological event horizon diverge leaving no length scale."}
{"id":"126014","title":"What's wrong with my calculation of gravitational potential for a uniform sphere?","text":"This is really embarrassing, but I'm not quite sure where I'm going wrong here... Why is this calculation of the gravitational potential inside a sphere with uniform mass distribution incorrect? **Set-Up** Let's say the sphere has mass $M$ and radius $R$ (and uniform mass density $\\mu$), and what we want to find is the potential at any distance $r$ from the center of the sphere, where $r<R$. We normalize the potential to zero at infinity. **Calculation** The potential $\\phi(r)$ is equal to the potential right outside of the sphere, plus the potential difference between some point inside the sphere and a point right outside. $$ \\phi(r)=\\phi_0-\\int_R^r \\frac{\\mu G}{r}dV $$ (Sorry for using $r$ for the upper limit of the integral as well as for the variable in the integrand. Hopefully this doesn't cause confusion.) Now to figure out the different aspects of the above equation equation. The potential right outside of the sphere is: $$\\phi_0=-\\frac{MG}{R}$$ The differential volume element can be expressed as the constant-potential spherical shell's surface area times the shell's differential width: $$dV=4\\pi r^2 dr$$ And one final detail, the mass density of the sphere: $$\\mu=\\frac{3M}{4\\pi R^3}$$ Using this information, $$\\phi(r)=-\\frac{MG}{R}-\\frac{3MG}{R^3}\\int_R^r r dr$$ $$\\phi(r)=-\\frac{MG}{R^3}\\left[R^2+\\frac{3r^2}{2}-\\frac{3R^2}{2}\\right]$$ $$\\phi(r)=-\\frac{MG}{2R^3}(3r^2-R^2)$$ **Conclusion** This result disagrees with a few places I've visited, like this one, which states that the correct result (in terms of the variables I've used) is $$\\phi(r)=-\\frac{MG}{2R^3}(3R^2-r^2)$$ Both results give the same potential at $r=R$, obviously, but my result starts to look ridiculous for values like $r=R\/2$. The only part in my calculation that seems sketchy to me is that first equation, where I talk about the potential difference at points inside and outside of the sphere; I don't know if it's correct to be dividing by $r$ in the integrand... Or maybe I just made a stupid algebra mistake somewhere in there. Where did I go wrong?"}
{"id":"126019","title":"What happens to \"large objects\" within very small time periods?","text":"What can we say about the state of \"large things\" within very small time periods? While reasonable (or \"useful\") divisions between classical and quantum physics are usually made in terms of size, can a similar division be made in terms of time? If so, what scale of time would that be? For example, suppose someone is \"sitting at a desk\". During very small time periods we cannot specify exactly where all the particles of the body are, right? Does that mean that the body (and everything else for that matter) is statistically \"spread out\" in small time periods?"}
{"id":"130069","title":"Understanding EM Fields","text":"I am an electronics engineering major and some questions arise when studying communications technology that utilizes wireless technology. In particular, I am more of a complete picture kind of person, as such I like to see how various things are connected. I want to understand, at least on the surface level, how the following things are connected: Phonons, photons, electrons, charge fields. As far as I understand things you have phonons. These are quasi-particles representing the quantized energy packets of vibrations through a lattice structure. This seems intuitive enough, and after some further reading I see that photons can be spontaneously emitted from photon interactions resulting from absorption and re-emission of the quantized energy of the phonons. I assume the photon emission frequency\/energy is directly related to the energy of the phonon. I believe that photons can also cause phonons when a photon is incident to a lattice structure, though I am not 100% on the details of this. Photons are actual particles of quantized packets of energy propagating through the EM field medium. As to what exactly constitutes the EM field I am not sure. I believe magnetic fields are due to electrons? I also believe that electrons have an electric field due to the inherent charge electrons pose? However, I believe the charges reach\/influence is based on the inverse square law. As such, is the electromagnetic field simply composed on the electron charges as the medium? If this is the case, then how can the EM field be so far reaching? I apologize for the large amount of questions, and I am sure there are similar questions around, but none that seem to tie all of these things together. EDIT: To add, am I correct to assume the EM Field is modeled as a continuous distribution of charge, as such the inverse square charge distance is a non- factor? And to answer one of the comments in the question, I have taken classes on partial differential equations, but my physics did not require them, thus I have only indirectly utilized Maxwell's Equations. I assume those will help connect a few of my questions mathematically, so I'll be sure to take a look at those."}
{"id":"1018","title":"Explanation: Simple Harmonic Motion","text":"I am Math Grad student with a little bit of interest in physics. Recently i looked into the wikipedia page for Simple Harmonic Motion. Guess, I am too bad at physics to understand it. Considering me as a _layman_ how would one explain:   * What a Simple Harmonic motion is? And why does this motion have such an equation $$x(t)= A \\cos(\\omega{t} + \\varphi)$$  associated with it? Can anyone give examples of where **S.H.M.** is tangible in Nature."}
{"id":"1019","title":"Common false beliefs in Physics","text":"Well, in Mathematics there are somethings, which appear true but they aren't true. Naive students often get fooled by these results. Let me consider a very simple example. As a child one learns this formula $$(a+b)^{2} =a^{2}+ 2 \\cdot a \\cdot b + b^{2}$$ But as one mature's he applies this same formula for Matrices. That is given any two $n \\times n$ square matrices, one believes that this result is true: $$(A+B)^{2} = A^{2} + 2 \\cdot A \\cdot B +B^{2}$$ But eventually this is false as Matrices aren't necessarily commutative. I would like to know whether there any such things happening with physics students as well. My motivation came from the following MO thread, which many of you might take a look into:   * http:\/\/mathoverflow.net\/questions\/23478\/examples-of-common-false-beliefs-in-mathematics"}
{"id":"129973","title":"Current in a circuit with resistors and capacitors","text":"Suppose we have a circuit with both resistors and capacitors and some voltage source. If we want to calculate the current in the circuit, do we apply Ohms law? My doubt is that this would only take into consideration the resistors and not the capacitors, so do the capacitors play any role in the current flowing through the circuit?"}
{"id":"114505","title":"Are states from an unpolarized source beam distinguishable or not","text":"I have seen this pop up twice so far in my reading of Feynman Lectures III. For example, in the first equation of section 5-8 (it may take up to a minute for the equation to load), where the beam is ejected from a furnace and so states are randomly produced, the probablities are added rather than first interfering the probability amplitudes and taking absolute-squared. But I thought nothing was done to distinguish the states $+S, 0S, -S$. Moreover, in the situation of the fancy \"extended\" Stern-Gerlach apparatus where I split the beam into three beams according to the $S$ states but recombine them without blocking any of the split beams, I perform future calculations by interfering the three amplitudes. I am convinced here that I have to do interference. But why is the first paragraph different?"}
{"id":"93215","title":"Special Relativity Time problem","text":"Suppose there are two observers, they each own a light clock which were synchronised in the beginning and are moving relative to each other. So each of them accuses that the other person's clock is running slow. Now they see a concrete event, like someone clapping or a bulb shining and note down the reading on their clock and send each other the information, now if the two pieces of information were different, which means one of the clock was actually running slow, so they will know who was moving actually. If the readings are same, then how will you justify Lorentz-Transformations for time ?"}
{"id":"61333","title":"Can dark matter be relativistic dust?","text":"As far as I know the mass of an observed object increases as it approaches the speed of light. Is it possible that the excess mass called \"dark matter\" is due to relativistic dust? Surely, stars cannot account for that excess mass since we can measure their speeds and they are not relativistic. However, can we observe and measure the velocity of dust in the galaxy? Maybe the dust could account for missing matter..."}
{"id":"110352","title":"Why can we simply absorb the positive coefficient of $i\\epsilon$ in a propagator?","text":"As far as I know, absorbing of the positive coefficient of $i\\epsilon$ in a propagator seems to be a trivial operation without even the need of justification. In Peskin page 286, he did this: $$k^0\\rightarrow k^0(1+i\\epsilon)$$ $$(k^2-m^2)\\rightarrow (k^2-m^2+i\\epsilon)$$ In M. Srednicki's Quantum Field Theory, page 51, > The factor in large parentheses is equal to > $E^2-\\omega^2+i(E^2+\\omega^2)\\epsilon$, and **we can absorb** the positive > coefficient in to $\\epsilon$ to get $E^2-\\omega^2+i\\epsilon$. Why and does this kind of manipulation affect the final result of calculation?   Although $\\frac{1}{k^2-m^2+i\\epsilon k^2}-\\frac{1}{k^2-m^2+i\\epsilon}$ is infinitesimal, but the integration of such terms may lead to divergences, and this is my worry. Also the presence of $k^0$ in the coefficient of $i\\epsilon$ could potentially influence the poles of an integrand and consequently influence the validity of Wick Rotation."}
{"id":"28863","title":"What is the function of the top point of a bouncing ball?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/fqSMt.png) A ball is thrown away as parallel to x axis from M(0,h) point with speed V . After each jumping on x axis , it can reach half of previous height as shown in the figure.(Assume that no any air friction and the ball is very small.) Question-$1$: Find f(x) that passes from of top points ? Question-$2$: I have just a claim about the system but I need to prove that if we throw away the ball from a point of $f(x)$ as parallel to x axis with the speed of $V_{initial}=V\\sqrt{\\frac{f(x)}{h}}$, the maximum points of the ball will be on same $f(x)$. How can the claim be proved or disproved? * * * My attempt to find f(x) $f(0)=h$ $f(x_1+x_2)=h\/2$ $f(x_1+2x_2+x_3)=h\/4$ $f(x_1+2x_2+2x_3+x_4)=h\/8$ . . . $x_1=vt$ $mgh=\\frac{1}{2}mV^2$ $h=\\frac{1}{2}g t^2$ After first bouncing , On first top point: N $(x_1+x_2,h\/2)$: $mgh\/2=\\frac{1}{2}mV_1^2$ $V_1^2=V^2\/2$ $V_1=\\frac{V}{\\sqrt{2}}$ $h\/2=\\frac{1}{2}g t_2^2$ $t_2^2=t^2\/2$ $t_2=\\frac{t}{\\sqrt{2}}$ $f(x_1+x_2)=h\/2$ $f(Vt+t_2\\frac{V}{\\sqrt{2}})=h\/2$ $f(Vt+\\frac{t}{\\sqrt{2}}\\frac{V}{\\sqrt{2}})=h\/2$ $f(Vt(1+1\/2))=h\/2$ After second bouncing, On second top point: O $(x1+2x2+x_3,h\/4)$: $mgh\/4=\\frac{1}{2}mV_2^2$ $V_2^2=V^2\/4$ $V_2=\\frac{V}{2}$ $h\/4=\\frac{1}{2}g t_3^2$ $t_3^2=t^2\/4$ $t_3=\\frac{t}{2}$ $f(x_1+2x_2+x_3)=h\/4$ $f(Vt+Vt+V_3t_3)=h\/4$ $f(Vt+Vt+Vt\/4)=h\/4$ $f(Vt(1+1+1\/4))=h\/4$ On 3th top point: $f(Vt(1+1+1\/2+1\/8))=h\/8$ On 4th top point: $f(Vt(1+1+1\/2+1\/4+1\/16))=h\/16$ . . On nth top point: $n--->\\infty$ $f(3Vt)=0$ How can I find $y=f(x)$ Thanks for answers * * * EDIT: I have found $f(x)$. MrBrody is right. The function is a line. my Proof: On nth top point: $f(Vt.a(n))=h\/2^n$ $a(0)=1$ $a(1)=1+1\/2=1+1-1\/2$ $a(2)=1+1+1\/4=1+1+1\/2-1\/4$ $a(3)=1+1+1\/2+1\/8=1+1+1\/2+1\/4-1\/8$ $a(4)=1+1+1\/2+1\/4+1\/16=1+1+1\/2+1\/4+1\/8-1\/16$ . . $a(n)=1+1+1\/2+1\/4+....1\/2^{n-1}-1\/2^{n}$ $a(n)=1+\\frac{1-(1\/2)^{n})}{1-1\/2}-1\/2^{n}=1+2(1-(1\/2)^{n})-1\/2^{n}=3-\\frac{3}{2^n}$ On nth top point: $$f(Vt[3-\\frac{3}{2^n}])=\\frac{h}{2^n}$$ where $t=\\sqrt{\\frac{2h}{g}}$ $Vt[3-\\frac{3}{2^n}]=x$ $3-\\frac{3}{2^n}=\\frac{x}{Vt}$ $1-\\frac{1}{2^n}=\\frac{x}{3Vt}$ $1-\\frac{x}{3Vt}=\\frac{1}{2^n}$ $$f(x)=\\frac{h}{2^n}=h(1-\\frac{x}{3Vt})=h(1-\\frac{x}{3V\\sqrt{\\frac{2h}{g}}})$$ Now I am trying to prove or disprove my claim in second question. If I find the solution I will post it."}
{"id":"82472","title":"Finding out the potential","text":"![enter image description here](http:\/\/i.stack.imgur.com\/v5LA9.png) > According to me, if we want to find out the potential the the equation will > be, $$dV = \\int \\frac{dQ}{4 \\pi \\epsilon_0 x}$$. But the answer is given is on the basis of $$dV = \\int \\frac{dQ}{4 \\pi \\epsilon_0 \\sqrt{(R^2+ x^2)}}$$. > Why the the distance x from the center of the radious is not allowed to get > the potential?"}
{"id":"82577","title":"is it possible to have magnetic flux density B not in the same direction of magnetic field intensity H?","text":"it is said that direction of magnetic flux density B in the same direction of magnetic field intensity H for isotropic media so what is isotropic media and is it possible to have B not in the same direction of H"}
{"id":"21897","title":"What is a \"measure equation\" as mentioned by this TeX Users Group guide?","text":"In this TeX Users Group (TUG) document, Typesetting mathematics for science and technology according to ISO 31\/XI by Claudio Beccari, the author makes various typesetting recommendations including: > 9\\. Measure equations should be absolutely avoided in professional > scientific texts; measure equations were somewhat popular before the SI was > universally adopted; now they should not be used any more. They survived in > those countries where the \"English system of units\" is being used, but, > since scientifically speaking this traditional system of units is \"illegal\", > measure equations have no reason to be used anymore. What are these \"measure equations\" he talks of? I take it that should he write say $7.25\\,\\text{cm}$, he would call $7.25$ the \"measure\" and $\\text{cm}$ the \"unit of measure\". From this and the context of the quoted paragraph I would guess that a measure equation is a relationship between different units of the same quantity, such as \" $1\\,\\text{in}\\equiv2.54\\,\\text{cm}$ \". However I find such things handy, or at least not unnecessary or obsolete, especially when I'm working on atomic scales where SI isn't the most convenient. Is this what the author means and if so, are they really that bad?"}
{"id":"21891","title":"Why are scattering matrices unitary?","text":"In Griffith's QM book, he introduces scattering matrices as an end-of-the- chapter problem. For a Dirac-Delta potential $V(x) = \\alpha \\delta (x - x_0)$, I've derived the scattering matrix and observed that it is unitary $S^{-1} = S^{\\dagger}$. I'm trying to explain why this is intuitively, but I don't really have an intuitive picture of what hermitian conjugation $S^{\\dagger}$ is doing here. Thoughts?"}
{"id":"10835","title":"Experimental evidence showing the kinetic energy of an electron changes in a static non-uniform magnetic field?","text":"In a previous question, Does a magnetic field do work on an intrinsic magnetic dipole?, one highly rated answer suggested that static magnetic fields do work on intrinsic magnetic dipoles in a non-uniform magnetic field. I can visualise the change in kinetic energy of the nucleus of an atom coming from a change in the configuration of the electrons around the nucleus. But for an electron, since it's truly fundamental, I'm scratching my head over where the energy comes from to change its kinetic energy. If it does, then it really must come from the static magnetic field. So what is the experimental evidence that shows the kinetic energy of an electron changes in a static non-uniform magnetic field?"}
{"id":"10833","title":"Michelson rotating mirror experiment","text":"Could someone explain the calculation required to answer this question. It is from a text book and the answer is recorded as 585Hz but I cannot replicate the answer. > In 1931 Michelson used a rotating prism with 32 faces to measure the time > for light to travel through a 1.6km evacuated tube to a distant mirror and > back. From his results he calculated that the speed of light is 299774000 > m\/s. > > Assuming the prism is rotated by only one face during the time of flight of > light calculate the rotation frequency of the prism. Is the book incorrect or am I? My attempt: $(1\/(1600\/299774000))\/32=5855\\:\\mathrm{Hz}$ which is suspiciously similar to the suggested answer 585Hz. But then I remembered about the round- trip (3200m) which threw all the numbers again."}
{"id":"104899","title":"Renormalizing with external momenta set to zero","text":"I've often seen in textbooks that authors renormalize diagrams by setting external momentum to zero. Under what conditions is this justified? An example of this is done in Manohar and Wise's book on Heavy Quark Physics after they renormalize QED and then calculate the operator renormalization, $ Z _S $, of \\begin{equation} S = \\frac{1}{ Z _S }\\bar{\\psi} _b \\psi _b = \\frac{ Z _\\psi }{Z _S } \\bar{\\psi} \\psi \\end{equation} where $ \\psi_b $ is the bare field. They calulate this through the diagram, $\\hspace{5cm}$ ![enter image description here](http:\/\/i.stack.imgur.com\/wWqzl.png) where the cross indicates an operator insertion. The authors then say ``The operator, $ S $, contains no derivatives (and $Z _S$ is mass independent in the $\\overline{MS}$ scheme), so $Z _S$ can be determined by evaluating (the diagram) at zero external momentum (and neglecting the (fermion) mass).'' Are these the two conditions necessary,   1. The operator doesn't have derivatives   2. The quantity of interest has no mass dependence and if so how do we know that the quantity you want to calculate (in this case, $ Z _S $) is mass independent, ahead of time?"}
{"id":"117362","title":"Nuclear fission mechanism: neutron capture","text":"Consider this nuclear fission reaction: $\\mathrm{^{235}U+{}^1n\\to{}^{236}U \\text{ (excited)}\\to{} ^{92}Kr+{}^{141}Ba+3{}^1n}$ I have not understand why a thermal neutron ($^1$n with $E\\simeq 0.025$ eV) has enough energy to start the reaction. I mean, to start the reaction the $^1$n should be captured by the nuclei of $^{235}$U. What are the conditions under which this neutron is captured? My reasoning is the following: the energy that the neutron must have to be captured by the $^{235}$U should be equal to the $S_\\mathrm n$ (separation energy for a neutron) of $^{236}$U which I found is 6.34 MeV, higher than the energy of the thermal neutron. So my reasoning must be wrong but I cannot understand where I make mistakes... I looked on the Povh Rith Particles and Nuclei but it does not explain the reaction well. Could anyone give me a reference or explain here the fission reaction ?"}
{"id":"61801","title":"phase transition by sublimation","text":"Why do sublime solids sublimate in the first place? Is it that their melting point and their liquid's boiling point are the same so that they manage to magically skip the liquid phase? (if so, why does it have to be so?)And is the reverse possible?"}
{"id":"36191","title":"Lagrangian and Equilibrium Points","text":"I'm wondering whether you can tell quickly just from looking at a Lagrangian whether a given point $q^0$ is an equilibrium point. Obviously all you have to do is verify it satisfies the E-L equations, but they are messy in complicated situations. Is there a conceptual shortcut I could take, or am I expecting too much? Many thanks in advance!"}
{"id":"129027","title":"Why does absorption cause seismic pulses to increase in length over distance?","text":"The specific question I'm trying to answer is \"How does the progressive loss of higher frequencies in a propagating seismic pulse lead to an increase in pulse length?\" I understand how the higher frequencies attenuate faster than lower frequencies (through the absorption constant), thus resulting in a change of shape but not so much how this increases the pulse length? I've been researching for a while and have come across multiple ideas including: 1) I've read that in non-porous media velocity = freq x wavelength, as the frequency components reduce in value, the wavelength would increase to balance this? Though I'm not sure I can assume the media is non-porous. 2) I've been reading about wave dispersion which discusses how higher frequency components move faster than lower frequency components in dispersive media, thus the wavelength increases. My issue with this though is that the question specifically refers to the loss of higher frequencies influencing the pulse length, and just talking about different frequencies having different velocities doesn't seem to cut it. 3) Geometrical spreading could be increasing the wavelength? Though again this doesn't really relate to attenuation as the question is asking. Any help letting me know what I'm missing here would be fantastic. This question has been bugging me for a couple of days now. Cheers"}
{"id":"21275","title":"Trying to understand Laplace's equation","text":"I'm struggling here so please excuse if I'm writing nonsense. I understand that the gravitational potential field, a scalar field, is given by $$\\phi=\\frac{-Gm}{r}$$ where $\\phi$ is the gravitational potential energy of a unit mass in a gravitational field $g$ . The gradient of this is (a vector field) $$g=-\\nabla\\phi=-\\left(\\frac{\\partial\\phi}{\\partial x},\\frac{\\partial\\phi}{\\partial y},\\frac{\\partial\\phi}{\\partial z}\\right)$$ And the divergence of this vector field is $$\\nabla\\cdot\\nabla\\phi=\\nabla^{2}\\phi=4\\pi G\\rho$$ and is called Poisson's equation. If the point is outside of the mass, then $$\\rho=0$$ and Poisson's equation becomes $$\\nabla\\cdot\\nabla\\phi=0$$ My question is, how do I express $\\phi=\\frac{-Gm}{r}$ as a function of $x,y,z$ so I can then end up with $\\nabla\\cdot\\nabla\\phi=0$ in empty space $r\\neq 0$? I would have thought that I could write $$\\phi=\\frac{-Gm}{r}=\\frac{-Gm}{\\sqrt{x^{2}+y^{2}+z^{2}}}$$ but when I try to calculate $\\nabla\\cdot\\nabla\\phi$ from this, I don't get zero for $r\\neq 0$. I'm confused."}
{"id":"2407","title":"Give a description of Loop Quantum Gravity your grandmother could understand","text":"Of course, assuming your grandmother is not a theoretical physicist. I'd like to hear the basics concepts that make LQG tick and the way it relates to the GR. I heard about spin-networks where one assigns Lie groups representations to the edges and intertwining operators to the nodes of the graph but at the moment I have no idea why this concept should be useful (except for a possible similarity with gauge theories and Wilson loops; but I guess this is purely accidental). I also heard that this spin-graph can evolve by means of a spin-foam which, I guess, should be a generalization of a graph to the simplicial complexes but that's where my knowledge ends. I have also read the wikipedia article but I don't find it very enlightening. It gives some motivation for quantizing gravity and lists some problems of LQG but (unless I am blind) it never says what LQG actually is. So, my questions: >   1. Try to give a simple description of fundamentals of Loop Quantum > Gravity. >   2. Give some basic results of the theory. Not necessary physical, I just > want to know what are implications of the fundamentals I ask for in 1. >   3. Why is this theory interesting physically? In particular, what does it > tell us about General Relativity (both about the way it is quantized and the > way it is recovered from LQG). >"}
{"id":"81001","title":"How can you calculate air resistances at different speeds?","text":"I've read that at 50mph air resistance to an average car is the equivalent of driving through water and at 80mph it's the equivalent of driving through oil. I can't find any references online to back up these figures. Is there a relatively simple way to explain the calculation to verify these figures, or are they incorrect?"}
{"id":"94602","title":"Is the thermal time constant simply the heat capacity divided by thermal conductivity?","text":"My professor gave me this temperature decay data to make an exponential fit and told me that the time constant tau should be heat capacity over thermal conductivity. I'm looking at the wikipedia page and that's not what I'm seeing. I don't know if there's some caveat to when we're doing 2D simulations but I just don't see how the time constant could be exactly heat capacity over thermal conductivity."}
{"id":"98950","title":"Physical interpretation of the change of diffusion term in navier stokes equations","text":"In the Navier-Stokes Equations, there is one term accounting for convective flow and one term for diffusive flow. At high flow rates, the diffusive term becomes much smaller compared to convective term, therefore can be neglected, leading to Euler's equations. I can understand this in terms of mathematics: There is a multiplicative factor $\\frac{1}{\\text{Re}}$ in the diffusive term. With increasing flow rates, the inertial forces become larger than the viscous forces, so the Reynolds number increases and the term for diffusive flow becomes much smaller. But what is the physical interpretation of this? Does the influence of random motions decrease when the flow rate increases?"}
{"id":"90739","title":"Lenz's Law and Eddy Currents","text":"You can determine the direction of eddy currents according to Lenz's law. E.g. If a metal sheet is losing flux into the page, it will experience induced eddy currents in a clockwise direction to replace it. However, this doesn't make sense to me logically. :S Consider a straight current-carrying wire. It has equal flux on either side of it's length. Now warp it into a circle (to simulate eddy currents). It should have equal flux on both outside and inside the the circle. Thus, in the above example, the eddy currents ARE NOT replacing the lost flux into the page, as it is also producing EQUAL flux out of the page (which \"cancels\" the other flux into the page). Am I misunderstanding something about Lenz's Law\/Eddy currents? :S"}
{"id":"3302","title":"Why is honeycomb lattice the ideal shape for solar cells?","text":"Quoting this web site (which I haven't thoroughly verified for credibility yet): > The material is graphene, also known as graphite, a naturally-occurring > mineral that forms a one-atom-thick sheet of honeycomb crystal lattice > material (the ideal shape for solar cells) that has surprisingly robust > electric characteristics. In what context is this ideal shape? In the nano or macro scale? Why?"}
{"id":"113996","title":"How could I determine the temperature at which an egg explodes","text":"A while back, some friends and I pondered \"what will happen to an egg if placed very near a campfire\". So we placed an egg on a grill, approximately 12 inches above a real campfire. Something like 7-10 minutes later, thankfully right as we walked away, the egg exploded in every direction in a gooey, golden glory. So I ask you, **how can I determine the internal temperature of an egg that will cause it to explode**?"}
{"id":"102196","title":"Did a racer on a \"restless\" course race at some particular average velocity? (2: Starting gate and finish line being rigid to each other)","text":"Consider two participants, a starting gate ($A$) and a finish line ($B$), in a (sufficiently) flat region, both undergoing uniformly accelerated motion, such that   * $A$ finds constant ping duration wrt. $B$: for any two (distinct) indications $A_J$ and $A_K$   $\\mathop{\\Delta}\\limits_{\\stackrel{\\large{\\text{until seeing}}}{\\small{\\text{the echo from }B}}} \\\\! \\\\! \\\\! \\\\! \\tau_A[ A_J ] = \\\\! \\\\! \\mathop{\\Delta}\\limits_{\\stackrel{\\large{\\text{until seeing}}}{\\small{\\text{the echo from }B}}} \\\\! \\\\! \\\\! \\\\! \\tau_A[ A_K ] = \\\\! \\\\! ({\\small{\\text{abbreviated as}}}) \\\\! \\\\! = \\mathop{\\Delta}\\limits_{\\small ABA} \\\\! \\tau_A,$   * $B$ finds constant ping duration wrt. $A$: for any two (distinct) indications $B_P$ and $B_Q$   $\\mathop{\\Delta}\\limits_{\\stackrel{\\large{\\text{until seeing}}}{\\small{\\text{the echo from }A}}} \\\\! \\\\! \\\\! \\\\! \\tau_B[ B_P ] = \\\\! \\\\! \\mathop{\\Delta}\\limits_{\\stackrel{\\large{\\text{until seeing}}}{\\small{\\text{the echo from }A}}} \\\\! \\\\! \\\\! \\\\! \\tau_B[ B_Q ] = \\\\! \\\\! ({\\small{\\text{abbreviated as}}}) \\\\! \\\\! = \\mathop{\\Delta}\\limits_{\\small BAB} \\\\! \\tau_B,$ and   * $A$'s ping duration wrt. $B$ is strictly less than $B$'s ping duration wrt. $A$: $\\mathop{\\Delta}\\limits_{\\small ABA} \\\\! \\tau_A < \\mathop{\\Delta}\\limits_{\\small BAB} \\\\! \\tau_B.$  (Evidently, therefore, $A$ and $B$ had not been at rest to each other; but they are rigid to each other.) Let there also be a racer ($R$) who passed the starting gate $A$ and (afterwards) passed the finish line $B$. For definiteness, suppose that $B$'s duration from having seen that $A$ and $R$ had passed each other until having been passed by $R$ equals $B$'s ping duration wrt. $A$.   (Consequently, $A$'s duration from having been passed by $R$ until having seen that $B$ and $R$ had passed each other equals twice $A$'s ping duration wrt. $B$.) **Question:** Can an \"average velocity\" of racer $R$ with respect to the race course, as far as it is specified by starting gate $A$ and finish line $B$ be defined, such that its magnitude could be expressed in terms of the given ping durations $\\mathop{\\Delta}\\limits_{\\small ABA} \\\\! \\tau_A$, $\\mathop{\\Delta}\\limits_{\\small BAB} \\\\! \\tau_B$, and the \"speed of light, $c$\"   (and where its direction is obviously \"from $A$ to $B$\") ?"}
{"id":"29384","title":"Anti-Matter for Neutrons","text":"The anti-particle corresponding to a proton or an electron is a particle with an equal mass, but an opposite charge. So what is the anti-particle corresponding to a neutron (which does not possess a charge)? And if it is just another neutron, will its collision with the original neutron be as destructive as the collision of a proton with an anti-proton or an electron with an anti-electron?"}
{"id":"622","title":"List of Scattering Phenomena","text":"While at lunch with my lab group we got into discussion of the different types of scattering phenomenon that we encounter in everyday life and physical experiments. We ended up listing about a dozen or so types of scattering that we see constantly, but I left unfulfilled. I want to compile a list of scattering phenomenon and a brief description. At the very least this will be helpful for those taking the physics GRE in the near future, otherwise its a good exercise for anyone interested in physics, even those in professional physics. I'll Begin to give you the basic jist of what I'm looking for, feel free to elaborate on the description as much as you want. An equation is useful if it's easily expressed **Compton scattering** : Scattering of a photon off of a particle, photon imparts momentum to the particle and loses some of its energy, the change in wavelength is described by $$\\Delta_{\\lambda} = \\frac{h}{mc}(1-\\cos{\\theta})$$ It is independent of the frequency of the incoming photon and inversely proportional to the mass of the particle scattered from. **Raleigh Scattering** : Elastic Scattering of photons off of molecules and other (micro and macro) particles, photons do not lose energy but change direction. Has a strong dependence on frequency and is the reason the sky is blue (and why you can see laser pointer beams in the dark). P.S. I don't have the reputation to create tags yet, and there are no tags relevant to this question (big-list scattering) please feel free to retag as necessary"}
{"id":"66129","title":"A Musical Pathway","text":"Using a small number of sound emitters, could you create a room where certain nodes emitted particular tones, but no meaningful sound was heard anywhere else. So, for example, by walking down a certain path, you could hear the tones for \"Mary Had a Little Lamb.\" Is there a generalized algorithm to make particular paths for particular tone sets?"}
{"id":"23125","title":"Shouldn't stars disappear from our view due to expansion?","text":"The currently accepted theory of the expansion of the universe would suggest that for any point in the universe, there is a 'boundary' (somewhat like the Schwarzschild radius) beyond which it cannot see the stars that may be there. (since beyond that radius the expansion must appear to be faster than light). However this also implies that stars that are currently visible should disappear when the expansion takes them across that boundary (as some must since the expansion is ongoing). Now even allowing for a very long delay in this being observed from earth say, it still means that after billions of years we must be seeing stars \"dropping\" out of our visibility radius. Is this an observed phenomenon? or am I misunderstanding something?"}
{"id":"72184","title":"How do you build a simple low cost single photon detector?","text":"Rather than buying a 4K-10K single photon detector, I was wondering if there were simple ways to build a simple single photon detector, of decent quality..."}
{"id":"73857","title":"Why do ice cubes stick together or to the edges of a drinking glass?","text":"I was drinking iced-water from a drinking glass (made of glass) at a restaurant yesterday when I was taking a drink, I noticed that there is very little ice water coming out and then suddenly, the ice water mixture comes crashing down. With my open mouth overfilled with ice water, my shirt gets wet – I hate when this happens. After drying myself off, I started thinking why this does happen:   1. How does ice that starts off loose and separated in water fuse together to form one large clump?   2. What causes the ice-water mixture to stop flowing momentarily and then suddenly starting flowing again? Is it that the ice fuses together in the glass and forms one large clump, and then the friction with the edges builds a \"dam-like\" situation? Or are there individual pieces of ice fusing with the edges of the drinking glass damming-up the ice?"}
{"id":"73851","title":"Will a diamond helm be more protective against bullets?","text":"Will a helm made entirely of diamond be more protective against bullets than standard steel or kevlar helmlets?"}
{"id":"25986","title":"What to show a beginner","text":"At my new job, it's soon going to be my turn for doing night\/graveyard shifts for a fair amount of weeks. Perfect excuse to buy a decent beginners telescope to replace the 4.5\" 15 y\/o Meade that doesn't work anymore with a 10\" Dobsonian reflector! My girlfriend wants to join me (which I'm all for), and having grown up on a (relatively poor) farm, she's never had the pleasure of looking through a telescope. When I asked her what she wanted to look at first, she shrugged and said \"I don't know. Show me everything!\" The problem is, the stuff I generally stare at are Nebulae and star clusters (since that's all I really could see), despite knowing that there is so much more out there that are equally enticing to the eye. So my question is this: For any category of objects, what are the most popular? If you have gone to many star parties, what are the objects people are generally asking to see most of the time? What were the things that when you first saw them through a telescope made you go \"Wow!\"? Note: Do understand that the objects generally have to be viewable during Late September (new moon), but having new objects I can show her in the coming months would be awesome as well. I've already told her the journey of the grand tour will probably take 6 months to a year (due to earth rotation and all that,) and will require the joining of us to the local astronomy club."}
{"id":"29655","title":"Why does spin have a discrete spectrum?","text":"Why is it that unlike other quantum properties such as momentum and velocity, which usually are given through (probabilistic) _continuous_ values, spin has a (probabilistic) _discrete_ spectrum?"}
{"id":"91956","title":"Measure mass difference of an object without a scale","text":"For a project i need to separate items based on their mass in real time. I like to explore measuring weight of an object (round metal ball) by taking photographs (several) during free fall and calculate mass from the pictures. Even if I cannot find the real mass I like to get a good estimate of the mass delta among many samples in real time. The basic logic is that I can measure velocity and acceleration from images with good accuracy. And use this data to calculate approximate mass. The density of the object is not controlled tightly neither the volume. Does this approach make sense?"}
{"id":"72453","title":"Euler number of the world sheet","text":"I have a question in the section 3.2 \"The Polyakov path integral\" in Polchinski's string theory p. 83. Given > $$ \\chi=\\frac{1}{4 \\pi} \\int_M d^2 \\sigma g^{1\/2} R + \\frac{1}{2 \\pi} > \\int_{\\partial M} ds \\,\\, k \\,\\,\\,\\, (3.2.3b) $$ It is said > We have noted $\\chi$ is locally a total derivative in two dimensions and > therefore depends only on the topology of the world-sheet -- it is the Euler > number of the world sheet I have two questions related to this sentence... 1) Where did the book note that $\\chi$ is locally a total derivative? 2) Does the Euler number for $\\chi$ come from the Gauss-Bonnet theorem? I never learned that theorem in my differential geometry course :( and Eq. (3.2.3b) is in different form with wiki   by different factors $\\frac{1}{4\\pi}$ and $\\frac{1}{2\\pi}$ (why? convention of normalization?). Would you recommend any reference for the proof of Gauss- Bonnet theorem for physicist?"}
{"id":"91950","title":"How to calculate the van der Waals force from the van der Walls equation?","text":"Given the van der Waals equation $$\\left(p+\\frac{n^2a}{V^2}\\right)\\left(V-nb\\right)=nRT$$ and the van der Waals constants $a$ and $b$, how can I find the van der Walls force between two atoms at distance $d$?"}
{"id":"3873","title":"How do I calculate the perturbations to the metric determinant?","text":"I am trying to calculate sqrt(-g) in terms of a background metric and metric perturbations, to second order in the perturbations. I know how to expand tensors that depend on the metric, but I don't know how to expand the metric determinant. Does anyone know how to to this or (even better) know of a source that does second order perturbation theory in GR?"}
{"id":"77871","title":"Can an asteroid have a magnetic field?","text":"Some M-type asteroids are believed to be metallic, even possessing Nickel- Iron. Given   * the proximity of the asteroid belt to Jupiter's magnetosphere,    * the regularity of motion of any given M asteroid relative to Jupiter   * the ease with which Nickel\/Iron may be magnetized, and    * the strength of Jupiter's magnetic field Is it reasonable to expect an M-type asteroid that fulfills such conditions to exhibit a magnetic field?"}
{"id":"92607","title":"Energy conservation at a the end of a hose pipe?","text":"I am trying to figure out the energy balance at the end of a hose pipe with water pouring out. Applying Bernoulli's equation inside the pipe we have three energy terms: Kinetic energy, \"pressure\" energy and gravitational potential energy. This total energy is conserved unless its converted to heat or whatever through pressure losses. Consider the situation just inside and just outside the end of the hose pipe. Assume the pipe is horizontal so we can ignore the gravitational component. Let say just inside the end of the hose we have 50psi and 2 feet per second. Then consider the situation just outside the end of the hose. We still have 2 feet per second because the fluid stream has a constant diameter (its not a nozzle) and is virtually incompressible. However, we now presumably have atmospheric pressure in the fluid stream, i.e, 14.7psi. So what happened to the energy contained in the pressure difference, i.e, the difference in energy between the water at 50psi and at 14.7psi? Or is the pressure inside the fluid stream suspended in the air still at 50psi? Or does the water expand very slightly to account for the difference between 50psi and 14.7psi and the energy is released by the water doing work against the air to perform this expansion? (And maybe the expansion is so small that it is not visible so the fluid stream appears to be a constant diameter?)."}
{"id":"100138","title":"is an electron gun dangerous?","text":"Is an electron gun dangerous & what would be the repercussions to human tissue? If I were to do a DIY electron gun and put my hand in the beam would I regret it?"}
{"id":"50035","title":"Clarifications about Poisson brackets and Levi-Civita symbol","text":"I need some clarifications about Poisson brackets. I know canonical brackets and the properties of Poisson Brackets and I also know something about Levi-Civita symbol (definition and basic properties), but I have some doubts. I don't know how I could apply Poisson brackets properties if I have a summation, for example if in the case of a system of N particles I have to solve $[L_i, x_{\\alpha j}]$. I know that a generical component of total angular momentum is given by $L_a=\\sum_{\\alpha=1} ^N l_{\\alpha a}$ and also know that the components of algular momentum af a particle is given by $l_a= \\epsilon_{aij} x_i p_j$. Now, if I have to calculate $[L_i, x_{\\alpha j}]$, I have these doubts: 1) how can I decide the indices of Levi-Civita symbol that I'm going to use for solving the problem? 2) how can I use the property of linearity of Poisson brackets in this case? and an other (general) question: 3) If I have a Levi-Civita symbol that multiplyes a sum of two terms and each term is mulplied for a Kronecker delta, I have to follow these steps: a) multiply the Levi-Civita symbol for each term b) impose the condition thanks to each Kronecker delta isn't equal to zero c) eventually, substitute these conditions in the two Levi-Civita symbol, but I have to substitute in each Levi-Civita symbol the condition that I found for that Kronecker delta that at the step a) was multipling just that one Levi- Civita symbol Is it correct this way to go on?"}
{"id":"22299","title":"Good references for specific heats","text":"Does anyone know of a good reference for specific heats? It seems that it is no longer popular to publish values. I am especially interested in the specific heats of polymers."}
{"id":"22295","title":"When should we use the concept of Impulse\/Momentum instead of Force?","text":"In my notes it says > \"The ideas of impulse and momentum is useful in solving problems where:- > > a) the force F is not easily calculable (e.g. sudden impact or blow) > > b) the impulse force is so large that the body suffers deformation (i.e. the > impact is no longer elastic) There is no explanation for what is said. When I think about it, force is the rate of momentum. How\/why could we calculate momentum for small amount of time (sudden impact), but not calculate force?"}
{"id":"10435","title":"Are scientists missing the point with distant cosmic objects, or is it just me?","text":"I was reading this article this morning: http:\/\/www.bbc.co.uk\/news\/science-environment-13539914 Scientists have discovered a gamma-ray burst whose light has taken 13.14 billion years to reach Earth. This much is cool and interesting. However, the assumption is then stated that this is \"the most distant single object yet spied by a telescope\". But hold on a minute. It is also known that galaxies are moving away from each other at incredible speeds, most faster than the speed of light, because the space\/time between them is expanding (or something like that... I'm no scientist!) So these so-called amazingly distant objects, well for starters, they don't exist any more... They are things that happened billions of years ago. But not only that, they are probably objects that were only a few million light years away from us when they actually took place. So surely then, the objects themselves aren't the most distant ones, but the light from them has been distorted such that the light has taken that long to reach us? Furthermore, if the galaxies are spreading out faster than the speed of light, who is to say this explosion actually happened 13.1 billion years ago? Isn't it possible that the light was created say 5 billion years ago, but has taken much longer to reach us because of the expanding space between the galaxies? I'm sure this stuff has already been considered by scientists, but I find it weird the way news articles always assume that just because light travels at a specific speed, that it's always going to take the same amount of time to reach us. Or am I getting it wrong? I'd love to know! EDIT: For people discussing the whole faster-than-light-speed thing, I came across this article: > As you look at galaxies further and further away, they appear to be moving > faster and faster away from us. And it is possible that they could > eventually appear to be moving away from us faster than light. At that > point, light leaving the distant galaxy would never reach us. When that > happens, the distant galaxy would just fade away as the last of the photons > reached Earth, and then we would never know it was ever there. And this one: > That mysterious dark energy force, which is accelerating the expansion of > the Universe is making the most distant galaxies move faster and faster away > from us. Eventually, they will cross an event horizon and appear to be > moving away from us faster than the speed of light. At this point, any light > emitted by the galaxy will cease to reach us. Any galaxy that crosses this > horizon will fade away from view, until its last photon reaches us. All > galaxies will disappear from view forever."}
{"id":"24639","title":"Looking for a way to simplify a physics formula","text":"I have the following physics formula: $$d = \\frac{1}{2} at^2$$ where d is equal to half (at) squared where:   * d is distance   * a is acceleration    * t is time I need to simplify this to get the acceleration. Can anyone think of an easy way to do this? I have been looking around but can't find a simple answer."}
{"id":"24633","title":"What is actually meant by 'sun set' and 'sun rise' times, when taking into account the mirage due to light bending in the atmosphere","text":"I’ve heard from the likes of Brian Cox that what we see of the sun during a sunset and sun rise is actually the mirage of the sun. The Sun has actually set\/risen and we see it due to the way light is bent across the atmosphere. Apparently due to coincidence of the size and distance of the sun, its exactly the same size - so if we see 50% of the sun, the sun is 50% below the horizon. So, I understand all this, so here is my question : If this is the case, then when we read things like what time sun sets and rises on websites, books, calendars, other official times, et al… does that mean when we see for example ‘sun set at 18:35’ is the time denoting the actual sun set taking into account of the mirage or what is visible to us. If I were to know the time and watched the sun against an accurate clock, would the sun not be visible before the actual sun set time? I would also like to know how this affects things like iphone apps that tells us what time sun sets are and where it is in relation to the horizon at a given time. So is the set\/rise time taking into account of the mirage and the time is recalculated to give us when it has ‘actually’ set\/risen or when the mirage has set\/risen. Thanks Repost from : http:\/\/astronomy.stackexchange.com\/questions\/2190\/what-is- actually-meant-by-sun-set-and-sun-rise-times-when-taking-into-accou"}
{"id":"73323","title":"Church-Turing hypothesis as a fundamental law of physics","text":"The Church-Turing hypothesis says one can not build a computing device which has more computing power (in terms of computability) than the abstract model of Turing machine. So, there is something in our laws of physics which prevent us from making devices which are more powerful than Turing machine, so in this respect it can be viewed as a law of physics. What is the physicists' view of Church-Turing hypothesis? Can the Church-Turing hypothesis be deduced from other fundamental law of physics?"}
{"id":"38971","title":"How to measure Projective Symmetry Group in spin liquid?","text":"Quasiparticles in spin liquid will no longer be the representation of symmetry group. So when group elements act on quasiparticles, there will be some phase factor. For example, in $\\pi$ flux state, $T_xT_y=-T_yT_x$, which means if we move the quasiparticle around a lattice, and there will be a minus sign. However, because the quasiparticle are always emerging in pairs, when we move one quasiparticle, it cannot be avoided that we will also move another. In this case, the phase factor (minus sign) will be cancelled out. If we wants to move only one quasiparticle, the symmetry operator must be a local operator. But what does it mean for local operator of the global symmetry? How can we measure the PSG in general cases?"}
{"id":"31832","title":"Lennard-Jones potential - equilibrium","text":"If I have Lennard-Jones potential, how can I calculate equilibrium? $$ U(r)=\\epsilon \\left ( \\left (\\frac{r_m}{r} \\right )^{12}-2 \\left (\\frac{r_m}{r}\\right )^6 \\right ) \\ $$"}
{"id":"101268","title":"Multivariable Dirac Delta and Faddeev-Popov Determinant","text":"From this mathstack page and in particular Qmechanic's answer: >   1. There exists an $n$-dimensional generalization $$\\tag{1} \\delta^n({\\bf > f}({\\bf x})) ~=~\\sum_{{\\bf x}_{(0)}}^{{\\bf f}({\\bf > x}_{(0)})=0}\\frac{1}{|\\det\\frac{\\partial {\\bf f}({\\bf x})}{\\partial {\\bf x}} > |}\\delta^n({\\bf x}-{\\bf x}_{(0)}) $$ of the substitution formula for the > Dirac delta distribution under pertinent assumptions, such as e.g., that the > function ${\\bf f}:\\Omega \\subseteq \\mathbb{R}^n \\to \\mathbb{R}^n$ has > isolated zeros. Here the sum on the rhs. of eq. (1) extends to all the zeros > ${\\bf x}_{(0)}$ of the function ${\\bf f}$. > Also from this page on the Faddeev-Popov procedure they say: > For ordinary functions, a property of the Dirac delta function gives: > $\\delta(x-x_0) = \\left|\\frac{df(x)}{dx}\\right|_{x=x_0}\\delta(f(x))\\,$ > assuming $f(x)\\,$ has only one zero at $x=x_0\\,$ and is differentiable > there. Integrating both sides gives :$$1 = > \\left|\\frac{df(x)}{dx}\\right|_{x=x_0}\\int\\\\!dx\\,\\delta(f(x))\\,$$. Extending > over $n$ variables, suppose $f(x^i) = 0\\,$ for some $x^i_0\\,$. Then, > replacing $\\delta(x-x_0)\\,$ with $\\prod_i^n \\delta^i(x^i-x^i_0)\\,$ :$$1 = > \\left(\\prod_i \\left|\\frac{\\partial f(x^i)}{\\partial x^i}\\right|\\right) > \\int\\\\!\\left(\\prod_i dx^i\\right)\\,\\delta(f(x^i))\\,$$. Recognizing the first > factor as the determinant of the diagonal matrix $\\frac{\\partial > f(x^i)}{\\partial x^i}\\delta^{ij}\\,$ (no summation implied), we can > generalize to the functional version of the identity: :$$1 = > \\det\\left|\\frac{\\delta G}{\\delta \\Omega}\\right|_{G=0} > \\int\\\\!\\mathcal{D}\\Omega\\,\\delta[G_a(\\phi^\\Omega)]\\,$$, where > $\\Delta_F[\\phi] \\equiv \\det\\left|\\frac{\\delta F}{\\delta g}\\right|_{F=0}\\,$ > is the Faddeev-Popov determinant. What I don't understand is that it seems their function $f$ seems to be $f:\\Omega \\subseteq \\mathbb{R}^n \\to \\mathbb{R}$. **How does the generalized Dirac formula $(1)$ work in this case?** I don't really understad their notation in: $$1 = \\left(\\prod_i \\left|\\frac{\\partial f(x^i)}{\\partial x^i}\\right|\\right) \\int\\\\!\\left(\\prod_i dx^i\\right)\\,\\delta(f(x^i))\\,$$ What does $$\\frac{\\partial f(x^i)}{\\partial x^i}$$ mean here?"}
{"id":"101261","title":"Intuitive understanding of wavelength","text":"Light is described as having wavelength. I can somewhat understand this in connection with for example the double slit experiment, that photons interact spatially in a wave like manner. But can the wavelength of light be described in a way which more intuitively relates to the waves of sound or oceans? Does \"the length\" have a specific orientation relative to the path of light? Doppler shifts seem to indicate that the wavelength is measured in the direction of light's movement, the speed stretches it out. But the double slit experiment, and the shapes of antennas, seem to indicate that the wavelength is meassured perpendicular to light's movement. How large is a photon and what shape does it have in terms of its \"length\"?"}
{"id":"4575","title":"neutrinos by formation of \"neutron pairs\"","text":"Here : http:\/\/www.newscientist.com\/article\/dn20084-neutron-star-seen-forming-exotic- new-state-of-matter.html are news on superfluidity in a neutron star. The necessary bosons they say are pairs of neutrons. So far, so good. But then they postulate the production of neutrinos in the formation of those neutron pairs. > When neutrons pair up to form a superfluid they release neutrinos which > should pass easily through the star,... What kind of reaction is that? n + n => neutrino + \"n-n\" + \"XYZ\" ?"}
{"id":"130109","title":"Why don't we experience dilatations in Minkowski spacetime?","text":"**The question** This question follows on from the use of projective coords for spacetime in Notation for Translation Group Generators . Under Felix Klein's Erlangen Program, Minkowski spacetime starts as a 4-d projective space. It is then made into an affine space (which has a notion of parallelism) by picking an invariant hyperplane at infinity; lines in the projective space which meet at infinity are parallel. An affine space admits dilatations which are expansions or contractions of space about the eigenpoint of the dilatation. When introduced projectively[1], dilatations are seen to be generalized translations because a translation is a dilatation in which the eigenpoint is on the hyperplane at infinity. Since one can smoothly change a dilatation into a translation by moving the eigenpoint onto the hyperplane at infinity, it is hard to avoid the conclusion that a dilatation is an active transformation on the same footing as translations, rotations and boosts. Yet we don't experience dilatations in everyday life. Why is this? **Some background observations** The general feeling in physics seems to be that dilatations are changes in the units of measurement, but this view is only possible because dilatations are introduced in an ad hoc way as scale transformations and the link with translations is missed. In order to see that dilations are active transformations like translations, the translation generators of Notation for Translation Group Generators become, $$ [P_{\\Gamma}]^{\\Phi}_{\\ \\Psi}=\\delta^{\\Phi}_{\\Gamma}\\delta^{4}_{\\Psi} $$ where indices $\\Phi,\\Psi,\\Gamma=0,1,2,3,4$. The translations are $P_{0},P_{1},P_{2},P_{3}$ and the dilatation generator is $P_{4}$. The attempt to construct classical relativistic Hamiltonian particle mechanics in Minkowski spacetime with dilatations fails unless the particles are massless. Massive particles only exist by removing the dilatations by hand in order to restrict the group to the ten parameter Poincare group. **Edit: An attempt to derive dilatations without having to read Coxeter [1].** Start with Minkowski spacetime as a 4-d projective space; points are rays in a 5-d vector space $x^{\\Phi}\\in V_{5}$ with upper case Greek indices like $\\Phi=0,1,2,3,4$. The spacetime group is GL(5,R). Introduce parallelism by picking an invariant hyperplane $\\delta^{4}_{\\Phi}\\in \\tilde{V}_{5}$. Invariance means, $$ [D(g^{-T})]_{\\Phi}^{\\ \\Psi}\\delta^{4}_{\\Psi}=\\xi \\delta^{4}_{\\Phi} $$ where $\\xi$ is a scale factor which is unimportant because the physical entities are rays. This implies that the group matrices are now have the block form, $$ \\begin{equation} D(g)= \\left[ \\begin{array}{cc} \\Lambda & a \\\\\\ 0 & 1+a^{4} \\end{array} \\right] \\end{equation} $$ where $\\Lambda^{\\mu}_{\\ \\lambda}$ is a 4x4 matrix and $a^{\\mu}$ is a 4x1 vector and lower case Greek indices run $\\mu=0,1,2,3$. These affine group matrices factor as, $$ \\begin{equation} D(g)= \\left[ \\begin{array}{cc} 1 & a \\\\\\ 0 & 1+a^{4} \\end{array} \\right] \\left[ \\begin{array}{cc} \\Lambda & 0 \\\\\\ 0 & 1 \\end{array} \\right] \\ . \\end{equation} $$ The first matrix is a dilation. In tensor notation, the pure dilatations are, $$ [D(g)]^{\\Phi}_{\\ \\Psi}=\\delta^{\\Phi}_{\\ \\Psi}+a^{\\Phi}\\delta^{4}_{\\Psi} $$ and a point transforms under a dilatation as, $$ x'^{\\Phi}=[D(g)]^{\\Phi}_{\\ \\Psi}x^{\\Psi}=x^{\\Phi}+a^{\\Phi}x^{4} \\ . $$ This shows that a dilatation acts like the identity on points in the hyperplane at infinity (the points which we can write as $x^{\\mu}$ because $x^{4}=0$) and the eigenpoint of the dilatation is $a^{\\Phi}$ with eigenvalue $1+a^{4}$. The last equation implies that a dilatation is an expansion or contraction of spacetime about the eigenpoint because a dilatation is a linear combination of the point and the eigenpoint. Consider two spacetime points $x^{\\Phi}$ and $y^{\\Phi}$. Transform them to $x'^{\\Phi}$ and $y'^{\\Phi}$ under a dilatation with eigenpoint $a^{\\Phi}$. The dilatation moves the line joining $x$ and $y$ parallel to itself so that the line $xy$ (in an informal notation) is parallel to line $x'y'$. To see this, the point where the line $xy$ intersects the hyperplane at infinity is, $$ b^{\\Phi}=\\delta^{\\Phi \\Psi}_{[\\Gamma\\Delta]}x^{\\Gamma}y^{\\Delta}\\delta^{4}_{\\Psi} =x^{\\Phi}y^{4}-x^{4}y^{\\Phi} $$ and the dilatation acts like the identity on $b$ which is in the hyperplane at infinity, so line $x'y'$ also intersects the hyperplane at infinity at $b$ and so $xy$ and $x'y'$ are parallel lines. Now consider the quadrilateral $x'y'yx$. We have just seen that the sides $xy$ and $x'y'$ are parallel. The quadrilateral can be made into a parallelogram if the sides $xx'$ and $yy'$ are also parallel. These lines are parallel if they intersect at a point in the hyperplane at infinity. But, these lines both go through the eigenpoint $a$ of the dilatation. Therefore, $xx'$ and $yy'$ are parallel if the eigenpoint $a$ is moved onto the hyperplane at infinity. With the eigenpoint $a$ on the hyperplane at infinity, we've made a parallelogram by starting with a line $xy$ and moving it parallel to itself to $x'y'$ with $xx'$ and $yy'$ also parallel. This is how to construct a parallelogram with a translation. In other words, a translation is a dilatation with its eigenpoint on the hyperplane at infinity. This construction is why I cannot see a reason to exclude dilatations from the spacetime group, it seems dilatations must be just like rotations and boosts. [1] Coxeter, \"Introduction to Geometry\", chapter 5."}
{"id":"122735","title":"Is proper time an intrinsic value of Minkowski space?","text":"What is proper time? Is it a part of Minkowski space (that is a mere spacetime interval)? Or is it an intrinsic characteristic of massive particles (a sort of \"aging\")? Example: In the following Minkowski diagram,   * A is the worldline of a particle moving with the observer's reference frame.    * B is the worldline of another particle with proper time between two events = 4 (space distance x= 3, time interval t = 5, proper time τ = 4, according to the equation $$ \\tau^2 = \\delta t^2 - \\delta x^2 $$)   * C is just 2 events without any worldline between, and if a particle would travel through both events, its proper time would be 4 ![enter image description here](http:\/\/i.stack.imgur.com\/3IKCG.jpg) Is there proper time in C even if no particle is travelling there? Or is in C only hypothetic proper time in case a particle is travelling this path?"}
{"id":"41404","title":"How many photons can an electron absorb and why?","text":"1. How many photons can an electron absorb and why?   2. Can all fundamental particles that can absorb photons absorb the same amount of photons and why?   3. If we increase the velocity of a fundamental particle, do we then increase the amount of photons it can emit?   4. At constant temperature and mass of an isolated fundamental particle that does not and will not move (constant speed and 0 vibration ), is emitting a photon the only way to loose energy?"}
{"id":"83950","title":"Under what conditions, will a projectile motion follow a half circle?","text":"What came to my mind is to throw an object far from the earth with a particular speed equal to $$V_{sat}=\\sqrt{\\frac{G\\times M_{central}}{R}}$$like satellites. However, it would follow a full circle."}
{"id":"39731","title":"Charge, velocity-dependent potentials and Lagrangian","text":"Given an electric charge $q$ of mass $m$ moving at a velocity ${\\bf v}$ in a region containing both electric field ${\\bf E}(t,x,y,z)$ and magnetic field ${\\bf B}(t,x,y,z)$ (${\\bf B}$ and ${\\bf E}$ are derivable from a scalar potential $\\phi(t, x, y, z) $and a vector potential ${\\bf A}(t,x,y,z)$), knowing that   1. ${\\bf E}=- \\nabla \\phi - \\frac{\\partial {\\bf A}} {\\partial t}$   2. ${\\bf B}= \\nabla \\times {\\bf A} $   3. $U=q \\phi - q {\\bf A} \\cdot{\\bf v}$ ($U$ is the velocity-dependent potential) the Lagrangian is $$L=1\/2 m v^2- q\\phi + q{\\bf A} \\cdot{\\bf v}.$$ Considering just the x-component of Lagrange's equation, how can I obtain $$m \\ddot{x}=q(v_x \\frac{\\partial A_x}{\\partial x} + v_y \\frac {\\partial A_y}{ \\partial x} + v_z \\frac{\\partial A_z} {\\partial x})-q(\\frac{\\partial \\phi }{\\partial x} + \\frac{d A_x}{dt}) ?$$"}
{"id":"95010","title":"Effect on length of day as the polar ice caps melt","text":"If the polar ice caps of the earth melt, how will the length of the day be affected?"}
{"id":"71363","title":"Scientist propose solution to quantum measurement problem, explain further","text":"Link to Article The scientists > In their recent opus magnum, Theo Nieuwenhuizen (Institute of Physics, UvA) > and colleagues claim to have found a solution to the so-called quantum > measurement problem Their claim > After reviewing the literature on measurement models, they focus on the so- > called Curie-Weiss model for quantum measurements. The joint dynamics of the > tested system coupled to the measurement apparatus now produces results that > explain the measurement postulates. It shows that the off-diagonal terms > (\"cat-terms\") disappear in a physical process due to the coupling to the > apparatus, on a basis selected by the interaction Hamiltonian. The > registration of the measurement outcomes by the pointer of the apparatus > occurs because the measurement triggers the transition from the initial > metastable state to one of the stable final states; the apparatus being > large, this means an amplification of the signal. The transition of the > pointer variable from its initial metastable state to one of the final > stable states is a process with classical features, such as the dumping of > excess free energy in the bath. I am by no means a physicist, but I do enjoy trying to understand some of the issues they face. From what I've read this quantum measurement problem, well is a problem. Could someone explain why it is a problem, and how these scientist from the Institute of Physics propose to solve it? (as layman as possible)"}
{"id":"133361","title":"N=4 SYM from Klebanov-Witten field theory","text":"This is with reference to M. J. Strassler's lectures on \"The Duality Cascade\" pg. 46. I want to see how $\\mathcal{N}=4$ SYM emerges when D3 branes, in the KW setup, are placed at smooth point of the conifold, purely from field theory point of view. That is: Given a $U(N) \\times U(N)$ quiver with $A_i$ and $B_i$ in the bi- fundamental and the anti-bi-fundamental respectively and an $SU(2) \\times SU(2)$ invariant quartic superpotential (which vanishes for $N=1$), in what manner do we higgs the theory so as to get $\\mathcal{N}=4$ SYM? For the case of $N=1$ there is a comment in the lectures. It says, > ... suppose we just have one D3 brane and we allow $A_1B_1$ to have an > expectation value, so that the D-brane sits at some point away from the > singular point of the conifold. Then the gauge group is broken to $U (1)$, > and six scalars remain massless — the six possible translations of the > D3-brane away from its initial point — exactly the number needed to fill out > an $\\mathcal{N}= 4$ $U(1)$ vector multiplet. I want to understand this statement. Please help me understand how to work out for case of $N=1$. I can try for the generic value of $N$."}
{"id":"12917","title":"how can you keep a gyroscope turning?","text":"I'm trying to understand how do gyroscopes and cmgs used for navigating keep on spinning even though there's friction. Any engine\/pushing mechanism I can think of will restrict the gyro's rotor to a fixed direction and defeat the purpose... how is it done without disturbing it?"}
{"id":"7027","title":"Amplitude of power spectral density","text":"Why is the amplitude of the Power spectral density higher for shorter period of time as compared to a longer period of time when calculated for any vibration data?"}
{"id":"123464","title":"Potential energy of a rotating bar","text":"I have a bar of length $L$, mass $m$, and negligible width. The bar can rotate along an axis through its center of mass (the rotation is in the $x$-$y$ plane), and also can move up and down (in the vertical axis). > My question is: What is the (gravitational) potential energy of this bar? I've found several exercises where $PE=mgy_{CM}$, where $y_{CM}$ is the position of the bar's center of mass, though I don't know why the assumption of just taking the CM. Since this is rigid body, I thought also in summing the potential energy of each particle (i.e, integrating the classical $mgy$ function with respect to the height $y$), but I don't know if this is correct, or necessary."}
{"id":"123465","title":"English translation of Heisenberg's paper ``Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik''","text":"Is there any english translation of Heisenberg's paper _Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik_ online? I have found not. Thanks so much."}
{"id":"71091","title":"How do we calculate the Higgs potential?","text":"What is the Higgs potential and how do we calculate it?"}
{"id":"71095","title":"vacuum expectation value and Casimir effect","text":"Is it correct to say that the VEV of the SM Higgs is 246 GeV? If so, is the VEV a reflection, or measure, of the Casimir Effect?"}
{"id":"71099","title":"Relativity and Current in Wire","text":"If an observer is stationary relative to a current-carrying wire in which electrons are moving, **why does the observer measure the density of moving electrons to be the same as the density of electrons if there were no current in the wire?** I read the explanation of magnetic force as a consequence of special relativity. That is, when the observer moves with respect to a current carrying wire in the same direction as the flowing electrons, then s\/he observes the density to decrease due to Lorentz expansion and observes the density of positive ions to increase due to Lorentz contraction. The imbalance of charges results in a force that can be explained by Coulomb's law. That explanation mentions that when the observer is stationary, the density of positive ions and the density of electrons appears to be the same. That last point is what I am stuck on. Why does the density of _moving_ electrons appear to be the same as the density of _stationary_ positive ions? **Clarification:** I know that when the electrons are not moving, their charge density cancels out that of the stationary positive ions. When the electrons are moving, how could the charge density _still_ cancel out that of the stationary positive ions? **Important:** My assumption is that a current carrying wire will neither attract nor repel a charge that is stationary with respect to the wire. I am now not sure if that is right :-("}
{"id":"37721","title":"Energy used to stop \/ slow an object","text":"I'm trying to workout how much energy (if any), I use (imagining me as an efficient machine rather than a complex bio-mechanical human) when I lower or catch a weight. I understand that when I push it up, I have used energy to do work as I have the force of the object (it's mass and gravity) and distance. So I've got two similar questions... If I am to stop a free-falling object, how much energy do I use? If I look at it from energy change I believe I need to use the same amount as the kinetic energy of the object at the moment before I catch? If I slowly lower an object how much energy do I use? In an ideal world would the total be the same as in the free falling situation, assuming the height at which the object fell was the same."}
{"id":"83376","title":"Hard disk rotational frequency","text":"I noticed the other day that the rotational frequency of typical consumer grade HDD are either 5400rpm or 7200rpm. These correspond to 90Hz and 120Hz respectively. Given that in parts of the world the grid frequency is 60Hz, and 120Hz is the first harmonic. What effects does this have on the platters. It would seem to me that this choice in rotational speed (7200 rpm) would be problematic in these certain countries."}
{"id":"30717","title":"What is the best material and design for building a parabolic dish?","text":"I'm currently using a homemade metallic spaghetti strainer with a wifi adapter inside it to increase my wifi adapter's signal reception, but would think that a different material, like the aluminum mesh used for the Arecibo Radio Telescope could be better? Why is this radio telescope's reflector spherical and not parabolic?"}
{"id":"91465","title":"Addition of spin?","text":"There is something I don't fully understand. When I have a problem like this: Let's look at a system with $j_1=1$ and $j_2=2$ spins. Let our system be in the state in which the total spin is $2$ and the projection onto the $z$ axis is 0 ($S_z$). What values can the $S_z$ of the $1st$ particle take? When I have a problem like this, I always start by creating this fig:![enter image description here](http:\/\/i.stack.imgur.com\/JfWRc.jpg) It represents the eigenstates. It's easy, I just have to find the simpliest way to the $(J=2,M=0)$. But when the problem is that I just have to add, for example, 1\/2 and 1\/2 spins, then how can I find the eigenstates? There must be a basic formula for it, I've been looking for it but I can't find it."}
{"id":"14549","title":"Particle antiparticle annihilation-do they have to be of the same type?","text":"I read that a particle will meet its antiparticle and annihilate to generate a photon. Is it important for the pairs to be of the same type? What will happen when for example a neutron meets an antiproton or a proton meets a positron? Are there any rules to determine what happens when such particles meet?"}
{"id":"15494","title":"How is the atmosphere composition of astronomical objects in solar system determined?","text":"How is the atmosphere composition of astronomical objects, for example triton, the moon of neptune, determined?"}
{"id":"38128","title":"What shape of track minimizes the time a ball takes between start and stop points of equal height?","text":"I was at my son's high school \"open house\" and the physics teacher did a demo with two curtain rail tracks and two ball bearings. One track was straight and on a slight slope. The beginning and end points of the second track were the same but it was bent into a dip with a decending slope and an ascending slope. The game was to guess which ball bearing wins the race. At first blush (\"the shortest distance between two points is a straight line\") one might think the straight wins, but in fact the bent track wins easily (because the ball is going very fast for most of the time). His point was that nature doesn't always work that way our antediluvian brains think it should. So my question (and I do have one) is what shape of track give the absolute shortest time? I think it's a calculus of variation problem or something. For simplicity, let's assume a) no friction b) a point mass and c) that the end point is a small delta below the start point. I played around with some MATLAB code and it seemed to give a parabola of depth about 0.4 times the distance between the start and stop point. The value of g and the mass didn't seem to matter. But I don't know if my code was right or wrong and I couldn't figure out the significance of 0.4. I Googled around for an answer but couldn't find one. Thanks for any pointers!"}
{"id":"103328","title":"Effect of linear terms on a QFT","text":"I was always told when first learning QFT that linear terms in the Lagrangian are harmless and we can essentially just ignore them. However, I've recently seen in the linear sigma model, \\begin{equation} {\\cal L} = \\frac{1}{2} \\partial _\\mu \\phi _i \\phi ^\\mu \\phi _i - \\frac{m ^2 }{2} \\phi _i \\phi _i - \\frac{ \\lambda }{ 4} ( \\phi _i \\phi _i ) ^2 \\end{equation} with $m ^2 =-\\mu^2 > 0$, adding a linear term in one of the fields $\\phi_N$, does change the final results as you no longer have Goldstone bosons (since the $O(N)$ symmetry is broken to begin with). Are there any other effects of linear terms that we should keep in mind or is this the only exception of the \"forget about linear terms\" rule?"}
{"id":"47112","title":"General Physics Short question","text":"Does any one know where this formula come from ? $\\alpha (t) = -\\omega^2\\ \\Theta(t)$ Thanks"}
{"id":"47113","title":"Asymptotic limit of the two kink solution of the sine-gordon equation","text":"I am reading a paper on the sine-gordon model. The solution for a two kink solution is given as: $$\\phi=4\\arctan\\left(\\frac{\\sinh\\frac{1}{2}(\\theta_1-\\theta_2)}{(a_{12})^\\frac{1}{2}\\cosh\\frac{1}{2}(\\theta_1+\\theta_2)}\\right)$$ where $\\theta_1=m\\gamma_1(x-v_1t)+\\delta_1$, and similarly for $\\theta_2$. Here $a_{12}, m, \\gamma, \\delta$ are all constants. The author says that the limit as $t \\to \\pm \\infty$, is $$\\lim \\phi(x,t) \\to 4\\arctan(\\exp [\\theta_1+\\eta_1^{\\pm}])+4\\arctan(\\exp[\\theta_2+\\eta_2^{\\pm}]) $$ where $\\eta_1^{\\pm}=-\\eta_2^{\\pm}=\\pm\\frac{1}{2}\\ln(a_{12})$ I am having a hard time proving this. I did the expansion of the hyperbolic functions in terms of the exponents, and tried to limit the appropriate terms to 1, but I am not getting an answer."}
{"id":"11497","title":"Why are generalized positions and generalized velocities considered as independent of each other?","text":"I'm confused how $$\\dot{\\mathbf{r}}_{j}=\\sum_{k}\\frac{\\partial\\mathbf{r}_{j}}{\\partial q_{k}}\\dot{q}_k+\\frac{\\partial\\mathbf{r}_{j}}{\\partial t}$$ leads to the relation, $$\\frac{\\partial\\dot{\\mathbf{r}}_{i}}{\\partial\\dot{q}_{j}}=\\frac{\\partial\\mathbf{r}_{i}}{\\partial q_{j}}$$ Sources suggest that while differentiating the first equation the generalized velocity and generalized positions are considered as independent of each other, i.e., $(\\dot{q_1},\\dot{q_2},...,\\dot{q_n})$ is independent of $(q_1,q_2,...,q_n)$. I don't understand how they are independent."}
{"id":"19793","title":"Feynman rule 4-point vertex WW -> ZZ","text":"I am looking for the Feynman rule of the 4-point gauge boson interaction of W+ W- -> Z Z. I am guessing it looks like the Yang Mills 4-point vertex for gluons, but with helicity included. Equation 8.81 in my professor's lecture notes seems to imply polarization, or am I misinterpreting?"}
{"id":"44316","title":"Thermal expansion is an expression of which conservation laws?","text":"Many objects get larger as they heat up and contract as they cool down. Which conservation laws are applied to describe this phenomenon? How do they interact with each other to produce this effect?"}
{"id":"88241","title":"Math needed for undergrad Statistical Mechanics\/Thermal Physics","text":"A professor recommended me to take a course on Statistical Physics as preparation for agent-based computing in social sciences. Now I have no experience in physics beyond basic highschool, and mathematics I have taken several discrete math courses (for computer science) and no calculus yet. I have had basic differentiation\/trigonometry\/integration in high school as well. What math\/physics do i need to be able to understand undergrad stat mechanics? Thanks a lot for your answer! (btw. I am already planning to take classical mechanics, because that's what the university states as prerequisite)"}
{"id":"64488","title":"What is three-photon interference?","text":"Whilst reading this paper on a quantum processor that performs a type of matrix computation, I came across the concept of 'three-photon interference'. A quick Google search shows that this process is frequently referred to in quantum optical experiments but unfortunately, no wikipedia article exists for it, can anyone yield some insight on this? My own guess is that it is exactly what it says: the interference pattern that results when 3 photons are combined. Why is this distinction so important?, why 3 photons and not 4, for example?"}
{"id":"95539","title":"Maxwells' equations and Coulomb's law","text":"Coulomb's law and Maxwell's equations should be consistant as one can be derived from the other. Say we have a point charge with such a charge that $-kq=1$, meaning that at any point the electric field will have a magnitude of $$|E|=\\frac{1}{r^2}$$ where $r$ is the distance from the origin (were we place our charge), and the vectors point towards the origin at all point. This would be equivalent to the following in cartesian co-ordinates: $$E=-\\hat{x}\\frac{x}{(x^2+y^2+z^2)^{3\/2}}-\\hat{y}\\frac{y}{(x^2+y^2+z^2)^{3\/2}}-\\hat{z}\\frac{z}{(x^2+y^2+z^2)^{3\/2}}$$ We can verify that $$|E|=\\frac{1}{\\sqrt{x^2+y^2+z^2}}$$ Gauss's law in its differential form allows us to calculate a charge distribution that would give rise to such a electric field using the divergence operator: $$\\nabla \\cdot E=-\\frac{1}{x^2+y^2+z^2}$$ from wolfram alpha: one, two Which absolutely doesn't make sense to me! Intuitively I would think it would be zero everywhere except (0,0,0). Or at least not to go of to infinity at any point. Could somebody please explain what's going on?"}
{"id":"53154","title":"Ideal gas temperature and pressure gradients?","text":"Consider an ideal gas in a $d\\times d\\times L$ box with the $L$ dimension in the $x$-direction. Suppose that the opposite $d\\times d$ sides of the box are held at temperatures $T_1$ and $T_2$ with $T_2>T_2$ and that the system reaches a steady state. According to these notes, the thermal conductivity of an idea gas scales as the square root of temperature; $k=\\alpha\\sqrt{T}$ in which case by Fourier's Law one gets that the temperature gradient in the $x$-direction is $$ T(x) = \\left[T_1^{3\/2}+(T_2^{3\/2}-T_1^{3\/2})\\frac{x}{L}\\right]^{2\/3} $$ What is the corresponding pressure gradient $P(x)$ in the steady state?"}
{"id":"69310","title":"Sound Propagation using Finite-Difference Time-Domain (FDTD) considering wind effects","text":"I am trying to plot the propagation of sound from a fixed source in a 2D environment using Finite-Difference Time-Domain (FDTD) Method taking into account the effects of the wind velocity. I came across the following set of equations for solving the equation for pressure with respect to time at different grid points in the 2-D environment. $$ \\frac{∂p}{∂t} = -( v_x\\frac{∂}{∂x}+v_y\\frac{∂}{∂y})p - \\kappa(\\frac{∂w_x}{∂x}+\\frac{∂w_y}{∂y}) +\\kappa Q $$ $$ \\frac{∂w_x}{∂t} = -(w_x\\frac{∂}{∂x}+w_y\\frac{∂}{∂y})v_x- (v_x\\frac{∂}{∂x}+v_y\\frac{∂}{∂y})w_x-b\\frac{∂p}{∂x}+bF_x $$ $$ \\frac{∂w_y}{∂t} = -(w_x\\frac{∂}{∂x}+w_y\\frac{∂}{∂y})v_y- (v_x\\frac{∂}{∂x}+v_y\\frac{∂}{∂y})w_y-b\\frac{∂p}{∂y}+bF_y $$ where, $ b = \\frac{1}{\\rho} $ is the mass bouyancy. But, I am unable to figure out what $\\vec{F}$ and $Q$ are? They are mentioned somewhere as dipole and monopole pressure sources respectively. What does they actually mean?"}
{"id":"64146","title":"The Preference for Low Energy States","text":"The idea that systems will achieve the lowest energy state they can because they are more \"stable\" is clear enough. My question is, what causes this tendency? I've researched the question and been unable to find a clear answer, so I was hoping someone could explain what's going on behind the scenes here (or if nothing is and it's just an observed law that isn't explainable)."}
{"id":"86299","title":"Hard circuit problem","text":"I want to find the p.d. between points A and B. I'm not sure how to approach this problem though, could I have a hint? ![enter image description here](http:\/\/i.stack.imgur.com\/chmiO.jpg)"}
{"id":"80191","title":"Is information about the speed of light hidden in its spectrum?","text":"Can the speed of light in the vacuum (c) be inferred from the spectrum of light? If that is not the case is it possible to tell from lights spectrum that it has entered a different medium, e.g. can the correct fraction of c be inferred from the spectrum then?"}
{"id":"1939","title":"Physics of a fixed wheel moving on a flat surface held at distance from a center point","text":"Ok, so my boss is trying to make a car turntable. In essence, he has a two boards that sit atop a rotating ring. He wants to put two wheels at the end of each board (8 wheels total). He thinks that you can angle wheels properly (2d) such that they can be flat (axis is parallel to the floor) and the whole contraption move smoothly. Take a look at this picture (Edit: copied below - Mark) ![alt text](http:\/\/i.stack.imgur.com\/YujYw.jpg) Ok, so the question, will this design move smoothly? My guess is \"no.\" I reason that if the wheels are unbanked then they will move in a straight line when driven, and the ring will realign them periodically to its circular path (probably causing a skip at the wheel as the stress is released). He thinks it will just work.. Please use technical terms with very easy layman definitions so I can reference wikipedia and texts and learn more about this. Sorry, for the utter lack of useful physics terms."}
{"id":"58166","title":"Eye sensitivity & Danger signal","text":"Why are danger signal in red, when the eye is most sensitive to yellow-green? You can check luminosity function for more details..."}
{"id":"47968","title":"Levi Civita Symbol and contravariance vs covariance","text":"I have a question regarding the Levi-Civita symbol and contravariance vs covariance. Some of this was asked in a previous post, but I think I need more clarification. Consider the magnetic field: \\begin{equation} B^k = \\tilde{\\varepsilon}^{ijk}\\partial_i A_j \\end{equation} Qn 1: In the formula I wrote it as the Levi-Civita symbol $ \\tilde{\\varepsilon}$, is it right or should it be the Levi-Civita tensor $\\varepsilon$ instead? Or does that not make sense since the formula $\\vec{B} = \\nabla \\times \\vec{A} $ is true only in cartesian coordinates and so we cannot use a tensor? (hope that made sense) Qn 2: I wrote $B^k$ with an upper index to indicate that it is a contravariant vector - so that means under coordinate transformations $\\tilde{\\varepsilon}$ transforms too. Except that I'm confused as to how it transforms? Carroll (in his book spacetime & geometry) says that the Levi-Civita symbol is defined not to change under coordinate transformations (so its entries remain +1 -1 0), yet goes on to derive a transformation law $\\tilde{\\varepsilon}_{i'j'k'\\cdots} = |g| \\tilde{\\varepsilon}_{ijk} \\frac{\\partial x^i}{\\partial x^{i'}} \\frac{\\partial x^j}{\\partial x^{j'}} \\frac{\\partial x^k}{\\partial x^{k'}} \\cdots$. So which is it? Qn 3: However in some books they write \\begin{align} B_k = \\epsilon_{ijk}\\partial_i A_j. \\end{align} In this case it seems like they are ignoring the tensorial nature of the object and just treating $\\tilde{\\varepsilon}$ as simply a symbol that gives the values +1, -1 or 0 in the summation. This occurs too, for example, in writing the commutation relations down for the spin operators: \\begin{align} [J_i, J_j] = i \\tilde{\\varepsilon}_{ijk}J_k, \\end{align} where I am pretty sure in this case the symbol is just thought of as just a number. (although $J_i$ are vectors of the Lie algebra...?). So, am I attributing too much meaning to the Levi-Civita symbol? Qn 4: In the course of my work I'm doing I have to contend with the angular momentum operator, which I wrote as \\begin{align} L^k = \\tilde{\\varepsilon}^{ijk}g_{ia}r^ap_j. \\end{align} In cartesian coordinates this reproduces $L^x, L^y, L^z$, since for example $L^z = x p_y - y p_x$. But is this formula right? Changing to spherical coordinates doesn't seem to give $L^z = \\frac{\\partial}{\\partial \\phi}$ and so on... Qn 5: Just a general question about the Levi-Civita symbol. Does it make sense to raise\/lower indices in the Levi-Civita symbol? I know that you can do it for the tensor because the Hodge star operator uses it. sorry for the lengthy post and thanks in advance for the help!"}
{"id":"1428","title":"Evolution of mass and velocity distributions under newtonian gravitation","text":"Let $\\rho(r,t)$ and $v(r,t)$ be mass and velocity distributions. Given $\\rho(r,0)$ and $v(r,0)$ (initial conditions) what is the differential equation that describes the evolution of $\\rho(r,t)$ and $v(r,t)$ in space and time, assuming Newtonian gravitation? You may assume $r$ to be one dimensional, if it makes writing the equations convenient. **EDIT** I think $v(r,t)$ should be replaced with $p(r,t) = \\rho(r,t)v(r,t)$ and name it as momentum distribution to avoid some technical faults. **EDIT 2** It seems that to completely characterize an inviscid fluid flow, Euler equations are needed. The law of conservation of mass is already given in the answer. A component of pressure is missing in the equation related to conservation of momentum.(first equation). Lastly a third equation corresponding to law of conservation of energy should be introduced.(3rd one of the euler equations of inviscid fluid flow).For the third equation we assume that the internal energy per unit volume of the fluid is zero."}
{"id":"68107","title":"Is light particle of wave?","text":"We know that Young's double slit experiment shows that light is a wave. On the other hand photoelectric effect shows that light is made up of photons. How can light be both at the same time?"}
{"id":"112808","title":"What does the notion of basis sets for photons in number of particle picture?","text":"Since $$|n.\\rangle=|u_{k_1}\\rangle\\otimes...\\otimes|u_{k_1}\\rangle\\otimes|u_{k_2}\\rangle\\otimes...\\otimes|u_{k_2}\\rangle\\otimes...\\otimes|u_{k_m}\\rangle\\otimes...\\otimes|u_{k_m}\\rangle$$(There are $n_{k_i}$ terms in $|u_{k_i}\\rangle\\otimes...\\otimes|u_{k_i}\\rangle$.) means in each modes there is a fixed number of photons. Does $$|\\\\{n\\\\}\\rangle$$ means the set of all the possible candidates of $|n.\\rangle$ ?"}
{"id":"122471","title":"Is there a unit system, where all physical laws looks nice?","text":"And by nice looking law I mean with no constants. I mean, what would we need to set, so all laws would without those nasty constants in front of them? (I mean all of them, also $\\pi$!) What would it mean? What I am trying to say with this question is: Is there a universal language, do particles and gravity live in the same space? If gravity and quantum world should go together, there should be one basis, where you don't need to use any constants."}
{"id":"122474","title":"Can an LC oscillator be used to generate visible light?","text":"The LC oscillator is most commonly used to generate radio waves for practical use and the frequency $\\omega$ of the LC oscillator equals that of the electromagnetic wave so produced. So, can they in principle be used to emit _visible light?_ The frequency of visible light is on the order of a few hundred terahertz, and the frequency of an LC oscillator is $$\\omega = \\frac{1}{\\sqrt{LC}}$$ I admit, the product $LC$ does become very small (on the order of $10^{-30}$) when the numbers are plugged in, but making an inductor and a capacitor with small values isn't difficult, is it?"}
{"id":"122476","title":"Deriving Lorentz transformation - Serway","text":"in Serway, the derivation of Lorentz transformation started from these two equations $$x′=\\gamma(x−ut),$$ $$x=\\gamma(x′+ut′).$$ The book derived the first equation and then deduced the second. Is there a proof that if the first is right then the second is also right? To be specific I don't understand:   1. How $t$ was converted to $t′$ in the second equation.   2. Why the position of $\\gamma$ didn't change in the second equation.  Note: I understand the first equation well but I don't understand how to reach the second from the first."}
{"id":"68108","title":"Why proper acceleration is $du\/dt$ and not $du\/d\\tau$?","text":"Wikipedia says: > In relativity theory, proper acceleration[1] is the physical acceleration > (i.e., measurable acceleration as by an accelerometer) experienced by an > object. and says: > In the standard inertial coordinates of special relativity, for > unidirectional motion, proper acceleration is the rate of change of proper > velocity with respect to _coordinate time_. Why coordinate time? As far as I know the moving observer itself measures the proper things. It's clock measures proper time, it's speedometer measures proper velocity. So if the observer accelerates with $10m\/s^2$ (and feels that proper acceleration), isn't the speed shown by the local speedometer changes $10m\/s$ per second on the local clock?"}
{"id":"22824","title":"Decoherence when no one is looking?","text":"I understand that in the single-electron-at-a-time double slit experiment, if a detector is placed before the slit, the interference pattern vanishes. Suppose I left the detector on, but put a bag over its screen (I can't tell what state the electron is before it passes the slits), does the interference pattern come back? If so, does that mean the electron \"knows\" I'm not \"looking\" and proceeds to interfere with itself? Edit: I had photons passing through the slit initially. I've updated it to electrons to reflect a more common setup."}
{"id":"59699","title":"Why is the Horizontal Force Constant in Deriving the One Dimensional Wave Equation","text":"My textbook in deriving the wave equation for a one dimensional elastic string stated that the horizontal direction force is constant.I understand that the horizontal components of the tensions on either side of the element have to be equal since it is assumed it moves only in the vertical plane but why does it have to be constant? Don't they just have to be equal?"}
{"id":"59695","title":"Probabilistic vs Statistical interpretation of Double Slit experiment","text":"Why is it assumed that the results seen in the double slit experiment are probabilistic and not just a statistical result of some unknown variable or set of variables within the system."}
{"id":"40874","title":"Using eigenvalues to determine the stability\/behaviour of the system","text":"first time I've been on physics.se but have used the math and cs before... Anyway, here's my question: If we have a damped pendulum described by the equation $$y'' + ay' + b = 0 , a>0$$ Using the conversion $x_0=y$ and $x_1=y'$ we can convert this into a set of first-order ODEs as follows: $$ x_0'=x_1$$ $$x_1'=-ax_1+b$$ or in matrix form : $$\\mathbf{x'}=\\begin{bmatrix}0 & 1\\\\\\0 & -a\\end{bmatrix}\\mathbf{x}+\\begin{bmatrix}0 \\\\\\ b\\end{bmatrix}$$ Now, the exercise where I'm doing this now says we can use the eigenvalues to determine the stability of the system the eigenvalues of the system. The eigenvalues are $\\lambda = 0$ and $\\lambda = -a$... I know that an eigenvalue of 0 means a neutral equilibrium such that a small change in the variable means it remains at the changed variable and that a negative eigenvalue means a small change will return to its original value, but I'm struggling to see how this relates to the physical situation of the damped oscillator, and what it means about the behaviour of the system.. Any thoughts would be appreciated"}
{"id":"29410","title":"Distance at which a 300m asteroid is visible","text":"Using the best available telescopes, and assuming a typical albedo of 10%, at what distance can a 300m asteroid be seen (well enough to start plotting its course) in 1) visible and 2) infrared wavelengths?"}
{"id":"59345","title":"Photon on null geodesic","text":"If given an FRW metric $ds^2 = -dt^2 + a^2(t)[dx^2+dy^2+dz^2]$ and for the trajectory followed by a photon (null geodesic; $ds^2=0$) with affine parameter $\\lambda$, know that $g_{\\mu\\nu}\\,\\frac{dx^{\\mu}}{d\\lambda}\\,\\frac{dx^{\\nu}}{d\\lambda} = 0.$ How does one find $\\frac{dt}{d\\lambda} = \\frac{w0}{a(t)}$? Already found the nonzero Christoffel coeffs and the remaining geodesic equations, $ 0 = \\frac{d^2x^i}{d\\lambda^2} + \\Gamma^i_{ti}\\,\\frac{dt}{d\\lambda}\\,\\frac{dx^i}{d\\lambda} $ and $ 0 = \\frac{d^2t}{d\\lambda^2} + \\Gamma^t_{ii}\\,\\frac{dx^i}{d\\lambda}\\,\\frac{dx^i}{d\\lambda} $ with $ \\Gamma^i_{ti} = \\frac{1}{a(t)}\\,\\frac{da(t)}{dt} $ and $ \\Gamma^t_{ii} = a(t)\\,\\frac{da(t)}{dt} $ I expect it to be pretty simple (yet not seeing it) and that the null geodesic in the $x$-direction come from something like (?): $ ds^2 = -(dt^2)+a^2(t)\\,dx^2 = 0 \\rightarrow \\frac{dt}{a(t)} = \\pm dx $ I also believe that $dx\/d\\lambda = c$ if the affine parameter is related by $\\lambda = t$. This is where I am confused and need it to move on to look at the cosmological redshift and derive the ratio of emitted and observed energies of a photon at $t1$ and $t2$. Anything to straighten me out would be great!"}
{"id":"78826","title":"Are CCDs approaching the sensitivity of photomultiplier tubes?","text":"I still see photomultiplier tubes (PMT) used for even benchtop spectroscopy experiments. How much more sensitive are they to CCDs? I found a post which claims about 1.5 x more sensitivity for PMTs, but any references or more complete understanding of the problem would be really appreciated. I'm also wondering if they need to be cooled or specialized chips or if commodity imaging chips like from a camera has reasonable S\/N characteristics for detection?"}
{"id":"119119","title":"Equations of graphs","text":"I am doing a practical write up of an experiment I did. Here is a graph using the raw data I got. ![enter image description here](http:\/\/i.stack.imgur.com\/n1JMW.png) What is the general formula of this graph, and what do I need to do in order to make that into a straight line graph? Take the natural log of the average speed and plot it against mass? Excel gives me the specific equation of the line, but I need to write down a general equation in order to get max marks. Or could it be that my empirical data is wrong with the theory? I'm having trouble understanding the maths here. Edit: Here are attempts at straight line graphs I have made. ln v vs ln M looks the straightest, is that the one I should therefore use? ![enter image description here](http:\/\/i.stack.imgur.com\/eZnE8.png) Can someone walk me through how I change the general formula of the first graph into the general formula of a straight line graph?"}
{"id":"102564","title":"Stresslet coefficient in fluid dynamics","text":"I encountered a parameter called stresslet coefficient in several literatures, which seems related with viscosity and stream function (also influencing the far field displacement). But there's no Wikipedia page for this concept. Can anyone explain what it is? Thanks!"}
{"id":"126201","title":"What is the difference between Rayleigh scattering and Thomson scattering?","text":"After reading the wiki articles I know, that both Rayleigh scattering and Thomson scattering are elastic processes. But what is the essential difference between those two processes, their cross sections and energy dependence?"}
{"id":"52949","title":"What is the Lagrangian from which the Klein-Gordon equation is derived in QFT?","text":"1. Is there a well-known Lagrangian that, writing the corresponding eq of motion, gives the Klein-Gordon Equation in QFT? If so, what is it?   2. What is the canonical conjugate momentum? I derive the same result as in two sources separately, but with opposite sign, and I am starting to suspect that the error could be in the Lagrangian I am departing from.   3. Is there any difference in the answers to that two questions if you choose (+---) or (-+++)? If so, which one?"}
{"id":"75429","title":"What is the 1977 Wow signal?","text":"Can someone explain about the 1977 Wow signal? I know that it is a signal received from space in 1977, in the form of a series of code \"6EQUJ5\". I would like to know if there is any thing or event which can generate this signal other than an intelligent life form? If yes then can you tell me which things or events can generate it??"}
{"id":"75426","title":"why does headlights dim when a starter motor is turned on","text":"okay first of all, what exactly does the starter motor do? does it increase the current flow and decrease voltage? and does starter motor have its own internal resistance? i understand how there is a voltage drop in the battery terminals, so does the starter motor lower the potential difference to become less than the voltage drop? so for example if the voltage is 12V and the voltage drop is 2V. The potential difference is now 10V. what is the voltage when the starter motor is used? does it become less than 10V? but then after the engine starts the starter motor turns off and the lights return to normal. So is normal 10V?"}
{"id":"92951","title":"Using centre of mass in gravitational field calculation","text":"When calculating the gravitational field of a continuous mass, like a rod, at some point outside it, why can't we just use the centre of mass of the rod and then directly plug in the distance between the point and the centre of mass in the gravitational field equation. Why do we need to integrate over the whole rod?"}
{"id":"55990","title":"Round bulb and Long bulb in Series Circuit","text":"If you have a series circuit with a battery, one round bulb (with thicker filament), and one long bulb (with thinner filament), the round bulb does not light up and only the long bulb lights up. Also, I understand that the voltage drop in the long bulb is greater, but I don't get how the charges distribute themselves like that. For example, if the round bulb comes first, how does the charges passing by the long bulb \"know\" to only give up a little bit of energy to save it up for the long bulb? Why does the round bulb not light up?"}
{"id":"55993","title":"Why do some particles have a greater mass than others?","text":"The property of mass that almost every particle possesses comes from the Higgs Field. It is this field, which permeates all of space, that particles interact with and hence obtain mass. But why do some particles have a higher mass if they are all interacting with the same Higgs Field? One would suspect that they'd all have the same mass, but there must be some property of a particular particle that makes it interact with the field in a different way, thereby giving it mass."}
{"id":"93022","title":"Exact definition of momentarily comoving reference frame (MCRF)","text":"Consider a particle $P$ in the framework of special relativity with position $r(t)=(ct,x(t),y(t),z(t))$ respect to an inertial reference frame $\\Sigma=(ct,x,y,z;O)$. I need to know if the following is a correct definition of a MCRF for $P$: > A momentarily comoving reference frame (MCRF) for P is a frame reference > $\\Sigma'=(ct',x',y',z';O')$ such that if $r'$ is the position of $P$ respect > to $\\Sigma'$, the following two conditions are satisfied: > > 1) $r'(t'_0)=(ct'_0,0,0,0) $ for every $t'_0$ fixed. > > 2) $\\frac{dr'}{dt'}(t'_0)=(c,0,0,0)$ for every $t'_0$ fixed. > > Moreover the motion of $P$ is uniformly accelerated if for every choice of a > MCRF $\\frac{d^2r'}{{dt'}^2}(t'_0)$ is constant $\\forall t'_0$. If the above definition is wrong I would appreciate a **formal** correct definition of a MCRF. Thanks in advance."}
{"id":"105097","title":"What are the the characteristics of a standing wave?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/s6Lzd.jpg) May someone explain #24 from this picture? B is apparently the correct answer. Can any of the other answers be correct here? Isn't D also correct?"}
{"id":"57596","title":"How is gravitational force is compared to flow of water?","text":"I listened to a lecture. The professor said that the gravitational field around the particle (spherical in shape) can be compared to a pond having a constant height and depth and water is constantly pumped in the center. The water flows radially outward which is similar to that of the gravitational field around a spherical object. Can anyone explain me how to compare gravitational field and this flow of water?"}
{"id":"24901","title":"How to create unusual sundial?","text":"I am considering small \"artsy\" project. I would like to create sundial by placing gnomon on the window and painting hour lines on the window facing wall. Since this is to be placed in bedroom I am constrained by my geographic location, wall, window placement and orientation. The esthetic and size (I have one wall only) of the project is gating factor (otherwise known as \"the wife\" factor). Due to above I am perfectly OK with the fact that this sundial will \"work\" by limited time of the day and even limited time of the year. However whenever it will work (that is the shadow of the gnomon will be cast on the \"said\" wall) I would like it to be as accurate as possible. Also all of the above make the calculations for creating hour lines quite challenging (at least for me) and to be honest I do not know where to start. Could you point me the resources that would help in calculating hour lines (software, tutorials, books, math equations)?   Could you describe how would you approach the task of calculating hour lines? (Resources that are little heavy on math side are OK for me. I am also capable of wring software on my own.)"}
{"id":"24904","title":"Resources and requirements for amateur spectroscopy","text":"If I wanted to move beyond just looking through my eyepieces and taking photographs of astronomical objects, what can I do do move into astronomical spectroscopy? Are there any good resources for this? Pitfalls? And perhaps most importantly, what kind of initial outlay would I have to make?"}
{"id":"64425","title":"How to find time taken for a spinning top to stop?","text":"The angular position of a spinning top is given by $\\theta = t^3 - 72t$, where $t$ is in seconds and $\\theta$ in \"radian\"."}
{"id":"134914","title":"QM interpretations","text":"I don't fully appreciate what the discovery of the decoherence phenomenon adds to the Copenaghen interpretation of QM. I will be more precise: the Copenaghen interpretation, if I am not wrong, is summarized by the following concepts:   1. QM can, and has to, predict _only the probabilities_ of measurement outcomes (\"has to\" because descriptions of nature such as the classical one, where every quantity always assumes a certain value is in disagreement with our current experimental observations, such as the violation of Bell's Inequality.).   2. In order to do QM, we use state vectors. However, these must not be considered as a real _physical entity_ , but rather _tools_ in a similar way as probability distributions in classical physics (This means that in an EPR experiment, the change of the vector state describing Bob's qubit after Alice has measured hers in a galaxy far away is not in contrast with relativity.).   3. The probabilities are computed via the Born rule: the state after measurement is an eigenstate of the observable measured, and the outcome is the corresponding eigenvalue. Now the decoherence: if one considers the universe divided into a qubit (i.e. a two level system) system (S), apparatus (A) and the rest (E) can give a QM dynamical description of a measuring process on the qubit. Initially, the universe is in the state $|\\psi\\rangle = (a |0 \\rangle + b |1 \\rangle)\\ |A_0 \\rangle | \\text{rest} \\rangle$. If $|A_i \\rangle $ are such that they can _imprint_ themselves in the environment, i.e. they do not entangle themselves with it, after the unitary evolution the SA system is described by the matrix: $\\rho = |a|^2 |0\\rangle\\langle0||A_1\\rangle\\langle A_1| + |b|^2 |1\\rangle\\langle 1||A_2\\rangle\\langle A_2|$. This analysis explains why at macroscopic scales classical physics works, i.e. we don't see superposition of states and probabilities don't interfere: the environment interaction rapidly evolves the vector states into those states that can _imprint_ in the environment, or in mixture of them. This is surely interesting on its own, but is there anything else to understand from decoherence? The fact that superpositions states turn into mixtures of the eigenstates of the measured observable _seems_ to explain the \"measurement problem\" (that in the Copenaghen paradigm is not a problem actually, but the most important assumption, as stated in item 1 above) but it does not: after the above evolution the system is a classical mixture, but the probabilities are still there. Explaining the measurement problem means to find an evolution that deterministically bring the system into the outcome state (in the Copenaghen interpretation, this is rejected). So decoherence does not explain the Born reduction rule. Additionally, in order to interpret the mixture above as something analogue to a classical mixture, we must have already adopted the Born probability rule, so decoherence does not explain it either. To sum up: having adopted the Copenaghen interpretation of QM, decoherence explains _in its framework_ the transition from quantum world (superposition always possible, quantum interferences) to classical world (no Schroedinger's cat states, bayes rule), but does not replaces or explain any of its axioms...or does it?"}
{"id":"134917","title":"Time Dilation Problem","text":"![Here's the formula...](http:\/\/i.stack.imgur.com\/K1zGw.gif) I'm having some trouble using the time dilation formula. Say an astronaut leaves Earth for 10 years, at 0.85c. How much time has passed according to an observer on Earth? I tried using the following formula: $$t = \\frac{1}{\\sqrt{1-(v^2\/c^2)}}$$ but couldn't seem to get an answer that made sense. Any help would be much appreciated! This concept is super confusing to me."}
{"id":"20950","title":"What is the relationship between Schrödinger equation and Boltzmann equation?","text":"The Schrödinger equation in its variants for many particle systems gives the full time evolution of the system. Likewise, the Boltzmann equation is often the starting point in classical gas dynamics. > What is the relationship, i.e. the classical limit, which connects these two > first order in time equations of motions? How does one approach this, or is there another way in which one sees the classical time evolution? Where are these considerations relevant?"}
{"id":"44233","title":"Why is electric field zero inside a hollow metal sphere ?","text":"The figure below shows a hollow metal sphere with a positive point charge $Q$ sitting outside it. What is the electric field at the center of sphere ? The answer is zero (look at here at the beginning of page 4), but I do not understand why? ![enter image description here](http:\/\/i.stack.imgur.com\/zYFZC.png)"}
{"id":"105540","title":"How to find full energy of field of an arbitrary half-integer spin?","text":"Let's have arbitrary half-integer spin $n + \\frac{1}{2}$ representation: $$ \\Psi_{\\mu_{1}...\\mu_{n}} = \\begin{pmatrix} \\psi_{a, \\mu_{1}...\\mu_{n}} \\\\\\ \\kappa^{\\dot {a}}_{\\quad \\mu_{1}...\\mu_{n}}\\end{pmatrix}, $$ $$ (i\\gamma^{\\mu}\\partial_{\\mu} - m)\\Psi = 0, \\quad \\gamma^{\\mu}\\Psi_{\\mu \\mu_{1}...\\mu_{n - 1}} = 0, $$ $$ \\Psi_{\\mu_{1}\\mu_{i}...\\mu_{j}\\mu_{n}} = \\Psi_{\\mu_{1}\\mu_{j}...\\mu_{i}\\mu_{n}}, \\quad \\partial^{\\mu}\\Psi_{\\mu ...\\mu_{n - 1}} = 0. $$ I need to find the general expression for the full energy of corresponding particles, but I don't know how to do that. I tried to do the following.   1. The solution of equations above may be given as $$ \\hat {\\Psi}_{\\mu_{1}...\\mu_{n}} = \\sum_{s}\\int \\left(\\hat {a}^{\\dagger}_{s}(\\mathbf p)u^{s}_{\\mu_{1}...\\mu_{n}}(\\mathbf p)e^{-ipx} + \\hat {b}_{s}(\\mathbf p)v^{s}_{\\mu_{1}...\\mu_{n}}(\\mathbf p)e^{ipx}\\right)\\frac{d^{3}\\mathbf p}{\\sqrt{(2 \\pi )^{3}2E_{\\mathbf p}}}, $$ where $s$ refers to the possible polarizations, $u, v$ are \"spinor-tensor\" waves which have some normalization law (as example) $u^{+}u = 2E_{\\mathbf p}$.   2. The Dirac equation for each $\\mu_{1}...\\mu_{n}$ may be rewritten in a form $$ i\\partial_{0}\\Psi_{\\mu_{1}...\\mu_{n}} = \\hat {H}\\Psi_{\\mu_{1}...\\mu_{n}} = E\\Psi_{\\mu_{1}...\\mu_{n}}. $$ So for finding the full energy I must construct integral like $$ E = \\langle \\hat {H} \\rangle = \\int \\Psi^{\\dagger} \\hat {H}\\Psi d^{3}\\mathbf r = i \\int \\Psi^{\\dagger} \\partial_{0}\\Psi d^{3}\\mathbf r. $$ The expression $$ \\int \\Psi^{\\dagger}\\Psi d^{3}\\mathbf r $$ must be lorentz invariant. But how to make it invariant? My first idea is convolute $\\Psi^{\\dagger}\\Psi$ as $\\Psi^{\\dagger}_{\\mu_{1}...\\mu_{n}}\\Psi^{\\mu_{1}...\\mu_{n}}$, and I think that it is the only possible way to construct the \"correct\" lorentz invariant object (in vector indices space), but it is not strictly. After that I only use conservation law $\\partial_{\\alpha}(\\bar {\\Psi}\\gamma^{\\alpha}\\Psi ) = 0$. But I'm not sure at this.  Can you help? I know that in general the expression looks like $$ E = \\sum_{s}\\int (\\hat {a}^{\\dagger}_{s}(\\mathbf p)\\hat {a}_{s}(\\mathbf p) - \\hat {b}_{s}(\\mathbf p) \\hat {b}^{\\dagger}_{s}(\\mathbf p ))E_{\\mathbf p} d^{3}\\mathbf p . $$"}
{"id":"24018","title":"How can something happen when time does not exist?","text":"I saw this documentary hosted by Stephen Hawkins. And if I didn't get it wrong, it says that there was no time before the big bang, time was created there. So how can anything happen when there is no time (eg: the creation of time and the universe)? Is this what he calls a 'singularity'? Which I could think like this: when there is no time nothing can happen, except during this singularity. Of course I don't know of a law that says 'when there is no time nothing can happen..', I just came up with it, as something intuitive."}
{"id":"67190","title":"Why does this state have a Schmidt rank of 1?","text":"A system is entangled if and only if the Schmidt rank is greater than 1. Why does this $$\\left[\\frac{1}{\\sqrt{2}}\\left(\\left|0\\right\\rangle+\\left|1\\right\\rangle\\right)\\right]\\otimes\\left[\\frac{1}{\\sqrt{2}}\\left(\\left|0\\right\\rangle+i\\,\\left|1\\right\\rangle\\right)\\right] = \\frac{\\left|00\\right\\rangle+i\\,\\left|01\\right\\rangle+\\left|10\\right\\rangle+i\\,\\left|11\\right\\rangle}{2}$$ have a Schmidt rank equal to one? or equivalently why is this not entangled? ($\\otimes$ denotes the tensor product)"}
{"id":"89292","title":"How is bound charge and free charge possible?","text":"I am studying Introduction to electrodynamics by Griffiths and I came along a concept I cannot seem to understand properly. The concept of free charge AND bound charge. I do not understand how we can have both. I understand bound is created by the presence of an E field and a dielectric. So where does the free charge come from if we are inside a dielectric ? Here is a quote from the book talking about it > In Sect. 4.2 we found that the effect of polarization is to produce > accumulations of bound charge, Pb = - V . P within the dielectric and CYb = > P . n on the surface. The field due to polarization of the medium is just > the field of this bound charge. We are now rejldy to put it all together: > the field attributable to bound charge plus the field due to everything else > (which, for want of a better term, we call free charge). The free charge > might consist of electrons on a conductor or ions embedded in the dielectric > material or whatever; any charge, in other words, that is not a result of > polarization."}
{"id":"47348","title":"Two-fluid description of superfluidity","text":"I'm trying to teach myself about superfluidity and I'm slightly confused on the ''two-fluid'' description. From what I understand, the superfluid is considered to be a mixture of two fluids, a superfluid component and a normal fluid component. Are these two fluids a true ''mixture'' or are is it just a model to break the behavior of the one _real_ fluid into its two different aspects? Could we separate them from each other as if they were truly separate fluids? For example, if it were truly a mixture, we could make it do something that only a superfluid could do, such as pass through an extremely small opening. This would separate the fluids and leave us with just the superfluid component. Would this remaining ''pure''-superfluid still be described by the two-fluid model?"}
{"id":"76307","title":"gamma-gamma collision seen from a relativistic rocket","text":"This is a thought experiment. Consider two photons traveling at right angles and intersecting at some point. As seen from Earth these photons will not interact with each other because their energy is too low for particle creation. Now consider a rocket traveling at high relativistic speed towards the intersection point. People on that rocket will \"see\" these photons extremely blue shifted. They would appear as gamma rays. Now from the rocket point of view there is a higher probability that particle creation would occur, because these photons appear so energetic. Any produced particles like electrons or positrons would be captured by the rocket and brought back to Earth as a proof. The experiment could be repeated to get good statistics. How many particles would that rocket bring to the Earth. None? Or maybe the amount predicted by observers on the rocket? Disclaimer. I am not a physicist. Probably this \"paradox\" has a very simple resolution just like other relativity paradoxes. I probably made some wrong assumptions."}
{"id":"4319","title":"Homework Help - Physics \/ Calculus III \/ Force-Work Problem","text":"I have tried to answer the problem correctly, and I have found similar (and even an identical) problem in my textbook, yet I still can't seem to yield the correct answer with this specific problem. The example problem in my textbook: A crate is hauled 8m up a ramp under a constant force of 200 N applied at an angle of 25 degrees to the ramp. Find the work done. W = F * D cos(25) = (200)(8)cos(25) = 1450J. Can anyone give me any advice on how to solve it? A woman exerts a horizontal force of 4 pounds on a box as she pushes it up a ramp that is 5 feet long and inclined at an angle of 30 degrees above the horizontal. Find the work done on the box. I would think that the work would simply be (4)(5)cos(30), but that is not the correct answer. (I've figured out how to accept answers now, so I'll accept the most helpful answer.)"}
{"id":"87792","title":"Noether current for the Lagrangian without Lorentz invariance","text":"I am reading an article by Watanabe & Murayama. It gives a proof on the counting of Nambu–Goldstone bosons without Lorentz invariance. I am trying to derive all the equations to get a better understanding. But I have met some trouble with Eqs. (5) and (6). How does the functions $e_i(\\pi)$ are determined so that $\\delta\\mathcal{L}_{\\mathrm{eff}}$ does not depend on $\\pi^a$ explicitly? Could anyone provide a derivation of (6) from (4) and (5)? Many thanks!"}
{"id":"2594","title":"Steps for going from Polyakov action to low-energy effective action (SUGRA?) in String Theory","text":"A lot of string-theory questions have been asked here. This one is, hopefully, different in that this inquiry is specific and the expected answer would be more mathematical than philosophical in character. To give some context, I'm familiar with LQG, QFT etc. I'm a beginner in CFT and string-theory. On reading papers on AdS\/CFT and String Theory the following sentence often appears: \" **The low-energy effective action for $x$ dimensional ST of type $y$ reduces to that of supergravity (SUGRA) in $z$ dimensions with matter fields of types $a$, $b$ and $c$** \". The $64,000 dollar question any noob, such as me, wants answered is what are the mathematical\/technical steps or arguments which lead us from the fundamental action of string-theory (the Polyakov\/Nambu-Goto action) to a low- energy effective description. Of course, I might be asking a question for which the only concise answer is: \" _go read GSW_ \". Here's hoping otherwise ;)"}
{"id":"66715","title":"Function for heating of a planet by its star","text":"I'm looking for an algorithmic model which will gives rough estimate of the average temperature of a planet's atmosphere (good enough to say whether there will be liquid water anywhere, or if metals and plastics will melt\/ignite) based on the luminosity of its star, the radius of its orbit, the constituents of its atmosphere, the mass of the planet, and the surface area of the planet. Specifically, I figure I need two mathematical models. The first would simulate the incoming thermal energy from the star, and the second would calculate how much is lost as blackbody radiation by the planet, taking into account the effects of the 'greenhouse effect'. Does anyone know how to model these things?"}
{"id":"130808","title":"What are the similarities and differences between speech and music sounds?","text":"From an acoustic perspective, I guess speech sounds are produced by varying\/manipulating:   1. resonance (shaping the cavity; something to do with harmony?)   2. f0 (the length of the string\/pipe\/whatever) However, in music, only (2) can be varied\/manipulated. That is, the resonance\/cavity doesn't change. As you probably guessed, I'm neither a physicist, a musician, or a speech pathologist. Can you help me understand better the similarities and differences of sound production for speech vs musical instrument?"}
{"id":"130806","title":"Weak localization and constructive interference","text":"To my knowledge weak localization is a result of constructive interference between scattering processes and their time reversed. Traveling a given path in a vector potential leads to the phase $q\\int d\\vec{x}\\ \\vec{A}(x)$. A particle traveling the time reversed path gets an additional \"$-$\"sign as the path is inverted. Somehow the spin flip seems to give an additional sign resulting in constructive interference. How is that? How is the phase of a particle in an vector potential connected to the spin?"}
{"id":"76876","title":"Key difference between 3D and 4D solutions","text":"In many papers in theoretical physics especially in the more advanced theories, I notice that physicists solve problems in 3 dimensions (2 spatial+ 1 temporal). In some specific papers (I can't seem to recall exactly which ones), some models solved in three dimensions gave finite solutions, whereas 4D (3+1) were either hard to solve or gave divergent results. I don't seem to understand why adding a 3rd spatial dimension should cause problems if 2 spatial dimensions gave no problems. Intuitively, it seems straightforward matter to extrapolate 4D solutions from 3D. Then what is the essential difference between the 2?"}
{"id":"121327","title":"What determines the speed of waves in water?","text":"While I was walking my dog this morning, I passed over a canal filled with boats, barges, and kayaks all of different masses and moving at different speeds. I noticed that all of these vessels left behind wakes, and the waves of these wakes moved through the water at different rates, though my ability to make good observations was limited. This made me wonder: what determines the velocity of these waves? I know that sound travels through water at 1,480 m\/s or so, but surface waves travel at different rates (tsunamis travel faster than the ripples I make in my bowl of soup when I blow on it). In particular, how is the velocity of wake waves determined?"}
{"id":"86902","title":"Superposition of waves hitting at right angle","text":"I'm considering, in general, a spherical wave coming from a pin hole and landing on a wafer, both directly and off of a mirror. The mirror and wafer are at right angle. If it weren't a pin hole, and the light was coherent and moving straight at the mirror\/wafer, I read that the period is $\\lambda\/(2\\sin(\\theta)$, but I have no idea why. I think the main part of this problem is just, in general, calculating the period of an interference pattern on a plane, of two light sources of the same wavelength, potentially in phase, hitting at two angles-one from the left, one from the right."}
{"id":"121491","title":"What interaction is responsible for the 21 cm Hydrogen line transition?","text":"The 21 cm Hydrogen line is from the transition between the hyperfine levels of the ground state of the hydrogen atom. So, what interaction is coupling the hyperfine levels? I suspect that it is not the usual EM interaction, which causes the 1s-2p transition."}
{"id":"36398","title":"Is it possible that nuclear fission contributes to climate change?","text":"This is probably a really stupid question, please forgive me. Is it possible that the use of nuclear fission on earth contributes to the increased energy in the Earth's system as according with the Laws of Thermodynamics regarding the Conservation of Energy?"}
{"id":"99130","title":"Is a magnetic field affected by the presence of air?","text":"Does air affect a magnetic field's ability to attract ferromagnets? If a magnetic field and a ferromagnet were placed in a vacuum would there be a better performance?"}
{"id":"133729","title":"Definition of force for a scalar field","text":"I have symmetry-breaking potential V that its reflection symmetry breaks in $\\phi_0$. ![enter image description here](http:\/\/i.stack.imgur.com\/qjjnz.jpg) The action in presence of gravity is: ![enter image description here](http:\/\/i.stack.imgur.com\/X2VKk.jpg) How we calculate the scalar mediated force $F_\\phi$ ? How can I obtain below relation? ![enter image description here](http:\/\/i.stack.imgur.com\/7gWpX.jpg) Thanks for your Help!"}
{"id":"133726","title":"Quartic terms as two particle interaction?","text":"In some lecture notes I found the sentence: > [...] considering two toy models: the classical and quantum harmonic > oscillator with quartic perturbation. The quartic terms $\\sim x^4$ and $\\sim > \\hat x ^4$, respectively, correspond to the two-particle interaction in > fermionic many-body models. Does the author only refer to a formal similarity (as in second quantization interaction terms are quartic) or can a perturbated harmonic oscillator really represent an interacting system? If so, how can one see this?"}
{"id":"133721","title":"Geodesic deviation on a unit sphere","text":"Very little interest in the original version of this question so I've rejigged it hoping for a more positive response. I'm trying to use the geodesic deviation equation$$\\frac{D^{2}\\xi^{\\mu}}{D\\lambda^{2}}+R_{\\phantom{\\mu}\\beta\\alpha\\gamma}^{\\mu}\\xi^{\\alpha}\\frac{dx^{\\beta}}{d\\lambda}\\frac{dx^{\\gamma}}{d\\lambda}=0$$ to show that on the surface of a unit sphere two particles separated by initial distance $d$ , starting from the equator and travelling north (ie on lines of constant $\\phi$) will have a separation $s$ given by$$s=d\\sin\\theta.$$ This is similar to Geodesic devation on a two sphere except that question was solved using simple spherical geometry. My plan is find $\\frac{D^{2}\\xi^{\\mu}}{D\\lambda^{2}}$ first by using the absolute derivative $$\\frac{DV^{\\alpha}}{d\\lambda}=\\frac{dV^{\\alpha}}{d\\lambda}+V^{\\gamma}\\Gamma_{\\gamma\\beta}^{\\alpha}\\frac{dx^{\\beta}}{d\\lambda}.$$ Then take the second derivative of this. Next find $\\frac{D^{2}\\xi^{\\mu}}{D\\lambda^{2}}$ by calculating the Riemann tensor part$$R_{\\phantom{\\mu}\\beta\\alpha\\gamma}^{\\mu}\\xi^{\\alpha}\\frac{dx^{\\beta}}{d\\lambda}\\frac{dx^{\\gamma}}{d\\lambda}.$$ And then try to juggle the results to show the separation $s=\\xi^{\\phi}$ as a function of $\\theta$. The line element for spherical coordinates $$l^{2}=dr^{2}+r^{2}d\\theta^{2}+r^{2}\\sin^{2}\\theta d\\phi^{2}$$ for a great circle of constant $\\phi$ on a sphere of unit radius reduces to $$dl^{2}=d\\theta^{2}$$ giving $\\frac{d\\theta}{dl}=\\frac{d\\theta}{d\\lambda}=1$ and $\\frac{d\\phi}{dl}=\\frac{d\\phi}{d\\lambda}=0$. The absolute derivative for $\\Gamma_{\\theta\\phi}^{\\phi}=\\Gamma_{\\phi\\theta}^{\\phi}=\\frac{\\cos\\theta}{\\mathbf{\\mathbf{\\sin\\theta}}}$ is$$\\frac{D\\xi^{\\phi}}{d\\lambda}=\\frac{d\\xi^{\\phi}}{d\\lambda}+\\xi^{\\theta}\\Gamma_{\\theta\\phi}^{\\phi}\\frac{d\\phi}{d\\lambda}+\\xi^{\\phi}\\Gamma_{\\phi\\phi}^{\\phi}\\frac{d\\phi}{d\\lambda}+\\xi^{\\theta}\\Gamma_{\\theta\\theta}^{\\phi}\\frac{d\\theta}{d\\lambda}+\\xi^{\\phi}\\Gamma_{\\phi\\theta}^{\\phi}\\frac{d\\theta}{d\\lambda}=\\frac{d\\xi^{\\phi}}{d\\lambda}+\\xi^{\\phi}\\frac{\\cos\\theta}{\\mathbf{\\mathbf{\\sin\\theta}}}.$$ And for $\\Gamma_{\\phi\\phi}^{\\theta}=\\sin\\theta\\cos\\theta$ is $$\\frac{D\\xi^{\\theta}}{d\\lambda}=\\frac{d\\xi^{\\theta}}{d\\lambda}+\\xi^{\\phi}\\Gamma_{\\phi\\phi}^{\\theta}\\frac{d\\phi}{d\\lambda}+\\xi^{\\phi}\\Gamma_{\\phi\\theta}^{\\theta}\\frac{d\\theta}{d\\lambda}+\\xi^{\\theta}\\Gamma_{\\theta\\phi}^{\\theta}\\frac{d\\phi}{d\\lambda}+\\xi^{\\theta}\\Gamma_{\\theta\\theta}^{\\theta}\\frac{d\\theta}{d\\lambda}=\\frac{d\\xi^{\\theta}}{d\\lambda}.$$ However, $$\\frac{D\\xi^{\\phi}}{d\\lambda}=\\frac{d\\xi^{\\phi}}{d\\lambda}+\\xi^{\\phi}\\frac{\\cos\\theta}{\\mathbf{\\mathbf{\\sin\\theta}}}$$ doesn't look right as it blows up when $\\theta=0$. Any suggestions where I might be going wrong?"}
{"id":"107247","title":"Dyson air blade principle as a propulsion system for an autonomous underwater vehicle?","text":"Any thoughts on applying this principle to an AUV (autonomous underwater vehicle) ? if the power source is a battery, would its efficiency be beneficial to that of a propulsion system?"}
{"id":"71118","title":"Why do spherical waves diminish as 1\/$r^2$?","text":"Seeing the derivation of a plane wave in Giancoli, I can't understand why a spherical wave would diminish in amplitude. Shouldn't the peaks still be mutually induced, and therefore nondiminishing?"}
{"id":"76464","title":"Classical models with unbounded particle number","text":"Is there any classical model which deals with the birth, life and death of particles? What application could it have? I am talking about a 'billiard-ball' kind of model, but the kind in which balls keep popping in and out. What would the configuration space of such a system be like if particle number was not conserved? How would the Hamiltonian evolve with time? I am not interested in a statistical models. I can't think of any real world application of such a model. But perhaps there is a mathematical apparatus to deal with the continuous evolution of such systems?"}
{"id":"76460","title":"Ship wake height","text":"It is well known that a ship traveling in a deep water produces wake. Far from the aft of the ship, these wake are called a Kelvin wake. I found the mathematical expression of the shape of the wake, but not its height vs. $r$ where $r$ is the distance from the ship's aft."}
{"id":"10012","title":"What are the viable non-anthropic explanations for the tininess of the cosmological constant?","text":"The cosmological constant is 10-120 times its natural value, but it is yet nonzero. Even TeV-scale supersymmetry breaking can't save it. The renormalization group would seem to imply it ought to be at the Planck scale, assuming naturalness, and no fine-tuning. The anthropic principle explains the size of the cosmological constant nicely, but are there any other explanations? Dynamical mechanisms run into problems with the renormalization group. The mechanism has to act in the infrared, but any ultraviolet definition will be modified by the renormalization group."}
{"id":"15514","title":"How to set up a very simple experiment in optics?","text":"This might come across as a very rudimentary question. My fundamentals of Optics are weak. In the optics chapter of my physics text book I saw diagrams each depicting an object on the left and a lens -- typically convex -- in the middle and an inverted image forming on the right side. Let's say I want to perform this experiment, how can I do it? I have always seen such lenses as parts of complex systems like cameras. Do I get such stand-alone lenses from the market off the shelf? And if I keep a candle on the left side, on what screen do I capture the image on the right side? A vertical plate, paper, or the like won't show an image. (Why is this so?)"}
{"id":"15516","title":"How are we able to view an object in a room with bulb..?","text":"This is a very basic question on optics. How are we able to view an object kept in a room with a bulb? From what I understand, light rays from bulb will hit the object and some colour will be absorbed and rest will be thrown off. but how does the thrown off light reach eyes? I thought a light ray will reflect with same angle back (as the angle between incident ray and normal) then how is it guaranteed that a reflected ray will reach eyes? ![enter image description here](http:\/\/i.stack.imgur.com\/X2En8.png)"}
{"id":"89955","title":"What causes there to be an offset between the light lensing through the space neighboring galaxy clusters and the galaxy clusters themselves?","text":"http:\/\/www.wired.com\/wiredscience\/2013\/01\/musket-ball-dark-force\/ > The reason this is strange is that dark matter is thought to barely interact > with itself. The dark matter should just coast through itself and move at > the same speed as the hardly interacting galaxies. Instead, it looks like > the dark matter is crashing into something — perhaps itself – and slowing > down faster than the galaxies are. But this would require the dark matter to > be able to interact with itself in a completely new an unexpected way, a > “dark force” that affects only dark matter. Lawrence M. Krauss > If you removed all of the particles, all of the radiation, absolutely > everything from space and all that remained was nothing that nothing would > weigh something.\" What weighs something has mass. If you remove everything else then all which remains is spacetime which means mass is associated with spacetime, correct? Is it necessary for physicists to conjure a 'new dark force' or is it more likely mass is associated with spacetime and it is the mass associated with spacetime which 'piles up' analogous to the bow waves of two boats which pass by each other very closely?"}
{"id":"123231","title":"Levi Civita covariance and contravariance","text":"I read some older posts about this question, but I don't know if I'm getting it. I'm working with a Lagrangian involving some Levi Civita symbols, and when I calculate a term containing $\\epsilon^{ijk}$ I obtain the contrary sign using $\\epsilon_{ijk}$. I always apply the normal rules: $\\epsilon_ {ijk}=\\epsilon^{ijk}=1$; $\\epsilon_ {jik}=\\epsilon^{jik}=-1$ etc. I believed that there is no difference between covariant and contravariant Levi-Civita symbol. What do you know about this?"}
{"id":"123236","title":"Interpretation of space time Minkowski diagram","text":"How to interpret the following space-time diagram in the image. I know how to interpret euclidean distance from Euclidean space diagram ![enter image description here](http:\/\/i.stack.imgur.com\/OdtPO.jpg) **omit the line \"whereas for Euclidean space\".**"}
{"id":"44849","title":"Phase and Group Velocity of Electromagnetic Waves","text":"Moving charges produce oscillating electric and magnetic fields -we have an electromagnetic wave.   1. In terms of moving charges or at the level of charges, what is phase velocity and group velocity of an electromagnetic wave?    2. What is the origin of these velocities?  I am to not able to relate the definitions of phase velocity(how fast the phase of a wave is moving) and group velocity(velocity of the envelope) to moving charges. I am trying to get an intuitive picture of this. Please help me."}
{"id":"86124","title":"Is Feynman talking about the Zeroth Law of Thermodynamics?","text":"In Volume 1 Chapter 39 of the Feynman Lectures on Physics, Feynman derives the ideal gas law from Newton's laws of motion. But then on page 41-1, he puts a caveat to the derivation he has just completed (italics in original): > Incidentally, when we say that the mean kinetic energy of the particle is > $\\frac{3}{2}kT$, we claim to have derived this from Newton's laws.... and it > is most interesting that we can apparently get so much from so little ... > How do we get so much out? The answer is that we have been perpetually > making a certain important assumption, which is that if a given system is in > thermal equilibrium at some temperature, it will also be in thermal > equilibrium with _anything else_ at the same temperature. For instance, if > we wanted to see how a particle would move if it was really colliding with > water, we could imagine that there was a gas present, composed of another > kind of particle, little fine pellets that (we suppose) do not interact with > water, but only hit the particle with \"hard\" collisons. Suppose that the > particle has a prong sticking out of it; all our pellets have to do is hit > the prong. We know all about this imaginary gas of pellets at temperature > $T$ - it is an ideal gas. Water is complicated, but an ideal gas is simple. > Now, _our particle has to be in equilibrium with the gas of pellets_. > Therefore, the mean motion of the particle must be what we get for gaseous > collisions, because if it were not moving at the right speed relative to the > water but, say, was moving faster, that would mean that the pellets would > pick up energy from it and get hotter than the water. But we had started > them at the same temperature, and we assume that if a thing is once in > equilibrium it stays in equilibrium - parts of it do not get hotter and > other parts colder, spontaneously. This proposition is true and can be > proved from the laws of mechanics, but the proof is very complicated and can > be established only by using advanced mechanics. It is much easier to prove > in quantum mechanics than it is in classical mechanics. It was first proved > by Boltzmann, but now we simply take it to be true, and then we can argue > that our particle has to have $\\frac{3}{2}kT$ of energy if it is hit with > artificial pellets, so it also must have $\\frac{3}{2}kT$ when it is being > hit with water at the same temperature and we take away the pellets; so it > is $\\frac{3}{2}kT$. It is a strange line of argument, but perfectly valid. My question is, what is the proposition of Boltzmann that Feynman is referring to? I can think of three possibilities:   1. It could refer to the Equipartition Theorem, which was independently proven by Maxwell and Boltzmann, since the title of the section is \"The Equipartition of Energy\".   2. It could refer to the Zeroth Law of Thermodynamics, because \"it will also be in thermal equilibrium with _anything else_ \" (especially when he emphasizes the \"anything else\") sounds like the fact that thermal equilibrium is an equivalence relation.    3. It could refer to Boltzmann's H-theorem, which was the derivation of the second law of thermodynamics from Newton's laws, because \"if a thing is once in equilibrium it stays in equilibrium - parts of it do not get hotter and other parts colder, spontaneously\" sounds like some formulations of the second law. So which of the three is it, if it's any of the three, and how would you prove it from Newton's Laws? Any help would be greatly appreciated. Thank You in Advance."}
{"id":"127429","title":"Practical (maybe naive) question on boiling water","text":"I put a glass bottle in a big pot, bottleneck up, I fill both pot and bottle with very hot water. The bottle is submerged except a few cm of the bottleneck and filled to the brim so some water inside the bottle is above the pot water. Then I light the fire. After a while the water in the pot boils but even if I keep it boiling for 30 minutes the water inside the bottle does not boil. All I see in the water inside the bottle are very tiny bubbles. The same happens if I put the bottle upside down (with some air at the top) What is the reason? Maybe the glass acts as insulator and water inside the bottle cannot reach 100 degrees? Or having a few cm of bottle (and water) above the pot surface cools the water inside?"}
{"id":"127420","title":"How does radiation shielding using absorbing materials work?","text":"I understand that, for example, a thick enough sheet of lead can absorb gamma radiation, but I want to understand what actually happens at the molecular\/atomic\/subatomic level. Also, can the same logic be applied to cosmic particles? I have tried Googling for an answer, but to no avail. Can someone enlighten me?"}
{"id":"69879","title":"Anyons without fractional spin?","text":"Is it possible to have particles obeying anyonic statistics but not having fractional spin? I am wondering, because while spin in quantum physics arises from the geometry\/topology of spacetime, statistics is connected to the geometry\/topology of configuration space. What would particles like this be called?"}
{"id":"89095","title":"What is the requirements from an optical lab in terms of darkness","text":"I am characterizing an optical lab and would like to know therefore what are the requirement from a standard optical lab in terms of ambient luminescence. In other words how dark should an optical lab be. The lab I am talking about should fit for experiments of photoluminescence and Raman spectroscopy."}
{"id":"89098","title":"Throwing a ball upwards in an accelerating train","text":"If I throw a ball upwards to a certain height in an accelerating train, will it end up in my hand? At the moment I release the ball, it will have a velocity equal to that of the train at that instant. But because the train is accelerating, the train will have a greater velocity and thus the ball will travel lesser distance and should fall behind me. Is my reasoning correct?"}
{"id":"32334","title":"Integration of partition-function over many momentum variables","text":"My integral looks like $$Z(\\beta) = \\frac{1}{h^3}\\int d^3p\\ \\exp{\\left(-\\frac{\\beta}{2m}\\sum^{3N}_{i=1}p_i^2\\right)}.$$ I'm confused about how to integrate over seemingly 3N variables in only a 3-dimensional integral."}
{"id":"5681","title":"Gravitational effect of the distant universe","text":"When we observe at close to the cosmological horizon (as close as we currently can) we see the universe as a much younger, denser place. As we feel the gravitational forces from these objects distant, will we measure a tidal force increasing the distance between an object less distant than the cosmological horizon and ourselves? Or for that matter between any two objects at different distances from earth? If so is it great enough to account for the accelerating expansion of the universe?"}
{"id":"116534","title":"Derive force from the pair-wise potential equation","text":"How can we calculate the force exerted on particle $i$ by particle $j$ given a general potential function $V(d)$? Let's discuss this genenal question with a concrete example below. To simulate the motion of a set of $n$ particles with equal masses, we need to calculate the resultant force for each particle, which is the sum of force vectors. These particles move along a one-dimensional axis and interact via a pair-wise potential, where the $$V(d) = \\frac{\\exp(-2d^3) - 1}{d},$$ where $d$ is the distance between particle $i$ and particle $j$. I don't have strong math and physics background, and I've been learning partial derivative and potential energy on wikipedia, but it makes me more confused. Simeon Carstens' answer below explains a lot!! It says $$F_i=−\\frac{\\partial}{\\partial x_i}V(x_1,...,x_n).$$ So my understanding is that the force exerted on particle i by particle j can be expanded as $Fi=-\\frac{∂}{∂xi}V(d(xi,xj))=-\\frac{∂}{∂xi}\\frac{exp(-2(xi- xj)^3) - 1}{xi-xj}$. By $(\\frac{u}{v})'=\\frac{u'v-v'u}{v^2}$, the above epxression can be derived as $Fi=\\frac{(6(xi-xj)^3+1)exp(-2(xi-xj)^3) - 1}{(xi-xj)^2}$. Am I correct? Please tell me if I'm wrong and which part is incorrect! Any help is apprieciated!"}
{"id":"79502","title":"Friction From an Object On-Top of a Sliding Object","text":"Consider a block $A$ lying on a flat and frictionless table, and a block $B$ lying on top of block $A$. A horizontal force $F$ is applied to block $A$. If there is no friction between blocks $A$ and $B$, then block $A$ will move in the direction of $F$ and $B$ will remain stationary until $A$ slides out from under it, at which point it will fall. If the friction between blocks $A$ and $B$ is nonzero, however, then block $B$ will move with block $A$ in the direction of $F$, as expected. While this result makes sense intuitively, and is consistent with out everyday experience, I've been unable to rationalize this behavior in terms of forces. If block $B$ is moving along with block $A$, then there must be some force acting on $B$ (it started out at rest, after all), and my teacher said this force is a frictional which is equal to $F$. This seems strange, considering my previous experience with friction, which I understand to be a force that _opposes motion_ , not one that causes it. How is it possible that there could be a frictional force acting on $B$ if there is no other force acting on it? Perhaps my understanding of friction is too narrow, but I do not see how we can mathematically or logically justify why friction would be causing motion in a body that is otherwise not subject to any forces."}
{"id":"81695","title":"Expansion of the Universe: Conversion of gravitational potential energy to kinetic energy?","text":"Suppose there is an object floating in space which over time begins to fall toward the source of a gravitational field. As it falls, its motion happens to be such that it gets locked in orbit around the source with a greater velocity than it had before it 'began to fall'. So it's gravitational potential has been converted to kinetic energy. According to relativity, this increased speed should increase the gravitational potential of the object (and therefore the object + the original source of the field), correct? Does this mean that as a result of this, the expansion of the Universe should slow down slightly (because there is now a slightly greater gravitational potential in the Universe)? And if so, would that imply that the conversion from gravitational potential to kinetic energy is in a sense a conversion between the bulk kinetic energy of the expanding Universe and the local kinetic energy of a test mass?"}
{"id":"81692","title":"Energy required to demagnetize a soft iron?","text":"How much energy is required to demagnetize a soft iron ferromagnetic material, that has a very low coercive force And a small hysteresis area? Also, would it be possible to demagnetize that same soft iron while it's aligned with an exterior magnetic field via vibrations from hammering\/dropping? **The hammering is constant so the vibrations should be constant as well.** That's why I assume it can be demagnetized."}
{"id":"11519","title":"How to make a small tokamak?","text":"$\\require{mhchem}$I made a fusor once, like the easy science project: deuterium-deuterium ones, but they're really inefficient. I was wondering if it would be possible to make a small tokamak; not one that will make a ton of power, but I just want to make one for the sake of making one, like a tokamak the size of a pizza box. What materials would I need and how would I go about assembling it? Assume it is deuterium-deuterium or $\\ce{p-^11B}$ fusion, maybe $\\ce{^3He-^3He}$."}
{"id":"11514","title":"Notational(?) Question in Whiting's Paper \"Mode Stability of the Kerr Black Hole\"","text":"I am a math grad student attempting to read Bernard Whiting's paper \"Mode Stability of the Kerr Black Hole.\" If you are in a university network, the paper should be easily found by a google search. At the beginning of the paper Whiting refers to the \"master equation of Teukolsky\" from the paper \"Rotating Black Holes: Separable Wave Equations for Gravitational and Electromagnetic Perturbations.\" However, the equation Whiting writes down appears (to me) to be different than Teukolsky's. I assume I am missing something silly and was hoping that someone else has read this paper and can easily point out my mistake. One needs to look at the paper for all the relevant definitions, but for ease of reference: Whiting writes down $\\\\{\\partial_r\\Delta\\partial_r - (1\/\\Delta)\\\\{(r^2+a^2)\\partial_t + a\\partial_{\\phi} - (r-M)s\\\\}^2 - 4s(r+ia\\cos\\theta)\\partial_t + $ $\\partial_{\\cos\\theta}\\sin^2\\theta\\partial_{\\cos\\theta} + (1\/\\sin^2\\theta)\\\\{a\\sin^2\\theta\\partial_t + \\partial_{\\phi} + i\\cos\\theta\\cdot s\\\\}^2\\\\}\\psi = 0$ where $\\Delta = r^2 - 2Mr + a^2$ while Teukolsky writes down $\\\\{(\\frac{(r^2+a^2)^2}{\\Delta} - a^2\\sin^2\\theta)\\partial^2_t + \\frac{4Mar}{\\Delta}\\partial^2_{t,\\phi} + (\\frac{a^2}{\\Delta} - \\frac{1}{\\sin^2\\theta})\\partial^2_{\\phi} - \\Delta^{-s}\\partial_r(\\Delta^{s+1}\\partial_r) - $ $\\frac{1}{\\sin\\theta} \\partial_{\\theta}(\\sin\\theta\\partial_{\\theta}) - 2s(\\frac{a(r-M)}{\\Delta} + \\frac{i\\cos\\theta}{\\sin^2\\theta})\\partial_{\\phi} - 2s(\\frac{M(r^2-a^2)}{\\Delta}-r-ia\\cos\\theta)\\partial_t$ $ + (s^2\\cot^2\\theta - s)\\\\}\\psi = 0$ When I expanded out Whiting's equation out I did not get Teukolsky's equation. For example, it is clear that the $\\partial_r$ terms are different. I also got that the constant terms (no derivative terms) did not agree."}
{"id":"89328","title":"Is a track in a cloud chamber made by one particle or many particles?","text":"Is the track in a cloud chamber made by one particle interacting with many particles each emiting a photon or is the track made by many particles interacting on the same trajectory? or is it a cascade of interactions?"}
{"id":"60524","title":"Implicit Differentiation, A doubt","text":"$v=v_c(\\tau, t)$ is a smooth function and suppose we have a relation $y_c(\\tau,v_c;t)=0$ when $x_c$ is written in the form $x_c=c+ty_c(\\tau,v_c;t)$, $c$ is real constant, $t$ is real number denotes time, could any one tell me how to find implicit differentiation of the relation $y_c(\\tau,v_c;t)=0$ and from there how we get $\\frac{\\partial}{\\partial t} v_c(\\tau;t)|_{t=0}=-\\frac{1}{2} a_c(\\tau)$ where $a_c(\\tau)$ denote the initial acceleration. page $2726,$ The paper from which I reading this Chillingworth, D. R. J. \"Dynamics of an impact oscillator near a degenerate graze.\" Nonlinearity 23.11 (2010): 2723."}
{"id":"60525","title":"I want the Saturn's position in terms of Declination and Right Ascension?","text":"I want the Saturn's position in terms of Declination and Right Ascension for a couple of month in the interval of 1 hour in a text file to do a simulation. Which site can provide me these data? Or, there may be some software which can give these data in txt file. I know Stellarium 0.12.0 which show data. But does it provide the options to save the data in the text file also?"}
{"id":"56992","title":"How to describe heat transfer between two solid materials?","text":"A general equation for dealing with heat transfer between one material and a region of insulating material. I've seen basic heat transfer equations for one material, but I'd love to see an explanation of how to do two."}
{"id":"52436","title":"Compton Scattering","text":"Compton Scattering essentially states that when a photon of a given wavelength hits an electron the energy level of the electron changes and the photon has its wavelength changed. This seems to be implying that it is the same photon that is reflected outwards. Do we know it is the same photon of a changed wavelength, or is it possible that the original photon merely pushed a 2nd photon out of the electron? (This would mean the original photon is not changing wavelength, that is meerly absorbed)."}
{"id":"52435","title":"Universal expansion faster than the speed of light","text":"If the universe is expanding faster than the speed of light, and force carrier particles move at the speed of light, wouldn't this cause infinite universal expansion? Since no forces would be acting upon the outer edge of the universe? (This doesn't account for the apparent acceleration of the universe but it does account for the lack of slowing down for the universe)."}
{"id":"90964","title":"Light bodies at high speeds behave like heavy objects when recorded by a high speed camera","text":"This is potentially a very dumb question, but recently I was looking at some slow motion footage of bullets hitting stuff and I realized something interesting. One of the clips showed two metal objects (approximately bullet sized) colliding in midair at high speeds. They didn't go directly at each other. In fact their approach was something like this               ------>           7          \/         \/        \/       \/      So as they approached eachother \"slowly\", when they finally connected the contact surface practically exploded. In any case, the deformation was much larger than what I expected considering the difference vector of their velocities was quite small (they were traveling at the same speeds, but slightly different angles). This reminded me of scenes in sci-fy movies where two large spaceships collide in space but at low speeds. Since their mass is very high, the lack of their speed doesn't matter. They rip through each other. Related to this, imagine you grab an ant or some small insect and throw it at the wall. It might grab the wall or fall off, but it won't be hurt. I imagine if you recorded it with a high speed camera you could apply the same principle. Like bullets, it behaves like a heavy object but since you can't throw it nowhere near as fast as a bullet, it's \"heavyness\" isn't enough because the camera \"slows it down\" enough so he doesn't get hurt. My question is, what's the correlation between those three factors? The mass, the speed of the object and the speed of the camera. Could we potentially make small models of spaceships, throw them at each other very fast so when you record them in slow motion it looks like big spaceships colliding slowly?"}
{"id":"90963","title":"Canonical equal time commutation relations in QED","text":"I understand that to quantize the classical electromagnetic field one needs to impose commutation relations and express the field in terms of creation and annihilation operators. I notice that the canonical equal-time commutation rule comes in to use here. How do you derive this? What is it meant to represent? ![enter image description here](http:\/\/i.stack.imgur.com\/pEpBe.png) Also, how do you prove the following commutation relation? ![enter image description here](http:\/\/i.stack.imgur.com\/BLWNa.png)"}
{"id":"121861","title":"How to find the amplitude of magnetic field of this plane wave","text":"Because my previous question was short and confusing, it was put on hold. This is my new question, which getes straight to the point. * * * I have an electromagnetic plane wave which has wavelength $λ$ = 1 mm which is traveling through air parallel with $z$ axis (to +$z$) .The amplitude of the electric field is perpendicular to the $z$ axis and is stated $\\vec{E}_o=A(\\hat x+\\hat y)$ where $A$ = 0,5 V\/m. How can I find the magnetic field amplitude of $\\vec{H}_o $ . I think I must use Maxwell's equation $ ∇×\\vec{E} = − j\\omega\\mu_0 \\vec{H} $, from which follows $ \\vec{H}_o = \\vec{k} \\circ \\vec{E}_o \/ \\omega \\mu_0$ , but where can the propagation constant $k$ can be derived?"}
{"id":"121867","title":"Can a difference in the \"speed of time\" introduce acceleration?","text":"Hypothetically, lets say we have a space divided equally into two adjacent areas where (somehow) in one of the areas time goes by at half the speed as the other area. Or specifically, when a clock in the fast area shows 1 minute having gone by, a clock in the slow area shows only 30 seconds having elapsed.   1. In order for the speed of light to be constant, would lengths in the fast half be compressed to half the size of the slow half? **My guess:** _Yes_   2. (assuming (1) is correct) Let's create two identical objects, one in each area, positioned as close together as possible without passing into the other area, each given an equal velocity in a direction parallel to the boundary between the two spaces. **Q:** Over time would the distance between the objects increase? **My guess:** _No_   3. OK, (2) was all kinds of lame. So to make it more interesting lets put a strip of space between (but not touching) the two objects where time flows at the average rate between the two areas (so 75% as fast as the \"fast\" area). Then lets connect the two objects with a rigid bar of negligible size and mass that goes right through the \"average speed\" area. Then at the exact center of the bar (equidistant between the two objects) we give the compound object a specific velocity - again in a direction parallel to the boundaries between the areas. **Q:** Would the compound object rotate? **My guess:** _yes - but it's just a hunch_   4. (if (3) is correct) Depending on the direction of the rotation, since you would constantly get the two objects changing which area they are in, would the compound object just keep spinning faster, accelerate toward the slow area, or accelerate toward the fast area?   5. I've totally been ignoring mass on purpose. I realize that the same things that cause time dilation also cause a change in mass. Would taking mass into account change the results of any of the above? (Edit below) I realize that this is all hypothetical and is a situation that cannot exist naturally (which is why I tagged it as a thought-experiment). What I want to know is whether the time-dilation portion of a space-time curvature introduces an acceleration above and beyond that caused by gravity, or whether it gets cancelled out by the length distortion, or if the curvature of space-time is what causes the acceleration of gravity and it would make no sense to talk about the effects of time-dilation separately. The thought experiment is just my effort to figure out exactly what the interactions between time and gravity are."}
{"id":"82920","title":"What is the depth in meters of the pond?","text":"A small spherical gas bubble of diameter $d= 4$ μm forms at the bottom of a pond. When the bubble rises to the surface its diameter is $n=1.1$ times bigger. What is the depth in meters of the pond? Note: water's surface tension and density are $σ= 73 \\times 10^{-3} \\mbox{ N}$ and $ρ= 10^3 \\mbox{ kg\/m}^3$, respectively. The gas expansion is assumed to be isothermal. My attempts: I used the equation of pressure: $P_1V_1=P_2V_2$ where $P_1$ is the pressure at the top and $V_1$ is the volume of bubble at the top and $P_2$ is the pressure at the bottom, and $V_2$ is the volume of the bubble at the bottom Because the bubble at the bottom, it received the hydrostatic pressure, so the equation became: $P_1V_1$= $(P_0+\\rho g d) V_2$ Since $V_1$ and $V_2$ is sphere, we can use the sphere volume. And $P_1$ is same with atmospheric pressure= $10^5 \\mbox{ Pa}$ $10^5 \\cdot (\\frac{4}{3} \\pi r_1^3) = (10^5 + 10^3 \\cdot 9.8 \\cdot d) \\cdot (\\frac{4}{3} \\pi r_2^3)$ Cancel out the $\\frac{4}{3}\\pi$ and we get: $10^5 \\cdot (r_1)^3 = (r_2)^3 \\cdot (10^5 + 9800d)$ Substitute $r_1 = 4 \\times 1.1= 4.4 \\mbox{ μm}= 4.4\\times10^{-6} \\mbox{ m}$ and $r_2 = 4 \\times 10^{-6} \\mbox{ m}$ $10^5 \\cdot (4.4 \\times 10^{-6})^3 = (4 \\times 10^{-6})^3 \\cdot (10^5 + 9800d)$ $10^{-13} \\cdot (4.4)^3 = 64 \\times 10^{-18} \\times 10^5 + 64 \\times 10^{-18} \\times 9800d$ $85.184 \\times 10^{-13} = 64 \\times 10^{-13} + 627200 \\times 10^{-18} d$ $21.184 \\times 10^{-13} = 627200 \\times 10^{-18} d$ $d= 3.37 \\mbox{ m}$ So, the depth of the pond is $3.37 \\mbox{ m}$. My question: what is the useful of $σ = 73 \\times 10^{-3} \\mbox{ N}$ ? I really confused about it. Thanks"}
{"id":"46143","title":"How do I calculate speed of air on each side of airfoil?","text":"I am doing a project on Bernoulli's principal for my High School physics course. I made an airfoil in accordance with NACA 2412 (same as a C-152). I am going to put it in a wind tunnel where the wind speed is going to be measured. How can I find the wind speed on top of and on the bottom of the airfoil, once I know what the outside wind speed (or velocity of the airfoil) is? My end goal is to find the pressure differences on the outside of the wing, and I can do that very easily once I know the speed on each side of the airfoil."}
{"id":"10651","title":"Is 3+1 spacetime as privileged as is claimed?","text":"I've often heard the argument that having 3 spatial dimensions is very special. Such arguments are invariably based on certain assumptions that do not appear to be justifiable at all, at least to me. There is a summary of arguments on Wikipedia. For example, a common argument for why >3 dimensions is too many is that the gravitational law cannot result in stable orbital motion. A common argument for <3 dimensions being too few is that one cannot have a gastrointestinal tract, or more generally, a hole that doesn't split an organism into two. Am I being overly skeptical in thinking that while the force of _gravity_ may not be able to hold objects in stable orbits, there most certainly exist sets of physical laws in higher dimensions which result in formation of stable structures at all scales? It may be utterly different to our universe, but who said a 4D universe must be the same as ours with one extra dimension? Similarly, isn't it very easy to conceive of a 2D universe in which organisms can feed despite not having any holes, or not falling apart despite having them? For example, being held together by attractive forces, or allowing certain fundamental objects of a universe to interpenetrate, and thus enter a region of the body in which they become utilized. Or, conceive of a universe so incomprehensibly different to ours that feeding is unnecessary, and self- aware structures form through completely different processes. While I realise that this is sort of a metaphysical question, is 3+1 dimensions really widely acknowledged to be particularly privileged by respected physicists?"}
{"id":"113808","title":"What is $\\hat{p}|x\\rangle$?","text":"Trying to solve a QM harmonic oscillator problem I found out I need to calculate $\\hat{p}|x\\rangle$, where $\\hat{p}$ is the momentum operator and $|x\\rangle$ is an eigenstate of the position operator, with $\\hat{x}|x\\rangle = x|x\\rangle$. It turns out that $\\hat{p}|x\\rangle$ should be equal to $-i\\hbar\\frac{ \\partial |x\\rangle }{\\partial x}$. But what does this even mean? I'm not sure what to do with an expression like that. If I try to find $\\langle x | \\hat{p} | x_0 \\rangle$, I end up with $-i\\hbar \\frac{\\partial}{\\partial x} \\delta(x-x_0)$, which I'm pretty sure is not allowed even in physics, where we would scoff at phrases like \"$\\delta$ is really a distribution and not a function\". Edit: Motivation. I have solved Heisenberg's equation and found, for a Hamiltonian $H = p^2\/2m + m\\omega^2x^2\/2$, that $\\hat{x}(t) = \\hat{x}_0 \\cos(\\omega t) + \\frac{\\hat{p}_0}{m\\omega} \\sin(\\omega t)$. I am given that the initial state is $|\\psi(0)\\rangle = |x_0\\rangle$, i.e., an eigenstate of position, and I have a hunch that by finding $\\hat{x}(t)|x_0\\rangle$ I might find a way to get $|\\psi(t)\\rangle$. But this is not what I need help with; I am now curious as to what $\\hat{p}|x\\rangle$ might be, never mind my original problem."}
{"id":"107300","title":"Statistical Mechanics - Distribution of Energies","text":"Consider a state space $\\mathbb{X}$. The probability density function under a canonical ensemble is given by the Boltzmann distribution $$\\pi_{\\mathbb{X}}(x)=\\frac{e^{-\\beta E(x)}}{\\mathbb{Z}(\\beta)}$$ where $E(x)$ is the energy of state $x\\in\\mathbb{X}$, and $\\mathbb{Z}(\\beta)$ the partition function or _Zustandsumme_. I see the marginal density of energy given as $$\\pi_{U}(E)=\\frac{\\Omega(E)e^{-\\beta E}}{\\mathbb{Z}(\\beta)}$$ where $\\Omega(E)$ is the density of states. My question is how do you derive the marginal density for energies from the density over the state space $\\mathbb{X}$?. $E:\\mathbb{X}\\rightarrow\\mathbb{R}_{+}$ is not a one-to-one function so one cannot apply the change of variables theorem."}
{"id":"54871","title":"meteor explosion, steam or shock\/vibration trigger","text":"Is the explosion of a meteor due to forces imparted to the meteor by entry into atmosphere, or is it due to heating of a meteor that contains water with conversion to steam triggering the explosion? Do chondrites explode more readily because of some water content than nickel\/iron meteorites?"}
{"id":"54875","title":"What is the meaning of phase difference?","text":"What do you mean by phase of a wave? And phase difference? Waves have always confused me as it's too difficult to visualize them. I am no good at waves mechanics, so if anyone could explain in simpler term? Also if anyone could direct me to complete solution to my problem as to study waves from where? How do you visualize it? How did you study and understand? I have been meaning to ask this question for quite a time but always thought it would be too dumb. Thank you and I am eagerly waiting for an answer and your valuable suggestion"}
{"id":"62967","title":"What materials focus EM radiation in the 2.4GHz range","text":"If glass and similar materials refract visible light effectively, what materials would be best for focusing lower frequencies of EM radiation, if any? If not, what other methods exist for focusing these ranges? The thought was inspired by wondering how you might build a camera that captures 'light' given off by Wifi routers"}
{"id":"10266","title":"\"Optically performed\" Fourier Transform","text":"This article says that they are only able to achieve such extremely high fiberoptic data rates because the multiplex light and then use a Fourier Transform to split it up again. But they say that they perform the Fourier Transform \"optically\". Now I understand the math behing a Fourier Transform, but I still can't wrap my mind around how this would be done in analog with light. Any ideas?"}
{"id":"10267","title":"Pressure inside a light bulb","text":"It is known that nowadays most light bulbs are filled with gas in order to minimise the evaporation rate of tungsten. Is e.g. argon put into the bulb with reduced pressure, so that the glass doesn't explode while heating the wire?"}
{"id":"104675","title":"What is sound in terms of acoustic sources?","text":"Sound is nothing more than small amplitude, unsteady pressure perturbations that propagate as a longitudinal wave from a region in space which created it (called the source region) into a quiescent (still) region where it is observed by hearing. It is at this point where I get slightly confused in that we may distinguish between two types of pressure perturbations: **hydrodynamic pressure perturbations** and **acoustic pressure perturbations**. The acoustic perturbations are what we term sound, as they are characterized by their ability to _propagate_ into the hearing region. The hydrodynamic pressure perturbations could be a consequence of fluid flow simply changing in the source region, and this may not necessarily propagate and become what we term sound. Perhaps evanescent waves can serve as some insight to what I am trying to explain: with reference to this link the pressure fluctations created by the subsonically pulled wavy plate decay exponentially in the upward direction and so do not constitute a sound. Is my interpretation so far correct? If so are there any rigorous methods for determining if pressure fluctations will propagate (thus becoming sound), or is sound simply that which we can hear outside a source region? Are there other cases besides the linked example for which pressure fluctations are created, but do not propagate as sound?"}
{"id":"63380","title":"Pendulum in an elevator","text":"![enter image description here](http:\/\/i.stack.imgur.com\/p5vwp.png) Suppose we have a pendulum tied to the ceiling of an elevator which is at rest. The pendulum is oscillating with a time period $T$, and it has an angular amplitude, say $\\beta$. Now at some time when the bob of the pendulum was at the mean position(equilibrium position\/center position), the elevator suddenly starts moving up with some acceleration $a$. I need to find out what is the new angular amplitude of the pendulum, in terms of $a$,$\\beta$ and if needed $T$, and $l$(length of the string). I'm completely stuck at this concept. I absolutely do not know how to proceed. My ideas are that we could use some conservation, maybe the energy of the bob. Any hints? P.S. I'm sorry if anything is too elementary."}
{"id":"21553","title":"Example in the book: A simple accelerometer","text":"# A simple accelerometer > _You tape one end of a piece of string to the ceiling light of your car and > hang a key with mass m to the other end (Figure 5.7). A protractor taped to > the light allows you to measure the angle the string makes with the > vertical. Your friend drives the car while you make measurements. When the > car has a constant acceleration with magnitude a toward the right, the > string hangs at rest (relative to the car), making an angle $B$ with the > vertical._ > >   * _(a) Derive an expression for the acceleration $a$ in terms of the mass > m and the measured angle $B$._ > >   * _(b) In particular, what is a when $B$ = 45? When $B$ = 0?_ > > I don't care about the answers, the important thing is the following:- The book says The string and the key are at rest with respect to the car, but car, string, and key are all accelerating in the +x direction. Thus, there must be a horizontal component of force acting on the key. That's the reason the book decided to consider a force in the $+x$ direction, but I'm looking for a better explanation: how would I find detect the force in the $+x$ direction in another way? To me, when I draw the free body diagram of the string, there looks to be no force acting on the $+x$ direction! I understand it starts with noticing that the string is attached to the ceiling of the car, and that the car has force causing acceleration in one direction, but I don't know how to go further than that."}
{"id":"21551","title":"How does one interpret the Dirac equation with a self-field potential?","text":"EVERY QFT text I've ever examined states that if there is an external vector potential, $A_\\mu$, then one writes the Dirac eq.(or Klein-Gordon eq.) using a covariant derivative to include this U(1) gauge field, $A_\\mu$. No problem. Wikipedia states that one must also include the contribution due to the electron self-field, $A_\\mu'$. Again, makes perfect sense. Thus if there is no external 4-vector potential, the Dirac equation should exhibit the internal one! However, all texts state the \"free\" Dirac equation, without an accompanying vector potential. If I now solve the free Dirac equation, the spinorial solutions can be used to construct the Dirac charged current. Using this current as the source, I can then solve the classical D'Alembertian eq. for the corresponding internal self-field $A_\\mu'$. If I insert $A_\\mu'$ back into the Dirac equation, the solution must be different than those obtained from the free Dirac equation. I am simply wondering what the interpretation of the solutions will be now?"}
{"id":"21555","title":"What happens when a supersonic airplane flies through a cloud?","text":"What happens when a supersonic airplane flies through a cloud? Will it punch a hole or is it more like a bullet through water (= hole closes immediately after the aircraft has passed)? Is there some special effect because of the supersonic speed? Or maybe the question should be: Does the airflow around an airplane change when the sound barrier is broken?"}
{"id":"45723","title":"Boundary conditions for fields in Kerr\/CFT","text":"I am reading a paper by Guica et al. on Kerr\/CFT correspondence (arXiv:0809.4266) and I'm not sure if I got this. They choose the boundary conditions, like a deviation of the full metric from the background Near- Horizon Extremal Kerr (NHEK) metric. Let's say that we can write that deviation like $$\\delta_\\xi g_{\\mu \\nu}=\\mathcal{L}_\\xi g_{\\mu\\nu}=\\nabla_\\mu\\xi_\\nu+\\nabla_\\nu\\xi_\\mu$$ And the most general diffeomorphism which preserve the boundary conditions given in the text is: $$\\xi=[-r\\epsilon'(\\varphi)+\\mathcal{O}(1)]\\partial_r+\\left[C+\\mathcal{O}\\left(\\frac{1}{r^3}\\right)\\right]\\partial_\\tau+\\left[\\epsilon(\\varphi)+\\mathcal{O}\\left(\\frac{1}{r^2}\\right)\\right]\\partial_\\varphi+\\mathcal{O}\\left(\\frac{1}{r}\\right)\\partial_\\theta$$ What my mentor told me, while briefly explaining this, is that we basically need to find the most general $\\xi$ such that $\\mathcal{L}_\\xi g_{\\mu\\nu}$ is within the class of the boundary conditions. But how do I find these $\\xi$? How can you find these boundary conditions and diffeomorphisms? Or better jet, how do I find diffeomorphism using those boundary conditions? :\\"}
{"id":"122887","title":"Non-additivity of energy (thermodynamics)","text":"In page 15, Chapter 1 (Laws of Thermodynamics) of the book \"An Introduction to the Study of Stellar Structure\" by Nobel laureate S. Chandrasekhar, it is mentioned (in context of total energy of two subsystems brought in contact): > In general when two bodies are brought in contact, the energy is not > additive; it is easy to see however, that the deviation is directly > proportional to common surface area of the bodies, hence, for the large > volumes the deviations from the additive law can be neglected I always thought that energy is additive as a fundamental property (or by definition). I did not find anything further to understand the statement. Please, help me understand this."}
{"id":"122886","title":"Numeric value of the electrons drift velocity in superconductors","text":"Somebody knows the numeric value of electrons drift velocity in superconductors? How this value depends from the used superconductor material? What's about the current? Since the electrical resistance tends to zero, how many electrons can pass an area (a layer) that is occupied e.g. by 100*100 atoms?"}
{"id":"44459","title":"Why does the Gallum-Arsenite bandgap narrow for higher temperatures and is this because of unavoidable impurities?","text":"I found this sentence: > GaAs at 300K contains 10^15 acceptor atoms per cubic centimeter. The bandgaps for Gallium Arsenite are: 0 Kelvin : 1.518 eV 300 Kelvin: 1.424 eV Now I am wondering:   1. Why does Gallium Arsenite have this impurity?   2. Is the bandgap at 300 Kelvin is stated for a _pure_ GaAs semiconductor or does it already incorporate the influence of the impurity?   3. If it doesn't what else does then cause the narrowing of the bandgap?"}
{"id":"79145","title":"How to derive following thermodynamic relation?","text":"I'd would like to derive the following $$\\bigg(\\frac{\\partial U}{\\partial V}\\bigg)_T = T \\bigg(\\frac{\\partial p}{\\partial T}\\bigg)_V - p$$ What I know is that the internal energy $U$ is a function of temperature and volume. Hence, a small change in $U$ can be related to changes in $T$ and $V$ by $$dU =\\bigg(\\frac{\\partial U}{\\partial T}\\bigg)_V dT + \\bigg(\\frac{\\partial U}{\\partial V}\\bigg)_T dV$$ But I'm not sure where to go from here."}
{"id":"68980","title":"Feynman diagram for $\\overline{K}\\,\\!^0$ antimeson production on the quark-level","text":"I've recently stumbled upon a physics problem concerning $\\overline{K}\\,\\\\!^0$ antimeson production. In this particular example, colliding a $\\pi^-$ meson with a stationary proton yields a $K^0$ meson and a $\\Lambda^0$ hyperon: $$\\pi^-\\,[\\overline{u}d] + p\\,[uud]\\rightarrow K^0\\,[d\\overline{s}] + \\Lambda^0\\,[uds]$$ This can be expressed in a Feynman diagram by letting the $u$ and $\\overline{u}$ quarks annihilate to a gluon, out of which a pair of $s$-$\\overline{s}$-quarks is generated. However, if a $\\overline{K}\\,\\\\!^0$ particle would be generated by the same method, in order to conserve the baryon number and the strangeness, more than just a particle must be produced. For example, the following reaction could take place, so that every quantum number is conserved: $$\\pi^-\\,[\\overline{u}d] + p\\,[uud]\\rightarrow \\overline{K}\\,\\\\!^0\\,[s\\overline{d}] + K^0\\,[d\\overline{s}] + n\\,[udd]$$ However, I can't seem to find a corresponding Feynman diagram for the reaction. I am guessing that the $\\Lambda^0$ hyperon decays weakly and somehow yields the antikaon and the neutron, but I can't figure out how... Does anyone have a clue what the Feynman diagram could be?"}
{"id":"62380","title":"is really an atom stable?","text":"Half filled and fulfilled atomic orbitals are stable because of : high exchange energy. The problem is with exchange energy. We have learnt that the half and fulfilled orbitals have maximum no. of electrons with the same spin so they can exchange their positions and thus release energy. BUT, we have also learnt that the electrons of the same orbitals have the same energy. Furthermore, the electrons of the same sub shell have the same energy in the absence of magnetic field. so, how on earth do they lose energy?"}
{"id":"13007","title":"Question concerning Isospin symmetry","text":"I'm currently taking an introductory course to particle physics and I'm now trying to understand the concept of isospin. However I do have some trouble. So let's write the up- and down Quark as a Flavour-Doublet (u,d). We then obtain $\\mid u \\rangle = \\mid \\frac{1}{2} \\frac{1}{2}\\rangle$ and $\\mid d \\rangle = \\mid \\frac{1}{2} \\frac{-1}{2}\\rangle$. Now I want to know what particles I can get by combining these two quarks. So let's write $\\frac{1}{2} \\otimes \\frac{1}{2} = 0 \\oplus 1$. So I have a singlett and a triplet. Most books now tell me that the triplet corresponds to the pion-triplet and the singlet to the $\\eta$-particle. Now I can for example write by using Clebsch Gordon coefficients that $\\mid 1,1 \\rangle = \\mid \\frac{1}{2} \\frac{1}{2}\\rangle \\mid \\frac{1}{2} \\frac{1}{2}\\rangle = \\mid u u \\rangle$. But this doens't correspond to any pion (They are mostly products of ups and anti-downs etc). However some books interpret this as the product of two proton wave function. This is quite confusing? Can someone help me? I'd me more than happy. Thanks in advance."}
{"id":"13008","title":"Inertia in an empty universe","text":"I was reading a recent article on Mach's Principle. In it, the author talks about inertia in an empty universe. I'll quote some lines from the article: > Imagine a single body in an otherwise empty universe. In the absence of any > forces, (Newton's second law gives): $$m\\mathbf{a} = 0$$ What does this > equation imply? Following Newton we would conclude from that $\\mathbf{a} = > 0$, that is, the body moves with uniform velocity. But we now no longer have > a background against which to measure velocities. Thus $\\mathbf{a} = 0$ has > no operational significance. Rather, the lack of any tangible background for > measuring motion suggests that $\\mathbf{a}$ should be completely > indeterminate. And it is not difficult to see that such a conclusion follows > naturally, provided we come to the remarkable conclusion that $$m = 0$$ In > other words, the measure of inertia depends on the existence of the > background in such a way that in the absence of the background the measure > vanishes! I don't see how the argument is complete. For example, in an empty universe, how is it possible to assign a value of 0 to a force? And wouldn't the existence of mathematics in such an empty universe be questionable?"}
{"id":"127526","title":"Nature as a calculator","text":"I wondered whether physical experiments can find the outcome of otherwise intractable mathematical problems. For example, solitons or other non-perturbative effects in QFT cannot be seen at any order in perturbation theory, and are difficult to calculate, but could in principle be measured. Is nature doing the whole path integral\/the whole perturbative expansion at once? Does that mean a computer could, in principle do it, in a finite time? or can nature solve math problems that are otherwise impossible?"}
{"id":"54126","title":"How does quantum jitter \"fix\" the masses of strings?","text":"So the masses of strings should be huge because of the high energy levels. Which means, in other words, the strings are vibrating with enough energy to have large masses as point particles. But quantum jitter offsets the high energy via negative energy, and makes the masses just right (what we observe). How does this occur? Where does quantum jitter come from? What is negative energy? Is quantum jitter changing the vibration\/resonance frequency of the string? If no, how is it acting upon the string?"}
{"id":"20207","title":"Can the expansion space time reverse itself and contract the same way?","text":"If there's a mechanism for space-time expanding faster than the speed of light, is there an example of It contracting in the same manner? If whatever mechanism is causing it to expand, can the underlying mechanism reverse itself?"}
{"id":"22891","title":"Are there any theories or suggestions for how the multiverse came into existence?","text":"I've just seen a documentary about the multiverse. This provides an explanation for where the big bang came from. But it leaves me wondering: how did the multiverse come into existence? Because this is a step further than something we don't even surely know the existence of, I'm only wondering if there are any theories or suggestions."}
{"id":"62039","title":"Perfect fluids in cosmology?","text":"In cosmology, it is often assumed that the equation of state of a cosmological fluid is of the form $p=w\\rho$. Why is this? Is it the equation of a _perfect_ fluid? Why does $w=0$ for matter $1\/3$ for radiation and $-1$ for a cosmological constant?"}
{"id":"91768","title":"Why is there no current flow when i connect two opposite terminals of two separate batteries","text":"I don't understand this..maybe there is something wrong with my understanding of battery or electricity. I am learning about electricity more deeply this time. So please spare with me... So lets begin with what i know:   * follow of current occurs when there is voltage drop.   * voltage drop happens when there is imbalance in charge.   * a battery is basically two separated chemicals. One side with electrons to spare and other side that needs electron to balance itself. which is imbalance in charge.   * When both sides are connected **\\+ side** snatches electrons from **\\- side**. (And if there is a medium, conductor, then first electrons are taken from conductor that in turn snatches from **\\- side** , which not being able to keep electrons lose them and give up those electrons, we label it as excessive.) The analogy i was taught with used marbles in pipe... When one side has excessive marbles And this pipe is connected to other side that has no marbles the marbles will instantaneously start falling out so even though marbles from container with excessive marbles haven't traveled there yet, they still start falling because filled side marbles are pushing marbles in pipe and which in turn push the last marbles in pipe out. 1st, Aren't electrons **pulled** by atoms that can use more electrons, and not move because one side is overfilled with electrons so they can't **wait to push** some out. Questions:   1. if i connect a conductor to positive side of a battery, shouldn't it make all electrons\/marbles fall into empty container as empty container will pull marbles as atoms need electrons?   2. if i connect two batteries' **only opposite sides** to each other shouldn't they drain (balance the each other out, cancelling all **-** & **+** charges)?"}
{"id":"91769","title":"What is opposite of electric field?","text":"A negatively charged molecule will generate a field, an electric field, and it will repel all negative ions. So what does positively charged molecule generates that causes other positive molecules to be repelled?"}
{"id":"67608","title":"Question regarding inverse relation of resistance with area of cross section","text":"It is said that resistance is inversely proportional to area of cross section. But greater area will have greater electric flux, and greater electric flux will have greater magnetic flux, and greater the magnetic flux will have greater eddy current, which is opposite to the current flowing. So this will oppose the current hence resistance increases. Am I right?"}
{"id":"51517","title":"Can light be canceled by merging with an inverted wave?","text":"Can light waves be canceled by merging them with their inverted waves? Seems like it would violate conservation of energy but waves are added together when they overlap, right? Where is the flaw in this logic? I'm thinking polarized laser light, added to its opposite, might becomes dark again. \\-- Appreciating your collective wisdom."}
{"id":"93721","title":"What creates black streams in fountain-type fireworks?","text":"For the New Year celebration of 2014, my nephews set off a series of fountain- type fireworks (as shown below). ![](http:\/\/i.stack.imgur.com\/BgQYd.jpg) When one looks carefully into the shower of light, one notices that a fair number of \"long, black and stringy streams of light\" are coming out of the fountain. Note: I do not see any black streams in this picture! After reading this online article from the American Chemical Society, the colors of light are, of course, created from electrons jumping orbitals. However, how are these “black streams” created? I assume that somehow visible light is being absorbed to create these \"black streams of light\". What mechanism is responsible for this to occur? Are there small bits of matter that are large enough and cooled enough to absorb visible light? I really don’t know. Any help would be much appreciated."}
{"id":"23574","title":"How can we model intrinsic curvature?","text":"Can it only be done in Euclidean space? Doesn't Euclidean space only model extrinsic curvature?"}
{"id":"23572","title":"How does an object regains its neutrality after being charged by rubbing?","text":"Objects (like comb) can be charged by rubbing as charged particles, particularly electrons, are transferred from one object to other. This can be seen as the object (comb) attracts small bits of paper. After some time, the charge on the body seems to disappear. How does this charge disappear without any external influence?"}
{"id":"129265","title":"Post Problem Analysis - Rod in a cylindrical time varying magnetic field?","text":"**A metal rod of length $l$ is placed (as shown in figure) in a cylindrical time varying magnetic field. Find the potential difference across it.** ![enter image description here](http:\/\/i.stack.imgur.com\/XgIjQ.png) My answer: As electric fields as forming circles around the centre with magnitude $$\\frac12r\\frac{dB}{dt}$$ whose magnitude is same along any chord at a fix distance from centre: $$\\Delta V=E.l=\\frac12r\\frac32at^{\\frac12}.l$$ $$\\Delta V=\\frac34arlt^{\\frac12}$$ **_My question:_ Is $\\Delta V =V_M-V_N$ or $\\Delta V=V_N-V_M$ $\\bf OR$ equivalently which side is at high potential?** Possible   * $(i)$ As Field lines are circular from $M$ to $N$, $M$ must be high potential.   * $(ii)$ As this field line apply a force on electrons- in the rod- towards left, then $M$ must have excess $e^-$ and $N$ must have deficiency; so $N$ would be high potential as $M$ is $(+)$ and $N$ is $(-)$."}
{"id":"74631","title":"How to distinguish high-energetic muons and electrons in the CMS and ATLAS muon detectors?","text":"At a typical energy of about 100 GeV, a muon has a Lorentz factor of about 1000, an electron about 200.000. The flight time to the detector should be around 30 ns (assuming d= 10m from the collision point). While e- would practically have the speed c, the muon's speed is about 0.9999995 c, causing a delay of about 15 fs, barely measurable I think. Is there a way to distiguihs e- from mu, unless we assume prior knowledge (that they have passed the electric calorimeter, about the penetrating power etc). Or, to put it in another way: what is the energy threshold below which they can be distinguished directly?"}
{"id":"74632","title":"Electric field outside a hollow cylindrical conducting shell","text":"Imagine that a point charge $+Q$ is located at radius $R$ and height $z=0$ inside of an infinitely long conducting cylinder shell of inner radius $a>R$ and outer radius $b$. From what we know of perfect conductors, we know that $Q_{induced}(r=a) = -Q$, $Q_{induced}(r=b) = +Q$, and the electric field inside the shell is $E(a<r<b)=0$. Is there a way to figure out the magnitude and direction of the electric field outside of the conducting shell as a function of position?"}
{"id":"110998","title":"Geometry in diagonal matrix and inertia tensor","text":"For this problem, can anyone explain to me why when $x_1$ axis is aligned with the diagonal of the cube, the resulting inertia tensor will become diagonal? How to interpret this result geometrically? I mean, is there some tricky way for some geometric shape that I don't have to compute the eigenvalue, eigenvectors to find the principal axises, but by observation, I can make a guess first. ![enter image description here](http:\/\/i.stack.imgur.com\/HyEC3.png) ![enter image description here](http:\/\/i.stack.imgur.com\/Rkpfz.png)"}
{"id":"92237","title":"What is the need to consider a singular spacetime?","text":"To have a consistent superstring theory (which is to avoid the conformal anomaly on the worldsheet CFT) we are forced to build our theory on the critical dimension $n=10$. However, the Standard Model, seen as an effective low energy theory lives on $n=4$. So we are forced to compactify the background spacetime for the string theory, as to be of the form $\\mathbb{R}^4\\times M$ where $M$ is a compact and \"very small\" riemannian manifold. $M$ could be a torus, for example. To save some of the supersymmetry after compactification we are furtheremore forced to take $M$ to be Kahler. Given this, my question is: From a physical viewpoint, why do we usually consider nowadays $M$ to admit singularities? In other words why should $M$ be an orbifold and not just a smooth manifold?"}
{"id":"73171","title":"Entropy of a naked singularity","text":"According to the wikipedia article http:\/\/en.wikipedia.org\/wiki\/Naked_singularity: \"Some research has suggested that if loop quantum gravity is correct, then naked singularities could exist in nature, implying that the cosmic censorship hypothesis does not hold. Numerical calculations and some other arguments have also hinted at this possibility.\" The entropy of a black hole is proportional to its horizon area. What is the entropy of the black hole if the cosmic censorship hypothesis fails to hold? Do we assume this hypothesis for the expression of the entropy?"}
{"id":"74188","title":"Special relativity and quantum mechanics","text":"I know how the Dirac's equation about the relativistic quantum mechanics works. Can anyone tell me how can one combine special relativity and quantum mechanics as a whole - special relativity is valid only under constant speed while the speeds of electrons in quantum mechanics vary with positions."}
{"id":"46923","title":"How many particles are created in the strong electromagnetic field?","text":"Consider a vacuum of charged massless scalar field. Then the uniform and isotropic electric field $E$ is turned on for a time interval $\\tau$. The question is, how many scalar particles are created?"}
{"id":"49757","title":"Elegant approaches to quantum field theory","text":"I have been reading _Quantum Mechanics: A Modern Development_ by L. Ballentine. I like the way everything is deduced starting from symmetry principles. I was wondering if anyone familiar with the book knows any equally elegant presentation for quantum field theory. Weinberg's books start off nice with the irreducible representations of the Poincare group\/algebra but the later chapters lose me with the notation. Also, most books I've read on QFT (Srednicki, Peskin and Schroeder, Mandl and Shaw) make a valiant initial attempt at a nice consistent framework but end up being a big collection of mathematical recipes and intuitive insights that seem to work but the overall structure of the theory seems to be sewn up. The relativistic equations crop out of rather flimsy arguments, the canonical anti\/commutation relations are imposed out of density indeterminate air, functional methods are developed cause we know no better, infinities come about with renormalization theory to the rescue but it seems very alien from the initial context. Is there any approach that ties all these things together in an elegant mathematical framework which accounts for all the patch up work that is needed? I am not talking about axiomatization just a global point of view that encompasses all the issues."}
{"id":"56561","title":"How to use Legendre polynomials in order to determine the (an)isotropy of an on-lattice cluster aggregate?","text":"I am currently testing various models of on-lattice (square lattice in two dimensions) cluster growth for anisotropy. I end up with a cluster, the boundary of which, in case of a truly isotropic model, should look like a circle. In anisotropic cases the cluster boundary will deviate from the circular shape. I've been told that the Legendre polynomials can be used in order to determine the anisotropy of a cluster, but I'm not exactly sure of how I can use these Legendre polynomials in order to say something about the isotropy or otherwise of the clusters. I gather that one can write the angular functional dependency of the cluster boundary in terms of a Legendre series due to the orthogonal nature of the Legendre polynomials, but once I've calculated the coefficients of the series I'm not sure how to interpret them in terms of anisotropy of the cluster or even if this is the correct thing to be doing. Any hints, tips or further reading would be appreciated. I am a humble mathematician, so please go easy on the physics jargon as I may not be well versed in it. Thanks for your time and help."}
{"id":"56560","title":"Pure state - density matrix - real life example of boxes in warehouse","text":"So if there are a billion boxes in a warehouse, I would like to know conceptually how to tell if it is in a pure state. I know that if it is in a pure state (not mixed) that the density matrix has only one nonzero diagonal element and that the trace is 1, whereas a mixed state would not be idempotent and the trace would be less than 1. I'm thinking that the warehouse resembles a microcanonical ensemble or a canonical ensemble since the box count is fixed. I also think it wise to imagine it is well-insulated (isolated) so that there is no entanglement with the environment. Would a good way to tell if it is in a pure state would be to measure each box's temperature? (Too many, right) Or instead measure a few times several boxes to make sure the measurement doesn't change and is the same for all in the sample? Or would one just look at the thermostate? ha ha please forget that last remark."}
{"id":"56565","title":"Charging an object by induction","text":"when charging a sphere by induction using a (-) charged object , and we put it to the right side of the sphere, electrons are pushed to the left side , so we ground the left side and the excess electrons escape. but what if we grounded the right side ( which has less electrons than usual ) wouldn't electrons flow from the ground to neutralise this side?"}
{"id":"49755","title":"How was the sinusoidal model for propagation developed?","text":"It's a little difficult to explain this question .. but I'll try anyway. To the best of my knowledge propagation models - audio, RF - are modelled as travelling in a sinusoidal form. Surely if a model was constructed based on observation, and then verified by experiment the experiment would more often yield the desired result than not. Is the model for sinusoidal motion\/waves deemed true because the equations are proved true by experiment? How do we know these waves are actually sinusoidal? How was this model developed?"}
{"id":"110444","title":"Solving coupled nonlinear ODEs, controlling numerical instability through numerical viscosity","text":"I'm solving a problem numerically that takes the form $ Q_{ij} \\ddot{y}_j +S_{ijk}\\dot{y}_j\\dot{y}_k +V_i=0$, where $(Q_{ij},S_{ijk},V_i) $ are all low order polynomial functions of the dependent variables $y_i$. The dependent variables are all functions of the variable t. The resolution of this spectral model is controlled by the number of $y_i$, denoted N. Now, when these equations become stiff, or I integrate for long times, numerical instability becomes apparent. If I increase the resolution, these spurious growths are attenuated, but it is not feasible to make N large enough so that they don't appear at all. If one looks at the time history of the higher modes, it seems like their large growth corresponds to the onset of the instability, so I want to attenuate these (un-physical) fast growing components of the solutions. To this end, I have added a term to the leading order term, so that the equation takes the form $Q_{ij}( \\ddot{y}_j +\\nu j^2 \\dot{y}_j)+S_{ijk}\\dot{y}_j\\dot{y}_k +V_i=0$, where $\\nu$ is a numerical viscosity to be prescribed later on. This does an OK job at attenuating the instability (e.g. I get accurate solutions for an twice as long), based on how I choose $\\nu$, and the constraints of the problem. I have played around with the power of j as well, and found varying success in different contexts. The issue is, for long times this scheme breaks down, and the solution greatly deviates from the expected solution, making me think I'm not doing addressing the issue very rationally. Indeed, I am doing all of this very naively, and am having trouble finding similar types of problems in the literature. Does anyone have experience damping numerical instabilities in equations that take this form? In particular, is there a way to essentially add a dissipation that removes these fast growing modes? References to relevant literature are also greatly appreciated. Thanks! Nick"}
{"id":"32189","title":"Does gravity slow the expansion of the universe?","text":"Does gravity slow the expansion of the universe? I read through the thread http:\/\/www.physicsforums.com\/showthread.php?t=322633 and I have the same question. I know that the universe is not being stopped by gravity, but is the force of gravity slowing it down in any way? Without the force of gravity, would space expand faster? Help me formulate this question better if you know what I am asking."}
{"id":"75585","title":"What does the Froude number represent?","text":"While reading on Wikipedia, I read the following > _The Froude number is defined as:_ > > $$\\mathrm{Fr} = \\frac{v}{c}$$ > > _where $v$ is a characteristic velocity, and $c$ is a characteristic water > wave propagation >velocity. The Froude number is thus analogous to the Mach > number. The greater the Froude >number, the greater the resistance._ While reading the following paper on shallow water waves, I could not understand how\/why the Froude number was given to be $$\\mathrm{F} = \\frac{gt_0^2}{L}$$ Where $t_0$ is the time scale, and $L$ is the length scale along the X and Z axes. Note that the scale for the amplitude of the surface above the mean depth is some other $A$ and not $L$. Please explain how the author has defined the Froude number. Is it the same as the one given on Wikipedia? If so, then please provide a step-by-step reasoning to show that they are indeed the same. If not, then please explain what the Froude number mentioned in the paper represents. EDIT: I'd like to add another minor question as it is related to the abovementioned paper. It mentions that the surface tension is accounted for but I do not understand how that is possible given that there is no term containing the surface tension $\\gamma$ explicitly."}
{"id":"122090","title":"Does a Force on the block $M_1$ in this diagram have an effect on the block $M_2$?","text":"Suppose we have two blocks: ![enter image description here](http:\/\/i.stack.imgur.com\/7ZmYU.jpg) There is no friction. Does $F$ has the same effect on $M_2$?"}
{"id":"26209","title":"Is the \"Great Attractor\" an indicator of the \"Multiverse\"?","text":"I have heard a bit about the Great Attractor (the gravitational anomaly that seems to be \"sweeping\" our universe in one direction). Someone (and forgive me, I do not recall the specifics) has hypothesized that this may be an indication of an other universe affecting us gravitationally. I am interested in people's opinions on this. Would this be an experimentally provable phenomenon of string-theory (in all its different incarnations)? Or are there more viable hypotheses that would explain this?"}
{"id":"21961","title":"From the perspective of an observer inside a black hole's horizon, where does the energy for Hawking radiation come from?","text":"Would energy be seen to \"flow\" to the outside of the black hole? Through what mechanism?"}
{"id":"3737","title":"What's a typical electron-hole recombination time of semiconductors?","text":"Have some trouble finding a useful reference to answer the following questions:   * What's a typical electron-hole recombination time of semiconductors? And how does it depend on the temperature and the band gap? If there's no formula, examples will do.   * What's the probability of an electron being kicked into the conduction band merely by thermal motion? Is it $\\exp(-\\Delta E\/T)$, where $\\Delta E$ is the band gap and $T$ the temperature? That thermal excitation should lead to noise in the system, is that correct?"}
{"id":"56389","title":"Does the unpredictability in the \"micro world\" means that everything is if we can look at it close enough imperfect?","text":"Does the unpredictability in the \"micro world\" means that everything is if we can look at it close enough imperfect? I mean, there is a saying \"You will never stand in the same river again\" or something like that. So, if nothing in our makro world is the same, if we look close enough (no two people or flowers are the same; and if they seem, it just means you are not looking carefully or cose enough ;) Is this also true in the micro world? If yes, does that mean that e.g no two atoms of hydrogen are the same and the sign $H$ is only some kind of reference for \"from-to\" or \"plus-minus- something-like-that\" tolerance? And the same principle can be applied for smaller particles or even strings etc.? Does that mean that there is no such thing as two identical proto particles and everything is slighty different? And my final concern is... Do we live in \"The Imperfect Universe\"? ;)"}
{"id":"56388","title":"Whirlpools and Tornados","text":"This may not be a great question. But whenever you drain water, a small whirlpool happens, obviously. This got me thinking... Can we model tornados with this effect, would it even be beneficial?"}
{"id":"3731","title":"Heterotic string as worldvolume theory of two coincident 9-branes in 27 dimensions?","text":"The heterotic string is a combination of right-moving excitations from a D=10 superstring and left-moving excitations from a D=26 bosonic string, with the left-movers behaving as if the extra 16 dimensions are compactified. The heterotic string is also derived from D=11 M-theory, as an open 2-brane stretched between two \"end-of-the-world\" 9-branes (spatial boundaries; this is M-theory compactified on a line segment, and the 9-branes lie at the ends). So I am led to imagine a 27-dimensional theory, containing branes. We compactify 16 dimensions, and consider the worldvolume theory of two parallel 9-branes. When they are coincident, we get the heterotic string; when they are slightly separate, we get \"heterotic M-theory\". A 27-dimensional fundamental theory has been discussed before (hep-th\/9704158 section 4; hep-th\/0012037; arXiV:0807.4899), but I don't see this particular line of thought discussed."}
{"id":"2131","title":"What experiments, other than Hubble Expansion, support the Dark Energy theory?","text":"Dark energy is introduced as a constant inside Einstein's equations. Its primary purpose, from what I understand, is to make Einstein's equations compatible with the _accelerating_ expansion of the universe. As a consequence, of the \"predictions\" of dark energy is the expansion of the universe according to Hubble's law. I know there are numerous experiments that verify this expansion (and its acceleration), and thus indirectly support the dark energy theory. My question is:   **Are there _other_ factors that give credit to the existence of dark energy?**   Are there any experiments that support this theory, but not only through the verification of Hubble expansion?"}
{"id":"79926","title":"combination of 5 identical capacitors","text":"Given 5 identical capacitors, each EMF. Draw the proper combination of all these 5 capacitors.. 1.) 2.5 2.) 4.17 I really do not know how to solve this thing and if anyone of you knows the title of this lesson please comment it here so I can study it further. We will be having our finals tomorrow."}
{"id":"105662","title":"Two ways of looking at weak current?","text":"A weak current of the form: $$j^\\mu=\\bar u_e \\gamma^\\mu \\frac{1}{2}(1-\\gamma^5)u_\\nu$$ implies that it automatically selects the left- handed neutrino. This current can also be written as: $$j^\\mu=\\bar u_e \\frac{1}{2}(1+\\gamma^5)\\gamma^\\mu u_\\nu$$ which seems to imply that only left- handed electrons take part in weak interactions. Which statement is right?"}
{"id":"14004","title":"How can it be that the beginning universe had a high temperature and a low entropy at the same time?","text":"The Big Bang theory assumes that our universe started from a very\/infinitely dense and extremely\/infinitely hot state. But on the other side, it is often claimed that our universe must have been started in a state with very low or even zero entropy. Now the third law of thermodynamic states that if the entropy of a system approaches a minimum, it's temperature approaches absolut zero. So how can it be that the beginning universe had a high temperature and a low entropy at the same time? Wouldn't such a state be in contradiction to the third law of thermodynamics?"}
{"id":"106009","title":"Would you see a rainbow from refraction when the sun is in front of you?","text":"I know how rainbows are formed, and why. Usually it is said that the Sun must be **behind** the observer, in order for its light to be totally reflected inside the droplet and then reach the observer. But surely not all the light is totally reflected? There must be some radiation refracting from inside the droplet to the air: since different wavelengths refract through different angles, shouldn't we also see a rainbow when the Sun is in front of us?"}
{"id":"36404","title":"Lever Mechanics - How to formulate an ideal lever launch","text":"Let's say I have a simple lever as shown below, and the lever is massless and the pivot is frictionless and there is no air resistance. I'm thinking the cradle for the projectile would have to have a specific shape to dictate the launch angle, but for the context of a general equation $\\theta$ will be the angle the lever makes parallel to the ground, and will have some value where the object will leave the lever. $\\theta_{launch}$ will refer to the angle the lever is at when the object 'launches', so the lever will have swept $\\theta_0 - \\theta_{launch}$. $t_0$ is the initial time the weight is released and $t_{launch}$ = launch time. ![Lever Launching](http:\/\/i.stack.imgur.com\/dN2fz.png) # Height Functions $h_{weight}(\\theta) = h_{pivot} + l_{shortArm}cos(\\theta)$ $h_{projectile}(\\theta) = h_{pivot} - l_{longArm}cos(\\theta)$ # Desired Equations   * Potential Energy of the system at Initial Conditions   * Velocity of Projectile at Launch   * KE of Projectile at Launch   * Efficiency of machine"}
{"id":"36407","title":"converting daily global radiation into hourly values","text":"I would like to convert daily global radiation into hourly solar radiation. While I do not know the terms, I tried using below equations I found in the papers but I have negative radiation values range between -75 to 100 j\/m2\/s. Any suggestions are welcome.               # radi(h) = r * radi(daily)      > r <\\- (pi\/24) * ( a + b * cos_d (w))* ((cos_d(w) - cos_d(ws)) \/ > (sin_d(ws)-((pi\/180) * ws * cos_d(ws))) ) > > a <\\- 0.409 + 0.5016 * sin_d(ws - 60) > > b <\\- 0.6609 - 0.4767 * sin_d(ws - 60)                # W = hour angle of the sun      > w <\\- (360\/24) * (h-12.0) # h = time of the day (hour) > > ws <\\- acos_d(-tan_d(phi)*tan_d(sigma))) # ws = sunset hour angle (degree) > > phi= 36.20 # phi = location latitude angle               # sigma= declination angle of the sun [degree]      > sigma = 23.45 * sin_d (360* ((284+day)\/365)) # day=DOY"}
{"id":"57389","title":"Surviving Free Fall with the Help of Mr. Bernoulli","text":"What is the best strategy to survive a free fall naked out of a jetliner at cruising altitude (ignoring temperature)? For instance, my strategy would be to streamline my fall so that my terminal velocity was very high. Then at some critical distance above the ground I would pitch upward and attempt to use my forward velocity to achieve lift. Presumably this would slow my fall to some survivable speed. (This is roughly what the space shuttle does after all.) Would it work? And idea if it would be survivable?"}
{"id":"113133","title":"Atomic nucleus consisting of only neutrons?","text":"Is it true that the nucleus of all atoms (including radioactive isotopes) contain at least one proton? Is there an atomic nucleus consisting entirely of neutrons? (Let's exclude neutron stars for the moment.) If so, how does one name them? (since Periodic Table starts from atomic number 1, not 0.)"}
{"id":"81574","title":"Infinite Wells and Delta Functions","text":"In considering a delta potential barrier in an infinite well, I can just enforce continuity at the potential barrier-it doesn't have to go to zero. Why then does it need to go to zero at the walls of the infinite well? These two cases seem to be very similar to be, I even feel like the well wall is equivalent to a summation of delta functions... Where is my logic faulty?"}
{"id":"53012","title":"What is the meaning of non-compactness in the context of $U(1)$ in gauge theories?","text":"In John Preskill's review of monopoles he states > Nowadays, we have another way of understanding why electric charge is > quantized. Charge is quantized if the electromagnetic U(l)em gauge group is > compact. But U(I)em is automatically compact in a unified gauge theory in > which U(l)em is embedded in a nonabelian semisimple group. [Note that the > standard Weinberg-Salam-Glashow (35) model is not \"unified\" according to > this criterion.] The implication of the third sentence is that, in some circumstances, the $U(1)_{em}$ gauge group may _not_ be compact. How could this be? Since $U(1)$ as a differentiable manifold is diffeomorphic to $S^1$ isn't it automatically always compact? The following paragraph: > In other words, in a unified gauge theory, the electric charge operator > obeys nontrivial commutation relations with other operators in the theory. > Just as the angular momentum algebra requires the eigenvalues of $J_z$ to be > integer multiples of $\\frac{\\hbar}{2}$, the commutation relations satisfied > by the electric charge operator require its eigenvalues to be integer > multiples of a fundamental unit. This conclusion holds even if the > symmetries generated by the charges that fail to commute with electric > charge are spontaneously broken. is OK, but I don't follow what that has to do with the compactness of $U(1)$."}
{"id":"83529","title":"How does energy transfer between B and E in an EM standing wave?","text":"I'm trying to understand how an electric field induces a magnetic field and vice versa, its associated energy, as well as relating it to my understanding of waves on a string. Using a standing wave as an example, I came up with the equations $\\vec{E}=A\\sin(\\omega t)\\sin(kx)\\hat{y}$ $\\vec{B}=\\frac{Ak}{w}\\cos(\\omega t)\\cos(kx)\\hat{z}$ I checked them against Maxwell's equations, and they're self-consistent. At time 0, this reduces to: $\\vec{B}=\\frac{Ak}{w}\\cos(kx)\\hat{z}$ Since the electric field is 0, based on the Poynting vector, there's no energy transfer at this time. At this time, at a node where $\\vec{B}=0$, there's neither electric field nor magnetic field. If there's no energy transfer, and no energy stored in either field, then how can an electric field exist at this point at some time later? How is the energy stored, or transferred from elsewhere?"}
{"id":"15757","title":"Line width of green DPSS laser","text":"Could someone suggest what is the line width\/mode stability\/coherence length of generic cheapo green (532nm) DPSS laser? I've found lots of nice info here: http:\/\/redlum.xohp.pagesperso- orange.fr\/laser\/modeanalysis.html but there are no any DPSS lasers."}
{"id":"6844","title":"How are light and time related?","text":"So, from what I understand: Special Relativity says that light is always observed moving at the speed of light (c). If some object had a velocity of (3\/4)c, and the object had some sort of clock attached to it, it would measure differently from a still clock. 1) What would a clock moving the speed of light measure compared to a still clock? 2) Why does it seem that the speed of light is a focal point of time? EDIT: I may need to rephrase this slightly. What I want to know is, why is it that observed time changes at high velocities?"}
{"id":"18250","title":"How do physicists justify using $c^2$ in equations?","text":"Excuse my ignorance but I see physicists use $c^2$ in equations and I was wondering what reasoning they use to justify $c^2$ since, as far as I understand, $c^2$ must be an absurd quantity because there is a speed limit imposed by physicists themselves at $c$. If there is no motion faster than $c$, how do you justify using $c^2$ in an equation?"}
{"id":"71466","title":"What routine tasks have to be performed on the outside of spaceships\/stations?","text":"What kind of routine tasks have to be performed on the outside of spaceships or space stations? Repairs, I presume. Anything else? Is there anything that needs to be cleaned off from the outside, like debris, or some other substance? Can anything on the surface freeze?"}
{"id":"33638","title":"Why is the probability that Kip could go back and kill his grandfather  $10^{-10^{60}}$?","text":"In _The Universe In A Nutshell_ , page 153, Hawking says that > _The probability that Kip could go back and kill his grandfather is > $10^{-10^{60}}$._ How do we get this result?"}
{"id":"76119","title":"Dirac equation in curved space-time with Torsion","text":"I am looking for pedagogical references in which Dirac equation in space-time with curvature and torsion were discussed."}
{"id":"31679","title":"What if the 126 GeV particle found at LHC that LOOKS like the Higgs is actually not the Higgs Boson?","text":"I am suspicious of the Higgs announcement. Press mania aside, the scientists seem careful to say only that \"we've found something which _looks_ like the Higgs Boson\". The difference between Higgs-containing theories and non-Higgs-containing theories is a tiny bump on the energy curve, smaller than the resolution of the detectors, which is why we have to perform statistical gymnastics on large data sets to discover or refute the deviation. In other words, the energy distribution of the reactions is consistent with there being a short-lived particle weighing 126GeV, and that's all the announcements seem to really say. Is there anything else in the results which tells us that the extra element is Higgs and not something else? Is there any identifying feature other than lifespan and mass?"}
{"id":"112772","title":"Given two surfaces, one that absorbs light and the other that reflects it, will there be a signficant temperature difference?","text":"My likely wrong understanding of light is this: By some way the molecules are structured, a photon hits an atom or a structure, is absorbed, then re-emitted and it's that photon wavelength that I see as light when my eyes absorb the photon\/wave. So if say something is painted black, does it follow that it would absorb light? And if so, when compared to an object that was painted white or not naturally black in color would it have a cooler temperature than the surface painted black?"}
{"id":"88432","title":"Is it possible that only one hemisphere of a planet has an atmosphere?","text":"Suppose there is a tidally locked planet orbiting a star. The planet's surface consists of a global ocean, that is, liquid water. At the inner hemisphere the temperature is so high that the water is constantly boiling, creating the atmosphere of water vapor. But the vapor does not reach the dark side of the planet, and precipitates around the terminator line. The further side has no atmosphere and an icy surface. Is this setup possible? Can it be such thing that one side of a planet has liquid (and boiling) water surface while the other side has no atmosphere? Is it possible at all that a part of an ocean was boiling while the other one was icy?"}
{"id":"34112","title":"number of microstates associated with two-level quantum systems","text":"this is a very simple question, but apparently one that has no simple answer, at least from standard quantum mechanics theory I'm trying to figure the number of simple quantum states (microstates) of the allowed configuration space of a two-level quantum system, ignoring for now mixed states. So if i have a state like: $$ \\Psi = \\psi_0 | 0 \\rangle + \\psi_1 | 1 \\rangle $$ removing all redundancies from the representation, we are left $$ \\psi_0 = \\rho_0 $$ $$ \\psi_1 = \\rho_1 e^{-i \\theta_1}$$ $$ \\rho_0^2 + \\rho_1^2 = 1$$ Now, this state space looks like $[0,1] \\times S_1$. I've seen discussions that quantum states take some volume in phase space, but i want to understand that in detail: so in this case, > **Question:** how many distinct states are in this configuration space? is > this a finite set? infinite but numerable set? nonnumerable? **Following Steve suggestion** , lets write the Shannon entropy of this wave function: $$ S( \\Psi ) = - \\rho_0 \\ln \\rho_0 - \\rho_1 \\ln \\rho_1 $$ There is some conflict when i see this entropy; it cannot derive from any counting state scheme, in particular because it completely ignores the relative phase factor between the basis states, and as we all we know by now, states with different relative phase factors are physically different; otherwise there would be no interference terms, and no quantum mechanics to speak of!"}
{"id":"47381","title":"How to handle Yukawa contractions in calculating SUSY beta functions?","text":"I'm reading Chapter 6 of Martin's introduction to SUSY http:\/\/arxiv.org\/abs\/hepph\/9709356, which is about RGEs in the MSSM. I tried to convince myself of some of the calculations, and I was particularly stuck with anything that has Yukawas in it. I don't know how to deal with them. Take for example the anomalous dimension formula, $$\\gamma^i_j =\\frac{1}{16 \\pi^2} [\\frac{1}{2} Y^{imn}Y^*_{jmn} - 2 g_a^2 C_a(i)\\delta^i_j]$$ where, $g_a$ is a gauge coupling corresponding to a group $a$, and $C_a(i)$ is the Casimir invariant. And $Y^{imn}$ denotes Yukawas. For example, $$\\gamma^{H_u}_{H_u} =\\frac{1}{16 \\pi^2} [3 Y^*_t Y_t - \\dots]$$ $$\\gamma^{\\bar{d}_3}_{\\bar{d}_3} =\\frac{1}{16 \\pi^2} [2 Y^*_b Y_b - \\dots]$$ $$\\gamma^{\\bar{Q}_3}_{\\bar{Q}_3} =\\frac{1}{16 \\pi^2} [Y^*_t Y_t + Y^*_b Y_b - \\dots]$$ the dots are for the second term which I know how to calculate. But I don't know how the results for the Yukawas were obtained! My understanding is that the indices $i,j,k$ are family indices for chiral superfields. Explicitly, $$ Y^{ijk} = Y^{\\Phi_i \\Phi_j \\Phi_k}$$ So if I wanted to calculate, the Yukawa part of $\\gamma^{H_u}_{H_u}$, I would try: $$Y^{H_u mn}Y^*_{H_u mn} = Y^{H_u \\bar{u}_m Q_n}Y^*_{H_u \\bar{u}_m Q_n} + Y^{H_u Q_m \\bar{u}_n}Y^*_{H_u Q_m \\bar{u}_n} + \\text{other permutations in superfields?} $$ If I chose third family approximation like in Martin's notes, then $m=n=3$ But I'm not sure this is correct or where it leads. Could someone help me with this. How should I handle $Y^{imn}Y^*_{jmn}$ to get those results?"}
{"id":"72341","title":"\"windows of order\" in the Bifurcation diagram","text":"When looking at the bifurcation diagram of a chaotic system, one observes \"windows of order\", namely short intervals where the system briefly leaves its chaotic state and then rapidly returns to chaos. These intervals are quite periodical. Does anyone have a good explanation for this? Thanks!"}
{"id":"52350","title":"Special Relativity: Transforming Maxwell's equations","text":"I'm working through Einstein's original 1905 paper*, and I'm having trouble with the section on the transformation of Maxwell's equations from rest to moving frame. The paper proceeds as follows: > Let the Maxwell-Hertz equations for empty space hold good for the stationary > system K, so that we have > > IMAGE: Untransformed Maxwell equations > > where (X, Y, Z) denotes the vector of the electric force, and (L, M, N) that > of the magnetic force. > > If we apply to these equations the transformation developed in § 3, by > referring the electromagnetic processes to the system of co-ordinates there > introduced, moving with the velocity v, we obtain the equations > > IMAGE: Transformed Maxwell equations I do not understand just where this second set of equations comes from. The only derivations of this transformation that I have been able to find involve either potentials, four-vectors, or both. However since four-vectors had not been invented in 1905, and because the statement of the transformation is so blunt, it seems that Einstein is using or appealing to a simpler method for finding the transformation. So my question is: **What is the simplest derivation of the transformation rules for Maxwell's equations in special relativity?** Is Einstein appealing to a pre-existing Lorentz method here, or is there a trick to accomplish all of this quickly? Or does the text here simply belie the true work involved in the derivation?   * I cannot give a proper link to the paper owing to link restrictions, but it's at hxxp:\/\/www.fourmilab.ch\/etexts\/einstein\/specrel\/www\/#SECTION21"}
{"id":"15997","title":"Orbital radius of Geo-stationary satellite","text":"Could you please tell, Why all the geo-stationary satellites are to be dropped at same height from earth? Why can't it be closer or away from its regular orbit(ie, 35,000 km)? If all satellites are dropped in the same orbit, then will not those collide one another?"}
{"id":"101754","title":"Information about newton's third law","text":"Well I am still small student of physics, but as far as I know, Newton's Third Law states that, > \"For every action there is equal and opposite reaction.\" Now let me tell you how I understood this, Suppose there are two bodies A & B initially at rest (w.r.t the earth), now if A makes collision with B, of course there will be some force acting on A->B (say $F_{ab}$) and according to the third law there would be some reacting force from B->A (say $F_{ba}$). And both would be numerically equal. But now just consider body A, it has some acting force on it i.e. $F_{ba}$ which is being given by body B, so won't reaction produce for that force? Is that reaction is nothing but $F_{ab}$? Now for both bodies, there should be some **static frictional force** (before movement) acting against the force which (tends to) move the body, so whenever two bodies act with each other, are there 4 force produced?"}
{"id":"15992","title":"Problem with an electricity \/ thermodynamics assignment","text":"I've been trying to figure this one out for a while on my own, so I'd like to ask for your help if you could offer some. The task states: > A heater made out of a wire with a diameter $R = 0.2\\text{ mm}$, length > $4\\pi\\text{ m}$ and electrical resistivity of $0.5\\times 10^{-6}\\ > \\Omega\\;\\mathrm{m}$ is connected to a voltage source of $220\\text{ V}$, > sinked in the water. > > Which mass of water will it heat up from $20^{\\circ}\\mathrm{C}$ to > $50^{\\circ}\\mathrm{C}$ in the time of 10 minutes? (C of water = $4200\\ > \\mathrm{J\\;kg}\/\\mathrm{K}$) I know I have the electrical properties of the wire and the thermodynamic properties of the water, but I don't know how to proceed from there. We've been studying electricity and I am not really aware how I can connect it with thermodynamics?"}
{"id":"102639","title":"Aurora Effect from Galactic Wind?","text":"The Solar Wind creates auroras due to interactions with Earth's Magnetic Field. Do we witness any similar effects from the Galactic Wind interaction with the Sun's Magnetic Field?"}
{"id":"135067","title":"How does this series expand the expression?","text":"How does $$\\sqrt{R^2 + |x|^2} = R + \\frac{|x|^2}{2R}+\\dots$$ when expanded around the point $x=0$? I tried using a Taylor expansion but it didnt work out."}
{"id":"135062","title":"What is a measurement in MWI","text":"I was reviewing the MWI, and couldn't figure what a measurement is in the MWI? Everett claims that split happens when you measure, but what is it for MWI (not for general QM)? Yes, I know about decoherence, but here I am talking specifically about the split, and how MWI considers the measurement. Also, decoherence is just a part of the problem. I am talking about actual collapse or a split. If an apparatus is of an intermediate size (neither a macroscopic nor a microscopic), how do we handle 'half-splits'? Since MWT is an interpretation and not just a postulate, this should be handled somehow. This question is not a duplicate of question about general measurements in QM like this one: What constitutes an observation\/measurement in QM?"}
{"id":"114576","title":"ket vector with two \"entries\"","text":"This is a very simple question. I am learning about angular momentum. In my lecture notes, the symbol $|\\lambda,m_l \\rangle$ was defined as a eigenfunction of a central potential. Two assumptions are introduced: $L^2 = \\lambda \\hbar^2$ and $L_z = m_l \\hbar$. So, $$ \\hat{L}^2 |\\lambda,m_l \\rangle = \\lambda \\hbar^2|\\lambda,m_l \\rangle \\\\\\ \\hat{L_z}|\\lambda,m_l \\rangle = m_l \\hbar|\\lambda,m_l \\rangle $$ But what does $|\\lambda,m_l \\rangle$ mean exactly? I am comfortable with $|\\psi \\rangle$, but I do not understand what having two variables in the ket means."}
{"id":"104027","title":"How much merit is there in the heuristic argument of bulk-edge relation for topological insulators?","text":"Take 2D quantum hall insulator for example. The typical argument goes like this: We have a Hamiltonian that has translation symmetry in both directions on a infinite lattice, and we assign a integer number(Chern number) to each of its band. The Chern number is a topological invariant(against homotopy, I suppose), hence remains constant if the Hamiltonian is varied continuously and as long as it is well defined. **Attach** a the quantum hall insulator(which has non-zero Chern-numbered bands) to a trivial insulator(of which all bands have 0 Chern number), and the band of the former must be concatenated with the latter. However this means the well-definedness of Chern number has to break down somewhere, or else one band could have two different Chern numbers. The place where the well-definedness breaks down must be a band gap closing, hence we have conducting (edge) states. My question is, the moment we decide to **attach** two insulators together, we must have broken the translation symmetry along at least one direction, so Chern number is already not well defined, how can we proceed with the argument that follows after? I am aware that there exists more serious arguments about bulk-edge relation like the ones by Yasuhiro Hatsugai, but I haven't got around to read it and I will in the near future. But can someone comment on the previous heuristic argument I phrased? Just how hard is it to make it a rigorous argument, if possible at all? Of course if I'm told Hatsugai's paper is exactly formalizing this heuristic argument I'll get to read it as soon as possible."}
{"id":"121514","title":"Slowing of time under gravity","text":"I am not calling this gravitational time dilation because that is a relativistic effect due to the equivalence principle. Now imagine two light clocks (a clock that ticks due to light) are placed above the surface of the earth. One is placed at sea level and one very high. You might be beginning to think that I am copying Einstein but please bear with me. Consider the following image: ![enter image description here](http:\/\/i.stack.imgur.com\/07OVX.jpg) The left part of the image shows the light clock placed closer to the earth's surface and the right part shows the clock placed very high. We do not need exact height as we are not doing any calculations. One light ray is emitted from the left end of the clock on the left of the image. Now since it is closer to the earth, it will bend more than the light ray in the other clock and take a longer time to reach the receiving end. The above clock (which is the clock on the right part of the image) is away from the earth and experiences less gravity. Therefore the light ray will bend less and reach the receiving end faster. Just imagine that the clocks are fixed in position and are not moving. They are only ticking when the light ray reaches the other end. Even GR predicts that a clock will run slower in high gravity but doesn't actually give a reason. Won't this also lead to the slowing of time in a higher gravitational area (where the force of gravity is more)? Can this be an actual reason for gravitational time dilation or not? **NOTE** The clocks are not falling. The light ray inside them will still bend as it is not being held in position by anything. Only the clocks are fixed. How can a light ray be fixed? So do not think that the clocks are falling."}
{"id":"121515","title":"How classical chaos can be described quantum mechanically?","text":"How can we describe the chaotic properties of classical systems using quantum mechanics when the Schrodinger equation that describes quantum dynamics is linear? How can we use quantum mechanics that basically underlies on such a linear equation to describe highly nonlinear chaotic properties of classical systems?"}
{"id":"45124","title":"If we have a current $I$ flowing down a wire, why must the net bound current be zero?","text":"Say we are dealing with a wire that has a current $I$ flowing through it, i.e. $I$ is the free current. Why must it then hold that the net bound current, that is, the bound volume current, $J_b$, and the bound surface current, $K_b$, adds up to zero? I cannot get an intuitive grasp on why this is true, especially considering the fact that if you, on the other hand, have a uniformly magnetized cylinder, the net bound currents do not add up to zero, since there's no bound volume current, but there is a bound surface current. Similarly, in a solenoid you also get a bound surface current, but there's no bound volume current, so the two again do not add up to zero. Any help with this would be greatly appreciated, since I'm just stuck in trying to wrap my head around this. I can calculate the damn things, but I have no idea why this would be the case on a qualitative level."}
{"id":"44290","title":"Playground of Forces","text":"Why is it that the gravitational force acts on large sized objects while the strong and weak nuclear forces act at subatomic levels only? What is that stops each other to enter each others domain?"}
{"id":"133576","title":"How can I split a resultant force into its $x$ and $y$ components?","text":"> Point charge 3.5μC is located at x = 0, y = 0.30 m, point charge -3.5μC is > located at x = 0 y = -0.30 m. What are (a)the magnitude and (b)direction of > the total electric force that these charges exert on a third point charge Q > = 5.0μC at x = 0.40 m, y = 0? ![Charges](http:\/\/i.stack.imgur.com\/4sdvH.gif) I first drew a diagram, showing that it can be broken down into two .3, .4, .5 triangles, leaving me with a theta of 36.87°. I then calculated that because the charges are equal, each individual force enacted on charge Q is .63N. I know I need to split the force into it's x and y components to find the resultant vector using R=√Rx^2 + Ry^2, but I do not know how to split these forces up."}
{"id":"133574","title":"Quantum Yang-Mills Theory and AdS\/CFT","text":"I just read the first chapter of Becker-Becker-Schwarz. To quote: > A remarkable discovery made in the late 1990s is the exact equivalence (or > duality) of conformally invariant quantum field theories and superstring > theory or M-theory in special space-time geometries. Can this AdS\/CFT duality be used to work in string theory or M-theory instead of quantum Yang-Mills theory, make the corresponding theory rigorous, and then go back to quantum Yang-Mills theory, thereby solving the open Millenium Prize question?"}
{"id":"103662","title":"Energy\/Momentum required to deflect earth from its orbit","text":"This question occurred to me when thinking about the firepower people have on earth. How hard is it to change the characteristics of the orbit of the earth when an explosion happens on its surface, using conservation of momentum? I know already that the orbit of the earth is not so accurate already and fluctuates a lot, causing the so-called \"leap second\". So my question is: How could such a thing be measured\/quantified? Can we say something like: The explosion energy required to deflect the eccentricity of the earth's orbit by 1%. And how much would that be? Any ideas?"}
{"id":"122575","title":"Solving the Schrödinger equation where the initial wave function is an energy eigenfunction","text":"I was watching Allan Adams' lecture on energy eigenfunctions, and there's one part (around 43 minutes into the lecture) that confuses me. Suppose we have the initial wave function $\\Psi (x,0)$ such that $\\hat{E}\\,\\Psi (x,0)=E \\,\\Psi (x,0)$ for some constant $E$. Then, plugging this into the Schrödinger equation, we'd get: \\begin{align} i \\hbar \\frac{\\partial}{\\partial t} \\Psi (x, 0) &= E \\, \\Psi (x,0) \\\\\\ \\frac{\\partial}{\\partial t} \\Psi (x, 0) &= \\frac{E}{i \\hbar} \\, \\Psi (x,0) \\tag{1}\\\\\\ \\therefore \\Psi (x, t) &= \\exp\\left({-i \\frac{E\\,t}{\\hbar}}\\right) \\Psi(x,0) \\tag{2} \\end{align} I'm a bit confused about how to go from $(1)$ to $(2)$. Now if we make the additional assumption that $\\hat{E}\\,\\Psi (x,t)=E \\,\\Psi (x,t)$ for all $t$, then the Schrödinger equation becomes: \\begin{align} \\frac{\\partial}{\\partial t} \\Psi (x, t) &= \\frac{E}{i \\hbar} \\, \\Psi (x,t) \\end{align} and I can solve this differential equation easily and get $(2)$. But from watching that part of the lecture, it seems we only need to assume a weaker statement - that the initial wave function is an energy eigenfunction. But then, it's not clear to me how I can get the solution $(2)$ from $(1)$. Am I missing something? Update: Thanks for all the answers. After reading through the accompanying lecture note, we indeed need to assume that the energy operator is a constant over time."}
{"id":"11081","title":"Physical Chemistry: What's the relationship between orbital overlap and barrier shape?","text":"This is a question for the physical chemists out there. For a given chemical reaction there is a barrier to traverse in order to proceed from reactants to products. Reactants e.g. an organic base abstracting a proton can approach each other at different orientations - attack angles. At times, this orientation will lead to better orbital overlap between the lone pair on the base and the s orbital of the hydrogen. I have heard people talk about better orbital overlap, lowering\/narrowing the barrier to reaction. Is this a valid way of conceptualizing this? Does better orbital overlap change the barrier? I personally have always associated a better overlap as starting further along the reaction coordinate\/barrier, not the actual barrier itself being altered? Any thoughts from the physchems?"}
{"id":"11087","title":"Are dangerous rays emitted during Solar Eclipse?","text":"It is said one should avoid staring at Sun as it can damage the eyes, but it is also said that one should not come out in sun during eclipse as it emits dangerous rays. Is that true? If yes, why?"}
{"id":"87152","title":"Finite Square Potential Well Solving","text":"I've got a question about solving the finite square well. Consider a 1D finite, square, potential well where: $$ V(x) = \\begin{cases} 0&\\,{\\rm if} x<0 \\\\\\ -V_0&\\,{\\rm if}\\,x\\in\\left(0,L\\right) \\\\\\ 0&\\,{\\rm if} x>L \\end{cases} $$ When applying the Schrodinger Equation to Zone II ($V(x)=-V_0$), it will be necessary to write a constant $k$, which my book has defined this way: $$k^2= - 2m\\frac{E+V}{\\hbar^2}$$ My Question is: Can we write this as $$k^2=2m\\frac{E+V}{\\hbar^2} $$ Shouldn't we get the same solution? If not, why not? The final solution should not depend on which way we write $k$, correct?"}
{"id":"83642","title":"B3LYP vs PBE functionals for conjugated organic systems","text":"Two of the most popular (exchange and correlation) functionals for density functional theory are B3LYP and PBE. Out of the people I've worked with \/ learned from, mostly the computational chemists were using B3LYP, and the physicists were using PBE. Now, different functionals have different strengths and weaknesses, but these two functionals seem to have a lot of overlap in their applicability. Right now I'm interested in ground-state properties of molecules with extended pi- frameworks in the gas-phase. For example, I'd like to compute the geometries, ground-state electron densities, and molecular polarizabilities of the acenes. Does anyone have concrete information or references on the performance of B3LYP or PBE in this situation, or in general?"}
{"id":"89429","title":"Integrating equations with units","text":"I was looking through an old copy of Barron's AP Physics and found this problem relating to impulse which I was initially confused about how to integrate. > **Example 6.1** During a collision with a wall lasting from $t=0$ to > $t=2\\text{ s}$, the force acting on a $2\\text{-kg}$ object is given by the > equation $\\mathbf{F} = (4\\mathrm{\\ kg\\ m\/s^4})t(2s-t)\\hat{i}$ They work out that the integral is equal to: $$\\frac{16}{3}\\hat{i}\\frac{\\text{kg m}}{\\text{s}}$$ I am confused about the role of the units in the problem. Looking at the answer, it seems that if I were to just ignore all the units and simply integrate $4t\\cdot(2-t)$ that would give me $16\/3$, and because that is a force I know it to be $\\mathrm{kg\\,m\/s}$ or $\\mathrm{N}$. Why it would be OK to ignore the units in the integral though is somewhat unintuitive to me (other than that I know the end result must be a force) and I feel like it may get me into trouble with other problems. Can someone explain how it is that the scalar values of the original unit quantities still give you the correct answer?"}
{"id":"126631","title":"Measurement in reciprocal metres","text":"I'm trying to name a measurement that is measured in reciprocal length, which is in a draft document for vehicle risk management. It currently says:               Crash rate = number of crashes \/ million kilometres driven      where the measurements are taken annually. So the units are in reciprocal length. I'm not happy with the word \"rate\" because that seems to imply the number of events over time, whereas the measurement is the number of events over distance. Existing names of units of reciprocal length in various disciplines don't seem to work - it's not a measurement of any cyclic phenomenon. Is there an better word for this measurement?"}
{"id":"82061","title":"Why does a CD work as a diffraction grating even with light from a light bulb?","text":"I have a really hard time solving the following problem I accidentally came across today. Looking at the back of a usual CD one sees coloured bands. This is explained by the fact that the surface (pits and bumps) is effectively a diffraction grating. But why is this effect already visible when one uses _incoherent_ light, e.g. from a light bulb, whereas the interference pattern of a single or double slit would be missing with such illumination?"}
{"id":"82064","title":"Writing a Wavefunction as a Linear Combination of Eigenstates","text":"We have the following wavefunction for the hydrogen atom: $$\\psi(r,\\theta,\\phi)=\\frac{1}{\\sqrt{4\\pi}}\\frac{1}{(2a)^{3\/2}}\\frac{r}{a}e^{-r\/2a}\\sin(\\theta)\\sin(\\phi)$$ where $a$ is the Bohr radius. `Question`: How can I express the wavefunction above as a linear combination of the eigenstates of the hydrogen atom's Hamiltonian? In other words, we need to express the above in terms of a linear combination of $$\\psi_{nlm}=\\sqrt{\\left(\\frac{2}{na}\\right)^3 \\frac{(n-l-1)!}{2n[(n+l)!]^3}}e^{-r\/na}\\left(\\frac{2r}{na}\\right)[L_{n-l-1}^{2l+1}(2r\/na)]Y_l^m(\\theta,\\phi)$$ For some $n$, $l$, and $m$'s. How can I find such a linear combination without just a simple guess and check; i.e., what is a systematic way to find the linear combination?"}
{"id":"82067","title":"What laws (formulas) govern forces between atoms?","text":"What laws (formulas) govern the fundamental forces of nature? For example, gravity is governed by the inverse-square law. I am thinking about how particles attract each other, but also repel. All matter attracts via gravity, but also attract and repel each other due to the electromagnetic force, also governed via the inverse-square law, and then there's the strong and weak force. I understand the idea that two hydrogen atoms repel, but when pushed together hard enough will attract and form helium and release energy. So this \"upwards slope\" of repulsion which is eventually \"snaps\" into attraction, like trying to roll a ball up a smooth hill with a hole in the center of it, is explained, likely, by varying forces governed by various laws. Can you help me understand it, **on the level of particles**? Where I am stuck is thinking about two particles, each being attracted and repelled by two different forces each governed by the inverse-square law. This doesn't explain the \"snapping\" together of atoms. Do I need different math in my forces, or is it ultimately explained by more than just two particles? Here is a diagram of what I mean, describing H + H -> H2: ![enter image description here](http:\/\/i.stack.imgur.com\/3YesO.gif) _Image source and article:http:\/\/employees.csbsju.edu\/hjakubowski\/classes\/ch111\/olsg- ch111\/equilibkinetics\/equilbkin.htm_ Please feel free to reword my question if you understand what I am asking but am not making myself clear."}
{"id":"23309","title":"Irreducible tensor representations with \"covariant\" indices","text":"As a follow-up of my question on the \"most general\" $\\mathrm{SU}(2)$-symmetric interaction of two spin 1\/2 particles, I ponder the following question: Consider an operator acting just on one particle of spin 1\/2, and just on the spin part. In second quantization, it can be written as $$\\sum_{\\alpha\\beta} \\psi^\\dagger_\\alpha V_{\\alpha \\beta} \\psi_\\beta$$ Now, under $SU(2)$, the operators will transform as $$\\psi^\\dagger_\\alpha = \\sum_{\\alpha'} \\psi^\\dagger_{\\alpha'} D_{\\alpha\\alpha'}$$ with $D \\in SU(2)$. I can move this transformation to the operator and conclude that it transforms like $$V_{\\alpha\\beta} \\rightarrow \\sum_{\\alpha'\\beta'} D_{\\alpha\\alpha'} V_{\\alpha'\\beta'} D_{\\beta\\beta'}^\\dagger$$. (I might have gotten some of the daggers and indices backwards, but the important point is that one matrix is the adjunct of the other). Now if I understand correctly, this allows me to conclude that $V_{\\alpha\\beta}$ transforms \"reducibly like a tensor product of two spin 1\/2 representations\", i.e., like $\\frac{1}{2} \\otimes \\frac{1}{2}$. From that, I conclude that $V$ should decompose into one component that transforms like a spin $0$ particle and one component that transforms like a spin $1$ particle. Indeed, as $V$ is a $2\\times 2$ matrix, I can write it as a linear combination of the unit matrix and the pauli matrices. The former transforms like a scalar, i.e., like spin $0$, whereas the latter transform like vectors. However, I have trouble relating this to what I know about combining spin $1\/2$ particles using Clebsh-Gordan coefficients, where I have, e.g., for the singlet $$|0 0\\rangle = \\frac{1}{\\sqrt{2}} \\left( \\uparrow \\downarrow - \\downarrow \\uparrow\\right)$$ Because of this conceptual problem, I also have trouble generalizing this to the case of two interacting spin 1\/2 particles, which then should lead to an interaction that transforms reducibly as $1\/2 \\otimes 1\/2 \\otimes 1\/2 \\otimes$ and gives rise to two different singlets. I'd appreciate it if someone could disentangle my misconceptions... EDIT: I forgot to specify what I mean with \"covariant\" in the title: I think an important thing to notice is that the matrix elements $V_{\\alpha\\beta}$ are **not** the 4 elements of a \"cartesian\" tensor of rank 2. The entire operator $V$ might be an element of a tensor of higher rank, or even the sum of elements of tensors of different rank. A simple example for such a thing would be an operator \"1 + x\", which is the sum of a rank zero tensor (scalar) and the element of a cartesian tensor of rank 1. Now, what I mean with covariant is that one of the indices of $V_{\\alpha\\beta}$ transforms with matrix $D$ and the other with matrix $D^\\dagger$. Also important is that the components of the operator transform with the actual $SU(2)$-matrices and not with some rotation matrix $R \\in SO(3)$. I guess this is why I have trouble translating the standard literature on tensor operators to my situation..."}
{"id":"110603","title":"Does a photon travel through space exactly at the speed of light?","text":"In practice, given that in the vacuum of the universe is not perfectly empty, does the photon actually reach exactly c? Is it even possible? If it travels at exactly c, then from the photon perspective, it arrives at the same time it leaves its point of origin, so for him the distance between the 2 points is zero? Many thanks."}
{"id":"108925","title":"Centrifugal force?","text":"Everyone calls it a pseudo force and that it is not really present, even there are so many machines listed to advantage of centrifugal force ( centrifuge, washing washing drier),and also it has the same magnitude as centripetal force, so that they cancel cancel each other out but i can definitely feel it every time a sharp turn is taken in the car. So can you explain what centrifugal force is exactly, i don't understand it."}
{"id":"117050","title":"How can Drift Velocity of an electron causes change?","text":"Assume we have a conductor that has current flowing and its placed in a magnetic field so that it experiences the Lorentz force, and it gains Kinetic energy due to that force. Would drift velocity change the electron's direction thus the current's total direction changes, therefore, changing the Lorentz force direction applied on the conductor?"}
{"id":"16054","title":"How is stable levitation possible?","text":"This question is with reference to the video in this blog post: http:\/\/www.universetoday.com\/90183\/quantum-levitation-and-the-superconductor\/ My question is the following: how is the disc stable in its levitated position? Specifically, 25 seconds into the video, the exhibitor turns the entire thing upside down, and the disc doesn't fall. This contradicts two intuitive ideas I have:   1. Right-side-up, gravity is counteracting the repulsive effect of the magnet. Upside down, gravity is working with it. Unless there's some adaptation going on somewhere else, shouldn't \"gravity switching sides\" be enough to tear the disc away from the magnet?   2. I remembered something about \"inverse-square\" forces not permitting stable equilibria from college physics - sure enough Wikipedia calls it Earnshaw's theorem. I don't see how this is exempt from that rule. I guess I don't understand diamagnetism."}
{"id":"88774","title":"Hamilton-Jacobi formalism and on-shell actions","text":"My question is essentially how to extract the canonical momentum out of an on- shell action. The Hamilton-Jacobi formalism tells us that Hamilton's principal function is the on-shell action, which depends only on the coordinates $q$. Therefore, if $S$ refers to the off-shell action with Lagrangian $L$, a result for the canonical momentum is $$ \\frac{\\partial L}{\\partial \\dot q} = p = \\frac{\\partial S^\\text{on- shell}}{\\partial q} $$ In theory, when evaluating the on-shell action, one plugs in the EOM into $S$ and all that's left are boundary terms. If we illustrate this with the harmonic oscillator in 1D, one gets \\begin{align} S^\\text{on-shell} &= \\int_{t_i}^{t_f} L dt \\\\\\ &= \\frac{m}{2} \\int_{t_i}^{t_f} \\left( \\dot q^2 - \\omega^2 q^2 \\right) dt\\\\\\ &= \\frac{m}{2} \\int_{t_i}^{t_f} \\left( \\frac{d}{dt} (q \\dot q) - q \\ddot q - \\omega^2 q^2 \\right) \\\\\\ &= \\frac{m}{2} \\left[ q(t_f) \\dot q(t_f) - q(t_i) \\dot q(t_i) \\right] \\end{align} Now, this depends on $\\dot q$, whereas it should depend only on generalized coordinates. Also, the on-shell action depends on the boundary values of our coordinate $q$, so we can't differentiate w.r.t. $q(t)$ to get a conjugate momentum $p(t)$ for all times, like it is usually done in the Lagrangian formalism. What am I not seeing?"}
{"id":"127580","title":"Fierz identity for Weyl spinors in tensor currents","text":"Using Fierz identity I found that certain four-fermion operator with left $l_i$ and right-chiral $r_i$ Weyl spinors vanish $\\bar{l}_1\\sigma_{\\mu\\nu} r_2 \\bar{r}_3 \\sigma^{\\mu\\nu} l_4 =$ $ -\\frac{3}{2} \\bar{l}_1 l_4 \\bar{r}_3 r_2 - \\frac{1}{2}\\bar{l}_1\\sigma_{\\mu\\nu} l_4 \\bar{r}_3 \\sigma^{\\mu\\nu} r_2$ $ - \\frac{3}{2} \\bar{l}_1 (-1) l_4 \\bar{r}_3 (+1) r_2 =$ $= -\\frac{1}{2}\\bar{l}_1\\sigma_{\\mu\\nu} l_4 \\bar{r}_3 \\sigma^{\\mu\\nu} r_2 = 0$ Why does this operator vanish? Is that true? Why is the product of tensor currents expressed in tensor and scalar currents only, while for such combinations of Weyl spinors $l_1$, $r_2$, $r_3$ and $l_4$ in the Fierz identity all of them vanish?"}
{"id":"61884","title":"Does a current carrying wire produce electric field outside?","text":"In the modern texts of electromagnetism in the presence of stationary currents the electric field is assumed conservative $\\nabla \\times E =0 $. Using this we get $E_{||}^{out}=E_{||}^{in}$ which means we have the same amount of electric field just outside of the wire! Is this correct? Is there any experimental proof?"}
{"id":"119400","title":"How to use an accelerometer to correctly quantify the magnitude of vibrations","text":"Consider the following setup, an accelerometer is placed inside a helicopter, measuring \"g\" values along 3 axes (x,y,z) every 10 ms (100 times a second), during a helicopter flight. If one, wants to obtain some measure of the maximum g force exerted on an object inside the helicopter, what would be the best way to do this? Make the analysis for each axis independently? Or treat x,y,z as components of a vector ? Is it correct to consider the maximum difference between two consecutive measurements as an approximation of the maximum g force exerted on an object inside the helicopter"}
{"id":"14855","title":"Naïve relativistic schrodinger equation","text":"> **Possible Duplicate:**   >  Why are higher order Lagrangians called 'non-local'? Bjorken and Drell presents the equation: $$i\\hbar\\frac{d\\psi}{dt}=H\\psi=\\sqrt{p^2 c^2+m^2 c^4}\\psi=\\sqrt{-\\hbar^2 c^2 \\nabla^2+m^2 c^4}\\psi$$ The squareroot can be expanded to obtain an equation with all powers of the derivative operator. What do they mean when they say this leads to a non-local theory? And is this equation incorrect or just impractical?"}
{"id":"2332","title":"Is there an equation for convective heat transfer?","text":"Is there an equation I can use to calculate the temperature (as a function of time) of an object which is gaining or losing heat by convection? Or equivalently, the rate of energy transfer from the object to the surrounding fluid (or vice-versa)? It can involve constants representing properties of the object and the surrounding fluid. I know about the heat equation for conduction and the Stefan-Boltzmann law for radiation, but I can't think of one for convection."}
{"id":"127568","title":"compact and non-compact gauge theory","text":"From the answer to this question, I see that the gauge theory has to be compact if the charges need to be quantized. I am not sure in what sense these are necessary and sufficient. So here is a question I had: In electromagnetism, the electric charges are quantized in units of electronic charge, so shouldn't the gauge fields $A$ have values on a compact manifold rather than be a real number ?"}
{"id":"46332","title":"What happens to orbits at small radii in general relativity?","text":"I know that (most) elliptic orbits precess due to the math of general relativity, like this: ![precession](http:\/\/i.stack.imgur.com\/H8l5b.png) source: http:\/\/en.wikipedia.org\/wiki\/Two-body_problem_in_general_relativity I also know that something is different for orbits at radii below a certain value. Wikipedia explains this as follows, I'm confused by this and just want to ask for clarification: > _If the particle slips slightly inwards from $r_{inner}$ (where all three > forces are in balance), the third force dominates the other two and draws > the particle inexorably inwards to $r=0$._ What does this mean? If you drew a graph of the path a particle takes in this unstable regime, what would it look like? Why is the stability transition point further than the Schwarzschild radius? Why does this graph show an unstable point beyond the Schwarzschild radius? For elliptical orbits that come very close to the event horizon, is there some kind of orbital decay? How is energy conserved? In short, do all orbits (with GR effects) look like the precession shown above, or is there another shape that we see if it gets closer to the Schwarzschild radius?"}
{"id":"46335","title":"Stability of balanced masses with different surface areas","text":"![enter image description here](http:\/\/i.stack.imgur.com\/Eze3s.png) Say I have this setup. The two round objects have equal mass and their centers of gravity at the same distance from the shaft. The objects only differ in that they have different surface areas (assume they are both flat-faced). If this apparatus is spun, will it fall over? I believe it won't. The centrifugal forces are balanced since they only depend on mass, and the aerodynamic drag will be in the direction of motion, so it won't pull the shaft in any outward direction."}
{"id":"56818","title":"What starts the movement of a photon","text":"Although a photon has no (rest) mass, it does have a measurable speed. Its movement can be altered by gravity. A photon \"travels\". If I turn on a flashlight, seen by someone at a distance, the photons from that flashlight traveled, at the speed of light, from the flashlight to the observer. But what \"force\" instantly starts the movement of a photon at the speed of light?"}
{"id":"2486","title":"Moving from point A to point B in zero-gravity","text":"Here's an proposal on how to get from point A to point B in zero-gravity without using any propellant and the question why it wouldn't work: A closed tube, filled with water and a round (solid) object. If you need equations, the volume of enclosed water is the same as the round object, but the round object is 10 times lighter. (imagine a glass with water plus a ping- pong ball). here's a picture: ![alt text](http:\/\/img210.imageshack.us\/img210\/6640\/sprite143.jpg) On earth the round object will float on the water inside the tube (subject to one G). In zero-gravity the round object has no preferred position. If we accelerate the tube in zero-gravity by one G, the situation is the same as on earth, the round object \"floats\". In this example we are accelerating the tube from the left side to the right side. The round object will consider floating to the right AS LONG AS THERE IS acceleration. Now consider adding a pipe to the bottom of the tube and connecting it to the top, a loop. Inside the pipe is a small water pump. If we give this apparatus a push, say one G in a zero-g environment, the round object will move \"up\", but now we start the water pump and spray the water on the round object, we try to submerge it. It will resist and impart a impulse on the water. like trying to hose down a air balloon floating on the pool. The pump will feel a resistance and hence the whole apparatus will move. Just running the pump at constant speed, same volume of water per second, will do nothing. but if we run the pump faster and faster the whole apparatus will start moving: the amount needs to be geometrical. The point is that we need to keep the apparatus feeling an acceleration, since only then will the round-object \"float\" and resist the incoming water at the top, hence we have something to \"push on\". Before you blow the \"foul whistle\": consider the situation if there were no round floating ball in the apparatus. (the pump runs on solar power or pre-charged battery) (disclaimer: i know standing on a sailboat and blowing into the sail will not get me anywhere, action<->reaction) thanks a bunch Sklivvz for the edit. sometimes the idea just needs to get out, never the mind how it looks like : P"}
{"id":"99559","title":"Rotation Group and Lorentz Group","text":"It is often stated that rotations in the 3 spatial dimensions are examples of Lorentz transformations. But Lorentz transformations form a group named the Lorentz Group, $O(1,3)$ which is a group a $4 \\times 4$ matrices, $\\Lambda$ having the following property: $$ \\Lambda^T g \\Lambda = g$$ where $g$ is the metric tensor. Now rotations matrices for the 3 spatial dimensions are $3 \\times 3$ matrices and form $SO(3)$. How can they be in the $O(1,3)$ ?"}
{"id":"57140","title":"Are electrons moving? If so relative to whom? and why is there no magnetic field?","text":"electrons on earth **moving** with us, due to rotation of earth, revolution of earth, sun and our galaxy right?   Then, why is there **no** magnetic field around a **piece** of copper wire?"}
{"id":"11706","title":"How to apply Andreev reflection formalism to ferromagnet ,normal metal interface?","text":"The traditional formalism for andreev reflection deals with what happens at normal metal, super conductor interface.http:\/\/en.wikipedia.org\/wiki\/Andreev_reflection (i.e when an electron from normal metal is incident on the super conductor interface, a hole is generated in the normal metal which lets the formation of a cooper pair in the SC which propagates in the appropriate direction) How do we apply a similar formalism when an electron of say up-spin is incident from a normal metal to half-metal (magnet) ? what happens when this electron comes spin polarized at an angle to the interface ? note that unlike in superconductivity, there is a direction associated with the half metal (which is basically a magnet with an orientation)"}
{"id":"80684","title":"Topological disorder in condensed matter?","text":"What is meant by topological disorder in condensed matter (both crystalline and amorphous)? For example, please see the following two papers from arxiv.org http:\/\/arxiv.org\/pdf\/0906.3848.pdf http:\/\/arxiv.org\/pdf\/cond-mat\/0308612v1.pdf Thanks"}
{"id":"20502","title":"Angular velocity $\\omega$ by $v$","text":"We have two girls, with mass ($M$). They become close to each other in speed of $V$. The distance between them is $3L$. ![](http:\/\/img2.timg.co.il\/forums\/1_159160419.jpg) I was asked to calculate the Angular velocity ($\\omega$) of the two girls. So I set the rotation axis in the middle (where the distance between the two girls and the axis is $1.5L$, and I calculate the Angular velocity using this equation: $\\omega=v\/r$, where $r=1.5L$, and I got that $\\omega={2v\\over 3L}$. As I understood, the answer is correct, but this is not the correct way. What is my mistake? (sorry about my english)"}
{"id":"20504","title":"What are the theoretical explanations behind the \"no-boundary condition\" in cosmology?","text":"In his book \"The Grand Design\" on page 167-173 Stephen Hawking explains how one can get rid of the so called \"time zero issue\", meaning by this the very special initial state needed for inflation to work (?), by applying the so called \"no-boundery condition\" which says that time behaves as a closed space dimension for times small enough such that the universe is still governed by quantum gravity. To see better what Hawking is talking or writing about I`d like to understand this a bit more \"technically\" and detailed, so my questions are: 1) How can situations where QG kicks in lead to a transition of the time dimension to a space dimension? I mean what is the physical reason for this \"ambiguity\" of the signature of the correspondng spacetime metric? 2) What are the triggering causes of the \"symmetry breaking\" between the 4 space dimensions when starting from the no-boundary condition such that one of the space dimensions becomes an open time dimension at the beginning of time? 3) Are there (quantum gravity) theories or models that can explain 1) and 2) and if so, how? Since I've just read about these issues recently in the \"Grand Design\" (and nothing else) I`m probably confused and hope that somebody can enlighten me by explaining some details ..."}
{"id":"95480","title":"Space and time translation of quantum field","text":"In book I am studying, the authors want to calculate the spectral representation of the g-lesser (and g-greater) functions and find a general expression for these. In order to do so they start with the expression for the fields in Wigner coordnates (translated in time and space) $\\hat \\psi \\left( \\mathbf R + \\frac{\\mathbf r}{2} , t + \\frac{\\tau}{2} \\right) $. They then claim that this may be written like $ \\hat \\psi \\left( \\mathbf R + \\frac{\\mathbf r}{2} , t + \\frac{\\tau}{2} \\right) = \\exp \\left[ \\frac{i}{2} \\left( \\hat H \\tau + \\hat{\\mathbf P} \\cdot \\mathbf r \\right) \\right] \\hat \\psi \\left( \\mathbf R , t \\right) \\exp \\left[ - \\frac{i}{2} \\left( \\hat H \\tau + \\hat{\\mathbf P} \\cdot \\mathbf r \\right) \\right] $ where $H$ is the Hamiltonian and $\\hat{\\mathbf P}$ is the total momentum operator. What are the underlying assumptions for the equation above to be true? One obvious thing is that the Hamiltonian must not be explicitly time dependent (since the time translation operator then gets a more complicated expression involving the time ordering operator). Is it for example understood that the total momentum operator and the Hamiltonian are commuting? If the Hamiltonian and the momentum operator do not commute then since $\\exp \\left[ \\frac{i}{2} \\left( \\hat H \\tau + \\hat{\\mathbf P} \\cdot \\mathbf r \\right) \\right] \\neq \\exp \\left[ \\frac{i}{2} \\hat H \\tau \\right] \\exp \\left[ \\frac{i}{2} \\hat{\\mathbf P} \\cdot \\mathbf r \\right] $ the order of which the time and space translations are made matter. Perhaps someone could also help me clarify what the meaning of this is..."}
{"id":"80350","title":"Is it possible to derive fermi-dirac or bose-einstein statistics using quantum operator formulations?","text":"I've been looking through theory on identical particles to get a better grasp of the uncertainty principle but it would be very interesting if these results could be extracted from the formalism as well. Does anyone have any leads on this?"}
{"id":"80352","title":"Motion of elevating airplane","text":"I wonder how the motion of an airplane which will not try stop to elevate should look like. Of course, at some point the plane will no longer be able to go to higher altitudes but is it going to stay on a circular trajectory or things are more involved?"}
{"id":"53530","title":"Why is ski jumping not suicidal?","text":"At least on television, ski jumpers seem to fall great vertical distances before they hit the ground - at least a few dozen meters, though I couldn't find exact distances via a quick search. And yet they almost always land on their feet as if they just fell two or three meters. (Here's a whole lot of footage from the Vancouver Olympics if you need to refresh your memory.) Without going into the level of equations (which I wouldn't understand), why are ski jumpers able to fall such great heights without seriously injuring themselves?"}
{"id":"99685","title":"Bose-Einstein-statistics out of fermionic many body system","text":"ok, let me try this again. How do I get to atoms obeying Bose Einstein statistics from considering the fermionic many body problem of a bunch of electrons, protons and neutrons forming these (bosonic) atoms. [ORIGINAL QUESTION] this keeps bugging me: Whenever there is a system of bosons (lets not consider photons\/phonons) then we use BE statistics to describe it. But shouldn't we also be able to describe it (more accurately) by FD statistics with regard to its constituent electrons, protons and neutrons? How is this consistent with the symmetry requirements on the wave function and phenomenas as Bose Einstein condensation?"}
{"id":"10549","title":"How to derive the Karman-Howarth-Monin relation for anisotropic turbulence?","text":"I find the derivation of the Karman-Howarth-Monin relation in the book Turbulence from Frisch (1995) a bit to short. Can someone point me to a more detailed derivation of that relation, if possible in a publicly available source (lecture notes, thesis, arXiv paper)? Or can someone explain to me how Frisch obtains the equation that follows the sentence on page 78: > Starting from the Navier-Stokes equation (6.6), we obtain $$\\begin{align} > \\partial_t \\frac{1}{2}\\langle v_i v'_i \\rangle =& \\- \\frac{1}{2} \\partial_j > \\langle v_i v_j v'_i\\rangle \\- \\frac{1}{2} \\partial'_j \\langle v'_i v'_j > v_i\\rangle \\\\\\ & \\- \\frac{1}{2} \\langle v'_i \\partial_i p \\rangle \\- > \\frac{1}{2} \\langle v_i \\partial'_i p' \\rangle\\\\\\ &\\+ \\frac{1}{2} \\langle > v'_i f_i \\rangle \\+ \\frac{1}{2} \\langle v_i f'_i \\rangle \\\\\\ & \\+ > \\frac{1}{2} \\nu \\left( \\partial_{jj} + \\partial'_{jj} \\right) \\langle > v_iv'_i\\rangle. \\qquad (6.11) \\end{align} $$"}
{"id":"53538","title":"Calculation of a bending moment","text":"I'd like to calculate the bending moment of a cantilever, fixed at its base, and submitted to a certain stress on a specific spot, but I can't find the proper definition of this bending moment (first time I'm encountering this name, and google didn't not help me so far)."}
{"id":"63226","title":"How can I interpret negative values of potential evapotranspiration?","text":"If I extract Potential Evaporation (PET, W\/m$^2$) from the National Centers for Environmental Prediction (NCEP), climate reanalysis data (downloadable as netCDF files here), there are some negative values, e.g. for summer in southern Manitoba (50N, 100W), I get the attached distribution of PET values.: ![enter image description here](http:\/\/i.stack.imgur.com\/G6Gct.png) Is this a real physical process, such as condensation (dew formation), or is it an error (e.g. should I average over these values or set them to zero)? * * * **update** requested metadata: The metadata (contained in the files) says that the valid range is -800, 5200; here is further description of the 'pevpr' variable (provided in the data header):   * long_name = \"Monthly Mean of Potential Evaporation Rate\"   * valid_range = -800,5200 ;   * units = \"W\/m^2\" ;   * add_offset = 0 ;   * scale_factor = 1 ;   * missing_value = -9.96921e+36 ;   * precision = 1 ;   * least_significant_digit = 0 ;   * var_desc = \"Potential Evaporation Rate\" ;   * dataset = \"CDC Derived NCEP Reanalysis Products\" ;   * level_desc = \"Surface\" ;   * statistic = \"Mean\" ;   * parent_stat = \"Individual Obs\" ;"}
{"id":"57859","title":"Is there a relation between supersymmetry and entropy?","text":"Considering that entropy denotes the level of order\/disorder in a system, would it be possible for entropy and supersymmetry to exist at the same time? Or, are they entirely unrelated?"}
{"id":"63223","title":"Excitation probability for 2 Level Atom","text":"I currently started learning Quantum Optics using the book \"Atom Physics\" of Christopher Foot. Currently I am working through Chapter 7 and a question occured concerning the intuitively physical understanding: Assume a two level atom with energy levels $E_{1}$ and $E_{2}$ (with $E_{1}$<$E_{2}$ is given). The eigenstates of the system are given by $|1\\rangle$ and $|2\\rangle$ respectively. Now assume that we perturbe the system by using a time dependent Electric field of the form \\begin{equation} \\vec{E} = \\vec E_{0} \\cos\\left(kz - \\omega t\\right) \\end{equation} and expand the solution of the Schroedinger Equation in terms of the unperturbed eigenstates \\begin{equation} \\Psi(\\vec r,t) = c_{1}(t) \\ e^{-i (E_{1}\/\\hbar) t} \\ |1\\rangle \\+ c_{2}(t) \\ e^{-i (E_{2}\/\\hbar) t} \\ |2\\rangle \\end{equation} Let's assume that $c_{1}(0)=1$ and $c_{2}(0)=0$. Using the dipole approximation as well as the rotating wave approximation as discussed for example in www.physics.ox.ac.uk\/Users\/lucas\/BIII\/QuantumNotes.pdf‎, we obtain that \\begin{equation} |c_{2}(t)| \\sim t^{2} \\ \\frac{\\sin^{2}(x)}{x^{2}} \\end{equation} with $x=(\\omega-\\omega_{0})t\/2$ and $\\hbar \\omega_{0}:=E_{2}-E_{1}$. Plotting this as a function of $\\omega-\\omega_{0}$ we obtain a \"Fraunhofer-diffraction\"-like graph. My Question: 1) As the interaction time increases the width of the maxima is decreasing. What is from a physical \/ intuitive perspective the reason for this. 2) There are also certain minima occuring for certain frequencies of the light. What is the physical interpretation (intuitively) for this? I am looking forward to your responses. Thanks in advance."}
{"id":"95153","title":"Finding the mass of 7cm string","text":"If a string has $120$ g and length $40$ cm. I found that the linear mass density is given by $$120g\/40cm=\\ linear\\ mass\\ density$$ How do I use this to find the mass of a $7$cm of the same kind of string? * * * What I have done so far: $$120g\/40cm = m\/7cm$$ $$m=7cm*120g\/40cm$$ $$m = 21$$"}
{"id":"29300","title":"Lorentz spinors of $SO(n,1)$ and conformal spinors of $SO(n,2)$","text":"* It would be great if someone can give me a reference (short enough!) which explains the (spinor) representation theory of the groups $SO(n,1)$ and $SO(n,2)$. I have searched through a few standard representation theory books and I couldn't find any.   * More specifically I would like to know how a Lorentz spinor of $SO(n-1,1)$ (say $Q$) is \"completed\" to a conformal spinor of $SO(n,2)$ (say $V$) by saying,  $V = (Q, C\\bar{S})$ where $C$ is a \"charge conjugation operator\" and $S$ is probably another $SO(n-1,1)$ spinor. Is there some natural Clifford algebra representation ($\\Gamma$) lurking around here with respect to which I can define the \"charge conjugation operator\" as $C$ such that $C^{-1}\\Gamma C = - \\Gamma ^T$? (...in general a representation of the Clifford algebra also gives a representation of $SO(n,1)$..I would like to know as to how this general idea might be working here...)   * Some of the other aspects of this group theory that I want to know are an explanation for facts like,      * $Sp(4)$ is the same as $SO(3,2)$, and the fundamental of $Sp(4)$ is the spinor of $SO(3,2)$     * $SU(2,2)$ is the same as SO(4,2), and the fundamental of $SU(2,2)$ is the spinor of $SO(4,2)$ (...just two \"facts\" hoping that people can point me to some literature (hopefully short!) which explains the systematics of which the above are probably two examples...)"}
{"id":"54627","title":"Flat space metrics","text":"This question concerns the metric of a flat space: $$ds^2=dr^2+cr^2\\,\\,d\\theta^2$$ where $c$ is a constant. Why is it necessary to set $c=1$ to avoid singularities and to restrict $r\\ge 0$? Thanks."}
{"id":"16963","title":"canonical and microcanonical ensemble","text":"What does one mean by **canonical** and **micro canonical ensemble** in statistical mechanics? Can one elaborate on this in a very simple way with examples? Pardon me, if it is a very simple thing; I am a math student."}
{"id":"34357","title":"Tips on teaching Dimensional Analysis?","text":"What's a good way to explain dimensional analysis to a student? Here's a simple question which this method would be useful: Let's say a truck is moving with a speed of 18 m\/s to a new speed of 13 m\/s over a distance of 48 meters. How long did it take for the truck to break to its new speed, and what was the acceleration of the truck? Edit @TimGoodman and @Laar, Hmm. I was under the impression that you could solve such questions by analyzing their dimensional properties. For example, based on V1 and V0 in the example above I know the person decelerated 5m\/s over a distance of 48 meters. If I want the time it took (first question) I can simply look at the units and see if I can eliminate all units except s (seconds). 48m\/5m\/s allows the meters to cancel out leaving 9.6 seconds. Now I have a new peice of information and I can now solve the second question \"What was the acceleration of the truck?\" Since I'm solving for acceleration I know my final units will be m\/s^2. I know the velocity of the deceleration was 5m\/s and it took him 9.6s. Dividing 5m\/s\/9.6s = .52 m\/s^2. My final units match and so by using dimensional analysis I've solved the problem. Maybe I have a misunderstanding of dimensional analysis. Any suggestions?"}
{"id":"109721","title":"Classical static equilibrium in atomic physics","text":"Consider a collection of $n+1$ mass weighted points in $\\mathbb{R}^3$. Suppose we have one mass located at the point (0,0,0) with mass $m\\in\\mathbb{N}$ and further suppose we have $n$ masses arranged arbitrarily each with mass -1. Now we want to assign to each mass weighted coordinate a collection of vectors in the following way:   1. There will be a total of $n$ vectors emanating from each weighted point.   2. The assigned vectors $\\vec v$ will be of two varieties: $$\\vec v\\in\\\\{\\frac{\\hat r_{ij}}{||r_{ij}||^2},\\frac{-m\\hat r_{i0}}{||\\vec r_{i0}||^2}\\\\}$$ where $||r_{ij}||$ is the norm of the vector between points $i$ and $j$, $\\hat r_{ij}$ denotes a direction away from both points $i,j$ and along the line between $i$ and $j$. $||r_{i0}||$ is the norm of the vector between a point $i$ and the origin and $-\\hat r_{i0}$ denotes a direction from point $i$ to the origin. I am interested in whether there exists a 3 dimensional configuration of the points where the sum of all the vectors is zero for each point. If this is not possible I hope someone would give some hints on how to prove it or point me towards a proof in the literature. In essence what I am seeking is whether there exist three dimensional equilibrium configurations for Coulomb's law of electrostatics as you come across the periodic table of the elements. I suspect not but who was the first to prove it?"}
{"id":"114938","title":"Why are there equations in physics with factors of 2, 3 and 5, but there aren't any with factors of 7 or 11?","text":"I noticed that there are a lot of equations in physics with factors of 2, 3 and 5 (either in the numerator or in the denominator), but there aren't any with factors of 7 or any prime number greater than 7. It's like the first three prime numbers (2, 3 and 5) are \"more important\" than the other prime numbers in physics. What is the reason for this?"}
{"id":"67834","title":"Relationships between different measure of opacity","text":"I'm reading some papers that compare different values for a materials opacity to a particular particle. The first is given as $\\frac{dE}{dX}$, a single particles energy loss per unit column depth (X = $x\\rho $) to a continuous process. Makes sense. So then the author goes on to compare his value to that of other authors, who have their opacities for the process expressed in $\\kappa $ given in $cm^{2}\/g $. They're both presented as measuring the same thing, but I usually think of $ \\kappa $ as being an opacity for a large group of particles that are removed discretely from a beam, so that the total energy in the beam dies of exponentially (since energy loss per distance is proportional to the number of particles left in the beam). $\\frac{dE}{dX}$ is due to a continuous process acting on all the particles, and so would have a roughly linear effect per distance. So my question is how does one compare the two values. What does $ \\kappa $ mean in a context of continuous energy loss. Thanks"}
{"id":"111215","title":"Why is scale factor inversely proprotional to temperature?","text":"In all lectures, books and papers I have read about, the scale factor of the universe is inversely proportional to temperature $$a \\propto \\frac{1}{T}$$ What is the reasoning behind this relation? People did mention stretching of wavelength proportional to scale factor and wavelength being inversely proportional to temperature by Wein's law, but somehow this argument doesn't seem convincing to me. How did we initially come about to this relation?"}
{"id":"92683","title":"Is the center of mass in general relativity equal to the center of mass in newtonian gravity?","text":"Consider 2 point masses $A,B$ a distance $d$ away from eachother without velocity or rotation spin. Is the center of mass in general relativity equal to the center of mass in newtonian gravity ? In newtonian gravity the center of mass is where the particles collide , if they collide. Is the center of mass in general relativity also the place where the particles collide , if they collide ? Or maybe the center of mass in general relativity is the place where the particles collide if they had no initial velocity or rotation spin ?"}
{"id":"107347","title":"Free energy device - QEG. Is this for real?","text":"I'm not a physicist but I ran across reports on the web of a so called over unity or free energy device that has been created that supposedly works. It's called the QEG. Does this seem legit or is it yet another failed attempt to prove that free energy exists? Of course this goes against the 1st law of thermodynamics. I suppose those who believe in such a device would say they aren't creating energy, they are pulling energy in from the ether?"}
{"id":"24505","title":"Electric potential due to a point charge in Gaussian\/CGS units","text":"I learned electrostatics in SI units. In SI, the electrostatic potential due to a point charge $q$ located at $\\textbf{r}$ is given by $\\Phi(\\textbf{r}) = \\frac{q}{4 \\pi \\epsilon_0 |\\textbf{r}|}$. Now, the Griffiths electrodynamics textbook says, \"Converting electrostatic equations from SI to Gaussian units is not difficult: just set $\\epsilon_0 \\rightarrow \\frac{1}{4 \\pi}$.\" So, in Gaussian\/CGS units, apparently $\\Phi(\\textbf{r}) = \\frac{q}{|\\textbf{r}|}$. However, one textbook (Understanding Molecular Simulation, by Frenkel and Smit) says that the potential due to a point charge is $\\Phi(\\textbf{r}) = \\frac{q}{4 \\pi | \\textbf{r} |}$. Did I make a mistake, or did Frenkel and Smit? Thank you."}
{"id":"72109","title":"Electric dipole, error in calculation","text":"Currently I am calculating the dipole moment of a metal sphere in a uniform electric field $E_0$ in z-direction. From here I know that the charge density look at page 15 is given by $ 3 \\epsilon_0 E_0 cos(\\theta)$ Now I wanted to calculate the resulting dipole moment by $ \\int_0^R \\int_0^{2\\pi} \\int_0^{\\pi} r^3 sin(\\theta) cos(\\theta) 3 \\epsilon_0 E_0 d\\phi d\\theta dr$ but in this case the integral over $\\theta$ is zero and therefore this whole term will be zero which is somewhat strange, since there should be a dipole moment. So what am I doing wrong?"}
{"id":"77550","title":"How to understand the Lagrangian of the standard model, effective or \"fundamental\"","text":"I have a question about understanding the Lagrangian of standard model, should we view it as a \"fundamental\" or effective theory? The \"fundamental\" theory here means the theory with physical cutoff (maybe Planck scale) $\\Lambda$; the effective theory here means the theory given by lowering the cutoff $\\Lambda'<\\Lambda$, (Wilsonian renormalization), and we pick up the leading terms as we saw in the action. **The reason to think the standard model Lagrangian is an effective action** We haven't tested the physics up to the physical cutoff (Planck scale?), this is an extra assumption that the standard model is valid up to physical cutoff. To the contrary, in the effective theory, the cutoff is much more reasonable (e.g. LHC scale). **The reason to think the standard model Lagrangian is a \"fundamental\" action** If we follow the renormalization flow, the coupling constant of QCD will grow in low energy scale, then the perturbation breaks down. We know from lattice calculations of hardron mass, that even in low energy, QCD works. The Wilsonian renormalization group transformation is formulated from perturbative expansion (may be this is just my limited knownledge, correct me if I am wrong). Since QCD agree with experiment beyond the regiom of perturbation, it has to be a \"fundamental\" action, than effective action. Similar attitude applies to the standard model. It seems the question is rather a mental setup. Since standard model works very well so far at LHC, it seems hard to distinguish these two opinions: whether viewing standard model action as \"fundamental\" or effective."}
{"id":"28161","title":"Is the Graveyard Really so Serious?","text":"Calculations in relation to black holes are solely in consideration of spacetime curvature and its effects. They are in total alienation with respect to the action of inertial agents[external forces].That world lines do not point outwards from a black hole is consistent with the fact that inertial agents are absent. Can the presence of inertial agents change the picture of the world lines? Let's consider the following situation: _An experimenter [source of \"external forces\" capable of producing non- geodesics] throws some mass into the black hole. Its surface area and volume increase.This is suggestive of outward world lines from the surface, causing the expansion----points on the surface should move outwards.The presence of \"external forces\" has modified the entire perspective of the situation._ The black hole picture takes into consideration only the blackhole. _Right from the outset the black hole is considered in isolation from the rest of the universe._ Any type of modification due to the presence of external agents--- artificial or natural ones do not figure in the blackhole calculations. Consideration of such factors can modify the picture of the black hole. Can technology prevent somebody near a black hole from falling into it? Can it remove the apprehensions of what we know to be a graveyard?"}
{"id":"55176","title":"Deriving infinitesimal time dilation for arbitrary motion from Lorentz transformations","text":"I'm trying to derive the infinitesimal time dilation relation $dt = \\gamma d\\tau$, where $\\tau$ is the proper time, $t$ the coordinate time, and $\\gamma = (1-v(t)^2\/c^2)^{-1\/2}$ the time dependent Lorentz factor. The derivation is trivial if one starts by considering the invariant interval $ds^2$, but it should be possible to obtain the result considering only Lorentz transformations. So, in my approach I am using two different reference frames $(t,x)$ will denote an intertial laboratory frame while $(t',x')$ will be the set of all inertial frames momentarily coinciding with the observed particle, i.e. the rest frame of the particle. These frames are related by $$t' = \\gamma \\left(t-\\frac{Vx}{c}\\right),\\quad x' = \\gamma \\left( x - V t\\right),$$ where $V$ is some nonconstant (i.e. time dependent) parameter which is, hopefully, the velocity of the particle in the laboratory frame. Treating $x$, $t$ and $V$ as independent variables (for now) and taking the differential of the above relations, I obtain $$dt' = \\gamma \\left(dt-\\frac{Vdx}{c}\\right) - \\frac{\\gamma^3}{c^2}(x-Vt)dV,$$ and $$dx' = \\gamma \\left( dx - V dt\\right) - \\gamma^3 \\left(t-\\frac{Vx}{c}\\right) dV.$$ Imposing either the definition of the rest frame $dx'=0$ or (what should be equivalent) $dx = Vdt$, the only way in which i obtain $dt = \\gamma dt'$ is if $dV=0$. So, the derivation breaks badly at some point or I must be wrong in using some of the above equations. Which one is it?"}
{"id":"56481","title":"Proof that flux through a surface is independent of the inner objects' arrangement","text":"$$\\Phi=\\iint_{\\partial V}\\mathbf{g} \\cdot d \\mathbf{A}=-4 \\pi G M$$ Essentially, why is $\\Phi$ independent of the _distribution_ of mass inside the surface $\\partial V$, and the shape of surface $\\partial V$? That is, I'm looking for a mathematical justification for the characterization of flux as a kind of \"flow\" of gravitational field through a surface. The reason I worry about that is this. I have encountered the following proof of Gauss' Law for gravity: $$\\iint_{\\partial V}\\frac{-MG}{r^2}\\mathbf{e_r} \\cdot \\mathbf{e_r}d A=\\frac{-MG}{r^2}\\iint_{\\partial V}d A$$ $$=-MG4 \\pi$$ However, two assumptions are made in the above proof that I've not convinced myself hold true for the general case:   * The Gaussian surface is a sphere.   * It is assumed that all mass is concentrated at the centre of the sphere. Returning to my original question:   * Why is it true that when the Gaussian surface is deformed (not adding or removing any mass to inside the surface, of course), the amount of flux, $\\Phi$, through the surface is equal to the flux in the spherical case?   * Why is it true that the positions of the masses inside the surface doesn't affect $\\Phi$? **Edit:** removed reference to Poisson's equation, changed much of question to make it clearer. **Further edit:** I am happy that $$\\iint_{\\partial V}\\mathbf{F}\\cdot d \\mathbf{A}=\\iiint_{ V}( \\nabla \\cdot \\mathbf{F})dV$$ However, my problem is with the following equation: $$\\nabla \\cdot \\mathbf{g}=-\\rho G 4 \\pi $$ Which can be used to get $$\\iint_{\\partial V}\\mathbf{g}\\cdot d \\mathbf{A}=\\iiint_{ V}(-\\rho G 4 \\pi)dV=-M G 4 \\pi$$"}
{"id":"130458","title":"The motion of the particle satisfies $\\textbf{v} = \\textbf{c}\\times \\textbf{r}$","text":"Why does a particle with position $\\textbf r(t)$ that satisfies $\\textbf{v}(t) = \\textbf{c}\\times \\textbf{r}(t)$ follow a circle that lies in a plane perpendicular to $\\textbf{c}$ with centre on a line through the origin in the direction of $\\textbf{c}$?"}
{"id":"130457","title":"Why does the sky suddenly look gray through this window?","text":"I am looking to the windowed roof of my building, and I notice that the sky, which has few white clouds, sometimes looks completely gray, as if there was a huge cover of gray clouds, but there are not. By observing it and waiting for the change to happen, I notice it takes a matter of seconds : when the sun shines directly on the building (but not directly on me), the blue sky turns gray. I wonder if this is just an optical illusion because my vision of colour could be relative to its environment, or if there is serious glass\/sun\/refraction effect or something. I haven't found anything on that matter on the internet or on Stack Exchange, so I'm asking you what you think about it. Additional information : The top window is a very peculiar glass, it's basically a strong glass window (because it's almost horizontal, it has to resist hits) and looks like it is covered by a layer of opaque black, with big holes (surface of holes is greater than surface of black, from a distance you don't see the holes). Also, I'm taking the pictures from a windowed office, you see the sky through two glass windows, but the one closest to me looks basic and I didn't notice a difference when looking through one layer. You will also notice there is a blue light, some neon-like directed at the roof, but during the day its effect is negligible. Here are the pictures, first the sky through top window without sun hitting the window, then with the sun (The colours\/ exposition differences don't seem to make a great difference compared to my naked eye observation. The pictures were taken only a few seconds apart, you can see the same cloud in both pictures. The gray sky isn't the most grayish I've observed, it gets even farther away from blue sometimes) : ![Sky through top window without sun hitting the window](http:\/\/i.stack.imgur.com\/fbIyO.jpg) ![Sky through top window with sun hitting the window](http:\/\/i.stack.imgur.com\/hBmW9.jpg)"}
{"id":"114066","title":"Is the universe expanding faster than we could catch up with it?","text":"Like the title says, could we ever catch up with the expanding universe?"}
{"id":"32687","title":"Parallel-plate capacitor","text":"I'm trying to grasp parallel-plate capacitors in a class of electromagnetic field theory. My book leaves out a lot and I couldn't figure it out from what I found on Google.   1. The book's procedure is to place a charge Q on the surface of the one conductor plate and -Q on the other. Now would this charge only be distributed over the surfaces of the plates that face each other, or on all four surfaces?   2. Is it possible to prove that any kind of source of potential difference connected to a capacitor can be represented as two opposite charges like that?   3. The book treats one of the charged surfaces as a conductor surface (or this is my understanding) and this yields, with $\\Delta S$ being area, $E_n=Q\/\\epsilon\\Delta S=\\rho\/\\epsilon$ when using Gauss' law (if you place say a thin box around the charged surface because the electric field within the conductor is zero when in equilibrium). I find this way of doing it very odd because first you place Q on the one surface and -Q on the other, and then disregard that you've put -Q on the other surface. And I actually don't understand how come the -Q does not matter. How can this work?   4. A different I think more intuitive way of doing it that I've found online is to treat each conductor as a sheet of charge. Then you have twice the area to spread out the charge over, so you get $E_n=\\rho\/2\\epsilon$ directed inwards from each surface but with different signs, the sum which is $\\rho\/\\epsilon$ in between the plates, just what we want. I like this because it uses the setup with Q and -Q, plus it's easy to see how the fields cancel each other out. But why is it OK to consider a conductor to be a sheet of charge? Is it because the electric field will not be changed by passing through a conductor even though it will temporarily be zero within it? Thank you in advance."}
{"id":"130184","title":"Maximum speed higher than the speed of light","text":"1. I wanted to ask if higher speed than the speed of light will be discovered, will scientists be able to adjust the special relativity to the new situation?   2. I read that informations transmitted faster then the speed of light, but According to special relativity, $c$ is the maximum speed at which all matter and information in the universe can travel. Is there a contradiction here?   3. How the fact that the speed of light is the maximum speed for all matter, affects special relativity?"}
{"id":"130182","title":"Gravitational Length Contraction","text":"General Relativity predicts that a clock at rest in a gravitational field will run slower than a clock in free fall. Similarly, will a vertical ruler on the earth's surface be shorter than a ruler in free fall? Why or why not and by how much?"}
{"id":"130180","title":"Why is it more convenient to consider space or time as a continuum?","text":"I often find that phisicists and cosmologists make use of Planck's units. I have read propositions that sound like > \" _...at the level of Planck's units many law of physics break down_ \" > > \" _...Planck time, the smallest observable unit of time...before which > science is unable do describe the universe_ \" > > \" _... it would become impossible to determine the difference between two > locations less than one Planck length apart_ \" even in string theory: > \"* Planck length is the order of magnitude of the oscillating strings that > form elementary particles, and shorter lengths do not make physical sense*\". Notwithstanding this and the fact that in QM (almost) everything is quantizied (discrete) I read that in mainstream they are still considered not discrete, cf. Phys.SE link. I am not particularly aware of any pros, I see only cons; can someone tell me what are the compelling reasons to consider space and time not discrete? Is it a requisite of relativity? As a corollary, I suppose that they must be both discrete or non discrete, right?"}
{"id":"27361","title":"Readable books on advanced topics","text":"I realise that there are already a few questions looking for general book recommendations, but the motivation and type of book I'm looking for here is a little different, so I hope you can indulge me. I enjoy reading quite a lot, and tend to prefer books that teach me something new, rather than straight up fiction. When I was younger I used to read a lot of popular science books, and at the time I thought these were teaching me something, but then I got older and went to university and studied theoretical physics, and that killed the genre for me. Now, I know that I can learn about a topic simply by seeking out review papers on it, or finding an appropriate textbook. That's what I do when I need to do something for work. However, most of the time this material does not make for light reading, and requires a significant amount of effort to work through, and is not necessarily the kind of thing I'd want to use as a way to relax. However, occasionally one comes across a very readable book on some aspect of physics. Nielsen and Chuang's Quantum Computation and Quantum Information strikes me as an example of this. There are probably better examples of this, but what are they? This brings me to my question: > Beyond the standard undergraduate topics, which areas of physics or > mathematics have books which both provide fairly comprehensive introductions > and which are actually enjoyable to read? To be clear, by \"enjoyable\" I mean something more than that they simply be accessible or well written. I mean that it should be something that I could read for relaxation rather than work. To some people this will probably sound like an insane thing to want to do. However modern physics is huge, and it bothers me that I don't know very much about, say, string theory beyond bosonic string theory, and if it is something I can learn more about in my spare time, then that would be great. * * * UPDATE: There appears to be some confusion in the answers over the type of book I am looking for. I am **not** looking for popularisations. Rather I am looking for something at the level of a graduate text, but only those which are particularly readable."}
{"id":"98651","title":"Speed of sound at temperatures below 0 °C","text":"How can the speed of sound be calculated for temperatures below 0 °C (down to -40 °C)? Does the calculation $v=331\\ \\frac{m}{s} + 0.6 \\frac{m}{s°C} \\times T$ still hold (where T's unit is °C)?"}
{"id":"25290","title":"What is a \"Trojan Moon\"?","text":"Today I read an article about Saturn's moon Helene, and it was described as a \"Trojan Moon\" but no further explanation was given. It was difficult to even get any context about the term from the article. What is a Trojan moon and how did the term \"Trojan\" come about?"}
{"id":"55829","title":"Do glass panes become thicker at the bottom over time?","text":"Is it true that a glass window, that has been placed in a wall for about 10 years or more, is thicker on the bottom than on the top? I can vaguely remember my physics teacher saying that this was true. So if this is true, how is this possible?"}
{"id":"101956","title":"Wave equation in the case of non-monochromatic light","text":"monochromatic light refers to electromagnetic radiation of a single frequency, $\\omega$ The wave equation in this case turns into the well known helmholtz equation.. How about for non-monochromatic light, how can we simplify the wave equation"}
{"id":"57464","title":"Product of exponential of operators","text":"in the context of non-relativistic quantum mechanics I want to show that, for any $A$ and $B$ operators $$e^{A}e^{B}=e^{A+B} $$ if and only if $$[A,B]=0$$ I remember my professor told use about looking for a differential equation but I don't remember the details, and I want to be able to prove it. Brute force the series doesn't seem to be a good idea. Any hint will be appreciated thanks."}
{"id":"132158","title":"Power to suspend a mass in air","text":"How much power is needed to suspend a mass in the air? The four parts below deal with the same problem. I post them all here, should there be some subtle things I didn't see. ## The textbook problem A 9th edition college physics textbook has this as an example problem in its energy chapter (details cleaned-up): An elevator car (\"a lift\") and its passengers has a mass of 1800 kg and negligible friction force in all its moving part. How much power must a motor deliver to lift the elevator car and its passengers at a constant speed of 3.00 m\/s? _Answer_ : Speed is constant, so $a=0$. Now let $F$ be the total force and $T$ be the force exerted by the motor. Then, $$ \\sum F = T-Mg = 0 \\qquad \\Rightarrow \\qquad T = Mg $$ with power = force x velocity, we then have $$ P = T\\cdot v = Mg \\cdot v $$ $$ P = 1800 \\,kg\\cdot9.8\\,m\/{s^2}\\cdot 3.00\\,m\/s=5.29\\times10^4\\,{\\textrm W} $$ And, what power must the motor deliver at the instant the speed of the elevator is $v$ if the motor is designed to provide the elevator car with an upward acceleration of 1.00 m\/s per second? _Answer_ : Now $a=1.00 \\,m\/{s^2}$, and then $$ \\sum F = T-Mg = Ma \\qquad \\Rightarrow \\qquad T = M(a+g). $$ Proceeding as before, we have $$ P = T\\cdot v = M(a+g) \\cdot v $$ $$ P = 1800 \\,kg\\cdot (1.00+9.8)\\,m\/{s^2}\\cdot 3.00\\,m\/s= 5.83 \\times10^4\\,{\\textrm W}. $$ That's the end of the example. **My questions**. We might as well ask these:   1. How much power must the motor deliver to suspend the elevator and its passengers so it's kept at the constant height? That is, zero velocity? Certainly it can't be zero right? Because otherwise the elevator car will free-fall. But how much power is required?   2. What power must the motor deliver at the instant the speed of the elevator is $v=3.00$ (as before) if the motor is designed to provide the elevator car with a _downward_ acceleration of 1.00 m\/s per second? I've been looking for an answer but to no avail. ## The helicopter problem This is _the_ real question that motivates this post. I've been asked by a friend who's going to build a model-helicopter. \"Ok, I have a 2000 W engine for my 50 kg model-helicopter. Assuming 100% efficiency, will it able to float 1 m above ground for at least 1 minute?\" I can't answer that. \"Ok then, what is minimum power required for the engine if I want it to float 1 m above ground, assuming 100% efficiency?\" I'll ask physics.se ## The pulley problem This is a simpler version of the above two problems. Using a rope and a pulley, what is the minimum power required for a man to suspend a 1 kg mass in the air 1 m above ground for 1 second? ## The table or string problem There's not much problem here. Suspending a 1 kg mass 1 m above the ground? Simple. Put it on a table, or hang it with a string. No movement. No power required. What's happening here?"}
{"id":"105253","title":"Immediately after the Big Bang, was the universe in a state of extremely low or extremely high entropy?","text":"Phase space theory suggests that the largest course-graining region, $p$, in a phase space, $P$, is the point in the phase space with the highest entropy. As such, it is in thermal equilibrium with the course-graining regions surrounding it. Now, when an intensity vs frequency graph is obtained for the CMB (Cosmic Microwave Background), a curve almost identical to Planck's curve representing black-body radiation is seen. The temperature of the CMB is observed to be almost uniform across the universe, and the curve of black-body radiation represents a point of thermal equilibrium for a system. Since the CMB has the same curve as black-body radiation, this would suggest that at the birth of the cosmic microwave background, the universe was in a state of thermal equilibrium. Referring back to the phase space spoken of in the first paragraph, since the universe was in a state of thermal equilibrium, it was at a point of highest entropy. However, surely due to the Second Law of Thermodynamics, the entropy of the universe would need to decrease - so it wouldn't be possible for the entropy of the universe to be in such a high state so soon after its birth - so it should surely begin in a low state of entropy. So what's the deal here? Did the universe begin in a really low state of entropy, or a really high one? Are all my explanations here logical or quite inaccurate?"}
{"id":"26011","title":"Has everything we see happened in the past?","text":"Whatever we see is basically based on the light that hits our eyes, right? When we look at the Moon we are looking at the Moon as it was couple of seconds ago, as the light takes some seconds (~2 seconds I believe) to reach us. In a way aren't we ALWAYS seeing the past as, no matter how close we are to something, we can never be at distance 0. Am I wrong in thinking about it like that?"}
{"id":"3505","title":"Free electrons and energy states","text":"Ok, background - studying for the physics part of my radiology exams, and came across a question that went something like this An electron fired through a tungsten target loses energy by: a) Bremsstrauhlung b) collisions with bound target electrons c) giving off characteristic radiation d) all of the above Now, by the process of elimination d must be right, because both a and b are My question regards c - characteristic radiation My understanding of this process is the excitation of the electron to a higher energy state, usually a more peripheral valence shell, which then releases that extra energy when it drops back (or another drops in to fill the hole) So, what I don't get is how a free electron (like one fired from the filament to the target) has energy states. By definition, it is unbound, and therefore there are no binding energies to create quantised energy states. Is there a property I am overlooking, is the question wrong, or is it a fudge? I apologise if this has been asked. I looked at the suggested answers, and did a google search of stack exchange, but my phone can't use the search bar here directly, so I could have missed something"}
{"id":"14639","title":"How is the Saddle point approximation used in physics?","text":"I am trying to understand the saddle point approximation and apply it to a problem I have but the treatments I have seen online are all very mathematical and are not giving me a good qualitative description of the method and why it's used and for what it's used. So my question is, how is the saddle point approximation used in physics? What is it approximating? When can it be used?"}
{"id":"18595","title":"Is decoherence even possible in anti de Sitter space?","text":"Is decoherence even possible in anti de Sitter space? The spatial conformal boundary acts as a repulsive wall, thus turning anti de Sitter space into an eternally closed quantum system. Superpositions remain superpositions and can never decohere. A closed system eventually thermalizes and undergoes Poincare recurrences. Whatever the dual is to an interacting conformal field theory, is it really a theory of quantum gravity, or does it only look like one superficially?"}
{"id":"90600","title":"How do voltage and voltage drops over a circuit relate to work done?","text":"I have a few remaining uncertainties when it comes to voltage. I numbered the paragraphs in my thought sequence for easier reference.   1. In a series circuit, I know voltage is the electric potential, and represents the potential for work, but that doesnt mean that any electron travelling on any 10 volt circuit does the same amount of work from start to finish, does it?   2. What i mean is, let's say you have a 10 volt series circuit with a 1 ohm wire. Tracking an electron in the current from start to finish, as work is done on the electron by the voltage, its potential drops, until it reaches 0 at the end of the circuit.   3. But if we increase the resistance of the wire to 5 ohms, it would take 5x longer for the electron to get to the end of the circuit, because the current would be 1\/5th, right?   4. But then that means that the electron in the 5 ohm circuit would have done 5x the amount of work (or work done on it) of the 1 ohm circuit over 5x the duration.   5. I know the work done per second is the same, but ultimately, if each electron in the 5 ohm circuit has more work done on it over the course of the circuit, then voltage cant be an expression of total work done from p1 (start of circuit) to p2 (end), right? I think i first thought that because voltage relates to potential for work, and the voltage always drops from full to 0 by the end of the circuit, so i assumed the work must be the same regardless of the circuit resistance.   6. So does this mean that when voltage is divided over the components of the circuit, it's really representing the ratio of where in the circuit the work is done?   7. i.e. in a 12v series circuit with a 3 ohm resistor and a 1 ohm resistor, there would be a voltage drop of 9v over the 3ohm resistor, and 3 volts over the 1 ohm, right? which would mean that 75% of total work done in the circuit was at the 3 ohm resistor?   8. But why is it that the more resistors you have in a series, the less the voltage drops over any one resistor? I know with more resistors the current would be lower. Does voltage drop over a resistor decrease as current decreases?"}
{"id":"30463","title":"Darcy law yields extreme speed for gas flow throgh packed spheres?","text":"The darcy law is as follows: $u=-\\frac{k}{\\mu}\\nabla p$. Assume we have a gas, then $\\mu$ is about $10^{-5}$. $k$ for packed spheres a few mm in diameter is of order $10^{-8}$ $m^2$. Say the pressure difference across a 0.1 m bed is 10 Pa. Then we get $u= 0.1$ m\/s which seems really high for such a small pressure difference. I mean, tens of Pascals is a variation which is comaparable to random pressure fluctations. Is this reasonable?"}
{"id":"76637","title":"EM vector potential","text":"We can write the electromagnetic field tensor as $$\\begin{bmatrix} 0 & -E_x\/c & -E_y\/c & -E_z\/c \\\\\\ E_x\/c & 0 & -B_z & B_y \\\\\\ E_y\/c & B_z & 0 & -B_x \\\\\\ E_z\/c & -B_y & B_x & 0 \\end{bmatrix} = F^{\\mu\\nu}.$$ Erick J. Weinberg, _Classical Solutions in Quantum Field Theory: Solitons and Instantons in High Energy Physics_ (p. 43), states: > For a static solution with vanishing electric fields $F_{0j}$, and hence > $A_0 = 0$, [...] How can this be proven?"}
{"id":"38654","title":"One-photon-at-a-time effect on double slit experiment with detectors?","text":"I do not have a access to a physics lab, nor have I come across this type of a double slit experiment, hence I am curious if someone has tried this. Given a double slit experiment setup with a detector. What happens if we have a detector at one of the slits, but we do not send one photon at a time (in other words, we send a bunch of photons without concerning ourselves about the count)? My assumption is that we will see a particle (bullet) pattern like we would if we sent one photon at a time, is this correct?"}
{"id":"83480","title":"Apparent size of objects under water","text":"A friend of mine has a homework question and we're having some trouble figuring out what physical mechanisms come into play for this. > An underwater swimmer sees a spherical air bubble that appears to have a > diameter $d=$ 1.5 cm. What is its actual diameter? We are having trouble, specifically, thinking of physical mechanisms that would change the apparent sizes of objects when seen underwater. Is it caused by refraction at the curved surface of the bubble?"}
{"id":"26849","title":"Values of SM parameters at one certain scale","text":"The general question is: What are the values of Standard Model parameters (in the $\\bar{MS}$ renormalization scheme) at some scale e.g. $m_{Z}$? As its parametrization in Yukawa matrices is not unique - what are the values of gauge couplings, fermion masses and CKM matrix? The background: I want to solve renormalization group equations of MSSM and in order to have initial conditions for them I need to know SM parameters at one scale - not at few different, which one can find at Particle Data Group webpage."}
{"id":"82130","title":"Acceleration of 2 bodies tied with a string","text":"Find the acceleration of the block of mass M shown in the figure . The co- efficient of friction between the 2 blocks is μ1 and that between the bigger block and ground is μ2. ![enter image description here](http:\/\/i.stack.imgur.com\/yzma2.png) Could someone help solve this problem using Lagrangian mechanics. I know the solution to this problem using Newtonian mechanics.However I am not finding the same answer when I am trying to do it using Lagrangian mechanics. Answer - $a=\\frac{[2m-\\mu_2(M+m)]g}{M+m(5-2(\\mu_1-\\mu_2))}$"}
{"id":"728","title":"Will a ball slide down a lumpy hill over the same path it rolls down the hill?","text":"Suppose I have a lumpy hill. In a first experiment, the hill is frictionless and I let a ball slide down, starting from rest. I watch the path it takes (the time-independent trail it follows). In the next experiment, the hill stays the same shape, but the ball now rolls without slipping down the hill. Assume there is no deformation of the ball or of the hill, no micro-sliding of the contact surface, and no other forms of rolling resistance. Energy is conserved. I release the ball from rest at same point on the hill. I track the path the rolling ball takes. Does the rolling ball follow the same path as the sliding ball, but more slowly, or does it sometimes follow a different path? Note: There may be some \"trick\" answers. For example, if the hill curves significantly on scales similar to the radius of the ball, the ball could get \"wedged in\" somewhere, or two parts of the ball could contact the hill at the same time. Let's assume that geometrically, the shape of the hill is such that it is only possible to have the ball contact the hill at one point, and that the ball always contacts the hill at a single point (i.e. it never flies off)."}
{"id":"15276","title":"Question about the parity of the ghost number operator in BRST quantization","text":"Given a Lie algebra $[K_i,K_j]=f_{ij}^k K_k$, and ghost fields satisfying the anticommutation relations $\\\\{c^i,b_j\\\\}=\\delta_j^i$, the ghost number operator is then $U=c^ib_i$ (duplicate indices are summed). The question raised in proving the BRST operator raising the ghost number by 1, given in the Example 6.1 on the page 116-118 of the book, String theory demystified. http:\/\/books.google.com\/books?id=S4JyPgw4ZlAC&lpg=PP1&pg=PA117#v=onepage&q&f=false On the second and the third lines of the formula derivation of $UC^iK_i$ $UC^iK_i=...=c^iK_i-c^i\\displaystyle\\sum_rc^rb_rK_i=c^iK_i+c^iK_i\\displaystyle\\sum_rc^rb_r=...$ where we need $c^r U K_r =-c^r K_r U$ The change of sign above is not manifestly obvious to me. \\---add--- Never mind. I find the second line cheated me. It should be $UC^iK_i=...=c^iK_i+c^i\\displaystyle\\sum_rc^rb_rK_i=c^iK_i+c^iK_i\\displaystyle\\sum_rc^rb_r=...$ Furthermore, in the proof followed in this example, there a few typos need to be corrected. But the final result is right. The BRST operator raises the ghost number by 1!"}
{"id":"35948","title":"degree of freedom of a rigid body 5 or 6?","text":"I'm confused here. I have a three particle (rigid) system. What would be the degree of freedom? I found out five. 3 coordinates for center of mass and 2 for describing orientation.   But we have only three constraints, i.e. three equations that reduce 9 coordinates by 3, 9 - 3 = 6, which gives 6 degrees of freedom?? Did I miss something above?"}
{"id":"35942","title":"Information relationship to Special Relativity","text":"How do we write mathematically that \"information\" cannot go faster than light? And along a similar line of thought, how do we relate \"information\" with special relativity. Lastly, what is the relationship between Special Relativity and the fact that the phase velocity of a wave packet can go faster than light (light speed here being the group velocity). Is there a reason we cannot consider the frame of reference of a specific phase in a wave packet?"}
{"id":"103943","title":"The average value of the the square of Dirac velocity operator","text":"Let's have Dirac velocity operator (the case of the free particle: $$ \\hat {\\mathbf v} = i [\\hat {H}, \\hat {\\mathbf r}] = \\hat {\\alpha}, \\quad \\hat {H} = (\\hat {\\alpha} \\cdot \\hat {\\mathbf p}) + \\hat {\\beta} m, $$ where $\\hat {\\alpha} $ is $\\hat {\\gamma}_{0}\\hat {\\gamma}$. I need to find $\\sqrt{\\langle (\\Delta \\hat {\\mathbf v})^{2}\\rangle} = \\sqrt{\\langle \\hat {\\mathbf v}^{2}\\rangle - \\langle \\hat {\\mathbf v}\\rangle^{2}}$. The second summand is equal to $\\frac{\\mathbf p^{2}}{p_{0}^{2}}$, and the first (maybe) $$ \\langle \\hat {\\mathbf v}^{2}\\rangle = \\langle \\hat {\\alpha}^{2} \\rangle = 3 \\langle \\rangle = 3. $$ So $$ \\sqrt{\\langle (\\Delta \\hat {\\mathbf v})^{2}\\rangle} = \\sqrt{3 - \\frac{\\mathbf p^{2}}{p_{0}^{2}}}. $$ But it seems that this result is incorrect, because it is unnatural. Can anyone help me? What is the correct answer?"}
{"id":"131445","title":"Color of cementite","text":"I have a question about micrograph of steel. In eutectoid steel, the pearlite exists as grains, within each colony the layers are oriented in essentially the same direction, which varies from one colony to another. The thick light layers are the ferrite phase, and the cementite phase appears as thin lamellae most of which appear dark. However in photomicrographs of a hypereutectoid steel (e.g.1.4 wt% C) , we can observe the proeutectoid cementite appears light. Why there is such a difference?"}
{"id":"69399","title":"Pressure derivatives of the bulk modulus of materials? (Solved)","text":"Isotherm bulk modulus is defined by $$K=\\rho\\left(\\frac{\\partial p}{\\partial\\rho}\\right)_T=-v\\left(\\frac{\\partial p}{\\partial v}\\right)_T$$ wherein $v=\\tfrac{1}{\\rho}$ is the specific volume. Studying around the Birch- Murnaghan equation of state and some other related topics I saw everyone talk about the pressure derivative of the isotherm bulk modulus very happily as if it is a function of e.g. pressure, like any other thermodynamic quantity usually is, so that one can calculate its nontrivial pressure derivative for it, even a nontrivial isotherm pressure derivative $\\left(\\tfrac{\\partial}{\\partial p}\\right)_T$ of it. See for example this link that introduces the quantity $$K'=\\left(\\frac{\\partial K}{\\partial p}\\right)_T$$ (the reference cited uses $B$ instead of $K$.) But to me its very odd. Let me compute the isotherm pressure derivative of $K$ so you understand why I'm confused: \\begin{align} K'&=\\left(\\frac{\\partial K}{\\partial p}\\right)_T\\\\\\ &=\\left(\\frac{\\partial}{\\partial p}\\right)_T \\biggl[\\rho\\left(\\frac{\\partial p}{\\partial\\rho}\\right)_T\\biggr]\\\\\\ &=\\left(\\frac{\\partial\\rho}{\\partial p}\\right)_T\\left(\\frac{\\partial p}{\\partial\\rho}\\right)_T+\\rho\\left(\\frac{\\partial^2 p}{\\partial\\rho\\partial p}\\right)_T\\\\\\ &=1+0\\\\\\ &=1 \\end{align} A constant, not a function of pressure at all. Oh I don't mean that the Bulk modulus is not an ordinary thermodynamic quantity, but its isotherm pressure derivative can not be function of pressure as the above calculations suggest, yet isentropic pressure dependence could be non-trivial and etc. No reference could I have found to have computed this derivative, everyone who I encountered have only reported to have measured the quantity experimentally. That everyone say it is a function of pressure and experiments validate their idea makes me think I have missed a point in my computations. But there is a greater point that suggests i am wrong, that using the specific-volume-based form of the bulk modulus when is differentiated will result in $-1$ instead of $1$: \\begin{align} K'&=\\left(\\frac{\\partial K}{\\partial p}\\right)_T\\\\\\ &=\\left(\\frac{\\partial}{\\partial p}\\right)_T \\biggl[-v\\left(\\frac{\\partial p}{\\partial v}\\right)_T\\biggr]\\\\\\ &=-\\left(\\frac{\\partial v}{\\partial p}\\right)_T\\left(\\frac{\\partial p}{\\partial v}\\right)_T-v\\left(\\frac{\\partial^2 p}{\\partial v\\partial p}\\right)_T\\\\\\ &=-1-0\\\\\\ &=-1 \\end{align} So that the contradiction $1=-1$ suggests that I must be wrong, but where is the point I am missing? Aren't the derivatives interchangeable in computing $\\left(\\frac{\\partial}{\\partial p}\\right)_T \\left(\\frac{\\partial p}{\\partial\\rho}\\right)_T$ ( _I understand the Thermodynamic's underlying geometry can be non-euclidean in different cases_ ) or else? So the simple question is this: where have I make a mistake in the above calculations? Regards. _**UPDATE_. Wrong calculation led into an apparent contradiction, but as could be guessed it was wrong so no contradiction at all. It was a math question which is now resolved, so maybe better to be deleted from ere as it had no physics questioned behind it.**"}
{"id":"13545","title":"Heuristic argument for the temeprature dependence of specific heat in the \"low\" temperature regimes","text":"Here by \"low temperature\" I meant it in the scale of the characteristic $\\hbar \\omega$ of the system. One can calculate and show that in the low temperature regime $C_V$ of phonons goes like $T^3$ for free fermions it goes as $T$ and for free bosons its like $T^{\\frac{3}{2}}$ In most of the above cases the power comes from some complicated calculation.   * I was wondering if there is a heuristic argument for \"understanding\" these powers.    * Further is the power of $T$ characteristic of the system? Like if in a substance one sees the low temperature behaviour of $C_V$ to be going as $T$ then does it says something like that the effective degree of freedom of the system is that of free fermions?   * Are there other examples of such calculation? Like typical systems where one can calculate the low temperature dependance of $C_V$?"}
{"id":"90530","title":"Too small error on the calculus of wavelenght","text":"I have this function: $$\\lambda=d \\sin(\\arctan(\\frac{x}{z}))$$ and I want to find its absolute error. $d$ is a constant ($10^{-6}$), $x =(0.716 \\pm 0.001)$ m, and $z=(1.000 \\pm 0.001) $ m. For the error of $\\lambda$ I have used $$\\Delta\\lambda=\\sqrt{\\left(\\frac{\\partial\\lambda}{\\partial x}\\Delta x\\right)^2 + \\left(\\frac{\\partial\\lambda}{\\partial z}\\Delta z\\right)^2}\\\\\\ \\Delta\\lambda=d\\sqrt{\\left(\\frac{z\\Delta x}{(z^2+x^2)(\\sqrt{\\frac{x^2}{z^2}+1})}\\right)^2 + \\left(\\frac{x\\Delta z}{(z^2+x^2)(\\sqrt{\\frac{x^2}{z^2}+1})}\\right)^2}$$ And I have obtained $6.6 \\cdot 10^{-10}$ m. I expect a much larger error. Is this wrong?"}
{"id":"7889","title":"conformal, Weyl transformations, apparent discrepancies and confusions","text":"Because of the apparent discrepancy of how some CFT and GR books define conformal transformation unlike in string theory area, I wanted to get rid of all the confusion from McGreevy's lecture notes: > http:\/\/ocw.mit.edu\/courses\/physics\/8-821-string-theory-fall-2008\/lecture- > notes\/lecture10.pdf I have understood and happy with the definition of conformal group and conformal transformation now, but I have the following two questions:   1. In page 2, where he defined conformal invariance, did he mean invariance under the diffeomorphism part only (which gives the conformal group)? The reason I think so is that he changed the metric which according to his definition of conformal transformation don't change. So, the reason I am asking is that I thought conformal invariance will be a suitable name for a theory, invariant under conformal transformation.   2. Just to make sure: the transformation in page 2: $\\delta g_{\\mu\\nu}=2\\lambda g_{\\mu\\nu}$ and $\\delta g_{\\mu\\nu}=\\delta\\Omega(x)\\delta g_{\\mu\\nu}$, these are results of diffeomorphism ..right? and not Weyl in any way. In other words, what are these transformations? All my questions are for general dimension and general space time (classically of course)."}
{"id":"101540","title":"Help in understanding the derivation for Fresnel Distance?","text":"An Aperture of size $a$ illuminated by a parallel beam sends diffracted beam (the central maxima) in angular width approximately $λ\/a$. Travelling a distance $z$, it aquires the width $zλ\/a$ due to diffraction. The distance at which this width equals the size of the aperture is called the **fresnel distance**. But as far as I know, the angular width of central maxima is given by $2λ\/a$ and not $λ\/a$. Help?"}
{"id":"20745","title":"Why electrons are relativistic in Graphene and non relativistic in vacuum?","text":"If a free region in space has a potential difference of one volt, an electron in this region will acquire kinetic energy of 1 eV. Its speed will be much smaller than the speed of light hence it will be a non relativistic electron. On the other hand conduction electrons in graphene are relativistic for the same potential difference. Question is how come that when the electrons are in vacuum they are non relativistic, and when they are inside Graphene they are relativistic (for the same potential difference)?"}
{"id":"43624","title":"Orbital momentum and spin transfer through gauge mediator","text":"I study the jellium model: conduction electrons free except for their mutual repulsion. We have the following vertex: $$\\hat{V}_{ee}~=~\\sum_{k,k',q}V(q)a^{\\dagger}_{k-q,\\sigma}a^{\\dagger}_{k'+q,\\sigma'}a_{k',\\sigma'}a_{k,\\sigma}. $$ One sees that the electron spins z-projections are unchanged after the interaction $(\\sigma,\\sigma')\\rightarrow(\\sigma,\\sigma')$ (consequence of the fact that the potential is scalar I guess) so or the mediating gauge boson (photon) isn't carrying any orbital momentum or helicity along the $z$-axis which is unphysical but maybe acceptable for a virtual particle, or there is on orbital momentum conservation at the vertex points. This point is a bit messy in my head, I'd appreciate if someone could share his interpretation or motivate the conservation of each electron's spin state."}
{"id":"43625","title":"Spin 3\/2 Statistical Mechanics Problem","text":"I am trying to solve a problem from the book 'Introductory Statistical Mechanics' (Bowley, Sanchez). The question reads: Calculate the free energy of a system of N particles, each with spin 3\/2 with one particle per site, given that the levels associated with the four spin states have energies e, 2e, -e, -2e.... What I want to know is how I use the face that each particle has a spin 3\/2? Does this add some kind of degeneracy I need to take into account?"}
{"id":"46647","title":"How did pre-Copernican astronomers accurately predict planetary position?","text":"Copernican elements (circular orbital elements) are not very accurate. But Copernicus simplified our understanding a great deal by placing the Sun at the center of the system. Im astonished by the historical accounts that, say, Mayan or Babylonians, or whoever may have had the ability to predict planetary position an accuracy that competes even with Kepler. Now, Im only curious about the reasoning and the mathematics behind making such predictions from pre-Copernican understanding. What approach did these ancient people take? I cant seem to find any online reference to this approach."}
{"id":"134164","title":"Hoop rolling inside a circular hole","text":"A hoop of radius $b$ and mass $m$ rolls without slipping within a stationary circular hole of radius $a > b$ and is subject to gravity. Use the generalized coordinates the rotation angle $\\phi$ of the hoop and the angular position of the hoop’s center $\\theta$. We have the rolling without slipping constraint $$b\\phi - a\\theta=0.$$ The Lagrangian of the system is $$L=\\frac{1}{2}m(a-b)^2\\dot{\\theta}^2+\\frac{1}{2}mb^2\\dot{\\phi}^2+mg(a-b)\\cos\\theta.$$ The Euler-Lagrange equations with Lagrange multiplier are $$m(a-b)^2\\ddot{\\theta}+mg(a-b)\\sin\\theta=\\lambda a, mb^2\\ddot{\\phi}=-\\lambda b$$ Solving for an equation of motion of $\\theta$, we have $$(2a^2-2ab+b^2)\\ddot{\\theta}+g(a-b)\\sin\\theta=0.$$ My questions are   1. how to find the generalized constraint force that makes the hoop roll without slipping?   2. how to find the constraint force that keeps the hoop’s CM moving on a circular path? ![enter image description here](http:\/\/i.stack.imgur.com\/JnRCG.jpg)"}
{"id":"10387","title":"Historical and philosophical reflexions about the concept of energy?","text":"Perhaps you know the books: Concepts of Mass in Contemporary Physics and Philosophy Concepts of Force: A Study in the Foundations of Dynamics by Max Jammer, which discusses mass and force from a perspective which include much historical as well as epistemological, conceptual and axiomatic elements. Are there any good books which discuss the energy concept in a similar way as Max Jammer discussed for example mass and force?"}
{"id":"108467","title":"Reason for the experiment","text":"Why is the independent confirmation by separate experiments such as important part of the development of the idea of the existence of electron wave?"}
{"id":"54422","title":"Does zero strain always imply zero stress?","text":"In solid mechanics, can I always assume that if an object undergoes no strain, then no stress is applied to it? I think it's true only because I can't seem to find a counter-example."}
{"id":"58076","title":"Inner Product Spaces","text":"I am trying to reconcile the definition of Inner Product Spaces that I encountered in Mathematics with the one I recently came across in Physics. In particular, if $(,)$ denotes an inner product in the vector space $V$ over $F$:   1. $(u + v, w) = (u, w) + (v, w) \\text{ for all } u, v, w \\in V$,   2. $(\\alpha v, w) = \\alpha(v, w) \\text{ for all } v, w \\in V$ and $\\alpha \\in F$,   3. $(v, w) = (w, v)^* \\text{ for all } v, w \\in V$, (* denotes complex conjugation) were some of the properties listed in my mathematics course. In physics, however, the inner product was said to be linear in the second argument and $(v,\\sum(\\lambda_i w_i)) = \\sum(\\lambda_i (v,w_i))$ where $v$ and $w_i$ are kets in Hilbert space and $\\lambda_i$ are complex numbers. To me, these properties of an Inner Product are not compatible. If the first definition of inner product is correct, then I think $(v,\\sum(\\lambda_i w_i)) = \\sum((\\lambda_i)^* (v,w_i))$ where $^*$ denotes complex conjugation."}
{"id":"41160","title":"Jumping on earth versus jumping on the moon","text":"Given the following problem: On the moon the acceleration due to gravity is $g_m = 1.62 m\/s^2$. On earth, a person of mass $m = 80 kg$ manages to jump $1.4 m$. Find the height this person will reach when jumping on the moon, if the person is wearing a spacesuit with mass $m = 124 kg$. I am a little bit confused as to whether or not the given information regarding mass is actually needed here at all. Assume that $v_0$ is equal in both jumps, and there is no rotational movement, can't we just use the formula for conservation of mechanical energy? $$\\frac{1}{2}m v_{f}^2 + mgh_f = \\frac{1}{2}m v_{0}^2 + mgh_0$$ And here we can cancel the mass, $m$ since it appears in all terms. So on earth we will have: $$gh_f = \\frac{1}{2} v_{0}^2$$ $$9.8 \\cdot 1.4 = \\frac{1}{2} v_{0}^2$$ $$v_0 = 5.2 m\/s$$ Then on the moon, since we know $v_0$, we can then find $h_f$: $$1.62 h_f = \\frac{1}{2} \\cdot (5.2)^2$$ $$h_f = 8.3 m$$ Would this not be an acceptable way to solve this? If this is wrong, can anyone please explain why this is wrong conceptually?"}
{"id":"32528","title":"Inefficiency Comparison of Car Air Conditioning vs. Open Windows","text":"On a recent long, hot journey in Spain, I was pondering which was the most efficient way of cooling the car. Which of these would be the most effective?   1. Switching on the air conditioning, thereby consuming more fuel   2. Winding down the (both driver and passenger) windows, thus disrupting the airflow around the car and ostensibly creating extra drag Let's assume I'm travelling at 100km\/h on level road, in a 1.2 litre Fiat 500. (It's a valid assumption, because it was true). Let's also assume that both methods cool me down equally as well. For what's worth, I went with option 2."}
{"id":"41165","title":"Schwarzschild radius in matrix models","text":"The Schwarzschild radius for 11D BHs is given by $l_{11}(l_{11}m)^{1\/8}$, which is the special case ($D=11$) of the general dimensional case of $(G_Dm)^{\\frac{1}{D-3}}$. Here $m$ is the BH mass and $G_D$ is $D$ dimensional Newton's constant. Now if M-theory is compactified on a torus $T^p$ of sizes $L$s, then I think $G_D$ is related to eleven dimensional Planck length $l_{11}$ as $G_D=\\frac{l_{11}^9}{L^p}$. I don't understand how can then the radius smoothly go to the 11 dimensional value if we start decompactifying the torus to get to the 11D case. It seems that it blows up. Where's my mistake? Thanks a lot."}
{"id":"41167","title":"How to prove Gegenbauer's addition theorem?","text":"I asked this at: http:\/\/math.stackexchange.com\/questions\/210153\/, but didn't get any reply, so I am trying here, since I actually need this in physics anyway. How can one prove the following identity: $$ V_k(r_1, r_2) = {2k+1\\over 2 r_1 r_2}\\int_{|r_1 - r_2|}^{r_1+r_2} e^{-{r\\over D}} P_k\\left(r_1^2 - r^2 + r_2^2 \\over 2 r_1 r_2 \\right) d r= $$ $$ =(2k+1) {I_{k+{1\\over2}}\\left(r_<\\over D\\right) \\over \\sqrt{r_<}} {K_{k+{1\\over2}}\\left(r_>\\over D\\right) \\over \\sqrt{r_>}} $$ ? Here $r_<=\\min(r_1, r_2)$, $r_>=\\max(r_1, r_2)$, $I_\\nu$ and $K_\\nu$ are the modified Bessel functions of the first and second kind, $P_k(x)$ are Legendre polynomials and $k=0, 1, 2, 3, \\dots$. I verified the identity numerically, so I know that it works, but I didn't figure out how one could prove it. I know it has something to do with Gegenbauer's addition theorem and the equation 10.23.8 at [1]. The $V_k(r_1, r_2)$ is just the coefficient in the Legendre expansion of: $$ {e^{{|{\\bf r}_1 - {\\bf r}_2|}\\over D}\\over |{\\bf r}_1 - {\\bf r}_2|} =\\sum_{k=0}^\\infty V_k(r_1, r_2) P_k(\\cos\\theta) $$ Note: when $D\\to\\infty$, then $V_k$ becomes ${r_<^k \\over r_>^{k+1}}$, which follows from the properties of the Bessel functions and this is just the well known multipole expansion [2]. [1] http:\/\/dlmf.nist.gov\/10.23#E8 [2] http:\/\/en.wikipedia.org\/wiki\/Legendre_polynomials#Applications_of_Legendre_polynomials_in_physics"}
{"id":"91425","title":"Did the Goudsmit-Uhlenbeck analysis of spin consider relativity?","text":"It's frequently mentioned in introductory quantum mechanics texts that Goudsmit and Uhlenbeck conjectured that the magnetic moment of an electron was due to angular momentum arising from the electron rotating around its own axis. But then when they tried to calculate how fast it would have to be spinning, assuming that the electron is a rigid sphere with radius equal to the classical electron radius, they found that a point on the equator would be moving with a speed greater than the speed of light, so they were embarrassed for publishing their work. My question is, did they do this calculation using Newtonian mechanics or special relativity? If we do take relativity into account, and consider a (Born-) rigid sphere with radius equal to the classical electron radius, and then we tried to find out what speed the sphere would need to rotate at in order to have an angular momentum that produces the magnetic moment of an electron, would we still get a speed faster than light? Momentum goes to infinity as speed approaches c, but what happens to angular momentum? I'm aware that angular momentum becomes really complicated in special relativity, with tensors and bivectors and the like, but is there a simple (or even approximate) expression that can give us some idea of what would happen in this case? This is of course just a curiosity, because there are other problems with the classical theory of spin, like the fact that a rotation of 720 degrees is required (for an electron) rather than a rotation of 360 to get you back to your initial state, due to the double cover property of SU(2). Any help would be greatly appreciated. Thank You in Advance."}
{"id":"129051","title":"Stability of D-Branes and Coupling to Fields","text":"I've been brushing up on my string theory using Becker, Becker, Schwarz (p. 208 in particular). They state that some D-branes can couple to R-R fields from closed string excitations. They then go on to say that these branes are exactly 1\/2 BPS and therefore stable. I understand that 1\/2 BPS objects are stable because they preserve enough of the supersymmetry. But I don't understand why the coupling to R-R fields ensures that the branes are 1\/2 BPS. Could someone possibly explain this point mathematically, and\/or direct me to a reference that makes it clear? Many thanks for your help! **Edit** : perhaps the answer is contained within Polchinski's seminal paper here? Still might be useful to get a second opinion!"}
{"id":"129055","title":"What are the conditions around Sgr A* for a space ship trying to plunge into the central black hole?","text":"The possible problems for a space ship near an active galactic nucleus are the following: tidal forces, hot plasma accreting into the black hole and \"jetting\" out, radiation, and magnetic fields. If I am missing something, please feel welcome to tell me. * * * **Tidal forces** Suppose we have a ship which has dimensions $< 1\\, \\rm km$, then tidal forces would surely not pose a problem as this length is much smaller than the Schwarzschild radius of Sgr A* which is presumed to be $r_s \\approx 10^7 \\rm km$. (You can also verify the negligibility explicitly through the equation of geodesic deviation in a Kerr metric.) * * * **Plasma** With plasma, I am not entirely sure. The accretion disc has many various models including \"slim\" and \"thin\" versions, which have a negligible extent beyond the equatorial plane, but also \"thick\" ones. Nevertheless, even with the \"thick\" Polish donut models, we see that the main density is concentrated around the equatorial plane (see image below) and there might be pathways passing through reasonably low densities of plasma. On the other hand, there surely is a means of transport of the plasma towards the axis to create jets. So for a perfectly axially symmetric situation, the horizon should be \"screened\" by a considerable flow of plasma from the disc towards the jets. On the \"other other\" hand, the typical length-scale of the accretion disc-hole system are proportional to the Schwarzschild radius $ \\sim 10^7\\rm km$ and this is also the typical length of the accuracy of the models we should expect. For the spaceship to safely pass, we would need an underdensity say of the lenght-scale $\\sim100 \\, \\rm km$ i.e. of order $10^{-5}$ with comparison to the length-scale of the system. Even though a mere \"bubble\" in a plasma would surely not guide the space ship all the way towards the black hole, it provides a possibility to cross thin regions where the model predicts the plasma to be denser. ![thick disk simulation](http:\/\/i.stack.imgur.com\/ybDqo.jpg) * * * **Radiation** We know Sgr A* to be a source of radiation all across the spectrum including X-rays. However, we see the radiation already vastly shielded by dust and factors such as the inclination of the disc with respect to us also play a role. Hence, estimating the actual intensity is not straightforward. If the radiation is too strong, the ship gets heated up beyond sustainability, not to mention the damage done on the pilots by the not-shielded X-rays. Once again, the jets can change the game - for sufficiently rapid jets, the radiation will gain much higher energies via inverse Compton scattering thus posing a much larger threat for the spaceship. Once again, I am not able to quantitatively assess the importance of such an effect especially due to the fact that from e.g. this article it would seem that Sgr A* is very much gamma- ray quiet. (Which does not mean the ship and it's crew wouldn't get fried by the \"weak\" gamma ray production from the vicinity of the jet.) Nevertheless, we should not forget that the ship would still be very close to a black hole and the radiation is not homogenous\/isotropic. The strongest radiation comes from the parts closest to the inner edge of the disc at $\\sim 3 r_s$. If the pathway was taken away from the equatorial plane, much of the radiation could be devoured or deflected by the black hole's shadow. * * * **Magnetic fields** Accretion is believed to be accompanied by _very_ strong magnetic fields. The two effects which could cause a problem are quickly varying fields causing electromagnetic induction on the electronic devices on the ship, and the effects of a very strong magnetic field on e.g. ferromagnetic parts of the spaceship. The variance of the field could be induced either by the ship moving through a space-varying field or by the inherent time-variance of the field itself. Once again, the variations of the field will be modeled on idealized models with length- and time-scales appropriate for the system itself, not the time- and length-scales relevant for the moving ship. There could be a \"noise\" in the field either spatial or temporal (on scales $\\ll r_s$ or $\\ll r_s\/c$) strong enough to cause critical damage to the ship through induction. * * * It would be a too broad question to theorize whether a ship could be able to pass through such conditions (I might ask this on Space.SE using the answers to this question). For now, I will ask only about the _conditions_ relevant for a space ship approaching the Black-hole horizon. **QUESTION 1:** What are the quantitative observational constraints on the density and temperature of plasma around Sgr A* and its spatial configuration? Is there a very-low-plasma-density pathway into the black hole allowed by the current knowledge? **QUESTION 2:** What is the intensity of different types of radiation in the very vicinity of Sgr A*? What would be the optimal angle to approach the presumed black hole in it's center to maximally evade the radiation? **QUESTION 3:** What is the presumed structure of the magnetic field in Sgr A*, it's strength and derivatives in both time and space? Is there a possibility of a \"strong noise\" in the magnetic field say at the time-scales $10^{-4}-10 \\, \\rm s$ and length-scales larger than $1\\, \\rm cm$? For the last question we do not know what the viable velocity of the space- ship would be, but assume at least $100 \\, \\rm m \\cdot s^{-1}$. For the moment, let us neglect the fact that the proper time of the background (defined by the quasi-exact Killing vector of the close-to-Kerr space-time) and the space-ship time might differ notably and always describe the situation from the point of view of an observer at spatial infinity. * * * This question is a followup trying to clear up the discussion of this question concerning itself with a spaceship plunging into an (im)possible worm-hole. EDIT: I have added a note on the gamma rays produced by the black hole as suggested by Kyle Kanos."}
{"id":"121708","title":"internal conversion and the electromagnetic force","text":"I have read that the mechanism behind internal conversion, in which a nuclear transition leads to the ejection of an electron in one of the lower atomic orbitals, is related to the fact that the wavefunctions of some of these electrons penetrate the nucleus. Because of this, there is a finite probability that the electron will be in the nucleus, and, when this occurs, the electron may couple to the ~~excited state of the nucleus~~ nuclear transition when it happens. Is it conjecture that the electron must penetrate the nucleus in order for internal conversion to occur, or is this a straightforward result? I would have figured that since internal conversion is mediated by the electromagnetic force (which has long reach), for any electron within range of a virtual photon there will be a nonzero cross section for this type of decay. What details am I missing in thinking this?"}
{"id":"121700","title":"Is general covariance a symmetry?","text":"Is general covariance a symmetry? If it is ,what is its symmetry group and corresponding generator?"}
{"id":"54398","title":"What happens if object is thrown in empty space?","text":"If I throw a object in empty space, I apply a force to throw that. Then it gains some acceleration and it's speed increases. So will it's speed keep on increasing, or it will get stable? If yes, what happened to the force I applied. No change in speed, so acceleration = 0, then $F = 0$."}
{"id":"119269","title":"If an area remains at temperature 0 Kelvin would the area be frozen in time?","text":"If an area is at 0 kelvin, I am guessing that there is no motion. Without motion in an area, will it not look as if time has stopped?"}
{"id":"111057","title":"Conjugate momentum is not gauge invariant","text":"The conjugate momentum of a charged particle moving in a uniform magnetic field is given by $$\\vec p=m\\vec v+q \\vec A$$ This expression is not unique because $\\vec A$ is not unique. $\\vec A$ is not a measurable quantity. But $\\vec p$ is a phase space variable and if it is not unique the prediction of future of the system is not unique either. Isn't this a problem?"}
{"id":"95011","title":"What is cp\/cv ratio?","text":"Recently in our chemistry class we read that the $C_p\/C_v$ ratio of the inert gases is 1.66 thus they show inert nature. I asked my teacher what was $C_p\/C_v$ ratio but he didn't answer me. So I want to know what the $C_p\/C_v$ ratio really is?"}
{"id":"116298","title":"Optical Waveguides with grating","text":"I have some problems about mode analysis in grating-coupled waveguides,Would you please introduce me a useful book about wave propagation in these structures."}
{"id":"123706","title":"Relativistic Effects Observed with Time Travel","text":"Disclaimer: My domain knowledge on these topics is pretty minimal. I'm a \"physics fan\". From what I understand about relativity. If there are two identical objects, A and B, and A is stationary and B is not. Then A will observe that B's time is elapsing more slowly than its own. It will also notice that B is shorter in the direction of travel than A is. Now lets assume that A and B could somehow see through time as well as space (this is potentially where the question breaks down). I suppose they would see something like a series of cubes, representing each instance in time, with all the objects in different positions. If both A and B were now stationary in space, but B was traveling twice as fast through time than A was. Would there be any relativistic effects A would see as it is observing B?"}
{"id":"73713","title":"Equivalent resistance in ladder circuit","text":"I have stumbled upon a given question I really have a hard time to solve. Basically I need to find an equivalent resistance in some form of \"ladder\" configuration. Where the chain is an infinite sequence of resistors. ![ladder](http:\/\/i.stack.imgur.com\/l7du7.png) I have really no good idea how to find this equivalent resistance. Trying the old fashioned rule of parallel and resistors in series I came to a very messy formula: I have really no good idea how to find this equivalent resistance. Trying the old fashioned rule of parallel and resistors in series I came to a very messy formula: $$R + \\frac{1}{\\frac{1}{R} + \\frac{1}{R+\\frac{1}{\\frac{1}{R}+....}}}$$ I know however that the solution should be much more simple. Now I tried using Kirchhoff's loop rule. Which states that the power difference in a closed loop must be 0. Naming the \"potential currents\" between AB $I_1$, between BC (through the single resistance) $I_2$ and the current \"from B to the right\" $I_3$ Considering the loop containing BC & the rest of the structure this rewrites to: $$I_1 = I_2 + I_3$$ $$I_3 \\cdot R - I_2 \\cdot R_{eq} = 0$$ The problem is, 2 variables, 2 functions doesn't really bring me closer to an answer :(. What am I missing?"}
{"id":"119190","title":"Why $SU(3)$ and not $U(3)$?","text":"Is there a good reason not to pick $U(3)$ as the colour group? Is there any experiment or intrinsic reason that would ruled out $U(3)$ as colour group instead?"}
{"id":"91103","title":"How to show that a general Galilean transformation in three dimensions is a conformal transformation?","text":"Suppose two particles with equal mass move and then collide. We can easily show that the angle of collision is ninety degree if we choose our frame of reference moving with velocity equal to one of the particles velocity. How can we prove that any inertial frame of reference preserve the angle of collision? In other words, how to show that a general Galilean transformation in 4-dimensional space-time is a conformal transformation?"}
{"id":"5524","title":"Wavefunction collapse in relativity","text":"It is well accepted that quantum theory has well adapted itself to the requirements of special relativity. Quantum field theories are perfect examples of this peaceful coexistence. However I sometimes tend to feel little uneasy about some aspects. Consider an EPR pair of particles light years apart. Suppose there are 2 observers moving relative to each other with constant relative velocity. Let us consider, there are spin detection mechanism at both end for each particle. Now suppose one of the observer is at rest w.r.t. the detector for the first particle. As soon as the detection made, the wave function of the 2 particle entangled system will collapse instantaneously and the second particle must realize a definite opposite spin value. Now due to relativity of simultaneity, the second observer may claim that the collapse of the wave function for the two particle system is not simultaneous. He may even claim that the second particle is measured first. In that case a special frame of reference will be privileged, the frame at which the wave function collapsed instantaneously. This will cause a significant strain on the core principle of special relativity. I am sure the above reasoning is flawed. My question is where?"}
{"id":"66984","title":"How to solve Boltzmann equation using monte carlo methods?","text":"I'm trying to solve for electron and hole distribution function using Boltzmann equation with various scattering mechanisms. Since I land up with an integro-differential equation, analytical soln. is very difficult. So, I need to understand how this problem could be solved using numerical methods. Please help. (Try suggesting some references with illustrative examples which could be useful) Thanks!"}
{"id":"87336","title":"Field of a parallel plate capacitor","text":"Pretty much in all the diagrams I have seen the electric field of the parallel plates is depicted as in this image ![enter image description here](http:\/\/i.stack.imgur.com\/YyLco.jpg) The field lines seems to start bending only after leaving the periphery of the plates, shouldn't the field lines be depicted as starting to bend as soon as we start to move away from the exact centre of the capacitor ?"}
{"id":"8496","title":"Spilling a cup of coffee","text":"I've observed this many times when I spill coffee, It gets attracted towards the cup or the vessel. why is this so? Which is the force acting? (Hope you understood my question)."}
{"id":"110014","title":"When the matter is heated does the photon absorbed by electrons of atoms or by atoms themself?","text":"When the matter is heated does the photon absorbs by electrons of atoms or by atoms themself?"}
{"id":"96090","title":"Perfectly centered break of a perfectly aligned pool ball rack","text":"Imagine the beginning of a game of pool, you have 16 balls, 15 of them in a triangle <| and 1 of them being the cue ball off to the left of that triangle. Imagine that the rack (the 15 balls in a triangle) has every ball equally spaced apart and all balls touching all other appropriate balls. All balls are perfectly round. Now, imagine that the cue ball was hit along a friction less surface at the center axis for this triangle O-------<| and hits the far left ball of the rack dead center on this axis. How would the rack react? I would imagine this would be an extension of newtons cradle and only the 5 balls on the far end would be the candidates to move at all. But in what way would they move? Which ball, for instance, would move the furthest? Thanks"}
{"id":"126867","title":"GPS Satellite - Special Relativity","text":"I'm going through an old relativity assignment, and I've been asked to calculate the time dilation for a satellite which orbits the earth in 12 hours at 26000km from the surface, and travels at a constant speed. The given radius of the Earth is 6400km. In the solutions, the author has calculated the velocity traveled by the satellite to be $\\dfrac{3.24\\times 10^7\\times 2\\pi}{12\\times 3600} \\text{ms}^{-1}\\simeq 4500 \\text{ms}^{-1} $. I'm fine with the calculation (if I've gone wrong with the numbers somewhere, I don't mind). What I'm not so OK with is that they proceeded to use $v=4500$ in the formula $\\gamma=\\dfrac{1}{\\sqrt{1-\\frac{v^2}{c^2}}}$, which is needed to calculate time dilation. As far as I am aware, this formula only holds for observers whose relative velocity is $v$. However, in this case where there is an observer on the Earth's surface and a satellite moving in circular motion around the center of the earth, their relative velocity isn't constant!"}
{"id":"126860","title":"How to calculate a collision which is partly elastic and partly inelastic?","text":"_(For the purpose of this question, \"calculating a collision\" means: given the velocities and masses of two objects in a collision, figuring out the new velocities of both objects after the collision)._ I know how to calculate a totally elastic collision, and how to calculate a totally inelastic collision. But I don't know how to calculate a collision which is part elastic and part inelastic. I don't know where to start. Guidance will be appreciated. (Go easy on the math please)."}
{"id":"82182","title":"Can quantum entanglement be a proof of a 4th spatial dimension?","text":"What I know about quantum entanglement is almost nothing. What I know can be resumed to: \"two particles created together have a correlated behaviour no matter the distance between them\". This is often interpreted as a faster-than-light phenomenon because it is instantaneous. But let's consider a 4th spatial dimension (we then have x,y,z,w)   * 2 particles are created in the point x=0,y=0,z=0,w=0   * Those 2 particles move apart in the 3D space such as: Particle 1 is @ x1,y1,z1,w and Particle 2 is @ x2,y2,z2,w Note that they are at the same w coordinate. Can this be an explanation for the fact that the correlation is instantaneous? I am not considering that this is a revolutionary thing. I am sure that a lot o people already thought about this, but I want to understand it better."}
{"id":"25057","title":"How would the night sky appear at the edge of the galaxy?","text":"In Isaac Asimov's Foundation Series, there is a planet named Terminus which is believed to be the planet farthest from the galactic center. > There are almost no visible stars in its sky, only the huge lens of the > Milky Way galaxy But what would we really see if we look at the opposite side from the galactic center (without a telescope or binoculars)? Would we see a black sky or a sky full of galaxies?"}
{"id":"82235","title":"Choice of X-ray scatterer in Compton effect","text":"I am going to perform an experiment on Compton Scattering, and I am going to use an X-ray scatterer to scatter the incident X-rays. I have been instructed that Acrylic Glass slab are the best for this purpose. I am not sure if it is because the acrylic glass slab does not have any free electrons present or any other reason. Please explain."}
{"id":"57260","title":"Electric potential of a spheroidal gaussian","text":"I'm looking for results that compute the electrostatic potential due to a spheroidal gaussian distribution. Specifically, I'm looking for solutions of equations of the form $$ \\nabla^2\\Phi=N\\exp\\left({-\\rho^2\/2\\sigma_r^2}\\right)\\exp\\left({-z^2\/2\\sigma_\\ell^2}\\right), $$ where $\\rho^2=x^2+y^2$, possibly with some polynomial factor of the form $\\rho^{|m|}\\cos(m\\phi)z^n$ in front. This has an easy solution when the two variances $\\sigma_r$ and $\\sigma_\\ell$ are equal, in which case there is spherical symmetry applies and Gauss's law readily yield $\\Phi$ in terms of error functions. However, I'm having trouble finding results for the general case. Using spheroidal coordinates doesn't seem to help, as the constant-$\\mu$ surfaces are confocal spheroids that taper out to spheres as they get larger, instead of maintaining their ellipticity as the charge density does. This looks standard enough that it ought to have been done before (right?), but it is also messy enough that I'm unsure it made it to any textbook. Has anyone seen the likes of this before?"}
{"id":"117240","title":"What is intrinsic gravitational entropy?","text":"What is intrinsic gravitational entropy? Does it have to do with dark matter or coarse graining in the universe? Is it unique to general relativity, or there are predictions from quantum mechanics as well? Please explain in detail."}
{"id":"117245","title":"Tritium decay is spontaneous even if the binding energy of tritium is higher than the binding energy of 3He. Why?","text":"Given this nuclear reaction: $^3_1\\mathrm H\\to {}^3_2\\mathrm{He}+e^-+\\bar{\\nu}$ and knowing the binding energies: $BE(^3_1\\mathrm H)=8.48 \\,\\mathrm{MeV}$ $BE(^3_2\\mathrm{He})=7.72 \\,\\mathrm{MeV}$ If I calculate the mass defect (obviously considering the binding energies in the mass calculation) I obtain a positive value: $M(^3_1\\mathrm H)c^2=2809.08 \\,\\mathrm{MeV} > M( ^3_2\\mathrm{He})c^2+M(e^-)c^2=2808.991 \\,\\mathrm{MeV}$ as expected for a spontaneous decay. Considering the binding energies I have written above I expect the $^3\\mathrm{H}$ to be more stable than $^3_2\\mathrm{He}$. My question is: why does this decay occur?"}
{"id":"16474","title":"Is there any anti-gravity material?","text":"I want to know if there is any anti-gravity material. I am thinking of making flying vehicles which are made up of anti-gravity material so that they will not experience any gravity on them and can easily take off and be more fuel efficient. Is there any such thing? Or any workaround?"}
{"id":"105018","title":"Does the transmission axis matter for sending polarized light through polarized glass?","text":"If I have polarized light and I send through only one polarized glass plane, does the transmission axis matter, or will the intensity be halved no matter what."}
{"id":"69982","title":"Variational Derivation of Schrodinger Equation","text":"In reading Weinstock's _Calculus of Variations_ , on pages 261 - 262 he explains how Schrodinger apparently first derived the Schrodinger equation from variational principles. Unfortunately I don't think page 262 is showing so I'll explain the gist of it: \"In his initial paper\" he considers the reduced Hamilton-Jacobi equation $$\\frac{1}{2m}\\left[\\left(\\frac{\\partial S}{\\partial x}\\right)^2 \\ + \\ \\left(\\frac{\\partial S}{\\partial y}\\right)^2 \\ + \\ \\left(\\frac{\\partial S}{\\partial z}\\right)^2\\right] \\ + \\ V(x,y,z) \\ - \\ E \\ = \\ 0$$ for a single particle of mass $m$ in an arbitrary force field described by a potential $V = V(x,y,z)$. With a change of variables $S \\ = \\ K\\log(\\Psi)$, (where $K$ will turn out to be $\\hbar=h\/2\\pi$) it reduces to $$\\frac{K^2}{2m}\\left[\\left(\\frac{\\partial \\Psi}{\\partial x}\\right)^2 \\ + \\ \\left(\\frac{\\partial \\Psi}{\\partial y}\\right)^2 \\ + \\ \\left(\\frac{\\partial \\Psi}{\\partial z}\\right)^2\\right] \\ + \\ (V \\ - \\ E)\\Psi^2 \\ = 0.$$ Now instead of solving this he, randomly from my point of view, chose to integrate over space $$I = \\iiint_\\mathcal{V}\\left(\\frac{K^2}{2m}\\left[\\left(\\frac{\\partial \\Psi}{\\partial x}\\right)^2 \\ + \\ \\left(\\frac{\\partial \\Psi}{\\partial y}\\right)^2 \\ + \\ \\left(\\frac{\\partial \\Psi}{\\partial z}\\right)^2\\right] + (V - E)\\Psi^2\\right)\\,\\mathrm{d}x\\,\\mathrm{d}y\\,\\mathrm{d}z$$ then Weinstock extremizes this integral which gives us the Schrodinger equation. Apparently as the book then claims on page 264 it is only after this derivation that he sought to connect his idea's to deBroglie's wave-particle duality. Thus I have three questions, 1) What is the justification for Feynman's famous quote: > _Where did we get that [Schrödinger's equation] from? It's not possible to > derive it from anything you know. It came out of the mind of Schrödinger._ > The Feynman Lectures on Physics in light of the above derivation. I note that all the derivations I've seen of the Schrodinger equation doing something like using operators such as $i\\hbar\\partial\/\\partial t = E$ to derive it always mention it's merely heuristic, yet what Schrodinger apparently originally did seems like a roundabout way of solving the Hamilton-Jacobi equation with no heuristic-ness in sight. What subtleties am I missing here? Why would I be a fool to arrogantly 'correct' someone who says Schrodinger is not derivable from anything you know? Note I've read a ton of the threads on this topic on this forum & none even go near the calculus of variations, in fact the above contradicts this explanation here that even refers to Schrodinger's Nobel prize lecture, so hopefully it's not a duplicate. 2) Is the mathematical trick Schrodinger has used something you can use to solve problems? 3) Why can't you use this exact derivation in the relativistic case? **Edit:** On the topic of complex numbers: On page 276, 14 pages after he explains what I posted, & after he goes through how Schrodinger linked his work with DeBroglie's work, it is only then that Weinstock says: > _In a more complete study of quantum mechanics than the present one the > admissibility of complex eigenfunctions $\\Psi$ is generally shown to be > necessary. If $\\Psi$ is complex, the quantity $|\\Psi|^2$ is employed as the > position probability-density function inasmuch as $\\Psi^2$ is not restricted > to real nonnegative values._ Apparently Schrodinger was able to do what I have posted using real-valued functions & have K as I have defined it, without i. If you're following what Weinstock is saying he shows how the hydrogen atom energy levels are explainable without complex numbers, i.e. he is able to derive a physical interpretation of the eigenvalues (discrete energy levels) of the Schrodinger equation that were in accord with experiment (see Section 11.3 Page 279 on). As far as I understand it it is in trying to find a physical interpretation of the eigenfunctions that one is forced into complex numbers, though apparently, according to the book, it can be shown to be necessary. That is something to take into account!"}
{"id":"9532","title":"Coffee cooling in a cup","text":"Will coffee in two differant size styrofoam cups in the same room conditions cool at the same rate? One cup is 8 oz. and the other would be 12 oz."}
{"id":"65516","title":"Proof of S-duality between Type IIB, IIB and Type HO, I string theories","text":"About every source on string theory I've read which do mention S-duality state that: $$\\begin{array}{l} \\operatorname S:\\operatorname{IIB} \\leftrightarrow \\operatorname{IIB}\\\\\\ \\operatorname S:\\operatorname{HO} \\leftrightarrow \\operatorname{I} \\end{array}$$ However, how does one prove that the Type IIB string theory is self-S-dual and even more bizzarely, that the Type HO string theory is S-dual to the Type I string theory?"}
{"id":"81768","title":"Shouldn't the addition of angular momentum be commutative?","text":"I have angular momenta $S=\\frac{1}{2}$ for spin, and $I=\\frac{1}{2}$ for nuclear angular momentum, which I want to add using the Clebsch-Gordan basis, so the conversion looks like: $$ \\begin{align} \\lvert 1,1\\rangle &= \\lvert\\bigl(\\tfrac{1}{2}\\tfrac{1}{2}\\bigr)\\tfrac{1}{2}\\tfrac{1}{2}\\rangle,\\tag{4.21a} \\\\\\ \\lvert 1,0\\rangle &= \\frac{1}{\\sqrt{2}}\\biggl(\\lvert\\bigl(\\tfrac{1}{2}\\tfrac{1}{2}\\bigr)\\tfrac{1}{2},-\\tfrac{1}{2}\\rangle + \\lvert\\bigl(\\tfrac{1}{2}\\tfrac{1}{2}\\bigr),-\\tfrac{1}{2}\\tfrac{1}{2}\\rangle\\biggr),\\tag{4.21b} \\\\\\ \\lvert 1,-1\\rangle &= \\lvert\\bigl(\\tfrac{1}{2}\\tfrac{1}{2}\\bigr),-\\tfrac{1}{2},-\\tfrac{1}{2}\\rangle,\\tag{4.21c} \\\\\\ \\lvert 0,0\\rangle &= \\frac{1}{\\sqrt{2}}\\biggl(\\lvert\\bigl(\\tfrac{1}{2}\\tfrac{1}{2}\\bigr)\\tfrac{1}{2},-\\tfrac{1}{2}\\rangle - \\lvert\\bigl(\\tfrac{1}{2}\\tfrac{1}{2}\\bigr),-\\tfrac{1}{2}\\tfrac{1}{2}\\rangle\\biggr),\\tag{4.21d} \\end{align} $$ where $F=I+S$, so this is the basis $\\lvert F m_F \\rangle = \\sum_m \\lvert\\bigl(I S\\bigr),m_I m_S\\rangle $. Now since adding angular momenta is commutative, the exchange between $I$ and $S$ shouldn't mathematically introduce any kind of difference. In other words, in the basis described in those equations, it shouldn't matter whether I write it as $\\lvert\\bigl(I S\\bigr),m_I m_S\\rangle$ or $\\lvert\\bigl(S I\\bigr),m_S m_I\\rangle$, right? Now the problem is the following: I have created the Hamiltonian matrix $H=-\\vec{\\mu}\\cdot \\vec{B} = -2 \\mu B_z S_z\/\\hbar$ in the $\\lvert F m_F \\rangle$ representation, and actually the result depends on how you call those angular momenta, so the result could be $$H = \\begin{pmatrix} \\mu_B B & 0 & 0 & 0 \\\\\\ 0 & \\- \\mu_B B & 0 & 0 \\\\\\ 0 & 0 & 0 &\\mu_B B \\\\\\ 0 & 0 & \\mu_B B & 0 \\end{pmatrix}$$ Or could be $$H = \\begin{pmatrix} \\mu_B B & 0 & 0 & 0 \\\\\\ 0 & \\- \\mu_B B & 0 & 0 \\\\\\ 0 & 0 & 0 &-\\mu_B B \\\\\\ 0 & 0 & -\\mu_B B & 0 \\end{pmatrix}$$ Depending on how you \"label\" them, $I$ or $S$... which is very confusing! This happens because the off-diagonal terms $$\\left\\langle 1 0 \\right| S_z \\left| 0 0 \\right\\rangle = \\frac{1}{2} \\left( \\left\\langle (\\frac{1}{2} \\frac{1}{2}) \\frac{1}{2} -\\frac{1}{2} \\right| + \\left\\langle (\\frac{1}{2} \\frac{1}{2}) -\\frac{1}{2} \\frac{1}{2} \\right| \\right) S_z \\left( \\left| (\\frac{1}{2} \\frac{1}{2}) \\frac{1}{2} -\\frac{1}{2} \\right\\rangle - \\left| (\\frac{1}{2} \\frac{1}{2}) -\\frac{1}{2} \\frac{1}{2} \\right\\rangle \\right)$$ will be either $\\hbar\/2$ or $-\\hbar\/2$ depending on your convention whether it's $\\lvert\\bigl(I S\\bigr),m_I m_S\\rangle$ or $\\lvert\\bigl(S I\\bigr),m_S m_I\\rangle$. How can I understand this physically and mathematically? Shouldn't the addition be commutative and the process be blind to which labels I use?"}
{"id":"24090","title":"Problem in a truss","text":"![enter image description here](http:\/\/i.stack.imgur.com\/9jFoH.png) For the truss above known are the following: l= 1 + 0.02*n [m], h = 2 + 0.01*n [m], F_1 = 200 + 2*n [N], F_2= 500 + 6*n [N], F3 = 400 + 5*n [N], Young's modulus E = 2*10^11 [N\/m2], allowable stress σ_a= 8*10^8 [N\/m2] In the amounts above, n a natural number. Find:   1. Reaction forces at A and E .   2. Tension forces in all members of the truss. Draw free body diagrams for all joints and members; indicate extension or compression for the member forces.   3. Diameters of all the members considering they all have circular cross-sections. The diameters have to be selected from the following sequence of diameters (in mm): 2, 5, 8, 10, 12, 16, 20, 24, 30, 36, 40, 50, 60, 70.   4. The axial deformation (Δl) of bars BC and FG."}
{"id":"24091","title":"Why is there a 90˚ phase angle between particle velocity and sound pressure in spherical waves?","text":"My text says that in a plane sound wave (or in the far field), particle velocity and pressure is in phase. As we move closer to the sound source (to near field and more spherical waves), the phase angle between these two quantities will gradually shift towards 90˚. Why does this phase shift occur? I haven't been able to find an intuitive explanation for why there is a phase shift in spherical waves, but not in plane waves, and vice versa."}
{"id":"68166","title":"Focal length, power and magnification","text":"In refracting (lens not mirror based) telescopes, to have a large magnification the objective lens needs to have a large focal length. if $\\text{Power} = \\frac{1}{\\text{focal length}}$ then that means telescopes that have a large magnication have a low powered objective lens? Also if a large focal length provides a greater magnification then a lens which refracts (bends) the light the least is better, so surely that's no lens at all (or just a pane of glass)? How does a longer focal length (and so low power lens) provide a greater magnitude? It seems to my common sense (obviously wrong) that a lens that bends the light more would provide a greater magnification? (As the rays would be spread over a greater area for a shorter length)"}
{"id":"60138","title":"Question about tide","text":"Let's say a canal is built in the middle of Kansas in the shape of a ring that stretches 100 miles in diameter. The band of water is 100 feet wide and 10 feet deep throughout the entire canal, and the land where the canal is built is flat in every way aside from conforming to the curvature of the earth. Would a current develop from tidal forces? If so, in what direction would it flow? How rapid would it be?"}
{"id":"90673","title":"Riemann curvature tensor symmetries confusion","text":"In the context of spacetime, reading Schutz, I'm confused about the symmetries of the Riemann curvature tensor, which I understand are: $$R_{\\alpha\\beta\\gamma\\mu}=-R_{\\beta\\alpha\\gamma\\mu}=-R_{\\alpha\\beta\\mu\\gamma}=R_{\\gamma\\mu\\alpha\\beta}.$$ But using the metric to contract the Riemann tensor can't I also say $$R_{\\gamma\\mu}=g^{\\alpha\\beta}R_{\\alpha\\beta\\gamma\\mu}=g^{\\alpha\\beta}R_{\\alpha\\gamma\\beta\\mu}?$$ Which leads me to think that $$R_{\\alpha\\beta\\gamma\\mu}=R_{\\alpha\\gamma\\beta\\mu}.$$ But $R_{\\alpha\\gamma\\beta\\mu}$ isn't one of the above listed symmetries. Where am I going wrong?"}
{"id":"90670","title":"Some small questions about quantum spin and rotations","text":"I'm studying about quantum-spin (in a syllabus about non-relativistic quantum- mechanics though), but I have some trouble understanding everything. So I would like to ask some small questions, which may perhaps clear up the picture as a whole. So they are trying to determine the influence of a spatial rotation, to the state in the two dimensional spin-space. So they try to determine the operator $Û_R$ responsible for this rotation (which acts on the spin-space and thus is represented by a 2x2 matrix). To do this they first try to determine $Û_{\\epsilon}$ which is the infinitesimal unitary operator for rotation about an infinitesimal angle $\\epsilon$ and they state that such an operator can always be written as: $$Û_{\\epsilon}= 1-i\\epsilonÂ$$ with $Â$ hermitic. So $Â$ is also acting on the two dimensional spin-space and can be represented by a 2x2 matrix. So far so good. But now they state that $Â$ has to be constructed in such a way that $<\\chi\\vert\\chi>$ and $<\\chi'\\vert\\chi'>$ and $<\\chi'\\vert\\chi>$ are rotation- invariant. Where $\\vert\\chi>$ and $\\vert\\chi'>$ are states of the two dimensional spin-space and $\\vert\\chi'> = Û_{\\epsilon}\\vert\\chi>$. What I don't understand is: 1) What is meant by rotation invariance? Because if we rotate $\\vert\\chi>$ and $\\vert\\chi'>$ by taking the action of a unitary rotation-operator $Û_{\\theta}$ , then these inproducts are necessarily conserved, whatever $Â$ may be. So I guess they mean something else. 2) Why must $Â$ be constructed so that this is true? Perhaps this will be clear if I understand 1) though. Furthermore they state that this can only be the case if: $$Â = \\vec{1}_n\\cdot\\vec{Â} = \\frac{1}{2}\\vec{1}_n\\cdot\\vec{\\sigma}$$ With $\\vec{Â}$ and $\\vec{\\sigma}$ hermitical vector operators. So I wonder: 3) Why is this only the case if this is true?   4) Somewhat further they also mention that $Tr(\\sigma_z) = 0$ but I don't really see here too why this has to be the case. I hope that someone can answer one (or more) of these questions, I think it will help a lot in understanding the other claims of the chapter about quantum-spin."}
{"id":"99185","title":"Why doesn't the potential energy of any object equal 0","text":"Consider a particle on the ground. This particle is raised by a force of magnitude $mg$ to a height $h$ above the ground. At this point, the work done on the particle by the force is $mgh$, which is equal to the potential energy of the particle. But, during this period, the force of gravity also acts on the particle and is displaced by $h$, and so does a work of $-mgh$ on the particle. Shouldn't the two cancel and no net work should be done on the particle? If they don't cancel, then where did the energy that came from the work done by the force of gravity go?"}
{"id":"2696","title":"Do all massive bodies emit Hawking radiation?","text":"It is known that any accelerated observer is subject to a heat bath due to Unruh radiation. The principle of equivalence suggests that any stationary observer on the surface of a massive body should also experience heat bath. Indeed by substituting surface gravity g into the formula for Unruh radiation one can find the black body temperature for a hypothetical hyper-cool planet: $$T = \\frac{\\hbar g}{2\\pi c k}$$ which is $3.9766×10^{-20}\\ K$ for Earth one even can find the time which it will take for Earth to evaporate: $5.69×10^{50}$ years. Since the heat in the super-cold Earth cannot come out of nothing one should assume that it will come from decay of particles due to a certain mechanism. Sometimes I heared an argument that an event horizon is needed for Hawking radiation to exist. But this can be countered by assumption of possibility of decay due to quantum virtual black holes (which inevitably should appear due to uncertainty principle, and the more massive and dense body is the greather concentration of virtual black holes inside it will be, eventually becoming similar to the concentration of bobbles inside a body of boiling water). Or just suggest that any massive body due to uncertainty principle can quantum tunnel into a black hole state so to emit Hawking radiation. So what is the conclusion here?   * Can we say that all massive bodies are surrounded by the atmosphere of heated vacuum? This is a weaker preposition: thermal state of surrounding vacuum does not mean energy transfer if the system is in thermodynamic equilibrium.   * Any body gradually evaporates, i.e. transfers its energy to the surrounding vacuum until completely vanishes? This is a stronger preposition and suggests emission of radiation al loss of mass."}
{"id":"102543","title":"Entropic force in polymers","text":"According to my textbook, the elastic force in a rubber is caused to the tendency of the polymers to return to their initial disordered state of higher entropy. But isn't this looking at entropy on a larger level (I'm hesitant to say macroscopic) than it is usually looked at? Generally, we consider the entropy of individual atoms in an ordered structure (a crystal lattice, for instance). But here, we are looking at the entropy of entire polymer _chains_. Also, it just doesn't seem to make sense with the non-statistical view of entropy. The book explains it in the following way. The first law of thermodynamics, $$dE = dQ - dW$$ Making some substitutions, $$dE = TdS + Fdx$$ where $F$ is the elastic force, $dx$ is the displacement, $T$ is the temperature and $dS$ is the entropy change. Next, approximating $dE$ as zero, and solving for $F$, $$F = -T\\frac{dS}{dx}$$ which is supposed to tell us the that the force is proportional to the rate $dS\/dx$ at which the rubber band's entropy changes. But how is there any chage in entropy when no heat is being added? And, for that matter, if I take a string, lying all entangled, and try to straighten it, why doesn't _it_ exert an entropic force and try to go back to its higher entropic state?"}
{"id":"2691","title":"Decompressing gas whilst driving a turbine - where does the energy come from?","text":"Here's the situation: a friend is working on a civil engineering project to make some equipment to decompress liquefied natural gas before pumping to households. I don't know what the proper term for this bit of kit is calld but I will call it the **decompresser** Apparently what happens in these stations is that the gas is decompressed and because the gas cools down during expansion, it has to be heated back to something like room temperature before going on its way through the gas pipes to homes. They do this by burning kerosene. Now my friend says that they have designed a new decompresser that, in order to be more energy efficient, will generate electricity by decompressing the gas through a turbine. It is a long time since I did any related physics but the simpleton-scientist- thought-process that I have going on in my head is something like the following: -   * if the pressurised gas is having to do work turning the turbine, then this energy has to come from somewhere    * thus the temperature of the resulting de-pressurized gas will be lower.    * Because the temperature of the gas is lower, they would have to put more energy than they would have done to reheat the gas and they would not have any net energy gain. They would have electricity but they would be burning an equivalent amount of kerosene.  Now assuming that these civil engineers know a thing or two about thermodynamics, I readily admit that my thinking is wrong. I would very much appreciate it if someone could put me right on whether or not they can generate electricity as well as have the gas come out at the same temperature and pressure and could they not just have used the extra kerosene to generate the same amount of electricity (ignoring the various differing efficiencies of energy conversion that are going on). **PS.** Because I am clueless as to the actual answer, I will be relying heavily on the upvoting of + comments about the answers when it comes to marking an answer as the right 1!"}
{"id":"58201","title":"Understanding bending light beam perpendicular to motion","text":"I'm just reading a book about gravity. An example it gives is a spaceship accelerating. A beam of light travelling at right angles to the direction of movement of the spaceship enters it via a small pinhole. The book states that the beam of light would appear to bend to an observer within the spaceship. Intuitively this makes sense - if the ship were travelling at $0.5\\,c$, for example, and was $10\\,m$ across (from the pinhole to the opposing wall) then I'd expect the light to hit the wall $5\\,m$ further back down (assuming the ship is accelerating \"up\") from the pinhole - the wall would have moved 5m further in the time it takes the light beam to cross the interior to the opposing wall I think I'm fundamentally misunderstanding though. The book states this is down to the acceleration of the ship and goes on to talk about how using equivalence the same bending would be a result of gravity. In my description above though, it's the velocity that causes the apparent bending, and the same would be visible if the ship was maintaining a constant $0.5\\,c$. I suspect that if this were the case (constant $0.5\\,c$) there would be no bending at all, but I'm not understanding why Can anyone enlighten me?"}
{"id":"108197","title":"How much smaller will be human body, when we hypothetically get of every space between particles","text":"I had an interesting dream, where advanced civilization compress their body with technology, that turn off space betweens atoms\/particles. They travel in small spaceship with billions citizens very effectively. Crazy idea, but my question is how much smaller will the human body be, when we hypothetically get of every space between particles."}
{"id":"102096","title":"Could the Sun be partly powered by cosmic rays?","text":"The Earth and the Sun has magnetic fields which shields us from cosmic rays, as a charged cosmic ray particle will loose kinetic energy when its direction is perpendicular to the magnetic field. So what happens to the kinetic energy of the cosmic ray particle? According to the first law of thermodynamics it can't just disappear. According to electro-mechanics it will induce an electric current. The particles also get a spiral motion and can induce a magnetic field like a current in a solenoid. Some radiation due to breaking of the particle (bremsstrahlung) may also occur. The vast magnetic field of our sun is bombarded with cosmic rays with far bigger energy intensity than the cosmic rays that hit our shielded Earth. On Earth the most energetic cosmic ray observed is a subatomic particle with kinetic energy equal to that of a baseball (5 ounces or 142 grams) traveling at about 100 kilometers per hour (60 mph). So I wonder if its a possibility that the Sun is partly powered by cosmic rays? How large can the total cosmic ray energy hitting the heliosphere be and where does the energy go?"}
{"id":"106731","title":"Local tension and pressure measurements","text":"We want to measure tensions and pressures on the surface of a tooth for an experiment in dental research. We want to do local measurements at different points on the tooth. The tooth is not a real one, but a model, so it would be possible to build a device into the tooth. The tensions and pressures we want to measure are of the order of a few kilo Pascal or potentially even less. Is there a reliable way to achieve such a measurement?"}
{"id":"67116","title":"When we talk about speeds in relativity theory, where are they measured?","text":"I recently asked a question here about if the direction we travel matters in relativity theory: Does it matter in which direction I travel in relativity theory? After I got answers and making more research about the topic, I found out that I was talking about speeds relative to the earth all the time. So, when I for example said that I travel away from earth at 50% of the speed of light, I automatically assumed that this is also the speed I can use to calculate the time dilation. But since the earth already travels at some always varying speed and thus is not standing still in space, I think I also need to put this into the calculation. I mean, if I would travel in the exact opposite direction the earth currently \"flies\" and at exactly the same speed, this would mean that I'm standing still and that the time for me would actually go as fast as possible, while the time on earth is still slowed down, as it always is due to its speed. So: What I wanted to show you is that it really seems to depend from where I actually measure the speed I travel and what I consider the \"root\". Moving away from earth at some speed can mean moving faster in space OR moving slower in space and thus also mean slower or faster time flow than the one on the earth. Am I right with this? Or am I totally wrong and it's valid to measure the speed I travel by how much the distance from me to the earth gets bigger? **UPDATE** Actually, I think I have understood the answer of @John Rennie, and it really seems to explain a lot of the issues I ran into before. His answer was, at least to my understanding, the following: If two objects in space stay together, they share the sime timeflow. If the distance between the objects changes over time, we cannot say which one will see the others clock going slower and which one will see the others clock going faster. To say that, we need to go back to the change of velocity that led to the increasing\/decreasing distance. Here, we need to determine which of the objects changed it's velocity by the method John mentioned. And this will lead us to the answer. This also explains the twin paradox perfectly. But after doing some thought experiments I came across a scenario that I cannot explain: Imagine we humans have indeed built a death star as a moving base in space. This death star has also a lot of smaller fighter ships on board. Now the death star starts to fly away from earth at, say, 0.1c. According to the people on earth, the time on the death start now goes slightly slower than the time on the earth. After a while, a fighter on the death star begins to fly twice as fast (0.2c relative to the earth, 0.1c to the death star) in the same direction to explore the forthcoming space. According to the earthlings, the time on the fighter goes even slower than the time on the death star. The guys on the death star confirm that and notice that the time on the fighter seems to go slower compared to their time. After a while, the fighter found some enemy and decides to return to the death star to alert them. Here, the fighter has 2 options: 1) since the death star is also on its way, he could just stop and wait until the deathstar comes across. 2) Stopping, and moving into the other direction to inform the death star faster and before he runs into the enemies. The fighter goes for option 2) and flies back at 0.1c. For the earthlings, the fighter is now getting closer at 0.1c, while the death star is still moving away at 0.1c. For the death star, the fighter is getting closer at 0.2c. And that is the point where this starts getting weird: For the earthlings, the fighters speed decreased from 0.2c to 0.1c, which means that the fighters clock is now going faster, while still being slower than that on the earth. Actually, since the death star and the fighter are moving at the same speed relative to the earth, they share the sime timeflow. For the deathstar, the fighters speed increased from 0.1c to 0.2c, which means to them that the fighters clock is now going even slower than that on the death star. Can it really be?"}
{"id":"99721","title":"Can the $\\vec H$-field have non-zero divergence? Does an $\\vec H$-field monopole exist?","text":"We know that, $\\vec \\nabla \\cdot \\vec B=0$ but $\\vec \\nabla \\cdot \\vec H\\neq 0$, if $\\vec \\nabla \\cdot \\vec M \\neq 0$. Does it mean that, in those cases $\\vec H$-field has poles although $\\vec B$-field does not?"}
{"id":"68837","title":"Does the Renormalization of QFT Contradict Canonical Quantization?","text":"Does the renormalization of QFT contradict canonical quantization? In canonical quantization, you take the classical fields and canonical momenta and turn them into operators, and you require that the commutators ( or anti- commutators) equal the Poisson brackets of the original fields. Also, you need to quantize the classical Hamiltonian (of course, since classical fields commute but quantum operators don't, you need to quantize the Hamiltonian according to a well defined prescription, such as normal-ordering). The form of the quantum Hamiltonian will resemble the form of the classical one, but it will be an operator rather than a number. In particular, if a physical constant appears in the classical Hamiltonian, it will also appear in the Quantum one, and have the same value. Now, in renormalization the constants that appear in the (bare) Lagrangian are not the physical ones, and so are different from the constants that appear in the \"classical\" Lagrangian (and therefore Hamiltonian). So the \"bare\" quantum Hamiltonian will have a similar form to the classical Hamiltonian, but the constants will be totally (and perhaps infinitely) different! So does the quantization scheme need to be reformulated?"}
{"id":"98005","title":"Modeling with First Order Equations","text":"A ball with mass 015kg is thrown upward with initial velocity 20m\/s from the roof of a building 30m high. There is air resistance of magnitude v^2\/1325 directed opposite to the velocity , where the velocity is measured in m\/s. a. Find the max height above the ground that the ball reaches. b. Find the time that the ball hits the ground. I am unsure how to set this one up."}
{"id":"29111","title":"Should any theory of physics respect the principle of conservation of angular momentum or linear momentum?","text":"Is it possible that a theory that can describe the universe at the planck scale can violate things that we now consider fundamental in nature?For example can it violate rotational and translational invariance and subsequently momentum wouldn't be conserved ?Should one consider these invariance principles to be fundamental that we must choose the lagrangian to respect them ?"}
{"id":"29116","title":"What forces act on a person sitting in a chair with wheels when they push off another stationary object","text":"1. What is slowing me down when I push off my desk?   2. What are the \"Major\" contributors? and   3. Does how hard I push with one hand (or with two hands) make a significant difference?"}
{"id":"31201","title":"Might a planet perform figure-8 orbits around two stars?","text":"Might a planet perform figure-8 orbits around two stars? I'm thinking that if the two stars were equal mass (and not orbiting each other) then a planet that were to go right between them would continue in a straight line, with no preference for either star. But since the two stars would in fact be orbiting each other, the system would be rotating and thus there would be a Coriolis preference for one of the stars. Might that preference be made to alternate stars? Another possibility would be if each star were in turn orbited closely by another planet, which would perform three orbits for each orbit of our planet of interest. Then things could be timed where on one pass star A's inner planet were aligned right to pull the planet of interest into an orbit, and on another pass star B's inner planet would be aligned right to pull the planet of interest into an orbit. So we have a system of five bodies, two massive (stars) each orbited by a minor, and one minor performing figure-8s. Is this at least plausible if contrived?"}
{"id":"91695","title":"Double-slit expirement fundamentals (half-silvered mirror version)","text":"In the double-slit experiment variation in which 2 half-silvered mirrors and 2 mirrors are used to illustrate the interference of a stream of photons or single photons at a given time step, how is it when the experiment is comprised of all 4 mirrors that the photon always knows to go to Camera 1\/A in every illustration of the experiment I've ever heard of? Why doesn't it ALWAYS go to Camera 2\/B instead with this setup? Seems so arbitrary and I have never come across the reason. Diagram to refresh you: (Assume all distances are equal in diagram.)                                         [CAM 1\/A]                                   |        regular mirror             |             \/\/--------------------\\     [CAM 2\/B]  (notice there is no beam to CAM 2\/B)             |                     | half-silvered mirror #2             |                     |             |                     |             |                     |      [XXX>--\\---------------------\/\/  regular mirror             ^half-silvered mirror #1      [XXX> is the photon emitter\/laser. The backslash to the right of the photon emitter represents one of the two half-silvered mirrors that I've designated as half-silvered mirror #1. So... why is it when interference occurs at half-silvered mirror #2 that the photon decides it prefers the camera designated CAM 1\/A when it could easily just go to the other CAM every single time it interferes?"}
{"id":"91698","title":"Diffraction of sound","text":"The sound waves, by the virtue of it being a wave, shows diffraction and interference. But in diffraction, I learnt that if the wave is allowed to enter through a small aperture, there is a central maxima, along with several minima (zero intensity) on the sides. > **Why then in everyday life do we not encounter a situation in which we are > at one of those minima and fail to here any sound? For example,in a > soundproof studio room, suppose the narrow open door is the only entry point > for sound, can we then find a spot where we hear nothing going on outside > but by shifting to either side, we can hear it (although with different > intensities)?** Why aren't such phenomenon encountered in daily life, not being able to hear a source because of diffraction (or interference)? My question also relates to **radio waves** whose wavelengths is comparable to that of sound waves."}
{"id":"105832","title":"Supercharge definition in Becker-Becker-Schwarz","text":"How does one interpret equation (4.20) in Becker-Becker-Schwarz \"String Theory and M-Theory\", particularly the $ \\frac{\\partial}{\\partial \\bar{\\theta}^A} $ term where the $\\theta$ is the Grassmann coordinates of the super-world-sheet {$\\sigma^\\alpha,\\theta_A$} and A is the spinor index {+,-}? The link to the text is: http:\/\/www.cambridge.org\/us\/academic\/subjects\/physics\/theoretical-physics-and- mathematical-physics\/string-theory-and-m-theory-modern-introduction?format=HB"}
{"id":"122535","title":"How to explain in simple terms why Entanglement is more than just complicated hidden variables","text":"I haven't taken a graduate level physics course on quantum mechanics so I get lost in the strange looking equasions. It's hard for me too see in any of the explanations of how quantum computers and entanglement work without thinking that it looks like a big parlor trick. It's like a magician taking a black and white marble shuffling them in a cup then taking one in each hand without anyone seeing which is in which hand and then revealing a white marble in his left hand and declaring: >   1. I can predict with absolute certainty that the marble in my right hand > is black. >   2. Until I had actually observed the white marble in my left hand, neither > marble actually had a definite color. They were just a superposition of two > possibilities. > A magic trick isn't very convincing unless the audience can understand it and not explain in simpler terms. Are there experimental results that the average person can understand and not explain using classical intuition?"}
{"id":"82413","title":"Big Bang and Spherical Symmetry","text":"If the universe did indeed start with the big bang why is the universe not spherically symmetric? As per Wikipedia entry on Big Bang, (and my understanding as well) big bang is the best explanation as of now. > The Big Bang is the scientific theory that is most consistent with > observations of the past and present states of the universe, and it is > widely accepted within the scientific community. Basically considering the universe as a sphere (all points equidistant from a center) for any given state at any given point, there should have existed an identical state at the other end (line passing through the center) of the sphere. Also, all points equidistant from the center should have had the same 'state'. What is the explanation for the 'asymmetry' that we have today?"}
{"id":"77298","title":"Anticommutatorrelation in Bogoliubov-de Gennes Hamiltonian","text":"I almost solved the problem Equivalence of Bogoliubov-de Gennes Hamiltonian for nanowire. In the next steps I used the notation by arXiv:0707.1692: $$ \\Psi^{\\dagger} = \\left(\\left(\\psi_{\\uparrow}^{\\dagger}, \\psi_{\\downarrow}^{\\dagger}\\right), \\left(\\psi_{\\downarrow}, -\\psi_{\\uparrow}\\right)\\right) $$ and $$ \\Psi = \\left(\\left(\\psi_{\\uparrow}, \\psi_{\\downarrow}\\right), \\left(\\psi_{\\downarrow}^{\\dagger}, -\\psi_{\\uparrow}^{\\dagger}\\right)\\right)^{T}\\text{.} $$ I'm trying to show that the Hamiltonian for a nanowire with proximity-induced superconductivity $$ \\hat{H} = \\int dx \\text{ } \\left[\\sum_{\\sigma\\epsilon\\\\{\\uparrow,\\downarrow\\\\}}\\psi_{\\sigma}^{\\dagger}\\left(\\xi_{p} + \\alpha p\\sigma_{y} + B\\sigma_{z}\\right)\\psi_{\\sigma} + \\Delta\\left(\\psi_{\\downarrow}^{\\dagger}\\psi_{\\uparrow}^{\\dagger} + \\psi_{\\uparrow}\\psi_{\\downarrow}\\right)\\right]\\text{,} $$ can be written as $$ \\hat{H} = \\frac{1}{2}\\int dx \\text{ } \\Psi^{\\dagger}\\mathcal{H}\\Psi $$ with $\\mathcal{H} = \\xi_{p} 1\\otimes \\tau_{z} + \\alpha p \\sigma_{y}\\otimes\\tau_{z} + B\\sigma_{z}\\otimes 1 + \\Delta 1\\otimes\\tau_{x}$ (here $\\tau_{i}$ are the Pauli matrix for the particle-hole space and $\\otimes$ means the Kronecker product). Here I calculate as example the first und third term of $\\Psi^{\\dagger}\\mathcal{H}\\Psi$. $$ \\tau_{z}\\Psi = \\left(\\left(\\psi_{\\uparrow}, \\psi_{\\downarrow}\\right), -\\left(\\psi_{\\downarrow}^{\\dagger}, -\\psi_{\\uparrow}^{\\dagger}\\right)\\right)^{T} = \\left(\\left(\\psi_{\\uparrow}, \\psi_{\\downarrow}\\right), \\left(-\\psi_{\\downarrow}^{\\dagger}, \\psi_{\\uparrow}^{\\dagger}\\right)\\right)^{T} $$ $$ \\Rightarrow \\left(\\left(\\psi_{\\uparrow}^{\\dagger}, \\psi_{\\downarrow}^{\\dagger}\\right), \\left(\\psi_{\\downarrow}, -\\psi_{\\uparrow}\\right)\\right)\\xi_{p}\\left(\\left(\\psi_{\\uparrow}, \\psi_{\\downarrow}\\right), \\left(-\\psi_{\\downarrow}^{\\dagger}, \\psi_{\\uparrow}^{\\dagger}\\right)\\right)^{T} = \\left(\\psi_{\\uparrow}^{\\dagger}, \\psi_{\\downarrow}^{\\dagger}\\right)\\xi_{p}\\left(\\psi_{\\uparrow}, \\psi_{\\downarrow}\\right)^{T} + \\left(\\psi_{\\downarrow}, -\\psi_{\\uparrow}\\right)\\xi_{p}\\left(-\\psi_{\\downarrow}^{\\dagger}, \\psi_{\\uparrow}^{\\dagger}\\right)^{T} = \\psi_{\\uparrow}^{\\dagger}\\xi_{p}\\psi_{\\uparrow} + \\psi_{\\downarrow}^{\\dagger}\\xi_{p}\\psi_{\\downarrow} - \\psi_{\\downarrow}\\xi_{p}\\psi_{\\downarrow}^{\\dagger} -\\psi_{\\uparrow}\\xi_{p}\\psi_{\\uparrow}^{\\dagger} $$ Now I use the anticommutatorrelation $\\\\{\\psi_{\\sigma}, \\psi^{\\dagger}_{\\sigma^{\\prime}}\\\\} = \\delta_{\\sigma,\\sigma^{\\prime}} \\Leftrightarrow \\psi_{\\sigma}\\psi^{\\dagger}_{\\sigma} = 1 - \\psi_{\\sigma}^{\\dagger}\\psi_{\\sigma}$ $$ \\Leftrightarrow 2\\psi_{\\uparrow}^{\\dagger}\\xi_{p}\\psi_{\\uparrow} + 2\\psi_{\\downarrow}^{\\dagger}\\xi_{p}\\psi_{\\downarrow} - 2\\xi_{p} $$ However, the term $-2\\xi_{p}$ here are wrong. For the third term I obtain $$ \\psi_{\\uparrow}^{\\dagger}B\\sigma_{z}\\psi_{\\uparrow} + \\psi_{\\downarrow}^{\\dagger}B\\sigma_{z}\\psi_{\\downarrow} + \\psi_{\\downarrow}B\\sigma_{z}\\psi_{\\downarrow}^{\\dagger} +\\psi_{\\uparrow}B\\sigma_{z}\\psi_{\\uparrow}^{\\dagger} = \\psi_{\\uparrow}^{\\dagger}B\\sigma_{z}\\psi_{\\uparrow} + \\psi_{\\downarrow}^{\\dagger}B\\sigma_{z}\\psi_{\\downarrow} - \\psi_{\\downarrow}^{\\dagger}B\\sigma_{z}\\psi_{\\downarrow} -\\psi_{\\uparrow}^{\\dagger}B\\sigma_{z}\\psi_{\\uparrow} + 2B\\sigma_{z} = 2B\\sigma_{z} $$ Does anybody see my mistake?"}
{"id":"110979","title":"Understanding wave functions","text":"I'm currently writing an essay on the measurement problem, and I'm not quite certain that I've fully understood the purpose of the wave function, in that does the following sentence make sense with respect to the wave function: > _Although the wave function itself is never physically observed, we are > given resultant macroscopic events which we can interpret, allowing us to > formulate a conclusion._ What I mean to say is that although we don't actually see the wave function, we can see the kind of after-math of what happens in the form of a physical macroscopic event..is that right?"}
{"id":"74923","title":"Casimir Effect and polarization of photons","text":"I have read Casimir's derivation of the Casimir fore between 2 parallel plates and have been told that in reality, the Casimir force should be twice as large due to the 2 polarization states of photons. I don't understand how\/why the 2 polarization states affects the force. Could anyone kindly explain?"}
{"id":"66790","title":"Action Reaction when I pushing a trolley?","text":"I tried to explain how those force work but I can hardly figure it out. I exerted a force on the trolley and there will be a force on trolley on me as well. This is the newton's third law. But why the trolley will move? Where's the force come from"}
{"id":"61226","title":"Has anyone studied a statistical scaling law for the universe?","text":"How do named objects in the universe scale? Is there a predictable curve for an ordered list, say {atom, animal, planet, solar system, galaxy, etc}? Can you then use the analysis to predict when the next larger structure will be seen?"}
{"id":"56671","title":"Sign Conventions for Dirac equation","text":"Is it possible to have the Dirac sign convention, (-,+,+,+) and at the same time use the metric $$dt^2-dx^2-dy^2-dz^2$$ i.e have opposing Dirac and metric tensor conventions?"}
{"id":"100929","title":"Rotation motion like the number 8","text":"I'm a college student majoring in culinology and I'm trying to find out the reason or method of the number 8 motion. Responses doesn't have to be in culinology examples, but that would be a great help if it is. From my observations of the circular motions v.s. the figure 8 motions, if I was to mix cake batter with big circular motions in a bowl; the center would not be affected. Mixing the cake batter with small circular motions would have the opposite effect as making big circles. Now mixing the cake batter in a figure 8 motion would hit all the spots, (center & outer area) but you'll have to turn the bowl to make this true. I think the amount of velocity and the force used to mix in these ways would change things."}
{"id":"4637","title":"Electromagnetic fields vs electromagnetic radiation","text":"As I understand, light is what is more generally called \"electromagnetic radiation\", right?   The energy radiated by a star, by an antenna, by a light bulb, by your cell phone, etc.. are all the same king of energy: electromagnetic energy, i.e. photons traveling through space. So far, so good? (if not please clarify) On the other hand, there are these things called \"electromagnetic fields\", for example earth's magnetic field, or the magnetic field around magnets, the electric field around a charge or those fields that coils and capacitors produce. Now here is my question:   * Are these two things (electromagnetic fields and electromagnetic radiation) two forms of the same thing? or they are two completely different phenomena?    * If they are different things, What does light (radiation) have to do with electromagtetism? I'm not asking a complex theoretical answer, just an intuitive explanation. Thanks."}
{"id":"69456","title":"Normal force at the inclined plane","text":"A particle mass $m$ moves down a frictionless inclined plane. Does a normal force cancels a normal component of $m$ **g**?"}
{"id":"100925","title":"Effective Field Theories of QCD","text":"Recently, I am studying the online course Effective Field Theory provided by MIT OCW. Prof. Stewart gives a nice picture to summarize the effective theories: ![Example Effective Field Theories derived from Quantum Field Theory](http:\/\/i.stack.imgur.com\/lXZ23.jpg) As a newbie in this field (I only have a knowledge of QFT on the level of Peskin & Schroeder's book), I am quite confused with so many terminologies. What do they stand for and Why do we need these different theories? Could someone explain this picture in depth (especially on their advantages and limitations)? Any help will be greatly appreciated."}
{"id":"101037","title":"Meaning of $k$ in Sachs-Wolfe formula for angular power spectrum","text":"I've seen the formula for the angular power spectrum of the CMB written as $$C_\\ell = \\frac2\\pi \\int\\left|\\Theta_\\ell(k) \\right|^2 k^2dk, $$ where $\\Theta_\\ell(k)$ is the temperature contrast at a given wavevector $k$ (the specific reference I'm using is the book \"Primordial Cosmology\" by Peter and Uzan (2009), pg. 313). The expression for $\\Theta_\\ell(k)$ is derived from linear perturbation theory in a cosmological background, using _conformal time_ $\\eta$, as opposed to cosmic time $t$. So it seems to me that $k$ in this formula is _not_ the physical wavenumber, but rather the conformal wavenumber: $$ k = ak_\\text{phys},$$ where $a$ is the scale factor. My issue is that if I'm using the conformal wavevector $k$ in the above formula for $C_\\ell$, it seems to be non-gauge invariant. I can make a transformation where I send $a\\rightarrow a\/\\lambda$, $k\\rightarrow k\/\\lambda$, $\\eta\\rightarrow \\lambda \\eta$, and $x^i\\rightarrow \\lambda x^i$. This transformation doesn't seem to affect $\\Theta_\\ell(k)$, but does rescale $k$ so that the transformation of $C_\\ell$ is $$C_\\ell\\rightarrow C_\\ell\/\\lambda^3 $$ This does not seem like a very good feature for a physical observable. My question is how do I get a scaling invariant observable, and more generally what is the correct interpretation of $k$ in that formula?"}
{"id":"101036","title":"Calculating Dynamics of a Thick Sheet of Material","text":"So I'm working on a computational neuroscience problem, and I'm attempting to model skin dynamics for use in some perception models. The problem is as follows: Imagine you had a two dimensional sheet with some thickness. Imagine stimulating it with a probe with negligible spatial extent (i.e. a point source of stimulation) which can deliver arbitrary time varying stimuli (let's say for our purposes it's a sinusoid). The question is basically how can you determine $z(x,y,t)$, the lateral position of the sheet as a function of time for each point on the sheet, given knowledge of $z(0,0,t)$ (which is just how the probe is stimulating the sheet). The idea is basically that the probe will induce travelling waves that will propagate to every other point on the sheet. Now, imagine instead that your probe has spatial extent, in other words, now you know $z(x,y,t)$ for $(x,y)\\in A\\subseteq \\mathbf{R}^2$ where $A$ is essentially the shape of the probe (which we can assume to be simple for now, like a circle or square). Then how can we determine $z(x,y,t)$ for $(x,y)\\in\\mathbf{R}^2$? My suspicion is that you could use the solution to the problem above and essentially integrate over $A$ (i.e. compute the response of the sheet elsewhere as the summation of all the contributions of the point sources of $A$). Finally, imagine instead of a sheet, you had some non-convex shape (like the hand for instance). How would you compute the sheet (skin) dynamics at another point given knowledge of how the probe is moving laterally. I think we've solved the first problem. Essentially, we experimentally derive propagation speeds and how sinusoids decay as a function of distance from the point at which the probe contacts the skin (for instance, we may decide that if you're stimulating the skin with a wave $z(0,t) = A\\cos(2\\pi f t)$, then $$z(r,t) = \\frac{1}{r^{\\gamma(f)}}z(0,t)$$ where the decay parameter as a function of frequency is experimentally derived. Then we extend this to arbitrary stimuli by decaying the wave in fourier space and then inverting it back into a time series. I'm wondering $(a)$ Is this solution correct even in the simple case? $(b)$ How does the solution change when you incorporate the spatial extent of your probe? $(c)$ How do you compute the skin dynamics at another point, where there isn't a straight line connecting contact point of the probe to the point in question? Any help would be greatly appreciated, or at least reference to some literature that might help me begin to think about how to solve this problem?"}
{"id":"93345","title":"What is the probability of ice in boiling water?","text":"Ice crystals are spatially ordered, and in every randomness there is a low possibility of temporarily order. If given enough boiling water, and sufficient time, could local clusters water molecules happen to be in a crystalized state? This may seem absurd, but I believe it must be possible, imagine dropping an ice cube in boiling water(and water vapor) in a perfect closed system, the ice cube melts while the water keeps boiling (because of the water vapor), then because of Poincare recurrence theorem, there will be an similar ice cube after a sufficiently long but finite time. EDIT: The temperature is 100 Celsius and 1 atm (not at triple point)"}
{"id":"160","title":"What are good mechanics experiments for 10 year olds?","text":"I'm trying to explain elementary mechanics - without the benefits of calculus or even algebra - and struggling. I'd like to find reasonable ways to demonstrate Newton's laws, minimally, and possibly continuing though basic projectile motion, although that may be too much to hope for."}
{"id":"74218","title":"Distinction between Larmor frequency and Rabi frequency","text":"Wikipedia said \"In the context of a nuclear magnetic resonance experiment, the Rabi frequency is the nutation frequency of a sample's net nuclear magnetization vector about a radiofrequency field. (Note that this is distinct from the Larmor frequency, which characterizes the precession of a transverse nuclear magnetization about a static magnetic field.)\" However the two definitions seem equivalent to me... Any clarifications?"}
{"id":"41360","title":"Are electrons on a conductor, in electrostatic equilibrium, stationary?","text":"Electrostatics basically means dealing with time independent electric fields (which was produced by stationary charges) Now consider a neutral conductor. We know that putting a net negative charge on the conductor, the charge will very quickly spread over the surface till electrostatic equilibrium is reached. What does this exactly mean? does it mean that electrons on the surface are not \"moving\" anymore and they become stationary? So in this situation we have 2 kinds of electrons: 1-electrons inside the meat of the conductor whose dynamics is described by quantum mechanics 2-electron on the surface which are not moving in the classical sense?"}
{"id":"109346","title":"Why the generators of boosts transform like a vector under rotation?","text":"$$\\left[J_i,J_j \\right]=i\\epsilon_{ijk}J_k$$ $$\\left[J_i,M_j \\right]=i\\epsilon_{ijk}M_k$$ $$\\left[M_i,M_j \\right]=-i\\epsilon_{ijk}J_k$$ where $J_i$ is the generator of rotation of Lorentz group, $M_i$ is the generator of boost of Lorentz group In many textbook of QFT, they say that the second one implies that the generators of the boosts transform like a vector under rotations. But I can't see it explicitly. Can anyone give me the explanation."}
{"id":"105356","title":"Aren't all collisions elastic at some level?","text":"The more I think about it, inelastic collisions produce heat and sound which imply motion at some scale, right? Are inelastic collisions macroscopic events that ignore motion at microscopic levels? Maybe I'm just misunderstanding the definition of an inelastic collision."}
{"id":"105357","title":"Anomalously broken conformal symmetry","text":"I'm trying to understand an argument made by Bardeen in On Naturalness in the Standard Model. The argument is about quadratic divergences in Standard Model. My notation is that the SM Higgs potential contains $\\mu^2\\phi^2$. Bardeen argues that   1. Setting $\\mu^2=0$ would increase the symmetry of the Standard Model by a conformal symmetry. > _But the things we can measure - which are basically the Green's functions - > are no more symmetric with $\\mu^2=0$ than with $\\mu^2\\neq0$! Because the > symmetry is anomalously broken._   2. The conformal symmetry is broken anomalously by non-zero beta-functions.   3. Quadratic divergences do not contribute to the beta-functions.  > _Whilst its true that they don't contribute to the running of the > renormalised parameters, they contribute to the running of the bare mass._ >   4. Quadratic divergences must therefore be a separate (explicit) source of > symmetry breaking. > _Not true if you consider that quadratic divergences contribute to the > running of the bare mass._   5. Quadratic divergences must therefore be an non-physical artefact of our re-normalisation procedure, and we must remove them with, for example, counter-terms, and this is not a fine-tuning. I have added my comments. My main question is, > If conformal symmetry is broken anomalously, why should our Lagrangian > respect conformal symmetry? The Green's function are not more symmetric with > $\\mu^2=0$. I am interested in this for any symmetry, but especially > conformal symmetry. Also, I don't understand 3. and 4. Quadratic divergences would contribute to the running of the bare mass. Wouldn't that break conformal symmetry anomalously? or is it only the renormalised parameters that must have vanishing beta-functions? The distinction seems artificial. I can't understand why quadratic divergences are an explicit source of symmetry breaking, whereas logarithms etc are anomalous. This is key to solving the naturalness problem, and I can't follow the argument. My feeling is that these arguments are faulty (which makes me think I must be making mistakes because Bardeen is a real expert who has surely thought a lot about it!). I'm certainly unconvinced. Have they been confirmed\/refuted at length in the literature?"}
{"id":"43273","title":"Imagining to speed up the Earth's revolution around the Sun until we feel its motion","text":"If the Earth were revolving around the Sun much faster than it does, let's say 100 or 1000 times faster or whatever big-enough multiple of its actual speed, would we feel it? What sensation would we feel? How could I estimate how big the revolution speed would be for us to feel it? Would it be possible to speed up the Earth motion around the Sun? How? What basic physics concepts can be used to explain what would happen?"}
{"id":"105358","title":"Are Negative Eigen Values of a Hessian Matrix physically acceptable?","text":"Suppose I have a Hessian Matrix of a System with 3N degrees of freedom, What are the physical significance of eigen values of the Hessian, Are negative Eigen Values physically acceptable?"}
{"id":"70824","title":"What do we mean when we say the QM wave function is a section of the $U(1)$ bundle?","text":"I have a couple questions here. To keep the discussion simple lets stick to the following case: what is the quantum mechanics of a single particle in the presence of a background EM field, such as that of a monopole, where multiple coordinate patches are required to define the vector field everywhere without singularity. I have learned about the geometric formulation of EM - where the field $B$ of a monopole is described as a real-valued two-form that is closed but not exact. I also know about using the vector potential as a connection for the complex wavefunctions, using the usual covariant derivative. The questions I have in mind are >   1. What (complex line) bundle is the wave function a section of? >   2. What are the bundle data (i.e. transition functions) that specify this? >   3. In what way can we see the monopole vector potential as a connection on > this bundle? >   4. People seem to talk about the monopole bundle only with the monopole > connection in mind - what would it look like to have a different connection > instead? > On a similar note, I've seen physicists refer to the classification of bundles, which there is one for every integer (based on the homology of $\\mathbf{R}^3-\\\\{0\\\\}$), and call that integer the monopole number. But I've also seen people call the flux integral of the magnetic field two form over some closed surface, which relates instead to the curvature of some _connection_ on the bundle - which should be quantized by Gauss-Bonnet theorem - the monopole number. Is there some reason that these are the same? I was reading a great set of lecture notes around 10 pages which explained the need to use a section of a bundle for the wave function and the relation to the Dirac quantization condition, but I can no longer find them. Links to related resources would be very welcome."}
{"id":"123542","title":"Question on circular motion — car turning on road","text":"Lets say a car is turning on a road that has friction. I made this small diagram ![enter image description here](http:\/\/i.stack.imgur.com\/3fhVd.png) where the two parallel lines indicate the tires of the car. In order for the car to keep turning in a circle, there must be an force directed to the center of curvature — the centripetal force. I have read that this force comes as a result of friction. However, in order for there to be a frictional force towards the center, doesn't there have to be a force directly opposite pulling away from the center, as frictional forces cannot exist by themselves — they have to act in order to counter an opposite force. Thus, how can there be a frictional force towards the center without a corresponding force pulling away from the center? Does this have something to do with the centrifugal pseudo-force? Further, where exactly does this frictional force come from? Also, why is the frictional force directed towards the center? Shouldn't it be opposite the direction the car is traveling at the moment -- thus in a direction opposite the tangential?"}
{"id":"18945","title":"Which physically acceptable quantum interpretations do not require the existence of any observer at all?","text":"Some interpretations of quantum mechanics -- like the Copenhagen interpretation in particular -- require the existence of an observer. The role of the observer is a bit mysterious. After all, observers are not needed in classical mechanics. Which physically acceptable interpretations of quantum mechanics do not require the existence of any observer at all? By that, I mean they would continue to make sense even in the absence of any observer. The many worlds interpretations appears to require an observer to subjectively determine which branch is selected."}
{"id":"98633","title":"Line of action force","text":"The task is to reduce the two forces into a single net force and then find the equation of its line of action. I chose to find the net force and momentum with respect to the origin. The magnitude of each force is $P$. ![asd](http:\/\/draw.to\/static\/d\/3yezUM.png?v=ec211) We can therefore write $F_{net} = (P,0,0) + (0,0,-P) = (P,0,-P)$. The distance vector from origin to $F_1$ is $(0,b,c)$ and from origin to $F_2$ is $(a,0,c)$. The net moment is then $$M_{net} = (0,b,c)\\times (P,0,0) + (a,0,c)\\times (0,0,-P) = (0, aP + cP, -bP ).$$ What I tried to do was to assume the attacking point of the net force to be $(x,y,z)$. This force must cause the net moment with respect to the origin. So I set up the cross product $$ (x,y,z) \\times (P,0,-P) = (0, aP + cP, -bP).$$ but I was unable to determine $x$, $y$ and $z$. So what can we do?"}
{"id":"132813","title":"Joint probability function for the values of a free field at two different points","text":"For a free real field $\\phi$ in its ground state, is there a way to find the probability distribution $p(\\phi_x,\\phi_y)$ for joint measurement of $\\phi(x)$ and $\\phi(y)$ at two spacelike-separated points $x$, $y$? Presumably the information needed to construct $p(\\phi_x,\\phi_y)$ would be contained in the expectation values of products of powers of $\\phi(x)$ and $\\phi(y)$, since we could invert the formula $$ E[\\phi(x)^n\\phi(y)^m] = \\int d\\phi_xd\\phi_y p(\\phi_x,\\phi_y)\\phi_x^n\\phi_y^m $$ but the expectation values are divergent for $n$ or $m$ greater than $1$."}
{"id":"132819","title":"Trying to make a realistic simulation of 'breaks' (Pool)","text":"I'm working on a 2D pool simulation project right now, and I'm trying to create good looking physics that are also performant. After collisions are detected, I update the velocities of the balls. So far, I've assumed collisions are elastic, which gives pretty good results. The formulas for the final velocities of both balls after a 2D elastic collision are easy to find, and I can follow the derivations of them well enough. The problem is that 'breaks' are not realistic: momentum is often passed along the top and bottom of the triangle of balls, and only the top and bottom corners fly off. I was wondering if changing the equations to use partially inelastic collisions would make the break look more realistic. I would use a coefficient of restitution that is close to 1. However, I'm not able to find many examples of 2D partially inelastic collisions that give final velocities. I'm not sure if I can derive these easily myself, and I'm a bit short on time. Would this approach be worthwhile? Will using slightly inelastic collisions give me the results I'm looking for? If so, are there any resources that can help me solve this problem?"}
{"id":"132818","title":"Aftermath of Plasma Arc","text":"You have a chamber with constant amount of gas (say either helium or argon). You create the plasma with an arc and turn it off. How much of the gas used to create the plasma returns back to helium or argon as opposed to dissipating into the chamber or creating new elements?"}
{"id":"83148","title":"Where I can find a torsional stiffness table for different types of stainless steels?","text":"I am trying to comparing the torsional stiffness of different types of stainless steels. I googled the keywords but I could not find any useful information. Would anyone please give me some links which provide those information?"}
{"id":"18367","title":"How does electron capture occurs?","text":"Electron capture is a kind of decay by which a nuclear transformation takes place. Below is an example of it. $$ _{29} ^{64} \\text{Cu} + e^- \\rightarrow _{28}^{64} \\text{Ni} $$ Of course, nucleus being positively charged and electron being negatively charged, there is an attraction between them, but electron generally don't fall in the nucleus. I am unable to understand that exactly how does nucleus is able to capture electron, using which interaction or force? Uncertainty principle won't allow electron in nucleus, though it is being captured. Exactly how?"}
{"id":"14468","title":"How long did inflation take to happen?","text":"If inflation happened, it ended about 13.7 billion years ago. But how long was inflation going on before that? Was it just a short burst of time so that it's really okay to say that the age of the universe is 13.7 billion years, or was it going on for a long time before it ended (maybe forever?)?"}
{"id":"121412","title":"Poynting's Theorem simplified?","text":"How can I apply Poynting's theorem to any system that has a magnetic field & electric field, to state conservation of energy? How does Poynting's theorem state conservation of energy in EM? I struggled to understand with the differential equations. Example: Work being done by the Lorentz force on a conductor by a supplied power source?"}
{"id":"24832","title":"Matter - Antimatter Reactory Practicality","text":"With **current technology** , would the energy released by a matter-antimatter annihilation be more than the energy needed to created the antimatter in the first place? Would it be worth it? Just curious, even if was on a small scale, Could one produce a machine that uses annihilation to produce a net energy gain?"}
{"id":"30613","title":"What is spontaneous symmetry breaking in QUANTUM GAUGE systems?","text":"Wen's question What is spontaneous symmetry breaking in QUANTUM systems? is cute, but here's an even cuter question. What is spontaneous symmetry breaking in QUANTUM GAUGE systems? There are some gauge models where the Higgs phase and the confinement phase are continuously connected, you know. Why are the Higgs phase and the confinement phase identical in Yang-Mills-Higgs systems? You mention cluster decomposition? Sorry, because bilinear operators have to be connected by a Wilson line."}
{"id":"15594","title":"Stablising the Higgs without SUSY","text":"Should the Higgs be found at the LHC, but no supersymmetry (assuming for the sake of argument that the LHC be capable of eliminating all versions of SUSY that are motivated by solving the hierarchy problem), what then? Are there alternatives for stablising the Higgs scale that don't involve SUSY (other than Randall-Sundrum)?"}
{"id":"15599","title":" Why will clay, when heated to high temperatures,  harden and become waterproof?","text":"I have a question > Why will clay, when heated to high temperatures, harden and become > waterproof?"}
{"id":"31561","title":"One-dimensional (1D) CdSe nanostructures","text":"I'm not a physicist, but I'm really interested in Nanotechnology. I've a question, and I hope that I can be provided by an answer. I was reading some articles about the one-dimensional CdSe nanostructures. My question is why it's called one-dimensional? Is it because of their geometrical shapes or something else? I truly appreciate any hint or a resource could lead to the answer. Thanks."}
{"id":"89014","title":"Using Lagrange's Equations with Generalized forces","text":"I am a bit confused on how this works. For instance if I wanted to look at an object moving in 2 dimensions only subject to gravity (and assuming that the potential is just mgy), I get that my Lagrangian is: $$\\frac12m(\\dot x^2+\\dot y^2)-mgy$$ I can easily solve the Euler-Lagrange equations, to see that $\\ddot y=-g$ but is there a way I could do this with generalized forces. I feel that this is an awful example. I vaguely remember my professor using generalized forces to find friction force. Any help with understanding generalized forces would be appreciated. The reason I want to know is that I am trying to model a curveball (baseball) and treating the lift and drag as generalized forces then using the E-L equations to solve for the motion."}
{"id":"89017","title":"Continuously feeding an evaporating micro-black hole?","text":"What would happen if you created a micro-black hole and could continuously feed it as quickly as it evaporates? Is it possible that it would remain relatively stable? If so, how might such a thing be continuously fed?"}
{"id":"38372","title":"What kind of cooling mechanism could be used in outer space?","text":"This question arises out of this question on Quora - Apollo 11: 1969 Moon Landing: Did Neil Armstrong really land on the moon? I'm convinced with most of the explanations provided in the first answer to that question. None of these conspiracy theories have bothered me. However one thing that confuses me is the heat. On earth close to 47% of heat is absorbed or reflected by the atmosphere. On moon an object would get struck directly with thermal radiation from sun acquiring about 50% more heat then it would on earth. I am wondering what kind of cooling device could we create for such an environment? A system like an AC or a refrigerator won't work in a vacuum. Off the top of my head the only method I could think of is circulating some liquid with high specific heat in the suits and letting it evaporate. But is this feasible? How much heat do we need to loose? What rate of evaporation would be needed? **What volume of liquid would be needed** to support this rate? Given the high amount of electricity it takes to makes our refrigerators work **how much electricity would such a system need?** Because of very large amount of heat and large surface area (the lunar module, the suits, and the camera casings) I'd assume a large quantity of water (say a few thousand liters) would be needed. How could they carry such a large amount of fluid?"}
{"id":"120042","title":"What can make signal triangulation tricky?","text":"Usually, when you're trying to find a source of whatever signal, you move around the source and then calculate the position using triangulation. I'm specifically talking about low frequency signals, such as radio waves. I was wondering what can make this process harder or completely impossible (the former seems very unlikely).   If there were two transmitters, would it be easy to tell that there are two and not one? Could the signal be dispersed unevenly in the area, making it appear somewhere else than it is?"}
{"id":"47011","title":"Witten's constrained S-matrix and Coleman-Mandula Theorem","text":"I remember reading somewhere that Witten argued that if the Poincaré symmetry of spacetime were nontrivially combined with internal symmetries, then the S-matrix would be so constrained that the scattering amplitude would be non- vanishing at discrete angles only. I would like to know more about this:   1. At _which_ angles would the scattering amplitude be non-vanishing   2. By crossing symmetry, wouldn't this mean that in the crossed channel, the amplitude be non-vanishing at discrete _energies_?   3. Is there a reference that contains more information on this?"}
{"id":"103590","title":"Thrust to Weight ratio in Space with an off set CoM","text":"With regards to this thread, Thrust center in space My question is, if the thrust to weight ratio was increased so that it was much higher than the weighted mass of the sphere (ship), would the sphere then not start to follow a straight trajectory along the axis of the thruster."}
{"id":"111246","title":"In what applications are lasers necessary vs. just convenient?","text":"I’m trying to understand, given the wide array of current laser applications:   1. Which applications actually require laser characteristics, as opposed to just use lasers because they are the most practical\/efficient source of light with characteristics _not unique_ to lasers   2. What alternatives exist, or may exist, to lasers for applications that don’t require laser characteristics My key assumption here is that the essential characteristic of a laser is that it produces temporally coherent light. Often, but not necessarily, lasers produce light with two other useful characteristics:   1. Monochromaticity   2. Collimation But these convenient characteristics can be produced using non-lasers. For example, LEDs produce monochromatic light. And lenses can take diffuse light and produce a beam collimated to near the diffraction limit, right? If this is the case, then the only applications that require lasers are those that depend on interferometry, and that’s pretty much just holography and measurement, right? All of the other applications – directed energy, medical therapies, optical discs, optronics – could be accomplished with non-coherent collimated and\/or monochromatic light, right? Presumably lasers are used in these cases only because they are currently the most efficient or practical way of producing light with those characteristics. But now I am wondering: What other technologies could supersede lasers in practice (and efficiency) for applications that require light with high energy, collimation, and\/or monochromaticity?"}
{"id":"12022","title":"Geometric\/Visual Interpretation of Virasoro Algebra","text":"I've been trying to gain some intuition about Virasoro Algebras, but have failed so far. The Mathematical Definition seems to be clear (as found in http:\/\/en.wikipedia.org\/wiki\/Virasoro_algebra). I just can't seem to gain some intuition about it. As a central extension to Witt Algebras, I was hoping that there has to be some geometric interpretation, as I can imagine Witt Algebras rather well. If anyone has some nice Geometric or Visual Interpretation of Virasoro Algebra, I'd greatly appreciate it!"}
{"id":"19906","title":"Effects of parallel superconducting plates","text":"Assuming the existence of virtual particle field ( zero point energy field) Casimir force is produced by 2 parallel conducting plates excluding some of the frequencies between the plates, if these 2 plates are superconducting does that change the amount of frequencies being excluded? One theory I've read postulates that it is interaction of matter with the ZPF (=zero point field) that is the origin of mass and inertia, if the parallel plates are suppressing the appearance of some of these virtual particles, would that not in some manner be modifying inertia for matter passing between the plates?"}
{"id":"119830","title":"Why is work in a constant electric field equal to the force times distance?","text":"Let's say, for example, that an infinite plate generates a constant electric field that exerts a force of 6N on a 1C charge. To move the charge from a distance of 5m to 1m from the plate, the work done is given by 6N * 4m = 24Nm. The way this has always been explained is that we apply a 6N force over the 4m distance. However, this doesn't make intuitive sense to me:   * If we apply a constant 6N force to the charge in the opposite direction of the electric field, won't the charge simply remain stationary? The force generated by the field and our applied force in the opposite direction should net to zero.    * If the electric field exerts a force of 6N on the charge at every single infinitesimal point along the path from 5m to 1m, when we apply an opposite force of 6N over that same distance, shouldn't the charge simply stay still?   * I think what's troubling me is that I don't see how applying a 6N force can \"overcome\" the electric field's force and actually move the charge in the opposite direction. I think the intuition is the same for the force of gravity. A 1kg object that is 1m above the ground is experiencing a force downwards of 10N. If I apply a constant upwards force of 10N, the object won't fall. But how is that if I apply a 10N upwards force over 1m, I can actually move the object upwards? Gravity is exerting a downwards 10N force on the object the entire time I am exerting an upwards 10N force over that 1m. If the forces are equivalent, shouldn't the mass stay still?"}
{"id":"97693","title":"Understanding Triplet And Singlet States","text":"We know, $2\\otimes 2=3\\oplus 1$. Thus we have a spin triplet of states and a spin singlet. Can we regard these states as the spin part of wavefunction for the excited states and the ground state of the deuteron nucleus? What is the difference between the members of a triplet?"}
{"id":"120151","title":"Connection between the Big Bang and Black Holes","text":"So... this is my first question here after a little while of lurking; so be gentle. I've been thinking lately about trying to wrap my head around the concept of spacetime and how it can be affected by gravity and the density of matter in space (i.e. near a black hole). I've been wondering. Is it possible that matter - which is broken down (spaghettified?) into it's constituent parts and into energy - is drawn to a point in spacetime which has already got a massive density? Would it be possible that any black hole, which has been or will ever be, causes energy to be drawn to a single point in space or time? I.E. all black holes are \"feeding\" into a single, infinitesimal time and location; causing (possibly the wrong word - it seems to me that it would be less cause vs. effect and more that it just _is_ ) the big bang? I tried to find some papers that would touch on the topic but couldn't find anything specific - which likely means I'm being a total dolt and there are many good reasons why this isn't the case :) I would appreciate any information which would correct my thinking and counter my naivety."}
{"id":"86392","title":"Current flowing through a square loop - Magnetic Field","text":"> _A square conducting loop of wire of side $2a$ carries a uniform current I. > Calculate the magnetic field at the centre of square due to the current in > the loop._"}
{"id":"86396","title":"What is the meaning of $U''(x)=0$?","text":"Most potentials with a minimum can be described approximately as a harmonic oscillator. So the procedure is to Taylor expand $U(x)$: $$U(x)=U(0)+U'(0)x+\\frac{1}{2}U''(0)x^2 +...$$ If we suppose that the potential is cero at the origin an it has a minimum there, we get: $$U(x)=\\frac{1}{2}U''(0)x^2$$ We take $U''(0)$ to be the _spring_ constant $k$. So the angular frecuency is given by: $\\omega=\\sqrt{\\frac{k}{m}}$ But what if $U''(0)=0$ and there is still a minumum at zero, like a potential $U(x)=x^4$? In this case if you blindly apply the formula you get zero frecuency, which is false. Does it just mean that to a small approximation a body will not oscillate?"}
{"id":"120157","title":"What is the rebound angle of a circle striking a concave surface?","text":"First, pardons to all because I don't have a physics background and I'm probably using laymans terms. What is the rebound angle of a circle striking a concave surface? Assume the simplest case:   * the circle (red) is not spinning   * the circle does not compress   * the arc (blue) does not deform   * there's no friction   * there's no energy lost in the impact Here's my attempt at an illustration with a red circle moving in a blue circle traveling towards the blue circumference along the gold path and rebounding on an unknown green angle. ![enter image description here](http:\/\/i.stack.imgur.com\/tt3gB.png)"}
{"id":"64627","title":"Why does increasing the temperature of a thermistor decrease it's resistance?","text":"Surely, upon an increase in temperature, the atoms within the thermistor would vibrate with more energy and therefore more vigorously, hence making the electrons flowing through the electric circuit more likely to collide with one of the atoms, so increasing resistance. However, the effect of temperature on a thermistor is contrary to this. I can't understand how it can be. It's analogous to running across a playground: if everyone is still you're less likely to collide with someone, however if everyone is constantly moving from left to right then a collision is more likely. So why does an increase in temperature decrease the resistance of a thermistor?"}
{"id":"100025","title":"Horizon problem in the standard cosmological model","text":"The horizon problem is usually stated as follows: Causally disconnected parts of the universe that we see today have the same temperature, so how could that be so if no information travels faster than the speed of light. But (without assuming inflation) wasn't the universe causally connected right after the big bag? And that those parts that are disconnected today only became disconnected due to expansion? Also, if temperature irregularities existed but to a very minute level, then why do we expect that the universe without inflation must look different than the homogeneous universe (on large scales) we see today?"}
{"id":"69216","title":"Clarity in Electric field Definition?","text":"The electric field at a point is defined as q\/$r^2$. How does this definition take the nature of source and test charge into consideration. If I bring any positive\/negative charge around the source charge, it brings a change in the distances involved due to forces of attraction or repulsion.how can I use this definition for practical purposes?"}
{"id":"69213","title":"antiferromagnetic spin wave","text":"I have a hamiltonian that is derived from a spin wave energy dispersion calculation for a nearest neighbor interacting cubic antiferromagnet. After a Holstein-Primakoff transformation and making a substitution of -q for q in one place, I get the following hamiltonian (J is the exchange energy, S is the magntiude of the spin on each sublattice, z=6 is the number of nearest neighbors of each site, and $\\mathop \\gamma \\nolimits_q = \\mathop e\\nolimits^{i*\\mathop q\\nolimits_x } + \\mathop e\\nolimits^{ - i*\\mathop q\\nolimits_x } + \\mathop e\\nolimits^{i*\\mathop q\\nolimits_y } + \\mathop e\\nolimits^{ - i*\\mathop q\\nolimits_y } + \\mathop e\\nolimits^{i*\\mathop q\\nolimits_z } + \\mathop e\\nolimits^{ - i*\\mathop q\\nolimits_z } $ : $$ \\sum\\limits_q {\\left( {\\begin{array}{*{20}{c}} {\\mathop a\\nolimits_q^\\dagger }&{\\mathop b\\nolimits_q^\\dagger }&{\\mathop a\\nolimits_{ - q}^\\dagger }&{\\mathop b\\nolimits_{ - q}^\\dagger }&{\\mathop a\\nolimits_q }&{\\mathop b\\nolimits_q }&{\\mathop a\\nolimits_{ - q} }&{\\mathop b\\nolimits_{ - q} } \\end{array}} \\right)\\left( {\\begin{array}{*{20}{c}} {JzS\/2}&0&0&0&0&0&0&{JS\\mathop \\gamma \\nolimits_q \/2}\\\\\\ 0&{JzS\/2}&0&0&0&0&{JS\\mathop \\gamma \\nolimits_q \/2}&0\\\\\\ 0&0&{JzS\/2}&0&0&{JS\\mathop \\gamma \\nolimits_q \/2}&0&0\\\\\\ 0&0&0&{JzS\/2}&{JS\\mathop \\gamma \\nolimits_q \/2}&0&0&0\\\\\\ 0&0&0&{JS\\mathop \\gamma \\nolimits_q \/2}&{JzS\/2}&0&0&0\\\\\\ 0&0&{JS\\mathop \\gamma \\nolimits_q \/2}&0&0&{JzS\/2}&0&0\\\\\\ 0&{JS\\mathop \\gamma \\nolimits_q \/2}&0&0&0&0&{JzS\/2}&0\\\\\\ {JS\\mathop \\gamma \\nolimits_q \/2}&0&0&0&0&0&0&{JzS\/2} \\end{array}} \\right)\\left( {\\begin{array}{*{20}{c}} {\\mathop a\\nolimits_q }\\\\\\ {\\mathop b\\nolimits_q }\\\\\\ {\\mathop a\\nolimits_{ - q} }\\\\\\ {\\mathop b\\nolimits_{ - q} }\\\\\\ {\\mathop a\\nolimits_q^\\dagger }\\\\\\ {\\mathop b\\nolimits_q^\\dagger }\\\\\\ {\\mathop a\\nolimits_{ - q}^\\dagger }\\\\\\ {\\mathop b\\nolimits_{ - q}^\\dagger } \\end{array}} \\right)} $$ I then, as outlined in this paper (http:\/\/jmp.aip.org\/resource\/1\/jmapaq\/v19\/i1\/p277_s1) solve the equation det(H-x*{{1,0},{0,-1}})=0 for x, where H is the 8x8 matrix above and 1 is the 4x4 identity matrix. This procedure gets the right answer for the energy x, as expected. However, if I transform sums over $ \\mathop b\\nolimits_{ - q}^\\dagger \\mathop b\\nolimits_{ - q} $ to $ \\mathop b\\nolimits_q^\\dagger \\mathop b\\nolimits_q $ and similarly for a_q, I get this hamiltonian: $$ \\sum\\limits_q {\\left( {\\begin{array}{*{20}{c}} {\\mathop a\\nolimits_q^\\dagger }&{\\mathop b\\nolimits_q^\\dagger }&{\\mathop a\\nolimits_{ - q}^\\dagger }&{\\mathop b\\nolimits_{ - q}^\\dagger }&{\\mathop a\\nolimits_q }&{\\mathop b\\nolimits_q }&{\\mathop a\\nolimits_{ - q} }&{\\mathop b\\nolimits_{ - q} } \\end{array}} \\right)\\left( {\\begin{array}{*{20}{c}} {JzS}&0&0&0&0&0&0&{JS\\mathop \\gamma \\nolimits_q \/2}\\\\\\ 0&{JzS}&0&0&0&0&{JS\\mathop \\gamma \\nolimits_q \/2}&0\\\\\\ 0&0&{0}&0&0&{JS\\mathop \\gamma \\nolimits_q \/2}&0&0\\\\\\ 0&0&0&{0}&{JS\\mathop \\gamma \\nolimits_q \/2}&0&0&0\\\\\\ 0&0&0&{JS\\mathop \\gamma \\nolimits_q \/2}&{JzS}&0&0&0\\\\\\ 0&0&{JS\\mathop \\gamma \\nolimits_q \/2}&0&0&{JzS}&0&0\\\\\\ 0&{JS\\mathop \\gamma \\nolimits_q \/2}&0&0&0&0&{0}&0\\\\\\ {JS\\mathop \\gamma \\nolimits_q \/2}&0&0&0&0&0&0&{0} \\end{array}} \\right)\\left( {\\begin{array}{*{20}{c}} {\\mathop a\\nolimits_q }\\\\\\ {\\mathop b\\nolimits_q }\\\\\\ {\\mathop a\\nolimits_{ - q} }\\\\\\ {\\mathop b\\nolimits_{ - q} }\\\\\\ {\\mathop a\\nolimits_q^\\dagger }\\\\\\ {\\mathop b\\nolimits_q^\\dagger }\\\\\\ {\\mathop a\\nolimits_{ - q}^\\dagger }\\\\\\ {\\mathop b\\nolimits_{ - q}^\\dagger } \\end{array}} \\right)} $$ But solving the equation det(H-x*{{1,0},{0,-1}})=0 (where H is the new 8x8 matrix above and 1 represents a 4x4 identity matrix) does not give me the same answer. I'd really like to know what is going on. Thanks!"}
{"id":"63075","title":"Escape velocity to intersection of two gravitational fields","text":"> Find the minimal velocity needed for a meteorite of mass $m$ to get to earth > from the moon. Hint: the distance between the center of earth and the center > of moon is $\\approx 60 R_E$, and the meteorite should reach a certain point > $O$ on that distance, where the gravitational forces of the moon and the > earth are canceling each other out. It is located $\\approx 6 R_E$ from the > center of the moon. I thought that the best way to solve this question is to find the difference between the initial enegy of the body and the final one at $O$. Beyond that point, the gravitational pull of the earth will 'overcome' the gravitational pull of the moon, and the object will gather speed itself by reducing the GPE in the earth's gravitational field. Therefore: $U_{G,i}=-G \\frac{M_{moon}m}{R_{moon}}$ $U_{G,f}=-G \\frac{M_{moon}m}{6R_E}-G \\frac{M_E m}{54R_E}$ $U_{G,f}-U_{G,i} \\approx m \\cdot 1.51 \\cdot 10^6 \\text{J}$ (there's no mistake in the calculation) $E_k=m \\frac{v^2}{2}=m \\cdot 1.51 \\cdot 10^6$ Therefore $v \\approx 1739.97 \\text{m\/s}$ However this answer is wrong. The correct magnitude of the speed should be somewhere around $2.26 \\text{km\/s}$ which is close to the escape velocity from the moon ($\\approx 2.37 \\text{km\/s}$). Where am I wrong? Why the difference in GPE is not satisfying the problem?"}
{"id":"90471","title":"Questions about voltage","text":"For some reason, I feel like the concept of voltage is escaping my grasp. I've done much research on these forums and through texts, and come across answers that seem quite well thought out, but still cant grasp the fundamental concept of what voltage is. I could explain (regurgitate) the definition of voltage in many ways, and plug it into equations, but when it comes to testing my fundamental understanding of it as a natural concept, I fall short. Ill explain what I know, and what I don't, and I have some theoretical questions which I hope will help give me a better understanding. **Here is what I think I know:** 1\\. Voltage is electrical potential energy. It represents an imbalance of charge distribution.   1. This means that given a sufficient conductor, the electrons will do work by moving to a state of equilibrium to correct this imbalance.    2. Voltage must be measured between two points, because it can only exist in relation to an imbalance between one point and another.   3. Measures of voltage determine the work which can be done as a result of the imbalance?   4. The quantity of voltage in a battery would not change if you increased the size of the battery. It would only change if you increased the average negativity charge density in the negative terminal? **Here is what I don't know or have questions about:** 1\\. So, increasing voltage increases current because higher voltage means a higher disequilibrium from the source, meaning more \"pressure\" on the electrons to move? Given a fixed wire diameter, does that means higher voltage would make electrons move faster?   1. Why do resistances in series add together? I don't understand. I picture a resistor as a bottleneck, in that the entire circuit is limited by the resistance of the most resistant component. But this is clearly wrong. Is it more correct to imagine that every time electrons travel through a resistor, they have to do more work and lose some of their ability to do future work, and have lower and lower energy the more resistors they go through?   2. If a current\/stream of electrons begins to flow and is traveling through a wire with 1 ohm resistance, and goes through a 5 ohm resistor, then back to a 1 ohm resistance wire, since resistance decreases, would it speed up again upon exiting the 5 ohm resistor? If so, I'm assuming its new speed (relative to its initial speed upon entering the circuit) would be proportional to the fraction of remaining voltage after the total voltage drop up to that point?   3. When a battery dies, I'm assuming that its voltage isn't at 0. Does a dead battery just no longer have the minimum voltage required to supply the circuit with enough power to power the device? In that case does that mean that dead batteries could still power a circuit with less resistance or lower power demands?   4. This quote from Wikipedia confuses me: \"In a series circuit, the current through each of the components is the same, and the voltage across the circuit is the sum of **the voltages** across each component\". What does it mean that components \"have voltage\". I'm assuming this means the drop in voltage from before the component compared to after, right? But what is the reasoning for saying the component \"has\" voltage? Is it trying to express the quantity of work done (i.e. heat emitted) from that component?   5. Last one, another wiki entry: \"In a parallel circuit, the voltage across each of the components is the same\". This confused me, but I think I just figured it out. If a current is flowing through multiple paths, it would take the path of least resistance to its destination, thus if a parallel circuit has a 1 ohm resistor and a 5 ohm resistor in parallel, both would have the same voltage drop, but the current flowing through the 5 ohm resistor would be 1\/5th the value of the 1 ohm? Thanks for taking the time to read my post. I hope I didn't get everything terribly wrong."}
{"id":"46850","title":"The state of Indefinite metric in Quantum Electrodynamics","text":"I faced difficulties to grasp why indefinite metric is introduced from no where in QED, after searching internet I found that this is a problem in QED, because one needs it to preserve theory's covariance, and to solve ultraviolet divergence problem, but in the same time it raise other different problems such as negative probabilities, and even from mathematical point of view that it's can't be introduced a proper topology for a Hilbert space with such metric. (those point I get from N. Nakanishi: Indefinite-Metric Quantum Field Theory. _Progress of Theoretical Physics Supplement_ No.51 (1972) pp. 1-95, doi:10.1143\/PTPS.51.1) Thus those information may be old, I would like to know the official (main stream) position of this problem, and if it considered as solved, especially that there is some papers that refuses this kind of metric and some usual books on QED doesn't mention it all what makes me even more confused."}
{"id":"58046","title":"Modeling incoming solar radiation","text":"I want to write a model for estimating incoming solar radiation for a specific latitude on earth but I am struggling to find an appropriate source which shows the required equations for doing so. Would anyone be able to provide me with a link to where I can find equations for estimating solar radiation (irradiance) given a specific cloud cover, latitude, time of day, and day of year?"}
{"id":"135403","title":"estimating the rotational speed of dc motor","text":"What is the best strategy to roughly estimate the RPM of simple DC motor as seen in this video. I want to estimate the max possible rotation of the coil that has $n$ number of turns and has moment of inertia $I$ for a uniform field $B$. The one side of the axle has been scraped so what current only flows along one half of the rotation.   ![enter image description here](http:\/\/i.stack.imgur.com\/8cNCH.png) There are factors like friction that slows down motion and back emf that reduces current. How do I put those into one equation for estimation. Or is there any better way? Any helpful links will be appreciated. Thanks for your time."}
{"id":"53742","title":"Frequency of an Electron","text":"My question is very simple. If frequency is defined as the cylces per unit time, Then what is meant by \"Frequency of an Electron\" ? If the rotation of electron around a nucleus is considered then, Which phenomena we consider for a free electron i.e: electron in a force field (unstable alone electron). EDIT: Is \"Frequency of an Electron\" a experimental quantity? **EDIT AFTER HAVING 2 ANSWERS:** My teacher told me how to calculate the frequency of an electron. We started from finding Energy of Electron then difference in energy, then we get this equation according to Bohr's Radius of Hydrogen Atom and his postulates: i.e: $$ f = \\frac{z^2e^42\\pi^2m}{h^3} (\\frac{1}{n_1^2} - \\frac{1}{n_2^2}) $$ Where:   * $z =$ Atomic number    * $e =$ Charge on Proton   * $m =$ mass of electron   * $h =$ Plank's constant    * $n =$ Orbit number From the last part of my equation, I am confused. The $n_1$ and $n_2$ shows that, that frequency will be the frequency of energy? or electrons?"}
{"id":"108500","title":"Inertia matrix of a rod rotating about an axis","text":"I'll provide a picture for clearer understanding. The problem is to calculate the angular momentum of the rod rotating about the z-axis. I have serious difficulties in deriving the inertia matrix, that's all I need help with. ![enter image description here](http:\/\/i.stack.imgur.com\/WGzDa.jpg) **Progress** Since the rod is rotating about the z-axis $\\Rightarrow \\omega _{x}=\\omega _{y}=0$ the angular momentum simplifies to $ \\boldsymbol{L}_{O}=-I_{xz}\\omega _{z}\\hat{\\mathbf{i}}-I_{yz}\\omega _{z}\\boldsymbol{\\hat{j}}+I_{zz}\\omega_{z}\\boldsymbol{\\hat{k}} $ Split the rod in three parts (SEE FIGURE) and calculate for each body. I have correct terms for body **A**. For **B** : $I_{xz}=\\overline{I}_{xz}+md_{x}d_{z} = 0 +\\rho b(0)(\\frac{b}{2})=0$ $I_{yz}=\\overline{I}_{yz}+md_{y}d_{z} = 0 +\\rho b(b)(\\frac{b}{2})=\\frac{1}{2}\\rho b^3$ **Wrong** $I_{zz}=\\overline{I}_{zz}+md^2 = 0 + (\\rho b)(b^2+(\\frac{b}{2})^2)=\\frac{5}{4}\\rho b^3$ (d is distance from midpoint of [B]B[\/B] to origin O) For **C** : $I_{xz}=\\overline{I}_{xz}+md_{x}d_{z} = 0 +\\rho b(\\frac{b}{2})(b)=\\frac{1}{2}\\rho b^3$ $I_{yz}=\\overline{I}_{yz}+md_{y}d_{z} = 0 +\\rho b(b)(b)=\\rho b^3$ **Wrong**$I_{zz}=\\overline{I}_{zz}+md^2 = \\frac{1}{3}(\\rho b)(b^2)+(\\rho b)((2b^2)^2+(\\frac{b}{2})^2)=\\frac{55}{12}\\rho b^3$ (distance d is from midpoint of [B]C[\/B] to origin O)"}
{"id":"80972","title":"If Renormalization Scale is Arbitrary, Why Do We Care about Running Couplings?","text":"**For the bounty please verify the following reasoning** [copied from comment below] Ah right, so the idea is that overall observable quantities must be independent of the renormalization scale. But at each order in perturbation theory the result can depend on renormalization scale, right? And they do so in exactly the right way to invalidate the use of a perturbation series when external momenta are of the wrong order. This is because the external momenta get caught up in loop integrals due to momenta conservation at vertices and the renormalization group equation comes along and means the right things cancel. [original question] **In Brief** People say that perturbation theory breaks down when the couplings run to high values. But given that this running depends on an arbitrary mass scale, how is this argument logical?! **A Longer Exposition (TL;DR)** It's obvious that Feynman diagram techniques work best when the coupling constant is small, since then you can neglect higher order terms. But various sources (e.g. Peskin and Schroeder) make claims about running couplings that seem incomplete to me. I often read sentences like   * if the renormalized coupling is small at low energy, then Feynman diagrams are good in that region   * for asymptotically free theories, Feynman diagrams are good for high energy calculations I understand exactly why these are true. But surely something much more general is true, namely   * if the coupling constant is small **at any renormalization scale** the Feynman diagrams are good for calculations My reasoning is as follows. Observable quantities are completely independent of the running of the coupling, so you can just choose your scale appropriately so that the coupling is small for the expansion. Hey presto, you can use Feynman diagrams. Is this true? I strongly expect the answer to be **no** but I can't convince myself why. Here's my attempt at a self-rebuttal.   * my argument above is incorrect, because I've assumed there's only a single coupling. In reality there's contributions from \"irrelevant\" terms at higher energies whose couplings can't be fixed from low energy observations.  This leads me to hypothesise that the following is true (would you agree?)   * if the coupling constant is small **at any renormalization scale above the scale of your observations** then Feynman diagrams are good This seems more plausible to me, but it would mean that Feynman diagrams would be good for low energy strong interaction processes for example. This feels wrong is a sense because the renormalized coupling is large there. But then again the renormalization scale is arbitrary, so maybe that doesn't matter. Many thanks in advance!"}
{"id":"13088","title":"AdS to dS uplifting and its opposite","text":"So as I understand it, localized structures in AdS can wick rotated to dS, the boundary has to be complixified as can be seen here. Also, uplifting is another technique that can be used to move from AdS to dS. (Thanks to Mitchell Porter for this suggestion) My question is that just as an exercise, is it possible to move from dS to AdS, I don't know the name for such a process, it is in some sense \"moving down\" from dS to AdS. Take the following example, suppose I have a mapping of a scalar field from boundary to bulk in AdS, now that can be uplifted to dS, it won't hold make much sense anymore since AdS allows for SUSY however dS doesn't but let's say for a moment that it can be done. Now consider the opposite case, we have a scalar field mapping in dS, can that be brought down to AdS? I think it can be, and this should be actually easier since AdS allows for SUSY which should make the mapping a little easier. The lack of a concrete example for dS\/CFT makes it (about the mapping which we considered in our hypothetical case for dS) difficult to ask this question but does it seem plausible that if such a mapping were to be constructed, it can be brought down to AdS? Thanks."}
{"id":"23177","title":"How to know what materials are good conductors of electricity?","text":"I'm not asking a question like \"Is the wood conductive?\". No. I'm asking what properties do they have to have to be good conductors. Theoretically I mean. Thanks."}
{"id":"1523","title":"Ising model for dummies","text":"I am looking for some literature on the Ising model, but I'm having a hard time doing so. All the documentation I seem to find is way over my knowledge. Can you direct me to some documentation on it that can be parsed by my puny undergrad brain? If the answer is negative, can you explain it right there, on the answer form?"}
{"id":"94574","title":"Gravitational field has no curl? What about gas discs around stars, black holes, etc.?","text":"So everybody says the gravitational field has no curl, and is not comparable to a liquid swirling around a drain. Observationally, of course, there are many examples of vector fields (which I think are gravitational fields) which look like they have some curl. A pair of stars, for example, one being slowly devoured by the other. The path of the swirling gas seems to trace out a field with a lot of curl in it. Just as if you were to pour dye into water that is swirling around a drain. How do you reconcile the observed vector field with the gravitational field which is not supposed to have curl? Is there some way to work them into a single model?"}
{"id":"40936","title":"Electric potential vs potential difference","text":"What is the difference between electric potential and potential difference? In our course book, they are given as separate topics but their definition is given the same."}
{"id":"1128","title":"Escape velocity from long ladder","text":"The escape velocity of earth is roughly $11 kms^{-1}$. However, what if a long ladder was built extending out of Earth's atmosphere and considerably more. Then if something was to climb up at much less than the escape velocity, what would happen when it reached the end? And what if the object that climbed the \"ladder\" then fired some kind of thruster\/rocket and was going fast enough so that it orbited the Earth. It would mean less energy required to get into orbit?"}
{"id":"97909","title":"Theoretical reasons for charge quantization","text":"I'm aware of Millikan's oil drop experiment and I've read that quarks have fractional eletric charge, but I was wondering if there's any theoretical argument that makes us believe charge is quantized. I've also read about Dirac's famous work in which he demonstrated that if a magnetic monopole exists then electric and magnetic charge have to be quantized. But if it doesn't exist? Is there any other theoretical reason for the charge to be quantized or is that the only way we know?"}
{"id":"22225","title":"Local Mach number for an airfoil","text":"I would like to know how to calculate the local Mach number on the upper surface of an airfoil given the ambient temperature, the local velocity on the airfoil surface, the freestream velocity, and the ambient pressure. I am assuming from my knowledge that the Mach number depends only the temperature, so I would need to calculate the local temperature on the upper airfoil surface, which is pretty trivial using the isentropic gas equations. So $$a ~=~ \\sqrt{\\gamma * R * T}.$$ And then just divide the local velocity by the local speed of sound. Is this the right way to do it?"}
{"id":"12849","title":"Which direction does a pendulum swing in a car going through a ramp?","text":"Suppose you have a car going clockwise around a ramp at constant speed. A pendulum is hanging from the rear view mirror. Does the pendulum swing outward or inward to the turn?"}
{"id":"130091","title":"Considering the theory of special relativity: Is torque still a vector?","text":"Considering the theory of special relativity: Is torque still a vector? In classical mechanics it is easy: You have 3 axes and thus 3 planes. Every plane has its own torque so torque has 3 dimensions and thus can be represented as a vector. How are torques represented in the theory of special relativity? You have 4 axes and thus 6 planes and each of them has its own torque."}
{"id":"28085","title":"Entropy of a black hole can be zero?","text":"Black holes are one of highest density objects in the universe. Because of this high density(m\/v) more matter is highly compressed in a very small volume( i am guessing that molecules cant freely move in black hole thus forming a perfect crystals). my question is whether entropy($\\bigtriangleup S$) of the black hole can be considered as zero or not? Then what is the free energy of the blackhole, whether it will like this $\\bigtriangleup G=\\bigtriangleup H$."}
{"id":"111573","title":"Is Space-Time a special form of energy?","text":"I know space-time can be influenced by matter and energy, so it must be somehow mingled in with the mix of it all, but does space-time have a fundamental particle? Can we make a little bit of space-time with enough energy? Might the Planck length & time quantize space-time?"}
{"id":"61020","title":"QM with complex eigenvalues","text":"What class of theories\/physical systems own finite\/infinite complex eigenvalues? I do know that e.g., quasinormal modes of BH do have complex eigenvalues, but are they finite or infinite in number? What about Conformal Field Theories with complex eigenvalues? What are the general spectral properties of CFT? By the other hand, what kind of non-hermitian operators could provide spectra with infinite complex eigenvalues in pairs ( complex- conjugated)? What about the problems in the \"new\"\/\"recent\" quantum graph theory?Is there some \"infinite\" graph with complex eigenvalues? Remark: I am not an expert on spectral graph theory but I do know a little bit about its methods."}
{"id":"52820","title":"Motivation For Definitions","text":"I noticed in my physics textbook that we define certain relationships to be true. I can see how this is considerably helpful in deriving other relationships from these definitions; for instance, take position: we define these quantities to be so, and from it we can define other quantities like velocity, acceleration, etc. Moreover, most of the time these definitions are well-grounded and intuitive. However, at other times they aren't. To serve as some examples: force, torque, and electric fields. How were these things defined? What was the reasoning used to define these quantities? What are the motivations for these definitions?"}
{"id":"52821","title":"For an isolated system, can the entropy decrease or increase?","text":"In any sizable system, the number of equilibrium states are much, much greater then the number of non-equilibrium states. Since each accessible micro state is equally probably, it is overwhelmingly likely that we will find the system in an equilibrium state. However, for a closed system, one that does not interact with any external system, the number of micro states is fixed. Therefore the entropy is fixed. At any given time, the system will be in one of its micro states. Even if the system is in a micro state that does not look like equilibrium from a macroscopic point of view, its entropy will remain the same, since entropy is a property of all the micro states, not just a given micro state. So, are the number of accessible micro states (and hence entropy) of an isolated system constant?"}
{"id":"96246","title":"Can we feel our own weight (can we feel normal reaction)?","text":"If we place something on our head, we **can** feel it's weight pushing us down. But we ourselves apply normal reaction which balances the weight of the object and stops it from actually pushing us down. Can we feel our own weight in a similar way ? For that I suppose we would have to feel normal reaction from something we stand on ! I tried 2 things:   1. Standing up, and I could not really experience any force either in upward or downward direction.    2. Pushing\/Punching a wall and at the instant I felt a force towards the wall and **not** towards my fist. I found it a lot weird that you can feel the weight of something kept on your head, but not the normal reaction of the body underneath us! Both the forces are completely balanced and neither is a pseudo but still we can feel only one! Is there a scientific reason for this observation? Maybe something about normal reaction which I do not know of."}
{"id":"26287","title":"What would a rotating black-hole look like to a \"geo-stationary\" observer orbiting the black hole","text":"A rotating black hole is believed to contain a ring singularity rather than a point. However, if an astronaut is orbiting the black hole at exactly the same angular velocity as the blackhole (in direction and magnitude), then it is no longer rotating to him. Would it appear as a point singularity? (if he could somehow see it)"}
{"id":"104247","title":"What does BICEP2's results tell us about gravitation waves and quantum gravity?","text":"The BICEP2 results, unless I am mistaken, are a measurement of CMB polarization, i.e. photon polarization. That is, taken at face value they say nothing about gravity directly. Now, we can start to make some assumptions in order to see what this result tells us about other physics like gravity. These assumptions fall into 2 very different categories: (i) very general statements about the nature of quantum mechanics, field theory, and quantum field theory and (ii) specific field-theoretic model dependent assumptions. Assuming this is true, (1) What does the BICEP2 results say about gravitational waves? And what assumptions did we need to make in order to make this statement(s) (general or model dependent assumptions)? (2) What do the BICEP2 results tell us about quantum gravity? And what assumptions did we need to make in order to make this statement(s) (general or model dependent assumptions)? (3) Lastly, are my questions (1) and (2) actually the same question?"}
{"id":"93109","title":"Work done by Lorentz Force in case of motional emf","text":"In the classical example of the slidewire generator where the rod slides on a U-shaped conductor in a magnetic field, we get a charge separation due to the Lorentz force. The way the induced emf is usually derived is the following: $\\mathcal{E}=W_l\/q=vBL$ where $W_l$ is the work done by the magnetic (Lorentz) force. But we know that work is defined as a **scalar product** of force and displacement. In case of Lorentz force, which is always perpendicular to the velocity vector, the scalar product should always be zero. So why then we allowed to use the above expression? It even confuses me more, because later on the book states: > The Joule heating produced in the slidewire generator is: $\\Delta Q=R > I_{induced}^2 \\Delta t=\\frac{(vBL)^2}{R} \\Delta t$ > > .... > > But **the Lorentz force does no work** , so where does this energy come > from? It turns out that there is another Lorentz force acting in the system > due to the relative motion of the charges... So, on the one hand it derives motional emf by assuming $F_l$ **does** work $W_l$ and on the other hand, it says later that it does **no** work (which can cause a confusion because heat is still being produced in the circuit). What's going on here? ![enter image description here](http:\/\/i.stack.imgur.com\/FifpA.jpg)"}
{"id":"55650","title":"Foucault pendulum","text":"The equations of motions for a Foucault pendulum are given by: $$\\ddot{x} = 2\\omega \\sin\\lambda \\dot{y} - \\frac{g}{L}x,$$ $$\\ddot{y} = -2\\omega \\sin\\lambda \\dot{x} - \\frac{g}{L}y.$$ What are the equations describing $\\dot{x}$ and $\\dot{y}$?"}
{"id":"55658","title":"How to evaluate commutators?","text":"I need to evaluate $[1\/x, p]$. Note: the $p$ is the momentum operator. So far this is what i have: $$= (1\/x)(p) - (p)(1\/x)$$ $$= (1\/x)(-ih*d\/dx)-(-ih*d\/dx)(1\/x)$$ Ii then factor out $-ih$ to get... $$=-ih(\\frac{1}{x}\\frac{d}{dx}-\\frac{d}{dx}\\frac{1}{x})$$ This is where im lost. Does the final answer just equal 0 because $(1\/x)(d\/dx) - (d\/dx)(1\/x) = 0$?"}
{"id":"51331","title":"Does quantum mechanics allow faster than light (FTL) travel?","text":"Let's suppose I initially have a particle with a nice and narrow wave function[1] (I will leave these unnormed): $$e^{-\\frac{x^2}{a}}$$ where $a$ is some small number (to make it narrow). Let's also suppose that the wave function travels along x axis with constant speed $v$ and it smears out with constant speed $v_s$. So after time $t$ wave function looks like $$e^{-\\frac{(x-vt)^2}{a+v_st}}$$ Now, if speed $v$ is close to $c$ and $v_s$ is big enough too, we could get a situation where the centre of mass moves at subluminal speed (this is the group velocity, I suppose), but the front[2] of the function moves at superluminal speed. If I measure either the particle's position after some time $T$ or time when it reaches some point $A$ i will mostly and averagely conclude that it has travelled with speed $v$. But in some (less probable) cases it will seem that it has travelled faster, even FTL. Is this a normal thing in quantum mechanics or I understand it all wrong? Maybe there is some constraint not only on $v$, but also on $v+v_s$? ## A little background When talking about superluminal tunneling speeds I usually hear explanations like this: > That speed is apparent. In those cases only some frontal part of the wave > function gets through the barrier. Although it's center of mass seems to > have travelled FTL, that function would still stay under the initial wave > function if it continued it's way without the barrier. The center just > shifted because of dropping the rear part. I have never actually understood why does it solve the problem, because the position of center doesn't change the fact that signal in some cases may arrive FTL. This is where my question comes from. **EDIT:** Sorry, the question was not about what some classic QM models allow, but actual quantum physics. I understand that Schrodinger's equation allows me any speed, I want to know if this situation is allowed in actual quantum physics. I guess the use of term QM was wrong. Sorry, my bad. ## Question reformulated Is it possible to send a particle that's slowly _spreads_ and with some small chance (in some rare cases) measure it arriving FTL? (provided that the mean arrival value still stays under $c$). I am aware that this is possible when tunneling through a barrier (and the mean value can even move to superluminal by _train dropping cars_ ), but is the same possible without any barrier? Maybe I could equivalently ask, if quantum uncertainties allow occasional exceeding of $c$. [1] If you say that I had some chance to measure it at any point in the very beginning and that it wasn't entirely localized, we can replace the Gauss' function with a square or triangle one that gets wider along the way. [2] If you need, we can define the _front_ of the wave, for example first of the points where second derivative is zero. It's position is $vt+\\sqrt{\\frac{a+v_st}{2}}$."}
{"id":"25390","title":"Where to find information on the pulsar map written to Voyager's golden record?","text":"Where might I find details about the pulsar map written on the Voyager's golden record? ![enter image description here](http:\/\/i.stack.imgur.com\/bFopN.png) > This diagram defines the location of our Sun utilizing 14 > > pulsars of known directions from our sun. The binary > > code defines the frequency of the pulses."}
{"id":"57656","title":"Inflation and scalar spectral index","text":"I've been reading that the results from the Planck satellite constrain a number called the \"scalar spectral index\" to be 0.96 rather than 1 at the 5-sigma level. This is supposed to be big news, but I don't understand why (my ignorance, of course). For instance, with an index of 0.96, can we say how many e-foldings occured during inflation? What else does it tell us?"}
{"id":"65231","title":"Questions about the Jeans length","text":"I have a couple of questions about the Jeans length. Suppose the universe has a homogeneous energy density, except there's a spherical region that is overdense.   * I understand that if the region is smaller than the Jeans length, then pressure, which travels at the speed of sound (compared to the speed of gravity $c$), can build up fast enough to counteract the gravitational collapse. But if the region is bigger than the Jeans length, it collapses. What exactly does it mean? Does the region get smaller and smaller until it vanishes? Wouldn't this create a black hole, since you're stuffing all the energy of the overdense region into a smaller and smaller place?   * Or, if the region is collapsing, wouldn't it eventually be smaller than the Jeans length, at which point it stops collapsing?   * And what happens if the region is underdense, rather than overdense?"}
{"id":"52179","title":"Hydrogen atom in quantum mechanics","text":"I have problems following the calculations in Griffiths' _Introduction to Quantum Mechanics_ (Chapter 4.2.1): If you apply the Schrödinger equation to the Coulomb potential you get the following differential equation: $$\\tag{4.56} \\frac{d^2u}{d\\rho^2} ~=~ [1-\\frac{\\rho_0}{\\rho} + \\frac{\\ell(\\ell+1)}{\\rho^2}]u. $$ A few steps later my books substitutes $u(\\rho)$ without explaining why this is allowed: The next step is to peel of the asymptotic behavior, introducing a new function $v(\\rho)$: $$\\tag{4.60} u(\\rho) ~=~ \\rho^{\\ell+1}e^{-\\rho}v(\\rho). $$ How is this function still related to the Hydrogen atom? To me it looks like something total random."}
{"id":"52176","title":"Could negative dimension ever make sense?","text":"After some quick check I found that negative dimensions are not used. But we have negative probability, negative energy etc. So is it so likely that we won't ever use negative dimension(s) ? ## Update I understand there're also dimensions that are not integers e.g. dimension 1½ (?) for fractals or so. Could there also be a dimension such as dimension i (imaginary)?"}
{"id":"52177","title":"Black holes and entropy","text":"1)I'm told that for Black Holes, when they radiate (Hawking radiation) particles and anti-particles) split at the event horizon, one going to infinity, the other into the BH. They then lose mass. How is that possible? Wouldn't their masses increase, since they are absorbing particle or anti- particles? 2)Also, their entropy varies as their masses, and since entropy increases, wouldn't their masses also be increasing, along with their area? How does that reconcile with case 1? Thanks"}
{"id":"60049","title":"What is the difference between the properties of Electron spin and Photon polarization\/helicity?","text":"What is the difference between a photon's polarization\/helicity and an electrons spin half? I know that the photon is spin 1 but isn't its polarization analogous to spin half? This question stems from wondering why there isn't a classical wave equation like maxwell's equation for the electron."}
{"id":"53882","title":"How is the direction of time determined in general relativity?","text":"In special relativity every frame has its own unique time axis, represented in Minkowski diagrams by a fan-out of time vectors that grows infinitely dense as you approach the surface of the light cone for a given rest frame. Viewed from any other frame, these same time vectors shift about exactly the same fan-out topology, so no one frame perspective is any more \"correct\" than any other one. In general relativity this does not quite seem to be the case. If nothing else, all those worldlines extend over billions of years and so presumably end up having a quite precise average time orientation for any given local region of space, even if you select a complicated overall topology. Conversely (and I assume equivalently), if space is viewed interpreted as manifold, the idea of an average \"flatness\" that is orthogonal to this long-term time axis appears to be assumed, since it would look quite odd to see the entire universe slanted relative to its own time axis, yes? So:   1. Am I understanding this correctly? Does GR applied over eons necessarily end up defining and distinguishing a unique and well-defined overall time axis?   2. If so, how is this unique time axis ascertained in experimental practice? Is it taken to be some sort of very large center-of-momentum average, as the billions-of-years bundle analogy would seem to suggest?   3. If a distinguished time arrow does exist in GR, does the existence of that axis imply any experimentally meaningful implications for its associated SR frame?"}
{"id":"102977","title":"Why does air circulate on an airfoil --- The Kutta Condition","text":"![enter image description here](http:\/\/i.stack.imgur.com\/flYBz.png) Why does the air circulate on a flowing airfoil, thus giving rise to increased velocity (circulation + relative airspeed) above the wing and hence decreased pressure."}
{"id":"102974","title":"Why friction force is force of constraint?","text":"My understanding about constraint force is that it is a force which limits the geometry of particle's motion. For example, situations such as the particle trapped in a track or limited in domain can be assumed constraint force. But in this point of view, I couldn't understand why friction is constraint force. In lagrangian formulation, we divide forces into two part, F= F(applied) + F(constraint) If particle moves in one dimension, and assuming there exists sliding friction, that particle can move anywhere. The sliding friction never restrict the domain that particle can move. so I think the sliding force is applied force, rather than constraint force. can anyone clarify why friction is constraint force??"}
{"id":"117162","title":"Milky way rotation data","text":"Can anyone tell me where I can download sample rotation curve data for the Milky Way in csv format?"}
{"id":"112903","title":"Magnetic hot air balloon","text":"Normally, hot air balloons are limited to 60,000 ft (related question), the limits being the mass of the physical structure containing the air, and the heat source. If one constructed a magnetic bubble and filled it with plasma (AKA mini- magnetospheric plasma propulsion) which was kept hot with a really powerful ground based microwave source (~3MW for a 1 ton payload, from my back of the envelope calculations), could it reach space (Kármán line, atmospheric pressure 32 mPa)? Or would something stop it working at a particular altitude, e.g. the plasma would cool down too fast, or become detached from the magnetic field, or the electrical conductivity of the air at some height would neutralise the plasma, or some other reason I don't know yet? If it is possible, how accurate are my estimates (I don't know how efficiently microwaves would heat a plasma, nor how much power would really be lost by something like this)."}
{"id":"68591","title":"Are fundamental forces conservative?","text":"I wonder whether fundamental forces are conservative. First of all, I'm not sure if we can talk about conservative forces, since to study electromagnetism, weak and strong interactions we need QFT. For gravity, I'd that it's not conservative, because energy conservation doesn't even apply."}
{"id":"93815","title":"How to convert RL Circuits?","text":"_**How do we convert a parallel RL circuit into its equivalent series RL circuit ?_**"}
{"id":"98680","title":"How is this process not quasi-static yet reversible?","text":"Consider a (adiabatic) canister with a piston containing some gas kept in a vacuum. There are two weights on the canister which equalize the pressure of the gas on the piston. Assume the system is at equilibrium. I remove one weight from the piston. The system will go out of equilibrium. After some flux, it will again settle into equilibrium. Clearly, this is not a quasi-static process. But its a reversible one, isn't it? If I put the weight back on the piston, the system will again achieve its initial state. How can a non-quasi-static process be reversible? **EDIT:** According to Wikipedia, a reversible process \"is a process that can be 'reversed' by means of infinitesimal changes in some property of the system without entropy production\". So, if a process can be reversed _without_ the means of infinitesimal changes without entropy production. Would that qualify as reversible? (as in the example given above)"}
{"id":"53224","title":"How reflected objects are composed and who is responsible for that?","text":"Please refer to this image. The scene contains an object close to a mirror in the wall and a window, note that the reflected object is receiving more light than the object itself. I read some articles about the Law of Reflection and the Ray Principle but did not get a straight answer on how the mirror(or my eyes) knows how to compose that virtual image in a way it appears to receive more light. According to the definition of the Plane Mirror, the light rays do not actually pass through the virtual image, then why the virtual image is being composed with more light and who is responsible for that? The mirror? My eyes?"}
{"id":"63536","title":"What would be the basis vectors for this 2D crystal structure?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/hvaL7.jpg) In the above image, I have a 2D crystal structure. The lattice vectors are described by:               a = {-1\/2, -Sqrt[3]\/2};     b = {1, 0};      and the location of atoms A and B are given by:               \\[Tau][A] = {2\/3, 1\/3};     \\[Tau][B] = {0, 0};      Now the problem I'm tasked with is to plot the locations of the atoms A and B, which I have done as seen above, where A is red and B is blue. I also have to plot the lattice vectors which I have done in blue arrows as seen above, and lastly I have to plot the basis vectors and this is where I don't know what to do. I don't know what the basis vectors are. Googling has led me to discover that a lot of people use lattice vectors and basis vectors interchangeably, and overall, I have no clear definition to work with. Sometimes people say basis vectors are orthonormal, sometimes they need to be linear combinations with integers, sometimes with any real numbers. I'm frustrated by the looseness with which all kinds of sources use these terms, and ultimately I don't have a clear reference to go with."}
{"id":"9702","title":"A lightning protection device physics","text":"Here is a description of the principle of the operation of a new lightning device: _During a storm the ambient electric field may rise to between 10 to 20 kV\/m. As soon as the field exceeds a thresold representing the minimum risk of a lightning strike, the OPR lightning terminal is activated. **It draw its energy from the ambient electric field the energy required to generate high voltage pulses, creating and propagating an upward leader.** No other power sources are required, and no radioactive components are used._ It remains unclear to me the physics of the drawing the energy from the electric field. Obviously, you can not draw the energy from the electrostatic field, no matter how strong it is. So i suppose that the electric field must oscillate more or less. The question is: Which is a minimal frequency of the oscillations so that such device can work properly? A full description of the device"}
{"id":"112263","title":"Relative Motion","text":"I read the following: http:\/\/www.feynmanlectures.caltech.edu\/I_15.html#Ch15-S4 > \"Now let us see what happens to the moving clock. Before the man took it > aboard, he agreed that it was a nice, standard clock, and when he goes along > in the space ship he will not see anything peculiar. If he did, he would > know he was moving—if anything at all changed because of the motion, he > could tell he was moving. But the _principle of relativity_ says this is > impossible in a uniformly moving system, so nothing has changed.\" In my point of view Feynman uses the principle of relativity as a kind of an _axiom_ for his derivation of time dilation. **Question:** How do we know that the _principle of relativity_ is true and can use it like an axiom for new phenomena? If we had treated Newton's laws of motion as axioms we would never have found out how nature actually works. In other words, what is the law behind the fact that you cannot know how fast you are moving? How can you use this assumption as axiom for e.g. the behavior of clocks as Feynman does further down in the link? P.S.: Feel free to edit since I'm not a native English speaker. Nevertheless I want to improve my English and it would be great if you pointed out my errors."}
{"id":"24115","title":"Feynman's sum over histories?","text":"The concept requires all possible path's to be mapped out, and any remaining paths not cancelled out represent the most probable path of the object. Considering this:      i) If \"infinite\" paths are mapped out (since there are infinite paths for every movement) , isn't it guaranteed for every path to be canceled out? Every path has an opposite, and if infinite paths are mapped, the opposites are guaranteed to be present. ii) How do you map out infinite paths? Nothing infinite can be completely mapped, on the basis that it is infinite."}
{"id":"67098","title":"What's the basic premise of General Relativity?","text":"What is the basic assumption(s) required to explore general relativity? For example, if one merely assumes that the speed of light $c$ is the same for all observers, and the laws of physics are the same for all observers, then it's possible using a mix of algebra and some calculus to derive every SR result from time dilation to length contraction, $E = mc^2$, and so on. What is the basic premise of general relativity? I assume that all of SR is included but is there anything else?"}
{"id":"108689","title":"What is the effect of total internal reflection on polarized light","text":"Total Internal Reflection causes phase change of a light beam. I searched for effect on a polarized light beam by Total Internal Reflection but could not find much. I am assuming polarization does not get altered. I am unable to understand the reason why atoms in glass are able to change phase but unable to change the orientation of electric field of an EM wave. I am re-studying optics after 20 years so please excuse me for being rather naive."}
{"id":"101316","title":"Optical absorption -- what are the common ranges and mechanisms?","text":"So let's say you do some reflection\/transmission spectroscopy of a material. It's clear that it's absorbing in some range. What would be your first step in identifying the source of the absorption? The two main clues I would look at would be   1. The range it occurs in (if your incident light is in the x-ray range, it's clearly not phonon absorption, for example)   2. The shape of the absorption (is it a peak? Or an absorption 'edge', where it isn't absorbing below some energy, and absorbs everything above, for example) But I'd like to know if there are other ways as well. Additionally, does anyone know of a 'map' of the ranges you'll typically find different types of absorption in? I saw one on wikipedia once, but it only had three types, and now I can't find it of course. For example, a few types of absorption I can think of off the top of my head are   * Interband electron transitions   * Intersubband electron transitions   * Impurities   * Phonons   * Vibrational modes   * Atomic transitions I understand that a lot of these will be material-dependent, but often the same mechanism in different materials will yield absorption in the same order of magnitude range. What ranges would one typically see these (and other) absorption mechanisms in? Thank you!"}
{"id":"107440","title":"Stationary wave on a string","text":"The problem says: > _A string lies on the x-axis, and is fixed at $ x=-a $ and $ x=a $. At $ t=0 > $ we impose a transversal velocity given by $ a^2-x^2 $. Assuming that the > velocity of the wave is $ c $, find the solution to the wave equation._ So here's what I did so far: I set up the solution in complex form as $$ \\tilde{y}(x,t) = \\tilde{A}e^{i(kx-\\omega t)}+\\tilde{B}e^{i(-kx-\\omega t)} $$   Then imposing the condition of fixed string, I got the values for $ k_n =\\frac{n \\pi}{2a} $ and $\\omega_n =\\frac{n \\pi}{2a}c $ In the process I got the relation $\\tilde{A}=-\\tilde{B}e^{2ka}$ Now my problem is when I plug back into the original equation, I get something like this: $$ \\tilde{y}(x,t)=\\tilde{B}e^{-i\\omega_n t}(e^{ik_nx} -e^{i(k_nx + 2k_na)})$$ I know that $2k_na=n\\pi$, but that gives me two possible results for $e^{i2k_na}$, 1 and -1. That confuses me a bit, and I don't know how to continue from here. I've seen the example of the plucked string that goes from 0 to $L$, and that's much simpler than this case. I guess I just don't know what to do with the constants introduced by the fact that the string goes from $-a$ to $a$. Any help is appreciated."}
{"id":"101234","title":"Do Dirac Gamma Matrices act like Tensors?","text":"Do Dirac Gamma Matrices act like Tensors? Is it true that $$ \\gamma_\\mu = \\eta_{\\mu\\nu}\\gamma^\\nu~? $$ Also what about $\\sigma_{\\mu\\nu}$, where $\\sigma_{\\mu\\nu}$ is defined to be: \\begin{align*} \\sigma_{\\mu\\nu} = \\frac{i}{2}[\\gamma_\\mu,\\gamma_\\nu]~? \\end{align*}"}
{"id":"59962","title":"how dense fluid affect the buoyancy force?","text":"I read a story regarding the Archimedes' principle in a magazine of popular science and I am thinking of the following question: how does the density of the fluid change the buoyancy force for the same object? As we know, the Archimedes' principle tell that for any object in a fluid, the buoyancy force equals to the weight of the fluid displaced by the object. It is pretty straightforward. Now, if I have an object partially floating on the surface of a liquid, so we have $$ F_b = \\Delta V \\rho g $$ where $\\Delta V$ is the volume of the displaced liquid and $\\rho$ is the density of the liquid. So what happen if we place the same object into a denser liquid? Physically or intuitively, since the liquid is denser, it is harder for the object to 'inject' into the liquid, so the buoyancy force should be bigger, so less part of the object submerge into the liquid. But if you look at the math, it seems not like this. Well, now $\\rho$ is bigger, but the volume of displaced liquid will be smaller because it is harder to submerge the object into a denser liquid too. So how do we that for denser fluid, the same object will experience bigger buoyancy force instead of being the same? So my question is from intuition, the same object in the denser liquid should submerse less than the case in less dense liquid. But from the math, it buoyancy force could stay the same or more. So how to prove from the math that our intuition is correct?"}
{"id":"53767","title":"How can I cheat an XRD sample (~10 mg) from a total 1g required? How can I use for the background.","text":"How much can affect (in terms of hump) my X-ray diffraction pattern the \"amorphous\" glass or maybe someone can suggest me other material to compensate for the small amount of the sample. Note: my sample is a powder."}
{"id":"4184","title":"Why should the Standard Model be renormalizable?","text":"Effective theories like _Little Higgs models_ or _Nambu-Jona-Lasinio model_ are non-renormalizable and there is no problem with it, since an effective theory does not need to be renormalizable. These theories are valid until a scale $\\Lambda$ (ultraviolet cutoff), after this scale an effective theory need a UV completion, a more \"general\" theory. The Standard Model, as an effective theory of particle physics, needs a more general theory that addresses the phenomena beyond the Standard Model (an UV completion?). **So, why should the Standard Model be renormalizable?**"}
{"id":"15546","title":"Could the acceleration of universe expansion be caused by gravity itself?","text":"Dark energy is suggested to be a repulsive force in the universe causing an accelerated expansion. If the amount of mass outside our observable universe is greater than inside (higher mass density), would it not cause an accelerated expansion from our viewpoint?"}
{"id":"111900","title":"How is time distributed through space?","text":"I was greatly confused when I saw video saying that if two wormholes were separated by _time_ instead of _distance_ , it would allow time travel. My question is what did they mean by this statement? What do they mean by time instead distance? How is time distributed through space?"}
{"id":"23231","title":"Conservation of Energy and Momentum Regarding Forces - clarification needed","text":"The other day, my teacher stated something along the lines of, \"Conservation of momentum is not violated by the actions of internal forces, but the conservation of energy is violated. Energy is conserved only when\/if internal forces are conservative.\" Can anyone explain this is simpler terms or reiterate it in some fashion?"}
{"id":"30949","title":"How to deal with conflicting \"no-slip\" Navier-Stokes boundary value constraints?","text":"The no-slip boundary value constraint for Navier-Stokes solutions was explained in my fluid dynamics class as a requirement to match velocities at the interfaces. Now that my class is done, I've been playing with solving an idealized fluid flow problem: figuring out the rate of \"spin-down\" of stirred fluid in a non- bottomless cylindrical container (ie: my coffee after being given a good stir). For the non-time dependent portion of the problem, I'm not able to apply the no-slip condition as taught in class to my cup of coffee. Matching velocities to my stir stick (I've idealized that as a cylinder as in Acheson's bottomless tea example (pg 45) ), I've got a problem at the base of the cup: If I match velocities of the fluid to the \"stir-stick\", I can't also match velocities to the bottom of the cup (zero velocity) where the stir-stick touches the base. I've been pointed to some journal articles to read, but won't be able to do so until the fall when I enroll in my next course (and get library access again). In the interim, I was wondering if I could get some pointers on an approach to deal with conflicting 'no-slip' conditions like this one (ie: any problem with a moving interface in contact with a static one has this trouble.) EDIT: I believe that I can work around the inconsistency problem with no-slip constraints for this problem by \"stirring without touching the bottom of the cup\" ... ie: set the stir depth to 1cm (say). This makes the non-time dependent solution uglier, since I have to treat three different regions:   1. Below the stir stick (zero velocity no-slip condition on the inside of the cup in this region ; velocity matching the stir velocity along the circle at the stir radius at the stir depth).   2. Between the center of the cup and the stir stick. \"no-slip\" conditions: match velocities below the stir stick. match the stir velocity at the stir radius.   3. Between the stir stick and the edge of the cup. \"no-slip\" conditions: zero velocity on the cup edge. stir velocity at the stir radius. This makes the problem much messier ... I already have Bessel functions to deal with, and now have to apply them separately in three regions. I'm still interested to see if there's a better way to deal with the boundary value conditions ... ie: without forcing an artificial distance between a static and a dynamic interface so that they don't touch."}
{"id":"134849","title":"Are double-slit patterns really due to wave-like interference?","text":"According to various sources on the web, it seems like the general concensus is that there isn't actually any wave-particle duality with quantum particles. For example, this article implies that diffraction patterns in double-slit experiments were interpreted as wave interference due to _apparatus limitation_ at the time they were first performed. Does this mean that all those sources and animations showing two waves interfering are simply incorrect, classical conclusions which don't have anything to do with (quantum) reality? What's actually the most confusing is that most sites which state that it is now possible to pass individual photons through these slits, also claim that these individual photons somehow _interfere with themselves_ resulting in the observed patterns. That seems like a rather thin explanation, doesn't it? So, is there actually any need to use wave interference to explain the phenomena, or can we simply state that the pattern is probabilistic in a certain way, without involving the \"spooky\" explanations?"}
{"id":"72523","title":"Pulley problem with spring balance","text":"![enter image description here](http:\/\/i.stack.imgur.com\/i73YD.jpg) Please excuse the crude drawings The question is, if I burn the thread t, how will the reading in the spring balance(S) change? Ie, increase, decrease, or remain the same? My approach: When we burn the thread, the objects will move, but the tension should increase a bit. ( I can't explain this, I only get this intuitively.) and, since the spring balance reads the double of the tension, I think this should make it a lot more than the initial reading. Since if the tension in the intact string changes by $\\triangle T$, the reading should increase by $2 \\triangle T$. That's all I can think of. So as you see, my approach is very intuitive, and hence I am asking this in order to get a logical and stronger reasoning"}
{"id":"72528","title":"Minimum required to lift a roller up a step","text":"Consider a roller of diameter $0.82m$ which is to be raised up a step measuring $0.32m$. Find the minimum force required to do so. The roller weighs $500N$. ![Diagram](http:\/\/i41.tinypic.com\/2qmlytu.png) The roller could be considered as a point mass $F$, and its body could be considered a lever with $F_{applied}$ acting at $G$ and $F_{perpendicular}$ acting perpendicular to diameter $GC$. Let $\\angle FCH = \\theta \\implies \\angle IFG = \\theta$. $F_{perpendicular} = mg\\sin\\theta$. After that's its a simple application of torque and moments. I'm not quite sure if this resolution of vectors into components is quite the right approach. At least, the value given in my textbook doesn't correspond to the one I got. EDIT: In response to the comment below, I decided to post my workings: $\\sin\\theta = \\frac{FH}{FC} = \\frac{0.09m}{0.41m} = 0.2195 \\implies F_{perpendicular} = 500N\\cdot 0.2195 = 109.756N$ The anticlockwise torque equals $\\tau_{anti-clockwise} = 109.756N\\cdot 0.41m = 45Nm$ This has to equal the clockise torque $\\tau_{clockwise} = F_{applied}\\cdot 0.82m$ $\\implies F_{applied} = \\frac{45Nm}{0.82m} = \\boxed{54.878N}$ According to the book, the answer is $110N$? Maybe the book is incorrect?"}
{"id":"20113","title":"Trinary System Arrangment?","text":"I'm writing a science fiction story and wondered if someone could help me with the setup of a trinary star system I have in mind. Basically I just wanted to know if: a) a system with a central, Sun-sized star could have two smaller (orange dwarf?) stars orbiting it in a circular trojan orbit, and... b) how much disturbance would this type of orbit cause to any planets orbiting the inner star? This may all sound ridiculous, but any input would be greatly appreciated. Thank you."}
{"id":"55518","title":"Emergent symmetries","text":"As we know, spontaneous symmetry breaking(SSB) is a very important concept in physics. Loosely speaking, zero temprature SSB says that the Hamiltonian of a quantum system has some symmetry, but the ground state breaks the symmetry. But what about the **opposite case** of SSB? The ground state of a quantum system possesses some kind of symmetry while the Hamiltonian does not have this symmetry. For example, the exactly solvable Kitaev-type model Hamiltonians explicitly break the spin rotational symmetry, but the ground states are spin liquids, which possess the spin rotational symmetry. I wonder whether this **opposite case** of SSB plays an important role like SSB in physics? **Note:** Examples with **exact emergent symmetries:** A simple example with _exact emergent $SU(2)$ spin-rotation symmetry_ can be found here A simple model that exhibits emergent symmetry? Another example with _exact emergent $U(1)$ symmetry_ is presented in the Supplemental Material of this paper, where it is appeared on page 2 under Eq.(A7). Examples with **approximate emergent symmetries:** A chiral spin-liquid phase with _emergent $SU(2)$ spin-rotation symmetry_ in the framework of Schwinger-fermion mean-field study of the strongly correlated Kane-Mele-Hubbard model. The example with _approximate emergent lattice 3-fold rotation symmetry_ is the existence of Ferromagnetic(FM) ground state in the Kitaev-Heisenberg model, where the model Hamiltonian explicitly breaks the lattice 3-fold rotation symmetry but the FM phase possesses the lattice 3-fold rotation symmetry. Another example with _emergent chiral symmetry_ was proposed by X.G.Wen in his paper, as seen on page 18, title **C**. A third example with _emergent time-reversal symmetry_ can be found here. An example with _an emergent global topological U(1) symmetry_ is presented here. **More examples with emergent symmetries are welcome.** **Erratum:** The example of \"Kitaev model\" I gave in my 2nd paragraph is not correct, please see Why we call the ground state of Kitaev model a Spin Liquid? for the reason."}
{"id":"32738","title":"What does symplecticity imply?","text":"Symplectic systems are a common object of studies in classical physics and nonlinearity sciences. At first I assumed it was just another way of saying Hamiltonian, but I also heard it in the context of dissipative systems, so I am no longer confident in my assumption. My question now is, why do authors emphasize symplecticity and what is the property they typically imply with that? Or in other more provocative terms: Why is it worth mentioning that something is symplectic?"}
{"id":"77464","title":"Electric field outside a sphere with a cavity","text":"I have a sphere of radius $2a$ centered at the origin and made of a nonconducting material that has a uniform volume charge density $\\rho$. A spherical cavity of radius $a$ eccentric to the right side (touching) with its center aligned with the x-axis is now removed from the sphere. Find the electric field produced by this distribution along the x-axis for $x > 2a$. I started by establishing expressions for $E$ in both the uniform section and the cavity: $E_u = \\rho 2a\/(3\\epsilon_0)$ and $E_c = -\\rho a\/(3\\epsilon _0)$ and $E_t$ inside the cavity is $ E_t = \\rho a\/(3\\epsilon_0)$ Is it correct that the $E$ along the x-axis would have the same magnitude as $E_t$?   I'm thinking it's not, but I'm not sure about the other idea I had which is that it would be $E = \\rho r\/\\epsilon_0$. $r$ now being the distance from the center of the solid\/enclosing sphere. This is the result of exchanging the $-\\rho$ that is required to make the net charge inside the cavity with $\\rho$ and adding the $E$-s together again."}
{"id":"77467","title":"Expectation Values in Quantum Mechanics","text":"Why is the expectation value what it is? Why don't you apply the operator, then multiply that by it's conjugate?"}
{"id":"43827","title":"Car accident, put in park or neutral?","text":"I was waiting on a red light the other day and was wondering. If I'm in my car, not moving and I see a car that's going to hit me from behind. Would I (my body) be safer if I put on the break or if I put the car in neutral? I assume there's no car in front of me and there's no posibility to be hit by an other car. Also, the car crashing into me is going at a good speed."}
{"id":"5062","title":"Noether's theorem vs. Heisenberg uncertainty principle","text":"In continuation of another question about Noether's theorem I wonder whether there exists some kind of relationship between this theorem and the Heisenberg uncertainty principle. Because both the principle and the theorem relate energy with time, momentum with space, direction with angular momentum. When this is a general fact then e.g. electrical charge and electrostatic potential(*) should be partners in an uncertainty relationship too. Are they? I feel that these results look so basic and general that I hope that a pure physical reasoning (without math or only with a minimal amout of math) exists. Also compare this question where again momentum and space are connected, this time through a Fourier transform. (*) i.e. electric potential and magnetic vector potential combined."}
{"id":"4407","title":"Why is everything floating in space so cold?","text":"Of course there is pretty hot stuff too. For example Suns. But isn't the vacuum of space the perfect containment for heat? And shouldn't the rare collision with particles even heat up an object that floats around in space? Why aren't they hot from time to time? Is it because they loose their heat by radiant heat? And if so does radiant heat emission always occur until the object is the same temperature as the surrounding or does it depend on other factors?"}
{"id":"36290","title":"In heterojunction problem, how to align the energy band in presence of bias voltage","text":"For example, SiO$_2$ barrier embeded between Fe magnet and 2-dimensional- electron-gas such as Si. How to align the energy bands of the three materials when an electric field is perpendicular to the interfaces? Do they have the same Fermi level? Could somebody draw the band offset for me? Where can I find the data of band offset?"}
{"id":"134590","title":"Entropy change at varying temperatures?","text":"Entropy change is defined as the amount of energy dispersed **reversibly** to or from the system at a specific temperature. Reversivility means that the temperature of the system must remain constant during the dispersal of energy to or from the system . But this criterion is only fulfilled during phase change & isothermal processes. But not all processes maintain constant temperature;temperature may change constantly during the dispersal of energy to or from the system . To measure entropy change ,say, from $300K$ & $310K$, the range is divided into infinitesimal ranges ,then entropy is measured in that ranges and then is integrated as said in this site . But I cannot understand how they have measured entropy change in that infinitesimal ranges as there will always be difference between the temperature however small the range might be . What is the intuition behind it? Change of entropy is measured at constant temperature,so how can it be measured in a range ? I know it is done by definite integration but can't getting the proper intuition . Also ,if by using definite integration to measure change, continuous graph must be there(like to measure change in velocity,area under the graph of acceleration is measured) . So what is the graph whose area gives change in entropy? Plz help me explaining these two questions."}
{"id":"134599","title":"Volume of Brillouin zone is the same as Fourier primitive cell?","text":"In Kittel's solid state text, problem 2.3, he says that the volume of the Brillouin zone is the same as a primitive parallelepiped in Fourier space. Somehow I can't see why this is true. Can someone help me see why this is true? Also, is the same relationship true between Wigner-Seitz cells and primitive parallelepiped in real space?"}
{"id":"128172","title":"The dimensional formula of distance travelled in $n$th second","text":"I read that the dimensional formula of distance traveled in $n^{th}$ second is same as that of velocity. Okay, the formula for the distance traveled in $n^{th}$ second is $s_t= u+\\frac{a}{2}(2t-1)$ where $u$ is initial velocity, $a$ is uniform acceleration and $t$ is the time. If we proceed and expand the aforementioned formula we will get $s_t = u + at-\\frac{a}{2}$. Now, the last term i.e. $-\\frac{a}{2}$ is not a velocity and isn't the principle of homogeneity violated?"}
{"id":"7154","title":"Chaos and quantum physics: How many ways can a bonfire burn?","text":"I'm interested in the extent to which quantum physical effects are seen at a macroscopic level. I might get some of the physics wrong, but I think I'll get it close enough that I can ask the question: Let's sat that we create a bonfire and let it burn until it burns out. As the smoke rises from the fire, turbulence takes over and the smoke particles and steam and hot air all mixed together. By the end of the night when the fire has burned out, the collection of molecules in the system are in some position\/velocity X. My question: Let's assume the multiverse interpretation of quantum physics. How many possible end state superpositions can there be in this situation? Ok, that's imprecise and incorrect because it would actually be an uncountable infinitude of possible end states. How about this: Given the end state that we observed, what percentage of the end state superposition would be \"visually\" indiscernable from the end state that we observed so that each molecule would be in nearly the same end state across that portion of the multiverse? Or put another way: Do quantum effects sneak into everyday life fast enough that we can observe them? If we are effected by quantum physics at all, I imagine this is roughly a function of the timescale of the chaos effects."}
{"id":"7153","title":"Band offset and strain-induced valence band splitting in semiconductor compounds","text":"My aim is to draw a plot of $GaAs_{1-x}Sb_{x}$\/InP (GaAsSb on InP substrate) band offset as a function of As\/Sb composition. The first part is easy if I'm doing this correctly: I found VBO (valence band offset) parameters for GaAs and GaSb, then the bowing parameter for $GaAs_{1-x}Sb_{x}$ VBO and finally the bowing parameter for $GaAs_{1-x}Sb_{x}$ band gap. This is what I got: ![Unstrained GaAsSb](http:\/\/i.stack.imgur.com\/XOAPX.png) However, when I put GaAs or GaSb on InP, strains appear and the valence band is split into heavy hole and light hole subbands. In some articles I found the way to calculate the splitting: $\\Delta E_{HH}=\\partial E_H^V + \\partial E_S$ $\\Delta E_{LH}=\\partial E_H^V - \\partial E_S$ $\\partial E_S=b(1-2\\frac{C_{12}}{C_{11}})\\epsilon_{||}$ $\\partial E_H^V=2 a^V (1-\\frac{C_{12}}{C_{11}})\\epsilon_{||}$ a, b and Cs are material properties and can be found in this article: http:\/\/jap.aip.org\/resource\/1\/japiau\/v89\/i11\/p5815_s1 (Vurgraftman, Meyer, Ram-Mohan) However, I'm not sure what the $\\epsilon_{||}$ is. Is it the direction of growth? Whatever it is, I can't see where I should consider the substrate, as I assume the strain to be substrate-dependent. Is it hidden in the $\\epsilon$?"}
{"id":"71786","title":"If kinetic energy is a function of the square of velocity, how can gravitiational acceleration be constant?","text":"Kinetic energy is defined as: \\begin{equation} E_k = \\frac{mv^2}{2} \\end{equation} Because of the squared velocity term in there, the faster you're going, the more energy it takes to further accelerate. (This is why it takes longer to get your car from 30 to 60 than from 0 to 30, for example.) However, unlike normal methods of acceleration that add (more or less) constant energy over time to an object, the force of gravity is defined as imparting a constant _acceleration_ over time to an object. How does that work? Does that imply that the faster an object is already moving (in the direction of the mass exerting gravity on the object), the stronger gravity pulls at it?"}
{"id":"76251","title":"How is the energy of a lightning calculated?","text":"I was wondering how is the energy of a lightning measured? Is it based on the luminous intensity and\/or the generated sound? What about the generated heat and energy spent in the production of ozone? Is it taken into account? I looked here and here, but I didn't find an answer that satisfies me."}
{"id":"43362","title":"Where does Computer Science background students fit in Theoretical Physics","text":"I am basically an Electronics student - background in computer science (that's where I want to work). I applied for an internship in USA in a research institute where the group is focused in Theoretical Condensed Matter Physics, Chemical Physics, Physical Chemistry, Materials Science. I mentioned my areas of interests as: Computational Science, Machine Learning, Web development My skills as: Python, C, Django, Java, etc.. I got selected. Now, I would like to know where could possibly a CS background guy would actually work on? I am looking for a detailed answer"}
{"id":"96442","title":"Relation between a Quasistatic and a reversible process","text":"Why is it that if a process is reversible, it is quasi-static? Does it mean that then the process is also non-dissipative if it is quasistatic?"}
{"id":"96443","title":"Is there any role of steering in turning?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/jqmeF.png) I am aware of the practical requirement of turning the steering wheel to turn a car while going on a straight and even curved road. But in the proofs of turning of car on leveled or banked curves, we see the tyres **straight**. Assuming a road where a car enters tangentially to a circle with some velocity satisfying the conditions for turning (no skidding), is there no reason in an ideal case to steer the car ? What all reasons in practical life make it necessary to steer ?"}
{"id":"96441","title":"Simple examples for exchange and correlation","text":"Is there an easy, in the best case intuitive, explanation of the difference of exchange and correlation? Is there a simple way to distinguish whether a certain contribution is due to exchange or correlation?"}
{"id":"89209","title":"Ideally speaking, will a rolling disk with no slipping come to a stop because of friction from the ground?","text":"Consider a rotating disk on a horizontal plane with static friction. The contact point of the disk with the plane has null instantaneous velocity. Assuming the center of the disk has fixed $v_0$ velocity at a time $t_0$, is it right to conclude that the disk will stop in a time $t_1 > t_0$ by effect of the momentum of friction force? (By the Euler second law we get a negative angular acceleration)"}
{"id":"5830","title":"What is the boundary condition of graphene flake with zigzag edges?","text":"It is a question about free carrier behavior in graphene flakes. (or may be called charge confinement) Say if we have a perfect hexagonal free standing graphene flake terminated with zigzag edges. Since there are two subatoms (let's call them A,B) in every unit cell. Edges will be terminated, let say, ABABAB (as shown in the figure). ![figure of graphene flake with zigzag edge](http:\/\/i.stack.imgur.com\/gt1tJ.png) My question is how to calculate (understand) electron wavefunction in this system (only considering relative low energy case, charge carrier can be seen as massless fermions). Another way to ask it what type of **boundary condition** one shall use for those two types of edges? According to this paper(pdf), scattering at the zigzag boundary dosen't couple the two valleys. I think we can only consider one valley picture(is this true?), which means the most simple Dirac equation $-i\\hbar \\nu_F\\nabla \\Psi_i=E\\Psi_i$, where $i=A,B$ (or Klein-Gordon eq). For simplicity, spin effect here can also be ignored(no magnetic field). I am not only asking about the edge states, also for higher energy states. What confusing me is how to distinguish this from the particle in a box case (relativistic particle, here I mean). Intuitively, I think electron will behave as billiard in side of the flake. But there could be two cases, either electrons bouncing between A,B edges, or they do it only in one type of edge, such as only bouncing at edges terminated with subatom A. Can it be understood this way? If yes, which of the two picture is the real one? Hope it is clear. Thanks."}
{"id":"11256","title":"What do we consider \"Perpetual Motion\"","text":"I know this is a bad question to most serious Physics but I have a question about what is considered “Perpetual motion.” The Foucault pendulum in the UN consists of sphere that passes directly over a raised metal ring at the centre that contains an electromagnet, which induces a current in the copper inside the ball. This supplies the necessary energy to overcome friction and air resistance and keeps it swinging uniformly. Now the swing of the pendulum is induced but the 36h 45m clockwise shift generated bay the earths rotation is perpetual or as long as the earth rotates. Is this assumption correct? Does a generator that works on tidal movements not fall in to the same assumption?"}
{"id":"65031","title":"Doubts about the definition of mass","text":"I'm having some problems understanding what are the possible definitons of mass and how they are related to each other. In Classical Mechanics, we can distinguish between inertial and gravitational mass (although they are supposed to be the same $m_i=m_g$): $$F=m_ia$$ $$F=G\\frac{Mm_g}{r^2}$$ On the other hand, in Special Relativity, mass is: $$p^\\mu p_\\mu=m^2$$ But in Quantum Mechanics, I think none of these definitions should be valid (Newton's second law doesn't hold, gravity can't be described in QM and QM isn't relativistic). So, what mass should we use in Schrödinger's equation, when we state that: $T=\\frac{p^2}{2m}$? In QFT, the correct mass is $p^\\mu p_\\mu=m^2$, isn't it? Finally, are there more possible definitions of mass, besides the description in String Theory (maybe in General Relativity, I've read that there are problems defining energy in a localised point, that we can only evaluate the total energy)? Are some definitons more fundamental than the rest? EDIT: What about this definition I've found in QM $m=\\hbar (\\frac{d^2E}{dk^2})-1$?"}
{"id":"19008","title":"Nanorobots. What stops us from producing them yet?","text":"If we can already predicts accuratelly motion on molecular levels, what stops us from developing small robots to, for instance, navigate through our blood vessels looking for cancerous cells and destroying them? What are the predictions for when we will able to do this? If someone is very interested on this goal and wants to participate on research, what should exactly he study? Programming, chemistry, physics, quantum mechanics? What could help at all? Books\/resources welcome."}
{"id":"19007","title":"Kinetic energy of Electron","text":"I had to find the kinetic energy of electron with wavelength $2$ pm. I used the formula $$ KE = \\frac{p^2}{2m} = \\frac{h^2}{\\lambda^2 2m}$$ which gave me result, $KE = 376.9$ KeV. But the answer given in the book is $292$ KeV. Am I making a mistake in calculating it?"}
{"id":"64780","title":"Comparing Static Frictions","text":"In this figure, which of the static frictional forces will be more? ![enter image description here](http:\/\/i.stack.imgur.com\/L9tHZ.png) _My aim isn't to solve this particular problem_ but to learn how is static friction distributed . Since each of the rough-surfaces are perfectly capable of providing the $-1N$ horizontal frictional force but why don't they ? This is kind of ambiguity that who will provide a bigger share in total static friction. And as the surface have different $\\mu$, so we can't even invoke symmetry."}
{"id":"60202","title":"Non-SHM oscillatory motion","text":"![enter image description here](http:\/\/i.stack.imgur.com\/muwVu.png) How to solve these kind of questions , where $|F| \\propto x^2$? How to find time period and velocity type related things to the oscillatory motion? $$m\\dfrac{d^2x}{dt^2}=F=-\\dfrac{dU}{dx}=-3kx|x|.$$ But after this $$m\\dfrac{d^2x}{dt^2}=-3kx|x|.$$ What is general solution of this ODE? I think it would give the $x$ in terms of $t$ and I would be able to get time period from it."}
{"id":"94355","title":"Vibrational noise for scientific apparatus on a satellite","text":"Say we're running an experiment on a satellite in _some_ earth orbit. For example, maybe we're imaging something with transmission electron microscope or we're catching atoms in ion traps --- really anything. In practice, and otherwise keeping things very simple on the satellite to avoid mechanical \/ etc. noises, what kind of vibrational noise profile will we have relative to what's possible on earth? What would be the primary cause of the main noise source (e.g. on earth this might be the building's ventilation system followed by geological sources, etc.)?"}
{"id":"43584","title":"What happens when a ferromagnetic object encounters a field too strong for it?","text":"Just thinking aloud ... It is possible for a star to grow so large it collapses under it's own gravity. Along a parallel path (so to speak) when a conductor carries a current too large for it to sustain, it burns out. Can a ferromagnetic object be imparted a field too strong for it? What happens then? I would begin to guess the dipoles within the magnet can only be magnetized so far; a stronger field wouldn't have an effect. But surely the same argument could have applied to a wire conductor?"}
{"id":"43588","title":"What makes thrust reversers different from fans on a sailboat?","text":"Why do thrust reversers on jet planes work? By Newton's third law, it seems like the force exerted by pushing air through the nozzle should cancel the force from redirecting the air forwards. The problem seems identical to the case of a fan on a sailboat, with \"fan\" replaced by \"jet engine\" and \"sail\" replaced by \"thrust reverser.\""}
{"id":"80605","title":"Calculate the latitude \/ longitude coordinates of the location where the Sun is at the zenith","text":"There are plenty of resources showing how to calculate zenith and azimuth of the Sun when the time and the location are given. However, I need to calculate the location where the sun is at the zenith for a given time and date, not the other way around. Is there a more or less easy way to do this calculation?"}
{"id":"17367","title":"Is it possible to calculate specific velocities from acceleration?","text":"I'm trying to understand a sample problem from a current (published 2011) high school physics textbook, but I really think the question is flawed. The task is to convert an acceleration-time graph (AT-graph) into a velocity-time graph (VT-graph). The AC-graph is given as: ![enter image description here](http:\/\/i.stack.imgur.com\/tTSbF.jpg) Then the book gives the following instructions for finding velocities from accelerations: ![enter image description here](http:\/\/i.stack.imgur.com\/kztRH.jpg) Here's my first problem. Where did they get the formula $v=(\\Delta a)(\\Delta t)$? Isn't $\\Delta a$ a more advanced concept called \"jerk\"? (a concept that this textbook never introduces, by the way). Shouldn't it be $\\Delta v = a\\Delta t$? And if so, we're not finding $v$ (which, correct me if I'm wrong, is impossible from the data given), but merely $\\Delta v$. Then they show the following calculations: ![enter image description here](http:\/\/i.stack.imgur.com\/ZzrCh.jpg) This chart is confusing to me. Is the time column an instant of time or a time interval? (I think it's a time interval). It seems to me that they're calculating $\\Delta v$, not $v$, in which case it's impossible to draw a VT graph. But that's what they do next anyway: ![enter image description here](http:\/\/i.stack.imgur.com\/TMsNF.jpg) Any feedback is appreciated."}
{"id":"17362","title":"What type of substances allows the use of the Ideal Gas Law?","text":"I know that I can use the ideal gas law with pure gases or pure liquids. But can I also use the ideal gas law at saturated gases and saturated liquids as long as they aren't two phase substances?"}
{"id":"17368","title":"Special conformal transformations and locality","text":"In the conformal symmetry, used in some QFT theories, the infinitesimal generators, applying to space-time, are all linear (translations, rotations, boosts, dilatation), except the special conformal transformations, which are not linear. I feel that this non - linearity could cause a conflict with locality, in the following sense:   * Let us consider two wave packets 1 and 2, separated by a space-like interval, corresponding to impulsions P1 and P2. If we make a linear transformation, for example a Lorentz transformation, we obtain impulsions P'1 and P'2. Now let us consider the total system 1 + 2; it has an impulsion P, which is equal to P1 + P2 because both systems are not causally related. By this same Lorentz transformation, the total impulsion P becomes P', but because the Lorentz transformation is linear, we have P' = P'1 + P'2, and P' is linear in P'1 and P'2, and is linear in P1 and P2. This is coherent with the fact that there is still no interaction between the systems 1 and 2.   * On the other hand, imagine we had taken, for example, a pseudo-Lorentz transformation , which would be, for example, quadratic, then, in the expression of P', we would have had cross-terms of type P1.P2, what would mean an interaction between systems not causally connected, which is incoherent. So, I need a detailed argumentation, to infirm or confirm my feeling : non- linear special conformal transformations could cause a conflict with locality."}
{"id":"12786","title":"Any interesting physics theroies\/equations to be used in a video game?","text":"I am designing a little game based on Newtons Graviational Law. Are there any theories that I can use to create games? thanks"}
{"id":"21184","title":"Speed of a dynamic hydraulic system","text":"We all have noticed that changing the temperature of the water in the shower is not instantaneous, rather the result is felt when the water that was in the tap works its way up to the showerhead. However, changing the pressure does feel instantaneous. I wonder at what speed the change in pressure propagates in the direction of flow in a flowing fluid. My guess would be at the speed of sound in the medium, based on the fact that it is faster than the speed of the flow of the water (like the temperature is), yet obviously at or below C. I am obviously missing the correct keywords to google for as I cannot find any references to this phenomenon. What is the answer to the query, or better yet, how could I have found this information (short of empirically)? Thanks."}
{"id":"107389","title":"Resultant wave in the Michelson-Morley experiment","text":"I was wondering when the two waves in the experiment, don't they lead to constructive interference? Consider the first case in the image. The two combining waves should lead to constructive interference. ![Wikipedia](http:\/\/i.stack.imgur.com\/u6zXO.gif) Source: Wikipedia. **Edit** Sorry for the confusion. What I wanted to ask is what John replied. But John how will they have half the intensity. The wavelength will be the same and so the frequency even if the wave splits. But after recombination the frequency should increase. Shouldn't it?"}
{"id":"21456","title":"Tubelights+power lines pictures?","text":"I've come across many pictures like these, sometimes in chain emails reporting the dangers of power lines. Another claim is that they run on \"wasted\" energy. The explanations given are that the wires set up a field, and thus a potential difference. This p.d. gives current in the wires. Now, I can't help but feeling skeptical. I can't see how the tubes would stay lit for the time it takes to set up the whole grid. Here's my logic: If current flows through the tubelight, there will be a net charge distribution on the caps. At one point, the charge distribution will be such that it creates its own potential, halting the flow of light. This works even if one end is grounded. An easier way to analyse this is a capacitor (GP: G for ground, P for power lines), with a bulb inside it. Each terminal of the bulb is connected to a plate (A and B). There's an optional grounding wire W. In this situation, it's obvious that eventually the plates AB will get enough charge (same charge density as PG incidentally) to create zero p.d. Current should be there for a brief moment in time. ![diagram](http:\/\/i.stack.imgur.com\/eJGEf.png) I understand that this is a DC situation, but for power lines, I think that something similar should happen. Or maybe not. I'm clear that the potential difference across the live and neutral switches polarities, but I'm not sure how the potential field is outside the wire. In fact, I feel that there shouldn't be a V\/E field, only a magnetic field. If it's true that the tubelights stay lit, then how is the energy \"wasted energy\"? I feel that even if the tubelight can stay lit, the energy is being drawn from the power lines. This looks just like how one can naively say that there is energy wasted in an open socket; when in reality plugging something in draws more energy from the system (and can dim other appliances). **Update** : So my questions are thus:   1. I'd like a clearer explanation of the potential and E field outside an AC power line.   2. Is the energy \"wasted energy\"?"}
{"id":"92861","title":"Could we run an electric car on electric eels?","text":"Could we run an electric car on a tank full of electric eels? I've heard electric eels are around 400 to 500 volts and can keep shocking for up to an hour. They also have a 10 volt system to sense with, which might be good for the headlights and cd player."}
{"id":"107383","title":"Sum of acceleration vectors","text":"If a point mass has some accelerations $\\mathbf{a_1} $ and $\\mathbf{a_2} $, why is mathematically true that the \"total\" acceleration is $\\mathbf{a}= \\mathbf {a_1}+\\mathbf {a_2}$?"}
{"id":"41985","title":"Inhomogenous schrodinger equation","text":"Please help me out in solving this inhomogeneous Schrodinger equation in Cylindrical co-ordinates [You may suggest if I have to go for mathematics]: $$ \\ddot R + \\frac1r\\dot R+\\left(b^2-d^2-\\frac{a^2}{r^2}\\right) R - \\frac{c^2}r \\delta(r-r_0) R ~=~ F e^{ikr}. $$ This is only the radial part."}
{"id":"20588","title":"Terminology question about energy","text":"I'm looking for the appropriate term to use for what gets \"used up\" as potential energy is converted to heat and work. For example, some of the the energy in solar radiation is converted by photosynthesis to chemical energy; much of the rest is lost as waste heat. The bio-chemical energy is used to power various biological functions until it has all been lost to waste heat. At the various steps along this path what term would you use for what there is less and less of? It can't be \"energy\" since that's conserved. It's potential energy, but is that really a standard term for what's lost when potential energy is converted to waste heat?"}
{"id":"41988","title":"Higgs stability in Standard Model","text":"I am a little unclear on what ramifications a negative quartic at high energies has on our world at low energies. (1) First of all, is it that there is a second, isolated minimum that appears at higher energies, or is it just that the original minimum at $246$ GeV was modified under RG? (2) Assuming its just a single minimum that was modified under RG: Now say the SM is correct at low energies with a 125 GeV higgs. Then we use RG to evolve up to some higher energies where the quartic is negative but the higgs potential is stabilized at some other (larger) field value by high-dimensional operators. Doesn't this new potential minimum just evolve back into the minimum at $v = 246$ GeV under RG evolution down to lower energies? I don't see why this is such a big deal, the vev shouldn't be RG invariant to begin with no? Measured Higgs mass and vacuum stability and http:\/\/motls.blogspot.com\/2012\/07\/why-125-gev-higgs-boson-isnt-quite.html?m=1 are related, but I don't think it addresses these particular questions (or at least in terms I understand). EDIT: Just to clarify my confusion. According to Terning's 'Modern Supersymmetry' pgs 78-79 in the MSSM the higgs gets a negative mass squared, and hence a vev, radiatively using the initial condition that all the masses are positive at high energies. This means at high energies the higgs potential is stable, and at low energies its unstable. So if you run the MSSM at low energies back up to high energies it will look like you are expanding around the wrong vacuum as well? Isn't this the same thing as what is happening in the Standard Model?"}
{"id":"44557","title":"Analysis of the impulse of 2 colliding carts under the effect of magnetic repulsion","text":"![enter image description here](http:\/\/i.stack.imgur.com\/IdJCw.jpg) Hi there! I have a question about an experiment that was conducted. It is related to momentum. 2 carts were put on a track on opposite sides. They were then propelled towards one another at approximately the same speed.Please note that on each end of each cart, there was placed a negative magnet, so as soon as they came close to each other, repulsion commenced. Two sensors were set on a frequency of 20 Hz to record their motion. After collecting the data, the data for the difference in momentum was graphed through the equation m2v2 - m1v1. The group did this to calculate the approximate average frictional force of the system by producing a linear fit. The slope of this line proved to be the average friction of the system. This was an extremely simple experiment and did not prove at all difficult. However, there is one question I have been asking myself that is driving me crazy. I can't seem to find the answer (Please note that our instructor did not ask us for this). Why are there spikes in the graph of Δp (change in momentum) very near the moment of collision (actually there is no \"collision\" because of the magnetic repulsion). Is it because the frequency was set only to 20Hz and thus since the instance of collision was so tiny, the sensors couldnt pick up data correctly? Is it because for the brief moment that the carts cease to move friction with the rail is static instead of kinetic and thus requires more force? Some useful information: mass of cart1: 255.9g mass of cart2: 251.1g Please note the equation: Δp= Δt*Favg and p = mv An answer would be greatly appreciated as this question has been driving me crazy."}
{"id":"26868","title":"What is the upper-limit on intrinsic heating due to dark matter?","text":"Cold dark matter is thought to fill our galactic neighborhood with a density $\\rho$ of about 0.3 GeV\/cm${}^3$ and with a velocity $v$ of roughly 200 to 300 km\/s. (The velocity dispersion is much debated.) For a given dark matter mass $m$ and nucleon scattering cross section $\\sigma$, this will lead to a constant collision rate of roughly $r \\sim \\rho v \\sigma \/ m$ for every nucleon in normal matter. The kinetic energy transferred to the nucleon (which is essentially at rest) will be roughly $\\Delta E \\sim 2 v^2 \\frac{M m^2}{(m+M)^2}$, where $M \\approx 1$ amu $\\approx 1$ GeV\/c${}^2$ is the mass of a nucleon. The limits for light ($m \\ll M$) and heavy ($m \\gg M$) dark matter are $\\Delta E_\\mathrm{light} \\sim 2 v^2 \\frac{m^2}{M}$ and $\\Delta E_\\mathrm{heavy} \\sim 2 v^2 M$. This leads to an apparent intrinsic heat production in normal matter $\\tilde{P} \\sim r \\Delta E \/ M$, which is measured in W\/kg. The limits are $\\tilde{P}_\\mathrm{light} \\sim 2 \\rho v^3 \\sigma m \/ M^2$ and $\\tilde{P}_\\mathrm{heavy} \\sim 2 \\rho v^3 \\sigma \/ m$. **What existing experiment or observation sets the upper limit on $\\tilde{P}$?** (Note that $\\tilde{P}$ is only sensibly defined on samples large enough to hold onto the recoiling nucleon. For tiny numbers of atoms--e.g. laser trap experiments--the chance of any of the atoms colliding with dark matter is very small, and those that do will simply leave the experiment.) * * * The best direct limit I could find looking around the literature comes from dilution refrigerators. The NAUTILUS collaboration (resonant-mass gravitational wave antenna) cooled a 2350 kg aluminum bar down to 0.1 K and estimated that the bar provided a load of no more than 10 $\\mu$W to the refrigerator. Likewise, the (state-of-the-art?) Triton dilution refrigerators from Oxford Instruments can cool a volume of (240 mm)${}^3$ (which presumably could be filled with lead for a mass of about 150 kg) down to ~8mK. Extrapolating the cooling power curve just a bit, I estimated it handled about $10^{-7}$ W at that temperature. **In both cases, it looked like the direct limit on intrinsic heating is roughly $\\tilde{P} < 10^{-9}$W\/kg.** However, it looks like it's also possible to use the Earth's heat budget to set a better limit. Apparently, the Earth produces about 44 TW of power, of which about 20 TW is unexplained. Dividing this by the mass of the Earth, $6 \\times 10^{24}$ kg, **limits the intrinsic heating to $\\tilde{P} < 3 \\times 10^{-12}$W\/kg.** Is this Earth-heat budget argument correct? Is there a better limit elsewhere? * * * To give an example, the CDMS collaboration searches for (heavy) dark matter in the range 1 to 10${}^3$ GeV\/c${}^2$ with sensitivities to cross sections greater than 10${}^{-43}$ to 10${}^{-40}$ cm${}^2$ (depending on mass). A 100 GeV dark matter candidate with a cross-section of 10${}^{-43}$ cm${}^2$ would be expected to generate $\\tilde{P} \\sim 10^{-27}$ W\/kg, which is much too small to be observed. On the other hand, a 100 MeV dark matter particle with a cross-section of $10^{-27}$ cm${}^2$ (which, although not nearly as theoretically motivated as heavier WIMPs, is not excluded by direct detection experiments) would be expected to generate $\\tilde{P} \\sim 10^{-10}$ W\/kg. This would have shown up in measurements of the Earth's heat production. * * * EDIT: So it looks like I completely neglected the effects of coherent scattering, which has the potential to change some of these numbers by 1 to 2 orders of magnitude. Once I learn more about this, I will update the question."}
{"id":"68668","title":"What is the domain of validity of the Omega equation?","text":"The Omega eqation is used in meteorology for estimate the value of the vertical velocity. Considering that the horizontal velocity field and density is given by observation. Under which condition this equation can be used for recover the vertical velocity and what is order of error between the real vertical velocity field and the one recovered by the Omega equation?"}
{"id":"13125","title":"How to make mirrors give more authentic image of myself","text":"My friend bought two normal mirrors from some shop probably IKEA, looking like this one: ![enter image description here](http:\/\/i.stack.imgur.com\/xmJ6w.jpg) My images are different from the two. One is a little fatter and shorter, and the other a little thinner and taller. I wonder what can possibly be the reasons?   1. They basically look the same, smooth and flat surfaces, almost the same size (one has a label saying 37x134cm), except one having a little thicker decoration border than the other. The one with thinker edge is the one giving me fatter and shorter image.   2. I don't have other things to place them, so I just make both of them stand on the floor, with a small angle lying against the wall. Although the two mirrors are placed in the same way, will different placement of the mirrors change my image?   3. What kinds of mirrors (size, ...) and how to place them will give me more realistic image of myself?"}
{"id":"13123","title":"Measuring concentration","text":"I am trying to measure the concentration of a Volatile Organic Compound (VOC) inside a liquid using Gas Chromatography (GC). The liquid contains particles that cannot be sent to a GC, so I am using the head-space method (allowing the organic compound to evaporate into the gas above the liquid--- see this link:http:\/\/www.chem.agilent.com\/Library\/applications\/5990-3285EN.pdf)-- I let the contents of the liquid equilibrate with the air in the vial at 50 C-- and then I use the air as feed for the GC. However, the concentration in the air is rather small ~ 10-20 ppm. Is there a technique that can be used to increase the concentration of the VOC's? I have wanted to use a smaller vial to collect my liquid, but the liquid splashes about and contaminates the cap of the vial, which makes me scared to inject a needle into it as I may wreck the needle. **Update** I tried to repeat the experiment with the vial at a higher temperature ~70 C. This lead to the needle clogging. I am unsure of the cause of the needle clogging but I suspect it may have something to do with water evaporating and taking some of the polymer particles with it. Once in the needle, the particles could get dry and solidify."}
{"id":"54007","title":"Number of microstates of discretized paths","text":"> _Let us consider a square grid, which has been rotated by 45deg. On this > grid we define a path, the directed polymer, which starts at the origin ($t = > 0$) and extends in the positive $t$-direction (at each grid point the path > goes either left or right; and steps in the negative $t$-direction are not > permitted). All paths of length $N$ end at the same $t$-position. The > distance of the endpoint to the $t$-axis is characterized by a number $m$. > Each path represents a microstate and the endpoint (i.e. $m$) the > macrostate._ How can I find the number of microstates, say, $W(m)$? I've tried that $W(m) = 0.5 N! m$, but I'm not sure. I do understand the problem, but I'm not sure how to solve it."}
{"id":"1599","title":"Can a web community write papers?","text":"the internet has changed science drastically. Not only in terms of distributing knowledge e.g. via online encyclopedias as wikipedia and freely available sources of publications as arXiv but also as a tool for communication through email and lively discussions in forums, newsgroups and communities like StackExchange. Scientific success nowadays greatly depends on the ability to build up teams with different scientific background and experience. To my mind, the ideal researcher has to seek for people helping him to tackle parts of his problems in which he is not an expert instead of spending months on something, someone else could have been done in days. So, can't **we** be this ideal researcher, or: Can a community on a webpage do research by openly presenting ideas and working them out? Or is this not possible due to \"stealing\" of ideas, lacking institutional structures and the like? What do you think? Robert"}
{"id":"48497","title":"Could temperature have been defined as $-\\partial S\/\\partial U$?","text":"When coming up with a definition of temperature, it's typical to start with an empirical definition that a system with a hotter temperature tends to lose heat to a system with a colder temperature. Combined with the second law of thermodynamics, that leads to the condition that for two systems in thermodynamic equilibrium, $$\\frac{\\partial S_1}{\\partial U_1} = \\frac{\\partial S_2}{\\partial U_2}$$ (consider it implicit throughout this question that volumes are constant). So temperature has to be some function of $\\partial S\/\\partial U$. Furthermore, we'd like the object that loses the heat to be the one with the hotter temperature, so temperature has to be inversely related to $\\partial S\/\\partial U$. Of course the conventional choice is $\\frac{1}{T} = \\frac{\\partial S}{\\partial U}$, but if we were rederiving thermodynamics and statistical mechanics from scratch, is there any reason we couldn't have chosen $$B = -\\frac{\\partial S}{\\partial U}$$ instead? Sure, it would lead to many common temperatures being negative, but suppose we're willing to accept that. Is there any other useful property that the standard definition of temperature has which wouldn't be shared by $B$? Would it _necessarily_ lead to nonlinear thermometers, for example? As Michael Brown pointed out in a comment, this alternative definition is just $B = -k\\beta$, so I guess another roughly equivalent way of saying what I'm asking would be, is there a practical reason not to use $\\beta$ or $B$ as temperature instead of $T$? I've already looked at this question and this one and some others, but any close-to-relevant answers there seem to be saying merely that it's conventional to define temperature the way it is defined, not that we couldn't have done it differently."}
{"id":"48495","title":"Relationship between height and velocity in conservation of mechnical energy","text":"I'm a high school physics student, and we recently did a lab on the conservation of energy where we measured the speed of a marble at varying heights on a rollercoaster track. We were supposed to graph the height and speed, and I ended up with something resembling a square root graph ![Original graph](http:\/\/i.stack.imgur.com\/Tp8ZP.png) We were required to linearize the data (so that we could have a single slope), so I squared the heights and got a slope of about $-0.11\\,cm^{-1}\\,s^{-1}$: ![Linearized graph](http:\/\/i.stack.imgur.com\/MJl1A.png) I don't understand what the slope is supposed to signify. I tried to take the derivative of velocity with respect to height in the equation $E = mgh + \\frac{1}{2}mv^2$, where E is constant, and got a cryptic answer of $\\frac{dv}{dh} = \\frac{-g}{v}$. Can someone help me understand what's going on here?"}
{"id":"48493","title":"Change of variables in path integrals","text":"I need to evaluate a path integral which involves a set of fields $X=\\left\\\\{ \\psi_i \\right\\\\}$: $$ I = \\int \\prod_i \\mathcal{D} \\psi_i e^{-S \\left[ \\left\\\\{ \\psi_i \\right\\\\} \\right] } $$ In order to simplify the treatment I'd want to make a change of variables, and express the integrand as a function of a new set of fields, let's call it $Y=\\left\\\\{ \\phi_i \\right\\\\}$. My problem is: how do I implement the change of variables? Can I extend the multi-dimensional case to the continuum and include the determinant of the Jacobian of the transformation in the integral, i.e. $$ J_{ij} = \\frac{\\delta X_i}{\\delta Y_j} $$ Moreover I'd want to defined one or more of the $Y_i$ variables as containing some derivatives of the original $X_i$ ones. My intuition leads me to write down the Jacobian in momentum space, where each derivative is then replaced by a $- \\mathrm{i} k_\\mu$ then evaluate it, also in momentum space. I would also say that as long as the Jacobian does not depends on the fields, it can be taken outside of the integral sign, i.e. it is a multiplicative constant. Is my intuition right? Any suggestion, or any reference to a textbook is welcomed. In particular I couldn't find in any textbook I own the cases in which one of the redefined fields depends upon the derivative of one the original fields. I know that this a mathematical question more than a physics question; however I decided to post it here because (usually) physicist are more used to manipulating path integrals."}
{"id":"79775","title":"Nature of frictional force","text":"I was thinking about a situation where a person in standing on the ground with some friction. The frictional force is directly proportional to the normal reaction acting on him by the ground. Assume that he leaned forward i.e his center of mass is not vertically up the point of contact of him with the ground.Then, does the frictional force change? My thoughts:Although the person has leaned forward the force $mg$ acting on him will be vertically downward. Since normal reaction is always normal to the surface of contact its magnitude will remain same and thus frictional force will remain same.But, I have seen athletes starting the running race with a leaning forward position which would be mostly for increasing the friction between their shoes and the track. So, I'm in a dilemma. Please help."}
{"id":"79772","title":"The double-trace deformation effect in AdS\/CFT","text":"Let me use this paper as the reference for this. I want to understand better the argument at the bottom of page 6. If the bulk $AdS$ metric is written as $\\frac{1}{r^2}(dr^2 + A(r)ds_{boundary}(x)^2)$ then the free massive scalars fields on it are of the form, $\\phi = r^{\\Delta_{+}}(\\alpha(x)) + r^{\\Delta_{-}}(\\beta(x))$,near $r=0$. (...and the numbers $\\Delta_{\\pm}$ depend on the dimensionality of the AdS and the mass of the field..)   * As per their notation their \"regular\" boundary condition in the bulk corresponds to setting $\\beta(x)$ (the coefficient of $r^{\\Delta_{-}}$) to zero on the boundary and \"regular\" boundary condition is setting $\\alpha(x)$ (the coefficient of $r^{\\Delta_{+}}$) to zero. How is this compatible with the fact that on using the regular boundary condition the dual CFT necessarily needs to have a term of the form $\\int d^d x \\beta(x)O(x)$ where $O$ has dimension $\\Delta_{+}$ and $\\alpha = \\langle O \\rangle$? I thought that they themselves said that in the regular scenario the $\\beta$ is being set to $0$ at the boundary - then what is this $\\beta(x)$ that is featuring in the boundary action?   * Below equation 3.19 they argue that the when the double-trace deformation of the boundary CFT is switched off it is seeing the \"irregular\" scenario and when it is turned to infinity it is seeing the regular scenario.  But how does one argue that the regular scenario is the the IR fixed point and the irregular scenario is the UV fixed point for the boundary CFT? Where is that argument?   * Rethinking the section 4 of this paper in a different way - Suppose one wants to calculate the determinant of the bulk theory $det(-\\nabla^2 + m^2)$ by taking product of eigenvalues. One wants to say calculate the determinant when the bulk has been quantized with the $\\Delta_{+}$ boundary condition. (...one can presumably ask the same question with $\\Delta_{-}$ as well..) If the bulk is $AdS_{d+1}$ then one can see that the small-r asymptotics ($r=0$ is the AdS boundary in the Poincare patch) of the harmonics are of the form, $# r^{a} + # r^{b}$ for some values $a$ and $b$ (which depend on the eigenvalue and $d$). Now knowing the above small-r asymptotics of the harmonics how does one pick out which of these will contribute to the above determinant with say the $\\Delta_{+}$ boundary condition? Can someone schematically sketch how the calculation looks like?"}
{"id":"111292","title":"Coordinate Systems in Loop Integrals","text":"Let us consider a two-point two-loop integral $\\int \\mathrm{d} ^ 4 k _ {2} \\int \\mathrm{d} ^ 4 k _ {1} \\, f(k _ {1}, k _ {2}, p)$, where $k _ {1}$, $k _ {2}$ and $p$ are four-dimensional vectors in Euclidean space and $p$ and $-p$ are the external momenta. There are three angles present in $f$: the angle $x$ between $k _ {1}$ and $k _ {2}$, the angle $y$ between $p$ and $k _ {1}$ and the angle $z$ between $p$ and $k _ {2}$. Which coordinate system is most suitable for consistently performing the integrals? For instance, for the one-loop integral $\\int \\mathrm{d} ^ 4 k \\, f(k, p)$, I take $p$ to be the z-direction. Then $p = k \\cos \\omega$, where $\\mathrm{d} V = |k| ^ {3} \\, \\sin ^ {2} \\omega \\, \\sin \\theta \\, \\mathrm{d} |k| \\, \\mathrm{d} \\omega \\, \\mathrm{d} \\theta \\, \\mathrm{d} \\phi$ is the four- dimensional spherical volume element and $|k|$ is the Euclidean norm of $k$. Is there a two-loop analog coordinate system which takes into account the three aforementioned angles?"}
{"id":"96045","title":"Idea of Covering Group","text":"1. $SU(2)$ is the covering group of $SO(3)$. What does it mean and does it have a physical consequence?    2. I heard that this fact is related to the description of bosons and fermions. But how does it follow from the fact that $SU(2)$ is the double cover of $SO(3)$?"}
{"id":"23742","title":"Why are we forced to choose a specific value for $\\pi$ field in Nambu-Goldstone phenomenon?","text":"In the sigma-model of spontaneous symmetry breaking, we have degenerate vacuum states. But if we don't pick up a particular value of VEV, we won't have any symmetry breaking. As I read from a book, in field theory at finite volume, there would be no problem; but at infinite volume, we are forced to choose a value of the $\\pi$ field. But why?"}
{"id":"23749","title":"Derivation of the Gauss-Codazzi equation","text":"I'm interested in the derivation of the Gauss equation (Gauss-Codazzi). Usually we consider the definition of the Riemann tensor on the hypersurface. $$^{(n-1)}R_{abc}^{~~~~~~~d}~w_d=[D_a,D_b]w_c$$ where $D$ is the connexion associated to the induced metric ($h$) on the hypersurface. We have \\begin{align} D_aD_bw_c=D_a\\Bigl[h_b^{~d} h_c^{~e}\\nabla_d w_e\\Bigr]=h_a^{~f}h_b^{~g}h_c^{~e}\\nabla_f\\Bigl[h_g^{~d}h_k^{~e}\\nabla_dw_e\\Bigr] \\end{align} where $\\nabla$ is the connexion of the metric ($g$) in the full space. Hence after some algebra we can arrive at the desired result. What I don't understand is why can we not do: $$ D_aD_bw_c=D_a\\Bigl[h_b^{~d} h_c^{~e}\\nabla_d w_e\\Bigr]=h_b^{~d} h_c^{~e}D_a\\Bigl[\\nabla_d w_e\\Bigr] $$ because $D_a h^b_c=0$. But in that case I would have $$ D_aD_bw_c=h_b^{~d} h_c^{~e}D_a\\Bigl[\\nabla_d w_e\\Bigr]=h_b^{~d} h_c^{~e}h_a^{~\\mu}h_d^{~\\nu}h_e^{~\\sigma}\\nabla_\\mu\\Bigl[\\nabla_\\nu w_\\sigma\\Bigr]=h_a^{~\\mu}h_b^{~\\nu}h_c^{~\\sigma} \\nabla_\\mu\\nabla_\\nu w_\\sigma $$ Therefore we would have $$^{(n-1)}R_{abc}^{~~~~~~~d}~w_d=[D_a,D_b]w_c=h_a^{~\\mu}h_b^{~\\nu}h_c^{~\\sigma} ~^{(n)}R_{\\mu\\nu\\sigma}^{~~~~~~~~d}w_d$$ which is not the desired result. So I understand the standard derivation, but I don't understand why in the way that I wrote, it doesn't work ?"}
{"id":"51784","title":"Time-ordering in QFT","text":"In Srednicki _QFT_ page 37. In the derivation of LSZ reduction formula, he introduces the time-order operator $T$, so no time-dependent creation\/annihilation operators are left in the transition amplitude. How can this be justified mathematically? And if I understand this correctly, if time ordering is not used, then a term like $$\\langle 0\\mid a_{1}\\left ( -\\infty \\right )a_{2}\\left ( -\\infty \\right )a_{1^{\\tilde{}}}^{\\dagger}\\left ( \\infty \\right )a_{2^{\\tilde{}}}^{\\dagger}\\left ( \\infty \\right )\\mid 0\\rangle $$ this mean that there's a contribution that depend on the amplitude of transition from the final momenta to the initial momenta but quantum mechanically the final momenta are not known in advance so such a term can't contribute to the process. Does this explain the use of time-ordering physically?"}
{"id":"51787","title":"Scattering in quantum wires","text":"I'm working on a project involving scattering of particles in networks of quantum wires off of point scatterers. The problem that I am having is that even in fairly simple configurations of quantum wires there is nontrivial scattering even in the absence of any scatterer. For example, if you consider a T-junction or a single wire with a bend in it there must be an S-matrix just from the particle scattering off of the walls of the wire itself. How can I calculate the transition amplitudes for this kind of scattering?"}
{"id":"35003","title":"Collision problem: Finding the final speed of the collider","text":"> A particle of mass $M$ moving in a straight line with speed $v$ collides > with a stationary particle of the same mass. In the center of mass > coordinate system, the first particle is deflected by 90 degrees. Find the > speed of the second particle after collision in the laboratory system. ## My effort: Considering the lab frame, we have, by conservation of linear momentum $$Mv=Mv_{1f}+Mv_{2f}\\tag1$$ which implies that $$v=v_{1f}+v_{2f}\\tag2$$ We want to know $v_{2f}$. Therefore, we have to find $v_{1f}$. From my textbook, I get $$v_{1f}=v+v_{1f}'\\tag3$$ where $v_{1f}'$ is the post-scattering velocity of the incident particle in the center of mass system and $v$ is the velocity of the center of mass in the laboratory system. Also from my text, I get $$V=\\frac{v_{1i}'}{2}=\\frac{v}{2}\\tag4$$ Therefore, it looks like I have everything I need to solve the problem except $v_{1f}'$. All I am given is that the post-scattering angle of the incident particle in the center of mass frame is 90 degrees. How do I find the velocity of the same particle in terms of the incident velocity in the lab system, $v$?"}
{"id":"95384","title":"Canonical transformations in Hamiltonian mechanics","text":"How to prove that in the new Hamiltonian, which is formed by any of the generator function will not contain $Q$ (transformed from $q$)? I.e. new Hamiltonian will only be a function of $P$ (transformed from $p$)."}
{"id":"8307","title":"Perturbation of a Schwarzschild Black Hole","text":"If we have a perfect Schwarzschild black hole (uncharged and stationary), and we \"perturb\" the black hole by dropping in a some small object. For simplicity \"dropping\" means sending the object on straight inward trajectory near the speed of light. Clearly the falling object will cause some small (time dependent) curvature of space due to its mass and trajectory, and in particular, once it passes the even horizon, the object will cause some perturbation to the null surface (horizon) surrounding the singularity (intuitively I would think they would resemble waves or ripples). Analogously to how a pebble dropped in a pond causes ripples along the surface. Is there any way to calculate (i.e. approximate numerically) the effect of such a perturbation of the metric surrounding the black hole?, and specifically to calculate the \"wobbling\" of the null surface as a result of the perturbation,maybe something analogous to quantum perturbation theory? Or more broadly, does anyone know of any papers or relevant articles about a problem such as this?"}
{"id":"56406","title":"Reconciling Units in Jeans' Criterion Formula","text":"In \"Physics and Chemistry of the Solar System\" Jeans' Criterion is given as: $\\frac{GmM}{R_c} = \\frac{3mkT}{2}$ ... To me this suggests that on the left we have Joules, and on the right we have kg$\\cdot$joules. I then went to some old lecture notes and found that the professor derived the Jean's Radius from: $\\frac{GM}{R_c} = \\frac{3kT}{2}$ which also has (as far as I can tell) an issue with units. Can anyone help my understand what I'm missing with this equation?"}
{"id":"21566","title":"Oil drop experiment--How was the result so accurate?","text":"I remember while learning about Millikan's oil drop experiment and being pretty skeptical about the setup. I know that there is a lot of controversy regarding manipulation of data, but the fact is; he still got a pretty close answer. I mean, we can get the correct answer by manipulation of data if we know the answer already. If we don't all we can do is 'reduce' the error box (which Millikan purportedly did). But I'm surprised that he still got an answer close to 1% of the actual value. I personally would have expected an error of maybe even an order of magnitude for these reasons:   * Viscosity of air: The viscosity is dependent upon humidity and other factors. I know he _did_ use the wrong value here.   * $g$: How was this known to the accuracy required for this experiment? It varies across the Earth, and besides, could it be measured to whatever accuracy using the equipment available at that time?   * Velocity of the oil drop: I don't know how fast the oil drops fell, but to me, terminal velocity looks pretty fast. How did he accurately measure the velocity? A micrometer eyepiece takes time to focus, and more time to take a reading. And anyways micrometer scales aren't that precise, are they?   * Potential difference: How did he accurately know this?   * \"Selecting one drop\": How was this done? If all drops had a charge as a small integral multiple of q, there should be more drops with the same charge which behave the same. How can you get rid of those?   * Densities of air\/oil: Air density varies. Oil density may not be known accurately.   * Electric field: Between plates, E is not exactly constant or equal to $V\/d$ There are fringing effects which may alter the value of E. IMO since the distance between the plates is comparable to the area (it look like it), fringing effects are pretty significant.   * Stokes law is only applicable for perfectly spherical bodies. Is a falling oil drop perfectly spherical? Basically I'm skeptical of how one can get such a tiny number with 1% accuracy with all those uncertainities. Could someone convince me how he _did_ get it so accurately? Were there any measures he took to minimise these errors (I can't find any)?"}
{"id":"21849","title":"Please recommend a good book about physics for young child (elementary school aged)","text":"I'm looking for a book that would be appropriate for an advanced elementary school aged kids (say, 6-11 YO) **describing the basics of physics (or sciences in general) in entertaining way**.   * The structure of the book should preferably (though not necessarily) be \"Curious question\" followed by entertaining-but-advanced for their-age physics answer. As a good example of a chapter\/story I had in mind: \"Why can the soap wash off dirt\/grease?\" \"Why can the water be poured into containers?\" \"How does a plane fly?\" \"How does a rocket fly?\"   * The topics that must be covered are atoms\/molecules, how they go together to make various matter etc...   * Doesn't have to be 100% limited to physics - could include chemistry, biology, other natural sciences.  P.S. For those fluent in Russian, i'm looking for an English equivalent of the soviet popular kid's physics\/science book \"почему вода мокрая\")."}
{"id":"99967","title":"Working of dry cell battery!","text":"please help me out in this query!! actually i need to know why when we use a battery or dry cell, As we know it consists of :- 1)electrolyte 2)cathode and anode. but i need to know as electrolyte is consisting of a salt (we take it as NaCl salt for it ) and the ions is formed from the salt due to potential we make on both ends of the battery, but why it cant happen as if the ions can directly form without connecting the battery into a wire , because the two terminals consists of positive and negative ions which can directly put force on NaCl compound and can break it ,but why their is a need to create a potential by attaching the battery with a wire externally? In other words:- Why molten salt as an electrolyte cant ionise normally in a cell without connecting the cell with a wire outside? regards :- Harjot singh"}
{"id":"133762","title":"Graph Theory and Feynman Integrals","text":"In Vladimir A. Smirnov's book _Analytic Tools for Feynman Integrals_ , Section 2.3, the alpha representation of general Feynman integral takes the form $$ F_{\\Gamma}(q_1,\\ldots,q_n;d) = \\frac{i^{-a-h}\\pi^{2h}}{\\prod_l\\Gamma(a_l)} \\int_0^{\\infty}\\mathrm{d}\\alpha_1 \\ldots \\int_0^{\\infty}\\mathrm{d}\\alpha_L \\prod_l\\alpha_l^{a_l-1} \\mathcal{U}^{-2} Z e^{i\\mathcal{V}\/\\mathcal{U} - i\\sum m_l^2\\alpha_l} $$ where $\\mathcal{U}$ and $\\mathcal{V}$ are defined as sums running over trees and 2-trees of the given Feynman graph. I know that $\\mathcal{U}$ is equivalent to $\\det{A}$ in the $4h$-dimensional Gauss integrals, but I can't figure out how it can be expressed in the language of graph theory. Could anyone provide some help? References on the topic of graph theory and Feynman integrals are also desired. Thanks a lot!"}
{"id":"116608","title":"How exactly does gravity work?","text":"The electromagnetic force and strong and weak forces require particles like photons and gluons. But in case of gravity there is no such particle found. Every _mass_ bearing object creates a gravitational field around it, and whenever another mass bearing object enters its field the gravitational force comes into operation. If all other forces of nature have some particles associated with them why should gravity be an exception? And if there is no such particle, what exactly is the gravitational field and how does it spread over an infinite distance and cause the gravitational force to operate? * * * **Note:** I am a high school student and have not studied quantum mechanics."}
{"id":"104780","title":"Choosing proper centripedal force in Newtonian gravitational forces","text":"I am trying to do an orbit simulation, I took Newton's law of Newtonian gravitational forces. It seems to work correct, but I can't really determine the velocity of an object needed to stay in free fall. I didn't want to use the masses in our planet systems, as the values are to high (in kg) So when my earth's mass is 1, what should be the masses be of the orbiting satellites and what should the initial velocity be to stay in freefall, when I want to altitude be 50 units (meters). It would be nice, if you could give me some hints!"}
{"id":"105432","title":"Is there a graphical representation of the Lorentz transformation equations?","text":"I always loved theoretical physics as a kid and when I came upon this site while seeking computer advice via superuser I had to stick my silly little head into an oasis of intelligence. I have often wondered what it would look like to view an object moving via the Lorentz transformation equations from a nominal speed of less than $c$ towards $c$ at an acceleration that a human could survive, let's say $3g$. And the space-time distance could be 100 light years. Would anyone be willing to produce a graph of this 1 dimension? I have tried to visualize it since I read \"A Sophisticate's Primer of Relativity\" 2nd Ed PW Bridgman when I was 19. As a novice I always wanted an answer from a higher power and now I think I can get a response."}
{"id":"111888","title":"Partition Function - TOTAL internal energy vs Average Energy","text":"Given a partition function Z, the books sometimes uses the terms 'total internal energy' and 'average energy' exchangeably. It confuses me to no end. On one hand they say that Internal energy is average energy: $$\\bar U = -\\frac{\\partial ln Z}{\\partial \\beta}$$ Then they use this to calculate specific heat: $C_v = (\\frac{\\partial U}{\\partial t})_V$. On the other hand they say internal energy is $$U = \\sum \\epsilon_p n_p = \\int \\epsilon \\space n_p g_{(\\epsilon)} d\\epsilon$$ where $n_p$ is mean number of particles with energy p, $g_{(E)}$ is density of states. Then they use this to calculate heat capacity. Which is which?!"}
{"id":"83402","title":"Problem in progresive wave","text":"In this progressive wave  we need to find out the phase difference. If we define it by $\\phi$ in the x distance, we get . $$\\phi = \\frac{2 \\pi}{\\lambda}x$$ But when we write the prograsive wave equation we , write as, $$y= A \\sin (\\omega t - \\phi)$$ A author says that this phase difference is less than the phase difference in the left side. My problem is, we can insert also + sign. But what is the physics for inserting the plus or minus sign. Thanks"}
{"id":"107495","title":"Examples to illustrate temperature dependant radiation by examples of temperature to color relation","text":"I'm looking for a set of examples to **illustrate** the relation of temperature and color of \"glowing\" bodies.   It should allow to build an intuitive understanding of this relation, so it's not about spectra, but just color in terms of visual appearance. The examples should cover infrared to somewhere above visual light if possible (or reasons why it's not possible). * * * Some good temparature\/color-relation-examples froom comments\/answers: An example for about **2500K** could be a **tungsten wire** of an incandescent lamp. It is not a good example because the wire in gas is not a good model for a black-body, as C. Towne Springer explains in his answer.   He provides an an **electric kiln** as an example for about **1400K** giving very good black-body approximation. As said, it's about the illustration, not about perfectly matching the spectrum of a black body - \"similar enough to look the same\" would qualify. From Robin's comment, the **sun** could be good as example for about **5800K**."}
{"id":"76015","title":"How long does it take for the speed of the train to decrease to $1~\\text{m\/s}$?","text":"Here's the full problem: > _The speed of a train coasting on a level track satisfies the differential > equation $\\mathrm{d}V\/\\mathrm{d}t = kv^2$. The initial speed of the train is > $v(0)=10~\\text{m\/s}$ and the speed is decreasing at the rate of > $-1~\\text{m\/s}^2$ when $v=5~\\text{m\/s}$. How long does it take for the speed > of the train to decrease to $1~\\text{m\/s}$? Under this model, when does the > train stop?_ I'm very bad with word problems and have been struggling quite a bit. I'd go to my school's tutoring center but currently we are on labor day vacation. -_- EDIT 1: I've attempted it a few times and here's my answer: $vknot=10~\\text{m\/s}, v=5~\\text{m\/s}, a=-1~\\text{m\/s}^2$ I found $k = -1\/25$ and the proceeded to integrate the $\\mathrm{d}V\/\\mathrm{d}t = kv^2$. The equation I then got was $v=kv^2t + vknot$ and plugged all my conditions in. My answer was $t=5~\\text{s}$. I have no idea if I am correct as word problems always throw me off. EDIT 2: I found k=-1\/25 by plugging into the dV\/dt equation. dV\/dt is the same as acceleration and it's stated that our acceleration is -1m\/s^2, leading to this: $-1 = k(5)^2$ ===> $-1 = 25k$ ===> $-1\/25 = k$. I don't know if this was valid to do, but I wasn't sure how to approach so made the assumption that it was valid. EDIT 3: After realizing the error with my integration, after integrating I got the formula $v=v(vknot)kt + vknot$. Time was then t=2.5 after plugging in my information. Now I just don't know how to tell when the train stops..."}
{"id":"39525","title":"Why are black hole singularities stable?","text":"The Friedmann equations says that huge matter densities lead to huge expansion rates. In Newtonian gravity, two massive point particles separated by an infinitesimal distance will experience an enormous force. So what is different about a black hole?"}
{"id":"78143","title":"Uniform Acceleration","text":"> _A steel ball is allowed to roll from rest down a 20.0 m long slop and over > a 25 m long level surface. It takes 4.0 seconds to roll down the slope. > Calculate the speed at the bottom of the slope._ I found the average speed on the slope to be 50 m\/s. But, I'm not sure what formula to use to find the speed at the bottom of the slope."}
{"id":"31730","title":"Minimizing the energy of a Slater determinant: why are the Lagrange multiplier elements of a Hermitian matrix?","text":"If I want to minimize the energy of a Slater determinant subject to the constraint that the spin orbitals are orthonormal (as in the Hartree-Fock approximation), I can use Lagrange's method of undetermined multiplier, i.e. $$L[\\\\{\\chi_{a}\\\\}] = E_{0}[\\\\{\\chi_{a}\\\\}]-\\sum_{a=1}^{N}\\sum_{b=1}^{N}\\varepsilon_{ba}([a|b]-\\delta_{ab})$$ where $\\\\{\\chi_{a}\\\\}$ are the spin orbitals, $E_{0}$ is the ground state energy, $[a|b]$ is the overlap integral between spin-orbitals $\\chi_{a}$ and $\\chi_{b}$ and $\\varepsilon_{ba}$ is a Langrange multiplier. By setting the first variation of $L$ to zero I can proceed to minimize the energy of a single determinant $|\\Psi_0\\rangle=|\\chi_{1}\\chi_{2}\\cdots\\chi_{a}\\chi_{b}\\cdots\\chi_{N}\\rangle$. A problem in Szabo and Ostlund's \"Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory\" asks me to prove that the Lagrange multipliers are elements of a Hermitian matrix, i.e. $\\varepsilon_{ba} = \\varepsilon_{ab}^{*}$, given that $L$ is real and $[a|b] = [b|a]^{*}$. How would one go about proving this? Where I am starting at the moment is that since $L$ is real, then $L = L^{*}$. If I find $L^{*}$ then making this equal to $L$ will show that it is necessary for the Lagrange multipliers to be elements of a Hermitian matrix. However, following through with this doesn't seem to get me to a point where I can say that $\\varepsilon_{ba} = \\varepsilon_{ab}^{*}$."}
{"id":"68506","title":"No well-defined frequency for a wave packet?","text":"There are similar questions to mine on this site, but not quite what I am asking (I think). The de Broglie relations for energy and momentum $$ \\lambda = \\frac{h}{p}, \\\\\\ \\nu = E\/h .$$ equate a specific frequency and wavelength to a particle, yet we know that a wave packet is a linear combination of an infinite range of frequencies and wavelengths. How is it that we (or nature) choose one frequency and wavelength out of the range? Does this have to do with the collapse of the wave packet when measured? And if so, is the resulting measured frequency a random outcome? Similarly, when an electron jumps from one energy level to another in an atom, it emits a photon of frequency $$ \\ \\nu = \\Delta E\/h .$$ Since the photon is not a pure sinusoidal wave, how can a single frequency be ascribed to the photon?"}
{"id":"31192","title":"Lorentz transformations of the polarization vector","text":"![enter image description here](http:\/\/i.stack.imgur.com\/jt63k.png) Let $\\bf{n}'$ be a unit vector in the direction of a wavevector in the plasma rest frame and $\\bf{B'}$ be a unit vector along the magnetic field in the plasma rest frame. The electric field of a linearly polarized electromagnetic wave is directed along the unit vector $\\hat{e}=\\bf{n'} \\times \\hat{B'}$, and the magnetic field of the wave is along the unit vector $\\hat{b'}=\\bf{b}\\times \\hat{e'}$, such that the Poynting flux along is $\\hat{e'}\\times \\hat{b'}$ directed along $\\bf{n'}$. We give a Lorentz boost to the explosion frame to find the electric field $e$ there, normalize it to unity, and project e$$ on some given direction (e.g. , along the projection of the flow axis on the plane of the sky). Fields in the wave expressed in terms of the direction of a photon in the explosion frame $\\bf{n}$ is \\begin{equation} \\hat{e'} ~=~\\frac{\\bf{b} \\times \\hat{B'}}{\\Gamma (1-\\bf{n}\\cdot\\bf{v})}+\\frac{1+\\Gamma (1-\\bf{n}\\cdot \\bf{v})}{(1+\\Gamma(1-\\bf{n}\\cdot \\bf{v}))} \\end{equation} where $\\Gamma=\\frac{1}{\\sqrt{1-v^2\/c^2}}$ is the bulk Lorentz factor. Would some one help to deduce the above equation?"}
{"id":"31194","title":"Higgs Boson mass in Electron volts?","text":"Im no physics genius here, I was just interested in the Higgs Boson so I was reading this article : How the Discovery of the Higgs Boson Could Break Physics I came across this > Furthermore, all indications are that scientists will find that the Higgs > **weighs 125 gigaelectronvolts (GeV)** – or about 125 times more than a > proton – which means that it sits exactly where the Standard Model expected > it to be. How can weight be measured in electron-volts?"}
{"id":"108343","title":"Question on dielectric sphere","text":"The question is: Suppose the field inside a large piece of dielectric is $E(0)$ so that $D(0) =$$\\epsilon(0)E(0)$ + $P$. Now a small spherical cavity is hollowed out of the material. Find the displacement($D$) in the center of the cavity in terms of $D(0)$ and $P$. Now, the question had a hint: carving out a cavity is the same as superimposing an object of the same shape but opposite polarization. Using that hint the field at center of sphere = $E(0)$ - field due to uniform polarization of sphere. So we get, $E=E(0)+P\/(3\\epsilon(0))$ To get $D$ it should be $\\epsilon(0)E$ and not $\\epsilon(0)E$ + $P$. Why is it so? We have a field $E(0)$ which gives rise to polarization $P$ of the dielectric material. So we get $D(0)$. So now while calculating $D$ at center why don’t we consider $P$? Is it because that, if I do consider $P$, then I would somehow consider its contribution twice?"}
{"id":"108346","title":"Gravity force between two objects with different mass","text":"We know thanks to Newton that: $$F=G\\frac{m_1\\cdot m_2}{r^2}$$ Where $G$ is the Gravitational Constant that is about $6.673\\cdot10^{-11}$ $m_1$ and $m_2$ are the masses of two different objects $r$ is the distance between them. We also know that $F=ma$ so can we write this? : $$ma=G\\frac{m_1\\cdot m_2}{r^2}$$ If so what is $m$ in the first term of the equation? $m_1$ or $m_2$? Thanks a lot for help!!!"}
{"id":"91366","title":"What happens when molten metal cools within a strong magnetic field?","text":"What happens when molten metal cools within a strong magnetic field? I don't know what more say, don't remember anything relevant to this in my uni text books."}
{"id":"72221","title":"Calculating an energy of an electron with known De Broglie wavelength (why can't we calculate it similar than we do it for a photon)","text":"Lets say we have an electron with known De Broglie wavelength $\\lambda$. Can anyone justify or explain why we calculate its energy $E$ using **1st** the De Broglie relation $\\lambda = h\/p$ to get momentum $p$ and **2nd** using the invariant interval to calculate $E$: \\begin{align} p^2c^2 &= E^2 - {E_0}^2\\\\\\ E &= \\sqrt{p^2c^2 + {E_0}^2} \\end{align} Why we are not alowed to do it like we do it for a photon: \\begin{align} E=h\\nu = h\\frac{c}{\\lambda} \\end{align} These equations return different results."}
{"id":"72229","title":"Uniqueness of the number of degrees of freedom","text":"As per my knowledge, degrees of freedom of any physical system are the number of independent quantities(coordinates) which need to be specified in order to specify the state of a system uniquely. However, my question is, why is there a unique number of degrees of freedom for any system? Or, to put it differently, if a set of $n$ independent coordinates describes the state of the system completely, why is it true that any other set of $n$ independent coordinates also describes the state of the system completely? And why can there not be a set of $m$ independent coordinated which also describe the state of the system completely, where $m>n$ ?"}
{"id":"73571","title":"How to solve bound states of 2D finite rectangular square well?","text":"I want to solve bound states (in fact only base state is needed) of time- independent Schrodinger equation with a 2D **finite** rectangular square well \\begin{equation}V(x,y)=\\cases{0,&$ |x|\\le a \\text{ and } |y|\\le b$ \\\\\\ V_0,&\\text{otherwise}}.\\tag{1}\\end{equation} $$\\Big[-\\frac{\\hbar^2}{2m}(\\partial_x^2+\\partial_y^2)+V(x,y)\\Big]\\psi(x,y)=E\\psi(x,y)$$ At first glance, this problem is simple. It seems that the solution is variable-separable and can be written as $\\psi(x,y)=f(x)g(y)$. Then $$ \\frac{f''(x)}{f(x)}+\\frac{g''(y)}{g(y)}+\\frac{2m}{\\hbar^2}(E-V)=0.$$ Let $E=E_x+E_y$ and $V=V_x+V_y$, then the problem is reduced to two 1D problems $$\\cases{f''(x)+\\frac{2m}{\\hbar^2}(E_x- V_x)f(x)=0\\\\\\g''(y)+\\frac{2m}{\\hbar^2}(E_y-V_y)g(y)=0}.$$ However, how to determine $V_x$ and $V_y$ in the 2D space? A definitely wrong method is making $$ V_x=\\cases{0,&$|x|\\le a$\\\\\\V_1,&$|x|>a$}\\text{ and }V_y=\\cases{0,&$|y|\\le b$\\\\\\V_2,&$|y|>b$}\\tag{2}.$$ In fact, the potential Eq. (2) is equivalent to two independent \"1D finite square well\" problems in $x$ and $y$ direction respectively. However, a careful reader will note that the potential Eq(2) is DIFFERENT from Eq(1), which means that the potential Eq(2) is NOT what we want. It's not a rectangular well, but as following ![Potential of Eq\\(2\\), but NOT a 2D square well.](http:\/\/i.stack.imgur.com\/IO5Xu.png). Then, I find that a variable-separable bound state for finite 2D square well does not exist. Although analytical solutions exist in each region with a constant potential, problems occur when matching boundary conditions to keep the continuity of $\\psi(x,y)$. Unlike matching boundary condition at descrete points in 1D, in 2D we have to match boundary conditions along lines, e.g., $$ f_1(a)g_1(y)=f_2(a)g_2(y)$$ in the boundary between $x<a$(region 1) and $x>a$ (region 2). This leads to $$ g_1(y)\/g_2(y)=f_2(a)\/f_1(a)=constant.$$ Matching all boundaries this way will lead to that $\\psi(x,y)$ have to be 0 outside the well. But this cronsponds to the case of INFINITE well. It's not the solution of finite well. Then I think no solutions exist under the separating-variable method. Then, the question is, beyond separating-variable method, how to solve this problem? BTW: Does anyone know that what kind (shape) of 2D well is solvable for bound states and how? (Potential with circular symmetry is excluded, because I know how to solve it. I want to find another shape of 2D well which is solvable.)"}
{"id":"76662","title":"Total angular momentum - single electron","text":"I have been dealing with total angular momentum of the single electron which is outside the closed shells in which sum of the angular momentums is zero. My book says that total atomic angular momentum is $\\mathbf{J}=\\mathbf{L+S}$ and its magnitude $J=\\sqrt{j(j+1)}\\hbar$ where quantum number $j$ can have half-integer values on an interval $|\\ell-s|\\leq j \\leq|\\ell+s|$. There is allso a quantum number $m_j$ which can have half-integer values on an interval $-j \\leq m_j \\leq j $. So lets say I have a single electron in an orbital $\\scriptsize\\boxed{\\ell=1}$ _(p orbital)_. In this case $j=\\tfrac{3}{2},\\frac{1}{2}$ while $m_j=\\tfrac{3}{2},\\frac{1}{2},-\\tfrac{3}{2},-\\frac{1}{2}$ and I can calculate two magnitudes for $J$: \\begin{align} J&=\\tfrac{\\sqrt{15}}{2}\\hbar\\\\\\ J&=\\tfrac{\\sqrt{3}}{2} \\hbar \\end{align} Now there is this weird vector sum image that I don't understand completely: ![enter image description here](http:\/\/i.stack.imgur.com\/vvecP.jpg) On the left image we have $J=\\tfrac{\\sqrt{15}}{2}\\hbar$ while on the right we have $J=\\tfrac{\\sqrt{3}}{2}\\hbar$. It seems that vector $\\mathbf{L}$ together with its magnitude $L=\\sqrt{\\ell(\\ell+1)}\\hbar = \\sqrt{2}\\hbar$ is the same in both cases - this makes sense as quantum number $\\ell$ doesn't change. But how do we explain the change in $\\mathbf{S}$?"}
{"id":"110715","title":"Is the cosmological redshift caused by the Planck mass increasing?","text":"The standard explanation for the cosmological redshift is that photons emitted from far away galaxies have their wavelengths lengthened as they travel through the expanding Universe. But perhaps the photons do not lose energy as they travel but rather the atoms in our detectors are more energetic in comparision with the atoms that emitted those photons a long time in the past leading to an apparent redshift effect? **Addition** (having had a comment exchange with @rob, see below) : My hypothesis is that the Planck mass $M_{pl} \\propto a(t)$ where $a(t)$ is the Universal scale factor. **Addition 2** Of course if the Planck mass $M_{pl}$ is changing then $G=1\/M^2_{pl}$ is changing so that we no longer have standard GR! I've asked this question before, see Cosmological redshift interpretation, but this time I'm including a little bit of theory to back up my hypothesis. For simplicity let us assume a flat radial FRW metric: $$ds^2=-dt^2 + a^2(t)\\ dr^2$$ Consider the null geodesic path of a light beam with $ds=0$ so that we have: $$dt = a(t)\\ dr\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (1)$$ Now at the present time $t_0$ we define the scale factor $a(t_0)=1$ so that we have: $$dt_0 = dr\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (2)$$ Substituting equation (2) into equation (1) we have: $$dt = a(t)\\ dt_0$$ In order for the interval of time $dt$ to stay constant as the scale factor $a(t)$ increases we must have the corresponding interval of present time $dt_0$ varying inversely with the scale factor: $$dt_0 \\propto \\frac{1}{a(t)}$$ Thus as cosmological time $t$ increases, and the Universe expands, equal intervals of cosmological time $dt$ correspond to smaller and smaller intervals of present time $dt_0$. Now the energy of a system is proportional to the frequency of its oscillation which in turn is inversely proportional to its oscillation period: $$E(t) \\propto \\frac{1}{dt}$$ The corresponding energy of the system in terms of the present epoch $t_0$ is given by $$E(t_0) \\propto \\frac{1}{dt_0}$$ $$E(t_0) \\propto a(t)$$ Thus an atom at time $t$ is a factor $a(t)$ times more energetic than the same atom at time $t_0$. As the energy scale is ultimately set by the Planck mass then the Planck mass must be increasing as the Universe expands: $M_{pl} \\propto a(t)$. This effect alone would account for the gravitational redshift of distant galaxies without the assumption that photons travelling from those galaxies lose energy due to wavelength expansion. Addition: I believe this hypothesis leads to a linear cosmological expansion $a(t)\\propto t$ (see comments below)."}
{"id":"87875","title":"Is there any usage of $x^x$ in physics?","text":"I am so excited about the function: $x^x$. I would like to know if there is any usage in the physics world of this function. Is there any formula that uses $x^x$?"}
{"id":"130710","title":"Flow velocity out of a pipe","text":"For a compressible homoentropic gas is the flow velocity out of the pipe the same as the velocity into the pipe?"}
{"id":"130716","title":"Why is $\\omega = \\sqrt{K\/m}$ valid for a quantum oscillator?","text":"I'm working in the 3rd edition of Modern Physics by Serway, Moses, and Moyer. In 6.6, it talks about a quantum oscillator. I don't fully understand how the definition of frequency works. Now, we assume that a particle feels a force $F = -Kx$ like the classical version, and we define $K$ as the curvature of $U(x)$ at the point of stable equilibrium ($x = 0$). That is, $$K = \\left.\\frac{d^2 U}{dx^2}\\right|_a$$ where $x = a$ is the general point of stable equilibrium. It then uses the formula $\\omega = \\sqrt{\\frac{K}{m}}$, though this is a classical equation. I don't understand why this is technically valid. I understand that that formula comes from the classical description $-Kx = ma = m \\frac{d^2x}{dt^2}$ where $x = A\\sin(\\omega t + \\phi_0)$. However, with quantum mechanics, why can we say that it oscillates by that description? We're not even dealing with a particle; it has wave and particle properties described by the wavefunction, so why can we immediately say that $U(x) = \\frac{1}{2}Kx^2 = \\frac{1}{2}m\\omega^2x^2$? When we solve the Schrodinger equation, we find that the particle isn't limited to the interval $[-A, A]$, so why is it valid to use a formula that is derived from assuming that? Any help is much appreciated!"}
{"id":"76901","title":"Explicit solutions to 2-d Dirac Equation","text":"The 2-d Dirac equation without any constants is represented usually as $$i*dt (\\phi) = D (\\phi)$$ where $D = m\\sigma_2-i\\sigma_1dx-i\\sigma_3dy$. Where can I find explicit closed form solutions to this equation? If there are none, are there perturbation-like non-exact solutions? Thanks alot."}
{"id":"74500","title":"Anomalies for not-on-site discrete gauge symmetries","text":"If a symmetry group $G$ (let's say finite for simplicity) acts on a lattice theory by acting only on the vertex variables, I will call it ultralocal. Any ultralocal symmetry can be gauged. However, in general there are discrete symmetries that cannot be gauged. For example, Freed and Vafa in http:\/\/link.springer.com\/article\/10.1007%2FBF01212418 discuss how in 1+1d one needs the pullback of a certain class in $H^3(G,U(1))$ to have some trivial periods. So, is the converse true--that if $G$ can be gauged then there is a formulation of the theory where $G$ acts ultralocally? In other words, is a symmetry with no ultralocal action necessarily anomalous? And if so, can we see this anomaly as an explicit class in $H^3(G,U(1))$ for 1+1d theories, for example? It seems to me like the answer is yes. If we have no anomaly and go ahead and gauge $G$, then we can put the result on a lattice where the $G$ gauge field will live on edges. These edge variables will have the flatness condition that the start and end vertex variables differ by the action of the edge variable (an element of $G$). It seems like $G$ shouldn't act anywhere else, since in some sense gauging $G$ is a type of \"fat quotient\" of the theory by $G$. Thus, if we take this lattice formulation and forget the gauge field, we end up again with the original theory, but now with an ultralocal action of $G$. What remains is how to quantify this anomaly in the cohomology group."}
{"id":"74504","title":"Twisted Nematic liquid crystal: extent of degradation due to DC drive?","text":"It is commonly accepted wisdom that a Twisted Nematic LCD must always be driven using AC instead of DC, because DC will supposedly denature the LC. I have two questions:   1. Does this effect only happen at typical drive voltages of 8V ~ 9V? If the LCD is driven with a lower voltage, around 3V, is the degradation not as bad?   2. At 3V, how many hours does it take to degrade the LC? I've run a test for 500 hours with no bad effects. Thanks!"}
{"id":"100831","title":"What is meant by \"net charge\"?","text":"Lets consider a system of two opposite charges separated by a certain distance (dipole), if we ask what is the net charge for this system? the answer would be zero. The net charge (what I have come to know until now) is said to be addition of sign of charges irrespective of the distance between the charges. Lets consider the following case:   According to Gauss's law the total electric flux through a closed surface enclosing a charge is equal to $\\frac{1}{\\epsilon_0}$ times the magnitude of charge enclosed. In the below figure we have a $+q$ charge enclosed in the guassian surface, so there is a flux linked with the guassian surface, which will be equal to $\\frac{1}{\\epsilon_0}$ times the $+q$ charge. ![enter image description here](http:\/\/i.stack.imgur.com\/eiu87.gif) Now, replace the $+q$ charge with the dipole in which charges are separated by a certain distance. Magnitude of electric field ($E$) due to an electric dipole at a distance $r$ from its centre in a direction making an angle $\\theta$ with the dipole is given by the equation,$$E=\\frac{1}{4\\pi\\epsilon}.\\frac{p\\sqrt{3\\cos^2\\theta+1}}{r^3}$$   where, $p=2aq$ ($2a$ is the distance of separation of the charges $q$). Electric field around the dipole will be zero if and only if distance of separation of the dipole charges is zero. Although we can't expect field lines to be perpendicular to the surface, we can expect some contribution due to the electric field components (perpendicular to the surface) of the enclosed dipole (with charges separated by a distance). From the dipole equation it follows that there exists electric field around dipole when charges are separated by distance, so there is flux through the surface which implies that there is charge enclosed in the surface (from gauss law). But net charge of dipole is said to be zero. By gauss law, net charge is not zero. Then what is net charge? From gauss law we can directly find the $q$. Can we say this as net charge? Then, is it that net charge can't be said directly just by adding the signs and is it a function of electric field? If I have misunderstood, pardon me and explain. Sometimes the problem might not be understood well due to communication error, if any correction required, comment on the part."}
{"id":"70084","title":"What object is quantized in quantum gravity?","text":"In theories of quantum gravity, which object is it that is quantized? Working on field theories, I expect the quantization to mean the promotion of a classical field to an operator valued field that works on some kind of Fock space. I heard in QG the metric gets quantized. This would imply, by my above reasoning, that the metric and thereby spacetime itself is an operator working on a Fock space of spacetime excitations. While I can accomodate this picture well with my idea of graviational waves, how does it rely to spacetime in general? Especially, how do the other quantum fields relate to this operator-spacetime, since they have an intrinsic dependence on spacetime?"}
{"id":"81829","title":"Force of exploding pumpkin","text":"Okay so in physics class we are going to explode a pumpkin by placing some kind of bomb inside of the pumpkin. Well us students have to come up with an apparatus to measure the force of the explosion, i am in desperate need of some help or some ideas of some solutions and what i need to measure to find the force of the explosion. I know that i need to find the mass of the chunks of pumpkin, the position of the pumpkin chunks from start to finish, and the time it took for the chunks to reach a certain point. What else do i need to know and\/or find? PLEASE HELP"}
{"id":"128034","title":"Straight line on 8-axis","text":"How would a straight line appear in a 8-axis co-ordinate system. Will it continue to look like a straight line or will it appear like zig-zag line. We can see that straight line appears like a dot in one-dimension, straight line in two-dimension graphs."}
{"id":"126229","title":"Modeling an Adiabatic Expansion (EES)","text":"How would one model an isentropic (reversible) adiabatic expansion using thermodynamic software in order to determine isentropic work? What I am looking for is an actual pressure-volume curve for the isentropic expansion of a gas. Approach using EES: 1) Start with a known gas state\/scale (Mass, Volume, Pressure, Temperature, Enthalpy, Entropy, etc.)   2) Update gas volume by a small increase in volume dV   3) Work done = Pressure * dV   4) Keep entropy constant   5) Recalculate new gas state based on the new Specific Volume (Volume\/Mass) and Entropy   6) Repeat until final state has been reached. Summing the work done in this method results in significantly less work (~85%) than is possible through isentropic expansion. The max isentropic work is calculated through   Wmax = Mass * (h_initial - h_final) Even with very small step sizes (thousands of steps), this approach doesn't work. Can anyone shed some light as to why these work values don't match up? Code:               Procedure Adiabatic Expansion(P, T, P_final : Work_max, Work)       Vol=1       v=Volume(CarbonDioxide,P=P,T=T)       s=Entropy(CarbonDioxide,P=P,T=T)       h_initial=Enthalpy(CarbonDioxide,P=P,T=T)       Mass=Vol\/v       Vol_step=1\/10000       Work=0            repeat       Vol:=Vol + Vol_step       v:=Vol\/mass       Work_step:=(P)*Vol_step       Work:=Work + Work_step       P:=Pressure(CarbonDioxide,v=v,s=s)       until (P <= P_final)            h_final=Enthalpy(CarbonDioxide,P=P,s=s)       Work_max=mass * (h_initial - h_final)       End"}
{"id":"862","title":"Does this statement make any sense?","text":"I am asking this question completely out of curiosity. The other day, my roommate, by mistake, used 'Light year' as a unit of time instead of distance. When I corrected him (pedantic, much), he said the following: \"Units are relative. And according to Fourier Transforms, units can be changed so Light year is a unit of time.\" That got me thinking and I read up Fourier Transforms on wikipedia but couldn't find anything about using a unit in one domain as a unit for another measurement. I do agree that units (particularly, base units are relative. eg: the meter), but does his statement make any sense? EDIT Thank you everyone for all the answers. It isn't so much to in it in or prove a point as it is to understand the concept better. Anyways this is his response after I showed him this thread. Any comments would be appreciated. His response: Nevermind, for the first time I accept I was wrong. BUT using lightyears to measure time is possible. My example didn't make sense bacause I was wrong when I meantioned that I'm still measuring dist. If you have a signal in time domain and ...take the FT, I get a signal which DOES NOT HAVE to be in frequency domain. Clarify this to the guy who posted last. Now the new signal is in a domain defined by me and so is its units. This signal although not equal to the original signal, still represents that if ya take an inverse FT. So, the idea of time will still be there. Now coming back to our case: lightyears here is not the lightyears you are used to read when dealing with distance. It represents time."}
{"id":"45378","title":"How to find exponential values","text":"I have a power equation for flow between two circular disks on a common axis as stator and rotor, which follows, $$P = \\frac{n^a}{G}\\tag{1}$$ Where:   * $P$ = power,   * $n$ = revolution (1200 rev\/min),   * $G$ = gap between two plates (0-2.5 mm) I have experimental data showing Power vs Gap curve at a constant revolution of 1200 rmp. How do I get the value of $a$ and how do I plot Eqn (1)?"}
{"id":"45371","title":"What causes a ring-like image around light of the moon?","text":"I just encountered an interesting image in sky. As you can see in following images there was a ring-like image around light of the moon. I don't know if it was clouds but it was looking like it is far above the clouds. I got curious and only place I know I could get some answers is here. It is much more clear in first photo, other ones are a bit blocked by clouds in between Here are some photos I took: ![Photo 1](http:\/\/i.stack.imgur.com\/s4cBL.jpg) ![Photo 2](http:\/\/i.stack.imgur.com\/zWijW.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/6c4tf.jpg)"}
{"id":"45372","title":"If acceleration causes relative time dilation does the eventual deceleration reverse it?","text":"If acceleration causes relative time dilation does the eventual deceleration reverse it? For example: traveling to Alpha Centauri Based on me reading this site: http:\/\/www.convertalot.com\/relativistic_star_ship_calculator.html -"}
{"id":"44310","title":"Solving a statically indeterminate truss","text":"I'm trying to solve the statically indeterminate truss shown below and I'm having a little trouble. ![truss](http:\/\/i.stack.imgur.com\/7tLWa.png) $H$, $P$ and $\\beta$ are given. The material is aluminum (density is 2700 ${kg\/m^3}$) and has a cross section of 0.002 $m^2$. We are instructed to take the weight of each member into account by distributing $\\frac12$ of each member's weight to its end joints. I can easily solve for $f_1$, $f_3$, $f_4$, $f_5$ and the support reactions using statics (method of joints), but I can't figure out how to go about solving for the remaining members. The main point of my question is, how do you solve statically indeterminate trusses? I assume I'll have to use some other property such as deformation. I've only posted the homework question to provide background as to why I am asking. Thanks for any help."}
{"id":"113165","title":"On characteristic X-ray spectra and relative intensity","text":"I am looking at the following image (from http:\/\/www.schoolphysics.co.uk\/age16-19\/Atomic%20physics\/X%20rays\/text\/X_ray_spectra\/index.html): ![enter image description here](http:\/\/i.stack.imgur.com\/Z270f.gif) and I want to know, of the two curves, which one represents the element with the higher atomic number. That is, I understand that X-rays are scattered and that the peaks are characteristic of a material. What I am less clear on i why one curve should be above another relative to their atomic numbers (I am going to assume the cutoff voltage on the left is identical even though in this picture it isn't). The reason I am confused is that I am told that that $\\frac{1}{\\sqrt{\\lambda}} \\propto Z$ which is all very well, but on this graph it would seem to indicate that the upper line is a heavier (higher Z) material. Is hat the case, or is intensity actually proportional to $\\frac{1}{\\sqrt{\\lambda}}$ as well? I would have thought the lower curve would be a higher Z since it usually take less energy to knock electrons off of elements further up the periodic table. Anyhow, what clarification might be offered is appreciated."}
{"id":"18070","title":"What is escape velocity? In reality, how can something no longer be under the gravitational influence of something else?","text":"Isn't G a continuous function and although you leave the immediate vicinity of the earth with an escape velocity won't it always exert a force, however small it may be. Won't that force eventually pull the object back to the earth (assuming the absence of other objects)"}
{"id":"122786","title":"Can horizontally and vertically polarized light combine to become circularly\/elliptically polarized light?","text":"Well, we know that circularly\/elliptically polarized light is made up from orthogonal components. So is it possible then to create circularly\/elliptically polarized light by combining horizontally and vertically polarized light? It seems to make perfect sense to me. EDIT: Clarification: I mean by creating circularly\/elliptically polarized light in the laboratory (though beam-splitters ?, I don't know) by combining different light beams of horizontally\/vertically polarized light."}
{"id":"70339","title":"A Cosmological horizon at the Hubble radius?","text":"I have calculated that if one extends a rigid ruler into space by a fixed proper distance $D$ then a clock at the end of the ruler, running on proper time $\\tau$, will run more slowly than time $t$ at the origin by a time dilation factor: $$\\frac{dt}{d\\tau} = \\frac{1}{\\sqrt{1 - H^2 D^2 \/ c^2}}$$ where $H$ is the Hubble parameter. If one substitutes in Hubble's law, $v = H D$ (the theoretical law which is exact), one finds the following satisfying result that $$\\frac{dt}{d\\tau} = \\frac{1}{\\sqrt{1 - v^2 \/ c^2}}.$$ Although this looks like a result from special relativity I derived it by combining the FRW metric line element and an equation for the path of the end of the ruler, $\\chi = D \/ R(t)$, where $\\chi$ is the radial comoving coordinate of the end of the ruler, $D$ is a fixed proper distance and $R(t)$ is the scalefactor. Does this prove that there is a cosmological horizon at the Hubble radius $D=c\/H$ where the proper time $\\tau$ slows down to a stop compared to our time $t$? This seems to be a general result which is true regardless of cosmological model. Details of Calculation The general FRW metric is given by: $$ds^2 = -c^2 dt^2 + R(t)^2\\left[d\\chi^2+S^2_k(\\chi)d\\psi^2 \\right]$$ where $d\\psi^2 = d\\theta^2 + \\sin^2 \\theta d\\phi^2$ and $S_k(\\chi)=\\sin \\chi$,$\\chi$ or $\\sinh \\chi$ for closed ($k=+1$), flat ($k=0$) or open ($k=-1$) universes respectively. The scale factor $R(t)$ has units of length. Consider a ruler of fixed proper length $D$ extending out radially from our position at the origin. The path of the far end of the ruler in comoving coordinates is $$\\chi = \\frac{D}{R(t)}$$ Differentiating this equation by proper time $\\tau$ gives us: $$\\frac{d\\chi}{d\\tau} = - \\frac{D}{R^2} \\frac{dR}{dt} \\frac{dt}{d\\tau}.$$ Using the FRW metric we can find a differential equation for the path of the far end of the ruler. We substitute in $ds^2=-c^2d\\tau^2$ (end of ruler has a time-like path), $d\\psi=0$ (the ruler is radial) and divide through by $d\\tau^2$ to obtain: $$c^2\\left(\\frac{dt}{d\\tau}\\right)^2 - R(t)^2 \\left(\\frac{d\\chi}{d\\tau}\\right)^2 = c^2.$$ Substituting the expression for $d\\chi\/d\\tau$ into the above equation we find: $$c^2\\left(\\frac{dt}{d\\tau}\\right)^2 - D^2 \\left(\\frac{\\dot R}{R}\\right)^2 \\left(\\frac{dt}{d\\tau}\\right)^2 = c^2.$$ Using the definition of the Hubble parameter $H=\\dot{R}\/R$ we finally obtain: $$\\frac{dt}{d\\tau} = \\frac{1}{\\sqrt{1 - H^2 D^2 \/ c^2}}.$$"}
{"id":"71061","title":"Are these data enough to solve this simulataneity problem","text":"observers: **S** and **S** '. S stands **in midway** between two events (say, 2 flashes) located **20 metres apart**. He sees two flashes occurring **simultaneously**. **S'** notes a **20ns difference** in them. **S'** is moving along x axis with uniform velocity What is this velocity in S frame? delta t equation leads to a 4th degree polynomial equation of v, so I must be wrong. Please help."}
{"id":"71067","title":"Educational physics stories with a funny punchline","text":"As a physicist, when I'm giving lectures or during colloquiums, I usually find it necessary to have some appropriate and related jokes in hand. All of the best teachers have some, and they use them when they have to; so I think that it is a necessity of good teaching. On the math side we have this gem(although being closed and reopened several times); however on the physics side I couldn't find much. Hopefully this topic will help me and many others with this regard."}
{"id":"71066","title":"RC Parallel Circuits","text":"I am trying to solve the following exercise: * * * A capacitor 27 and $\\mu$F is connected to a battery of 21 V through a resistance of 14 M$\\Omega$. 1) What 's the charge on his parallel-plate? 2) To reduce such a charge of 16%, I connect a wire in parallel with the condenser. What will be now the ddp across the parallel-plate capacitor? 3) What resistance the new wire must present to ensure this outcome? Solutions: q = 576$\\mu$C; $V_c$ = 17.64 V; $R_2$ = 2.666667K$\\Omega$ * * * I succeeded to find the solution for the question 1) and 2) but not the 3). Here is what I tried: the picture should be like this: ![enter image description here](http:\/\/i.stack.imgur.com\/YVOob.jpg) $R_1$ = 14 M$\\Omega$ $V_f$ = 21 V c = 27 $\\mu$F * * * q = c * $V_f$ = 576 $\\mu$ C $V_c$ = $\\frac{q}{c}$* (1 - 16\/100) = 17.64 V $R_2$ resistance in parallel to the capacitor (1) $R_2$ * I = $V_c$ (2) $V_f$ = ( $R_2$ + $R_1$ ) * I I tried to solve (1) and (2) to find the $R_2$ but the result I found is different from the solution. Any hints? Thanks."}
{"id":"18659","title":"complex numbers in optics","text":"I have recently studied optics. But I feel having missed something important: how can amplitudes of light waves be complex numbers? I suppose this is quite fundamental, but I do not find the answer in my course ... Thanks, Isaac"}
{"id":"87057","title":"Angular Displacement","text":"If something is rotating about a point and it covers a complete circle, should we take its angular displacement as 360 degree or 0? Please give link to some established material on this subject which defines your answer, whether it is that it should be taken as 0 or 360 degrees. Question that led to this problem : The angular speed of a motor wheel is increased from 1200 rpm to 3120 rpm in 16 seconds. 1. What is its angular acceleration, assuming the acceleration to be uniform, 2. How many revolutions does the wheel make during this time. In the solution to above question the author(s) for solving part 2 have used equation of motion and mentioned that they have obtained angular displacement for it, it basically implied that the equations of motion used for rotation provide angular displacement which does not become zero on returning to original point which is not followed when we consider it analogous to displacement of linear motion. Addendum : Even here ( Angular displacement and the displacement vector ) the selected answer says that on completing the circle, angular displacement is 360 degrees, is there some established text to support this ? Similarly here ( http:\/\/www.ask.com\/question\/calculating-angular-displacement ) something else is said in the answer, is then angular displacement ambigous and hence has no correct definition, if there is please guide me to some established text."}
{"id":"45955","title":"Splitting light into colors, mathematical expression (fourier transforms)","text":"I am trying to solve a problem that includes a function of the light hitting a certain area. My question is, how would I change a function $G(x)$ of photons hitting a certain area to include just photons of a certain wavelength, say red light. I feel like this could be accomplished using a Fourier transform and de Broglie's law, but I'm not sure. Can someone please help, just for a general Gaussian function $G(x)$? More information: Basically, given a function that gives the number of photons hitting a certain area, I want a mathematical way to determine how many of those photons are of a specific frequency (such as red light). G(x) is defined as the integral in a gaussian slit experiment (that is, a double slit experiment with the path integral of a gaussian probability) such as in Feynmann's Path Integrals and Quantum Mechanics or in arxiv.org\/pdf\/1110.2346."}
{"id":"94505","title":"picture and isomorphisms","text":"I found a paper by A. Belopolsky : \"Picture changing operators in supergeometry and superstring theory\" where he says there exists a possible physical state in one picture that doesn't exist in another picture. What A.B. does is essentially defining the picture formalism on the BRST complex and the de-Rham complex saying that the main idea would be that there is an even and an odd ghost number, picture changing involving the change in the odd ghost number. What strikes me here is the splitting of the complex on the odd number. I am not sure I get the full idea about what this means? Is it a sort of split-exact complex? And then why are the isomorphisms on the cohomology such a big deal? Shouldn't that be natural?"}
{"id":"123788","title":"Surface potential resulting from the charge transfer between insulator and conductor","text":"Common non-contact electrostatic voltmeters are used to measure the surface potential. Imagine we place a charged insulator on top of the conducting plate (insulated from the surrounding) and measure the resulting potential from the bottom of the plate. How is the measured potential related to the charge transferred from the insulator? To my understanding, the charge will be transferred from the insulator to conductor only at their interface. Therefore, the potential detected by the voltmeter at conductor surface should be only due to this portion of charge transferred. Is that correct? Does the conductivity affect the surface potential detected? If yes, what is the relationship? I do not have much experience in electrostatic events, I am a chemist trying to understand the above situation."}
{"id":"118534","title":"Why is electric current dangerous to humans?","text":"How does a strong electric current harm our body? A strong electric current will posses a great charge. But how does that charge injure us?"}
{"id":"118532","title":"Calculating the components of angular momentum of a rigid body","text":"You have a rigid body with 1 fixed point in space (the origin). It's self-explanatory how to get the following equation for the angular momentum: > $\\vec L = \\sum_n m_n\\vec r_n\\times\\vec v_n$ Where you take the sum of all \"$n$\" indicates all \"$n$\" points of mass.   This can be transformed into: > $\\vec L = \\sum_n m_n(\\vec\\omega(r_n^2)-(\\vec\\omega.\\vec r_n)\\vec r_n)$ Now, out of this, I have to get the following: > For component \"$i$\" of this equation, you get:   >  $L_i = \\sum_n m_n(r_n^2\\omega_i-x_{ni}\\sum_j\\omega_jx_{nj})$ I understand that for component \"$i$\", the scalars $m_n$ and $r_n^2$ are the same as for the other components, I also understand that for component \"$i$\" I need to take the $\\omega_i$ component.   But what is meant with everything that follows after that? What does the \"$x$\" indicate for example? Does the \"$j$\" indicate the other components?"}
{"id":"41701","title":"How was the universe created?","text":"I do not know much beyond high school Physics. Thus, I am asking this question from almost layman's perspective: > _What, as per the best of our existing knowledge and widely accepted among > the scientific community, are the models of creation of the universe, and > what is their core argument?_"}
{"id":"47704","title":"Scattering Amplitudes in Centre of Mass Frame","text":"I'm reviewing page 59 of the QFT notes here and am a little confused by a reference frame argument. You can compute the second order probability amplitude term for nucleon-nucleon scattering to be $$-ig^2\\left[\\frac{1}{(p_1-p_1')^2-m^2+i\\epsilon}+\\frac{1}{(p_1-p_2')^2-m^2+i\\epsilon}\\right](2\\pi)^4\\delta(p_1+p_2-p_1'-p_2')$$ in a scalar field theory approximation. Now the author argues that we may remove the $i\\epsilon$ terms by moving to the centre of mass frame. Here he says that $p_1=-p_2$ in this frame and that $|\\vec{p_1}|=|\\vec{p_1'}|$ by conservation of momentum. He continues to claim that the four-momentum of the meson is hence $k=(0,\\vec{p}-\\vec{p'})$ so $k^2<0$. Quite what $p,p'$ are I don't know exactly. I don't understand this argument at all. Surely in the centre of mass frame, the sum of all momenta $p_i$ and $p_i'$ is zero (.)? Also where has the second constraint come from? I don't see how morally you could get more than my claim (.). Could someone explain this argument to me? Very many thanks in advance."}
{"id":"47709","title":"Thermal equilibrium and non correlations","text":"I read in a book on quantum fluctuations and quantum noise that, at thermal equilibrium the classical canonical variables are uncorrelated, ie: $$\\langle xp\\rangle=\\langle x\\rangle\\langle p\\rangle$$ But I am not sure to understand the sense of _at thermal equilibrium_ , for me it just means $$\\langle x^2\\rangle=\\langle p^2\\rangle=\\frac{T}{2}$$ in the correct units with $T$ the temperature. On the other hand what I can derive easily in the canonical ensemble is: $$\\langle xp \\rangle = Z^{-1}\\int {\\cal{DxDp}}\\;\\text{e}^{-\\beta H(x,p)}xp$$ and $$\\langle x\\rangle\\langle p \\rangle = Z^{-2}\\int {\\cal{DxDpDx'Dp'}}\\;\\text{e}^{-\\beta H(x,p)}\\text{e}^{-\\beta H(x',p')}xp'$$ which are equal if we have the following equality $$H(x,p) = H_1(x) + H_2(p)$$ separability of $x$ and $p$ in the Hamiltonian, which seems unrelated to the previous hypothesis of thermal equilibrium (used here by taking the mean over the canonical distribution of course)."}
{"id":"126221","title":"How far can a voice carry through the desert at night?","text":"Let's use these conditions: temp = 7C, relative humidity = 10%, air pressure = 85 kPa, elevation = 1500m, shouting at 100dB (not sure if this is a reasonable volume). Given these conditions, what distance could a human voice travel and still be mostly understandable? How far to be heard at all? I haven't done any physics for a long time, and I really don't know how to go about this. Thanks."}
{"id":"15426","title":"Tidal acceleration for a retrograde rotation?","text":"Consider two nonelastic spherical bodies with uniformly distributed density, a small such body in a circular orbit around the bigger one. And consider the smaller body's rotation is matched (as if \"tidally locked\") to its orbit so the same hemisphere always faces the larger body. Now, here's my question: a particle on the daylight surface of the orbiting body is orbiting the \"sun\" with a smaller radius than a particle resting on the \"night\" side. If the two particles weren't attached to the body, the daylight particle would have a faster solar orbit and the nighttime particle a slower orbit. Would this not impart a retrograde rotational force on the orbiting body? If this is an already understood concept, what is it called? I read up on tidal locking and tidal acceleration on Wikipedia, and this dynamic wasn't mentioned. I think it's kind of interesting because I never thought about tidal forces doing anything with inelastic objects."}
{"id":"116813","title":"AdS Space Boundary and Geodesics","text":"I'm new to working with AdS space and am primarily concerned with black holes. I'm just playing round with the metric for AdS$_4$ $$ds^2=-f(r)dt^2+f^{-1}(r)dr^2+r^2d\\zeta^2$$ for $f(r)=r^2+m $, $\\space\\space\\space\\zeta=d\\theta^2+\\sin^2\\theta d\\phi^2$. My problem is trying to understand the boundary; specifically when considering particle trajectories:   1. For null geodeiscs, I've read that they reach the boundary of AdS space, which seems to commonly expressed as saying they are represented as straight lines. I don't understand how these two phrases are the same and how to show that this is the case starting from the metric I've stated. Using constants of motion etc, and assuming a radial path, I find the equation $\\frac{dr}{d\\lambda}=k$, for $k$ constant.   2. For timelike geodesics, I know they do not reach the boundary and equivalently I read that they are represented by the boundary of slices of the hyperboloid i.e. ellipses. Again, how do I show that this really represents timeline geodesics? As above (but $ds^2=1$ in this case) I find the equation $\\Delta\\tau=\\log(r+\\sqrt{k^2+m+r^2})\\space \\vert^b_{r_0}$ where $b$ is the boundary and $r_0$ the initial $r$. I've been reading (as much as I can using the fairly limited coherent literature on the topic) and I can only find discussions on this matter, with some diagrams. None seem to go about this question the way I have above and consequently I'm thinking there must be something wrong with what I've done."}
{"id":"114014","title":"How does one prove that the current of a spontaneously broken symmetry generates a particle?","text":"I am having a hard time arguing that, after spontaneous breaking of a continuous symmetry of a field Lagrangian, local fluctuations around the vacuum can be interpreted as particles (without referring to analogies from condensed matter physics). I encountered a treatment which stated that, when the symmetry is unbroken, the corresponding current annihilates the vacuum $$J^{\\mu} | 0 \\rangle = 0$$ while after spontaneous symmetry breaking, the current creates a state out of the vacuum with some momentum $k^{\\mu}$ $$J^{\\mu} | 0 \\rangle = k^{\\mu} | k \\rangle $$ My question is why this second equality holds only after spontaneous symmetry breaking."}
{"id":"7727","title":"formal framework for talking about 'minimal couplings'","text":"usually on physical theories one would have Lagrangians or Hamiltonians with multiple fields; say, a vector $A_{\\mu}$ and a scalar $\\phi$ and one would postulate ad hoc a coupling between the fields which will be used to derive some physical states and then, retroactively support the coupling from a match with the physics Usually one reads about 'minimal couplings' and for scalars with vectors on might see a factor like $\\frac{ \\partial{\\phi}}{ \\partial{ x_{\\mu}} } A_{\\mu}$ But my question goes on in what general or abstract sense such factors represent a minimal coupling? minimum of what? if is the total degree of powers of the fields? do we have some variational principle for lagrangian couplings that these terms can be called a stable point in such variations?"}
{"id":"19218","title":"Certain material heating water in a recipient","text":"I don't know how to resolve a problem, but I don't want the answer since I'm almost going to have it resolved. What the problem says is we have 85 liters of water at 7ºC in an iron pot of 29kg. We want the water to be at 86ºC. The temperature of the iron pot is 12ºC. The water is heated by fire wood (65% of the thermal energy is wasted in combustion), and has a heating value of 12 MJ\/kg. We need to determine the amount of wood needed to heat the water. I would apply the normal formula of energy transfer: m1*c1*(t2-t1)=m2*c2*(t2-t1). But since we are using fire wood there, I don't know how to resolve this problem, probably because I'm missing something there... and I don't know what should I do. Which formula or principle would I need to use when having the material and the heating value of the material used?"}
{"id":"13744","title":"Do all event horizons emit radiation?","text":"So, the event horizon around a black hole emits radiation, and Rindler space is full of thermal energy. I guess I have two questions- does the Unruh effect have anything to do with radiation from the apparent horizon in Rindler space? And what about the cosmological horizon- any radiation there?"}
{"id":"13742","title":"What is the energy functional for $\\nu=5\/2$ Moore-Read state?","text":"I am trying to do some Monte Carlo simulations for Pfaffian state from Fractional Quantum Hall effect. I am wondering what is the energy functional for $\\nu=5\/2$ Moore-Read state?"}
{"id":"43287","title":"How would one calculate the amount of water contained in a cloud?","text":"So I was looking out the sky one day and I wondered how I would go about calculating how much water was contained in a cloud. I figured the following simple outline 1) We need to roughly know how big it is. We'll also be figuring out how high the cloud is. Use some sort of geometric method? Triangles? 2) We need to relate its color to its density. (Darker clouds more dense) (White fluffy clouds less dense) 3) Correction factors due to lighting conditions during different parts of the day. Can some body help me figure out the relevant calculations needed? Or how much water does a cloud contain based on its category?"}
{"id":"65596","title":"What does the wavevector $\\textbf{k}$ mean?","text":"In Ashcroft, Mermin Solid State Physics, Eq. 17.43 is $$ \\epsilon(\\textbf{k}) = \\frac{\\hbar^2 k^2}{2m} - e\\phi(\\textbf{r}) $$ where $\\textbf{k}$ is the wavevector and all other symbols have their usual meaning. What does the wavevector tell me and why do we use it instead of position $\\textbf{r}$?"}
{"id":"132166","title":"Bekenstein bound for electron?","text":"Using the Wikipedia version of the Bekenstein bound, and substituting the Wikipedia values for electron mass and radius, one obtains 0.0662 bits. Does this really mean that a system, any system, placed inside a sphere the size of an electron, and weighing no more than an electron does, is almost determinate? How about an electron itself? Wouldn't one need at least a few bits to characterise the behavior of an electron in magnetic space? (I am a professional mathematician but I know very little about physics, I'm sure I'm missing something obvious here...)"}
{"id":"122801","title":"Would half of electromagnetic wave be absorbed on first sensor?","text":"If I would set up electromagnetic emitter that would emit one wave, but while it would be emitting it I would spin emitter 180 degrees at such speed that one half of wave would be emitted in that time (or it could be 1 degree, but the speed of turn would have to match the half of wave's oscillation) and then would set up a lot of sensors on the path of the wave that are able to measure strength of electromagnetic field (so that it would hit multiple sensors with that partial wave, sensors can be placed as far away as it needs be to allow the small turn in angle allow to have bigger affect). Would all the sensors show the value of the field going up to the peak and back down to zero, or would first sensor absorb the wave and all other sensors would show 0 values. **Update:** Reason I am asking is that I can't wrap my head around - for sensor to register something it has to be affected by electromagnetic field - to affect sensor some energy has to be used, if wave would affect all the sensors it would imply that it has infinite energy as increasing the amount of sensors would still 'do to the work' register a value, but there has to be more 'work' done with each additional sensor, so in my understanding 1st sensor (or some other) would have to take all the energy of the wave and rest would not get anything. Please forgive me if question is unclear or silly, I am not a physicist. If you need additional information please ask I will do my best to clarify."}
{"id":"116444","title":"Why does solar cell will create as a Multiple cells instead of one large cell area?","text":"Why is a solar module assembled from multiple cells instead of one large cell area?"}
{"id":"94256","title":"absorption length of thermal neutrons in a medium","text":"My book states that $$N(x)=N_0\\cdot e^{n\\cdot \\sigma \\cdot x}$$ for $n$ absorbing particles per volume element with a cross section $\\sigma$. Can somebody explain where this comes from?"}
{"id":"43138","title":"Widom-Larsen Theory","text":"Does the Widom-Larsen theory pose a credible underpinning for Low Energy Nuclear Reactions, often incorrectly referred to as \"Cold Fusion\"? http:\/\/arxiv.org\/abs\/cond-mat\/0505026"}
{"id":"2563","title":"What happens if a black hole sucks up a massive amount of oxygen and hydrogen?","text":"Suppose a black hole were to suck up a massive amount of surrounding oxygen and hydrogen. Would this ignite inside the black hole and turn it back into a star?"}
{"id":"11429","title":"Evaporation from a capillary tube","text":"Consider a capillary tube (say from a liquid \/ capillary thermometer), that means a tube of small internal diameter which holds liquid by capillary action . The tube is fulfilled with water and closed at one end. How long it would take for water to completely evaporate out through the open end of the tube? To be specific, let the diameter of the tube be $d=1$mm, the length of the tube $L=1$m, the environmental temperature $T=300$ K and pressure $P=10^5$ Pa, the relative humidity $f=60$ %, the contact angle a) $\\theta=0^{\\circ}$ and b) $\\theta=180^{\\circ}$. Does it have any effect on the time when the tube is positioned horizontally or vertically(open end above)? ![A capillary tube with a liquid](http:\/\/i.stack.imgur.com\/1akdv.gif)"}
{"id":"17117","title":"Explanation of Cardy's \"a theorem\"","text":"There seems to have been some discussion of Cardy's \"a-theorem\" recently: > “It is shown that, for d even, the one-point function of the trace of the > stress tensor on the sphere, Sd, when suitably regularized, defines a > c-function, which, at least to one loop order, is decreasing along RG > trajectories and is stationary at RG fixed points, where it is proportional > to the usual conformal anomaly.” said Cardy. “It is shown that the existence > of such a c-function, if it satisfies these properties to all orders, is > consistent with the expected behavior of QCD in four dimensions.” I would appreciate it if someone could give a reasonbly concise explanation of what the theorem states and implies, at a level for someone with a superficial understanding of quantum field theory. For example, what is the sphere in the above quote and how does the existence of the c-function lead to the main conclusion of the a-theorem which is purportedly: > \"... a multitude of avenues in which quantum fields can be energetically > excited (a) is always greater at high energies than at low energies. \" Edit: I think I found what I needed. Section 4.4 of David Tong's string theory notes gives a nice explanation of the trace anomaly and the \"a\" and \"c\" theorems. I assume the sphere they were talking about in the Nature article is just the $S^4$ of Euclidean compactified spacetime, in which case \"a\" is its Euler characteristic."}
{"id":"52508","title":"Can spacetime exist in the absence of matter and energy?","text":"I'm pretty sure Ernst Mach would have said that spacetime cannot exist without matter in it. But I'm also pretty sure that a black hole can be described as a self- sustaining gravitational field, which is (I guess?) a case of spacetime sustaining itself without the matter meaning much anymore. So, what is the current theoretical view on this?"}
{"id":"48139","title":"How many ways are there to distribute M excitations of N identical particles among K=3 quantum harmonic oscillators?","text":"I'm trying to numerically calculate a partition function of N non-interacting but identical particles in a 3D SHO. To do this, I'd like to know the degeneracy of $M$ excitations, $N$ _indistinguishable_ particles, in $K=3$ _distinguishable_ oscillators. In other words, what's the degeneracy of $E = \\hbar \\omega (M + 1\/2)$? I want to capture the Bose statistics and calculate something like a few-particle Bose-Einstein condensate. For example, if I have two excitations, two particles, and three oscillators, the degeneracy is nine (please correct me if I'm wrong!):   * Both excitations in one oscillator: 3 ways   * One excitation in each of two oscillators: 6 ways My current approach is to write an algorithm that counts all of these combinations, but I feel like there must be a simpler way. Alternatively, say we only have one oscillator. How many ways can I distribute M excitations among N identical particles? Edit: I originally miscounted the degeneracy of my example. I think it's 9, not 6."}
{"id":"93333","title":"What is the basic difference between absolute pressure and gauge pressure?","text":"What is the basic difference between absolute pressure and gauge pressure? I have a great confusion regarding these two pressures. Please someone explain it to me"}
{"id":"27654","title":"Why isn't there heterotic holographic QCD?","text":"I think the question speaks for itself... Top-down holographic QCD, like Sakai-Sugimoto, always involves the Type II string. There are one or two papers on hQCD using the Type 0 string. But I can't see anyone making hQCD models using the heterotic string. Why not? I may as well explain how the question arose: I was thinking about two forms of conventional string phenomenology - M-theory on G2 manifolds, and E8xE8 heterotic models - and was looking at possible beyond-standard-model symmetries, like flavor symmetries, family symmetries, etc. Somehow I stumbled into a time warp and ended up in 1965, looking at \"The Covariant Theory of Strong Interaction Symmetries\" by Salam, Delbourgo, and Strathdee, in which \"A classification of particles is suggested based on a U(12) symmetry scheme\". They want to make hadrons out of a \"12-component (Dirac) quark\" (i.e. _u_ , _d_ , _s_ as a single \"quark\"), and propose representations for vector and pseudoscalar mesons, and spin 1\/2 and spin 3\/2 baryons. It occurred to me that it would be rather eccentric to think about ways to embed a variation of their scheme into E8xE8, but that someone working on holographic QCD might well be interested in forgotten approaches to the strong interactions; and yet, top- down hQCD employs Type II string theory, which is dual to the heterotic string. Why haven't insights from heterotic string theory played a role in holographic QCD so far?"}
{"id":"29262","title":"Lepton Number Conservation","text":"What is the global symmetry of the electroweak Lagrangian that gives rise to lepton number conservation? As I understand it, electric charge is some linear combination of the conserved quantities corresponding to the $SU(2)$ symmetry and the $U(1)$ symmetry of the electroweak theory. If something similar is true for lepton number, then perhaps it would not be easiest to give the global symmetry directly. This is perfectly fine of course."}
{"id":"63145","title":"Coulomb interaction and conservation laws","text":"In many-body solid-state physics, the Coulomb interaction term in the Hamiltonian usually implies the momentum conservation law in indicies: $$H_c=\\frac{1}{2} \\sum_{\\mathbf{k},\\mathbf{k}',\\mathbf{q} \\neq 0} V_{\\mathbf{q}} a^{\\dagger}_{\\mathbf{k}'-\\mathbf{q}} a^{\\dagger}_{\\mathbf{k}+\\mathbf{q}}a_{\\mathbf{k}'} a_{\\mathbf{k}},$$ where $\\mathbf{k},\\mathbf{k}',\\mathbf{q}$ are quasi-momenta and quantum numbers for the continuum spectrum of electron gas simultaneously. In quantum chemistry textbooks, the Coulomb term usually looks like: $$H_c=\\sum_{i,j,k,l} V_{i,j,k,l} a^{\\dagger}_{i} a^{\\dagger}_{j}a_{k} a_{l}$$ Numbers $i,j,k,$ and $l$ are running over some discrete energy spectrum. Is it possible to state any conservation laws for quantum numbers $i,j,k,$ and $l$ in the expression above? Should not they obey any conservation law, selection rules or any additional restrictions? I will appreciate any references to textbooks or papers."}
{"id":"54904","title":"A rod filled with water and charged particles is rotated in magnetic field","text":"Assume we have a rod of diameter $d$ and length $l$. It is filled with a mixture of water and, say, sodium chloride. Thus there are positively charged sodium ions and negatively charged chloride ions in the water. Now, assume that the rod is rotated perpendicular to the ground. I should add that the rotation happens at the mid point, not at one of the ends. Earth's magnetic field $B$ thus asserts force to the moving particles according to $q (v \\times B)$ (Lorentz force). This means (I think) that the particles should start to move to the ends of the rod. If the rotation is in counterclockwise direction, the right hand rule states that positive particles should move to the right side and opposite for negative. (I think) Question is, does this really happen or is the centrifugal force going to push the particles to the sides? If so, how strong magnetic field $B$ or the velocity $v$ should be in order for this to happen?"}
{"id":"34258","title":"If a pendulum is on a rotating table, will a torque be generated?","text":"Here is the set up. Very simple. A flat (i.e. horizontal table, there is no gravity) and rounded table that spins on its axis (through the center of the table). A spring mass system is now put on the table inside a slot along the radius of the table. One end of the spring is attached to the origin of the table, and the other end is the small bob mass. Lets assume the relaxed length of the spring is half way along the radius. So the above is the physical set up. Here is a picture, showing the spring with a little bit of tension in it. ![enter image description here](http:\/\/i.stack.imgur.com\/u5ijO.png) So, the pendulum can only move along the radius, as it is inside a slot in the table. Now I spin the table, say clockwise with some initial angular speed, and at the same time release the bob so it also starts to vibrate back and forth on the radius as the table rotates. What I am not sure about is that whether a torque be generated due to the reaction of the bob on the edge of the slot as the disk spins or not? This reaction force has an arm length from the center, hence a torque will be present. In otherwords, will this system's angular momentum be constant or not? If there is no torque generated, then the angular momentum is ofcourse constant. Thanks **Edit** May be I should explain why I thought at first there might be a torque generated. When the bob is rotating along with the table, it will have 2 components of accelerations. One component is along the radius direction, which is `(x'' - x (theta')^2)` where `x` is the bob coordinate along the radius, and `theta(t)` is the angle of rotation. In the above, the term `x(*theta')^2` is the centripetal acceleration and `x''` is the acceleration of the bob along the radius. There is also an acceleration perpendicular to that, which is `(x theta'' + 2 x' theta')`. In the above, the second term is the coriolis acceleration and the first term is the standard Euler acceleration. I made this diagram ![enter image description here](http:\/\/i.stack.imgur.com\/GVb2S.png) Now, whenever I see something accelerating, then there must be a force along that direction. Right? This is the force that I took as generating a torque on the disk (actually it is the reaction to this force, ie. back to the disk). This is why I thought there is a torque. But when I did the same derivation using Lagrangian method `L=T-V`, I got different equation of motion. (no torque). Hence I thought to ask the experts here. **update 4** I have finally I think sorted this whole thing out!. I have updated the derivation. Solved this problem using Newton and Lagrangian methods, and both give the same equation of motions, and verified that the angular momentum remain constant. I have a PDF and HTML page and the applet here http:\/\/12000.org\/my_notes\/mma_demos\/slot_on_disk\/index.htm This is not as easy as it first seemed. There is actually a torque involved. Due to Coriolis force. This torque however changes in value such that the angular momentum remain constant. The critical thing for me to see was that the spring in infinitly rigid against rotation. This was mentioned below by Shaktyai. This is critical, as it means a slot is not even needed to keep the spring stright, and hence there is no Euler acceleration term perpendicular to the radial direction. The only side acceleration is the Coriolis one. This does generate a torque. I am not sure what this torque is called ( Coriolis torque?). It all now fits nice. I removed the slot from the applet since it is not needed, and updated it. If someone can verify it for me it will be appericated. The applet runs in the browser. Written in Mathematica. Wiki has a nice illustration of the Coriolis effect here http:\/\/en.wikipedia.org\/wiki\/Coriolis_effect"}
{"id":"34253","title":"Symmetries, Generators, Commutators and Observables","text":"I'm learning about generators and conservation laws and have derived the equation (1) $$[Q,A]=-i\\hbar f(A)$$ which is satisfied by the observable generator $Q$ for a transformation group with elements of form $$g_a(A)=af(A)+\\textrm{O}(a^2)$$ The lecture notes I'm reading say that this equation (1) defines $Q$ provided we know $f(A)$ for all observables $A$. Why is this true mathematically? And what does all observables mean? The example in the notes applies it to transformations along the $k-$axis for a system of $r$ particles, obtaining $$[Q,\\hat{x}_i^r]=-i\\hbar\\delta_{ik}$$ $$[Q,\\hat{p}_i^r]=0$$ It then states that $Q = \\hat{P}_k=\\sum_r \\hat{p}_k^r$. This is obviously a solution, but do I know that it's the only one? Many thanks in advance."}
{"id":"29193","title":"Would a Helium powered balloon vehicle work on Mars?","text":"Would a Helium powered balloon vehicle work on Mars? Things that might need to be answered:   1. How much liquidHelium would you need to pack in order to fill a balloon with enough helium gas to travel the planet.   2. How much energy does it take to re-compress Helium per\/litre at low temperatures on the surface of mars and would a solar cell be enough?   3. Would a \"Balloot\" (balloon\/parachute) be enough to complete spacecraft re-entry on arrival to mars without really needing to immediately \"touch down\"?   4. Any other things to think about by anyone else?"}
{"id":"29190","title":"How exactly does mass affect speed?","text":"I konw that mass affects weight (force), so how does that relate to speed? F=ma. so how does all this affect speed?"}
{"id":"29199","title":"Commutators with a density matrix","text":"The equation describing the evolution of our system is as follows: $ \\dot{\\rho} = u_1(t)(a^\\dagger a \\rho - 2a\\rho a^\\dagger +\\rho a^\\dagger a) + u_2(t)(a a^\\dagger \\rho - 2a^\\dagger\\rho a +\\rho a a^\\dagger) $ where $\\rho$ is the density matrix and $a$ is the photon annihilation operator. I need to rearrange this equation to look something akin to: $\\dot{\\rho} = u_1(t) \\hat{A} \\rho + u_2(t) \\hat{B}\\rho $ How do I handle the commutators and the like in order to rearrange it?"}
{"id":"49936","title":"What is the \"foamy space\" hypothesis that has been debunked recently?","text":"In \"Space-Time Is Smooth, Not Foamy\", a Space.com article, it is stated: > In his general theory of relativity, Einstein described space-time as > fundamentally smooth, warping only under the strain of energy and matter. > Some quantum-theory interpretations disagree, however, viewing space-time as > being composed of a froth of minute particles that constantly pop into and > out of existence. What are these \"quantum-theory interpretations\" the article is talking about? Can you quickly summarize the physics behind this experiment?"}
{"id":"23896","title":"What is the exact definition of center of gravity?","text":"I've come across many definitions. Is it 1) The point from which the weight of the body acts, i.e., the point at which if the entire mass of the body is assumed to be concentrated, the **gravitational force** acting on the body remains the same 2)The point at which if the entire mass of the body is assumed to be concentrated, the **gravitational potential energy** of the body remains the same 3)The point at which if the entire mass of the body is assumed to be concentrated, the **gravitational torque** acting on the body about the origin remains the same 4) All\/some of the above? I came across this while trying to answer this question."}
{"id":"131608","title":"Positivity for the level of Chern-Simons theory","text":"In many classical papers about Chern-Simons theory (see, e.g. [1]), it is claimed that the Chern-Simons theories with gauge group $G$ are classified by an element of $k\\in H^4(BG,\\mathbb Z)$, the so-called \"level\". (I'm not assuming here that $G$ is simply connected, or even connected) Now, I have reasons to believe that this description is not quite accurate. Rather, I suspect that only the levels subject to a certain **positivity condition** yield actual quantum field theories. The exact positivity condition I have in mind can be formulated by saying that the bilinear form on $\\mathfrak g$ which is the image of $k$ under the Chern-Weil homomorphism should be positivie definite . (Here, the Chern-Weil homomorphism goes from $H^*(BG)$ to $Sym(\\mathfrak g^*)$) **Question:** Is there any reference on Chern-Simons theory in which the above positivity requirement on the level is noted? * * * **Reference:** [1] R. Dijkgraaf and E. Witten, Topological Gauge Theories And Group Cohomology, Commun. Math. Phys. 129, 393 (1990)."}
{"id":"72020","title":"minimizing the Higgs potential equivalent to finding the minimum?","text":"When my advisor tells me to \"minimize the Higgs potential\", is she asking me to find the minimum (take the derivative of the potential and set it equal to zero)?"}
{"id":"72028","title":"A question about Virasoro algebra","text":"(skip disclaimer) I have a question in Polchinski's string theory book volume 1 p54, related to the Virasoro algebra. Introducing complex coordinates $$w=\\sigma^1 + i \\sigma^2 $$ $$z=\\exp (-i \\omega) =\\exp( -i \\sigma^1 + \\sigma^2) $$ and $$T_m=L_m - \\delta_{m0} \\frac{c}{24}$$ $$ \\tilde{T}_m=\\tilde{L}_m - \\delta_{m0} \\frac{\\tilde{c}}{24}$$ It is said > The Hamiltonian H of time translation in the $w=\\sigma^1+i\\sigma^2$ frame is > $$H=\\int_0^{2 \\pi} \\frac{ d \\sigma^1}{2 \\pi} T_{22} = L_0 + \\tilde{L}_0 - > \\frac{ c +\\tilde{c}}{24} (2.6.10) $$ My questions are (sorry ask two questions in a thread, since they are closely related) (i) Does $T_{22}$ mean $T_{zz}$ or $T_{\\sigma^2 \\sigma^2}$? (ii) How the anti holomorphic operator $\\tilde{L}_0$ are obtained in Eq. (2.6.10)?"}
{"id":"72029","title":"Do all impacts create a wave-like disturbance in the medium through which they travel?","text":"There is a scene in the first Matrix movie, where a helicopter strikes a skyscraper. The most interesting part is the 'slow-motion' bit where, as the helicopter strikes the building, a wave first propogates from the impact zone, outward along the facade, before the glass finally shatters. I was wondering whether this same thing happens across the board. Is a wave produced _every time_ two surfaces come into contact (AKA collide)? This is the clip I'm referring to: https:\/\/www.youtube.com\/watch?v=2E3JDtqHaVM&t=3m56s"}
{"id":"77211","title":"Free neutrons pass through most materials but how do they damage materials?","text":"Neutrons has no charge so they are hard to interact with materials. That's what I'm thinking. But they collide with atoms and it causes nuclear fission. And they can also be a dangerous radioactive which bring out biological damage. Why? You know that the most of the atom is empty space."}
{"id":"73777","title":"Does the Lennard-Jones force equation give its answer in Newtons?","text":"I'm trying to do the dimensional analysis of the Lennard-Jones force to work out what units are being used in my MD simulation. The lennard Jones force is given as the negative derivative of the potential: $F = 24e\/r*[2*(s\/r)^{12}-(s\/r)^6]$ This article: http:\/\/chemwiki.ucdavis.edu\/Physical_Chemistry\/Quantum_Mechanics\/Atomic_Theory\/Intermolecular_Forces\/Lennard- Jones_Potential suggests that the units of r and sigma are in angstroms and epsilon is in kj\/mol. Upon performing dimensional analysis of the bracket I get 0 implying that the force is dimension-less? or 0? This cannot be correct; could someone tell me where I am going wrong? Thanks"}
{"id":"91619","title":"Geometric quantization AND nuclear physics","text":"Classical mechanics has a natural mathematical setting in symplectic geometry and it may be asked if the same is true for quantum mechanics. Geometric quantization is one formalization of the notion of quantization of a classical mechanical system\/classical field theory to a quantum mechanical system\/quantum field theory. Geometric quantization play an important role in Nuclear Collective Model. see here or restricted symplectic model of collective nuclear motion. I am looking for recent applications of geometric quantization theory in nuclear physics."}
{"id":"106950","title":"'There shall be reheating' and other inflation-related questions","text":"I've some basic questions about inflation:   1. What is reheating exactly? What triggered the reheating? Is it something that one pulls out of a hat? Through which force did the decay of the inflation occur ?    2. How can I picture the inflaton? It is a scalar field, like the Higgs, could we ever hope to observe the inflaton? Can we assign to the field excitation a mass?    3. On the blog Quantum Frontier I read the following: > \"A possible solution the the super-Planckian value of the change of the > inflaton field during the inflation is that the field didn't go in a > straight line but somehow turned in circles.\" (Preskill's post on Quantum > Frontiers) What is meant by \"going into circles\" ? That it oscillated around a minimum of the potential (but isn't that the phenomenon associated to reheating ?) Or is it an identification, e.g. $\\phi(x)= \\phi (x+2\\pi)$? (This seems even stranger :S.) Thanks in advance."}
{"id":"114100","title":"How does tempered glass work?","text":"I've read that tempered glass work by heating and then rapidly cooling the glass so that the outer layer 'freezes' (stops expanding\/contracting) but the core contracts further as it cools. I'm told that this causes compression in the outer layer and tension in the core but I can't see how this could be. Surely, if the still-contracting core were 'pulling' on it, the outer layer would be under tension. For the outer layer to be compressed, surely the core must continue to expand after the outer layer 'freezes', pushing on the outer layer. Any help would be greatly appreciated."}
{"id":"90379","title":"Experiment dropping electrons into glass of protons","text":"So, when you drop dye into a glass of water the dye spreads out. Now I realize you cant simply replace the water in the glass with protons (or a pure concoction of electrons) but I am wondering... What is the closest real world experiment to doing just that: mixing a drop of electrons with a glass of protons, or vice versa."}
{"id":"51425","title":"How large would the electric field need to be to split the mj -\/+1 626.6 nm line of neon?","text":"I am an undergraduate considering an extension on a quantum beats experiment with neon. The premise is to excite neon atoms coherently from a metastable 'ground' state into a superposition of excited states (transistion wavelength 626.6 nm) which have been zeeman split by a magnetic field. The decay of the coherently excited states shows beats in a time domain analogue of Young's double slit experiment. I would like to do essentially the same experiment but with the Stark effect instead of the Zeeman effect but I have been unable to find any literature which I can understand that would help me get a feeling for the electric fields involved. If i can get away with something below the kV range then I can probably do it."}
{"id":"51421","title":"Energy spectrum of a tight-binding model","text":"Consider the one-dimensional tight-binding Hamiltonian $$\\mathcal{H}=t\\sum_m\\left(a^\\dagger_m a_{m+1}+a^\\dagger_{m+1} a_{m}\\right).$$ With the lattice constant set to 1, the energy spectrum is given by $$\\epsilon(k)=2t\\cos{k},$$ for $k\\in[-\\pi,\\pi]$ in the first Brillouin zone. However, if we repeat the calculation with each unit cell containing two sites, we would obtain an energy spectrum of $$\\epsilon(k)=\\pm2t\\cos{k}$$ instead, with the size of the first Brillouin zone reduced by half. Since the energy spectrum of a system can be determined unambiguously by experiments such as neutron scattering, it seems that we have a contradiction. What am I missing here?"}
{"id":"61739","title":"Bose-Einstein condensate for general interacting systems","text":"There is Bose-Einstein condensate (BEC) for non-interacting boson systems. Can we prove the existence of BEC for interacting systems?"}
{"id":"27797","title":"Would the arms of a rotating ice skater still move outwards if there was no other object in the universe?","text":"If there is no other object in the universe apart from a rotating ice skater, then nothing can be used as a reference frame. Would it make any sense to say that the skater is rotating? If so, rotating with respect to what? Would she feel an outward pull on her arms? Does rotation makes sense if there is no background reference frame?"}
{"id":"104086","title":"Conductivity of a crystalline solid","text":"In a crystalline solid each atomic level 'splits' into _n_ levels ( _n_ = number of atoms in the system). When the number of atoms is large each level becomes replaced by a band of closely spaced levels. In a semi-conductor we have an empty \"conduction band\" and a fully occupied \"valance band\". Conductivity arises because electrons get excited to the conduction band. **Question:** Why can't electron in the valence band freely move around and therefore conduct electricity? My question also applies to metals where the conduction band is already half-filled. What's special about this conduction band that allows electron to move around freely?"}
{"id":"32345","title":"Determining Orbital Velocity","text":"Is there any way to determine the orbital velocity of a point around another stationary point, if I don't know the mass of either of the points but know the force that gravity exerts and the distance between them. I'd like to know at what velocity I should push the non-stationary point perpendicular to the stationary point, so that it will maintain a circular orbit."}
{"id":"52997","title":"Degree of Time Dilation At a Distance From the Sun where acceleration = g?","text":"At higher altitudes above a body, clocks tick more slowly, and gravitational field is weaker. But what is the relationship? It is tempting for a GR newbie such as myself to think that anywhere that the gravity is equal to that of the earth's surface that time-dilation would be the same, i.e. dilation as a simple function of local gravitational acceleration. I have been assured that this is not the case. I wonder if someone can calculate the following situation, which strikes me as the simplest 2nd case to compare with earth's surface: At the surface of the Sun, gravity is roughly 28 times that at the Earth's surface. If we move out to a suitable distance, we can find a place where the gravitational acceleration is the same as the surface of the Earth - 1g - and we can hold a clock there with a rocket. What is the time dilation there with regard to a \"Far away\" observer? I understand to the dilation on the Earth's surface to be 0.0219 seconds per year compared with the distant observer."}
{"id":"104621","title":"Specific heat of aluminium as a function of temperature","text":"I am trying to find the curve describing the specific heat of aluminum as a function of temperature between about 50K to 300K(or room temperature) for an experiment I am doing. Ultimately I need to integrate between those bounds. Google doesn't seem to provide any useful links. I was able to find some values at specific temperatures, but not enough data to produce a curve. Any references to places that might contain this sort of data are welcome."}
{"id":"104629","title":"Entropy before and after collapse of the wavefunction\/ and interpretation?","text":"Seems like it might be pretty rudimentary but I want to see if my thinking is on the right track as well as what the result means. The question is, is the entropy of the collapse of a wavefunction or entangled system more before the collapse? My reasoning is that as Entropy is a measure of the \"unpredictability\" of information content (in this context), the more we know the less entropy there is. So for example if we have N fair coin tosses in which we don't know the outcome, we would have N bits of unknown information as $ \\sum_{i = 0;N} 2^N = N$ but the first we flip and once the answer is known we would have an entropy of N-1 as we don't know those N-1 states. Once all is known we have an entropy of 0. Of course we would deal with a continuous function for waves, but am I on the right track of thought? If so, then how does the second law of thermodynamics fit in in that the Universe is supposed to go to a higher state of entropy? Just an aside question, but is the wavefunction collapse the same as the collapse of an entangled system? Seems that the wavefunction itself is just a superposition of several different eigenstates, is it safe to say that these eigenstates are \"entangled\"? Thanks, Brian"}
{"id":"26110","title":"Could dark energy be the effect of gravity at great distances?","text":"This may be a silly question, but is it possible that dark energy and gravity are related to each other? Space-time is deformed everywhere in space by objects with mass. The more massive the object, the more deformation in the local area, and, as a result, the more concentrated space-time is within that deformation. However, where is the \"extra\" space-time coming from to be able to form these super-concentrated regions of space-time? Is that what we observe as dark energy? Is it possible that at great enough distances, gravity could be pulling space- time apart due to the lack of mass in the great voids of space? Not only that, but if there was enough mass in a part of space to have a little more gravity than the surrounding area, it could prevent that area of space from being \"stretched out.\" Thus, might dark matter just be regions of space where space- time is denser than the surrounding regions? I'm sure these aren't new ideas, but I'd like to know what premises and assumptions are incorrect."}
{"id":"26114","title":"How could they tell things weigh a sixth of their \"Earth weight\" on the moon?","text":"I frequently keep hearing stuff about \"gravity on the moon is only a sixth of that as Earth's\" and \"of course you'll weigh more on Jupiter\". I do know the relative sizes of those bodies allow for gravitational acceleration that's different from the $9.8 \\mathrm{\\; m\\cdot s}^{-2}$ we're accustomed to here, and I'm pretty sure nobody's ever bothered to bring a scale to Mars or something. How can one reckon how much you'll weigh on the moon if you don't know the \"one-sixth\" bit? How about on other planet(oid)s and satellites?"}
{"id":"63806","title":"Yang-Mills instanton","text":"How can instanton solution to Yang-Mills theory with gauge group $SU(3)$ or $SU(N)$ be obtained? For $SU(2)$ it is explained in textbooks but what about more general color gauge groups? EDIT: How does $A^\\mu$ look like for SU(3) or SU(N). Wikipedia only gives for SU(2). It is given by for SU(2) $$A^\\mu_a = 2\/g \\frac{\\eta^a_{\\mu\\nu}(x-z)}{(x-z)^2+\\rho^2}$$"}
{"id":"79856","title":"Transmission Lines + DC Current","text":"I have a problem where I'm given a circuit that looks like: ![enter image description here](http:\/\/i.stack.imgur.com\/YxYR1.png) It contains a transmission line with a characteristic impedance $Z_0$, a source resistance of $R_s$, and a load impedance of $R_L$. If the voltage source provides a constant DC voltage source of $V_0$, what are the voltages at points A and B? I was considering using the voltage divider equation, but I don't think that's the right approach."}
{"id":"79850","title":"Super cooled steam and entropy change","text":"I was thinking about a situation where I have some super cooled steam which suddenly freezes to water.What are the entropy changes(positive or negative) for the system and the universe? My thoughts:On transition from steam to liquid water the molecules gets more packed together which reduces their giggling around and hence, the entropy of the system would decrease. Since it is assumed to be a spontaneous process the entropy of the universe should increase.Is my thinking correct?"}
{"id":"61478","title":"Electric Field in Circular waveguide in TE mode","text":"I know that the magnetic field inside a circular waveguide is $B_z=B_0\\dfrac{\\rho}{R}\\left(1-\\dfrac{\\rho}{2R}\\right)\\cos{\\phi}\\sin{\\frac{\\pi z}{d}}$ How can I find the two components of the electric field, $E_\\rho, E_\\phi$ that is?"}
{"id":"2000","title":"Can a multipass x-ray absorption cell be constructed?","text":"I've been trying to understand the various concepts behind x-ray optics compared to standard visible\/IR optics like mirrors and such. However, the x-ray mirrors I've been finding typically have optimal angles of reflectance on the order of a few milliradians, particularly for hard x-rays. I think I mostly understand the reasons for this (having something to do with Bragg diffraction?), but, say, for absorption experiments with low sample density, it may be necessary to use something analogous to a multipass absorption cell. Since x-rays can't be reflected at anywhere near 90° with high efficiency, can a multipass absorption cell be constructed for x-rays (say, with fiber optics or something else)? Has anyone done so? Multiple Google searches for such a thing have turned up nothing, but maybe I'm just searching for the wrong thing."}
{"id":"2001","title":"Reflected and refracted wave sphased","text":"When we derive refraction and reflection laws for a generical plane wave on a surface, we say that reflected and refracted must be in phase with the incident wave. Why a medium cannot do a sphased reaction beside incident wave? Sorry for my trivial question."}
{"id":"18490","title":"Free surface of inviscid fluid flow","text":"The following problem seems like it should have a definite solution, but I've been thinking about it for months and haven't got anywhere. It might not be a well-posed problem, but if it isn't I'd like to understand why. An incompressible, inviscid fluid of density $\\rho$ flows continuously (in a steady state) as shown in the following diagram: ![diagram](http:\/\/i.stack.imgur.com\/Y1qJz.png) We know the height of the fluid (and hence its pressure) at points $x_1$ and $x_2$, but we don't know the velocity of the fluid or its height at any other value for $x$. The top of the fluid is a free surface, i.e. it's determined by the properties of the flow rather than being specified as part of the problem. I've drawn it as slightly concave but I've no idea if that's right. Let us assume that the velocity profile at $x_1$ is vertical (i.e. velocity does not vary with height above point $x_1$). Because the fluid is inviscid it seems to me that the constant vertical velocity profile should be maintained as the fluid travels to the right. So if we were to dye a vertical line of the fluid a particular colour, it would remain a vertical line as it travelled to the right, because the pressure differential across the line is constant with depth. If this is correct it means we can think of the velocity component in the $x$-direction, $v_x$, as a function of $x$ rather than $x$ and $y$. Because the flow is incompressible we know that $h(x)v_x(x)$ must constant over space, and this is the value I want to solve for (although it might not have a unique value - in that case I just want to know the function $h(x)$). If we need to we can also assume we know the initial and final velocities, $v_x(x_1)=v_1$ and $v_x(x_2)=v_2$. It seems like Bernoulli's equation should have some relevance here. That would certainly be the case if the fluid were confined to a pipe instead of having a free surface. (In this case the pressure difference would be independent of the difference in height, so we'd need to know that as well.) But every time I try to solve this problem using the Bernoulli equation I get into a terrible mess. I'm really not sure of the best way to approach this problem, so any insight anyone can offer would be much appreciated."}
{"id":"18498","title":"Are proton, antiproton, electron, positron the only observed subatomic particles that can freely exist and don't decay, i.e. are stable?","text":"Are proton, antiproton, electron, positron the only subatomic particles that can freely exist (i.e. I don't want particles that only exist in bound state as constituents such as quarks) and don't decay, i.e. are stable? What about muons? Are there other particles\/hadrons that can exist freely (i.e. not in some bound states) and don't decay?"}
{"id":"102664","title":"How was this dot identified as a mystery planet originally (Pluto in this case)?","text":"I was watching a documentary on a mission to study Pluto. At 21:11, an image is presented with an arrow mark pointing a small dot which is now identified as Pluto. As there are hundreds of dots on the image (I'm not sure what kind of an image this is as I'm not an astrophysicist), how can stars, planets, asteroids and other objects be distinguished in this image? And what kind of an image is this? ![Enter image description here](http:\/\/i.stack.imgur.com\/KXHRF.png)"}
{"id":"112176","title":"Are Normal Stress Differences in Newtonian Fluids Equal?","text":"I was reading about normal stress differences in the context of Non-Newtonian fluids. There was a remark that the normal stress differences in Non-Newtonian fluids are not equal whereas the normal stress differences in Newtonian fluids are equal. I can't figure out why the normal stress differences in a Newtonian fluid should be equal. Can anyone highlight a proof?"}
{"id":"33926","title":"Could the Higgs boson field be responsible for the dramatic mass increase when an object begins to travel close to the speed of light?","text":"could the Higgs boson field be responsible for the dramatic mass increase incurred when an object nears the speed of light?"}
{"id":"33928","title":"What allows new born suns to travel away from each other?","text":"What allows massive new born suns to move away from each other, as they have been observed. I would think that their massive gravity would prevent this and cause them to slam into each other."}
{"id":"112179","title":"Number of possible entangled states","text":"Suppose we have four single particle states: $|u,\\pm\\rangle$ and $|v,\\pm\\rangle$. How many lineary independent entangled states can we have for a pair of particles? I know that if A and B are entangled, the state AB cannot be written as a product of state A and state B, i.e. $|AB\\rangle \\neq |A,i\\rangle |B,j\\rangle$. Does this mean that the state of AB has to have at least two terms? Is $|\\psi\\rangle = |u,+\\rangle|v,-\\rangle + |v,-\\rangle|u,+\\rangle$ such a state?"}
{"id":"38042","title":"Can anyone analyze this perpetual machine that a Norwegian built?","text":"Take a look at this video: http:\/\/www.youtube.com\/watch?v=tlx2PgESXhs It shows a machine that a Norwegian artist built. It seemingly keeps going forever. They filmed it over 3 days and it kept going at the same time. Can someone analyze the energy losses in this machine? I'm seeing several losses at the moment:   1. The ball is losing energy due to friction while going around the track.   2. The pendulums are losing energy due to air friction. What other sources of energy losses can you notice?"}
{"id":"38047","title":"Could tidal effects make matter hot enough for nuclear fusion?","text":"There was a recent question about cosmological firewalls. Putting aside questions of entanglement and information and looking at the problem from a pure mechanical viewpoint, we know that although there is little perceptible radiation emitted from the blackhole, the tidal forces approaching the event horizon are enormous leading to spaghettificaton. Although the tensile forces will eventually tear a person apart, there are also compressive forces for material located in the same plane. A human body contains a considerable amount of hydrogen and other lighter elements. Would the compressive effects of spaghettification be sufficiently high to cause any of the hydrogen or other elements to fuse prior to reaching the event horizon?"}
{"id":"83239","title":"what does \"medium-induced gluon radiation\" mean","text":"I've often run across phrases such as \"medium-induced gluon radiation\" from some papers,what is \"medium-induced gluon radiation\"?"}
{"id":"133141","title":"What is the difference between induced current and eddy current?","text":"I was reading today about electromagnetic induction and eddy currents. But while reading I was unable to differentiate between induced current and eddy current. What is the difference?"}
{"id":"83237","title":"Measuring voltage drop from induced current","text":"I'm having trouble connecting voltage drops and induced current. Imagine you have a triangular loop make up off three resistors. You place this loop in a constantly changing magnetic field. This changing magnetic field will cause an EMF and a resulting induced current in the loop. Now imagine I place a voltmeter across one of the resistors. What does the voltmeter read? $I\\times R$ or $I\\times 2R$? If you look at the situation one way, you are measuring the voltage drop across one resistor, so V would equal $I\\times R$. But if you look at in another way, you're also measuring the voltage drop across the other half of the circuit which contains TWO resistors (hence a voltage drop of $I\\times 2R$). I have tried to model this situation in my schools physics lab and I do see that there is a voltage drop across the resistor so the answer is not \"there is no voltage drop\". My model wasn't perfect, but it seemed to imply that the voltmeter read $I\\times 3R$? How could that be?"}
{"id":"102338","title":"Do free electrons really not interact with photons?","text":"If free electrons don't interact with photons, why are free electrons accelerated by electromagnetic fields?"}
{"id":"47094","title":"Convert latitude of lowest altitude to argument of perigee?","text":"I am designing an orbit around Mercury. I know the values I want for the semi- major axis, eccentricity, inclination, and RAAN. I want the altitude of closest approach (periapse) to occur at $60.0^{\\circ}$ N latitude above Mercury's surface. **My Question** : How do I convert this constraint for the _latitude of closest approach_ into an _argument of periapsis_ , $\\omega$? Specifically, this is modeled after the orbit below: ![enter image description here](http:\/\/i.stack.imgur.com\/pPllt.jpg)"}
{"id":"87441","title":"Counting D0-D4 Bound States","text":"I have a slightly technical combinatorics question. Consider the degeneracy $D_n$ of bound states of $n$ D0 branes and one D4 brane. This is given in Polchinski by (13.6.24), \\begin{align} \\sum_{n=0}^{\\infty}q^n D_n=2^8\\prod_{k=1}^{\\infty}\\left(\\frac{1+q^k}{1-q^k}\\right)^{8}. \\end{align} I was able to verify this up to $n=3$, basically you have to count all ways to form bound states of the D0 branes and then bound them to the D4 brane. However, I wasn't able to verify it in general, since brute forcing it is a bit of a mess. Is there some clever mathematical formalism that allows one to deal with combinatorical problems like this?"}
{"id":"111536","title":"What would the gravity be like if Earth and the moon were touching?","text":"What would the gravity be like if Earth and the moon were touching? ![enter image description here](http:\/\/i.stack.imgur.com\/qLzZG.png) Assuming the moon keeps its form and doesn't collapse on its own weight, would the area between it and the Earth (point B) have less gravity? What about point A? Would gravity in that area stay the same?"}
{"id":"73980","title":"Why does electricity need wires to flow?","text":"If you drop a really heavy ball the ball's gravitational potential energy will turn into kinetic energy. If you place the same ball in the pool, the ball will still fall. A lot of kinetic energy will turn into thermal energy because of friction, but the gravitational potential energy will still be converted. Similarily, why doesn't electricity flow without a good conductor? Why won't Electrons won't flow from the negative terminal to the positive terminal without a wire attaching them? I know how electricity flows like a wave and that metals have free electrons floating around that allows the wave to propagate. But when these free electrons aren't available to propagate the wave, why don't the electrons just \"move\" like the ball? Why don't the electrons just \"move\" through the air to the positive terminal. A slow drift speed means that tell probably take a long time, but they should still get there. Shouldn't they?"}
{"id":"112249","title":"If temperature is average KE per particle, and heat is total KE of all the particles, how can molar heat capacity vary?","text":"If temperature is defined as the average kinetic energy per particle, and heat energy is defined as the total kinetic energy of all the particles (or more strictly, heat transferred is the total kinetic energy transferred to those particles), surely we can get heat by just multiplying temperature by the number of particles? If so, shouldn't molar heat capacity be constant across all substances? I'm fairly sure I'm missing something that should be obvious here..."}
{"id":"43733","title":"Interpretation of an \"interaction\" term","text":"In QFT a polynomial (of degree >2) in the fields is said to be an interaction term, Ex.: $\\lambda\\phi^4$. **Question** Is it possible to give an interpretation to terms like $\\frac{1}{\\phi^n}$? (for $n\\in\\mathbb{N}$) Cheers"}
{"id":"19451","title":"Can the quartic oscillator Hamiltonian be made quadratic? ","text":"I'm interested in turning a quadratic + quartic oscillator Hamiltonian, $$H = \\frac{p^{2}}{2m} +\\frac{kx^{2}}{2} + \\lambda x^{4},$$ into a purely quadratic problem. One of the simplest things I can think of is replace the quartic term by $$x^{4} \\rightarrow \\langle x^{2}\\rangle x^{2}.$$ Has anyone come across such an approximation? If yes could you give me some reference to the work where this approximation has been used. Is there any other simple approximation which can turn the above Hamiltonian into a quadratic problem?"}
{"id":"19452","title":"Pohlmeyer reduction of string theory for flat- and AdS- spaces","text":"The definition of Pohlmeyer invariants in flat-space (as per eq-2.16 in Urs Schreiber's  DDF and Pohlmeyer invariants of (super)string) is the following: $ Z^{\\mu_1...\\mu_N} (\\mathcal{P}) = \\frac{1}{N} \\int\\limits_0^{2\\pi} d\\sigma^1 \\int\\limits_{\\sigma_1}^{\\sigma_1+2\\pi} d\\sigma^2... \\int\\limits_{\\sigma_{N-1}}^{\\sigma_1} d\\sigma^N \\ \\mathcal{P}^{\\mu_1}(\\sigma_1)\\ \\mathcal{P}^{\\mu_2}(\\sigma_2)...\\mathcal{P}^{\\mu_N}(\\sigma_N) $ Another reference is eq-3.29 in Thomas Thiemann's  The LQG -- String: Loop Quantum Gravity Quantization of String Theory I. Flat Target Space. These formulations of the Pohlmeyer invariants is used to show its equivalence to DDF invariants for flat-space (cf. eq-2.53 in Schreiber's paper) Now, what is the equivalent invariant for the Pohlmeyer reduction in AdS case (as in Miramontes' Pohlmeyer reduction revisited)? I cant seem to find one that resembles the flat-space expression."}
{"id":"19983","title":"Frame dragging around a black hole","text":"If you have one stationary nonrotating black hole, being orbited by a 2 or more black holes just outside each other's event horizons does this alter the size of the central black hole's event horizon? what's the answer to this thought experiment, if you put a spaceship into orbit around the central black hole. and it Falls into the event horizon just barely, and then introduce the multiple black holes orbiting just outside the central black hole's event horizon would it enable the spaceship to escape central black hole's event horizon via the pull of the exterior orbiting black holes. (I don't know if rotating or nonrotating would have any effect on the event horizon"}
{"id":"19981","title":"How are particle simplices associated into complex particles?","text":"Nonfundamental particles are seen as made up of fundamental particles (in whatever specific theory). consider the simple case of 2 simplex particles (subscript 1 and 2) which form a complex particle (no subscripts):   * Are $\\phi(x_1), \\phi(x_2)$ _enough_ to determine $\\phi(x_1,x_2)$ or is _additional interaction information necessary_?   * How is $\\phi(x)$ related to $\\phi(x_1,x_2)$? i.e. how is the wavefunction of the complex particle as a function of the _complex particle position_ related to the wavefunction of the complex particle as a function of the _individual subparticle positions_? I would wildly guess to use barycentric centering: $\\phi(x)=\\int \\phi(x_1,x_2)\\delta(x - \\frac{m_1 x_1+m_2 x_2}{m_1+m_2}) d x_1 d x_2$. But does that really make any sense and does it violate normalization of the resulting wavefunction etc? now consider a collection of N simplex particles   * In classical physics one could use the energy of the particle to say if it is bound or not, how does this translate to conditions on the wavefunctions?   * One can envision 2 seperate solar systems in classical mechanics, where solar system 1 is not bound to solar system 2, but within each solar system the planets are bound to their star. Given only the positions and velocities of the stars and planets, is boundedness association possible in classical physics? how does one recognice that they form 2 stellar systems which are not bound, but withing stellar systems planets are bound?   * How does one do a similar association given wavefunctions of subparticles to identify which subparticles form a complex particle? Or does none of the above make any sense?"}
{"id":"13376","title":"Voltage drop over a cell membrane","text":"Again, a problem from exam preparation: > [A] cell's membrane allows sodium ions to pass through it, but not chlorine > ions. The cell is placed in a salty solution with a ten times higher > concentration of salt outside than in the cell. As a result, sodium diffuses > into the cell. Calculate the voltage difference across the cell membrane as > a result. Well, in equilibrium, the chemical potentials of the solution and the cell should be the same. Let $\\mu_A$ be that of the solution and $\\mu_B$ that of the cell. Assuming that the salt is in a dilute solution, I can write its chemical potential as $\\mu_A = f(T,P) + kT \\ln(n_A)$ where $f(T,P)$ is some function independent of the number of particles, and $n_A$ is the number-concentration of sodium ions in the solution. I'd assume that for the cell, I have $\\mu_B = f(T,P) + kT \\ln(n_B) + U$ where now I include the voltage that exists across the cell's membrane. Problem is, now I have two unknowns: I don't know the final concentration of ions in the cell and I don't know the final voltage of ions in the cell. **Alternative idea** but not sure if that makes sense: Osmotic pressure is defined as the pressure you must bring up to prevent osmosis from occurring, and it is also the pressure that finally builds up when you allow osmosis from happening. Could I analogously obtain the final voltage as the voltage that I'd have to apply manually to prevent any diffusion from happening at all?"}
{"id":"80251","title":"Where can get channel waveguide dimension data ny nx na ns","text":"I need channel waveguide dimension to test my waveguide mode solver. Can someone help me?"}
{"id":"80259","title":"A naive question on the Quantum Hall Effect(QHE) and the confinement in gauge theory?","text":"The non-interacting 2D lattice QH system is described by the Hamiltonian $H=\\sum t_{ij}e^{iA_{ij}}c_i^\\dagger c_j+H.c$ My confusion is: Does this imply that the $2D$ lattice QHE is described by the $2+1$ compact lattice $U(1)$ gauge theory? If it is, then according to the general conclusion \" the $2+1$ compact lattice $U(1)$ gauge theory is always confinement \", thus, does this mean that the low-energy quasiparticles of $2D$ lattice QHE are not the free electrons? I'm confused on this point, thank you very much."}
{"id":"48607","title":"Why photo cannot be captured while facing toward the Sun \/","text":"When the Camera is facing toward the Sun and a person is standing to and We're capturing his photo , all we get is a black shadow even when we capture with the flash light or anything. Why does it happen ?"}
{"id":"48605","title":"Canonical transformation and Hamilton's equations","text":"I was trying to prove, that for a transformation to be Canonical, one must have a relationship: $$ \\left\\\\{ Q_a,P_i \\right\\\\} = \\delta_{ai} $$ Where $Q_a = Q_a(p_i,q_i)$ and $P_a = P_a(p_i,q_i)$. Now to do the proof I started with $\\dot{Q_a}$:   1. Chain rule and Hamilton's equation for initial coordinates $q_i,p_i$ $$ \\dot{Q_a} = \\frac{\\partial Q_a}{\\partial q_j} \\dot{q_j} + \\frac{\\partial Q_a}{\\partial p_j} \\dot{p_j} = \\frac{\\partial Q_a}{\\partial q_j} \\frac{\\partial H_a}{\\partial p_j} - \\frac{\\partial Q_a}{\\partial p_j} \\frac{\\partial H_a}{\\partial q_j} $$   2. Then I apply chain rule for the Hamiltonian derivatives: $$ \\dot{Q_a} = \\frac{\\partial Q_a}{\\partial q_j} \\left( \\frac{\\partial H}{\\partial Q_i} \\frac{\\partial Q_i}{\\partial p_j} + \\frac{\\partial H}{\\partial P_i} \\frac{\\partial P_i}{\\partial p_j} \\right) - \\frac{\\partial Q_a}{\\partial p_j} \\left( \\frac{\\partial H}{\\partial Q_i} \\frac{\\partial Q_i}{\\partial q_j} + \\frac{\\partial H}{\\partial P_i} \\frac{\\partial P_i}{\\partial q_j} \\right) $$   3. Now reordering the terms yields us: $$ \\dot{Q_a} = \\frac{\\partial H}{\\partial Q_i} \\left\\\\{ Q_a,Q_i \\right\\\\} + \\frac{\\partial H}{\\partial P_i} \\left\\\\{ Q_a,P_i \\right\\\\} $$ Now here the problem, for the transformation from a coordinate system $(q_i,p_i)$ to a coordinate system $(Q_a(q_i,p_i), P_a(q_i,p_i))$ to be canonical we require: $$\\left\\\\{ Q_a,P_i \\right\\\\} = \\delta_{ai}$$ But why we have as well the following requirement, or is it just too obvious or true because of some property of any coordinate transformation? $$\\left\\\\{ Q_a,Q_i \\right\\\\} = 0$$ * * * The problem I am having is as follows. I agree, that the following two are true (if I used the covariant notation correctly): $$ \\left\\\\{ q^i,q_j \\right\\\\}_{q,p} = \\frac{\\partial q^i}{\\partial q^k} \\frac{\\partial q_j}{\\partial p_k} \\- \\frac{\\partial q^i}{\\partial p^k} \\frac{\\partial q_j}{\\partial q_k} = 0 $$ $$ \\left\\\\{ Q^i,Q_j \\right\\\\}_{Q,P} = 0 $$ But why its the case that the following is also true? $$ \\left\\\\{ Q^i,Q_j \\right\\\\}_{q,p} = 0 $$"}
{"id":"6324","title":"Paramagnetism what about Paraweakism or Parastrongism?","text":"Ok, I was just curious but the electromagnetic force can allow paramagnetism macroscopically in some objects. Can this be done microscopically to the subatomic level? Also, what about other forces (besides gravity I'm guessing) meaning the weak and strong force? Strong force being the strongest, would its para- also be and always be stronger?"}
{"id":"70274","title":"Kinetic energy of a rotating rod","text":"I have no idea where to even begin with this... Find the kinetic energy of a record of uniform density, mass 50 gm and radius 10 cm rotating at 33 and 1\/3 revolutions per minute. Normally I would do: $$ KE = \\frac{1}{2} m v^2, $$ but when I do that, I get a different answer then the one that is in the book, which is: 15231. I don't know how to use the fact that the radius is 10cm, but I'm thinking I need to find some type of integral and taking the piece into slices."}
{"id":"59719","title":"What's the real fundamental definition of energy?","text":"Some physical quantities like position, velocity, momentum and force, have precise definition even on basic textbooks, however energy is a little confusing for me. My point here is: using our intuition we know what momentum should be and also we know that defining it as $p = mv$ is a good definition. Also, based on Newton's law we can intuit and define what forces are. However, when it comes to energy many textbooks become a little \"circular\". They first try to define work, and after some arguments they just give a formula $W = F\\cdot r$ without motivating or giving intuition about this definition. Then they say that work is variation of energy and they never give a formal definition of energy. I've heard that \"energy is a number that remains unchanged after any process that a system undergoes\", however I think that this is not so good for three reasons: first because momentum is also conserved, so it fits this definition and it's not energy, second because recently I've heard that on general relativity there's a loss of some conservation laws and third because conservation of energy can be derived as consequence of other definitions. So, how energy is defined formally in a way that fits both classical and modern physics without falling into circular arguments?"}
{"id":"134751","title":"Questions on quantum mechanics","text":"First Question I have problem with little alternative form of Dirac equation. This problem refer to a matrix $\\alpha$ in this equation. I mean detailed general contravariant form of equation in which appeared matrices $\\alpha$ and $\\beta$. I need to change traditional place of matrix $\\alpha$ and take it in new place in end of equation. Primary, original place of this matrix in this form of equation is instantly behind $\\sigma$ sign of Hamiltonian. I want placed matrix $\\alpha$ now beside matrix $\\beta$ which is located beyond mass element of Dirac equation. The equation then takes the form where matrices $\\alpha$ and $\\beta$ are next to themselves. Will be correct this modification of equation? Second Question I have trouble with shape of equation element of massless particle in arbitrary equation of wave arbitrary of second rank. This equation will be similar to Klein-Gordon equation. In my opinion this massless element can be in expressed in this shape? Planck's constant on second power, angular frequency on second power, speed of light on second power. I understand this problem that forms of elements must be in coincidence with rank of equation. Then Planck's constant and circle frequency in numerator and speed of light in denominator will be all on (square) Will be this equation in right form?"}
{"id":"121427","title":"What does Feynman mean when he says that $F=ma$ is not exact?","text":"Chapter 12-2 in Feynman Lectures Vol. 1 states: > In fact the law, $F=ma$ is not exactly true; if it were a definition we > should have to say that it is _always_ true; but it is not ... First, > because Newton's Second Law is not exact, and second, because in order to > understand physical laws, you must understand that they are all some kind of > approximations.   1. What does he mean by _approximation_?    2. Also, how is $F=ma$ _not_ a definition of force?"}
{"id":"97982","title":"CMB radiation and holography","text":"If in principle someone have the exact data of the CMB of our universe and unlimited computability power, to what extent it is possible to determine how the universe should look like? Does it fully determines the later time structure of the universe or the same information could be matched to more than one universe?"}
{"id":"97986","title":"Convective derivative of oscillating fluid with bulk motion?","text":"I have a fluid $u$, which comprises a zeroth order constant motion $u_0$ as well as a first order oscillatory motion $u_1$. The convective derivative is $$\\frac{\\partial u}{\\partial t} +u\\frac{\\partial u}{\\partial x}$$ where $u=u_0+u_1$ We are assuming a linear solution, so that second order terms (any products of 1st order terms) are neglected. I know that the time derivative will be $\\partial(u_1+ u_0)\/\\partial t$ but I have no idea how to linearized the flow part: $$(u_0+u_1)·\\frac{\\partial(u_0+u_1)}{\\partial x}$$ Because we are assuming linear solution, we can move into frequency domain, and above expression is: $(u_0+u_1)·ik(u_0+u_1)$ - I'm afraid that I still don't get how to go about linearizing. My guess is FOILing it out and dropping $u_1u_1$ terms, but I want to be confident before moving forward."}
{"id":"119094","title":"Finding the total energy in centre of mass frame","text":"I'm working through a problem in a special relativity textbook (Woodhouse) and I'm having some difficulty. I have to show that if I have a particle of rest mass $M$, total energy $E$ colliding with a stationery particle of rest mass $m$ then the total energy, $E'$ in the frame where their centre of mass is at rest is given by: $$E'^2 = (M^2 + m^2)c^4 + 2Emc^2$$. Now I understand that the 4 velocity of the centre of mass frame is $V$, and is given by $V = (c,0,0,0)$, I'll call the momenta $P$ and $Q$ (of M and m respectively). Now in the solutions it says that the centre of mass frame is defined so that it's 4 velocity is proportional to the sum of the momenta of the two particles (Say $V = k(P+Q)$) it then goes on to say that hence the total momentum in the centre of mass frame is $(E'\/c,0)$ but how can this be, because the particles are still moving in the centre of mass frame? I can see that if this were the case then $g(V,P+Q) = E'$ and then if I could work out the form of $V$ I'm pretty sure I could calculate $E'$ okay. Any help is very much appreciated"}
{"id":"5402","title":"Meaning and types of singularity in case of string or any cosmological model (Mathematical description)","text":"What is actual meaning of singularity can we use this term for conclusion in any research paper( related to cosmological models ).what r the types ."}
{"id":"116884","title":"Height of rebound of a projectile from a sphere","text":"![enter image description here](http:\/\/i.stack.imgur.com\/ss5ZM.png) My work here is: First, I use conservation of energy: (taking the plane as zero potential) $$\\frac{1}{2}m(\\frac{2ga}{5})+2amg=\\frac{1}{2}mv^2+a(1+\\cosθ)mg$$ $$\\frac{1}{5}ga+2ga=\\frac{1}{2}v^2+ga(1+\\cosθ)$$ $$v^2=(\\frac{12}{5}-2\\cosθ)ga \\tag1$$ Now resolving forces radially at the point of falling ($R=0$) $$mg\\cosθ-\\frac{mv^2}{r}-R=0$$ $$mg\\cosθ=\\frac{mv^2}{a}$$ $$v^2=ag\\cosθ\\tag2$$ Equating $(1)$ and $(2)$ $$\\cosθ=\\frac{4}{5}$$ As required. I am having problems in the second part. Let $v=v_1$, the velocity of arrival at the plane, $v_2$, and the velocity of rebound $v_3$. I considered the velocity of arrival equating energies. $$\\frac{1}{2}mv_1^2 + mga(1+\\cosθ)=\\frac{1}{2}mv_2^2$$ $v_1=\\sqrt{\\frac{4ag}{5}}$ $$v_2^2=\\frac{22ag}{5}$$ Using Newton's Law of Restitution: $$e=\\frac{-v_3}{v_2}=\\frac{5}{9}$$ $$v_3=\\frac{5v_2}{9}=\\frac{5\\sqrt{\\frac{22ag}5}}{9}$$ Using the dynamics equation $v^2=u^2+2as$ $$(v_3\\sinθ)^2=2gh$$ From earlier, $\\sinθ=\\frac{3}{5}$ $$h=\\frac{(\\frac{3v_3}{5})^2}{2g}$$ But this gives a wrong result. I am unsure what I did wrong and, am I wrong in assuming that the particle will bounce of with the same angle $θ$ that it came out of the ball?"}
{"id":"8599","title":"Chern-Simons Theory in 3D","text":"For the CS theory on a three manifold $M$ with a gauge group $G$, it is said that the gauge field $A$ is a connection on the trivial bundle over $M$. Why the bundle should be trivial? I know that space of classical solutions is made of flat connections but why should we assume the bundle is trivial from the beginning? Another question: Can anyone explain the meaning of farming of a three manifold in simple terms? Thanks."}
{"id":"89515","title":"Isotropic harmonic oscillator in polar versus cartesian","text":"I read another Phys.SE post here: 3D Quantum harmonic oscillator that I believe says the wave function in Cartesian coordinates for a 3D harmonic oscillator is the product of the 3 one dimensional wave functions. If so, are the angular and radial equations combined within it? Is it possible to separate the radial part from the angular part? The wavefunction for 1D in the ground state is $$\\psi_{0}(x)=(\\frac{m\\omega}{\\pi\\hbar})^{\\frac{1}{4}}e^{-\\frac{m\\omega x^{2}}{2\\hbar}}.$$ I think this product would then be: $$\\psi_{0}(r)=(\\frac{m\\omega}{\\pi\\hbar})^{\\frac{3}{4}}e^{-\\frac{m\\omega r^{2}}{2\\hbar}}.$$ Is this the radial equation for the ground state? Would I then need to multiply by the angular equation for the ground state to get the full wave equation?"}
{"id":"127218","title":"Do the laws of physics that apply to all observers also apply to a non-observer?","text":"The Timelessness of a photon https:\/\/www.youtube.com\/watch?v=5ELA3ReWQJY An observer's laws of physics are time based. \"When you're traveling at the speed of light, time does not exist\" - Neil deGrasse Tyson. So a photon's laws of physics can not include time because it does not exist. How does the Photon exist in \"space-TIME\", if Time doesn't exist for the Photon? The Photon would not exist in \"space time\", it would exist only in \"space\". The physics of \"space time\", which is where we are, would not apply to \"space without time\", where the Photon is.   1. The rules of physics are the same for all observers in uniform motion relative to each other.   2. The Photon is never in uniform motion relative to any or all observer.   3. A Photon in an observer's frame is not the same as a Photon in a Photon's frame.   4. The Photon must have \"a frame\" (although it is not compatible with our own), because the Photon occupies coordinates in space as a particle.   5. The Photon's frame would not define Time.   6. The Photon does not exist in any observer's frame, because the Photon is never in uniform motion relative to any observer, but the Photon does exist.   7. If the Photon does not exist in any observer's frame, then the Photon must have a frame of its own that it does its existing in.   8. The laws of physics for all Observers includes a definition for time.   9. The Photon's frame has no time coordinates, so time coordinates from any observer's frame can not be meaningfully associated with the Photon or the Photon's frame. Time is relative to all \"observers\" but light is not an observer. Time can't be meaningfully applied to light because time doesn't exist to light. Time is not relative between an observer and light because they are not both \"observers\". If one physicist says that a particle is instantaneous And another physicist says the same particle takes 8 minutes to travel from the Sun to the Earth That's a contradiction 8 Minutes > Instant All physical laws should remain unchanged under a Lorentz transformation. An observer's physical laws include time laws. A Photon's physical laws do not include time laws. All physical laws can't remain unchanged when transforming an observer's physical laws into a photons physical laws. This would cause the application of time laws to the photon, but time laws and photons are incongruent. Assume that \"space time\" is a prison. In the prison there are guards and inmates. Everyone in the prison follows rules, but the guards follow one list of rules and the inmates follow a different list of rules. Some of the rules for the inmates and the guards are the same, some of the rules for the inmates and the guards are not the same. Among the inmates, all inmates must follow all of the inmate rules, no inmate receives special treatment. Also among the guards, all guards must follow all of the guard rules, no guard receives special treatment. The two lists are written in different languages. All of the inmates speak different languages, but one inmate speaks the language that the inmate rules are written in, and all of the languages that the inmates speak. All of the guards only speak the language that the guard rules are written in. It's possible for the one inmate who can read the inmate rules, to translate those rules into all of the languages that the inmates speak. And because all of the inmates follow exactly the same set of rules, the number of rules would be the same in each of the inmate languages. It's also possible for the inmate to translate the list of guard rules entirely into the language of the inmates, and the list of inmate rules into the language of the guards. This doesn't matter though, because the inmates and the guards still follow different rules. Writing the rules of the guard in the language of the inmate does not allow the inmate to follow the rules of the guard, and writing the rules of the inmate in the language of the guard does not subject the guard to the rules of the inmate. The laws of physics for any observer include laws that define the behavior of time. The entire system of physics for all observers then, is inherently based on the existence of time, and time is one of the defining features of the \"existence\" or \"reality\" that all observers occupy. It is the presence of time that allows an observer's laws of physics to establish a frame of reference. Because all observers follow exactly the same list of rules, like all inmates, all of the rules of \"observer physics\" can be translated (by the one inmate, \"man\") from one observer to the next, even though all observers speak a different language, or have differing personal frames of reference. However, the laws of physics for \"light\" or \"the speed of light\", do not recognize the existence of time. The entire system of physics for \"light\", is inherently without time, and this lack of time is one of the defining features of the \"existence\" or \"reality\" of \"light\" or, \"anything 'traveling' at the 'speed' of light\". It is the lack of \"time\", and therefore the lack of a definition between \"at rest\" and \"moving\", which is THE REASON why a photon can't be assigned a frame of reference when trying to translate the photon's physics to the observer's physics. It only APPEARS that photons exist by the same laws of physics as an observer because some of the laws cross over from one reality to the next. It appears that the physics of photons and the physics of observers are the same because the existence of time in our reality prevents you from observing the photon in it's \"natural habitat\" to see how we different we really are."}
{"id":"129847","title":"Gravitational force and time dilation","text":"Suppose the radius of the earth is reduced by half but the mass is same, then how long will it take to complete one rotation, 24, 48, 12 or 6 h.? please give the mathematical relations and solution."}
{"id":"129842","title":"Why the initial spectrum of perturbation of inflation must be Gaussian?","text":"Why the initial spectrum of perturbation of inflation must be Gaussian? I heared it was because of Central Limit theorem. But I don't know how use this theorem explicitly in this question to get this result."}
{"id":"101577","title":"Bernoulli's principle on a curve ball","text":"I've seen a few excellent answers here on the Magnus force, which explains why balls with a spin will curve. However, my intuition is still telling me that the Bernoulli's principle would push it the opposite way and I need help understanding why my reasoning is flawed. Imagine that you kick a soccer ball on the left side so that it's spinning clockwise (viewing the ball from above) and the ball will curve to the right. Since the left side of the ball is spinning against the air, wouldn't this mean faster relative speeds and thus a lower pressure than the right side? And wouldn't this lower pressure on the left side cause it to curve _left_ instead of right?"}
{"id":"95801","title":"How to measure the mass of the electron?","text":"I've done a little bit of research and it seems Millikan was able to measure the ratio between the charge of the electron and its mass. But how can one measure one of the two constants to get the value of the other?"}
{"id":"95809","title":"Question about creation operator and its Hermitian adjoint","text":"The definition of creation operator for bosonic system is $$a^{\\dagger}|... n_i ...\\rangle = \\sqrt{n_i + 1} |... n_i + 1 ...\\rangle $$ If I take Hermitian adjoint of this I will get $$(a^{\\dagger}|... n_i ...\\rangle)^{\\dagger} = |... n_i ...\\rangle^{\\dagger} (a^{\\dagger})^{\\dagger} = \\langle... n_i ...| a = \\sqrt{n_i + 1} \\langle... n_i + 1 ...|$$ But this can't be correct since $$\\langle... n_i ...| a = \\sqrt{n_i} \\langle... n_i - 1 ...|$$ Can someone explain to me why my Hermitian adjoint equation is wrong. From the answer I got from JeffDror and the comment left by user26431. I infer that in dual space $A_1 = a$ acts as creation operator and $A_2 = a^{\\dagger}$ as annihilation operator. I am a bit confused by what JeffDror wrote \"if you are acting backwards with the operator then you need to use the Hermitian adjoint\". In my mind it sounds like that to act with operator A on some bra-vector I must first take Hermitian adjoint of it and than act on bra. It make sense to me if imagine that bra-vector is row vector that I get for transposing ket-vector and non hermitian operator A in ket space is not represented by square matrix. Then in order to multiply matrix by column vector from the left I must transpose matrix that represents A. Is this correct way of thinking about operator how operator A acts in dual space?"}
{"id":"96577","title":"Why are oxygen and hydrogen compressible, but water is barely compressible","text":"What makes the two gases compressible, but does not apply to a liquid like water?"}
{"id":"96316","title":"Isotropic systems and homogeneity","text":"In isotropic systems, the atomic arrangement is homogeneous in all directions. In the case of glass, which has the atomic structure of a liquid and, therefore, a random atomic structure that is definitely not homogeneous, is it that the atomic arrangements in each direction are equally disordered and therefore homogeneous in that sense?"}
{"id":"64890","title":"Lattice model completely constrained by boundary data","text":"I am dealing with a lattice model that has the peculiar property that if I specify all the spins on the boundary, by local conservation laws, the whole lattice configuration (throughout the whole volume) is fixed. This trait is true for all $N\\times N \\times N$ lattices with this model. I am interested if in the thermodynamic limit this behavior holds and if this fixing prohibits phase transitions. Any ideas on these two matters? Edit: These are things I think I can add. Since the internal variables are fixed by the boundary configurations, then for the partition function, there are bulk volume configurations where boundary spins are different but the internal spins are the same (compared to another configuration). Thus, I can define an equivalence between boundary configurations for two boundary configurations that yield the same interior. Then I only need to sum over equivalence class of surface configurations, and the model is somewhat 2D."}
{"id":"64892","title":"Easy question about magnetism?","text":"I have to build a simple electric motor by attaching a magnet to a battery, extending the terminals of the battery (with stiff wires so they could act as supports), and placing a coil of wire on top of it. I know that the more loops you have in the coil the stronger the magnetic field and therefore the faster the coil will rotate. However, how does the diameter of the magnetic coil affect speed of the wire coil rotating? A smaller diameter might make the magnetic field stronger, but the magnetic force which produces torque would also be be applied closer to the center of rotation, therefore creating less torque."}
{"id":"116873","title":"Can the hot combustion products from a large flame be in \"non-local thermal equilibrium\"","text":"**Question:** Does it take some time for the hot combustion products from a flame to reach local thermodynamical equilibrium (i.e. for the energy state populations to follow the Boltzmann distribution)? If so, on what type of time scales might this departure from local thermodynamical equilibrium occur? **Background:** I am trying to model infrared emission spectra in the mid-infrared region to reproduce emission spectra of hot exhausts from the flame of an industrial flare stack measured with an FTIR-spectrometer. The emitted spectral radiance is calculated using the Planck function and the absorption crossections of different molecular species. The absorption crossections are calculated for different species using the HITRAN and HITEMP line databases. This seems to work fine for some species and some regimes. However, I seem to have problems especially when the exhaust is really hot. For instance, in the region 900-1000 cm$^{-1}$ I can qualitativly see that I have strong water emission lines, but with only varying the temperature parameter I cannot get the relative intensities of the different lines right. I keep thinking that one possible problem might be that the exhaust product from the combustion are not in local thermodynamical equilibrium. In that case the state populations for the species would not follow the Boltzmann distribution which is assumed for calculating the line intensities and their temperature dependence."}
{"id":"10944","title":"Number density of LO and LA phonons as a function of temperature?","text":"I'd like to know the how the number density of longitudinal optical (LO) and longitudinal acoustic (LA) phonons varies as a function of temperature of the material. Is there a simple expression for these two cases? I'm guessing that this would work, $N_{LO} = \\int g_{LO}(E) f(E, T) dE$ $N_{LA} = \\int g_{LA}(E) f(E, T) dE$ where $g(E)$ is the density of states for LO and LA phonons and $f$ is the Bose-Einstein distribution. What would be appropriate limits for the integrals. Does anybody know a reference where the density of states for these two modes is given? **EDIT** To improve the question, I'm interested in semiconductor 3D crystals. But maybe I left this too long, sorry. Best regards,"}
{"id":"65417","title":"Cosmic Inflation: Lower Expansion Rate with than without?","text":"When I read the Cosmic Inflation diagram at https:\/\/en.wikipedia.org\/wiki\/File:Horizonte_inflacionario.svg which I rotated to make it readable: ![Plot 1](http:\/\/yukterez.enabled.io\/inflation.png) where the Hubbleradius, and therefore also the Hubbleparameter, which is c\/Hubbleradius, is constant over inflation, then it seems to me, that with inflation the Hubbleparameter would have been much lower than without: ![Plot 2](http:\/\/yukterez.enabled.io\/Inflation_Plot.gif) If the Hubble Parameter would go with Ω-Radiation (to the 4th Power), it would reach Infinity at 0, and about 10^103 m\/sek\/m at Planck size, which is way more than the expansion rate postulated for inflation: ![Radiation dominated Expansion Rate near 0](http:\/\/yukterez.enabled.io\/Hp.PNG) Did I get this right, or do I have the wrong data? And if I have it right, is it more like the green curve with a hard inflation to radiation edge, or the blue one with a smooth blending; or is there all another way to get a plottable Hubble Parameter including inflation as a function of the scale factor? PS: the y Axis is the Hubble Parameter (c\/Hubbleradius) in m\/sek\/m and the x Axis is the Scale Factor in m\/m (dimensionless)."}
{"id":"43077","title":"Can wind currents form in the mesosphere?","text":"Probably not the right forum for this question. How high in the atmosphere may an air-current form? Can a wind current form up in the mesosphere?"}
{"id":"2249","title":"What's the explanation for the Giant Magnetoresistance effect?","text":"I've been reading this review on Giant Magnetoresistance, and something about it is bothering me. The basic effect is that, using a special \"stack\" of layers (alternating between ferromagnetic and not magnetic) you can cause great variations on the electrical resistance of the material by applying a small magnetic field.   ![alt text](http:\/\/i.stack.imgur.com\/jJQRA.png)   The explanation given both on the review and on Wikipedia is the following. The configuration on the left will cause scattering for all spin down electrons that cross it, but will present very low resistance for the spin up electrons (thus a small resistance overall). The one on the right, will scatter all electrons and thus have very large resistance overall. **Here's my problem** (quoting the review) > These experimental results were all obtained for the ’Current In the Plane > of the layers’ (CIP) geometry. The current is then parallel to the plane of > the layers. How can this be? CIP in the above image would mean the current is horizontal, with probes positioned on the left and right of the middle layer. But, if you don't force the current to cross the ferromagnetic layers, why would there be an increase in resistance? I checked with a professor, and he confirmed that the standard method is indeed to apply the probes horizontally, but he couldn't tell why the effect still worked. One option that came up is that electrons don't follow a straight line from one probe to the other, and thus they would still \"flow\" through the other layers, but I know that's not the case. Using the resistors analogy seen in the image above, applying a CIP is equivalent to setting the resistors in parallel for each of the spin orientations, and parallel resistors _always_ have less (or equal) resistance than the weakest resistor. To be clear, my question: How can the above explanation be compatible with the CIP case? If it's not, then why does it still work?"}
{"id":"46247","title":"Spin-orbit coupling constant for rubidium","text":"I have come across the following question in my course notes: The $5s\\to 5p$ transition in rubidium is split into two components with wavelengths of 780nm and 795nm respectively. For the $5p$ state, calculate the spin-orbit coupling constant $a$. I have calculated the spin orbit splitting energy to be $E=\\frac{3}{2} a$, calculated the energy of the 795nm component to be $E=\\dfrac{hc}{\\lambda}=2.5\\times 10^{-19} J$, suggesting that $a=1.67\\times 10^{-19}J$. I'm not convinced by this answer as it gives a very large value for the magnetic field of the electron due to its orbital motion. Is the method I'm using correct, or does something seem to be out of place?"}
{"id":"46240","title":"Is the proper interpretation of temperature missing in this book?","text":"In Randall T. Knight’s textbook “Physics for Scientists and Engineers” in the first chapter on thermodynamics (Ch. 16: A Macroscopic Description of Matter) one of the first conceptual questions is stated like so: > The sample in an experiment is initially at $t_1$°C. If the sample’s > temperature is doubled, what is the new temperature in °C? The answer to that implies the reference to absolute temperature (for simplicity, to one significant figure, as the book required), i. e.: $t_2°C = (T_1 × 2)K - 273 = (t_1°C + 273) × 2 - 273$. I understand that the whole idea of this question to reiterate to the pupil that we ought to think of temperature as a specific physical property, and not in an everyday sense, but I am confused about the wording. In the section on temperature author writes: > We will begin with the commonsense idea that temperature is a measure of how > “hot” or “cold” a system is. These are properties that we can judge without > needing an elaborate theory. As we develop these ieas, we’ll find that > **temperature** _T_ is related to a system’s _thermal energy_. […] We’ll > study temperature more carefully in Chapter 18 and replace these vague > notions of hot and cold with a precise relationship between temperature and > thermal energy. That is, as yet we are still at the stage of “vague notions”. Few paragraphs later Knight introduces Kelvin scale but nowhere there or in between does he specifically insists on that idea I mentioned after the formula. At least not that I can see. Am I missing something that should be obvious to me, and I fail to grasp it instantly because my brain is depleted by 50+ hours a week of studies? Am I just plain stupid, because to my eternal shame it took me way too long to figure it out? Or was there a better way to home onto that idea by an author: what do other textbooks say on that topic, how do they word this concept? Is the question logically flawed or confusing, or is it the common way of describing the situation in scientific literature?"}
{"id":"46249","title":"Delta Dirac Charge Density question","text":"I have to write an expression for the charge density $\\rho(\\vec{r})$ of a point charge $q$ at $\\vec{r}^{\\prime}$, ensuring that the volume integral equals $q$. The only place any charge exists is at $\\vec{r}^{\\prime}$. The charge density $\\rho$ is uniform: $$\\rho(\\vec{r}) = \\delta(\\vec{r} - \\vec{r}^{\\prime})\\rho$$ But if I evaluate the total charge, I get $$ q = \\int dq = \\int^{\\infty}_{-{\\infty}}\\delta(\\vec{r} - \\vec{r}^{\\prime})\\rho ~dV $$$$= \\rho\\int^{\\infty}_{-{\\infty}}\\delta({x} -{x}')dx\\int^{\\infty}_{-{\\infty}}\\delta({y} -{y}')dy\\int^{\\infty}_{-{\\infty}}\\delta({z} -{z}')dz$$ The Dirac delta functions integrate to one each, but what becomes of the charge density $\\rho$? For that matter, how does one integrate a zero dimensional point over 3 dimensinal space? Any help greatly appreciated. EDIT: So it seems that the charge density is just the charge itself ($\\rho = q)$?"}
{"id":"35833","title":"Single photon and double-slit experiment","text":"Laser fires single particles of light, called photons, through the slits. Even though only single photons of light are being fired through the slits and They create three pattern again. How single particles of light can create this wave pattern?"}
{"id":"22921","title":"Is there an explanation for the 3:2:1 ratio between the electron, up and down quark electric charges?","text":"I understand that the NNG formula relates $Q$, $I_3$, and $Y$ and can be derived in QCD; does this unambiguously predict the electric charge ratios without making assumptions about the definitions of isospin and hypercharge? If so, this is unintuitive to me! Why does a particle carrying $SU(3)$ color charge care what charge it has under the totally separate electroweak $U(1)\\times SU(2)~$ symmetries? If not, is there a name for the \"problem\" of explaining the charge ratios?"}
{"id":"91158","title":"Why would the reflection of a lamp by a bowl look like a heart?","text":"I have this bowl roughly the shape of a circular truncated cone, and the lamp shines on the bowl at an angle. The reflected lamp light hit the bottom of the bowl and created a light ring that looks just like a cardioid. Could someone help me, or give me a hint on how to really solve the equation for the light ring?"}
{"id":"60594","title":"Energy of particle in electric field","text":"I'm taking a physics class and the professor teaches us really basic things in lecture and then gives homework way beyond what he taught in lecture. Obviously I need to find some resource other than Stack Exchange, but in the mean time, perhaps you could explain some of the terms in this problem and give me a starting point for how to solve it. Regarding the problem, given below, I'm not sure what the significance of \"exactly one mean free path through the air before hitting a molecule\" is. I know eV is an electron volt, which is a unit of energy (from high school) but I'm not sure how this is related to the strength of a field. I would assume this energy is related to the force on the particle, which I can relate to the strength of the field and the force on the particle, as well as the distance the particle travels, but that doesn't seem right because no figures are given for distance or charge. I haven't got a clue what cross-sectional area might have to do with this. I do, however, know what a Maxwell distribution is from chemistry class. > (a) Assume that an ion moves, on average, exactly one mean free path through > the air before hitting a molecule. If the ion needs to acquire approximately > $1.11\\ \\mathrm{eV}$ of kinetic energy in order to ionize a molecule, > estimate the minimum strength of the electric field required at standard > room pressure and temperature ($300\\ \\mathrm{K}$). Assume that the cross- > sectional area of an air molecule is about $0.095\\ \\mathrm{nm^2}$. (Assume > the velocities of the particles have a Maxwell distribution of velocities.) > > (b) How does the strength of the electric field in Part (a) depend on > temperature? (Use the following variable as necessary: $T$.) > > (c) How does the strength of the electric field in Part (a) depend on > pressure? (Use the following variable as necessary: $P$.)"}
{"id":"60590","title":"Is it ever necessary to extend an analysis of Grover's algorithm beyond $k\/N = 1\/2$?","text":"Is it ever necessary to extend an analysis of Grover's algorithm beyond $k\/N = 1\/2$, where $k$ is the number of \"hits\" in a total of $N$ possible values for $|\\,x\\rangle$? If we know $k$, and know that it is greater than $1\/2$, can't we just redefine hits as misses by modifying the oracle to produce $$-(-1)^{f(x)}\\,|\\,x\\rangle$$ (note the additional negative sign) and then measure according to whatever strategy for $k<1\/2$ produces satisfactory hit probabilities, $p_h$, since we can then say with confidence $p_h$ that one of the _unobserved_ outcomes is a hit?"}
{"id":"57035","title":"Formulation of the Three-Body Newtonian problem","text":"I am trying to understand three body problem in Newtonian space. I want to make formulation of differential equations for known initial conditions for the case with:   1. Identical three masses    2. Periodic solution    3. Zero Angular momentum The equations in vector form are: $$\\frac{d^2 \\overset{\\rightharpoonup }{r}_1}{\\text{dt}^2}=\\frac{k^2 m_2 \\left(\\overset{\\rightharpoonup }{r}_2-\\overset{\\rightharpoonup }{r}_1\\right)}{r_{12}^3}+\\frac{k^2 m_3 \\left(\\overset{\\rightharpoonup }{r}_3-\\overset{\\rightharpoonup }{r}_1\\right)}{r_{13}^3}$$ $$\\frac{d^2 \\overset{\\rightharpoonup }{r}_2}{\\text{dt}^2}=\\frac{k^2 m_1 \\left(\\overset{\\rightharpoonup }{r}_1-\\overset{\\rightharpoonup }{r}_2\\right)}{r_{12}^3}+\\frac{k^2 m_3 \\left(\\overset{\\rightharpoonup }{r}_3-\\overset{\\rightharpoonup }{r}_2\\right)}{r_{23}^3}$$ $$\\frac{d^2 \\overset{\\rightharpoonup }{r}_3}{\\text{dt}^2}=\\frac{k^2 m_1 \\left(\\overset{\\rightharpoonup }{r}_1-\\overset{\\rightharpoonup }{r}_3\\right)}{r_{13}^3}+\\frac{k^2 m_2 \\left(\\overset{\\rightharpoonup }{r}_2-\\overset{\\rightharpoonup }{r}_3\\right)}{r_{23}^3}$$ where $k$ denotes the gravity constant. I want to formulate (not solve) this problem properly. I've found this reference can help understand zero angular momentum. I don't understand it well."}
{"id":"102463","title":"Is it possible to detect fake Tungsten aka Wolfram gold bars with a strong magnet?","text":"Tungsten aka Wolfram is paramagnetic so it is weakly attracted to magnets. A guy devised the following to test for Tungsten in gold bars: http:\/\/www.youtube.com\/watch?v=foELQ7T8_90 But he is using a paperclip and not real Tungsten. Question: How strong a magnet do I need in order to achieve a significant attraction to Tungsten. By significant I mean that it should exert a force of several (1\/100)*grams so that it can be measured with a scale as in the YouTube video. Will a Neodymium magnet do? If yes, what should be the strength of the magnet in kg?"}
{"id":"17707","title":"What \"I\" should use in Rotational Energy formula $(I \\omega^2)\/2$","text":"$\\text{Rotational Energy} = \\frac{1}{2} I \\omega^2$. What $I$ should be used? $I$ as a inertia tensor matrix = stepRotation * inverse moment of inertia * inverse stepRotation; Or I as moment of inertia? Inertia tensor matrix gives changing energy due to rotation changes. So I don't believe its the right answer, but I'm not sure. Moment of inertia remains the same in simulation because it only represents the mass resistance to rotation in a fixed axis calculated by integration of massparticles * radius from object center."}
{"id":"27449","title":"Majorana-like representation for mixed symmetric states?","text":"Is there a generalization of the Majorana representation of pure symmetric $n$-qubit states to mixed states (made of pure symmetric $n$-qubit)? By Majorana representation I mean the decomposition of a state $$|\\psi\\rangle = \\text{normalization} \\times \\sum_{perm} |\\eta_1\\rangle |\\eta_2\\rangle \\cdots |\\eta_n\\rangle,$$ where $|\\eta_k\\rangle$ are uniquely determined (up to a global phase in each and the permutation) qubit states. Typically, they are presented as points on the Bloch sphere. When it comes to a desired generalization - by it I mean a unique set of invariants and covariants of $\\text{SU}(2)$ assigned to each density matrix. The uniqueness (up to permutations) is crucial, otherwise one can just spectrally decompose the density matrix $$\\rho = \\sum_i p_i |\\psi_i\\rangle\\langle\\psi_i|$$ and apply the standard Majorana representation to each of its eigenvectors (obtaining invariants $\\\\{ p_i \\\\}_{i\\in\\\\{1,\\ldots,n\\\\}} $ and covariants $\\\\{ |\\eta_{i,j}\\rangle \\\\}_{i,j\\in\\\\{1,\\ldots,n\\\\},}$ ). However, for the case of the eigenvalue degeneracy it is no longer unique. A naive dimension counting $(n+1)^2-1=2n + n^2$ gives a hint that there may be $n$ covariant points plus an invariant $n\\times n$ matrix. An auxiliary question is if there is a unique representation of $k$-dimensional subspaces of pure symmetric $n$-qubit states? (If there is one, then it is possible to 'fix' the mentioned approach with the spectral decomposition.)"}
{"id":"60621","title":"Coupling constant problem","text":"In the scalar $φ^4$ theory we write Lagrangian as $$\\mathcal{L}=\\frac{1}{2}(\\partial_t\\phi)^2 -\\frac{1}{2}\\delta^{ij}\\partial_i\\phi\\partial_j\\phi - \\frac{1}{2}m^2\\phi^2-\\frac{g}{4!}\\phi^4. $$ I want to know details about the coupling strength g. How will I start? I studied some technique from Feynman diagram, that it represents the strength of a scattering. But I need to find out the value of this constant and need guideline how will I study so I can understand clearly. Is there any relationship which gives us the enough evidence for putting the coupling constant into Lagrangian."}
{"id":"9786","title":"Violations of Dulong-Petit rule as an upper limit to heat capacity","text":"Does any known substance have a heat capacity at constant volume ($C_V$) **per mole of atoms** greater than $3k_B$ ~ 24.94 J\/(mol K)? In order to count, the substance must actually be made of atoms, that is, ordinary nuclei and electrons. If so, what are the extra degrees of freedom responsible for this unusually large heat capacity?"}
{"id":"9788","title":"Why doesn't string theory have a mass gap?","text":"Gravitons are string modes which can have arbitrarily small energies. Just make the wavelength arbitrarily large. String theory has no mass gap, but why? A classical string can only have positive energies. The string tension makes a positive contribution. The string kinetic energy also makes a positive contribution. A classical string can have an arbitrarily small mass because it can be arbitrarily small in length. But quantum strings can't shrink beyond the string scale. The lack of a mass gap can be traced to a shift factor to the string energy in the Virasoro constraint. For a single string, this shift is unconstrained and arbitrary. Without interactions, this is none other than the \"string chemical potential\". But we can create\/annihilate strings after second quantizing them. This shift becomes physical. Modular invariance of worldsheets of genus one fixes the shift, and it turns out to be negative. String theorists explain this as the Casimir effect on the worldsheet, but this is just a name, and not an explanation. Why is this Casimir shift negative? Only a negative value can cancel the mass gap. Don't say $ζ$-function regularization, because it's purely formal and doesn't explain a thing."}
{"id":"37938","title":"Quantum $n$-body problem","text":"Is the quantum $n$-body problem as difficult as the classical $n$-body problem? Or quantum mechanics allows to get a simpler exact solution? Suppose there are 3 particles with uniform potential field. Can their wave functions be found exactly?"}
{"id":"112401","title":"Dark Matter and Modified Gravity","text":"Could someone please explain briefly or refer me to an article or manuscript that shows how f(R) modified gravity theories can be used to explain the problem of Dark Matter, particularly Galaxy rotation problem?"}
{"id":"48164","title":"Transform hamiltonian","text":"I have got the following Quantum Hamiltonian: $$H=\\frac{p^{2}}{2m}+k_{1}x^{2}+k_{2}x+k_{3}$$ Which transformation can I use to change this Hamiltonian into an harmonic oscillator hamiltonian? Note: $k_{1}, k_{2}$ and $k_{3}$ are constants."}
{"id":"20773","title":"Matlab package: graphical calculus for quantum operations including beam splitters and polarizing beam splitters","text":"I need a matlab package that will make my life easier. I have quantum circuits with optical beam splitters, polarizing beam splitters and photodetectors. These circuits are getting very complicated and I am wondering if anyone knows of a matlab package that would make my life easier. A graphical package would be excellent and I suppose any software will do."}
{"id":"48162","title":"Origin of Laue equations?","text":"The Bragg condition (by Bragg in 1913) can be derived by the Laue equations that is making use of the Miller indices and all the latice\/crystal stuff (so basically it's bringing Bragg's law to more fundamental ground). But what is interesting to me: When did the \"Laue way\" came up? Was it only named after Max von Laue or did he himself publish a paper introducing this formalism? Neither Wikipedia nor the books I have are citing it. The Laue diffraction\/condition\/law is always just presented without origin. But I'd like to find it's real origin."}
{"id":"27429","title":"$\\pm$ (light-cone?) notation in supersymmetry","text":"* I would like to know what is exactly meant when one writes $\\theta^{\\pm}, \\bar{\\theta}^\\pm, Q_{\\pm},\\bar{Q}_{\\pm},D_{\\pm},\\bar{D}_{\\pm}$.  {..I typically encounter this notation in literature on $2+1$ dimensional SUSY like super-Chern-Simon's theory..}   * I guess that when one has only half the super-space (i.e only the $+$ of the above or the $-$) it is called the $(0,2)$ superspace compared to the usual $(2,2)$ superspace. In this case of $(0,2)$ SUSY I have seen the following definitions, $Q_+ = \\frac{\\partial}{\\partial \\theta^+} + i \\bar{\\theta}^+\\left( \\frac{\\partial}{\\partial y^0} + \\frac{\\partial}{\\partial y^1} \\right)$ $\\bar{Q}_+ = -\\frac{\\partial}{\\partial \\bar{\\theta}^+} - i \\theta^+\\left( \\frac{\\partial}{\\partial y^0} + \\frac{\\partial}{\\partial y^1} \\right)$ which commute with, $D_+ = \\frac{\\partial}{\\partial \\theta^+} - i \\bar{\\theta}^+\\left( \\frac{\\partial}{\\partial y^0} + \\frac{\\partial}{\\partial y^1} \\right)$ $\\bar{D}_+ = -\\frac{\\partial}{\\partial \\bar{\\theta}^+} + i \\theta^+\\left( \\frac{\\partial}{\\partial y^0} + \\frac{\\partial}{\\partial y^1} \\right)$   * I am guessing that there is an exactly corresponding partner to the above equations with $+$ replaced by $-$. Right?  How does the above formalism compare to the more familiar version as, $Q_\\alpha = \\frac{\\partial}{\\partial \\theta^\\alpha} - i\\sigma^\\mu_{\\alpha \\dot{\\alpha}}\\bar{\\theta}^{\\dot{\\alpha}}\\frac{\\partial}{\\partial x^\\mu}$ $\\bar{Q}_\\dot{\\alpha} = -\\frac{\\partial}{\\partial \\bar{\\theta}^\\dot{\\alpha}} + i\\sigma^\\mu_{\\alpha \\dot{\\alpha}}\\theta^{\\alpha}\\frac{\\partial}{\\partial x^\\mu}$ which commute with, $D_\\alpha = \\frac{\\partial}{\\partial \\theta^\\alpha} + i\\sigma^\\mu_{\\alpha \\dot{\\alpha}}\\bar{\\theta}^{\\dot{\\alpha}}\\frac{\\partial}{\\partial x^\\mu}$ $\\bar{D}_\\dot{\\alpha} = -\\frac{\\partial}{\\partial \\bar{\\theta}^\\dot{\\alpha}} - i\\sigma^\\mu_{\\alpha \\dot{\\alpha}}\\theta^{\\alpha}\\frac{\\partial}{\\partial x^\\mu}$ {..compared to the above conventional setting, in the $\\pm$ notation among many things the most perplexing is the absence of the Pauli matrices!..why?..} I would be very grateful if someone can explain this notation. {..often it turns out that not just the Qs and the Ds but also various superfields also acquaire a $\\pm$ subscript and various usual factors of Pauli matrices look missing..it would be great if someone can help clarify this..}"}
{"id":"67016","title":"SUSY (Supersymmetric) Quantum Mechanics","text":"I have seen some books, e.g. by Fred Cooper (Supersymmetry in Quantum Mechanics), define: $A = \\frac{\\hbar}{\\sqrt{2m}} \\frac{d}{dx} + W(x)$, $A^\\dagger = \\frac{-\\hbar}{\\sqrt{2m}} \\frac{d}{dx} + W(x)$, $V_1(x) = W^2(x) - \\frac{\\hbar}{\\sqrt{2m}} W'(x)$, $V_2(x) = W^2(x) + \\frac{\\hbar}{\\sqrt{2m}} W'(x)$. In other places, I have seen it defined: $A = \\frac{\\hbar}{\\sqrt{2m}} \\frac{d}{dx} + W'(x)$, $A^\\dagger = \\frac{-\\hbar}{\\sqrt{2m}} \\frac{d}{dx} + W'(x)$ $V_1(x) = W'(x)^2 - \\frac{\\hbar}{\\sqrt{2m}} W''(x)$, $V_2(x) = W'(x)^2 + \\frac{\\hbar}{\\sqrt{2m}} W''(x)$. It seems like these two definitions will give you completely different Hamiltonians and partner potentials. Could someone comment on if these two conventions are equivalent, and if so, how?"}
{"id":"19911","title":"Nanotech - zero dimensional fullerene","text":"Not really a physics junkie, and I think this is a chemistry question, but since there's no chemistry.stackexchange.com (yet) I think here is the best place. I was just reading up on nanotech and came across the phrase \"can construct 0D fullerenes, 1D graphenes...\". Flags went up at \"0D fullerenes\" and I can't find further information about it in many places. A fullerene is apparently any pure-carbon atomic structure, in various forms, but the zero-dimensional part is throwing me. What is a 0D fullerene?"}
{"id":"91342","title":"Why does the coefficient of performance need to be calculated with $W + Q$?","text":"> How much work must a heat pump with a COP of 2.50 do in order to extract > 1.00 MJ of thermal energy from the outdoors (the cold reservoir)? The first formula that came into my mind after reading this question is the $$COP=\\frac{\\lvert Q(\\text{cool})\\rvert}{\\lvert W\\rvert}$$ However, when I used the equation above, the answer that I got was 0.4 MJ instead of the answer given which is 0.67 MJ. And then I tried various other ways to try and get the 0.67 MJ and only when I used this equation that I finally get the answer 0.67 MJ. $$COP=\\frac{\\lvert W+Q(\\text{cool})\\rvert}{\\lvert W\\rvert}$$ What I don't understand is that why do we need to add $W$ with $Q$? Or is it not the correct way after all? Would somebody explain this to me, please?"}
{"id":"68973","title":"What sort of thermodynamic process is this?","text":"I have to calculate the work done in this thermodynamic cycle: LINK. ![enter image description here](http:\/\/i.stack.imgur.com\/9l79q.jpg) But the bottom bit of the cycle is something I've never seen before. I guess the curve in $pV$ space is given by $$\\left( V- \\frac{V_2 -V_1}{2} \\right) ^2 + \\left( p-p_1 \\right) ^2 = \\left( \\frac{V_2 -V_1}{2} \\right) ^2$$ Which is physically meaningless because of units..."}
{"id":"132567","title":"Is there any difference between massless Dirac fermions and Weyl fermions?","text":"In graphene we call the low energy excitations around the Dirac point Dirac fermions, which are massless. Is this just by convention or is there any further differences between massless Dirac fermions and Weyl fermions"}
{"id":"109551","title":"W + jets at NLO","text":"I would like to calculate $pp \\rightarrow W (\\rightarrow \\ell \\nu)$ + n-jets (n=2, maybe also 3) at NLO, with some cuts and plot some distributions. I used MadGraph extensively for LO processes and I was wondering how reliable it is at NLO? Does anyone know any reference comparing multiple NLO codes? I have tried MCFM and I am not very experienced with it but the calculation seems to take a lot of time (obviously since it's NLO). Is there a way to make it run in parallel? I know that you can separate the real emission and virtual parts and then combine them later, but that doesn't seem to give a huge speed increase. Any help is appreciated. Thanks."}
{"id":"109556","title":"Does the induced charge on a conductor stay at the surface?","text":"My textbook says that when a conductor is placed in an electric field, the electrons in it realign so that the net electric field inside the conductor is zero. There isn't a proof for this. It merely states that the electrons drift opposite to the applied electric field and create an opposite electric field inside the conductor canceling the external one. I imagine Gauss's Law would provide a simple proof. If we try enclose a point inside the conductor in a gaussian surface, and note that the enclosed charge is zero, so the flux through the surface is zero and so is the electric field at the surface. When I looked back at my reasoning, I realized that I had assumed that the charges induced on the conductor always stay at the surface, which I had no way of proving. How do I prove this formally?"}
{"id":"29010","title":"How can a Photon have a \"frequency\"?","text":"I picture light ray as a composition of photons with an energy equal to the frequency of the light ray according to $E=hf$. Is this the good way to picture this? Although I can solve elementary problems with the formulas, I've never really been comfortable with the idea of an object having or being related to a \"frequency\". Do I need to learn quantum field theory to _really_ understand this?"}
{"id":"29019","title":"When a high speed neutrino just misses an old neutron star, why isn't it trapped?","text":"Suppose a neutrino is seen travelling so fast that its Lorentz gamma factor is 100,000. It races past an old, no longer active neutron star, narrowly missing it. As far as the neutrino is concerned, it is the NEUTRON STAR that is moving at extreme speed, & its mass is 100,000 times larger than 2 solar masses. Therefore, from the speeding neutrino's perspective, the neutron star should appear to be a black hole definitely large enough to trap the neutrino. So how come the speeding neutrino continues its travel right past the old stellar remnant? Is there an agreed name for this question or paradox?"}
{"id":"461","title":"Why is it thought that normal physics doesn't exist inside the event horizon of a black hole?","text":"A black hole is so dense that a sphere around it called the event horizon has a greater escape velocity than the speed of light, making it black. So why do astronomers think that there is anything weird (or lack of anything Inc space) inside the event horizon. Why isn't simple the limit to where light can escape and in the middle of event horizon (which physically isnt a surface) is just a hyper dense ball of the matter that's been sucked in and can't escape just like light. Why is it thought that the laws of physics don't exist in the event horizon?"}
{"id":"92440","title":"Moving Between Degenerate Vacua?","text":"In spontaneous symmetry breaking, moving round the circular valley of Mexican hat potential doesn’t cost energy. These angular excitations are called Goldstone bosons. But doesn't the angular excitation imply that the system moves from one vacuum to the other because all points on the circular valley are degenerate vacua? If Goldstone excitations are like this then how can a symmetry remain broken?"}
{"id":"114802","title":"Are combined masses in space, such as galaxies, considered to be uniform bodies? *In addition, a related question about force","text":"If we were to calculate the force that one galaxy exerted onto another, would we consider the individual masses within the galaxies, or the masses of the galaxies as a whole? Do the individual stars within one galaxy have control over the individual stars within another? Or, do we consider the galaxies to be single bodies, each with it's own mass, and the distance between them as two uniform bodies? If so, could we call galaxy clusters uniform bodies as well? And if a galaxy cluster were to all of a sudden lose a star (hypothetically), would the net force exerted onto the other galaxy cluster be the same if you counted each star individually? As another curiosity, I would like to pose this question: if a galaxy cluster were to lose one fourth of its mass, $2.0\\times10^3$ light years away from the other galaxy cluster's centre of mass, how would the effect be different than if the mass was lost only $0.50\\times10^3$ light years away? What I mean is, the distance would be different, so the force would be different (**(if you can address this as well please)and information would have to travel faster than light for this to occur, yet with a mass loss this great, the other cluster would be sure to notice it almost immediately)."}
{"id":"114805","title":"Does causality alone resolve the mathematical ambiguity of expressing physical systems?","text":"Newton's 2nd law of motion is most often written in the differential form $\\sum F = {dp \\over dt} $ but can also be expressed in an integral form $ p = \\int\\sum F dt $ Each form of expressing the connection between force and momentum is mathematically correct, but if we adhere to causality we are limited to the second expression (Forces lead to momentum (movement) - not vice versa). This limitation is further exemplified when one attempts to simulate the equations in a digital computer. If feedback is involved the first expression can lead to algebraic loops, while the second gracefully evolves simulated motion. Algebraic loops do not represent physical reality (even the fastest of all physical systems are limited by the speed of light!) So other than what is defined to be the independent\/dependent variables, and\/or the operation of either differentiation vs integration, there is nothing in the mathematics that tells us which is the 'correct' formulation of the laws of motion. The presence of differentiation in the mathematics implies prediction - knowing the future and therefore expresses the physical system in a non-causal manner. Rewriting the system in an integral form expresses the system in a causal way. The integral form in a sense reveals the roles of each factor in cause and effect. Are there perhaps other ways (missing equation(s), other types of mathematics) that can help to resolve the ambiguity that calculus can cause? Is $F = ma$ incomplete? Perhaps some expression involving entropy is required for completeness. Physical reality favors memory and shuns prediction. BTW Newton's law just served as a simple example towards my question. The ambiguity can be realized in any other mathematics involving calculus that model physical systems (including Maxwell's equations)."}
{"id":"101140","title":"Distribution of Kinetic energy after an Impact","text":"How does the huge amount of kinetic energy of some fast object like a meteor or rocket etc. gets distributed into various kinds of energy like kinetic, sound, light etc after making an impact with another body. Is there a rule under which this happens or is it TOTALLY random, and if it is random then what kind of random process does it form?"}
{"id":"111321","title":"Can a person at a photon sphere of a black hole decide where the black hole is?","text":"A person at the photon sphere of a black hole will observe the following:   * The black hole surface will cover exactly half of the visible sky (say, the left half), and cosmic horizon will cover the other half.   * Looking in any direction between these hemispheres the person will see oneself.   * The person will observe no centrifugal force however fast he moves. Any object the person thowing to the left will experience the centrifugal force pulling it further left, while any object thrown to the right also will experience the centrifugal force pulling it to the right. Approaching the black hole in such circumstances may look quite like actually escaping it to the crew.   * The space to the right as well the space to the left seemingly includes objects of any length and dimensions, the further objects are, the more they are time-dilated and length-contracted. Is there any way for a spacecraft commander to decide (by an experiment conducted in a small$^1$ amount of time) which hemisphere is the sky and which hemisphere is black hole in such circumstances? \\-- $^1$ Here _small_ means small compared to characteristic time scales of the problem."}
{"id":"34779","title":"On Aharonov–Bohm effect","text":"Aharonov–Bohm effect in brief is due to some singularities in space. In books it's infinite solenoid most of the time, which makes some regions of space not simply connected. What intrigues me is the fact that in real experiment we can't use infinite solenoid. So even if we use one and say that locally it's good approximation it doesn't change the fact that whole space is still simply connected. But the fact is that the effect was experimentally observed. So the question arises - how one should describe this effect in more rigorous manner (or maybe not rigorous but possible in real world)?"}
{"id":"111329","title":"Do Quantum Energy Generators really produce more energy than they consume?","text":"I'm not a science expert, just someone whose interested. I was taught that you cannot produce more energy than the amount put in. I received a thing on facebook asserting that a Quantum Engery Generator can do this, so I'm naturally very cynical about the claims being made on www.collective- evolution.com. However, as someone without technical knowledge, I am unable to critically examine the claims being made. I'd be grateful if someone with more knowledge than me could take a look and let me know. Thank you."}
{"id":"107235","title":"How will a particle with energy less than $V_{\\rm min}$ behave?","text":"Consider e.g. the finite square well: $V = -V_o$ between $x=-a$ and $x=a$, $V=0$ elsewhere Now for scattering states, $E$ must be $> 0$. For normalizable bound states, $E$ must be $< 0$ and $> V_{\\rm min}$ (=$-V_o$ in example). But if a particle in a lab has Energy which $< 0$ and $<V_{\\rm min}$, is it bound or will it scatter? And I don't know why I haven't thought about this before but what does it mean for a photon to have negative energy?"}
{"id":"67900","title":"What precisely, is the string theory landscape in 10 dimensions?","text":"I was reading this Physics.SE thread. The OP said, (changed the last word to type HO) > > > For D = 10, there are 5 vacua. Or maybe it's more correct to say 4, > since type I is S-dual to type HO. This lead me to the 2 closely related questions:   1. When considering the landscape of string theories in some $D$ dimensions, are the S-dual theories considered as the same theory, as suggested by the quote? What about the T-dual theories? I don't think the T-dual theories should, because in that dimension itself, they are different until and unless one dimension is compactified, when they become approximately equal.   2. When it is said that the string theory\/M-theory landscape in $4$ dimensions in $10^{520}$, are the S-dual theories counted as the same? I don't think they are, because there is no classification of this landscape yet."}
{"id":"22100","title":"2d or 1d conduction in this scenario?","text":"There is a rectangular fin attached to a heat exchanger with a base temperature of 350K. The fin has uniform properties and experiencesa uniform heat generation. It also experiences heat transfer with the surroundings through convection and radiation. Dies the fin experience 1D or 2D conduction?"}
{"id":"61507","title":"Gravity in stone","text":"I want to ask is it possible for a stone to have some magnetic power different from power that we see in ordinary metal magnets (something like earth gravity that attract everything). Someone told me that he saw a stone that attract other smaller stone but did not attract metals. Is it practically possible in science?"}
{"id":"2480","title":"Bubbles in zero gravity","text":"What happens if you blow bubbles into a glass of lemonade on the international space station? Since you are weightless in orbit, there's no up, down, left nor right. We define down on the Earth surface as the direction things tend to fall, down. On Earth bubbles rise if blown into a glass of lemonade (with a straw and a cover)."}
{"id":"3986","title":"Newton's Bucket","text":"Newton's Bucket This thought experiment is originally due to Sir Isaac Newton. We have a sphere of water floating freely in an opaque box in intergalactic space, held together by surface tension and not rotating with respect to the distant galaxies. Now we set the box and water to rotate about some axis and we notice that the sphere flattens into an oblate spheroid. How does the water know it’s spinning? NOTE: Newton thought this proved the concept of absolute rotation with respect to a preferred spatial frame of reference. Perhaps these days we can do better, or different?"}
{"id":"35426","title":"What's the meaning of dimensionality?","text":"I got amazed with this answer and as you can see, I had a little talk with joriki. He suggested me some examples such as the 2DEG. I'm stuck with some aspects on dimensionality, see this line: > A two-dimensional electron gas (2DEG) is a gas of electrons free to move in > two dimensions, but tightly confined in the third. In mathematics, it seems that there could be $n$ dimensions, but how's it on the \"real world\"? I thought that such $n$-dimensional objects on the real world could exist on the absence or presence of a specific dimension, but at least in this wikipedia article, it seems that this object _have_ the third dimension - at least I conceive \"confined\" as something different of \"abscence\". Can you clarify this to me? I am open to both answers and references on this topic, I just hope that asking both is not being too broad."}
{"id":"73257","title":"Why is there a sharp cracking sound when we break a plastic ruler?","text":"Why is there a sharp cracking sound when we break a plastic ruler? The question appears to be simple and common but I need a apropriate answer."}
{"id":"73254","title":"Wave Particle duality because of discrete time?","text":"If time is discrete, such as the Planck's length, would the transition from one frame of time to the next explain why it appears matter changes from a particle to a wave? During that infinitely small space between each frame we can not measure the particle and it appears as a wave?"}
{"id":"93428","title":"Can matter leave the cosmic horizon?","text":"Cosmic horizon in the de Sitter space is a sphere, centered at the observer with finite radius where the red shift due to cosmic expansion becomes infinite. Given that no information can be transmitted from behind the cosmic horizon, I wonder whether any matter can ever pass behind the event horizon, because this will give rise to an information paradox, similar to the black hole information paradox. The matter approaching the cosmic horizon seems to experience time dilation, so will that time dilation prevent it from crossing the horizon? To work around the paradox, one have to postulate that the information about outgoing matter should be encoded in the de Sitter radiation."}
{"id":"27166","title":"Chern-Simons theory","text":"In Witten's paper on _QFT and the Jones polynomial,_ he quantizes the Chern- Simons Lagrangian on $\\Sigma\\times \\mathbb{R}^1$ for two case: (1) $\\Sigma$ has no marked points (i.e., no Wilson loops) and (2) $\\Sigma$ has marked points and each point has attached a representation of the gauge group. In case (1), Witten shows that the vector space should be the space of holomorphic sections of a determinant line bundle over the moduli space of flat connections. For the second case he states that the vector space should be the $G$-invariant subspace of the tensor product of all the representations associated to the marked points; by this I mean, if $\\Sigma$ has $r$ marked points and each point has a rep $R_i$ then the quantum Hilbert space is $(\\bigotimes_{i=1}^r R_i)^G$. Does anyone know how to interpret this second case in terms of sections of some bundle? I mean, shouldn't the second case reduce to the first when you remove the marked points? Also, Witten states, immediately following case (2), that in the presence of no marked points the quantum Hilbert space is 1-dimensional. How can one see that from the formula for the quantum Hilbert space, $(\\bigotimes_{i=1}^r R_i)^G$?"}
{"id":"55598","title":"Estimate number of hairs on human head","text":"A technique of vital importance at all levels in physics is estimation. This is obvious from the first chapter in any introductory physics textbook, but is also related to the working physicist. Checking orders of magnitudes during research presentations is common practice - I've seen many good questions with good followup answers that started with \"If I estimated that value I would get something much different\". In general, the actually result is not the interesting thing - it's what individual things will affect the result. There are even famous examples of this: Fermi's piano tuner problem, and the Drake equation. Apparently, Fermi was so good at this that he estimated the size of the Trinity nuclear bomb test to within a factor of 2 (see the wikipedia article for a discussion of that). In this spirit, I would like to see someone try and estimate the number of hairs on the human head. The answer must include the basic assumptions so we can see where the major unknowns lie, and the best answer is one which requires no specific knowledge"}
{"id":"60486","title":"Why isn't this capacitor charging?","text":"Let's say you have a parallel plate capacitor and you connect one plate to the positive terminal of a battery and the other plate to the negative end. So this is like a static situation, you have a large positive charge (the cathode in the battery, which keeps recharging)) connected to a metal plate, and a big negative charge (the anode) connected to another metal plate. From what I've learned about static situations, the charges should equalize so that the plate connected to the positive charge should have half the charge of the cathode, and the same for the negative side. However, my physics teacher said this would not happen and he even did an experiment to show me. However, I don't understand why my reasoning doesn't work. Could someone explain to me why I am wrong? Thanks!"}
{"id":"56024","title":"Galilean invariance of the Schrodinger equation","text":"I am only asking this question so that I can write an answer myself with the content found here: http:\/\/en.wikipedia.org\/wiki\/User:Likebox\/Schrodinger#Galilean_invariance and here: http:\/\/en.wikipedia.org\/wiki\/User:Likebox\/Schrodinger#Galilean_invariance_2 I learned about these pages in a comment to this answer by Ron Maimon. I think Ron Maimon is the original writer of this content. This is creative commons, so it's ok to copy it here. It is not on any textbook on non-relativistic quantum mechanics that I know of, and I thought it would be more accessible (if to no one else, at least for myself) and safe here. I hope this type of question is not in disagreement with site policy."}
{"id":"64101","title":"Is there any phenomenon in physics which is sensitive to irrational numbers?","text":"We can measure only rational numbers by our scale. Here is an example where irrational numbers does makes sense. If so then this question may have some theoretical importance. Is irrational numbers important for physics?"}
{"id":"36228","title":"Boundary conditions for crystals","text":"As students on solid state physics, we are all taught to use the periodic boundary condition, taking 1D as an example: $\\psi(x)=\\psi(x+L)$ where $L$ is the length of the 1D crystal. My question is:   1. Why this boundary condition is acceptable? Only because we think we are dealing with the bulk, and the surface are not relevant too much?   2. How about other boundary condations, such as zero boundary condtion, grain boundary condition etc? If ignoring their complexity, can they get the same answer as the periodic boundary condition?   3. How about other artificial boundary conditions, for example, if I let $\\psi(x)=2 \\psi(x+L)$, could I get the same energy band?"}
{"id":"128245","title":"How was Newton able to guess that gravitational force is inversely proportional to distance squared?","text":"This question is puzzling me since I learnt about the gravitation law in school. Why did Newton guess\/assume that gravitational force is inversely proportional to the square of distance? Did he verify that experimentally? (I remember hearing that the first experimental verification of the law of gravitation was after Newton's death.) If the answer to the above question is no, is it for example more plausible to suppose that $F\\propto1\/r^2$ than to suppose that $F\\propto1\/r^4$? Did Newton carry out a thought experiment that makes $F\\propto1\/r^2$ a plausible guess? So in summary: Why did Newton choose exponent of $-2$ instead of any other exponent? Was it a guess that depended on pure luck or an educated guess?"}
{"id":"74105","title":"How does one get these definitions of the energy momentum tensor?","text":"I was just reading a book - Mirror Symmetry by Clay Mathematics Institute, and on Page 402 of the book, the writer says that energy momentum tensor is defined classically by $$\\delta S = -\\frac{1}{4 \\pi} \\int \\sqrt{h} d^2 x \\delta h^{\\mu\\nu}T_{\\mu\\nu}$$ and quantum mechanically by $$\\delta_h \\langle O\\rangle = \\left\\langle O \\frac{1}{4\\pi} \\int \\sqrt{h} d^2 x \\delta h^{\\mu\\nu}T_{\\mu\\nu}\\right\\rangle$$ **Here is my question:** I know that by Einstein-Hilbert action one can get $T_{\\mu\\nu} = 2 \\frac{\\delta \\mathcal{L}}{\\delta h^{\\mu\\nu}} $. But how can I get the exact form of $\\delta S$ as above? Or $\\delta_h\\langle O\\rangle$."}
{"id":"87220","title":"Are $G$, $F$ and $H$ (thermodynamics potentials) extensive quantities?","text":"Internal energy $E$ is an extensive quantity for most systems. But energy extensivity is not valid in systems with long-range interactions, like gravity (e.g. in astrophysical systems). For extensive systems: $$E=TS+\\bf J. \\bf x +\\bf{\\mu}.\\bf N\\tag{1}$$ where $\\bf J$ and $\\bf x$ are intensive and extensive thermodynamic variables. (like pressure and volume, respectively) Some other thermodynamic potentials like $G$, $F$, and $H$ are defined as : $$F=E-TS$$ $$G=E-TS-\\bf J.\\bf x$$ $$H=E-\\bf J.\\bf x$$ For extensive systems, physical meanings are obvious, as is written in Wikipedia: > we can say that ΔU is the energy added to the system, ΔF is the total work > done on it, ΔG is the non-mechanical work done on it, and ΔH is the sum of > non-mechanical work done on the system and the heat given to it. Apparently these interpretations are deduced from equation $(1)$, which doesn't hold in non-extensive systems. So it seems that these potentials are ,inherently, extensive quantities, and have no meaning in non-extensive systems. Do these potentials have a physical meaning (if applicable at all) in non- extensive systems like galaxies?"}
{"id":"87222","title":"Definition and Measurment of Energy and Heat in Thermodynamics","text":"The concept of work is clear from mechanics, where $dW=F\\cdot dx$, leading to $dW=PdV$. Also different forms of energy are defined in mechanics, e.g., $E=\\frac{1}{2}m v^2$, etc. How are the energy and heat defined and measured in thermodynamics? Because once we define them independently, we can state the first law of thermodynamics, $d E=\\delta Q+\\delta W$."}
{"id":"36220","title":"Why does your reflection suddenly turn upside down while slowly walking away from a curved mirror?","text":"Many people should have realised, when looking into a concave curved mirror (or even a rather reflective spoon in that fact) at a close distance, you will see a slightly distorted reflection. But as you move further away, the image will suddenly become upside-down. why is this?"}
{"id":"39685","title":"energy production in the nuclear fission","text":"![enter image description here](http:\/\/i.stack.imgur.com\/FEj0S.jpg) according to this image of nuclear fission, before fisson reaction, the total mass of uranium and the slow neutron is 236 amu. and after adding the masses of Barium, krypton and the 3 neutrons after the fission reaction, you will again get 236 amu. Question: if we get back all the mass, where did energy come from? is it the binding energy that is released?"}
{"id":"5483","title":"Is Mach's Principle Wrong?","text":"This question was prompted by another question about a paper by Woodward (not mine). IMO Mach's principle is very problematic (?wrong) thinking. Mach was obviously influenced by Leibniz. Empty space solutions in GR would result in a Minkowski metric and would suggest no inertia. Mach's principle seems incompatible with GR. Gravitational waves could also be a problem. I had thought that papers like one by Wolfgang Rindler had more or less marginalised the Mach Principle, but I see lots of Internet discussion of it. Is it correct? Wrong? Is there evidence? (frame dragging experiments)? Let's use this definition from ScienceWorld.Wolfram.com: > _In his book The Science of Mechanics (1893), Ernst Mach put forth the idea > that it did not make sense to speak of the acceleration of a mass relative > to absolute space. Rather, one would do better to speak of acceleration > relative to the distant stars. What this implies is that the inertia of a > body here is influenced by matter far distant._"}
{"id":"41243","title":"What came first, Rice Crispy or \"Snap,\" \"Crackle,\" and \"Pop\"?","text":"The fourth, fifth, and sixth derivatives of position are called \"Snap\" \"Crackle\" and \"Pop\". What came first, the rice crispy characters, or the physics units?"}
{"id":"41249","title":"Alpha Centauri Bb: Comparing astrometric precision vs doppler precision","text":"How do you compare astrometric precision vs doppler spectroscopic precision? In particular will the Gaia spacecraft to be launched next year be able to confirm the existence of Alpha Centauri Bb? I.e. is Gaia's astrometric precision at least as great as the precision of the doppler spectroscopy that was used to discover the planet?"}
{"id":"113415","title":"How can the electromagnetic stress energy tensor be restricted to flat space-time","text":"The Wikipedia article describing the electromagnetic stress energy tensor seems to suggest that this tensor can only be defined in flat space-time. How is it possible to define an electromagnetic stress energy tensor this way since any available electromagnetic energy\/momentum must render the space-time curvature nonzero? How in practice would someone extract useful information with this stress energy tensor?"}
{"id":"81597","title":"Energy density in Magnetic fields?","text":"I don't understand what does magnetic field energy density stand for? The possible work that the magnetic field can do? So we can calculate the amount of \"work\" it can do just from the energy density? Example: A magnetic field is created by a solenoid, there is a energy density of 10J\/m3. From that can I derive the magnitude of magnetic force using work = F*d Thus, it would be F = W\/d?"}
{"id":"123376","title":"Synchronization phenomenon: A simple explanation?","text":"Being from a mathematical background, physicists' intuitive arguments always seemed challenging for me to follow. I am currently reading a book called \"Synchronization: A Universal Concept in Nonlinear Sciences\". The book is well-written but from time to time, I have hard time following the intuitive arguments. In its most simple case the authors discuss the effect of a periodic external force on a self-sustained quasilinear oscillator. I have to emphasize that these are arguments in chapter 3 of the book. Things get more formal in chapter 7. The only problem is that in chapter 7 there is no argument about potentials anymore. In chapter 3 the argument mainly relies on a few parameters, $\\overline\\phi_e$ being the initial phase of the forcing oscillator which has the following phase relation $\\phi_e=\\omega t+\\overline\\phi_e$ and the driven oscillator is based on $x(t)=A\\sin(\\omega_0 t+\\phi_0)$ and the external force varies based on $\\epsilon \\cos(\\omega t+\\overline\\phi_e)$. The following points are the main things that I don't understand:   1. What does potential mean?(I know that potential function defines an invariance most of the times, but how it is defnied for oscillators case?    2. What are big parallel vectors and the small ones in figure 3.3(a) representing?   3. Why they phase locking take place in $\\phi_0$   4. Why point 1 is stable and point 2 is not? I imagine stability means the point with minimum level of potential. I hope the answer to this question becomes clear if I get an answer for the first question. The following figures might be helpful for this clarification: ![enter image description here](http:\/\/i.stack.imgur.com\/pHyPj.png) ![enter image description here](http:\/\/i.stack.imgur.com\/AH8ju.png)"}
{"id":"6856","title":"Readings of the detectors at Japan and Izu-Bonin-Mariana Trenches","text":"Given the relevance of this subduction system, I would expect that a wide range of detectors (temperature, vibration, seismometers, whatever) are deployed in the depth of these trenches. What would be the canonical source at which one could access the readings of these detectors?"}
{"id":"86577","title":"Properties of material affecting magnetic refrigeration","text":"I am reading about magnetic refrigeration at the moment - I see that two factors affecting the applicability of the material for use in magnetic refrigeration is that they must react strongly (i.e the magnetic moments) with temperature at the operational temperature of the refrigerator and that to maximize cooling, necessarily a small specific heat. My book says there are other properties that could be advantageous for this application, but I am not sure what they could be. Any ideas? Many thanks."}
{"id":"132917","title":"Direction of current in concentric cylinders","text":"Example 7.2 in David Griffiths E & M book (3rd edition) has a side view of 2 concentric cylinders, with smaller radius $a$ and larger radius $b$. The region in between $a$ and $b$ has conductivity $\\sigma$. \"If they are maintained at a potential difference $\\textit{V}$, what current flows from one cylinder to the other for a given length $L$?\" The E field is pointing radially outward along $\\textit{s}$. My question is: what direction is the current? Do electrons flow in the opposite direction of an E field? If so, does that mean the current is flowing radially inward, along $\\textit{-s}$, from $b$ to $a$?"}
{"id":"18287","title":"The implication of anti-commutation relations in quantum mechanics","text":"All the textbooks I saw are very clear about the implications of commutating operators in quantum mechanics. However, much less is said about anti- commutation relations. Does it have a general implication if two operators anti-commutate?"}
{"id":"33686","title":"Why does physics not attempt to explain observations in addition to describing them?","text":"Physics are there to describe observations of things' behaviours to recreate them or basically use nature for your goals. Which is not enough. Why do physicists not try to explain, why things are the way they are? In other words: > Why does it work? instead of > How does it work?"}
{"id":"18289","title":"Can we describe the classical laws of physics in a frame-of-reference-independent way?","text":"First of all, I am not a physicist, so I cannot guarantee things I say will make sense. I will try my best, though. In classical mechanics we have the notion of inertial frame of reference. If my understanding is correct, such frames are exactly those for which Newton's laws of motion have the usual form. I believe I have also heard, that they are the frames that are \"moving at a constant speed\" with respect to some distinguished point in space. So we have to be careful whether our frame is accelerated or not. Which makes me wonder: > Does there exist a set of physical laws (of same descriptive power Newton's > laws have, but perhaps expressing relations in some other set of physical > quantities) such that the laws from this set look the same, no matter how > the frame of reference moves? > > Does such a set of physical laws exist if we restrict to polynomially moving > frames of reference? (By this I mean that the motion of the frame is > described by a polynomial in $t$, where $t$ is time.) I imagine this could simplify some calculations."}
{"id":"47971","title":"Empirical meaning of relativity in the context of QM","text":"In special relativity an event E is mapped to coordinates (x,t) in one inertial frame, and to coordinates (x',t') in another, and SR provides the relation between (x,t) and (x',t'). What is the empirical content of this theory (e.g. how would you operationally test it)? Since there are two frames, two observations are required - one that measures the (x,t) of E in one frame, and another that measures (x',t') of E in the other frame. Classically this presents no problem, since we can assume an ideal measurement exists that would not disturb whatever is being measured (the event E), and so two of these measurements can be made in rapid succession, one in each frame. But in QM, the result of measurement is intrinsically tied to the measurement apparatus and also the very act of measurement affects what is being measured. So when you observe an event E (say, particle position x at time t) in the first frame, using the apparatus in that frame, you can no longer observe that same event E in the second frame. Your second observation will at most be measuring the position of the particle after it has been observed in the first frame. So, in effect, a single event E does not have well defined (empirically\/operationally) coordinates in all frames of reference at once. How then can we even talk about a relation between (x,t) and (x',t'), when these are not well defined?"}
{"id":"31686","title":"Can colliders detect B violation?","text":"I think there is some theoretical uncertainty whether high-energy collisions can violate B. It is known that at high temperature (higher than the Higgs scale) you violate B by SU(2) instantons. But in a situation where you have a very energetic 2-particle collision at arbitrarily high energy, I am not sure if there is a non-negligible probability of producing a Baryon violating configuration. I don't know of any calculation of B violation expected in accelerators, although there might be an argument that it should be very close to zero, because of the non-thermalizing nature of 2-particle collisions. Can you detect standard model B violation in colliders? Does LHC look for rare B violating events, or would such rare events be indistinguishable from a proton or neutron escaping undetected?"}
{"id":"13957","title":"Solid objects inside the event horizon - can they remain \"solid\"?","text":"So, once something is inside a black hole's event horizon, it can only move towards the center. This is fine for a point-object. But 3D solid objects rely on molecular forces to stay in one piece. These forces act in all directions inside the solid. But there could not be any interaction between atoms at different distances from the singularity, right? So, what happens to a solid (let's assume a crystalline lattice, for simplicity) hypothetically placed inside the event horizon? Does it get cleaved in ultra-thin concentric layers, centered on the singularity?"}
{"id":"14239","title":"Functional Derivative of Convolution","text":"How to carry out the following functional derivative? $$\\frac{\\delta F}{\\delta n(r)}$$ where $$F=\\int dr n(r) \\int C(|r-r'|) n(r') dr'$$ is it simply: $$2 \\int dr' C(|r-r'|) n(r')$$?"}
{"id":"5523","title":"Why do car keys have longer range when held next to your head?","text":"I first saw this on Top Gear, and assumed it was a load of rubbish, but today I tried it out and it actually works, if you walk away from the car continuously unlocking\/locking it until it stops, and go a bit further away so that the car will not un\/lock. Then out the car keys next to your head and point them directly at the car it will un\/lock. How does your head\/brain\/etc increase the range of the keys? And why does it only work next to your head if you point the keys directly at the car, when you do not have to point towards the car when in normal range?"}
{"id":"89751","title":"Does coffee with cream cool slower than black coffee?","text":"At Modernist Cuisine, Nathan Myhrvold states that \"Coffee with cream cools about 20% slower than black coffee\". He then gives three reasons, two of them related to radiation and the third one to the influence of viscosity on the evaporation rate. My question is twofold:   1. **Is his conclusion correct?**   2. **Are those reasons relevant?** The question is somewhat open-ended, because \"20% slower\" could refer to the instantaneous cooling rate at $t=0$ or to the time needed to reach some lower temperature. It would be great to consider both parameters in the answer (will they behave the same?)"}
{"id":"99229","title":"Commutator of parity and Hamiltonian operators under even potential function","text":"I need to show what is $[H,P]$ where $H$ is the Hamiltonian and $P$ the parity operator. $V(\\underset{\\sim}x) = V(-\\underset{\\sim}x)$ in this case. I start off with $$ \\langle \\underset{\\sim}x|HP|\\psi\\rangle = \\langle \\underset{\\sim}x|(\\frac{p^2}{2m}+V)P|\\psi\\rangle = \\langle \\underset{\\sim}x|(\\frac{p^2}{2m}+V)P|\\psi\\rangle = \\langle \\underset{\\sim}x|\\frac{p^2}{2m}P|\\psi\\rangle+\\langle \\underset{\\sim}x|VP|\\psi\\rangle $$ and since $\\langle \\underset{\\sim}x|V = V(\\underset{\\sim}x)\\langle \\underset{\\sim}x|$ (is this step valid?) and $ \\langle \\underset{\\sim}x|P = \\langle -\\underset{\\sim}x|$, the above equation becomes $$ \\langle \\underset{\\sim}x|HP|\\psi\\rangle = \\langle \\underset{\\sim}x|\\frac{p^2}{2m}P|\\psi\\rangle + V(\\underset{\\sim}x)\\psi(-\\underset{\\sim}x) $$ Similarly I have $$ \\langle \\underset{\\sim}x|PH|\\psi\\rangle = \\langle \\underset{\\sim}x|P\\frac{p^2}{2m}|\\psi\\rangle + V(-\\underset{\\sim}x)\\psi(-\\underset{\\sim}x) = \\langle \\underset{\\sim}x|P\\frac{p^2}{2m}|\\psi\\rangle + V(\\underset{\\sim}x)\\psi(-\\underset{\\sim}x) $$ Taking the difference of the two, I find that $$ \\langle \\underset{\\sim}x|HP-PH|\\psi\\rangle = \\langle \\underset{\\sim}x|\\frac{p^2}{2m}P-P\\frac{p^2}{2m}|\\psi\\rangle = -\\langle \\underset{\\sim} x|\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x_i^2}P|\\psi \\rangle +\\langle -\\underset{\\sim} x|\\frac{\\hbar^2}{2m}\\frac{\\partial^2}{\\partial x_i^2}|\\psi\\rangle $$ which I had trouble evaluating. Any hints?"}
{"id":"30081","title":"Was Sun a part of a binary system?","text":"I've read that many stars and other celestial bodies are found to constitute binary systems where the two bodies spin around each other. But our Sun is one of the exceptions. Could it be possible that the Sun was also a part of a binary system long ago ?"}
{"id":"89287","title":"Can exergy and exergy destruction be understood through thermodynamical and\/or statistical-mechanical principles?","text":"My textbook _Fundamentals of engineering thermodynamics, Moran and Shapiro_ states: > The exergy is the maximum theoretical work obtainable for an _overall_ > system consisting of a system and the environment as the system comes into > equilibrium with the environment (passes to the dead state). The exergy of a system (which I'll call $\\chi$), at a specified state is given by the expression: $$ \\chi = (U-U_o)+p_o(V-V_o)-T_o(S-S_o) +KE +PE$$ Where $S_o, U_o$ etc. denote the dead states of a system. An Exergy change is : $$\\chi_2 - \\chi_1 = (U_2-U_1)+p_o(V_2-V_1)-T_o(S_2-S_1) +(K.E._2-K.E._1) +(P.E._2-P.E._1)-^?T_o\\sigma_o$$ Where $\\sigma_o$ is the \"entropy production\" and I'm not sure if it belongs there or not. The equation is clearly related to the Gibbs free energy, but the gibbs free energy is for an isothermal and isobaric process _(natural variables are T and P)_. It seems obvious to me that if a system is passing to the dead state it's temperature and pressure could change, for example: as a cup of coffee comes into equilibrium with the environment it's temperature will eventually be the same as the temperature of the \"reservoir\".(Note that these equations can also be expressed as rates) I haven't really been satisfied with the depth of the physics in my textbook and have had trouble finding good conceptual and mathematical explanations of exergy and exergy destruction. Can someone present a thermodynamic or statistical-mechanical description of exergy and exergy destruction? The intuitive mechanical engineer explanation I've come across is that it is about the economics of energy use ( _for example: heating your house with a radiator would be a poor use of exergy_ ), but I'd like something more mathematically rigorous if possible. Online references or a generalized example would be great too."}
{"id":"89282","title":"What is the area in Faraday's law if we have only a piece of metal moving in a magnetic field?","text":"If a piece of metal of length $l$ is moving with a speed $v$ in a region where there is a uniform magnetic field $B$ perpendicular to it, there will be a potential difference across its terminals equal to $lvB$ which is known as motional EMF. This can be shown and understood in terms of magnetic and electric forces on the free charges in the metal. How can one calculate such EMF from Faraday's law, $\\displaystyle\\mathcal{E} = \\left|\\frac{d\\Phi_B}{dt}\\right|$? (where $\\Phi_B$ is the magnetic flux $\\int \\bf{B}\\cdot d\\bf{a}$) (If $B$ is not changing, then the change in the magnetic flux must be due to change in an area, but the area of what? What are the boundaries of this area?)"}
{"id":"132001","title":"Tight binding operators in 2D lattice system","text":"I have a very naive problem about lattice system, how to translate common operators defined in the bulk ($\\hat{x}$, $\\hat{p}$...) into their lattice analogues. In a single- band tight-binding description, the hopping Hamiltonian is taken in the form $$\\hat{H}_0=-J(\\sum_{m,n} a^{\\dagger}_{m+1,n}a_{m,n}+a^{\\dagger}_{m,n+1}a_{m,n}+H.c)$$ Then in the limit of lattice constant $a\\rightarrow0$, how can we recover the usual kinetic energy term? $$\\hat{H}_0=\\frac{1}{2m^*}(\\hat{p}_x^2+\\hat{p_y}^2),$$ with a effective mass $m^*=1\/(2Ja^2)$."}
{"id":"132007","title":"Quantum Entanglement - an illusion based on a wrong assumption?","text":"Almost all resources I've read about Quantum Entanglement speak about how 'amazing' it is that two entangled particles are bound over any distance, and that the state of one particle determines the state of the other. I believe that there is possibly a profoundly wrong assumption here that doesn't get addressed properly. The assumption is that when the state of one particle is observed, it is then, and only then, determined, exiting it's super position state (and thus the state of the other particle also being determined, over any distance, instantly - which is where most of the focus lies when talking about entanglement). But here is my problem - why is the assumption that the state of the first particle is being _determined_ on observation so easily accepted ? It seems to me much more logical and absolutely free of unexplained voodoo that:   1. the particles are entangled (have opposite symmetrical states).   2. The state of the first and second particles is _unknown_ and unknowable until observed, but is predetermined from the moment of the particle's inception.   3. Upon observation, nothing in the particle changes, except that our knowledge of the first particle's state leaves a \"super-positioned\" state into a specific one.   4. Basic logical consistency dictates that we \"instantly\" know the state of the second particle, without the need for \"spooky action at a distance\". So I guess that my basic premise is - it seems much more reasonable that our (my?) understanding of superposition is wrong, rather than that particles exchange state information instantly across any space. Please help me understand where I am wrong ?"}
{"id":"47350","title":"Why do cars gain lift while going at a high speed?","text":"I've been researching the physics involved with spoilers and I've just been turning up the same information about spoilers giving the back wheels of a car more traction by acting as an up side down airfoil. So my question is, what exactly causes the car to apparently gain lift while moving quickly, so as to merit the use of a spoiler, and is this entirely due to the aerodynamics of the vehicle in question? Edit: Clarity of question."}
{"id":"6545","title":"Why are Saturn's rings so thin?","text":"Take a look at this picture (from APOD http:\/\/apod.nasa.gov\/apod\/ap110308.html): ![Saturn's rings at APOD](http:\/\/apod.nasa.gov\/apod\/image\/1103\/titansaturn_cassini_900.jpg) I presume that rocks within rings smash each other. Below the picture there is a note which says that Saturn's rings are about 1 km thick.   Is it an explained phenomenon?"}
{"id":"95572","title":"Resampling spectral data?","text":"I want to use the CIE CMFs (Color Matching Functions) ![CMFs](http:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/8\/8f\/CIE_1931_XYZ_Color_Matching_Functions.svg\/446px- CIE_1931_XYZ_Color_Matching_Functions.svg.png) provided here: http:\/\/www.cvrl.org\/cmfs.htm With data from my spectrometer in 10 nm increments. However, the CMFs are only provided at 5 nm increments. What would be the best way to do this? Would it be okay to resample the spectral data with a linear interpolation? Any ideas?"}
{"id":"18242","title":"What's the Standard Model width of a 125 GeV Higgs?","text":"There's a fairly broad mass spread in the new results out of Atlas and CMS. I'm curious how this fits with the expected SM width."}
{"id":"98010","title":"Noether Charge For Scalar Fields Under Lorentz Transformations","text":"The conserved charge associated with the Lorentz transfomation of a **scalar field** is given by $Q^{\\alpha\\beta}=\\int d^3x\\frac{1}{2}(x^\\alpha T^{0\\beta}-x^\\beta T^{0\\alpha})$. The quantities $Q^{ij}$ is recognized as the components of orbital angular momentum but what physical quantities does $Q^{0i}$ correspond to for $i=1,2,3$? This cannot be intrinsic angular momentum or spin because we are considering a scalar field. What is the significance of these quantities?"}
{"id":"74836","title":"When is temperature not a measure of the average kinetic energy of the particles in a substance?","text":"I had always thought that temperature of a substance was a measure of the average kinetic energy of the particles in that substance: $E_k = (3\/2) k_bT $ where $E_k$ is the average kinetic energy of a molecule, $k_b$ Boltzmann's constant, and $T$ the temperature. (I'm not sure of the 3\/2 coefficient.) Then I heard from several folks that this is a simplistic notion, not strictly true, but they didn't explained what they thought was flawed with this idea. I'd like to know what (if anything) is objectionable about this idea? Is it that the system must be macroscopically at rest? Is it that it ignores the quantum mechanically required motion of particles that persists at low temperatures? When is it not valid?"}
{"id":"74834","title":"The Einstein-Klein-Gordon (EKG) equations","text":"I am a little confused about a few papers I read on the Einstein-Klein-Gordon (EKG) equations. From what I understood one takes the energy-stress-tensor of the scalar field: $$T_{\\mu\\nu } = −\\partial_\\mu \\varphi ∂_\\nu \\varphi − \\frac12 g_{\\mu\\nu}∂_\\alpha\\varphi∂_\\alpha\\varphi − V(φ )$$ $$V(φ) = −\\frac12 (mφ)^2 + \\fracκ4φ^4$$ Where $κ$ is the usual self-interaction coupling constant . Then this stress-tensor is plugged into Einstein's equation and solved usually with the Schwarzschild or other convenient metrics. Now from what I know about QFT, isn't $φ$ an operator that maps the Hilbert space $H$ of particle states to $H$ itself? Doesn't that make the components of the energy-stress-tensor observables and thus operators as well? If so, then how can one equate the components of the Einstein tensor (which are purely geometric tensor fields) to operators? I once read that people (as of today) usually plug in < $T_{μν} $ > in Einstein's equation, but in the papers I read they directly used the operator itself and not it's expectation value. I also wondered how people plug the Maxwell stress tensor into Einstein's equations in a similar way? I get that the EM and scalar field are real valued fields, but shouldn't we use the expectation values in Einstein's equation? What am I missing? Here is a link to one such paper: http:\/\/arxiv.org\/abs\/0805.3211"}
{"id":"12703","title":"Maximal Gravity","text":"I found this interesting problem in Introduction to Classical Mechanics with Problems and Solutions by David Morin: > Given a point $P$ in space, and given a piece of malleable material of > constant density, how should you shape and place the material in order to > create the largest possible gravitational field at $P$? Any ideas?"}
{"id":"88152","title":"What does it mean to be stationary?","text":"I'm looking for a simple answer. What do we regard a stationary. Do we mean an object that is not moving noticeable from the viewers perspective because then a parked car would be considered stationary, but (assuming it's on earth) the earth is rotating and those at the circumference where the spin is greatest are traveling 1 038 mph and the earth itself is moving with the Milkyway at 600 000 mph. I have a bunch of other questions but I rather not put them all in one post, but do we really account for theses other speeds or do we ignore them."}
{"id":"88151","title":"What's the typical material used as a low-k ILD in semiconductors?","text":"Semiconductor companies (namely IBM, Intel, TSMC and etc.) may use different low-k materials as inter-layer dielectric (ILD) in their fabrication process. I'm wondering what is the most typical one. Where can I find their thermo- electro-mechanical material properties? I need these information for FEM simulations."}
{"id":"88158","title":"If nothing weighs something then doesn't that mean there is mass associated with spacetime?","text":"Lawrence M. Krauss says, \"If you removed all of the particles, all of the radiation,absolutely everything from space and all that remained was nothing that nothing would weigh something.\" What weighs something has mass. If you remove everything else then all that is left is spacetime. Meaning, there is mass associated with spacetime. Mass physically occupies three dimensional space. A particle moving through the mass associated with spacetime would displace the mass analogous to the bow wave of a boat. If mass is associated with spacetime then wouldn't it be the mass associated with spacetime which waves in a double slit experiment?"}
{"id":"81754","title":"Does Hamilton Mechanics give a general phase-space conserving flux?","text":"Hamiltonian dynamics fulfil the Liouville's theorem, which means that one can imagine the flux of a phase space volume under a Hamiltionian theory like the flux of an ideal fluid, which doesn't change it's volume. But is it possible to reproduce every phase space conserving flux with an appropriate Hamiltonian? So can I simply imagine the entity of all possible Hamiltonian dynamics as all possible phase space conserving fluxes? Or are Hamiltonian dynamics a special case for phase space conserving fluxes? If they are a special case, what would be an example for a phase space conserving flux for which there is no Hamiltonian that can produce it?"}
{"id":"69550","title":"Is the Ground State in QM Always Unique? Why?","text":"I've seen a few references that say that in quantum mechanics of finite degrees of freedom, there is always a unique (i.e. nondegenerate) ground state, or in other words, that there is only one state (up to phase) of the Hamiltonian with the minimum eigenvalue. My questions: Q1) Is it true? Q2) Under what condition is it true? Q3) I can easily construct a Hermitian operator, in a finite dimensional space, which has two lowest eigenvectors. For example, if $ \\left\\\\{ {\\left| a \\right\\rangle ,\\left| b \\right\\rangle ,\\left| c \\right\\rangle } \\right\\\\} $ is an orthonormal basis of a 3-dimensional Hilbert space, define a Hamiltonian $H = 1 \\cdot \\left| a \\right\\rangle \\left\\langle a \\right| + 1 \\cdot \\left| b \\right\\rangle \\left\\langle b \\right| + 2 \\cdot \\left| c \\right\\rangle \\left\\langle c \\right|$ . Then $\\left| a \\right\\rangle $ and $\\left| b \\right\\rangle $ are two ground states. How is that consistent with the answer to Q1 (if that answer was positive...)?"}
{"id":"64707","title":"Do laser beams interfere noticeably with each other?","text":"I want to arrange some red lasers along a 90 degree arc of a circle, aimed towards the center of the circle. When their beams hit the opposite side of the circle, will I notice any interference effects? What if I use dozens of lasers? ![Laser arc](http:\/\/i.stack.imgur.com\/Ok0m9.png) If they do interfere, how do I calculate the interference pattern?"}
{"id":"46487","title":"Identification of extended quantum objects?","text":"In some physics theories like string theory we have notion of spatially extended quantum objects - strings, membranes etc. Assuming that such objects exist, how would they appear in experiments ? More precisely are there any thought experiments for determining whether a quantum thing being observed has spatial extension or not, and if it has spatial extension then what is its dimension, topology etc."}
{"id":"40789","title":"Why can radio wave travel through walls while imfrared can't?","text":"> **Possible Duplicate:**   >  If both radio waves and gamma rays can travel through walls Does it have anything to do with the materials used to build the wall? What determines if a material can absorb an (x) wavelength?"}
{"id":"63735","title":"Mirror image at relativistic speeds","text":"Imagine moving parallel to the surface of a very large flat mirror at relativistic speeds. What would be the effect of viewing yourself in the mirror? At non-relativistic speeds your image would be similar to what you would see when stationary. Assume you are moving to your right with respect to the mirror at high speeds. Without doing the math I would expect your image would start moving to your left. Also, your image would start to distort because the path lengths of the photons from various parts of your body would take different times to arrive back at your eye. So what arrives simultanously to your eye would be different regions of yourself at different times in the past. Also, assuming that the light source that is illuminating you is moving with you, the photons would transfer some of their momentum and energy to the mirror, and so the light from the image would be red-shifted. The last part I am not sure about, is reflection a process of absorption and re-emission from the surface atoms of a mirror?"}
{"id":"43504","title":"Minimal Maxwell's Demon","text":"I would like to understand where the waste heat is generated in the Maxwell's demon problem. To this end I've come up with the simplest scenario I can think of. If my scenario is workable I am hoping someone will be kind enough to apply the right math to it to show what is going on and why it does not work to provide a \"free lunch.\" My scenario is as follows: Take a 1 dimensional container of length l. Start with an evenly distributed population of particles; slow red ones and fast blue ones, traveling at +v or -v, and +2v or -2v respectively. The demon sits at the boundary at l\/2. He lets slow red particles at +v and fast blue particles at -2v through and keeps the opening between the two halves of the container closed otherwise. The idea of using a one dimensional container is to keep the math as simple as possible. In order for this to work the particles have to be able to go right through one another without interacting. I am imagining that if one can work out what happens in this simplest case then it should be possible to extend the scenario to a 2-dimensional container in which the line forming the barrier between the two halves of the container is made up of discrete copies of the Maxwell's Demon in my 1-dimensional scenario."}
{"id":"49764","title":"Does photon possesses no time to cover any arbitrary distance?","text":"Photon travel 8 minutes (with speed $c$) from the sun to reach the earth. Any particle (or space-ship) with velocity $0.99 c$ covers the same distance (93 millions km) within less than 2 minutes (according to SR). Does this make sense?"}
{"id":"49760","title":"What happens when you heat vodka in a microwave?","text":"Since ethanol has a lower dielectric constant than water would the water heat up and boil before the ethanol? Would the water transfer heat to the ethanol and, since ethanol has a lower boiling point, bring it to a boil before the water?"}
{"id":"10638","title":"Statistical physics of molecular dissociation of a diatomic gas","text":"Say there are $N$ atoms of type $A$ in a box of volume $V$ and they are undergoing a reversible association-dissociation reaction $A + A = A_2$. Let an $A$ atom have mass $m$, and hence the molecule $A_2$ has mass $2m$. Assume that the temperature $T$ is high-enough that Boltzmann statistics applies. Say $N_1$ be the number of $A$ atoms and say $N_2$ be the number of $A_2$ molecules at equilibrium. Then $N_1 + 2N_2 = N$. If there had to be $N$ $A$-type atoms in the the box then I guess the partition function $Z$ would be given by, $$Z = \\frac{1}{N!} [ \\int e^{-\\frac{\\beta p^2}{2m}} \\frac{d^3p d^3q}{h^3} ] ^N = \\frac{1}{N!} [ \\frac{V}{h^3} (2\\pi m kT)^{\\frac{3}{2}} ] ^N .$$ Say that the dissociation energy of the reaction is $E_0$. Say $N_1$ be the number of $A$ atoms and say $N_2$ be the number of $A_2$ molecules at equilibrium. Then $N_1 + 2N_2 = N$. I want to write the partition function for the above mixed system in equilibrium. Using the expression for $Z$ as stated above, I am guessing that this partition function (say $Z_N$) would look like, $$Z_N = \\sum _ {N_1 + 2N_2 = N} e^{-\\beta N_2 E_0} \\frac{1}{N_1 !} [ \\frac{V}{h^3} (2\\pi m kT)^{\\frac{3}{2}} ] ^{N_1} \\frac{1}{N_2 !} [ \\frac{V}{h^3} (4\\pi m kT)^{\\frac{3}{2}} ] ^{N_2}. $$ (In the above I am using the idea that for $N_2$ molecules to have formed $N_2$ $E_0$ (binding) energy would have been released and hence that much of extra energy has to be accounted for which is not captured in the partition functions over the $N_1$ atoms and $N_2$ molecules...)   * Is the above expression for the partition function correct ?  One further introduces the grand canonical partition function (say $Z_{\\mu}$ ) for the above system with a chemical potential $\\mu$ to \"take care of the constraint $N_1 + 2N_2 = N$\", $$Z_{\\mu} = \\sum _ {N=0} ^\\infty e^{\\beta \\mu N} Z_N.$$   * Firstly I don't understand the meaning of the $\\mu$ in the above. (I can't relate it to what I would usually think as chemical potential) I mean..isn't the above expression for $Z_{\\mu}$ independent of $N_1$ and $N_2$?    * How does the $Z_{\\mu}$ help find the expectation value for $N_1$ and $N_2$?   * If my expressions for $Z_N$ is correct then does $Z_{\\mu}$ have a closed form expression?  All this is to understand the claim that as expectation values the following relationship apparently holds, $$\\frac{N_1 ^2} {N_2} \\sim e ^{-\\frac{|E_0|}{kT}} .$$ I would be glad to know of other insights along the above."}
{"id":"10636","title":"QM interpretation of Larmor precession","text":"Typically when I think of a spin moment in an external field, I visualize it classically in terms of a magnetic moment precessing around the external field vector. Now how is this described quantum mechanically? My guess would be that the external field causes transitions between the up-state and down-state at the Larmor frequency. However if the Zeeman energy was much greater than the thermal energy, wouldn't the spin need to stay in the low energy state? Does that mean the spin ceases precession in the classical picture?"}
{"id":"53020","title":"How to interpret the magnetic vector potential?","text":"In electromagnetism, we can re-write the electric field in terms of the electric scalar potential, and the magnetic vector potential. That is: $E = -\\nabla\\phi - \\frac{\\partial A}{\\partial t}$, where $A$ is such that $B = \\nabla \\times A$. I have an intuitive understanding of $\\phi$ as the electric potential, as I am familiar with the formula $F = -\\nabla V$, where $V$ is the potential energy. Therefore since $E = F\/q$, it is easy to see how $\\phi$ can be interpreted as the electric potential, in the electrostatic case. I also know that $F = \\frac{dp}{dt}$, where $p$ is momentum, and thus this leads me to believe that $A$ should be somehow connected to momentum, maybe like a \"potential momentum\". Is there such an intuitive way to understand what $A$ is physically?"}
{"id":"11948","title":"Is it theoretically possible for the orientation angle of a projectile to remain exactly equal to the orientation of velocity?","text":"This question is sparked by my answer to this question: Is this simulation following real physics? After examining the math, I don't see how it is theoretically possible for the situation simulated to happen. The situation in question is whether or not a spear in projectile motion can stay oriented over the entire path such that the angle is equal to the angle of the velocity vector. I start with proposing that the rate of change of the angle of the velocity vector must be a constant - there can be no torque applied during the projectile motion, only the initial angular velocity imparted at the release. So taking a projectile, with say an initial velocity of <20, 20> - this would be at a 45 degree angle, the case where the condition intuitively seems most likely: $\\frac{dy}{dt} = -9.8t + 20$, $\\frac{dx}{dt} = 20$ We find $\\frac{dy}{dx}$: $\\frac{dy}{dx} = \\frac{-9.8t + 20}{20}$ And an expression for the angle, then: $\\theta = \\arctan{\\frac{dy}{dx}}$ Graphing the derivative of this function over the time span of the projectile shows that it is NOT a constant. This implies that there would have to be some sort of torque being applied mid-air if the angular displacement is directly proportional to $\\theta$. However, it IS a very shallow bowl. First of all, is this analysis correct? Second, if this implies that the condition is impossible, what happens when one throws a javelin or spear perfectly? Why is it so natural to do so?"}
{"id":"54556","title":"How to calculate the intensity of the interference of two waves in a given point?","text":"There are two different point sources which produce spherical waves with the same power, amplitude, ω, wavenumber and phase. I can calculate the intensity of each wave in a point: $$ I_1 = P \/ (4 \\pi r_1^2) $$ $$ I_2 = P \/ (4 \\pi r_2^2) $$ But how can I calculate the intensity of the resulting wave in that point?"}
{"id":"54557","title":"How would I explain a farad being a second per ohm?","text":"The farad has many alternative representations in SI. I can comprehend some of them. For example, a capacitance of 1 farad means that it will take a single coulomb to create a single volt between the capacitor plates. It could also mean that it will take one second of continuous 1A current to create that volt between the plates. In this manner, how would I imagine the second\/ohm representation?"}
{"id":"54554","title":"Permanently charging a sphere by induction using a high voltage","text":"I want to charge a sphere using a van de graff generator , it would sound easy , but using a 1 MV or a 900 KV is really hard, it is enough to break like 1 meter of air so whenever i try to ground the sphere there is spark . So i wanted to know if it will help if i covered the sphere with a layer of plastic or an insulator , or if there was a vacuum between the the generator and the sphere. And another thing, can a graphite rod be charged like a metal one ?"}
{"id":"62468","title":"Light Ray Reflection from concave mirror","text":"Suppose a ray of light hits a concave mirror and is parallel to principle axis but far away from it such that it doesn't follow _paraxial ray approximation_. Will it pass through focus or between focus and radius of curvature or between pole and focus? Here pole, focus and radius of curvature mean the same thing as in paraxial ray approximation ."}
{"id":"118930","title":"Will a black hole eventually turn into a neutron star?","text":"As far as i understand, black holes radiate away energy in form of Hawking Radiation. Thus, they lose mass, i suppose. Is there a point where the mass becomes too small for the object to still be a black hole? What happens then, will it turn into a neutron star supported by degeneracy pressure? Or is it all about density, and once a black hole is formed, it will stay a black hole (i guess until it completely radiates away)?"}
{"id":"118932","title":"Double decay $\\beta\\beta$ observation and neutrino mass","text":"If the double decay $\\beta\\beta$ will be detected this means the neutrino is a Majorana particle coincident with its antiparticle. At the moment the half life of this decay is put to $\\tau\\ge10^{25}$years. The limit of the Majorana neutrino mass is about 0.2–0.4 eV. What is the relation between the probability of $\\beta\\beta$ decay and the neutrino mass? Thanks."}
{"id":"122544","title":"Magnetic field and AC current","text":"If I run an ac current under a plate of iron filings, what pattern should the filings form? If I use the plug for an appliance, will the magnetic fields in the two wires cancel each other out (more or less)?"}
{"id":"122545","title":"Steel filings just like iron filings?","text":"I am having problems with an experiment involving field lines. I filed a piece of steel to get \"iron filings.\" Ought these steel filings be vastly different from the iron filings typically used in this sort of experiment?"}
{"id":"122540","title":"Why are anti-de Sitter spaces so interesting when we believe the universe is expansionary?","text":"Perhaps this is a naive question, but in my recent (admittedly limited) readings about AdS spaces, I keep wondering why they seem to be such a hotbed for theoretical research (AdS\/CFT correspondence, etc.). To my understanding, an AdS space has constant negative curvature in a vacuum, which should yield an attractive universe, not one with accelerating expansion. An AdS space can be thought of as having a negative cosmological constant, while a universe with accelerating expansion would imply that such a constant be positive. Since we observe that our universe's expansion is accelerating, it seems that if anything, we should be seeking to model it as a de Sitter space. Am I mistaken? What aspects of our universe do AdS spaces attempt to model?"}
{"id":"10097","title":"Does this equation mean anything?","text":"I saw this equation on a coffee mug, and I am wondering what it means. I am not far enough along in physics to understand it. Does it actually mean anything or is it gibberish? ![enter image description here](http:\/\/i.stack.imgur.com\/ifd4q.jpg) I can't embed the image because I am a new user. (But I can, L.M.)"}
{"id":"10098","title":"1-form formulation of quantized electromagnetism","text":"In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps to cite. Taking the electromagnetic field $\\hat A$ to be an (operator-valued distributional) 1-form, we can write a smeared operator $\\hat A_H$ in terms of the inner product on the exterior algebra as $\\bigl<\\hat A,H\\bigr>$ (taking the test function $H$ also to be a 1-form, but satisfying Schwartz space-like smoothness conditions in real space and in fourier space instead of being a distribution). $\\hat A$ projects into annihilation and creation parts, $\\hat A^+$ and $\\hat A^-$ respectively, $\\hat A=\\hat A^+ +\\hat A^-$, for which the action on the vacuum is defined by $\\hat A^+\\left|0\\right>=0$, and we have the commutation relations $\\Bigl[\\bigl<\\hat A^+,H\\bigr>,\\bigl<\\hat A^-,J\\bigr>\\Bigr]=\\left<H,E(J)\\right>$, where $\\widetilde{E(J)}(k)=2\\pi\\delta(k^2)\\theta(k_0)\\tilde J(k)$ projects to the positive frequency forward light-cone. The commutation relations are not positive semi-definite for arbitrary test functions $H$, $\\Bigl[\\bigl<\\hat A^+,H^*\\bigr>,\\bigl<\\hat A^-,H\\bigr>\\Bigr]\\not\\ge 0$, which is fixed by the Gupta-bleuler condition, which can be stated in this formalism as $\\delta\\hat A^+\\left|\\psi\\right>=0$, for _all_ states $\\left|\\psi\\right>$, not just for the vacuum state. By the Hodge decomposition theorem, we can write the test function $H$ uniquely as $H=d\\phi+\\delta F+\\omega$, where $\\phi$ is a 0-form, $F$ is a 2-form, and $\\omega$ is a harmonic 1-form, so we can write $$\\bigl<\\hat A^+\\\\!,H\\bigr>\\left|\\psi\\right>\\\\! =\\bigl<\\hat A^+\\\\!,d\\phi+\\delta F+\\omega\\bigr>\\left|\\psi\\right>\\\\! =\\Bigl(\\\\!\\bigl<\\delta\\hat A^+,\\phi\\bigr>+\\bigl<\\hat A^+\\\\!,\\delta F+\\omega\\bigr>\\\\!\\Bigr) \\left|\\psi\\right>\\\\! =\\bigl<\\hat A^+\\\\!,\\delta F+\\omega\\bigr>\\left|\\psi\\right>.$$ The harmonic 1-form $\\omega$ has to be zero to satisfy the Schwartz space condition, and by the left action of $\\hat A^-$ on arbitrary states we have the same projection of the arbitrary test function $H$ to $\\delta F$, so all we are left with is the electromagnetic field observable $\\hat\\Phi_F=\\hat A_{\\delta F}=\\bigl<\\hat A,\\delta F\\bigr>$. Unsurprisingly, in the free field case, in the absence of charges, the electromagnetic potential observable is exactly equivalent to just the electromagnetic field observable, for which we can verify that $\\Bigl[\\bigl<\\hat A^+,\\delta F^*\\bigr>,\\bigl<\\hat A^-,\\delta F\\bigr>\\Bigr]\\ge 0$, using which we can use the GNS construction of a free field Hilbert space. So this is a reference request. Is there any literature that uses this kind of mathematical formalism for the quantized electromagnetic field? Even _vaguely_ the same! My sense is that AQFT has moved to more abstract methods, while more practical interacting QFT has become historically committed to index notations that are little changed from 50 years ago, even though the methods of interacting quantum fields have changed in many other ways, and that mathematicians who take on the structures of QFT, as Folland does in http:\/\/www.amazon.com\/Quantum-Theory-Mathematical-Surveys- Monographs\/dp\/0821847058, make strenuous efforts not to let their notation and methods stray too far from the mainline."}
{"id":"29569","title":"Loop of wire in a varying magnetic field","text":"Let us suppose we have a loop of wire with some definite resistance in a magnetic field. Let the magnetic field be varying. This varying magnetic field will st up an EMF in the wire. My questions are as follows: 1). Energy is dissipated across the wire resistance. Where does this energy come from? 2). The current generated in the loop will also generate a varying magnetic field. Will this magnetic field induce another EMF in the loop setting up another varying magnetic field? Will this go on indefinitely?"}
{"id":"15914","title":"What happens to speed and frequency of a light beam moving in transparent medium when observed from different inertial frame of reference?","text":"Suppose a transparent medium where speed of light is $c\/n$, an inertial frame of reference $K$ which is stationary relatively to the medium and an inertial frame of reference $K'$ which is moving with speed $V$ according to $K$ along the $x$ axis. Also suppose that a light beam is travelling through the medium and its frequency measured from $K$ is $f$ and from $K'$ is $f'$ and the angle between the $x$ axis and the bean is $\\theta$ and $\\theta'$ respectively. If we know $n,f,\\theta'$(or $\\theta$ or both) and $V$ how can we calculate $f'$? * * * From a book I have read the transformation was the following: $$f'~=~\\frac{f}{\\gamma (1+\\beta n\\cos\\theta')},$$ where $$\\gamma~=~\\frac{1}{\\sqrt{1-\\beta^2}},\\qquad \\text{and}\\qquad \\beta~=~v\/c.$$ I can prove it with momentum-energy Lorentz transformations only if I suppose that light is moving with same speed ($c\/n$) both in $K$ and $K'$ which (I think) is wrong according to speed transformation. Does the speed of light when moving through a transparent medium change when observed form a different inertial frame of reference or not?"}
{"id":"95304","title":"Any difference between \"Mueller matrix\" and \"Scattering matrix\"?","text":"I find in some references 4x4 Mueller matrix and in other references 4x4 Scattering matrix. Are they different or identical? If they are different, I would like to know the book or any research paper describing the relation between them."}
{"id":"123631","title":"Would a craft travelling increasingly close to the speed of light appear to be decelerating?","text":"I've been pondering the implications of time dilation. Is it right to assume that if a craft was travelling at a speed very close to the speed of light (>= 0.9999c for example), that to an external observer it would appear to be moving very much slower? Example: at 0.9999c, time dilation is 70.71, so while the craft is travelling at 186,263.37 miles\/s, doesn't that mean to a (stationery) observer it's only moving at 2,634.10 miles\/s? And at 0.99999c, time dilation is 223.61, so while the craft is now travelling at 186,280.14 miles\/s, doesn't that mean to the observer it's now only moving at 833.07 miles\/s? The obvious implication is that as the craft approaches the asimptote of the speed of light, external observations (beyond around 0.9c) see it decrease in velocity towards the asimptote of being stationary. Is this reasoning correct?"}
{"id":"95303","title":"How about retrodiction, if climate prediction is difficult?","text":"What I mean by the question is if _retrodiction_ has been attempted back to, say, 50 or 100 years starting with the presently known state of climate as initial conditions using the same equations and methods that are popularly used to _predict_ climate 50 or 100 years ahead? If yes, how well retrodiction turned out? If not, why not?"}
{"id":"131365","title":"What if all the electrons leave a metal?","text":"I was studying photoelectric effect. Then I thought that what will happen if all the electrons from a metal piece come out as photoelectrons by using a light source of particular frequency? Will the metal still exist?"}
{"id":"66614","title":"Cycling down a mountain","text":"I am Dutch, so I cycle a lot, hence the following question. Say I am standing at the top of a mountain on my bicycle. I have an amount of energy $E$ available. Say that after the slope of the mountain ends, there is an infinite long, straight, smooth road ahead. How do I get the furthest on this flat road? Options: a) using all my energy $E$ to peddle while I am coming down the mountain b) roll gently down the mountain and start peddling precisely when I am at the end of the mountain c) roll gently down the mountain and down the straight road and only when I am starting to get to a hold, I start cycling very slowly, keeping my acceleration and speed to a minimum d) some kind of hybrid of the above. In this story I would like to keep it as realistic as possible: there's friction, both from my tires and from air resistance. I once heard that air resistance goes $\\propto v^2$, such that I am tempted to opt for option c). I am interested in your opinions\/thoughts. Help me spend as little energy as possible while cycling!"}
{"id":"37570","title":"State-dependent diffusions: Fick's law vs. Fokker-Planck's, which and why?","text":"Consider a \"state-dependent diffusion\": a diffusion process for which the diffusion coefficient $D(x)$ depends on the (stochastic) state $x$ of the system. (An example is provided by the diffusion of tracers in a spatially inhomogeneous bath, e.g. with varying viscosity or temperature.) What is the associated flux? If $c(x)$ is the probability to find the system in the state $x$ at a given time, should the flux read $j(x)=-D(x)\\nabla c(x)$, as in Fick's law? Or rather $j(x)=-\\nabla(Dc)(x)$, as in the Fokker-Planck equation? (I'm using the terms \"Fick\" and \"Fokker-Planck\" here only as tags for each alternative.) It is well known that this question does not have a definite answer, and should be answered case by case. My actual question is threefold:   1. Do you know examples of physically relevant state-dependent diffusions? What category do they fall in? Fick, or Fokker-Planck?   2. Do you have an intuition for the physics underlying this alternative?    3. The Fokker-Planck case has the peculiarity to lead to equilibrium distributions which are not of the Boltzmann-Gibbs form (the steady-state $c(x)$ depends on $D(x)$ and not just on the state's energy). What does this tell us about the foundations of statistical mechanics?"}
{"id":"119683","title":"Dimensionial Regularization (integrating over extra dimensions)","text":"I tried to replicate a result about dimensional regularizations in the review article of Howard Georgi, _Effective Field Theory_ at: people.fas.harvard.edu\/~hgeorgi\/review.pdf. On page 17, he writes about Integrals of the form: $$I = \\int \\frac{d^4l}{(4\\pi)^4}\\frac{1}{(l^2 + A^2)^\\alpha}$$ He then regularizes the integral in the form $$I = \\int \\frac{d^{4 + \\delta}l}{\\mu^\\delta(4\\pi)^{4+\\delta}}\\frac{1}{(l_\\delta^2 +l^2 + A^2)^\\alpha}$$ and integrates seperately over the extra dimensions $\\delta$ getting: $$I = \\int \\frac{d^4l}{(4\\pi)^4} \\left(\\int \\frac{d^{\\delta}l_\\delta}{(4\\pi\\mu)^{\\delta}}\\frac{1}{(l_\\delta^2 +l^2 + A^2)^\\alpha}\\right) = \\\\\\ \\int\\frac{d^4l}{(2\\pi)^4}\\left(\\frac{1}{(l^2 + A^2)^\\alpha}\\frac{\\Gamma(\\alpha - {\\delta\\over 2})}{\\Gamma(\\alpha)}\\left(\\frac{l^2 + A^2}{4\\pi\\mu^2}\\right)^{\\delta\\over 2}\\right)$$ This seems to make perfect sense to me, but when I try to calculate the integral over the extra dimensions $\\delta$, I run into problems. First of all, the angular part of the Integration over $\\delta$ is $$\\frac{\\pi ^{\\delta \/2} \\delta }{\\Gamma \\left(\\frac{\\delta }{2}\\right)}$$ The radial part of the integral is: $$\\int \\frac{d^{\\delta}l_\\delta}{(4\\pi\\mu)^{\\delta}}\\frac{l_\\delta^{\\delta-1}}{(l_\\delta^2 +l^2 + A^2)^\\alpha} = \\frac{\\Gamma \\left(\\frac{\\delta }{2}\\right) \\Gamma \\left(\\alpha -\\frac{\\delta }{2}\\right) \\left(A^2+k^2\\right)^{\\frac{1}{2} (\\delta -2 \\alpha )}}{2 \\Gamma (\\alpha )}$$ Together I get: $$\\frac{\\pi ^{\\delta \/2} \\delta \\Gamma \\left(\\alpha -\\frac{\\delta }{2}\\right) \\left(A^2+k^2\\right)^{\\frac{1}{2} (\\delta -2 \\alpha )}}{2 \\Gamma (\\alpha )}$$ All in all I get a extra factor of $\\delta$ which makes the integral zero in the limit $\\delta\\rightarrow 0$. But it should not give zero? I can calculate dimensionally regulated integrals the usual way, integrating over all dimensions d at once, reproducing all the known results. But I fail at the Integration over the extra dimensions $\\delta$. Where is my error? (How) Is it possible to integrate over extra dimension $\\delta$ seperately?"}
{"id":"29445","title":"Rotation in an 'empty' universe","text":"> **Possible Duplicate:**   >  Is rotational motion relative to space? Assume a universe with the same physics as ours, but containing only one rotating (charge-free) body - let's say the size of the Earth. Would the rotation cause a bulge at the equator of the rotating body?"}
{"id":"28218","title":"Apollo and orbital mechanics: orbital decay if the Trans Earth Injection (TEI) burn had failed","text":"I'm reading Jim Lovell (Apollo 8 and 13) and Jeffrey Kluger's book _Apollo 13_ , which is a fantastic read about a long past era I only have kindergarten memories of. On page 54 there is a paragraph that reads (emphasis mine): > Also, unlike the LOI burn, during TEI there would be no free-return > slingshot to send the ship home in the event that the engine failed to > light. If the hydrazine, the dimethylhydrazine, and nitrogen tetroxide did > not mix and burn and discharge just so, Frank Borman, Jim Lovell, and Bill > Anders would become permanent satellites of Earth's lunar satellite, > expiring from suffocation in about a week and then **continue to circle the > moon, once every two hours, for hundreds -- no thousands; no, millions -- of > years.** Apollo 8 was in an elliptical orbit of 169.1 × 60.6 miles. I am skeptic about how long such an orbit would be stable. My skills in orbital mechanics are rudimentary at best and I know there is no atmosphere to cause orbital decay due to atmospheric drag. But I have heard about other effects causing orbits to decay over time, such as variations in the moon's gravitational field (density anomalies, mascons). This may cause lunar satellites to eventually crash into the moon. Can someone estimate how long this would be for the case of the Apollo 8 lunar orbit?"}
{"id":"26367","title":"Is there any way to anticipate exceptional Aurora Borealis viewing conditions?","text":"I've always wanted to see the Aurora Borealis, but never had the opportunity, since it is so rarely visible in my area (Pennsylvania). However, last week was apparently a rare opportunity of exceptional visibility. Unfortunately, I missed my opportunity, as I did not find out about it until the next day. Are there any good resources or strategies for anticipating opportunities like this? Are coronal mass ejections like the one that precipitated this event predictable? If so, are such predictions available to the amateur public?"}
{"id":"93223","title":"Show that the electric field, $\\mathbf{\\vec{E}}$ cannot be generated by any static distribution of charges","text":"> Show that the electric field $$\\mathbf{\\vec{E}}=\\begin{pmatrix}0 \\\\\\ E_0x > \\\\\\ 0\\end{pmatrix}$$ where $E_0$ is a constant, cannot be generated by any > static distribution of charges. I understand that an electrostatic field is irrotational and the divergence of the electric field is the charge density over epsilon 0 but I have no idea why that makes the answer."}
{"id":"93225","title":"How 4-vector nature of the value is connected with it's conservation law?","text":"In electrodynamics Poynting vector and energy flux of field don't create 4-vector. Also they aren't conserved independently from substance (conservation law includes summand connected with current density). In linearized gravity mass density and mass current density as components of stress-energy tensor also aren't conserved. And they also aren't components of 4-vector. Is facts of non-conservation of values above connected with the absense of 4-vector nature?"}
{"id":"133976","title":"Can a super conducting magnet & gravity create a cheap source of energy?","text":"I have a question about whether this would produce a cheap source of energy. If you created two superconductive magnets configured in a figure of eight layout, in a cold enough environment so you needed no extra power for cooling and the magnets would be in persistent state (so after initial charge would need no extra power). If you then made these magnets levitate a very heavy piece of iron could you then use the gravity of the earth (or moon if the experiment was set up on the cold moon) to generate power when then power was switched from one magnet to the other magnet. E.g. The iron block 1 would rise, power would move to magnet 2, iron block 1 would fall (maybe turning a flywheel on the way down), iron block 2 would rise etc. In my mind this configuration would not cost any additional energy ( as long as the environment was cold enough) and you could get virtually free energy by utilising the gravitational field of the earth or moon... Has this idea got legs?"}
{"id":"56226","title":"Choosing the right point when calculating moments","text":"A body is composed of two straight pins that are joined at a right angle. They have lengths $a$ and $b$ and the mass per unit length is $\\rho$. When the body is balanced on a flat surface, as shown, how large is the normal force against the ground in the right point of contact? 4 options as can be seen in the picture. ![picture](http:\/\/i.imgur.com\/Hr1RBRF.jpg) Let me point out that this is a conceptual question. When I first tried to solve this problem I decided to choose to calculate the moment around the left contact-point in order to reduce one term(left normal force). This seems like a natural way but gives a false answer($N_2= \\rho g$. If I instead calculate about the vertex of the triangle and use Newton's second law I get the correct solution.(answer is D). So how should I choose the \"correct\" point? **Around left contact point:** By dropping an altitude **h** at the right angle we get that: Notice that: $\\cos\\alpha=\\frac{a}{\\sqrt{a^2+b^2}}$, $\\cos\\beta=\\frac{b}{\\sqrt{a^2+b^2}}$ and $m_ag=\\rho ag$ and $m_b=pbg$ $-\\frac{\\rho a^2g}{\\sqrt{a^2+b^2}}-\\frac{\\rho b^2g}{\\sqrt{a^2+b^2}}+N_2\\sqrt{a^2+b^2}=0$. Simplifying gives $N_2=\\rho g$"}
{"id":"110306","title":"Poseuille number for flat channel?","text":"For a flat channel with fully developed flow, the Poseuille number when using the hydraulic diameter is $$Po=C_fRe_{D_{H}}=24$$ But when using 2b as the length scale, the Poseuille number is $$Po=C_fRe_{2b}=12$$ Why is this? I thought that the hydraulic diameter of a flat channel is 2b. Shouldn't they be the same? In the derivation used, b is half the channel width"}
{"id":"114685","title":"How to measure vector potential such as that of magnetic field?","text":"If we want to measure gravitational potential energy of a body-earth system, we can simply do it by measuring the body's height from ground. How could we can measure vector potential such as that of magnetic field?"}
{"id":"104474","title":"What happens if you let a cable roll slip into a black hole?","text":"1. Does the cable roll spin faster the more cable goes into the black hole in reference of a observer standing next to it?   2. Can gravity pull the cable that it exceed the speed of light inside a black hole in reference to the center of the black hole?"}
{"id":"104475","title":"Translation invariance in crystals","text":"**Context** : solid state physics - dynamics of atoms in crystals. Coupling constants are defined as: \\begin{align*} \\frac{\\partial^2\\Phi}{\\partial r_{nai}\\partial r_{mbj}} = \\Phi_{nai}^{mbj} \\end{align*} where $\\Phi$ is the total energy of the crystal expressed as a function of nuclei coordinates. $r_{nai}$ is the position vector of the $n$-th cell, $a$-th atom at $i$-th spatial direction. My book says that coupling constants have some properties. One of them is the \"translation invariance\", which is stated as: \\begin{align} \\Phi_{nai}^{mbj} = \\Phi_{0ai}^{(m-n)bj} \\end{align} I don't understand the zero index in RHS. Since the $n$ or $m$ represent the cell, what does it mean to refer to zero-th cell ?"}
{"id":"104470","title":"Where do our 4 macroscopic spacetime dimensions reside in multidimensional models of the universe?","text":"In models such as M-theory with 7 'higher dimensions' plus the 4 macroscopic spacetime dimensions, where do our 4 macroscopic spacetime dimensions reside _ordinally_? My reason for asking is TV shows such as the 'Fabric of the Cosmos' that alludes to 'lower' dimensions in string theory. I can understand if our 4 are, say the 7th, 8th, 9th and 10th so there are dimensions lower than us but that we cannot readily detect."}
{"id":"104471","title":"Are we so sure about superposition?","text":"Apparently particles can be anywhere when not observed. How strong is this theory really? Okay the wave-function can be collapsed through observation but how are we so sure that when an object is not being observed that it is in potential states? How are we so sure whether or not the wave function is collapsed by a conscious observing force other than us? And is superposition an assumption due to the fact that when we observe waves they act as particles, so we assume that everything has different potential until measurement; when really we don't know for sure whether it is that an object is in potential states or a conscious observance by a higher power already puts things in order already eliminating this potentiality?"}
{"id":"127020","title":"matrix elements of the electronic molecular Hamiltonian between a hartree product and a Slater determinant","text":"This may belong in Chemistry, but I thought I might try my luck here first. In Szabo's book, an exercise requires a proof that = (N!)^(1\/2) * given that |K(HP)> is the Hartree product wave function corresponding to the N-electron Slater determinant |K>. I've been trying for a couple days now, but the only idea I have is to evaluate both integrals explicitly by throwing away all terms arising from the Slater determinant that would vanish by orthonormality (all but one for the one) then evaluating the integrals over the sums of one and two electron operators as a sum over allowed orbital permutations. This would defeat the point of having the theorem to simplify the derivation. Does anyone have a better more clever way or a hint to point me in the right direction? Thanks"}
{"id":"127028","title":"How to measure rainfall? And how is it determined for a specific location?","text":"How is rainfall measure in a pinpoint location? This is what I think:   There are automatic rainfall measuring stations at: A, B, C. But say I want to know point D, located between A and B, its rainfall level. How do I figure that out? Do I take the average? Do I go to the nearest neighbor on the map? Thanks in advance!"}
{"id":"113425","title":"Time reversal symmetry in the presence of friction","text":"I was reading a paper on time reversal symmetry, and came across an example of a pendulum swinging in the presence of friction: > When we consider the more realistic physical situation of a swinging > pendulum in the presence of friction, we can tell the difference between a > forward and a reverse film of this pendulum. Namely, the original (forward) > film will show the swinging pendulum losing amplitude until it comes to a > standstill. However, the film in reverse direction shows a swinging pendulum > whose amplitude is increasing in time. The latter film is clearly unphysical > as it does not satisfy the natural laws of motion anymore (assuming there is > no hidden source of energy feeding the pendulum). The presence of friction > breaks the time-reversal symmetry of the ideal pendulum. I don't understand this. If I am not wrong, we are only reversing the direction of time, not that of the frictional force (the nature and cause of this force is not relevant either). In the movie played backwards, the frictional force will be along the direction in which the bob swings. This force would be proportional to $v^2$, and would give the bob more momentum consistent with the observation. So what am I missing? How does the presence of friction break time reversal symmetry?"}
{"id":"60123","title":"Energies and numbers of bound states in finite potential well","text":"Hello I understand how to approach finite potential well (I learned a lot in my other topic here). However i am disturbed by equation which describes number of states $N$ for a finite potential well ( $d$ is a width of a well and $W_p$ is potential ): $$ N \\approx \\dfrac{\\sqrt{2m W_p}d}{\\hbar \\pi} $$ I am sure it has something to do with one of the constants $\\mathcal L$ or $\\mathcal K$ defined this way: \\begin{align} \\mathcal L &\\equiv \\sqrt{\\tfrac{2mW}{\\hbar^2}} & \\mathcal{K}&\\equiv \\sqrt{ \\tfrac{ 2m(W_p-W) }{ \\hbar^2 }} \\end{align} and the transcendental equations for ODD and EVEN solutions: \\begin{align} &\\frac{\\mathcal K}{\\mathcal L} = \\tan \\left(\\mathcal L \\tfrac{d}{2}\\right) &&-\\frac{\\mathcal L}{\\mathcal K} = \\tan \\left(\\mathcal L \\tfrac{d}{2}\\right)\\\\\\ &\\scriptsize{\\text{transc. eq. - EVEN}} &&\\scriptsize{\\text{transc. eq. - ODD}} \\end{align} **QUESTION:** Could anyoe tell me where does 1st equation come from?"}
{"id":"2451","title":"Why does a car engine not do work if the wheels don't slip?","text":"I saw this mind boggling result that if the tires don't slip then the work done by an engine to move a car is zero. Why is this true? Moreover, what does this truly mean? **Update:** Sorry about not being clear, but I was talking about an idealised case where air resistance is negligible and the tires are perfectly rigid and there is no internal friction at play."}
{"id":"93914","title":"Does the covariant derivative of a derivative of the metric vanish?","text":"The title holds the main question. For a little more backgroud: In a calculation I keep coming across terms of the form $$ D_M \\partial_N e_P^A \\overset{?}{=} 0$$ where $e_P^A$ is the vielbein and $D_M$ is the covariant derivative _including_ both the spin _and_ the affine connection, i.e. the vielbein postulate reads $D_M e_P^A = 0$. Generally, I would not expect the expression to vanish, but the again I know little about the more involved machinery of differential geometry."}
{"id":"65575","title":"Phase Space dimension of Lorenz Strange Attractor","text":"It is often discussed in 3 spatial dimensions and the need for third dimension to prevent self intersection is mentioned. But shouldn't the phase space of the Lorenz system be 6 dimensional, i.e., the 3 momenta also?"}
{"id":"61835","title":"What does the sum of two qubits tell about their correlations?","text":"> How much can I learn about correlations between two quits by measuring the > sum of their values? What is the best way to formalize such a question? Below is my original, longer formulation of the same question in spin language. I have an ensemble of pairs of spin-$1\/2$ particles. All I can do is to measure total $S_z$ for each pair, each time learning one of the three standard outcomes: $+1$, $0$ or $-1$. Based on the knowledge of this probability distribution $P(S_z)$, what can be said about the density matrix of my ensemble? In particular, what is best measure for correlations between the two particles in these circumstances? I understand that I won't be able to distinguish whether the correlations are classical or necessarily quantum-mechanical, but struggle to come up with a measure better than a classical correlation coefficient $$r=\\frac{\\langle s_1 s_2 \\rangle-\\langle s_1 \\rangle \\langle s_2 \\rangle}{\\sqrt{\\langle (s_1- \\langle s_1 \\rangle)^2 \\rangle \\langle (s_2- \\langle s_2 \\rangle)^2 \\rangle }}$$ where $s_1$ and $s_2$ are the $\\pm 1\/2$ spin projections (or $0$, $1$ qubit values) of the 1st and the 2nd particle, respectively, so that my only observable can be expressed as $S_z=s_1+s_2$. I then minimize $r$ over all possible $\\langle s_1 \\rangle$ and $\\langle s_1 \\rangle$ consistent with measured $P(S_z)$ and declare that the outcome is the minimal possible correlation in the pair. Is there a more elegant way? **UPDATE:** Classically, if we take the definition of zero correlations as $\\langle s_1 s_2 \\rangle =\\langle s_1 \\rangle \\langle s_2 \\rangle$, then it implies that $s_1 = +1\/2$ and $s_2=+1\/2$ with some _independent_ probabilities $p_1$ and $p_2$, so that the sum is distributed bimomially $P(+1)=p_1 p_2$ and $P(-1)=(1-p_1)(1-p_2)$. These two equations have a solution with $0\\leq p_1, p_2 \\leq 1$ only if $$P(+1) < 1-2 \\sqrt{P(-1)}+P(-1)$$ Graphically, this condition is satisfied within the region bounded by a black line in the plot below. The rest of the available probability space is the region of excess correlations ($r_{\\text{min}}>0$, depicted in red). ![enter image description here](http:\/\/i.stack.imgur.com\/BjYIe.png) **Updated question:** is there a quantum loophole in such analysis? I suspect not but would really like to come up with a shorter\/cleaner argument. **UPDATE-2** My selected spin\/qubit degree of freedom is one of the many variables characterizing the state of the two particles. Can we be sure there exists no pure product state that can lead to $r_{\\text{min}}>0$? See a related active question on fermionic entanglement of two particles."}
{"id":"117333","title":"Is phosphorene a topological insulator?","text":"All of the other \"-ene\" materials (silicene, germanene, stanene, and even graphene) are topological insulators. Phosphorene and these other materials all have honeycomb lattices and would appear to have similar properties. However, I don't see any papers referring to phosphorene as a topological insulator, so is phosphorene just a semiconductor? Thanks."}
{"id":"61838","title":"How did the Bardeen-Brattain point-contact transistor work?","text":"How did the Bardeen-Brattain point-contact transistor work? What are the conditions under which a Schottky barrier injects minority carriers? In a bipolar junction transistor the forward-biased emitter injects minority carriers into the base, which are then swept across the reverse-biased base- collector depletion layer. At first glance a point-contact transistor might work the same way: the rectifying Schottky barrier might inject minority carriers from the metal emitter into the doped silicon base which would then be swept across the reverse-biased base to metal-collector depletion layer. Except that all the textbooks I can find say things about Schottky barriers like, > Under normal operating conditions, the minority carrier current is orders of > magnitude smaller than the majority carrier current. (Sze, _Semiconductor > Devices_ , 1985). Or, > the forward current ... is due to the injection of majority carriers from > the semiconductor into the metal. The absence of minority carrier injection > and the associated storage delay time is an important feature of Schottky > barrier diodes. Although some minority carrier injection occurs at high > current levels, these are essentially majority carrier devices.\" (Streetman, > _Solid State Electronic Devices_ , 1980). So what gives? The point contact transistor could only possibly work if a gold to n-type germanium point contact is injecting. Is it perhaps the case that at high current _densities_ minority carrier injection is actually quite common? (But point-contact diodes should have very high current densities, and thus if this is the explanation then we don't have a good explanation for their good capacitance.) There is a lot of erroneous information about how point-contact transistors work floating around on the web. In the \"maker\" community the suggestion (probably from the article Henderson, PB; Home made transistors, _Wireless World_ , Jan 1954, pp. 20-23) is that somehow melting phosphor-bronze point contacts into N-type germanium does a P-type doping, so that point contact transistors are actually just simple PNP junction transistors. But phosphor is an N-type dopant, so this seems an unlikely explanation. Jeri Ellsworth repeats this assertion in her Youtube video. The Wikipedia page, the PBS page and the Nobel prize page(see frames 19-26 of the animation) all claim that the point contacts somehow magically create a hole based inversion-layer _across the entire surface of the crystal_. (To be fair to all of them, Bardeen\/Brattain's original papers in Physical Review (74(2):230-232, July 1948) made extensive arguments about surface states that may have confused the issue). In Bardeen's Nobel acceptance speech, (as published in Science, 126:105-112, Jul 19, 1957 (paywall), however, he gives the traditional explanation of a Schottky rectifying contact and then says (page 109) > _If $\\chi$ (the energy barrier) is sufficiently large, the Fermi level at > the interface may be close to the valence band, implying an inversion from > n-type conductivity in the bulk to p-type near the contact. The region of > hole conduction is called, following Schottky, an \"inversion layer.\" An > appreciable part of the current flow to the contact may then consist of > minority carriers in this case, holes. An important result of the research > program at Bell Laboratories after the war was to point out the significance > of minority carrier flow._ So I've got a Nobel laureate saying one thing, and two eminent IEEE fellows saying something that seems contradictory."}
{"id":"52033","title":"Does the Earth rotate the same encased in ice during the height of an Ice Age as it does when the bulk of it's water is liquid and always in motion?","text":"Ice Age vs. Now. Does the Earth rotate at the same rate when encased in ice during the height of an Ice Age as it does when the bulk of it's water is liquid and always in motion?"}
{"id":"53343","title":"Does weather obstruct microwaves and radiowaves?","text":"I read an article ('How airplanes use radar' at Bright Hub.com) that stated that airplanes use radio waves for communication because they aren't obstructed by weather. Is this true?"}
{"id":"63473","title":"Does nature tetrate?","text":"We see addition, multiplication and exponentiation in the natural formulae that make up physics. However, do we ever see tetration (repeated exponentiation) or higher hyper- operators in nature? (Not counting cases where the tetration step could be trivially eliminated.)"}
{"id":"9625","title":"Mechanical energy problem - please help","text":"Quite new to all this - simple question. \"A ball starting from rest rolls down a hill and reaches 10m\/s at the bottom. If the same ball starts rolling down the hill from an initial velocity of 5m\/s, will it's velocity at the bottom be: (A) Less than 15m\/s; (B) Equal to 15m\/s; (C) Greater than 15m\/s.\" The answer I have marked in the book this question comes from says (A). However, I am unable to reason to that conclusion from the way the question is stated. Here is my reasoning, please point out where it is wrong. If the same ball (the same mass) traveses the same distance down this same hill (the same change in height), the gravitational potential energy must be the same at the point on top of the hill, regardless of its initial velocity. Therefore, on rolling down the hill, by the time it reaches the bottom, this potential energy would have transferred to kinetic energy - indicating (B)? Thanks for your time. Tim."}
{"id":"112301","title":"Calculating the rate at which a car turns","text":"This is going to be difficult to describe, as I have no real concept of this sort of physics, so please bear with me. If this is too much or beyond the scope of this website, I apologize. I am making a simple-ish driving simulator and I need a formula that I can use to determine the rotation of the vehicle for every frame. If the car's rotation is 90.0 degrees and the wheels are pointing 20.0 degrees to the right, that should mean I can use a simple polar offset to send it a distance of _a_ toward 70.0 (90.0 - 20.0). This does not work when the car is not moving, or when the air resistance is large, like when the car is turning at high speeds, and I am not sure how I should be dealing with such factors. What I have works for now, but I need a more accurate solution. **What I have** What I have barely turns the car at low speeds, and allows it to turn on a dime at high speeds. This is the algorithm that I am currently using in pseudo-code.                   if (vehicle.engineCurrentRPM > vehicle.engineMaxRPM) vehicle.engineCurrentRPM = vehicle.engineMaxRPM;         if (vehicle.engineCurrentRPM < -vehicle.engineMaxRPM) vehicle.engineCurrentRPM = -vehicle.engineMaxRPM; \/\/ This allows it to go full speed in reverse, which I can fix myself.          \/\/ The engineMaxRPM is from a Bugatti Veyron and is equal to 6000.0.     \/\/ That is 600.0 lower than the redline, and I did that on purpose.          vehicle.engineCurrentTorque = (vehicle.engineCurrentRPM * (922.0 \/ 2200.0));     \/\/ A simple formula that approximates the torque-rpm relationship. I only include this because torque _might_ be relevant.          differentialRatio = 4.0;     drag = 0.39;     gearRatio = 3.64;     grip = 1.0;     wheelRadius = (2.0 * 0.0254);     \/\/ 20 inch wheels converted to meters and set to 10%. I don't know why this is necessary, but it seems to be in all of the formulas I have seen     weight = 1995.806; \/\/ 4400 pounds * 0.453592 to get kilograms.     power = (vehicle.engineCurrentTorque * vehicle.engineCurrentRPM) \/ 5252.0; \/\/ HP = Torque x RPM ÷ 5252          \/\/ The variable \"power\" is unused, but there if you need it.          airResistance = 0.05 * (1.0 - drag);          tWheels = vehicle.engineCurrentTorque * gearRatio * differentialRatio;     f = (tWheels \/ wheelRadius);     vehicle.acceleration = (f \/ weight) * grip;          vehicle.object.rotation.z -= vehicle.wheelAngle * vehicle.acceleration * time;     float angle = vehicle.rotation.z;     angle *= M_PI; \/\/ The pi constant.     angle \/= 180.0;          vehicle.velocity.x -= vehicle->acceleration * sin(angle) * time;     \/\/ The world is upside-down for some reason, so subtracting is necessary. This will be fixed!     vehicle.velocity.y += vehicle->acceleration * cos(angle) * time;     vehicle.velocity.z += 0.0;          vehicle.position.x += vehicle.velocity.x;     vehicle.position.y += vehicle.velocity.y;     vehicle.position.z += vehicle.velocity.z;          vehicle.velocity.x *= (1.0 - airResistance);     vehicle.velocity.y *= (1.0 - airResistance);     vehicle.velocity.z *= (1.0 - airResistance);"}
{"id":"112304","title":"Electric Field Homework Question","text":"Can a charged particle move through an electric field that exerts no force on it?"}
{"id":"98923","title":"Stress calculations in a perforated paper","text":"![Unperforated Paper](http:\/\/i.stack.imgur.com\/y1H0E.jpg) You have a sheet of paper (torn out of a good quality foolscap notebook) as shown above, and you start pulling it apart with both your hands (forces indicating by the blue arrows). Its difficult to tear the paper apart this way --- it has a very high tensile strength, I assume. If you try this at home, you are bound to fail. ![Perforated paper](http:\/\/i.stack.imgur.com\/75tzW.jpg) Now if you make tiny perforations in the paper (indicated by red circles), pull it apart the same way, you'll notice that its very easy to tear apart the paper. And the line of tearing will definitely have a few circles on them. Is this happening because of the high stresses that are formed around the circle? Can someone give a good mathematical as well as verbal explanation of this phenomenon."}
{"id":"98925","title":"Would sea water moderate a nuclear submarine meltdown?","text":"My son asked me: > If nuclear submarines get sunk or blown up, what is to stop them going into > nuclear meltdown? I thought about it and came to the conclusion that because they're in the sea the water itself would act as a coolant, stopping nuclear meltdown. Is this correct, as I wouldn't want to teach him nonsense?"}
{"id":"123694","title":"What is Getting in the Way of Testing D-Wave?","text":"I know there are other questions i.e. Do quantum computers manufactured by D-Wave Systems, Inc. work? , What can the D-Wave quantum computer do? , etc. But I can't seem to find my answer. What is getting in the way of determining if D-Wave is a quantum computer? Why can't we just analyze the way they built, and see if it's quantum? Is there a problem with patents getting in the way? Why can we _still_ not tell?"}
{"id":"59831","title":"What's the avarage temperature of the Universe?","text":"I'm sorry if this was asked before but with all the stars and dark matter and all the other stuff, curently in the Universe, what's the avarage temperature of the Universe? Is it like extremely high or extremely low?"}
{"id":"123270","title":"If a window in a plane cabin breaks will a bunch of water vapor form?","text":"If a plane has some windows broken mid-flight (ie open) will the plane suddenly be full of water vapor because of the difference of pressure?"}
{"id":"68580","title":"Commutator of Lorentz boost generators : visual interpretation","text":"I have always struggled to visualize the correctness of the commutation relation for the generators of the boost in the Lorentz group. We have $$[K_i,K_j] = i \\epsilon_{ijk} L_k$$ I fail to picture this. For definiteness' sake, let's take a point $\\vec{x}$ in my coordinate system, lying in the $O_{xy}$ plane. The difference between boosting first in the $x$ and then in the $y$ direction, and boosting first in the $y$ and then in the $x$ direction ammounts simply to a rotation in the $Oxy$ plane, since $$[K_1,K_2] = iL_3$$ I must be doing something wrong when I draw this on a paper. I realize that this holds at the infinitesimal level, but a simple drawing should be able to realize this principle 'at first order' shouldn't it ? My problem is that when I draw it, I end up with the same situation regardless of which direction I boost into first."}
{"id":"78723","title":"How do you calculate the radiant for Kreutz Sungrazer comets? where in the sky do I look?","text":"According to wikipedia: > All known members of the [Kreutz Sungrazer] group up until 1965 had almost > identical orbital inclinations at about 144°, as well as very similar values > for the longitude of perihelion at 280–282°, with a couple of outlying > points probably due to uncertain orbital calculations. This longitude of 280 deg seems to point in the direction of the Orion constellation. However, the radiant would seem to be 180 degrees away (opposite) since I expect the radiant to point to the location of the apehelion. I'm also wondering if there is any reason behind the apparent mirror image similarity (or otherwise some connection) between the sungrazers and the orionids, I found this image on pinterest http:\/\/www.pinterest.com\/pin\/129197083031621626\/ ![enter image description here](http:\/\/i.stack.imgur.com\/Z6JZr.jpg) and this picture is actually what peeked my interest in wanting to know where in the sky the radiant for the sungrazers is. FWIW the approximate RA of the orionids radiant seems to be roughly that of Saiph's, or 05h 47m 45 seems to translate to a longitude of -93 deg = 267 deg which, as I noted above, lies close to the perihelion of the sungrazers (http:\/\/math.mercyhurst.edu\/~griff\/courses\/webmapping\/RA_convert.php). [--edit: oops it appears the orientation of the pic is \"upside\" down, the red star I mistook for Saiph there in Orion constellation actually seems to be Betelguese whose longitude is -91 deg = 269 deg. Comparing with the stars shown in Stellarium, the radiant seems to lie right in the middle part of Gemini, at RA of 7 hours = longitude of -75 deg = 285 ] [--edit 2: http:\/\/ssd.jpl.nasa.gov\/sbdb.cgi?ID=dK11W030;orb=1;cov=0;log=0;cad=0#orb contains a jpl orbit diagram for comet lovejoy--an example of a Kreutz sungrazer; the orbit diagram shows that the RA of the APHELION--not perihelion-- of lovejoy is aligned roughly with gemini (or jupiter currently), so now I am completely confused and am wondering if there is a typo in the wiki sungrazer page, or what the misunderstanding on my part is. ] If anyone can point me to where in the sky the sungrazer radiant (such as nearest star\/s) is it would be really appreciated from me."}
{"id":"96656","title":"Why does a tail of a balloon poodle not collapse?","text":"I was just wondering. How can it not collapse? There is pressure in the tail top and in tail itself. How can that little \"bridge\" (with that I mean the part that in not inflated, see image 2) and tail top sustain and not collapse?      ![enter image description here](http:\/\/i.stack.imgur.com\/xj6kj.jpg)   **1\\. Image of balloon poodle** ![enter image description here](http:\/\/i.stack.imgur.com\/hj9lY.png)   **2\\. Scheme of the balloon tail with a schematic \"pressure\"** As you can see in _Image 2_ I guess that the pressure is something like that. My best guess is that the surface tension or strenght of the \"bridge\" is strong enough to sustain. But I would like to know what you guys think!"}
{"id":"103411","title":"Calculate trajectory of projectile with gravity from disk planets in 2D?","text":"I'm making a 2D game where you control a spaceship and fly between disk-formed planets of differing size (mass\/pull). The planets are small and you can fly near several at once. I need to calculate the gravity produced by each of these planets pulling on the ship including the velocity of the ship. I figured out a way to approximate it:   1. Calculate the position of the rocket in the next frame based on the angle and velocity   2. From this position, calculate the position of a point pulled towards the planet where the larger the planet, the further this point is towards said planet   3. Do this for each planet then average each of these points to find the new location for the rocket this frame   4. Rotation doesn't matter in my case, but I would calculate it by comparing this point's location and the previous point's location.   5. Repeat for every frame. This may be a good approximation, and accuracy isn't a huge deal in my case, but I am wondering if there is a better way to do this with physics formulas. I haven't taken a physics class yet. The other main problem with this solution is that it must calculate it step by step or frame by frame. If there is a formula to do this the path could be calculated from the very start before."}
{"id":"43118","title":"Ultracold atoms and General theory of Relativty","text":"I am looking for good reviews for the subject of Ultracold atoms and it's application in test of General theory of relativity. I am planning to pick up this topic as a semester project, can somebody guide me ? I am basically looking forward to some good books and papers which gives me a flavor of theoretical aspects of this subject. I came across an website which says about Quantum gauge theories and Ultracold atoms and that has interested me a lot. http:\/\/cordis.europa.eu\/search\/index.cfm?fuseaction=proj.document&PJ_RCN=10276613"}
{"id":"111828","title":"Understanding how clouds form: warm air near the earth's surface?","text":"I'm trying to understand how clouds are formed. I've read that it begins with air near the earth's surface heating up. But I'm hearing two different reasons for how this happens:   1. Air pressure near the earth's surface is high and this causes water vapor near the earth's surface to heat up, since the high pressure causes the kinetic energy of the water vapor molecules to increase.   2. The sun heats up the earth's surface, and this heats up the water vapor near the earth's surface. So what is the truth?"}
{"id":"67206","title":"Nonlinear combination of velocities implies no absolute time?","text":"Landau 1961 begins with a brief presentation of special relativity. This question is about the validity of a certain argument that they use in building up the foundations of the subject from scratch, so I think I should first sketch what they consider foundational. They define an inertial frame as one in which Newton's first law holds. They telescope Einstein's two 1905 postulates into a single principle of relativity, which is that the laws of physics are form-invariant w.r.t. changes of inertial frame. They assert (based on unspecified experiments) that forces are propagated at a finite speed. They also seem to assume (presumably also based on experiment) that this speed is a maximum and universal for all interactions. Given a maximum velocity, they argue that time is not absolute. The argument is that in Galilean relativity, velocities combine linearly, which is incompatible with a maximum velocity and with the Michelson-Morley experiment. Also, Galilean relativity posits absolute time. Therefore absolute time must be wrong. This seems like a clear logical fallacy to me. (A implies B, and A also implies C. B is false, so C must be false.) [EDIT] Oops, crucial mistake in my original statement of the question. The parenthetical originally ended with \"C must be true.\" Should have ended with \"C must be false.\" Is there any way of salvaging this argument? Am I misunderstanding it? Landau and Lifshitz, The classical theory of fields, 2nd ed., 1961"}
{"id":"111824","title":"What is the infinitesimal work done when the force is given by the gradient of a scalar function that depends both on position AND time?","text":"The title is slightly confusing but I didn't know how else to phrase my question. Basically, this is the situation: When the force applied to a particle is given by the gradient of a scalar function that depends explicitly on JUST the position of the particle, that force is conservative and $T+V$ is conserved.   However, if it depends explicitly on both position AND time, then $T+V$ is still defined but not conservative. In my notes, this is proven: $$F(r(t),r)\\cdot \\dot{r} dt = -\\nabla V(r(t),t)\\cdot \\dot{r}dt = \\left( -\\frac{dV(r(t),t)}{dt}+\\frac{\\partial V(r(t),t)}{\\partial t}\\right)dt$$ What I don't understand is the second line, working it out a few times but I never get the correct result."}
{"id":"96652","title":"d=2 O(3) sigma model becomes \"conformal antiferromagnet\"","text":"In Advanced topic in quantum field theory \/ M. Shifman on page 251 the author discusses the fact that the theta term is topological and does not affect the equations of motion. Then he said: > \"In particular, at $\\theta=\\pi$ the O(3) sigma model becomes conformal and > describes a ferromagnet rather than antiferromagnet. Unfortunately, I cannot > dwell on the aspect in this text\" Where he refers to the term $\\frac{\\theta}{8\\pi}S^{a}(\\partial_{\\mu}S^{b})(\\partial_{\\nu}S^{c})\\varepsilon^{\\mu\\nu}\\varepsilon_{abc}$. I would like to see here more details about this point. Is it a uniqe phenomena for $\\theta=\\pi$?"}
{"id":"96650","title":"What does quantum phenomena exist as prior to observation?","text":"It's been said that according to the Schrodinger equation, independent of observation, particles exist in a state of a wave function, which is a series of potentialities rather than actual objects. Then the act of observing, causes a wave of potentiality to collapse to a state of matter. Is this true? Also, are these quantum experiments anomalies, and not how nature works, or do these quantum experiments, like double slit and quantum erasers, actually reflect how nature works even in the absence of human manipulations."}
{"id":"29607","title":"Cases in which angular velocity and angular momentum point into same direction","text":"I know that angular momentum $\\vec{L}$ and angular velocity $\\vec{\\omega}$ of a rigid body doesn't point into the same direction in general. However if your body spins around a principal axis, $\\vec{L}$ and $\\vec{\\omega}$ point into the same direction. In which other situations is this is exactly or at least approximately true? Please give reasons or a mathematical derivations why it is true in a given situation."}
{"id":"29600","title":"Intuitive explanation for the de Broglie \/ Planck relations","text":"A friend asked me to explain \"why\" a particle's energy is proportional to it's frequency, i.e: $$E=h\\nu$$ The reason this result is so un-intuitive, is that in the macroscopic world, A wave's energy (e.g electric power dissipated through a resistor) is not frequency dependent but rather amplitude dependent. Of course a \"pure\" photon doesn't really have an amplitude due to it's description as a plane wave, but that's beside the point. **So in short** : Is there an easy explanation of \"why\" the energy is only dependent on frequency? can this be \"derived\" from other principles (say, the particle-wave duality)? * * * **N.B.** I'm aware there are no real \"reasons\" for any law of physics other then the fact that they agree with experiment, but laying those laws on a smaller number of basic principles is always nice when possible."}
{"id":"34632","title":"Can two electrons never touch each other?","text":"The Coulomb's force is given by $$ F = {k q^2 \\over r^2} $$ When $ r \\rightarrow 0 $, $ F \\rightarrow \\infty $ Does this mean two electrons never touch each other?"}
{"id":"109079","title":"What determines the speed required to pull a table cloth?","text":"I was watching this show \"Street Genius\" on National Geographic and the host Tim Shaw demonstrated an experiment about Inertia, What he did was, He tied one end of a table cloth to a car through a long rope and started driving the car, when the rope was taut, the cloth was pulled and the things on the table were still on the table. What are the calculations involved in it and what determines the speed required to pull the table cloth without disturbing the things on it?"}
{"id":"76986","title":"Probability density of detection of collinearly emitted photons in two detectors","text":"**Update** :   As proposed by @dmckee, I added equation numbers and improved the display of some equations. The answer by @Trimok inspired me to look at coordinate systems which are not specific to the detector locations, but more general, and this indeed made things clearer since $\\delta(\\cos \\theta + 1)$ is a $\\delta(\\cos \\theta + \\cos 0)$ in disguise. We have a relation between a Dirac delta of a ray in spherical coordinates, $\\delta^2$, and a Dirac delta of a point, $\\delta^3 = \\delta(r-r_0)\\delta(\\theta-\\theta_0)\\delta(\\phi-\\phi_0)\/(r^2 \\sin \\theta)$, such that $\\delta^2 = \\int_0^\\infty \\delta^3 \\operatorname{d}r_0 = \\delta(\\theta-\\theta_0) \\delta(\\phi-\\phi_0) \/ (r^2 \\sin\\theta) \\tag{E1}$ (Using this definition, one can easily show that the sifting property of $\\delta^2$ holds and the result is a line integral of some $f$.) The major problem (of understanding) which I am having is now reduced to this: We have two expressions for $\\delta^3$, which are $\\delta(r-r_0)\\delta(\\theta-\\theta_0)\\delta(\\phi-\\phi_0)\/(r^2 \\sin \\theta) \\tag{E2}$ and $\\delta(r-r_0)\\delta(\\theta-\\theta_0)\\delta(\\phi-\\phi_0)\/(r_0^2 \\sin \\theta_0) \\tag{E3}$ These are totally symmetric in $x$ and $x_0$ (note that $\\delta$ is an even function). However, for $\\delta^2$ we have $\\delta(\\theta-\\theta_0)\\delta(\\phi-\\phi_0)\/(r^2 \\sin \\theta) \\tag{E4}$ and $\\delta(\\theta-\\theta_0)\\delta(\\phi-\\phi_0)\/(r^2 \\sin \\theta_0) \\tag{E5}$ which are not symmetric any longer. So what makes the $r$ stand out compared to $r_0$? One explanation I worked out with the help of a colleague is directed towards the fact that there is no $r_0$ after integration, so having any $r_0$ would \"overdetermine\" the set of rays through the origin. Does anyone have a better (for example, more intuitive) explanation why the symmetry is lost here? Thank you! **Original question** :   This is a long one. I have been working on this for quite a while now, so I summarized most of what I found out. I am stuck at a certain point, however, and it would be great if someone could help me out. Assume the following situation: Two perfectly collinear photons $p_1$ and $p_2$ are emitted in the origin of some Cartesian coordinate system. We consider two infinitesimal detectors $d_1$ and $d_2$, with surfaces $\\operatorname{d}\\\\!A_1$ and $\\operatorname{d}\\\\!A_2$ directed towards the origin so that we do not have to deal with angles. The solid angles of these detectors be $\\operatorname{d}\\\\!\\Omega_1$ and $\\operatorname{d}\\\\!\\Omega_2$. I now express some basic detection probability densities, where $p_id_i$ means that \"$p_i$ is emitted towards $d_i$\".   $\\frac{\\operatorname{d}\\\\!P(p_1d_1)}{\\operatorname{d}\\\\!\\Omega_1}=\\frac{1}{4\\pi} \\tag{1}$ $\\frac{\\operatorname{d}\\\\!P(p_1d_1)}{\\operatorname{d}\\\\!A_1}=\\frac{1}{4\\pi (r_1)^2} \\tag{2}$ $\\frac{\\operatorname{d}\\\\!P(p_2d_2)}{\\operatorname{d}\\\\!\\Omega_2}=\\frac{1}{4\\pi} \\tag{3}$ $\\frac{\\operatorname{d}\\\\!P(p_2d_2)}{\\operatorname{d}\\\\!A_2}=\\frac{1}{4\\pi (r_2)^2} \\tag{4}$ I think there is no doubt that these probabilities are well-defined and the integrated detection probabilities in a spherical $4\\pi$-detector equals 1, no matter what its radius is. So a photon is surely emitted towards a detector surrounding the origin. Now, for some reason we want to calculate $P(p_2d_2 | p_1d_1)$, the conditional probability of $p_2$ emitted towards $d_2$, given that $p_1$ has been emitted towards $d_1$. Due to collinearity of the photons, the individual events are not independent: $P(p_2d_2 | p_1d_1) \\neq P(p_2d_2)$. I now propose that $\\frac{\\operatorname{d}\\\\!P(p_2d_2 | p_1d_1)}{\\operatorname{d}\\\\!\\Omega_2} = \\frac{\\delta(\\cos \\theta_2(d_1)+1)}{2\\pi}, \\tag{5}$ where $\\theta_2(d_1)$ is the polar angle of the position of $d_2$ expressed in spherical coordinates with polar axis from the origin through $d_1$. $\\delta$ is the Dirac function, so obviously, this expression equals 0 if $\\cos \\theta_2(d_1) \\neq -1$, that is, when $\\theta_2(d_1) \\neq \\pi$. We can verify that the integral of this probability over a spherical $4\\pi$-detector $d_2$ equals 1: with $\\operatorname{d}\\\\!\\Omega_2 = \\operatorname{d}\\\\!\\cos \\theta_2(d_1) \\operatorname{d}\\\\!\\phi_2(d_1)$, we find $\\int \\frac{\\operatorname{d}\\\\!P(p_2d_2 | p_1d_1)}{\\operatorname{d}\\\\!\\Omega_2} \\operatorname{d}\\\\!\\Omega_2 = \\int \\frac{\\delta(\\cos \\theta_2(d_1)+1)}{2\\pi} \\operatorname{d}\\\\!\\Omega_2 \\tag{6}$ $= \\int_0^{2\\pi} \\frac{1}{2\\pi} \\operatorname{d}\\\\!\\phi_2(d_1) \\cdot \\int_0^\\pi \\delta(\\cos \\theta_2(d_1)+1) \\operatorname{d}\\\\!\\cos \\theta_2(d_1) = 1. \\tag{6}$ So $p_2$ is still surely emitted towards a detector surrounding the origin, no matter where $p_1$ is emitted towards. Further, we would like to verify that the law of total probability is fulfilled by integrating $P(p_1d_1 \\wedge p_2d_2) = P(p_2d_2 | p_1d_1) \\cdot P(p_1d_1)$ over a spherical $4\\pi$-detector $d_1$. This works regardless of which formulation of $\\operatorname{d}\\\\!P(p_1d_1)$ (with respect to $\\operatorname{d}\\\\!\\Omega_1$ or $\\operatorname{d}\\\\!A_1$) we use -- I chose $\\operatorname{d}\\\\!A_1 = (r_1)^2 \\operatorname{d}\\\\!\\cos \\theta_1(d_2) \\operatorname{d}\\\\!\\phi_1(d_2)$ here. Note, however, that integration over $d_1$ changes the polar axis used in the formulation of $\\operatorname{d}\\\\!P(p_2d_2 | p_1d_1)$ above, so we need a different, but equivalent formulation. This is more than a change of coordinate system (from one polar axis to another), because $\\theta_2(d_2)$ would not help us a lot (it's 0). Instead, one can use $\\frac{\\operatorname{d}\\\\!P(p_2d_2 | p_1d_1)}{\\operatorname{d}\\\\!\\Omega_2}=\\frac{\\delta(\\cos \\theta_1(d_2)+1)}{2\\pi}, \\tag{7}$ which uses both a different variable in the argument of the $\\delta$ as well as a different coordinate system: $\\theta_1(d_2)$ is the polar angle of detector $d_1$'s location in spherical coordinates with polar axis from the origin through $d_2$. **Note that both expressions of $\\frac{\\operatorname{d}\\\\!P(p_2d_2 | p_1d_1)}{\\operatorname{d}\\\\!\\Omega_2}$ are symmetric in the indices 1 and 2.** Finally, we arrive at $\\int \\frac{\\operatorname{d}\\\\!P(p_1d_1 \\wedge p_2d_2)}{\\operatorname{d}\\\\!A_1 \\operatorname{d}\\\\!\\Omega_2} \\operatorname{d}\\\\!A_1 = \\int_0^{2\\pi} \\int_0^\\pi \\frac{1}{4\\pi (r_1)^2} \\cdot \\frac{\\delta(\\cos \\theta_1(d_2)+1)}{2\\pi} (r_1)^2 \\operatorname{d}\\\\!\\cos \\theta_1(d_2) \\operatorname{d}\\\\!\\phi_1(d_2) \\tag{8}$ $= \\frac{1}{4\\pi} \\int_0^{2\\pi} \\frac{1}{2\\pi} \\operatorname{d}\\\\!\\phi_1(d_2) \\cdot \\int_0^\\pi \\delta(\\cos \\theta_1(d_2)-1) \\operatorname{d}\\\\!\\cos \\theta_1(d_2) = \\frac{1}{4\\pi} = \\frac{\\operatorname{d}\\\\!P(p_2d_2)}{\\operatorname{d}\\\\!\\Omega_2} \\tag{8}$ So, the law of total probability is fulfilled: the probability of $p_2$ being emitted towards $d_2$ can be partitioned into all conditional probabilities of $p_2$ being emitted towards $d_2$ _given that_ $p_1$ has been emitted towards $d_1$, weighted by the probability that $p_1$ is _actually_ emitted towards $d_1$. So far, so good. Congratulations for reading up to this point :) As expressed above (and verifiable relatively easily), it does not matter which formulation of $\\operatorname{d}\\\\!P(p_1d_1)$ I use -- both $\\operatorname{d}\\\\!\\Omega_1$ and $\\operatorname{d}\\\\!A_1$ work fine. However, when I try to base my calculations on $\\frac{\\operatorname{d}\\\\!P(p_2d_2)}{\\operatorname{d}\\\\!A_2}$ instead of $\\frac{\\operatorname{d}\\\\!P(p_2d_2)}{\\operatorname{d}\\\\!\\Omega_2}$, hence, I try to calculate $\\frac{\\operatorname{d}\\\\!P(p_2d_2 | p_1d_1)}{\\operatorname{d}\\\\!A_2}$ instead of $\\frac{\\operatorname{d}\\\\!P(p_2d_2 | p_1d_1)}{\\operatorname{d}\\\\!\\Omega_2}$, I arrive at the following problem. So I started with $\\frac{\\operatorname{d}\\\\!P(p_2d_2 | p_1d_1)}{\\operatorname{d}\\\\!A_2}=\\frac{\\delta(\\cos \\theta_2(d_1)+1)}{2\\pi (r_2)^2}, \\tag{9}$ and (6) is still easily verified (integrating over $\\operatorname{d}\\\\!A_2$) since the two additional $(r_2)^2$ simply cancel out. However, working out (8) is now more complex when it comes to changing from one variable\/coordinate system to another. I figured that $\\frac{\\operatorname{d}\\\\!P(p_2d_2 | p_1d_1)}{\\operatorname{d}\\\\!A_2}=\\frac{\\delta(\\cos \\theta_1(d_2)+1)}{2\\pi (r_2)^2} \\tag{10}$ is the expression which fulfills total probability (I can show this, if necessary), but now I wonder why this is not symmetric with the first expression in the indices 1 and 2 (as it was above). Specifically, I wonder where the connection to a Dirac of a ray along Oz in spherical coordinates is, which is $\\frac{\\delta(\\cos \\theta - 1)}{2\\pi r^2}$ (compare http:\/\/www.physicsforums.com\/showthread.php?t=215263) More generally, I am working on expressing the probability density in 3D, so I expect to obtain something like $\\operatorname{d}\\\\!\\Omega_2 \\cdot \\delta_{L(x_1, x_2)}(x_0)$ or $\\operatorname{d}\\\\!A_2 \\cdot \\delta_{L(x_1, x_2)}(x_0)$, where $L(x_1, x_2)$ is the line segment from $x_1$ and $x_2$, and $x_0$ is the point of emission. So my specific questions are:   * Why are the expressions involving $\\operatorname{d}\\\\!A_2$ not symmetric, while those involving $\\operatorname{d}\\\\!\\Omega_2$ are?   * Specifically, where is the connection to the Dirac of a ray in spherical coordinates, which involves an $r^2$ and which I would expect to be changed when converting from one coordinate system to another.   * How (else) can I obtain a 3D probability density involving the three locations, which can be used to integrate over all three volumes in order to describe the number of coincidences detected from emissions from a volume in two finite detectors?"}
{"id":"112844","title":"Energy Required To Push Up a Slope","text":"I'm having a bit of trouble with this problem: A $5.0\\:\\text{kg}$ block is moved up a slope with constant velocity of $2.5\\:\\text{m\/s}$. The amount of force required to push the block up the slope is $20.0\\:\\text{N}$. The slope is $45.0\\:\\text{cm}$ high, $150\\:\\text{cm}$ horizontally on the ground, and its hypotenuse\/slope is $125\\:\\text{cm}$. How much energy is required to push the block up the slope? Angles are not supposed to be used in order to obtain the answer. (I initially thought that the energy required would just be equal to the potential energy at the top of the slope, i.e. $mgh$. However, I'm not sure if there are other factors that are needed to be taken into consideration?)"}
{"id":"35328","title":"Why does observation collapse the wave function?","text":"In one of the first lectures on QM we are always taught about Young's experiment and how particles behave either as waves or as particles depending on whether or not they are being observed. I want to know what about observation causes this change?"}
{"id":"55252","title":"Is angular momentum always conserved in the absence of an external torque?","text":"Consider either the angular momentum of the earth around the sun or equivalently swinging a ball horizontally on a string. I know that with respect to the point of rotation of the swinging ball, angular momentum is constant. Consider now an origin outside of the 'orbit'. It seems to me that the angular momentum is different as the orbit progresses but there is no external torque (the physical scenario is unaltered!). The easiest way to see this is to consider when the ball is at opposite ends of the circle - the radius changes but the velocity only flips sign. In addition to this, how about we consider an event where we cut the string. Now place our reference point on the line of motion when the ball flies away in a straight line. By definition the ball will have zero angular momentum with reference to any point on this line ($m|r||v|sin\\theta=0$) but certainly before the event it had some angular momentum. The main question is in the title, but essentially what am I missing conceptually about this problem?"}
{"id":"74586","title":"Can one prove the full spin-statistics theorem from the spin 0, 1\/2 and 1 cases?","text":"Using second quantization for scalar field, spinor field and vector fields, we can get commutation and anticommutation relations for the birth and destruction operators of the fields, which leads us to the Bose or to Fermi statistics. Is it possible to expand these results on a field of arbitrary spin (integer or half-integer), using the basic idea that each field can be built by combination of spinor $\\frac{\\hbar }{2}$ fields?"}
{"id":"55258","title":"What is a two-photon process?","text":"I am reading some introductory materials on modern optics, in which they mention two-photon processes everywhere. I know fundamental optics and a bit on quantum mechanics. Can anyone explain in a plain way what is a two-photon process or any simple introduction to a related topic."}
{"id":"6174","title":"If I take a handful of salt and wait for an infinite time will it become a single crystal?","text":"That pretty much says it. Suppose I have some powder of $NaCl$. It is kept in contact with itself in vacuum. You are free to remove all the disturbances that bother you. Is that true that, well, there exist a ($\\mbox{very}^\\mbox{very}$ large) amount of time $T$ that for every moment $t>T$ you will have a single beautiful crystal with 99% probability."}
{"id":"77583","title":"How do I build an alpha emitter?","text":"I would like to build an alpha emitter. I suppose I had some vague idea of knocking the electrons off of some helium. But upon further examination the idea does strike me as...naive? My question is then, how do I go about building an alpha emitter, cheaply, and with household\/hardware store items. (Besides springing for a smoke detector.)"}
{"id":"24605","title":"Infinite acceleration?","text":"Why is acceleration regulated by mass? In a frictionless environment, why doesn't an object move at infinite acceleration if force is applied on it? Force causes movement, so unless there is an opposing force there shouldn't be any reason for the force to cause infinite acceleration. Why does mass act like an opposing force that limits the acceleration caused by an force to less than infinity?"}
{"id":"24601","title":"Does the earth's gravity affect horizontal movement?","text":"In a frictionless environment, would the weight of an object affect how hard it is to move it horizontally? Since without friction, there is no horizontal force resisting your applied force, would the object be easy to move? I am aware of inertia but don't see any reason for it to play a role considering there is no opposing force to your applied force."}
{"id":"24600","title":"Why does the force of gravity get weaker as it travels through the dimensions?","text":"Some theories predict that the graviton exists in a dimension that we of course can't see, and that is why the force of gravity is so weak. Because by the time gravity has got from the dimension in which the graviton exists to our dimension it has lost much of its strength. For a better understanding of this theory, I think this might help: ![](https:\/\/lh4.googleusercontent.com\/-M3Z7HBllZys\/T5JT- dGobZI\/AAAAAAAAFIE\/I_XJUA3M0pI\/randall_750.jpg) Bigger: https:\/\/lh4.googleusercontent.com\/-M3Z7HBllZys\/T5JT- dGobZI\/AAAAAAAAFIE\/I_XJUA3M0pI\/randall_750.jpg?imgmax=1600 Why does the force lose strength as is travels through the other dimensions?"}
{"id":"130798","title":"Normal ordering","text":"If I understood correctly there are two terms called normal ordering:   1. $:c c^\\dagger: = c^\\dagger c \\hspace{.5cm}$so shifting all creation operators to the left and all annihilation operators to the right.   2. $:c^\\dagger c:=c^\\dagger c- \\langle c^\\dagger c \\rangle\\hspace{.5cm}$ so subtracting the contraction Both of these are called normal ordering (I think) and use the same notation. But what is the connection? If I got something wrong, please correct and explain!"}
{"id":"79022","title":"How do you determine the value of the degeneracy factor in the partition function?","text":"In the partition function, expressed as $$Z = \\sum_j g_je^{-\\beta E_j}$$ I'm wondering what determines the $g_j$ factor. I've been trying to look around the internet for an explanation of it but I can't find one. I guess it is the number of degenerate states in a given energy level? How do you determine how many degenerate states there are? A simple example involving how to determine the $g_j$ factor would be great. Thanks for the help!"}
{"id":"77635","title":"Microscopic Theory of superfluidity","text":"In comparison with superconductivity what is the microscopic theory of superfluidity? Who has done it?"}
{"id":"17134","title":"Inertial Mass of a scalar field","text":"Does it make sense to talk of the inertial mass of a scalar field? By the equivalence principle, it must be equal to its gravitational mass. We know that the scalar field contributes towards the stress-energy tensor, so, shouldn't it have an inertial mass too?"}
{"id":"130017","title":"How could we define speed without time?","text":"This is a bit of a brain twister but it's a very serious question. Please be precise in your answer. I do not believe time exists. I believe it is simply an illusion of our perception, the perception of seeing things around us change. The faster things change the more time seems to go by. In some sort, change is the result of the dissipation of energy. Thus what I believe is that what we are really observing is the energy being dissipated and created all around us and we call it time. Since time doesn't exist (for me). That would mean that the equation of [ speed = distance \/ time ] is impossible. But nevertheless the concept of speed is fundamental for physics. What could we replace this equation with to no longer have speed dependent of time but of energy?"}
{"id":"75678","title":"Can a sound like a sonic boom change a object like a wall? And how the light on the walls will be changed after a sound like this?","text":"Can a sound like a sonic boom change a object like a wall? And how the light on the walls will be changed after a sound like this?"}
{"id":"75675","title":"What is ``thermal\" about a thermal quotient of EdS and EAds?","text":"This is in continuation of my previous question and is in reference to this paper. I guess that the authors are interested in $S^n$ and $\\mathbb{H}^n$ since these are the Euclideanized versions of $dS^n$ and $AdS^n$ respectively. But I am not clear about the physical motivations of the quotients that they take. They take the following quotients,   * The sphere $S^{2n+1} = SO(2n+2)\/SO(2n+1)$ can be thought to be given as the zero-set in $\\mathbb{C}^{n+1}$ of the equation, $\\sum_{i=1}^{n+1} \\vert z_i \\vert ^2 = 1$. Here they want to identify $\\phi \\sim \\phi + \\frac{2\\pi}{n}$ for some $n \\in \\mathbb{N}$ where $\\phi$ is the phase of $z_1$. And this quotient is given by $\\mathbb{Z}_n\\backslash SO(2n+2)\/SO(2n+1)$   * On $\\mathbb{H}^{2n+1} = SO_0(2n,1)\/SO(2n)$ they take the metric as, $cosh^2\\rho dt^2 + d\\rho^2 + sinh^2 \\rho d\\Omega_{2n-1}$ and take the quotient as $t \\sim t + \\beta$ and then the quotient space is given as $\\mathbb{Z} \\backslash SO_0(2n,1)\/SO(2n)$ . Here $\\frac{2\\pi}{n}$ and $\\beta$ are connected by the same analytic continuation as what takes one from the metric on $S^{2n+1}$ to the metric on $\\mathbb{H}^{2n+1}$ Clearly in the paper they keep referring to \\beta as the inverse temperature but I can't clearly see what is the thermal picture which will correspond to evaluating the partition function on the these quotients as above. If there were a particle at inverse temperature $\\beta$ on spatial $\\mathbb{H}^{2n}$ then I would think that one would calculate the partition function on the manifold $\\mathbb{H}^{2n}\\times S^1$ such that the $S^1$ is of radius $\\beta$. Is this somehow related to the quotient of $\\mathbb{H}^{2n+1}$ described above in the second bulled point? (...the same could have been asked about $S^{2n} \\times S^1$ also..)"}
{"id":"70005","title":"At what point does everything become nothing?","text":"I understand that the universe, which I'll call \"everything\", is expanding and it used to be much smaller. But I keep hearing assertions about a universe coming from nothing. If you rolled the clock back, at what point would everything become nothing?"}
{"id":"33180","title":"Lepton masses in the Standard Model","text":"Some simple questions regarding leptonic masses in the Standard Model (SM):   1. Why there is not an explicit mass term in addition to the effective mass term that arises from the Yukawa terms after spontaneous symmetry breaking (SSB).   2. The lepton mass term is: $$m_{l_i}\\sim y_i\\cdot v \\sim y_i\\cdot \\frac{m_H}{\\sqrt{\\lambda}}$$ with $m_{l_i}$ the lepton mass, $y_i$ the Yukawa coupling, $v$ the vacuum expectation value of the Higgs field, $m_H$ the Higgs mass, and $\\lambda$ the self-coupling of the Higgs.   So the running of leptonic masses in the SM are given by the running of $y_i$, $m_H$, and $\\lambda$. Thus the lepton masses are not protected to radiative corrections $\\beta(m_{l_i})\\neq 0$ for $m_{l_i} = 0$. This is in contrast with explicit lepton mass terms in theories without SSB where Chiral symmetry protect fermions' masses from radiative corrections. Is this correct? If not, why?"}
{"id":"33181","title":"Relation among anomaly, unitarity bound and renormalizability","text":"There is something I'm not sure about that has come up in a comment to other question: Why do we not have spin greater than 2? > It's a good question--- the violation of renormalizability is linked > directly to violation of unitarity, which was exploited by Weinberg > (surprise, surprise) to give an upper bound of something like 800 GeV on the > fundamental Higgs mass from the W's and Z's unitarization. The breakdown of > renormalizability is a wrong one-loop propagator correction to the gauge > boson, and it leads to a violation of the ward-identity which keeps the non- > falling-off part of the propagator from contributing. It's in diagrammar (I > think), it's covered in some books, you can ask it as a question too. I know what unitarity bound is the user talking about, but I don't know what is the violation of the Ward identity that he mentions. I guess that it is the _global_ $SU_L(2)$ symmetry but I have never seen anything relating the unitarity bound and this anomaly. The general issue is the following: Assume a Yang-Mills term and the coupling of the charged vector field to a fermionic conserved current under a global symmetry. Then one adds an explicit mass term to the vector field so that one breaks the gauge symmetry by hand, but not the global part that gives the conserved current (the gauge symmetry that goes to the identity in the boundary entails constraints). Then, according to the user (at least what I understood), when one takes into account loops effect the global part is also broken. Therefore, the mass term is breaking the redundancy part of the symmetry by hand (at the classical level) and it is also breaking the global part at the quantum level. I would be grateful if somebody is able to clarify this to me. References are also welcome."}
{"id":"36005","title":"When and how were relative distances to the planets first measured?","text":"I understand that the absolute distance to a planet can be measured using earth-baseline (e.g., diurnal) parallax, and that the first reasonably accurate such measurement was made for Mars by Cassini (and his assistant Richer) in 1672, and then, famously (with some modifications) for Venus by Halley, Cook, and others and others during its 1769 transit; but when were earth-baseline _relative_ distance measurements first made for each of the planets? Interpretation of such measurements requires assumptions about the nature of the orbits of planets, but some of the most compelling evidence for the validity of these assumptions is Kepler's third law of planetary motion, the persuasiveness of which itself rests on observations of relative distances. In fact, all of my texts say, in effect, that the third law \"fit\" the observed pattern of orbital period and relative distance. Where did these distance measurements come from? What relative distance measurements were available to Kepler in 1619? When was earth-orbit-baseline parallax used to make these measurements for each of the planets?"}
{"id":"71353","title":"What is Si-delta doping?","text":"I want to know what the delta means in this case. I know the Si means the element used, by some way to doping. I guess the delta means that using some elements to create holes in semiconductor made of Si. Am I right?"}
{"id":"37759","title":"Could Dark Matter be a manifestation of tachyons tunnelling between black holes?","text":"Among the many Dark Matter candidates, I wondered if there are any along the lines of the title. The inner horizon of a black hole, with its mass inflation, and\/or the high spacetime curvature, seems a violent enough place for the strangest things to happen that can anywhere, and tachyons (if these exist) seem to have enough counterintuitive properties that they might perceive far greater space volumes accessible inside black holes than outside, so that any created inside and straying out would readily tend to find their way back into the same or another black hole, with some energy change while in transit that would manifest itself outside the hole(s) as \"dark mass\" (assuming this was a fairly steady state process). Piling speculation on speculation, one might further envisage that a tachyon which exited a black hole and lost enough energy outside (and hence gained speed) for long enough might greatly increase its chance of staying outside and thus manifest itself as dark energy. The latter would also explain why the expansion of the universe is accelerating: If the average distance between black holes (between galaxies for example) is increasing with cosmic expansion, then on the preceding assumption slightly more tachyons can be expected to remain outside black holes, and so on. It seems all this would make most sense (if at all) if tachyons appeared somehow \"spread out\" in normal space, like a field, as opposed to conventional sub-luminal particles. Apologies if this is all too vague and speculative for some peoples' liking. I am not axe-grinding, quite the opposite, just curious to hear any informed comments on an idle fancy which is most likely untenable."}
{"id":"30769","title":"Fracture because of high-speed rotation","text":"I was watching a rerun of an early MythBusters episode, where they look at whether CDs in high-speed drives can explode \/ fail simply because of being rotated too fast. The following are some calculations I did: Let the thickness of the CD be $t$, made of material with density $\\rho$. Consider a small element of the CD at a distance $r$ from the center. The element subtends an angle $d\\theta$ at the center, and is of length (along the radius) $dr$. The centripetal force needed to pull this in at angular velocity $\\omega$ is $m r \\omega^{2}=\\rho \\left(r d\\theta dr t\\right) r \\omega^{2}$. The CD has a central spindle hole of radius $r=1.5\\ cm$, and is itself of radius $R=12\\ cm$. The force then exerted on an elemental area subtending angle $d\\theta$ at the inner edge of the CD is (integrating the previous quantity from $r=r$ to $R$) $\\frac{\\rho \\left(R^3 - r^3\\right) d\\theta t \\omega^2}{3}$, which means the inside edge has pressure approximately $\\frac{\\rho R^3 \\omega^2}{3r}$ (dividing the previous quantity by the area $t r d\\theta$, and assuming $R^3$ dominates $r^3$). Fracture now happens when $\\frac{\\rho R^3 \\omega^2}{3r} > T$, the fracture strength of polycarbonate plastic (which being brittle, is approximately its ultimate tensile strength). Substituting $\\rho = 1.22\\ g\\ cm^{-3}$, and $T = 75\\ MPa$ (all approximate values obtained from Wikipedia), I got $\\omega > 17000\\ rpm$. My question is, is the calculation correct? Or is there something I'm missing or misinterpreting? The value is the same order of magnitude as what MythBusters reported -- fracture at $\\approx 25000\\ rpm$. This is heartening, but it would be nicer to know that the number is off only because of incorrect inputs (strength of plastic, density etc.). As nice as MythBusters sometimes is, it would be nicer still if they did a little more hard science. * * * Correction: As akhmeteli points out below, the CD dimensions are incorrect. The central spindle hole has radius $7.5\\ mm$, and the CD itself has radius $60\\ mm$. There's also an arithmetic mistake in the above calculation. The updated answer is therefore $\\approx 24000\\ rpm$. That would ordinarily make me very happy, but as both user758556 and ja72 argue, this calculation would be useful as nothing more than a sanity check. What's really happening is a little too complicated for these arguments to properly explain."}
{"id":"47839","title":"Elastic potential","text":"I have a doubt: elastic potential energy is given by: $U=\\frac{k}{2}x^2+K$ but does elastic potential exist? (for example: potential gravitational energy is given by $U=mgz+K$ and gravitational potential is given by $V=-U$) ($K$ is the constant of integration)"}
{"id":"14517","title":"What are the implications for quantum gravity if the LHC sees no higgs?","text":"Following the news in serious \"non-hype\" physics blogs I`ve learnd that as things are now one needs a lot of patience and more data to learn what happens in the higgs sector. There are already a number of good questions and discussions about what the absence of a higgs would mean for the standard model or more general for theoretical physics on this site. So I want to \"extrapolate\" this a bit and ask about what would be the implications for the different quantum gravity approaches if no higgs boson(s) (are) seen at the LHC ?"}
{"id":"72870","title":"Mass of empty AdS$_5$","text":"Five dimensional empty AdS$_5$ space has mass $$ E = \\frac{3 \\pi \\ell^2}{32 G}. $$ > Is the above equation correct? Let's do some dimensional analysis to confirm. In natural units, in 5 dimensions $[G] = -3$ where $[...]$ is the mass dimension. Also $[\\ell]=-1$. Therefore $\\left[ \\frac{\\ell^2}{G} \\right] = 1$. So the dimensions seem to work out fine. Here's my second question: > The limit of $\\ell \\to \\infty$ of $AdS_5$ space is flat space. Isn't it > weird that the mass diverges in this limit? I would have presumed that mass > should vanish in this limit, since flat space has vanishing mass? Are we > using two different definitions of mass? EDIT: Due to some requests in the comments, I will include the derivation of the formula above. I use the boundary stress tensor given by Brown-York (derived from the Einstein action alongwith the Gibbons-Hawking boundary term. $$ t_{ij} = \\frac{1}{8\\pi G} \\left[ K_{ij} - \\gamma_{ij} K + \\frac{2}{\\sqrt{-\\gamma}} \\frac{\\delta S_{ct}}{\\delta \\gamma^{ij}} \\right] $$ Here $K_{ij} = \\nabla_{(i} n_{j)}$ is the extrinsic curvature. $n^\\mu$ is the unit normal vector to the boundary. $\\gamma$ is the boundary metric and $K = \\gamma^{ij} K_{ij} $. $S_{ct}$ is the counterterm action included to make all the B-Y charges finite, which are defined as $$ Q_\\xi = \\int_{\\cal B} d^d x \\sqrt{\\sigma} u^i \\xi^j t_{ij} $$ Here $\\sigma_{ab}$ is the metric on a spatial hypersurface and $u^i$ is a time-like unit vector normal to the hypersurface. $\\xi^j$ is a Killing vector of the boundary metric. Now, for $AdS_5$, the counterterm action is given by $$ S_{ct} = -\\frac{3}{\\ell} \\int d^4 x \\sqrt{-\\gamma} \\left( 1 + \\frac{\\ell^2}{12} R(\\gamma) \\right) $$ The B-Y tensor is then $$ t_{ij} = \\frac{1}{8\\pi G} \\left[ K_{ij} - \\gamma_{ij} K - \\frac{3}{\\ell} \\gamma_{ij} + \\frac{\\ell}{2} \\left( R_{ij} - \\frac{1}{2} \\gamma_{ij} R \\right) \\right] $$ We can now work in the Fefferman-Graham coordinates for $AdS_5$ space where the metric is $$ ds^2 = \\frac{\\ell^2 d\\rho^2}{4\\rho^2} - \\frac{(1+\\rho)^2}{4\\rho} dt^2 + \\frac{\\ell^2 ( 1 - \\rho)^2}{4\\rho} d\\Omega_3^2 $$ Thus $$ t_{ij} = - \\frac{ \\rho }{4\\ell \\pi G} \\left( \\gamma_{ij}^{(0)} + \\gamma_{ij}^{(2)} \\right) + {\\cal O}(\\rho^2) $$ where $$ \\gamma_{ij}^{(0)} dx^i dx^j = -\\frac{1}{4} dt^2 + \\frac{\\ell^2}{4} d \\Omega_3^2 $$ $$ \\gamma_{ij}^{(2)} dx^i dx^j = - \\frac{1}{2} dt^2 - \\frac{\\ell^2}{2} d \\Omega_3^2 \\\\\\ $$ We also have $$ u = \\frac{2 \\sqrt{\\rho}}{1+\\rho} \\partial_t,~~ \\xi = \\partial_t,~~\\sqrt{\\sigma} = \\frac{\\ell^3 (1 - \\rho)^3 }{8 \\rho^{3\/2} } \\sin^2\\theta \\sin \\phi $$ Plugging all this in, we find that the B-Y charge corresponding to the Killing vector $\\partial_t$ is $$ Q_t = \\frac{3 \\pi \\ell^2 }{32 G} $$ This is where I got the formula from. I interpret this as the mass of $AdS_5$ space. Disclaimer - I have intentionally left out several computations to reduce the length of the problem. I have not referred any paper and all computations have been done by me."}
{"id":"38170","title":"What is the longest distance that can be jumped after swinging from a rope?","text":"In the movie _Mission Impossible 3_ , the main character Ethan Hunt tries to enter a building in Shanghai by swing through the sky, as shown below: ![enter image description here](http:\/\/i.stack.imgur.com\/1CxqB.png) The jump consists of 2 sections, the red part, which is an arc, and the orange part, which is a parabola. A question naturally arises: **_For a target building with a given height, what is the furthest horizontal distance it can be placed such that we can jump to its roof using this method?_** Before answering this question, we must consider the following fact: ![enter image description here](http:\/\/i.stack.imgur.com\/Vppya.png) Observing the path when we release at different heights, we see that if we release very late, we can reach great heights, but with reduced horizontal distance. If we release early, we increase horizontal distance, but with reduced height. Because of this trade off, the area that can be reached is restricted within the yellow line。 The question is then: (1) **What is the equation x(z) that describes the orange line above?** (2) **For a given point (x,z) that is inside the region bounded by the orange line, what is the proper release height h(x,z) such that the parabola will go through (x,z). The answer should be a set of h. This is the same as asking what is the proper release height if we want to reach a building with height z and is located at x distance away from the center building.** For the ease of discussion, I propose that we use the follow coordinate system and notation : ![enter image description here](http:\/\/i.stack.imgur.com\/82StP.png) * * * Here are some calculations that are already done: At the release point, $$\\text{Upward shooting angle}=\\psi$$ $$v=\\sqrt{2gh}$$ $$h(\\psi)=r\\cos\\psi-H$$ $$\\psi(h)=\\arccos\\frac{h+H}{r}$$ Upward and horizontal speeds are: $$v_{u}=v\\sin\\psi$$ $$v_{h}=v\\cos\\psi$$ Time for Ethan to fly through the orange part is: $$t=\\frac{2v_{u}}{g}=\\frac{2v\\sin\\psi}{g}$$ Thus $$d_{h}=v_{h}\\cdot t$$ $$=v\\cos\\psi\\frac{2v\\sin\\psi}{g}$$ $$=\\frac{v^{2}}{g}\\sin2\\psi$$ If we were to use $h$ as the variable, then $$d(h)=\\frac{v^{2}}{g}\\sin\\left(2\\arccos\\frac{h+H}{r}\\right)$$ $$d(h)=\\frac{\\left(\\sqrt{2gh}\\right)^{2}}{2g}\\left(\\frac{h+H}{r}\\right)\\sqrt{1-\\left(\\frac{h+H}{r}\\right)^{2}}$$ $$d(h)=h\\left(\\frac{h+H}{r}\\right)\\sqrt{1-\\left(\\frac{h+H}{r}\\right)^{2}}$$"}
{"id":"47783","title":"What is the exact gravitational force between two masses including relativistic effects?","text":"I was wondering if there is a closed-form formula for the force between two masses $m_1$ and $m_2$ if relativistic effects are included. My understanding is that the classic formula $G \\frac{m_1 m_2}{r^2}$ is just an approximation (which is good enough for probably even going to the Moon), but what would the \"correct\" formula be according to the general theory of relativity? Does a closed formula even exist? For example for an idealized situation of just two spherical masses with homogenous mass distribution?"}
{"id":"88831","title":"Emf of a rod sliding on two conducting rails","text":"The following is the question. ![enter image description here](http:\/\/i.stack.imgur.com\/2v9Ro.png) Actually, I do not know how to find the answer. I have tried to find out the solution or hints from the lecture note given to me but it is quite difficult to understand. The following is what I have tried. ![enter image description here](http:\/\/i.stack.imgur.com\/YBctA.jpg) Please give me some hints. Thank a lot. (Ans : 0.14V)"}
{"id":"103356","title":"Physical applications of the mathematical curvature","text":"I was studying multivariable calculus last semester and had one of the topics talking about a curvature, but we had no applications on it. So how does it help in physics? E.g. curvature of curve: $k=|T(t)|\/|r'(t)|.$"}
{"id":"19291","title":"Why doesn’t gravity break down in a large black hole?","text":"By popular theory gravity didn’t exist at the start of the Big Bang, but came into existence some moments later. I think the other forces came into existence a little latter. When a black hole crushes matter to a singularity (infinite density), at some point shouldn’t the forces cease to exist including gravity? Kent"}
{"id":"82725","title":"At what frequency would a quartz crystal shatter?","text":"Would it be possible to emit a sound at such a frequency that a quartz crystal inside of a watch would shatter? (Rendering the watch useless)"}
{"id":"65666","title":"Using the Normalization Condition with Wavefunction","text":"I'm very confused with this problem and I was looking for some guidance. $$\\psi(x) = Ae^{ikx}e^{-x^2\/2a^2}$$ Use the normalization condition to find A. So I understand that you use the normalization condition where $$\\int_\\infty^\\infty|\\psi(x)|^2 dx = 1$$ But the integral is really complicated and doesn't really lead me anywhere. Can anyone suggest a hint? Am I missing something? $$\\int_\\infty^\\infty (Ae^{ikx}e^{-x^2\/2a^2})^2dx$$ $$\\int_\\infty^\\infty (A^2 e^{2ikx}e^{-x^2\/a^2})dx$$"}
{"id":"80768","title":"Photons hitting the Black Hole Singularity","text":"I was watching one of Dr. Susskind's Lectures on String theory and he was talking about black hole singularities (see the 40 minute mark of this lecture http:\/\/www.youtube.com\/watch?v=SkEvsxg5Tu4&list=PL3E633552E58EB230). I'm curious, given that the singularity becomes a 'time' and not a 'place', if at the moment you cross the horizon, you emit a photon, according to the explanation given, does this mean that it would take an infinite amount of time for the photon to hit the singularity? And this would also mean that the faster you move after crossing the horizon, the longer it would take to hit the singularity (which seems kind of cool)? I'm just curious to know if I'm interpreting the explanation correctly."}
{"id":"82894","title":"Reciprocating vs. rotating machines","text":"Why is the RPM achieved on a reciprocating machine in general lesser than that obtained in a rotating machine? For example, an IC engine typically provides an RPM of about 2500 while a turbine can go as high as 80,000 RPM. The inertia and the repetitive change in direction of motion in case of reciprocating machine is one aspect, but is there an analytical justification\/limitation on the RPM?"}
{"id":"46073","title":"Can you speed up radioactive decay of plutonium?","text":"We all know the issue of deep geological repositories for fuel rods. Is there a currently feasible way to speed up the rod's decay to render them harmless in less than 10 years?"}
{"id":"46076","title":"Logic behind topological orders","text":"Long-range entanglement (LRE) is the main feature of topological orders. The string-net condensation model was constructed to exhibit LRE. But the many-body systems of such models do not look like any earthly materials at all but are closer to quantum gravity models. In quantum gravity, nobody sees the detailed structure beyond the Planck scale. But in condensed matter, the microscopic structures are \"crystal\" clear. Then how can such strange models be used to explain LRE in laboratory materials?"}
{"id":"103281","title":"How to calculate the force acting on a magnet due to a solenoid","text":"Can I assume both magnet and solenoid as magnetic dipole and use Coulombs law to find the force acting the magnet?"}
{"id":"52236","title":"Where to learn Temperature Dependent Conductivity induced by Electron-Phonon Interaction?","text":"I want to learn how to calculate the temperature dependent conductivity induced by electron-phonon interaction. I know in low temperature, the resistance in metal $\\rho$ is proportional to $T^5$, $T$ is the temperature. In high temperature region, $\\rho$ is linear to $T$. Could someone tell me which text book and which chapter can I find related calculations? Thanks in advance."}
{"id":"99237","title":"Entropy of a chain","text":"A chain has N segments which can be oriented in either the x or y directions. For each segment oriented along y, there is an energy penalty of $\\epsilon$. We also know the end segment is at $(L_x, L_y)$. How can we define the entropy S as a function of E,N,Lx, and Ly? I know $S=k*ln(\\Omega)$, but is can you actually solve for the possible arrangements of the chain? I'm assume overlapping and crossing cannot be done."}
{"id":"132072","title":"Is light so important in special relativity?","text":"I'm an amateur physics enthusiast with no formal university education in Physics. So my question might sound very naive, so forgive me. I had this question in the back of my mind since the wrong discovery of neutrinos travelling faster than light. My understanding of special relativity is as following. You can measure the physical parameters like length, time, mass of objects in a frame of reference (A) from another frame of reference (B) having a constant relative velocity with A by only looking into the information signals coming from A to B. Since the light(electromagnetic radiation) is the fastest known carrier of information between two frames and since light has a constant relative velocity with any moving inertial frame, the length\/time measurements between moving inertial frames are formulated based on its velocity. Back in 2011 when the 'faster than light' neutrinos were reported one point I heard in popular science reports was that it will invalidate the special theory of relativity. But why is it so? Any particle faster than light would simply interchange the constant $c$ in the special relativity with some new constant(provided this particle also has a constant relative velocity with inertial frame). Doesn't the special relativity theory still holds with the information being exchanged with this new fast particle instead of light? As long as these particles don't travel with infinite velocity all measurements from moving frames will have the time dilation and length contraction. Is my understanding right? Edit 1: I see another question here and the answer to that seems like my understanding is right. Special relativity theory can be derived from any particle travelling with a constant speed."}
{"id":"46697","title":"Is there Pair production in between charged plates","text":"In classical electromagnetic theory, If parallel plates are charged oppositely and placed close to each other, there will be no charge will not flow from one plate to another. How does this situation change if one considers Quantum electrodynamics? Can the electric field in between the plates cause pair production? What is the probability, if it happens? How does one apply the formalism of quantum field theory to such a question ? I am rather new to the subject."}
{"id":"46691","title":"Confined in a box, what is the average distance between a particle hitting a side?","text":"This particle, alone in the box, at a constant velocity $v$, travels in straight lines until hitting a side of the hollow cube, bouncing off in a _random_ direction independent of incident angle (I know it's not momentum conserving, but I'm looking for a statistical average, not a description of a deterministic system). Let the experiment go on for (essentially) infinite time, and (I assume) the average time between collisions (and so distance) wil converge to a value. _What is it_? And what about a hollow sphere? **Edit** :   How to begin this confuses me, so I'll consider the simpler scenario of the particle having to hit on an edge of the original square (that is, its velocity had no $z$-component) from its original position $(x_0,y_0)$. Consider each of the 4 edges in turn, by trigonometry deduce the angle between $(x_0,y_0)$ and the vertices ($\\theta_1$ and $\\theta_2$), and express that edge in polar form: $r(\\theta)=\\frac{y}{sin(\\theta)}$, if $y$ is the shortest distance between $(x_0,y_0)$ and the edge. Integrate $\\int_{\\theta_1}^{\\theta_2}|r(\\theta)|d \\theta$ to find the sum of the distances, and divide by the edge's length to find the average. My problem is that I think I'm overcomplicating to the extent that when I extend this to 3 dimensions, it won't spit out a meaningful answer: even in the 2-dimensional case there's $ln(\\arccos(...)$ in there that makes me think there's a much more elegant solution. I'm sorry if I'm still being vague, but I've never encountered ways to get around these problems in the first place."}
{"id":"45524","title":"Intensity in rainfall and global warming","text":"Why we should observe an increment on the mean intensity in rainfalls and an increment on mean dry days with global warming?"}
{"id":"48335","title":"Is Keshe's technology for real?","text":"http:\/\/www.youtube.com\/watch?feature=player_embedded&v=UrN99RELqwo They clain they can create free energy using plasma reactors. Is this fake? > M.T. Keshe said that several entities, individuals, groups, and governments, > have been able to replicate the energy technology phenomenon by referring to > the books and patents available from the KesheFoundation.com website. One > scientist allegedly could power three towns pounds with his device."}
{"id":"48334","title":"Charge conjugation in Dirac equation","text":"According to Dirac equation we can write, \\begin{equation} \\left(i\\gamma^\\mu( \\partial_\\mu +ie A_\\mu)- m \\right)\\psi(x,t) = 0 \\end{equation} We seek an equation where $e\\rightarrow -e $ and which relates to the new wave functions to $\\psi(x,t)$ . Now taking the complex conjugate of this equation we get \\begin{equation} \\left[-i(\\gamma^\\mu)^* \\partial_\\mu -e(\\gamma^\\mu)^* A_\\mu - m \\right] \\psi^*(x,t) = 0 \\end{equation} If we can identify a matrix U such that \\begin{equation} \\tilde{U} (\\gamma^\\mu)^* ( \\tilde{U} )^{-1} = -\\gamma^\\mu \\end{equation} where $ 1 =U^{-1} U$. **I want to know that, why and how did we do the last two equation. More precisely, I want to know more details and significance of the last two equations**."}
{"id":"26948","title":"Supersymmetric Nonrenormalization Theorems","text":"I'm looking for approaches to nonrenormalization theorems in supersymmetric QFT which are as much as possible mathematical, elegant and involve few heavy straightforward computations"}
{"id":"26945","title":"Gauge invariance for electromagnetic potential observables in test function form","text":"This is a reference request for a relationship in quantum field theory between the electromagnetic potential and the electromagnetic field when they are presented in test function form. $U(1)$ gauge invariance becomes a particularly simple constraint on test functions for smeared electromagnetic potential operators to be gauge invariant observables. This is such a simple constraint that I think it must be out there, but I have never seen this in text books or in the literature, presumably because we mostly do not work with test function spaces in QFT; instead we use operator-valued distributions directly, where, however, gauge fixing is a perpetual nuisance. For the electromagnetic potential operator-valued distribution smeared by a test function $f^\\rho(x)$ on Minkowski space, $\\hat A_f=\\int_M \\hat A_\\rho(x)f^{\\rho*}(x)\\mathrm{d}^4x$, to be an observable that is invariant under $U(1)$ gauge transformations $\\hat A_\\rho(x)\\rightarrow\\hat A_\\rho(x)-\\partial_\\rho\\alpha(x)$, we require that $\\int_M \\partial_\\rho\\alpha(x)f^{\\rho*}(x)\\mathrm{d}^4x$ must be zero for all scalar functions $\\alpha(x)$. Integrating by parts over a region $\\Omega$ in Minkowski space, we obtain, in terms of differential forms, $$\\int_\\Omega d\\alpha\\wedge(\\star f^*)=\\int_{\\partial\\Omega}\\alpha\\wedge(\\star f^*)-\\int_\\Omega \\alpha\\wedge(d\\\\!\\star\\\\! f^*),$$ which will be zero for large enough $\\Omega$, and hence for the whole of Minkowski space, for any smooth test function $f^\\rho(x)$ that has compact support and is divergence-free, $d\\\\!\\star\\\\! f=0$. [If we constrain the gauge transformation function $\\alpha(x)$ not to increase faster than polynomially with increasing distance in any direction, it will be enough for the test function $f^\\rho(x)$ to be Schwartz and divergence-free.] So we have proved: > **Theorem:** The smeared electromagnetic potential $\\hat A_f$ is a $U(1)$ > gauge invariant observable if the test function $f^\\rho(x)$ is smooth, of > compact support, and divergence-free. The divergence-free condition on $f^\\rho(x)$ ensures that the commutator for creation and annihilation operators associated with the electromagnetic potential $\\hat A_f=\\mathbf{\\scriptstyle a}^{\\,}_{f^*}+\\mathbf{\\scriptstyle a}^{\\dagger}_f$ , $$[\\mathbf{\\scriptstyle a}^{\\,}_f,\\mathbf{\\scriptstyle a}^\\dagger_g]=-\\hbar\\int \\tilde f^*_\\rho(k)\\tilde g^\\rho(k)2\\pi\\delta(k_\\nu k^\\nu)\\theta(k_0)\\frac{\\mathrm{d}^4k}{(2\\pi)^4},$$ is positive semi-definite (which is necessary for us to be able to construct a vacuum sector Hilbert space), and that because $\\delta f=\\delta g=0$ we can construct, on Minkowski space, $f=\\delta F$, $g=\\delta G$, where $F$ and $G$ are bivector potentials for the electromagnetic potential test functions $f$ and $g$. In terms of $F$ and $G$, we can write $a^{\\,}_F=\\mathbf{\\scriptstyle a}^{\\,}_{\\delta F}$ , $a_G^\\dagger=\\mathbf{\\scriptstyle a}^\\dagger_{\\delta G}$ , which satisfy the electromagnetic _field_ commutator $$[a^{\\,}_F,a_G^\\dagger]=-\\hbar\\int k^\\alpha\\tilde F_{\\alpha\\mu}^*(k) k^\\beta\\tilde G_\\beta{}^\\mu(k)2\\pi\\delta(k_\\nu k^\\nu)\\theta(k_0)\\frac{\\mathrm{d}^4k}{(2\\pi)^4}.$$ Consequently, turning around the usual relationship because we are working with test functions instead of directly with quantum fields, we can regard test functions for the electromagnetic field as potentials for test functions for the electromagnetic potential. Because of the restriction that electromagnetic potential test functions must have compact support (or that gauge transformations must be constrained if electromagnetic potential test functions are taken to be Schwartz), electromagnetic potential observables are less general than electromagnetic field observables if electromagnetic field test functions are taken to be Schwartz (as is most commonly assumed), or equivalent if electromagnetic field test functions are taken to be smooth and to have compact support. So, references? EDIT (October 24th 2011): Noting the Answer from user388027, and my comment, a decent reference for what constraints are conventionally imposed on gauge transformations would be welcome. I would particularly hope for a rationale for the constraints from whatever theoretical standpoint is taken by the reference."}
{"id":"1942","title":"Does the cube stay on the loop-de-loop?","text":"Suppose a cube of mass $m$ enters a loop-de-loop of radius $r$, travelling halfway around and reaching the top. At the top, it has velocity $v$, and the second half of the loop is frictionless. How can I determine whether or not it will continue around the loop, or fall off? According to the notes I took during class I can check to see if $v^2\/r < g$, but I don't understand why that works. Can someone explain this? How would I determine if the cube stays on the loop if the second half of the loop isn't frictionless?"}
{"id":"13242","title":"Pressing a fluid through leaks ","text":"Say I have a cylindrical container of some volume $V$, that I fill with water or some other fluid with dynamic viscosity $\\mu$. Here, the bottom of the container is milled such that some fraction, $p$, of its surface area, $A$, is removed to allow passage of fluid. One could, for example, drill pores until this criterion is met. After positioning a plate at the top of the container, I press down on the plate with some force $F$ to compress the fluid. As a function of $F$, how quickly will I drain the fluid through the bottom of the container? To calculate this, what additional measurements\/parameters, such as surface hydrophobicity\/etc., will I require, and how does this change with the minimum cross-sectional area of the pores I mill in the bottom of the container?"}
{"id":"28538","title":"High school double lens optics question","text":"This is a first year high school homework question (in the Finnish high school), and I'm having serious trouble solving it. I apologize for possible non-standard terms: I'm doing the translation from Finnish to English and all of the words may not be how they would be in an English physics book. This was originally posted to math SO. The following paragraph no longer applies after the move: (I am aware that this is the Math and not the Physics stackexchange. I decided to ask this here, because I believe that my trouble lies mainly in the mathematical execution and not the physical understanding. Math also seems significantly more active, thus improving my chances of getting an answer. If that is required or strongly suggested, I will copy this to Physics and either remove it from here or leave it, whichever is desired.) The question: > A convex lens creates a real image from an object on the main axis, $85.0$ > cm from the object. When a concave lens is placed between the object and the > convex lens, $65.0$ cm from the object, the image moves to $140.0$ cm from > the object. What is the focal length of the concave lens? No other information is given. Knowledge of the equation $1\/a + 1\/b = 1\/f$ is assumed, as well as the way multi-lens systems work (the image of a previous lens is the object of the next lens) I have approached the problem by creating three lens equations: $1\/a_1 + 1\/b_1 = 1\/f_1$, where $a_1$ is the distance from the object to the convex lens, $b_1$ is the distance from the convex lens to the first image, and $a_1 + b_1 = 85$ cm $1\/a_2 + 1\/b_2 = 1\/f_2$, where $a_2 = 65.0$ cm is the distance from the object to the concave lens and $b_2$ is the distance from the concave lens to the fake image produced $1\/a_3 + 1\/b_3 = 1\/f_3$, where $a_1$ is the distance from the fake image to the convex lens, $b_3$ is the distance from the convex lens to the (second) real image and $f_3 = f_1$, since it's the same lens In addition, there are a few geometrical equations that can be extracted: $$\\begin{align*} &a_1 + b_3 = 140 \\text{ cm}\\\\\\ &a_2 - b_2 + b_3 + b_3 = 140 \\text{ cm} \\end{align*}$$ leading to: $$b_2 = a_2 + a_3 - a_1$$ I know that the key is to calculate $b_2$, but it's proving to be difficult. The farthest I have got by combining the $b_2$ equation above and $f_1 = f_3$ is $$b_2=\\frac{a_3(140\\text{ cm}-a_1)}{a_1(85\\text{ cm}-a_1)}\\cdot 85\\text{ cm}-75\\text{ cm}$$ However, there are still $2$ unknowns, which is $2$ too much. I have the feeling that I am not seeing some geometrical equation that would somehow allow me to calculate the values for $a_1$ and $a_3$. It is also possible that the wording allows the assumption that the second real image is equal in size to the first real image, but 1) I can't see that helping too much and 2) from the way it's worded in Finnish I wouldn't dare make that assumption in an exam, for example. The correct answer, according to the book, is $f_2 = -27.3$ cm. Any hints what to try are welcome, as well as a more complete solution. I have tried to solve this for at least two hours, checking every geometrical relations I can think of and just generally trying to manipulate the various equations in a way that would help me get forward."}
{"id":"16275","title":"Are there 'higher order moments' in physics?","text":"This may be a rather noob question but please let me clarify: I'm struggling to understand the use of the word 'moments' w.r.t., probability distributions. It seems after some research and poking around it seems to have been derived from physics when trying to solve\/prove something about\/related to binomial distribution and the method was called method of moments. I've asked the corresponding question here: http:\/\/stats.stackexchange.com\/q\/17595\/4426 Now 'Pearson' (one of the very famous statisticians) comments: > We shall now proceed to find the first four moments of the system of > rectangles round GN. If the inertia of each rectangle might be considered as > concentrated along its mid vertical, we should have for the sth moment round > NG, writing d = c(1 + nq). Here are some of the details of the proof (as in the above post): ![enter image description here](http:\/\/i.stack.imgur.com\/VtJ6d.png) Now Pearson talks about calculating the 'rth' moment and uses a derivative function to do so: ![enter image description here](http:\/\/i.stack.imgur.com\/07Cfx.png) Question: I'm not aware of such a function from my knowledge of elementary physics. What kind of moments are being calculated here? How do you calculate 'higher order moments'? Is there any such thing? Basically looking to clarify something in statistics but was historically alluded to physics and hence just want to get it ironed out :) **UPDATE:** Intent of question: What I want to know is does the above derivation have anything at all to do with the concept of moments in physics and how is it related? Since the 'word' moment (and its intent) seem to be borrowed from physics when the author is making the derivation. I personally want to know if something like this does exist in the field of physics and how are these two derivations (and 'moments' related)"}
{"id":"20041","title":"How to interpret the appearance of time units in the units of a physical quantity?","text":"Or phrased more abstractly, how to interpret the appearance of time dimension $[time]$ in the dimension of a physical quantity? For example, the dimension of pressure is $[mass] [length]^{-1} [time]^{-2}$ corresponding to the SI-unit _Pascal._ When you calculate pressure, you do not have to know any time. It is simply _force_ per _area_ – a division. There are many units that have time as component in them."}
{"id":"20046","title":"How to find the electric field at a point based on a uniformly charged surface","text":"What is the general solution to finding the electric field at a point based on some (or multiple) charged surfaces. I know that we can perform a line\/surface integral if a charge is close to a wire or a surface. Furthermore, seeing as the surface is uniformly charged, we can use a Gaussian surface to eliminate the integral and write it as: $$ E_{point} = \\frac{\\lambda}{2\\pi\\epsilon_0 R} $$ Where $ \\lambda $ is the constant charge and $ R $ is the distance of that charge from the wire. Does this formula change when we have a charged _surface_ instead of a wire? And what if there is more than one wire\/surface acting on the charge? Are these formulas additive?"}
{"id":"20045","title":"projective camera: back-projecting a point on the image plane into 3-space","text":"suppose I got a projective camera model. for this model I would like to back- project a ray through a point in the image plane. I know that the equation for this is the following: $$ y(\\lambda) = P^+ \\pmb{x} + \\lambda \\pmb{c} $$ where $P^+$ denotes the pseudoinverse of the camera matrix P. P has a dimensionality of 3 by 4. $x$ is the point on the image plane in homogenous coordinates. Hence it's dimensionality is 3 by 1. $c$ the center of the camera in 3-space in homogeneous coordinates. (note that this equation is taken from the book \"Multiple View Geometry in Computer Vision\" page 162.) Now I don't fully get this equation. I get that $P^+ x$ results in a point on the line we are looking for. Hence we have two points that we can use for constructing a line. However I don't get the parametrization using $\\lambda$. Why is the equation not in the form like: $$y(\\lambda) = (1-\\lambda) \\pmb{a} + \\lambda \\pmb{b}$$ Any help in understanding the original equation of the resulting ray would be appreciated! :D"}
{"id":"79034","title":"How should I think about reciprocal lattice and Miller indices?","text":"When I hear someone talking about a (100) plane or a (111) plane or an (hkl) in general, my first thought is, is the system cubic. The reason I think this is because I tend NOT to think of the planes associated with their miller indices, but the vectors that are normal to planes. In cubic systems, [hkl] vectors are normal to their corresponding (hkl) planes. Hence, direct space (hkl) planes are parallel to their reciprocal space counterparts. In cubic systems, these relationships are true. I think this is a bad way to think of the relationship between Direct and Reciprocal space, as not all systems are cubic and I fumble when dealing with other sorts of lattices. Does anyone have some advice of a more fundamental way to view the relationship between the two spaces? Preferably, one that is quite intuitive. I realize they are Fourier Transforms of one another. I am sorry if this is a silly question. For example, in an HCP system, how should I envision the (hkl) plane? It is not normal to the [hkl] vector? is there no simple intuitive way to see such things? I can certainly do the math I suppose."}
{"id":"79036","title":"In black hole complementarity, does Bob literally observe Alice's death?","text":"I'm interested in the picture of black hole complementarity where an observer (Bob) far away from the black hole watches another observer (Alice) fall in, and develops a very different picture of events. I want to stay within the framework of black hole complementarity, and limit this to _Bob's reference frame_. Mainly:   1. Can Bob practically watch Alice die from her emitted photons?   2. Alternatively, is time dilated so much in Alice's reference frame that Bob would have to wait until the end of the universe to watch this? I _think_ there are arguments against #2, and I might only have this confusion because of arguments I've read which were not fully informed, which I may now be repeating. There's also the matter of cause of death for Bob to write on the coroner's report. It seems that conflicting accounts exist, although they ultimately go back to the firewall concept. > The asymmetric mass, from the point of view of the exterior observer, smears > itself symmetrically on the surface when it gets close. second source: > The AMPS proposal is that what Alice encounters at the horizon does not look > like empty space at all — rather B and A are unentangled, which means that > Alice sees many energetic particles. Monogamy of entanglement is rescued, > but not poor Alice. She is incinerated by an intense wall of fire as she > attempts to pass through the event horizon. While these graphic accounts of Alice's death sound concrete, they're not _necessarily_ inconsistent with point #2 above. If time dilates within certain asymptotic properties, then it could look somewhat like Zeno's paradox. There are ways this could prevent Alice from ever _fully_ merging with the surface of the black hole. But could it prevent Bob from ever being able to confidently able to say that Alice died? Of course Bob has to recalibrate his video camera as it shifts ever-more toward the IR part of the spectrum, but does that video ever show Alice boiled to death or dismembered and smeared into the surface?"}
{"id":"129547","title":"White bread toasted seems hotter than brown bread?","text":"Ok this is a bit of a random question but I 99% of the time use white bread for toast and I always toast my bread the same amount and every time I take it out it's too hot to handle and the butter melts instantly on it. But this morning I used brown bread for the first time in forever and I noticed that even after getting it to the same amount of \"toastiness\" it was a lot cooler and I could easily handle it. Is it something that's in the bread white bread or the fact that it's more dense that causes it to be hotter?"}
{"id":"95670","title":"Is the gauge fixing $\\partial_\\mu A^\\mu + \\gamma A_\\mu A^\\mu=0$ used in the literature and does it have a name?","text":"In an exercise for a course on Gauge Theories, I was asked to derive the action of QED with the method by Faddeev and Popov, using the following gauge- fixing function: $$F(A) = \\partial_\\mu A^\\mu + \\gamma A_\\mu A^\\mu, $$ where $\\gamma$ is constant. I was wondering whether this function is simply a construction for the exercise or it is actually used in QFT, and how it is referenced in literature."}
{"id":"67774","title":"How does phonon cause two electrons to attract each other?","text":"We know that like charges repel each other. But my professor claimed that two electrons can attract each other as well. What he said was that due to screening an electron traveling at some speed wont repel another electron but will in some case they attract each other due to weak phonon exchange. What does that mean? What is phonon exchange? Do two electrons really attract each other?"}
{"id":"28394","title":"U(1) Charged Fields","text":"I don't quite understand what is actually meant by a field charged under a $U(1)$ symmetry. Does it mean that when a transformation is applied the field transforms with an additional phase? More specifically, why is it called \"charged\"?"}
{"id":"89824","title":"Is There Any Work Done By Normal Reaction Force even if point of application of force is not getting displaced?","text":"Suppose that I am standing on the ground and I suddenly jump and rise up to any certain height.So accordingly I would have gained some speed i.e. Kinetic Energy during the process of jumping.According to Work Energy Theorem Total work done on by body is equal to this kinetic energy.Now it is obvious that gravity has no tendency to provide me upside velocity and besides my body's center of mass has not been so much displaced during the process of just jumping.The only other force acting on me is Normal Reaction Force which will provide me the upward change in momentum but here at every instant of the action of the normal reaction force point of application of normal reaction force ( my feet's point of contact with ground ) are at rest during jumping so according to the defination of work done Normal reaction should not have performed the work.What is the funda here ? Please get me out of this confusion."}
{"id":"23668","title":"How do I calculate output voltage from a given supply voltage and resistance?","text":"I have searched everywhere but I can't seem to find out how I can calculate the output voltage (or voltage over a component) from the resistance of a resistor (or other component) and the supply voltage. Everything seemed to be about Ohm's Law, which looks like it could help but I don't know how. In the case I need to solve, there is a supply of 5v and I need to get the resistance needed to lower the voltage for an LED (the current and resistance at optimum voltage I can get if needed) to 3v. I will definitely need to do this operation again so if you could give something like an equation, rather than the necessary resistance."}
{"id":"66427","title":"Finding the terminal velocity of a magnet dropped in a solenoid","text":"> We have to find proportionality of the terminal velocity with the factors of > the system: ![enter image description > here](http:\/\/i.stack.imgur.com\/OItlm.png) > > Plot: a small dipole(mass $m$) with dipole moment $\\mu$ is dropped in a long > solenoid (radius $r$, Resistance $R\\: \\mathrm{\\Omega}$) along it's axis. The > gravity is acting downwards. I found it to be $$v\\propto \\dfrac{mR}{\\mu^2r}$$ Here is how: > The force in downward direction $F_{down}$ must be $mg$ and in upward > direction $F_{up}$ must be magnetic force. $$F_{up}\\propto \\dfrac{\\mu > I_{induced}}{r}$$ > > $$I_{induced}\\propto\\dfrac{ \\text{electric flux change}}R \\propto\\dfrac{\\mu > \\pi r^2 v}R$$ Flux change depends directly on the velocity, so added $v$ in the numerator. But the expression is not correct. The proportionality on the $r$ radius is not correct. I think there is a problem in the induced current factor."}
{"id":"89823","title":"Likelihood of the creation of a single unbound quark in the collision of very high energy particle beams","text":"I am going over old exam and am not understanding the logic behind the answer given in the mark-scheme. > A beam of protons and antiprotons attain energies of 1400 GeV in a > synchrotron. Why is it that, even at the very high energies of this particle > beam, the collisions are unlikely to lead to the creation of a single > unbound quark. The mark-scheme, very confusingly, answer this as follows: > as quarks separate the force between them is constant; > > (so) energy required to release single quark increases with distance; > > these energies exceed the threshold for meson (pair) production; Each part, separated by a semicolon, gives one point (so that you understand why its so oddly formatted). I get that we should simply know that the energy transferred between the quarks gets bigger with distance (work = force * distance), and that force is constant, but why does it jump to the conclusion that these energies exceed what we already have? And why is the meson (pair) production relevant?"}
{"id":"27719","title":"Hilbert-Schmidt basis for many qubits - reference","text":"Every density matrix of $n$ qubits can be written in the following way $$\\hat{\\rho}=\\frac{1}{2^n}\\sum_{i_1,i_2,\\ldots,i_n=0}^3 t_{i_1i_2\\ldots i_n} \\hat{\\sigma}_{i_1}\\otimes\\hat{\\sigma}_{i_2}\\otimes\\ldots\\otimes\\hat{\\sigma}_{i_n},$$ where $-1 \\leq t_{i_1i_2\\ldots i_n} \\leq 1$ are real numbers and $\\\\{\\hat{\\sigma}_0,\\hat{\\sigma}_1,\\hat{\\sigma}_2,\\hat{\\sigma}_3\\\\}$ are the Pauli matrices. In particular for one particle ($n=1$) it is the Bloch representation. Such representation is used e.g. in a work by Horodecki arXiv:quant-ph\/9607007 (they apply $n=2$ to investigate the entanglement of two qubit systems). It is called decomposition in the Hilbert-Schmidt basis. The question is if there is any good reference for such representation for qubits - either introducing it for quantum applications or a review paper? I am especially interested in the constrains on $t_{i_1i_2\\ldots i_n}$."}
{"id":"65993","title":"error propagation and collision in ideal gas","text":"When dealing with gas, a statistical approach is needed because   1. For N particles, you have to solve 6N equations which cant be done analytically. To know our time step for numerical solving, you can estimate the frequency of a collision. For an ideal gas, this frequency is around $10^9$Hz which gives $10^{32}$ equations per second for 1mole of gas at room temperature.   2. Errors propagate quickly and soon it is completely impossible to know anything about the particles you are studying. To show the second point, consider a collision between 2 molecules as shown on this picture ![collision of 2 particles in a gas](http:\/\/i.stack.imgur.com\/JCZO0.png) We have the relation $\\frac{sin\\theta_1}{l_0} = \\frac{sin\\theta_0}{2r_0} \\approx \\frac{\\theta_0}{2r_0}$ where $l_0$ is the mean free path and $r_0$ is the radius of the particle knowing the uncertainty $\\delta\\theta_0$ , we can estimate the uncertainty on $\\delta\\theta_1 \\approx \\delta\\theta_0\\frac{d\\theta_1}{d \\theta_0} \\approx \\delta\\theta_0(\\frac{l_0}{2r_0}) \\approx \\delta\\theta_0 * 10^3$. After n collisions, the error is $\\delta\\theta_0*10^{3n}$ which rapidly explode. Now see how important this effect was, I tried to apply it to the smallest angle I could find. The observable universe being about $10^{25}m$ in radius and the planck length $10^{-35}m$, the smallest angle one can observe is about $10^{-60} rad = 10^{-3*20} rad$. To get an error of $\\pi$, it would therefor take 20 collisions. This doesn't seem possible, is this result correct? If not, where is the problem? I can't find it but I can't believe that 20 collisions are enough to lose all information about the particle."}
{"id":"59377","title":"Sign of acceleration","text":"I'm developing an application using accelerometer sensor. I'm not good at physics so forgive me if the question is trivial. If I have 3 values of acceleration: $x$, $y$, $z$, I find acceleration magnitude by taking square root of $x^2+y^2+z^2$, but how do I find it's sign? Example reading: x: -0.010020584 y: 0.010257386 z: -0.04910469 The magnitude will be around 0.05115, but how do I know if it is deceleration or acceleration?"}
{"id":"65996","title":"Non-existence of gluon singlets: Any recent theoretical progress?","text":"An unanswered question from last year (2012) on gluon singlets asked whether there is any theoretical explanation for the experimental absence of the ninth or colorless (singlet) gluon. This is the gluon that, if it existed, would give allow the strong force to extend far beyond the range of atomic nuclei, with catastrophic results. (Non-experts can find an explanation of the singlet gluon in the Addendum below.) Qmechanic partially answered the question with a link to this 1996 paper by J.J. Lodder, which proposed that the singlet gluon exists but is so massive that its effects are negligible. However, Lubber's massive-singlet idea seems to have gone nowhere. On an cursory search I could not find any references to the 1996 draft, even though it was written seventeen years. I also do not find Ludder's approach very plausible. Theoretically he had to do a bit of mayhem to Standard Model symmetries to come up with his model. More importantly, though, its seems unlikely that the idea holds water experimentally. The mass of the singlet gluon would have to be astronomical indeed for it not to have shown up in high-energy experimental results, especially in the post-Higgs era. So, my version of the singlet question is this: Do there exist any plausible, Standard Model compatible theories for the observed absence of the singlet gluon, other than the apparently non-starter idea that the singlet exists but is very massive? Or alternatively, have searches for the Higgs boson produced any evidence that the singlet gluon state may in fact exist and have a very large mass? * * * **Addendum for non-experts: What is a \"gluon singlet\" and why is it important?** One way to understand the \"colorless\" gluon singlet is through an analogy with how photons work for the electric force. Photons interact with electrically charged electrons, but do not carry any electric charge themselves. Because they have no charge, even low-energy photons can easily escape an atom or electron and travel infinite distances. Imagine, however, what would happen if photons did carry electric charge. Such photons would have the same energetic difficulties leaving a neutral atom as an electron, dramatically altering and limiting how they behave. (I should probably mention that electrically-charged photons in one sense really do exist: They are more-or-less the $W^\\pm$ particles of electroweak theory.) Curiously, the charged situation is reversed for the otherwise photon-like gluons. It is the gluons that convey the strong force, and thereby hold quarks together to form protons and neutrons, as well as secondarily binding protons and neutrons together within atomic nuclei. There are eight types of gluons instead of just one, due to there being more than two types of charge in the strong force. However, in sharp contrast to photons, all of the eight gluons _normally_ carry strong (color) charge. The color charges of gluons cause them to interact strongly with the quarks that emit them and with each other. As in the earlier hypothetical example of how electrical charges would dramatically limit the range of photons, the presence of color charges on gluons similarly limits the distances over gluons can convey the strong force. Consequently, the strong force has almost no impact beyond the scale of atomic nuclei. However, the same mathematical model that predicts the eight color-charged gluons also predicts a ninth neutral or strong-charge-free gluon, called the singlet gluon, that has never been seen experimentally. Its lack of color would make its impact far greater than that of any of the eight other gluons. In particular, just as charge-free photons can carry the electric force far beyond the range of atoms, a charge-free gluon, if it existed, would allow the strong force to extend far beyond the range of nuclei. The repercussions would be huge. In fact, the non-existence of the singlet gluon is best demonstrated by the fact that we exist at all. My best guess (only that, since I have not seen any papers on it) is that if the colorless singlet quark really did exist, every clump of two or more atoms in the universe would melt together into an amorphous sea of quarks. Even if that is not correct, I assure you it that the consequences of the existence of singlet gluons would be... very bad indeed! That is also why I find I think the non-existence of the singlet gluon is a truly interesting question, one that probably deserves more theoretical attention than it has received over the decades since strong force theory was first codified."}
{"id":"55491","title":"What's the difference between 'least count' and 'smallest division'?","text":"I'm wondering the difference between the 'least count' and the 'smallest division' of an instrument. Can you explain that for me, plz?"}
{"id":"55495","title":"a metal bar on a pair of conducting rails that carries a current","text":"This is a homework question, and I solved it already, but something bugs me. So the problem is stated as following: A metal bar of mass M sits on a pair of long horizontal conducting rails separated by a distance $\\mathcal{l}$ and connected to a device that supplies a constant I to the circuit, also the bar has length l. There is a constant magnetic field $\\vec{B}$ that points into the screen where this whole setup is. The current is following in clockwise direction. Now if there is no friction on the rail, the bar will move to the right because of $$F = I\\vec{L} \\times \\vec{B}$$ This is easy. And since there is no friction, the acceleration will be $$a = \\dfrac{F}{M}$$ and hence the magnitude of the velocity at any later time $t$ will be $$v(t) = at = \\dfrac{IlB}{M}t$$ The question is asking for the velocity at time $t$. However, what bugs me is that since the bar is moving to the right, thus the flux through the closed loop of the powersource, rails and the bar will be increasing, which means that there will be an induced electric potential trying to push back the current `I`. However, the problem does not give the resistance of the bar, and I assume that in this hypothetical situation, the whole thing has 0 resistance. So how does the electric potential make sense in this case? Since I am not given a resistance, I cannot calculate the \"pushing back\" current caused by the change in flux. Thank you!"}
{"id":"78818","title":"Parallel Transport of a 4-vector","text":"![enter image description here](http:\/\/i.stack.imgur.com\/aEqXf.jpg) Why does the parallel transported $4$-vector change from $X^a(x)$ to $X^a(x) + \\bar{\\delta}X^a(x)$ ? This is also discussed in D'Inverno's relativity book [page - 72]; but the reason is not clear."}
{"id":"111776","title":"Modulus of action-reaction forces","text":"I read somewhere that action-reaction forces satisfy the form $$ \\mathbf{F}_{12}=-\\mathbf{F}_{21}=f(|P_1-P_2|)(P_1-P_2) $$ meaning that the modulus can only depend on the distance between the two points $P_1$ and $P_2$. Is that right?   Cannot the modulus depend on, say, $|\\mathbf{v}_1-\\mathbf{v}_2|$?"}
{"id":"8467","title":"Is the quantum state or information of a particle is all that differentiates it from another particle?","text":"So I have learned in my QM classes that you can't tell one electron from another electron. They are indistinguishable. I also learned that the wavefunction of a particle includes the spacial part and the spin part. (There might be other parts I don't know! Please don't fault me for not knowing.) I have also learned that it is possible to use the quantum teleportation protocol to retrieve exactly, 100%, the spin state of one particle in another particle after particular measurements. http:\/\/en.wikipedia.org\/wiki\/Quantum_teleportation My question is, if you can teleport the spatial state and the spin state and whatever other state left there is that describes the particle, then wouldn't you be **effectively teleporting the original particle?** A sub question directly related to this is; Is matter\/particles just a changeable glob of stuff that acts differently, thus being a different particle, depending upon what quantum state its in? (This kinda sounds like string theory to me... Oh no!!) Also, what's never answered is, **What state does the original particle assume when its state is teleported to the new particle?** Does the two particles, the original particle and the particle retrieving the quantum information, switch states?"}
{"id":"92346","title":"Josephson effect - fractional or not","text":"It is well known that a current proportional to $J\\propto\\sin(\\phi_1-\\phi_2)$ flows when two superconductor with phases $\\phi_1$ and $\\phi_2$ are connected. Also, a current which depends on the phases according to $\\sin[(\\phi_1-\\phi_2)\/2]$ is claimed to be fractional Josephson effect (which is believed to be a signature of Majorana fermions). But I see that even with two s-wave superconductors connected through a barrier, the current could be proportional to $\\sin[(\\phi_1-\\phi_2)\/2]$ (see for instance - http:\/\/arxiv.org\/abs\/cond-mat\/9811026) when the barrier is zero. I do not understand this well. Does it mean that $J\\propto\\sin(\\phi_1-\\phi_2)$ is true only in the large barrier limit ? Then, what is holy about fractional Josephson current?"}
{"id":"114187","title":"Electric potential of spherical water droplet","text":"I'm trying to answer the following: > 1000 spherical water droplets, each of radius $r$ and each carrying a charge > $q$, coalesce to form a single bigger spherical drop. If $v$ is the electric > potential of each droplet and $V$ that of bigger drop, then find > $\\frac{V}{v}$. It is not mentioned here whether $v$ is potential at a point outside or on the surface of sphere, where should I take the potential? As water is a good conductor of electricity so water droplet can be regarded as spherical shell.Potential due to spherical shell is *** $\\frac{kq}{r}$.*** where **_k_** is constant and **_q_** is charge on the shell and **_r_** is the distance between the centre of spherical shell and point on which **_P_** is to be calculated and at a point inside or on the surface of shell **P** is constant and is equal to *** $\\frac{kq}{r}$.*** here **_r_** is the radius of shell. So **_P_** depends on the distance between the centre of spherical shell and point on which **_P_** is to be calculated.So in the above problem **_P_** on which point should I need to consider. I am unable to understand what is meant by the potential of water droplet.I know potential at a point due to point charge,system of charge and due to continuous charge distribution.But they are calculated at point,I mean we calculate potential at point due to different charge distributions.How can we calculate potential of a body! I want to know what is meant by( \" $v$ is the electric potential of each droplet\" )this line of the question."}
{"id":"109430","title":"\"+\" and \"-\" sign in Maxwell Stress tensor","text":"I have trouble in determining the \"+\" and \"-\" sign of momentum per unit time, per unit area of the following question. Why in the second part, $d\\vec{a}$ is pointing in the $ -\\vec{z} $ direction? I mean, the normal vector is pointing upwards, so the area vector should point upwards, so $d\\vec{a}$ is pointing $ +\\vec{z} $ direction, right? And for (b), why it is negative? Is it because $ \\vec{T_{zz}}$ and $d\\vec{a}$ point in the different directions? And for (c), this is the most confusing part for me. $ -{T_{zz}}$ should be minus, becasue in (a), we get the component of the tensor matrix to be \"+\". I just get really confused about the sign \"+\" or \"-\" of this problem; yet, the calculation is pretty easy for me. ![enter image description here](http:\/\/i.stack.imgur.com\/2StCb.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/w5zr0.jpg) ![enter image description here](http:\/\/i.stack.imgur.com\/cwK23.jpg)"}
{"id":"119157","title":"What would be the resulting effect of two ships passing near each other while using a Alcubierre drive","text":"So, I don't have any formal background in physics or space. but I read about the Alcubierre drive, and now I'm curious. If two ships passed one another, what sort of interference would be generated? How dangerous would it be? I'm thinking it would actually neutralize the effect of warp for either ship since their fronts would be generating opposite forces. Also, am I correctly interpreting what a warp drive travel path would look like? If not, my question is pretty irrelevant. The arrows are supposed to indicate the ships travel path. ![Ship traveling in a warp. Arrows indicate direction of travel](http:\/\/i.stack.imgur.com\/N7vtd.jpg)"}
{"id":"119156","title":"Thermodynamic potentials and approach to equilibrium","text":"I have seen some material which tries to show that when approaching equilibrium upon different constraints, the changes in some thermodynamic potential is always positive or negative (for example, that Helmoltz free energy decreases for constant temperature and volume). I cannot make sense of these arguments, however, because the quantities these thermodynamic potentials depend on are only defined at equilibrium. I think, that what one should show is that if there are several equilibria, the one which minimizes\/maximizes the appropriate potential is the one which maximizes entropy. Is this the right way to think about thermodynamic potentials?"}
{"id":"52912","title":"Why does 'proper length' exist as a notion?","text":"$$\\text{proper time}= \\tau= \\sqrt{dt^2-d\\mathbf{s}^2}$$ $$\\text{proper length}= L= \\sqrt{-dt^2+d\\mathbf{s}^2}$$ What tangible benefit is brought about by calling $i \\tau$ 'proper length' (applying when $\\Im(L)=0$ (the spacetime intervals are spacelike))? Could one extend the notion of the interval to simulatneusly cover both proper time and distance? Call this $\\text{Interval}=dt^2-d\\mathbf{s}^2$, _which can be imaginary_. Is it impossible to merge proper time and length because they are fundamentally different things, as proper time is a explicitly measurable quantity, but proper distance must be determined implicitly? My problem is that I don't understand the rationale behind having proper length, rather than a cheap mathematical trick to make the interval real when $ dt^2-d\\mathbf{s}^2<0$."}
{"id":"25629","title":"How to measure the diameter of a star?","text":"I am thinking about something I read somewhere (if only I could find it again) in a textbook. It is about the size of a star and its ER peaks. It has to do with the waves coming off the edge (maybe) and arriving later than those from \"head on\" and therefore you can know something about its diameter. It has been puzzling me but I can't quite remember it. Just yesterday I was reading about a black hole that pulses at a minimum of 10 minutes and so it is at least 10 light-minutes across. (I probably am not getting it just right, but help me out!) Is this the same principle? Anyone know what I'm talking about? I would love to have this explained and\/or would like to know what it is called, so I can look it up."}
{"id":"25622","title":"How fast will the sun become a red giant?","text":"I've read many accounts of our sun's distant fate, but what I've never heard is on what time scale these events occur. For instance, when the sun runs out of hydrogen, I presume it doesn't just WHAM! become a red giant suddenly, engulfing the inner planets in seconds. But how long would it take for the sun to expand? Would anyone left on Earth have days, weeks, months, years to escape? Centuries? What kind of time scales _are_ we talking about, here?"}
{"id":"25626","title":"Why isn't sunset time in sync with solstice?","text":"The winter solstice in the Northern Hemisphere in 2011 is on December 22. But if I look at the sunset times for a location such as Washington, DC on the USNO site, the sunset time starts reversing much earlier (around December 6th). Shouldn't the sunset time start reversing (from earlier to later) on the same date as the solstice? **Update:**   Omega Centauri's explanation is correct.   * The Royal Observatory gives a similar account of this oddity on this page under the section titled \"The apparently odd behaviour of sunrise\/set times near the winter solstice\".   * Cornell University provides more details here."}
{"id":"75473","title":"Behavior of black holes in higher- and lower-dimensional space-times","text":"The behavior of black holes in 3+1 dimensional space-time as our own is rather well known: formation, event-horizon size, mass, spin, radiation etc. However, my question is what would black holes behave like in universes with different number of space-time dimensions, such as 2+1 dimensional or 4+1 dimensional space-time? Would they be able to form? Would the event horizon radius be smaller or larger? Would their densities be higher or lower? Would they spin, and how quickly? Would they emit Hawking radiation? How would the space-time behave close to it compared to in 3+1 dimensional space-time?"}
{"id":"96041","title":"Visible light through a single-mode optical fiber?","text":"If I understand things correctly, the optical fibers used for (long-range) data transmissions are generally single-mode fibers, transmitting light in the 1300-1500 nm spectrum. Now, could such a fiber transmit visible light (~400-700 nm) a short distance, say a few meters? Or does the fiber have a property that prohibits transmission of shorter wavelengths?"}
{"id":"106248","title":"Are small speakers inherently limited to higher frequencies?","text":"I am hoping to build a subwoofer using multiple smaller speakers (165mm) instead of a single larger speaker (380mm). My theory is that the displaced air volume is what matters, not the individual speaker's diameter.   The surface area of a 380mm speaker is $\\approx900cm^2$.   The surface area of a 165mm speaker is $\\approx140cm^2$.   Presuming the speakers are capable of being driven to the same $X_\\max$, six or seven 165mm speakers should be roughly equivalent to a 380mm speaker.   Taken to an extreme, an array of 100 $9cm^2$ speakers should be equivalent to a single $900cm^2$ speaker. Is this correct, or do smaller speakers have some other limiting factor that limits their low-frequency reproduction?"}
{"id":"106244","title":"How is barometric light porduced?","text":"A shaken sealed glass tube containing liquid mercury in a dark room will glow. This has been know since the 17th century and is called barometric light. How does the addition of potassium permanganate crystals to the mercury influence the production of light?"}
{"id":"106246","title":"Does Lyapunov exponent equate to exponential inflation?","text":"Physics can be modeled by dynamical systems $f^t(x)$ as well as by PDEs. The most common dynamical system has hyperbolic fixed point and can be an attractor or a repellor. The dynamics at repellors and attractors are simple exponential expansion or contraction and are represented by a Lyapunov exponent. My question is how could one prove or disprove that exponential inflation is just an example of a repellor with a Lyapunov exponent?"}
{"id":"2088","title":"Will a wave packet undergo dispersion when traveling down a hanging rope?","text":"Suppose I tie one end of a rope to my ceiling and the other end to a spot on my floor directly underneath it. Because the rope has some mass, the tension varies along the rope, from highest at the ceiling to lowest at the floor. If a wave packet begins propagating down the rope, will its shape change? If so, it is possible to calculate the shape of a Gaussian wave packet as it travels down the rope? Intuitively, it seems to me that the wave packet will change shape for two reasons. First, the portions higher up the rope have higher tension, and therefore higher speed. They will \"catch up\" to portions of the wave packet further down. Second, because the tension is changing, the wave equation now has a term related to the first derivative of the displacement of the rope. Specifically, I tried assuming that the displacement of the rope is only horizontal and that the slope of the rope is never far from vertical. I made $y$ a coordinate measuring up from the floor and $x$ a coordinate to the right. Letting the tension be $T(y) = T_0+\\lambda g y$, with $\\lambda$ the mass per unit length, I got the wave equation ( **edit:** important typo corrected) $$\\frac{\\partial^2{x}}{\\partial t^2} = g \\frac{\\partial x}{\\partial y} + \\frac{T_0+\\lambda g y}{\\lambda} \\frac{\\partial^2 x}{\\partial y^2}$$ but I don't know what to do with it."}
{"id":"134940","title":"A vector field $\\vec{F}$ from cartesian to spherical","text":"I have a vector field F that has the same magnitude and directions at all points in space. If I choose $z$-axis parallel to $\\vec{F}$, so the cartesian coordinate of $\\vec{F}$ is $\\vec{F}=F\\hat{z}$, How can I express $\\vec{F}$ in spherical coordinates? My guess = $\\vec{F}=\\cos(\\theta)\\hat{r} - \\sin(\\theta)\\hat{\\theta}$ I think?"}
{"id":"63888","title":"Frequency Modulation","text":"If FM radios work by modulating the frequency, how is it that we can tune into a specific channel, and hear a song or station? Wouldn't the channel need to be modulated along with the varying frequency?"}
{"id":"63882","title":"Limit on velocity in Minkowski Spacetime geometry","text":"Let A be a rocket moving with velocity v. Then the slope of its worldline in a spacetime diagram is given by c\/v. Since it is a slope, c\/v = tan(theta) for some theta > 45 and theta < 90. Does this impose a mathematical limit on v? If so what is it? As in, we know tan(89.9999999999) = 572957795131. And c = 299792458. Using tan(89.9999999999) as our limit of precision, the smallest v we can use is: c\/v = tan(89.9999999999) => 299792458 \/ v = 572957795131 Therefore, v = 1911.18 m\/s What is the smallest non zero value of v? Is there a limit on this?"}
{"id":"19614","title":"Electron Fermi gas","text":"My question is about 2-dimensional Fermi gas of electrons.   1. What is magnetic susceptibility when $T<<T_F$ (where $T_F$ is Fermi Temperature) and,   2. What is the ratio between Pauli and Curie spin susceptibilities?"}
{"id":"20903","title":"Why is it wrong when I try to use Newton's Law?","text":"> A 25 kg block is connected to a 30 kg block by a light string that passes > over a frictionless pulley. The 30 kg block is connected to a light spring > of force constant 195 N\/m, as in Figure P13.59. The spring is unstretched > when the system is as shown in the figure, and the incline is smooth. The 25 > kg block is pulled 15 cm down the incline (so that the 30 kg block is 35 cm > above the floor) and is released from rest. Find the speed of each block > when the 30 kg block is 20 cm above the floor (that is, when the spring is > unstretched). ![image of blocks on ramp and spring](http:\/\/www.webassign.net\/sf\/p13_59.gif) I wrote out all the equations For the 25kg mass $$\\sum F_x = T- mg\\sin\\theta = ma$$ $$\\sum F_y = n = mg\\cos\\theta$$ For the 30kg mass $$\\sum F_y = kx + Mg - T = Ma$$ Using energy, I got v = 0.91m\/s, but using Newton's Law I got 0.95m\/s EDIT: **Energy** $E = Mgh + \\frac{1}{2}kh^2$ $E' = mgh\\sin\\theta + \\frac{(M + m)}{2}v^2$ $E = E'$ Solving, $v = \\sqrt{\\frac{2Mgh + kh^2 - mgh\\sin\\theta}{m + M}}$ **Forces** $T- mg\\sin\\theta = ma$ $kh + Mg - T = Ma$ Add the two equations $kx + g(M - m\\sin\\theta) = (M + m)a$ $\\frac{kh + g(M - m\\sin\\theta)}{M + m} = a$ $v^2 = 2\\frac{kh + g(M - m\\sin\\theta)}{M + m}h$ $v = \\sqrt{2\\frac{kh + g(M - m\\sin\\theta)}{M + m}h}$ The source of the mistake seem to lie with the 2 in front of the kh"}
{"id":"20900","title":"storing energy (as mass)","text":"When chemical energy is released mass is reduced, if only by a negligible amount. Presumably that's true for all energy. And presumably that works in reverse as well: storing energy involves an increase in mass. It seems to follow that moving some object against a gravitational gradient, increases the mass of the object -- potential energy is being stored. Somehow that's difficult to understand. Is it really true?"}
{"id":"107595","title":"Why is copper a metal?","text":"**Context** : Solid state physics **Question** : Can it be reasoned in the context of solid state physics (perhaps within the band theory of solids?) that copper is a metal ?"}
{"id":"33430","title":"Does a coronal mass ejection change solar neutrino emission rates?","text":"Does the CME and neutrinos have any relation? CME is measured by Corona graphs.. How do they measure neutrinos coming from the sun? Does any of these have effects on earth's magnetic field or atmosphere..?"}
{"id":"33439","title":"Why do rolling disc (coin) move in circular path?","text":"We have a coin that is rolled such that it's tilted at an small angle $ \\theta $. **Question::** What turns around rolling disc so that it traces circular motion (spiral as it's speed decreses)? **Problem statement::** In the following figure, what turns the disc so that it follows the circular path? ![image 1](http:\/\/i.stack.imgur.com\/0Mp7R.jpg)   **Solution::**   I interpreted the problem incorrectly. Only a sliding body would have a torque like that."}
{"id":"15629","title":"Is pressure distribution affected by shape","text":"We have two iron (assume real-life stiffness) manhole covers resting on friction-less, perfectly smooth shims on flat ground. One is circular and the other square. If a force F is applied vertically downwards at the center of each manhole cover, is the pressure on the manhole shim distributed uniformly in each case? i.e. is the downward force at any given cover-contact-point on the shim the same for both manholes."}
{"id":"30165","title":"expectation vaules of commutator and anti commutator (momentum and position)","text":"what is expectation vaules of commutator and anti commutator when the case is momentum and position. when the case is commutator: $$\\langle i\\hbar\\rangle=?$$ when the case is anti commutator: $$\\langle - i\\hbar \\rangle=?$$"}
{"id":"15623","title":"Why Are Maxwell's Equations Preferred Before Those Proposed by H. Hertz?","text":"In a recent exchange it was pointed out that in certain situations Maxwell's equations need Lorentz force as a patch (the latter not being part of those equations neither is it derivable from them). It appears that the equations proposed by H. Hertz lack that deficiency while retaining Lorentz invariance. What might be the reason for using Maxwell's equations instead?"}
{"id":"15622","title":"What are typical values of the critical thickness of insulation?","text":"As most people who've had any elementary heat transfer course are aware, when insulating a pipe, wire, etc, there is a critical thickness for the insulation below which it causes greater heat transfer than no insulation would. In actuality, since the critical radius is a minimum, one needs to go even farther before it catches up to the uninsulated case. This effect is due to the thermal resistance from the added insulation (a function of material and thickness) being less than the decrease in thermal resistance to convection brought about by the increased surface area due to the thicker insulation. A more detailed explanation is found here What I wonder is how large this critical thickness typically is for realistic cases such as an electrical wire, a steam pipe, or a water pipe (all cases in STP air)? I realize exact answers aren't possible given the fact that it will vary with material type, temperature diff., and size. The motivation for this is that it theoretically could prevent wasting insulation or provide a practical avenue for energy dissipation, but Fundamentals of Heat and Mass Transfer 6e by Incropera et al. indicates that the effect is usually too small to be of practical importance."}
{"id":"113847","title":"Principle of Caratheodory and The Second Law of Thermodynamics","text":"**Background** _Constantin Carathéodory formulated thermodynamics on a purely mathematical axiomatic foundation. His statement of the second law is known as the Principle of Carathéodory, which may be formulated as follows:_ > In every neighborhood of any arbitrary initial state $P_{0} $ of a physical > system, there exists neighboring states that are not accessible form $P_{0} > $ along quasi-static adiabatic paths. The above statement is taken from 'Heat and Thermodynamics' 8th Ed by Zemansky and Dittman and it provides a very concise discussion on the topic which I did not find very illuminating. Moreover, Wikipedia states it slightly differently as: > In every neighborhood of any state S of an adiabatically enclosed system > there are states inaccessible from S. _With this formulation he described the concept of adiabatic accessibility for the first time and provided the foundation for a new subfield of classical thermodynamics, often called geometrical thermodynamics._ **My questions are:**   * What is the exactly meant by _Adiabatic Accessibility_ and how is this relevant to the formalism of The Second Law?   * How is this formalism equivalent to the Kelvin Planck and Clausius statements of the Second Law?   * Except for the satisfaction of having an axiomatic approach, does this provide any advantage over the Kelvin-Planck formalism using heat engines? P.S: The text in _italics_ has been copied from Wikipedia."}
{"id":"107628","title":"Covariant derivative of a vanishing tensor component","text":"Is the covariant derivative of a vanishing tensor component necessarily zero?"}
{"id":"24028","title":"How do I calculate DC motor speed for a given load?","text":"Suppose I have a robot of a given mass, and I'm choosing between 2 different wheels and 2 different motors to put on it. For each wheel I have the diameter, and for each motor I know the stall torque and free speed. How would I figure out which motor and wheel combination will make the robot move the fastest? My calculations show to use the big motor with the big wheels, but the small motor with the small wheels goes faster than the big motor with the small wheels. I am not sure my calculations are correct, I need to know the correct way to work this problem."}
{"id":"100734","title":"Diffussivity of gases, molecular weight and Lewis number","text":"I am trying to understand the following figure of laminar premixed flames: ![enter image description here](http:\/\/i.stack.imgur.com\/hCKz4.png) The Lewis number is defined as $Le = \\frac{\\alpha_{mix}}{D_{fuel}}$ where $\\alpha_{mix}$ is the thermal diffusivity of the mixture and $D_{fuel}$ is the mass diffusivity of the fuel. The propane fuel has larger molecular weight than air and methane has lower molecular weight compared to air. It is not clear to me why mass diffusivity increases as the flame gets richer in the case for propane (Lewis number drops) but the converse is true for methane. I know it has to do with the difference in molecular weight, but why does diffusivity increase for propane if you introduce more propane into the system but decrease in the case of methane? I am therefore not sure why Le decreases in case of propane but increases in the case of methane as the mixture gets richer. Maybe the thermal diffusivity is changing?"}
{"id":"86622","title":"What is the correct Hamiltonian for a system of coupled quantum oscillators?","text":"The Hamiltonian (see Eqn. 1 in Appendix 2 of this paper) for a system of coupled quantum oscillators is given as $$H=\\frac{1}{2}∑_{i}p^{2}_{i}+\\frac{1}{2}∑_{j,k}A_{jk}q_{i}q_{k}$$ Yet, in my QM course, the Hamiltonian for such a system was given as $$H=\\frac{1}{2m}∑_{i}p^{2}_{i}+\\frac{k}{2}∑_{i}x^{2}_{i}+\\frac{K}{2}∑_{i}(x_{i}-x_{i+1})^{2}$$ where the third term represents the coupling between the oscillators. Why aren't these equations equivalent?"}
{"id":"4345","title":"Entangled particles","text":"So we have two particles (A and B) that are entangled. From what I understand, entanglement isn't destroyed, it is only obscured by subsequent interactions with the environment. Particle A goes zooming off into outer space. 10 years later, Particle B becomes incorporated into my brain. The day after that, an alien scientist measures the entangled property on Particle A. This will have some immediate non-local effect on Particle B won't it? And since B's state has been altered (in some sense), and it is part of my brain, then my brain state has been altered as well, hasn't it? Maybe only a tiny amount, obscured by the many environmental interactions that the two particles have been subjected to since the initial entanglement, but in a way that is real and at least conceivably significant. And if that is true, then to the extent that mental states supervene on brain states, my mental state would also have been altered by non-local effects. Or is that wrong? Thanks!"}
{"id":"4340","title":"Other possible theories (other than string theory) which are generalizations of the standard model with incorporation of gravity","text":"The only finite mathematical framework that incorporates both the standard model of particle physics and gravity under one umbrella that I am aware of is string theory. I would like to know whether there are any other mathematical possibilities exist which do not depend on supersymmetry and still consistent with the standard model and gravity and produce finite answers. In a nutshell my question is: can there be any alternative to string theory? (Remember, I am not talking about only gravity. I am talking about gravity as well as other phenomena)."}
{"id":"108265","title":"$\\mathrm{d} \\Omega_{CM}$ for a $1\\rightarrow 2$ particle decay?","text":"The differential solid angle is described in e.g. Srednicki's QFT text but only for the case of scattering. Because in the case of scattering it's defined with respect to the incoming three-momentum ${\\bf k_1}$ and it's scattered three-momentum ${\\bf k_1'}$. This definition would not make much sense for a particle decay (I can't at least see how it would easily carry over). What about the case of **particle decay**? What is $$\\mathrm{d} \\Omega_{CM}$$ in the case of particle decay? Any good references where it is properly defined?"}
{"id":"109793","title":"Electric field inside a metal nanoparticle?","text":"In localized surface plasmon resonance, the conduction electrons of a metal nanoparticle oscillate in tune with the frequency of the incident electromagnetic wave . The effect is dominated by electrons present on the surface of the particle as it is a nanostructure and surface atoms constitute a major portion of total number of atoms. As a dipole mode is created , electric field strength nearby the particle on outside is increased but what happens inside the volume of the nanoparticle ? For a small metallic nanoparticle , let us say 20 nm , the field (considereing a 500nmEM wave)totally penetrates the particle and the electrons of the whole particle (volume + surface ) will resonate at plasma frequency, then there will be a field gradient generated inside the particle too but inside metals , a field cannot exist. I am a little confused in this aspect."}
{"id":"79370","title":"Questions on electric field formed in conductor and the magnetice field it moves through","text":"When a conductor (e.g. a wire) is moving through a uniform magnetic field, it will create a electric field in the wire as the magnetic field exerts a force on the moving conduction electrons in the wire and create a potential difference. ![the wire is so thick!](http:\/\/i.stack.imgur.com\/2kBWr.gif) And here is my questions:   1. (If we can get the magnetic field to exert a super strong force on the electrons) Is it ever possible for the electrons in the wire to get separated from the wire like what happens when high energy photon hits an atom and knock off its electrons?   2. Since $F_e$ and $F_b$ always act in the opposite direction, why don't they cancel off one another (so the net force is 0) when the charges are in the electromagnetic equilibrium ($qE = qvB$) ?   3. What's so significant about the this equilibrium ($qE = qvB$)? The formula for motional emf $emf = vBl$ is derived from the fact that $E = vb$ and that was because the charges are in the electromagnetic equilibrium. But wouldn't this formula $emf = vBl$ be inaccurate if we have a motional emf that are caused by charges not in the electromagnetic equilibrium ($qE = qvB$)? Why when there is a motional emf we assume that the chargers are in this equilibrium? I'm still learning secondary school physics and I'm very new to this topic and so please explain it in laymen terms if possible. Thank you so much."}
{"id":"91087","title":"Determine whether the light will reach certain points or not","text":"I've got an interesting question which goes like that: suppose we have a long cylindrical rod made of a transparent material (see illustration below). A point source $O$ which sits on rod's axis emits light in all directions. The light is partially reflected and partially refracted (i.e. goes to the air). The question is - which points will the light rays reach? (1) A only, (2) A and B only (3) All the four points (4) Not enough information ![enter image description here](http:\/\/i.stack.imgur.com\/ARRM0.jpg) According to the book the right answer is (3), but I'm not so sure that this is correct - one the one hand, the light is emitted in all possible directions but on the other hand, my friend thinks there could be a total internal reflection at some point. In both cases there's no strict proof for either answer. By the way, there's no information about positions, indices of refractions and so on."}
{"id":"91085","title":"Radiative cooling time for a black body","text":"Okay, so this is confusing me a bit. How can I calculate the time it takes to cool a perfect black body from an initial temperature $T$ down to equilibrium temperature (say, 3 K for space)? I know that $$\\frac{P}{A}=\\sigma T^4,$$ but I'm stuck at getting $t$ out of there."}
{"id":"79378","title":"speed of light observation","text":"I was on a ship at night travelling parallel to the shore watching a lighthouse beam hitting the concrete sea wall. As the beam hit the wall at ninety degrees and continued to rotate I saw what appeared to be the light splitting into its spectrum and streaking away at incredible speed. Was I seeing the actual speed of light as the beam's direction changed?"}
{"id":"91556","title":"Is the Earth negatively or positively charged?","text":"The Earth carries a negative electric charge of roughly 500 thousand Coulombs. Does that mean the Earth is negatively charged?"}
{"id":"23346","title":"Twin paradox - observers counter orbiting Earth","text":"Imagine three observers - one (A) stationary on the surface of Earth (latitude 0 deg) and two others orbiting the planet in the same circular equatorial orbit just in the opposite direction. When the orbiting observers B and C meet each other just above the observer A all theisr clocks are synchronized. When their meet again what will their clock show? They move relative to each other, but experience equal acceleration and gravitational potential. Will the orbiting observers have their clock synchronized all the time? If the time intervals differ who measures greater time and why? Suppose Earth in non- rotating itself and with perfect spherical symmetry."}
{"id":"101664","title":"Why is radiation at the relativistic limit characterized by 1\/gamma angle?","text":"Trying to think of a reason as to why? Also, the factor that radiation is zero on axis? I haven't been able to resolve these two fundamental principles in my head :("}
{"id":"23342","title":"How do I know an initial speed of a thrown object using the max height","text":"> The simulation being referred to is in box2d An object is thrown to the max height of $h$ with gravity of $g$, what is it initial speed?   I tried the following:   $v = v_0 - g t$   $0 = v_0 - g t$   $t = \\frac{v_0}{g}$ $h = v_0 * t - \\frac{1}{2}g * t^2$   $v_0 = \\sqrt{2 * g * h}$ But putting it into physical simulation gives different max height, is the equation wrong or its a simulation artifact?"}
{"id":"76827","title":"Cylindrical charge distribution","text":"I'll just list the problem first and then explain what I need help with. > Consider a cylindrical charge distribution extending from $r = 0$ to $r = > 8.2 $cm of charge density $\\rho = r\/a_0$, where $a_0 = 21 cm · m^3\/µC$. This > cylindrical charge distribution is surrounded by a dielectric shell of > dielectric constant $6.72$ whose inner radius is $17.7 cm$ and outer radius > is $25.9 cm$. I cannot seem to find a formula for finding the electric field at a given distance. Any suggestions are where to even start?"}
{"id":"76824","title":"Proper Time with a non-zero Initial Velocity","text":"I have a question regarding how to find the proper time for a body with an initial velocity to slow down to 0. For example, the equation I have been working with looks like: $$\\int^\\tau_0 d\\tau=\\int^t_0\\frac{1}{\\sqrt{1+\\frac{(\\alpha t+v)^2}{c^2}}}dt$$ Where $t$ is equal to the coordinate time for the rocket to slow down, $\\alpha$ is the proper acceleration and $v$ is the intial velocity. Integrating gives: $$\\tau=\\frac{c}{a}arcsinh(\\frac{\\alpha t+v}{c})$$ I believe, however, that this is incorrect as the values it returns are far too small to be a legitimate value for proper time. I do believe that this is on the right track to deriving the correct value."}
{"id":"21134","title":"Light Rays that are Perfectly Parallel","text":"I just heard this simple reasoning in a documentary film: > Light rays from distant stars are perfectly parallel. This is pretty interesting thought. In nature, it is hard to find something really perfect. For example, there are no isolated systems -- only quasi- isolated. But here it looks like indeed the light rays are perfectly parallel. **Maybe this can be confirmed by calculations?** \\-- e.g. that angle between two light rays cannot be infinitely small (Planck constant will be probably involved?)."}
{"id":"74467","title":"Why is renormalization necessary in finite theories?","text":"Michael Brown made the following comment here: > The modern understanding of renormalization (due to Kadanoff, Wilson and > others) is hardly controversial and has nothing really to do with > infinities. It is necessary even in completely finite theories, but the fact > that it fixes infinities in QFT is a bonus. Can anyone explain what this means?"}
{"id":"74462","title":"Optimal slit width in Young's double slit experiment","text":"I'm trying to do Young's double slit experiment at home. Note that I don't have a laser, only a torch. I could get a bulb or use a candle though, if it helps I built the slits by cutting into a black chart paper with a knife. I tried to build a setup by placing a single slit, double slits and a screen one after the other, and shine a torch through the single slit. I tried varying distance between single-slit\/double-slit and double-slit\/screen which did not help. **_I did not observe any interference pattern._** All I got was two parallel bright fringes instead, like the ones you would get by shining a torch through two very thick slits. > I think that is precisely the problem, that the slits are pretty thick. > **What is the optimal slit width I should have to observe an interference > pattern? How do I build that with a chart paper?** I have seen videos of this experiment online, where people use pencil leads and _hair_ (!!), but they both use a laser. _I don't want to use a laser_ (just because I don't have one, and I'm probably too lazy to go and get one!). Note that you may suggest using any other materials that might be easily available at home."}
{"id":"135157","title":"Rigorous distinction between quasiparticles and collective excitations","text":"I would like to hear your opinion on the question whether there is an accepted distinction between both concepts in condensed matter physics. I would tend to use the word quasiparticle for dressed fermionic particles and the term collective excitation in the case of bosonic excitation. But considering for example magnons in ferromagnets makes it very arbitrary kind of definition since magnons can be viewed (if I am not wrong) both as quasiparticles and also as collective excitations."}
{"id":"116398","title":"force of hail falling straight","text":"There is a hail storm and it's falling straight down (0 wind factor) will the hail meet a moving car with greater force than a car standing still?"}
{"id":"134209","title":"Dirac's remark that inspired Feynman when formulating path integral","text":"When Feynman was trying to formulate path integral of quantum mechanics, he was inspired by Dirac's remark which roughly states that $e^{i\\frac{S}{\\hbar}}$corresponds to the transition amplitude, where $S$ is the action. I'm wondering what was Dirac's original argument behind this statement."}
{"id":"134201","title":"Gauss' law and ions?","text":"My text book says that with we have a singly ionized sodium atom net charge +e and if we choose a spherical surface centered on the ion and large enough to contain it all we do not need to know the distribution of charge inside the ion. And using Gauss' law can find the the electric field outside the sphere radius R as $E(R)=-\\frac{e}{4\\pi \\varepsilon_0R^2}$. Here is my problem surly we need to know that the charge is distributed evenly around the center?? If we this was not the case then all the charge could, say, be all at one side of the sphere and thus the electric field at that side will be greater then the electric field at the other side of the sphere will it not??"}
{"id":"134207","title":"Work and transfer of energy","text":"A businesswoman is applying force of 12 N [or more] to carry her briefcase for a horizontal distance of 200 m. How much work is she doing on the briefcase?. I tried multiplying force by distance but the answer I got is different from the answer in the answers sheet. The correct answer is 0. I don't know why We have to use this formula: W = F * D"}
{"id":"14234","title":"Formula for getting energy required to accelerate to a certain speed?","text":"I am curious how many newtons it takes to accelerate an object with a given coefficient of drag, reference area, weight, and air pressure? I know this may sound like a simple equation but I have no idea where to find this formula. Thanks."}
{"id":"14230","title":"Eigenfunctions v.s. eigenstates","text":"Is there a difference between \"eigenfunction\" and \"eigenstate\"? They seem to be used interchangeably in texts, which is confusing. My guess is that an \"eigenfunction\" has an explicit representation\/argument, e.g. in terms of position, $x$, whereas an \"eigenstate\" is abstract. Thanks in advance for any clarification! :) But then again, some texts violate this rule, so I'm prolly wrong..."}
{"id":"33378","title":"Was Planck's constant $h$ the same when the Big Bang happened as it is today?","text":"Was Planck's constant $h$ the same when the Big Bang happened as it is today? Planck's constant : $$h= 6.626068 × 10^{-34}\\, m^2 kg \/ s,$$ $$E=n.h.\\nu,$$ $$\\epsilon=h.\\nu$$"}
{"id":"122110","title":"Totally antisymetric wavefunction: clarification about terminology","text":"Pauli's Principle says: \"The wavefunction of two identical fermions must be totally antisymmetric\". I know that, for a antisymmetric wavefunction, $(-1)^L*(-1)^{S+1}*(-1)^{I+1}=-1$ \"totally antisymmetric\" means this relation or it means that these 3 relations: $(-1)^L=-1$ and $(-1)^{S+1}=-1$ and $(-1)^{I+1}=-1$ must be verified simultaneusly?"}
{"id":"122116","title":"With these two elements how can the isotope half-life be determined using first order kinetics?","text":"With these two elements how can the isotope half-life be determined using first order kinetics? nobelium-259 half-life recorded @ 58 minutes   fermium-253 half-life recorded @ 3 days Although I do not know, there is probably an average starting concentration that can probably be determined by the name of the element. Maybe in the comments someone can hook me up with some estimate averages so the expansion of the math could be given in the answer."}
{"id":"87393","title":"How to quote answer when error is smaller than significant figures of data?","text":"We have these rules:   1. Can't quote answer to more significant figures than original data.   2. Error quoted to one significant figure unless first significant digit is a one. In which case quote error to 2 significant figure. But with original data to 3 significant figure, I get the answer **13.15(6)**. i.e. **13.15 +\/- 0.06** How do I quote this to 3 significant figure without breaking the rules? If I round the error to 1, I still have to have a trailing 0: **13.20(10)**"}
{"id":"123861","title":"Verlet timestep","text":"I am using velocity Verlet in molecular dynamics. Is just a simple question, if a simulation using time-step femtosecond, in velocity Verlet is just necessary $dt = 10^{-15}$ to use femtosecond?"}
{"id":"87644","title":"Tension in vertical circular motion","text":"In vertical circular motion we conserve energy for calculating velocities at a point (if initial velocity given). But, energy can only be conserved when forces are conservative. Tension is not a conservative force. Does it not affect the particles velocity? Is the tension's role only to provide centripetal acceleration?"}
{"id":"47668","title":"Extremal black hole with no angular momentum and no electric charge","text":"A black hole will have a temperature that is a function of the mass, the angular momentum and the electric charge. For a fixed mass, Angular momentum and electric charge are bounded by the extremality condition $$M^2 - a^2 - Q^2 \\gt 0$$ Exactly at the extremality boundary, both entropy and temperature are zero. Suppose i create a black hole with a spherically symmetric, incoming wavefront of electromagnetic radiation **in a pure quantum state** (that is, the density matrix satisfies the property $\\rho^2 = \\rho$). The wavefront is shaped in a way such that the whole energy of the packet will be inside the Schwarzschild radius, which will form an event horizon. Since the wavepacket is as nearly as pure as it is physically possible to create, the quantum (Von Neumann) entropy is zero or nearly zero. But the formation of the black hole does not create nor destroy entropy, so the black hole must contain zero or nearly zero entropy as well. So the black hole seems to be extremal (it has zero temperature) but it nonetheless does not have any angular momentum (it is formed from a wavefront with zero net polarization over the whole sphere) and it does not have any charge (electromagnetic radiation is neutral). > **Question:** what \"hair\" does have a black hole formed from such a pure > state, so that it can be extremal and still do not have angular momentum or > electric charge (which are the _classical_ hair that we come to expect from > classical general relativity) _This question is a mutation ofthis question, but while that specific question tries to look what input black hole states create specific output (Hawking) radiation states that are far from thermal from a statistical point of view, this question is specific about extremality that is unrelated to angular momentum and charge_"}
{"id":"94430","title":"Sound - for purposes of vibration","text":"What is the best way to distribute noise from more than one source (I'm envisioning a system with many), within a dome, with the ground as its primary target, at optimal frequencies and volumes to create maximum vibrations on the ground? Think, this picture - http:\/\/hyperphysics.phy- astr.gsu.edu\/hbase\/acoustic\/imgaco\/foc3.gif \\- but with more than one source. ![enter image description here](http:\/\/i.stack.imgur.com\/Zt4xK.gif) I'm assuming you would want the dome to be perfectly circular, and have the point at which the dome wall meets the ground at a slight incline (as opposed to vertical) so all the sound stays directed at the ground. Imagine a tennis ball which is cut in half, but was split just before the equator. You take the side that is smaller, and flip it over. Thats what I am imagining. I'm just judging from the above drawing that you would want the sound sources crammed as close as possible in the center of the dome, in as perfect as a circle as you could design it, and maybe directed so that they are all pointed to the center of their own, independent, equally sized areas (the dome being the total area of the surface that the sound is being pointed at). It seems like if you could cram all the sound into a point of singularity and put it at the center of the dome, and then release it somehow - that would be ideal (although that sounds pretty complicated). Not sure about frequencies or volumes."}
{"id":"8265","title":"Is there a good Modern English translation of Newton's \"Philosophiæ Naturalis Principia Mathematica\" available for free viewing or download?","text":"I have found several nice versions available for purchase, but all of the free ones are either Latin, old English, or pixellated. I searched google and was only able to locate the free ones linked to from the Wikipedia page on Philosophiæ Naturalis Principia Mathematica."}
{"id":"95885","title":"Which one conducts heat better: water, or air?","text":"If we were to assume room temperature of 20C, which of the fluids conducts heat better, and why?"}
{"id":"126742","title":"Do all the electrons form Cooper pairs at absolute zero?","text":"At $T=0K$, do all the electrons form Cooper pairs, or just the electrons near the Fermi Surface do?"}
{"id":"89590","title":"Why there is a $\\frac{1}{2}$ in the distance formula $d=\\frac{1}{2}at^2$?","text":"I'm preparing for my exam, but I have difficulties in perceiving why there is a $\\frac{1}{2}$ in the distance formula $d=\\frac{1}{2}at^2$?"}
{"id":"126747","title":"Cross-Section Compendium?","text":"Are there any books or references that exhaustively list cross-sections for various processes in QED or particle physics? I think such a book would make for a very useful reference, and in particular I would like to practice deriving cross-sections."}
{"id":"65149","title":"Is the fine-structure constant a parameter of the standard model?","text":"According to the wikipedia entry on the fine-structure constant: > In fact, α is one of the about 20 empirical parameters in the Standard Model > of particle physics, whose value is not determined within the Standard > Model. but, the wikipedia list of parameters does not mention α:               me  Electron mass       511 keV     mμ  Muon mass       105.7 MeV     mτ  Tau mass        1.78 GeV     mu  Up quark mass   μMS = 2 GeV     1.9 MeV     md  Down quark mass     μMS = 2 GeV     4.4 MeV     ms  Strange quark mass  μMS = 2 GeV     87 MeV     mc  Charm quark mass    μMS = mc    1.32 GeV     mb  Bottom quark mass   μMS = mb    4.24 GeV     mt  Top quark mass  On-shell scheme     172.7 GeV     θ12     CKM 12-mixing angle         13.1°     θ23     CKM 23-mixing angle         2.4°     θ13     CKM 13-mixing angle         0.2°     δ   CKM CP-violating Phase      0.995     g1 or g'    U(1) gauge coupling     μMS = mZ    0.357     g2 or g     SU(2) gauge coupling    μMS = mZ    0.652     g3 or gs    SU(3) gauge coupling    μMS = mZ    1.221     θQCD    QCD vacuum angle        ~0     v   Higgs vacuum expectation value      246 GeV     mH  Higgs mass      ~ 125 GeV (tentative)      Is α one of the basic parameters of the Standard Model? If not, then is there a simple formula for α in terms of these other parameters? (My guess is that α can be derived from g1\/g2\/g3. However, I have been unable to find an explicit formula so far.)"}
{"id":"69828","title":"Equivalence Theorem of the S-Matrix","text":"as far as I know the equivalence theorem states, that the S-matrix is invariant under reparametrization of the field, so to say if I have an action $S(\\phi)$ the canonical change of variable $\\phi \\to \\phi+F(\\phi)$ leaves the S-matrix invariant. In Itzykson's book there is now an exercise in which you have to show, that the generating functional $$Z^\\prime(j)=\\int \\mathcal{D}[\\phi] \\exp{\\\\{iS(\\phi)+i\\int d^4x\\hspace{0.2cm} j(x)(\\phi +F(\\phi)\\\\}}$$ gives the same S-matrix as the ordinary generating functional with only $\\phi$ coupled to the current due to the vanishing contact terms. He then writes, that this proves the equivalence theorem, which I do not fully understand. Suppose I take this canonical change of variable, then I get a new action $S^\\prime(\\phi)=S(\\phi+F(\\phi))$ and a generating functional $$Z(j)=\\int \\mathcal{D}[\\phi] \\exp\\\\{iS(\\phi+F(\\phi))+i\\int d^4x\\hspace{0.2cm} j(x)\\phi\\\\} $$ If I now \"substitute\" $\\phi+F(\\phi)=\\chi$ I get $$Z(j)=\\int \\mathcal{D}[\\chi] \\det\\left(\\frac{\\partial \\phi}{\\partial \\chi}\\right) \\exp\\\\{iS(\\chi)+i\\int d^4x\\hspace{0.2cm} j(x)\\phi(\\chi)\\\\} $$ with $\\phi(\\chi)=\\chi + G(\\chi)$ the inverse of $\\chi(\\phi)$. Therefore comparing $Z(j)$ and $Z^\\prime(j)$ I get an extra jacobian determinant. Where is my fallacy, or why should the determinant be 1? Thank you in advance"}
{"id":"96281","title":"Srednicki's book chapter 8","text":"Reading first page in chapter 8 of Srednicki's it reads: > To employ the $\\epsilon$ trick, we multiply $H_0$ with $1-i\\epsilon$. The > results are equivalent to replacing $m^2$ with $m^2-i\\epsilon$. Now, our Hamiltonian is $H_0=\\frac{1}{2}\\Pi^2+\\frac{1}{2}(\\nabla\\phi)^2+\\frac{1}{2}m^2\\phi^2$ so making the replacement he suggests ($m^2\\to{}m^2-i\\epsilon$) we would have $H_0(1-i\\epsilon)=\\frac{1}{2}\\Pi^2+\\frac{1}{2}(\\nabla\\phi)^2+\\frac{1}{2}m^2\\phi^2-\\frac{i\\epsilon}{2}\\phi^2$ while on the other hand, in reality $H_0(1-i\\epsilon)=\\frac{1}{2}\\Pi^2+\\frac{1}{2}(\\nabla\\phi)^2+\\frac{1}{2}m^2\\phi^2-\\frac{i\\epsilon}{2}\\Pi^2-\\frac{i\\epsilon}{2}(\\nabla\\phi)^2-\\frac{i\\epsilon}{2}m^2\\phi^2$ And we see that both terms don't agree. How can both expressions be equivalent?"}
{"id":"65498","title":"Sackur-Tetrode equation - clarification required - problem with units","text":"I'm a 2nd year physics undergraduate and recently I've volunteered to give a short presentation on the Sackur-Tetrode equation derivation and its use at removing the Gibbs paradox. I've looked on the Internet and in some books and found many methods for deriving this result. The one I'm going with is the derivation from the book _Thermodynamics and Statistical Mechanics_ by W. Greiner. However, there are some things I'm still not certain about (I dared to expand the argument a little) and in order to seem like I know what I'm talking about, I need some clarification. Here's more or less how I see it going: * * * In order to obtain an expression for the entropy of an ideal gas with $N$ (identical) particles of mass $m$ occupying a volume $V$ and with internal energy $E$, we use Boltzmann's entropy formula: $$S(N,V,E)=k\\ln \\Omega(N,V,E).$$ This reduces the problem to finding $\\Omega(N,V,E)$, the number of microstates of our system which realise the macrostate $(N,V,E)$. A microstate of the system gives information about the entire system at a microscopic level, that is, it specifies the exact momentum and position of each constituent of the system. There are $N$ particles and each one has three directions of position and of momentum, therefore our system has $6N$ degrees of freedom. We imagine a $6N$ dimensional space, called the phase space, where each axis corresponds to a degree of freedom or the system. Each point in this space corresponds to a specific microstate: $(r_x^{(1)},r_y^{(1)},r_z^{(1)}, p_x^{(1)}, p_y^{(1)}, p_z^{(1)},...,r_x^{(N)},r_y^{(N)},r_z^{(N)}, p_x^{(N)}, p_y^{(N)}, p_z^{(N)})$. The microstates which correspond to our particular macrostate $(N,V,E)$ form some sort of high-dimensional hypersurface. The \"volume\" of this surface, $\\phi(N,V,E)$, would give us a quantity related to the number of microstates. Of course in phase space, \"volumes\" of surfaces have dimension of action raised to some power, and since we're interested in a number, the idea is to introduce some sort of elementary bit of \"volume\", $\\phi_0$ and then divide to produce a number $\\frac{\\phi(N,V,E)}{\\phi_0}$ which we claim to be the number of relevant microstates. Fortunately, it physically makes sense for such an elementary \"volume\" to exist, since from the uncertainty principle we know that a point in phase space really cannot be specified with infinite precision, and that there is a quantum of action, which is nice because our \"volumes\" have dimension of action to some power. We already used the fact that we have $N$ particles. Now it's time to take our other givens into account. Let's start with $E$. Classicaly, for an isolated system of noninteracting particles, the total energy is purely kinetic and given by the sum of the kinetic energies of each particle: $$E=\\sum_{k=1}^{N} \\frac{ [p_{x}^{(k)} ]^2 +[p_{y}^{(k)} ]^2 +[p_{z}^{(k)} ]^2 }{2m}$$ That gives a bound for our momenta coordinates. The position coordinates can be likewise bounded using our last given: the volume $V$. We may assume that the system occupies a cube of length $l$, then for every $k=1,2,...N$ we have: $$0\\le r_{x}^{(k)} \\le l ; \\quad 0\\le r_{y}^{(k)} \\le l ; \\quad 0\\le r_{z}^{(k)} \\le l ; $$ This seems like cheating a bit (why assume a particular shape of the container), but the assumption can be later dropped because in further calculation we don't really care about the bounds on the position coordinates, thanks to the fact that the total energy is independent of them. So this works for arbitrary $V$. The two previous equations specify the desired \"volume\" in our phase space. The dimension of this object is $6N-1$ because we only have one constraint, namely the energy being constant. To calculate its \"volume\" we need to consider the wonderful looking integral: $$\\phi(N,V,E)=|A|=\\int_{A} dr_{x}^{(1) }dr_{y}^{(1)} dr_{z}^{(1)} dp_{x}^{(1) }dp_{y}^{(1)} dp_{z}^{(1)} ... dr_{x}^{(N) }dr_{y}^{(N)} dr_{z}^{(N)} dp_{x}^{(N) }dp_{y}^{(N)} dp_{z}^{(N)}$$ Where $A$ is the set given by the previous equations. We can simplify this notation by writing $$\\phi(N,V,E)=|A|=\\int_{A} d^{3N}r \\ d^{3N}p$$ Notice that the position coordinates are independent of momentum coordinates, so really, the integral factors: $$\\phi(N,V,E)=|A|=\\int_{A_r} d^{3N}r \\int_{A_p} d^{3N}p$$ Where $A_r$ is the set given by the position conditions and $A_p$ by the momentum conditions. The first factor is simply $V^N$, or $l^{3N}$. The second factor is slightly trickier to calculate, but it's just the surface of a 3N dimensional sphere with radius $\\sqrt{2mE}$. This already implies that the units will be $\\mbox{[momentum]}^{3N-1}$, and so the units of $|A|$ are $\\mbox{[momentum]}^{3N-1} \\mbox{[length]}^{3N}$. The actual derivation of the surface of an $n$-dimensional sphere of radius $r$ will be left out of this, and we just quote the result: $S_n = \\frac{2 \\pi ^{n\/2}}{\\Gamma (n\/2)}r^{n-1}$ Anyway, we have obtained a solution for our \"volume\" $|A|$: $$\\phi(N,V,E)=|A|=2 V^N \\frac{\\pi ^{3N\/2}}{\\Gamma(3N\/2)}\\sqrt{2mE}^{3N-1} \\quad \\mbox{[momentum]}^{3N-1} \\mbox{[length]}^{3N}$$ * * * Here I have a big problem. As said before, the number of states is $\\frac{\\phi(N,V,E)}{\\phi_0}$. I would _like_ for $\\phi_0$ to be something like $h^{3N}$ or $h^{3N-1}$ but this gives wrong units! Ignoring this for the moment and just taking $\\phi_0 = h^{3N-1}$ here's the rest of the derivation: * * * We thus obtain an expression for the number of microstates: $$\\Omega(N,V,E) = 2 V^N \\frac{\\pi^{3N\/2}}{\\Gamma(3N\/2)} \\left( \\frac{\\sqrt{2mE}}{h} \\right) ^{3N-1}$$ Taking the log of this and using appropriate approximations etc.(I skip this here but won't during the presentation ofc.) we arrive at: $$ S(N,V,E) = kN \\left( \\frac{3}{2} + \\ln V \\left( \\frac{4 \\pi m E}{h^2} \\right) ^{\\frac{3}{2}} \\right)$$ This unfortunately leads to the Gibbs paradox. The reason for this is that we've assumed the particles to be disinguishable when in reality they are not. In order to account for this, we need to introduce the Gibbs correction factor into the total amount of states $\\Omega$: $$\\Omega_C(N,V,E) = \\frac{1}{N!} \\Omega(N,V,E) $$ Dividing by the amount of permutations accounts for indistinguishability. Going through the same derivation leads to the proper formula now. Thus the paradox is fixed. * * * I would love to hear some feedback about this presentation and particularly with the problem I had with $\\phi_0$. Additionally, the introduction of the Gibbs correction factor seems kind of shady to me, I read somewhere that it's again just an approximation and that proper counting would produce a more complicated result than just diving by the factorial, but once again the error is negligible."}
{"id":"82911","title":"Any known relationships among quark masses?","text":"Here is some numerology: Gen 1 Mass(d)\/Mass(u) = 2 = 2 * 1 Gen 2 Mass(c)\/Mass(s) = 12 = 4 * 3 Gen 3 Mass(t)\/Mass(b) = 40 = 8 * 5 Does standard model make any predictions of relations? Do any GUT or string theories make any predictions of relations?"}
{"id":"65495","title":"How to find distance of closest approach for a Schwarzschild geodesic?","text":"What is the distance of closest approach in this Wikipedia article? I can't seem to find its definition, and this other question doesn't have an answer I can understand."}
{"id":"69790","title":"Interpreting the equation $\\vec F = R\\vec v$","text":"> The Aristotle's law of motion, which is incorrect, states that > > _The velocity of an object $\\vec v$ is directly proportional to the force > $\\vec F$ acting on it_ > > or $\\vec F \\propto \\vec v$ > > or $\\vec F = R\\vec v$, > > where $R$ is the constant of proportionality which is the resistance to > motion. I want to know how can we interpret this equation $\\vec F = R\\vec v$ ? Does this equation imply that $\\vec F$ and $\\vec v$ have the same direction? If it does, then how do we know this through this equation? Does equality mean that they have the same direction? The book in which I actually saw this equation, wrote this equation in this way: $\\vec v = \\frac {\\vec F}{R}$."}
{"id":"119575","title":"Do the same experiments yield different results due to the principle of uncertainty?","text":"When thinking about small particles and their uncertainity, I've allways rather seen them being _all over the place_ rather than randomly changing location. I would think that, in the same time, you'd always have same result, have you tried to do something to them (eg. shoot another particle at them). However, in certain situations, like the evaporation of a black hole, the particle just either is here or it is not. This makes me wonder: if you try something with uncertain particles, does every experiment end up differently because, event though all conditions are the same? (The last requirement seems to be unfeasible so consider it just a thought experiment). Similarly, if there was a separated Big bang would the other universe look different than ours because particles ware randomized differently? Out of the question, I must say that I find it very confusing for particles to really have random locations. Wasn't there ever a suspicion that their location (as well as the probability of it) has a rule we can find out?"}
{"id":"116562","title":"Air vs. Water attraction of Oxygen Molecules","text":"The air we breathe is made up of nitrogen and oxygen gases. The water in a pond is made of a single hydrogen\/oxygen molecule. If it wasn't for the surface tension on top of the water, oxygen molecules can enter the water and occupy space next to the hydrogen\/oxygen molecules. My question is this: All things being equal, is oxygen more attracted to water or air?"}
{"id":"16160","title":"Difference between momentum and kinetic energy","text":"From a mathematical point of view it seems to be clear what's the difference between momentum and $mv$ and kinetic energy $\\frac{1}{2} m v^2$. Now my problem is the following: Suppose you want to explain someone without mentioning the formulas what's momentum and what's kinetic energy. How to do that such that it becomes clear what's the difference between those two quantities? From physics I think one can list essentially the following differences:   1. momentum has a direction, kinetic energy not   2. momentum is conserved, kinetic energy not (but energy is)   3. momentum depends linear on velocity, kinetic energy depends quadratically on velocity I think is is relatively easy to explain the first two points using everyday language, without referring to formulas. However is there a good way to illustrate the 3. point?"}
{"id":"51955","title":"How exactly, or whats the process, rather, of energy changing into matter?","text":"$E=mc^2$ this is the equation by Einstein claiming energy can change from energy to mass. this would happened at the big bang I assume, when electrons and protons were made to create hydrogen and some helium."}
{"id":"28311","title":"Can lightning be used to solve NP-complete problems?","text":"I'm a MS\/BS computer science guy who is wondering about why lightning can't (or can?) be used to solve NP complete problems efficiently, but I don't understand the physics behind lightning, so I'm posting here. What seems peculiar to me about lightning is that it seems to follow the \"easiest\" path to a given destination, as opposed to just the shortest path. This seems analogous (identical?) to the weights in the traveling salesman problem. Since the lightning process ionizes the air and seems to pick the easiest route quickly, I have 2 questions: 1) Does lightning use a method similar to Dijkstra's approach (such that a \"good enough\" path is found in polynomial time, but not necessarily the \"best of all options\")? Is it basically an NP \"solver\"? If the answer to the above question is \"no\" and it is indeed the best pick: 2) Could the selection process of lightning be used in a computer algorithm to solve a traveling salesman-type problem in polynomial time, and 3) If not, could an arbitrary series of weighted nodes be physically set up somewhere and have electricity passed through it and have the results fed back into some kind of computer mechanism for every time a user wants to know a shortest path? If you could use this approach to solve NP-hard problems quickly, you would have your name on the news and maybe in the history books. So why hasn't anyone used lightning to solve this problem? UPDATE: Another aspect I find interesting: when lightning strikes it ionizes the air to find the shortest path. If the world were totally flat (and the clouds were also) this would ionize all the air equally. Which leads me to believe the route \"picked\" by the lightning incorporates all points, which seems to be a requisite for finding the best of all solutions."}
{"id":"11674","title":"Flux compactification","text":"How flux compactification solves the moduli space problem in string theory? Please provide some details and, if posible, an example. Thanks in advance."}
{"id":"121839","title":"Why is string theory a two dimensional quantum (conformal) field theory on its worldsheet?","text":"In string theory, we quantize the two dimensional field theory on the string's worldsheet. I have a question about this kind of quantization of string theory: did we have similar theory for point-like particle theory, i.e., we quantize the classical theory on worldline, then we get the usual quantum mechanics? If the string theory is the two dimensional quantum field theory on its worldsheet, then what does the momentum operator measure? Does it measure the expected momentum from initial time to final time represented by the worldsheet (because usually the moment operator measure the moment of particle at some specified time)?"}
{"id":"27936","title":"Meaning of the phrase \"dipole moment of the combination\"","text":"Here is a question I came across in a book: > Three point charges $-q$,$-q$ and $2q$ are placed on the vertices of an > equilateral triangle of side length $d$ units.What is the **dipole moment of > the combination**? I had learnt that the dipole moment is defined for 2 point charges only with equal magnitudes but opposite character.I am in a bit of a fix regarding what the author means by that phrase in bold.Thank you in advance! Edit:I just noticed theoretical physics has been merged here;I sincerely apologise if the question is unbearably stupid."}
{"id":"52799","title":"Supersymmetric Sigma Model","text":"I was working through the Mirror Symmetry book by Clay Math Institute. It deals with supersymmetric sigma model in 10.4 section. It doesn't derive how the action is invariant under the variation. I am trying hard, but stuck at few places. Does any one have any references for such a calculations ? The Lagrangian and variation is given in the following image. ![I have given the lagrangian, if one doesn't have the soft copy of the book.](http:\/\/i.stack.imgur.com\/mu6Ta.jpg)"}
{"id":"52793","title":"Where to find probability density plots for all elements?","text":"Does anyone know where I can find something similar to this, but for all elements? I would love to find something with the same image quality. Also, is there any software that can produce images like these?"}
{"id":"62915","title":"Singularities in Schwarzchild space-time","text":"Can anyone explain when a co-ordinate and geometric singularity arise in Schwarzschild space-time with the element $$ ds^{2}~=~(1-\\frac{2GM}{r})(dt)^{2}-(1-\\frac{2GM}{r})^{-1}(dr)^{2}-r^{2}(d\\theta)^{2}-(r^{2}\\sin^{2}\\theta)(d\\theta)^{2}$$"}
{"id":"64361","title":"Do electrons need specific energies to excite electrons","text":"Photons need specific energy levels, equal to the difference between two energy levels to excite an electron in an atom. Is this the same case with electrons that collide with atoms?"}
{"id":"16785","title":"What final states can proton-proton collision generate?","text":"What final states can proton-proton collision generate? Where to find listing of possible final states of a particle interaction, for example proton colliding with proton?"}
{"id":"16786","title":"What is the dominant interaction in elastic collision pi- + neutron -> pi- + neutron?","text":"What is the dominant interaction in elastic collision pi- + neutron -> pi- + neutron? Is it strong nuclear interaction or electromagnetism? Edit: sorry I meant to say \"elastic\". There is no matter - energy transformation."}
{"id":"12090","title":"Small scale water power, how does water volume and hight convert into electric energy?","text":"I was playing a little bit with the basic physics behind water power production but I can't get the numbers right. Let's say that I put a windmill that pumps water into a watertank on the top of my house, then I connect some kind of pipe with generator and starts to drain the watertank. How much electric power (kWh) can I get out from a watertank with size $X\\text{ m}^3$ placed $Y\\text{ m}$ above the ground? How does the formulas look like? * * * Let's put some numbers on this problem, and see where we end up: Let's say the tank is $1\\text{ m}^3$, and it is $10\\text{ m}$ off the ground so the water will fall $10\\text{ m}$ to the generator. Let's connect the generator with a standard garden hose that has a 1 inch diameter, with an area of $2.54\\text{ cm}\/(2\\pi) \\sim 5.1\\text{ cm}^2$. And then I guess we would get a $10\\text{ m}$ column of water pressure, that could be transformed with the area into the force the hight is putting on the system. Something like the earths gravity (9.82)*density*height = 9.82*1*10 ~ 98 Newton (???). And then maybe use that we can find that pressure=Force\/Area, but how to move from pressure to energy? * * * Thanks David Zaslavsky for the example, and in theory that would mean that to store 1kWh I need like 40m3 at 10m height. That more or less mean that if one would try to build something like this in real life things need to be quite big. Also thanks Fortunato for illustrate the practical problem in extracting the energy, and that even thou it is hard to get hight numbers it can be worth the effort anyway."}
{"id":"45777","title":"observable quantities are gauge invariant?","text":"I have a simply question, that is whether spatial velocity is gauge invariant. It is seems that under a infinitesimal coordinate transformation the velocity is just transform as other vectors, and it is not invariant. on the other hand the velocity surely can be measured. Best regards @Chern it might be adding a sentence containing this issue to your question, so the essence of the question is \"I've read that observable quantities are gauge invariant, but if GR is a gauge theory under Diff(M), then why aren't things like particle velocities counterexamples?\" – twistor59 yesterday"}
{"id":"95714","title":"Effect of spin on gravitational field?","text":"If a large sphere of uniform density is spinning around one axis, is its gravitational field uniform? If yes, then what if it's not a sphere or not uniform? For example if   * we remove a chunk of the exterior?   * it's made of lead on one side of the equator and aluminum on the other?   * 50\/50 lead\/aluminum, symmetric around the axis?   * it's slightly pancake-shaped?"}
{"id":"21693","title":"Lorentz force law in Newtonian relativity","text":"I know that in special relativity Electric and Magnetic fields mix together in different reference frames, but my question is about classical mechanics. It seems weird to me is that the Lorentz Force law has velocity in it, it doesn't make much sense in classical mechanics, and I assume the Lorentz force law was derived before Special relativity. In classical mechanics acceleration should be the same in all reference frames, so let's take two examples:   1. There's a constant magnetic field and a particle moving, if I'm moving with the particle it looks to me as if it is stationary so it should not have any magnetic field acting on it. so why is it accelerating?    I assume the problem here is that it is impossible to have a constant magnetic field in both frames, but I'm not sure why.   2. There are two charged particles moving in parallel to each other, meaning they have the same velocity, again if I'm moving together with the particles it looks to me as if they are stationary and should not have any magnetic field acting on them. I realize the real answer is to use special relativity, but my question is how did Lorentz think about this before special relativity was discovered and how is this problem \"solved\" in classical mechanics? I assume this also has some connection with what the definitions of Electric Field and Magnetic Field actually are."}
{"id":"12324","title":"Commutation of operators in quantum theory","text":"I have always written the commutation rules of quantum theory as , $[q,p] = i\\hbar\\delta _{ij}$ But seems that some people write this as, $[q^i,v_j]= \\frac{i\\hbar}{M}\\delta^i _{j}$ (..this is often done in the context of taking the Galilian group limit of the Poincare group…though I am not sure which aspect of it does it emphasize-- the non-relativstic aspect or the non-quantum aspect?..)   * But somehow dimensionally the second form doesn't look okay. Am I missing something? In the same strain, it seems that the operators for \"finite boost by $v$ \" and is done by the operator $exp(\\frac{iK.v}{\\hbar})$ and the \"finite translation by $q$\" is effected by the operator $exp(\\frac{iMv.q}{\\hbar})$. (..where $v$, $q$ and $K$ are all $3-vectors$..)   * I would like to know how the above is rationalized. To ask again - is the above taking just the non-relativistic limit or is it also a non-quantum limit ?"}
{"id":"53447","title":"The physical laws of thermodynamics tell us that electrical impulses must produce heat?","text":"> Health - Controversial New Idea: Nerves Transmit Sound, Not Electricity - By > Robert Roy Britt, LiveScience Managing Editor posted: 14 March, 2007 1:00 pm > ET Rendering shows a biological membrane at its melting point. The green > molecules are liquid, and the red are solid. Molecules of anesthetic reduce > the number of red areas so that the sound pulse can no longer transport its > signal. The nerve is anesthetized. Credit: Heiko Seeger, Ph.D. Niels Bohr > Institute Nerves transmit sound waves through your body, not electrical > pulses, according to a controversial new study that tries to explain the > longstanding mystery of how anesthetics work. Textbooks say nerves use > electrical impulses to transmit signals from the brain to the point of > action, be it to wag a finger or blink an eye. \"But for us as physicists, > this cannot be the explanation,\" says Thomas Heimburg, a Copenhagen > University researcher whose expertise is in the intersection of biology and > physics. \"The physical laws of thermodynamics tell us that electrical > impulses must produce heat as they travel along the nerve, but experiments > find that no such heat is produced.\" How should I read: > \"The physical laws of thermodynamics tell us that electrical impulses **must > produce heat** as they travel along the nerve, but experiments find that no > such heat is produced.\" ? I learned: If only resistive [being coupled to an electrical resistance] electrical signal transmission takes place, there is dissipation of electric energy in the relevant resistance and hence development of heat. But also, I learned: If the signal is purely capacitive [a circuit of ideal capacitors] or inductively [ideal coils], no energy is dissipated. Under ideal capacitors and inductors current and voltage differ always 90 ° in phase. Therefore φ = 90 °. If the current _I_ and the voltage _V_ , then the energy dissipation is always given by: _I_ × _V_ × cosφ And because cos 90 ° = 0, is also the dissipation 0. Is \"purely capacitive\" or \"inductively\" not the explanation why no such heat is produced?"}
{"id":"21522","title":"How are functional determinants of Laplace-type operators used in physics?","text":"Many mathematical papers concerning the $\\zeta$-regularized Determinant of Laplace-type operators refer for motivation to the broad use of such determinants in mathematical physics, especially in quantum geometry\/string theory. Can someone give further details (or may refer to good literature) how the determinant of the Laplacian is used in that kind of settings? What is the \"meaning\" of calculating the determinant? For example, what is the correct physical interpretation of the following example: Consider a surface $(M,g_0)$ (a Riemannian Manifold of dimension 2) with a conformal structure, so I have the conformal class related to $g_0$. Naively I think: The surface represents a world sheet of a propagating string. What role plays the determinant of the Laplacian? In another context I read about using such determinants to calculate path Integrals (?) Thanks for illuminating comments!"}
{"id":"21527","title":"Equivalence principle and radiation from falling particle","text":"I am currently having a hard time solving a problem of GR from Lasenby's book. I can't make it more clear than by quoting the exercise: 7.2 A charged object held stationary in a laboratory on the surface of the Earth does not emit electromagnetic radiation. If the object is then dropped so that it is in free fall, it will begin to radiate. Reconcile these observations with the principle of equivalence. Hint: Consider the spatial extent of the electric field of the charge. Could someone give me a second hint, currently I am stuck because I try to think about an energetic argument: from the laboratory the particle is losing energy from radiation and potential energy from falling, but in the particle none of them is lost. And I am stuck there."}
{"id":"78591","title":"Orbital speed for a circular orbit?","text":"I wanted to look up the formula for orbital speed for a circular orbit on Wikipedia, and I found 2 formulas: > All bounded orbits where the gravity of a central body dominates are > elliptical in nature. A special case of this is the circular orbit, which is > an ellipse of zero eccentricity. The formula for the velocity of a body in a > circular orbit (orbital speed) at distance r from the centre of gravity of > mass M is $ v = \\sqrt{\\dfrac{GM}{r}}$. I found this weird, because this leaves out the mass of the body orbiting $M$. I would think that this would have some effect on the orbital speed. I figured that perhaps the radius of the orbit indicates the mass of the orbiting body through some physical relation, but I wasn't sure so I continued looked up stuff on Wikipedia and I found: > The relative velocity is constant: $v = \\sqrt{\\dfrac{G(M+m)}{r}}$. This was the equation I intially expected to see, but now I'm confused because these are 2 formulas for the same situation, right? Or does the word 'relative' indicate a difference?"}
{"id":"91938","title":"Vacuum expectation value in QFT","text":"In QFT, one writes the VEV of a field $\\psi$ as $\\langle0|\\psi|0\\rangle$. But as I understand it, the fields in QFT are not operators, but just some functions which we use to calculate cross-sections. What does one mean by bra- keting it in such a way as if $\\psi$ is an operator? (Edit: a field in QFT is also a solution of the equation of motion, of course.)"}
{"id":"122876","title":"Self Stirring Boiling Liquid","text":"A couple of weeks ago, a friend and I were brewing a new batch of beer. We boil our wort in a turkey fryer. Pretty standard setup for beginners. Anyway, our wort was getting pretty close to boiling (not quite bubbling on the top yet.) I started stirring the wort and then stopped. However, the wort continued spinning and the spinning was accelerating. The \" **pit** \" of the whirlpool was continually getting deeper until it was a few inches deep. Then, while the whirlpool continued spinning at a seemingly constant rate, the wort started shooting up from the middle of the whirlpool in spurts like a tiny geyser. This went on for about 30 seconds or so as we gawked and thought it was cool. But it started making a mess, so we stuck our stirring spoon in there to stop the wort from stirring, which successfully stopped the whole process. A few minutes later when I stirred it again, the same thing happened. I tried using Google to find what this effect might be, but I was unsuccessful in turning up any info."}
{"id":"68468","title":"Why doesn't an electrometer's capacitance influence the measurement of voltage?","text":"I've read on the subject in several books, and none of them mentioned whether we can neglect electrometer's influence on the measurement of the voltage or not. Maybe my question sounds a bit stupid, but I really can't understand why they do not address the fact that electrometer has a certain capacitance, and when we connect its parts to the desired points, the electrometer and the conductor become one equipotential conductor, and certainly its capacitance no longer depends only the conductor we're testing but also on the electrometer. And we know that $q=CV$ so it influences the voltage as well. So why the measurements are still fine? Can we really neglect the capacitance of an electrometer? Or am I completely wrong here? Thanks in advance."}
{"id":"134957","title":"Definition of pressure in Thermodynamics","text":"In Mechanics we define _pressure_ as \"the amount of force acting per unit area\". It appears naturally in fluid mechanics where we consider a volume of fluid $W$ contained in a region filled with fluid and we ask about the force due to the rest of the fluid on $W$ accross its boundary $\\partial W$. Now, in Thermodynamics, pressure is defined by $$P = -\\dfrac{\\partial U}{\\partial V}$$ That is the rate of change of internal energy with respect to the volume. Now why is that? I mean, what is the motivation to define pressure in that way in Thermodynamics? If it's still force per unit area, what force per area are we talking about here? And where is this force applied? In fluid mechanics, for instance, pressure is defined _pointwise_ , that is, if $D\\subset \\mathbb{R}^3$ is the region of the fluid, then pressure is a field $p : D\\times \\mathbb{R}\\to \\mathbb{R}$ such that if $S$ is a surface in $D$ with normal $\\mathbf {n}$ then the stress force accross $S$ at $a\\in S$ is $p(a, t)\\mathbf{n}(a)$. So, $p$ is a force per area which we know _where_ it acts and by _whom_ it is applied. In Thermodynamics, however, it seems the pressure is the same for the entire system. So if we pick a thermodynamic system, pressure is a characteristic of the state of the entire system. This confuses me, because I can't see what this thermodynamic pressure really is and why it is defined as the partial derivative of internal energy with respect to volume."}
{"id":"122878","title":"Why is more angular momentum required to stabilize the orientation of a heavier bullet?","text":"It's often stated that \"heavier\" bullets require more \"spin\" to stabilize vs. \"lighter\" ones. My question is whether this is really a matter of _mass_. I'm assuming that the destabilizing forces\/torques are primary aerodynamic and that those aerodynamics are almost entirely a function of bullet shape, not bullet mass. If two bullets are identical in shape and differ only in density, would the heavier one _really_ need more angular momentum to keep it stable during its flight?"}
{"id":"68463","title":"Method of images tutorial?","text":"I'm having an exam in Electrodynamics soon. I think I have most of it under control, but the method of images I'm not quite sure about. There is not much in my book about, so I was thinking some of you might knew a website, article or something like that, maybe even with examples, that could help be get that last piece of the puzzle in my head? I have tried Google, but I didn't find much to be honest. That's why I ask here."}
{"id":"129478","title":"Is time not a universal parameter, really, in special relativity?","text":"I want to check that I get it right. The way that I have it in my mind is that, in my frame of reference and from my point of view, SR associates only one time parameter for the whole space, and not distinct one for each point in space. And that it's only when we want to translate the view of the world from a frame of reference to another, do we need to use Lorentz transformations which mix space with time, depending upon the usual parameters ($v$, $x$ and $t$). And that when we say moving clocks evolve more slowly, we don't mean that time depends on the velocity of that clock, because in that case we would be associating time with objects and not with the points of space itself. From our frame of reference, the point at which the clock exists has a time parameter equal to all other points in my frame. Am I wrong\/correct?"}
{"id":"67635","title":"Speed of Light Question and Einstein","text":"Einstein's clock experiment. What would happen if you were to travel at the speed of light or faster than the speed of light towards the clock? Can we translate this into traveling large distances in space? By that I mean, if light is traveling towards us while we are traveling towards it at the speed of light or faster, shouldn't we be able to arrive at that Galaxy (Sun\/planet) in no time at all? Meaning that we would be able to reach a Star before their light has even hit our planet or hit us."}
{"id":"24347","title":"What happens when the string slackens during vertical circular motion?","text":"Consider a particle attached to one end of a string of length $l$ moving anti- clockwise in a vertical circle whose centre is $O$. What exactly happens physically when the string becomes slack and leaves circular motion? I'm guessing that the particle falls downwards until its distance from $O$ is again $l$, whereupon it will re-enter circulation motion but now in a clockwise direction. But I think that this is either incorrect or incomplete."}
{"id":"2951","title":"What are zero modes?","text":"What are zero modes in quantum field theory, and what are they used for? Or, where can I read about them? I was never able to find a good introduction on the subject. I am particularly interested in zero modes that appear in the context of Yang- Mills, QCD and QCD-like theories, at zero or finite temperature. My doubt consists on the nature of these zero modes in these theories: are they special field configurations, or quantum states, or something else? Basically, I'd like to know what's their physical significance."}
{"id":"79170","title":"How big an egg can be?","text":"Please hold on to flag as off-topic. Few days ago I received a question - how big an egg can possibly big? Can it be as big as 1 meter, as a human? (Imagining that there could be really huge animals which would lay these eggs) My thoughts are that an egg should have an upper limit and basically you can find out the biggest possible egg for Earth. By big I assume the biggest volume. I started to think this way - the thing inside the egg should breath, therefore the egg should allow CO2 and O2 to go in and out. The thing inside the egg consumes constant amount of O2 which is related to the volume of the egg. On the other hand the surface of the egg is related to the amount of air the baby can breath. But with the increase of the size of the egg - the volume is growing much faster then the surface. So at some point of time the amount of air going inside of the egg would not be enough. I think that I am on the right track, but can not finish it to receive the answer. P.S. I hesitated about the Math.stackexchange and physics, but assumed that this one is more applicable but please feel free to move it to math if you think it is better"}
{"id":"67983","title":"Conformal Compactification of spacetime","text":"I have been reading Penrose's paper titled \"Relativistic Symmetry Groups\" where the concept of conformal compactification of a space-time is discussed. My other references have been this and this. In case you cannot see the paper above, let me describe what I understand so far: The idea of conformal compactification is to bring points at \"infinity\" on a non-compact pseudo-Riemannian manifold $M$ (equipped with metric $g$) to a finite distance (in a new metric) by a conformal rescaling of the metric ${\\tilde g} = \\Omega^2 g$. This is done so that $(M,{\\tilde g})$ can be isometrically embedded into a compact domain ${\\tilde M}$ of another (possibly non-compact) pseudo-Riemannian manifold $M'$. This allows us to discuss the asymptotic behaviour of the manifold under consideration. In the references above, they mention the following without explanation: > Then observe that any regular extension of $\\phi$[$=\\Omega^2$] to the > conformal boundary $\\partial {\\tilde M} \\subset M'$ must vanish on said > boundary. This reflects the property of a conformal compactification that > “brings infinity to a finite distance”. This I do not understand. Firstly, what is meant by conformal boundary? Secondly, why should $\\Omega = 0$ on the conformal boundary? Is there any good reference for this material?"}
{"id":"77310","title":"Why is it easier to measure the specific charge of an electron over the charge?","text":"The electron was discovered in 1897 and the $e\/m$ ratio was measured at that time ,but the charge $e$ itself was measured in 1911. Why was it not possible to measure it earlier?"}
{"id":"34442","title":"Can a combination of two or more x-ray type beams be made to burn tumours at their intersecting focal points inside a body?","text":"A single x-ray or similar type beam can pass through a human body with little or no damage. Has anyone ever tried to use a second beam that is phased to have no harmful reactions with the tissue except at the exact spot where two or more beams collide. I know they use sound waves to smash gall stones but I have never heard of using two waves that interact on each other. Is it even possible to do this? They also use chemicals collected in a tumour that will react when hit by a beam."}
{"id":"34440","title":"Dimensions of obstacle needed to block light?","text":"How many dimensions are required minimum for an object to block light? in 1d I doubt it's possible. in 2d I think it could do it if only the light was limited to a dimension perpendicular to the 'thickness' dimension in 3d I think it could do it no matter what dimension the object was looked at in. Clarification: 3d -> A block of wood can \"block light\" (make a shadow) no matter how you shine light at it. Edit: What about what are the minimum dimensions needed to block light assuming you are dealing with an $n$ dimensional space?"}
{"id":"66587","title":"Conductors connected with a wire","text":"Let's say I have two spherical conductors with different radii and different amount of positive charge on them. The spheres are far enough from each other. I connect them with a conducting wire. I'm told that the charge is going to flow until both of the conductors gain the same potential (on top\/inside of them). Now, intuitively it seems fine but I do not quite understand a few things. 1) The intuition is that the electric field should be zero everywhere when they reach the same potential, because then the charge is not going to flow. But it is not always the case. 2) Shouldn't the wire also contain some charge after the rearrangement? Because if no, then its potential will be zero, and so there will be a difference in potentials between one of the conductors and the wire, so the charge should still flow into the wire. I'll try to rephrase the question, because it seems people keep getting my question wrong - in general, I can't understand why charges should always flow until both of the conductors reach the same potential. I mean, the fact that the conductors have same potential doesn't really tell us that the field which drives charges is zero. What's the **proof** that connected conductors reach the same potential?"}
{"id":"32848","title":"Is hydrogen the same everywhere?","text":"Silly thought. Feel free to shoot it down Does a hydrogen atom undergo any kind of change subject to it's environment? If one were to study a hydrogen atom on the surface of Mercury, another above Earth, and a third in interstellar space - would they exhibit any difference\/s?"}
{"id":"80015","title":"Moment of inertia about a tilted axis","text":"I apologize if it seems foolish, I am a beginner. Suppose I am given an object, whose moment of inertia along the x,z axes are known. Suppose it rotates around a tilted axis, say at angle $\\theta$ with the x axis. Can I find out the moment of inertia by the following method? I resolve the rotatory motion of the object into its x and z components- $\\omega \\cos \\theta$ and $\\omega \\sin \\theta$. I know its MOI along the x and z axis, so I find out the work done by the rotation along the x and z axis- $2W_x= I_x \\omega^2 \\cos^2 \\theta$ and $2W_z= I_z \\omega^2 \\sin^2 \\theta$. If the axis passes through the CM, I know the values of $I_x$ and $I_y$ readily. If not, I use the parallel axis theorem to find out $I_x$ and $I_z$. I think the total work done will be $W= W_x + W_y$, so from here I can find the total work done (I am not sure about this, but in analogy to translational motion, it makes sense. But the total work done is also $\\frac12 I \\omega^2$. So making them equal, I can find I (note- the $\\omega^2$ term will cancel out. Will my method give the correct answer here, or do I just have to use the classic irritating method of integration? To be precise, I claim $$I^2= I_x \\cos^2 \\theta + I_z \\sin^2 \\theta$$"}
{"id":"61670","title":"Physics behind the flow of gas coming out of a balloon","text":"I'm working with stratospheric balloons (latex ones) and I want to put a valve on it so it can float for a longer time. I'm trying to define which valve I should use, which demands I estimate the flow os gas it can address. Also, it would be great to be able to model the problem mathematically, so I can design an altitude control system. Thinking about this, I got some doubts related to the physics of the latex balloon membrane and the gas inside of it. I'm trying to understand what exactly makes the gas inside a balloon come out when we open a valve placed on its \"mouth\". I know the pressure inside the ballon must be greater than the external in order to the gas flow. But when the system is in equilibrium, the internal pressure must be equal to the external, so the ballon doesn't expand\/contract. Or ALMOST that. I think most people thinking about this just ignore or deem negligible the force the membrane of the balloon exerts in the gas inside of it. I came to the following conclusion: $$p_i = p_o + p_b$$ where: $p_o$ = external pressure $p_i$ = internal pressure $p_b$ = pressure exerted by the balloon membrane into the gas in its interior * * * This justifies why the internal pressure would be greater than the external and the system would, still, remain in equilibrium. If this is correct, I can conclude that what will make the gas come out is pb, at any altitude. So, if I have pb, I could use Bernoulli's principle to calculate the velocity with which the gas come out and, from this, find which is the flow rate I need to model the control problem and guide me on the valve specification. I have two concerns with this approach:   1. Is it correct? Physically speaking, this is what really happens? Can someone give me more insights on the dynamics of mechanical tension on the membrane and gas pressure inside the balloon?   2. I can't find a way to calculate the pressure the membrane would exert on the gas (pb). I know that I can calculate the tensions that appears in the membrane surface accordingly with the internal pressure (with pressure vessels theory). I have a feeling that I can derive the force with which the membrane compress the gas from this tensions, and it seems to me it would be only a geometrical\/calculus problem, but I'm struggling getting it forward. I have idealized a simplified model in which I apply Hook's Law, but I'm stuck with infinitesimal areas. Any one can guide me here?"}
{"id":"61677","title":"Calculation of the quantized Hall coefficient in the Integral Quantum Hall Effect","text":"I have been reading about the QHE over the past couple of days. I am facing difficulty understanding a calculation in this review. www.nimt.or.th\/nimt\/upload\/linkfile\/sys-metrology-248-434.pdf In this paper, equation 16 gives the energy levels of a 2d electron gas as $$E_k =\\frac{\\hbar^2}{2m}[k_x^2+k_y^2]$$ which I have written as $$E_k=\\alpha k^2$$ The author then says (eqn 17) that the number of states with energy less than $E$ is $$N(E)=\\frac{S_k}{(2\\pi)^2}=\\frac{\\pi k^2}{(2 \\pi)^2}$$ $S_k$ is the area in k space. I don't know what the $k$ in this equation is, and if it is the same $k$ that I have taken. I am not being able to understand how they got this equation. I integrated the equation for $E_k$ from $-\\sqrt{\\frac{E}{\\alpha}}$ to $\\sqrt{\\frac{E}{\\alpha}}$. But I am getting it as $\\frac{2}{3}\\sqrt{\\alpha}k^3$. How do I get that equation? Why is that equation true in the first place? What is the $(2\\pi)^2$ in the denominator for?"}
{"id":"74666","title":"Quantum Anomalies in Non-Gauge Theories?","text":"I'm reading about quantum anomalies in QFT and all the examples seem to arise in gauge theories. Is it true that theories without a local gauge invariance don't have quantum anomalies? I can't think of examples of symmetry violation in any non-gauge theories, particularly the free theories. Then again, there might be a well-known example I'm not aware of!"}
{"id":"73120","title":"energy efficiency, transformation from chemical to thermal energy, for district heating plants","text":"If I have a system in which I introduce 100e.u.F (energy units) derived from chemical energy (i.e. fuel) and generate thermal energy to heat water what are e.u. absorbed by water? I.e. the efficiency of the transformation? Basically I would like to know the average\/general efficiency of energy transformation for a district heating plant. Considered that the only energy input is fuel and the only energy output is heat (no electricity)? What are the energy loss? Thanks, Luther"}
{"id":"22182","title":"Can the mass of longitudinal and transverse W bosons be measured separately?","text":"Some higgsless unified models of particle physics predict that the mass of longitudinally polarized W bosons and the mass of transversely polarized W bosons are different. In those models, a difference in masses is also predicted to appear for the Z boson. Is there a way that the existing data from the LHC, the LEP or the Tevatron can be used to check this prediction? Or maybe it has already been checked?"}
{"id":"73129","title":"Why do Nuclei lose mass?","text":"When it comes to things like gravity and the electromagnetic force, masses aren't reduced-but with nuclei the mass difference is noticeable. What about nuclear forces makes them capable of putting mass into binding energy?"}
{"id":"22187","title":"Infinite universe - Jumping to pointless conclusions","text":"I watched an episode of thee BBC Horizon series titled 'To infinity and beyond'. In this program a number of respected physicists and mathematicians were talking about the nature of infinity and an infinite universe. They argued that if the universe is infact infinite then every possibility must play out somewhere, even going so far as to get a physicist to calculate how far you would have to travel to find an exact replica of the earth. They also used the classic example of infinite monkeys producing the works of Shakespeare. Now to me the monkey example makes sense considering that the works of Shakespeare are finite and so are the number of possibile outcomes of a monkey mashing away at a keyboard for a given period of time (although very large still finite). But the logic of applying this to the universe seems flawed to me, surely in the context of the universe the possibilities are also infinite as it is with a sequence of numbers. the numbers 1,2,3,4... go on infinitely without repeating themselves so to say because something is infinite every possibility must play out seems to me incorrect. Now perhaps you could argue that a finite number of atoms can only assemble themselves in a finite number of ways - again I can see this although I am not entirely sure that two lots of atoms arranged in the same way are necessarily the same since we currently rely on probablity calculations to predict the outcome of subatomic processes (this could arguably be due to slight variations at a more fundamental level...) but even if this were true. They went further to suggest that in an infinite universe there is an exact copy of you or I doing exactly the same thing and indeed every possible thing. Now it seems to me at this point things have become ridiculous, even by the time you have calculated the possibility of an exact atomic copy, the number of possible interactions I could have at any time with the universe, their affects on my atomic makeup and indeed future interactions, surely at this point we are dealing with infinite possibilites. Any feedback on any of these points welcome but my central assumption here is this: Infinite possibilities will not play out even over an infinite period of time. Is this correct?"}
{"id":"98764","title":"Equivalent temperature: laser and cell containing Rb","text":"What's the meaning of \"equivalent temperature\" related to a cell containing rubidium and crossed by laser?"}
{"id":"107812","title":"Do gases have a general upper limit of density?","text":"Is there some limit for the density of gases, at which no change in condition could make it more dense without making it fluid, or solid - or something 'in between'?"}
{"id":"100842","title":"Eigenenergies and eigenkets given the Hamiltonian","text":"For a two level system the Hamiltonian is: $$ H=a(|1\\rangle \\langle1|-|2\\rangle\\langle2|+|1\\rangle\\langle2|+|2\\rangle\\langle1|) $$ where $a$ is a number with the dimension of an energy. I need to find the energy eigenvalues and the corresponding eigenkets (as a combination of $|1\\rangle$ and $|2\\rangle$). I used: $$ H |ψ\\rangle=E|ψ\\rangle $$ And using the fact that : $|a\\rangle = \\sum_i c_i |a_i\\rangle$ I wrote $|ψ\\rangle$ as a combination of the two system kets $|ψ\\rangle=c_1|1\\rangle + c_2|2\\rangle$ ($c_1$,$c_2$ are complex numbers). so $$\\begin{aligned} H|ψ\\rangle =& a(|1\\rangle\\langle1|-|2\\rangle\\langle2|+|1\\rangle\\langle2|+|2\\rangle\\langle1|)\\cdot(c_1|1\\rangle+c_2|2\\rangle) \\\\\\ =& a(c1|1\\rangle-c_2|2\\rangle+c_2|1\\rangle c_1|2\\rangle)=a ((c1+c2)|1\\rangle+(c1-c2)|2\\rangle) \\\\\\ =& E|ψ\\rangle. \\end{aligned}$$ How do I continue?"}
{"id":"56536","title":"Electron Relaxation\/Polarization for and n-type Semiconductor","text":"Please help me understand the following (general) statement, referring to electrons in a full valence band of an n-type semiconductor: \"An electron filling up the last empty state in the valence band will in doing so prevent the other valence electrons from reducing their energy through polarization and relaxation around the impurity center\" I don't understand what is meant by the last part of this. I know that in a full band, electrons have no empty state to move into, but I don't know what is meant by polarization and relaxation \"around the impurity.\" The impurity referred to is a donor (n-type SC). Another statement (which I think is basically the same) says: \"In the case of a completely filled valence band, no (impurity, I assume) scattering can take place, because all states are occupied. Now, if a hole is created at, for example, the top of the valence band, the remaining valence electrons will relax around the impurity ions... this new flexibility of the system makes the system lower in its energy.\" What does this mean \"relax around impurity ions\"? To sum it up, what I'm not understanding here is: a) what is meant by \"relax around the impurity\" (I have no idea what they mean by \"relax\"), and b) reducing energy by polarization or relaxation (What\/how\/why? What do they mean by polarization?) Edit: I spoke with a professor about this, and I _think_ that this all has to do with screening. In other words, if there are empty states in the valence band, the charge density is modified to a new equilibrium to screen the impurity, \"relaxing\" around the impurity. I think the point on polarization follows the same logic (screening). Any input on my thoughts would be appreciated."}
{"id":"104542","title":"Properties of an object whose collision with Earth would completely disintegrate Earth","text":"This picture got me thinking, whether such collision would be possible, and would it really look like that from the moon. What would be the properties (mass, size, velocity etc.) of an object, whose collision with Earth would completely destroy it in the manner of that picture? That is, after the collision, a large portion (if not most) of Earth would continue straight along the same path as that of the colliding object, but from the opposite side of the planet. If such an event were to happen, would the sight in that picture be an accurate depiction when observed from the surface of the moon?"}
{"id":"104543","title":"Black hole (classical or quantum?)","text":"Theoretically we can convert a body into black hole by compressing its mass below some radius (known as Schwarzschild Radius). Suppose such object after becoming black hole has a radius which is comparable to atom and nucleus radius. Now my question is: what happens if we run this object into double slit experiment setup (single black hole at a time)? Shall we get interference pattern on the detector wall because its radius is in quantum domain (say we somehow know when on detector screen this black hole will hit) or simple-particle pattern (i.e. classical way) because of the mass associated with that black hole? If you have a really tiny black hole, will it behave as a classical particle, or will it exhibit quantum behavior?"}
{"id":"32159","title":"Is 4-volume element a scalar or a pseudoscalar in special relativity?","text":"In general relativity 4-volume element $\\mathrm{d}^4 x = \\mathrm{d} x^0\\mathrm{d} x^1 \\mathrm{d} x^2\\mathrm{d} x^3$ is clearly a pseudoscalar (or scalar density) of weight 1 since it transforms as $\\mathrm{d}^4 x \\to \\mathrm{d}^4 x' = J \\,\\mathrm{d}^4 x$, with $J$ Jacobian determinant of the coordinates transformation. In special relativity we consider only Lorentz transformations, which have $J = \\pm 1$, so shouldn't $\\mathrm{d}^4 x$ be a pseudoscalar also in special relativity? Landau states (\"The Classical Theory of Fields\", § 6) that «the element is a scalar: it is obvious that the volume of a portion of four-space is unchanged by a rotation of the coordinate system», but this doesn't proof that $\\mathrm{d}^4 x$ is a scalar rather than a pseudoscalar since a pseudoscalar looks like a scalar under a proper rotation of the coordinate system ($J = 1$), differences arise only for an axis inversion ($J = -1$)."}
{"id":"26252","title":"What are the differences in using an eyepiece with a barlow instead of a shorter eyepiece?","text":"I intend to start making visual observations, and I want to know more about the difference between using barlow lenses and one or two eyepieces and using a complete kit of eyepieces, both for planetary and deep space objects."}
{"id":"75205","title":"Physical intepretation of nodes in quantum mechanics","text":"I am taking my second course in QM, and my head is starting to spin as it probably should. But I would very much like to clear up my head about a few details regarding the wave function. As I know it is impossible to predict where particles are and one can only give a probability of where it _should_ be. The simplest case is a \"frictionless\" particle \"bouncing\" back and forth inside a infinite square well. Eg a particle in the following potential $$ V(x) = \\left( \\begin{array}{cc} 0 \\ , & \\text{for} \\ 0 \\leq x \\leq a \\\\\\ \\infty \\ , & \\text{elsewhere} \\end{array} \\right) $$ Which gives rise to the following normalized solution $$ \\psi_n(x) = \\sqrt{\\frac{2}{a}} \\sin\\left( \\frac{\\pi n}{a}x \\right) $$ My problem is what the **nodes** in the square function represents. If I draw $|\\Psi_2(x,0)|^2=|\\psi_2(x)|^2$ I obtain a graph similar to the one below. ![enter image description here](http:\/\/i.stack.imgur.com\/5Z2US.png) What is the physical explenation that finding the particle around a small region around $a\/2$ is close to zero? Or why is it so much less likely to find it near $a\/2$ than $a\/4$? Eg why is $$ P(a\/2 -\\varepsilon \\leq X \\leq a\/2+\\varepsilon) = \\int_{a\/2-\\varepsilon}^{a\/2+\\varepsilon} \\left| \\psi_2(x) \\right|^2 \\,\\mathrm{d}x \\sim 0 $$ for small $\\varepsilon$"}
{"id":"98186","title":"What would happen if a supermassive blackhole at the center of a galaxy evaporates?","text":"Well the title says it all but I will just explain it in detail a bit. We know that a blackhole eventually has to evaporate due to the hawking radiation. It is also said that the blackholes at the center of galaxies are the galaxies' source of energy. So what would happen if that blackhole evaporates?"}
{"id":"98182","title":"Does gravity cause spacetime to bend?","text":"Does gravity cause spacetime to bend even though space has no mass?"}
{"id":"79954","title":"What are the demographics of stars visible to the naked eye?","text":"Of the stars that can be seen with the naked eye, what are the distributions of each type? For example, how many are main sequence? how many are Super Red giants\/white dwarfs\/neutron stars...etc. What's the distribution by mass and distance?"}
{"id":"79958","title":"Coulomb interaction as virtual particles exchange?","text":"I've been reading about virtual particle exchanges in physics books and in Physics SA posts, where a particle interpretation of gravity and Coulomb interaction is established. The Feynman Diagram picture (combined with the snowball picture that a fellow contributor provided in a related post) serves to account for the repulsion part: _grosso modo_ two particles in motion, a virtual photon\/graviton, momentum change induces opposite motions ($\\Delta \\vec p \\ $ and $ \\ -\\Delta \\vec p$). Still, how can we account for the attraction (as a layman's analogy if not in formalism)?"}
{"id":"2160","title":"Physical vapor deposition of SiO2","text":"Could you suggest any if it's possible to deposit a layer of SiO2 via PVD? Why doesn't it widely used? Is that correct that I need ~1600C in vacuum? Could you suggest any books\/papers about that? Unfortunately cannot tag this properly :-)"}
{"id":"14039","title":"Lean angle of a turning bicycle","text":"I'm asked to derive a relationship for the leaning angle of a bicycle with the following specs:   * Center of gravity for bike and rider is a distance $L$ above the ground when vertical, and the total mass of bike and rider is $M$.   * The wheel radius is $r$ and the wheel mass is $m$   * The radius of curvature for the turn is $R$, the speed is $v$ I am to show that $$\\tan \\theta = \\frac{v^2}{R} \\frac{2mr + ML}{MLg}$$ but I am unsure. If I take $M$ to be the total mass, including the wheels, then I would guess that the condition is that centrifugal force and gravity act in such a way that the total torque is zero, i.e. $$M g L \\sin \\theta = M \\frac{v^2}{R} \\cos \\theta L$. But then the mass of the wheels and their radius doesn't enter. If, on the other hand, I exclude the wheels from $M$, wouldn't I still have to add them to both sides, i.e. $$(MgL + 2mgr) \\sin \\theta = \\frac{v^2}{R} \\cos \\theta (ML + 2mr)$$ or where does the extra term 2mr come from?"}
{"id":"3297","title":"does current product push or pull effect","text":"Since i was a child i was told that current has pushing or pulling effect. However, i never got a dc shock and i don't remember ac shock (i got it very long before). The day before yesterday, an experienced electrician told me that dc has pulling effect ( most of the batteries supply only 12V it's not so much to worry but if supplied with very high voltage it is very deadly comparison to ac and) and ac has pushing effect. Logically i can't find any explanations. electricity is just flow of charge, and in our body, we have more electrons than protons (from C-14 decay) but still very less than enough to give shock or pushing or pulling effect."}
{"id":"106037","title":"Is there a finite frequency of physical interactions?","text":"If particles use gauge bosons to convey information about various forces then is there something like a finite \"refresh rate\" at which they update information about themselves?"}
{"id":"57371","title":"How to add magnitude data in an ENU coordiante system","text":"I have water velocity data taken in an ENU (East, North, UP) or XYZ coordinate system. The data is contained in 3 columns like this:               E   N   U     1   3   .1     2   5   .2     2   6   .1     -1  4   .1     -2  4   .1      So the first column represents velocity (in m\/s) in the East (or X) direction (negative numbers represent velocity in the West (or -X) direction. The second column indicates velocity in the North (or Y) direction. And the 3rd column represents velocity in the Up (or Z direction). The data is scalar, but I'm assuming I want to end up with speed and directional data. For ease of understanding, if I were to just consider 2 dimensional data (ignore the Z or up data), is it acceptable to add the E and N data to get useful magnitude and directional data? For example if in the last row East is -2 and North is 4. So I've got 2 m\/s West and 4 m\/s North. But does the number 2 (-2 + 4) mean anything? Or do I need to use $a^2 +b^2= c^2 = 4 +16= 20 \\rightarrow \\sqrt{20}= 4.47$ But If I were to do this, then I loose the - sign (West direction) information, and it would not have mattered if the data was originally West (-2) or East (2). In the end, I'm looking to get useable data on current speed. But I'm not even sure if the end result of this kind of data should be (A) magnitude + direction (ie 2m\/s at 20 degrees) or just (B) a single number that is only a combined magnitude with no reference to direction(ie 2m\/s). I know for my calculations I do need an absolute magnitude that would represent current speed (option B) but am not sure how to calculate this."}
{"id":"83574","title":"Does the moon affect the Earth's climate?","text":"So, this morning I was talking to a friend about astronomical observations, and he told me that lately there has only been good weather when there was a full moon in the sky, which was a shame. I jokingly said: 'maybe there's a correlation!', but then I started thinking: wait, if the moon can affect the oceans, why shouldn't it also make an impact on the atmosphere, which is just another fluid. So... are there atmospheric tides? Does the moon affect the weather or the climate in a significant way?"}
{"id":"47465","title":"Charging using a CRT TV set","text":"I was doing an experiment with my CRT TV where I put aluminum foil on the screen wire it to a can and open the TV and a ring between 2 can will swing if I ground the other can. http:\/\/scitoys.com\/scitoys\/scitoys\/electro\/electro4.html#franklin Anyway I tried something out, I used the same way to charge a metal sphere while the tv was on, the sphere was charged and everything seemed to be alright only when removed the wire (connecting the sphere and the foil) from the foil on the TV the sphere lost its charge, why is that? If the charge just decayed, is that normal to decay so fast (in less than a minute) while it is not grounded?"}
{"id":"6899","title":"Applicability of Baxter's method for IRF models","text":"In a interaction-round-a-face model of $n^2$ particles in a lattice, a weight $W(a,b,c,d)$ is assigned to each face in the lattice based on the spins $a,b,c,d$ (listed say from the bottom-left corner in counter-clockwise fashion) of the particles at the corners of the face. Based on this, a partition function is formed $Z=\\sum\\prod W(a,b,c,d)$, where the the sum is over all possible spins of all particles and the product is over all the faces in the lattice. The well-known text _Exactly Solved Models in Statistical Mechanics_ by Baxter shows how to obtain exact solutions for $Z$ in a couple cases, like the six-vertex model and the eight-vertex model, using \"commuting transfer matrices.\" In reading Baxter's book, I am uncertain of the applicability of this method. For instance, I am working on a particular application where particles in a square lattice can have either up spin (+1) or down spin (-1) and the weight of any face $W(a,b,c,d)=1$ unless all four spins around the face alternate, in which case $W(1,-1,1,-1)=W(-1,1,-1,1)=4$. (The boundary condition here is that the all the spins on the boundary must be up.) Does anyone know if the commuting-transfer-matrix method (or any other method) should yield an exact solution for this particular weighting of the spins? More generally, should the commuting-transfer-matrix method yield exact solutions for _any_ weighting of the spins (even in the two-spin case)?"}
{"id":"6898","title":"Giant string in space","text":"I saw an interview with Ed Witten, where he said one way to confirm string- theory is to observe a giant string floating in space, left over from the Big- Bang. How does one observe such a string, does it have thickness? Is this string any different then the strings that (hypothetically) make up elementary particles? What is required to produce such a large string, why is this string so much bigger then others? What happends if one tries to interact with this string, can one stretch\/cut it etc? Can we have a stable mega-string on earth, and can it be used for anything?"}
{"id":"30197","title":"Has the use of the holographic principle of string theory in condensed matter physics silenced the skeptics?","text":"It seems to me that the use of string theory in calculations of strongly- interacting matter in condensed matter physics is an example of the theory being on the right track. And then there's the application of string theory to black holes. Have these quieted the skeptics? If not, what's the deal? Could a theory be used in making accurate calculations and still be wrong? Aren't the odds of that happening rather small?"}
{"id":"71432","title":"What keeps a gas giant from falling in on itself?","text":"There is not enough gravity at the center to start nuclear fusion, but it seems that there would be plenty enough to collapse the planet."}
{"id":"132085","title":"Electric Field in Conductor Zero?","text":"My textbook claims that the electric field in a conductor is zero in a static condition, as otherwise, a current would flow. But what if I go infinitely close to a proton; there will be an electric field right? Is my idea wrong or is the textbook flawed?"}
{"id":"39456","title":"The Z-Torque: how can it be shown intuitively that it does not work?","text":"There is a new kickstarter project that claims to increase torque and power compared to a normal crank on a bicycle (Z-Torque on kickstarter). If this patented (US Patent Number 5899119) approach works it would be a revolution for personal transport. ![enter image description here](http:\/\/i.stack.imgur.com\/TWcjx.jpg) The idea behind this is that a Z-shaped crank should allow somehow better leverage compared to a straight crank. There are quite a few comments that this should not work as intended but as in a lot of cases it sounds plausible at first but it is incompatible with physics. As the inventor and some of the supporters are convinced that this actually works I am looking for a clear and simple argument why this is not possible. Using vectors and cross products ($\\tau = \\vec{r}\\times\\vec{F}$) this is easy to show but can it be shown with as few abstractions as possible?"}
{"id":"39451","title":"On the double slit experiment with 4 slits","text":"I'm not a physicist nor a science savvy person, but I was wondering if this experiment was ever performed in a simultaneous fashion on screens with fixed references(marks) and firing different wavelengths particles. So, to recap, 4 slits, 2 emitters firing at different times (where the $\\Delta T$ is known and certain) on two different screens. I hope this question is not too stupid. P.S. I'm sorry, I've been too vague. I ask this because I've recently learned that this interference pattern emerge even firing large atom clusters instead of single particles. This is astonishing and it's clearly stating that there's a limit in mass aggregation where it stops behaving in duality. Cause, you know, firing bricks will hardly produce interference. And, in the end, I was wondering if it's not possible that it's spacetime itself vibrating at a certain $f$ and if there's a limit when mass stop to be affected (relatively) by this vibration. Like pinning a vibrating rubber sheet. I'm so confused."}
{"id":"31604","title":"pattern of moving electrons in wire under AC","text":"when we use a DC battery into a circuit the flow of electrons in wire creates a clear picture as they flow from negative terminal to positive terminal. under AC(Alternating current) we know how energy flows and what does it mean by $I_{rms}$ or average current etc. But what about the pattern in which electrons move?"}
{"id":"34127","title":"Can an electric motor force angular momentum not to be conserved in an isolated system?","text":"An ice skater is in a spin, she pulls her arms in and she spins faster, she lets her arms extend outward and then she starts to slow down. She will probably weigh on a weigh scale about the same weight during this spin cycle. Now let us connect an electric motor to her to make her spin faster as she extends her arms outward. Would angular momentum still be conserved? It seems to me that it would not. Also would she still have the same weight? I think she will weigh less as she is being accelerated by the electric motor."}
{"id":"34126","title":"Why the breath sometimes warm and sometimes cold? (2 different explanations!)","text":"If you blow air against your hand with your mouth open, you feel warm breath. If you do with with your lips closed except for a small opening, you feel cold breath. One explanation from here says \"With your mouth open most of the air which reaches your hand has come out of your mouth. With pursed lips lots of cold air is entrained so the air reaching your hand is cold and moving quickly.\" Another explanation from Paul Hewitt video,(go to min 24:00), is that air comes out cold it because it expands coming out of small opening. So which explanation is right?"}
{"id":"35836","title":"difference between battery and capacitor in urdu languege","text":"## Heading   1. List item   2. List item`enter code here` > Blockquote*emphasized text*"}
{"id":"100471","title":"Why should a holomorphic function be expanded in Laurent series rather than Taylor series?","text":"In 2d free conformal field theory, there is an operator equation: $$ \\partial\\bar\\partial\\hat{X}^\\mu\\left(z,\\bar z\\right)=0 $$ Why can it have Laurent expansion like this below rather than Taylor $$ \\partial X^\\mu\\left(z\\right)=-i\\left(\\frac{\\alpha'}{2}\\right)^{1\/2}\\sum_{m=-\\infty}^{\\infty}\\frac{\\alpha_m^\\mu}{z^{m+1}}, \\\\\\ \\bar\\partial X^\\mu\\left(\\bar z\\right)=-i\\left(\\frac{\\alpha'}{2}\\right)^{1\/2}\\sum_{m=-\\infty}^{\\infty}\\frac{\\tilde{\\alpha}_m^\\mu}{\\bar z^{m+1}}. $$"}
{"id":"100474","title":"Inside a conductor?","text":"My textbook says the field inside a conductor must be zero in order for the system to be equilibrium and therefore there must be no excess charge inside. Their proof: 1) Place a gaussian surface inside the conductor. Since the system is at equilibrium, all points on the surface must have an electric field of zero. 2) Therefore the net flux is zero, implying the charge inside is zero. 3) If there is no charge inside, all excess charge must lie on the surface. My problem: \"Since the system is at equilibrium, all points on the surface must have an electric field of zero.\" Why? Consider an conductor with no excess charge. To simplify things, say the conductor consists of a 100 free electrons and a 100 ions of charge 1e+. The conductor is neutral and at equilibrium. Now consider the field on one specific electron. There a total of 99 other electrons and 100 ions. So there _would_ be a net field acting on that specific electron. So even though the system is at equilibrium, the net field on the individual particles is non-zero. So a system can be at equilibrium, even if the net field inside is non-zero. What am I misunderstanding? **Edit:** Another example: Consider a system consisting of one electron and one proton. If nothing is holding them back they will attract each other until they touch. After they touch, the force acting on them is still non-zero, yet they wouldn't accelerate."}
{"id":"103873","title":"A bird problem and trains","text":"So that the problem: > _There are 2 trains at speed of 30km\/h in the same trail and opposite > directions and moving for a collision at 60km of distance of each other. One > bird in front of one of the trains fly at 60km\/h toward of other train. When > he reaches he flies back to the other train until the trains collide._ I got a question: as the bird travels in all the way to the collision?"}
{"id":"86055","title":"Value of of gravity 9.8m\/sec^2 and centrifugal force","text":"I had a doubt regarding centrifugal force on earth is not accounted in any equations as object falling towards earth.eg f=ma.i read questions asked by few about the same and got to a conclusion that value of g(9.8m\/s^2) is a relative number derieved.but I still cant find it in gravitational force equation.why is that.or is my interpretation wrong.this might be a repetitive question. But please clarify."}
{"id":"100780","title":"free energy and entropy","text":"my understanding of free energy and entropy is that as entropy of a system increases its free energy decreases. As sun has free energy and this energy is being converted to useful work, and its entropy is increasing. Its entropy will continue to increase till its material is exhausted to cease free energy production. Are permanent magnets source of free energy? Because of permanent dipoles they are magnetic and do they lose their orientation very slowly to become disordered and increase its entropy?"}
{"id":"131575","title":"How damaging is light?","text":"On _Surely You're Joking Mr Feynman_ , when talking about the Trinity test, the author states: > the only thing that could really hurt your eyes (bright light can never hurt > your eyes) is ultraviolet light. I got behind a truck windshield, because > the ultraviolet can't go through glass, so that would be safe, and so I > could see the damn thing. This is also clear in this video, where the glasses are seen as black by the UV camera. If that's the case, why are sunglasses said to `prevent bright sunlight and high-energy visible light from damaging or discomforting the eyes`? _(from Wikipedia)_ Is it for marketing purposes or was Feynman wrong?"}
{"id":"19668","title":"Does a perfect mirror behave the same as a blackbody radiator?","text":"If I put a perfect mirror(i.e. reflects with no attenuation) next to a blackbody radiator its spectra should be the same as the blackbody radiator. Looking only at the spectra - is there any difference between a blackbody radiator and a perfect mirror? For instance, suppose the mirror is accelerating as in the dynamic Casimir effect - does the spectra change in the same way as the blackbody radiator?"}
{"id":"101437","title":"How can the product of two real linear operators be not real?","text":"I'm puzzled about a statement from Diracs book \"The principles of quantum mechanics\" (§8, p.28): > As a simple examples of this result, it should be noted that, if $\\xi$ and > $\\eta$ are real, in general $\\xi\\eta$ is not real. This is an important > difference from classical mechanics. However, $\\xi\\eta + \\eta\\xi$ is real, > and so is $i(\\xi\\eta - \\eta\\xi)$. Only when $\\xi$ and $\\eta$ commute is > $\\xi\\eta$ itself also real. Here $\\xi$ and $\\eta$ are linear operators, so (I think) these could be represented as matrices. However how can a product of two real matrices be \"not real\"? What does Dirac mean when he says \"is not real\"? Is Dirac maybe talking about eigenvalues? So does Dirac mean, that the product of two matrices with real eigenvalues could have imaginary eigenvalues? And does he want to say, that a real symmetric matrix like $\\xi\\eta + \\eta\\xi$ and a purely imaginary antisymmetric matrix like $i(\\xi\\eta - \\eta\\xi)$ always have real eigenvalues?"}
{"id":"135059","title":"Mie theory \"upper limit\"?","text":"I have read most often that objects experimented with in Mie theory are on a scale not much larger than the incident wavelength (usually a fiber of diameter $5$ microns with an incident wavelength of $630$ nm or so). Can Mie theory be used to describe the scattering pattern of an object any size, as long as a sufficient number of terms in the Mie solution can be calculated?"}
{"id":"135056","title":"Is air resistance affected by both mass and surface area?","text":"If so, which has the biggest impact?"}
{"id":"45179","title":"Impedance Matching","text":"When viewing fast pulses on an oscilloscope, why is impedance matching required? I'm not totally clear on why. I'm guessing it is because I don't want 'ringing' and because I want signal integrity."}
{"id":"45171","title":"Determining Water Level","text":"Given a cylinder of a constant shape and size, how can the water level be calculated using the inside surface of the container? For instance, could the inside lining of the container be a pressure sensitive foil of some sort, and using that pressure could the water level be calculated?"}
{"id":"134010","title":"Coriolis effect misunderstanding","text":"My wife asked me what the Coriolis Effect was, so I did some research. I think I get it now but I'm surprised at the number of \"explanations\" people have uploaded that actually seem wrong. This one is an example. http:\/\/www.classzone.com\/books\/earth_science\/terc\/content\/visualizations\/es1904\/es1904page01.cfm This is wrong, right?"}
{"id":"26132","title":"What are the facts that allow accepting the Oort cloud theory?","text":"I admit without any fact that the Oort cloud (comet reservoir) should exist, and it seems to be accepted by far by the large majority of astronomers. But it is still a theory without any direct observation. Except the fact that we guess that comets come from this cloud (a cloud of comets). So what are the facts or indirect observations that may validate this comet cloud theory? Collateral to this question, is it easy to observe these indirect facts, and is there any special way to observe these Oort comets to validate the theory? Is it acceptable to say there is no cloud and just a few solar and most inter- solar objects?"}
{"id":"100359","title":"About muons, taus and the number of generations","text":"there! Why cannot consider the muon or the tau just like excited states of the electron (and similarly, with the (u,d) (c,s) (t,b) pairs? I guess that it is related to the absence of decays like muon-> electron+photon and similar stuff. But, is there any other reason?"}
{"id":"70427","title":"Why do we feel a sudden chill?","text":"When we are in a room, especially in some old houses, we feel this sudden rush of cold air. How does a strong air current arise and pass away so quickly?(lasts for about a second) I know air-currents are caused by pressure differences. But can pressure difference of the magnitude required to generate an air current be created within a second? If so , please answer with equation that relates the pressure and Temperature with time interval. Thanks..i hope I am now much clearer with my question. (I am inclined to think in the direction of something similar to a weather situation)"}
{"id":"121671","title":"Orbital plot to explain bonding and anti-bonding molecular orbitals","text":"Related is this question: https:\/\/chemistry.stackexchange.com\/questions\/2586\/molecular-and-atomic- orbitals-and-antibonding. In my textbook on organic chemistry they represent an orbital as a 2d wave function to explain the formation of bonding and anti-bonding molecular orbitals (I displayed it here as a simple $\\sin x$ function, just for illustration. I understand real wave function may be quite different). ![Example wave function, sin x](http:\/\/i.stack.imgur.com\/ktOWh.png) They say the bonding atomic orbitals can either combine in phase, resulting in a bonding molecular orbital (the electron density becomes larges between the molecules, creating an electrostatic astringent force exerted by the electron in the center on the nuclei of the combining atoms), here represented as $2\\sin x$ ![Example bonding molecular orbital wave function, 2sin x](http:\/\/i.stack.imgur.com\/ctM61.png) or combine out of phase resulting in a nodal plane between the nuclei: an anti-bonding molecular orbital. Here represented as $\\sin x + \\sin (x+\\pi)$ ![Example anti-bonding molecular orbital wave function, sin x + sin \\(x + pi\\)](http:\/\/i.stack.imgur.com\/DMLtU.png) Now the question: how can a single orbital, that is either in phase or out of phase with the orbital of the bonding atom (i.e. the other atom) form both a bonding molecular orbital and an anti-bonding molecular orbital, while it is clear that an orbital is either in one phase or an other and not both? Related is, what this 2d diagram actually represents (related question, but not quite the same: https:\/\/chemistry.stackexchange.com\/questions\/12397\/meaning-of-depiction-of- atomic-orbitals). Please ask if anything is unclear. Your help is appreciated!"}
{"id":"119352","title":"How is the direction of Magnetic\/Electric Lines of Force Known?","text":"It is shown that the direction of magnetic line is from north to the south and that of the electric line is from positive to negative. How do we\/scientists know that the imaginary lines of force or the field goes from this to that, what is the meaning of this direction? Do not give me mathematical definitions which I already know; I want the fundamental meaning of this."}
{"id":"29871","title":"How do particle scattering cross sections scale with energy in colliders?","text":"How do particle scattering cross sections scale with energy in colliders? Particularly photons, electrons, protons, and gold or lead nucleii? (If necessary, break this into four separate questions.) It is stated that due to the Heisenberg uncertainty principle, it takes more energy to measure a smaller distance and the inverse wavelength energy- wavelength relationship for photons is well known.   Therefore, I would conclude that the (total scattering) cross section for photons at least would vary as the inverse square of the energy. On the other hand, general relativity says the the longitudinal dimension shrinks with increasing velocity, but the transverse dimensions do not.   Therefore, I would conclude that the cross section of say a gold atom in RHIC or a lead atom in the LHC would basically remain constant as the energy increased. But how about the electron, supposedly a point particle?   Due to PEP and LEP, we should have good data on this. Looking at PDG figure 41.6,   http:\/\/pdg.lbl.gov\/2011\/reviews\/rpp2011-rev-cross-section-plots.pdf   I see lots of big spikes for resonant particles, but the underlying bacground is clearly a falling powerlaw that I think looks like it could be an inverse square law. The proton cross sections in the next few figures, e.g. 41.7 and 41.11, show a falling elastic scattering cross section, but they also show an approximately constant total cross section. What little i've found for RHIC supports the approximately constant cross section. Therefore I conclude that the cross sections of photons and electrons (neglecting resonances) scale as the inverse square of the energy, but the cross sections of protons and nucleii are roughly constant with energy. Comments or corrections? So is it true that the photons and electrons are different from the protons and nucleii? And why don't the protons scatter as three or six point particles (quarks and gluons) instead of one big blob?"}
{"id":"119356","title":"What is the nature of electric field? is it quantized? is it a wave?","text":"What I seek here is to understand whether the electric field in its pure form as in between the electron and the proton is uniform or does it have some kind of wave\/particle nature or both, does it have frequency or wavelength, or is it quantized?. I do not want any explanations of the effects of electric field or thereof, I want to understand the field itself. Please tell me something I all ready do not know from my engineering education."}
{"id":"1042","title":"How would I calculate the work function of a metal?","text":"In the photoelectric effect, the work function is the minimum amount of energy (per photon) needed to eject an electron from the surface of a metal. Is it possible to calculate this energy from the atomic properties of the metal (atomic number, atomic mass, electron configuration, etc.), and the properties of the lattice structure, in a way that doesn't require a complex computational model? (If so, how?) Or, if not, is there a simple formula in terms of the metal's atomic\/lattice properties that can approximate it to within a few percent?"}
{"id":"122095","title":"Could there be a planet just as the Earth, but bigger?","text":"Could there be a planet twice or thrice the size of Earth but while being the same? and by the same I mean, hosting the same life as Earth, having a day last 24 hours and a year 365 days. Basically, could there be a bigger Earth? What would be needed to replicate Earth but in a bigger planet?"}
{"id":"122092","title":"Which is more efficient Water heating method : Electric heating coil OR Induction stove?","text":"Which is more efficient Water heating method : Electric heating coil Vs Induction stove"}
{"id":"5364","title":"Why isn't pressure a measure of energy?","text":"Hey guys, I'm having a problem in understanding the 1st Law of Thermodynamics. If i would increase the pressure of a closed system by compressing a gas in a cylinder isothermal, the 1.Law states, that all the work i put in to the system, i get out by heat. So my 2.state would be the gas at the same temperature (same internal energy), but compressed (at a higher pressure). But isn't this compressed gas able to do more work, than the not compressed?   If i drilled a hole in the cylinder my high pressure gas could do some work. Where does this energy come from, since all my input energy i put in by work, came out in form of heat?"}
{"id":"82690","title":"How can one make sure that one had understood the material studied?","text":"I do not fully understand the process of understanding of a material one reads. Suppose someone reads a chapter from a physics book. How does one make sure that one had really understood what she\/he had read? I must confess that at times I read from text and I even feel that I had understood the content but later (may be at the time of exams or solving assignments or when somebody raises a question from that topic), I realize that the feeling was illusory and I feel so stupid. At times I feel that my understanding is incomplete even at basic levels. I am eager to know from experienced people in this forum if there is any method to test whether one had really understood the content of what one had read. I am talking of advanced level physics courses. May be one can also suggest useful skills one can develop for better grasp of items one studies. Thanks. I have some other related thoughts. Perhaps this is not just a problem with me but with many others, including even researchers. I have seen during one seminar given by an well-known faculty member of an institute, another well- known researcher from audience started asking questions and making comments which revealed lack of understanding of the speaker and he was almost in tears! Not that the speaker was dishonest but this shows that even experts like him can have \"illusory\" sense of understanding of a topic (not only what he\/she had read but done some research work). So may be there is nothing like full understanding of a topic but deeper and deeper understanding of topics. So the question now becomes, how can one attempt to understand a topic deeply and keep going deeper? And for students, how can they be sure their understanding of a topic is good enough?"}
{"id":"8653","title":"Is there the smallest particle that can be guaranteed to be unable to be broken down into smaller particles?","text":"Is there the smallest particle that can be guaranteed to be unable to be broken down into smaller particles?"}
{"id":"110650","title":"Information or Matter or Energy","text":"**Which one is the most fundamental of the universe? What it is made of?** Physicists are searching for unique and fundamental form. The question is , information is more fundamental than energy ? According to the thermodynamic Physics **information is any kind of event that affects the state of a dynamic system that can interpret the information.** But as far as I know its a bit of energy as I heard from the Physicist Lineard Susskind's Lectures. Will you please clarify according to the article ? Actually I was confused reading the head line article.http:\/\/www.pbs.org\/wgbh\/nova\/blogs\/physics\/2014\/04\/is-information- fundamental\/ I want to know that why are they referring the information as fundamental unity? Isn't it that Information is a mathematical concept or its a bit of energy i mean quantized energy?"}
{"id":"127312","title":"Introductory derivations of Heisenberg uncertainty principle","text":"I'm not an expert when it comes to quantum mechanics, so correct me wherever I'm wrong, but: I've always been a little bit bothered by introductory derivations of the Heisenberg uncertainty relations (for, for example, position and momentum) that involve carrying out an imaginary \"observation\" of a particle by bonking it with photons. The argument goes (at least the way I've heard it) that the more precision you want in the position measurement, the higher frequency (and so energy) the photon must have. Higher energy bonking leads to more uncertainty in the momentum afterward, and so there's a tradeoff between position and momentum precision. The derivations then manage to work out that the precisions satisfy the Heisenberg uncertainty relations. The problem I have with those kinds of derivations is that they start out assuming that a particle actually _has_ a position and momentum to speak of, and if you were lucky enough to guess them, you'd be able to predict the particle's motion exactly - it's entirely classical reasoning. The way I understand it, the position and momentum representations of the state are fourier transforms of each other, and so it's just a simple mathematical fact that as one becomes more localized, the other one becomes less localized. Another way of saying it (and please feel free to comment on the validity\/value of this way of thinking about it), that I think makes it qualitatively more obvious, is that each observable corresponds to an eigenbasis. It's pretty intuitive that, in general, if one aligns the state vector closely with any one basis vector in one eigenbasis (corresponding to a nearly definite value of that observable), then the state vector _won't_ be aligned any one basis vector of the other observable, but will instead be a mixture of several basis vectors, and so will have an uncertain value of the other observable. If this sort of reasoning is correct, then it'd seem like the classical derivations of the Heisenberg uncertainty principle are complete gibberish - rather than having to do with a particular way the measurement is made, the limits on measuring simultaneous, definite values of position and momentum are a result of the fact (which the vector space representation makes plain) that a state cannot even have simultaneous definite values. If it really is true that the classical derivations of the Heisenberg uncertainty relations is whack, is there any value in them at all? Thanks!"}
{"id":"129927","title":"Is everything moving at c in a c unit circle","text":"I was trying to explain special relativity to a few friends in a simple way and wound up with an analogy using a c unit circle. I was using y as travelling in time, x moving in space; move in space and you are borrowing from your clock-speed. E.g. the twin paradox: Your travelling twin has borrowed clock-speed. Twins clock was slower -> twin is younger. What is the problem with representing relativity using a (c) unit circle from an observers point of view? Anything Lorentz or Minkowski would be out of the question of course, I'm looking for a layman friendly description that is close to true. The circle discussed, was in my mind a half circle $y>=0$. Where $y=0$ would be a photon."}
{"id":"129928","title":"What type of magnet to use?","text":"I am asking this question to find out practical sizes of magnets, but this will be in a fictional setting. Say I have a person dressed in iron armor, medieval knight-style. Are there types of magnets that weigh on the order of 500 kg or so that would be strong enough to pull the armor to it? I'm not thinking of super-strength magnets like neodymium. Maybe a kg of force per kg of the magnet? Sorry, I'm not really familiar enough with magnets to know how to express their strength. What material would such a magnet be made of? Would the costs of making such a magnet be overly inhibitive, or is a magnet of that size even feasible?"}
{"id":"126953","title":"If there turns out to be matter with negative mass, would there also be antimatter with negative mass?","text":"I just heard of the concept of **negative mass**. Googling led me to see that the concept of negative mass is often contrasted with **antimatter**. This made me to wonder whether there would then also be a fourth category of **exotic matter** being **antimatter with negative mass**. Of this I can find no mention. If it turns out that we have **1)** normal matter **2)** antimatter and **3)** normal matter with negative mass, would that imply that we would also have **4)** antimatter with negative mass?"}
{"id":"126952","title":"How many atoms does one need to get reliable crystal information with X ray or electron diffraction?","text":"I know that, in theoretical derivations of X ray diffraction (XRD) or electron diffraction (ED), we always assume an infinite lattice, and, as you make the crystal smaller, edge effects start to matter more. As you make it even smaller, you start to lose information about actual orientation of the crystal and such (obviously, the diffraction from two atoms really tells you literally nothing). I know it's a continuum, so \"reliable\" is a little subjective, but at what point can you not get any useful information from XRD or ED? I've seen ED images inset in TEM pictures of ~10nm structures, which comes out to about 100 atoms."}
{"id":"28838","title":"Wick rotation and the arrow of time","text":"It is well known that we can switch from a statistical system to a quantum mechanical system by a Wick rotation. Has this rotation some implication on the way the time flow? namely, this is an accident or has a deep meaning?"}
{"id":"3443","title":"A flyby of orbiting supermassive black holes","text":"Consider two supermassive black holes of equal mass orbiting about their common centre of mass. Is it the case that a free-fall trajectory along the axis of rotation would be outside of either event horizon at all black hole separation distances > 0 (based on the symmetry of the situation)? To rephrase this, would you be able to navigate a rocket along a path at right angles to the orbital plane and bisecting the line between the two black holes with no ill effects whatsoever even when the black holes were very close to each other? Supplementary question: What is the shape of each of the event horizons prior to coalescence? _Note: I’m assuming that tidal forces would be small until the two singularities were extremely close. Note also that the point midway between the two BHs is the L1 Lagrangian point._"}
{"id":"100665","title":"Practice AP Physics B Exam Question regarding Momentum","text":"I am trying to review momentum for the AP exam coming up. I will be taking the AP Physics C exam for Mechanics, but I was just practicing on any free response questions I could find and I came across this one. It is from 1983, free response number 2. It says: > _A block of mass $M$ is resting on a horizontal, frictionless surface, and > it is attached as shown above to a relaxed spring of spring constant $k$. A > second block of mass $2M$ and initial speed $v_0$ collides with and sticks > to the first block. Develop expressions for the following quantities in > terms of $M$, $k$, and $v_o$._ > > _a.) $v$, the speed of the blocks immediately after impact._ > > _b.) x, the maximum distance that the spring is compressed._ ![The diagram that came with the problem](http:\/\/i.stack.imgur.com\/tP5l8.png) So I got (a) pretty easily, just by using conservation of momentum. I found the result of $$v=\\frac23 v_0$$ For part b, I began by finding the change in Kinetic Energy of the 2 blocks from immediately after impact to the point where their velocity is zero as a result of the negative work that the Spring does on the blocks. I found that $$K_i=\\frac12(3M)\\left[\\frac23v_0\\right]^2 = \\frac23Mv_0^2$$ and that $$K_f=0$$ I then equated this change in $K$ with the Work done by the spring and got $$-\\frac23Mv_0^2 = \\frac12kx_i^2 - \\frac12kx_f^2 = \\frac12kx_0^2 - \\frac12k(x_0-x)^2$$ I then tried to solve this equation for x, which is the distance that the spring is compressed, and found that $$x=\\frac12*(x_0+sqrt({x_0^2+\\frac43\\left(\\frac{Mv_0^2}k\\right)})$$ from the quadratic formula. Now, in theory, I don't really see anything wrong with this, but according to solutions I found online, people have been equating the change in K with the change in momentum, and having this equation: $$\\frac23Mv_0^2 = \\frac12kx^2$$ Can anyone explain why this equation would be true, or failing that, where I may have gone wrong in my own calculation? I appreciate the help."}
{"id":"82033","title":"Asymptotic symmetry algebra","text":"So after a lot of research, and tons and tons of papers that I've went through, I finally have some idea how to solve the equations that will give me candidates for the asymptotic symmetry group for Kerr\/CFT correspondence. One has boundary conditions $h_{\\mu\\nu}$, and metric that behaves like $g_{\\mu\\nu}=\\bar{g}_{\\mu\\nu}+h_{\\mu\\nu}$, where $\\bar{g}_{\\mu\\nu}$ is the background metric (in this case it's taken as near horizon extreme Kerr metric), and $h_{\\mu\\nu}$ are perturbations (given). So the task is finding diffeomorphisms $\\xi$ for which this metric will transform according to $$\\bar{g}_{\\mu\\nu}+\\mathcal{L}_\\xi \\bar{g}_{\\mu\\nu}=g_{\\mu\\nu}=\\bar{g}_{\\mu\\nu}+h_{\\mu\\nu}$$ And one solves that by solving equations $$\\mathcal{L}_\\xi g_{\\mu\\nu}=\\mathcal(r^m)$$ where $\\mathcal{O}(r^m)$ are boundary conditions, given in terms of radial coordinates. From this, and the fact that $h_{\\mu\\nu}=h_{\\nu\\mu}$ I get ten equations. Now my issue is this: What to take as ansatz for $\\xi$? In some papers, they take this: $$\\xi^\\mu=\\xi^\\mu_0(t,\\theta,\\phi)+r\\xi^\\mu_1(t,\\theta,\\phi)+\\mathcal{O}(r^2)$$ But in certain papers I've found that authors take this form (see after eq. 4.4 here, or here eq 5.1) $$\\xi^\\mu=\\sum\\limits_{n=-1}^\\infty \\xi^\\mu_n r^{-n}$$ such that we have subleading contributions (a falloff in radial coordinate). Now, I'm inclined to use the second one, since the diffeomorphism given in Guica et. al is given with subleading terms. I should solve this by putting a certain few terms in, and see what the boundary condition says. For instance we have $$\\mathcal{L}_\\xi g_{tt}=\\mathcal{O}(r^2)$$ That means that all $\\mathcal{O}(r^2)$ contributions will cancel, and only equations with $r$, or smaller $r$ contributions will survive. But the answer is drastically different if I take first or second ansatz. So what to take? Am I on the right track? If I take second one, what n should I start from? $n=-1$ or $n=-2$? Oh and I finally got how to solve this by looking at this article and following how they got it in appendix A **EDIT:** I've been thinking. Does the fact that in their article, the metric, and the ansatz goes with powers of $v$, means that since their equations are limited by $\\mathcal{O}(v)$ onward all higher power will cancel each other? In that case, since I'm taking only subleading terms in metric, and boundary conditions (different for each of 10 terms), and if I take that my ansatz falls off as $r^{-n}$, that means that all lower order correction cancel out? The question is still: what n to take? Since it's not the same if my $\\xi$ starts with $r^2$, or $r$ :\\ **EDIT2:** I'll put one component to show what I'm doing. Although I think something is wrong, since I'm getting complicated equations (when things get too complicated, it's kind of a show that things might not be going in the right direction, at least that's what experience showed me) So, my metric is $g_{\\mu\\nu}=\\bar{g}_{\\mu\\nu}+h_{\\mu\\nu}$, as explained above. The first question that comes to my mind: do I put manually $h_{\\mu\\nu}$ in this expression, or do I use the given boundary conditions? I could say that $h_{\\mu\\nu}$ is arbitrary, and my metric has to have a power fall off perturbations, so that my $tt$ component would be $$g_{tt}=-\\Omega^2(\\theta)(1+r^2(1-\\Lambda^2(\\theta)))+r^{-1}h_{tt}(t,\\theta,\\phi)+\\mathcal{O}(r^{-2})\\quad (\\star)$$ I removed $2GJ$ since it's just a factor in front of the metric, not depending on any of the variables ($t,r,\\theta,\\phi$), and it doesn't contribute to diffeomorphisms. I will try this after I finish what I did. However, I just put the given boundary conditions. For $tt$ component I have $$\\mathcal{L}_\\xi g_{tt}=\\mathcal{O}(r^2)=\\xi^\\sigma\\partial_\\sigma g_{tt}+g_{\\sigma t}\\partial_t\\xi^\\sigma+g_{t\\sigma}\\partial_t\\xi^\\sigma$$ Since $g_{tt}$ depends on $r$ and $\\theta$, I have two components in the first argument. In the second and third arguments are the same since $g_{t\\phi}=g_{\\phi t}$. So I have $$\\xi^\\theta\\partial_\\theta g_{tt}+\\xi^r\\partial_r g_{tt}+2(g_{tt}\\partial_t\\xi^t+g_{t\\phi}\\partial_t \\xi^\\phi)=\\mathcal{O}(r^2)$$ $$(\\xi^\\theta_{-1}r+\\xi^\\theta_0+\\xi^\\theta_1r^{-1}+\\mathcal{O}(r^{-2}))(-2\\Omega\\Omega'+2\\Omega(\\Lambda\\Lambda'\\Omega-\\Omega'(1-\\Lambda^2))r^2+\\mathcal{O}(r^2))+(\\xi^r_{-1}r+\\xi^r_0+\\xi^r_1r^{-1}+\\mathcal{O}(r^{-2}))(-2\\Omega^2(1-\\Lambda^2)r+\\mathcal{O}(r))+2((-\\Omega^2-\\Omega^2(1-\\Lambda^2)r^2+\\mathcal{O}(r^2))(\\partial_t\\xi^t_{-1}r+\\partial_t\\xi^t_0+\\mathcal{O}(r^{-1}))+(\\Lambda^2\\Omega^2r+\\mathcal{O}(1))(\\partial_t\\xi^\\phi_{-1}r+\\partial_t\\xi^\\phi_0+\\partial_t\\xi^\\phi_1 r+\\mathcal{O}(r^{-2})))=\\mathcal{O}(r^2)$$ After sorting all out, and saying that all the corrections of $\\mathcal{O}(r^2)$ vanish, I end up with $$(\\Lambda\\Lambda'\\Omega-\\Omega'(1-\\Lambda^2))\\xi^\\theta_{-1}-\\Omega(1-\\Lambda^2)\\partial_t\\xi^t_{-1}=0$$ But, this one is relatively ok, the other ones sometimes have six or more components ($\\xi^\\mu$). I'll work all of them out, and then try with $(\\star)$ as an assumption and see if things simplify... **EDIT 3:** So things aren't simplifying. But one thing I've learned: I need to solve asymptotic Killing equation. My next try, the one I think is a fast and relatively good try, is to just see what the leading orders of components should be, such that they satisfy the asymptotic Killing equations, given by boundary conditions. Then I'll try to do explicit calculation. For instance (and I hope I'm on a right track) for $tt$ component I would have $$\\mathcal{L}_\\xi g_{tt}=\\mathcal{O}(r^2)$$ $$\\xi^\\theta\\partial_\\theta g_{tt}+\\xi^r\\partial_r g_{tt}+2(g_{tt}\\partial_t\\xi^t+g_{t\\phi}\\partial_t \\xi^\\phi)=\\mathcal{O}(r^2)$$ $$\\xi^\\theta\\mathcal{O}(r^2)+\\xi^r\\mathcal{O}(r)+\\mathcal{O}(r^2)\\partial_t\\xi^t+\\mathcal{O}(r)\\partial_t\\xi^\\phi=\\mathcal{O}(r^2)$$ So for this to be true (LHS=RHS) I should have $$\\xi^r\\to\\mathcal{O}(r)$$ $$\\xi^\\theta\\to\\mathcal{O}(1)$$ $$\\partial_t\\xi^t\\to\\mathcal{O}(1)$$ $$\\partial_t\\xi^\\phi\\to\\mathcal{O}(r)$$ I hope I'm on a right track. I got a hint how Guica et. al did it. First we need to guess what the symmetries are and then generate boundary conditions by acting with those diffeomorphisms, and then check that the charges are finite and integrable. So it's more a guessing game than exact solving :S **EDIT 4:** So far (by solving asymptotic Killing equations) I only get the $\\xi^\\theta$ component right. That is I have $$\\xi^\\theta_0=\\xi^\\theta_{-1}r$$ which, when I put in the ansatz I get $$\\xi^\\theta=\\xi^\\theta_{-1}r+\\xi^\\theta_0+\\xi^\\theta_1r^{-1}+\\mathcal{O}(r^{-2})$$ $$\\xi^\\theta=\\xi^\\theta_1 r^{-1}+\\mathcal{O}(r^{-2})$$ which is exactly the one in the article by Guica et. al. But no luck with other components. If only there was some info on what I need to put in asymptotic Killing equation after the equal sign... **EDIT 5:** I'll write the equations I got by putting $\\mathcal{L}_\\xi g_{\\mu\\nu}=\\mathcal{O}(r^n)$ where $\\mathcal{O}(r^n)$ are given by boundary conditions: $$tt:\\quad (\\Lambda \\Lambda' \\Omega+(\\Lambda^2-1)\\Omega')\\xi^\\theta_{-1}+(\\Lambda^2-1)\\Omega \\partial_t\\xi^t_{-1}=0$$ $$t r:\\quad (\\Lambda^2-1)r(\\xi^t_{-1}+\\xi^t_0r^{-1})+(\\Lambda^2-1)r^3(\\xi^t_{-1}+\\xi^t_0r^{-1}+\\xi^t_1r^{-2}+\\xi^t_2r^{-3})-$$ $$-r(\\xi^t_{-1}+\\xi^t_0r^{-1})+\\Lambda^2\\xi^\\phi_{-1}+\\Lambda^2r^2(\\xi^\\phi_{-1}+\\xi^\\phi_0r^{-1}+\\xi^\\phi_1r^{-2})+\\partial_t\\xi^r_{-1}=0$$ $$t\\theta:\\quad (\\Lambda^2-1)r^2(\\partial_\\theta\\xi^t_{-1}r+\\partial_\\theta\\xi^t_0+\\partial_\\theta\\xi^t_1r^{-1}+\\partial_\\theta\\xi^t_2r^{-2})-(\\partial_\\theta\\xi^t_{-1}r+\\partial_\\theta\\xi^t_0)+$$ $$+\\Lambda^2r(\\partial_\\theta\\xi^\\phi_{-1}r+\\partial_\\theta\\xi^\\phi_0+\\partial_\\theta\\xi^\\phi_2r^{-1})+(\\partial_t\\xi^\\theta_{-1}r+\\partial_t\\xi^\\theta_0)=0$$ $$t\\phi:\\quad (2\\Lambda(\\Lambda'\\Omega+\\Lambda\\Omega'))(\\xi^\\theta_{-1}r+\\xi^\\theta_0)+\\Lambda^2\\Omega\\xi^r_{-1}+$$ $$+(\\Omega(\\Lambda^2-1))(\\partial_\\phi\\xi^t_{-1}r+\\partial_\\phi\\xi^t_0+\\partial_\\phi\\xi^t_1r^{-1})-\\Omega\\partial_\\phi\\xi^t_{-1}+$$ $$+\\Lambda^2\\Omega(\\partial_\\phi\\xi^\\phi_{-1}r+\\partial_\\phi\\xi^\\phi_0)+\\Lambda^2\\Omega(\\partial_t\\xi^\\phi_{-1}r+\\partial_t\\xi^t_0)+\\Lambda^2\\Omega\\partial_t\\xi^\\phi_{-1}=0$$ $$rr:\\quad \\xi^\\theta_{-1}r+\\xi^\\theta_0=0$$ $$r\\theta:\\quad \\partial_\\theta\\xi^r_{-1}+(\\xi^\\theta_{-1}r+\\xi^\\theta_0)=0$$ $$r\\phi:\\quad (\\xi^t_{-1}r+\\xi^t_0)+\\xi^\\phi_{-1}=0$$ $$\\theta\\theta:\\quad \\Omega'(\\xi^\\theta_{-1}r+\\xi^\\theta_0)+\\Omega(\\partial_\\theta\\xi^\\theta_{-1}r+\\partial_\\theta\\xi^\\theta_0)=0$$ $$\\theta\\phi:\\quad (\\partial_\\phi\\xi^\\theta_{-1}r+\\partial_\\phi\\xi^\\theta_0)+\\Lambda^2r(\\partial_\\theta\\xi^t_{-1}r+\\partial_\\theta\\xi^t_0+\\partial_\\theta\\xi^t_1r^{-1})+$$ $$+\\Lambda^2(\\partial_\\theta\\xi^\\phi_{-1}r+\\partial_\\theta\\xi^\\phi_0)=0$$ $$\\phi\\phi:\\quad (\\Omega\\Lambda'+\\Lambda\\Omega')\\xi^\\theta_{-1}r+\\Lambda\\Omega r(\\partial_\\phi\\xi^t_{-1}r+\\partial_\\phi\\xi^t_0)+$$ $$+\\Lambda\\Omega\\partial_\\phi\\xi^\\phi_{-1}r=0$$ The problem is, the only equation I can get some information, about how my vector should behave in components is $rr$, from which I get the general form of $\\xi^\\theta$ component of diffeomorphism. From $r\\theta$ I get that the $\\xi^r_{-1}$ term is a constant that doesn't depend on $\\theta$, and $\\theta\\theta$ equation will just confirm that $rr$ one is correct. I guess from the fact that $\\xi^r_{-1}$ is not zero I could say that the $r$ component will start at that term (like in the article). **EDIT 6:** I have added the bounty to raise awareness to this, if anyone knows if what I did so far is correct, and what to do next, please do tell :)"}
{"id":"7433","title":"Equations describing the liquifaction of gases","text":"I would like to find out how much energy is required to liquify gases (oxygen or nitrogen, for example). **What are some relevant equations?** I'm curious whether creating small quantities of dry ice or cryogenic fluids is feasible at home. If James Dewar was able to liquify hydrogen in 1898 surely it must not be that difficult!"}
{"id":"70553","title":"Where does the reverse current come from in a organic semiconducting diode?","text":"Unlike traditional inorganic diodes, organic semiconducting materials hold few thermally exited free carriers under room temperature, so the reverse current should be small if pinholes of the films is well avoided. However the fact is that the reverse current varies a lot between devices using different types of organic materials. Is it because there are other sources of reverse current? Or is it mostly dominated by charge transport layers which have relatively higher conductivity compare to active layer?"}
{"id":"117009","title":"Use microwave cavity in atomic clock","text":"In most of the papers regarding atomic clocks, the author talks about a microwave cavity. In this box, all the unwanted frequencies of the electromagnetic radiation are absorbed and the other frequencies are maintained. But how does this work with an atomic clock? In this case, one uses atoms rather than radiation. What is the difference between these applications of the cavity? I thought this might have to do something with the duality between atomic particles and waves, although I couldn't find any sources that could verify this."}
{"id":"16081","title":"What is a Free Body Diagram?","text":"I have this problem: > A 10 kg box is resting on a frictionless table and someone drags it by a > string at a 30 degree angle to the horizontal with a force of 40 Newtons. > Find the acceleration of the box. Draw a free body diagram of the system. What is a free body diagram?"}
{"id":"98096","title":"If electrons behave as standing waves when they are bound to an atom then how do they carry charge?","text":"Today in my physics lesson we learnt that the best way of describing the behaviour of an electron that is bound to an atom is to treat it as a standing wave. I understand that this is the wave-particle duality however I was taught that in the wave-particle duality something either acts as a wave or a particle, not both at the same time. With this in mind I'm curious as to how electrons can still carry charge when they are behaving as a wave. I asked my teacher this question and he wasn't able to answer it so hopefully someone on here can explain it to me :)"}
{"id":"117003","title":"Are there any astronomical objects at night that are dangerous to look at (through a telescope)?","text":"Are there any astronomical bodies that would be dangerous to my vision to view through a telescope? Obviously the sun is dangerous, but are there other bodies at night I should avoid?"}
{"id":"10866","title":"2nd order variation of Hilbert-Einstein action + Gibbons-Hawking-York boundary term","text":"While the first order metric variation of Hilbert-Einstein action plus Gibbons-Hawking-York boundary term is well-known and takes the form: $\\delta S_{HE}+\\delta S_{GHY}=-\\frac{1}{16\\pi G}\\int d^3x \\sqrt{-\\gamma}~(K^{\\mu\\nu}-K\\gamma^{\\mu\\nu})\\delta \\gamma_{\\mu\\nu}$ ($K_{\\mu\\nu}$ and $\\gamma_{\\mu\\nu}$ are the extrinsic curvature and the projection metric), does anybody know the second order variation of this combined action? Thanks to anyone who can help me!"}
{"id":"133128","title":"Integral over a product of two Green's functions","text":"Need some help here on a frequently encountered integral in Green's function formalism. Forgive me since I am a junior student. I have an integral\/summation as a product of a retarded and advanced Green's functions, looks simply like $$\\sum_{p^{\\prime}}\\frac{1}{p^{2}-{p^{\\prime}}^{2}-i\\epsilon}\\frac{1}{p^{2}-{p^{\\prime}}^{2}+i\\epsilon}$$ where I have omitted the mass $m$ to make the notation simple; $\\epsilon$ is a positive infinitesimal. I can convert the summation to an integral over momentum $p$ or energy, this is standard. Then I follow the complex analysis and identify 4 poles (if the integration is over momentum) and further apply the residue theorem. But, what I get is an expression proportional to $\\frac{1}{\\epsilon}$. This means it is divergent since $\\epsilon\\rightarrow 0$. Could any one please point out to me what have I missed in this calculation? What mistakes did I make? Many many thanks."}
{"id":"16333","title":"How do we perceive hotness or coldness of an object?","text":"Some objects, especially metallic ones, feel cold on touching and others like wood, etc. feel warm on touching. Both are exposed to same environment and are in their stable state, so some kind of equilibrium must be being reached. What is this equilibrium? And how do we perceive hotness or coldness of an object? Does skin have some kind of heat sensors, etc., which transmit signals to brain? Like, how do the eye transmit\/convey an image formed on the retina to brain?"}
{"id":"19503","title":"How long does it take to weigh an uncooperative mass?","text":"(This is curiosity, not a practical question. It was inspired by standing still for a digital scale, and considering whether it would be possible to make a scale that could handle being jumped on, juggled on, etc.) Setup: An impermeable rigid box with arbitrary contents on top of (fixed to) an ideal scale, in a known uniform gravitational field and no electromagnetic field. Assume classical mechanics (i.e. no relativistic effects or uncertainty). Question: How long must data (i.e. the force on the scale) be collected in order to know the mass of the box and contents, given that the contents of the box are _doing their best to make it difficult_ (which I think can only consist of accelerating masses vertically)? I think that it is not possible to get an exact answer after a finite time; for example, consider if the box contains a mass being accelerated downward arbitrarily slowly — thus reducing the total weight by a small fraction, until it touches bottom. Is there an upper bound on the error, given a time interval and some property of the collected data (e.g. the maximum or mean force)?"}
{"id":"63960","title":"Holonomy twisting","text":"There is Witten's topological twist of standard SUSY QFTs with enough SUSY into Witten-type TQFTs. What is a holonomy twist?"}
{"id":"65750","title":"Wigner characteristic function","text":"I came across the \"representation of a Gaussian state by its Wigner characteristic function\". I don't know what Wigner characteristic function is and google results are not precise enough. Neither have I been able to find this in some quantum mechanics book. It will be great if you could explain what it means, but even better would be citing a source from where I could read it up. I came across this term in this paper: \"Inseparability criterion for continuous variable systems\". Duan, Giedke, Chirac and Zoller. Physical Review Letters (year 2000)"}
{"id":"2364","title":"Polar vs non-polar fluid","text":"In the book \"Vectors, Tensors, and the Basic Equations of Fluid Mechanics\" by Rutherford Aris I read the following: > If the fluid is such that the torques within it arise only as the moments of > direct forces we shall call it nonpolar. A polar fluid is one that is > capable of transmitting stress couples and being subjected to body torques, > as in polyatomic and certain non-Newtonian fluids. Can someone help me understand this? Say by giving me another definition of polar and nonpolar fluids. Thanks."}
{"id":"21226","title":"How do we know that nonperturbative canonical quantum gravity is wrong?","text":"In these forums and elsewhere it is routinely agreed that \"we do not have a theory of quantum gravity.\" My question is, how do we know that canonical quantum gravity is \"wrong\"? I understand that the theory is perturbatively nonrenormalizable, but doesn't that just mean that we can't apply perturbation theory to it? It seems the theory is nonperturbatively renormalizable. A theory being nonperturbative doesn't make it \"wrong\". My understanding is that in modern QFT nonrenormalizability is not anymore considered such a big deal, and that even the Standard Model is expected to be an effective theory modified by nonrenormalizable terms appearing at higher energies. So barring practical considerations, why is canonical quantum gravity considered \"not a theory of quantum gravity?\" EDIT: I am not sure that the term \"Canonical Quantum Gravity\" is correct. I am referring to what is called \"Quantum Einstein Gravity\" in this paper. If someone knows better please help me correct my terminology. I was not meaning to imply anything about LQG because I thought that LQG \"brought more to the table\" than just a nonperturbative approach to the same QFT, but I could be wrong about that."}
{"id":"1862","title":"Which areas in physics overlap with those of social network theory for the analysis of the graphs?","text":"I am studying social networks in terms of graph theory and linear algebra. I know that physicists have published and worked a lot in this field. This causes me to assume that there are sub-fields in physics which overlap in the essence of their problems with those of small world networks. Which natural phenomena exhibit these kind of features similar to small world networks? I would like to know that, so maybe I can look at those problems to get inspiration that can be taken to social network theory."}
{"id":"1869","title":"What reasons are there to suspect string theory may be an incorrect theory?","text":"It's a truism that in science it is at least as important to state the reasons why a theory or idea might be wrong, as to state the reasons why it is might be correct. For example, early renormalization methods made experimental predictions that were very accurate, but the inventor's famously worried for decades about the conceptual problems of divergences; and many of their papers\/articles\/books on the subject include a long discussion about how acutely worried they were that their method for handling divergences was deeply flawed somehow. My intention with this post is not to engage any user in a debate about whether string theory is true or not. I think anyone who takes string theory seriously should be happy to acknowledge as clearly as possible all the potential theoretical issues\/problems with the theory. Therefore I would like to ask, what are the major theoretical problems which would lead a reasonable person to be worried that string theory is not an accurate theory? I am excluding the fact that string theory has made no (major) predictions that have been confirmed, I am only interested in conceptual and theoretical ideas which might discredit the theory. Please keep all answers to one issue per post."}
{"id":"81075","title":"Current division in ideal inductors?","text":"The Question is:- > Find the current distribution in the inductors at steady state (i.e. > assuming $t\\rightarrow \\infty$ after the switch is closed and circuit > established and consider the inductors without any resistance and ideal). ![enter image description here](http:\/\/i.stack.imgur.com\/noa0t.jpg) **My Efforts**   Since both of the are in parallel, the equations for current at any time $t$ after establishing the circuit is given by ($i_1$=corrent in $5mH$ inductor and $i_2$ in the other, and the total current $i$):- $$i(t)=i_1(t)+i_2(t)=4(1-e^{-\\frac{5t}{L_{net}}})$$ where $L_{net}=\\frac{10}{3}mH$. But I do not know how this current divides into two in the inductances at any time $t$ or in the steady state. Had the inductors not been ideal, i.e. if they had different resistances, the current could divide in the ration of their resistances in the final steady state, but since they are ideal, this method won't work. How can I solve this? the answers given are $\\frac83A$ in the $50mH$ inductor and $\\frac43A$ in the other."}
{"id":"57170","title":"Is the momentum operator diagonal in position representation?","text":"The matrix elements of the momentum operator in position representation are: $$\\langle x | \\hat{p} | x' \\rangle = -i \\hbar \\frac{\\partial \\delta(x-x')}{\\partial x}$$ Does this imply that $\\langle x | \\hat{p} | x' \\rangle = 0 $ whenever $x \\neq x'$? Is the momentum operator diagonal in position representation? I know that the momentum operator shouldn't be diagonal in position representation (otherwise solving for the eigenergies and eigenfunctions of most Hamiltonians would be trivial). I am obviously confused here. I need more than just yes\/no answers to these questions. I need some explanation, or some intuition."}
{"id":"17062","title":"Relativistic transformation of the wave packet length","text":"Let us suppose we have an excited atom at rest. It has a certain mean lifetime $\\tau_0$. If we wait sufficiently long time $t>>\\tau_0$, we will find a deactivated atom and a (spherical) electromagnetic wave function of photon with about $\\tau_0\\cdot c$ long layer with non zero probability to find a photon within. Something like a fast expanding probability \"ring\" with a $\\tau_0\\cdot c$ width of the ring. Now, let us consider this system in a moving reference frame. It seems to me that this width $\\tau_0\\cdot c$ is relativistic invariant: it is a difference between two \"fronts\" of electromagnetic wave rather than a length of a material body subjected to the Lorentz contraction. Is it correct? In other words, whether this picture relativistic invariant? ![Wave function evolution](http:\/\/i.stack.imgur.com\/vtenD.jpg)"}
{"id":"94675","title":"Is the Copenhagen interpretation merely an approximation to quantum mechanics?","text":"So, I'm reading Max Tegmark's _Our Mathematical Universe_ (Knopf edition, p. 229). He's discussing Everett\/MWI for a bit and I'm not really paying attention and then I wake up to this: > [I]t's time to update the quantum textbooks to mention decoherence (many > still don't) and to make clear that the Copenhagen interpretation is better > thought of as the _Copenhagen approximation:_ even though the wave function > probably doesn't collapse, it's a very useful approximation to do the > calculations as if it does collapse when you make an observation. Now, I believe I have read (elsewhere, e.g., on this site) that these interpretations do not make any difference _whatsoever_ * when it comes to predictions. But here, apparently (at least that is how I read the above), it is claimed that MWI and Copenhagen actually _differ_ in predictions (and that MWI has the _right_ predictions). **Is that correct?** (This would make it slightly more difficult to be agnostic about choosing your favourite interpretation for the time of day, right?) NB: I'm not specifically asking whether the wavefunction collapses or not. *As in: zip, nada, zilch, _exactly_ $0$."}
{"id":"68052","title":"Problem from Sakurai about a delta-function potential","text":"Can you help me with this problem from Sakurai: A particle of mass m in one dimension is bound to a fixed center by an attractive delta-function potential: $$V(x) ~= ~-a\\delta(x) , \\qquad a>0.$$ At $t = 0$, the potential is suddenly switched off (that is, $V = 0$ for $t > 0$). Find the wave function for $t > 0$."}
{"id":"52618","title":"Who first provided a string realization of dual resonance models?","text":"After the $N$-particle generalization of the Veneziano amplitude was written down and studied, who was the first (or who were among the first) to realize that the amplitudes could be understood in terms of a relativistic string in the early 1970s?"}
{"id":"68054","title":"Does shadow have mass?","text":"I know it sounds like a foolish question but I have a reason for asking and I'm hoping someone here, can give a convincing response. Here is why I pose the question...it seems to me that all this discussion of photons, and other sub-atomic particles possessing mass or negative mass is really a discussion about the mass of shadows. Although we can't really weigh shadow we can see it, we can measure its area, observe its outline. a photon- which is massless - may seem heavier than shadow. It is really driving me crazy when I think about it."}
{"id":"90745","title":"Gravity on the International Space Station - General Relativity perspective","text":"My question is an extension to this one: Gravity on the International Space Station. If all the outside views of the ISS was sealed, then the crew inside would not be able to tell whether they were in orbit around the earth in orbital speed or free floating in space beyond the orbit of Neptune, right? How would time dilation due to gravitation fields be affected? Supposing you have three atomic clocks: 1 - One on surface of the Earth, at Sea Level, 2 - One in the ISS, 3 - One in deep space beyond the orbit of Neptune. At what speed would each clock run compared to the other two?"}
{"id":"90740","title":"Calculating Forces via Feynman diagrams?","text":"How would one go about calculating forces that test objects feel using Feynman diagram methods? For example, say we have a massive object in GR so that the metric takes on the standard Schwarzschild form, $g^S_{\\mu\\nu}$. Now add some test matter in which is described by some lagrangian $\\mathcal L'$ and has corresponding stress tensor $T'^{\\mu\\nu}$. In the action, the first order perturbation will then be $S\\supset \\int d^4x\\, \\sqrt{-g^S}\\, h_{\\mu\\nu}T'^{\\mu\\nu}$ where $h_{\\mu\\nu}$ is the perturbation of the metric away from the Schwarzschild metric. I know how to calculate the force on the test particle in GR, but how would one do so by using field theory techniques like Feynman diagrams? Relatedly, in some modified gravity theories you get some action like $\\mathcal L\\sim \\frac{1}{2}(\\partial\\phi)^2+\\mathcal{L}_{\\rm int}(\\phi)+\\phi T\/M_{\\rm pl}$ where $\\phi$ is related to the modification of gravity. If $\\phi$ takes on a non-zero profile, say $\\phi\\sim1\/r$, then this results in a fifth force on test particles, according to the literature. How do I see this by using Feynman diagrams? I know that usually to figure out the potential generated by an interaction you can used the Born approximation, but I dont' quite see how that will work out in this case."}
{"id":"60700","title":"Static discharge out of 3 volts button battery","text":"I am starting to get interested in static electricity I have found this which is very cool but I do not know how I can control the output voltage. http:\/\/www.instructables.com\/id\/How-to-build-the-Worlds-Smallest-Electronic- Shock\/#intro What I need to do is to use a 3 Volts Button Battery and get a 50-500 volt static charge from it. And I need the device to be very small. What can I change in order to control the voltage? and is a 3 volts button battery good to keep this thing alive for months? and lastly, can I flip the battery to control the out put charge (negative or positive)?"}
{"id":"113657","title":"Is it possible to increase the electromagnetic repulsion between two particles?","text":"Given the fact that the force stopping me falling through the ground is the electromagnetic repulsion between the electrons in my feet and the ground, would it be possible to increase this repulsive force? Also is friction generated due to this repulsion of angular faces at a very small scale ? if so could a material be made so smooth it had zero friction because the electromagnetic force simply kept the two apart instead of causing friction? Sorry if its a stupid question but im very interested to find out and why not if i am wrong."}
{"id":"104948","title":"Renormalizing composite operators","text":"Consider the QED Lagrangian, \\begin{equation} {\\cal L} = \\bar{\\psi} ^{(0)} ( i \\partial_\\mu \\gamma^\\mu - m ) \\psi ^{(0)} - e A _\\mu ^{(0)} \\bar{\\psi} ^{(0)} \\gamma ^\\mu \\psi ^{(0)} - \\frac{1}{4} F ^2 _{ (0) } \\end{equation} If we apply the typical renormalization procedure, \\begin{align} & A _\\mu = \\frac{1}{ \\sqrt{ Z _A } } A _\\mu ^{ ( 0 ) } \\\\\\ & \\psi = \\frac{1}{ \\sqrt{ Z _\\psi } } \\psi ^{ (0) } \\\\\\ & e = \\frac{1}{ Z _e } e ^{(0)} \\\\\\ & m = \\frac{1}{ Z _m } m ^{(0)} \\end{align} Then we get a mass counterterm, \\begin{equation} m ( 1 - Z _m Z _\\psi ) \\bar{\\psi} \\psi \\end{equation} Up to here, I have no issues. Now suppose we want to consider the renormalized of some operator, $ {\\cal O} ( x ) $. To be explicit, lets take the operator to be $ \\bar{\\psi} \\psi (x) $. The way I would naively think about it is that we should insert this operator into the Lagrangian in terms of the bare fields and then allow its coupling to get renormalized in the same way that we did above. So we would have, \\begin{equation} \\Delta {\\cal L} = g ^{(0)} {\\cal O} ^{(0)} ( x ) = g ^{(0)} \\bar{\\psi} ^{(0)} \\psi ^{(0)} \\end{equation} then we would say \\begin{align} & g = \\frac{1}{ Z _g } g ^{(0)} \\end{align} and we get a counterterm, \\begin{equation} \\Delta {\\cal L} = g ( 1 - Z _g Z _\\psi ) \\end{equation} However Wise and Manohar in their book, _Heavy Quark Physics_ , get a slightly different result. They get a counterterm of, \\begin{equation} \\Delta {\\cal L} = g \\left( 1 - \\frac{ Z _\\psi }{ Z _g } \\right) \\end{equation} What is wrong about my understanding of composite operators?"}
{"id":"54610","title":"Non-covariance of the higher rank propagator (from Weinberg's QFT textbook)","text":"In chapter 6.2 of Weinberg's  QFT Vol1 , he gave the general form of Wick contractions of all possible fields(scalar, spinor, vector, etc.), he showed $\\Delta_{lm}(x,y)=\\theta(x-y)P^{(L)}_{lm}(-i\\frac{\\partial}{\\partial x})\\Delta_+(x-y)+\\theta(y-x)P^{(L)}_{lm}(-i\\frac{\\partial}{\\partial x})\\Delta_+(y-x)\\cdots(6.2.8)$ where $\\Delta_+(x)=\\int d^3p(2p^0)^{-1}e^{ip\\cdot x}$, and $P^{(L)}_{lm}$ is a covariant polynomial if its argument is a on-shell 4-momentum(i.e. $P^{(L)}_{lm}(\\sqrt{\\mathbf{p^2}+m^2},\\mathbf{p})$), but may not be covariant for off-shell 4-momentum. Then by a series of mathematical identities, he was able to show $\\Delta_{mn}(x,y)$ can be written in a 4-momentum integral: $\\Delta_{lm}(x,y)=\\int d^4q\\frac{P^{(L)}_{lm}(q)\\exp(iq\\cdot (x-y))}{q^2+m^2-i\\epsilon}\\cdots(6.2.18)$ From here he argued since the 4-momentum $q$ in $(6.2.18)$ is not always on- shell, $P^{(L)}_{lm}$ may not be covariant, and in turn the $\\Delta_{mn}(x,y)$ may not be covariant. It's indeed not for a vector field as what he showed immediately after, and that's why we need to add a non-covariant term in the Hamiltonian to make the propagtor covariant ect.etc. I did follow the steps he showed, but something strange occured to me when I looked back at the expression $(6.2.8)$: $(6.2.8)$ looks completely covariant, since $\\theta$ is invariant, $\\Delta_+$ is invariant and is written in 3-momentum integral so that $P^{(L)}_{lm}(-i\\frac{\\partial}{\\partial x})\\Delta_+(x-y)$ must be covariant, yet $(6.2.18)$-which is supposed to be equivalent to $(6.2.8)$- is not covariant. I am wondering if there's any subtlety I missed. Also there's another mystery to me: in chapter 3.5 (page 144), he gave a perturbative proof of Lorentz invariance based on Dyson series, the invariance of Hamiltonian density and micro-causality condition, and he did mention Lorentz invariance could be disturbed by the reasonings given in chapter 6.2, but the proof in chap 3.5 is completely formal and I can't see exactly how the reasonings in 6.2 can jeopardize it. (PS: I wish I could have formulated my question in a more self-contained way, but I couldn't unless I copied some whole pages of Weinberg, so I apologize for the potential vagueness in advance)"}
{"id":"54613","title":"Height of a piston in a heated cylinder containing H2O","text":"I'm a first time user and I hope I won't be too enigmatic asking the following question: I have a cylinder (radius= $6$ $\\text{cm}$) with a frictionless piston on top of it and inside $30$ $g$ of $H_2 O$ (molar weight is $M=30$). External pressure is $1 \\text{ atm}$ and initial temperature is $T_1=273$ $K$. $Q=24$ $KJ$ of heat are put into the system. What height does the piston reach? I'm pretty sure I didn't get this right, but here is what I did: $$Q = Q_\\text{water} - Q_\\text{Latent evaporation heat} + Q_\\text{steam}= mc \\times (373-273) - m L + (m\/M)Cv \\times (T_2-373)$$ from this equation I found $T_2$ and knowing that pressure is constant, I found the volume of the steam and thus the height of the piston which resulted something crazy like $27 \\text{m}$. What have I done wrong?"}
{"id":"54612","title":"Dividing two star spectra","text":"I am doing some work that involves dividing two stellar spectra from the same star. Those stellar spectra are constructed by summing random samples of multiple spectra from the same star to improve the signal to noise ratio (SNR). I know that for a sum, the $SNR$ will vary with $\\sqrt N$ where $N$ is the number of spectra I am adding to construct each one of my spectra. My problem is: Assuming $SNR_1$ and $SNR_2$ are the signal to noise ratios of the spectras I am dividing by each other, how can I obtain the $SNR$ of the resulting spectrum? Do you recommend any documentation where thse operations are explained in detail? Thanks!"}
{"id":"48283","title":"Polarization photon and Stokes parameters","text":"I have the following situation: About the polarization of the photon, I introduce the basis: Horizontal polarization $|\\leftrightarrow>=\\binom{1}{0}$ Vertical polarization $|\\updownarrow>=\\binom{0}{1}$ The density matrix in this problem is: $$\\rho =\\frac{1}{2}\\begin{pmatrix} 1+\\xi _{1} & \\xi_{2}-i\\xi _{3}\\\\\\ \\xi_{2}+i\\xi _{3} & 1-\\xi _{1} \\end{pmatrix}$$ The Stokes parameters are: $\\xi _{1}, \\xi _{2}, \\xi _{3}$ The probability that if the photon has got lineal polarization whose axis forms an angle $\\theta$ with de horizontal is: $$|w>=cos\\theta |\\leftrightarrow>+sin\\theta|\\updownarrow>$$ $$ P_{\\theta}=<w|\\rho|w>=\\frac{1}{2}\\left ( 1+\\xi_{1}cos(2\\theta)+\\xi_{2}sin(2\\theta) \\right )$$ Is there any value of the Stokes parameters for which this probability is zero?"}
{"id":"62707","title":"A Type of Pendulum","text":"![http:\/\/i.stack.imgur.com\/vAnpP.png](http:\/\/i.stack.imgur.com\/Z9V2R.png) Is there any chance that $$rtl(\\ddot\\omega+\\ddot\\phi)\\cdot\\sin{(\\phi+\\omega t)}- gl\\dot\\phi \\cdot \\sin{\\phi} + ltr\\dot\\omega(\\dot\\phi^2-\\dot\\omega)\\cdot \\cos{(\\phi+\\omega t)}-gtr\\cos{(\\omega t)}+l^2\\ddot\\phi-t^2r^2\\ddot\\omega=0$$ is equation of motion for the ball in the picture? * * * To describe x and y coordinates I wrote $x = l\\cdot \\sin{\\phi} + r\\cdot \\cos{\\omega t} $ and $y = l\\cdot \\cos{\\phi} + r\\cdot \\sin{\\omega t}$ Than express kinetic energy as $E_k = \\frac{1}{2} m \\cdot (\\dot x^2 + \\dot y^2)$ and potential energy as $E_p = mgy$. Next step was Langrangian $\\mathcal{L} = E_k - E_p$ I put $\\frac{d}{dt}\\left( \\frac{\\partial{\\mathcal{L}}}{\\partial \\dot \\phi} \\right) - \\frac{\\partial \\mathcal{L}}{\\partial \\phi } = \\frac{d}{dt}\\left( \\frac{\\partial{\\mathcal{L}}}{\\partial \\dot \\omega} \\right) - \\frac{\\partial \\mathcal{L}}{\\partial \\omega}$ and that lead to the equation above, but I might do some mistakes. I hope it's ok to put the two equations together like this. Is it? * * * So if $q$ is only $\\phi$ than $$\\dot x = \\dot \\phi l \\cos{(\\phi)}-r \\omega \\sin{(\\omega t)}$$ $$\\dot y = - \\dot \\phi l \\sin{(\\phi)} + r\\omega \\cos{(\\omega t)}$$ $$E_k = \\frac{1}{2}m(\\dot \\phi^2 l^2 \\cos^2{(\\phi)} - 2r \\omega l \\dot \\phi \\cos{(\\phi)} \\sin{(\\omega t)}+r^2 \\omega^2 \\sin^2{(\\omega t)} + r^2 \\omega^2 \\cos^2{(\\omega t)} - 2r \\omega l \\dot \\phi \\sin{(\\phi)} \\cos{(\\omega t)}+l^2 \\dot \\phi^2 \\sin^2{(\\phi)})$$ $$Ek = \\frac12 m(\\dot \\phi^2 l^2 (\\cos^2{(\\phi)}+\\sin^2{(\\phi)}) + r^2 \\omega^2 (\\sin^2{(\\omega t)}+\\cos^2{(\\omega t)})-2rl \\omega \\dot \\phi (\\cos{(\\phi)}\\sin{(\\omega t)}+\\sin{(\\phi)}\\cos{(\\omega t)}))$$ $$E_k = \\frac12 m \\dot\\phi^2 l^2 + \\frac12 m r^2 \\omega^2 - mrl \\omega \\dot \\phi \\sin{(\\phi + \\omega t)}$$ So it would be kinetic energy of the ball. Than I need potential energy. $$E_p = mgl\\cos{(\\phi)} + mgr\\sin{(\\omega t)}$$ Therefor Lagrangian should be: $$\\mathcal{L} = \\frac12 m \\dot\\phi^2 l^2 + \\frac12 m r^2 \\omega^2 - mrl \\omega \\dot \\phi \\sin{(\\phi + \\omega t)} - mgl\\cos{(\\phi)} - mgr\\sin{(\\omega t)}$$ It seems like right time for Lagrange equation. I'll compute every part of equation on the new line. $$\\frac{\\partial \\mathcal{L}}{\\partial \\phi} = -mrl\\omega \\dot \\phi^2 \\cos{(\\phi+ \\omega t)} + mgl \\dot \\phi \\sin{(\\phi)}$$ $$\\frac{\\partial \\mathcal{L}}{\\partial \\dot \\phi} = ml^2 \\dot \\phi - mrl \\omega \\sin{(\\phi + \\omega t)}$$ $$\\frac{d}{dt} \\left( \\frac{\\partial \\mathcal{L}}{\\partial \\dot \\phi} \\right) = ml^2\\ddot \\phi - mrl \\omega^2 \\dot \\phi \\cos{(\\phi + \\omega t)}$$ If this was right, the equation would look like: $$ml^2\\ddot \\phi - mrl \\omega^2 \\dot \\phi \\cos{(\\phi + \\omega t)} + mrl\\omega \\dot \\phi^2 \\cos{(\\phi+ \\omega t)} - mgl \\dot \\phi \\sin{(\\phi)} = 0$$ If I put cos together and multiply whole equation by $\\frac{1}{ml^2}$ I get $$\\ddot \\phi - \\frac rl \\omega \\dot \\phi \\cos{(\\phi + \\omega t)} (\\omega + \\dot \\phi) - \\frac gl \\dot \\phi \\sin{(\\phi)}$$ which is not right."}
{"id":"95105","title":"How many how many planck areas would capture of a resting electron add to the surface of a black hole?","text":"In other words, what is the 'information content' of an electron (or of other supposedly elementary particles for that matter) as revealed by their effects on black hole horizon area?"}
{"id":"95101","title":"Raising and lowering operators of orbital angular momentum","text":"For the orbital angular momentum, the raising and lowering operators are given by, $$ L_+ = e^{i\\phi} \\bigg(\\frac{\\partial}{\\partial\\theta} + i\\: cot\\theta\\frac{\\partial}{\\partial\\phi}\\bigg) $$ $$ L_- = -e^{-i\\phi} \\bigg(\\frac{\\partial}{\\partial\\theta} - i\\: cot\\theta\\frac{\\partial}{\\partial\\phi}\\bigg) $$ With this I obtain $$ L_+^\\dagger = - L_- $$ But with the actual definition in terms of $ L_x $ and $ L_y $ with $$ L_+ = L_x + i L_y $$ $$ L_- = L_x - i L_y $$ $$ L_+^\\dagger = L_- $$ How do I reconcile between these two results ? Or is there any mistake I have committed ? PS : My professor hinted saying it had something to do with compactness of angular momentum, but I didn't understand !! (My problem is not in obtaining the result of $L_+$ and $L_-$, but in reconciling these two facts)."}
{"id":"109342","title":"Can a wave possess spin?","text":"Since a matter wave is associated with a particle in quantum mechanics, does the wave spins? I mean, can we visualize the spinning of wave or is it possible that the wave spins?"}
{"id":"30821","title":"Snooker\/billiards cueing","text":"I understand the 2 body dynamics very well, but I can't see how the professionals apply their **chalked** cue and hit the way they do. As a few for instances:   1. How much does the chalk come into play in terms of friction?   2. How off center is it possible to play?   3. How do they aim the off center? Also in a situation like this http:\/\/www.youtube.com\/watch?v=McThDshEgU0 Ronnie O'Sullivan's fantastic play. What useful measurements are there to grade his skill?"}
{"id":"672","title":"What is the relation of sound propagation to air pressure?","text":"Hi there i am wondering, air is making sound propagate. So in vacuum there is no sound. but what is the relation of pressure to sound volume? is it linear? if i have a source of sound and lets say i am in a room with air pressure 0.5 of the pressure outside would the sound be 0.5 lighter? or does air pressure have no impact on the sound volume? Is there a comprehensible formula for that? thanks a lot!"}
{"id":"114965","title":"Why the trap is needed in cold atom experiment?","text":"In ultracold atomic gas experiments, the optical lattice provides a periodic optical potential to trap the atoms, why an extra trap, usually a harmonic trap is needed to trap the atoms?"}
{"id":"29681","title":"Set theory, category theory, realism and the recent \"reality of the wavefunction\" papers","text":"I will add a better phrased question here. Do we need to consider quantum foundations to form a quantum theory of gravity? The kind of foundational question I am thinking of is expressed in the recent \"reality of the wavefunction\" paper. I have become convinced that if we move physics into a category theoretic foundation (one that eschews the presentation of the theory of categories IN SET) then we might avoid problems of ontic and epistemic states for quantum theory. Does anyone strongly disagree? I have been working on a presentation of the theory of categories that is more like type theory and would avoid things like \"The category of groups consists of the set of all groups\". The reason we avoid problems of ontic\/epistemic is because the mathematics is grounded in the concept that \"morphism\" equals \"cuasal relation\" rather than \"my set of states is ...well...a set of elements which exists by virtue of the fact that the elements exist\". We might trade \"causal relation\" for a more generic relation that reflects \"causally neutral probability theories\". Does anyone strongly disagree?...or agree :)"}
{"id":"108423","title":"Which solution to the electromagnetic wave equation is the most accurate model of monochromatic light?","text":"When a photon is modeled as a monochromatic electromagnetic wave its electric and magnetic components are usually taken to be sine waves (for example here http:\/\/hyperphysics.phy-astr.gsu.edu\/hbase\/waves\/emwv.html). I believe the practical reason for this is that any solution of the electromagnetic wave equation can be expressed as a sum of sine waves. But physically, when a photon can be interpreted as a wave, how is it best modeled? Do we have empirical evidence to think that it is best modeled by a single sine wave for the E and B fields, or if not which solution of the electromagnetic wave equation would best model it? Are there experiments that could show that light waves resemble more say square waves than sine waves? **Update** To rephrase the core of my question more properly: If a photon \/ electromagnetic wave travels through a point in space, in vacuum, and we measure the electric and magnetic fields at this point with a very high temporal resolution, would we measure the electric and magnetic fields to fluctuate exactly as sine waves, or as something else? Has such an experiment ever been made?"}
{"id":"91909","title":"Vector Equilibrium","text":"I'm trying to understand more about this 'vector equilibrium' apparently coined by R.Buckminster Fuller. In this site (http:\/\/cosmometry.net\/), it claims: > \"The Vector Equilibrium, as its name describes, is the only geometric form > wherein all of the vectors are of equal length and angular relationship (60° > angles throughout). This includes both from its center point out to its > circumferential vertices, and the edges (vectors) connecting all of those > vertices.\" I'm wondering, how can this be the case, when even in the image on that site, it is clear that some sides of the shape are squares => 90 angles between the vectors connecting the vertices. Is this site incorrect then?"}
{"id":"107373","title":"Interaction energy between dipole and potential","text":"It is known that interaction energy = $-\\vec{p}.\\vec{E}$ where $\\vec{p}$ is dipole moment and $\\vec{E}$ is the electric field. I have to calculate the interaction energy of a system whose dipole moment and potential are the only available data. My assumption is since the negative gradient of potential is electric field, can we calculate the dipolar component of the potential as an alternative to electric field, i.e. $$\\begin{align} \\phi_{dx}&=\\phi_x(x-x_c) \\\\\\ \\phi_{dy}&=\\phi_y(y-y_c) \\\\\\ \\phi_{dz}&=\\phi_z(z-z_c) \\end{align}$$ where $x$, $y$, $z$ are the positions of atoms and $x_c$, $y_c$, $z_c$ is the geometrical centre, and then calculate the dot product with the dipole moment $(p_x,p_y,p_z)$. Does it sounds meaningful?? Kindly provide some suggestions in solving my problem."}
{"id":"67869","title":"How can contracting dimensions lead to cosmological inflation?","text":"Using the Kasner metric, given by $$ ds^2 = -dt^2 + \\sum_{j=1}^D t^{2p_j}(dx^j) $$ it is possible to not only describe the cosmological expansion of some space directions (the ones with positive Kasner exponents $p_j$, but this metric allows for some dimensions to contract too, those have negative $p_j$. The two Kasner conditions $$ \\sum_{j=1}^{D-1} p_j = 1 $$ and $$ \\sum_{j=1}^{D-1} (p_j)^2 = 1 $$ say that there have to be contracting and expanding dimensions at the same time, as the $p_j$ can not all have the same sign. In a comment I have read, that in models with for example 3 expanding and $n>1$ contracting dimennsions, the contracting dimensions _drive_ the inflation in the other directions by leading their expansion to accelerate without a cosmological constant. This is interesting and about this I'd like to learn some more. So can somebody a bit more explicitely explain how such inflation models work? For example what exactly would the vacuum energy from a physics point of view be in this case? Up to now I only heard about inflation models where the vacuum energy density is the potential energy of some inflaton field(s) in a little bit more detail."}
{"id":"107378","title":"What is fidelity in experimental QM?","text":"I often comer across the term high fidelity in QM papers. Does fidelity imply ratio of entangled photons \/ total photons? Is there some other metric to measure how good the source is?"}
{"id":"61422","title":"What would happen to the Moon if Earth is turned into a black hole?","text":"Assume that all of sudden the Earth is turned into a black hole. And the moon revolves around the Earth (before turning into a black hole). What would happen to the Moon after earth changes to black hole will it be sucked to the black hole or continue to revolve the black hole?"}
{"id":"22224","title":"How to find replacement function of a mass?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/8OAWc.png) I wonder how I can find the replacement function of the center of the blue mass? The center of mass of the blue mass is $(0,0)$ and the blue mass is homogeneous. The masses do not move at t=0 in the system (Their initial speeds are zero). There is no friction in the system and the red mass is a point mass. $u$ is the angle of the red mass with $y$ axis. I need help in learning the theory behind solving such systems. Thank you. EDITED: I would like to show what I did till now to solve the problem. I followed the hints that Manishearth offered $u=\\alpha _0$ The definations of the blue mass center replacement: $X_b(t)$ The replacement function that I try to fnd $Y_b(t)$ The definations of the red mass Center replacement: $X_r(t)$ $Y_r(t)$ 1- Mass of center of entire system is in the beginning: $$r_{0x}=\\frac{m. r. sin(\\alpha _0)}{m+M}$$ $$r_{0y}=-\\frac{m. r. cos(\\alpha _0)}{m+M}$$ Mass center will not depend on time. If So, $$mX_r(t)+MX_b(t)=(m+M)r_{0x}=m. r. sin(\\alpha _0)$$ (1) $$mY_r(t)+MY_b(t)=(m+M)r_{0y}=-m. r. cos(\\alpha _0)$$ (2) 2-Energy is conserved $$\\frac{m (\\dot X_r(t))^2 } {2}+ \\frac{m (\\dot Y_r(t))^2 } {2}+\\frac{M (\\dot X_b(t))^2 } {2}+\\frac{M (\\dot Y_b(t))^2 } {2} +mg(a+Y_r(t))+Mg(a+Y_b(t))=mg(a-r.cos(\\alpha _0))+Mga$$ (3) I dont know how to apply Hint 3 and Hint 4. Till now I have 4 unknown and 3 equations. Am I in correct way to solve $X_b(t)$ ? Thanks in advice"}
{"id":"22227","title":"Evidence that \"space exists rather than just particles\"","text":"What is (theoretical) evidence that you need to define all of space with properties rather than just stating where all particles are? I mean, does every single coordinate x, y, z have a meaning or would it be sufficient to keep track of particle positions and type. In other words, to specify reality, do you need all of $\\Psi(x,y,z,a)$ where $a$ is another dimension (charge, spin, etc.) or is there any framework where you just specify a _countable_ number or particles with coordinates $(x_{P_i}, y_{P_i}, z_{P_i})$ and you do not care about coordinates inbetween particles (obviously there would be distant interaction laws). Can you disprove sufficiency of the second representation?"}
{"id":"22222","title":"M-theory and many-worlds interpretation","text":"I am getting some confusion on whether M-theory accepts many-worlds interpretation. Can anyone show me the reasons or rebuttals for the possibility of the many-worlds interpretation in M-theory? Thanks."}
{"id":"73682","title":"Where the extra power comes from?","text":"Suppose we have two radio waveforms each has amplitude of 1, then the total power is 2. Suppose these two waveforms add up some where constructively, then the amplitude become 2, and the total power is 4, so where is the extra power comes from?"}
{"id":"73687","title":"How many watts of light do instruments in modern telescopes need?","text":"I just calculated that if the European extremely large telescope(978 m^2 area) would be pointed at the Andromeda galaxy once it is built, it will only get mere 1.2*10^-6 Watts of power from Andromeda. And I didn't account for atmospheric factors or the fact that the telescope is only going to read visible and infra red light. That is to me a really small number that we achieve even after using the highest aperture that would be available to us. Are my calculations correct(I have double checked but I still really doubt it.)? I used the relation between intensity, luminosity and distance. Please, I am just an amateur. If you guys find an embarrassing arithmetic mistake, let me know so. If my calculations are correct, do the scientific instruments on say, Hubble work with such a small amount of light? How many Watts of light does the Camera on these telescopes need to work properly?"}
{"id":"24686","title":"Mathematical question on Collisions","text":"> A `2.5`kg ball travelling with a speed of `7.5`m\/s makes an elastic > collision with another ball of `1.5`kg and travelling at the speed of > `3.0`m\/s in the same direction. > > \\-- What are the velocities of the balls immediately after collision?"}
{"id":"28158","title":"Does friction act in the same way in different directions?","text":"> A physics textbook is launched up a rough incline with a kinetic energy of > $X$ joules. When the book comes momentarily to rest near the top of the > incline, it has only gained ($X-20$) joules of gravitational potential > energy. About how much kinetic energy will it have when it returns to the > launch point? (Hint: Remember friction and conservation of energy). **Attempt** : So when the textbook goes up and reaches the point where it is at rest its GPE is $X$ joules. When it goes down and reaches the starting point its kinetic energy should by $X$ joules, I guess. Thats what I am not sure about. Does the friction act with the same force when the textbook goes down?"}
{"id":"98689","title":"Resources for QCD colour factors","text":"I'm looking for resources on the computation of (I think) very simple Feynman diagrams for QCD processes. The processes are q'qbar' -> qqbar and gg->qqbar (only at tree level, no loops involved) and my main problem is the computation of the colour factors. I'm not interested in the absolute value of the cross-section, but in the angular distribution (in terms of cos theta). At least for the gg->qqbar process I think I need the correct factors though, to correctly add the diagrams. I've already looked into the Quantum Field Theory book by Peskin, but there I only found the final cross-sections but not the calculation."}
{"id":"98684","title":"Can frequency be equal to 0?","text":"Is correct to speak about frequency equal to 0 ? $$f= \\frac{1}{t} $$ If $t\\rightarrow\\infty$ can I consider that the frequency is equal to 0 ?"}
{"id":"31888","title":"Flow of freezing liquid or a melting solid","text":"If a liquid is freezing, is equation of continuity violated? As the liquid flows, some portion of it is getting frozen. The mass of the fluid thus keeps dropping. Similarly, when a molten fluid flows over a solid, its mass may increase because the solid over which it flows also melts. In either of these flows, can we assume that div **u** = 0, where **u** is the velocity vector."}
{"id":"35501","title":"Kaluza Klein Particles and Mini Black Holes","text":"Are the Kaluza Klein Particles and Mini Black Holes associated with the ADD and Randal-Sundrum models different names for the same class of particles or are they distinct."}
{"id":"56708","title":"Should a 1D Guassian wave oscillate?","text":"I wrote a few lines that numerically solve Maxwell's equations. The result is a moving wave that looks like a single pulse. This looks strange to me because I expect waves to move in oscillator fashion, perhaps like cos(x). Why doesn't the wave appear to oscillate? How does this kind of Gaussian wave work? Can it be broken into solutions? Do similar solutions exist in 2D? Are they still Gaussian?               %%Super basic FDTD     clear all;     gsz=100;     ez=zeros(gsz);hy=zeros(gsz);     imp=377.0;     %     steps = 100;     for i=1:steps         for j=1:(gsz-1)             hy(j)=hy(j)+(ez(j+1)-ez(j))\/imp;         end         for j=2:gsz             ez(j)=ez(j)+(hy(j)-hy(j-1))*imp;         end         ez(gsz)= ez(gsz)+exp(-((i-30)^2) \/100);         subplot(2,1,1);         plot(ez);         subplot(2,1,2);         plot(hy);         M(i)=getframe;     end"}
{"id":"56700","title":"What properties are entangled in quantum entanglement?","text":"When two particles are entangled, one can measure the properties of one of the particles and instantaneously know the properties of the other. This is because the two particles possess the same properties when entangled, no matter how far apart. Which properties are entangled? Is it just fundamental properties, such as spin, or can it be instantaneous\/changing properties, such as the speed of the particles? It may seem very odd to suggest that speed could be entangled, but I think it's a reasonable question to ask as quantum entangles is plenty weird anyway."}
{"id":"56705","title":"Why doesn't a marble rolling on a table ever reflect back at the edge?","text":"In light of the fact that when encountering a potential step there is a non- zero probability of a particle reflecting back, I was just wondering why it is that a marble doesn't \"reflect\" back from the edge of a table, but rather rolls off. Is it because the potential difference is extremely insignificant when compared to the total energy of the marble (so not just kinetic, but also internal energy), so that the reflection coefficient essentially evaluates to 0?"}
{"id":"56707","title":"Atomic physics through classical resonance?","text":"I have a rather general question regarding the theory of Quantum Mechanics. To preface this question, consider a violin string. When a violinist exposes the string to a bow, this is exposing the string to a wide range of frequencies. In response to this excitation, the string resonates at rather distinct frequencies. In other words, the energy spectrum of a violin string can be understood as a linear combination of distinct energies(frequencies). Why would it not be possible to develop an accurate atomic physics model based on resonant orbital electronic frequencies including possibly spin-orbit resonances as well?"}
{"id":"95798","title":"Mass generation by Chern-Simons theory","text":"Why the mass generation via a Higgs mechanism is different from that of Chern- Simons theory? I haven't done any formal course in Quantum field theory,so how do I understand this just having some basic knowledge in Classical field theory?"}
{"id":"128521","title":"Why does Pluto's orbit cross Neptune's orbit?","text":"Why does Pluto's orbit cross Neptune's orbit? Is this a fault in Newton's law of gravitation?"}
{"id":"75906","title":"Realistic Ocean Shears","text":"I was wondering if anyone could point me to published data on the vertical structure of strong ocean currents, such as the Gulf Stream. I am primarily interested in the mean current profile as a function of depth (in the core of the current) near the surface, which go on to estimate vertical shears."}
{"id":"27339","title":"Tip of a spreading wave-packet: asymptotics beyond all orders of a saddle point expansion","text":"This is a technical question coming from mapping of an unrelated problem onto dynamics of a non-relativistic massive particle in 1+1 dimensions. This issue is with asymptotics dominated by a term beyond all orders of a saddle point expansion (singular terms of an asymptotic series), like in the problem of the lifetime of a bound state in 1+0 negative coupling $\\phi^4$ toy model. Consider a particle with an initial (normalized) wave-function $$\\psi_0(x) = e^{-(x+e^{-x})\/2}$$ This specific shape defines a natural unit for $x$, note the double-exponential asymptotics of $\\psi_0(x)$ as $ x \\to -\\infty$. Time evolution under the Hamiltonian $\\mathcal{H}=-\\frac{1}{2}\\partial_x^2 $ transforms the wave-function to (using the textbook propagator) $$\\psi(x,t) = (2 \\pi i t)^{-1\/2} \\int e^{i (x-x')^2\/(2t)} \\psi_0(x') d x'$$ **My question is about the asymptotics of this integral** , especially the leading front propagating to the left. Here is where I've hit the wall: The saddle point expansion in $t^{-1}$ gives $$t |\\psi(x,t)|^2 \\sim e^{-e^{-x}} \\left [1 + (e^{3x} -2 e^{2x}) \\, t^{-1} \/8 + O(t^{-2}) \\right ] $$ which converges nicely (checked numerically) for $x \\gtrsim 1$, but fails to capture the terms of order $e^{x\/t}$ that dominate over the double exponential at negtavie $x$. For $t \\to +\\infty$ the solution becomes symmetric, $$|\\psi(x,t \\to \\infty)|^2=\\frac{1}{t \\cosh (\\pi x\/t)}$$ Any ideas\/hints will be appreciated."}
{"id":"27335","title":"Matrix geometry for F-strings","text":"A stack of N D-branes has the strange property that the traverse D-brane coordinates are matrix-valued. When the matrices commute, they can be interpreted as ordinary coordinates for N indistinguishable objects. But in general they correspond to something classical geometry doesn't describe. In particular this leads to a non-perturbative description of string theory on asymptotically Minkowski spacetime: Matrix Theory. S-Duality exchanges F-strings and D1-strings. This means this odd \"matrix geometry\" should happen for F-strings as well. The question is, how can we see it directly, without invoking S-duality?"}
{"id":"75908","title":"Tension on pulleys Physics Question","text":"Collar A is connected to a 50lb load on a frictionless horizontal rod. Determine magnitude of P to maintain equilibrium when x=4.5. ![enter image description here](http:\/\/i.stack.imgur.com\/mylsw.png) Im confused on the concept of Tension I originally got the answer 11.25 lb like the images below but my solution manual is different. ![enter image description here](http:\/\/i.stack.imgur.com\/405Nx.png) The solution manual says: $$\\tan \\alpha= 20\/4.5 = 77.3 d^\\circ$$   sum of F sub X=0   $$-P + T \\cos 77.3 $$   $$P= 50lb (cos 77.3)$$   $$= 10.98lb$$ (answer manual) Is this because the tension in the rope is equivalent to the mass of the hanging weight? Is there a way to solve for 10.98lb using method like the one below ...i.e. adding the vector components and solving for the unknowns? confused especially on how the solution manual is using 50lb for magnitude of T and when using vector addition getting the 11 so lbs. Thanks! but this one below is what makes sense to me. http:\/\/imageshack.us\/photo\/my-images\/15\/yci9.png\/   http:\/\/imageshack.us\/photo\/my-images\/534\/y53e.png\/"}
{"id":"94416","title":"Why doesn't a typical beam splitter cause a photon to decohere?","text":"In many experiments in quantum mechanics, a single photon is sent to a mirror which it passes through or bounces off with 50% probability, then the same for some more similar mirrors, and at the end we get interference between the various paths. This is fairly easy to observe in the laboratory. The interference means there is no which-path information stored anywhere in the mirrors. The mirrors are made of 10^20-something atoms, they aren't necessarily ultra-pure crystals, and they're at room temperature. Nonetheless, they act on the photons as very simple unitary operators. Why is it that the mirrors retain no or very little trace of the photon's path, so that very little decoherence occurs? In general, how do I look at a physical situation and predict when there will be enough noisy interaction with the environment for a quantum state to decohere?"}
{"id":"74366","title":"No magnetic dipole moment for photon","text":"Electrically neutral particles such as neutrinos can have nonvanishing magnetic dipole moments. Spin-1 particles, e.g., deuterium nuclei, can also have dipole moments. Googling seems to show that the Z boson has a magnetic moment. So is there an elementary argument that explains why the photon has a zero magnetic moment? By analogy with the Bohr magneton and nuclear magneton, we might actually expect that zero mass would produce _infinite_ dipole moment. Altschul 2007 has some discussion of empirical bounds and difficulties in creating a theory in which the moment doesn't vanish, but I'm having trouble translating anything in the paper into an elementary argument for why it's so much easier to have the moment vanish. > Altschul, \"Astrophysical Bounds on the Photon Charge and Magnetic Moment.\" > _Astropart. Phys._ **29** no. 4, pp. 290–298 (2008), arXiv:0711.2038."}
{"id":"74363","title":"Why Weyl invariance is important for consistent string theory","text":"This post is related to this link. I know there is a Weyl invariance for the Polyakov action at least in classical level. My question arises from obtaining effective action in string theory, such as section 7.3 in this lecture note > A consistent background of string theory must preserve Weyl invariance, > which now requires $\\beta_{\\mu\\nu}(G)=\\beta_{\\mu\\nu}(B)=\\beta(\\Phi)=0 $ later the lecture note tried to find effective action by requiring vanishing beta function. Also in p110 in Polchinski's string theory vol I, > We have emphasized that Weyl invariance is essential to the consistency of > string theory. Why Weyl invariance is so important for consistency of string theory?"}
{"id":"74368","title":"Path of least resistance vs. short circuit","text":"Some sources on the web claim that \"electricity follows the path of least resistance\" is not true, e.g. this physics SE question. However, in every explanation of \"short circuits\", the author says that current flows through the short because it's following the path of least resistance. How do I reconcile these two facts?"}
{"id":"29338","title":"How to prove Galilean invariance?","text":"**Thinking this would be obvious, I was trying to prove the Galilean invariance of Newton's second law of motion, but I failed. This is what I've got so far:** If we define a world line in Galilean space-time $\\mathbb{R}^{4}$ as the following curve $$\\bar{w}\\colon I\\subset\\mathbb{R}\\to \\mathbb{R}^{4}\\colon t\\mapsto (t,\\bar{x}(t))$$ $$\\bar{x}\\colon I\\subset\\mathbb{R}\\to \\mathbb{R}^{3}\\colon t\\mapsto (x(t),y(t),z(t))$$ where $\\mathbb{R}^{3}\\subset\\mathbb{R}^{4}$ Euclidean, then the acceleration is given by $$\\bar{a}\\colon I\\subset\\mathbb{R}\\to \\mathbb{R}^{4}\\colon t\\mapsto \\frac{d^{2}\\bar{w}(t)}{dt^{2}}=(0,\\frac{d^{2}\\bar{x}(t)}{dt^{2}})\\equiv(0,\\tilde{a}(t))$$ where $\\tilde{a}$ the classical acceleration and the force field that causes the acceleration $$\\bar{F}\\colon\\mathbb{R}^{4}\\to\\mathbb{R}^{4}\\colon\\bar{w}(t)\\mapsto m\\bar{a}(t)=(0,m\\tilde{a}(t))$$ So if Newton's second law of motion is written as $\\bar{F}(\\bar{w}(t))=m\\bar{a}(t)$ then a Galilean transformation causes $\\bar{F}=G\\bar{F}'$ and $\\bar{a}=G\\bar{a}'$ since they both live in $\\mathbb{R}^{4}$. Therefore $$\\bar{F}(\\bar{w}(t))=m\\bar{a}(t)$$ $$\\Leftrightarrow G\\bar{F}'(\\bar{w}(t))=mG\\bar{a}'(t)$$ $$\\Leftrightarrow \\bar{F}'(\\bar{w}(t))=m\\bar{a}'(t)$$ which is what we needed to prove (if F=ma in the unprimed frame then F'=ma' in the primed frame). Note that this is analogue to how Lorentz invariance is shown in special relativity. **However $\\bar{a}$ doesn't transform like $\\bar{a}=G\\bar{a}'$** A general Galilean transformation $G$ in $\\mathbb{R}^{4}$ is given by $$t=t'+t_{t}$$ $$\\bar{x}=R\\bar{x}'+\\bar{u}t+\\bar{t}_{\\bar{x}}$$ The relation between velocity and acceleration before and after a Galilean transformation is given by $$\\bar{v}(t)=\\frac{d\\bar{w}(t)}{dt}=\\frac{d\\bar{w}(t)}{dt'}\\frac{dt'}{dt}=\\frac{d\\bar{w}(t)}{dt'}= (1,R\\frac{d\\bar{x}'}{dt'}+\\bar{u})$$ $$\\bar{a}(t)=(0,\\tilde{a}(t))=\\frac{d\\bar{v}(t)}{dt}=(0,R\\frac{d^{2}\\bar{x}'}{dt'^{2}})=(0,R\\tilde{a}'(t'))$$ $$\\Leftrightarrow\\tilde{a}(t)=R\\tilde{a}'(t')$$ This is not the same as $\\bar{a}=G\\bar{a}'$ because $$\\bar{a}(t)=G\\bar{a}'(t')$$ $$\\Leftrightarrow\\begin{pmatrix} 0 \\\\\\ \\tilde{a}(t) \\\\\\ 1\\end{pmatrix} = \\begin{pmatrix}1&0&t_{t}\\\\\\ \\bar{u}&R&\\bar{t}_{\\bar{x}}\\\\\\ 0&0&1 \\end{pmatrix}\\cdot\\begin{pmatrix} 0 \\\\\\ \\tilde{a}'(t') \\\\\\ 1\\end{pmatrix}$$ So the acceleration of a world line transforms not with $G$ but with the linear part of $G$, meaning that the translation part must be zero: $(t_{t},\\bar{t}_{\\bar{x}})=\\bar{0}$. Can someone help me out of this mess? **Edit:** Let me try again, this time forgetting that we're talking about forces and just consider a 3D vector field. Of course Galilean space-time is still 4-dimensional and a general Galilean transformation $G$ in $\\mathbb{R}^{4}$ is still given by $$t=t'+t_{t}$$ $$\\bar{x}=R\\bar{x}'+\\bar{u}t+\\bar{t}_{\\bar{x}}$$ and the relation between the classical acceleration in inertial frames (primed and unprimed) related by a Galilean transformation $G$ is still given by $$\\tilde{a}(t)=R\\tilde{a}'(t')$$ One could say that the acceleration transforms with $R$ if the frame transforms with $G$ because $\\tilde{a}$ lives in the associated vector space of Galilean space time (therefore the affine translation $(t_{t},\\bar{t}_{\\bar{x}})$ doesn't apply) and moreover lives in the 3D Euclidean subspace of this vector space $\\mathbb{R}^{3}\\subset\\mathbb{R}^{4}$ (therefore the boost $\\bar{u}$ doesn't apply). Suppose now that we define a 3D vector field on a world line as $$\\bar{F}\\colon C\\subset\\mathbb{R}^{4}\\to\\mathbb{R}^{3}\\colon\\bar{w}(t)\\mapsto m\\tilde{a}(t)$$ In this case, by the same reasoning as for the acceleration, we can say that the vector field (defined in the unprimed frame) transforms with $R$ if the frame transforms with $G$ $$\\bar{F}(\\bar{w}(t))=R\\bar{F}'(\\bar{w}'(t'))$$ If we use this together with $$\\bar{F}(\\bar{w}(t))=m\\tilde{a}(t)=mR\\tilde{a}'(t')$$ it follows that $$\\bar{F}'(\\bar{w}'(t'))=m\\tilde{a}'(t')$$ So if we define a vector field as $\\bar{F}(\\bar{w}(t))=m\\tilde{a}(t)$ (forget that we're talking about force) and since both sides live in the same space ($\\mathbb{R}^{3}$), then they also transform in the same way under a Galilean transformation (whatever this way is, in this case $R$). Therefore it doesn't matter in which frame we define the vector field on a world line as the acceleration multiplied by the mass, it will have the same form in all frames. Therefore we could say that the definition of the vector field is Galilean invariant. The problem I'm having is that one says that \"Newton's second law of motion is Galilean invariant\". This implies that it is invariant, regardless the nature of the force. So if we make the nature of the force abstract, we can just forget that we're talking about force and consider a 3D vector field. Then everything boils down to showing that the definition has the same form in all inertial frames, which it has as shown above. Is this a valid point of view?"}
{"id":"101908","title":"Problem evaluating $C^{-1}M^\\dagger C$","text":"How can I show the following? $$\\overline{\\psi_L}M^\\dagger (\\psi_L)^c=\\overline{\\psi_L}CM^\\dagger\\overline{\\psi_L}^T$$ where $\\psi^c=C\\overline{\\psi}^T$ and $C=i\\gamma^2\\gamma^0=-C^T=-C^\\dagger=-C^{-1}$. In particular, what is $C^{-1}M^\\dagger C$? I'm stuck at this point. If I can show, $$C^{-1}M^\\dagger C=M^\\dagger$$, the job is done. EDIT: http:\/\/arxiv.org\/abs\/hep-ph\/0001264 see equation 24 of this review"}
{"id":"101907","title":"Lagrangian approach to spinning thread reel","text":"I am trying to better understand Lagrangian dynamics and am struggling to complete the following question: > A reel of thread of mass $m$ and radius $r$ is allowed to unwind under > gravity, the upper end of the thread being fixed. Find the initial > acceleration of the reel.                                  ![Diagram of spool and thread](http:\/\/i.stack.imgur.com\/EWSJq.png) I believe there are three generalised co-ordinates here $(x,y,\\theta)$, as shown in the diagram and the constraint that $x=0$ (as there is no acceleration in the $x$-direction). We therefore have kinetic energy: $$T=\\frac{1}{2}m\\dot{y}^{2}+\\frac{1}{2}m r^{2}\\dot{\\theta}^{2}y \\cos(\\theta)$$ And potential energy given by: $$U=-mgy$$ We therefore can define the Lagrangian: $$\\mathcal{L}(y,\\theta,t)=\\frac{1}{2}m\\dot{y}^{2}+\\frac{1}{2}m r^{2}\\dot{\\theta}^{2}y \\cos(\\theta) + mgy$$ We therefore have the Euler-Lagrange equations: \\begin{align} \\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\frac{\\partial \\mathcal{L}}{\\partial \\dot{y}}\\right) &= \\frac{\\partial \\mathcal{L}}{\\partial y} \\\\\\ \\frac{\\mathrm{d}}{\\mathrm{d}t}\\left(\\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\theta}}\\right) &= \\frac{\\partial \\mathcal{L}}{\\partial \\theta} \\end{align} Expanding these we get: $$m\\ddot{y}=\\frac{1}{2}mr^{2}\\dot{\\theta}^{2}\\cos(\\theta) + mg$$ And: $$mr^{2}\\ddot{\\theta}y\\cos(\\theta)-mr^{2}\\dot{\\theta}^{2}\\sin(\\theta)+mr^{2}\\dot{\\theta}\\dot{y}\\cos(\\theta)=0$$ Solving both of these simultaneously gives: $$\\ddot{\\theta}=\\frac{\\dot{\\theta}^{2}\\tan(\\theta)-\\dot{y}\\dot{\\theta}}{y},\\quad \\ddot{y}=\\frac{1}{2}\\left(2g + r^{2}\\dot{\\theta}^{2}\\cos(\\theta)\\right)$$ Which I am unable to solve to yield anything useful; and therefore am assuming I am on the wrong track. I would be grateful for any pointers as to where I have misunderstood things. * * * Just to clarify what the issue is, given the expression above for $\\ddot{y}$ and the boundary conditions $y_{0}=0$, $\\dot{y}_{0}=0$, $\\theta=0$ and $\\dot{\\theta}=0$ we get: $$\\ddot{y}_{0}=g$$ Which is what I expected, however the answer to the question states that $\\ddot{y}_{0}=\\frac{2g}{3}$."}
{"id":"101902","title":"about orthogonal catastrophe","text":"I am reading Wen's book, QFT of many-body systems ( @Xiao-Gang Wen ). I am a little confused about the orthogonal catastrophe introduced in Chap.5. Below Eq.(5.1.6), it is stated that ``the influence of the core electron on the _conduction electron_ Green's function is not important...''. If I understand correctly, it is exactly the (almost) vanishing overlap between the initial and final state wavefunctions of the _conduction electrons_ that causes the orthogonal catastrophe. Why does the asumption that ``the influence of the core electron on the _conduction electron_ Green's function is not important'' works? Thanks in advance for any help."}
{"id":"134659","title":"Vector decomposition validity","text":"**Is force or field decomposition into component vectors always valid?** Lets say a constant electric field $\\vec{F}$ is acting in space such that it makes an angle $\\phi$ with respect to the horizontal direction. The component along x axis (horizontal direction) is $F\\cos(\\phi)$ and along the vertical $F\\sin(\\phi)$. Is there any assumption that the space is homogeneous or any other such (uniform space) condition while applying decomposition? OR is it like decomposition of vectors always refers to a single point in space, and doesn't matter upon the nature of medium\/space?"}
{"id":"133387","title":"What does a supernova look like at its peak luminosity?","text":"I know that in some types of supernovae, the cause of the increased luminosity is the radioactive decay of certain elements ejected during the explosion, so a question came to my mind. If the ejected material carrying the isotopes that decay to give the electromagnetic radiation is expelled at velocity of say 5% the speed of light, and given the fact that some supernovae stay extremely luminous for more than 4 weeks, then by that time the radioactive isotopes will have traveled more than 30 billion kilometers from the exploded star. So, does that mean a supernova at 4 weeks can be expected to look like a star with a radius of 30 billion kilometers and luminosity of $10^8$-$10^9$ times the solar luminosity ? Or I am getting the idea of the radioactive decay as the source of the supernova luminosity wrong ?"}
{"id":"96192","title":"Is it possible to create an audible sound source in mid air by intersecting ultrasonic sound beams?","text":"In the book _Daemon_ by _Daniel Suarez_ a technique gets described, that enables a device to create an acoustical illusion of a moving person inside a house. > Just then a voice called out clearly from the end of the hallway upstairs. > ‘Who’s there?’ […] No reply. But they heard walking again. The footsteps > came down the marble stairs to their right, some distance away from them. > They could clearly see the staircase, but no one was there. They could hear > the sound of a hand sliding down the metal railing. […] The footsteps were > moving across the floor to them now. […] Then, **in midair not five feet in > front of them** , a man’s voice shouted, ‘You don’t belong here!’ _(page > 94)_ As I searched the internet for more information on the mind-shaking technology described by Suarez, I just encountered papers (and videos) on the use of a single hypersonic sound beam, that creates audible sound where it hits a surface. This leads to an effect that most people describe as a 'god voice' similar to the experience of the use of bone conducting speakers. _However what I am searching for is an answer to the question, if it is possible to create sound in mid air (at a chosen position in 3D space) as the author is shaping the picture of the technology, cited below._ > A HyperSonic Sound system – or HSS – does not use physical speakers. HSS > pulsates quartz crystals at a frequency thousands of times faster than the > vibrations in a normal speaker – creating ultrasonic waves at frequencies > far beyond human hearing. Unlike lower-frequency sound, these waves travel > in a tight path – a beam. Two beams can be focused to intersect each other, > and where they interact they produce a third sonic wave whose frequency is > exactly the difference between the two original sounds. In HSS that > difference will fall within the range of human hearing – **and will appear > to come from thin air**. This is known as a **Tartini Tone** – in honor of > Giuseppe Tartini, the eighteenth-century Italian composer who first > discovered this principle. * * * PS: The reason why I even bother to search for information about a technology mentioned in a novel, is the author's concern, to just describe technology, that is in development in some lab and not placed in science fiction. He has a website for the description of some technology used in his book."}
{"id":"3554","title":"What's a good reference for this classical picture Feynman's talking about?","text":"I have a mathematics background but am trying to educate myself a little about physics. At the beginning of Feynman's QED book (not the popular one) is the following: > Suppose all of the atoms in the universe are in a box. Classically the box > may be treated as having natural modes describable in terms of a > distribution of harmonic oscillators with coupling between the oscillators > and matter. I guess this is something that physicists learn, but I have never heard of it. What is Feynman talking about and where can I learn more about it? The Wikipedia article on harmonic oscillators gives no indication that physicists do this."}
{"id":"38685","title":"Hamiltonian of oscillators quantized proof","text":"https:\/\/docs.google.com\/open?id=0BxrBcN1-BZWUOXNxR1l4S0l2MjQ http:\/\/www.2shared.com\/complete\/Qjy1_uzp\/Quantum_Mechanics_in_Simple_Ma.html (I uploaded a pdf file that contains the parts of the textbook that I am having some difficulty understanding. Seven pages.) (two links point to the same pdf.) A matrix $N$ has all entries except diagnoal ones zero, and first row, first column also zero, other diagonal entries depending on its column, row place; for example, second row, second column has one as its entry, third row, third column has two as its entry and so on. It is known that $H = \\hbar\\omega (N + \\frac{1}{2}) $. The question is, my textbook says that if there are a continuous range of possible values, then for some states, $\\langle (N - \\langle N \\rangle )^2 \\rangle \\leq [\\frac{1}{2} (n - \\langle N \\rangle)]^2$ , where $n = 0,1,2,3,...,$ then progresses to say that since $\\langle (N - \\langle N \\rangle )^2 \\rangle \\geq (n - \\langle N \\rangle)^2 (\\langle I_0 \\rangle + \\langle I_1\\rangle + ...)$ then $\\langle (N - \\langle N \\rangle )^2 \\rangle \\geq (n - \\langle N \\rangle)^2$. Then it says that this proves that there cannot be a continuous range of possible values, as $(0-\\langle N \\rangle)^2$, $(1 - \\langle N \\rangle)^2$ and so on cannot be smaller than $(n - \\langle N \\rangle)^2$. I am not sure how one gets $\\langle (N - \\langle N \\rangle )^2 \\rangle \\geq (n - \\langle N \\rangle)^2 (\\langle I_0 \\rangle + \\langle I_1\\rangle + ...)$. Can anyone show me the proof? Also, how does one get the part - $(0-\\langle N \\rangle)^2$, $(1 - \\langle N \\rangle)^2$ and so on cannot be smaller than $(n - \\langle N \\rangle)^2$?"}
{"id":"30452","title":"Continuum model of the n-body problem?","text":"Consider a collection of $n$ point masses evolving under the influence of gravity. For very large $n$, one might be interested in making a continuum approximation of this system (for cosmology, say). However, my naïve attempt to derive a continuum model directly from particle-particle interactions fails. In particular, consider a function $\\rho: \\Omega \\rightarrow \\mathbb{R}$ that describes the mass density over some domain $\\Omega$. To compute the gravitational force at a point $x \\in \\Omega$, I might try to simply integrate the force contributed by every other point $y$: $$ f(x) = \\int_\\Omega \\frac{\\rho(x)\\rho(y)}{|x-y|^2}\\frac{y-x}{|y-x|} $$ Even if we exclude the point $x$ (i.e., we integrate over $\\Omega \\backslash x$ instead of all of $\\Omega$) this integral diverges -- we effectively have particles arbitrarily close to $x$. So the question is: what's the right way to compute the gravitational force in this situation? It would also be nice to have some pointers to continuum models used in cosmology **_and their derivation from the particle model_** (many references simply start with a continuum model and do not explain its origin). By the way, I really only care about classical models -- let's forget about relativity, etc., for now. Thanks!"}
{"id":"87314","title":"Do higher frequency EM waves travel longer than lower EM frequency waves?","text":"In radio communication, the distance traveling seems to be an important parameter before we choose the frequency band of communication. There is also a path-loss model that describes a distance $\\leftrightarrow$ frequency dependence of the attenuation with an isotropic antenna. However, I fail to find any specific physical law that demonstrates the frequency dependence of the attenuation. Some even say that higher the frequency, higher the energy content. What are the factors for propagation distance?"}
{"id":"33817","title":"The appearance of volume $V$ in the Fourier series representation of a periodic cubic system","text":"In the textbook _Understanding Molecular Simulation_ by Frenkel and Smit (Second Edition), the authors represent a function $f(\\textbf{r})$ (which depends on the coordinates of a periodic system) as a Fourier series. I quote from page 295 of the text: > Let us consider a periodic system with a cubic box of length $L$ and volume > $V$. Any function $f(\\textbf{r})$ that depends on the coordinates of our > system can be represented by a Fourier series: > > $$f(\\textbf{r}) = \\frac{1}{V} \\sum_{\\boldsymbol{\\ell} = -\\infty}^{\\infty} > \\tilde{f}(\\textbf{k}) e^{i \\textbf{k} \\cdot \\textbf{r}} \\; \\; \\; \\; > \\textbf{(12.1.6)}$$ > > where $\\textbf{k} = \\frac{2\\pi}{L}\\boldsymbol{\\ell}$ with $\\boldsymbol{\\ell} > = (\\ell_x, \\ell_y, \\ell_z)$ are the lattice vectors in Fourier space. The > Fourier coefficients $\\tilde{f}(\\textbf{k})$ are calculated using > > $$\\tilde{f}(\\textbf{k}) = \\int_V d\\textbf{r} \\; f(\\textbf{r}) > e^{-i\\textbf{k} \\cdot \\textbf{r}} \\; \\; \\; \\; \\textbf{(12.1.7)}$$ Now, the authors use equation **(12.1.6)** to write the electric potential $\\phi(\\textbf{r})$ in Fourier space: $$\\phi(\\textbf{r}) = \\frac{1}{V} \\sum_{\\textbf{k}} \\tilde{\\phi}(\\textbf{k}) e^{i\\textbf{k} \\cdot \\textbf{r}}$$ The authors write: > In Fourier space, Poisson's equation has a much simpler form. We can write > for the Poisson equation: > > $$-\\nabla^2 \\phi(\\textbf{r}) = -\\nabla^2 \\left( \\frac{1}{V} > \\sum_{\\textbf{k}} \\tilde{\\phi}(\\textbf{k}) e^{i\\textbf{k} \\cdot \\textbf{r}} > \\right) = \\frac{1}{V} \\sum_{\\textbf{k}} k^2 \\tilde{\\phi}(\\textbf{k}) > e^{i\\textbf{k} \\cdot \\textbf{r}} \\; \\; \\; \\; \\textbf{(12.1.8)}$$ My question is, why is the $\\frac{1}{V}$ factor present in equations **(12.1.6)** and **(12.1.8)**? What is the significance of the $\\frac{1}{V}$ factor in $\\phi(\\textbf{r}) = \\frac{1}{V} \\sum_{\\textbf{k}} \\tilde{\\phi}(\\textbf{k}) e^{i\\textbf{k} \\cdot \\textbf{r}}$? In contrast, the article on Wikipedia does not include this prefactor. I realize that that article is dealing with the general case, whereas here we are considering a system with a cubic box of volume $V$. But shouldn't the units of $\\phi(\\textbf{r})$ be the same as those of $\\tilde{\\phi}(\\textbf{k})$? The $\\frac{1}{V}$ seems to preclude $\\phi(\\textbf{r})$ and $\\tilde{\\phi}(\\textbf{k})$ having the same units. Do you have any advice? Thanks."}
{"id":"76595","title":"What does imaginary number maps to physically?","text":"I am taking undergraduate quantum mechanics currently, and the concept of an imaginary number had always troubled me. I always feel that complex numbers are more of a mathematical convenience, but apparently this is not true, it has occurred in way too many of my classes, Circuits, Control Theory and now Quantum Mechanics, and it seems that I always understand the math, but fail to grasp the concept in terms of its physical mapping. Hence my question, what does imaginary number maps to physically? Any help would be much appreciated"}
{"id":"83303","title":"Why does aperture size affect depth of field in photography?","text":"Why do small and big holes (the aperture) in a lens change the depth of field in the image? (if you have big hole you have smaller depth of field). I wondered if I should ask this on photo.stackexchange.com, but then realize that I want optic based answer which I think will get here."}
{"id":"31419","title":"Higgs field questions","text":"The Wikipedia article about Higgs field poses some questions to me.   * The article says that the Higgs field is a \"nonthermal\" field, a field whose energy does not decrease as the universe expands, yet it claims that in the early universe Higgs field was stronger which led to inflation. How the field's strength could decrease then?   * If the Higgs field was stronger in early universe, why the article claims that elementary particles had no mass at early stages of the universe?   * The article claims that Higgs field is repulsive and is behind the inflation. Yet it does not interact with photons. Would not that make the photons not to follow the universe's expansion so that we would see distant stars not actually in the direction where they should be?   * The article claims that all electrons in neutron beta-decay are left-handed only for a stationary observer while for a moving observer this changes. Does it mean that one can measure his speed relative to the Higgs field by counting the right-handed electrons in neutron's decay?   * The article claims that in absence of Higgs field electrons would have no rest mass. But I cannot imagine a charged particle without rest mass because the electromagnetic field has potential energy. Does it mean that in absence of Higgs field electrons are chargeless?"}
{"id":"5776","title":"Observation of cosmological topological defects","text":"Are there any projects\/experiments running for the observation of topological defects (domain walls, monopoles, etc)? Are there any _past\/canceled_ or _future_ such projects?"}
{"id":"87531","title":"Trying to rhyme Peskin and Schroeder with Weinberg","text":"This is a follow up question of this one. In the Vol 1, Weinberg derives how a unitary operator $U(\\Lambda)$ acts on one-particle states, which is given by equation (2.5.2): \\begin{equation} U(\\Lambda)|p,\\sigma\\rangle = \\sum_{\\sigma'} C_{\\sigma' \\sigma}(\\Lambda,p)|\\Lambda p, \\sigma'\\rangle \\end{equation} Am I correct in thinking that Peskin and Schroeder use this result to derive the Lorentz transformation on a annihilation operator of the quantized Dirac field (given by equation (3.106)): \\begin{equation} U(\\Lambda) a^s_{\\mathbf{p}}U^{-1}(\\Lambda) = \\sqrt{\\frac{E_{\\Lambda\\mathbf{p}}}{E_{\\mathbf{p}}}} a_{\\Lambda \\mathbf{p}}^s \\end{equation} However, Peskin and Schroeder are _only_ considering Lorentz transformations that do not mix states with different spin indices! In other words, the first equation I wrote becomes: \\begin{equation} U(\\Lambda)|p,\\sigma\\rangle = C|\\Lambda p, \\sigma\\rangle \\end{equation} where $C$ is normalization constant which can be determined to be $C=1$. If this is not correct, can you please let me know how they (Peskin and Schroeder) have derived equation (3.109)? I hope this question is not too confusing. Edit: for you to see my reasoning. We can now write (for $C=1$): \\begin{equation} U(\\Lambda)|\\mathbf{p},s\\rangle = \\sqrt{2 E_{\\mathbf{p}}} U(\\Lambda)a^{s\\dagger}_\\mathbf{p}U^{-1}(\\Lambda)|0\\rangle=|\\Lambda \\mathbf{p},s\\rangle = \\sqrt{2 E_{\\Lambda \\mathbf{p}}} a^{s\\dagger}_{\\Lambda \\mathbf{p}}|0\\rangle \\end{equation} which implies: \\begin{equation} U(\\Lambda)a^{s\\dagger}_\\mathbf{p}U^{-1}(\\Lambda) = \\sqrt{\\frac{E_{\\Lambda\\mathbf{p}}}{E_{\\mathbf{p}}}} a^{s\\dagger}_{\\Lambda \\mathbf{p}} \\end{equation} Subsequently, using the unitarity of $U(\\Lambda)$, we can take the adjoint of the above equation: \\begin{equation} U(\\Lambda)a^{s}_\\mathbf{p}U^{-1}(\\Lambda) = \\sqrt{\\frac{E_{\\Lambda\\mathbf{p}}}{E_{\\mathbf{p}}}} a^{s}_{\\Lambda \\mathbf{p}} \\end{equation} which is equation (3.109)."}
{"id":"87539","title":"Some hints for special case of metric tensor in GR","text":"Let's have metric $$ ds^2 = dt^2 - dx^2 - dy^2 - dz^2 - 2f(t - z, x, y)(dt - dz)^2. $$ I need to prove that it is an exact solution for Einstein equations in vacuum for $\\partial_{x}^{2}f + \\partial_{y}^{2}f = 0$. The straightforward method is obvious, but does some method especially for this metric exist? Edit. As I see, first is replacing the variables as $$ u = (t - z), \\quad v = (t + z). $$"}
{"id":"15245","title":"Dirac paper quoted at Wikipedia","text":"From Wikipedia's http:\/\/en.wikipedia.org\/wiki\/Uncertainty_principle: > In 1936 Dirac offered a precise definition and derivation of the time-energy > uncertainty relation in a relativistic quantum theory of \"events\". The citation to the paper is missing. Can anyone provide the title of this paper and where can I find it? Thanks."}
{"id":"39229","title":"Do strong and weak interactions have classical force fields as their limits?","text":"Electromagnetic interaction has classical electromagnetism as its classical limit. Is it possible to similarly describe strong and weak interactions classically?"}
{"id":"12620","title":"Mixed state after measurement","text":"I'm looking at Section 2.4.1 of Nielsen and Chuang's _Quantum Computation and Quantum Information_ were they derive the density operator versions of the evolution and measurement postulates of quantum mechanics and something is bugging me. Let $$\\\\{(p(i), | x_i \\rangle) \\colon i=1,2,...,n\\\\}$$ be an ensemble and suppose you perform a measurement on this ensemble that results in outcome $m$. Then the post-measurement ensemble is $$\\\\{(p(i|m), | x'_i \\rangle) \\colon i=1,2,...,n\\\\}$$ where $p(i|m)$ is the probability that the state of the system was originally $| x_i \\rangle$ given that the measurement outcome was $m$. It seems natural enough to use $p(i|m)$ for the probabilities of the post-measurement ensemble. However, mathematically, I do not see why that should be the case. My question then is this: Given only the definition of what an ensemble is and the state vector version of the postulates of QM, is there a way to derive the rules to compute the post-measurement probabilities of an ensemble?"}
{"id":"12625","title":"How does Hubble's constant affect the Earth's orbit","text":"If Hubble's constant is $2.33 \\times 10^{-18} \\text{ s}^{-1}$ and the earth orbits the sun with average distance of 150 million kilometers; Does that mean the earth's orbital radius increases approximately $11\\text{ m}\/\\text{year}$? Does the earth's angular momentum change? If so, where does the torque come from? If the angular momentum doesn't change, does the earth's orbital velocity (length of a year) change? If so, where does the lost kinetic energy go? * * * Aside: the 11 meters per year figure comes from Hubble expansion of space the distance of the earth's orbital radius integrated over an entire year. $$(2.33 \\times 10^{-18}\\text{ s}^{-1}) (1.5 \\times 10^{11} \\text{ m}) (3.15 \\times 10^7 \\text{ s}\/\\text{year}) = 11 \\text{ m}\/\\text{year}$$"}
{"id":"21024","title":"Young's double-slit experiment with detectors","text":"_Related:Accuracy of various optical instruments_ In many books, it's written that knowing which slit a photon passes through (by placing a detector before the slit) in a Young's double-slit setup beforehand destroys the interference (producing two bright fringes). My question is, does this require there to be only one photon in the double slit experiment at a time? The one-photon interference is explained by there being \"half a photon\" coming from each slit, and these interfere. If we know which slit it comes through, then it becomes a full photon in one slit and it can't interfere with itself. But can _other_ photons interfere with it? I'm quite sure that light can be either a particle or a wave in such experiments, never both. Aside from this, how was the photon detected without absorbing it? If I keep a detector behind a slit, yes, it will detect all the photons, but it won't let any past it to the main apparatus. Unless the detector was something else.. UPDATE: A clearer version of this question:   1. If you know which slit a photon went through, its wavefunction is collapsed. One can say that it ceases to be wave like. This collapsed photon cannot interfere with itself   2. Can it interfere with others? It is, after all, not really a wave anymore (it is a wave, but collapsed to a point).   3. In a normal double slit experiment (beam of light, no doodads), are photons interfering with themselves, each other, or both? (See comment: Is Time Significant in the Double Slit Experiment)"}
{"id":"46616","title":"Parity of a decay","text":"If a particle of unknown intrinsic parity decays into 2 particles each with negative intrinsic parity, does that necessarily imply that the original particle also has negative parity?"}
{"id":"46613","title":"A foundational question about entanglement","text":"The recent Nature article http:\/\/www.nature.com\/news\/data-teleportation-the- quantum-space-race-1.11958 prompts me to ask this question, which is of the same tenor as that asked at the recent Perimeter Institute meeting attended by Zeilinger et al., i.e. \"Do the pairs somehow communicate though some still- unknown information channel?\" It seems to me that entanglement could be described as behaving as if in some sense the coordinate separation is described by an extra degree of freedom (dimension) in which either a) the two entangled components are co-located in the extra dimension, allowing instantaneous communication; or b) in the extra dimension, the speed of light is no longer a limitation, and so co-location is not a requirement. ... or some mixture of the two. This could all be rephrased in terms of the concepts of \"brane\" and \"bulk\", of course. Are there many other such \"unconventional\" interpretations of the nature of entanglement out there?"}
{"id":"135240","title":"What is the difference between a vortex ring and a circular line vortex?","text":"Can someone explain the difference between a vortex ring and a circular line vortex? Or are they the same?"}
{"id":"41863","title":"How do I find the initial velocity in this problem?","text":"> An X-ray tube gives electrons constant acceleration over a distance of > $20\\text{ cm}$. If their final speed is $2.0\\times 10^7\\text{ m\/s}$, what > are the electrons' acceleration? I know this equation is relevant: $$v_\\text{final}^2 = v_\\text{initial}^2 + 2a(x-x_\\text{initial})$$ Also I have that $x = .2\\text{ m}$ (after conversion from cm), and $v_\\text{final}$ is given. How do I find $v_\\text{initial}$? I need that so I can find acceleration."}
{"id":"81896","title":"Dipping a Dyson Ring below the event horizon","text":"The basic assumption about inescapability beyond the Event Horizon is that the necessary escape speed (orbital speed) would exceed speed of light, therefore no object can achieve it. Now, would it be possible to escape it by other means, say, by mechanical push against the gravity, through application of force without increase of speed? * * * First, short introduction: What is a Dyson Ring? (feel free to skip or skim this section if you know it): Dyson Ring is a hypothetical, artificial, ring-shaped structure surrounding a massive body (e.g. a star), rotating - essentially orbiting it - rotating at speed slightly higher or lower than orbital speed necessary to maintain orbital equilibrium - state of freefall near its surface. This way, its rotation creates centripetal force simulating gravity for inhabitants. It would create a habitat much larger than any planet, given some \"rim\" it could hold atmosphere, and essentially is a neat sci-fi alternative for a planet - similar conditions, vastly more surface. The one property of it is that tensile strength of its construction allows it to exist at distance from the central body (star) that is off from normal orbit matter moving at this speed would take. Most theoretized Dyson Rings move slightly faster than needed, stretched out, with the inner side inhabited, and the Sun in zenith for all of their surface. It is possible though (if not recommended due to buckling risks) to make one that moves slower, the star's gravity not fully overcome by rotation, compressive force applied to the construction. * * * Now - technological problems aside - of building a Dyson Ring (not habitable and much smaller; a scientific device of ~30km radius.) and bringing it close enough to a Black Hole. Let's imagine a Dyson Ring that can change its circumference (say, built of segments connected with expanding\/contracting actuators) - and, as effect, radius within certain range - specifically between outside and inside of Schwarzschild radius of a selected Black Hole. It is also supplied with power sources that allow to vastly increase its rotary speed. Now, with the actuators expanded the ring is placed around the black hole, outside the even horizon, rotating so fast that its segment orbit the black hole. Its angular velocity so high the linear velocity of its surface is near to speed of light. Normally the centrifugal force would tear it apart, but it remains in equilibrium with centripetal force of gravity of the black hole. Now, actuators contract. The angular velocity rises a little, with accordance of law of conservation of momentum, but since we're at relativistic speeds, the increase of angular speed is minimal; it's mostly the ring gaining some mass. Now, the ring is below the Event Horizon. Since at this altitude orbital speed would exceed the speed of light, any object there would quickly spiral into the black hole. Still, the ring does not, due to its own tensile strength preventing further decrease of its own radius. It moves slower than the orbital speed, but the mechanical strain overcomes the gravity. (note, while below the Event Horizon, we're still pretty far from the singularity, so the forces exerted are not yet extreme enough to cause collapse of matter, and the rapid rotation overcomes great most of the compressing gravitational force) Expand the actuators, increase the radius of the ring - and it is forced back outside the Event Horizon, able to broadcast its findings. Possible physically, or am I missing something?"}
{"id":"81891","title":"Intuitive understanding of the entropy equation","text":"In thermodynamics, entropy is defined as $ d S = \\dfrac{\\delta q_{\\rm }}{T}$. This definition guarantees that heat will transfer from hot to cold, which is the second law of thermodynamics. But, why do we denote entropy as$\\dfrac{\\delta q_{\\rm }}{T}$ other than $\\dfrac{\\delta q_{\\rm }}{T^2}$,$\\dfrac{\\delta q_{\\rm }}{e^T}$,or something else? Is there an intuitive explanation for this $\\dfrac{\\delta q_{\\rm }}{T}$?"}
{"id":"20774","title":"quantum optics of a polarizing beam splitter","text":"I would like to use the Heisenberg picture in quantum field theory to model a polarizing beam splitter. Is there an easy way for someone to show me how the field operators ($a^\\dagger_{input1}$, $a^\\dagger_{input2}$etc) transform across a polarizing beam-splitter?"}
{"id":"116907","title":"What's the relationship between glow and arc discharge?","text":"I'm trying to understand the differences between an arc and glow discharge. Most texts I have found on the Internet only describes the observed differences but doesn't really explain what happens on the atomic level in the two types of the discharge. So what I know:   * Glow discharge happens on higher voltage in the mA current range. It has recognizable structure (eg. the Faraday dark space and the Crookes dark space). It's faint and cool.   * Arc discharge happens on lower voltage, and it's in the Amp current range and higher. It runs from electrode to electrode without dark spaces. It's often blindingly bright and hot.  A characteristic of the normal glow discharge is that the voltage is stable, only the current changes, hence it's used as a voltage stabilizer. As you increase the current through the glow, it tends to cover more area of the cathode. When it covered all of it, the voltage starts to increase to supply more and more energetic electrons, this is the abnormal glow. **What happens at point`I` in the graph below that causes the voltage drop again?** ![enter image description here](http:\/\/i.stack.imgur.com\/6CYxA.png) Glow discharge is often demonstrated with discharge tubes. Countless of youtube videos show amateur experimenters pumping down their tubes, causing initially an arc discharge which turns into a glow discharge as the tube pumped down, **what causes this mode transition?** Whenever you turn on a compact fluorescent bulb it glows for a second then the discharge turns into an arc as the cathodes heat up and the bulb brightens up. I know hot electrodes emit more electrons, but **why more electrons cause transition into the arc mode?** I suppose it's due to the same reason for all cases. Can anybody help me put the things together, so I can get the point?"}
{"id":"128643","title":"Quantum mechanical proof of Conway's SPIN and TWIN axioms?","text":"I think I understand correlations of EPR-state and GHZ-state which deal with spin-1\/2 particles and (-1, 1) measured values. Conway's state is spin-1 particle state with (-1, 0, 1) measured values. Which quantum mechanical state vector correspond to Conway's state? And how to prove SPIN and TWIN axioms of Conway's free will theorem from quantum mechanics?"}
{"id":"89632","title":"What is limiting line in series spectrum?","text":"> The wavelength of first line in the Balmer Series is 'whatever(in nm)' . > Calculate the wavelength of the second line and the 'limiting line' in the > Balmer Series. I found this question in an ancient question paper in the library. Now, I have solved the first part by calculating the atomic number from the first relation and then applying it while calculating the wavelengths of the second line in the Balmer series which must mean the line after Balmer (which is paschen). **But for the second part 'what is limiting line', and that's exactly what my question is. Also since the given wavelength is in 'nanometer' what whould be the unit of energy if found.** I certainly don't need any solution for the above noted question( from the question paper), so this question cannot be declared as an 'homework question' due to some weird rules, (ofcourse they are meant for a mass good, but the tag name 'homework' hurts, anyway don't want that to be a topic of discussion here, but please consider it.)"}
{"id":"122429","title":"Microscopic definition\/expression of\/for the heat current","text":"Often I see the following microscopic definition\/expression of\/for a heat current due to an external field $$ {\\bf j}_Q = 2 \\int \\frac{\\text{d}{\\bf k}}{(2\\pi)^3} \\frac{\\hbar {\\bf k}}{m} (\\epsilon_{{\\bf k}} -\\mu) f(\\epsilon_{{\\bf k}}), $$ where $f$ is the solution the the appropriate Boltzmann equation. Note that the factor of two is there for the spin of electrons. I do not really understand where this expression comes from. Especially the factor $(\\epsilon_{{\\bf k}} -\\mu) $ bothers me. Does it tell us that only electrons near the Fermi level contribute to the heat current? Or does it have another meaning?"}
{"id":"98040","title":"Propagator of Chern-Simons Abelian gauge theory","text":"I need to compute the \"topologically massive photon\" propagator. I've started with : $$ \\mathcal{L}=-\\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu} + \\frac{\\mu}{4}\\epsilon^{\\mu\\nu\\lambda}A_\\mu\\partial_\\nu A_\\lambda $$ $$ =A_\\mu\\underbrace{[\\frac{1}{2}g^{\\mu\\lambda}\\partial^2+\\frac{\\mu}{2}\\epsilon^{\\mu\\nu\\lambda}\\partial_\\nu]}_{(\\Delta^{-1})^{\\mu\\lambda}}A_{\\lambda} $$ So how can I invert The under braced part which will yield the topologically massive photon propagator?"}
{"id":"44617","title":"Inner product of particle-anti-particle spinor components","text":"Suppose I have four-component spinors $\\Psi$ and $\\bar \\Psi$ satisfying the Dirac equation with $$\\Psi(\\vec x) = \\int \\frac{\\textrm{d}^3 p}{(2\\pi)^3} \\frac{1}{\\sqrt{2 E_{\\vec p}}} \\sum_{s = \\pm \\frac{1}{2}} \\left[ a^s_p u^s_p e^{i \\vec p \\cdot \\vec x} + \\tilde b^s_p v^s_p e^{-i \\vec p \\cdot \\vec x} \\right] $$ $$ \\bar \\Psi(\\vec x) = \\int \\frac{\\textrm{d}^3 p}{(2\\pi)^3} \\frac{1}{\\sqrt{2 E_{\\vec p}}} \\sum_{s = \\pm \\frac{1}{2}} \\left[ \\tilde a^s_p \\tilde u^s_p e^{-i \\vec p \\cdot \\vec x} + b^s_p \\tilde v^s_p e^{-i \\vec p \\cdot \\vec x} \\right] \\gamma^0$$ with these definitions: $$ \\tilde a \\equiv a^\\dagger \\quad ; \\quad \\gamma^0 = \\begin{pmatrix} 0 & 1_2 \\\\\\ 1_2 & 0 \\end{pmatrix} \\quad ; \\quad u^s_p = \\begin{pmatrix} \\sqrt{p \\cdot \\sigma} \\xi^s \\\\\\ \\sqrt{p \\cdot \\bar \\sigma} \\xi^s \\end{pmatrix} \\quad ; \\quad v^s_p = \\begin{pmatrix} \\sqrt{p \\cdot \\sigma} \\eta^s \\\\\\ -\\sqrt{p \\cdot \\bar \\sigma} \\eta^s \\end{pmatrix}$$ $$ \\sigma = (1_2, \\vec \\sigma) \\quad ; \\quad \\bar \\sigma = (1_2 , - \\vec \\sigma) \\quad ; \\quad p = (p_0 , \\vec p)$$ where $\\vec \\sigma$ is the usual vector of Pauli matrices and $\\xi, \\eta$ are two-component spinors. The lecturer then goes on to choose an appropriate basis for $\\xi$ $$ \\xi^1 = \\begin{pmatrix}1 \\\\\\ 0\\end{pmatrix} \\quad ; \\quad \\xi^2 = \\begin{pmatrix} 0 \\\\\\ 1\\end{pmatrix}$$ and similarly for $\\eta$ such that $\\tilde \\xi^r \\xi^s = \\delta^{rs}$ and $\\tilde \\eta^r \\eta^s = \\delta^{rs}$. This appears sensible and using this, we can compute various inner products of $u$ and $v$. In the examples discussed in the notes, mixed terms, that is, those containing products of $\\xi^r$ and $\\eta^s$ never occur (multiplied by 0, hence irrelevant) or cancel out. In an example assignment, however, we are meant to calculate $\\bar \\psi(-i \\gamma^i \\partial_i + m) \\psi$ which then leads me to terms such as $$ \\sum_{s,t} 2 m \\tilde a^t_p \\tilde b^s_{-p} (p_i \\sigma^i) \\tilde \\xi^t \\eta^s$$ which also cancel out in the end, but pose the question in the meantime how $\\tilde \\xi^t \\eta^s$ is actually defined\/can be calculated. The identities gained from an appropriate choice of basis for $\\eta$ and $\\xi$ ‘feel’ wrong here, since, after all, $\\xi$ comes from a particle-spinor and $\\eta$ comes from an anti-particle spinor. I take it my question could be rephrased as to whether $\\xi$ and $\\eta$ live in the same space or belong to two different spaces (which would make their inner product an even more thrilling exercise)."}
{"id":"7250","title":"Gravitational lensing or cloud refraction?","text":"**My current understanding of gravitational lensing follows** When a star or other massive body passes between us and another star, the phenomenon generally labeled 'gravitational lensing' occurs. The term 'gravitational lensing' refers to the bending of light rays due to the gravitational influence of the massive body. Since the bends concentrate light for the observer, they act like a magnifying lens. **OTOH** But since I was young I have pondered other explanations for such observations. One thought about what (else) might cause the observed effect is a cloud of gas around the massive body itself. The assumption being that any change in density between the thin gas of interstellar space and the thicker gas surrounding a massive object should form a refractive boundary. In the case of an active star, that gas & might be the heliosphere of the star, and for any massive object that is _not_ creating an heliosphere, it could be due to clouds of gas that the massive object is sucking in. I raised this idea with a physics teacher in high school & he assured me that any possible refraction from clouds of gas had been ruled out in observations of gravitational lensing. I never got to see papers on the subject, so I cannot confirm that (and am not convinced). My questions are:   1. Does anybody know of papers on gravitational lensing that discuss the refraction from clouds of gas around the lensing body?   2. How could an observer\/experimenter go about quantifying the likely refractive index of such a gas cloud."}
{"id":"53521","title":"What's the average position of oscillating particles in a box with periodic boundary conditions?","text":"Imagine an open box repeating itself in a way that a if a particle crossing one of the box boundary is \"teleported\" on the opposite boundary (typical periodic boundary position in 3D). Now put a hundred of oscillating particles inside. Each particle position is identified in a uniq fashion through time. **I want to get the average position of each of these particles.** **EDIT:** Thanks for your input. But I think I did not explain myself correctly. Please forgive me for that. That's why I removed most of my wordy description and created the following picture as a 1D reduction of my problem instead. ![1D simplification](http:\/\/i.stack.imgur.com\/McuWL.png) _**Both pictures:_**   * This is a 1D projection of the 3D problem.   * The axis is infinite.   * The `blue` line represents the _1D unit cell_ repeated by translation along the axis without overlapping.   * The `red` dot represents the position of one unique and identified particle.    * The `green` dots represents the equivalent particles to the `red` one obtained by translation. These are really equivalent to the `red` one: same nature, velocity, force field, etc.   * As an input, I only have particle positions inside of the _1D unit cell_ (the `blue` line). _**Top picture:_**   * The `red` particle is present in the unit cell for each time step.   * I take the particle position $x(t)$ and average it over $t$ to get $\\bar{x} = \\frac{1}{4}$ of the unit cell in this case (`black arrow`). _**Bottom picture:_**   * The `red` particle is **crossing** the boundary of the _unit cell_.   * However, $x(t)$ is only defined in the _unit cell_ , therefore in this case at:      * $t=0$, the position of the `green` particle (on the right) is taken into account ;     * $t=1$, the position of the `red` particle (on the left) is taken into account     * _and so on_ ;   * If the average $\\bar{x}$ is calculated without much care, it is false since this average is done between two different particles (`red arrow` instead of `black arrow`). I developed a small algorithm that tries to correct this assuming that $x(t+\\Delta t) - x(t) < \\frac{\\Delta x}{2}$, with $\\Delta x$, the size of the _unit cell_. However this algorithm is quite slow. I know I can improve the algorithm itself and my implementation of it. But I also know that this is a trivial (and inefficient) way to do it. I also hate to reinvent the wheel again and again... Do you know any smart implementation or algorithm able to calculate this kind of \"periodic\" average?"}
{"id":"59668","title":"FWHM in resonance amplitude square derivation","text":"Consider a linear harmonic oscillator subject to a periodic force: $$ \\ddot x + 2 \\beta \\dot x + \\omega _0 ^2x = f_0\\cos \\omega t$$ The solution tends to: $$A \\cos (\\omega t - \\delta)$$ where: $$A^2=\\dfrac{f _0 ^2}{(\\omega _0 ^2-\\omega ^2)^2+4\\beta ^2\\omega ^2}$$ The maximum of this function is reached when: $$\\omega ^2 = \\omega_0^2-2\\beta ^2$$ which is: $$A^2 _{max}=\\dfrac{f_0 ^2}{4\\beta ^2\\omega_0 ^2-4\\beta^4}\\approx \\dfrac{f_0 ^2}{4 \\beta ^2\\omega_0 ^2}$$ if $\\beta$ is small compared to $\\omega _0$. **I need to find theFWHM of $A^2$**. I put: $$\\dfrac {A_{max}^2}{2}=\\dfrac {f_0 ^2}{8 \\beta ^2\\omega_0 ^2}=^!\\dfrac{f _0 ^2}{(\\omega _0 ^2-\\omega ^2)^2+4\\beta ^2\\omega ^2}$$ $$\\iff (\\omega _0 ^2-\\omega ^2)^2+4\\beta ^2\\omega ^2=8\\beta^2\\omega_0^2$$ My book gives the hint \"Is it ok to consider $\\omega + \\omega _0 \\approx 2\\omega _0$ or $\\omega \\approx \\omega _0$, it is _not_ ok to consider $\\omega - \\omega _0 \\approx 0$\". With this hint it's not hard to find the solution: $$( \\omega - \\omega _0)^2(\\omega + \\omega_0)^2+4\\beta^2\\omega^2\\approx 4\\omega _0 ^2[(\\omega-\\omega_0)^2+\\beta^2]$$ $$\\therefore (\\omega-\\omega_0)^2\\approx\\beta^2\\iff\\omega=\\omega_0 \\pm\\beta $$ solved. Now my question is: it's pretty clear that the approximation $\\omega -\\omega _0 \\approx 0$ wouldn't work. It is also clear to me that it is not contradictory to assume $\\omega \\approx \\omega _0$ , for we use this approximation for the other term $\\beta ^2 \\omega ^2 $. _Is there a more precise way to see this (i.e. that the approximation $\\Delta \\omega \\approx 0 $ is not correct)??_"}
{"id":"129066","title":"relating spinor and fundamental representation for $E_8$","text":"While proving a very important relation which is satisfied both by $SO(32)$ AND $E_8$, which makes it possible to factorize the anomaly into two parts. The relation is $Tr(F^6)=\\frac{1}{48}TrF^2TrF^4-\\frac{1}{14400}(TrF^2)^3$, where trace is in adjoint representation. I am able to prove this relation but while doing so, I have some identities which relates the spinor representation $128$ of $SO(16)$ to fundamental representation of $SO(16)$ which I must show but this is not working out. The simplest one being $TrF^2=16trF^2$, where $Tr$ is in spinor representation $128$ and $tr$ is in fundamental representation. There are other relations showing the equality between $TrF^4$ and $tr(F^2)^2$ and $trF^4$. I am aware that spinor representation would be $\\sigma_{ij}$ which is $128$ dimensional. While trying to prove these identities, I have noticed that if $F^2$ in the fundamental representation is diagonal with only two elements -1 and -1 and if $\\sigma_{ij}^2=\\frac{-I}{4}$ where $I$ is $128$ dimensional identity matrix then I can get the result. But I can not convince myself why it should be true. Any details would be appreciated of how to prove it. The identities can be found in GSW chapter 13 last section (VOL.2)."}
{"id":"29447","title":"Lenses (refractor) or mirrors (reflector) telescope?","text":"What differentiates, in terms of practical quality, not technical implementation, a refractor from a reflector telescope? Why would one prefer a refractor over a reflector, when reflectors come with such large diameters at a smaller price?"}
{"id":"110895","title":"Does water have surface tension in a vacuum?","text":"I could be totally wrong here but I was thinking about water surface and what creates that. My thought is it is the thin mixture of water and air separating the two. This mixture creates the boundary between water and air that has the property of surface tension. So does the surface of water in a vacuum act in the same way? Is there a surface with surface tension when there is no air to make the mixture?"}
{"id":"94992","title":"Spatial light modulation for mode multiplexing, how does it work?","text":"If a laser light becomes spatial phase modulated (by phase only spatial light modulator) what happens to the intensity distribution and frequency spectrum of the laser light ? I know that this technology is used to perform mode multiplexing. But how affects the spatial phase modification the mode field distribution (e.g. transform a SMF mode to a MMF mode)? I suppose the Fourier Transformation can help here? But how exactly ?"}
{"id":"119258","title":"Interpretation of decay associated difference spetrum","text":"We measured transient absorption spectra and we got $\\Delta A(\\lambda,t)=A(\\lambda,t)-A(\\lambda,0)$. If we use DAS (decay associated spectra) for interpretation, we assume that our data have this dependance: $\\sum_{i}A_i(\\lambda)e^{-t\/\\tau_i}$, where $\\tau_i$ is a lifetime of a component $i$. Decay associated spectra are then $A_i(\\lambda)$. If we focus on one of the $i$ spectra ($A(\\lambda)$), is there any easy way how to interpretate it? I am confused with mixing decacy associated spectra together with transient absorption."}
{"id":"119252","title":"Speed of light in different parts of the universe","text":"Do we have enough evidence to claim that speed of light remains the same at 'c' even at the most distant parts of the visible universe?"}
{"id":"32041","title":"Why is the second order perturbative correction to the ground state energy always down?","text":"What is the physical\/deeper reason for the second order shift of the ground state energy in time independent perturbation theory to be always down? I know that it follows from the formula quite straightforwardly, but I could not find a deeper reason for it even if if I took up examples like the linear Stark effect. Would you please tell me why?"}
{"id":"129682","title":"Flaw in the radius of a spherical electron model (classical electron radius)","text":"My textbook asks me to derive an equation for the potential energy ($U$) of sphere ($r_0$) filled with an electric charge of uniform density ($\\rho$), expressed in terms of the total charge $Q$. The equation can be derived by treating the potential energy ($U$) as the total work it would require to build the sphere and computing an integral [which came from the standard charge through a sphere formula (if you feel the need for a diagram or intermediate mathematical steps let me know)]: ($Work$) = $\\int_0^r ((4\\pi\\rho)^2r^4\\cfrac13)~dr$ which means that: ($Q$) = $4\\pi\\cfrac13r_0^3\\rho$ , and from the relationship between potential energy and Work, we obtain: $$U = \\frac{3Q^2}{5r_0}$$ The next question (based off of the equation we just derived) asks us to set our answer equal to $mc^2$, and attempt to calculate the proper radius of an electron (the value we receive is, I now know is the \"classical radius\" of the electron, or the \"Lorentz radius\" $\\approx$($2.8 *10^{-15}\\;\\rm m$). Finally we arrive at our problem : It asks us to explain why this theory does not adequately analyze the proper radius of an electron, or to find out what the flaw in this theory is. I really do not know, I wouldn't have used this method in the first place. Any thoughts here? Is looking at an electron as a sphere inncorrect? I actually attempted to research this matter on my own, and could not ascertain the reason that the classical electron's radius is a poor theory, only that it does not work well in quantum mechanics."}
{"id":"129689","title":"Could symmetric molecular stretching cause a physical object to become amorphous?","text":"I understand very little of this, but my google-foo has yielded me next to nothing, I thought I might then just ask. My real question here is if it might be possible through the use of vibration to cause a solid object (rock) to become amorphous\/polymorphous by causing the bonds between particles within a molecule to weaken. If the bonds that made up each molecule were weakened and the particles that it were made up of were somehow dissipated or transferred temporarily, it makes sense to me that the object might become polymorphic and manipulatable with ease. Possible? Yes\/no would be helpful and then how\/why. Thanks in advance! _Edited to ask a single more specific question_"}
{"id":"9876","title":"How can a point-like particle \"feel\" gravity, if locally the curvature of spacetime is always flat?","text":"I imagine a point-like particle can only experience the local properties of spacetime. But locally there is no curvature and no gravity, as it is often stated that > _Locally, as expressed in the equivalence principle, spacetime is > Minkowskian, and the laws of physics exhibit local Lorentz invariance._ > (Wikipedia:General relativity) But if this is the case how does a point-like particle know, which direction it should follow in a gravitational field? Can this be a seen as a hint that point-like particles cannot be used as underlying concept for a field theory like general relativity?"}
{"id":"9874","title":"Video documentaries about physics?","text":"Does anyone know about good video documentaries about physics, possibly on YouTube? I've seen a couple of them about string theory, but I could not find many others."}
{"id":"11158","title":"Is there a four dimensional form of Born's Rule -redub","text":"Generalizing Born's Rule for 4-dimensions $x_4$, write $$\\langle a\\rangle = \\int\\Psi A\\Psi^* \\mathrm{d}x_4$$   1. Is this consistent with quantum mechanics?    2. Is this a generalized form of the Born's Rule for the phase space of the wave function? @DavidZas I was thinking of a four dimensional space where Time is not a parameter. There been recent work on Time Reversed light pulses. http:\/\/prl.aps.org\/abstract\/PRL\/v106\/i19\/e193902 and the PHYSORG description http:\/\/www.physorg.com\/news\/2011-05-physicists-time-reversed-pulses.html And this leads me to wonder about abstract phase space, and I mean phase both in the epoch angle of a wave function and the phase space of a Hamiltonian (which yes, is very confusing). The way I see it, the wave function implictly has the Action of phase space built into it. $\\psi = e^{i(\\omega t - k x)} = e^{\\frac{i}{\\hbar }(\\mathbb{E} t - \\mathbf{p} x)}$ Where the phase and the action are connected by $\\frac{\\mathbb{E} t - \\mathbf{p} x}{\\hbar }$ Imagine dividing the phases by the action of the particle It's a bit like there's a blob (note the very precise language) of phase space wobbling about like the liquid drop model of the nuclear physics. Only this is a more abstract phase space that's oscillating. So, again, would Born's Rule apply to such a system? Actually after rereading what I just written, I'll be surprised if anyone replies this time! :)"}
{"id":"78978","title":"Where can I get latest full updated list of supernovae candidates from the SDSS-II Supernova Survey?","text":"I am doing a project where I need the list of supernovae candidates from the Sloan Digital Sky Survey II (SDSS-II) supernova survey. **One list** is available as part of the data-release-7 (DR7) website. The list contains 902 confirmed candidates. On the other hand, **there is a list** on another website for the supernova- survey project, where the list is supposedly confirmed _and_ updated. This list has 1030 candidates. Not only is is the number of candidates on each list different, but each list has candidates that are not on the other list. **The question at first, then, was which list is correct one?** There is no information on the SDSS DR7 website on when the list was last updated, but on the main site for the external project site, the footer says that the page was last modified in 2009, which is after DR7. I am not sure if this date is the same for the page with the actual list, though. When I inspected the overlapping entries (those sharing the same internal SDSS id), I noticed that the larger list at least for the ones that I looked at, in some places had information in a field that was blank in the smaller list. This was usually the value of the redshift for the given supernova candidate. I took this as an indication that the larger list, i.e. the one that is not on sdss.org, is actually the newer of the two lists. **But what about the confirmed candidates that are still listed in the older (smaller) list? Are they no longer considered supernovae candidates?** I do not want to have to guess which list is better one to use, and I am not sure that merging the candidates from the older confirmed list that are not on the newer into the newer list, since there must be a reason why they are not included in the latter in the first place. But this rationale for this decision is not given alongside the list or anywhere else on the project webpage that I looked. _What I really want is the latest complete and updated list of supernovae candidates available._ **So the question is, where do I obtain such this list? And when was it made?**"}
{"id":"28314","title":"How should I be thinking about tides?","text":"I am working on a project for physics that involves tides. This is my current mind set when thinking about tides: The earth applies a gravitational force on some mass `M`. The moon & sun apply a gravitational forces on the mass `M` away from the earth. The bodies of water on the earth have some acceleration towards the moon & sun due to their forces (F=ma). The bodies of water accelerate till they reach a sort of equilibrium between forces. I would like you to poke holes in my understanding of tides and try and answer my naive questions bellow. How would you describe tides effected by the moon using forces? How can that be used to calculate the height of tides?"}
{"id":"55411","title":"Speed of the Moon","text":"Why the motion of the Moon looks very slow in the sky? Doesn't it need the high speed in order to escape the earth's gravity?"}
{"id":"126852","title":"Boltzmann Distribution of Electrons in Confining Potential","text":"I have a particle simulation wherein many non-interacting electrons are trapped in a electric potential well. I would expect, and therefore I initialize according to this, that the electrons would take the shape of a Boltzmann distribution. However, when the simulation is allowed to run for a long time, the charge density converges to a slightly different shape: ![enter image description here](http:\/\/i.stack.imgur.com\/kAjFj.jpg) In this chart, the horizontal axis denotes position. The red line is the negative of the confining potential. I flipped it just for readability. This is on one vertical axis. On the other vertical axis is both the charge density as the system is initialized (green), and the charge density that the system settles into (blue). The colors are somewhat hard to see, so to clarify, the initialized (green) state is the one that is more negative in the region from around z=20cm to z=60cm Any ideas as to why the Boltzmann distribution is not the correct initialization? UPDATE: This is the graph with the electron-electron interaction turned on. It has pretty much settled into equilibrium by now ![enter image description here](http:\/\/i.stack.imgur.com\/6icd6.jpg)"}
{"id":"127954","title":"Calculate water flow rate through orifice","text":"I'm not very good with fluid physics, and need some help. Imagine the following setup with water contained in-front of a wall with an opening on the bottom: ![Fluid image](http:\/\/i.stack.imgur.com\/GjPyZ.jpg) How do I calculate the water flow $Q$?. I have made some re-search and found I need to (partially) calculate the pressure across the opening (orifice). But I don't know the pressure on the back side of the orifice. Can this be solved in any way? **Note:** I'm not saying _\"please give me the solution, I'm lazy\"_. I want to figure it out myself. But since, in this case, I only found formulas involving calculating pressure drop, I canno't use them to solve the problem. Therefore I'm turning my face to you, to see if there's another way to solve this problem. **Update:** The \"tank\" holding the water is actually `a big lake`, and the opening is how much the water gate have opened. I need to very precisely calculate how much water flows through the opening."}
{"id":"127109","title":"Geophysics with earth science","text":"I am a communications engineering student. I am interested in doing a project on geophysics. The concept is for a hole-digging robot with a camera and sensors and wireless transmitter that is made of high mp alloys. It is intended to make a hole down to 3000 km and perform a sample-return for any interesting things that could be found at that depth. This could help scientists learn more about the interior of Earth. Is this something that could be done?"}
{"id":"65389","title":"Categorizing solutions to Hierarchy problem","text":"We know that no gauge symmetry can prevent a term $m_\\phi^2|\\phi|^2$ for a scalar field, and that, given the quadratic loop corrections, the natural scale is $m_\\phi \\sim M_P$. This is related to the hierarchy problem. Now, I was thinking, is it fair to categorize the solutions to the hierachy problem into two sets:   1. **Solutions which _transfer_ a symmetry to the scalar:**     * Supersymmetry, in which $\\phi$ is protected via a symmetry transferred from a fermion.     * Little Higgs, in which $\\phi$ is a Goldstone-boson protected via a shift symmetry linearly realized from the original scalar that precipitated SSB.     * Composite Higgs, in which the fundamental degrees of freedom at high energies are not scalar, but fermionic, and thus $m_\\phi$ is protected.   2. **Solutions which lower the true $M_P$**. i.e. the true $M_P \\sim M_W$, but on the brane $M_P^\\textrm{eff}\\sim 10^{19}$ GeV, e.g. ADD and extra-dimensions. Is this division of solutions accurate? Are there any solutions that don't fall into these categories? And are there any other ways to transfer symmetries to scalars?"}
{"id":"119626","title":"Particle accelerators by focused laser beam?","text":"My (layman) estimate is that such accelerators could relatively easily and cheap override the current high-end, multimilliard-$ accelerators. Some years ago I've read some interesting breakthrough in the news, but practically nothing since that. What is the current focus of the research\/development? Is there any obstacle found? What is the current estimate to the practical energy limit of such accelerators?"}
{"id":"122128","title":"Proof the superstring action is Weyl invariant","text":"The superstring action is: $$ S = k\\int \\mathrm d\\sigma \\sqrt{-h} \\left [ h^{\\alpha \\beta} \\partial_\\alpha X^\\mu \\partial_\\beta X_\\mu + 2i {\\bar{\\psi}} ^\\mu \\rho^\\alpha \\partial_\\alpha \\psi_\\mu - i {\\bar \\chi}_\\alpha \\rho^\\beta \\rho^\\alpha \\psi^\\mu \\left (\\partial_\\beta X_\\mu -\\frac{i}{4}{\\bar \\chi}_\\beta \\psi_\\mu \\right)\\right ]$$ This is Weyl invariant, but the symmetry isn't manifest. All the books and lecture notes I've seen just state this outright, without proving it. Is there a reference where I can see the proof done in detail? Even when they give the action in the superconformal gauge ($ h_{\\alpha \\beta} = \\eta_{\\alpha \\beta}$ and $\\chi_\\alpha = 0$), they don't bother to prove it."}
{"id":"51869","title":"Resistance of Light Dependent Resistor (LDR)","text":"Is there a mathematical expression relating the resistance of Light Dependent Resistor (LDR) with light intensity?"}
{"id":"105068","title":"How does a ducted fan suck in air?","text":"I am thinking of a simple ducted fan with a single set of rotating blades, no stator vanes behind. If the effect of the fan is to raise the pressure of the air, how is there flow into the fan against this pressure gradient? I understand that the rotating fan blades have the effect of increasing the stagnation enthalpy of the flow. For an incompressible flow, this is equivalent to the stagnation pressure increasing via the fan blades imparting kinetic energy into the flow. I know that static pressure and velocity increase across the fan, so how can there be flow against this pressure gradient? I think the explanation will have something to do with the different frames of reference chosen to analyse the problem."}
{"id":"105063","title":"Where and how exactly does string theory and Q.E.D. use zeta function regularization?","text":"In the video they mention it being used in many fields of physics inclusing String and QED theory. https:\/\/www.youtube.com\/watch?v=w-I6XTVZXww But I remember reading somewhere that 1+2+3..=-1\/12 is obviously a \"mathematical trick\" (something about stupidly equating incompatible sets), and if so, how does this turn out to be true for things that are real (like QED)?"}
{"id":"102590","title":"Spring Damper System for a Vibrating Motor","text":"Good day people of SE I have a friend that has a final year project and is stuck. He has a motor with a small weight at the end of the shaft that causes vibrations. This motor is on a thin plate. Beneath the plate, in each corner, is a spring and damper. He has to formulate a model that describes the power absorbed by the spring damper system. The aim is to minimize or negate the vibrations to the platform under the spring-damper system. The weight, spring and damper can be adjusted. The level of the student is about 3rd year engineering. The standard of the mechanical engineering department is quite low compared to the other engineering practices, so a purely mathematical solution will be sufficient. I have no information regarding the position of the weight on the shaft. I assume it will be a rectangular weight positioned slightly to one side to allow some serious vibrations, but this is a guess. Where do you suggest we start researching this? Thank you EDIT : Thank you Tobias for the comment. I changed the question as instructed."}
{"id":"107519","title":"When a wire is stretched by a load, is there any change in its volume?","text":"When a wire, or a rod(whose diameter is not negligible) is subjected to a tensile stress, is there any change in its volume? If yes, does the volume increase or decrease? Take into consideration both the longitudinal elongation and the lateral contraction. If no, then what is wrong with the following derivation? $V=\\pi d^2 l\/4$$\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ (V$ is the volume of the wire) $\\implies\\Delta V\/V=2\\Delta d\/d +\\Delta l\/l$ $\\implies\\Delta V\/V=(1-2\\sigma)\\ \\Delta l\/l$ $\\ \\ \\ \\ \\ \\ \\ (\\Delta d\/d=-\\sigma \\Delta l\/l)$ where $\\sigma$ is Poisson's ratio."}
{"id":"57951","title":"Quantum Mechanics: Show that the expectation value of angular momentum does not change with time","text":"The potential is given by $V\\left(\\left\\|(x,y,z)\\right\\|\\right)$, so $[\\hat{L}, \\hat{H}] = 0$. Using the definition of $\\langle \\hat{L} \\rangle$ and the time-dependent Schrödinger equation, show that the expectation value of angular momentum does not change with time. Expectation value of angular momentum, $\\langle \\hat{L} \\rangle$ = $\\langle \\psi \\mid \\hat{L} \\mid \\psi \\rangle $ where $\\psi$ is a generic wavefunction."}
{"id":"9099","title":"The chiral ring of N=2 WZW models","text":"Does anyone knows if the chiral ring of the Kazama-Suzuki models have been found? If yes, please send me the reference. Thanks"}
{"id":"63401","title":"QED photon propagator to one-loop order gets different answers","text":"I'm a self-studying 14-year-old who has a passion for particle physics. I'm currently trying to calculate the QED photon propagator to one loop. However, in all the places I've looked, even with the same renormalization scheme, the results are different. I don't get it. Also, I need a step-by-step tutorial on how to calculate that trace in the numerator, specifically, the one with those gamma matrices."}
{"id":"68156","title":"Why does it take so long to make a nuclear bomb?","text":"So as I know nuclear bombs are derived from fission reactions: By providing the nucleus with enough power to trigger a chain reaction. If uranium was present why does it take so much to make a nuclear bomb."}
{"id":"98801","title":"Is it possible to clearly receive weak signal from space?","text":"Imagine there is communication satellite on geostationary orbite. It transmits signal in a defocused beam, \"illuminating\" whole Earth hemisphere it's floating over. For example, signal is amplitude-modulated with 10GHz carrier frequency. What if satellite transmitter power is very, very low, like 25mW, is it possible to receive this signal on Earth with temporal resolution high enough for high-speed data transfers by just using sufficiently large or huge satellite antenna dish; or will noise get amplified too and this approach will be fruitless? What if two identical satellites are transferring signals of their own, and floating close enough (say, 1° away) to each other to be visible at once from surface, is this possible to receive separate signals from both satellites, or interfence will cause total mess? Are there ways to overcome this?"}
{"id":"2667","title":"Can we measure the \"Tilt-A-Whirl\" effect of the Earth+Sun orbiting the galactic hub?","text":"You've probably ridden on the fairground ride called the \"Tilt-A-Whirl\", or-- as Disneyland calls it--the \"Spinning Teacups\", as well as other fairground rides that employ epicycles. You can really feel the centrifugal force strongly when your car spins complementary to the spin of the main rotor. Now, can that extra force be measured on Earth as our orbit around the Sun complements the Sun's orbit around the galactic hub? Has anyone done it?"}
{"id":"108777","title":"Transformers rate equation","text":"Going back to transformers after some years I realize they hide a possible missconception I'm not sure I really understand. It is not easy to find a full derivation of the well known relation $\\frac{\\Delta V_2}{\\Delta V_1}\\\\!=\\\\!\\frac{n_2}{n_1}$, as if this question were being circumvented, and that's why in an aim to give an intuitive idea of it I fall in a nonsense. ![Typical transformer arrangements...](http:\/\/i.stack.imgur.com\/UfNmA.png) I start thinking in the secondary winding: attending to Maxwell-Faraday-Lenz's Law, $\\varepsilon=\\Delta V_2=-\\frac{d\\phi}{dt}$, and given a certain primary winding signal frequency, the higher the maximum flux $\\phi$ is, the higher the induced e.m.f. $\\varepsilon$ will be. Then we want a secondary coil plenty of loops (high $n_2$), as this increases its effective surface, and it is _pierced_ by the same field $\\vec{B}$ lines $n_1$ times. Ok, so $\\Delta V_2\\varpropto n_2$. On the other hand, to increase that flux, we need a strong field due to primary solenoid. And this, **for a given intensity $I_1$** (let $R_1$ be $n_i$-independent), is fulfilled if we have high $n_2$. Then we would say that, at the end, $\\Delta V_2\\varpropto n_1 n_2 \\Delta V_1$, which is not true, and which would produce some symmetry problems. What physics am I missing? Something related to induction in the primary too? Is due to not considering that $I_i=\\frac{\\Delta V_i}{n_i R_o}$? What makes, physically, that by decreasing $n_1$ one increases $\\Delta V_2$? How to understand that? Usual derivations, as in Wikipedia, apply that $\\Delta V_1=-\\frac{d\\phi}{dt}$ should be verified, but I find it a bit misleading, as the _cause_ and _effect_ roles symmetry is not clear. Given the true relation, in fact, what is the reason not to set always just $n_1=1$? Just to have a more precise ratio control and\/or to have a more efficient system with $\\vec{B}$ lines better guided inside the iron or is there any other reason?"}
{"id":"132360","title":"Why batteries in serial don't discharge each other?","text":"A question that I've asked myself for long time about why batteries in series work as they do. I'll try to explain: If you take two batteries, let's say A and B, and you put A's anode touching the B's cathode (only that, without closing the circuit). It turns out that not only A and B poles don't discharge each other but the resulting device sums the voltages of both batteries. When I think of it my idea is that A and B poles should get \"lost\" (they have cancelled) but the poles that remain unconnected still have a different potential so it shoud work at that voltage, not the sum. After all you are connecting a + pole to a - one and it's supposed that one pole has a different charge that goes to the other until they are balanced. So, why batteries discharge when closing the circuit but not when it's open if poles of different batteries are in touch? Why the charge from one pole doesn't flow to the other? And why the voltages sum? I'm afraid that the explanation is quite simple but...which is? Thanks in advance! **NOTE: I edit my own question.** As Murphy states, once published I've found there can be a possible explanation here: Why two connected 1.5 volt battery become 3 volt I'll see if it tells me what I want. Nevertheless, feel free to give me your own answer. **NOTE 2:** Ok, guys, thank you very much for your answers. Finally I think I got it! The point is in the electrons that cause the potential difference. The accumulation of electrons in one pole makes impossible that the electrons of the other battery enter the possitive pole. They are rejected electrostatically. Allowing it you would have the two poles charged but the battery doesn't work that way. So, when you close the circuit, both accumulations of electrons sum because one battery would help expelling the other one's charge pushing with it's electrons. Am I right?"}
{"id":"91003","title":"How to calculate velocities after collision?","text":"I'm currently writing a program for a particle simulator. One of the requirements is that the particles collide in a realistic way. However, I don't know how to calculate the final velocities. For each collision, I have the $x$-component and $y$-component of each velocity, as well as the displacement and mass of each particle. Is it possible to calculate the direction and magnitude of their velocities after the collision? If so, how?"}
{"id":"106743","title":"Why is the objective function used in Nudged Elastic Band method reasonable?","text":"In Nudged Elastic Band (NEB) method, which is used to find reaction pathways when both initial and final states are know, an objective function is first constructed and then minimized to find reaction pathways. NEB creates a string of replicas (or 'images') of the system between initial and final states, and connects them with springs to represent a path connecting initial and final states. Initially, the images may be interpolated between reactant and product linearly. Then an optimization algorithm is applied to relax the images down towards the Minimum Energy Path (MEP). The objective function is constructed as a combination of potential energy of each image and \"spring energy\" between each image as equation below shows. $$ S(\\vec R_0,...,\\vec R_N)=\\sum_{i=1}^{N-1}E(\\vec R_i)+\\sum_{i=1}^N\\frac{k}{2}(\\vec R_i-\\vec R_{i-1})^2 $$ Where $\\vec R_i$ is the coordinates of i-th image. NEB solved \"corner-cutting\" and \"sliding-down\" problems in this model by considering vertical and parallel components of potential force and string force. I will not introduce here, you can look at this article for more details. My question is why the objective function above is reasonable. Does minimization of this objective function sufficiently leads to the MEP? If yes, how to prove it? If no, why is this \"chain-of-states\" model widely used? Thank you very much for answering."}
{"id":"66296","title":"Could we use the coriolis effect to travel","text":"Given a platform that is floating independent of Earth's gravity, would it be possible to put a platform over the equator and over the day, would the Earth rotate from under it allowing you to travel around the world?"}
{"id":"66290","title":"Gauss's Law - Electric Field outside a Shell?","text":"I'm somewhat confused as to why the electric field outside a spherical shell is $\\frac{Q}{4\\pi r^{2}\\epsilon_{0}}$ Going through the work: $$ Q = 4 \\pi r^{2} \\sigma ; \\frac{4\\pi r^{2}\\sigma}{ε_{0}} = \\int \\int_{s} \\vec{E} \\cdot d \\vec{A} $$ Which, doing the integral leaves = $$\\frac{4\\pi r^{2}\\sigma}{ε_{0}}= 4\\pi R^{2}E$$ which means that $$E = \\frac{r^{2} \\sigma}{ε_{0}R^{2}}$$ Is this correct, or am I making some silly mistake?"}
{"id":"74978","title":"Will the absorption of high energy gamma rays end up heating the absorping material?","text":"By high energy I mean 100+ MeV gamma rays. I think that at this energy of photons, pair production is the dominating absorption process. So, as known, in pair production an electron and a positron are produced and any excess energy of the photon over the rest mass of the electron and the positron will be in the form of the kinetic energy of the new electron and positron. So now, how will the electron and positron end up in a dense material ? like will the electron come to a stop releasing its in energy in the material and thus heating it up ? or will it escape the material ? I am assuming using the attenuation length of Lead at this gamma ray energy as the absorbing material."}
{"id":"74976","title":"Is there a thermodynamic law or theorem that expresses how systems \"break\" or \"change\" when enough energy is added?","text":"I have a simple question about thermodynamic laws, and I am hoping you can help me. Let's say that I have a sphere container with some pressurized gas in it. I can slowly increase the pressure over time, and eventually the container will have to rupture (no matter what it is made of). More generally, if energy is slowly added to a system in a consistent manner, there will be some point in time when the system either changes or breaks as it can no longer contain the energy. Which of the thermodynamic laws would explain this thought experiment? Could you point me in the right direction? Thanks"}
{"id":"74974","title":"Can magnetic energy be used to create a magnetic motor which can power vehicles and machines?","text":"Can magnetic energy be used to create a magnetic motor which can power vehicles and machines?"}
{"id":"74970","title":"Thermodynamics: Are pressure and temperature enough to calculate enthalpy?","text":"![enter image description here](http:\/\/i.stack.imgur.com\/VG8Km.gif) From the chart above, it looks like it is possible to calculate enthalpy from just pressure and temperature. After a few hours of Googling my question, I'm not able to find anything helpful. My company is interested in finding the effectiveness of an air conditioning system by measuring the change in enthalpy from one side to the other. As a software person, I need a formula that I can use to calculate the enthalpy. I know this formula would need to include different properties of the gas, can you elaborate on what those constants are? Thank you very much."}
{"id":"86330","title":"Finding potential difference (PD) between two points","text":"How do I find the potential difference between two points in a circuit without using Kirchhoff laws?"}
{"id":"81100","title":"Time to travel distance given acceleration\/deceleration","text":"> A body accelerates at 0.2 m\/s$^2$ & retards at 0.3 m\/s$^2$. Maximum velocity > obtained by it is 30 m\/s. Find the least time in which it covers 6 km Well, if I do not assume that the body starts from 0 m\/s and stops at 0 m\/s. I get stuck with six equations, three from each condition. And if I do assume it, the condition of body crossing 6 km doesn't meet. All I wish to know is how to begin with this problem."}
{"id":"105802","title":"First-order wave equation: Why is its presence not common?","text":"The (one-dimensional) **wave equation** is the second-order linear partial differential equation $$\\frac{\\partial^2 f}{\\partial x^2}=\\frac{1}{v^2}\\frac{\\partial^2 f}{\\partial t^2}\\tag{second order PDE}$$ that admits as its solutions functions $f$ of the form $$f=f(x\\pm vt),\\tag{solution}$$ as can be verified in a straight-forward manner. These solutions have a convenient interpretation that justifies the phrase _wave equation._ I noticed there are **first-order** partial differential equations which have as solutions functions of the form $f(x\\pm vt)$: $$\\frac{\\partial f}{\\partial x}=\\pm\\frac{1}{v}\\frac{\\partial f}{\\partial t}\\tag{first order PDE}.$$ A quick Google search shows this is indeed called the first-order wave equation, but it usually shows up in the context of math classes. **So now the question:** Why is the usual second-order PDE favored over these first-order ones if both admit the same solutions? Is there a physical reason? Are this first-order equations useful in their own right? Perhaps there other solutions that one admits that isn't desired, or maybe it just looks cleaner since one doesn't have to carry around the $\\pm$ symbol in the differential equation."}
{"id":"105803","title":"From Minkowski to Euclidean Time in Path Integrals","text":"I'm trying to prove the following equality: $$ <x_{f},\\, it_{f}|x_{i},\\, it_{i}>=\\mathcal{N}\\int_{\\left\\\\{ x\\in\\mathbb{R}^{\\mathbb{R}}:\\, x\\left(t_{f}\\right)=x_{f}\\wedge x\\left(t_{i}\\right)=x_{i}\\right\\\\} }\\mathcal{D}x\\exp\\left\\\\{ -\\frac{1}{\\hbar}\\int_{t_{i}}^{t_{f}}dt\\left\\\\{ \\frac{1}{2}m\\left[x'\\right]^{2}-\\left(-V\\left[x\\right]\\right)\\right\\\\} \\right\\\\} $$ where the definition of $ <x_f, t_f|x_i, t_i>$ is given by: $$ <x_{f},\\, t_{f}|x_{i},\\, t_{i}>\\equiv\\mathcal{N}\\int_{\\left\\\\{ x\\in\\mathbb{R}^{\\mathbb{R}}:\\, x\\left(t_{f}\\right)=x_{f}\\wedge x\\left(t_{i}\\right)=x_{i}\\right\\\\} }\\mathcal{D}x\\exp\\left\\\\{ \\frac{i}{\\hbar}\\int_{t_{i}}^{t_{f}}dt\\left\\\\{ \\frac{1}{2}m\\left[x'\\right]^{2}-\\left(V\\left[x\\right]\\right)\\right\\\\}\\right\\\\} $$ What I have done so far:   * Assume that the domain of integration, that is, $\\left\\\\{ x\\in\\mathbb{R}^{\\mathbb{R}}:\\, x\\left(t_{f}\\right)=x_{f}\\wedge x\\left(t_{i}\\right)=x_{i}\\right\\\\}$ is such that all the functions in this set can be analytically continued to a new domain of integration $ \\left\\\\{ x\\in\\mathbb{C}^{\\mathbb{C}}:\\, x\\left(it_{f}\\right)=x_{f}\\wedge x\\left(it_{i}\\right)=x_{i}\\right\\\\} $. Is this valid?   * Plug in the definitions: $<x_{f},\\, it_{f}|x_{i},\\, it_{i}>=\\mathcal{N}\\int_{\\left\\\\{ x\\in\\mathbb{C}^{\\mathbb{C}}:\\, x\\left(it_{f}\\right)=x_{f}\\wedge x\\left(it_{i}\\right)=x_{i}\\right\\\\} }\\mathcal{D}x\\exp\\left\\\\{ \\frac{i}{\\hbar}\\int_{it_{i}}^{it_{f}}dt\\left\\\\{ \\frac{1}{2}m\\left[x'\\right]^{2}-\\left(V\\left[x\\right]\\right)\\right\\\\}\\right\\\\}$   * Now to compute $ \\int_{it_{i}}^{it_{f}}dt\\left\\\\{ \\frac{1}{2}m\\left[x'\\right]^{2}-\\left(V\\left[x\\right]\\right)\\right\\\\} $ make a change of variable (is this valid?? Don't you need Cauchy's theorem and also to assume that the time boundaries go to infinity?) $t \\mapsto -it$ to get: $i \\int_{t_{i}}^{t_{f}}dt\\left\\\\{ \\frac{1}{2}m\\left[x'\\right]^{2}+V\\left[x\\right]\\right\\\\} $ so you get the correct exponent.    * But how do you prove that $\\mathcal{N}\\int_{\\left\\\\{ x\\in\\mathbb{C}^{\\mathbb{C}}:\\, x\\left(it_{f}\\right)=x_{f}\\wedge x\\left(it_{i}\\right)=x_{i}\\right\\\\} }\\mathcal{D}x = \\mathcal{N}\\int_{\\left\\\\{ x\\in\\mathbb{R}^{\\mathbb{R}}:\\, x\\left(t_{f}\\right)=x_{f}\\wedge x\\left(t_{i}\\right)=x_{i}\\right\\\\} }\\mathcal{D}x$? Is it even the same $\\mathcal{N}$?"}
{"id":"105807","title":"Is There an Equation To Calculate Water Evaporation Taking Into Account Environmental Variables","text":"I am starting to learn more in my area of science...Meteorology and the fact that I own a pool, I have the perfect testing environment. I just so happened to have a water level drop from an actual leak that I learned how piping systems work, and will continue to learn more there. The leak was plugged yesterday. But, it got me thinking about my good old weather schooling and physics days. So, here is my question guys: Is there an equation for the following: Calculation of water level evaporation with following variables (X):   water volume, water flow\/speed, ambient air temperature, dew point\/humidity, chemical composition of water (pool), temperature of water, wind speed. Constants and Assumptions: No leaking or structural damage that holds the water, contained to a finite amount of gallons. I understand that there may be other environmental variables to control, but this is a good start. If there is a standard deviation to follow in all of the above variables, that can be measured, can or is there an equation for this? What I am trying to learn: Is there correlation in any of the variables to the measurement of the change in the water level (Y). Then I would like to apply the formula and change some of the constants to measure on larger scales. Any answers related to the subject at hand is most helpful. Thank you."}
{"id":"99425","title":"Why does mixing every paint colour produce gray instead of white?","text":"I'm trying to reconcile these facts,   * If I had paints of the primary colours and I mixed them together, I would produce gray.    * White light contains every colour.  My most convincing attempt to answer the question: The colours of objects that don't produce light are the colours those objects don't absorb. A white object doesn't absorb any colour, a blue object absorbs all colours but blue, a green object absorbs all colours but green, etc. So by mixing paints of the primary colours I mix influences that collectively absorb all colours and reflect all colours; whereas, if I had paint that only absorbed all colours I would have black paint, and if I had paint that only reflected all colours I would have had white paint. Gray is a combination of white and gray, so I found this answer intuitively satisfying. However, I don't know if it's an accurate explanation of the observation. Why does mixing every paint colour produce gray instead of white? Thank you."}
{"id":"130855","title":"Force applied to an inclined plane","text":"Below is a picture of the problem. Any guidance would be helpful. This problem isn't actually from any assignment, per se. I'm hoping that, by understanding this, it'll help me to understand a more complicated system where one inclined plane is resting on another inclined plane, and there's a mass on the top of the second plane, and the bottom incline is being accelerated by an applied force--but since I don't understand how this much simpler system works, I don't seem to have a chance at the more complicated system. ![Diagrams](http:\/\/i.stack.imgur.com\/gHLO3.jpg) Forgot to include what the question is asking for. Since this isn't really a question I've been asked, I guess what I want to know is the acceleration of the system."}
{"id":"110926","title":"Why does a spurion analysis work independently of the UV physics?","text":"In short, my question is why does a spurion analysis work to produce the correct symmetry breaking terms regardless of the high energy physics? The context that this question arose is from an Effective Field Theory course (for more context, see here, Eq. 5.50). Consider the QCD Lagrangian, \\begin{equation} {\\cal L} _{ QCD} = \\bar{\\psi} \\left( i \\gamma^\\mu D_\\mu - m \\right) \\psi \\end{equation} The kinetic part is invariant under a chiral transformation: \\begin{equation} \\psi \\rightarrow \\left( \\begin{array}{cc} L & 0 \\\\\\ 0 & R \\end{array} \\right) \\psi \\end{equation} however, the mass term is not. Now the claim I don't understand is as follows. Suppose the mass transformed as, \\begin{equation} m \\rightarrow L m R ^\\dagger \\end{equation} In that case the mass term would be invariant under such a transformation. To write down the correct chiral symmetry breaking terms in our Lagrangian we find the terms invariant given this transformation for $ m $ and then make $ m $ a constant again. The way I understand this physically is that the breaking arises from a high energy spurion field, $ X $, which gets a VEV, $ m $. When we write down all possible chiral symmetry preserving terms using the transforming $m$, we are writing down all the terms that the spurion couples to. The VEV is then inserted and is equal to $ m $. But this procedure assumes that the spurion obeys the chiral symmetry, $ SU(2) _L \\times SU(2) _R $, and transforms as, $ X \\rightarrow L X R ^\\dagger $. How do we know this assumption is true? In fact it seems to fail for the case of QCD since the ``spurion field'' is really the Higgs field, which is a singlet under $ SU(2) _R $."}
{"id":"3612","title":"Binary Black Hole Solution of General Relativity?","text":"This is rather a technical question for experts in General Relativity. An accessible link would be an accepable answer, although any additional discussion is welcome. GR has well known solutions relating to single Black Holes: Schwarzchild, Rotating & Rotating with Charge. These solutions demonstrate some non-trivial GR behaviour. However do there exist any (corresponding) binary star\/black hole solutions? Because of the non-linearity of GR such a solution could well demonstrate additional properties to a \"solution\" that simply consisted of a pair of \"distant\" Schwarzchild solutions glued together."}
{"id":"22075","title":"Deriving Planck's radiation law from microscopic considerations?","text":"In the usual derivation of Planck's radiation law, the energies or frequencies $\\omega$ of the oscillators depend on the measurements $L$ of the black body. The model is such that the only characteristic energy is given by these oscillator excitations and in terms of the temperature $T$. Also, the oscillation amplitude vanishes on the walls. Specifically, the structure of the atom walls doesn't enter the computation. After all, that's pretty much the definition of a black body. My question: > Are there computations (maybe QED + statistical physics?) for more realistic > systems, which might model the interaction of photons with the wall atoms > and which give results such that one can see the limiting case to an ideal > black body? The question came up when I wondered about the theory behind emission and absorptions of photons on the walls of a black body."}
{"id":"31922","title":"What parameters determine the acceleration a spacecraft can acquire using the gravity of a celestial body?","text":"Contemporary spacecraft frequently rely upon gravity to accelerate\/decelerate. Given a spacecraft already in space - What factors determine the acceleration it can acquire from the gravity of another body? Is there an upper limit to such gravitational acceleration?"}
{"id":"77772","title":"Does proton make neutron charged by the process of induction\/friction\/conduction?","text":"If two bodies undergo friction, the one of the bodies which has electrons less tightly bound than the other loses them. Here the protons do also have charged quarks which they could exchange with neutrons to make them charged either by the process of conduction (may not be possible by friction), but it does not happen I think so. If so, Why? Edit: Can the neutron develop induced charge due to protons inside the nucleus? Though we can't expect exchangeable isolated quarks, can we expect them to orient or get settled to one side, so that neutron can develop induced charge? I might have misunderstood sometimes, if so, pardon me and explain."}
{"id":"98479","title":"Find the position of the final image","text":"I was going through Geometrical Optics and came across the following question in my textbook: Q. An object is placed $21$ cm in front of a concave mirror of radius of curvature $20$ cm. A glass slab of thickness $3$ cm and refractive index $1.5$ is placed close to the mirror in the space between the object and the mirror. Find the position of the final image formed. The distance of the nearer surface of the slab from the mirror is $10$ cm. Assume only paraxial rays. Brute force led me to the answer-at the position of the object itself . I have the following questions to ask: $1.$Is my answer correct ? $2.$What can be the best possible way to solve such a problem? $3.$Is the data about the distance of the slab from the mirror really required? Thanks for the help... **NOTE:** I posted this on Math S.E. because Math S.E. **does** have a **physics** tag."}
{"id":"75739","title":"Does the Cosmic Microwave Background obey $\\sigma T^4$?","text":"The fact that the CMB has a temperature of 2.7 degrees Kelvin is well publicized, but I have a harder time finding a measure of the wattage per unit area. The Wikipedia article on the CMB is totally silent on the subject. Then when I do find a source for W\/m^2 for the CMB, they use the blackbody formula. > A flux of radiation has a Kelvin temperature. We see the temperature of the > CMB as 2.735 Kelvin. We can convert this to watts\/meter2 for blackbody > radiation with the Stephan and Boltzmann law. K4*5.5698E-8_W\/(m2*K4) = _W\/m2 > > 2.735 Kelvin = 3.11E-6_W\/m2 > > When one sees something, it is in terms of W\/m2 and the inverse square law. That source does mention the fact that _expansion of space decreases photon density_. But I don't actually see a revision of the W\/m^2 figure. I can't understand if they mean to say that the combination of temperature and density change balances it out so that it still follows the blackbody radiation law. There's also the matter of whether the gas from which the CMB came was a true blackbody or had some reflectivity to it. Surely we have a measured value for this. Does the value come anywhere close to the Stefan–Boltzmann law prediction? Does the expansion of space predict a deviation from that intensity or not? Do non-blackbody properties of the primordial gas predict a deviation from that?"}
{"id":"100978","title":"Electrical Potential Energy And forces Experienced By Particles","text":"I am really stuck on a question and would appreciate a helping hand. A particle has a charge of $+3.5$ $\\mu$$C$ and moves from point $A$ to point $B$, a distance of $0.5m$. The particle experiences a constant electric force and it's motion is along the line of action of the force. The difference between the particles electrical potential energy at point $A$ and point $B$ is $EPE_{A}$ $-$ $EPE_{B}$ $= +7.0x10^{-3} J$. (a) Find the magnitude and direction and direction of the electric force that acts on the particle. (b) Find the magnitude and direction of the electric field the particle experiences. The fact that the work done is a positive quantity, leads me to think that the electric force acts in the direction from $A$ to $B$ but I don't know how to quantify it. Thank you for any help you can give!"}
{"id":"101009","title":"Numerical solution of the BCS gap equation, with Coulomb potential","text":"I'm interested in the excitons condensation, which is described by an equation which is very similar to the standard BCS gap equation. I will be referring to this article: http:\/\/arxiv.org\/abs\/cond-mat\/9501011. The only differences w.r.t. the standard BCS theory are:   1. The momentum dependency of the gap cannot be neglected, so we write $\\Delta = \\Delta(k)$.   2. The potential must be model realistically, in this case after a Coulomb potential. In the end the gap equation reads: $$ \\Delta (k) = \\int d^2 k' \\underbrace{\\frac{e^{-d \\left| k - k' \\right|}}{\\left| k - k' \\right|}}_{V_{k k'}} \\frac{\\Delta(k')}{\\sqrt{\\Delta^2 + (k'^2 - \\mu)^2}} $$ Knowing all the the parameters ($d$, $\\mu$), clearly this equations needs to be solved numerically for $\\Delta (k)$. My first guess (and this approach does work, indeed, in many similar cases) would be of choosing a \"trial\" $\\Delta$, iterating over and over, hoping sooner or later to converge to a solution which is not the banal $\\Delta=0$. This approach does not work here, because the potential $V_{k k'}$ has singularities. The article I am referring to is quite cryptical about it. They mention an unpublished paper as far as the numerical calculations are involved, and simply say that \"the equation is converted in matrix form\", as: $$ \\Delta_i = M_{i j} \\Delta_j $$ as expected, and then \"the logarithmic singularities are treated separately\". Clearly the singularities are on the diagonal of the matrix when the equation is discretized. This process is completely obscure to me. I suppose I cannot just separe the divergent diagonal part and the finite part. I tried parametrizing the divergency as $\\frac{1}{\\epsilon}$ and letting $\\epsilon$ go to zero at the end of the calculation but to no avail. Is there some standard techniques to numerically solve my equation? Thank you in advance."}
{"id":"101007","title":"2 torches with the same power - More lumen?","text":"I didn't find an answer for this. A friend and me argued about the brightness-impact of multiple bulbs\/torches. I told him, that 2 torches with the same power (the same amount of lumen and everything) hold side by side, won't be a difference in lumen, if you place the measure-device directly in the center of the \"beam\". So: The brightness is the same. Means: If I point to the lumen-measure device with one torch (maybe ~0.5m distance), adding another would have no impact on the result, if it has exactly the same power. He denied and said, that the more torches\/bulbs you add, the more brightness you will get. He said, that if you add another torch, you see 2 rings of light: The center beam contains the power of both, but at the outside (edges), the beams divide and there would be lesser power, where because there is just the beam of one torch. I draw a picture to show the Idea: ![enter image description here](http:\/\/i.stack.imgur.com\/5pwnT.png) I said, that you don't have more brightness, but you have a better illumination, which let you think that the light is stronger. But if somebody \"beams\" to your eyes with one torch, adding another with the same attributes won't make a difference in brightness, if you hold them close together. If you would put 1000 torches next to each other, it won't be much brighter, but the illumination is of course bigger and this may let you think, that there is stronger light. Who of us was right? I couldn't get an evidence for my theory, because I lack the required items. I hope you understand what I mean, my english isn't the best. :) I hope there are some of you, who can give me a good answer to this. We \"fight\" about this point since months. :D I couldn't find an evidence on the internet."}
{"id":"122624","title":"Calculate minimum intensity of a point source visible to a camera","text":"I have two cameras and know the specified minimum faceplate sensitivity for both: Camera 1: Minimum faceplate sensitivity 4 x 10-6 Lux   Camera 2: Minimum faceplate sensitivity 2 x 10-7 lux I also know the minimum intensity (1 x 10-7 W m-2) of a point source (470 nm) visible to Camera 2 at a distance of 0.5 m from the front of the camera (ambient light levels: zero). I would like to be able to calculate the minimum intensity of a point source visible Camera 1 through the same medium (seawater). (Camera 1 is no longer functional and so cannot be calibrated) Although there would obviously be a number of factors that would add error into this calculation, is there a way to calculate this (supposing ideal conditions)?"}
{"id":"5156","title":"Question regarding the validity of the big bounce","text":"I have several questions regarding the \"big bounce\" theory. It appears to be popular among LQG researchers. My questions are as as follows. 1) How one reconciles it with the fact that it is now experimentally established that the universe is expanding in an accelerating manner? 2) The entropy of the universe should have been very very low (maybe zero) at the time of the big bang\/ big bounce. What was the entropy \"before\" that event? Was it even lower? If it was lower then doesn't that mean entropy decreases indefinitely? On the other hand if we accept the entropy was higher then doesn't that violate the second law of thermodynamics? 3) What observational evidence can confirm pre-bounce physics? According to GR there are some points near the bang\/bounce where one simply can't easily define time in a meaningful way. Even for a quantum gravity theory, is it possible for any theory to extrapolate itself \"before\" the \"bounce\"? If there are no observational consequences after the bounce shouldn't one apply Occam's razor? ![enter image description here](http:\/\/i.stack.imgur.com\/HeKrp.jpg)"}
{"id":"77541","title":"Does effective theory have the same meaning in particle and condensed matter physics","text":"I have a naive question about the meaning of effective theory in particle physics and condensed matter physics. In particle physics, from what I know, the effective theory comes from the Wilsonian renormalization. We start from a theory with a fundamental cutoff $\\Lambda$, e.g. Planck scale, and consider a smaller cutoff $\\Lambda'$, e.g. LHC scale, $\\Lambda'<\\Lambda$. The Lagrangian then reshuffle into an infinity series including all possible powers of fields. The theory with $\\Lambda'$ cutoff is called effective theory (possibly only a few leading terms) In condensed matter theory, I heard the effective theory means neglect some details, like the BCS theory. Roughly speaking, \"the effective\" in both fields has similar meaning, that we ignore something ($>\\Lambda'$ in particle physics or uninteresting detail in condensend matter). My question is, strictly speaking, does the two \"effective\" in particle and condensed physics have the same meaning? Can we start from QED, lower the cutoff, lead to BCS theory? (perhaps such crazy attempt will never work out...) or any deeper connection than heuristic similarities?"}
{"id":"45851","title":"Classical (or semi-classical) interpretation of photoelectric effect?","text":"This site says that \"it has recently been proven that the photoelectric effect can be interpreted classically (or at least semi-classically) in non-particle, wavelike terms\". Is anyone familiar with this other explanation of the photoelectric effect?"}
{"id":"87315","title":"Yan - Drell process","text":"I'm looking for more information about Yan - Drell process but didn't manage to find a good place to learn from (still looking for one). I understood that this process can help us to understand the inner structure of hadrons \/ mesons but I would love to get more information what is going on. In which accelerator this process is actually being researched?"}
{"id":"45858","title":"Root mean square value for DC","text":"The mean value of alternating current comes to be zero because of half of the cycle being positive while the other half negative. so, take the root means square value of Alternating current given by: $I_{rms} = 0.707\\times I_{max}$ But why do we take \"rms\" for direct current? if we take the simple average of direct current, we would come up with a value which is not zero (it is zero in AC)."}
{"id":"73700","title":"Why does a cathode have to be heated to emit electrons?","text":"Considering that electrons are highly mobile inside of a metal, why do they have such a tough time getting out at the edge of it and continuing their trip ballistically?"}
{"id":"15894","title":"Motivation for Potentials","text":"This is a hypothetical question about \"pedagogy\". Let's say I am trying to take someone who has just a very small amount of knowledge about Newtonian mechanics and convince them that the Lagrangian formulation of mechanics is a _natural_ thing to try. First of all, to convince them that this is _natural_ , at the very least, I would have to show how one can reproduce Newtonian Mechanics from Lagrangian Mechanics, in particular, Newton's Second Law. However, to do this, I would need to introduce the notion of a potential. However, I am having trouble justifying this. I would like to be able to say \"most forces we encounter in nature are conservative, and hence, we are not losing too much by assuming all of our forces are conservative . . .\". However, of the two _fundamental_ forces that come to mind in _elementary_ physics, the gravitational force and the electromagnetic force, only one is conservative. I hardly feel that one out of two is a sufficient justification for the introduction of a potential. But if it is not even _natural_ to assume that our forces always come from a potential, then it certainly won't be _natural_ to define the Lagrangian in a way that reproduces Newton's second law. Is there a nice way of convincing such a student that, even though the electromagnetic force is not conservative, there are still ways of deriving it from some sort of potential? In particular, the difficulty here is that I do not want to assume they have any specific knowledge of electrodynamics."}
{"id":"26538","title":"Why doesn't helium start to fuse while there is still hydrogen fuel?","text":"In all the descriptions of the stellar life cycle it seems as though helium doesn't start being fused until all (most?) of the hydrogen is gone. Is this true? Why is this? It seems counter intuitive. Consider a fire consisting of gasoline (hydrogen), and wood (helium). Sure the gasoline would burn up quicker, but the wood would still be consumed while the initial gasoline was being consumed."}
{"id":"30648","title":"Are the higher-order terms in the series for energy really negligible?","text":"To show that energy in special relativity reduces to $E=m+mv^2\/2$ for low velocities, if we make a Taylor expansion of $m\\gamma$ around $v=0$ we get $$E=m+mv^2\/2+3mv^4\/8+\\cdots$$ But why can we cutoff after the second term? How does one know all the other terms don't add up to something significant? We get $E = \\text{rest energy}+\\text{kinetic energy}+\\frac{3mv^4}{8} + \\cdots$ What kind of energy is the $3mv^4\/8$ term?"}
{"id":"133193","title":"How do you obtain the fluctuation spectrum of a tubular membrane?","text":"I am reading through a paper. A tubular membrane, submitted to tension $\\sigma$ acting as a Lagrange multiplier to conserve area, fluctuates around a cylindrical shape of length L and radius R. Parametrisation is as follows, \\begin{equation} \\mathbf{r}(\\phi,\\zeta)= (x,y,z) = R([1+u(\\phi,\\zeta)]\\cos{(\\phi)}, [1+u(\\phi,\\zeta)]\\sin{(\\phi)}, \\zeta) \\end{equation} where $0 \\leq \\phi \\leq 2\\pi$, $0 \\leq \\zeta \\leq L\/R$ and $u(\\phi,\\zeta)$ is the deformation field. Using the Canham-Helfrich Hamiltonian, $F= \\int dS\\left( \\frac{1}{2} \\kappa H^2 + \\sigma \\right)$ one can obtain expressions, to second order, for the excess energy. Long story short, the excess energy can be represented in Fourier modes from which you obtain the mean squared amplitude using the equipartition of energy: \\begin{equation} \\langle \\lvert u_{m,n} \\rvert ^2 \\rangle = \\frac{k_B T}{\\kappa} \\frac{1}{(m^2 -1)^2 + \\bar{q}^2 (\\bar{q}^2 + 2m^2)} \\end{equation} The correlation function can also be found: \\begin{align} C((\\phi - \\phi '),( \\zeta - \\zeta')) &= \\langle u(\\phi,\\zeta)u(\\phi ', \\zeta ') \\rangle \\\\\\ &= \\frac{k_B T R}{2 \\pi \\kappa L} \\sum_{m,\\bar{q}} \\frac{\\cos(m(\\phi - \\phi '))cos(\\bar{q}(\\zeta - \\zeta '))}{(m^2 -1)^2 + \\bar{q}^2 (\\bar{q}^2 + 2m^2)} \\end{align} So my questions are: 1) What is the meaning of the mean squared amplitude and the correlation function? 2) How can I obtain a fluctuation spectrum from them? The paper can be found here: http:\/\/www.msc.univ-paris-diderot.fr\/~jbfournier\/publi\/critical_tubes.pdf Many thanks in advance."}
{"id":"104412","title":"Atwood machine with spring","text":"I'm just beginning to learn about Lagrangian mechanics, and I am asked to find the kinetic energy of this Atwood machine (See figure). ![Atwood machine](http:\/\/i.stack.imgur.com\/SZIt4.png) I am told, that the kinetic energy should be: $$T=2m\\dot{x}^{2}+\\frac{1}{2}m\\dot{y}^{2}-m\\dot{x}\\dot{y}$$ I am also told, that the movement is so slow, that the spring is always stretched, and $x$ and $y$ are my generalized coordinates. So what to do? My first though was, that since the spring is always stretched, all masses are dependent of each other, and I could maybe look at it as a whole string with one mass on the left, and two on the right. But that didn't give me the right answer. So I was hoping someone could give me a hint or something?"}
{"id":"132826","title":"Mathematical model of a fluid: which viewpoint is the standard one?","text":"I've been studying fluid mechanics and I have some doubts on the mathematical model of a fluid. First we let $D\\subset \\mathbb{R}^3$ be a region in three- space filled with a fluid. This region has as many points as the real line. Since in Classical Mechanics when studying systems with a finite number of particles we consider each particle as a point in $\\mathbb{R}^3$ I thought that _each_ point of this region $D$ should be considered a particle of fluid. In that setting the fluid would be composed of uncountably many particles. With this logic, I thought on labeling each particle by it's position at time $t = 0$, so that a state of the fluid would be a bijection from $D$ onto $D$. The time evolution would be then $t \\mapsto \\varphi_t : D \\to D$ and we could summarize this with a map $\\varphi : D\\times \\mathbb{R}\\to D$ given by $\\varphi(a,t) = \\varphi_t(a)$. This is exactly the Lagrangian description of the fluid, so I thought I was right in considering each particle of the fluid as a point in $D$. Now, I've been told it's wrong to think it this way, because if it were so, each particle would have zero mass. I've also been told that the right way to think is to imagine each particle as a infinitesimal volume $dV$ containing _some_ real molecules of fluid with mass density $\\rho : D\\times \\mathbb{R}\\to \\mathbb{R}$ so that each particle would have mass $\\rho dV$. Since each piece of volume is infinitesimal we could still identify them with points. My doubt is, when modeling a fluid with math which viewpoint is the standard one and why? The view point as in Newtonian and Lagrangian Mechanics of considering the fluid particles as points of $D$ or the view point of considering the fluid particles as infinitesimal chunks of fluid with volume $dV$ one at each point of $D$?"}
{"id":"44892","title":"Artificial planetary magnetic field","text":"I wonder how difficult it is to create an artificial planetary magnetic field with generators? What power they would need? The question is inspired by thinking about possible colonization of Jupiter's moons Io and Europa which are located inside the Jovian radiation belt. Is it possible to create with easy means an artificial magnetic field such that it to shield the surfaces of these moons from radiation? Or it would require astronomical amount of power? By easy means I mean a device that would not require power greater that normal industrial power plant, best of all, solar-powered or based on once-charged superconductor coils."}
{"id":"103099","title":"Is a capacitor in an open circuit charged?","text":"Say I have a circuit consisting of a battery, a wire, an open switch, and a capacitor. The circuit is open since the switch is open. My book says that the capacitor will only be charged when the switch is closed, but I don't see why this is true. I would expect the capacitor to be charged a little - not as much as if the circuit is closed, but still charged none the less. To further illustrate my point consider this: If the circuit is open, the current must be zero. Consequently the field must be zero. For the field to be zero, the capacitor's field must cancel out the battery's field. Therefore the capacitor _must_ be charged. Generalizing this concept, shouldn't capacitors be charged (to a lesser degree) in open circuits? EDIT: In other words, if the field is zero, the capacitor _must_ be charged to cancel out the field of the battery."}
{"id":"13829","title":"Why is radiative forcing from CO2 logarithimic and not a decreasing exponential?","text":"I have a question about climate science, but this is relatively straight forward and (I hope) easy. I've been reading that the radiative forcing caused by CO2 in the atmosphere (or any gas) is proportional to the logarithm of the concentration. I don't understand why. I'm familiar with basic attenuation, so if I assume that the gas captures or reflects some light, which equates to the heat per unit area (the units for radiative forcing), then I would write an equation like the following, with $F$ being the forcing in $W\/m^2$ and $C$ being the ppm concentration. _The $1-e^x$ form is appropriate because we need the reflected quantity, versus the $e^x$ transmitted quantity_ $$F \\propto \\left( 1- e^{-\\lambda C} \\right)$$ What is fundamentally different about the greenhouse effect? I hesitate to give a specific reference because it seems to occur so frequently, but I will note that IPCC 2007 report is said to make such a statement. Wikipedia gives the following equation. $$\\Delta F = 5.35 \\times \\ln {C \\over C_0}~\\mathrm{W}~\\mathrm{m}^{-2} $$ I would love to see just a simple and clean takedown of why $ln$ and not $exp$."}
{"id":"13822","title":"How feasible is it to use the dielectric constant of a liquid to determine specific gravity?","text":"I homebrew my own beer and I'm looking for way to cheaply monitor the specific gravity of my brew during the fermentation process. I came across this thread that seems to indicate capacitance may be used to determine specific gravity. Is there a direct relationship between the dielectric constant and\/or capacitance of a liquid and its specific gravity?"}
{"id":"96205","title":"Charging a Capacitor?","text":"How is the charge +q given to the one plate to the capacitor with the help of a battery, does the battery remove electrons from the plate and +ve charge gets accumulated on one plate?Does it? And one plate gets charge induction of other plate occurs which creates potential difference between the two plates and then the +ve charge of the 2nd plate is grounded and passes through to the ground. Is this process correct?"}
{"id":"38343","title":"Why is there a factor of 1\/2 in the interaction energy of an induced dipole with the field that induces it?","text":"In this paper, there's the following sentence: > ...and the factor 1\/2 takes into account that the dipole moment is an > induced, not a permanent one. Without any further explanation. I looked through Griffiths' electrodynamics to see if this was a standard sort of thing, but couldn't find anything. I was thinking it might be because the field of the dipole itself opposes the inducing field, but that doesn't quite seem right for some reason."}